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B) the crack faces are traction-free. Note that by taking this approach we are implicitly making the assumption that the effect of the crack on surface displacements is small, so that surface stresses remain unchanged by the presence of the crack. A simplified single crack initiated at the contact trailing edge (X
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=
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x/a
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= −1), and normal to the surface is considered (). Note this assumption has been extensively considered in many fretting–fatigue investigations and was confirmed by examination of the cracking damage.The stress components along the crack face are collected in order to determine the unsatisfied tractions (σN, σS). Since the crack faces have to be traction-free, we distribute both climb and glide displacement discontinuities (or “dislocations”) along the crack so that the stresses induced (σ˜yy,σ˜xy) cancel σN and σS. The integral equations expressing the requirement that the crack faces be traction-free are:σS(xˆ)+G4π(ν+1)∫0bBxˆ(e)KxˆS(xˆ,e)de+∫0bByˆ(e)KyˆS(xˆ,e)de=0σN(xˆ)+2G4π(ν+1)∫0bBxˆ(e)KxˆN(xˆ,e)de+∫0bByˆ(e)KyˆN(xˆ,e)de=0with G and ν the shear modulus and the Poisson coefficients,where σS(xˆ) and σN(xˆ) are the resolved shear and normal components of the stress tensor σ(x,y) in the (xˆ,yˆ) coordinate of the system, KxˆN,KyˆN,KxˆS,KyˆS are the kernels established by the above method detailed in It is not possible to solve the equation analytically, but powerful numerical quadratures are given in references The crack propagation analysis requires the determination of a pertinent effective stress intensity range parameter (ΔKeff). This latter is not easy to determine, particularly under complex and high stress gradient conditions like imposed by fretting–fatigue loading. The adopted strategy will consist to consider different formulations to estimate ΔKeff and to compare the experimental crack arrest conditions. The best fitting will permit us to establish the most relevant ΔKeff assumption. It has been shown that contact loading induces a mixed mode stress condition The full range of mode I and mode II contributions are considered. The effective stress range is expressed by:where KII max is defined at the loading state when KI
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=
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KI max (i.e. open crack condition) and KII min computed at the unloading state (i.e. Q
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= −Q*, σFATIGUE
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= −σa) when KI
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=
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KIm in inducing a closed crack situation (). This formulation, neglecting both closure and crack face friction effect, provides an upper bound estimation of the effect stress intensity range parameter (ΔKeff) and therefore a pessimist estimation of the crack arrest condition (i.e. smaller predicted crack arrest domain).Because pure alternating loading conditions are imposed (R
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= −1), the usual Elber's assumption that the effective mode I stress intensity range can be reduced to the maximum stress intensity value (i.e. ΔKI eff
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=
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KI max) is considered Many investigations confirm that friction phenomena within the crack interface, reduce the mode II contribution. One approximation is to neglect the mode II contribution when the crack faces are under compression state (i.e. at the unloading state). The effective SIF range is therefore estimated by:The mode II contribution is neglected and the effective stress intensity range is identical to the mode I component:where KI max is defined at the loading state (i.e. Q
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= +Q* and σFATIGUE
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= +σa).The examination of the crack suggests that crack propagation may take place in the short crack regime when the actual stress intensity factor range is less than the long crack threshold, ΔK0. We therefore intend to apply short crack approaches, to predict the crack arrest conditions. Two strategies are here considered. The first, based on the Kitagawa and Takahashi diagram, consists in a discontinuous description of the transition from short to long crack domains and the second, introduced by El Haddad and co-authors First introduced by Araujo and Nowell for fretting–fatigue conditions We choose to examine the propagation of the crack in the modified K–T diagram (i.e. where ΔK rather than stress is plotted against b) (). Hence, the threshold SIF range related to the crack arrest condition (ΔKth) is given bywhere b0, is the transition crack length between short and long crack regimes, approximated by:For the alloy used here, this gives b0
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= 170 μm (since ΔK0=7MPam). The crack arrest condition is assumed when the following inequation is satisfied:Like for the previous description, the transition from short to long crack behavior is related to the threshold crack length b0. However, rather than considering a discontinuous transition, we assume the continuous El Haddad approximation of the short crack arrest threshold ( confirms that whatever the crack length, El Haddad's approximation provides a lower value of the threshold crack arrest condition. The difference is particularly important in the transient domain when b
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=
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b0. This suggests that El Haddad's approximation is more conservative than K–T's approach.To establish the experimental crack arrest boundary, the following methodology has been defined. Three levels of fretting loading have been selected, respectively Q*/μP
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= 0.5, 0.63 and 0.73. For each contact loading, the alternated fatigue stressing was adjusted from σa/σd
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= 0.4–0.6. The studied loading conditions selected above the crack nucleation boundary and the corresponding damage evolutions are reported in By contrast with the nucleation phenomenon, propagation failure can be observed after several million cycles. Therefore, to estimate the stabilized crack arrest conditions, the test duration was increased up to 10 million cycles. Hence, for each test, cracking damage is characterized by reporting either if the specimen is broken after 107 cycles or if the maximum crack length for unbroken specimens is found. All the experimental results are reported in , defining the so-called crack arrest fretting–fatigue map (CA-FFM). Like for the crack nucleation analysis, the experimental crack arrest boundary can be extrapolated from failure and non-failure conditions. The experimental crack arrest boundary displays a quasi vertical evolution, which suggests that the crack arrest process is only a function of the fatigue loading and relatively unaffected by contact stress. This result confirms the conventional idea that under fretting–fatigue, crack nucleation is controlled by contact loading, whereas crack arrest is mainly controlled by fatigue bulk stressing. This conclusion is however restricted to the medium fatigue stress range (0.4 <
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σa/σd
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< 0.6) and must be tempered regarding the relative contact size and the small loading range studied.As described previously, crack modeling was carried out assuming a single crack normal to the surface and located at the trailing edge (X
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= −1), where the maximum value of crack nucleation risk has been found. Then, for each fretting loading condition, the modeling strategy consists in identifying, by iterative computations, the maximum fatigue stressing below which the crack arrest condition is achieved. This analysis is applied for the different approximations of the effective SIF range parameters and the two short crack arrest approaches.Comparison with the models suggests that the mixed mode taking into account the full loading range (ΔKeff_A) is too pessimistic whatever the short crack arrest approximation. By contrast, pure mode I (ΔKeff_D) but also the mixed mode taking into account the closure effect and the crack face friction (ΔKeff_C) approximations are too optimistic. Both ΔKeff_D and ΔKeff_C formulations display a quasi superimposed evolution which infers that the mode II contribution during the loading step (i.e. KII max) is quasi negligible.The best approximation of the experimental crack arrest boundary is achieved by combining both the mixed mode, taking into account the closure effect but neglecting the crack face friction (ΔKeff_B), and the Haddad's approximation of the short crack arrest condition.All the failure conditions are predicted. The model is even able to discriminate between the non-failure condition (FF12) and the failure situation (FF15), which are characterized by a fatigue stress difference of less than 10 MPa. This investigation also suggests that a plain mode I description and K–T's approximation can induce dangerous optimistic crack arrest predictions. The better approximation of the crack arrest condition provided by Haddad's approximation is confirmed in , where the maximum crack length at the crack arrest condition is plotted versus the corresponding effective SIF range value (ΔKeff_C=KImax2+KIImax2).K–T's approximation provides a rather good estimation of the crack arrest condition in the short crack arrest domain (i.e. b
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<
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b0) but unfortunately appears too optimistic in the intermediate short/long crack domain. El Haddad's approximation, providing a smoother evolution toward the long crack domain, leads to a more conservative prediction of the crack arrest condition (A significant discrepancy between the experimental crack arrest boundary and the safe El Haddad – mixed mode approximation is nevertheless still observed. This infers that more elaborated formulations, taking into account plasticity, closure and representative friction effects in the crack faces should be introduced into the model. Besides, more complete short crack arrest descriptions have to be considered for future optimization of the modeling. Moreover, the practical difficulties which are experienced in obtaining an experimental value for the long crack threshold must be underlined. Perhaps a better estimation of this value combined with an optimized description of the effective stress intensity factor can allow the application of the less conservative K–T's formulation, as can be expected from which shows that only one experimental point is below K–T's boundary. However, some cautions need to be exercised regarding K–T's approximation. a, shows that two specimen failures are not predicted by it.The main objective of this work is to provide realistic and safe estimations of the cracking response. Therefore, in spite of its limitations, the current pessimist strategy which combines an El Haddad description of the short crack arrest condition, an elastic description of the stress field and the application of a mixed mode formulation taking into account the closure effect but neglecting the crack face friction appears to be a good compromise to approximate the crack arrest boundary in the CA-FFM initially.Both crack nucleation and crack arrest boundaries are reported on the same graph, defining the so-called material response fretting–fatigue map (). The experimental results are compared respectively with the model of crack nucleation defined from Crossland's multiaxial fatigue criterion, by taking into account the stress gradient effect through a 3D process volume description, and using the crack arrest boundary defined from the El Haddad-mixed mode approximation (ΔKeff_B).The studied alloy displays a comparatively low fatigue limit but a high crack arrest threshold. This implies a rather small safe crack nucleation domain but a large crack arrest domain. Hence, for this specific alloy, the safe crack nucleation domain is systematically bordered by the crack arrest domain (). This favors a conservative prediction of cracking risk. Indeed, assuming a loading state located in the safe crack nucleation domain, even if a crack is accidentally nucleated, the structure will be safe because the crack is expected to be stopped whatever the fatigue stress applied (i.e. σa
