text
stringlengths 0
1.19M
|
|---|
⋅ |
S), where ε0 is the dielectric constant of vacuum i.e. 8.85 × 10−12 |
F m−1, and d is the thickness of the gap between the electrodes and S is the contact area of the electrodes.The average values of the surface area, pore volume, and pore size of the material were measured on an ASAP 2010 Accelerated Surface Area & Porosimetry made by the US Micromeritics Company. The average grain size of the particle material was determined on a JEM-200 CX electron microscope. The contact angle between the pellet and the water drop was measured on a Video-based contact angle measuring device (model: OCA20), the surface energy of the material was obtained using the SCA21 software and the EOS method.The X-ray diffraction (XRD) pattern (see ) shows that the material is composed of NH4Al(OH)2CO3 that belongs to the orthorhombic crystal system and space group Cmcm(63) (JCPDS No. 42-0250), AlO(OH) that belongs to the orthorhombic crystal system and space group D16 2H-PBNM (JCPDS No. 5-355) and (NH4)2SO4 that belongs to the orthorhombic crystal system and space group Pna21(33) (JCPDS No. 40-0660).7NH4HCO3+2NH4Al(SO4)2=NH4Al(OH)2CO3+AlO(OH)+4(NH4)2SO4+6CO2+2H2OThe CO2 and H2O in the product can be removed by a volatilization during drying of the material. shows the dependency of the static yield stress (τy) on electric field strength (E) for the suspensions with various concentrations. Like many other ER fluids, samples 1–5 also possess the property that the yield stress increases as the electric field strength increases, as the result of an increase in the polarization interactions between particles. From we can see that: samples 1 and 2 exhibit similar yield stress; the yield stress increases with increasing concentration from sample 2 to sample 5. The yield stress of sample 5 reaches 22.8 kPa at E |
= 4 kV/mm. In order to learn the effect of suspension concentration on the ER activity of the material, the relative yield stress, τr (=τE/τ0, τE and τ0 are the static yield stress with and without an electric field, respectively), is used to describe the magnitude of the ER activity. As the results shown in , the τr values (12.7–33.3) of all samples are large, indicating that the material has good ER activity that is favorable for practical application. Here, the relative magnitude of the τr for different samples is note worthy, the τr of sample 1 is the largest and that of sample 5 is the smallest, which can be attributed to an increase of τ0 with increasing suspension concentration. It is notable that more concentrated ER fluids tend to have larger τE but τ0 as well, which may cause τr to decrease.The higher yield stresses and ER activities of the ER suspensions can be related to the composition of material. The (NH4)2SO4 that belongs to ionic crystal can effectively enhance dielectric constant and dielectric loss tangent of the material, the NH4Al(OH)2CO3 containing NH4+,CO32- and OH− and the AlO(OH) containing OH− can increase the interfacial polarization of the ER suspension in an electric field In order to understand a power law relationship between τy and E, the different α values (0.82–1.75) of the five samples are all smaller than 2. The shapes of the particles were irregular, and a broad particle size distribution (5–100 nm) was also observed (see ), thus, the dipole moment of the particles is not uniform, which may have resulted in the observed α |
< 2; the applied electric field in this study is relatively high (1.0–4.2 kV/mm), which may also be related to our observation of α |
< 2 The uncommon value of α |
< 1 of the sample 5 may be related to its high concentration. The particles in the suspension of an ER material can turn along the direction of an electric field to form a fibrillation structure. However, the turn of the particles would delay for a high concentration suspension. As a result, the increment of the yield stress would decrease gradually with increasing the concentration of the suspension when increasing electric field strength.In order to know the influences of the shear rate and electric field strength on the shear stress, the relationships between the shear stress, shear rate and electric field strength have been researched for various ER fluids . The shear stress increases linearly with increasing shear rate when no electric field is present. This shows that the suspension exhibits the behavior of a Newton fluid without an applied electric field. However, when the electric field strength increases, the shear stress increases with the electric field strength at all shear rates. In addition, we can see that the shear stress increases slowly with increasing shear rate under four different applied electric fields of E |
= 1.8–4.2 kV/mm. Therefore, it is obvious that the interfacial polarization between the dispersed phase and the medium in the suspension is more important factor than shear rate in influencing the shear stress for the studied ER fluid. The electrostatic interaction force between the particles, which originated from the induced dipole moment caused by the interfacial polarization, dominates the shear force, that is to say, the chain-like structure, which is established by the electrostatic interaction between the particles, is not fully broken even at higher shear rates and lower electric field strength.Moreover, the leaking current density was very low (<10 μA/cm2) in the suspension under an applied electric field. The properties of the material, including high and stable τE and τr and high breakdown strength in an electric field, are advantageous in its application as an ER material.The dielectric property of materials would affect ER performance in different manners, because the relaxation time for interfacial polarization of ER fluids, which would be influenced by the dielectric property, is known to be related to yield stress and stress enhancement under the applied electric field , the ε and tan |
δ values of the sample are large over the entire frequency range, especially the tan |
δ values (see b) are larger than 0.1, except for the tan |
δ values (tan |
δ≈ |
0.1) at f= |
1, 1000 and 2000 kHz, and the tan |
δ value reaches a peak with a value of 0.19 at f= |
40 kHz in the dielectric loss tangent spectra.a, we can see that the ε value of the sample is much larger at lower frequencies than at higher frequencies, which could also be a reason why the material has high ER activity The surface properties of a material, which include surface energy and surface area that is proportional to the pore volume and pore size in the particle and grain size, are important factors in influencing the ER performance of a material ). The surface area of a particle material is expected to increase with decreasing particle grain size. Therefore, the observed grain size is consistent with the result from the surface area analysis. The surface energy of the material reaches a value of 67.17 mN/m. Considering corresponding results in literatures A nanocomposite composed of polar inorganic compounds is synthesized. The suspensions of 40, 45 and 50 wt% in silicone oil have higher yield stress at a DC electric field. The ER activity of the material is also higher. The dielectric property, surface area and surface energy are all important factors in influencing the ER performance of the particle material. The impact of the composition on ER property of the material is dominant.Evaluation of mechanical weakening of calcarenite building stones due to environmental relative humidity using the vapour equilibrium techniquePrevious studies have evaluated the water-weakening effect of rock materials after immersion in water during different periods of time. However, the water transference between the environment and the building stones frequently occurs in vapour form when changes in the relative humidity of the air involve little variations in moisture content of stones. In this sense, the novelty of this work lies in assessing the impact of the environmental relative humidity (RH) on Unconfined Compressive Strength (UCS), Young's modulus (Est), Brazilian Tensile Strength (BTS) and Point Load Strength Index (Is(50)) of three calcarenite building stones. To this aim, calcarenite specimens were exposed to five different environments with RH values ranging from 10 to 93% which were devised in laboratory through a novel modified Vapour Equilibrium Technique (VET) by using water-glycerol solutions at different concentrations and then, they were mechanically tested. The results indicated that, despite the water content (w) and the degree of saturation (Sr) inside the pore network of calcarenites were relatively small for all RH environments (w = 0.03–1.79% and Sr = 0.2–35.6%), important reductions of UCS (28.2–34.7%), Est (20.0–31.3%), BTS (17.0–41.3%) and Is(50) (23.9–37.6%) were found when the RH varied from 10 to 93%. Furthermore, negative linear relationships were established between the values of the mechanical properties and the environmental