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s−1, whereas only one result is available for the other two strain rates. The NTS value obtained at the lowest strain rate is the least of all the NTS values obtained in the present study. There is high scatter of NTS values, as can be seen from the two test results available for the intermediate strain rate, emphasizing that it is necessary to test more than one specimen for each combination of test parameters, to arrive at an average/representative value. However, for want of adequate test material, this aspect could not be taken care in the present study. While this limitation exists, it seems likely that the NTS in synthetic sea water tests decreases with decreasing strain rate. That increasing degree of environmentally assisted damage takes place with decreasing strain rate during the SSRT runs is only to be expected. It is worth noting in this context that the results of Bradhurst and Heuer shows the NTS values obtained with two different Kt values, when testing was done at the strain rate of 4.17 × 10−6 |
s−1. The NTS value corresponding to Kt of 3.5 (1468 MPa) is much smaller, compared to the NTS value (1092 MPa) when Kt is 2.82. These are only two results; it is possible that the effect of the severity of the notch on the NTS value in tests is similar to that noticed in 30% RH air tests. It has been reported that as with high strength steels, the more severe the notch, the greater is the susceptibility to SCC of 18%Ni maraging steels brings out a comparison of the NTS and time to failure values obtained in air and synthetic seawater environments for the strain rate 4.17 × 10−6 |
s−1, where a one to one comparison is possible with the data available. The values obtained in seawater are much smaller. While the data is very limited, it appears that seawater environment leads to a drastic reduction of NTS and tf. Ratios in the range 0.8–1.0 normally denote high resistance to EAC. The ratio in the present study is 0.53 for the NTS and 0.57 for tf at the highest strain rate used. This shows that the maraging steel grade tested is highly susceptible to corrosive damage in seawater environment.Fracture surfaces were studied using the SEM technique. Crack initiation occurred in all cases at the root of the notch. The fracture surface consisted of essentially two regions: (i) the region next to the crack initiation point, where material separation occurred as a result of EAC and (ii) region where failure occurred as a result of overload fracture. illustrates this for the case of specimen No. 13 tested at the strain rate of 8.33 × 10−7 |
s−1. Region A failed by EAC and region B by overload fracture. shows the SEM picture of typical fracture appearance of the region A at high magnification. It can be seen that material failed by intergranular fracture. The figure also shows incidence of secondary cracking along grain boundaries. For all the specimens tested in synthetic seawater, regardless of the strain rate, intergranular fracture dominated the EAC region. Microvoid coalescence and quasicleavage comprised the fracture modes in the region B as illustrated by , showing the region B of the fracture surface of specimen No. 13 at a high magnification.There is strong evidence that the resistance of high strength steels to SCC in many corrosive media decreases as the strength increases. Maraging steels, like other ultra high strength steels, are susceptible to SCC in most aqueous environments. A yield strength of about 1400 MPa has been mentioned as the threshold value, above which the vulnerability to catastrophic cracking can be expected To study the tendency of aged 18Ni2400-maraging steel to EAC, slow strain rate tensile tests were carried out on notched round specimens. The experiments were conducted in 30% RH air and synthetic seawater at three different strain rates. Testing was conducted in the peak-aged state. It appears that there is a mild HE occurring in the steel in 30% RH air environment at the strain rate of 2.10 × 10−7 |
s−1. It appears that the EAC is sensitive to the severity of the notch.Testing in synthetic seawater appears to, based on the very limited data available, lead to drastically reduced NTS and time to fracture (tf) values compared to testing in air. The very high strength of the steel in the tested condition, it is believed, has importantly contributed to the high susceptibility to cracking in seawater. Testing in synthetic seawater promoted intercrystalline fracture in the region in which fracture occurred by EAC.A comparative study of the structure–property behavior of highly branched segmented poly(urethane urea) copolymers and their linear analogsThe solid-state structure–property behavior of highly branched segmented poly(urethane urea) (PUU) copolymers and their linear analog was investigated. A limited study of their solution rheological behavior was also undertaken. The linear PUUs were synthesized by the two-step prepolymer method, whereas the oligomeric A2+B3 methodology was utilized to synthesize the highly branched materials. The soft segments (SS) were either poly(tetramethylene oxide) (PTMO) or poly(propylene oxide) (PPO). All copolymers utilized in this study, with one exception, contained 28 wt% hard segment (HS) content. DMA, SAXS, and AFM studies indicated that the linear as well as the highly branched PUUs were microphase separated. The SS Tg of the highly branched PUUs was nearly identical to that of their respective linear analogs. However, the linear copolymers exhibited broader and less temperature sensitive rubbery plateaus, both attributed to one or both of two reasons. The first is better hydrogen bonding organization of the HS phase as well as greater HS lengths than in the highly branched analogs. The second parameter is that of a potentially higher chain entanglement for the linear systems relative to the branched analogs. Tapping-mode AFM phase images confirmed the microphase morphology indicated by SAXS and DMA. Ambient temperature strain-induced crystallization was observed in the PUU based on PTMO 2040 g/mol at a uniaxial strain of ca. 400%, irrespective of the chain architecture. Stress–strain, stress relaxation, and mechanical hysteresis of the highly branched copolymers were in general slightly poorer than that of their linear analogs. Ambient temperature solution viscosity of the highly branched materials in dimethyl formamide was substantially lower that that of the linear samples of nearly equal molecular weight.Hyperbranched polymers are typically synthesized by the polymerization of ABx monomers, where x is 2 or higher. Another approach is the copolymerization of A2 and B3 monomers, which is often more attractive due to commercial availability of many A2 and B3 monomers. Since their initial discovery, the synthesis of hyperbranched counterparts of almost all common linear polymers has been reported The hyperbranched polyurethanes, polyureas, or poly(urethane urea)s discussed above consist only of hard segment (HS) moieties and thus are not segmented, that is, they do not consist of alternating hard and soft segments along the chain backbone. Recently, with the aim of producing branched polymers with a high number of terminal functional groups that also possess useful structural properties, our laboratories reported the synthesis of segmented, highly branched poly(urethane urea)s (PUU) based on the oligomeric A2+B3 approach Bis(4-isocyanatocyclohexyl)methane (HMDI) (Bayer MaterialScience) and cyclohexylisocyanate (CHI) (Aldrich) with purities greater than 99.5% were used. Poly(tetramethylene oxide)glycol (DuPont) with 〈Mn〉 of 2040 g/mol, and poly(propylene oxide)glycols (Bayer MaterialScience) with 〈Mn〉 values of 2030, 4040, 8000 and 11,800 g/mol were all used as received. The respective hydroxyl numbers were utilized to calculate 〈Mn〉 of the PTMO and PPO oligomers. 2-Methyl-1,5-diaminopentane (DY) (DuPont), tris(2-aminoethyl)amine (TRIS) (Aldrich), HPLC grade isopropyl alcohol (IPA) (Aldrich) and tetrahydrofuran (THF) (Aldrich) were used as received.The copolymers utilized in this report, except one, are based on a constant HS content of 28 wt%; HMDI+DY constitute the HS in linear PUUs whereas HMDI+TRIS+CHI comprise the HS in the highly branched analogs. The samples are listed in and identified by the nomenclature: SS type and MW-Chain architecture. For example, a linear sample based on PTMO of 〈Mn〉 2040 g/mol is identified as ‘T2-L’ whereas a highly branched sample based on PPO of 〈Mn〉 8000 g/mol is identified as ‘P8-H’. The molar ratios of precursors, the absolute 〈Mw〉, and the MWD of these copolymers are also listed in . Equimolar amounts of TRIS and HMDI terminated PPO or HMDI terminated PTMO were utilized to synthesize P2-H and T2-H. However, to maintain a constant HS content of 28 wt%, excess HMDI and TRIS had to be added as the PPO SS MW was raised above 2030 g/mol. The required excess () increased with increasing PPO MW. Thus, the extent of HS branching level in a given HS will somewhat increase with PPO MW in the highly branched PUUs.The copolymer films (0.3–0.5 mm thick) utilized for solid-state analyses were cast from IPA solution into poly(tetrafluoroethylene) molds and dried at room temperature overnight. Films were further dried at 65 °C until a constant weight was reached, monitored thermogravimetrically. These films were stored at ambient temperature under vacuum until they were analyzed.The absolute molecular weight of the samples () was determined according to the procedure presented in detail elsewhere A Seiko Instruments model DMS210 was used for dynamic mechanical analysis (DMA). The film samples, 3–4 mm wide and 10 mm long, were quenched from room temperature to −150 °C using liquid nitrogen and immediately thereafter subjected to a 2 °C/min heating scan under a dry nitrogen atmosphere. tan |
δ and storage modulus, E′, data were obtained at a frequency of 1 Hz.Ambient temperature tensile testing was conducted using an Instron model 4400R equipped with a 1-kN tension load cell and calibrated with a 2 kg standard (19.61 N). ‘Dog-bone’ shaped film specimens, 2.9 mm wide and a grip separation distance of 10 mm were used to generate the stress–strain curves at a crosshead speed of 25 mm/min. Three samples were tested for each copolymer and the average Young's modulus of these three runs is reported. Only representative stress–strain responses are presented in the appropriate figure. Stress relaxation studies were also performed on similar dog-bone shaped specimens at strain levels of 25 or 500%. Mechanical hysteresis experiments were also performed at strain levels of 25 or 500% for two uniaxial deformation cycles each and at a crosshead speed of 25 mm/min.Pin-hole collimated small angle X-ray scattering (SAXS) data were collected at ambient temperature using a Rigaku Ultrax18 rotating anode X-ray generator operated at 40 kV and 60 mA. A pyrolytic graphite monochromator was used to filter out all radiation except the Cu Kα doublet, with an average wavelength of 1.5418 Å. The camera used 200, 100 and 300 μm pin-holes for X-ray collimation. Two-dimensional data sets were collected using a Molecular Metrology 2D multi-wire area detector, located approximately 65 cm from the sample. After azimuthal averaging, the raw data were corrected for detector noise, absorption, and background noise A Digital Instruments (now Veeco) Dimension 3000 scanning probe microscope controlled by a Nanoscope IIIa controller, was utilized for tapping-mode atomic force microscopy (AFM). Images of the free surface of thoroughly dried solution cast films were captured at a magnification of 2×2 μm2 and a set-point ratio of approximately 0.6. Nanosensors' TESP 7 or Veeco's TAP 300 tips having a spring constant of 35–40 N/m were utilized for imaging.TA Instruments AR1000 rheometer with 2°, 40 mm diameter cone-and-plate attachment was used to measure the viscosity of 15 wt% copolymer solutions in dimethyl formamide at room temperature. The solution viscosity of a sample was determined from the initial slope of shear stress versus shear rate within the Newtonian or linear region of this plot. The reported viscosity numbers are averages of 3–4 repeat runs per sample.The storage modulus, E′ and tan |
δ responses as functions of temperature for both linear and highly branched PUUs based on PTMO and PPO are presented in . The DMA behavior of PTMO-2000 based copolymers presented in (a) is addressed first. Focusing on E′ of the linear sample, T2-L behaves as a rigid solid up to ca. −100 °C. Thereafter, E′ begins to decrease due to the glass transition of the PTMO SS (Tg−78 °C, tan |
δ peak value). Not surprisingly, the SS Tg is slightly higher than that of the pure PTMO oligomer (−85 °C The DMA behavior of the PTMO based highly branched analog, T2-H is also presented in (a). Interestingly, below 50 °C the E′ and tan |
δ responses of T2-H are very similar to T2-L. The former sample also displays a drop in E′ from ca. −100 to −50 °C due to the SS glass transition and a rubbery plateau extends thereafter. The average plateau moduli of both samples are also comparable up to 50 °C. However, the rubbery plateau is significantly narrower in T2-H than it is in T2-L. The narrower rubbery plateau of T2-H is expected to arise due to (1) the lower 〈Mw〉 of T2-H as compared to T2-L (recall ); this will lead to a lower number of entanglements per chain, (2) difficulty in the hydrogen bonding/organization (or packing) of branched HS in T2-H, and (3) slightly shorter average length of the HS in the branched architecture. The shorter average HS length, and the branched structure of HS in T2-H is expected to promote increased intersegmental mixing and therefore contribute to a narrower rubbery plateau. This last observation follows from the expectation that T2-H should have slightly shorter HS than T2-L because the HS content of the highly branched samples is calculated by assuming the terminal end-capping moieties, namely CHI, to be part of the HS. From the tan |
δ response of T2-H, it can be again noted that the breadth of the SS glass transition and the peak transition temperature, Tg (−78 °C) are very similar to T2-L. However, due to reasons noted above, the rise in tan |
δ, which corresponds to the decrease in E′ occurs at a lower temperature (ca. 50 °C) than in T2-L (200 °C).(a), the DMA response of the CHI end-capped T2-H (28 wt% HS content) is compared with its counterpart, that is, without the end-cap moiety (T2-H no CHI, 25 wt% HS content). Despite the slightly lower HS content and the likely reduced, if any, intermolecular bidentate hydrogen bonding of the latter, no significant differences are noticeable between the two highly branched PTMO-2040 based samples.(b), the DMA behavior of PPO-2030 based linear and highly branched samples is very similar to that of the corresponding PTMO-2040 based samples presented in (a). However, some noteworthy differences can also be observed. First, the PPO SS Tg occurs at −57 °C as compared to the PTMO SS Tg at −78 °C, principally due to differences in the chemical structures of the two types of SS and also due to potential differences in the extent of microphase separation (see below). The Tg of the pure PPO oligomers is reported to be approximately −73 °C Considering the series of copolymers based on PPO, note that the overall 〈Mw〉 of these copolymers (), with the exception of P2-L and P12-H, is comparable and in the range of 100,000 g/mol. Schollenberger and Dinbergs (c)–(e)), a systematic decrease in the SS Tg and also the breadth of the SS glass transition are observed (from both the E′ and tan |
δ responses) with increasing PPO MW. This trend results because when the HS content of the copolymers is maintained constant, the longer SS experience fewer restrictions, which therefore form fewer junctions with the HS. O'Sickey et al. (c)–(e), as the PPO SS MW in increased, the rubbery plateaus become less temperature sensitive and extend to higher temperature (ca. 200 °C) primarily due to an improved extent of microphase separation (discussed below). In comparison to the linear PUUs, the rubbery plateau of the highly branched analogs based on PPO-4040 and PPO-8000 begins to decay earlier due to their branched architecture, shorter HS length as compared to their linear analogs, and potentially lower extent of microphase separation. Of particular interest is the observation that the breadth of the rubbery plateau of P12-H is nearly comparable to its linear analog, P12-L. The fact that the SS MW in these two copolymers is well above the Mc of PPO may be responsible for such behavior. Another important reason why the rubbery plateau of P12-H is comparable to P12-L is the low 〈Mw〉 of P12-H (53,200 g/mol), which means that the degree of branching in this sample is only modest.The pin-hole collimated ambient temperature SAXS profiles of PPO and PTMO based linear and highly branched PUU copolymers are presented in . The corrected intensities are plotted as a function of the scattering vector, s, where s=2sin(θ/2)/λ, θ is the radial scattering angle and wavelength λ=1.542 Å. The scattering profiles of all 11 samples exhibit distinct first-order interference peaks. The SAXS data suggest that all the samples in this study possess a microphase separated morphology, consistent with the respective ambient temperature DMA responses. The interdomain spacing of a given sample, approximately 1/smax, according to Bragg's Law, is noted in the legends in The SAXS data for the linear PUUs based on PPO are presented in (a). From this figure it can be observed that with increasing SS MW, the interdomain spacing increases from approximately 80 Å for P2-L to 145 Å for P12-L. The absolute intensity of the first order interference peak also increases with SS MW, although this behavior is not readily apparent because the SAXS profiles have been translated vertically for clarity. Simultaneously, the breadth of the first-order interference peak decreases systematically with increasing SS MW. However, direct comparison of the peak breadth is not warranted since the abscissa (namely, `s') is non-linear with respect to the structural scale length (Bragg spacing).Focusing on the SAXS data for the microphase separated highly branched PUUs based on PPO, presented in (b), several remarkable features can be noted. First, a marked systematic narrowing of the breadth of the first order interference peak is observed. This trend is consistent with the behavior of the linear PUU samples. Second, the position of the first-order interdomain interference peak shifts generally to lower angles with increasing SS MW, as observed for the linear PUU samples. The only exception to this trend is found for sample P12-H, with a domain spacing of 179 versus 196 Å for sample P8-H. The underlying reasons for such behavior are not understood at this point but the data have been reconfirmed.Finally, a distinct apparent weak higher order interference shoulder occurs in the scattering profiles of P8-H and P12-H (see arrows in (b)). In both samples, the position of the higher order reflection corresponds to a ‘d’ spacing that is half the spacing of the corresponding first-order interference peak. Kinning and Thomas The SAXS profiles of the three PTMO based linear and highly branched PUUs are presented in (c). These three samples also exhibit first-order interference peaks, which again indicate microphase separated morphologies. The interdomain spacing of these PTMO-2000 based samples is higher than that of the PPO-2030 counterparts in (a) and (b). This difference can be attributed to the difference in the average number of bonds along the backbone of PTMO-2040 (142) and PPO-2030 (105) and the unperturbed chain dimensions between PTMO and PPO of comparable average MW, as well as the broader MWD of the PTMO segments (1.87) as compared to the PPO segments (<1.03). O'Sickey et al. Both DMA and SAXS data indicate that the segmented PUUs addressed in this study possess a microphase morphology at ambient temperature. In addition, the ambient temperature morphology of these copolymers is especially amenable to investigation by tapping-mode AFM. The phase images of the free surface of solution cast PPO-based PUUs captured at ambient temperature are presented in . The dark and the light regions correspond to soft and hard domains, respectively, and all eight PPO-based highly branched PUU samples show separate regions of light and dark indicative of a microphase separated morphology, in agreement with the SAXS and DMA data. Beginning with samples P2-L and P2-H, in (a) and (b), respectively, the morphology seems best characterized as microphase separated but disordered. The phase image of the P4-L ((c)) is very similar to P2-L. On the other hand, the hard domains in sample P4-H ((d)) appear to adopt a more short rod-like shape. While the hard phase in this sample may at first appear to possess considerable degree of hard phase percolation, as will be shown later, its corresponding stress–strain curve does not exhibit any yielding behavior, strongly suggesting that the hard phase in P4-H is not percolated throughout the matrix.(e)–(h) show AFM images for samples P8-L, P8-H, P12-L, and P12-H, respectively. In general, the hard domains in these samples appear to be spherical or particulate-like. The phase image of the sample P8-H ((f)) is particularly striking. For clarity, the same image as (f) is reproduced at higher magnification in (b)) as an inset. The spherical, particulate nature of the hard domains is apparent, and the average spacing between the domains is approximately 200 Å, which corresponds well with the first-order SAXS interference peak (196 Å). Moreover, from this image the presence of limited short-range order in the hard domain packing can be observed. The sample P12-H also exhibits similar morphology as seen from In this section, the stress–strain response, the extent of stress relaxation, and the mechanical hysteresis behavior of the linear and highly branched PUU copolymers are presented and compared. The stress–strain results of the PTMO and PPO-based samples are presented in (a) and (b). The Young's moduli of these samples are listed in (a), the samples T2-L and T2-H display no distinct yield point, which strongly suggests that in these particular systems, 28 wt% HS is insufficient to enable extensive percolation of the hard phase. The AFM images of the PPO-2030 based samples also indicated insufficient percolation of the hard phase. The Young's modulus of T2-L (listed in ) is, surprisingly, lower than T2-H. However, the highly branched copolymer without end-capping, T2-H-no CHI, displays the lowest Young's modulus amongst the three PTMO-based copolymers due to the potentially lower extent of intermolecular bidentate hydrogen bonding and the slightly lower HS content in this sample (25 wt%) as compared to T2-L and T2-H (28 wt%). The strain hardening at higher extensions noted in the linear sample T2-L, undoubtedly occurs due to the well know ability of PTMO (MW>ca. 1500 g/mol) to strain-induce crystallize at greater than approximately 400% strain. WAXS was utilized to investigate the similar strain hardening behavior exhibited by the highly branched sample T2-H. The 2D ambient temperature WAXS pattern () of T2-H uniaxially deformed to 400% strain demonstrates two distinct equatorial reflections at ‘d’ spacings of ca. 4.5 and 3.3 Å, which are due to the crystalline PTMO phase In addition, the tensile strength of T2-L is higher than that of T2-H and T2-H-no CHI, possibly due to the substantially higher overall 〈Mw〉 of T2-L. However, both highly branched samples display a slightly higher elongation at break. Furthermore, while the DMA and SAXS behavior of the samples T2-H and T2-H-no CHI are very similar, the Young's modulus and tensile strength of T2-H are higher than those of T2-H-no CHI. This behavior highlights the benefit of end-capping the chain ends with a moiety that can facilitate intermolecular bidentate hydrogen bonds for structural applications.The stress–strain behavior of samples P2-L and P2-H is presented in (a). As expected, these two samples, like their PTMO counterparts, do not display a distinct yield point. In addition, the inability of the atactic PPO SS to strain induce crystallize and their comparatively lower overall 〈Mw〉 () results in copolymers with lower tensile strength as compared to the PTMO based copolymers. However, both P2-L and P2-H display a remarkably high strain at break, approximately1200%. The PPO-4040 and 8000 based materials ((a) and (b)) display generally superior Young's modulus (values listed in ), tensile strength, and elongation at break amongst the series. While the overall 〈Mw〉 of the linear copolymers influences the ultimate properties of the samples, the improved stress–strain behavior of the PPO-4040 and 8000 based samples in comparison to the PPO-2030 and 11,800 based counterparts is not fully understood at this point. However, as expected, the tensile performance of the linear copolymers is distinctly superior to that of their highly branched analogs.Stress relaxation of the copolymers is another mechanical property of practical interest for systems intended for structural applications. In light of the highly branched copolymers' greater difficulty to entangle, the extent of stress relaxation in these materials is expected to be higher than in their linear analogs. The extent of stress relaxation of the copolymers addressed in this report is presented in . Stress relaxation is quantified as the ratio of the absolute decrease in the stress at t=10,000 s (ca. 3 h) to the stress recorded immediately after the sample was first stretched. The samples were uniaxially deformed to 25 or 500% strain and different samples were used for each test. A crosshead speed of 150 mm/min was utilized to reach the required strain. As expected, at both strain levels the extent of stress relaxation of T2-H and T2-H-no CHI is higher than in the linear analog, T2-L. In addition, due to the absence of the end-capping moieties (CHI), which results in a reduced level of intermolecular hydrogen bonding in T2-H-no CHI, it displays higher stress relaxation than the CHI end-capped T2-H. A comparison of the samples T2-L and T2-H with P2-L and P2-H reveals that due to the ability of PTMO to strain induce crystallize both, T2-L and T2-H relax a smaller extent relative to P2-L and P2-H. Amongst the PPO based samples, at 25% strain the extent of stress relaxation exhibited by the highly branched PUUs is generally higher than the linear analogs. At 500% strain, a similar trend can be observed in the PPO-2030 and 4040 based copolymers. The sample, P8-H failed ca. 1000 s after it was stretched to 500% strain, and the stress relaxation of P12-H could not be measured because its strain at break is less than 500% (recall The mechanical hysteresis behavior of materials is also of practical importance and it was utilized to further compare the linear segmented PUUs with their highly branched analogs. The mechanical hysteresis of the PTMO and PPO based copolymers during two consecutive cycles to 25 or 500% strain is presented in (a) and (b), respectively. In general, at both strain levels the mechanical hysteresis during cycle 1 is greater than during cycle 2. Such behavior is not surprising because once the original microstructure becomes partially disrupted during the first cycle it does not have enough time to completely ‘heal’ before the next cycle is initiated. The mechanical hysteresis of linear PTMO-2040 and PPO-2030 based PUUs at 25% strain is, as expected, distinctly lower than their highly branched analogs. However, for the other PPO based materials the difference in the mechanical hysteresis between the linear and highly branched samples appears to be similar. Surprisingly, at 500% strain ((b)) the percent mechanical hysteresis of the linear materials and their highly branched analogs are comparable. Such behavior indicates that the substantial disruption of the microstructure upon deforming the copolymers to 500% strain apparently overrides differences in the entanglement density between the linear and highly branched materials. At 500% strain, despite the ability of the PTMO to strain-induce crystallize, the samples T2-L and T2-H display mechanical hysteresis that is comparable to P2-L and P2-H—this seems a bit surprising to the authors, but the results have been reconfirmed.The above discussion on the comparison of the mechanical properties of the copolymers indicates that the tensile performance of the highly branched PUUs is generally lower than analogous linear PUUs with identical SS MW and overall HS content. The difficulty of the highly branched architecture to entangle to the same degree as their linear analogs is undoubtedly a major reason for such a difference in tensile properties. However, it is well known that in dilute solutions, the hydrodynamic volume of a branched polymer is lower than that of a linear polymer of comparable MW. In addition, when the concentration of the polymer solution is increased such that it is in the semi-dilute regime where chain overlap is expected, the lower entanglement density of branched polymers also lowers the viscosity of the polymer. Since a material of lower viscosity can be more easily processed, controlled incorporation of branching can be utilized to improve the processability of polymers. The effect of architecture on viscosity was investigated by using a cone-and-plate rheometer to measure the ambient temperature solution viscosity of selected PUUs addressed in this report. The results are presented in The Newtonian (zero shear) solution viscosity of a sample was determined from the initial slope of the linear region of the shear stress versus shear rate plot. The upper shear rate limit of this linear region (before the onset of shear thinning) is also listed in . The PTMO based linear sample was too viscous for the torque bar of the instrument to measure. In light of its substantially high overall 〈Mw〉 such behavior is not surprising. However, the solution viscosity of T2-H was measured to be 410 mPa s. The lower solution viscosity of T2-H is believed to be primarily due to the reduced entanglements of its highly branched molecules. The lower overall 〈Mw〉 of T2-H is also expected to be a contributing factor. On the other hand, the solution viscosity of the linear samples based on PPO-2030 as well as PPO-4040 is higher than the viscosity of the respective highly branched analogs despite the significantly higher 〈Mw〉 of the branched systems. Such behavior provides further support to the argument that polymer chains with a highly branched architecture encounter considerable difficulty to entangle in the semi-dilute regime and consequently the reduced entanglement density of the highly branched polymers lowers their solution viscosity.The solid-state structure–property behavior of highly branched segmented PUUs was compared with their linear analogs. DMA, SAXS, and AFM were used to demonstrate that all the copolymers investigated in this report possessed a microphase separated morphology. From the DMA results it was also noted that despite the incorporation of branching, that is, HS branching, the breadth of the SS glass transition and the SS Tg (and therefore, the extent of microphase separation) of the highly branched PUUs was very similar to their linear analogs. Such behavior is consistent with the results presented in an earlier publication from our laboratory The strain-induced crystallization of the SS phase in the PTMO-2040 based highly branched PUU at ambient temperature was another noteworthy result. Due to such behavior, the stress–strain response of the highly branched PUU was comparable to its PTMO-based linear analog. The stress–strain response of the sample not end-capped with CHI was markedly poorer than its linear analog, even though strain hardening could still be observed in the sample that was not end-capped.The stress–strain response of the PTMO or PPO based copolymers did not exhibit any yield point, thereby suggesting that at 28 wt% HS content there was insufficient percolation of the hard phase through the soft matrix, a fact confirmed by AFM. The stress–strain response, the extent of stress relaxation, and the mechanical hysteresis of the highly branched PUUs were poorer in comparison to their linear analogs, there being one exception. Surprisingly, the mechanical hysteresis of the highly branched PUUs at high deformation (500% strain) was comparable to the linear samples.Limited ambient temperature solution rheological experiments indicated that the highly branched PUUs had significantly lower viscosity than their linear analogs of nearly equal molecular weight.Form-invariant representation of fatigue crack growth rate enabling linearization of multiscale dataMultiscaling of fatigue crack growth rates for metal alloys is revealed by the spread of the data in three different regions known as I, II and III. They are referred to, respectively, as the small, long and fast crack regions. A fourth and perhaps fifth region can also be identified for cracks that are even smaller although the morphology of the line crack configuration may become dubious as the size scale is reduced. For metal alloys, the approximate crack size range is a |
≈ 10–80 mm (macromechanical) for regions II and III, a |
≈ 10−3 to 10−1 |
mm (microstructural) for region I and presumably a |
≈ 10−4 |
mm or smaller (nanochemical) for the other regions. The division among these regions can be made to disappear when the same data are re-interpreted by using a dual scale micro/macro line crack model that consists of three essential parameters d∗ (micro/macro length). μ∗ (micro/macro material constants) and σ∗ (external/internal stress ratios). The model can also accommodate two incidental variables (φ1, |
φ2) for the double singularity representation to account for specific microstructural behavior. More complex models involving more than two incidental variables can be developed if the need arises. Otherwise, the function with (φ1, |
φ2) can be absorbed into one of the empirical parameters used for correlating the da/dN relation. That is the microscopic effects are reflected indirectly although not sensitively by the macroscopic parameter. This will be demonstrated by the use of the dual scale crack model.Through crack fatigue data for the 7075-T6 and 2024-T3 aluminum panels of the three regions I, II and III are shown to lie on straight lines with different slopes and y-intercepts that correspond to different mean stress σm and stress amplitude σa when the volume energy density range is applied in the dual scale model. The slopes of these lines ranged from 2.890 to 3.019 while the spread of σm was from 9 to 12 kg/mm2 and the spread of σa was from 2.5 to 6.5 kg/mm2. The slopes of the straights obtained had a wider scatter of 3.532–4.224 when the energy density range was replaced by the stress intensity range. This was expected because the stress intensity approach does not account for the mean stress effect. The method of approach established for dual scaling applies equally well to micro/nano cracks such that when connected with the macro/micro cracks it can be further extended to the development of triple scale models involving macro/micro/nano cracks.Crack initiation and propagation are recognized as two separate processes in conventional fracture mechanics analyses and they are treated accordingly The concept of scaling is not new as it is well known in relating material and kinetic parameters of the model to the full size structure. But these techniques apply mostly to rigid bodies. They do not address the change of the inhomogeneity of the deformable and damageable solid. Specimen size effects are also well known and they have been traditionally attributed to statistical effects. Continuum mechanics has attempted to introduce length parameters into theories that depend not only on the local strain but also strains everywhere in the domain, thus involving the gradients and hence length parameters. The strain gradient approach, however, still invokes vanishing of the rate of change of volume with surface as the continuum element size is made to approach zero. The addition of length via strain gradients and the exclusion of size by assuming the element to be a point are two different propositions that have not been made clear. The higher order or non-linear theories of continuum mechanics by in large stopped short at quantifying physical processes that involve damage at both the macro- and micro-scales. A case in point is the theory of elastoplasticity. Loosely speaking, non-linearity is said to be associated with the advent of microscopic damage or with plastic deformation. Such a notion has handicapped the development of other models to account for energy dissipation at different time and size scales. This is particularly true in the failure of material in fatigue, where crack initiation and propagation is caused by damage accumulation even though the practice relies on the use of elasticity and not plasticity for good reasons. Up to now, the threshold of failure can only be determined by empirical means. All of the existing theoretical approaches stop short at the onset of global instability since testing was first set foot to the characterization of material strength.The realization that the final failure of solids involve the separation of material in the form of fracture led to the testing of pre-cracked specimens In recent years, there has been a resurgent interest to examine fatigue crack growth in view of the different crack size scale involved in the correlation of the da/dN data Quantitative assessment of the crack growth requires an analytical representation of the crack configuration as it propagates. The process of fatigue entails the transition of a microcrack changing into a macrocrack. Self-similarity must therefore be excluded. The difference has to involve some fundamental change when a crack enlarges from one scale to another since the microstructure of the material recognizes such changes. For polycrystals, the grain size would be a good land mark to use for size distinction. This necessitates the use of a length parameter for comparison. Let this be d∗. The stiffness of the material or modulus can be a factor. It is relative involving a comparison between the micro- and macro-modulus, say μ∗. Moreover, damage by cracking cannot leave out the external load with reference to the internal resistance. This ratio is accounted for by σ∗. Three minimum essential parameters d∗, μ∗ and σ∗ are thus identified. When the crack is large or macroscopic, it suffices to consider the bulk material properties, microstructure effects can be averaged out. The same will not hold when the crack is small where d∗, μ∗ and σ∗ need to be accounted for. Additional consideration is needed because materials possess special microstructure effects that stand out as being dominant. The α and β phase in titanium alloys are important and their effects should be reflected. Hence, incidental variables will also be needed to contrast the microstructure-dependent behavior of one alloy to another. These variables can be denoted as (φ1, |