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<
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σd).Materials displaying an elevated fatigue limit but low damage tolerance are more complex to interpret. plots the estimated crack nucleation and crack arrest boundaries for an equivalent loading spectrum (i.e. similar contact and fatigue loading ranges) but considering a 30NiCrMo8 low-alloy steel whose estimated fatigue properties are respectively σd(R
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= −1) = 420 ± 10 MPa and ΔK0=4±1MPam ( we deduce a very short long crack length transition around b0
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= 23 μm. This alloy is characterized by a tempered martenstic structure displaying an original austenitic grain size around 10 μm diameter. The latter dimension is currently adopted for the stress averaging analysis (i.e. ℓ3D=10μm), which, combined with Crossland's multixial fatigue formulation, allows us to estimate the crack nucleation boundary.), the safe crack nucleation domain is increased but not as expected (). Indeed, the fatigue limit of the 30NiCrMo8 alloy is significantly higher but the process volume applied is smaller, which reduces the stress averaging effect; therefore the crack nucleation boundary remains quasi unchanged.Simultaneously, the sharp reduction of the long crack threshold (ΔK0) promotes a severe reduction in the crack arrest domain. It is interesting to note that above a given bulk stress (σa_CAth), the so-called fretting–fatigue crack arrest bulk stress limit, the crack arrest boundary passes below the crack nucleation boundary. Therefore, above this bulk stress limit, if a crack nucleates, it will systematically propagate until failure because the intermediate crack arrest domain no longer exists. Hence, by contrast to the lower fatigue loading range (σa<σa_CAth) or the application of damage tolerant materials like AISI 1034 steel, such a contact configuration is greatly corrupts the stability and the safety of a cracking risk design based on a safe crack nucleation approach. Higher security factors, to reduce cracking risk, are required. compares the global response of the two alloys. The upper limits between the crack nucleation and crack arrest boundary are plotted as a function of the applied fretting and fatigue loadings. The upper limit of safe cracking damage systematically corresponds to the crack arrest boundary for the AISI 1034 steel but alternatively to the crack arrest or the crack nucleation limits, for the 30NiCrMo8 alloy, depending on whether the fatigue stress is above or below σa_CAth. One important conclusion is that, despite its very low mechanical properties, the AISI 1034 steel alloy appears more resistant against fretting–fatigue cracking due to higher damage tolerance properties (i.e. higher crack threshold (ΔK0)).In the previous paragraph, the link between the fatigue properties of the bulk material and the fretting–fatigue cracking evolution was discussed. However, in most fretting problems, surface palliatives are usually applied to reduce the cracking risk ). The top surface fretting stresses are fully accommodated by these layers, and the bulk material is thus protected from crack nucleation.A second strategy is to apply an in-depth surface treatment to limit and even block crack propagation (). Shot peening and laser peening are the most common treatments of this category. Compressive residual stresses deep below the surface are introduced by the application of plastic deformations. By contrast to conversion treatments or thin hard coatings, these compressive stresses are not stable, and, when a cyclic loading give beyond the plastic yield, the compressive residual stresses are partly or fully erased. Fretting loading tends to relax the surface compressive stresses and explains why it is usually admitted that shot peening treatment does not improve the crack nucleation response of a fretting–fatigue contact (a). However, below the surface, where contact stress reduces, the residual compressive stresses are maintained and will play a determining part in blocking crack propagation, inducing a sharp increase of the fretting–fatigue limit (b). For critical systems, these two palliative strategies are combined () either to extend the crack nucleation domain by applying a thick coating like WC–Co HVOF layers, or to increase the crack propagation resistance by introducing a very deep residual compressive stress field using shot peening and now laser peening., the given fretting–fatigue mapping concept appears as a useful approach to quantify and compare the relative benefits of palliative strategies against fretting–fatigue damage. Surface crack nucleation palliatives, by improving crack nucleation resistance, obviously extend the safe crack nucleation domain but also promote a shift of σa_CAth towards lower fatigue stresses, and consequently induce a reduction of the intermediate crack arrest domain (a). Surface treatments improving the crack propagation resistance extend the intermediate crack arrest domain and consequently increase the threshold value σa_CAth (b). Combined safe crack nucleation and crack arrest palliatives extend both safe crack nucleation and crack arrest domains, and smoothly shift the threshold stress σa_CAth toward highest fatigue stress values (An experimental methodology has been developed to identify, respectively, crack nucleation and crack arrest conditions as a function of the applied fretting and fatigue loadings. Synthetic fretting–fatigue maps have been introduced, defining respectively, the safe crack nucleation, crack arrest and catastrophic failure material responses. It is shown that a basic elastic approximation of the fretting–fatigue loading combined with a Crossland multiaxial fatigue analysis is able to predict the crack nucleation boundary if the contact stress gradient effect is taken into account. This can be achieved by using a process volume methodology, where the stress state considered for the multiaxial fatigue analysis is averaged over a representative volume. It has also been shown, that for the peculiar stress gradient imposed by a contact fretting loading, the different stress averaging approaches like the 3D stress process volume, the 2D stress process surface, and the critical distance method converge to similar results. This work shows that the process volume approach can be calibrated by using plain fretting tests, and extrapolated to more complex fretting–fatigue configurations providing a conservative prediction of the crack nucleation risk. Alternative multiaxial fatigue criteria like Dang Van and McDiarmid formulations lead to similar conclusions, suggesting the “over-conservative” prediction in the medium fatigue stress range is not related to the fatigue criterion formulation but seems to depend on the stress averaging strategy. Hence more advanced stress averaging approaches must be considered if more realistic but less conservative predictions of the crack nucleation boundary are required.It is also shown that a plain El Haddad short crack arrest methodology combined with a mixed mode SIF approximation allows a conservative description of the crack arrest domain. Based on these different approximations, material response fretting–fatigue maps of various alloys can be compared. Taking into account the material properties and the applied contact configuration, different palliative strategies (i.e. surface treatments) can then be adopted, focusing either on crack arrest or safe crack nucleation strategies.Competing fatigue failure behaviors of Ni-based superalloy FGH96 at elevated temperatureFatigue experiments were performed on a polycrystalline P/M processed nickel-based superalloy, FGH96 at 600 °C to investigate competing fatigue failure behaviors of the alloy. The experiments were performed at four levels of stress (from high cycle fatigue to low cycle fatigue) at stress ratio of 0.05. There was large variability in fatigue life at both high and low stresses. Scanning electron microscopy (SEM) was used to analyze the failure surfaces. Three types of competing failure modes were observed (surface, sub-surface and internal initiated failures). Crack initiation sites were gradually changed from the surface to the interior with the decreasing of stress level. Roles of microstructures in competing failure mechanism were analyzed. There were six kinds of fatigue crack initiation modes: (1) surface inclusion initiated; (2) surface facet initiated; (3) sub-surface inclusion initiated; (4) sub-surface facet initiated; (5) internal inclusion initiated; (6) internal facet initiated. Inclusions at surface were the life-limiting microstructures at higher stress levels. The probability of occurrence of inclusions initiated is gradually reduced with decreasing of stress level, simultaneously the probability of occurrence of facets initiated is increasing. The existence of the inclusions resulted in large life variability at higher stress levels, while heterogeneity of material caused by random combinations of grains was the main cause of fatigue variability at lower stress levels.Powder metallurgy nickel-base superalloy is the primary material choice for gas turbine disk because of its exceptional strength and oxidation resistance at the service temperatures. It has been known that the processing of these alloys by powder metallurgy route introduces defects inside the material, leading to microstructural heterogeneities. A number of researchers The main target of the work presented in current paper is to understand the roles of microstructure in competing fatigue failure behaviors of FGH96, a polycrystalline powder metallurgy nickel-based superalloy. In addition to the supplementary data, the present research work was intended to investigate the effect of stress level on competing failure modes. Fatigue experiments in a large sample were performed at 600 °C at four different stresses ranging from 900 MPa to 1200 MPa (from high cycle fatigue to low cycle fatigue). Scanning Electron Microscope (SEM) analysis were conducted to investigate the roles of defects and characteristics of microstructures in competing modes.The material in this study was a polycrystalline P/M processed nickel-based superalloy, FGH96. This alloy is used for turbine discs in aircraft engines. The 0.2% yield strength and the ultimate tensile strength of this material at the test temperature of 600 °C were about 1030 MPa and 1500 MPa, respectively.Electron backscatter diffraction (EBSD) was used to quantitatively γ grain size and the nature of grain boundaries. The pole figure is shown in . No significant texture was observed in the microstructure. The grain size distribution of FGH96 is shown in . As shown, the average grain size of the alloy excluding twin is about 23 µm. Annealing twin are prevalent in the microstructure. As shown in , the red lines represent ∑3 twin boundaries. No significant texture was observed in the microstructure.All fatigue specimens were machined from the same disk. The shape and size of the specimens are shown in . The specimen was rough polished using increasingly fine abrasive paper up to 2000 grit. The diameter of the specimen was further reduced by approximately 200 µm by electropolishing in order to minimize the effect of surface residual stress produced by machining, as well as to produce a “nominally-uniform” surface condition. Electropolishing was conducted using an electrolyte of 55% ethanol with 35% butyl cellusolve and 10% perchloric acid at 25 V and −20 °C.The fatigue tests were conducted at 600 °C using a SHIMADZU servo-hydraulic test system according to the test standard The fatigue life variability behavior of the FGH96 superalloy at four levels of stress is shown in . The mean lifetime is shown by black block symbols in the figure. It shows that the mean lifetime increases with decrease of stress level, varying from about 3×104 to about 3×106. The coefficients of variation (CV), a characterization parameter of fatigue variability, are shown in . CV is equal to standard deviation divided by the mean. The scatter will be significantly larger with a higher value of CV. As shown in , at stress of 900 MPa, the value of CV is highest. This result was similar to some other researches The fatigue lives were plotted on a probability plot in . Fatigue life variability is commonly described by the lognormal probability density function. However, at 1200 MPa, it was apparent that they did not correspond to a single lognormal distribution. A “Stair” phenomenon was observed. It means that they did not obey the linear lognormal distribution. This result was similar to the researches by Jha et al. In order to understand the distribution of the fatigue life