RH while negative tri-parametric exponential functions were proposed between the mechanical parameters and w. In addition, the mechanisms involved in the observed behaviour were discussed and relationships between the different mechanical properties were also proposed.Natural stones are currently used in several civil engineering and architectural elements such as masonry walls, rockfills, dams, bridges, facades or ornamental features. Also, these geomaterials have been traditionally utilised in a large number of heritage buildings that usually require maintenance or restoration work and in which the use of similar materials to the originals is essential to preserve their historical character (). These constructions components and buildings are frequently located in outdoor environments and therefore exposed to cyclic water content variations due to rain, condensation, atmospheric humidity or ground capillary uprise (). In this sense, preceding works have demonstrated that the presence of moisture plays a key role in the durability of porous rock materials (), being responsible for many decay processes.Particularly, previous studies on porous limestones widely used in emblematic building around Europe have proved that these geomaterials can suffer important degradation after few freeze-thaw cycles () and that porosity and pore size distribution control this type of weathering process (), which mainly occurs when the degree of water saturation exceeds 80–85% (). Furthermore, porous limestones are also very sensitive to salt crystallization damage (), especially those which exhibit more porosity and larger quantities of small pores (< 0.1 μm) () and therefore, parameters derived from mercury porosimetry curves are the most adequate for evaluating the durability against this kind of decay (). In addition, physical properties and durability of repair mortars and protective products and their compatibility with limestones need to be carefully considered (). Also, other weathering processes such as biological growth, corrosion of metallic elements, chemical attack by acid rain, air pollutants or differential hydric dilation can significantly affect to porous stones (Apart of the reduction of the durability associated with water transport inside the microstructure due to the abovementioned weathering processes, the mechanical parameters of porous building stones are frequently subjected to important variations depending on their water content (). In this connection, several researches have shown that water saturation of rock materials can cause significant decreases of some of their most important mechanical properties such as unconfined compressive strength (UCS) or static Young's modulus (Est) (). Furthermore, it has been recently observed that other mechanical parameters like Brazilian Tensile Strength (BTS), Point Load Strength Index (Is(50)), shear strength parameters (cohesion and internal friction angle) or dynamics properties are also significantly reduced when the effective pore network of geomaterials is completely filled of water (). This water-weakening effect varies considerably according to the type of rock material: it can be small, in the case of hard igneous geomaterials (reductions of 0–20%); moderate, in the case of cemented medium strength rocks (reductions of 30–40%), or very large, in case of soft clay sedimentary stones (reductions of more than 70%) (). As a consequence, a detailed understanding of this behaviour is particularly crucial for the design and safety assessment of stone structures and construction elements that may be in contact with water.Regarding the quantification of the variation of UCS and Est with the moisture content of sedimentary stones, laboratory data from previous studies has suggested that these properties undergo a sharp decrease for low water content and then they remain almost constant for larger water content. Concretely, the mathematical function frequently used to describe this phenomenon is a negative exponential curve of the following type:where σ(w) is the corresponding mechanical property, w is the water content and, a, b and c are three characteristic parameters of the rock. Particularly, this type of function allows to express the mechanical properties values at zero water content as a+c, the mechanical properties values at high water content (fully saturated state) as c and the rate of the mechanical properties losses with increasing water content as b (In the previous cited researches, the water-weakening effect has been mainly performed determining the mechanical properties of specimens after their immersion in distilled water during a specific time or after applying several freeze-thaw cycles. In both cases, the saturation mechanism of the specimens is carried out through the penetration of water in the liquid form within its pore network, and therefore the nearby decay processes to this mechanism could be the saturation by rainwater infiltration or by groundwater level rise. However, the water transference between the environment and the stone materials frequently occurs in vapour form when changes in the relative humidity of the air involve variations in the moisture content of stones until reaching the corresponding new equilibrium ( the transport and distribution of fluid water and vapour in porous stone occur by means of the following steps. Firstly, the surface of stone begins to adsorb single water molecules from the atmosphere in vapour form to engender a layer. As air humidity rises, the quantity of layers of water molecules on the surface increases. Subsequently, when a certain water content is reached, the layers merge into the pores due to capillary condensation forming drops and thereafter the capillary forces begin to govern the water flow. Lastly, the pore network is filled of more water and the saturation flow happens.Although a great deal of research on the effect of liquid water on mechanical behaviour of geomaterials has been performed to date, very limited number of papers have evaluated the impact of the environmental relative humidity surrounding the stones materials on their mechanical properties. prepared sandstone specimens with different water content by placing the samples in several controlled atmospheric environments with relative humidities between 0 and 96% created by using different salt saturated solutions. They observed that the largest variations of UCS occurred at moisture content lower than 1% and suggested that this behaviour was due to the stress corrosion phenomenon, that is, the presence of water rose the velocity of crack propagation in silicates by substituting silica‑oxygen bonds with weaker hydrogen bonds within the silicate lattice. When this phenomenon happened at the tip of a microcrack propagating under tension, it diminished the stress necessary for failure at the tip of the crack by weakening the strength of the crystal lattice that lied in the path of failure (). Nevertheless, the fracture propagation velocity and crack opening also depend on other parameters such as toughness, activation energy, temperature or stress (apart of the relative humidity) ( analysed the strength values of specimens of Bath limestone conditioned in controlled atmospheric environments of known relative humidities and found that its UCS values decreased from 28 to 18 MPa when the relative humidity increased from 0.0 to 75.5% while for higher relative humidities the UCS remained almost constant, suggesting that at low moisture contents suction contributed to rock strength in a similar way to soils. evaluated the strength properties of Tournemire chalk with different degrees of saturation prepared by using controlled humidity atmospheres with RH values of 5, 36, 50, 76 and 98% through saline solutions. They obtained that Est, triaxial compressive strength and cohesion decreased and Poisson's coefficient rose when water content increased. Complementarily, studied the effect of desaturation-resaturation processes on mechanical properties of a mudstone imposing different RH values through saline solutions. They reported that UCS and Est were doubled when RH decreased from 98 to 32% (i.e., when the suction increased from 3 to 155 MPa). Furthermore, analysed the influence of suction on the hydromechanical behaviour of partially saturated chalks by applying matric suction procedures. Concretely, they utilised the osmotic technique by using dialysis membranes and a polyethylene glycol solution for obtaining low suction values and the vapour equilibrium technique using salt solutions for obtaining high suction values. They found that UCS and Est significantly decreased when water content increased (in an approximately exponential and linear way, respectively). obtained a linear relationship between the RH, varying from 39 to 85%, and the deformation during the dehydration and rehydration cycles at macro and microscale for argillaceous rocks. Furthermore, they proposed a negative linear