φ2) in Additional incidental variables can be accommodated by using a higher order eigenfunction expansion technique which is in the stage of being developed. But the real advantage of the method is that it need not consider anymore variables or parameters than it is necessary. Attention can be restricted to small segments of time and size scales to limit the description of a single physical event of interest. It will also be shown that dual scale crack model is sensitive to the crack growth rate relation assumed. The difference between the ΔK and ΔS model in fatigue will become obvious in the work to follow. Here, K stands for the stress intensification and S for the volume energy density.(σij)micromacro=(KS)micromacro2πr0.75fij(φ1,θ)+(KW)micromacro2πr0.25gij(φ2,θ)+⋯The method of scale multiplier can be used can be expressed in terms of (KS)micromacro alone as(σij)micromacro=(KS)micromacro2πr0.75fij(φ1,θ)+rdgij(φ2,θ)+⋯The notation ()micromacro applies to the dual scale model to denote that the crack can be micro or macro depending on the ratio d/r (or d∗) and the restraining stress σ0 to applied stress σ∞ ratio is given by σ∗ which is contained in (KS)micromacro. Large σ∗ would close the crack making it invisible and the opposite would result if the restraining stress is small. Refer to . This effect is particularly important in fatigue where loading and unloading take place to open and close the crack continuously. Such a mechanism is inherent in the mechanics of the dual scale model. As it can be seen from Eq. , the local crack tip stresses depend on the relative distance r/d. Such a character is necessary for addressing micro/macro effects. Since both (KS)micromacro and (KW)micromacro are related by Eq. , it suffices to consider (KS)micromacro and superscript S can be dropped such thatKmicromacro=16π(1-νmacro)μmicroσ∞c2-a2f(φ1,φ2)μmacrod0.251-2σ0πσ∞sin-1achas been made. Also, the superscript S in (KS)micromacro will be dropped in the work to follow. The relation between c and a can be found in Once the local stresses are known, the local energy density follows and it will also have the character of dual scale. SinceSmicromacro=(1-νmacro)2(c2-a2)128f2(φ1,φ2)μmicrod0.5r0.5μmicroμmacroσ∞21-2σ0πσ∞sin-1ac2F(θ), S depends on the radial distance r from the micro/macro crack tip and the angle θ, not to mention the other micro/macro parameters. That is the energy density intensification is non-uniform around the crack. The quantities fr(θ), fθ(θ) and frθ(θ) are complicated functions of θ and φj with j |
= 1, 2. They can be found in may be derived where Smicromacro can possess other inverse r dependencies. The choice will be dictated by the type of physical process need to be modeled.There is an inherent hierarchy of size and time coupling effects associated with material damage by fracture. Crack initiation in metal alloys is usually a slow process involving small defects which will grow in size from micro to macro and pick up speed until rapid fracture takes place. These events can be reflected by the fatigue data illustrated in where A and n can be found empirically. By tradition, ΔK in Eq. is taken as the macro-stress intensity factor such that it depends on the stress amplitude range and the square root of the half crack length a for a panel with a central crack with the full length 2a. The understanding is that Eq. marked for long cracks for a from 10 to 40 mm in aluminum alloys To begin with, there are obviously as many choices as variance in the test data if the correlation of da/dN to ΔK or the equivalent is purely empirical. A possible alternative would be As a priori assumption, the repeating action of opening and closing of the crack surface as illustrated in will be regarded as the major consideration such that a crack can change from micro and macro or vice versa as loading and unloading takes place in fatigue. This effect can be controlled by the tightness ratio σ∗ of the contacting crack surfaces in addition to the micro/macro interaction μ∗ from the material and a relative length parameter d∗. These are the requirements to be included in a dual scale model based on the local stress intensification Kmicromacro, the difference of the maximum and minimum values of which can be denoted byΔKmicromacro=(Kmicromacro)max-(Kmicromacro)min and letting σ∞ equal to σmax at maximum stress and to σmin at minimum stress, it is found that(Kmicromacro)max=16π(1-νmacro)μmicroc2-a2f(φ1,φ2)d0.25μmacro×1-2σ0πσ∞sin-1acσmax(Kmicromacro)min=16π(1-νmacro)μmicroc2-a2f(φ1,φ2)d0.25μmacro×1-2σ0πσ∞sin-1acσminΔKmicromacro=16π(1-νmacro)μmicroc2-a2f(φ1,φ2)d0.25μmacro1-2σ0πσ∞sin-1ac(σmax-σmin)Note that σ∗ |
= |
σ0/σ∞ and μ∗ |
= |
μmicro/μmacro. The parameter d∗ is d |
= |
c |
− |
a in normalized with respect to some length dimension. The incidental variables φ1 and φ2 have been identified with different microcrack branching patterns by determining the appropriate φs which would be beyond the scope of the present discussion. The use of Eq. yields a fatigue crack growth rate formally similar to Eq. accounts for the transition of crack growth from microcracking in region I to macrocracking in region II. If the microstructure details are not of direct interest, then the incidental variables φ1 and φ2 can be absorbed into the macro-parameters. To this end, Eq. may be multiplied by the factor 3f(φ1, |
φ2)/40 to renderΔKmicromacro=6π(1-νmacro)μmicroc2-a25d0.25μmacro1-2σ0πσ∞sin-1ac(σmax-σmin)That is a new parameter C can be defined asEffectively, the microstructural variable (φ1, |
φ2) has been absorbed into the newly defined empirical constant C in Eq. . The other microscopic material constant such as μmicro in Eq. may be further absorbed by introducing another parameter. It is apparent that microscopic effects do enter into C or A in Eq. . For macroscopic analyses, it suffices to include microscopic effects via the Young’s modulus and yield strength. This explains why Eq. using the classical ΔK for macrocrack has worked well in many engineering applications.The fundamental difference between using ΔKmicromacro in Eq. is not only scaling for crack growth but also to include non-local effects that have been excluded in the classical ΔK approach of the stress intensity factor that invoked the limit of the distance r to approach zero or the crack tip. This limitation does not exist in the ΔSmicromacro approach because the distance r in Eq. is required to remain finite. In this sense, ΔSmicromacro can be regarded as the energy released when the crack extends by the amount r |
= Δa. Now let (Smicromacro)max be associated with σmax and (Smicromacro)min associated with σmin such thatΔSmicromacro=(Smicromacro)max-(Smicromacro)minΔSmicromacro=(1-νmacro)2(c2-a2)128f2(φ1,φ2)μmicrod0.5r0.5μmicroμmacro21-2σ0πσ∞sin-1ac2F(θ)(σmax2-σmin2)A two-parameter fatigue crack growth rate relation can thus be obtainedThis is reminiscence of the classical strain energy density approach to fatigue in is replaced by ΔS for macrocracking only. Again, a simplified version of Eq. may be used by leaving out the details of the incidental parameters such that,The simplification lies in reducing Eq. ΔSmicromacro=(1-2νmicro)(1-νmacro)2(c2-a2)4μmicrod0.5r0.5μmicroμmacro21-2σ0πσ∞sin-1ac2(σmax2-σmin2)This completes the formulation of the dual scale line crack model. Discussion of the numerical results will follow.The two-parameter crack growth rate relation expressed by Eq. depends on the experience and knowledge of the user, particularly in recognizing the restrictions of A and n. The present fracture control methodology relies heavily on tests and analyses where Eq. remains as the corner stone. Hence, preservation of the use of the two parameter da/dN relation becomes an issue. Referring to for the fatigue crack growth rate data of metal alloys, three distinct regions I, II and III can be identified. They refer to the growth of short, long and fast cracks, respectively. The two parameter fit applies to region II where the equationcan be used to find A being the y-intercept of the line with slope n in . The three regions I, II and III are also made reference to the crack growth data in (a) and the invariant form of ΔKmicromacro and ΔSmicromacro. The straight line representation in (b) is made possible by the use of ΔKmicromacro for the dual scale model. The classical ΔK in Eq. . It is thus required to rotate regions I and III clockwise so as to arrive at the straight line in (b). Linear interpolation can then be used to connect the data for scaling purpose. For microcrack and macrocrack growth in addition to those for rapid crack propagation, the re-interpretation of the fatigue data in As mentioned earlier, accommodation for the opening and closing of the crack surfaces is a major consideration for modeling the fatigue crack growth behavior. This implies that the restraining stress σ0 in relation to the applied stress σ∞ or the ratio σ0/σ∞ should vary with the crack half length a. Assuming thatσ0σ∞=0.65fora=3mm0.48421fora=10mm0.01049fora=65mmthe bilinear relationship between the stress ratio σ0/σ∞ and the crack half length a is adopted.σ0σ∞=-0.023684a+0.72105for1⩽a⩽10mm-0.008613a+0.57034for10⩽a⩽65mmThe unit of length a is millimeter. The two stage bilinear relationship is displayed in . The corresponding numerical values of the curve in As the micro/macro crack extends, the tip opening segment length d can also have an effect. That is local intensification will depend on the relative distance r/d which will increase with a, as the microcrack becomes macroscopic. This ratio is not sensitive for very small cracks. The behavior is shown in and the numerical results can be found in is independent of the radial distance r, d |
= 1 mm will be used. The curve in applies only to the ΔSmicromacro of Eq. . This difference is similar to the different microcracking patterns predicted It suffices to know the trends of the curves in for application of the dual scale model. Moreover, only is needed to make the form of the crack growth rate relation in Eq. invariant. However, better correlation of fatigue data is obtained using Eq. is also used. The results become less sensitive to mean stress effects that Eq. will be used such that data in regions I, II and III as shown in would lie on a straight line without any bends. The shear modulus ratio μmicro/μmacro will be set at 2 because its value is not known to change with the crack length. The parameters σ∗ and d∗ will account for the transition of microcracking to macrocracking where the property of crack similitude that was first discussed in with R being the mean stress ratio σmin/σmax.Making use of the numerical results provided earlier, the log of da/dN versus log of ΔKmicromacro may be plotted. The curves for 2024-T3 and 7075-T6 aluminum alloy are displayed in . Note that the data in all three regions I, II and III can now be adequately represented by straight lines for 2024-T3 and 7075-T6 although they have different slopes. The slope signified by n for the former with 3.532 is smaller than that of the latter with 4.179. This means that the crack growth rate of 2024-T3 is slower than that of 7075-T6 which is to be expected. The x-coordinate in has been shifted and did not start from zero. It is clear from that the y-intercepts give log |
C |
= − 6.967 for 7075-T6 and −6.710 for 2024-T3. When the stress amplitude σa is raised from 2.5 to 4.0 kg/mm2 for the same mean stress σm |
= 12 kg/mm2 although the stress ratio R is reduced from 0.655 to 0.500, a slight increase in the crack growth rate da/dN for both materials are seen. This is the result of increase in the slopes of the straight lines for 2024-T3 (n |
= 3.632) and 7075-T6 (n |
= 4.297). The straight line fit in is good because most of the data points pertained to half crack length in the range 10–40 mm corresponded to region II.A further increase of the stress amplitude to σa |
= 6.5 kg/mm2 with σm |
= 12 kg/mm2 accompanied by a further reduction in R to 0.297 show that the slopes of da/dN acquire further gain such that n |
= 4.224 for 2024-T3 and n |
= 4.742 for 7075-T6 although the data gained more scatter, . The detrimental effect of increase stress amplitude on fatigue life is exemplified as in the use of the classical S–N curve where no consideration to crack growth was made. Reduction of the maximum stress also has a noticeable effect on the crack growth rate. Consider Case IV in where σa |
= 2.5 kg/mm2 with σm= 9 kg/mm2 and R |
= 0.565 being similar to that in Case I. A complete set of data points for all three regions were obtained as exhibited in . Decrease in the slopes of the curves confirm a reduction of the crack growth rate when the stress amplitude is decreased. The effect of stress amplitude starts to take over in Case V when σa |
= 4.0 kg/mm2 with σm |
= 9 kg/mm2 and R |
= 0.385. shows that the slopes of the curves are increased to n |
= 3.725 for 2024-T3 and n |
= 3.781 for 7075-T6. Further aggravation of the crack growth rate is seen by the curves in when the stress amplitude is raised to σa |
= 6.0 kg/mm2. Now the slopes attain even higher values of n |
= 3.812 for 2024-T3 and n |
= 4.511 for 7075-T6. for a comparison of the data for the 2024-T3 and 7075-T6 material, particularly with reference to the parameters C and n in Eq. for the ΔKmicromacro model. The stress amplitude is seen to play a dominant role by the fatigue data in . Cases I–III show that the increase in stress amplitude tends to increase the crack growth rate by raising both the y-intercept log |
C and the slope n. The reduction of maximum stress reflected by lowering the mean stress σm for Cases IV–VI did lower the da/dN as shown by Case IV but the further increase in the stress amplitude again raised the absolute log |
C and n values as summarized in The difference between ΔKmicromacro in Eq. is not insignificant and will vary for different test conditions and materials. Based on the same crack growth data as those used for obtaining the curves in displays a plot of log(da/dN) versus logΔSmicromacro. All data points fell on straight lines, one for 2024-T3 and one for 7075-T6 with the respective slopes of m |
= 2.890 and 3.394. These values are smaller than those of n |
= 3.532 and 4.179 in for Case I of the ΔKmicromacro model. The difference in slopes for the two materials of the two models is about the same. shows similar trends for Case II where the data are more scattered as the stress amplitude was increased from 2.5 to 4.0 kg/mm2. Much more scatter of the data resulted in Case III where σa is increased to 6.5 kg/mm2 with σm kept at 12 kg/mm2. This is also accompanied by an increase in the crack growth rates as illustrated in , however, are much closer together when the same data are represented by the ΔKmicromacro model shown by the curves in inclusive. A good fit of the data is found in for Case IV. The two straight lines for 2024-T3 and 7075-T6 are nearly parallel with slopes of m |
= 3.036 and 3.055. The superiority of the fatigue crack growth resistance for 2024-T3 does not show up for the allied mean stress and stress amplitude. The 7075-T6 aluminum would have served equally well. This illustrates the location specificity when selecting the use of materials in non-uniform stress and/or energy density fields. The crack growth behavior of the two materials start to deviate from one another as the stress amplitude is raised to 4.0 kg/mm2 with a mean stress of 9 kg/mm2 as in Case V where the plot of ΔSmicromacro is given in . Case VI for σa |
= 6.0 kg/mm2 correspond to the results in . Data for large stress amplitudes tend to have more scatter. It is worthwhile to note the slopes m in . An important conclusion is that the variations of the m values are much less than those for n as the mean stresses change. This is to be expected because the ΔSmicromacro model accounts for the mean stress effect whereas the ΔKmicromacro model does not. This concludes the presentation of the numerical results for the fatigue crack growth rate where the data were transformed onto a straight line relationship. The position of the x-axis in has been shifted. The difference in reference can be seen from the curves in should now be clear. Moreover, the logΔSmicromacro in are negative while ΔSmicromacro in Eq. Metal fatigue has intrigued the material engineers for more than a century and the problem remains not understood. By in large, the effort has been concerned with relating the onset of rapid fracture to the initiation of cracks from small defects in the material. In more recent years, a resurgent of interest can be evidenced in the open literature concerning the fatigue crack growth rate da/dN models with reference to the commonly known regions I, II and III using the stress intensity factor range ΔK as the indicator. The concerns are whether data for micron size and smaller cracks could still be adequately correlated by the two-parameter relation. Hence, a fourth region for crack nucleation has been proposed Can the two parameter da/dN versus ΔK, ΔKmicromacro or some other forms be still used regardless of the proposed form of ΔK? What is at stake is whether the form-invariant assumption used in this work has any limitations?Can the same ΔK be used for cracks at the different time and size scales? The answer is likely to be “no”, because fatigue data in regions I, II, III, etc. do not seem to follow any consistent treatment in physics nor in correlation.What has been learned from experience is that through crack specimens used by aircraft structure engineers pertain to a |
≈ 10-80 mm (macromechanical) known as region II. Material engineers are more concerned with surface cracks that are associated with region I where a |
≈ 10−3 to 10−1 |
mm (microstructural). The fourth region would presumably fall into the scale range of dominance of chemical reactions. This corresponds to a |
≈ 10−4 mm or smaller (nanochemical) Noteworthy of mentioning is that the relation between the descriptive damage mechanisms and the corresponding analytical representation of the empirical models is not always clear or speculative at best. To begin with, length alone is not adequate for distinguishing micro- and macro-cracks as all aircrafts are known to fly with micro-cracks several centimeters or longer. According to the traditional theory, these length dimensions would have exceeded the critical values for trigger rapid crack propagation. Tightness of the crack mouth is an important factor that should be accounted for. The physical model of a microcrack should differ fundamentally from that of the macrocrack. The clue that there prevails a very small segment of the micro-crack tip that does not close led to the discovery of a double singularity The Theory of Critical Distances to estimate the static strength of notched samples of Al6082 loaded in combined tension and torsion. Part I: Material cracking behaviourThe present paper is the first of a two-part series reporting an experimental and theoretical study of the fracture of circumferentially notched samples of a commercial aluminium alloy, i.e. Al6082, subjected to tension, torsion and mixed tension/torsion loading. The overall aim of the work was to investigate the use of a particular method of failure prediction, known as the Theory of Critical Distances. This first part reports the experimental data – load–deflection curves and observed material failure modes – and discusses the consequences of these findings for the development of the theory, which is covered in the second part. It was observed that relatively blunt notches loaded in tension failed by a conventional ductile fracture mode similar to plain (unnotched) specimens. However, in tensile specimens containing sharp notches, failure occurred via the initiation, stable propagation and, finally, unstable propagation, of circumferential ring cracks. Under torsional loading, and independent of the notch root radius, static failures of the tested samples always occurred by the formation and stable propagation of ring cracks. Under mixed-mode loading there was a gradual transition between the ductile and brittle modes and between stable and unstable cracking. For all types of loading, it was observed that crack initiation always coincided with peak loading conditions, and that cracks invariably grew on the plane perpendicular to the specimen’s longitudinal axis.net stress concentration factor under tensile loadingnet stress concentration factor under torsional loadingDue to their well-known weakening effect, stress concentration phenomena are always a matter of concern to structural engineers engaged in performing the static assessment of real components. According to this fact, since the pioneering work done by Galileo Galilei by considering plain materials, researchers investigating such a problem have made a big effort to formalise sound methodologies suitable for estimating static strength when mechanical assemblies experience stress concentration phenomena.Examination of the state of the art shows that, since about the middle of the last century, a lot of work has been done in this area of the structural integrity discipline to take full advantage of emerging Linear Elastic Fracture Mechanics (LEFM). As to its accuracy and reliability, even if the use of this powerful theory was seen to be successful, strictly speaking, it applies only to those situations involving sharp notches, that is, to those stress concentrators which behave like long cracks.On the contrary, as far as conventional geometrical features are concerned, the standard methodology suggests following different strategies according to the type of material to be assessed. In more detail, when brittle materials are involved, static strength is usually estimated by using the maximum principal stress criterion together with linear-elastic peak stresses.The problem of performing the static assessment of notched ductile materials is instead much more tricky. Nominal net stresses are commonly used, rather than the estimated elastic peak stresses at the notch root, due to the fact that the stress fields in the vicinity of stress raiser apices can redistribute themselves as soon as the material begins to deform plastically. Further, when multiaxial loadings are involved, care must be taken to choose the proper equivalent stress (i.e. Von Mises’ equivalent stress, Tresca’s equivalent stress, etc.) according to the actual fracture behaviour of the material to be assessed. Unfortunately, this approach has several difficulties, including the definition of a reference section to be used to calculate nominal stresses for components having complex geometry.The above problems could partially be overcome by taking full advantage of either elasto-plastic Finite Element (FE) models or classical rules suitable for estimating elasto-plastic notch root stresses and strains (as, for instance, the well-known rule devised by Neuber In this complex scenario, and according to the authors’ understanding of the static assessment issue, any reliable engineering method suitable for estimating static strength of notched ductile materials should be capable of efficiently and simultaneously modelling both the damaging effect of the stress/strain field acting on the process zone, including the effect of multiaxial stresses, and also the role of stress concentration phenomena. At the same time, due to the needs of industrial reality, the calibration of such a method should be based on pieces of experimental information which can easily be generated by using standard testing equipment. Finally, to reduce the time of the design process itself, the stress analyses needed to estimate the necessary engineering quantities should, ideally, be done by directly post-processing simple linear-elastic FE models.The present investigation ambitiously aims to devise and validate an engineering method which meets all the requirements mentioned above. The specific hypothesis to be investigated is that the Theory of Critical Distances, which has previously been applied to other problems in failure prediction The material investigated in the present study was commercial aluminium alloy Al6082. The material was supplied in bars having diameter, dg, equal to 10 mm. The plain samples used to determine the static properties of such an aluminium alloy were machined to obtain a gauge length of 5 mm with a diameter equal to 6 mm. The material was found to have an ultimate tensile stress, σUTS, equal to 367 MPa, a yield stress, σY, equal to 347 MPa and a Young’s modulus, E, equal to 69,090 MPa. a shows one of the stress–strain curves generated from plain (i.e. unnotched) specimens, plotted in terms of engineering quantities.The tested V-notched cylindrical samples () had gross diameter, dg, equal to 10 mm and net diameter, dn, ranging between 6.1 mm and 6.2 mm. The notch opening angle, α, was equal to 60° and four different values of the notch root radius, rn, were investigated, i.e. 0.44 mm (tensile stress concentration factor Kt |
= 2.94; torsional stress concentration factor Ktt |
= 1.71); 0.50 mm (Kt |
= 2.76; Ktt |
= 1.64); 1.25 mm (Kt |
= 1.92; Ktt |
= 1.32) and; 4.00 mm (Kt |
= 1.33; Ktt |
= 1.12). Static tests, conducted according to standard methods Examination of the state of the art shows that the most efficient approaches suitable for predicting static strength of notched ductile materials make full use of the degree of triaxiality of the stress fields in the vicinity of crack initiation sites plot the degree of triaxility, H, against the distance, r, from the notch tip. In more detail, the above charts were built by post-processing the linear-elastic stress fields along the notch bisector (calculated, under uniaxial fatigue loading, through refined FE models) and by defining the degree of triaxility as follows fully confirm that, as expected, the degree of multiaxiality of the linear-elastic stress fields increases with increasing of the sharpness of the tested notch.For tests conducted under pure tensile and pure torsional loading, the failure force, Fu, and failure torque, Mu, were defined as the maximum value recorded during each test. shows the profile of some load vs. extension and torque vs. twist angle curves generated by testing the investigated notched geometries: the above charts make it evident that, under simple loading paths, it was straightforward to determine Fu and Mu according to the above definition.On the contrary, much more tricky was the determination of the ultimate tensile loading and the ultimate torque under mixed tension/torsion loading. As an example, a shows the loading vs. extension and the torque vs. twist angle curves generated by testing, under a σnom to τnom ratio equal to 0.23, a notched sample having notch root radius equal to 4 mm. In the above chart the ultimate tensile force, Fu, and the ultimate torque, Mu, are not aligned along a unique vertical line. This is due to the fact that, to keep the ratio between the two nominal stress components constant during the test (see b), the axial strain rate had to be different from the corresponding torsional strain rate. Moreover, for any controlled channel, the deformation rate had to vary to keep the σnom to τnom ratio unchanged when the material began to deviate from the linear-elastic behaviour. The direct analysis of all the gathered channels showed that, under combined loading, the maximum point was always reached sooner in the torque vs. twist angle curve than in the corresponding load vs. extension curve; this was true independently of both sharpness of the notch and ratio between the applied nominal loadings. For this reason, the time corresponding to the maximum value in the torque vs. twist angle curve was used to determine both Fu and Mu as illustrated in b. It is worth noticing here also that the difference between the Fu value estimated under mixed-mode loading according to the above definition and the maximum value of the tensile loading recorded during each test was always less than 8%, with an average value equal to approximately 3.5%. Lastly, the reanalysis of the gathered channels proved that as soon as the maximum value of the torsional loading was reached, the material became unstable, rapidly coming to final breakage (see b). These considerations should fully confirm the validity of the assumptions made to define the ultimate strength of the tested samples under complex loading paths. shows two other examples of the static curves obtained by testing the notched specimens with rn |
= 0.44 mm under a σnom to τnom ratio equal to 0.23 and 1.0.Three tests were carried out for every investigated geometry/loading configuration and the obtained results are listed in Finally, in order to investigate in detail the cracking behaviour of this material, several tests were conducted during which the specimens were closely observed using a optical microscope with digital camera, allowing the physical appearance of the specimens to be correlated to the applied stress and strain. Finally, crack initiation was defined as the formation on the specimen surface of a technically detectable crack, that is, a crack having length of about 0.5 mm. shows the fracture profile of a plain specimen tested under tensile loading. This picture confirms that failure was characterised by a moderately ductile fracture following some conventional necking. In other words, the maximum value in the stress vs. strain curve corresponded to the onset of necking. No stable crack propagation was observed before final breakage occurred, therefore it is very probable that failure of these specimens occurred in a conventional manner for ductile materials, beginning internally with the formation and coalescence of small cavities A similar cracking behaviour, producing a conventional cup-and-cone fracture surface, was observed also in the notched tensile samples having rn |
= 4 mm (a): again, it should be emphasised that no cracking was observed before final breakage occurred, so that, the maximum load in the load-extension curve corresponded again to the conventional onset of necking.On the contrary, in the specimens having a sharper notch root radius of 0.5 mm, the observed reduction of the tensile load was seen to coincide with the formation of a circumferential, ring crack which initiated in the notch at the point of maximum stress. Observing both the specimen surface during testing and the fracture surface afterwards, it can be concluded that this crack propagated in a stable manner in the radial direction during the time that the applied load was decreasing. Final breakage was seen to take place without any evident deformation of the net section by rapid unstable crack growth occurring on a material plane perpendicular to the applied force (b). Finally, it is interesting to observe that, since the tested aluminium was characterised by a moderate ductility, the yield stress under uniaxial loading (but also the corresponding ultimate tensile force) tended to increase with increasing of the notch sharpness due to the well-known “notch strengthening” phenomenon The observed material cracking behaviour under torsional loading is shown in (for the blunt notch of rn |
= 4 mm) and (for the sharp notch of rn |
= 0.5 mm). These two figures make it evident that, independent of the sharpness of the tested notch, the maximum torque recorded during the test corresponded to the initiation of a ring crack, similar to that seen in the sharply notched tensile specimens. These torsional cracks were also seen to propagate radially on a plane perpendicular to the specimen axis, though the resulting fracture surface was somewhat different in appearance from that in the tensile tests, being characterised by circular concentric marks created by the rubbing action of the two emerging fracture surfaces (see ). Unlike the tensile testing, however, no unstable crack propagation occurred in this case: rather, there was a gradual reduction in the applied torque with increasing twist angle, corresponding to a gradual decrease in the remaining cross-section as the ring crack grew.It is interesting to recall here that the fracture surfaces observed in the above notched samples loaded in torsion were different to that commonly observed in brittle materials subjected to Mode III loading Contrary to the cracking mechanisms observed in brittle materials, the direct inspection of our samples of Al6082 tested under torsion suggests that the initiation and propagation phenomena occurred on planes perpendicular to the specimen axis simply because the most important mechanisms leading to initiation and growth of cracks were mainly shear stress governed, the tested material being characterised by a relatively ductile behaviour (Finally, it is interesting to observe that, since the tested aluminium displayed a negligible level of plastic deformation when loaded in pure torsion, the notch strengthening effect b, the measured yield shear stress was practically independent of notch sharpness.Cracking behaviour was also monitored in the mixed tension/torsion experiments carried out under a σnom to τnom ratio equal to unity. Two different notch geometries were considered, i.e. rn equal to 4 mm and to 0.5 mm. shows, for this mixed-mode loading condition, failure in the blunt notch was similar to that previously observed when this notch was tested in pure tension, i.e. no crack was detected before final breakage occurred and the fracture surface was seen to be characterised by two different areas: a shear-like fracture surface at 45° to the specimen axis and a brittle-like fracture surface perpendicular to the axis of the sample itself.On the contrary, failure from the sharper notches () was preceded by the formation of the ring crack at the time of maximum torque, which propagated first in a stable manner and then, at final failure, in an unstable manner. summarises the different fracture surfaces obtained by reducing the notch root radius value from 4 mm down to 0.44 mm as well as by increasing the ratio between the applied nominal loadings from zero up to infinity: the above matrix makes it evident that, as the geometry/loading configuration changed, the morphologies of the fracture surfaces varied according to the different mechanisms summarised in the above paragraphs. For the sake of clarity, the above complex behaviour is summarised also by the schematic chart reported in It is evident that the introduction of a notch, and the orientation of this notch with respect to the applied loading, plays a crucial role in determining the mode of failure as well as the critical load. Here we have demonstrated transitions between three mechanisms of failure – ductile failure (with no cracking); stable crack growth and unstable crack growth – as a result of changing notch sharpness and loading mode, for the particular case of the circumferential notch. It is well-known that notches have the effect of “embrittling” a ductile material, especially one which has a relatively low fracture toughness, such as this aluminium alloy; here we show that the embrittling effect is stronger, i.e. it occurs for blunter notches, if a torsional loading component is present. On the other hand, we see that, in this geometry at least, torsion has the effect of stabilising crack growth, causing a more gradual loss of strength with increasing deformation.Some of these findings have important consequences for the development of a suitable theory for the prediction of failure in this situation – a theory which will be developed in full in the second part of this two-part series of papers. The first important finding is that crack initiation did not begin until peak loading conditions occurred. As noted above, the definition of peak loading is not a straightforward matter when both tension and torsion are being applied simultaneously; nevertheless, using the definition developed above, it is clear that no cracks appeared until the maximum stress was reached. This is a very useful observation: since the aim of the theory is to predict the strength of the specimen, i.e. its maximum load-bearing capacity, then it is clear that one does not need to take crack propagation into account: rather, failure can be defined in terms of crack initiation. This simplifies the theoretical model considerably as it means that it can be based on a stress analysis of the initial, uncracked, specimen geometry.The second important finding is that, whatever the degree of tension or torsion applied, crack propagation always took the form of a circumferential crack which lay in a plane perpendicular to the specimen axis. The initiation of this crack is obviously dictated by the need to form in the notch root, where stresses are highest. In high-cycle fatigue of metals, and in the static fracture of some brittle materials, torsional loading causes cracks to grow at 45° to the specimen axis, but this behaviour did not occur in the present case. This finding is important for theoretical developments because it confirms that the crack does not follow the path which maximises the Mode I crack-opening stress, implying that a maximum principal stress criterion will not be appropriate to describe this type of failure.A complicating factor for the development of any theoretical model is the change in mode from ductile to brittle failure. This suggests that a single unified approach may not be able to predict failure loads for all the notch types and loading modes used here. Approaches based on LEFM might be expected to work well when cracking precedes failure: the notches might, for example, be modelled as pre-existing cracks. But such an approach would clearly not be expected to apply when ductile failure occurred: this type of failure might be controlled by an effective stress such as that of Von Mises or Tresca. Despite these reservations, we decided to proceed on the basis of developing a single approach to predict all types of failure, at least as an initial hypothesis to be tested.The most important mechanisms leading to final static breakage of the tested notched samples made of Al6082 can be summarised as follows:In the presence of blunt notches subjected to uniaxial tensile loading, failure occurs due to a ductile-like fracture without any initial stable crack propagation.In the presence of sharp notches subjected to uniaxial tensile loading, failure occurs due to a brittle-like Mode I fast fracture preceded by the growth of initial stable cracks.Under torsional loading, and independent of notch sharpness, failure occurs due to the propagation, on material planes perpendicular to the specimen axis, of stable shear cracks.Under mixed Mode I and III loading, failure occurs under a combination, depending on the mutual interaction between sharpness of the notch and nominal tensile stress to nominal shear stress ratio, of the above three fundamental mechanisms.Crack initiation always coincides with the peak loading conditions, as defined from the tensile and shear force/deflection curves.Crack propagation always follows the plane perpendicular to the specimen axis.Hyper-pre-stress vs. strain-gradient for surface relaxation in diamond-like structuresIn this paper, based on