of FGH96, the fatigue fracture surfaces of all the tests were investigated by SEM analysis. There were three types of fatigue initiation sites: (a) surface initiated; (b) sub-surface initiated; (c) internal initiated. As shown in , the fatigue initiation site of (b) was at the subsurface and partial crack growth phase was in the interior of the material. Internal initiation refers that the fatigue initiation site and the crack growth phase were both in the material interior. In the situation of (c), the fatigue lifetime was longer than that of the other two, because the crack growth occurred in vacuum-like environment. It has been shown that fatigue crack growth life for specimens tested in vacuum were longer than in air The distribution of fatigue crack initiation sites was shown in . As shown in the figure, the crack initiation sites were gradually changed from the surface to the interior with the decreasing of the stress level. At the higher stress level, σmax of 1200 MPa and 1100 MPa, a mix of surface and subsurface failure was observed. At the σmax of 1000 MPa, three types of initiation sites were observed, surface, sub-surface and internal. At the σmax of 900 MPa, all the tests were controlled by internal initiated mechanism. The fatigue lives increased with the increasing of distance from the surface of the initiation positions. All the fatigue data were divided into two districts: short and long fatigue lifetime district. The shorter lifetime district contained all the surface initiated data except one surface- initiated point at 1200 MPa, as shown in the black rectangular box in . Although its initiation site was on the surface, the lifetime was close to that of the sub-surface initiated at 1200 MPa. The long fatigue lifetime district includes sub-surface and internal initiation. S-N fatigue curves for short and long lifetimes were presented in . Although distinct surface and internal crack initiated regimes were found, obvious duality of the S-N curve was not seen.In order to investigate the roles of microstructures in the fatigue variability, all the fatigue fracture surfaces were analyzed. As shown in , all the fatigue failures occurred with inclusions or cleavage. There were no pore-initiated failures.The probability of occurrence of surface inclusions initiated is gradually reduced with decreasing of stress level, simultaneously the probability of occurrence of sub-surface inclusions initiated failure is increasing. At higher stress level, σmax of 1200 MPa and 1100 MPa, surface inclusions initiated mode was observed. At lower stress level, σmax of 1000 MPa, only sub-surface inclusions initiated failure mode was observed. At σmax of 1000 MPa, only internal inclusions initiated failure mode was observed. The fracture surfaces of surface sub-surface and internal initiated failures are shown in . The sizes of long axis of the equivalent ellipse of the inclusions are about 37 µm, 35 µm and 48 µm respectively, slightly larger than the average grain size.Inclusions at surface were the controlling microstructure of the life-limiting at higher stress levels. As shown in , the life of surface facets initiation, the point in the black rectangular box, is about 5 times larger than the worst surface inclusions initiation. This is the reason that why the life scatter was larger at 1200 MPa with a “step” characteristic. If there were no inclusions at the surface, the fatigue life would be extended and the fatigue variability would be reduced at higher stress level. At lower stress levels (1000 MPa, 900 MPa), the crack initiation occurred with internal inclusions. Internal inclusions initiated fatigue lives did not make the fatigue life variability larger at lower stresses., there were three kinds of crystallographic facet initiated failure modes: (1) surface facet initiated; (2) sub-surface facet initiated; (3) internal facet initiated. The probability of occurrence of internal facet initiation increases gradually with decreasing of stress level. is the typical characteristic of fracture surface of facet initiation failure. The initiation sites of facet-initiated fatigue failures contained a group of small crystallographic facets. With the decrease of stress level, the initiation location contains more facets. For the internal failure mode, “fish eye” fracture surfaces were observed. In the center of the “fish eye”, a bigger facet was surrounded by many small facets, as shown in (c). The radius of equivalent circle of the center facet is slightly larger than the average grain size. The equivalent circular areas of internal facet initiation at 900 MPa are shown in . The area decreases with the increase of fatigue life. The random combination of grains caused local inhomogeneity resulting lager life scatter.Fatigue experiments of FGH96 were conducted at elevated temperature. SEM analysis was performed to investigate the mechanism of fatigue variability. The following main conclusions can be drawn:Three competing failure modes were observed, surface, sub-surface and internal initiated failures. At higher level of stress, failures were controlled by surface mechanism. At lower level of stress, fatigue failures were controlled by internal mechanism. At the middle stress level, a mixture of surface, subsurface and internal initiated failures was observed.For FGH96 at 600 °C, there were six kinds of fatigue crack initiation modes: (1) surface inclusion initiated; (2) surface facet initiated; (3) sub-surface inclusion initiated; (4) sub-surface facet initiated; (5) internal inclusion initiated; (6) internal facet initiated.Evaluation of microstructural features of the fatigue initiation sites indicated that inclusions were the life-limiting of high levels of stress. The existence of the inclusions resulted in large dispersion of the high stress levels. Sub-surface and internal inclusions initiated fatigue failures did not make the fatigue life variability larger at lower stresses.Heterogeneity of material caused by random combinations of grains was the main cause of fatigue variability at lower stress levels.Application of texture simulation to understanding mechanical behavior of Mg and solid solution alloys containing Li or YThe viscoplastic self-consistent model was used to interpret differences in the mechanical behavior of hexagonal close packed magnesium alloys. There are only subtle differences in the compression textures of magnesium and its solid solution alloys containing lithium or yttrium. However, the plane strain compression textures of the alloys showed an increasing tendency for the basal poles to rotate away from the “normal direction” towards the “rolling direction”. Texture simulations enabled these distinctions to be attributed to the increased activity of the non-basal slip mode. The alloys had improved compressive ductilities compared to pure magnesium, and the increased slip mode activity provides a satisfying explanation for this improvement, since it can accommodate c-axis compression within individual grains. Accounting for individual deformation mode hardening enabled the flow curves to be simulated and the anisotropic plastic response of textured wrought alloys to be mechanistically understood and predicted.Research on magnesium alloys is currently enjoying a renaissance due to increased interest in application in the transportation industry for improving fuel efficiency through vehicle weight reduction (e.g. Polycrystal plasticity simulation may be used to help determine the physical processes that cause different alloys within a class to exhibit distinct deformation behavior. The basic premise is outlined in the following points:crystalline solids deform by mechanisms of simple shear, namely dislocation glide and mechanical twinning;these simple shears yield a crystallographic rotation as well as a strain; and, therefore,crystals within a polycrystalline aggregate tend to rotate to preferred orientations reflective of the symmetry of the deformation modes of the crystals and the imposed deformation.In order to understand the distinctions in the plastic behavior of alloys, the following points must also be considered:alloying can change the relative strengths of the contributing deformation modes (by inducing changes in the lattice parameters and dislocation dynamics); as a consequencethe contribution to deformation of the various deformation modes will be altered; thusthe crystallographic texture after deformation may be altered.It is suggested that this indirect approach for interrogating the microstructure may yield new understandings, which were formerly unattainable relying solely upon the traditional experimental techniques of single crystal slip trace analysis and TEM based dislocation analysis. This approach has the advantage that it relies entirely upon experimental data obtained from polycrystals, which is invariably the case of greatest interest. Furthermore, the approach yields statistically relevant information, as opposed to TEM studies, which often rely heavily upon scattered observations of dislocation images that represent “hard” deformation modes. This work seeks to develop further understanding of the operation of secondary deformation modes, such as pyramidal When compared to crystal systems like face-centered or body-centered cubic, hexagonal close packed (hcp) metals exhibit a wider variety of deformation textures. Historically, hcp metals have been categorized in terms of c/a ratio and the observed rolling textures were similarly categorized. The distinct textures are actually due to the combined effects of c/a ratio and the fact that different hcp metals deform by different slip and/or twinning modes (e.g. beryllium, titanium, zirconium, magnesium), the {} twin is activated by c-axis tension. During compression, grains are favorably oriented if their c-axis is perpendicular to the compression axis; and twinning reorients the c-axis of the twin nearly parallel to the compression axis. This is a major reason why basal or near-basal textures are common for cold-rolled hcp metals, as has been documented by Phillippe and collaborators } twinning mode is activated by compression along the c-axis slip is active. Hence, the common tendency for hcp metals to develop a basal compression texture is precluded in the case of Zn. Finally, it is often suggested that Mg will develop an ideal basal texture, since it has a nearly ideal c/a ratio of 1.624 During the past two decades the simulation of texture has become an established technique, due to advances in microcomputer technology and in plasticity modeling (relaxed constraints Noncubic materials typically have a limited number of primary slip systems, which do not satisfy the Taylor criterion for generalized strain (5 independent slip systems). In other words, the single crystal yield surface is highly anisotropic. “Hard” secondary systems are only observed to be active in regions of stress concentration, such as grain boundary triple points or dislocation pile-ups.Noncubic crystals show a greater tendency to mechanically twin than cubic materials. Mechanical twinning may relax the requirement for 5 independent slip modes Methods of deformation texture simulation are summarized elsewhere (e.g. Kocks et al. A point often overlooked when doing polycrystal simulations of plastic deformation concerns the macroscopic hardening response of the aggregate. This response depends on the evolution of the CRSS associated with individual deformation modes. As a consqeuence, it reflects the interaction and hardening of slip and twinning systems. Correctly reproducing the macroscopic hardening behavior is, therefore, as important as correctly reproducing the final texture of the aggregate. In this paper, we focus on deriving such a comprehensive understanding of Mg alloys deformed in compression.The Mg–Li system is of great interest because it offers the potential to substantially lower the density (ρLi=0.58 g/cc vs ρMg=1.74 g/cc) and improve the room temperature ductility of magnesium. With above 5 wt% Li, a duplex alloy containing the body centered cubic β-phase is obtained, and above 11 wt% Li the alloy is entirely βThe Mg–Y system is representative of another class of magnesium alloys, including a variety of rare earth additions, such as Ce and Th. It has been noted that small additions of rare earth elements and thorium, in particular, improve the ductility of magnesium Magnesium solid solution alloys with different amounts of lithium and yttrium () were cast in a glove box with a cover gas (Ar+1.5%SF6+0.5%O2). A resistive furnace was used to melt the alloys in mild steel crucibles at 750 and 850°C for additions of lithium and yttrium, respectively. The castings were made in a 25 mm diameter × 75 mm long copper mold.Microstructural characterization was performed on all of the alloys