correlation between RH and Est. In addition, studied the deformability characteristics of mudstones combining hydric and mechanical tests using environmental scanning electron microscopy (ESEM) and digital image correlation (DIC) techniques. They obtained that Est decreased when RH increased (specifically, Est values of 16, 7 and 4 GPa for RH values of 21, 80 and 99%, respectively) and that the specimens with more water content showed a more marked microcracking and a more ductile behaviour. observed that 90% of UCS and BTS losses occurred when limestones specimens were equilibrated with a 97% relative humidity atmosphere and noted that the clay mineral content could govern the weakening mechanism. Recently, obtained that the BTS of slate was reduced from 6.1 to 2.1 MPa when the environmental relative humidity rose from 55 to 92%, despite the maximum value of water content in the slate was very low in comparison with its saturation content.The last cited papers suggest that very little variations in the moisture of some stones due to the water vapour transference caused by the increase of the environmental relative humidity can lead significant reductions in their mechanical properties. The practical relevance of this phenomenon, its omission in existing standards and the relatively scarce literature existing on this issue for calcarenites have motivated the present experimental study. This work evaluates the influence of the environmental relative humidity on mechanical properties such as UCS, Est, Is(50) and BTS of three porous calcarenite building stones widely used in southern Europe. Particularly, in order to simulate five different environments, containers with relative humidities ranging from 10 to 93% were prepared applying the vapour equilibrium technique (VET), and more specifically, by using glycerol solutions with different concentrations. Calcarenite specimens were introduced in the containers until they reached the corresponding equilibrium with the environment and then the samples were mechanically tested.As a consequence, this study lays the bases for a standard procedure to evaluate the effect of environmental relative humidity on the mechanical properties of porous rocks and provides an in-dept knowledge of the vapour hydro-mechanical behaviour of calcarenite building stones. The results could be used to predict the durability of the studied rocks and to assess the safety of heritage buildings subjected to different relative humidity atmospheres during its lifetime and also to optimise the selection of the most suitable stone at quarry level depending on the environments in which is planned to use it.Several types of carbonate porous rocks from the Iberian Peninsula have been traditionally used as construction material. Nevertheless, special mention should be made here of the use Bateig calcarenites from the Alicante province, which due to its easy carving and availability have been widely used during the last seven centuries within the heritage buildings of Spain (). In this sense, a great quantity of monuments of the Valencian Community such as the City Hall and Post Office of Valencia, as well as emblematic constructions of other regions like the Almudena Cathedral, the National Library or the City Hall of Madrid were built with these stones (). Furthermore, nowadays Bateig calcarenites lead the way in the major international market and they are present in buildings, residences and other civil and architectural works around the world, and specially in southern Europe. As a consequence, these materials could be exposed to very different environments.Temperature and RH control the change of state of water and its movement and therefore, these parameters are essential for evaluating the retention of water vapour by condensation within the stones and the water flow between these geomaterials and the environment (). With the aim to establish the range of RH values used in this work, climatic data from representative places in which the studied calcarenites are commonly utilised was compiled from the Meteorological State Agency of Spain ( shows the location of the selected meteorological stations (i.e. Alicante, Madrid, Santiago de Compostela and Tenerife) and the monthly values of RH and temperature in these cities during the 1980/2010 period. As can be seen, the mean RH values in the selected cities ranged from 25 to 86% while the mean maximum and minimum temperature values varied from −4 to 35 °C (absolute minimum and maximum temperature values were − 8 and 42 °C, respectively).This data shows that Bateig calcarenites are usually exposed to a wide range of RH and temperature values derived from the water vapour condensation on the rock surface or its penetration into pore network by convection or diffusion movements. This partial saturation could theoretically induce mechanical debilitation () and decay processes by ice and salt crystallization () On this basis, it was decided that Bateig calcarenite specimens would be exposed to simulated atmospheres in laboratory with theorical RH values close to 20, 40, 60, 80 and 98% by using VET and then, their mechanical properties would be assessed, as described in the following sections.Building stones used in this work are three fine- and medium-grained calcarenites from the province of Alicante (Spain), which belong to a Middle-Late Miocene unit from the Betic-Balearic domain. Specifically, intact blocks of the calcarenites varieties commercially known as Diamond, Beige and Blue Bateig stones (hereinafter referred as to G-1, G-2 and G-3, respectively) were taken from an active quarry just located in the side of Bateig Hill, close to the municipality of Elda, and then, they were transported to the laboratory in order to prepare the specimens. The rock blocks were visually inspected and carefully chosen to ensure the absence of microcracks, joints and zones with different grades of weathering.With the aim of performing the UCS and Est tests in accordance with the International Society for Rock Mechanics Suggested methods (), 15 cylindrical core samples from each type of stone, 28 mm in diameter and 70–75 mm in length were drilled perpendicular to bedding using a diamond drill rig. Furthermore, to carry out the BTS, 15 cylindrical core specimens, 52 mm in diameter and 26 mm in thickness, were drilled from each block. In addition, to perform the Is(50), a minimum of 10 cylindrical core specimens, 28 mm in diameter and 30 mm in length, were drilled from each block. Finally, before to the physical characterization, each specimen was heated in an oven (70 °C for 48 h) in order to measure the initial water content, as well as the mass corresponding to the solid fraction.To determine chemical and mineral composition the following analyses were carried out: thin-section petrographic studies, using an OPTIKA B600POL petrographic microscope with the X4 objective lens; X-ray fluorescence (XRF), conducted using a PHILIPS MAGIX PRO X-ray sequential spectrometer equipped with a rhodium X-ray tube and beryllium window and a single goniometer-based measuring channel covering the whole measurement range, in accordance with ; carbonate content determination, using Bernard calcimeter in accordance with UNE 103–200-93 () and, lastly, X-ray diffraction (XRD), performed using a Bruker D8-Advance X-Ray diffractometer with a KRISTAL- LOFLEX K 760-80F X-ray generator and XR tube with a copper anode.Physical properties were determined by using traditional techniques. Specifically, apparent and saturated densities (ρb and ρsat) were obtained by dividing the dry or saturated weight of specimen by its bulk volume, which was calculated from its dimensions, in accordance with UNE-EN 1936 standard (). Water absorption (Wa) was calculated as the relation between the saturated mass after immersion for 48 h and the dry mass of the sample, in accordance with UNE-EN 13755 standard (). Real density (ρr) was obtained through the pycnometer method following the UNE 103–302-94 standard (). Open porosity (po) was measured using dry, submerged and saturated weights and the volume of the specimens while total porosity (p) was computed from apparent density and real density following the UNE-EN 1936 standard (). Additionally, pore-size distribution was obtained through the mercury intrusion porosimetry (MIP) technique using a porosimeter (POREMASTER-60 GT) equipped with low- and high-pressure stations. In addition, MIP tests also allowed to determine other properties of pore network such as specific surface area (SSA) and pore tortuosity (τ). SSA is a property of materials that can be defined as the area of solid surface per unit mass of material (). Pore tortuosity is a dimensionless parameter that describes the geometry of the flow paths widely used to quantify the complexity of porous media. It represents the departure of a porous structure from being formed of straight pores, and can be conceptually defined as the ratio between the actual or effective flow