the discrete valence force field model for diamond-like crystals, we construct and compare two anisotropic models with cubic symmetry containing an intrinsic internal length. The first model is constructed starting from the discrete valence force field model via lattice dynamics and the classical polynomial approximation of the acoustic branches near the Γ point. We show that the strain-gradient continuum approximation obtained in this way enforces a result obtained previously in and leads to internal lengths which are unphysically small. The second model follows a line studied in the one-dimensional context in . It accounts for incompatible reference lengths in the discrete setting and is able to model both cohesion, surface energy and defects. We study the ground state of an infinite lattice and we conclude that, as expected, for finite structures a boundary-layer effect occurs. We explore numerically using the model with hyper-pre-stress: (a) the average lattice parameter of bulk-like, plate-like and beam-like nano-structures, and (b) the lattice parameter variations as a function of porosity in porous silicon. We conclude that the model with hyper-pre-stress provides a realistic model for the mechanics of nano-structures including size and shape effects.It is well accepted that the macroscopic properties of objects with at least one very small characteristic length (ultra-thin films, nano-wires, nano-tubes, etc.) are governed by the competition between bulk and surface effects and thus may deviate significantly from the bulk properties. The review presented in shows several generic experimental facts: for instance, 〈100〉 oriented Ag nano-wires exhibit a dramatic increase of Young's modulus, while for the 〈110〉 oriented Si nano-wires a softening phenomenon occurs. Most of these experimental facts can be explained using atomistic calculations but, according to the conclusion in there is still a large gap between computational predictions and experimental measurements of the Young modulus both for metallic and semiconductor nano-wires, and more generally for nano-structures.From a theoretical point of view, experimental evidence about the dependence of the elastic moduli on specimen size, commonly called size effects, requires, at the continuum level, a theory containing at least one intrinsic internal length. The simplest generalization of the classical linear elasticity that includes an internal length is to include the strain gradient in the elastic energy density (). The theoretical predictions of the resulting theory allow the identification of higher-order constants from a phenomenological basis and have been widely discussed in recent years (see for example and references within). The most common applications discuss mainly isotropic materials (), where the coupling between the first and the second gradient of the displacement field vanishes, leading to a non-local model including five new material parameters (see also for strain gradients in plasticity). Generally, including strain-gradients in the continuum description leads to some boundary layer effects associated with manifestations of surface and/or interface effects.A different perspective on size-effects can be obtained by noting that if one of the characteristic sizes of the specimen is very small (several nm) the surface energy and the bulk elastic energy are of same order of magnitude. It follows that extending the classical theory of elasticity so as to account for surface energy also introduces an intrinsic length scale. This line was largely developed following the theoretical frame proposed by ) which includes surface strains, and further extended by Steigmann and Ogden to include curvature-dependent surface energy ). The importance of the Steigmann–Ogden correction and the size-dependent elastic modulus for nanostructures undergoing nonhomogeneous deformations was recently studied by , following a line previously developed in , an atomistic approach based on the density functional theory and molecular dynamics was used in order to determine strain-gradient elasticity constants. The strain-gradient theory obtained in this way possesses higher-order material parameters which can be estimated by DFT and MD calculations, via dispersion relations. As mentioned in the simplest approximation of the dispersion relations near the Γ point lead to the linear theory of elasticity. Higher order polynomial approximation lead to more complex theories and among these, the strain-gradient theory is a second-order approximation. From an analytical point of view, very simple discrete models face an apparent paradox: the material parameters of the strain-gradient theory will depend on the first-order elasticities and some geometric characteristics of the discrete structure, i.e. the lattice parameter. Thus an important conclusion in is that, for a large class of materials including both metals and semi-conductors, the second-order effects computed manifest only at unphysical small sizes.A different theoretical approach designed to include an internal length in the discrete setting was presented in . A detailed analysis of the mechanical response of the one dimensional chain including NN and NNN interactions with incompatible reference lengths shows that, in the generic case, boundary layers occur. However, in this case, in contrast with the continuum setting, the reference configuration of the chain is non-trivial, and so, at very small scales, the elasticity of the chain may deviate significantly from the classical predictions of the homogenization theory. This is in some sense expected since elasticity reflects only material behavior at very large scales with respect to the inter-particle distance.In this paper, motivated by the experimental evidence of size-dependent lattice parameters in nano-porous silicon, we explore the two lines presented above for materials with a diamond-like structure. The starting point is Keating's valence force field (VFF) model that fits the bulk elasticity of group IV elements in the diamond-like structure remarkable well. Using lattice dynamics we explicitly derive the second-order approximation, i.e., the continuum strain-gradient approximation of the discrete model following the method of . The key point on this line is the approximation of the dispersion relations near the Γ point in the multidimensional case with cubic symmetry. The analytical results obtained in this paper complete and enforce the conclusions obtained by . We show that the internal lengths obtained are too small to predict correctly the size-dependent lattice parameter observed in nano-porous silicon, which is a serious drawback of the strain-gradient theory, already noted using a different approach in Back to the discrete setting and following the ideas in , we explore a second alternative that includes incompatible reference lengths in the VFF model. The key ingredient in the construction of a model with hyper-pre-stress is to account for incompatible reference lengths of the NN pair interactions and NNN angular-interactions. The analytical complexity of the resulting three-dimensional discrete model is prohibitive for a complete qualitative study comparable with that of the one-dimensional chain () but, using numerical simulations, we explore several size effects induced by the presence of the incompatible reference lengths associated to NN and NNN interactions. The discrete model is thus able to predict both (a) the variations of the lattice parameter during porosification and (b) the correct macroscopic elastic constants. The first application of the model with hyper-pre-stress is the numerical study of the average lattice parameter of beam-like, plate-like and bulk-like structures of silicon. As expected, the numerical results reveal the enhanced relaxation of beam-like structure with respect to plate-like structures and that of plate-like structures with respect to bulk-like structures. The second application addresses the problem of the vertical relaxation of porous silicon during porosification, for which experimental data are available in the literature. Using periodic homogenization on a super-cell we estimate the effect of porosification on both macroscopic elastic constants and pre-strain. The obtained results show a remarkable fit between the predicted and measured vertical strain induced by porosification.The paper is organized as follows: the next section introduces the valence force field of Keating its quadratic approximation and shows the ability of the discrete model to predict the elastic properties of group IV elements (C, Si and Ge). As already indicated, the prediction of the macroscopic elasticity indicates only that near the center of the first Brillouin zone the discrete model fits well the experimental dispersion data. In order to extend the comparison to the entire first Brillouin zone we compute in the third section the lattice dynamics associated with the discrete model. As a by-product we show that the predicted dispersion relations agree well with experimental data () providing a good departure point for the approximation of the dispersion relations. The third section presents two successive polynomial approximations of the dispersion relations near the Γ point. As expected, the first one recovers the classical elasticity theory and the second one is the strain-gradient theory we look for. We obtain analytical values of the dispersive elastic constants in terms of stiffnesses of the discrete model and the lattice parameter and recover numerical values in very good agreement with The fourth section introduces the hyper-pre-stress in the quadratic approximation of the discrete model. Assuming positive stiffnesses, incompatible reference lengths for the NN and angular NNN interactions lead to: non-trivial reference configuration and a boundary layer effect in the finite lattice. We show that the case of interest is that of incompatible reference lengths and we show that for finite structures accounting for incompatible reference lengths is equivalent to subjecting the structure to a force system acting only on incompletely coordinated particles. As a first application we explore numerically the average lattice parameter of bulk-like plate-like and beam-like finite structures and show a size effect induced by the presence of the hyper-pre-stress. The second application concerns the porosity-dependent lattice parameter in nano-porous silicon. In the later application we are able to fit the single internal length of the model with hyper-pre-stress so as to predict correctly the porosity-dependent (average) lattice parameter.In a fundamental paper on discrete interactions in covalent crystals, divides interaction potentials into two major groups: the first one contains pair interaction potentials (Lennard-Jones, Morse, etc.) which apply to arbitrary configurations but give poor results for strong covalent bonds. Born's the pioneering work in this direction was extended by up to fifth-shell in an attempt to fit both elastic moduli and phonon dispersion data for silicon. The second group of potentials, intended to describe small distorsions from a ground state, includes three-body potentials. The most famous example in this group is Keating's valence force field model (Formally, for a system containing N particles, the total interaction energy can be expanded formally asW=∑1≤i≤NV1(ri)+∑1≤i<j≤NV2(ri,rj)+∑1≤i<j<k≤NV3(ri,rj,rk)+⋯Models accounting only for pair interactions (Lennard-Jones, Morse, etc.) consider only approximations accounting for the first two terms, usually including a cut-off radius. Models in the second group include at least the third term and are able to provide a fairly good description of the elastic properties of diamond-like structures of group IV elements, i.e., C, Si and Ge. We note here that accounting for NNN interactions (or longer range pair interactions) is obviously not equivalent to accounting for three-body terms. Moreover, the role of angular interactions can be investigated only in the multidimensional setting.Among the models designed to extend the models of Born and Keating we cite here the most commonThe Stillinger–Weber model: (which falls into the second group) for which the interaction energy is given byW=∑i,jV2(|rj−ri|)+∑i,j,kλeγ1/(rij−a)+1/(rik−a)(cosθijk+1/3)2,where λ and γ are material constants θijk is the angle between rij and rik and a is the lattice parameter. The model includes nonlinearities able to give a fairly realistic description of crystalline silicon but encounters transferability problems (i.e., is unable to cover non-tetrahedral polytypes).): The interaction energy is assumed to have the formW=∑i,jfc(rij)[Ae−λ1rij+B(1+βnξijn)1/2ne−λ2rij],where ξij=∑k≠i,jfc(rij)[1+c2/d2+c2/(d2+(h−cosθijk)2)]eλ33(rij−rik)3 and fc is a cut-off function and A,B,λ1,λ2,λ3,β,n,c,d and h are material parameters. The model emphasize the fundamental role of coordination and uses only pair interactions whose bond strength depends on coordination. It covers a broad spectrum of situations but fitting to real situations is a hard task. However, this model predicts () for the elastic constants of silicon: C11=121GPa, C12=86GPa and C44=10GPa in disagreement with experimental data in and f(ρi) is the electron density around atom i and V2 is a pair potential, both calibrated from quantum mechanical calculations.The form of the cohesive energy potentials in may be justified by results from band theory (), giving a matrix description of an atom's local environment (The simplest valence force field (VFF) model () assumes that, in the harmonic regime near a ground state, the total elastic energy of a covalent system can be decomposed as a sum between central and non-central interaction energies aswhere the pair potential Wij and the three-body potential Wijk are given, respectively, by we use a for the lattice parameter and xi for the actual (spatial) positions of atom i. Using ui for the displacement of atom i with respect to its position Xi on the reference configuration (the diamond lattice with lattice parameter a), we haveIt follows that, up to third-order terms in the displacement, the interaction energies in the VFF model are expressed as:Wijk≃B2[(u⇒j−u⇒i)·n⇒ki+(u⇒k−u⇒i)·n⇒ji]2,, A and B are stiffnesses for, respectively, central and non-central interactions.Thus the model with an interaction energy given byW=∑(i,j)A2[(u⇒j−u⇒i)·n⇒ij]2+∑(i,j,k)B2[(u⇒j−u⇒i)·n⇒ki+(u⇒k−u⇒i)·n⇒ji]2,is the quadratic approximation of the VFF model.We notice here that both the VFF model and its quadratic approximation possesses a trivial reference configuration which is the diamond type structure with lattice parameter a. Thus, in the regime of small strains, near the ground state of the diamond lattice with lattice parameter a, both the VFF model and its quadratic approximation give identical results. is its ability to accurately describe the elasticity of diamond-like crystals. This shows that, near the ground state, the simple NN (pair) interactions and NNN (three-body angular) interactions are the dominant terms in the expression of the elastic energy of the diamond structure. For sake of completeness, we shall briefly present here the computation of the macroscopic elasticity predicted by the VFF model. The discrete homogenization method we use is well accepted and can be rigourously justified at the analytical level ( under the suitable hypothesis fulfilled here).We consider the unit cell (represented in ) in an infinite crystal and containing two identical atoms. Under imposed macroscopic strain E, the individual displacements of the atoms in the unit cell are given by a homogeneous part and a fluctuation, i.e., As the elastic energy is invariant with respect to translation of the cell, assuming periodicity and without losing the generality, we can suppose that u˜1=0 and use U instead of u˜2. Collecting together the contributions of the four terms accounting for the pair interactions of a unit cell and the twelve terms accounting for three-body interactions, we obtain per unit cellW=124[a2(A+12B)(E112+E222+E332)+2a2(A−4B)(E11E22+E22E33+E11E33)+4a2(A+4B)(E122+E132+E232)+16a(A−4B)(E23Ux+E13Uy+E12Uz)+16(A+4B)(Ux2+Uy2+Uz2)].The position of atom 2, under imposed macroscopic deformation, is provided by the minimization with respect to U in and represents the local equilibrium of the lattice. Substitution of this result in Wmacro.(E)=1vol(Y)minUWmicro(E,U)=16a(A+4B)[A2(E11+E22+E33)2+16AB(E112+E222+E332)+4(E122+E132++E232)+16B2(3(E112+E222+E332)−2(E11E22+E11E33+E22E33))],Wmacro(E)=12C11C12C12000C11C11C12000C11C12C11000000C44000000C44000000C44E11E22E332E232E132E12·E11E22E332E232E132E12,C11=A+12B3a,C12=A−4B3a,C44=16AB3a(A+4B).As expected, the result is the macroscopic elastic energy of a material with cubic symmetry, inherited from the (point group) symmetry of the lattice and its interactions.At a qualitative level, it is important to note that the three macroscopic elasticities only depend on the two microscopic parameters (stiffnesses A and B) so that the model also predict the relationwhich is verified remarkably well for all group IV elements in the diamond structure. gives the values of material parameters A and B obtained using experimental data for C11 and C12, the predicted elastic constant C44, the experimental value for C44 from and the value of the ratio at left-hand side of is unable to predict surface relaxation of finite structures as it does not contain an internal length. In other words, the different local environment (coordination) of atoms situated far from or near to a free-surface in a finite domain does not affect the size of the structure. The reason for this is the fact that all forces acting on particles vanish simultaneously in the ground state, which is also the current reference configuration.Among the different methods leading to models with internal lengths we shall explore a strain-gradient continuum approximation of the discrete VFF model in the next section. To this end, we shall follow the classical line initiated by . The next section also offers a different perspective on the result in This section presents the lattice dynamics for diamond structures based on the discrete VFF model. Its main goal is to obtain the dispersion relation in order to determine successive approximations of the acoustic branches near the center of the first Brillouin zone (the Γ-point). As the elastic energy is invariant to translations, the lower-order polynomial approximation leads to linear elasticity theory and corresponds to the quadratic polynomial approximation of the acoustic branches near the center of the first Brillouin zone, the point Γ. Successive higher-order polynomial approximations correspond to higher-order partial derivatives in the continuous setting () and, of these, the simplest one is the strain-gradient model we look for.For the lattice dynamics calculations and notations we shall follow the general presentation of The diamond structure is a non-primitive lattice with two atoms per unit cell, further denoted by i and j. The dynamics for each atom is governed bywhere the total elastic energy is the sum of central and non-central parts in . Taking into account pair and three-body interactions in the right-hand side of (), the displacement of atom i contributes to∂W∂ui=∑j∈NN(i)∂Wij∂ui+∑{j,k}⊂NN(i)k≠j∂Wijk∂ui+∑j∈NN(i)k∈NN(j)k≠i∂Wjik∂ui,where we have used NN(i) for subscripts j corresponding to atoms which are near-neighbors of atom i. If we introduce Sijk=nij⊗nik we obtain:Contribution of central forces (atom j is NN of atom i) Contribution of non-central forces of the type Wijk (atoms j and k are NN of atom i) −∑{j,k}⊂NN(i)k≠j∂Wijk∂ui=B∑{j,k}⊂NN(i)k≠j[(Sijj+Sikj)(uk−ui)+(Sikk+Sijk)(uj−ui)]Contribution of non-central forces of the type Wjik (atom k and i are NN of atom j) −∑j∈NN(i)k∈NN(j)k≠i∂Wjik∂ui=−B∑j∈NN(i)k∈NN(j)k≠i[Sjkk(ui−uj)+Ejki(uk−uj)]Since the diamond structure is a non-primitive lattice, we have to distinguish, at the right-hand side of , the contribution of particles from different FCC-sublattices. We first note that: subscripts i and j correspond to atoms situated in different sublattices, while k correspond to an atoms situated in the same sublattice as j; subscripts i and j correspond to atoms in different sublattices, while k corresponds to an atom in the same sublattice as i.The next step is to look for particular solutions in the form of progressive plane waves for each sublattice, i.e.using U for the amplitude in one of the sublattices and V for the other one.At this point we need to describe the coordination around a fixed atom in one of the lattices. To this end, we introduce four vectors v1, v2, v3 and v4 giving the positions of the four near neighbors of an atom of one of the sublattices and note that ∑i=14vi=0. Obviously, the coordination around an atom of the second lattice is given by the opposite vectors. It follows that, formally, the dynamics of both sublattices are given by similar equations but the connectivity of each sublattice is described by opposite vectors. after simplifications, we obtain for one sublattice−ω2mU=−A∑j∈NN(i)Sijj−B∑{j,k}⊂NN(i)k≠j(Sijj+Sikj+Sikk+Sijk)+∑j∈NN(i)k∈NN(j)k≠i(Sjkk+Sjkieik·(vj−vk))U+A∑j∈NN(i)Sijjeik·vj+B∑{j,k}⊂NN(i)k≠j[(Sijj+Sikj)eik·vk+(Sijk+Sikk)eik·vj]+∑j∈NN(i)k∈NN(j)k≠i(Sjkk+Sjki)eik·vjV.We shall denote −AUU and −AUV respectively the factors of U and V in the above equation. Since coordination in the second lattice is obtained by changing vi to −vi, the dynamics of the second lattice is governed byAVV=A∑j∈NN(i)Sijj+B∑{j,k}⊂NN(i)k≠j(Sijj+Sikj+Sikk+Sijk)+∑j∈NN(i)k∈NN(j)k≠i(Sjkk+Sjkie−ik·(vj−vk)).Finally, the system has the following structurewhere, A(k) denotes the dispersion matrixWe are interested in the structure of the spectrum of A(k) near the Γ-point (i.e., k=0) which is also the center of the first Brillouin zone. Invariance of the cell translation as a whole implies () that, with respect to a suitable basis, A(k) has the following structure:A(k)=Aaa(k)Aao(k)Aao⋆(k)Aoo(k)=O(|k|2)O(|k|)O(|k|)O(1).A straightforward computation shows that this is obtained using the collective cell variables shows that the elasticity tensor C^ defined asC^[v⊗k1]k2=[12D2Aaa(0)[k1,k2]−(DAao(0)[k1])Aoo−1(0)(DAao⋆(0)[k2])]vdefines the macroscopic elasticity of the lattice in the long wave-length approximation.This result also shows that in order to obtain the macroscopic approximation it is sufficient to obtain A(k) at O(|k|3) as |k|→0. However, exact dispersion relations are needed in order to plot the dispersion curves in the whole first Brillouin zone and to compare them to experimental data. In other words, if a discrete model is able to fit the experimental elastic constants then the structure of the spectrum of the dispersion matrix near Γ is correct, but this does not imply that the dispersion curves predicted by the model fit the experimental dispersion data.We defer the details of the computation (at O(|k|3) as |k|→0) of the dispersion matrix to Appendix but note here that the final result obtained using C^11=a2A+12B24,C^44=a22AB3(A+4B),C^12=a2A−4B24.As the volume of the unit cell is a3/4 and the total mass in the cell 2ma, we recover the result of . Thus, the first-order approximation of the discrete model leads to the macroscopic equations of the elastodynamics with elasticities given by Cij=C^ij2/V(cell) which gives, as expected, the result in . Once again, the cubic symmetry of the macroscopic elasticity is inherited from the microscopic symmetry of the lattice. Moreover, using a result of (since we fit assumption A of Section 2 in ) the resulting macroscopic law satisfies the generalized Legendre-Hadamard condition. show experimental and computed dispersion relations along the high symmetry lines in the first Brillouin zone, respectively. Experimental data obtained by and the predicted dispersion relations obtained using the exact dispersion relation we plot the lines with high symmetry, in the first Brillouin zone (in the reciprocal space) for diamond-like crystals. They are given bythe segment (k,0,0) (for k∈[0,2π/a]) denoted ΓΔX,the segment (k,k,0) (for k∈[0,3π/2a]) denoted ΓΣK andthe segment (k,k,k) (for k∈[0,π/a]) denoted ΓΛL.Computing higher-order polynomial approximations of the acoustic branches near the center of the first Brillouin zone needs a different approach. The main reason is that higher-order approximations of the dispersion matrix affect the splitting between the acoustic and optical branches. Two solutions to this problem exist in the literature: the first one is presented in ) and we shall compute the third-order and the fourth-order approximations of the dispersion relations near the Γ point. In particular, we shall recover, as expected, the well-known fact that for centro-symmetric crystals the coupling constants between the deformation and its spatial derivatives vanish.The starting point is the splitting of the exact dispersion matrix A(k) in Y(k)=Aaa(k)−EaI3×303×303×3Aoo(k)−EoI3×3,and consider X(k) as a small parameter. We summarize the result in as follows: em the splitting between the acoustic and optical branches for polynomial approximations of degree m≤4 are obtained as K(4)=12[X,{[[Y,XH]H,Y]H+23[K(2),XH]H}]−112[[K(2),XH],XH].Above, the commutator [X,Y] denotes XY−YX while XH is the unique solution A of the equation [H,A]=X.We note that along the line presented above, computations at O(|k|m+1) as |k|→0, still hold in order to obtain K(m). From the previous section we haveor, otherwise stated, Ea=0 and Eo=83(A+4B). We obtain successivelyDaa(2)=a224(A+4B)f(2)(k1,k2,k3)g(2)(k1,k2,k3)g(2)(k1,k3,k2)g(2)(k2,k1,k3)f(2)(k2,k3,k1)g(2)(k2,k3,k1)g(2)(k3,k1,k2)g(2)(k3,k2,k1)f(2)(k3,k1,k2),Since the unit cell has volume a3/4 and two atoms we identify a224(A+4B)(A2+48B2)k12+16AB|k|2=a38[C11k12+C44(k22+k32)]a224(A+4B)(A2+16AB−16B2)k1k2=a38C12k1k2.This identification gives, as expected, the elastic constants of the previous section, i.e. C11=A+12B3a,C12=A−4B3a,C44=16AB3a(A+4B).where Daa(4) is the fourth-order homogeneous polynomial with the same structure as Daa(2) and f(4)(k1,k2,k3)=f^122122(k24+k34)+6f^122133k22k32+6f^211222k12(k22+k32),g(4)(k1,k2,k3)=(4f^112222(k12+k22)+12f^112233k32)k1k2.A long but straightforward computation of coefficients in f(4) and g(4) leads to: f^122122=−a4(A−4B)2(7A2+40AB+112B2)4608(A+4B)3,f^122133=−a4(A−4B)3(13A+44B)13824(A+4B)3,f^211222=−a4(A−4B)2(13A2+88AB+208B2)27648(A+4B)3,f^122222=−a4(A−4B)2(5A2+32AB+80B2)9216(A+4B)3,The result K(3)=0 means that there is no coupling between the first derivatives of the strain and the strain itself in the free-energy density of the continuum approximation. This can be obtained directly from group representation theory using the symmetry properties of the centro-symmetric lattices (see As for the previous approximation, we define f=(2/V(cell))f^. The two qualitative results: f111111=0 and f112233=f122133 are specific to the discrete VFF model. By inspecting we conclude that all second-order coefficients are negative.The material parameters of the second-gradient approximation of the discrete model can be computed numerically using the values of A and B from the discrete VFF model. Using We notice here that they are one order of magnitude lower than those obtained using the ab initio and molecular dynamics simulations in . This result enforces the conclusion that the internal length introduced in the continuum model using the approximation of the dispersion relation near theΓpoint is insignificant unless the specimen size is unphysically small. As already noted, we interpret this result as a consequence of the fact that near the Γ point the dispersion relations are almost linear (Fig. ). It follows that corrections near the center of the first Brillouin zone will not improve significantly the previously obtained approximations of the dispersion curves. A different alternative is proposed in where, in the one dimensional case, approximate models based on a different qualitative method (such as the Padé interpolation, or approximation of first roots) were studied.The dynamics of the continuum model obtained above is governed byIt follows that the square of the ratio between fijklmn and Cijkl provides one (or several) estimates for the characteristic (internal) lengths of the approximate model. In our case f≃10−10N and C≃1010N/m2 which predicts internal length scale ≃1Å. we have obtained analytical expressions for the coefficients of the dynamic equation of motion which represents the continuum approximation of the discrete model. We recall that, as shown in , in the multidimensional case, the coefficients in the equation of the motion do not determine in a unique manner the analytical expression of the free-energy. Thus, as already discussed in , the sign paradox in the multidimensional case can be avoided. we have plotted the theoretical dispersion relations (i.e., real part of acoustic and optical branches) along the high symmetry lines in the first Brillouin zone. Topological characteristics (multiplicity, etc.) are conserved and, as expected, the first-order approximation leads to linear dispersion relations for the acoustic branches. The second order-approximation illustrates a well-known behavior of the acoustic branches (also discussed in ): it crosses the horizontal axis at finite values of the wavenumber and ω(k) becomes purely imaginary. Thus, small wavelength perturbations will grow exponentially and this leads to the instability of the elastic response for continuum models which represent second-order approximations of discrete systems. We note that this result is an artifact introduced by the method used for the approximation. As experimental dispersion data show that the acoustic branches curve downward the sign of the second derivatives near the origin is negative and this is captured by the approximation method. Otherwise stated, in this approximation the gradient part of the computed high order energy is not positive definite. Different discrete-to-continuum methods able to correct this unphysical behavior were proposed in An alternative to the higher order continuum theory, which also introduces an internal length into the discrete model was presented in in the one dimensional setting. The key ingredient is the competition between two force systems with different reference lengths. The analysis of the simplest one-dimensional model with near-neighbor (NN) and next-to-near neighbor (NNN) interactions with incompatible references lengths shows that surface energy is associated with free-boundaries and a size-effect. In this section we shall adopt the ideas in and modify the discrete VFF model in order to predict both (a) the bulk elasticity of crystalline diamond-like structures and (b) surface effects associated to relaxation near free-boundaries. The constructed model incorporates an internal length that can be calibrated so as to fit the observed lattice parameter evolution determined experimentally.The starting point is the quadratic approximation of the VFF model including two equilibrium lengths, further denoted l1 and l2 and associated to central and non-central interaction energies. The total elastic energy of the model becomesW=∑(i,j)A2[(u⇒j−u⇒i)·n⇒ij−l1]2+∑(i,j,k)B2[(u⇒j−u⇒i)·n⇒ki+(u⇒k−u⇒i)·n⇒ji−l2]2. we obtain, in an infinite crystal, the macroscopic constitutive relation in the formWith respect to the reference configuration used to write , the macroscopic elastic constants are still given by . It follows that in an infinite lattice (or far from the boundary in a finite domain) the average lattice parameter for a free bulk material isbut the physical meaning of a is not the ground state lattice parameter but just an arbitrary reference length, with respect to which displacements in Following a parallel to the one-dimensional case presented in we have to distinguish between the two cases of compatible and incompatible reference lengths.In the following, two reference lengths l1 and l2 will be called compatible ifOtherwise, the couple (l1,l2) will be called incompatible. shows that when the reference configuration is subjected to a macroscopic deformation ε0I (pure dilation or compression, depending on the sign of Al1−2Bl2) the lattice parameter changes from a to a^ and the lattice reaches its ground state. With respect to this new (reference) configuration, with lattice parameter a^, the elastic energy isW=∑(i,j)A2[(u⇒j−u⇒i)·n⇒ij−L1]2+∑(i,j,k)B2[(u⇒j−u⇒i)·n⇒ki+(u⇒k−u⇒i)·n⇒ji−L2]2Note that the couple (L1,L2) is the only couple with the properties shows that the incompatibility is conserved by a change of the reference configuration, while the second one reflects the fact that at lattice parameter a^ is such that we are in the ground state. Without losing the generality, we shall use in the following so that the bulk macroscopic material behavior is described by:C11=A+12B3a^,C12=A−4B3a^,C44=16AB3a^(A+4B),and the elastic energy density is, up to an additive constant, W=12C[ϵ]:ϵ. including incompatible references in the lattice is equivalent to a self-equilibrated force system acting on each particle. The following remarks provide more insight into the role of the reference lengths:Assume (L1,L2)compatible and consider a finite free-lattice. Then the minimum of the total elastic energy is attained (up to a rigid body displacement) by the homogeneous displacement given byThis result can be easily verified since in the case of compatible reference lengths the minimum of the total elastic energy is zero.In the following, a particle of a finite-lattice will be called completely coordinated if all its NN and NNN neighbors belong to the structure. Otherwise, the particle is called incompletely coordinated. Then, in the reference configuration, for a fully coordinated particle the force system acting on it is self-equilibrated. In order to prove this assertion let i be a fixed atom in the lattice and jα (α=1,2,3,4) its four near-neighbors. Taking into account the expressions of the central and non-central forces acting on particle i we obtain:−A∑j∈NN(i)[(uj−ui)·nij−L1]nijui=uj=0=AL1∑j∈NN(i)nij=0,due to the geometry of the diamond structure.