in the as-cast and thermo-mechanically processed conditions. Standard sample preparation techniques were used, including polishing to a 0.3 μm finish and etching with either a dilute (3%) solution of nitric acid in methanol or a picral–acetic etch ) to obtain grain sizes between 25 and 100 μm. In addition to similar grain size, a nearly random texture was also desirable to make direct comparisons between the deformation textures of the different alloys later in the study. Originally, it was observed that the as-cast pure magnesium developed a very strong basal texture when compressed 20%, so a process involving small upsets in three orthogonal directions was used. The resulting grain sizes for the different alloys are shown in . Three of the alloys had as-cast grain sizes within the targeted range, so only a homogenization anneal was performed in those cases.Computer controlled servo-hydraulic load frames were used to produce a uniform microstructure by cold-working, for deforming the alloys for deformation texture comparisons, and for mechanical testing. After preparing the alloys as specified in , samples were deformed by uniaxial and plane strain compression, to a true strain of approximately 0.3, in order to develop deformation textures. Plane strain compression was performed in a channel die with a width of 12.7 mm, on samples 3 mm thick and approximately 25 mm long. Teflon tape was used as a lubricant between the samples and the channel die. To determine the mechanical behavior of the alloys, electro-discharge machined (EDM'd) compression samples with an aspect ratio of 1.5 (φ6.4 × h9.6 mm) were used. Additionally, tensile samples were EDM'd with a reduced section of 25.4 × 3.4 × 3 mm, shoulder radii of 3.4 mm, and a grip section 10.2 mm wide. A clip-on extensometer with 12.7 mm gage length was used to measure the strain. All of the tests were performed at constant crosshead speeds, which gave a nominal initial strain rate of 5×10−3 s−1.The crystallographic texture was measured with X-ray diffraction using the reflection method Bragg peaks. The pole figures were measured for one second at each location in 5° tilt steps from 0 to 80° and azimuthal steps of 1.667° over the entire 360°. The azimuthal data were averaged every 5° for the subsequent analyses. Corrections for defocusing of the peak and background intensities were made by experimentally measuring pole figures from a fine grained random sample and developing a correction curve. Complete orientation distribution functions were calculated from the experimental pole figure data using the WIMV method of the Preferred Orientation Package of Los Alamos (popLA) The polycrystal plasticity modeling approach was used to simulate the observed textures by varying the CRSS values of the possible deformation modes (). Guidance in selecting the CRSS values for each deformation mode was obtained from single crystal literature. For instance, the ratio of the CRSS for basal and tensile twinning is on the order of 1:2 glide is on the order of 1 to 6 at 200°C, and higher at lower temperatures slip and compression twinning are obviously much harder than the primary modes of basal slip and tension twinning. Because the models for twinning are less developed, the initial simulations were performed without compression twinning. The strengths of the different deformation modes were normalized to and the only “fitting parameter” adjusted to match the experimental textures was the relative Each of the deformation modes were assumed to be rate insensitive . However, the ability to make an unambiguous determination of active systems afforded by a rate sensitive model is retained. VPSC contains a viscoplastic formulation of the inhomogeneous ellipsoidal inclusion embedded in a homogeneous effective medium (HEM). The relationship between the strain rates and stresses within the inclusion and the surrounding aggregate is given bywhere the superscript c denotes the strain rate or stress within the inclusion and the overbar indicates the average value within the HEM as a whole. The accommodation tensor, , is a function of the secant compliance tensor of the HEM at the stress , which is related to the shape of the ellipsoid. is the identity matrix and neff is a parameter related to rate sensitivity that allows the strength of interaction between the inclusion and the HEM to be tailored. (neff=0 would produce a rigid upper bound solution, while neff=n would result in a softer tangent self-consistent approximation.) In the current study, the value of the interaction parameter was set to 10, rather than n, reflecting a somewhat stiffer interaction between the individual grains and the homogeneous effective medium The contribution of twinning to the texture evolution was treated according to the predominant twin reorientation (PTR) scheme is the characteristic shear of the twinning mode, t). In addition, a sum of the twinned volume fractions is made over all of the crystallites,After each straining step, the crystallites are polled randomly to test whether or not the volume fraction of the predominant twinning system exceeds a threshold value. If so, the entire crystallite is reoriented to a new orientation reflective of that twinning system and the volume fraction of this reoriented crystallite is then added to a quantity called the “effective twinned fraction” or Veff,mode. This polling process continues until either all the grains are checked in a random order or the value of Veff,mode exceeds the total accumulated twinned volume fraction, Vacc,mode. The aforementioned threshold value typically starts at 10% and increases rapidly, saturating at 50%. This approach statistically solves the practical problem of monitoring a continually growing number of orientations that would develop if each twinning system activated within each grain was represented by a new orientation. Additionally, it maintains the twinned volume fraction at a level that is consistent with the shear activity of the twins contributing to the deformation.Texture simulations can broadly identify the appropriate ranges of the relative slip system strengths; however, no attempt was made to account for the roles of self or latent hardening during the texture simulations. More exacting tests of the model are to simulate the polycrystal's stress–strain response during plastic deformation and θs are CRSS values and hardening rates for the slip system s, and Γ is the accumulated strain within the orientation. In order to minimize the number of parameters to be fit, only two limits of the equation were employed in this work: the simple Voce law (the final hardening rate θ1 set to zero) and linear hardening (both τ1 and θ0 are set to zero). The actual hardening of each mode is accomplished through the following equation at the end of each straining step, which allows for the incorporation of latent hardening effects.In other words, the incremental increase in the CRSS of the slip system s, Δτs, depends on the amount of shear strain accumulated by each slip system s′. The latent hardening matrix, hss′, allows the strength of interaction between different slip systems to be modeled. Again, with a view towards simplicity, all the values of hss′ were set equal to 1 during this study except for the following case: if s′ is a twinning mode hss′=Y, a value greater than 1. This reflects the fact that twin boundaries will introduce potent barriers to subsequent slip and twinning.The simulations were conducted on aggregates of 1000 discrete orientations representative of the initial texture. The assumption of a random starting texture was justified in most cases and was made throughout this study, unless stated otherwise. The simulations were conducted for both uniaxial and plane strain compression boundary conditions and were performed using incremental straining steps of 0.01 von Mises equivalent true strain up to a level of strain dictated by the experiments that were being modeled.In the as-cast and homogenized conditions, all of the alloys had random textures. After uniaxial compression to a true strain of approximately 0.3, the alloys exhibit a slightly off-basal texture. The inverse pole figures in show the peak texture component is rotated away from the basal pole by ∼20°. The primary deformation modes of basal slip and tension twinning would generate a sharp basal texture. Therefore, other modes are necessary to explain the final off-basal components. These results highlight the importance of further developing our understanding of secondary deformation modes in hcp magnesium alloys, but they are insufficient to demonstrate significant alteration of the slip behavior due to alloying.The plane strain compression textures show distinctions between the different alloys. Although there is a tendency to align basal poles with the “normal direction” (ND) there is a spreading of the basal poles towards the “rolling direction” (RD) that increases from pure Mg to the Y alloys to the Li alloys (). In the case of the 3 wt% Li alloy, the basal poles have clearly rotated away from the ND towards the RD by ∼20°. demonstrate that the alloys under investigation are more ductile (in compression) than pure magnesium. The ultimate strengths were all greater as well, despite moderate hardening rates. All of the samples failed catastrophically along a single shear-type crack. The lithium containing alloys continued to harden up to failure, indicating a resistance to plastic instability that is critical for forming applications.The flow curves for the 3% Li, 1% Y, and 3% Y show a similar appearance, with three distinctive stages of deformation. First, there is a rapid increase in the flow stress during the elasto-plastic transition, followed by a sharp transition to a lower hardening rate. The second stage, which extends to about 10% strain, is characterized by a slightly increasing hardening rate typical of twinning activity and the creation of barriers to the propagation of dislocations The flow curves of the 5% Li alloy and of pure Mg show a similarity during the initial stages. Both exhibit rapid hardening up to a rather high stress before undergoing a single transition to lower hardening and flow softening, respectively (). Neither material shows the twinning signature on hardening discussed above. Only these two alloys required grain refinement after casting and homogenization in order to fall within the objective grain size range (). As a result, they retained a weak texture after cold-work and recrystallization, which had a significant impact upon their compressive flow curves. Indeed, testing of the as-cast and homogenized (albeit coarse grained) material exhibited mechanical behavior quite similar to the other three alloys (). In fact, the Mg–5 wt% Li alloy had a flow curve nearly identical to that of 3 wt% Li (see ). The very coarse grained pure Mg samples showed substantially more variability than the others tested during this study, so multiple test curves are shown. The marked twinning signature in the hardening response is to be expected, since twinning is favored in coarse-grained aggregates. further highlights the distinctions in the hardening behavior observed during the compression testing of the different alloys. It is obvious from the pure Mg flow curve that the initial yielding of these alloys occurs at an extremely low stress, making it impossible to observe a purely elastic portion of the stress–strain curve. This behavior is reflective of the easy activation and motion of basal dislocations, as observed recently by Gharghouri et al. Based upon the texture simulations performed, the following conclusions can be made about the influence of CRSS ratios on compression textures (either uniaxial or plane strain).If only basal slip and tension twinning are active, a sharp basal texture results ( slip will allow c-axis compression and, hence, the basal poles may rotate away from the compression axis. The CRSS ratio of basal to slip will determine how far the basal poles rotate away from the compression axis (e.g. towards the “rolling direction” in plane strain compression). This ratio was the main parameter adjusted to simulate the experimental textures (cf. Prismatic slip generates rotations about the basal pole without reorienting the latter. Therefore, accommodating deformation with extensive prismatic slip leads to weaker basal texture development. For instance, including prismatic slip during simulations of uniaxial compression can stabilize orientations with basal poles in the transverse direction if the CRSS is lower than that of tensileon twinning, although this is unlikely to be the case for cold deformation of magnesium. If prismatic slip is similar in strength or weaker than slip, a ring will form about the normal direction during plane strain compression (). If the