path length (Leff) and the straight distance among the ends of the flow path (Ls) (). Both properties were calculated using PoroWin version 8.1 software (PoreMaster, Quantachrome Instruments) (). Finally, the P- and S-wave velocities (Vp and Vs, respectively) were obtained in the three calcarenites by using a signal emitting-receiving machine coupled with a computer. Particularly, the corresponding wave velocities of each specimen were calculated as the ratio between its length (i.e. the distance between transducers) and the travel time of each wave, following the UNE-EN 14579 standard (The relationship between the environmental relative humidity and the corresponding equilibrium water content inside the pore network of geomaterials for a determined temperature is known as the sorption isotherm curve. It can be obtained in laboratory preparing atmospheres with different RH by using the vapour equilibrium technique (VET). This method consists of placing the material samples in a desiccator or a hermetically sealed container with constant temperature, promoting water exchange and transfer between the samples and the surrounding atmosphere, in the form of vapour (). The control of the relative humidity within the container can be done using saturated salt solutions (KNO3, NaCl, KCl or K2SO4 among others) as well as unsaturated solutions of chemical compounds such as sulphuric acid of glycerol (Due to the wide geographical areas where Bateig calcarenites are frequently used and the large range of typical RH values to which this type of rocks can be exposed during its lifetime, different glycerol solution concentrations were used to simulate environments with RH values close to 20, 40, 60, 80 and 98%. In particular, the concentration of the glycerol solutions to achieved these RH values were obtained from the DIN 50008–1 standard (The used methodology can be summarized as follows. Firstly, dry specimens of each calcarenite variety were divided into five groups and introduced into transparent plastic containers with hermetic closure (labelled as CT-1, CT-2, CT-3, CT-4 and CT-5) together with the corresponding glycerol solution. Then, the containers were put into an oven at a constant temperature of 50 °C. Subsequently, the evolution of the RH values with time and the temperature control were daily recorded. To this aim, each container was equipped with a thermohygrometer (). Finally, two weeks after achieving the RH equilibrium value inside the containers (i.e. the RH changes are lower than 1% in five consecutive days) the wet mass of specimens were measured and the mechanical tests were immediately performed under laboratory conditions (i.e. 25 °C and a RH of 50%).The variations of the RH values into the containers with respect the time are shown in . Generally, the equilibrium takes longer to reach the higher RH values (around four days). Additionally, the equilibrium RH values in the containers were slightly smaller than those indicated by DIN 50008–1 standard, which can be attributed to the fact that the temperature inside them (50 °C) was greater than that one specified in the DIN 50008–1 standard (25 °C) (After three specimens of each calcarenite were exposed to the corresponding environment, UCS and Est were determined by using a servo-controlled testing machine with a maximum load of 40 kN according to ISRM Suggested Methods (). Specifically, specimen strain was measured using a specific device consisting of two joined metal rings positioned in parallel along the specimen axis and two diametrically opposed LVDTs that recorded variations in the axial relative distance between rings during unloading–reloading cycles. Axial strain was measured up to a value approximately equal to 50% of the failure load of specimens to calculate the secant Est. Once the Est test was concluded, the rings were detached from the sample and the loading test was repeated until failure to determine the UCS.Similarly, three specimens of each stone type subjected to the corresponding RH environment were mechanically tested to obtain the BTS. The tests were carried out following the . In particular, the specimens were placed in direct contact with the machine bearing plates during the tests and the load on the specimens was applied at a constant rate such that failure in the samples happened within 15–30 s. BTS (MPa) was calculated through the Eq. Where P (N), D (mm) and L (mm) are the failure load, the diameter and the thickness of specimens, respectively.Furthermore, six determinations of the Is(50) for each stone type exposed to each previously-defined RH environment were performed by using a Point Load Testing device. Specifically, the specimens were diametrically loaded to failure by application of a concentrated load using two spherically-truncated conical platens. The loading rate was adjusted to make sure that specimen failure happened between 10 and 60 s after starting the test, as required by ISRM suggested method () and the parameter was calculated through Eq. Where F, P (N) and D (mm) are the size correction factor, the failure load and the diameter of specimens, respectively.According to the XRD analyses, the studied stones are limestones mainly constituted of calcite (75–80%), quartz (5–15%), dolomite (5–10%), and little amounts of phyllosilicates, ankerite and feldspar (5–10%).In this connection, the thin-section microphotographs confirm that (G-1 is a biocalcarenite principally constituted of fossils (65%) with a size fluctuating between 0.3 and 0.6 mm, such as Turborotalia, Heterostegina, Globigerinidae, Rotalidae. 10% are terrigenous components with a crystal size ranging among 0.2–0.5 mm, such as monocrystalline quartz, schist, potassium feldspar and clay galls. Regarding ortochems (15%), micrite is the main component. Furthermore, minor amounts of phyllosilicates (chlorite and illite) were identified. This variety exhibits a high total porosity (21%), with quite similar interparticle and intraparticle porosities (9 and 12%, respectively) (G-2 is a biocalcarenite mostly composed of fossils (50%) with a size mostly varying between 0.1 and 0.6 mm, such as bryozoans, molluscs and foraminifera (Rotalidae, Globigerinidae and Textularidae). 20% are terrigenous components with a crystal size of 0.1–0.5 mm, such as microcrystalline quartz, dolomite and small quantities of feldspar and muscovite. 18% are ortochems, particularly micrite matrix (14%) and sparitic cement (4%) exhibiting equicrystalline mosaics of calcite spar. Also, authigenic elements such as glauconite and small quantities of phyllosilicates (palygorskite, chlorite and smectite) were recognised. Its total porosity is approximately 18% and mostly intergranular (15%) while its intragranular porosity is scarce and close to 3%. An abundant bioturbation was observed in this variety (G-3 is a biocalcarenite principally composed of fossils (60%) with a size mainly fluctuating among 0.1–0.4 mm, such as bryozoans, echinoderms and foraminifera (Textularidae, Globorotalia, Globigerinidae and Rotalidae). 15% are terrigenous components with a crystal size ranging between 0.1 and 0.3 mm, such as dolostone extraclasts, monocrystalline and polycrystalline quartz, potassium feldspar, muscovite, tourmaline and rock fragments (slate and metacuarcite). In addition, small amounts of clay minerals filling fossils and phyllosilicates (palygorskite and smectite) were recognised. 15% are orthochems, specifically micritic matrix (10%), sparry cement (5%) and minor siliceous fibrous cement confined to small areas. Its total porosity is approximately 13%, with an interparticle porosity (12%) much higher than intraparticle porosity (1%) (From a chemical point of view, the XRF analyses revealed that calcarenites were mostly composed of CaO (41.9–43.7%), SiO2 (13.8–14.3%), MgO (1.7–2.5%), SO3 (1.4–6.8%), Al2O3 (1.1–1.8%) and Fe2O3 (0.7–1.1%). In addition, the loss on ignition (LOI), that is the loss in weight that results from heating calcarenite samples to a high temperature (after preliminary drying at a temperature just above the boiling point of water) and that corresponds to their organic matter and carbonate contents, varies from 30.4 to 36.6% (). These results are in agreement with those obtained with the Bernard calcimeter, which indicated that the calcite contents ranged from 73.5 to 76.1%.Apparent densities of the calcarenites ranged from 2106 to 2288 kg/m3 while saturated densities varied from 2311 to 2404 kg/m3. Particularly, the greatest values were obtained in G-3 and the smallest values were found in the G-1. In line with this finding, G-3 showed the lowest porosities (i.e. an open porosity of 11.63% and a total porosity of 15.47%) and water absorption capacity (i.e. 5.07%) while the G-1 showed the highest mean values of these physical properties (i.e. 20.55, 22.27 and 9.74%, respectively). Therefore, G-2 showed intermediate values of density and porosity. Concerning ultrasonic wave velocities, the highest values were found in G-3 