∑j,k∈NN(i)j≠k∂wijk∂ui+∑j∈NN(i)∑k∈NN(j)k≠i∂wjik∂uiui=uj=uk=0The explicit computation of the first sum in BL2∑j,k∈NN(i)k≠j(nij+nik)=3BL2∑j∈NN(i)nij,and vanishes due to the symmetry of the diamond lattice (as in −BL2∑j∈NN(i)∑k∈NN(j)k≠injk=−3BL2∑j∈NN(i)nij=0after rearrangement. It is interesting to notice here that the right-hand sides of vanishes even without the symmetry of the diamond structure.A different perspective on the second remark is obtained by noting that for a free finite-lattice (and up to a rigid displacement) the minimum of the total energy given in is achieved by a displacement that minimize alsowhere the first two sums represent the total elastic energy in the original VFF model and the third one may be regarded as the work of external forces fi acting only on incompletely coordinated particles. To prove this we note that the development of provide three terms: a quadratic part with respect to the displacement, denoted W2, which is exactly , a constant term, denoted W0, that does not affect the minimum and a term that is homogeneous of degree one with respect to the displacement, denoted W1. But our previous remark shows that in the general expressionthe factors of ui vanish for completely coordinated particles, which proves the assertion.Assertions (1) and (2) show that when the reference lengths are incompatible the self-equilibrated force system introduced by the hyper-pre-stress produces a boundary layer effect in the finite lattice..2) with respect to the ground state of an infinite lattice, the elastic energy isW=∑(i,j)A2[(u⇒j−u⇒i)·n⇒ij−L1]2+∑(i,j,k)B2[(u⇒j−u⇒i)·n⇒ki+(u⇒k−u⇒i)·n⇒ji−L1A2B]2.The lattice parameter in an infinite lattice is a^. fit the experimental data; dispersion curves are those presented in figures (4).The model includes an adjustable internal length so that, in a finite domain, it has a size-dependent lattice parameter.A straightforward consequence of the last property is that the model will also predict a size-dependent Young's modulus. Due to the analytical complexity of the multidimensional case we are not able to present a complete analysis following the line in but instead we shall illustrate the main features of the model using numerical simulations.We note that a positive (negative) sign of L1 induces, in a finite domain, an average lattice parameter larger (smaller) than that of an infinite lattice. As these effects are similar we shall focus on the numerical study of the L1>0 case (average lattice parameter of finite structure larger that of the infinite lattice) as indicated by the experimental data obtained using Raman spectroscopy for porous silicon. The reverse phenomenon (average lattice parameter of finite structures smaller than that of an infinite lattice) can be obtained by changing the sign of L1.In order to illustrate the size-effect induced by the hyper-pre-stress we have performed numerical computations of the average lattice parameter for silicon free-structures with different characteristic lengths. For simplicity, we have considered only parallelipipedic structures oriented along the crystallographic axes.For the material parameters A and B we choose the values given in . The lattice parameter a^ (used as the reference configuration) equals 5.431 Å and L1 is fixed at 1% of the interatomic distance of the infinite lattice, i.e. (3/4)a^. We choose L2=(A/2B)L1 in order to estimate the deviation of the average lattice parameter in finite structures from that of the ideal infinite lattice. In other words, the reference configuration is that of the ground state of the infinite lattice.For l=10 nm (which corresponds to 20 mono-layers) and α between 2 nm and 10 nm we have considered nano-structures with lateral dimensions (α,l,l) – also called in the following plate-like structures, (α,α,l) – also called beam-like structures and (α,α,α) - also called bulk-like structures. shows the numerical results obtained for the deviation of the average lattice parameter with respect to that of the infinite lattice. As expected, for L1>0 the average lattice parameter increases when one characteristic dimension of the structure decreases, but tends to a^ (which is the average lattice parameter of an infinite lattice) when all dimensions of the structure are very large. also indicates, as expected, that for fixed l, the average lattice parameter increases faster for bulk-like structures than for beam-like structures, and the lattice parameter for beam-like structures increases faster than for plate-like structures when α varies. This is obviously the effect of the specific surface available for these different shapes. illustrates the average lattice parameter for a bulk-like structure and α=10 nm..Porous silicon is a nano-structured material obtained using electrochemical anodization in a solution of fluorhydric acid (HF) and ethanol (), one can speak of nanoporous silicon when the size of crystallites is less than 5 nm, of mesoporous silicon when the pore sizes range between 5 nm and 50 nm and macroporous silicon when the pore sizes are larger than 50 nm. From an experimental point of view, porosity is controlled by the anodization current density, anodization time and HF/ethanol ratio. The pores obtained are inter-connected as they have been obtained by the action of the HF/ethanol solution on bulk Si.In the case of nanoporous silicon the specific area can reach very high values like, for example, 100 m2/g. In such situations the mechanical behavior of the material can be expected to be largely dominated by the “surface effects”. It follows that porous silicon is a particularly interesting object to understand deviations from classical bulk properties and size effects in elastic (covalent) crystals.Generally, real three-dimensional crystal surfaces present an additional difficulty due to the recombination (of electrostatic nature) of surface bonds. Thus, surface reconstruction always induces additional deformation near the free-surfaces so that, generally, both the size-effect mentioned above and the surface reconstruction contributes to the elasticity of nano-scale objects. In very special circumstances one can distinguish between these two sources of deformation near the free-surfaces of crystals. Electrochemical anodization used to obtain porous silicon avoids surface recombination since the chemical reactions implicated in the process result in a Si–H terminated surface.During anodization the lattice parameter is measured using X-ray diffraction (XRD) and/or Raman spectroscopy. When anodization is performed on part of a bulk specimen the porous silicon conserves the crystallographic directions of the original bulk specimen but:the lattice parameter in the [001] direction is greater than the bulk lattice parameter,the lattice parameter in the [100] and [010] directions are the same as in the bulk.This behavior is commonly presented as a relaxation of porous silicon due to porosification, and is interpreted as the manifestation of a modification of the average lattice parameter under the constraint of “no lateral motion”. Experimental data concerning the modification of the (average) [001] lattice parameter induced by anodization as a function of porosity were presented in . The experimental data show an almost linear behavior in the range between 30% and 90% porosity.In order to explore the porosity-dependent lattice parameter predicted by the model with hyper-pre-stress, we have considered a cubic super-cell oriented along the crystallographic axes and containing a one-parameter pore. The volume of the porous super-cell is obtained by eliminating the atoms inside the union of the three cylinders Ci={(x1,x2,x3),max(|xj|,|xk|)≤α} for i=1, 2, 3. A typical super-cell obtained is illustrated in The computational procedure involves the following steps:Computation of the macroscopic elasticity: At fixed super-cell size and porosity (i.e. fixed α) we solve the discrete homogenization problem with periodic boundary conditions and deduce the macroscopic constitutive relation in the formwhere E0=e0I due to the orientation of the pore with respect to crystallographic directions. We underline here that both macroscopic elasticity and pre-strain are porosity dependent. As expected, using we obtain E0(0)=0 since the bulk material is self-equilibrated, and C(0) is given by . Obviously, the computation of the macroscopic coefficients for the bulk material (i.e., vanishing porosity) is independent on the cell-size.Estimate of the vertical relaxation: Since the porous material is constrained in the lateral directions but free in the vertical one, we conclude that the vertical strain induced by the pre-strain, porosity and lateral constraint E11=E22=0 is given byWe fit the single new material parameter L1 so as to obtain one of the experimental data, i.e. the vertical deformation at 50% porosity (see ). The value of L1 obtained using only this data was used to predict the average lattice parameter at different porosities following the procedure described above. The numerical results obtained are illustrated in a linear regime at small porosities in agreement with experimental data;an increasing average lattice-parameter at larger porosities in slight disagreement with experimental data.We note that to interpret Raman spectroscopy data (methods used in experiments) a macroscopic behavior law for porous material is needed—and this is partially available in the literature (). In this context, the Vegard law overestimates the macroscopic elasticities which in turn leads to a underestimate of the lattice parameter in the high porosity range. Moreover, the particular geometry we have used (for simplicity) induces a macroscopic law with cubic symmetry. This is not the case if one considers only one vertical pore in the super-cell, in which case the macroscopic law will possess transversal isotropy. The actual method also answer also question of a porosity-dependent lattice parameter for nano-porous silicon. However, for very large systems, as is the case of macro-porous silicon, the modification of the average lattice parameter induced by E0 will be insignificant as bulk behavior dominates the surface effects.In this paper we have investigated two alternative solutions intended to include an internal length is the discrete model of Keating in order to fit both (a) the macroscopic elasticity of crystals with diamond-like structure and (b) a size-dependent lattice parameter (measured by Raman spectroscopy).The first solution consists in considering successive continuum approximations of the discrete model. The second-order polynomial approximation of the dispersion relation near the center of the first Brillouin zone leads to the continuum elasticity theory but does not incorporate an internal length. The next approximation leads to a second-gradient model which includes several internal lengths but the model obtained has some intrinsic drawbacks. Starting with a realistic discrete elastic model that includes pair interactions and three-body interactions, we explicitly compute all fourth-order material parameters of the strain-gradient theory obtained using a polynomial approximation of the acoustic branches, and show that, in agreement with previously obtained results (), the internal lengths obtained are too small to apply to realistic situations. This conclusion can be anticipated from the dispersion relations which show almost linear dependence on the wavenumber near the Γ-point.The second solution we present includes incompatible reference lengths in the valence force field model. This is equivalent to a self-equilibrated force system acting on each atom in the reference configuration. In agreement with results obtained in the one-dimensional case in we show that the average lattice parameter of finite structures deviates from the ideal lattice parameter of an infinite lattice.The fundamental difference between the strain-gradient theory and the model with hyper-pre-stress is that the strain gradient theory includes an intrinsic length which manifests itself only under charge. Otherwise stated, the reference configuration is not affected by the inclusion of the strain gradient (at least, as long as residual stresses are not accounted for) so that, for our particular application, the porous material and the bulk material will not exhibit different lattice parameters. This drawback is overcome by including hyper-pre-stress, which is a modified description based on a force system acting on the reference configuration. In this case the reference configuration of the porous material in not trivial and as a consequence the model is able to describe size and shape effects for both finite and infinite free-structures.We study numerically the size-dependent average lattice parameter for stress-free finite beam-like, plate-like and bulk-like nano-structures and show that, as expected, both a size effect and a shape effect occur. In the particular case of nano-porous silicon, we can fit the new material parameter of the model with hyper-pre-stress so as to fit the experimental results obtained in in the range of moderate porosities. At very large porosities (>80%), due to the limited size of our computation, the size-effect of small cross-section beam-like crystallites dominates the numerical results so that the predicted values diverge from the measured data. Applying the model with hyper-pre-stress to the prediction of hardening/softening in nano-structures will be discussed in a future work.We present here the details of the computation leading from . For convenience, we shall introduce the following notations:P=0k1k2k1k3k1k20k2k3k1k3k2k30,T=k12000k22000k32,S=0k3k2k30k1k2k10.B∑{j,k}⊂NN(i)k≠j(Sijj+Sikj+Sikk+Sijk)=8B3I,B∑j∈NN(i)k∈NN(j)k≠iSjkk=3B∑j∈NN(i)Sijj=4BI,B∑j∈NN(i)k∈NN(j)k≠iSjkieik·(vj−vk)≃−B34I−a2k⊗k.The block AUV at O(|k|3) as |k|→0 the sum ofA∑j∈NN(i)Sijjeik·vj≃−4A3I+iaA3S+a2A24k⊗k+S2,B∑{j,k}⊂NN(i)k≠jSijj+Sikjeik·vk=B∑{j,k}⊂NN(i)k≠jSijk+Sikkeik·vj≃−8B3I+i2aB3S+a2B12k⊗k+S2,B∑j∈NN(i)k∈NN(j)k≠i(Sjkk+Sjki)eik·vj≃−B34I−a2k⊗k.As already noted AVV is identical to AUU where vectors vi are replaced by −vi. Then, a straightforward computation shows that AVV=AUU so that finally, the dispersion matrix is at O(|k|3) as |k|→0AUV=(A+4B)−43I+a224k⊗k+iA3(A−4B)S+(A−4B)a212P.The acoustical and optical branches are separated by changing to collective variables, i.e. from (U,V) to (1/2)(U+V),(1/2)(U−V)). This leads toAaa(k)=112a2AP+124a2(A+4B)(k·k)I+13a2BT,Aoo(k)=(83(A+4B)−124a2(k·k))I−112a2(A−8B)P+13a3BT,We note that Aaa, Aao and Aoo are respectively O(|k|2), O(|k|), and O(1). Using relation Relationship between fluidity and stability of self-consolidating mortar incorporating chemical and mineral admixturesIn this study, the effect of chemical and mineral admixtures, including superplasticizer, viscosity modifying agent (VMA), limestone powder and fly ash in different W/C on fluidity, viscosity, and stability of self-consolidating mortar is investigated and proper workability regions for the prepared mixtures are presented. The obtained results indicate that W/C is the most significant parameter influencing the rheological properties of cementitious mixtures, specially their stability. Furthermore, the maximum allowable W/C for preventing inhomogeneity could not be a fixed value for all the mixtures and should be adjusted for the target fluidity. On the other hand, using VMAs is an effective method for stabilizing self-consolidating mortars and preventing any kinds of instability while limestone powder and fly ash mainly affect bleeding and aggregate blockage. Besides, these mineral admixtures improved the fluidity of the mixtures to some extent.Achievement of a super workable concrete, which could be cast easily without any signs of instability, has been desired by engineers for the last decades A highly fluid concrete is also prone to segregation during the mixing process, transporting, pumping, placement and also during the dormant period. Segregation can be defined as separation of granular particles from mortar, which is often associated to static sedimentation. Segregation of fresh concrete may cause several problems in concrete production and cause a non-uniform mixture A proper workability, therefore, may not achieve only by increasing the concrete fluidity. Stability of fresh concrete is also an influencing parameter that should be mentioned The basic characteristics of acceptable SCC are high fluidity and adequate stability These parameters affect the fresh properties of SCC in different ways and usually improve a property while suffering another. For example, increasing W/C or superplasticizer content of a SCC mixture seems to be very effective to increase the fluidity while suffering its stability. On the other hand, VMA or additional fine graded fillers may be used for improving the stability of a SCC mixture which tends to a reduction in its fluidity in some cases. Therefore, to obtain a proper mix design of SCC mixtures the influence of various effective parameters should be respected simultaneously. The effect of several parameters such as W/C, chemical and mineral admixtures on fluidity has been reported A total of 54 mortar mixtures were prepared to evaluate the influence of chemical admixtures and mineral additives on the yield stress, viscosity and stability of fresh mortars. Relationships between the properties of concrete and corresponding mortar ASTM C 150 Type I ordinary portland cement was used for all the mixtures, and its chemical composition and physical properties are given in . Limestone riverbed sand with a specific gravity of 2.7, fineness modulus of 2.67 and water absorption of 2.81% was employed for all mixtures. The sand had a maximum aggregate size of 2.36 mm and particle-size distribution within the standard ASTM C33.A polycarboxylic-based superplasticizer (SP) was used. Its solid content and specific density are 36% and 1.07, respectively. Viscosity-modifying admixture (VMA) was a polysaccharide type used as a suspension in an aqueous solution with a concentration of 20%. The mineral additives used in this study were limestone powder and fly ash. Their chemical compositions and physical properties are also summarized in , four mortar test series were investigated. All the investigated mixtures were proportioned with a fixed cement content of 700 kg/m3 and various W/C of 0.35, 0.45, and 0.55. The amount of sand in the mixtures was varied in the range of 1250–1450 kg/m3 depending on W/C and powder content. Different dosages of SP, VMA, limestone powder, and fly ash were added in each series. The SP dosage was kept constant at 1% of binder mass for the mixtures incorporating VMA or mineral admixtures.Mortar mixtures were prepared in accordance with ASTM C 305 “Standard Practice for Mechanical Mixing of Hydraulic Cement Pastes and Mortars of Plastic Consistency” The static stability of the mortar was evaluated using a mini-column segregation test. The apparatus used was similar to the one described in ASTM C1610 where SI = static segregation index, percent, CAT |
= mass of aggregate in the top section of the column, CAB |
= mass of aggregate in the bottom section of the columnThe acceptable segregation index for fresh concrete in most applications obtained from standard column segregation test All the above tests performed twice, and the average value was reported. The test was repeated if the difference between the results was more than 20%.The results of mini-slump flow test are presented in . The results show that the W/C and superplasticizer content have a great influence on fluidity of the mixtures (a). Linear regression analysis of the results shows that increasing the W/C by about 10% can increase the mini-slump flow by about 18%. Furthermore, the addition of 1% superplasticizer, by mass of binder, can improve the fluidity of the mortar by up to 30%. In this way, the effect of 1% superplasticizer on fluidity is almost the same as 17% increscent in the W/C.b shows the fluidity of mortar mixtures incorporating limestone powder. The results indicate that using limestone powder may increase the fluidity of mortar mixtures. The improving effect of limestone powder on fluidity is more pronounced in lower W/C. The effect of fly ash on fluidity of mortar mixtures is also illustrated in c. Based upon the obtained results, using fly ash improves fluidity of mortar mixtures. The improving effect of fly ash on the fluidity may be due to its spherical shape which tends to reduce friction at the interface of aggregate and paste and producing “ball-bearing effect” at the contact point On the other hand, using VMA has a deteriorating effect on fluidity. The obtained results show that using VMA up to 1.5% decreases the slump flow of the mixtures linearly (d). The addition of 1% VMA tends to a 30% reduction in the slump flow of the mixtures. The obtained results indicate that increasing VMA dosage greater than 1.5% does not have significant effect on the fluidity of the mixtures. Mortar mixtures with high dosages of VMA may suffer from poor rheological properties. Therefore, in that case, a higher content of superplasticizer or higher W/C should be applied in order to decrease yield stress and improve self-consolidating properties. Based on these test results, to maintain the target slump flow when 1% VMA is added, the addition of 1% superplasticizer or increasing W/C by 17% is required. presents the results of mini V-funnel test. As can be seen in a, the effect of W/C on the flow times is more dominant than the effect of superplasticizer dosage. The relationship between W/C and viscosity of the mixtures is not linear. Increasing W/C from 0.35 to 0.45 decreases the flow time by about 77%, while increasing W/C from 0.45 to 0.55 resulted in 11% reduction in the flow time.d show that the addition of fly ash also affects viscosity of the mixtures and increases it to some extent. There is not much published study on the viscosity properties of self-consolidating mortar containing fly ash, and there is not a general agreement on the effect of fly ash on the viscosity of flowable mortars. Some authors state that incorporating fly ash in the mixtures decreases the viscosity Obtained results clearly show that viscosity of the mixtures is mainly influenced by the W/C while the other investigated parameters have a negligible effect on the viscosity. Moreover, it is worthwhile to note that the used VMA has a negligible effect on the viscosity of cement based mixtures. This is in accordance with the report of Koehler et al. Stability of the mixtures was also evaluated by determining the segregation resistance and resistance to bleeding and risk of blockage across narrow sections.The results of mini-column segregation test are summarized in . Based on these results, increasing W/C in a constant SP or increasing the SP content in a constant W/C, both, lead to an increase in segregation index of the mixtures. However, the effect of SP dosage is not linear and depends upon the W/C (a). For example, mixtures with a W/C of 0.35 have a segregation index less than 30% and they are all stable even if SP = 2%. On the other hand, increasing the SP content from 0% to 2% for the mixtures with a W/C of 0.55 changes a very stable mixture (SI = 20%) to an unstable mixture (SI = 158%).Moreover, using VMA seems to be very effective in enhancing segregation resistance of the mixtures. The effect of VMA on stabilizing cementitious mixtures seems to be more dominant at higher W/C. For example, the addition of VMA by 1% of cement mass in the mixtures with a W/C of 0.55, led to a decrease in segregation index from 159% to 83%. However, all the mixtures containing more than 1.7% VMA have a segregation index less than 30% and are very stable (b). It should be mentioned that using VMA decreases the fluidity of the mixtures which necessitates greater SP or W/C to maintain the target fluidity. c and d indicate that the effect of W/C on the aggregate segregation is more important than limestone content and fly ash content. However, increasing the powder content, slightly increase the risk of segregation.Fluidity of the mixtures may be achieved either by the addition of superplasticizer or increasing the W/C. However, the increased fluidity may cause instability. The relationship between fluidity and segregation index for the mixtures with variable dosages of SP and W/C is shown in . The solid line shows the mixtures in which the W/C was kept constant and the fluidity was increased by the addition of SP. On the other side, the dashed line shows the mixtures with constant SP and variable W/C. It is clearly evident that the slope of solid lines is much lower than those of dashed lines. This means that the SP increment is preferable over W/C increment for achieving the required fluidity since it will result in more stable mixtures.Based on these results, the most significant parameter influencing aggregate segregation is the W/C. shows the segregation index versus W/C of the mixtures. As can be seen in the figure, mortar mixtures with a W/C of 0.35 are stable in all the cases. On the other hand, mixtures with a W/C of 0.55 often exhibit sever segregation. Some SCC guidelines Bleeding and blockage of the aggregates are evaluated indirectly in the mini-slump flow test and mini V-funnel test. Bleeding of the mixtures is summarized in . As shown in the figure, increasing the W/C or SP content, especially at higher W/C, both, lead to an increase in bleeding of the mixtures. However, the comparison of a indicates that for a constant fluidity, mixtures with a lower W/C and higher SP dosage seem to exhibit less signs of instability. On the other hand, by using VMA, limestone powder or fly ash, even at low dosages, bleeding of the mixtures was completely disappeared. For example, in the case of W/C of 0.55, the addition of only 0.5% VMA by cement mass resulted in a decrease in the mixtures bleeding from 9 cm to 0 cm. In a similar way, no bleeding was observed in the mixtures incorporating 10% limestone powder or fly ash.Blockage of the mixtures is presented in . As shown in the figure, mixtures with higher W/C are prone to aggregate blockage while flowing narrow section of mini V-funnel nozzle. The obtained results indicate that using limestone powder or fly ash and also applying VMA, even at low dosages, could be very effective in enhancing flow properties of cementitious mixtures and prevent aggregate blockage while flowing across narrow sections. Therefore, for preventing bleeding and aggregate blockage of cementitious mixtures reducing W/C or using VMAs, limestone powder or fly ash could be suggested.The obtained results in this study also indicate that using VMAs is a very effective and efficient way for preventing any types of instability, including segregation, bleeding or blockage of the aggregates while using limestone powder or fly ash only improve the mixtures resistance to bleeding or aggregate blockage. indicates that there exists a close relationship between fluidity and stability of the mixtures. As the former increases, the latter greatly decreases. However, a proper self consolidating concrete should exhibit high fluidity and stability characteristics simultaneously. To reach a stable highly flowable mixture, the adjustment of mixture proportions and obtaining the optimum values of fluidity and viscosity are necessary. The relationship between segregation index and fluidity of the mixtures is plotted in . As the fluidity increases, the segregation index increases exponentially. Khayat and Assaad divided into four regions. Mixtures in the first region (mixtures with fluidity greater than 24 cm and segregation index less than 30%) correspond to stable and highly flowable mixtures. Mixtures in the second region exhibit high segregation resistance while suffering from poor fluidity. These mixtures do not have sufficient fluidity to flow readily under their own weight. To increase the fluidity of these mixtures, the higher W/C or SP dosages should be applied. For example, in , point 1 (W/C of 0.4 and no SP dosage) corresponds to the fluidity of 12 cm and segregation index of 6%. An alternative for increasing the fluidity to 30 cm is to use 1.5% superplasticizer. By this method, segregation index will be equal to 33% (point 2) that indicates a relatively stable mixture. By increasing W/C up to 0.55 and using 0.35% superplasticizer the target fluidity may also be provided (point 3). This increases the segregation index up to 95% and resulted in a mixture which is more prone to segregation. Based on these results, for obtaining high fluidity, increasing the W/C should not be applied due to instability aspects. Moreover, higher W/C may, also, lead to poor mechanical behavior of hardened concrete. So it is more efficient to apply higher SP dosages for obtaining super fluidity of the mixtures. When the lack of fluidity is not too much, the use of limestone powder or fly ash may be also effective.Mixtures in the third region are highly fluid while suspecting to instability and segregation. Using VMAs is a very effective tool for improving the stability of such mixtures. But it should be noticed that the use of VMAs may decrease the fluidity of the mixture and an increase in SP content is necessary for retrieving the initial fluidity. The obtained results indicate that increasing the W/C may compensate the stabilizing effect of VMAs, so it is not recommended to increase W/C to regain the reduced fluidity. The use of VMAs may lead to mixtures with low fluidity and addition of water or SP content may result in poor mechanical behavior of hardened concrete or non-feasible mixtures due to economical aspects. Consequently, it is suggested that SCC mixtures, if they are not prone to instability, may prepare with no use of VMAs. VMAs can be used for SCC applications to improve segregation resistance, increase cohesion, reduce bleeding, allow the use of a wider range of materials such as gap-graded aggregates and manufactured sands, and mitigate the effects of variations in materials and proportions Three stability response workability regions are defined in as a function of fluidity and viscosity. These regions correspond to segregation index ranging less than 30%, between 30% and 130% and greater than 130%. The fluidity of mixtures in the first region (stable mixtures) is lower than 30 cm while mixtures in the third region (unstable mixtures) exhibiting the fluidity greater than 30 cm and flow time lower than 5 s. Based on these results, neither the fluidity less than 30 cm nor the flow time lower than 5 s, by itself, can ensure the stable mixture. On the other hand, mixtures with a flow time greater than 10 s may suffer from poor workability. There are also some intersection regions observed in which indicate that mixtures with the same flowability characteristics may exhibit various stability properties due to different mixture proportions.To obtain a proper SCC with super workability, the effect of different concrete ingredients including W/C, type and dosage of chemical and mineral admixtures on the fluidity and also stability of the mixtures should be exactly studied. These parameters affect the rheological behavior of SCC in different manners and the only way to achieve self-compacting characteristics is the correct adjustment of these parameters. In total, 54 mortar mixtures were prepared to investigate the effects of different parameters on the rheological properties of cementitious mixtures. Based on the results of this study, the following conclusions can be made:The most significant parameters affecting mixtures fluidity are the W/C and superplasticizer content. The use of limestone powder and fly ash also improves fluidity to some extent. On the other hand, the use of VMA can decrease fluidity.The effect of W/C on the viscosity of mixtures is more dominant than other studied parameters. The W/C affects viscosity exponentially such that increasing W/C up to 0.45 strongly decreases the viscosity. Furthermore, using fly ash increases viscosity, but other parameters did not significantly influence the viscosity.The increase in fluidity of the mixtures tends to have a significant reduction in the stability. To minimize the unpleasant effects of high fluidity on the stability, addition of superplasticizer instead of increasing W/C is suggested. Based on the obtained results, in a constant W/C, increasing the mini-slump flow by one centimeter tends to increase segregation index by an average of 2.6%. For a constant superplasticizer content, using higher W/C to increase mini-slump flow by one centimeter can result in an increase in segregation index by about 7.5%.The W/C is the most significant parameter influencing rheological parameters of cementitious mixtures. Higher W/C tends to inhomogeneous mixtures, while reducing the viscosity.Maximum allowable W/C of a cementitious mixture for preventing heterogeneity mainly depends on the target fluidity. A mortar mixture with mini-slump flow less than 20 cm is stable in all of the investigated W/C. For a mini-slump flow of 25 cm, W/C should limited to 0.46. On the other hand, all the mixtures with a mini-slump flow greater than 35 cm are prone to severe segregation.Using VMAs is a very effective tool of stabilizing self-consolidating mortar and preventing bleeding, segregation and blockage of flow through narrow sections, while limestone powder and fly ash are shown to be only effective in reducing bleeding and aggregate blockage.Fatigue behavior assessment for steel fiber reinforced concrete beams through experiment and Fatigue Prediction ModelThe fatigue flexural behavior of steel-fiber-reinforced concrete (SFRC) beams was assessed through the experiment on SFRC beams subjected to fatigue loading and the analysis of Fatigue Prediction Model (FPM). The stress level, tensile reinforcement strength, and steel fiber volume fraction were considered in the program. Test results indicated that steel fibers can improve the fatigue performance of the beams. The fatigue life of the beams increased significantly with the increase of steel fiber volume fractions from 0 to 1.0%, and slightly decreased for 1.5% steel fiber volume fraction because of the fiber cluster. A FPM of SFRC beam was proposed based on the fatigue properties of constituent materials and cross-section analytical method. This model can simulate the development of strains in concrete and steel bars of beams under fatigue loading, and can also be extended to assess the effects of reinforcement ratio upon the fatigue life and failure mode of SFRC beams.Reinforced concrete (RC) structures, such as bridge girders, crane beams, airport pavements, and machine foundations, often suffer from repeated loadings during their service lifetime. These structures often failed under serviceability limit state in which the applied load is smaller than their ultimate capacity The fatigue performance of SFRC in structural level has also been paid attention to. Kormeling For under-reinforced concrete beam, the fatigue failure often happens in the tensile reinforcement. The fatigue life of RC structures is mainly dependent on the fatigue performance of steel bars. The fatigue design is based on the stress-life S-N properties of the reinforcement in practical application In this paper, 12 SFRC beams were tested subjected to fatigue loading to evaluate the influences of stress level, steel fiber volume fraction, and tensile reinforcement strength on the fatigue performance of the beams. The concrete strains in the compression zone were monitored and used to analyze the fatigue behavior of SFRC. Considering the fatigue degradation of the constituent materials of SFRC beams, an analytical FPM based on the fatigue properties of constituent materials and cross-section analytical method was proposed and validated by the test results. The model was also extended to assess the fatigue life and failure mode of SFRC beams with different tensile reinforcement ratio.Concrete mixes were designed with the target compressive strength of 80 MPa. The mixes was made of Portland cement with a 28-d compressive strength of 42.5 MPa, crushed aggregates with the average particle size of 12.5 mm, river sand with a maximum particle size of 5 mm, and potable water. A polycarboxylate superplasticizer with a water-reducing rate of 20% was used to improve the workability of concrete. The mix proportion of concrete was 1:1.2:2.1:0.008 (cement: river sand: coarse aggregate: superplasticizer by weight) with a water/cement ratio of 0.31. Hooked-end steel fibers with volume fractions of 0.5%, 1.0%, and 1.5% were added into the concrete. The mechanical properties of the steel fibers are listed in . The hooked-end steel fiber used in the concrete was the diameter of 0.55 mm, the length of 35 mm, and had an ultimate tensile strength of 1345 MPa and elastic modulus of 200 GPa. The shape and size of steel fibers are presented in . 12-mm-diameter deformed bars with the yielding strength of 400 MPa (HRB400), 500 MPa (HRB500), and 600 MPa (HRB600) were used as the tensile reinforcements, respectively. 8-mm-diameter plain bars with the yielding strength of 300 MPa (HPB300) were used as stirrups and the compression reinforcements. The mechanical properties of steel bars are listed in A total of 12 beams were tested in this study. Two beams (BJ1.0-4, BJ1.0-5) reinforced with HRB400 and HRB500 tensile reinforcements were tested under static loading to determine the ultimate bearing capacity (fu), respectively. The other ten beams subjected to fatigue loading were divided into three groups. Group 1 with HRB500 tensile reinforcement and 1% fiber content were tested under the different stress levels S (S = fmax/fu, where fmax is the maximum fatigue load applied to the beam) of 0.5, 0.55, 0.6, 0.7, and 0.8 (BS1.0-5, BS1.0-55, BS1.0-6, BS1.0-7, and BS1.0-8). Group 2 with HRB500 tensile reinforcement and different fiber contents of 0, 0.5%, 1.0%, and 1.5% (BF0-6, BF0.5-6, BS1.0-6, BF1.5-6) were tested under the same stress level S of 0.6. Group 3 with 1.0% fiber content and the different tensile reinforcements of HRB400, HRB500, and HRB600 (BH1.0-4, BS1.0-6, BH1.0-6) were tested under the same stress level S of 0.6. The minimum fatigue load was set as 10% of ultimate bearing capacity of each beam.All beams were tested under four-point bending load, the dimensions of the beam were 1200 mm (length) × 120 mm (width) × 200 mm (depth). The concrete cover thickness was 15 mm. The loading schematic and reinforcement details are presented in . For each beam, six 150 mm × 150 mm × 150 mm cubic concrete blocks were cast to test the compressive and splitting tensile strength of concrete, six 150 mm × 150 mm × 300 mm prism concrete blocks were cast to test the axial compressive strength and elastic modulus of concrete. Five 3 mm × 2 mm electronic strain gages were used to measure the strains of steel bars as shown in , in which two were on the stirrup and three were on the tensile reinforcement. The displacement of the beam were measured through five Linear Variable Displacement Transducers (LVDTs), one at the midspan, two at loading points, and another two at support points. Seven 100 mm × 10 mm electronic strain gages were placed on the concrete surface in the constant moment region to measure the concrete strains. A hand-held strain gage was also used to validate the results tested by electronic strain gages. The layouts of LVDTs, electronic strain gages, and mark of hand-held strain gage are presented in The tests were conducted on 500 kN MTS testing machine. The beams BJ1.0-4 and BJ1.0-5 were tested under a static load by displacement control at a rate of 0.2 mm/min until the beam failed. For the fatigue test, the beam was first loaded to the upper limit of fatigue load with a loading rate of 0.2 kN/s, and then reduced to zero. After that, a sinusoidal load with a frequency of 5 Hz was applied between the upper and lower limit of fatigue load up to beam failure. During the fatigue test, the MTS actuator was programmed to stop at the certain fatigue cycles to make a static test. The static tests were conducted at 1 × 103, 3 × 103, 5 × 103, 1 × 104, 1.5 × 104, 2 × 104, 5 × 104, 1 × 105 and every 105 cycles until the failure happens. The beam was considered as run-out if the number of fatigue cycles exceeds 2 million, and then a static test was conducted until the beam failed. The data of electronic strain gages and LVDTs were collected through a dynamic data acquisition device. The crack widths measured by an optical microscope and the strains measured through the hand-held strain gage were recorded during each static test. The residual strains of concrete in the compression zone were also recorded when the load uninstalled to zero. MTS and acquisition system worked continuously until the failure of beam.The beams BJ1.0-4 and BJ1.0-5 were monotonically tested under static loading. In the fatigue test, the beam BS1.0-5 did not fail after undergoing 2 million fatigue cycles, so it was also tested under a static load until the failure happened. As shown in , the failure mode of beams BJ1.0-4, BJ1.0-5, and BS1.0-5 was the same as the tension-controlled failure of under-reinforced beam. The deflections of the beams increased almost linearly with the increase of load before the tensile reinforcement yielded. After reaching to the ultimate bearing capacity of the beams, the load gradually decreased while the deflections increased rapidly. Finally, the beam was failed with the crushing of concrete in compression zone. shows the measured fatigue life of each beam. As shown, the fatigue life of the SFRC beams decreases with increasing stress level and increases with fiber content. The fatigue life is mainly determined by the applied stresses and fatigue properties of the tensile steel bar, the influence of tensile reinforcement strength on it is not significant. Compared with non-fiber RC beam, the fatigue life of the beams with fiber contents of 0.5%, 1.0%, and 1.5% is increased by 38%, 106%, and 61% under the same stress level of 0.6, respectively. It is attributed to the fiber bridging effects in the tensile region which reduces the stress ranges of tensile steel bar, thus increases the fatigue life of the SFRC beams significantly. It should be noted that the fatigue life shows a decrease from 1.0% fiber content to 1.5%. Zhang . It can be seen that there has an obvious fiber cluster for the specimen with 1.5% fiber content. The fiber cluster decreases the bonding properties between steel fiber and concrete matrix, which will directly affect the fatigue performance of the specimen. Hence, for SFRC beams casted with coarse aggregate, the optimum fiber content should not be more than 1.0%.All the beams subjected to fatigue loading were failed by the fracture of tensile steel bar. During the fatigue loading, the stress concentration of the tensile steel bars occurred at the crack location of the beam. With increasing loading cycles, the fatigue damage accumulated in the tensile steel bars, and the fatigue crack nucleation formed gradually and initiated in one of the tensile steel bars. The beam began to fail as soon as the residual strength of tensile steel bar could not bear the applied fatigue load. shows the failure sections of tensile steel bar, it can be seen that one section is smooth owing to the micro-crack propagation and the other section is typically necking failure owing to the applied load.The concrete surface strains in constant moment region of the beam were measured using a hand-held strain gage during each static test. The strains under maximum fatigue load at the different fatigue cycles for each beam are presented in . It can be seen that the depth of neutral axis for each beam has a slightly fluctuation during fatigue loading, while the plane-section assumption is still suitable for SFRC beams subjected to fatigue loading.During the fatigue loadings, the increasing of beam deflection is attributed to the fatigue deterioration of constituent materials and the crack propagations in the depth of cross section, both of them will cause the stiffness degradation of the beam. The measured midspan deflections for each beam under the maximum fatigue loads are compared