CRSS for prism slip is higher than that of slip, prismatic slip will be nearly inactive. suggest that the significant difference between the experimental textures of the alloys in this study is the activity of the slip mode. In other words, the tendency to split the basal intensity towards the rolling direction is enhanced primarily by an increase in the activity of slip (i.e. reducing X from 12 to 3), rather than prismatic slip. If the role of prismatic slip becomes significant, there are features in the simulated texture not observed experimentally, such as the “ring” of intensity in the basal pole figure centered about the normal direction.In order to demonstrate that this approach includes the necessary physics and is useful for predictive modeling of material behavior under arbitrary loading conditions, the following test case was modeled. A hot-rolled AZ31B (Mg–3 Al, 1 Zn, 0.2 Mn in wt%) plate was tested in compression along the rolling direction (in-plane) and along the normal direction (through thickness). A single set of hardening parameters was obtained to simulate the plastic response of each of these cases using the VPSC model (see ). The material used in this study had a simple basal texture with a peak intensity of 4 times random along the normal direction (see inset in The presence of an initial texture can give rise to anisotropic and/or asymmetric mechanical behavior. As shown in , the through-thickness yield strength in compression is approximately twice the compressive yield strength along the rolling direction. Throughout the magnesium literature, this phenomenon is typically expressed as a yield strength asymmetry along the rolling (or extrusion) direction where the strength in tension along the rolling (or extrusion) direction is often twice the compressive yield strength. From a textured polycrystal perspective, such an asymmetry can be clearly connected to the single crystal yield strength asymmetry associated with twinning: the relative alignment of the compressive axis and the main basal component determines whether or not tensile twins will be activated. From the point of view of modeling, the textured aggregate case study is useful because it allows the parameters of certain deformation modes to be tuned more independently than the random samples allow. For instance, the basal texture of the AZ31 plate is such that through-thickness compression tests preclude either extensive tensile twinning or basal slip, therefore slip parameters may be tuned using this configuration (). Likewise, the grains of in-plane compression samples are again unfavorably oriented for basal slip, but tension twinning is favored. The tensile twinning mode is responsible for the low in-plane compressive yield stress. The final hardening parameters employed are given in The AZ31B experimental data reflects the fact that deformation twinning may introduce strong barriers to subsequent slip and twinning. During in-plane compression, a large amount of twinning during the early stages of deformation () leads to a higher final flow stress than that observed in the through-thickness compression test, even though the initial yield stress of the latter was much higher (). This effect was modeled by setting the latent hardening parameter of the twins to be higher than that of slip. The hss′ value which accurately simulated this effect was Y=1.4. The same latent hardening parameter was employed during the subsequent modeling of the experimental alloys.For the case of the Mg–Li alloys, Mg–3 wt% Li was chosen as representative (the single crystal hardening parameters are reported in shows the stress–strain profile for both the “best fit” simulation as well as the case of no hardening, which was employed during the modeling of texture. In general, it can be observed that tension twinning is restricted to the first few percent of deformation and then there is rapid hardening of the basal slip mode and a gradual increase in the activity of the slip mode. Two specific points may also be made from these simulations. First, even the case of no individual mode hardening shows some variation in the flow stress and this directly reflects the role of texture upon hardening. Secondly, incorporating hardening allows more realistic evolution values of the CRSS to be employed. For example, the starting CRSS ratio for slip is 1:8, while the final value is close to 1:3 appropriate for modeling the major texture features. This has the effect of delaying the onset of slip and actually improves the texture simulation by allowing basal poles to accumulate about the normal direction. The simulations that did not include hardening allowed slip to clear out all orientations with basal poles near the normal direction (For the Mg–Y alloys, Mg–1 wt% Y was chosen as representative (). In this case, the transition from the initial rapid hardening to a nearly saturated flow stress profile is not gradual like the previous two alloys, AZ31 and Mg–3% Li. In fact, there is evidence of a small peak at the transition in hardening (). Conventional wisdom regarding a peak followed by saturation suggests some dynamic recovery process, however, this test was conducted at room temperature like all the other tests in this study. The gradual transition to a lower hardening rate at about 10% strain exhibited by AZ31 and Mg–Li was modeled effectively by incorporating pyramidal slip with a Voce hardening law. This approach was not effective for modeling the behavior of Mg–Y. Instead, modeling suggests that tensile twinning is responsible for the initial rapid hardening and, afterwards, compressive twinning is required to maintain a steady contribution to deformation from basal slip and tensile twinning ( slip mode with a reduced axial ratio, c/a. shows that the change is due to a decrease in c, not an increase in a. Therefore, the Peierl's resistance should increase for basal slip and this was verified experimentally by Quimby et al. Although non-basal slip had already been observed in pure Mg slip planes. This latter solution is a theoretically more satisfactory explanation for the improved ductility of Mg–Li alloys over pure Mg because the slip mode satisfies the von Mises criterion by itself, while the basal and prismatic modes are still insufficient to accommodate strains along the c-axis.In a recent study of Mg–Li single crystal deformation at room temperature and below by Ando and Tonda slip by lithium additions was found. Single crystals were strained in tension along a direction. In this orientation, the Schmid factors for slip modes are nearly identical (0.449 and 0.433, respectively). Therefore, the fact that slip traces only indicated slip is strong evidence that its CRSS is lower than that of prismatic tensile twinning was effective for simulating the plastic response of AZ31 and Mg–Li, the rapid saturation of the flow stress exhibited by Mg–Y requires another approach. One possibility is that another twinning mode is playing an important role in the deformation of this alloy. In order to simulate the flow curve of this alloy, the compressive twinning mode was employed. This twinning mode has often been observed to occur in Mg and its alloys twin mode reorients crystals such that they may continue deforming by the soft modes of basal slip and/or second order tension twinning. After a preliminary modeling attempt (see ), the rapid saturation of the flow curve can be reproduced due to the sudden onset of compression twinning followed by increased activity of the soft modes. Like slip, compression twinning has the effect of rotating the basal poles towards the “rolling direction”. However, no serious attempt has been made to concurrently match the flow curves and experimental textures. In the future, experimental assessments of this twinning mode's activity will be made using electron backscattered diffraction and/or transmission electron microscopy.The use of qualitative observations of deformation texture to extract information about slip mode activity is not new. In fact, such an observation was made in the written discussion section of the original report by Hauser et al. As verified by the current simulations, basal slip and tension twinning alone will cause a strong basal texture to develop during plane strain compression. Secondary slip or twinning is necessary to rotate the basal poles away from the sheet normal direction. Couling twinning and basal slip within primary compression twins. The current explanation, based on activity has not, to the authors' knowledge, been proposed before.The qualitative observation of a “Ti or Zr-type rolling texture” after hot strip extrusion of Mg–14.8 at% Li at 425°C was proposed as evidence of increased prismatic slip in Mg–Li alloys, since the primary deformation mode in Ti and Zr is prismatic slip has been shown to be a thermally activated deformation mode in magnesium slip mode may be more important than any possible enhancement of prismatic Another mode frequently referred to in the magnesium literature is the pyramidal . However, there is experimental evidence that the prismatic slip could also be induced by a combination of basal and prismatic slip. Hence, it seemed judicious to reduce the number of deformation modes under consideration during our study.This study demonstrated the utility of using polycrystal plasticity models as a means for interpreting the differences in the mechanical behavior of different alloys, in particular those with a noncubic crystal structure. Differences in the deformation textures of alloys are strong evidence that the balance of deformation mode activity was altered by alloying additions. The results of the study are outlined below.There are only subtle differences in the uniaxial compression textures of magnesium and alpha solid solution alloys containing lithium or yttrium. However,neither the alloys nor the pure base metal exhibit the ideal basal texture commonly ascribed to magnesium subjected to compressive type deformation. Rather, the basal poles are rotated away from the compression axis by ∼20°, which suggests the activity of some secondary deformation mode(s).The plane strain compression textures of the yttrium and lithium alloys showed an increased tendency for the basal poles to rotate away from the normal direction towards the rolling direction. The simulations of the deformation textures using a self-consistent viscoplastic polycrystal plasticity model enabled the distinctions in the textures to be attributed to the increased activity of the nonbasal All of the alloys tested (3 and 5 wt% Li, and 1 and 3 wt% Y) had dramatically improved compressive strains to failure over pure magnesium. Although the strengths of these binary solid solution alloys were modest compared to commercial wrought alloys, they all were stronger than pure magnesium. The increase in slip mode activity helps explain their improved ductilities since c-axis compression can be accommodated.The macroscopic flow curves of the alloys could be simulated by including individual deformation mode hardening, within each grain. Simultaneously reproducing the flow curves along with the deformation textures is a stringent test of the model. The result is more reliable parameters, which enable us to mechanistically understand and predict the anisotropic plastic response of textured wrought alloys.The main conclusion of this study, that Li additions to Mg enhance the activity of the pyramidal slip mode, is corroborated by a recent investigation of the dislocation structures in these alloys using transmission electron microscopy Ductile necking behavior of nanoscale metallic glasses under uniaxial tension at room temperatureGlasses are normally brittle materials with no tensile ductility at room temperature. Using in situ, quantitative nanomechanical tests inside a transmission electron microscope, we demonstrate that certain nanoscale metallic glass samples are exceptions to this general rule. Such metallic glasses can be intrinsically ductile, capable of elongation and necking under uniaxial tension, in lieu of catastrophic fracture caused by severe shear banding. Beam-off tests confirm that the ductile behaviors are not artifacts due to electron-beam effects during the in situ tests. Additional experiments indicate that ductile necking gives way to fast shear banding failure at increased samples sizes and elevated strain rates. The observed spread-out shear transformations delaying strain localization and severe shear banding are explained in terms of the propensity for participation in deformation, while the tendency towards necking is attributed to the lack of strain hardening mechanism and inadequate strain rate hardening.Monolithic glasses are generally brittle at room temperature, displaying cleavage fracture upon deformation. Their plastic elongation in uniaxial tension is practically zero. Metallic glasses (MGs), however, are projected to be possibly different, as their non-directional metallic bonds and densely packed atomic structures may tolerate profuse local shear transformations that can mediate plastic flow without failure Therefore, one possible way for MGs to be capable of exhibiting some tensile ductility is to promote spread-out shear transformations in lieu of severe shear banding. Molecular dynamics (MD) simulations Whether the two conditions above would indeed render MGs ductile in tension has not been settled experimentally so far. Although there have been several reports of significant tensile ductility due to the suppression of catastrophic shear bands in nanoscale samples (e.g. Refs. The Cu49 Zr51 MG tested was prepared using melt spinning. A schematic of the tensile sample and tungsten grip is shown in . Tensile samples with nominal diameter (defined as D