variety (i.e. a Vp of 4574 m/s and a VS velocity of 2638 m/s) while the smallest values were obtained in G-2 (i.e. a Vp of 3429 m/s and a VS velocity of 2068 m/s Therefore, intermediate values of both waves velocities were found in G-1. The full data can be seen in With regard to the pore network of the calcarenites, the MIP technique revealed that G-3 has the smallest pore size, with diameters mostly ranging from 0.1 to 1 μm (52.9%) and from 0.01 to 0.1 μm (25.3%). Conversely, G-1 has the greatest pore size, with diameters mainly varying from 1 to 10 μm (50.4%) and from 0.1 to 1 μm (24.2%). Therefore, G-2 has intermediate pore diameters, specifically 42.9% ranging from 0.1 to 1 μm and 35.2% varying from 1 to 10 μm (b). In line with this, G-3 showed the highest value of the specific surface area (5.5 m2/g) and tortuosity (2.1) while G-1 and G-2 exhibited similar values each other (3.2 m2/g and 2.0, respectively).The sorption isotherm curves of calcarenites are depicted in . Specifically, they were obtained by fitting the experimental data to the following moisture storage function proposed by where we (kg/m3) is the equilibrium moisture content, wf (kg/m3) is the moisture content at free saturation, bw is the approximation factor and, RH is the relative humidity. They show a conventional shape with a significant rise in moisture content above 90% RH. This upper RH range corresponds to the setting up of capillary interfaces where the liquid phase begins to be considered as “free” water (). The three stones exhibited different behaviours since G-1 is weakly hygroscopic while the sorption capacity in G-3, which has the smallest pore size, is much more significant. Specifically, G-3 reached a maximum Sr value of 35.12% (w = 1.77%) after being subjected to an environment with a RH value of 93%, while, in the case of G-1, the maximum Sr value obtained was 9.86% (w = 0.97%). For its part, in G-2, the maximum Sr value measured was 17.87% (w = 1.51%). In this connection, the obtained moisture storage function revealed that G-3 exhibited the highest wf value (63.79 kg/m3) while considerably smaller wf values were found in G-1 and G-2 (29.40 and 49.98 kg/m3, respectively). These results are in line with the findings of the MIP tests in which the greatest quantity of pores with diameters lower than 0.1 μm was exhibited by G-3. In addition, the obtained bw values were 1.27, 1.31 and 1.19 in G-1, G-2 and G-3, respectively. Therefore, the modified VET used in this work is a suitable method to prepare samples with a small and specific water content.The evolution over the time of Sr (or w) in the three calcarenites when exposed to 98% RH environment is shown in . A strong time dependence to reach the equilibrium can be observed for the different stone varieties. In particular, close to 140 h were required in G-3 while considerably shorter periods of time were required in G-1 and G-2 (about 30 and 40 h, respectively). These differences can be attributed to the lower pore size and the greater tortuosity and specific surface area of G-3 in comparison with the others. Due to this finding, the specimens were exposed to the different RH environments by using VET during two weeks in order to ensure that the equilibrium was reached (assumed when water content is constant) and then, mechanical tests were performed.In these water retention curves, three clearly differentiated stages could be distinguished: a first stage in which the relationship between the storage time and the water content absorbed by the calcarenites was mostly linear; a second stage in which the water absorption rate decreased gradually; and a third stage in which the water content absorbed by the sample had already reached its maximum value and therefore remained constant. This behaviour has been modelled by using exponential functions (b, c and d). Furthermore, from the first stage it is possible to calculate the water absorption coefficient (Aw), defined as the gradient of the straight line obtained by plotting the cumulative mass of water absorbed per unit area versus the square root of storage time. In this connection, despite Aw is frequently calculated by capillary tests (), it can be also applied for the samples exposed to VET with the purpose of represent the velocity of water absorption, which is related with the permeability to water of the geomaterial, considering its non-saturated condition (). Specifically, the obtained Aw values were 0.26, 0.40 and 0.27 kg∙m-2·h-0.5 for G-1, G-2 and G-3, respectively. These values are considerably lower than those found in similar stones through capillary tests by (0.86–2.48 kg∙m-2·h-0.5) and could be attributed to the great differences in the kinematic of the processes to saturate the samples by water vapour diffusion and by capillary conduction.The results showed that environmental relative humidity caused a significant reduction of UCS in the three tested calcarenites. Specifically, when RH increased from 12 to 93%, the UCS reduced from 28.6 to 20.5 MPa in G-1 (i.e. a drop of 28.5%), from 23.8 to 15.5 MPa in G-2 (i.e. a drop of 34.7%) and from 45.8 to 32.8 MPa in G-3 (i.e. a drop of 28.2%). The full UCS values for the five different RH environments and for the three stone varieties are summarized in . The statistical analysis suggested that UCS values diminished linearly with RH increment (). In this sense, if data is extrapolated for RH = 0% (completely dry specimens), the UCS values would be 30.2 MPa in G-1, 25.6 MPa in G-2 and 46.3 MPa in G-3 while for RH = 100% (atmosphere fully saturated of water vapour) the corresponding UCS values would be 19.5 MPa in G-1, 15.4 MPa in G-2 and 30.3 MPa in G-3, which means important differences between both opposite RH environments.Furthermore, because of an increase in RH implies an increase in water content inside the pore network of the calcarenites, this finding is consistent with the well-known behaviour in which the UCS value reduces as water content rises. In this connection, the abovementioned negative exponential tri-parametric function (see Eq. ) was used to model the variation of UCS with water content. This type of function enables the estimation of UCS values for high water content through the parameter c (20.0 MPa in G-1, 11.4 MPa in G-2 and 33.3 MPa in G-3), the UCS value in fully dry conditions through the sum of a and c parameters (29.7 MPa in G-1, 24.2 MPa in G-2 and 50.2 MPa in G-3) and the rate of UCS variation with water content through the b parameter (3.47 in G-1, 0.73 in G-2 and 3.21 in G-3) (In a similar manner to UCS, the results indicated that environmental relative humidity produced a substantial decrease of Est in the three tested calcarenites. Particularly, when RH rose from 12 to 93%, the Est decreased from 21.5 to 14.8 GPa in G-1 (i.e. a reduction of 31.3%), from 10.6 to 8.5 GPa in G-2 (i.e. a reduction of 20.0%) and from 31.0 to 24.2 GPa in G-3 (i.e. a reduction of 21.9%). Also, the statistical analysis indicated that the Est values decreased linearly with RH increase (). In this connection, the extrapolation of the data allows to estimate that for RH = 0% (completely dry samples), the Est values would be 22.6 GPa in G-1, 10.8 GPa in G-2 and 30.9 GPa in G-3 while for RH = 100% (atmosphere fully saturated of water vapour) the corresponding Est values would be 13.6 GPa in G-1, 8.2 GPa in G-2 and 23.2 GPa in G-3, which means significant differences among both extreme values of environmental RH.The relationship between Est and water content within the calcarenites has been modelled by using the negative exponential tri-parametric function (b, d and e). In this case, the Est value for high water content would be theoretically higher in G-3 (c = 25.0 GPa) than in G-1 (c = 14.5 GPa) and G-2 (c = 8.4 GPa). Nevertheless, the rate of reduction in G-2 (b = 2.12) was smaller than in G-1 (b = 4.69) and G-3 (b = 6.72). Est values for the different water contents in the three calcarenites are summarized in Negative linear relationships between environmental RH and BTS have been proposed for the three stones (). In this case, the greatest decrease of BTS when RH ranged from 10 to 90% was found in G-2 (41.3%) while the smallest change was obtained in G-1 (17.0%). BTS values for the five different RH environments in the three calcarenites are shown in . Furthermore, if data is extrapolated for RH = 0% (fully dry specimens), BTS values would be 4.2 MPa in G-1, 3.8 MPa in G-2 and 7.1 MPa in G-3 while for RH = 100% (atmosphere entirely saturated of water vapour) the corresponding BTS values would be 3.3 MPa in G-1, 2.0 MPa in G-2 and 4.4. MPa in G-3, which represents important variations between both contrary RH environments.Negative exponential tri-parametric functions correlating BTS and the water content inside the pore network of calcarenites are depicted in b, d and f. In this case, a-parameter varied from 3.17 to 1.14, b-parameter fluctuated from 0.74 to 2.86 and c-parameter ranged from 0.99 to 4.1 depending on the stone