against the fatigue cycles, as shown in . It can be seen that the deflections of beams increase with the increase in stress level, while decrease with the increase in fiber volume fraction. The midspan deflections under the maximum fatigue load for SFRC beams with 0.5%, 1.0%, and 1.5% fiber contents were decreased by 7.9%, 13.3%, and 17.9% at 200,000 fatigue cycles compared with the non-fiber RC beam. It indicates that adding steel fibers into concrete can greatly increase the stiffness of SFRC beams. also shows a rapid increase of the beam deflections before the beam failure, the main reason is the sudden fracture of tensile steel bar.The final crack patterns of the beams with different fiber contents subjected to fatigue loading are given in . In the figure, the red line indicates the failure location of the beam, the blue number indicates the number of fatigue cycles when the crack develops to this height, and the circled numbers below the beam indicate the appearance sequence of the cracks. It can be noted that the number of cracks increases with the increase in fiber content, especially in the constant moment region of the beam. The cracks of the beam become thin and dense with the addition of steel fibers. For RC beam, the tensile stresses in the crack location are transferred by the tensile reinforcements. However, for SFRC beams, the tensile stresses will be evenly transferred by the steel fibers after concrete cracking, the stress concentrations of concrete and the crack patterns of beams can be improved.The average crack widths of the beams with different fiber contents under the maximum fatigue loads versus the fatigue cycles are plotted in . As shown, the average crack widths of the beams decrease with the increase in fiber content. Compared to the non-fiber RC beam, the average crack widths of the beams with 0.5%, 1.0%, and 1.5% fiber contents were decreased by 1.3%, 26.5%, and 35.3% at 300,000 fatigue cycles, respectively. The fiber bridging effect could decrease the average crack widths of the beams significantly. Moreover, the rapid increase of the average crack widths was found before the beam failure owing to the sudden fracture of tensile steel bar, which was consistent with the deflection development trend.It has been recognized that the fatigue performance of structural members is determined by the fatigue properties of its constituent materials. The member will be failed if one of the constituent materials reaches its fatigue limit. For SFRC beams subjected to the fatigue loading in this paper, there had two failure modes, one was the fracture of tensile reinforcement, and the other was the crushing of concrete in the compression zone of the beam. Therefore, the fatigue properties of SFRC and tensile reinforcement used in this experiment are discussed, then the cross section analysis and a step-by-step procedure is presented to implement the developed model.where σc is the stress applied on the specimen, Ec the initial ealstic modulus of concrete.The strain increment εcn refers to the creep deformation of concrete or residual strain. Feng where t is a constant value, n the number of fatigue cycles; ασ is closely related to the applied fatigue stress.According to the tested residual strains in the compression zone of 10 SFRC beams in this study, it was found that there also has a linear relationship between Lgεcn and Lgn for SFRC beams, the stress level and fiber volume fraction have the significant influences on the residual strain, as shown in . The parameters ασ and t are mainly influenced by the stress level and fiber volume fraction, respectively. By the regression analysis of the experimental data, the expressions forασ and t are obtained as follows:, the elastic modulus for SFRC in the compression zone of the beam at the maximum fatigue load of n fatigue cycles can be derived as:Ecfn=σc,max/σc,maxEc+Lg-1(0.0337σc,max-5.3415)∗n(0.1148-0.0371λf) is the irrecoverable part within the total strain. The failure of concrete will be occurred when the residual strain εcn comes to a threshold criterion. The same fatigue failure criterion for plain concrete where fc is the compressive strength of concrete.During fatigue loading, the strains of the tensile steel bars accumulated with the increasing fatigue cycles, which lead to the initiation of cracking on the surface of the tensile steel bar where N is the fatigue life of the tensile steel bar, σs,max the maximum nominal stress applied to the tensile steel bar, As the initial section area of the tensile steel bar, fy the yield strength of the tensile steel bar.Since the applied fatigue load is lower than the yield strength of the beam in this experiment, the tensile steel bar is in its elastic fatigue stage during the fatigue loading. Thus, it can be assumed that the residual area of tensile steel bar reduces linearly with the increase of fatigue cycles. Therefore, the residual area of tensile steel bar subjected to n fatigue cycles can be calculated as follows:The fatigue life N of tensile steel bar can be obtained from stress-life S-N curve. According to the S-N curves of HRB400 , the equation including N can be obtained as follows:LgN=34.19-11.05LgΔσ(HRB400)52.04-17.39LgΔσ(HRB500),where Δσ=σs,max-σs,min is the nominal stress range.For the tensile steel bars reinforced in SFRC beam, the other fatigue variables should be considered such as the stress level and the degradation coefficient of tensile stress which was undertaken by steel fibers in the crack location during the fatigue loading. Hence, the stress-life expressions for tensile reinforcements in SFRC beam should be modified as follows:LgN=34.19-11.05LgKs(Kρ-ρ)σs,max(HRB400)52.04-17.39LgKsKρ-ρσs,max(HRB500)where ρ=σs,min/σs,max is the stress ratio, Kρ the fatigue strength coefficient influenced by stress ratio, Ks the degradation coefficient due to the fiber pullout in the crack location during fatigue tests.Kρ=0.938/1-0.6165ρ(HRB400)0.629/1-2.185ρ(HRB500)For normal RC beam, the tensile stress of concrete is often ignored in the fatigue calculation, the tensile stress of steel bar in crack location is consistently 20%-40% higher than the average stress of steel bar owing to the stress concentration The localized stress of tensile steel bar increases with the decrease of residual area of tensile steel bar under fatigue loading, when the localized fatigue stress σs,maxf reaches the yield point of tensile steel bar, the fracture failure of tensile steel bar happens. The fatigue failure criterion of tensile reinforcement is estimated as:The strains of tensile reinforcement measured by electrical strain gages were compared with the strains of concrete measured by hand-held strain gage. Results showed that the measured data of tensile reinforcement strain are fairly close to those of concrete strain in the same place of beams. This indicated that the excellent bonding properties between steel bar and SFRC. Hence, it is assumed that no bond-slip between steel bar and SFRC happened during the fatigue tests. Furthermore, the strains measured by hand-held strain gage indicated that the plane-section assumption is still suitable for SFRC beam subjected to fatigue loading as shown in shows the stress and strain distribution in the cross section of SFRC beams under the fatigue flexural moment Mf. Based on the above discussion and , the following equations can be obtained:where εc is the concrete strain in the compression zone; εs is the strain of tensile steel bar; xn is the depth of the compression zone; h0 is the effective depth of the beam section; b and h are width and height of the beam section, respectively; Es is the elastic modulus of the tensile steel bar; fftu is the equivalent tensile stress of SFRC in the tensile region. According to the linear model in CEB-FIP where fR1 and fR3 are the residual flexural tensile strengths corresponding to the crack mouth opening displacement (CMOD) of 0.5 mm and 2.5 mm of the notched beam, respectively;CMOD3 is the crack mouth opening displacement and is taken as 2.5 mm; wu is the maximum crack opening accepted in structural design and is taken as 1.5 mm herein.where φ=1+lf/1002; fc is the axial compressive strength of the concrete prisms in this experiment; R is the number of hook-ends in steel fibers and is taken as 1.The step-by-step fatigue analysis to assess the fatigue life and failure mode of SFRC beams is given in . The procedure was implemented in a computer program and the details are provided below:Input the initial material parameters (Ec,Es, As, fftu), specimen configuration (b, h, h0), and fatigue flexural moment (Mf).Calculate the strain and stress of SFRC and tensile steel bar using Eqs. (15)–(17).Determine the fatigue states of SFRC and tensile steel bar using Eqs. If both the failure of SFRC and tensile steel bar do not happen, update the elastic modulus of SFRC and the residual area of tensile steel bar using Eqs. , respectively. Return to step (2) and make next calculation.If either the SFRC or tensile steel bar fails, end the program and output the data.9 SFRC beams in this study and 12 SFRC beams conducted by Parvez . The average value of the ratios of experimental to predicted fatigue life is 0.992 with the coefficient of variation of 0.23. It indicates that the predicted fatigue life is in a good agreement with the experimental results.The experimental and predicted strains of concrete in the margin of the compression zone and tensile steel bars for each beam at different fatigue cycles are plotted in that the FPM results correlate with the experimental results at all stages of fatigue loading up to failure. Both of them show a rapid growth of the strains before the beam fails, which indicates the effectiveness and practicability of the FPM. The scatter of the strain values for some beams is owing to the test measures, and the cross-section analysis of FPM is in the midspan section of the beams, although the fracture location in the experiment may deviate from the midspan section.To investigate the effect of tensile reinforcement ratio on the fatigue performance of SFRC beams, the supplementary beams with HRB500 tensile reinforcements of 2Φ14, 2Φ16, 2Φ18, and 2Φ20 for different fiber volume fractions of 0, 0.5%, and 1.0% were calculated by the proposed model, respectively. The basic parameters such as beam section and material behavior were taken the same as the beams with different fiber contents series in this experiment, and the stress level S was taken as 0.6. The predicted fatigue life for the beams with different reinforcement ratios is presented in . As shown, the fatigue life of the beams shows a consistent tendency for different fiber contents series, it increases with the increase of tensile reinforcement ratio up to 2.12%, while decreases obviously when the reinforcement ratio reaches to 2.62%. The main reason is the failure of concrete in the compression zone prior to the tensile reinforcement. For the beams with the reinforcement ratio of 1.28%, the failure mode is fatigue fracture of tensile reinforcement. When the reinforcement ratio reaches to 1.68%, the failure mode of the beams with 0% and 0.5% fiber volume fractions is the rupture of tensile steel bar, but the beam with 1.0% fiber volume fraction failed by the concrete crushing in the compression zone. When the reinforcement ratio reaches to 2.12%, all of the beams are failed by the concrete crushing in the compression zone. The optimum reinforcement ratio should not exceed 2.0% for both RC and SFRC beams subjected to fatigue loading according to the calculation results.The proposed model is based on the fatigue analysis of SFRC beams subjected to constant-amplitude fatigue loading. It is not applicable to the specimens subjected to random-amplitude fatigue loading. Moreover, it was assumed that there is no slip between steel bar and SFRC during the fatigue tests, and the stress-strain behavior of SFRC with hooked-end steel fibers was implemented to represent the fatigue behavior of SFRC in the compression zone. Therefore, for the SFRC beams reinforced with FRP (Fiber Reinforced Ploymer) bars or corrosion steel bars, the slip should be considered in the model, and the Eq. should be modified for using other fibers such as PVA fibers, glass fibers, and so on.In the presented work, the fatigue performance of SFRC beams with different design parameters was experimentally studied. An analytical FPM was proposed to assess the fatigue behavior of SFRC beams. The following conclusions can be drawn from the experimental results and FPM analysis:The fatigue life of SFRC beams decreased with the increase in stress level and increased with the increase in fiber volume fraction. All of the SFRC beams subjected to fatigue loading were failed by the rupture of tensile steel bar.The stress range of tensile steel bar was the controlling factor for the fatigue life of the beams. The bridging effect of steel fibers in tensile region reduced the stress ranges of tensile reinforcement, which increased the fatigue life of SFRC beams directly.Compared with the 1.0% fiber volume fraction beam, the fatigue life of the beam with 1.5% fiber volume fraction showed a slightly decrease owing to the fiber cluster. The fiber volume fraction should not be more than 1.0% to obtain the optimal fatigue performance of SFRC beams cast with coarse aggregate.Steel fibers could improve the stiffness of beam and bonding properties between the steel bars and concrete, which lead to the narrower crack widths and reduced deflections during the fatigue loading. The crack patterns of beams were also improved with the addition of steel fibers.High reinforcement ratio lead to the failure of concrete in the compression zone for both normal RC beams and SFRC beams subjected to fatigue loading. The optimum reinforcement ratio should not exceeds 2% according to the analysis by the proposed model.The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.Resonant acoustic evaluation of mechanical properties of masonry mortarsThis study aims to establish relationships between dynamic and static mechanical properties for masonry mortars containing lime. An experimental research was conducted with four mortar mixes of different compositions. Acoustic and compression tests were performed at different ages. Acoustic tests showed lower dispersion and higher elastic moduli than static tests. A good relationship between them was found assuming a power law (R2 |
= 0.966). Dynamic elastic modulus showed a mutual correlation with compressive strength and static elastic modulus (R2 |
= 0.982). A new equation was proposed to represent mechanical properties vs. age considering the coupling between the hydration and carbonation hardening mechanisms.Dynamic behavior of masonry structures has been on focus of recent research, involving NDT evaluations using sonic Different approaches can be employed to simulate masonry panels with FEM models. Rekik and Lebon Although the importance of the dynamic properties of masonry mortars, very limited research is found in the literature on this topic. Swamy and Rigby However, their results cannot be direct applied to masonry mortars because of the composition of the mortars tested that has not included lime. It is well known that lime is frequently added in a great amount to masonry mortar mixes because of the significant increase in workability. When lime is included, several properties of mortars change as a consequence of the different hardening mechanisms in comparison to Portland cement. The hydrated lime hardens due to carbonation (Paiva et al. , while Portland cement hardens mainly due to hydration of calcium silicates (Mehta and Monteiro . In masonry mortars both mechanisms are frequently present.2(CaO)2·SiO2+4H2O→(CaO)3(SiO2)2(H2O)3+Ca(OH)22(CaO)3·SiO2+6H2O→(CaO)3(SiO2)2(H2O)3+3Ca(OH)2In a previous study of the authors (Haach et al. The consistent results obtained in this previous research stimulated the authors to expand the analysis in this paper. A greater number of cylindrical samples (240) were now produced allowing determination of dynamic and static properties at 7, 14, 28, 60, 90 and 120 days. The evolution of mechanical properties over time could be evaluated. Besides this, taking into account the different mixes and ages of testing considered in the present work, a very wide range of static and dynamic properties was generated, allowing more robust regression analysis in comparison to the previous paper from the authors. Prismatic specimens (50 mm × 50 mm × 340 mm) were also produced, making possible the determination of the dynamic transverse modulus (Gd) and Poisson’s ratio (νd), consequently allowing a more complete determination of dynamic properties of masonry mortars.Thus, the objective of this study is to characterize the dynamic elastic properties of masonry mortars (Ed, Gd and νd), their evolution with time and establish a robust correlation with mechanical properties obtained from static tests (fm and Estatic).The mortar compositions evaluated in this study were specified by BS 5628-1 The mixes of mortars were prepared using Portland cement, lime and sand. Four mixes of mortar were prepared keeping the same binder/aggregate proportion: 1:3 (Portland cement:sand), 1:0.5:4.5 (Portland cement:lime:sand), 1:1:6 (Portland cement:lime:sand) and 1:2:9 (Portland cement:lime:sand). The water/binder ratios for the each mix of mortar were 1.00, 1.31, 1.90 and 2.00, respectively.Portland cement type II specified according to the classification of ASTM C150 ). Some physical properties of materials are indicated in Mixing of mortars was performed in accordance with the recommendations given in ABNT NBR 13276:2002 Sixty cylindrical specimens with 50 mm of diameter and 100 mm of height (height-diameter ratio of 2:1) were cast according to ABNT NBR 13279:1995 Static tests of cylindrical specimens were performed according to ABNT NBR 13279:1995 Compressive strength (fm), elastic modulus (EStatic), axial strain at peak stress (εp), fracture energy (Gc) and ductility factor (µ) were the variables evaluated through static tests. Static elastic modulus was obtained in accordance with ASTM C469 A modal analysis was performed with the Finite Element Method software Abaqus® 6.13 to identify all natural frequencies and modes of vibration of interest. A three dimensional finite element with 20 nodes, quadratic interpolation and reduced integration was used (C3D20 R). For cylindrical specimens a mesh of 2300 hexahedral elements and 10578 nodes was used. In prismatic specimens a mesh of 2205 hexahedral elements and 10976 nodes was used. shows flexural, torsional and longitudinal modes of prismatic specimens. The same types of modes appeared in the simulations with cylindrical specimens. It was found the natural frequency of the i-mode is given by Eq. where Ed is the dynamic elastic modulus (N/m2), ρ is the density of the material (kg/m3) and Ci is a constant related to a specific mode of vibration and geometry of the tested specimen. The results of the numerical analysis allowed determination of the constants (Ci) for the prismatic and cylindrical specimens tested shown in Acoustic tests were conducted as described schematically in . The specimen was supported by steel wires connected to a rigid frame in order to simulate free-free boundary conditions. A small impact was manually applied to the specimen. A regular microphone was used to capture the sound transmitted by the surface of the specimen. The microphone was connected to a standard sound card in the computer. A special purpose software (SONELASTIC® Typical frequency spectra obtained in cylinders and prisms acoustic tests (mix 1:1:6 at 7 days of age) are shown in experimental curves from , respectively. Estimates of frequencies obtained by Eq. for these specimens are also shown in these figures. The estimates were important to assure a correct identification of natural frequencies during acoustic tests.In cylindrical specimens a longitudinal impact was applied slightly misaligned with the longitudinal axis so that the first flexural and longitudinal modes were clearly identified (). In prismatic specimens two measurements were necessary. In the first case a longitudinal impact was applied in the center of the prism cross section allowing determination of the first longitudinal mode (a). In the second measurement a transverse impact was applied closed to the edge of the prism exciting simultaneously the first flexural and torsional modes (b). In this way, the dynamic shear modulus (Gd) could be calculated. Dynamic elastic moduli, Ed,L and Ed,F, were calculated from natural frequencies of the 1st longitudinal and 1st flexural vibration mode, respectively, according to ASTM C215 In spite of the focus of this research is the results of acoustic tests, static tests were performed in order to obtain reference results for comparison with mechanical properties measured from those tests. show the results of static tests for all ages and mixes and shows the dynamic elastic moduli obtained from acoustic tests using cylindrical specimens.The dispersion of the mechanical properties evaluated with static tests was very distinct. Compressive strength showed the lowest coefficient of variation (CV) with values around 7.6% whereas, static elastic modulus around 12.2%, axial strain at peak stress around 19.2%, ductility factor around 18.2% and fracture energy, which showed the highest values of CV, around 22.5% on average. Acoustic tests provided very low coefficient of variation when compared with static tests. Dynamic elastic modulus showed an average CV of approximately 4.5% in acoustic tests which, is almost three times lower than the CV of the static elastic modulus. Elastic moduli obtained from 1st longitudinal and 1st flexural vibration modes were almost the same as showed in . Due to this similarity all next analyzes are performed only with the 1st longitudinal vibration mode. shows representative stress–strain curves of cylindrical specimens under compression for the mortar mixes tested (120 days of age). Mixes with a greater amount of lime (1:1:6 and 1:2:9) had lower strength and much higher deformability. On the other hand, mixes with lower lime content (1:0.5:4.5) or without lime (1:3) showed a comparatively more brittle behavior. evidenced the observed behavior of axial strain at peak stress as a function of compressive strength. For mortars of low compressive strength (large lime content), a reduction of axial strain at peak stress was caused by the increase of compressive strength. This behavior is related with the ductility of the material generated by the presence of lime (Tables ). On the other hand, the axial strain at peak stress increased together with the compressive strength when small lime content or no lime was used. The second type of behavior also occurs in concrete as shown by fib Lime had also a large influence in variation of the mechanical properties with age, changing the usual shape of the curve compressive strength or elastic modulus vs. age as shown by . This behavior occurs due to the different hardening mechanisms for cement and lime, hydration and carbonation, respectively, as commented in Section 1. The carbonation occurs much slower than hydration mechanism mainly due to the enormous differences between the water and carbon dioxide concentrations available for the reactions (Morales-Flórez et al. ) dependent of three parameters: A, B and C. Where, A is variable y in the infinite or in terms of design in 28 days and B and C are constants that define the increase in variable y from 0 to A.Based on the evolution curve of the mechanical properties with time proposed by Haach et al. ) was performed with the experimental results as showed in . This new function considers a coupling between the hydration and carbonation mechanisms, which implies in a contribution of the lime in the mechanical property. Both elastic modulus and compressive strength were related to age by the exponential law as follows:where y is compressive strength or elastic modulus, A1 is variable y after complete hydration, A2 is the contribution of lime to variable y due to the carbonation in advanced ages, B1 and C1 are constants that define the increase in variable y from 0 to A1 due to the hydration mechanism and B2 and C2 are constants that define the increase in variable y in advanced ages due to the carbonation mechanism.In general, a reasonable correlation was obtained between Eq. and the experimental results, as showed in . This correlation could be improved if more tests were performed in early and advanced ages in order to enrich the representation of the increase of the variable y due to the hydration and carbonation mechanisms. Here, the null values for A2, B2 and C2 in case of mortar mix 1:3 should be highlighted due to the absence of lime in this mortar. Static elastic modulus was the variable which showed the worst coefficients of determination, R2, as expected due to the high dispersion of this mechanical property as discussed previously. In the opposite side was the dynamic elastic modulus with the best values.Static and dynamic elastic moduli were related through a linear and a power law as shown by with coefficients of determination (R2) equal to 0.9392 and 0.9663, respectively. In spite the good correlation, results showed the relation between the elastic moduli is influenced by mortar mix () however, it does not significantly changes with the age (Both static and dynamic elastic moduli showed reasonable correlations with compressive strength of mortar with coefficients of determinations equal to 0.9239 and 0.9608, respectively (). The lowest dispersion of the results of elastic modulus from acoustic tests enabled a correlation with compressive strength with the highest R2.Dynamic elastic modulus obtained from prismatic specimens were a little lower than those obtained from cylindrical specimens. shows the dynamic elastic moduli obtained from acoustic tests using prismatic specimens. This behavior occurs probably due to the different cast direction of cylinders and prisms and also due to the size effect, as observed by other researchers (Gonnerman . Dynamic elastic moduli, Ed,L and Gd, are related to compressive strength of mortar follow a power law with the same exponent indicating proportionality between the two mechanical properties, see . As in case of cylindrical specimens, this relation seems to be independent of age or mortar mix.One advantage of to perform tests in prismatic specimens is the possibility of to evaluate the Poisson’s ratio by the relation between shear and longitudinal elastic moduli. In general, concrete is assumed as an isotropic material, for simplification, with dynamic Poisson’s ratio ranging from 0.10 to 0.25 Once dynamic elastic modulus showed reasonable correlations with compressive strength and static elastic modulus, an analysis was conducted aiming evaluate a mutual correlation between the three mechanical properties. Eq. was found with a coefficient of determination equal to 0.9815. Dynamic elastic modulus was estimated from Eqs. using experimental compressive strengths and static elastic moduli. The value obtained from equations was denominated theoretical elastic modulus and compare with experimental values of dynamic elastic modulus, see EDynamic=146.14fm0.30EStatic0.44(in MPa)Results showed in spite the high value of the R2 of the Eq. , the relation EExperimental/ETheoretical reached an average value of 1.1422 with a CV equal to 20.93%. That relation estimated dynamic elastic modulus more than two times higher than the experimental value for some specimens. Mortar mix 1:2:9 showed the highest discrepancies in the estimated dynamic elastic modulus. The Eq. generated relations EExperimental/ETheoretical with average value of 0.9800 with a CV equal to 12.48%. Although the R2 of this equation was lower than R2 of Eq. , the dynamic elastic modulus estimated from it was not higher than 30% of the experimental value. The same behavior was observed for Eq. , which had a R2 equal to 0.9663 (lower than R2 of Eq. ) and an average value of EExperimental/ETheoretical equal to 1.0078 with a CV equal to 11.52%. Eq. presented the best results with an average value of the relation EExperimental/ETheoretical equal to 1.0037 with a CV equal to 7.52%. The estimated values of dynamic elastic modulus did not overcome 20% of the experimental values and the relation EExperimental/ETheoretical shows a uniform distribution through the different mortar mixes. shows the ratio EExperimental/ETheoretical for the mortar mixes individually. In general the mix 1:2:9 had the highest coefficients of variation and the most distant values of the unit. On the other hand, the mix 1:3 showed the lowest dispersion. Regarding the ratio for the estimation of dynamic elastic modulus, Eq. presented the lowest coefficients of variation, except for mix 1:2:9. Regarding the average values of EExperimental/ETheoretical ratio Eqs. for dynamic values showed the best results for all mortar mixes.An experimental study was carried out to evaluate mechanical properties of masonry mortar based on resonant acoustic tests. Four mixes with significantly different compositions were considered. Mixes with greater amount of lime had lower strength and much higher deformability while mixes with lower lime content or without lime showed a comparatively more brittle behavior.Lime had also a strong influence in the variation of mechanical properties with age. Mortar with lime showed a behavior over time clearly influenced by hydration and carbonation hardening mechanisms. Based on the experimental results, a new equation was proposed to represent the variation of compressive strength and elastic modulus with time considering the coupling between the hydration and carbonation mechanisms. The average coefficient of determination obtained for compressive strength, static and dynamic elastic moduli was equal to 0.7449, 0.5751 and 0.8807, respectively.Compression tests were also conducted for all mixes and obtained results of static elastic modulus with higher dispersion (average CV = 12.2%) than acoustic tests (average CV = 4.5%). This is explained because compression tests are typically more susceptible to errors caused by small eccentricities of loading, misalignment and irregularities of the specimens in comparison to acoustic tests.The elastic modulus obtained by acoustic tests was significantly higher than the static (56–125% higher). Although this systematic difference, static and dynamic modulus could be correlated by a linear (R2 |
= 0.988) or a power law (R2 |
= 0.966). Both static and dynamic elastic modulus showed reasonable correlations with compressive strength of mortar (R2 equal respectively to 0.924 and 0.961). Alternatively, a more robust equation was proposed to evaluate the dynamic, from static elastic modulus and compressive strength even with lower dispersion (R2 |
= 0.982).The dynamic shear modulus (Gd) obtained with prismatic specimens was consistently equal to 45.3% of the dynamic elastic modulus (Ed,F) obtained from the flexural mode resulting in Poisson’s ratio equal to 0.103. This value corroborates with the range 0.10–0.25 for the Poisson’s ratio found in the literature.The authors believe that the experimental results found in this research can support future investigations on dynamic behavior of masonry structures where estimation of dynamic mechanical properties of mortar is often necessary.Mechanisms of dislocation multiplication at crack tipsWhether a stressed material fractures by brittle cleavage or ductile rupture is determined by its ability to convert elastic strain energy to plastic deformation through the generation and motion of dislocations. Although it is known that pre-existing dislocations play a crucial role in crack tip plasticity, the involved mechanisms are unclear. Here it is demonstrated by atomistic simulations that individual pre-existing dislocations may lead to the generation of large numbers of dislocations at the crack tip. The newly generated dislocations are usually of different types.The processes involved are fundamentally different for stationary cracks and propagating cracks. Whereas local crack front reorientation plays an important role in propagating cracks, the multiplication mechanism at stationary cracks is connected with cross-slip in the highly inhomogeneous stress field of the crack. Analysis of the forces acting on the dislocations allows to determine which dislocations multiply and the slip systems they activate. These results provide the necessary physical link between pre-existing dislocations and the generation of dislocations at crack tips.The resistance of a material against crack propagation is undoubtedly one of the most important properties of structural materials. It is quantified by the fracture toughness KIc. The temperature dependence of the fracture toughness and the strain-rate dependence of the brittle-to-ductile transition (BDT) have both been shown to correlate well with dislocation mobility Fracture toughness is determined by the competition of the dynamics of the atomic bond-breaking processes Although it has long been known that predeformation and the availability of dislocations in a material strongly influence fracture toughness, only recently has attention been directed towards the mechanisms of interaction of existing dislocations with the crack tip. In situ experiments by X-ray topography The atomistic simulations reported here reveal several mechanisms by which pre-existing dislocations interact with an atomically sharp crack and multiply. It is shown here that such multiplication necessarily requires cross-slip at the crack tip, which is substantially aided by a sign-change in the driving force of some dislocations in the field of the crack. For propagating cracks, the interaction with dislocations can lead to a local reorientation of the crack front, which enables the activation of additional slip systems.Because of the wealth of experimental details available on dislocation sources in silicon, this seems like the material of choice for any such simulation. However, dislocation motion in silicon is thermally activated and very sluggish at reasonable temperatures due to the relatively high activation energy for dislocation motion a. The simulations are performed in the so-called γ-orientation b). The dislocations in each class are listed in the table in c. Screw dislocations can change between classes I and II by cross-slip, whereas dislocations of class III cannot change to another class. Dislocations of class III have never been observed to be emitted from γ-oriented cracks in silicon under mode I loading The present study therefore focuses on dislocations of class I and II (for the dislocations of class III, see Ref. a) corresponds to a cubic crystal with side lengths of 75 nm (about 38 million atoms). Fixed boundary conditions are used for the atoms in the outermost boundary layers, except along the crack front direction, where the motion of the atoms in the boundary layers is only restricted in the z-direction. In this configuration a crack is introduced by application of the displacement field of an atomically sharp crack in a rigidly clamped thin strip The interaction between the crack and the dislocation is studied in microcanonical MD simulations starting from an initial temperature close to 0 K. During the simulations, the increase in temperature due to conversion of the elastically stored energy of the starting configuration into the motion of atoms was less than 10 K. For visualization only atoms with a potential energy deviating by more than 2% from the bulk energy are displayed. The Burgers vectors of the dislocations were determined using a slip vector analysis The dislocations of class I were shown previously to just cut the crack front without causing any dislocation emission, cross-slip or dislocation reactions If an initially straight screw dislocation with Burgers vector DB is placed on the (a)-plane Δx |
= 6 nm in front of the crack tip, a segment of the screw dislocation immediately cross-slips onto the (c)-plane (see a). This segment is classified as a class I dislocation. It is attracted towards the crack tip and cuts it like other class I dislocations without any further dislocation reactions. The dislocation line in front of the crack also cross-slips onto the (c)-plane. The remaining initial dislocation above the crack surface does not cross-slip but bows down and intersects the crack surface where it stays pinned. This segment then acts as a pole around which the cross-slipped dislocation revolves, forming a spiral source. The boundary conditions do not allow the newly generated dislocations to escape, which causes a back-stress on the source and eventually stops its operation. The entire process is shown in Video 1 in the The situation is different when a dislocation with the same Burgers vector and identical glide plane but non-screw line orientation is placed close to the crack tip. show an example in which an initially straight 60° dislocation with Burgers vector DB(a) placed in front of the atomically sharp crack tip (Δx |
= 6 nm) approaches a crack, loaded at 0.995 of the Griffith strain. As can be seen in a, parts of the dislocation close to the crack tip are strongly curved. Once the dislocation intersects the crack front, the upper part of the leading partial dislocation changes its glide plane to the (b)-plane by creating a stair-rod partial dislocation between the (a)- and the (b)-plane, according to the reaction Dα |
→ |
Dβ |
+ |
βα. This essentially pins the upper part of the dislocation at the crack front, while the lower part of the dislocation proceeds to intersect the lower crack surface. On the upper side, however, a new leading partial dislocation Dγ is generated on the (c)-plane from the intersection of the stair-rod dislocation βα and the crack front. The emission of further leading partial dislocations on the (c)- and (d)-planes lead to the complex dislocation structure in c. The cross-slip process of an individual partial dislocation is usually referred to as the Fleischer mechanism Other simulations under different loading conditions as well as in smaller simulation boxes have all confirmed these processes. In particular, for the kind of dislocations studied here, stimulated emission of dislocations was always observed for strains larger than 0.9ϵG and was always connected with partial dislocation cross-slip and the formation of a stair-rod dislocation at the crack tip.It is, however, interesting to note that the partial dislocation cross-slip at the crack tip seems to be a necessary but not sufficient condition for the stimulated emission of dislocations from the crack tip. We investigated this with the 60° dislocation described above. When the dislocation is pinned further away from the crack so that it can barely reach the crack tip or is subjected to low loads, e.g. 0.82ϵG (which can only be reached with blunted crack tips), partial dislocation cross-slip takes place without the occurrence of stimulated dislocation emission. To elaborate on this aspect further we also performed static simulations of dislocations impinging on cracks. However, in no case did the static simulations lead to stimulated emission of dislocations. Even a dislocation placed directly at the crack front, relaxed and subsequently equilibrated at 100 K, showed cross-slip but no new dislocation was nucleated. The stress field and interaction of the dislocation and the crack therefore does not seem to be sufficient to stimulate the dislocation emission process. The dynamics of the process which is captured in the MD simulations where the dislocation reaches the crack tip with velocities of the order of 1–2 nm ps−1 seems to be important.The interaction of a propagating atomically sharp crack with the 60° DB(a) dislocation placed Δx |