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≡
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A1/2, where A is the cross-sectional area) ranging from 70 to 120 nm were fabricated using FIB. This is the sample size regime where tensile ductility of MGs may possibly be present The uniaxial tensile tests were carried out using the quantitative capability of a Hysitron PI95 TEM PicoIndenter For metallic glass samples with larger diameters, the dominant deformation mode under tensile loading is generally shear banding. One typical example is shown in a displays a stress–strain (σ–ε) curve of a sample with D
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= 122 nm (aspect ratio ∼5, b), pulled at a strain rate of ∼1.5 × 10−3
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s−1. The curve shows that, for the entire deformation process, the stress increased approximately linearly with strain, before fracturing at an ultimate tensile stress of ∼3.2 GPa. The fractured end, as seen in the projection view in c, exhibits a straight edge that is inclined at an angle of θT= ∼55° to the length direction of the sample (similar to the θT reported for bulk Cu–Zr based MGs a is caused by the instability of the system in the transient right after shear banding has occurred. The system experienced unexpected displacement jump when the instability set in. The force turned negative when the system forced the tip to go back to the programmed position.However, when the sample size was reduced further, a significant change in the deformation mode was observed. illustrates the deformation behavior of a D
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= 80 nm sample, in situ monitored during tensile pulling at a strain rate of ∼1.3 × 10−3
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s−1. With the e-beam continuously on the sample throughout the test, we were able to videotape the entire elongation process (see ), and several snapshots from the movie are displayed in a–g. Rather than shear banding, clear and gradual necking is observed, starting at a strain around 4.6% (see c). After the eventual fracture, the fractured region displays a cone-like shape, typical of ductile metals that have experienced necking in a uniaxial tensile test, as shown in h. Our high-resolution TEM observation (i), as well as the corresponding electron diffraction pattern of the fracture surface (inset of i), finds no obvious crystallization in the fractured sample. The engineering stress–strain curve of this sample is shown in red in . There is a clear indication of the non-uniform strain during the necking stage (beyond the stress peak), and the total elongation to failure is about 10%. These findings are consistent with the observed morphological evolution in that directly correlates the stress-strain curve with the sample morphological changes (Movie 1 in Appendix A, obtained from the video tape recorded during the in situ test).Two additional samples, with D
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= 82 nm and D
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= 86 nm, were tested under exactly the same conditions. Their stress–strain curves are also displayed in in green and blue, respectively. Upon loading, all three samples exhibit a similar initial slope (Young’s modulus ∼80 ± 10 GPa), consistent with previous reports of this MG ; these will be discussed in more detail later.Our experiments confirm that the size of the sample is indeed important for observing tensile ductility. Apparently, a smaller D is necessary for observing the ductile necking behavior in this Cu–Zr-based MG. This trend is consistent with that reported earlier for a similar transition We next consider the possible existence of the electron-beam-introduced artifacts discussed earlier. We should first point out that the e-beam effect depends on the actual electron beam density used, as well as the structure and property of the sample. Note that the e-beam current density used in our tests, approximately 2 × 10−2
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A cm−2, was within the range that would be expected for regular imaging in a TEM. Because of the relatively low e-beam intensity and the good thermal conductivity of the metallic samples, the plastic flow we observed is unlikely to be caused by e-beam heating, or by the radiolysis effects that are significant in the case of covalently bonded ceramics Specifically, three samples, with D
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= 80, 81 and 73 nm, were tested with the beam-off condition at a strain rate of ∼1 × 10−3
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s−1, and their stress–displacement curves are shown in a. In this figure displacement from the loading apparatus is plotted instead of the strain actually experienced by the sample (gauge length), because it was not possible to monitor the latter accurately in the gauge section under beam-off conditions (there was no movie to record the elongation of the marker-specified gauge length along with pulling). It should be noted that the displacement recorded by the PicoIndenter assembly contains not only the elongation of the gauge length but also the elongation outside the gauge length (e.g. the contact interface), which would vary from test to test, even for the nominally same testing conditions. In other words, every sample/test is different, such that the displacement is not the same for any given stress level. This is one of the main reasons why the three curves in a do not have the same appearance. Negative stress will be generated during the unloading process when the two fractured parts meet each other again (a). For well-aligned samples, the displacement corresponding to the fracture point should be the same as that where the negative stress occurred during the unloading process. This feature can be used to confirm if the sample under testing is aligned well with the tungsten grip. A side note here is that the stress–displacement relationship of the D
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= 73 nm sample (bluea) has a curved shape in the early stage of straining. This is very likely the result of some misalignment, as was evidenced by the fact that the displacement corresponding to the fracture point is different from that of the occurrence of the negative stress during the unloading process.Even though the displacements are inaccurate for the aforementioned reasons, the engineering stresses measured under the beam-off condition were as accurate as those measured under the beam-on condition. As shown in a, the peak stresses achieved under the beam-off condition are also at σ
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∼ 2.5 GPa, similar to those reported under the beam-on condition (see ). In addition, post-mortem TEM observation found that all three samples had similar cone-like fracture geometry. A typical example is shown in b. The initial diameter of this sample was 80 nm. Similar to those achieved under the beam-on condition, no indication of crystallization was seen (The fact that the shape of the stress–strain/displacement curves, the magnitude of the peak stress, the fractured sample geometry and the microstructure of fractured region are not different for the beam-on and beam-off conditions suggests that the ductile behavior we observed is not due to the electron beam illumination, but is an intrinsic property of the as-fabricated MG samples.However, we did find that an excessive e-beam can indeed alter the mechanical behavior of the as-fabricated samples in an obvious way. One typical example is shown in . Two samples, with D
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= 118 and 124 nm, were tested at the same strain rate 1.0 × 10−3
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s−1. Using the normal imaging condition with an electron current density of 2 × 10−2
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A cm−2, the sample with D
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= 118 nm failed at a major shear band, similar to the sample shown in . For the sample with D
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= 124 nm, the beam current density used was even larger than 0.1 A cm−2, at least five times the density used before. Both the stress–displacement curve (red curve in a) and the geometry of the fractured end (c) demonstrate considerable plastic flow. This is in sharp contrast with that observed under regular imaging conditions, under which samples of this size would only deform (and fail) via shear banding. The mechanism of e-beam enhanced plasticity in this case needs further study, but likely involves beam-induced heating, surface diffusion and irradiation damages. In any case, our work indicates that, while the normal imaging e-beam density, e.g. 2 × 10−2
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A cm−2 or less, appears to be safe (as shown above), caution should be exercised to avoid an excessively high beam intensity. demonstrate that MG samples with nanometer diameters can elongate and neck to an extent reminiscent of ductile metals, rather than quickly fracture at the onset of plastic deformation due to severe shear banding. This supports the assertion at the beginning of this article that at least certain MG samples (in our case, the nanoscale samples that have been subjected to FIB processing) are indeed intrinsically ductile in uniaxial tension. In other words, as long as the MG internal structure contains adequate fertile sites that can be activated to undergo shear transformations, ductile and necking behavior can be expected and are in fact observed, as predicted in previous molecular dynamics simulations of rapidly quenched MG samples under uniaxial tension Another property revealed by the quantitative tensile tests is the large elastic strain achievable in the nanoscale MGs; this aspect has been discussed in detail in an earlier publication One additional factor we discovered to favor ductility is that the strain rate needs to be sufficiently slow, such that the shear transformations throughout the sample can catch up with the imposed displacement rate. Conversely, even for moderately increased strain rates, the deformation mode shifts towards shear banding. This trend is demonstrated in with a series of TEM micrographs showing the fracture morphology of samples deformed at various strain rates. For example, for the samples with D
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= ∼80 nm, the shear banding mode became controlling when the strain rate was increased to >∼2 × 10−3
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s−1. Moreover, even for “necking samples”, i.e. samples with stress–strain curves featuring peak stress and necking down, those tested at relatively higher strain rate tend to end with shear-like fracture (e.g. samples with D
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= 86 nm and D