type.The greatest decrease of Is(50) was exhibited by G-2 in which a drop of 37.6% was obtained when RH rose from 15 to 92% (associated to an increment of water content of 1.45%). By contrast, the lowest reduction was exhibited by G-3 in which a drop of 23.9% was measured when water content increased from 0.16 to 1.79%. In the case of G-1, Is(50) diminished by 28.0% when water content rose from 0.04 to 0.97%. The complete Is(50) values for the five different RH environments (and the corresponding w and Sr values) obtained in the three stones are summarized in The relationships between Is(50) and environmental RH or water content inside the pore network of the tested geomaterials are shown in . In this sense, in a similar manner to the other mechanical parameters, negative linear fitting functions were stablished between Is(50) and RH, while negative exponential tri-parametric functions were proposed for the IS(50)-w data pairs. In this sense, if data is extrapolated for RH = 0% (completely dry specimens), the Is(50) values would be 2.9 MPa in G-1, 2.1 MPa in G-2 and 4.9 MPa in G-3 while for RH = 100% (atmosphere fully saturated of water vapour) the corresponding Is(50) values would be 1.9 MPa in G-1, 1.1 MPa in G-2 and 3.6 MPa in G-3, which also means important differences among both opposite theoretical RH environments.UCS and Est are considered the most relevant properties for characterising geomaterials in engineering practice. However, the tests for their determination are relatively time-consuming, expensive and require well-prepared cores which are difficult to obtain in soft or highly weathered rocks. For these reasons, index such as Is(50) and BTS are frequently used to indirectly predict UCS and Est and empirical correlation functions between all these mechanical parameters are often proposed by researchers for the different geomaterials.In order to establish relationships between the mechanical parameters for the studied calcarenites, a statistical analysis was performed using the pairs of values obtained under different RH conditions. Specifically, different fitting equations were tested (linear, exponential, power, etc.) for each pair of properties and the functions with the highest R2 were chosen. In this sense, the best fits were obtained by using through-the-origin linear functions (This paper analyses the influence of environmental RH on the mechanical properties of porous calcarenite stones broadly used as construction materials in south-eastern Europe. For that, dry specimens of three calcarenite varieties were exposed to five different atmospheres with a constant RH ranging from 10 to 93%. Specifically, these RH environments were simulated in sealed containers through the proposed modified VET by using glycerol solutions at different concentrations. The dry specimens were stored inside the containers to promote the water vapour transference from the surrounding environment to the porous network of calcarenites until the corresponding equilibrium water content was reached. Subsequently, the specimens were mechanically tested. A discussion of the main results of this experimental study and their links or differences compared to previous works is included below.The dissimilar mechanical behaviour of tested calcarenites can be attributed to their different petrological and physical characteristics (). Particularly, the higher UCS, Est, BTS and Is(50) values exhibited by G-3 could be linked to: (a) the considerably much smaller porosity caused by the existence of sparry and siliceous fibrous cement that fills the interparticle and intraparticle pores and makes the grain well cemented (); and (b) the much greater apparent density and P- and S-wave velocities () in comparison with G-2 and G-1 varieties. However, despite G-1 exhibited lower apparent density and greater total porosity than G-2, the achieved mechanical properties values were higher. This result may be explained by the fact that G-2 showed smaller ultrasonic velocities values (), a less homogeneous micro-fabric and texture (e.g. a larger variety of mineral crystal sizes of the terrigenous components and good sorted carbonate grains) (). For these reasons, the typical negative correlation functions between porosity and the mechanical strength values found by others authors for several rock types () could not be established for the whole dataset of tested calcarenites.The fact that the tested geomaterials exhibit diverse porous structures and textural characteristics also suggests that their durability could be significantly different (). In this connection, despite the three lithotypes exhibit high open porosity values (>11%) and, therefore, they are potentially vulnerable to salt crystallization and frost actions (), the greater presence of small pores in G-3 compared to the other varieties (and specially of micropores with diameter less than 0.1 μm), may mean that it would be the most susceptible lithotype against freeze-thaw (). Nevertheless, the differences in durability are also linked to other microfabric properties such as the size of carbonate grains, the cement type () or the presence of expandable clay minerals (). In this connection, G-2 lithotype showed the largest variety of size of calcite grains, a high bioturbation and presented several clay minerals (chlorite and smectite). Therefore, this variety could also display an important weathering when subjected to moisture environments (). In any case, the estimation of durability of stones from their physico-mechanical properties and petrological characteristics presents some limitations and hence, complementary approaches such as accelerated laboratory durability tests, complex environmental testing or exposure site testing are usually required to obtain a more reliable evaluation (The used modified VET has proven itself as an adequate method to prepare calcarenite specimens with small and accurate values of water content (w = 0.03–1.79%) or degree of saturation (Sr = 0.2–35.6%). Nevertheless, the maximum w or Sr values achievable and the time required to obtain them were highly dependent on the microstructure of the geomaterials (i.e. porosity, pore size and tortuosity). Similar finding was pointed out by for different types of limestones. The equilibrium was assumed to be achieved when the water content inside the pore network of geomaterials was constant. This time was generally high and varied largely from one calcarenite variety to another. Particularly, the calcarenite variety with the lowest porosity and the highest tortuosity and specific surface area (G-3) required more time than the other varieties (G-1 and G-2), which exhibited quite similar values for all these properties. In this connection, future studies aimed at shortening the equilibrium time required in stone specimens by improving the traditional VET would be highly recommended to perform by following the steps taken by soil mechanics researchers (Mechanical properties of calcarenites were significantly reduced when environmental RH increased from 10 to 93% because it implied an increment of the water content inside their pore network. In this line, despite the facts that the specimens exposed to these RH atmospheres were only partially saturated and that their Sr values were considerable small (less than 36%), the percentage of decreases were 28.2–34.7% in UCS, 20.0–31.3% in Est, 17.0–41.3% in BTS and 23.9–37.6% in Is(50). In this sense, a previous work performed by showed that the reductions of UCS, Est and Is(50) after the full saturation of the specimens by immersion under vacuum conditions were 29.8–61.9%, 22.6–68.3% and 34.6–55.6%, respectively. Therefore, an important weakening of calcarenites happens far from saturation, indicating that it is not necessary to reach a full saturation to significantly debilitate these materials. These results are in agreement with those found in chalks by , who reported that triaxial compression strength, Est and cohesion values approximately dropped by 40–60% when RH increased from 5 to 98%. Similar results were obtained by , who noted that UCS and Est decreased by 50–70% when water content increased from 0 to 30% due to the suction reduction. Significant decreases of Est (around 50–75%) were also reported in mudstones when RH increased from 21 or 32 to 99% (). In addition, the reductions of BTS found in calcarenites were in consonance with those obtained in slates by , who reported a drop of 67% when RH increased from 55 to 92%. Additionally, the fact that the existing regulations for the commercialization of these materials do not consider this phenomenon and are mainly focused on determining its vulnerability to weathering processes (like salt crystallization, atmospheric pollutants or freeze-thaw cycles) suggests the need for developing specific standards to assess the sensitivity of the mechanical parameters of building stones to the environmental RH.The variations of the abovementioned mechanical properties (UCS, Est, BTS