= 8 nm in front of the crack tip at an overload of 1.04ϵG is shown in ). The simulation is equivalent to the one shown in just at a higher load at which the crack propagates. The initial processes (a) were similar to the stationary crack. However, the evolution of the system was quite different since the propagating crack was arrested at the point where it interacted with the dislocation. The rest of the crack front, however, kept propagating. Due to the emission of blunting dislocations from the point of interaction, the crack front was blunted along the Bα and βA directions and locally aligned with these directions. This enabled the emission of multiple dislocation half-loops on the (a)- and (b)-planes, which were coupled by stair rod dislocations to partial dislocations on the (c)- and (d)-planes.The simulation of the interaction of different types of dislocations with a γ-crack showed that only dislocations of class II (see ) interact with the crack tip in a non-trivial way All the characteristics of the dislocation sources observed in this study can be related to the few experimental investigations of the detailed mechanisms of dislocation nucleation and multiplication at crack tips.First, the rare case of a single Burgers vector source observed by George and Michot The more commonly observed dislocation source according to Ref. The crack arrest experiments by Gally and Argon ), as well as similar results of the interaction of a propagating crack with tiny pores ). While the initial cross-slip event appeared similar, the propagating crack then generated dislocations with predominantly blunting character and clear V-shaped traces along the crack front. If one imagines that the dislocations generated from several such V-shaped sources meet, they would block each other and produce the same configuration of etch pits as shown in Fig. 11 of Ref. The above dislocation source mechanism leading to the V-shaped crack fronts is specific to propagating cleavage cracks and cannot occur at a stationary crack. The observation of distinctly different nucleation processes at a stationary crack front as in compared to a propagating crack front as in is clearly linked to the fact that blunting dislocations can only be generated when the crack front lies within an inclined glide plane. Only then does the emission of dislocations with edge character lead to the blunting of the crack front. The blunting, i.e. the increased curvature of the crack tip, is caused by the formation of a ledge along the crack front direction by the emitted dislocation. Since in the γ-orientation the stationary crack front is not part of an inclined glide plane, the nucleation of blunting dislocations is suppressed, and only shielding dislocations are emitted, mostly on the highest stressed set of glide planes. Propagating cracks, however, can locally adopt different crack front orientations, e.g. when a part of the crack front is held back by an obstacle while the remaining crack front keeps propagating. When the local crack front orientation is part of an inclined slip system, blunting dislocations can be generated, which hinder the further propagation of the reoriented crack front, while expanding in the forward direction of the propagating crack front which retained the initial orientation.The possibility of activating new slip systems by local reorientation of the crack front direction of propagating cracks should be a general feature which applies to different materials and crystal structures, as long as multiple blunting slip systems exist for the given crack orientation, which intersect the crack front direction at an angle attainable by local variations of the crack front orientation. These observations also solve one of the puzzling details in the literature, namely that different fracture experiments on the same cleavage system, such as the loading of a stationary crack Investigating the mechanisms of dislocation multiplication in our simulations in more detail, it is found that dislocations of class II generally show strong local curvature in front of the crack tip and react with the crack tip in a non-trivial way. The reaction can be decomposed into three different elementary processes which may occur individually or in combination with one another:the cross-slip of a dislocation segment that attains screw orientation while approaching the crack tip,the partial cross-slip of dislocation segments directly at the crack tip, usually followed bythe stimulated emission of other dislocations.The combination of these elementary processes leads to different types of avalanche multiplication mechanisms of dislocations at the crack tip. In all cases of multiplication one of the two cross-slip processes (a) or (b) was involved. Cross-slip of dislocations at or near the crack tip therefore appears essential for the generation and multiplication of dislocations at cracks.The observed behavior of the dislocations in the stress field of the crack—and particularly the crucial cross-slip processes—can be rationalized from the glide-component of the Peach–Koehler (PK) force FPK on a dislocation segment of unit length b. Dislocations of class I generally experience higher driving forces than dislocations of class II. The force on class I dislocations is mostly of the same sign and only shows one sign change behind the crack tip. This can explain the observation that class I dislocations usually just pass the crack tip without significant reactions The cross-slip propensity of screw dislocation segments can be rationalized as the difference in the magnitude of FPK on the screw dislocations in different slip planes (a). For example, screw dislocations of class II on the (a)-plane see large driving forces in front of the crack to cross-slip to the (c)-plane. However, the sign of the forces changes on the upper surface just behind the crack tip. This is the reason why only a part of the screw segment cross-slipped in . This change of sign is thus directly related to the occurrence of the spiral source in . It can also explain the observation of the cross-slip of a dislocation emitted on the (c)-plane onto the (a)-plane behind the crack tip reported in Ref. Our simulations show that cross-slip always takes place by the Fleischer mechanism . It is worth recalling that dislocations of class II cannot cross-slip between the (a)- and the (b)-planes. Dislocations of class II can only cross-slip from the (a)- or (b)-plane to the (c)- or (d)-plane, thereby changing their classification to class I dislocations (see ). The change of glide planes requires the creation of a stair-rod dislocation, which is possible when the leading partial dislocation is oriented along the line of intersection between the two glide planes. If the leading partial is part of a screw dislocation whose Burgers vector is parallel to the line of intersection of the glide planes, e.g. the (a)- and the (c)-plane, the cross-slip process can be completed as in . However, in all other cases the Burgers vector of the full dislocation is not parallel to the direction of the stair-rod dislocation. Then the trailing partial dislocation cannot react with the stair-rod dislocation and the cross-slip process by the Fleischer mechanism cannot be completed. The consideration of the driving force for the change of the glide plane is thus only relevant for the leading partial dislocation. b shows that at the crack tip the leading partial dislocation Dα(a) of has a large driving force above the crack plane to change its glide plane to the (b)-plane, whereas below the crack plane the (a)-plane is favored. Because the cross-slip process cannot be completed, the stair-rod dislocation and the trailing partial dislocation remain pinned at the crack front.Stimulated emission of dislocations from the crack front directly upon impinging of a lattice dislocation of class II was always observed in conjunction with this sort of partial dislocation cross-slip. In all cases stimulated dislocation emission took place on the (c)- or (d)-planes, which again is in excellent agreement with the observations of Scandian and Michot that only dislocations of class II stimulated the emission of dislocations which were mostly of class I Our static simulations furthermore show that structural defects and the stress field caused by the presence of a dislocation at the crack front alone do not seem to be sufficient to stimulate the dislocation emission observed in the dynamic simulations. We conclude that the dynamics of the dislocation, the inertia of the dislocation Due to the short time-scales of MD simulations, thermally activated processes are usually prohibited. On experimental time-scales, however, the pinning of the impinging dislocation at the crack front by the partial cross-slip process increases the probability of thermally activated dislocation nucleation in the stress field of the dislocation core. Dislocations which just cut the crack front, on the contrary, might not remain at the crack tip for a sufficiently long time for thermally activated processes to take place.Although we have simulated the fracture behavior of nickel we have compared most of our results to experimental observations in silicon. This seems to be far-fetched since the fracture and deformation behavior of silicon and nickel are very different. In particular nickel in experiments is found to be ductile even at low temperatures. This, however, may be a consequence of the very processes we study here, namely that pre-existing dislocations which cannot be avoided in metals interact with the crack front and generate sufficient plasticity to prohibit crack propagation. Our atomistic model for nickel, however, can support a brittle propagating crack if the crystal is defect free Because of the relatively small simulation cell, the emission of a few dislocations already leads to a significant reduction of the load on the crack tip and the evolution of a macroscopic plastic zone is prohibited. The generation of dislocation avalanches from stimulated emission as described in Refs. Model experiments and modeling of the BDT have largely focused on dislocation-free silicon or other single-crystalline materials. Taking these as a basis, macroscopic simulations of the BDT have hitherto only considered dislocation motion within a homogeneous plastic zone. In contrast, the fracture properties of technical materials are of course dramatically influenced by changes in the materials microstructure. Initial steps to add information about the microstructure into the modeling of the BDT behavior In summary, we have performed MD simulations of the interaction of pre-existing dislocations with atomically sharp cracks in the γ-orientation. The simulations revealed fundamentally different interaction mechanisms for stationary cracks compared to propagating cracks. The intersection of a propagating crack by a dislocation can lead to the local reorientation of the crack front which enables the emission of blunting dislocations on initially not available slip planes. This observation can explain the differences observed in experiments on stationary vs. propagating cracks.Our simulations furthermore show that only dislocations of class II lead to the stimulated emission and multiplication of dislocations at the tip of a stationary crack. Such stimulated dislocation emission and avalanche-like dislocation multiplication have also been observed in in situ experiments, and the dislocation products observed in the simulations agree in detail with the experimental results. The simulations show that cross-slip of dislocations at or near the crack tip is essential for stimulated emission and dislocation multiplication to take place. The analysis of the cross-slip propensity of dislocations in the stress field of a crack shows that a sign change in the driving forces on the dislocation at the crack tip is essential for the multiplication and allows to explain which incoming dislocations lead to dislocation multiplication and which do not. This opens new ways to include the interaction between dislocations and crack tips in physical models for the generation of a plastic zone near a crack tip and subsequently the BDT.Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.actamat.2012.11.016Vibration analysis of single-walled carbon nanotubes using different gradient elasticity theoriesThe present work aims at investigating the vibrational characteristics of single-walled carbon nanotubes (SWCNTs) based on the gradient elasticity theories. The small-size effect, which plays an essential role in the dynamical behavior of nanotubes, is captured by applying different gradient elasticity theories including stress, strain and combined strain/inertia ones. The theoretical formulations are established based upon both the Euler–Bernoulli and the Timoshenko beam theories. To validate the accuracy of the present analysis, molecular dynamics (MDs) simulations are also conducted for an armchair SWCNTs with different aspect ratios. Comparisons are made between the aforementioned different gradient theories as well as different beam assumptions in predicting the free vibration response. It is shown that implementation of the strain gradient elasticity by incorporating inertia gradients yields more reliable results especially for shorter length SWCNTs on account of two small scale factors corresponding to the inertia and strain gradients. Also, the difference between two beam models is more prominent for low aspect ratios and the Timoshenko beam model demonstrates a closer agreement with MD results.Iijima’s discovery paper on multi-walled carbon nanotubes in 1991 The vibrational behavior of CNTs is of great importance to the researchers in many nano-devices such as oscillators, charge detectors, field emission devices and sensors. In recent years, many elastic continuum models have been developed for studying the vibrational characteristics of CNTs More recently, attention has focused on higher-order gradient elasticity theories as alternative formats of Eringen’s elasticity to investigate the size effect on different problems of nano- and micro-structures. Up till now, different formats of higher-order gradient elasticity theories have been developed to solve boundary-value problems by the concentration on the elimination of singularities from crack tips and dislocation lines, size effects, and wave dispersion In an early investigation by Askes and Aiftantis where σij and ɛij denote the stress and strain tensors, respectively and Cijkl is the fourth order elasticity tensor; ℓ is an internal length scale obtainable by experiment or microscopic models.The strain gradient combined with inertia gradients counterpart of Eq. in which ρ is the mass density and ui stands for displacement. It should be remarked that when ℓm= 0 the theory reduces to the strain gradient theory. As follows from Eq. , the stress gradient and the combined strain/inertia gradient theories differ by their number of length scales. The stress gradient format has only one length scale, while the combined strain/inertia theory has two length scales, ℓ and ℓm which are corresponding to the strain gradients and inertia gradients, respectively. The strain gradient length scale ℓ is related to the Representative Volume Element (RVE) size in elastostatics and the inertia gradient length scale ℓm is related to the RVE size for elastodynamics , it can be found that the dynamical behavior of CNTs can be described in a more accurate way due to the presence of extra material length scale parameter ℓm .Consider an SWCNT of length L, radius R, Young’s modulus E, Poisson’s ratio ν, shear modulus G, mass density ρ, cross-sectional area A, and cross-sectional moment of inertia I as shown in . Using the Euler–Bernoulli and Timoshenko beam models in conjunction with different gradient elasticity theories, the governing equations for the free vibration of SWCNTs are derived as follows.The simplest beam theory is the Euler–Bernoulli beam theory (EBT) in which it is assumed that the straight lines which are vertical to the mid-plane will remain straight and vertical to the mid-plane after deformation. Based on the Euler–Bernoulli beam theory, the displacement field at any point can be written aswhere w is the transverse displacement. The strain–displacement relations corresponding to EBT can be given asIn the case of stress gradient (nonlocal) theory the axial equivalent of Eq. where ɛ is the axial normal strain. Using Eqs. The standard equation of motion reads as one can derive the following governing equation-EI∂4w∂x4=ρA∂2w∂t2-(ℓ2ρA+ρI)∂4w∂x2∂t2+ℓ2ρI∂6w∂x4∂t2On the other hand, the axial stress corresponding to strain gradient elasticity is given byThus, one can arrive at the following equation of motionThe constitutive equation for the case of combined strain/inertia gradient elasticity reads as-EI∂4w∂x4+ℓ2∂6w∂x6=ρA∂2w∂t2-ρI∂4w∂x2∂t2-ℓm2∂6w∂x4∂t2The other type of beam theory is the Timoshenko beam theory (TBT) in which the effects of shear deformation and rotational inertia are taken into account, so the straight lines will no longer remain vertical to the mid-plane of the beam after deformation. The displacement field in this theory can be written asu1(x,z,t)=-z∂w(x,t)∂x+z∂w(x,t)∂x+ψ(x,t)u2(x,z,t)=0u3(x,z,t)=w(x,t)where ψ is the angular displacement of the beam. The nonzero strains according to the Timoshenko beam theory are expressed aswhere τ |
= |
σxy and κs is the shear correction factor. The shear equivalent of Eq. The bending moment and shear force in the stress gradient theory have the following formThe equation of motion on the basis of stress gradient TBT is given by-κsGA∂w∂x+ψ+EI∂2ψ∂x2=ρI∂2ψ∂t2-ℓ2ρI∂4ψ∂x2∂t2TBT based on the gradient elasticity formulation with strain gradients can be formulated asκsGA∂2w∂x2+∂ψ∂x+κsGAℓ2∂4w∂x4+∂3ψ∂x3=ρA∂2w∂t2-κsGA∂w∂x+ψ-κsGAℓ2∂3w∂x3+∂2ψ∂x2+EI∂2ψ∂x2+ℓ2∂4ψ∂x4=ρI∂2ψ∂t2Also, with the combined strain/inertia gradients one can obtainκsGA∂2w∂x2+∂ψ∂x+κsGAℓ2∂4w∂x4+∂3ψ∂x3=ρA∂2w∂t2-κsρAℓm2∂4w∂x2∂t2+∂3ψ∂x∂t2-κsGA∂w∂x+ψ-κsGAℓ2∂3w∂x3+∂2ψ∂x2+EI∂2ψ∂x2+ℓ2∂4ψ∂x4=ρI∂2ψ∂t2+κsρAℓm2∂3w∂x∂t2+∂2ψ∂t2-ρIℓm2∂4ψ∂x2∂t2For the case of EBT models, exact solutions for free vibrations of the SWCNTs are developed. Explicit formulas are proposed to obtain the natural frequencies corresponding to each type of gradient model. The displacement w can be considered in the following generalized form which satisfies the simply-supported boundary conditions into the constitutive relations of different gradient EBT models for SWCNTs and solving the resulting eigenvalue problem, the natural frequencies of SWCNTs can be obtained as followsωEBT2=π4m4EI(L2-π2m2ℓ2)/ρL2(AL4+Im2π2L2+π4m4Iℓm2)For TBT, the angular displacement of the beam can be assumed asSubstituting the above equation along with the Eq. where K and M for different gradient theories areK=κsGAm2π2/L2κsGAmπ/LκsGAmπ/L(κsGAL2+EIm2π2)/L2,K=κsGAm2π2/L2κsGAmπ/LκsGAmπ/L(κsGAL2+EIm2π2)/L2,K=κsGAm2π2(L2-m2π2ℓ2)/L4κsGAmπ(L2-m2π2ℓ2)/L3κsGAmπ(L2-m2π2ℓ2)/L3κsGA(L2-m2π2ℓ2)/L2+EIm2π2(1-m2π2ℓ2)/L2,K=κsGAm2π2(L2-m2π2ℓ2)/L4κsGAmπ(L2-m2π2ℓ2)/L3κsGAmπ(L2-m2π2ℓ2)/L3κsGA(L2-m2π2ℓ2)/L2+EIm2π2(1-m2π2ℓ2)/L2,M=ρA+(κsρAm2π2ℓm2)/L2κsρAnπℓm2/LκsρAnπℓm2/LρI+κsρAℓm2+(ρIm2π2ℓm2)/L2By solving the eigenvalue problem given by Eq. , the natural frequencies of simply-supported SWCNTs can be obtained.In the current study, the effective thickness of the SWCNTs is assumed to be equal to the spacing of graphite (h |
= 0.34 nm). In addition, Poisson’s ratio ν, mass density ρ and Young’s modulus E are assumed to be 0.3, 2300 Kg/m3 and 1.1 TPa, respectively.To validate the present approach, MD simulations are conducted for a simply-supported (8, 8) SWCNT with different aspect ratios raging from 8.3 to 39.1. The NanoHive simulator Calibration of the small length scales applied in the gradient continuum models plays an important role in predicting the response of nanotubes. In the current analysis, the appropriate values of ℓm/ℓ related to different beam theories are extracted by matching the fundamental frequencies obtained from the continuum models with those calculated by the MD simulations. The ratio of ℓm/ℓ are found to be 40 and 8 for EBT and TBT, respectively. It is observed that the small length scale related to the inertia gradient for Timoshenko beam model is significantly lower than that in the Euler–Bernoulli beam model. present the values of fundamental frequency obtained from MD simulations and also the Euler–Bernoulli and Timoshenko beam models based on the classical, stress, strain and combined strain/inertia gradient elasticity theories. The results predicted by the present gradient models are found to be in excellent agreement with the ones obtained from MD simulation which indicates the capability of the present approach in accurately predicting frequencies of SWCNTs.These tables show that for all gradient models the frequency of SWCNTs decreases with increasing length-to-diameter ratio. From , it is observed that the results obtained by applying all of the Euler–Bernoulli beam models leads to the identical results for aspect ratios higher than 24.5. According to , the same conclusion holds for Timoshenko beam models for aspect ratios higher than 17.3. Additionally, it is observed that employing the strain gradient elasticity theory with inclusion of inertia gradients leads to highly accurate results which are comparable to those obtained by MD simulations particularly for low aspect ratios because of two small scale factors corresponding to the inertia and strain gradients. The distinction between the frequencies predicted by the EBT and TBT is more prominent for lower aspect ratios. Moreover, comparison of the results from the EBT and TBT reveals that the Timoshenko beam models developed here give predictions closer to the MD results for shorter SWCNTs.To show the difference between different gradient models at low aspect ratios, the fundamental frequencies of SWCNTs against aspect ratios ranging from 2 to 6 are graphed in for the EBT and TBT, respectively. As it can be seen, for both the EBT and TBT, the classical model tends to overestimate the frequencies of shorter length nanotubes. For instance, for the aspect ratio equal to 2, the amounts of overestimation of classical model in the case of the Euler–Bernoulli beam model are 13.57%, 18.69% and 43.25% relative to the stress, strain and combined strain/inertia gradient theories, respectively. It should be noted that there are the same values of overestimation relative to the gradient Timoshenko beam models except in the case of combined strain/inertia gradient model where the overestimation becomes equal to 48.10%. These figures also reveal that the difference between different gradient models becomes more pronounced for very short SWCNTs. show that the results of combined strain/inertia beam model are lower than those of other gradient beam models and therefore this gradient beam theory can accommodate the results of other ones. As the ratio of length-to-diameter increases, the discrepancy between various gradient theories decreases so that the frequency envelopes tend to converge at an aspect ratio of approximately 5.As would be expected, the difference between the results of EBT and TBT is relatively more prominent at low values of aspect ratio. For example, from , at L/D |
= 2 the percent of relative error between the results predicted by the Euler–Bernoulli and Timoshenko beam models is 16.46% for the classical, stress and strain gradient theories. However, the percent of relative error reaches at the value of 20.40% in the case of combined strain/inertia gradient theory. The primary reason is that in contrast to the EBT, the effect of rotational inertia is taken into account in the TBT. Clearly, the percent of relative error at higher values of the aspect ratio is considerably smaller than that at lower values of aspect ratio. For example, there is a percent of relative error of approximately 3.65% at L/D |
= 5.In this work, the free vibration analysis of SWCNTs was investigated based upon different gradient elasticity theories. The Euler–Bernoulli and the Timoshenko beam theories are employed to model the SWCNTs. To validate the present analysis, MD simulations were also conducted for an armchair SWCNT with different aspect ratios. Explicit formulas are proposed for Euler–Bernoulli model relevant to each type of gradient theory to evaluate the natural frequencies of SWCNTs. Also, the mass and stiffness matrices were obtained for different gradient Timoshenko beam models. It was concluded that applying the strain gradient elasticity with inclusion of inertia gradients lead to highly accurate results which are comparable to those predicted by MD simulations particularly for low aspect ratios because of two small scale factors relevant to the inertia and strain gradients. Furthermore, the difference between two Euler–Bernoulli and the Timoshenko beam models is more pronounced for shorter SWCNTs and the Timoshenko beam model indicates a closer agreement with MD results.Stochastic vibration of axially loaded monosymmetric Timoshenko thin-walled beamAn analytical method is presented to perform the flexure–torsion coupled stochastic response analysis of monosymmetric axially loaded Timoshenko thin-walled beam subjected to various kinds of concentrated and distributed stochastic excitations with stationary and ergodic properties. The effects of warping stiffness, axial force, shear deformation and rotary inertia are included in the present formulations. First, the damped general governing differential equations of motion of axially loaded Timoshenko thin-walled beam are developed and the free vibration analysis is performed. Once the natural frequencies and mode shapes are obtained, mode superposition method in conjunction with receptance method is used to compute the mean square displacement response of the axially loaded thin-walled beam. Finally, the method is illustrated by its application to two test examples to investigate the effects of warping stiffness, axial force, shear deformation and rotary inertia on the stochastic response of the thin-walled beams.The thin-walled beam structures are commonly found in the design of the aircraft wings, propeller blades, bridge decks, vehicles axles and so on, due to their outstanding properties. Since the thin-walled beam members are widely used in aerospace, automobile and civil architecture industries, it is important to ensure that their design is reliable and safe. The dynamic analyses of the thin-walled beam structures also help to optimize the design and avoid future investments on repairs. It is, therefore, essential for design engineers to evaluate the dynamic characteristics of the thin-walled beam structures accurately.It is well known that when the cross-section of the beam has two symmetric axes, the shear center and the centroid of the cross-section coincide, and all flexural and torsional vibrations are independent of each other, this case represents no coupling at all. Then the classical Bernoulli–Euler and/or the Timoshenko beam theory are valid. However, for a large number of practical beams of thin-walled sections, the centroid and shear center of the cross-section are obviously non-coincident, the above assumption is not valid. When the cross-section of the thin-walled beam has only one symmetrical axis, the flexural vibration in the direction of the symmetrical axis is independent of the other vibrations. But the flexural vibration in the perpendicular direction of the symmetric axis is coupled with torsional vibration.In spite of the practical interest of the flexure–torsion coupled thin-walled beam problems, particularly in the context of aerospace, civil and mechanical applications, the main body of the available investigations has devoted entirely to study the dynamic response of beams having double symmetrical axes and structures composed of this kind of beams such as Refs. In order to accurately predict the stochastic response of the thin-walled beam structures, comprehensive structural models have to be used. In particular, shear deformation, rotatory inertia, warping stiffness as well as flexure–torsion coupling and axial force must be included in their modelling. The necessity of incorporating transverse shear effect arises from the fact that it is usually important when the depth-span ratio of the thin-walled beam is relatively large. In addition, torsion related non-uniform warping occurs when a section is restrained against out of plane deformation and/or when a non-uniform distributed torque is applied along the length of the beam. Therefore, the free warping assumption may result in erroneous predictions of the behavior of cantilevered type structures. Consequently, the warping stiffness effect is incorporated in this work.In the previous studies of the stochastic response analysis of the flexure–torsion coupled thin-walled beams, it seems that there is no work investigating the effects of axial force, warping stiffness, shear deformation and rotatory inertia on the dynamic behavior of the thin-walled beams simultaneously. This problem is presented in this paper. The stochastic flexure–torsion coupled vibration of elastic axially loaded thin-walled beams with monosymmetrical cross-section subjected to various kinds of concentrated and distributed stochastic excitations with stationary and ergodic properties is investigated. The effects due to axial force and warping stiffness on the stochastic response of the thin-walled beams are especially interested. Furthermore, the effects due to shear deformation and rotatory inertia are also of interest here. First, an analytical method for determining natural frequencies and mode shapes of the flexure–torsion coupled vibration of the axially loaded beams with thin-walled monosymmetrical cross-section is developed by using the general solution of the governing differential equations of motion. This method takes into account the effects of axial force, warping stiffness, shear deformation and rotatory inertia in a unified way. Once the natural frequencies and mode shapes are obtained, normal mode method in conjunction with receptance method is used to compute the mean square displacement response of the axially loaded thin-walled beam. Finally, the method is illustrated by its application to two test examples to investigate the effects of warping stiffness, axial force, shear deformation and rotary inertia on the stochastic response of the thin-walled beams.Under consideration is a thin-walled structure, modelled as an open or closed thin-walled beam with monosymmetrical cross-section. Considering a uniform and straight thin-walled beam with length L shown in , it is assumed that the terms associated with secondary warping and warping inertia which are negligibly small can be discarded. The shear center and centriod of the thin-walled beam are denoted by s and c, respectively, which are separated by distance yc. In the right handed Cartesian co-ordinate system in , the x-axis is assumed to coincide with the elastic axis (i.e., loci of the shear center of the cross-section of the thin-walled beam). The flexural translation in the z direction and the torsional rotation about the x-axis of the shear center are denoted by v(x,t) and ψ(x,t), respectively, where x and t denote distance from the origin and time, respectively. The rotation of the elastic axis due to flexure alone is denoted by θ(x,t). A constant compression axial force P is assumed to act through the centroid of the cross-section of the thin-walled beam. P can be positive or negative, so that tension is included. The external excitations acting on the thin-walled beam are represented by a force f(x,t) per unit length that parallel to sz-axis and applied to the shear center together with a torque m(x,t) per unit length about sx-axis, respectively.The damped governing equations of motion for the forced vibration of the axially loaded Timoshenko thin-walled beam exhibiting flexure–torsion coupling and including warping stiffness effect can be written as following three coupled differential equations, which may be derived using Hamilton's principle (for details of the derivation, see where E is Young's modulus of elasticity of the thin-walled beam material, G is the shear modulus of the thin-walled beam material. EI, kGA, GJ and EΓ are flexural stiffness, shear stiffness, torsional stiffness and warping stiffness of the thin-walled beam, respectively. μ is mass of the thin-walled beam per unit length, I is the second area moment of inertia of the beam cross-section about y-axis, Is is polar mass moment of inertia of per unit length thin-walled beam about x-axis, superscript primes and dots denote the derivative with respect to position x and time t, respectively. ρ is the density of the thin-walled beam material, A is the cross-section area of the thin-walled beam, k is the effective area coefficient in shear. The damping coefficients c1, c2 and c3 are the linear viscous damping terms of per unit length thin-walled beam in flexural deformation, torsional deformation and rotatory deformation due to flexure, respectively.For undamped free vibration of axially loaded Timoshenko thin-walled beam, the external excitations f(x,t) and m(x,t) are set to zero, as are the damping coefficients c1, c2 and c3, in order to determine the natural frequencies and mode shapes of the thin-walled beam. A sinusoidal variation of v(x,t), θ(x,t) and ψ(x,t) with circular frequency ωn is assumed to be of the formswhere n=1,2,3,…,Vn(x), Θn(x) and Ψn(x) are the amplitudes of the sinusoidally varying flexural translation, flexural rotation and torsional rotation, respectively. gives the three simultaneous differential equations for Vn, Θn and Ψn. can be combined into one equation by either eliminating all but one of the three variables Vn, Θn and Ψn to give the following eighth order differential equation:Note that r, s and p above describe the effects of rotatory inertia, shear deformation and axial force, respectively. Any one or all of these parameters can be set to zero so that corresponding effect(s) can be optionally ignored. can be obtained by substituting the trial solution Xn=eκnξ to give the characteristic equation are real, two of them negative and the other two positive. Suppose that the four roots are χn1, χn2, −χn3, −χn4, where χnj (j=1,…,4) are real and positive. Then the eight roots of the characteristic where c1*–c8* is a set of constants which can be determined from the boundary condition, and in conjunction with the boundary conditions yield the eigenvalues (natural frequencies) and eigenfunctions (mode shapes) of axially loaded Timoshenko thin-walled beam. Also based on and the boundary conditions, by following the procedure described in Ref. is the generalized mass in the nth mode, δmn is the Kronecker delta function.With the free vibration modes, natural frequencies and orthogonality condition described above, it is now possible to investigate the general stochastic vibration problem of the damped axially loaded Timoshenko thin-walled beam.For forced stochastic vibration of the axially loaded Timoshenko thin-walled beam, assume v(x,t), θ(x,t), ψ(x,t) can be expanded in terms of the eigenfunctions to give the following three equations:where qn(t) is the generalized time-dependent co-ordinate for each mode. Substituting where superscript dot denotes derivative with respect to time. by Vm, Θm and Ψm, respectively, then summing up these three equations and integrating from 0 to 1, and using orthogonality condition (17) giveswhere Fn(t) and Mn(t) can be expressed aswhere ζn is a non-dimensional quantity known as the viscous damping factor. Here the assumption c2=c1Is/μ, c3=c1ρI/μ has been made to take advantage of the orthogonality condition (17) in order to avoid having coupling terms In this paper, the stochastic response of the axially loaded Timoshenko thin-walled beam to stationary, ergodic stochastic excitations with zero initial conditions is investigated in the frequency domain by using the receptance method. of the generalized time-dependent co-ordinate qn(t) is related to the cross-spectral density functions of the flexure load Fn(t) and torsional load Mn(t) by the following relation: is the cross-spectral density function between Fn(t) and Fl(t), is the cross-spectral density function between Mn(t) and Ml(t). Since it is assumed that the stochastic excitations f(ξ,t) and m(ξ,t) are stationary in time, then so are the generalized forces Fn(t) andMn(t). Furthermore, Fn(t)and Mn(t) are assumed to be independent stochastic processes so that the cross-spectral density functions between Fn(t) and Mn(t) can be excluded.Based on the expressions of the generalized forces Fn(t) andMn(t), the cross-spectral density functions can be obtained explicitly as, respectively is the distributed cross-spectral density function between the stochastic flexural loads f(ξ1,t) and f(ξ2,t), is the distributed cross-spectral density function between the stochastic torsional loads m(ξ1,t) and m(ξ2,t).For the flexural load f(ξ,t) and torsional load m(ξ,t), the corresponding cross-spectral density functions are related to the cross-correlation functions Rf(ξ1,ξ2,τ) and Rm(ξ1,ξ2,τ), respectively, by the following Fourier transform pairs:The cross-correlation functions Rf(ξ1,ξ2,τ) and Rm(ξ1,ξ2,τ) of the flexural load f(ξ,t) and torsional load m(ξ,t) are defined as denotes the ensemble average of the stochastic process. of the flexural translation v(ξ,t), torsional rotation ψ(ξ,t) and flexural rotation θ(ξ,t) can be written asFor ξ1=ξ2=ξ, the cross-spectral density functions reduce to the spectral density functions The mean square values of the flexural translation, torsional rotation and flexural rotation can be found by integrating the corresponding spectral density functions over all frequenciesIf the external stochastic excitations are assumed to follow the Gaussian probability distribution, the response probability will also be Gaussian, and therefore the response can be fully described by its spectral density function.As an example, two kinds of loads are considered here as applied on the thin-walled beam. The first one is that there is only one stochastic varying concentrated flexural load acting on the thin-walled beam at ξ=ξf. In this case, The spectral density functions of the flexural translation, torsional rotation and flexural rotation are then given by The next type of load consists of only distributed flexural load acting on the thin-walled beam, and the stochastic varying load is assumed to be following form:where g(t) is a stochastic process. Note the load considered here is stochastic with respect to time only. The extension to cover spatially varying stochastic load is easy.The cross-correlation function for the above flexural load is given byThe corresponding cross-spectral density function isThe spectral density functions of the flexural translation, torsional rotation and flexural rotation are then expressed asSome numerical results are given to demonstrate the theoretical formulations derived in last section, which can be directly applied to compute the stochastic response of the axially loaded Timoshenko thin-walled beam.The first example is a cantilever thin-walled beam with monosymmetrical semi-circular cross-section. The geometrical and physical properties of the thin-walled beam shown in are given as follows: I=9.26×10−8 |