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= 81 nm in suggests a correlation, i.e. a higher strain rate favors a higher likelihood of shear banding (at least involvement of shear fracture near the end of the tensile test). This is consistent with predictions by computer molecular dynamics simulations We note here that, unlike most conventional metals, when an MG exhibits ductile behavior, necking starts very soon after the onset of plastic deformation. As seen in the stress–strain curves and TEM videos (see )/images, the subsequent plastic strains concentrate in the necking region till the failure of the sample. This indicates a high propensity for the geometric instability in uniaxial tension, even though the more severe shear localization mode, shear banding, has been abated. The necking instability is presumably due to the lack of a strain hardening or strain rate hardening mechanism in MGs.In crystalline metals, dislocation multiplication and accumulation provides obstacles for dislocation motion, and thus a potent microstructural strain hardening mechanism. In the amorphous MGs, in contrast, the shear transformations generate disordering and excess volume, catalyzing further deformation in the already deforming region. This lack of strain hardening promotes strain localization – in our case, necking during tensile elongation – even though spread-out shear transformations delocalize the severe plastic instability in the form of narrow shear bands.At elevated temperatures under low stresses and low strain rates but ample thermal activation, due to the presence of significant strain rate hardening, viscous flow in a glass can sustain large and uniform strains Finally, we note that our nanoscale samples, like all the previously used small samples that showed tensile ductility In summary, tensile ductility can indeed be realized in a metallic glass at room temperature. Our in situ experiments have successfully demonstrated the gradual necking morphology as well as the characteristic stress–strain curve in a quantitative manner. The findings are also corroborated by tests in the complete absence of electron beam irradiation/heating effects during straining. Our results indicate that elongation and necking are an intrinsic deformation mode, favored over severe shear banding and rapid fracture, for the Cu–Zr MG under certain sample/test conditions. The necessary conditions favorable for a glass to exhibit ductile behavior include small sample size, amorphous structure with atomic configurations prone to shear transform (possibly facilitated by FIB-induced disordering and generation of excess volume) and relatively slow strain rate, all of which are in line with the trends expected from prior predictions/simulations. Necking comes in rather early during the ductile elongation, in lieu of severe shear banding, as a manifestation of strain localization due to the absence of microstructural mechanisms for strain hardening and strain rate hardening. Increasing the sample size and strain rate tend to change the deformation mode from necking to shear banding. Our study emphasizes that several important factors need to be carefully considered and controlled simultaneously when drawing conclusions about the ductile behavior of metallic glasses.Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.actamat.2013.05.001In situ tension test of 82 nm diameter Cu–Zr metallic glass inside transmission electron microscope. There are several occasional jerky shifts in the video which are due to discharging off the sample holder, with no mechanical movement in the sample assembly, as confirmed by the smooth stress–strain record.Second-order elasticity of soft multilayer capsules: Universal relations and parametric studiesComposite gels are of key interest in bioengineering, e.g., as replacement tissues and substrates for stem cell cultures. Although gel elasticity plays a pivotal role in many bio-functions, there is insufficient knowledge on the effect of elastic nonlinearity, multilayer configuration and chemical concentration. To address such issues, multilayer capsules subjected to dilatation fluctuations are investigated using second-order elasticity, validated by universal curves for uniaxial tension and swelling of homogeneous gels. Detailed simulations show that the material inhomogeneity, elastic nonlinearity, shell thicknesses and fluctuation profile all have significant influence on the elastic fields, and hence may be selected to induce favorable microenvironment such as a tensile or compressive stress state appropriate for the targeted functionalities.Many biological processes are significantly influenced by mechanical forces. Recent works have shown that stem cell lineage specification, morphology, proliferation, adhesion, apoptosis and other properties are extremely sensitive to the elasticity of the cell microenvironment. For instance, human mesenchymal stem cells (hMSCs) cultured on polyacrylamide gels mimicking the elasticity of brain, muscle and osteoid collagen indicated the greatest expression of neurogenic, myogenic and osteogenic transcripts on these gels, respectively (). Increased proliferation of hMSCs was also observed on the stiffer methacrylated hyaluronic acid hydrogels as compared to the softer ones over the stiffness range of 3–100 kPa (). Similarly, the gradient elasticity of photodegradable hydrogels was shown to influence the differentiation of valvular interstitial cells into myofibroblasts (). The substrate stiffness was also shown to influence the phenotype of embryonic chicken cardiac myocytes, a conclusion reached by culturing the myocytes on laminin-coated polyacrylamide substrates with stiffness in the range of 1–50 kPa (). The contractile stress of the cardiac myocytes also increases with the substrate stiffness. These and other investigations have established that elasticity mechanics should be taken into consideration when designing biomaterials. Hydrogels with stiffness varying over several orders of magnitude can be realized by varying the degree of cross-linking and by photochemical modulation (In a recent work, the differentiation of stem cells into bone cells using polymeric multilayer capsules (PMLCs) consisting of alternating layers of poly-L-lysine and poly-L-glutamic acid was demonstrated in vivo (). The polyelectrolyte multilayer films were embedded with bone morphogenetic proteins and transforming growth factors. In general, drugs, nucleic acids, proteins and peptides of various concentrations can be encapsulated within the capsule cores or embedded within the polymer layers (). The PMLCs are usually fabricated by a layer-by-layer technique in which polymers are sequentially adsorbed onto spherical substrates with dimensions ranging from ∼10 nm to >10 μm. The multilayer configuration offers advantages in that multiple physico-chemical properties and functions may be achieved within a single coherent material (). Furthermore, many biological tissues such as cardiac muscles are multilayered. Hence, for applications in regenerative medicine it may be inherently advantageous to develop biomimicking materials which are similarly multilayered.The review above emphasizes that mechanical properties play an essential role in the biomaterial design of multilayer capsules for applications in drug delivery, stem cell technology and regenerative medicine. Such diverse applications have motivated our investigation into the elasticity of multilayer capsules. However, in many previous works soft materials are characterized by linear rather than nonlinear elasticity. The stresses and displacements, which have a direct bearing on the biological processes, remain to be studied in detail. Also, the multilayer nature of many soft materials has yet to be considered analytically. The effect of the chemical concentrations of proteins, growth factors, etc. on the mechanical response of the hydrogels has likewise received little attention, although the authors and their associates have initiated research in this area for homogeneous and bilayer capsules (Much theoretical work on the elasticity of soft materials has been done by the mechanics community, e.g., the third- and fourth-order incompressible elasticities of biological soft tissues (), the instability of an elastic shell modeled as an incompressible Neo-Hookean solid (), the electric field-induced displacement of a charged spherical colloid in a linear elastic compressible hydrogel (), and the simulation of tumor growth in its microenvironment, accounting for factors such as nutrient transport, cellular velocity, proliferation and mechanical pressure (). Others have investigated non-affine deformations of polymer gels, as opposed to affine deformations in which the microscopic and macroscopic strains are the same (). However, little work has been reported on the mechanics of multilayer soft hydrogels, which form a large class of natural and bio-inspired materials.In this paper, it is intended to study the stresses and displacements in a multilayer hydrogel subjected to various dilatation profiles arising from chemical concentrations. Second-order elasticity (also classified as third-order since the energy density is third order in the strains) is adopted in this study since it represents the next level of complexity beyond first-order or linear elasticity. This choice is also motivated by the availability of analytical solutions which can be readily used for parametric studies with little computational cost. Alternative nonlinear elastic models such as various modifications of the Neo-Hookean model and microstructurally-based models may also be used, but more extensive numerical computations are expected. We also restrict our investigation to elasticity and neglect other effects such as viscoelasticity.The paper is organized into the following sections. In Section , the second-order elastic model is reviewed, and new solutions for multilayer spherical and cylindrical capsules are presented. In Section , the second-order elasticity is validated against experimental data for a diverse range of synthetic and biological gels subjected to uniaxial tensile loading and swelling under different pH values. In Section , the multilayer solutions are presented and parametric studies are conducted to investigate the dependence of the mechanical response on the gel elasticity, the multilayer configuration, and the dilatation profile. Further discussion is made in Section , and a set of concluding remarks is given in Section The second-order elastic theory was developed by , who gave analytical solutions for the loading cases of uniaxial tension, simple shear and torsion of bars and the compression of spherical shells and cylindrical tubes. This () and other higher-order elasticity theories () have recently been employed for the investigation of biological gels. In the following, it is first shown that the previously obtained analytical solutions for uniaxial tensile loading of soft gels can be recast into a universal form when certain modified stress and strain parameters are defined. Subsequently, the governing field equations for multilayer spherical and cylindrical capsules subjected to a stepped dilatation profile are solved, yielding analytical expressions for the stresses and displacements. It is also shown specifically that a universal form for the displacement (and hence swelling) solution can be obtained if a modified pH parameter is defined. Experimental data are fitted against these universal relations.As a starting point, higher-order elasticities employ a strain energy density W of the invariants of the Green–Lagrange strain E (). The choice of invariants can be I1=trE, I2=[(trE)2-tr(E2)]/2 and I3=detE. In second-order compressible elasticity, W is expanded as a linear combination of the invariants up to the third order in the strains, i.e.:where λ, μ are the second-order and l, m and n the third-order elastic constants, respectively (). A total of five elastic constants are thus needed. Other choices of W may be employed, e.g., W for third-order incompressible isotropic elasticity (Consider an isotropic homogeneous cylindrical bar under a uniaxial tensile stress σ, defined as the applied force per unit initial cross-sectional area (engineering stress). Using a perturbation analysis, the engineering strain ɛ takes the form:where E is the Young’s modulus related to λ and μ via the well-known relationand δ is dimensionless and related to the five elastic constantsδ=-32-μ2l(3λ+2μ)(λ+μ)2-(3λ+2μ)2(λ+μ)2m-3λ24μ(3λ+2μ)(λ+μ)n. contains two parameters E and δ (rather than five), implying that different values of the five elastic constants can lead to the same stress–strain response provided they share the same values of E and δ according to Eqs. . Moreover, δ can be positive or negative. A negative value implies a nonlinear stiffening effect as the stress produces a strain less than that predicted by linear elasticity. Conversely, a positive δ implies a softening effect. by δ and defining a modified strain ε̃=δε and a modified (dimensionless) stress σ̃=δσ/E, the following simple universal relation is obtained:where no material parameters appear explicitly. The values of E and δ can be obtained by fitting Eq. against the stress–strain data, and can be used to calculate the modified strain and stress. The modified σ̃ vs. ε̃ data can be compared to the universal form of Eq. . A universal representation is a convenient way to collapse different data sets into a single reduced data set, as illustrated in Section Consider next an N-layer spherical (or cylindrical) capsule subjected to a fluctuating spherically (or radially) symmetric dilatation profile (). The interfaces are located at the radial coordinates r