and Is(50)) with the environmental RH were well fitted through linear functions for the three calcarenites, which is in line with the results reported in some cited previous works (). Concerning the relationships between the mechanical parameters and water content inside the pore network of calcarenites, tri-parametric negative exponential functions fitted very well to the data, which is in consonance with the findings obtained in several rock types like sandstones (The mechanisms used to explain the reduction of the mechanical properties of geomaterials produced by water (in liquid and vapour form) are diverse and complex. Specifically, the following causes are pointed out in the literature: fracture energy decrease (surface energy loss), capillary tension and suction decreases, pore pressure increase, frictional reduction and chemical and corrosive deterioration (). Out of these possible causes, the capillary suction decrease and the chemical and corrosive deterioration could be the two main weakening mechanisms in the partially saturated calcarenites since pore pressure increase and frictional reduction would be very limited in this scenario because the specimens present very small Sr values.Regarding to the first main mechanism, it should be reminder that total suction s (MPa) is defined as the adherence force per unit of area between the water and the surface of the material. It consists in an internal confinement stress which provides cohesive strength to the geomaterials that is linked to the RH through the Kelvin formula (where T is the temperature (K), ρw is the density of pure water at 273 K (998 kg/m3), R is the gas constant (8.314 J∙mol−1· K−1) and Mw the molecular mass of water (0.018 kg/mol). This expression indicates that s decreases as RH increases. Therefore, this could justify the obtained reduction of the mechanical properties of calcarenites when RH rises.). Furthermore, the water vapour condensation may also cause other microscopic physicochemical changes such as the reduction of the cement quality, the deterioration of the intergranular bonds, the dissolution of some minerals (calcite) and the swelling of the clay minerals (). In this connection, it should be recalled that the tested calcarenites have small quantities of phyllosilicates (smectite, chlorite, glauconite and palygorskite) which could be expanded by adding interlayer water molecules into their structure and subsequently generate microcracks due to unequal local pressures.Another important contribution of this work is the proposal of through-origin linear correlations between the different mechanical parameters of partially saturated calcarenites. These ratios allow the indirect evaluation of mechanical parameters such UCS and Est through less expensive and rapid methods such as Brazilian or Point Load tests, which do not require so careful preparation of the specimens and could even perform at field (PLT) (The results showed that the ratio between UCS and Is(50) (kUI=UCS/Is(50)) was 9.97 for the whole dataset. This value is significantly lower than the range of values specified in the ISRM Suggested Method () for several geomaterials (15 to 50). Furthermore, the obtained result was in line with those reported in soft rock materials such as porous chalks (kUI = 8–18) () or harbour dredge materials (kUI = 8–15) (Regarding the ratio between UCS and BTS (kUB=UCS/BTS), the obtained value was 6.77. This ratio is frequently used as an indicator of rock brittleness and ranges widely depending on the type of rock ( reported that kUB = 19–28 for coal, kUB = 6–24 for argillaceous and shales or kUB = 12–17 for sandstones and siltstones. However, the kUB value obtained for calcarenites by was considerably lower (kUB = 4.9–6.2), which is in agreement with the findings of this work. With respect to the ratios between Est and Is(50) (kEI = Est/Is(50)) and between Est and BTS (kEB = Est/BTS) the obtained values were kEI = 6.72 and kEB = 4.51, respectively. These numbers were smaller than those proposed by for limestones and sandstones (kEI = 11.4) but greater than those found by Concerning the ratio between Est and UCS (kEU = Est/UCS), the measured value was kEU = 0.67 for the whole dataset, which is an intermediate number between the results reported by (kEU = 0.81) for limestones. With regard to the ratio between both indirect index (kBI=BTS/Is(50)), the obtained value was kBI = 1.47, which is quite similar to those reported by for weak conglomerate (kBI = 1.6) and by for several rock types (kBI = 1.5). The different ratios (k) were calculated by expressing Est in GPa and UCS, Is(50) and BTS in MPa.The most relevant findings derived from this study are summarized below:A novel modified Vapour Equilibrium Technique is used to expose stone specimens to controlled relative humidity and temperature environments in laboratory. This methodology consists on entering the stone samples together with a water-glycerol solution with a specific concentration within sealed containers hosted inside a temperature-controlled oven, promoting the water vapour exchange between the specimens and the surrounding atmosphere until the equilibrium is reached. This procedure enables to successfully introduce small and accurate quantities of water inside the pore network of stones, which can be used to evaluate their mechanical vulnerability to water vapour. Specifically, in the tested calcarenites, equilibrium values of Sr ranging from 0.2 to 35.6% could be reached when the environmental RH varies from 10 to 93%. Furthermore, these equilibrium values of Sr (or w) and the time necessary to reach them are greatly dependent on microstructural properties like porosity, pore diameter or tortuosity.The mechanical parameters of calcarenites significantly decrease when RH of their surrounding environment increase. Particularly, the reductions of UCS, Est, BTS and Is(50) generally fluctuate from 17 to 41% when the environmental RH ranges from 10 to 93%. As a consequence, significant mechanical weakening of calcarenites occurs even when their water content is small, indicating that the fully saturation of their pore network is not required for their debilitation. In this sense, the relationship between the mechanical properties outlined above and the environmental RH can be well modelled though linear functions, while the variations of them with the water content inside their pore network can be well fitted by using tri-parametric negative exponential functions.This work offers an in-depth insight into the effect of relative humidity on the mechanical properties of porous calcarenite stones through a proposed novel modified VET procedure. The obtained results will lead to precisely assess the vulnerability of construction elements made of these materials when exposed to different RH environments during their lifetime and also to optimise the selection of the most appropriate calcarenites at quarry level depending on the atmospheres where they are utilised. In addition, the obtained ratios between the mechanical properties of the partially saturated tested calcarenites could be used to indirectly assess time-consuming and expensive parameters (UCS and Est) through Brazilian or Point Load Tests, which do not require so meticulous preparation of the samples and even could perform out of the laboratory.The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.Determination of the Paris law constants in round bars from beach marks on fracture surfaces► The Paris law constants are determined using a reverse engineering technique. ► Beach marking method is used to evaluate the shape and depth growth of fatigue cracks. ► Numerical simulations of fatigue crack shape for different Paris law constants are performed. ► The constants are found by comparing the numerical predictions with the experimental results. ► The technique is successfully applied to 12 mm-diameter round bars with errors less than 5%.In this article, a 3-step reverse engineering technique capable of determining the Paris law constants from the analysis of final fracture surfaces of fatigued round bars is proposed. Initially, at least two crack shapes and the number of cycles between them are obtained experimentally. Then, a 3D-FE automatic fatigue crack growth technique, able to predict the crack shape evolution and the number of fatigue cycles, is employed. Finally, a comparison between numerical and experimental results in terms of crack shapes and fatigue lives is carried out in order to find the Paris law constants that best fit the numerical predictions to the experimental results. This technique has been successfully applied to circular cross-section specimens made of S45 steel. Differences between experimental and predicted C and m constants less than 5% and 3%, respectively, were found.crack growth increment of ith node for jth iterationmaximum crack growth increment for jth iterationrange of SIF value of ith node for jth iterationdisplacement components in x, y, z