m4, J=1.64×10−9 |
m4, Is=0.000501 kg m, yc=0.0155 m, L=0.82 m, Γ=1.52×10−12 |
m6, μ=0.835 kg m−1, E=68.9×109 |
N m−2, G=26.5×109 |
N m−2, k=0.5, A=3.08×10−4 |
m2, ρ=2711.04 kg m−3, P=1790 N.The natural frequencies and mode shapes of the above axially loaded thin-walled beam for undamped free vibration are computed by setting the damping coefficients c1, c2, c3 and the external excitations f(x,t) and m(x,t) in to zero. The first five natural frequencies of the axially loaded semi-circular section thin-walled beam are shown in . The corresponding first five normal mode shapes are shown in . The first five normal mode shapes with the warping stiffness ignored are shown in . The first five normal mode shapes with the Timoshenko effect omitted are shown in . The first five normal mode shapes including the warping stiffness and Timoshenko effect are shown in . All the first five normal modes are coupled modes, i.e., the modes show coupling between flexural displacement and torsional displacement, although they are dominant torsional modes. The first four mode shapes ignoring the warping stiffness or Timoshenko effect are similar to the ones accounting for the warping stiffness and Timoshenko effect, although there is some difference between them. The fifth mode ignoring the Timoshenko effect is similar to the one accounting for the warping stiffness and Timoshenko effect.Based on the natural frequencies and mode shapes, the mean square values of the flexural translation, flexural rotation and torsional rotation due to a stochastic varying concentrated flexural load can be computed without any difficulty. The stochastic flexural load is assumed to be an ideal white noise, so the Sf(ω) in can be replaced by a constant, i.e., Sf(ω)=S0 (S0 is a constant). In , respectively, are shown the mean square parameters of flexural translation, flexural rotation and torsional rotation along the length of the cantilever thin-walled beam subjected to an ideal white noise concentrated flexural load acting at the tip of the beam. The value of the damping coefficient used in computation is 0.01. The mean square flexural translation accounting for the Timoshenko effect is only a little different from the one excluding the Timoshenko effect. But the effect of Timoshenko on the flexural rotation and torsional rotation is noticeable. The mean square flexural and torsional displacements including the warping stiffness are remarkably different from the ones excluding the warping stiffness, as can be seen from . Compared to the effects of Timoshenko and warping stiffness, the axial force has a more significantly effect on the mean square flexural and torsional displacements. The percentage errors for mean square values of the flexural and torsional response at the tip of the cantilever thin-walled beam without the warping stiffness or Timoshenko effect or axial force included are shown in . The numerical results show that it is necessary to consider the effects of the axial force, warping stiffness and Timoshenko in order to obtain the mean square displacement response accurately.A cantilever thin-walled uniform beam with a monosymmetrical channel cross-section is considered next. The geometrical properties and physical properties of the beam shown in The first five natural frequencies of the axially loaded channel section thin-walled beam are shown in . The corresponding mode shapes of the first five normal modes with an axial force . The first five normal mode shapes disregarding the warping stiffness effect are plotted in . The first five normal mode shapes with the Timoshenko effect ignored are plotted in . The first five normal mode shapes taking into account the effects of warping stiffness and Timoshenko are plotted in that the first five normal modes omitting the effect of warping stiffness are absolutely dominant torsional modes. The warping stiffness and Timoshenko effect make great difference to all of the first five mode shapes. It can be seen from that all of the first five modes are coupled modes. The mode 1, mode 3, mode 4 are dominant torsional modes when the Timoshenko effect is excluded, and the mode 2 and mode 5 excluding the Timoshenko effect are strongly coupled modes. When the warping stiffness and Timoshenko effect are considered, the mode 1, mode 2 and mode 3 are dominant torsional modes, but the mode 4 and mode 5 are strongly coupled modes.According to the procedure discussed before, the stochastic response of the above axially loaded thin-walled beam can be computed without any difficulty based on the natural frequencies and mode shapes. To compare the results obtained from the present theory including the effects of warping stiffness, Timoshenko and axial force with those given by the theory excluding the effect of warping stiffness or Timoshenko effect or axial force effect, the mean square values of the flexural translation, flexural rotation and torsional rotation due to a stochastic varying concentrated flexural load are calculated. In , respectively, are shown the mean square parameters of the flexural translation, flexural rotation and torsional rotation along the length of the channel section thin-walled beam subjected to an ideal white noise concentrated flexural load acting at the tip of the beam. The value of the damping coefficient has been taken as 0.01. It can be seen from , the mean square values of the flexural displacement and torsional displacement predicted by the present theory considering the effects of axial force, Timoshenko and warping stiffness are distinctly different from those obtained from the theory excluding the effect of axial force or warping stiffness or Timoshenko effect. So, it is absolutely necessary to include the effects of axial force, warping stiffness and Timoshenko effect when the mean square displacements of this particular thin-walled beam are computed. The percentage errors for mean square values of the flexural and torsional response at the tip of the cantilever thin-walled beam without the warping stiffness or Timoshenko effect or axial force included are shown in A method has been presented to perform the stochastic response analysis of the axially loaded Timoshenko thin-walled beam. The thin-walled beam is assumed to be uniform, straight, damped, and subjected to an axial force. The effects of shear deformation, rotatory inertia and warping stiffness, which are usually important for open cross-section thin-walled beam, are included in the present formulations. Once the natural frequencies and mode shapes of the axially loaded thin-walled beam are obtained, mode superposition method in conjunction with receptance method is used to compute the stochastic response of the beam such as the flexural displacement and torsional displacement. Although the illustrative examples given in this paper are that of two simple thin-walled beams, the developed theory can be applied to other types of boundary conditions of the thin-walled beams or beam assemblages and can be used to other kinds of stochastic excitations.The damped governing equations of motion for the forced vibration of the axially loaded Timoshenko thin-walled beam exhibiting flexure–torsion coupling and including warping stiffness effect can be derived using the Hamilton's principle as follows.The total strain energy U of an axially loaded Timoshenko thin-walled beam shown in where all the variables and symbols are defined in The total kinetic energy T of an axially loaded Timoshenko thin-walled beam is given byThe governing equations of motion and the boundary conditions can be derived conveniently by means of Hamilton’s principle, which can be stated in the formHerein T is the kinetic energy, U the potential energy, δW the virtual work of the non-conservative forces, which can be written as and carrying out the usual steps yield the governing equations of motion and the boundary conditions.(b) The boundary conditions at the ends (x=0,L)Finite element analysis of bonded insulated rail jointsInsulated rail joints are installed at regular intervals to create signal blocks for detecting train positions and sometimes breaks in rails. Their lifetime may be significantly shorter than that of the rail. Failure may occur in the adhesive between the rail and the joint bars on the sides of the rail. A standard butt joint is considered, with the center of the joint supported by a tie. The finite element program ABAQUS is used. A static wheel load is applied, and the rail is also subjected to a tensile load. The vertical displacement of the rail and the shear and peel stresses in the epoxy are determined. The effects of the location of the wheel, size of the ties, and length and thickness of the joint bars are investigated.Rail joints hold sections of rail together. Electrically insulated joints are used in automatic block signaling systems Due to increased axle loading under heavy freight trains, bonded insulated rail joints have become increasingly problematic and may fail in as little as 1 or 2 years. Mechanical failure may be initiated in the adhesive connecting the rail sections with the joint bars on the sides Rail joints are either “suspended” or “supported” Large forces may occur at rail joints and cause deterioration and damage Stresses in the adhesive connection between the rail and the joint bars are investigated in the present paper. Davis and Akhtar , the joint and its finite element model are described. A standard case is defined, and results for this case are presented in with the wheel either at the center of the joint or midway between the center and an adjacent tie. The effects of increase in (a) the size of the ties, (b) the length of the joint bars, and (c) the moment of inertia of the joint bars are discussed in A photo of two insulated joints is shown in . These joints are suspended and are staggered (rather than being located opposite each other on the two rails). A cross section of the rail and joint bars is sketched in , the joint bars are 0.9 m (3 ft) long and contain six bolts. The bolts (not shown in ) compress the rails and joint bars together, and are electrically isolated (often by insulating thimbles). The shear stresses due to wheel loads are mainly transferred between the rail and bars by the epoxy. A layer of insulating fabric is usually included in the epoxy layer on each side of the rail. The gap between the two sections of rail is also insulated, as described before.The tie under the end post and two ties on each side are modeled with solid elements. The effective length along the tie (perpendicular to the rail) is assumed to be one-third of the total tie length under each rail An elastic foundation is placed under these five ties to represent the vertical resistance of the ballast. Based on an elastic foundation stiffness per unit length k depicts the top view of the contact areas of the elastic foundations for the central nine ties.). Also, in the case of the wheel located at the center of the joint, since there is symmetry about that location, only one side (7.62 m long) was analyzed.The contact area and pressure distribution between a wheel and a rail were examined in Refs. where z is the coordinate along the rail, z and y are in cm, and p is in MPa.The “standard case” is defined with the standard ties, standard length 0.9 m (3 ft) of the joint bars, and standard moment of inertia of the joint bars. The wheel is placed either at the center of the joint or halfway between the central tie and the adjacent tie on the left. depicts the standard case with the off-center wheel. The joint bars overlap one-third of each of the ties that are adjacent to the central tie. For the case with longer joint bars of length 1.2 m (4 ft), the joint bars overlap the entire width of the adjacent ties.For the central wheel location, the deflection of the bottom of the rail in the region of the joint is plotted as a dashed curve in for the standard case. The vertical axis is greatly exaggerated. The joint bars cover the distance from −0.45 to 0.45 m on the abscissa. The maximum downward deflection is 0.310 cm (0.122 in.). Normal to this figure, the end of the effective length of the tie under the load deflects 0.292 cm (0.115 in.), i.e., there is a change of deflection of 0.018 cm (0.007 in.) over a length of 43 cm (17 in.). Hence the tie does not exhibit much curvature If the joints were suspended rather than supported, previous work The remaining figures involve the shear and peel (normal) stresses in the epoxy adhesive layer on the side attached to the rail. With respect to ) and τyz. The shear stress τxz in the “plane” of the epoxy layer is not computed, since it is typically small because it is directly tied to shear strains within the steel itself. The mesh in the epoxy layer is not fine enough to accurately model stress concentrations, but the results indicate the distributions of the stresses and give approximate extreme values of the stresses. Many results not shown here (e.g., further contour plots) can be found in Ref. illustrate the stresses for the standard case with a central load. The center of the distributed wheel load is represented by an arrow. The half-joint bar lies over half the central tie and one-third of the adjacent tie on the left. The shear stress τyz in has its maximum value of 32.1 MPa (4650 psi) at the center of the circle, i.e., at the gap between the rail sections and at a height at which the epoxy layer has a high curvature (see ). Along the rail, the minimum shear stress τyz occurs near the midpoint of the layer. is not found at the gap, but near the bottom of the other end of the epoxy layer (at the left end of the joint bar), and is 8.00 MPa (1160 psi). In , it is seen that the maximum compressive peel stress occurs under the load, and its value is 25.6 MPa (3710 psi), whereas the maximum tensile peel stress is located at the left end of the layer and is 11.2 MPa (1620 psi).The off-center load is considered next with the standard case. The shear stress τyz shown in has its maximum value at a similar location to that in for the central load, and the value here is somewhat lower at 28.8 MPa (4180 psi). shows the shear stress τxy, which is largest at the left end of the epoxy (as before in ). The maximum value is 8.34 MPa (1210 psi), slightly higher than for the central load. Finally, the peel stresses are presented in . The maximum compressive peel stress of 15.6 MPa (2260 psi) occurs under the load, and is lower than that for the central load. The maximum tensile peel stress of 11.4 MPa (1660 psi) is observed in the epoxy attached to the rail section that is not loaded by the wheel, near the top of the layer at the gap. It is slightly higher than that for the central load.In this section, the standard case is modified by changing one feature at a time, to see if joint designs can be improved. The individual effects of (a) the larger ties, (b) the longer joint bars, and (c) increased widths of the joint bars are considered. Maximum values of stresses and of deflections of the bottom of the rail are presented. The stress distributions are similar to those of the standard case with the same wheel location , the deflection under a central load is given by the solid curve in . The maximum downward deflection is 0.236 cm (0.093 in.), which is 24% smaller than that for the standard ties. For the off-center load, the maximum deflection is 0.300 cm (0.118 in.) for the standard ties and 0.231 cm (0.091 in.) for the larger ties, and occurs close to the location of the wheel.The effect of the larger ties on the maximum stresses is depicted in for the case of a central load. The shear stresses τxy and τyz decrease by 11% and 8%, respectively, the tensile peel stress is essentially the same, and the compressive peel stress decreases by 18%. shows corresponding results for the off-center load. For the larger ties, the shear stress τyz decreases by 4%, and the other stresses are essentially unchanged.For the standard joint with a central load, a change of length of the joint bars from 91 cm (36 in.) to 122 cm (48 in.) is considered. The maximum deflection decreases by about 2%. The effect on the maximum stresses is illustrated in . The shear stress τxy decreases by 20%, the shear stress τyz hardly changes, the tensile peel stress increases by 7%, and the compressive peel stress increases by about 4%. The increase in the tensile peel stress could cause problems for bonded insulated joints by increasing the likelihood of epoxy failure.Now the effect of the thickness of the joint bars (see ) is considered. If I denotes the moment of inertia of the standard joint bars, then the cases 2I and 3I are considered here. The maximum deflections for moments of inertia I, 2I, and 3I, respectively, are 0.310 cm (0.122 in.), 0.300 cm (0.118 in.), and 0.292 cm (0.115 in.) for the central load, and 0.300 cm (0.118 in.), 0.295 cm (0.116 in.), and 0.290 cm (0.114 in.) for the off-center load. Thus the thicker joint bars do not substantially reduce the maximum deflection in the cases considered here.Maximum stresses for the central load and the off-center load are shown in , respectively. The standard ties and standard joint-bar length are used. In most cases, the increases in joint-bar thickness cause significant increases in the stresses in the epoxy. For example, in , the maximum tensile peel stress increases by 79% and 110% if the moment of inertia is doubled and tripled, respectively. Similarly, the maximum tensile peel stress for corresponding joint-bar moments of inertia increases by 73% and 110% in The finite element program ABAQUS has been used to analyze an adhesively bonded insulated rail joint. Attention was focused on the deflection of the rail and on the stresses in the epoxy layer between the rail and the joint bars. Failures of this adhesive layer in joints subjected to heavy wheel loads have been suspected of causing many joints to exhibit a short lifetime compared to that of the rail. Therefore, it is important to know the stresses in the adhesive. Supported joints were studied, in which a tie is located directly under the gap between the two sections of rail (i.e., at the center of the joint). In practice, the gap is often suspended between two ties. A supported joint tends to have lower deflections, but may be subjected to higher impact loads under moving trains The magnitudes and locations for maximum shear stresses and peel stresses (compressive and tensile) were determined for a number of cases. The standard case involved ties that were 23 cm (9 in.) wide, and joint bars that were 91 cm (36 in.) long and had a combined moment of inertia of 1040 cm4 (24.9 in.4), with the rail moment of inertia being 3920 cm4 (94.2 in.4) for bending in a vertical plane. The effects of larger ties, longer joint bars, and thicker joint bars were investigated, with the wheel being located either at the center of the joint or midway between the center and an adjacent tie.Increasing the tie size tended to decrease the maximum epoxy stresses slightly. Increasing the joint bars’ length tended to decrease the maximum shear stresses and increase the peel stresses. Increasing the joint bars’ thickness tended to increase the maximum stresses significantly. Finally, moving the wheel from the joint center to the off-center location yielded a smaller maximum displacement and similar maximum stresses, with the location of maximum tensile peel stress switching from the end of the joint to the center of the joint.A study on the combined effect of in-situ (TiC-TiB2) reinforcement and aging treatment on the yield asymmetry of magnesium matrix compositeThe present work aims to study the individual as well as the combined effect of in-situ particles and precipitates on the tension-compression asymmetricity of AZ91 magnesium (Mg) alloy and AZ91 + TiC-TiB2 Mg matrix composites. Investigations have been done on four different material conditions: (a) alloy without particles or precipitates, (b) alloy with precipitates, (c) composite with particles, (d) composites with both particles and precipitates. Both the particle and precipitate significantly contribute towards lowering of tension to compression yield asymmetry. These particles/precipitates increase the critical stress for twinning by exerting a back stress which hinders the twin propagation. Due to decrease in the twinning propensity, the tension to compression asymmetry reduces from 1.30 (at base condition) to 1.04 (peak aged condition of the composite). This phenomenon has been experimentally validated by means of tension and compression tests and the results have been correlated with the pre and post-deformation microstructures.Magnesium alloys are one of the most attractive materials used in the field of automotive and electronic industries due to its high specific strength and low density. However, the potential engineering applications of commercial magnesium alloys are limited due to its low strength and high tension-compression yield asymmetry behavior Tension-compression yield asymmetry of a material is defined as the ratio between tensile yield stress (TYS) and compressive yield stress (CYS). The higher yield asymmetry in magnesium (Mg) and its alloys is due to HCP crystal structure which have c-axis to a-axis ratio (c/a) of around 1.633 The tension-compression yield asymmetry of Mg alloys can be minimized mainly in four ways: (a) by weakening the texture, (b) by grain refinement, (c) by heat treatment and (d) by adding aluminum as an alloying element. Reduced texture is one of the major reason for lower yield asymmetry in cast alloys as compared to wrought alloys Although the yield asymmetry behavior of magnesium and its alloys are well studied by the research community, the influence of reinforcement on the material's asymmetry is not established. The reinforcement can be added to the magnesium matrix by either ex-situ or in-situ process Therefore, the present work is intended to study (i) the effect of age hardening on tension-compression yield asymmetry behavior of base and composite materials for both tensile and compression test and (ii) effect of precipitates and in-situ reinforcement on deformation behavior through microstructure observation by electron backscatter diffraction (EBSD) analysis and scanning electron microscopy (SEM).The starting materials used in the present research work are as-cast AZ91 magnesium alloy and TiC-TiB2/AZ91 in-situ composite. The in-situ composite plate was fabricated at a temperature of 900 °C with a holding time of 2 h for in-situ reaction. The detailed fabrication process was described in author's previous work Universal testing machine (UTM) was used to study the tension to compression asymmetry of various material conditions at a cross-head velocity of 1 mm/min at ambient temperature. The specimens for tension and compression test were machined as per the dimensions are shown in . All the samples were cut in three different directions with the longitudinal axis (Loading axis) of the samples along the rolling direction (RD), 45° to the rolling direction (RD-45) and transverse direction (TD). Tensile specimen with dimensions of 25 mm length, 6.35 mm width and 2 mm thick in the gauge section (ASTM-E8 standard) were cut with the length aligned along the three directions mentioned (RD, RD-45, TD). Similarly, for compression test, the samples were machined from the plate in three directions in rectangular shape with a height of 25 mm, the width 10 mm and thickness 4 mm. The 0.2% offset yield strength, ultimate tensile strength and total elongation were obtained based on the average value of three tests per material condition. To study the deformation mechanism under compression loading, compression tests were conducted to partially deform the samples up to a strain level of 0.05.The microstructural features of various material conditions were characterized by using EBSD analysis before and after 5% compression. The samples for EBSD characterization were prepared by mechanical polishing up to 1 μm surface finish, followed by colloidal silica polishing (0.02 μm). The EBSD measurements were performed at the center of the samples with a scan area of 200 μm × 200 μm and step size of 1 μm for the base and in-situ composite in all materials conditions. The data acquisition and analysis were carried out by using the TSL OIM analysis 7.0 software. The average indexing percentage for the base and composite samples were 85% and 70% respectively.Tension and compression tests were carried out to study the tension-compression yield asymmetry of base and composite materials at different heat treatment conditions. The engineering stress-engineering strain curves under the tension and the compression along RD, RD-45 and TD of the base and composite materials with different heat treatment conditions are shown in . The quantified information of mechanical properties for different material conditions is presented in shows the engineering stress-engineering strain curve at ambient temperature along RD during tensile and compression test. The yield stress under tensile deformation of B material is 171 MPa (a), whereas the yield strength after solution treatment (BST) decreases to 95 MPa and increases to 225 MPa in BSTPA material. The yield stress in compression test (b) for B, BST and BSTPA materials is 131 MPa, 76 MPa and 184 MPa respectively which is lower than that of respective tensile test. The similar trend is observed in composite materials too under all heat treatment conditions. Yield stress of C, CST and CSTPA in tensile test is 203, 104 and 228 MPa respectively and in compression test, the yield stress of C, CST and CSTPA decreases to 177, 88 and 217 MPa respectively. The conclusion from above observations shows that the yield stress in compression test is lower than that of tensile test in all heat treatment conditioned materials.Engineering stress-engineering strain curves for tension and compression test in RD-45 is shown in a, the yield stress of B, BST and BSTPA are 163 MPa, 93 MPa and 200 MPa respectively in tension (a) and 120 MPa, 71 MPa and 160 MPa respectively in compression test (b). Whereas the yield stress of C, CST and CSTPA materials increased to 195 MPa, 102 MPa and 218 MPa respectively in tension and 160 MPa, 86 MPa and 191 MPa respectively in compression test. The conclusions for these observations are: (i) yield stress for base materials in all heat treatment conditions is observed to be lower than that in composite materials and (ii) the yield stress for tensile test is higher than that for the compression test in all six conditions. shows the tension and compression test results in TD. The yield stress shows the same trend as that of RD and RD-45 i.e. yield stress in tension test is higher than that of compression test. However, in comparison to RD and RD-45°, the yield stress of TD is lower in all materials conditions. The yield stress of B, BST and BSTPA are 176 MPa, 91 MPa and 173 MPa respectively in tension (a) and 123 MPa, 62 MPa and 146 MPa respectively in compression test (b). Whereas the yield stress of composite material in all heat treatment conditions are 175 MPa, 95 MPa and 209 MPa in tension and 142 MPa, 78 MPa and 181 MPa in compression test for C, CST and CSTPA materials respectively. shows the tension-compression yield asymmetry (R = TYS/CYS) values along three different loading directions i.e., RD, RD-45 and TD. In RD (a), the R value for B, BST and BSTPA materials is 1.30, 1.25 and 1.22 respectively, whereas for composite material, R value are 1.14, 1.18 and 1.04 for C, CST and CSTPA materials respectively. With this, it can be concluded that the value of R is lesser in composite materials than that of base material at all heat treatment conditions. Similar trend is also observed in RD-45 (b) loading direction for both base and composite materials. The R value of base materials are 1.36, 1.30 and 1.25 for B, BST and BSTPA materials respectively. Similarly, for C, CST and CSTPA materials, the R value is found as 1.22, 1.18 and 1.14 respectively. c shows the R value of base and composite material along TD. It can see from figure that the R value for B, BST and BSTPA materials is 1.43, 1.46 and 1.18 respectively and 1.23, 1.22 and 1.15 for C, CST and CSTPA composite materials respectively. It is concluded from the above observation that, the tension-compression yield asymmetry value is less along RD compared to other two loading directions i.e., RD-45 and TD. shows the comparison of ductility of different material conditions with respect to loading direction. The ductility along RD for all materials conditions is higher than that of RD-45 and TD both in case of tension (b) test, whereas the materials in TD shows low ductility in all conditions for tension and compression test.The objective of current work is to reduce the yield asymmetry. It is observed that the value of the yield asymmetry (R) is lowest along RD in comparison to other two loading directions i.e., RD-45 and TD. Therefore, further investigation is carried out along RD only.EBSD orientation maps of the base and composite materials with all heat treatment conditions before and after compression testing along RD direction are exhibited in . The twins fraction was measured from the EBSD analysis for all material conditions and are listed in , the twin fraction of B materials before deformation was found to be 5.4% and after 5% partial deformation, the twin fraction increases to 20%. Similarly, the twin fraction of C materials before and after deformation were 4.5% and 17.5% respectively. This indicates that, the presence of twin fraction in C materials is lesser than that of B materials both before and after deformation. shows the EBSD orientation maps of the BST and CST materials before and after deformation. A clean microstructure with 0% twin density is observed in BST material before deformation, but after 5% partial deformation in compression, the twin fraction was found to be 35%. Similar trend is found in CST material i.e., the twin fraction in CST material was 0% before deformation and 23% after 5% partial deformation. Whereas in BSTPA material (), the twin fraction decreases to 2.8% after deformation and there are no twins in CSTPA materials after 5% deformation.The present research is intended to study the synergetic effect of precipitates and in-situ reinforcement particles on the yield asymmetry of magnesium alloy. For this purpose, both the base (B) and composite (C) materials are subjected to solution treatment at 400 °C for 24 h and the aging at 175 °C for 100 h. Four different microstructures are developed to study the deformation behavior on yield asymmetry during compression deformation in RD loading direction: (a) magnesium alloy without any precipitate and in-situ reinforcement i.e. BST material, (b) magnesium alloy with in-situ reinforcements i.e. CST material, (c) magnesium alloy with presence of precipitate i.e. BSTPA material and (d) magnesium alloy with both in-situ reinforcements and precipitates i.e. CSTPA material. SEM and TEM microstructural characterization were carried out to study the influence of precipitate and in-situ reinforcement on twin nucleation and growth. shows SEM microstructure of BST material after 5% of compression testing along with the schematic diagram which describes the twin deformation mechanism in magnesium alloy. The SEM microstructure (a) shows the presence of different types of twins inside the grains and these twins are mainly tensile twins (b). The compression curve of BST material along RD (b) shows a concave shape, which is reported to be a typical feature of {101¯2} extension twinning b). In BST material, twin is formed during deformation due lack of slip systems in HCP crystal structure of magnesium alloy. b and c shows schematic diagram for deformation behavior and shear behavior of {101¯2} twin. shows the point to point misorientation angle analysis between parent matrix and twin of BST material after 5% compression test. The angle between {0002} basal plane and {101¯2} twinning plane is 43.15° (86.30°/2) (). Because of symmetry condition required for twinning, the plane perpendicular to {101¯2} (i.e. T2 plane {1¯012}) shifts counterclockwise in magnesium lattice as shown in b. Due to this rotation of T2 plane, the crystal inside the twinned volume gets shortened in a direction parallel to the basal plane b. Due to this reason, in magnesium alloy twinning occurs under tension parallel to the c-axis or under compression perpendicular to the c-axis. In BST material, twins are observed after 5% compression test, which indicates that the compression load applies perpendicular to c-axis. Most of these twins are tensile twin as the CRSS of tensile twin is lower than that of the compression twin.The formation of tensile twin can also be analyzed by theoretical calculation of Schmid factor (SF) and yield stress. The yield stress (activation stress) of a deformation mode along a particular load direction is represented as where σ is yield stress of the material, τ is the CRSS and m is the Schmid factor. shows the slip and twin system of magnesium alloy. The relationship between loading axis with four slips and two twin systems are shown in a is angle between loading axis and c-axis of HCP crystal. The SF analysis was made based on the following assumptions: (a) parent grain has a perfect basal texture with the c-axis aligned parallel to the ND and the a-axis randomly oriented in the RD-TD plane as shown in c and (b) the prismatic plane including the rotation a-axis is placed on the RD-TD plane (i.e., angle between loading axis and a-axis α = 0°). For the calculation of yield strength, the CRSS value of basal slip, prismatic slip and twinning were used as 10 MPa, 55 MPa and 30 MPa respectively where φ is angle between the load axis and slip plane; and λ is angle between the load axis and slip direction. The loading directions along σ1 with α = 0° is given by 〈112¯c1〉.where c1 is c-index of relative loading directions σ1, the angle α = 0° indicates the projective location of relative loading direction on the basal plane (For a particular deformation system, hkiluvtw , the slip or twinning plane normal with 3-index is given by:The values of cosφ or cosλ can be calculated as per equation cosφor cosλ=u1u2+v1v2−12u1v2+u2v1+ca2w1w2u12+v12−u1v1+ca2w12u22+v22−u2v2+ca2w2212The calculated values of SF and yield stress as per Eq. for different deformation modes such as basal slip, prismatic slip, and {101¯2} tensile twinning at α = 0° are shown in . When the applied loading direction is along RD, angle θ varies from 0° to 30°. Initially when θ varies from 0° to 9°, the yield stress of {101¯2} extension twinning is much lower than that of basal slip and prismatic slip (b). The activation stress of {101¯2} twinning mode varies from 60 MPa to 117 MPa at θ = 0° to θ = 30° respectively. As θ increases the yield stress of basal slip reduces rapidly and it approaches to that of {101¯2} twinning at θ = 9°. So, the contribution of basal slip in deformation along RD is less as θ varies from 0° to 30°. The conclusion from the above theoretical calculation of schmid factor is that the main deformation mechanism along RD is combination of tensile twin and basal slip. In this study, the yield strength asymmetry (R) of BST material in RD was found to be 1.25. This is due to the presence of {101¯2} tensile twin (in addition to basal slip) which reduces the yield strength during compression deformation depicts SEM microstructure of CST material after 5% compression along with schematic diagram showing twin interaction with in-situ reinforcement. After 5% compression deformation in CST materials (a), different types of twins are observed inside the grains and the twin percentage is less than that of BST material. In general, grain boundary act as a twin nucleation site and these twin nucleations are influenced by the modification of grain boundary structure and chemistry. The presence of these reinforcements can modify the twin nucleation at grain boundaries. There are two stages in the formation of twins during compression: (a) twin nucleation and (b) twin growth. In-situ TiC-TiB2 particles (in CST material) are responsible for the suppression of twinning during compression, which further strengthened {101¯2} twinning dislocations due to particles-twin dislocation interaction. This increases the critical resolved shear stress (CRSS) of {101¯2} twinning, which leads to suppression of twin nucleation. The mobility of twin boundaries can be reduced by the (i) loss of coherency during the deformation, (ii) segregation of in-situ reinforcement and (iii) pinning effect from TiC-TiB2 particles. b shows the schematic diagram of twin and in-situ particles interaction. In b, the Y-axis is parallel to the c-axis [0001] of an hcp unit cell in the magnesium matrix and X-axis represent the a-axis in the direction of {101¯0} which positioned in the basal plane. Then the in-situ TiC-TiB2 reinforcement are plotted as per the experimental result is shown in a. The in-situ particles act as a barrier of twin nucleation as well as twin growth which give additional path to activate (0002) basal slip systems as the CRSS of basal slip system is lower than other non-basal slip system i.e, prismatic and pyramidal slip system. Due to the presence of both twin and slip system, the yield strength increases in CST material and hence reduces the yield asymmetry. In CST material the yield asymmetry value (R) along RD decreases to 1.17 as compared to BST (1.25), which is attributed to the effect of in-situ TiC-TiB2 reinforcements. These reinforcements hinder the growth of twinning. shows the TEM microstructure of BSTPA material and schematic diagram showing precipitate arrangement with the HCP crystal. After aging treatment of material B, there is a development of Mg17Al12 precipitate. These Mg17Al12 precipitates are formed in BSTPA material with different shapes, such as disc, circular and niddle/rod shapes as shown in a. The rod-shaped precipitates are parallel to c-axis whereas disc shape precipitates are parallel to the basal plane b. When the compression load is applied perpendicular to basal planes, the rod-shaped Mg17Al12 precipitates can act