|
=
|
rj, j
|
= 1, N. The layers are nonlinearly elastic, isotropic, homogeneous and perfectly bonded to one another, but can be of dissimilar material compositions and thicknesses. Fluctuations such as the chemical concentration c(r) result in local dilatation field ϑ (r), e.g., the stepped profiles ϑp1 and ϑp2 indicated in . The relation between c and ϑ is given by the change in lattice parameter per unit concentration and depends on the specific polymer gel and the chemical species. An analogous relation exists between temperature and dilatation, in which the change in thermal strain per unit temperature is defined by the coefficient of thermal expansion. For the elasticity problem, however, it is not necessary to specify this relation and ϑ (r) enters the governing equation directly. In general, the fluctuation field may be controlled, as in the embedding of drugs, nucleic acids, nutrients and growth factors of selected concentrations within the layers of sub-micrometer thicknesses (A total of five elastic constants λj, μj, lj, mj and nj, j
|
= 1, N are needed to describe the elasticity of each layer j. The dilatation is effectively treated as the loading associated with a body force density (). The governing equilibrium equation in the radial direction can be written as:Tr′+2rTr-2rTθ=kλ+23μd(3ϑ)dr-k2λ+23μ-2l-2n9(3ϑ)d(3ϑ)dr,for multilayer spheres, where Tr and Tθ are the radial and meridional stresses (the azimuthal component Tϕ
|
=
|
Tθ for spherically symmetric loading) in a spherical coordinate (r, θ, ϕ) system, andTr′+1rTr-1rTφ=kλ+23μd(3ϑ)dr-k2λ+23μ-2l-2n9(3ϑ)d(3ϑ)dr,for multilayer cylinders, where Tr and Tφ are the radial and circumferential stresses in a cylindrical coordinate (r, φ, z) system. The prime denotes differentiation with respect to r, and k and k2 keep track of the order of approximation (). For convenience, the subscripts of the elastic constants indicating the layers are dropped. Eqs. can be transformed into equations involving the radial displacement ur. Specifically, the stresses are derived from W via the relationwhere T is the matrix of stresses and J is the matrix relating current to reference coordinates. For multilayer spheres with spherically symmetric loading, J and E are 3 × 3 diagonal matrices:E=12(J∗J-I)=kdiagur′,urr,urr+12k2diagur′2,ur2r2,ur2r2,where I is the identity matrix and ∗ denotes the transpose. For multilayer cylinders with radially symmetric loading, they are 2 × 2 diagonal matrices:The cylinder problem is essentially a plane problem. By the perturbation procedure, ur is decomposed into a linear (u) and a nonlinear (w) part such that ur
|
=
|
u
|
+
|
kw, where k keeps track of the order of approximation. The total displacement is the simple sum of u and w. By Eq. , the stress components can be expressed in terms of u and w, resulting in Tr=kTrL+k2TrNL, where TrL and TrNL denote the linear and nonlinear components of the radial stress, respectively, and similarly for Tθ and Tφ. The stresses are the simple sums of their respective linear and nonlinear parts. For multilayer spheres, Eq. separates into two parts, associated with k and k2, respectively:(λ+2μ)w″+2w′r-2wr2=-2(λ+3μ+2m)u″+2u′r-2ur2u′-1ru′-ur2-(λ+2l)u″+2u′r-2ur2u′+2ur-λ+23μ-2l-2n99ϑdϑdr.(λ+2μ)w″+w′r-wr2=-2(λ+3μ+2m)u′u″+12ru′2-u22r3-(λ+2l)u″+u′r-ur2u′+ur-λ+23μ-2l-2n99ϑdϑdr. are the classical equilibrium equations for linear thermoelasticity, if ϑ is interpreted as the dilatation due to thermal expansion. Mathematically, they are second-order differential equations with the unknowns u and w.Assume that ϑp1 or ϑp2 is represented by the constants ϑj, j
|
= 1, N. Solving Eqs. for each layer in spherical coordinates and Eqs. for each layer in cylindrical coordinates, the displacements uj and wj, j
|
= 1, N can be obtained in analytical form:uj=Ajr+Bjrj-13r2,wj=Cjr+Djrj-13r2+λj+3μj+2mjλj+2μjBj2rj-16r5,1≤j≤N,uj=Ajr+Bjrj-12r,wj=Cjr+Djrj-12r+(λj+3μj+2mj)2(λj+2μj)Bj2rj-14r3,1≤j≤N,for multilayer cylinders, where B1
|
=
|
D1
|
= 0 and there remain 4N
|
− 2 unknown dimensionless parameters A1, C1, and Aj, Bj, Cj, Dj, j
|
= 2, N. It should be noted that Aj, Bj, Cj, Dj for spheres and cylinders are different. These parameters depend on λj,μj,lj,mj,nj, the layer thicknesses rj-rj-1, j
|
= 1, N (the inner radius of the core r0
|
= 0) and the dilatation ϑj. Using Eq. , the linear and nonlinear components of the radial and meridional stresses for multilayer spheres in each layer j can be written as:where H1j, H2j and H3j are parameters related to Aj, Bj, Cj, Dj for multilayer spheres. For multilayer cylinders, the linear and nonlinear components of the radial and circumferential stresses in each layer j can be written as:where H1j, H2j and H3j are parameters related to Aj, Bj, Cj, Dj for multilayer cylinders. Eqs. are all power-law expressions. To solve for the 4N
|
− 2 parameters, the continuity of ur across the interfaces at rj, j
|
= 1, N
|
− 1 requires uj(rj) =
|
uj+1(rj), wj(rj) =
|
wj+1(rj), while the continuity of Tr at the same interfaces requires TrjL(rj)=Trj+1L(rj), TrjNL(rj)=Trj+1NL(rj), and finally the traction-free condition at the outermost surface requires TrNL(rN)=TrNNL(rN)=0 for both multilayer spheres and cylinders. These constitute a set of 4N
|
− 2 equations which permit the solution for the same number of parameters. The expressions for the constants H1j,
|
H2j and H3j in terms of the elastic constants, the thicknesses and the dilatations are rather cumbersome and are omitted here for brevity. for uniaxial tension is validated against fourteen data sets labeled 1–14 in . The materials include biopolymer networks (neurofilaments, collagen, vimentin, fibrin and actin) (), covalently crosslinked networks based on poly(propylene glycol) bis(aceto-acetate) with either neopentyl glycol diacrylate (PPG BisAcAc-NPGDA) or with 2-hydroxyethyl acrylate derivatized bis(4-isocyanatocyclohexyl) methane (PPG BisAc-Ac-EA-HMDI-EA) (), extremely soft poly(acrylamide) (PAAm) hydrogels (), physical and chemical gels of poly(vinyl alcohol) (), and a longitudinal strip of human abdominal aorta (). In addition, molecular dynamics simulations of semiflexible networks ( lists the materials and the corresponding parameters E and δ obtained through data fitting of Eq. . Two examples of σ−ε plot are shown in and the test data for a physical gel of poly(vinyl alcohol) (labeled P-GEL-15 in the original paper of to the data for a longitudinal strip of human abdominal aorta (). The value of δ is negative in these two cases, implying a stiffening effect associated with second-order elasticity. shows that most of the materials display the stiffening effect, whereas the PPG BisAcAc-NPGDA networks, the soft PAAm gels and the chemical gels of poly(vinyl alcohol) display a softening effect with corresponding positive δ. plots the universal relation, i.e., Eq. , in the form of -σ̃ vs. -ε̃. It can be seen that the fourteen data sets collapse into a narrow data band, through which the universal line passes. The agreement between the nonlinear elastic model prediction and the data is reasonably good, given that the materials include both biological and synthetic gels of various types. The σ̃ vs. ε̃ behavior is significantly nonlinear and it is insufficient to describe the gel elasticity using only a second-order elastic constant such as E. A universal relation for uniaxial tension has also been derived from a microscopic model of the polymer chains as worm-like chains ( shows that a simple second-order elastic continuum model is also capable of describing such a universal relation. Indeed, as the next section shows, a universal relation can also be developed for the swelling behavior.In this sub-section, the data for the swelling dependence on the pH of homogeneous polymer gels are fitted against a universal relation. The swelling is characterized by the weight ratio Q, defined as the ratio of the mass of water absorbed to the mass of the dry polymer. For spheres,Q=ρwater[(R+ur)3-R3]ρpolymerR3=ρwater1+urR3-1ρpolymer,Q=ρwaterh[(R+ur)2-R2]ρpolymerhR2=ρwater1+urR2-1ρpolymer.In the above definitions, ρwater and ρpolymer are the densities of water and polymer, respectively, R is the radius of the gel at dry state, ur
|
≡
|
ur(R) is the radial displacement at the outermost surface after swelling, and h is the thickness of the cylinder gel.It can be shown that the parameters Aj, Bj, Cj, Dj in Eqs. are quadratic functions of the dilatation ϑj, after applying the boundary conditions. Thus, ur/R is a quadratic function of dilatation ϑj. For a single-layer homogeneous sphere or cylinder, the following linear (and simplest) assumption is made between the dilatation and pH (which is a logarithmic measure of the concentration of hydrogen ions):for the region 0.1R
|
<
|
r
|
<
|
R, and ϑ
|
= 0 for a small pH-insensitive core r
|
< 0.1R. The constants a and b can be obtained through fitting the displacement model to the swelling data. Hence, ur/R is a quadratic function of pH in the second-order elasticity model:where a′, b′ and c′ are parameters related to the elastic constants as well as a and b of Eq. for cylinders. Note that Q is either a sextic or a quartic function of pH. By fitting Eqs. to the swelling data, the parameters a′, b′, c′ can be determined. In a similar way as for the uniaxial tension data, universal relations for Q can be developed. Defining a modified swelling ratio Q̃=ρpolymer/ρwater•Q and a modified pH P̃h=a′pH2+b′pH+c′, Eqs. can be recast in the following universal forms:where no material parameters appear explicitly. It should be noted that universal relations can be similarly derived for a multilayer capsule, following the steps of Eqs. Twenty sets of swelling vs. pH data, labeled 15–34 in , are used to validate the universal relations. The polymer gels include: styrene and 4-vinylpyridine copolymers (St/4VP copolymers) (), poly(N-isopropylacrylamide-co-sodium acrylate) gels (NIPA/SA) (), poly(ethylene glycol) methyl ether methacrylates-poly(acrylic acid) double-network hydrogels (PPEGMA-PAA) (), poly(N-isopropylacrylamide-co-acrylic acid) gels (NIPA/AAc) (), polyacrylamide/laponite clay nanocomposite hydrogels (NC) (), poly(methacrylic acid)/Poly(N-isopropylacrylamide) interpenetrating polymer networks (PMAA/PNIPAAm IPN) (), and hydrophobically modified poly(acrylic acid) gels (HM PAA gels) ( lists the materials tested and the corresponding values for the parameters a′, b′, c′ appearing in Eq. . The St/4VP copolymers are spherical whereas the rest of the polymers in (a) shows Q vs. pH for the (NIPA/AAc) gels (). The weight ratio increases with pH, reaches a peak around a pH value of 8, and decreases with pH thereafter. This non-monotonic Q dependence on pH is fairly well-reproduced by Eq. , indicated as a solid curve. In contrast, (b) shows that the weight ratio of HM PAA gels C8–20% () increases monotonically with pH up to the value of 9. Here the theoretical prediction of Eq. is acceptable, although the point of inflection is not well-captured.Taking ρpolymer/ρwater
|
≈ 1.4 (a reasonable estimate for the polymers) and the values of a′,
|
b′,
|
c′ in for the different polymers, the modified data can be computed. plots the modified weight ratio against the modified pH for the case of spherical capsules (). The universal curve predicted by Eq. assembles the Q̃-P̃h data for cylindrical capsules in two separate plots for clarity. The upper (a) and lower (b) plots show the data labeled 18–27 and 28–34 in , respectively. The modified pH value in (a) is less than 5, while it is mostly between 4 and 20 in (b). Good agreement is obtained between the reduced data and the universal curve in . Hence, the second-order elastic model captures fairly well the swelling behavior of a diverse range of polymers and composites.On the basis of the second-order elastic model, which has been validated against uniaxial tension and swelling data, theoretical predictions can now be made via simulations for multilayer configurations of polymer gels. Multilayers are gaining increasing importance in the fields of regenerative medicine, stem cell technology and drug delivery, as reviewed in Section . Focus will be placed on the influence of the elastic constants, the interface positions and the dilatation profile on the displacement and stresses in spherical multilayers.Consider a periodic four-layer spherical capsule of equal layer thickness (rj
|
−
|
rj−1)/r1
|
= 1, j
|
= 1, 4 and alternately identical layers (1, 3) and (2, 4) subjected to the stepped dilatation profile ϑp1 shown in 1. In this profile, the dilatation steps up from a value of 0.1 in Layer 1 to 0.4 in Layer 4, simulating varying chemical concentrations in the layers. The elastic constants are taken to be λ1
|
= 35.7, μ1
|
= 10.3, l1
|
=
|
l2
|
= −35.6, m1
|
= −24.2, and n1
|
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