as semi-fibers and strengthen Mg alloy.The TEM microstructure of BSTPA material and schematic diagram showing twin interaction with Mg17Al12 precipitate and in-situ reinforcement after 5% compression is shown in . In BSTPA material, the {101¯2} twin volume fraction percentage are smaller (b) as compared to BST materials. These observations suggest that the presence of different form of Mg17Al12 precipitates affect the twin nucleation and twin growth. In other words, the {101¯2} twin nucleation and growth is limited due to the presence of precipitates. The schematic diagram of twin and different type of Mg17Al12 precipitate interaction during deformation in BSTPA was shown in c. In BSTPA material, the interaction of {101¯2} tensile twins with Mg17Al12 precipitates can be explained in three ways: (i) if the Mg17Al12 precipitates are much wider than deformed twins, then the twins may consume the precipitates as shown in a, (ii) if the Mg17Al12 precipitate size is much smaller than deformed twin, the precipitate will act as a barrier toward the migration of twin a and (iii) if twin and Mg17Al12 precipitate width are similar or in niddle shape as shown in b, then the twin tends to bypass the precipitate by bowing around it. In BSTPA material more precipitates are present in different forms (a and b), which makes them efficient in resisting the migration of twin boundaries and hinders the rate of twin propagation.The combined effect of in-situ reinforcement and precipitates on yield asymmetry are studied under TEM microstructure and schematic diagram showing twin interaction with precipitate and reinforcement of CSTPA material after 5% compression is shown in a), both precipitates and in-situ reinforcement particles (TiC and TiB2) are present which acts as a barrier towards the migration of twin and hinders the rate of twin propagation. To better understand this mechanism, a schematic diagram showing the twin, particle and precipitate interactions is illustrated in c, the Y-axis is parallel to the c-axis [0001] of an hcp unit cell in the magnesium matrix and X-axis represent the a-axis in the direction of {101¯0} which is positioned in the basal plane. Then the β-phase Mg17Al12 precipitate and in-situ reinforcement can be plotted randomly in different orientations, as it was observed to form irregularly in magnesium matrix (The presence of hard in-situ TiC-TiB2 particles and Mg17Al12 precipitates in CSTPA material, limit the lateral growth of the twins. The in-situ TiC-TiB2 particles and Mg17Al12 precipitates will give some back stress, when the twin approach towards precipitates. This back stress obstructs the motion of migrating twin boundary. If the twin width and the in-situ particle/precipitate are similar in size, the twins bow around the in-situ particle/precipitate (being harder/stronger, the in-situ particle/precipitates will not able to shear), which results initiation of twinning shear. When the twinning shear reaches the interface between the matrix and the precipitates/particles, the movement of atoms towards the precipitate would be accommodated by the surrounding material, which is likely accomplished by the basal slip of the matrix near the particle/matrix interface. However, the basal slip in the matrix along the fine horizontal dashed lines (c) would also be restricted by the Mg17Al12 precipitates and in-situ particles, which enable the accommodation of twinning shear by the (0001) twin basal slip (i.e., along the fine parallel dashed lines inside the twin), as indicated in c. Due to these accommodation twinning shear by the basal slip inside twin, the further movement of twin is restricted. As a result, the presence of more Mg17Al12 precipitate and in-situ particles in CSTPA material would significantly modify the potential of twin nucleation sites and obstruct the rate of twin formation. In CSTPA material the yield asymmetry value (R) along RD decreases to 1.04 (a), which is attributed to the combined effect of in-situ TiC-TiB2 reinforcement and precipitate. These reinforcements hinder the growth of twinning leading to a reduction in yield asymmetry.Tension and compression test is carried out in three different directions i.e., rolling direction, 45° to rolling direction and transverse direction of AZ91 magnesium alloy and AZ91+TiC-TiB2 in-situ composite. Deformation behavior during compression test is studied through EBSD analysis. The individual effect of reinforcement and precipitate on tension-compression yield asymmetry was investigated in details and the significant outcomes are summarized below:Both the alloy and in-situ TiC-TiB2 composite materials shows improved mechanical properties under tensile test along rolling direction compared to other two direction i.e, 45° to rolling direction and transverse direction.After age hardening the yield strength and tensile strength of AZ91 alloy (BSTPA) in rolling direction were enhanced by 31% and 48% respectively due to formation of different size of precipitate. However, in TiC-TiB2 composite material (CSTPA) the yield strength and tensile strength were improved by 33% and 65% respectively than that of AZ91 after peak aging.The tension-compression yield asymmetry of alloy after solution treatment (BST) is due to the presence of {101¯2} tensile twin, formed due to the c-axis extension during compression test and cause decrease in yield strength of the material.During solution treatment, the presence of in-situ TiC-TiB2 reinforcement in composite (CST) material suppresses the nucleation and growth of twin and reduces the tension-compression yield asymmetry as compared to AZ91 alloy.After peak aging TiC-TiB2 reinforced in-situ composite (CSTPA) material shows lowest tension-compression yield asymmetry (1.04) due to presence of both TiC-TiB2 in-situ reinforcement and Mg17Al12 precipitate. The reinforcement particles and precipitate give a back stress to the twin and oppose the twin nucleation and growth.The present research is intended to study the individual as well as the combined effect of in-situ TiC-TiB2 reinforcement and precipitates on the tension-compression yield asymetricity of AZ91 magnesium (Mg) alloy and AZ91 + TiC-TiB2 Mg matrix composites.The novelty of the present work is as follows:The influence of in-situ TiC-TiB2 reinforcement on twin nucleation and growth of magnesium composite has been established.The combined effect of the precipitates and in-situ reinforcements on tension-compression yield asymmetry of magnesium composite has been studied for the first time.Computationally implemented modeling of creep of composite materials caused by phase dissolutionMultiphasic composites with time-evolving, transient microstructures exhibit time-dependent behavior under load. One mechanism leading to this behavior is the dissolution of load-bearing phases within such materials. When the dissolution process occurs inside one phase of the composite material, the stress transmitted by the dissolving phase transfers into the neighboring phases, resulting in additional deformation of the whole composite. This additional deformation of the macroscopic material manifests as time-dependent creep of the composite. The purpose of this paper is to provide a general kinematic framework for such dissolution induced creep for virtually any material class, and to mechanistically model the same utilizing a computationally-implemented approach that is consistent with the kinematic framework. Utilizing this modeling approach, two distinct processes are computationally simulated, namely (1) creep of a water saturated, porous, frozen body upon warming and (2) creep of cement paste due to hydration-induced dissolution of cement grains. The simulation results show that microstructure evolution induces significant creep behavior of the whole composite in these two simulated cases.Many natural and man-made materials are multiphasic composite materials, such as polymers, mortar and concrete, alloys, and porous media. Most of these materials exhibit notable time-dependent viscoelastic/viscoplastic (VE/VP) behavior under long-term external mechanical loading, and many computational algorithms and numerical simulation techniques have been developed to study the creep/relaxation behavior of composite materials, such as mesoscale finite element simulations of concrete The objective of this paper is first to establish a holistic kinematic framework linking the microstructure evolution to the evolution of stress and strain fields to better understand the evolution of composite constitutive properties and deformation mechanisms. The second objective is to demonstrate the utility and robustness of the framework and affiliated modeling approach via prediction of the dissolution induced creep of two disparate classes of materials. In particular, the computational scheme is applied to both a porous, water-saturated, frozen body when subjected to warming and hydrating cement paste, such that the effects of phase dissolution on creep can be demonstrated according to the simulation results.The computational scheme integrates two different models: a time-evolving microstructure model and a finite element based mechanical model. The microstructure models generate time-evolving microstructures for different composite materials, which are then utilized in the finite element calculation model to predict the time-evolving creep and elastic responses of these composites.To account for the time dependence (as well as the strain and stress history dependence) of the elastic and viscoelastic properties, the computational scheme is discretized in time. At each time step in the computational scheme, the microstructure models provide a 3D snapshot of the composite materials for the corresponding age of the time step. All the 3D snapshots of the materials are meshed into voxels utilizing a spatially aligned numerical discretization under a stress-controlled boundary condition.The concept of evolving natural configurations this difference in motion yields “apparent strain” in the composite natural configuration, as shown in shows the overview of the kinematical framework of the dissolution-formation process utilizing the concept of time-evolving natural configurations. Note that conceptually, this framework is analogous to that utilized in the analysis of mixtures or other materials with multiple natural configurations , the framework at each step corresponds to one time step shown in . The motions χκ show the paths of a material point between different configurations. The motions χκp and χκp(t) denote the path of a point in the composite body from the initial, reference configuration to the natural configuration of the composite, and then to the current configuration. The motions χκpv and χκp(t)v denote the motion of a given voxel (occupying the same location in the composite as the point referred to above) from the initial reference configuration to the natural configuration, and then to the current configuration. In (a), all the motions (both the composite and voxel) start from the initial reference configuration. At t=t1, as shown in (b), external or internal stimuli are applied on the material, deforming the composite body from the reference configuration to the current configuration, and removal of the stimuli will not necessarily make the composite return back to the initial reference configuration. If the inherent phases that comprise the composite are VE/VP themselves, the natural configuration of the whole composite may evolve with time, and with the removal of the stimuli, the body will return to the evolved natural configuration instead of the reference configuration; likewise for the individual voxels inside the microstructure. Once phase dissolution occurs in a voxel at t=t2, as shown in (c), any stress carried by this voxel is forced to be zero and the current configuration becomes the voxel’s new natural configuration. At this time, before the formation of new phases, the natural configuration of the composite stays the same as the configuration shown in (b), leading to a difference between the composite natural configuration and the voxel natural configuration. This difference in deformation is a mapping χκapparentv that leads to “apparent strain” in the composite body. From the perspective of continuum mechanics, apparent strain is defined here as the strain that exists independent of any applied stimuli (the state of stress specifically in this paper). At t=t3 in (d), the natural configuration of the composite changes again with the formation or precipitation of new solid phases. At the last time step in (e), with any future change in the stress field within the composite (which may occur due to further dissolution, intrinsic creep in the voxels, and/or change in boundary conditions), voxels filled with newly formed phases deform from the individual voxel natural configuration to the new current configuration; likewise, the whole composite body deforms to the new current configuration., it can be seen that the existence of apparent strain is a key reason behind the dissolution-induced VE/VP behavior and the irreversible nature of much of the dissolution induced creep. To computationally predict the overall deformation behavior of the composite material, the effect of apparent strain on the inherent stress and strain fields inside the composite must be quantified. shows a more complete version of the framework demonstrated in (e), and in this figure, the motion of a material point inside continuous body is defined aswhere x is the position vector in the current configuration of the motion, X is the position vector in the reference configuration of the motion, and the boldface indicates a vector (or, in general, a higher order tensor). The deformation gradient of the body is defined according toAs stated previously, a primary goal here is to quantify the stress relaxation or creep that occurs due to phase dissolution in a composite microstructure. Such stress relaxation or creep is determined by quantifying the time evolution of the spatially averaged strain field and the spatially averaged stress field under a particular boundary condition. For a composite material body input into the computational scheme, spatially averaged strain is calculated according to the deformation from the reference configuration to the current configuration (i.e., motion χκ). The spatially averaged stress field is calculated according to the difference between the composite natural configuration and the composite current configuration, which is mapped through the motion χκp(t). Unfortunately, when stress relaxation or creep is induced by dissolution of load-bearing phases, one cannot directly determine the deformation gradient (and thus the strain and stress) associated with the motion χκp(t). However, the motion χκp(t) can be decomposed into two different computable motions: one from the composite natural configuration to the voxel natural configuration χκapparentv, and one from the voxel natural configuration to the current configuration χκp(t)v. For these three motions, according to Eq. motionχκp(t):Fκp(t)=∂xκ∂xκpmotionχκapparentv:Fκapparentv=∂xκpv∂xκpmotionχκp(t)v:Fκp(t)v=∂xκv∂xκpv=∂xκ∂xκpvwhere xκ and xκv are the position vectors of a point in the composite body and the corresponding voxel in the current configuration, xκpv is the position vector of the voxel in the voxel natural configuration, and xκp is the position vector of the point corresponding to the location of the voxel in the composite natural configuration. From Eq. with the Green-Lagrangian strain expressionwhere I is the identity tensor, one finds that for the motion χκp(t),Eκp(t)=12FκapparentvFκp(t)vTFκapparentvFκp(t)v-I=12Fκp(t)vTFκapparentvTFκapparentvFκp(t)v-I,where Eκp(t) is the strain tensor of a particular point in the composite body generated during the motion χκp(t) from the composite natural configuration to the current configuration. Similarly, according to Eq. Eκapparentv=12FκapparentvTFκapparentv-IandEκp(t)v=12Fκp(t)vTFκp(t)v-I,where Eκapparentv is the strain tensor for the voxel that exists at the same, particular point in the composite, generated during the motion χκapparentv, and Eκp(t)v is the strain tensor of the particular voxel generated during the motion χκp(t).Eκp(t)=Fκp(t)vTEκapparentvFκp(t)v+Eκp(t)v.is the displacement of a given voxel from the natural configuration of the voxel to the current configuration. Sincewhere one should carefully note that Gradκpv is the gradient operation in reference to the position in the voxel natural configuration, we find that Eκp(t)=GradκpvuκpvT+IEκapparentvGradκpvuκpv+I+Eκp(t)v.If the strains are small in magnitude (as in the simulations considered in this work), then where ε is the infinitesimal, linearized strain. Eq. where εapparent is the apparent strain of a particular voxel generated during the dissolution-formation process (and associated with the mapping of a point from the composite natural configuration to the corresponding voxel natural configuration), and εmechanical is the mechanical strain of the voxel as a result of the stress state within that voxel. Note that the specific requirement for the use of the linearized strain here is that the individual voxel strain between the natural and current configurations must be small along with the product of that strain and the apparent strain that occurs between the composite body as a whole and the individual voxels. It is important to be aware that one might have a composite exhibiting small strain at the macroscopic length scale while yet exhibiting locally large strains at the voxel level (depending on choice of length scale for the voxel); in such a case, the linearization in , it can be seen that, from a book-keeping perspective, apparent strain can be treated in the same fashion as free strains or eigenstrains To implement the above conceptualization and kinematic framework of the dissolution process into the computational scheme, the phases inside each voxel at all time steps should be tracked and all the voxels that participate in the dissolution process should be recorded. In the model, under the applied strain controlled periodic boundary condition over the entire time scale, different microstructures of composites are input into the scheme. Microstructures from two neighboring time steps would be compared first at the beginning of each time step, so that once the phases in one single voxel changes between the two time steps, the location and the configuration of this voxel would be noted for further calculation, while its historical mechanical responses would be expunged. The newly formed phases would be assigned with new constitutive properties for the subsequent time steps. Finite element analysis is carried out on the new microstructure to calculate the new internal stress and strain fields inside the material, and this procedure repeats for all time steps. Apparent VE/VP relaxation moduli are determined by quantifying the volume averaged stress at each time step (recall that the volume averaged strain is fixed as zero to simulate a relaxation experiment). More detailed information regarding the conceptualization of the model can be found in A computationally implemented homogenization model utilizing the finite element method was developed previously to predict the elastic moduli of 3D composites Once the mechanical equilibrium state is reached, the volume averaged stress may be predicted through solving a series of elastic equations on the finite element mesh At any time step, since all the phases inside the microstructure voxels are purely linearly elastic, when the composite is subject to infinitesimal strains, the total elastic mechanical energy stored in one single voxel can be calculated bywhere En is the total stored elastic energy, εpq is the infinitesimal strain tensor, Cpqrs is the elastic moduli tensor; p, q, r, s |
= 1, 2, or 3, and the integral is over the volume of a single unit voxel. By expressing the strain tensor in terms of displacement components, Eq. where Drpsq is the stiffness matrix and urp is the p’th component of displacement at r’th node. From Eq. , the strain tensors for any voxel in the microstructure can be approximated as the sum of the mechanical strain and the apparent strain, and so may the displacement vector of the nodes,where Urp is the displacement vector determined by the surrounding voxels, δrp is the correction vector determined by periodic boundary conditions, and δrpapparent is the correction vector determined by the apparent strain produced in the microstructure during the dissolution process. Defining δrpcombined=δrp+δrpapparent, Eq. En=12urpTDrp,squsq+2δrpcombinedDrp,squsq+δrpcombinedDrp,sqδsqcombinedwhere δrpcombined is the total correction vector for the displacement of a node. Eq. where A is the Hessian matrix comprised of the stiffness matrices, u is a vector of all the displacements, b is a global vector, and C is a global constant; b and C are determined by both the external strain controlled, periodic boundary conditions, and the dissolution and precipitation process of phases that occurs in voxels, which generates apparent strain. Since in Eq. there is no term with an order higher than quadratic, to derive the exact displacement solution leading to the minimum elastic energy, Simpson’s rule may be used such thatIn the computational scheme, the apparent VE/VP bulk modulus and VE/VP shear modulus are calculated directly from the simulation results when the composites are subjected to prescribed, periodic volumetric strain and prescribed, periodic shear strain.Due to a constraint in the current version of the computational scheme, microstructures that are input directly into the model can only be subjected to strain-controlled periodic boundary conditions. To predict the time-evolving strain of the macroscopic composite under constant external stress, Boltzmann’s superposition principle ifi=1,ε(ti)=-σ03Kti(ti);ifi≠1,ε(ti)=σ0-∑k=1i-1[ε(tk)Ktk(ti)]3Kti(ti),where σ0 is the maintained constant boundary stress, ε(ti) is the additional linear strain to be applied on the composite at time ti to achieve the desired boundary stress history, and Ktk(ti) is the apparent VE/VP bulk modulus of the composite at time ti when loaded at age tk. The resultant strain history of the composite (e.g., the sum of the creep and elastic strains) is calculated throughwhere εtotal(ti) is the strain of the composite at time ti. The preceding application of Boltzmann’s superposition principle was utilized in similar fashion (but with a constant applied, boundary deviatoric stress) to determine the time-dependent shear strain associated with dissolution of load bearing phases.In order to illustrate the robustness and breadth of applicability of the kinematical framework and computationally implemented modeling approach, the apparent VE/VP creep behavior of two composite materials was simulated: (1) a porous solid containing ice in the pores that is gradually warmed and (2) hydrating cement paste. In the first case, creep occurs as a result of the melting process of the solid ice phases into the liquid water phases, which is functionally equivalent to the dissolution process in terms of the effect on stress redistributions. In this simulation case, the distribution of pore sizes plays a big role. Due to thermodynamic considerations (i.e., Gibbs-Thomson equation The second simulation involves quantifying the creep that occurs in young cement paste that is undergoing the hydration reaction. When cement grains are mixed with water, a chemical reaction proceeds whereby the grains gradually dissolve and hydration products are precipitated out of the ion-rich aqueous solution. These products fill the space between the remaining cement grains and ultimately bond them together to form a solid-like matrix. If that matrix is stressed, both hydration products and remaining cement grains transmit a portion of the stress; as cement grains dissolve, the stress carried by those phases is transmitted to the remaining solid microstructure, which induces creep behavior in the composite.Based on an algorithm previously utilized to generate random distributed impenetrable objects . These porous microstructures are solid matrices filled with randomly distributed pores, which are in the shape of spheres and are assigned with various radii. In this brute force algorithm, first, the largest pores are placed in the matrix in a cubic box with the dimension of 1 μm. Then, the second largest pores are randomly located in the matrix while avoiding any overlap with the previous existing pores. This procedure is repeated until a particular pore volume fraction is reached. The 3D images of the microstructures generated from the Mathematic file are then converted to grayscale 2D images. The domains are then subdivided into 100 pixels along each axis perpendicular to the faces of the cube (to form a total of 100×100×100=106 voxels per microstructure per temperature step). Depending on the grayscale level of each voxel, every voxel inside the microstructure contains a single phase (solid/ice/water), which would be assigned mechanical properties in the finite element calculations.Two different series of microstructures were generated utilizing the aforementioned algorithm. Microstructure 1 (MS1) has a relatively narrow pore size distribution with the maximum pore size of 0.006 μm and minimum pore size of 0.0062 μm. Microstructure 2 (MS2) has a wider and more uniform pore size distribution with the maximum pore size of 0.0034 μm and minimum pore size of 0.0227 μm. It can be easily seen that the pore sizes in MS1 are much larger than MS2. The porosity in MS1 is 0.2, and the porosity in MS2 is 0.1.To determine the fraction of pores containing ice or water at each temperature for each microstructure, thermodynamic equilibrium was presumed to exist temporally at each time step. That is, the heating rate was slow enough so that each individual ice crystal could establish local equilibrium with its melt. As a consequence of the equilibrium between a solid crystal (e.g., ice) and its melt (e.g., liquid water), when temperature is increased, to stop ice from melting, the pressure in the crystal has to go down according to the relationwhere pC is the crystal pressure, T is the current temperature, and the liquid pressure, pL, remains at atmospheric pressure. Tm and Σm are the melting temperature and the melting entropy, respectively. The pressure difference, pC-pL is related to the curvature radius, R, of the solid-liquid interface through the mechanical equilibrium (Laplace’s equation),Here, θ is the contact angle between the liquid and the pore wall, and γCL defines the crystal-liquid interface energy. Combining Eqs. of the melting pore at a particular temperature can be determined asThat is, when the temperature increases to the current temperature, T, as given by , ice in all pores equal to and larger than the radius R will melt. Eq. is widely known as the Gibbs-Thomson equation.Simulations are carried out on porous materials with time-evolving microstructures as ice melts inside its pore network utilizing the computational scheme. Two different factors are analyzed in the scheme to examine their effects on the overall creep behavior of ice-filled composites. One factor is different heating rates, which affects the melting/dissolution rate of the load bearing ice crystals, and the other one is different pore size distribution inherent to the microstructure. We choose to use the term ‘dissolution’ loosely here for the transition of ice to liquid water, since, from a stress-redistribution modeling perspective, removal of a solid due to phase change or actual dissolution in a solvent are functionally equivalent. In the simulations of the composites with frozen pores, the Young’s modulus of ice phases was prescribed as 8.7 GPa, and the Poisson’s ratio was set to be 0.31 , an increasing axial strain of the composite as ice melts is quantified utilizing the new computational routine described in this paper. When the temperature increases from −10 °C to 0 °C, the evolution of axial strain of the composite structure is strongly affected by the different melting rates. The solid lines show the evolution of the stress-normalized axial creep strain of the composite under a slower rate of temperature increase (1 °C/h) and thus a slower ice melting/dissolution rate, while dashed lines indicate a higher rate of temperature increase (2 °C/h) and thus a higher melting/dissolution rate. From , a higher melting rate (dissolution rate) would lead to faster creep of the composite. After the temperature reaches 0 °C, all the ice inside the microstructure has transitioned into water, and further increase in the temperature would not induce any additional creep behavior in the porous body.The axial creep strain of porous bodies with different pore sizes is shown in when the composites are subjected to warming (and thus dissolution of ice phases). From , when warming starts, the two composites establish similar instantaneous elastic deformation (because of similar elastic compliance) at time zero. As temperature increases, composites with different pore sizes establish different creep rates. The porous body with small, well distributed pore sizes creeps at a relatively constant rate over the time scale from −10 °C to −2 °C, but the strain stays constant after the temperature reaches −2 °C. The porous body with rather uniform, large pore sizes shows variable creep rate over the ice melting process. When the temperature rises from −10 °C to −2 °C, the composite with large pore sizes creeps at a very slow rate. Once the temperature reaches −2 °C at the 8th hour, suddenly, the large pore sizes composite shows a dramatic increase in the creep strain. The differences in creep rates are due to thermodynamic considerations; when an ice-filled porous body is subjected to increases in temperature, the melting rate of the ice phases is not purely linearly related to the rate of temperature change, but affected by the pore size distribution as well. Those materials with relatively well distributed pore sizes will have a roughly linear dependence of creep rate on rate of temperature increase. In contrast, those materials with a narrow, rather uniform pore size distribution will have a sudden large increase in creep strain when the temperature passes the threshold that induces melting of several of the pores simultaneously, which is what happens when the temperature reaches −2 °C (at about 9 h in ) in the material with larger pore sizes.One thing to notice here is that, in reality, as ice melts inside a porous body, the combined ice/liquid mixture would require less volume in the structure due to its increasing density. This may produce an inherent negative pressure inside the porous body, leading to potential contraction of the whole body as ice melts. Since this paper focuses on the investigations of the effect of solid phase dissolution (and thus stress redistribution) on the overall creep behavior of a frozen porous composite, a simplification is made in this paper that the creep behavior associated with the inherent pore pressurization (i.e., poromechanical effects To generate the time-evolving microstructures for hydrating cement paste, a hydration-based microstructural evolution model THAMES (Thermodynamic Hydration And Microstructure Evolution) The creep behavior of 0.40 water to cement mass ratio (w/c) cement paste at 298 K was simulated when subjected to a stress controlled boundary condition. All the phases in the cement paste were assumed to be purely elastic with assigned elastic properties taken from shows the predicted creep of cement paste in axial strain while excluding the instantaneous elastic deformation of the composite under loading. In , cement paste continues to creep under constant applied stress as a result of the dissolution effect of load-bearing solid cement grains, and this effect is substantial. An older specimen would establish a slower creep rate because the hydration rate of cement grains at later ages is lower (and thus the dissolution rate). This prediction results successfully demonstrates the well-known aging effect of cement paste whereby older specimens creep/relax slower than younger specimens (see, e.g. A kinematic framework was established to quantify the effect of dissolution of load-bearing phases on the overall stress and strain fields in evolving composite materials. The framework accounts for the differing evolutions of natural configurations in a particular phase (existing in a particular voxel) in the discretized microstructure and in the composite material body. Based on the developed framework, a microstructure evolution model and a time-stepping finite element method capable of tracking phase formation were integrated to create a computational scheme. The kinematic framework and complementary computational approach provide a methodology by which stress relaxation or creep of any composite body with dissolving phases may be modeled. To demonstrate this utility, the approach was applied to two different composite materials to predict their apparent time-dependent creep behavior when subjected to stress controlled boundary conditions. When simulating the apparent creep strain of a water saturated, frozen, porous body due to dissolution of ice phases upon warming, the model captures the effect of heat rate and the effect of pore size distribution. And when predicting the apparent VE/VP behavior of early-age cement paste, the well-known aging effect is successfully demonstrated. From both the simulation results, the effect of dissolution of solid, load-bearing phases in leading to apparent time-dependent behavior is substantial.While the kinematic framework is established with no restrictions regarding the magnitude of the deformation gradient, the computational scheme utilizing the framework presented here is limited to small deformation gradients. Thus, the computational scheme is currently limited to predicting composite behavior at stress magnitudes below that which induce locally large strains. For the cementitious materials simulated in this paper, such a limit restricts the simulations to stress levels that do not induce local damage.Nonlinear free and forced vibration behavior of functionally graded plate with piezoelectric layers in thermal environmentIn the present study, finite element formulation based on higher order shear deformation plate theory is developed to analyze nonlinear natural frequencies, time and frequency responses of functionally graded plate with surface-bonded piezoelectric layers under thermal, electrical and mechanical loads. The von Karman nonlinear strain–displacement relationship is used to account for the large deflection of the plate. The material properties of functionally graded material (FGM) are assumed temperature-dependent. The temperature field has uniform distribution over the plate surface and varies in the thickness direction. The considered electric field only has non-zero-valued component Ez. Numerical results are presented to study effects of FGM volume fraction exponent, applied voltage in piezoelectric layers, thermal load and vibration amplitude on nonlinear natural frequencies and time response of FGM plate with integrated piezoelectric layers. In addition, nonlinear frequency response diagrams of the plate are presented and effects of different parameters such as FGM volume fraction exponent, temperature gradient, and piezoelectric voltage are investigated.Functionally graded materials (FGMs) have received considerable attention in many engineering applications since they were first reported in 1980s. FGMs are composite materials, microscopically inhomogeneous, in which the mechanical properties vary smoothly and continuously from one surface to the other. This is achieved by gradually varying the volume fraction of the constituent materials. FGMs are usually made from a mixture of metals and ceramics using powder metallurgy techniques. The ceramic material provides the high-temperature resistance due to its low thermal conductivity, while the ductile metal constituent prevents fracture due to its greater toughness. Unlike fiber–matrix composites, in which cracking and debonding may occur at high temperatures due to the material property mismatch at the interface of two discrete materials, FGMs have the advantage of being capable of withstanding severe high temperature while maintaining structural integrity. Birman and Byrd FGM plates are often susceptible to failure from large deflections, or excessive stresses that are induced by large temperature gradients and/or mechanical loads. Hence, it is of prime importance to understand the large amplitude (geometrically nonlinear) vibration behavior of FGM plate structures to ensure more accurate and reliable structural analysis and design. In recent years, the dynamics of FGM plates have attracted increasing research effort, most of which has been devoted to linear vibration and dynamic response. However, investigations on the large amplitude vibration and transient response of FGM plates are limited in number.Based on the first order shear deformation plate theory (FSDT), Praveen and Reddy Vibration control of plate structures using smart materials such as piezoelectrics, shape memory alloys, and rheological fluids are increasingly important in practical applications. In recent decades, many studies have been carried out on the modeling and controlling of the vibration behavior of isotropic and composite plates by means of piezoelectric sensors and actuators. For example, several variational statements for the vibration analysis of multilayered piezoelectric plates and shells are presented by Carrera et al. To the best of the author’s knowledge, the finite element formulation to investigate geometrically nonlinear vibration behavior (nonlinear natural frequency, time and frequency response) of FGM plate with integrated piezoelectric layers has not been presented in any literature. In the present study, finite element formulation based on HSDT is developed to analyze nonlinear natural frequencies and dynamic response of FGM plate with surface-bonded piezoelectric layers under thermal, electrical, and mechanical loads. The geometric nonlinearity using von Karman’s assumption is introduced. The material properties of FGM are assumed temperature-dependent and are graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. The temperature field is assumed constant in the plane and varies only in the thickness direction of the plate and the electric field considered only has non-zero-valued component Ez.Consider an FGM plate with surface-bonded piezoelectric layers. The length, width and total thickness of the hybrid laminated plate are a, b and h. The thickness of the FGM layer is hf, while the thickness of each piezoelectric layer is hp. The coordinate system to analyze vibration behavior of plate is shown in . The origin of this coordinate system is considered in the mid plane of the plate.It assumed that the top surface of FGM layer (z |
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