directionsMechanical components with circular cross-sections, such as shafts, bolts, screws, wires, etc. play an important role in engineering. In this kind of components, fatigue, as a consequence of crack initiation and propagation, is recognised as one of the most common failure modes. As is well-known, fatigue crack propagation life can be estimated using the Paris law,where C and m are material constants; and ΔK is the stress intensity factor range. These two material constants are usually calculated using a well-established procedure, based on standard specimens, notched and pre-cracked (BS ISO 12108:2002 and ASTM 647-08e1). The specimens commonly employed (the middle-crack tension and the compact tension geometries) have mode-I through cracks with nearly straight stable shapes without significant shape changes during the propagation. Nevertheless, when considerable shape changes occur In this article, a 3-step reverse engineering technique, able to determine the Paris law constants in fatigued round bars from the observation of fatigue crack shapes on fracture surfaces, is presented. Firstly, an experimental test is performed in order to obtain, at least, two different crack shapes on fracture surface of the round bar as well as the number of cycles between them. Secondly, the crack shape evolution is predicted by employing a 3D-FE automatic crack growth technique The experimental fatigue crack growth tests were conducted using 12 mm-diameter and 190 mm-long round bar specimens (a). The initial crack was a straight edge crack with 1 mm-depth which was created by using a linear cutting machine (b). Both ends of the specimen, with a diameter of 15 mm and length of 50 mm, were fixed to the grips of the machine using button-head connections. In order to minimise bending strains, the major axis of specimen was aligned with the load axis. The material used in this study was the carbon steel S45. Its mechanical properties are given in A MTS809 servo-hydraulic test machine was used to apply the cyclic loading to the specimen. The tests were performed at room temperature, in load control, under a maximum cyclic tension of 25 kN, with a 15-Hz sinusoidal-waveform and stress ratio R |
= 0.1. The shape and depth growth of fatigue cracks were monitored using a zoom stereomicroscope and beach marking. The beach marks were produced by changing the stress ratio, i.e. the applied load was reduced to one-half for several cycles.a exhibits the typical beach marks obtained in the fatigue crack growth tests. From the experimental data, several visible crack shapes, at different places of cross-section, were selected. This selection was based on a careful analysis of the crack shape in terms of symmetry. The experimental crack shapes used in the determination of C and m constants (1, 2, A and B) are presented in shows the polar coordinates (r,θ) of each one of them measured according to the referential schematised in a. Only half of the crack front was analysed from which seventeen points were obtained. The numbers of load cycles applied experimentally between the various crack shapes analysed are presented in . Based on the experimental data, the fatigue crack growth rate calculated in the depth direction is given by Yang et al. i.e., m |
= 3.256 and C |
= 1.9037 × 10−9 (da/dN [mm/cycle] and ΔK [MPa m0.5]). The crack length was measured by analysing the beach marks. For further details of the experimental determination of the Paris law constants, please see Yang et al. The numerical simulation was performed using Lynx, a specific software capable of simulating fatigue crack growth of planar cracks under mode-I constant amplitude loading . Firstly, a numerical model representative of the cracked body is created (a). This step encompasses the definition of the geometry, boundary conditions, loading, crack shape, elastic constants and fatigue crack growth rate. Secondly, the displacement field of the crack front nodes is calculated (b). The pre-processor GeoStar®, included in the commercial FEM package Cosmos/M®, is used to solve the FE model. Thirdly, the stress intensity factors along the crack front are computed (c) using the two-point extrapolation method where Vp is the crack opening displacement, r is the radial distance from the crack tip, E′ is the modified Young modulus, being E’ |
= |
E/(1 − |
v2) in plane strain state and E’ |
= |
E in plane stress state, and v is Poisson’s ratio. Fourthly, an adequate crack growth model is applied in order to define the crack front advances (d) and to estimate the corresponding number of fatigue cycles. The propagation at each crack front node, under remote mode-I loading, is assumed to occur along the direction normal to the crack front. The crack increment, at an arbitrary node along the crack front, derived from the Paris law, can be expressed asbeing Δai(j) the crack growth increment of ith node for jth iteration, Δamax(j) the maximum crack growth increment for jth iteration, ΔKi(j) the SIF range of ith node for jth iteration, ΔKmax(j) the maximum SIF range for jth iteration, and m the Paris law exponent. According to Eq. , it becomes clear that the m exponent of the Paris law is the unique material parameter required to calculate the crack increment. The number of loading cycles can be calculated by employing the following expression.N(j+1)=N(j)+ΔN(j)⇔N(j+1)=N(j)+Δamax(j)C[ΔKmax(j)]mFinally, the positions of corner and mid-side nodes of the crack front are moved to their final positions (e) by employing a third order cubic spline function that passes through the provisional corner nodes. This approach gives more realistic crack front shapes and therefore more accurate numerical results. The final crack front is then used directly as the input data of the next iteration and the procedure is repeated as long as no critical values of fracture toughness or crack length are reached.The physical model considered in this research is shown in . Due to material, loading and geometry symmetries, only a quarter of the specimen was simulated. The geometry of ends was simplified taking into consideration their remote position relatively to the crack front. Movements along x and z (δx |
= δz |
= 0) were restrained to simulate the constraint imposed by the high rigidity of the loading machine grips. The crack was assumed to be plane, normal to the axis of the specimen, and existing in its middle-section. Therefore, mode-I loading is expected along the whole crack front. The material was assumed to be homogeneous, isotropic and with linear elastic behaviour. presents the finite element model developed. The crack front was divided into seventeen corner nodes (d) and sixteen mid-side nodes. A refined region nearby the free surface of round bar was considered. A spider web mesh consisting of three concentric rings with five elements arranged surrounding the crack front was created (e). The collapsed 20-node isoparametric element was used in the first concentric ring. The mid-side nodes were moved to quarter-point positions (c). The standard 20-node isoparametric element (a) was employed in the other two concentric rings of the spider web mesh as well as in the transition and regular meshes. The assembled model (g) had 71743 nodes and 7232 elements. The stress intensity factors at corner nodes were computed using the points A and B identified in e. A plane strain condition was assumed for all positions except at the free boundary where plane stress state prevailed.The numerical procedure was carefully optimised. Several modelling decisions were made taking into account literature results and authors experience e) centred on the crack front; the use of singular elements (c) to simulate the stress singularity; the use of a cubic spline to define the crack front; and the use of a well-known K calculation method. Nevertheless, other important variables were optimised through a specific parametric study. The radial size of crack front elements (L1) plays an important role in the simulation of crack tip stress singularity and its optimum value depends on the average size of the singular region at the crack tip The crack shape and the crack length were analysed by using the well-known dependent parameters crack aspect ratio (a/b) and dimensionless crack length (a/D). The variables a and b represent the semi-axes of an ellipse whose centre is coincident with the origin of the coordinate system schematised in . The prediction of the m constant was based on the accumulated difference (ad) defined by the following expressionbeing di the difference between the numerical and the experimental coordinates at the ith node of crack front and n the number of nodes. This global parameter analyses the entire crack front and is equal to zero only when both experimental and numerical crack shapes are superimposed. presents typical fatigue crack shape developments obtained after the optimisation of the numerical procedure (m |
= 3). Three different surface crack configurations subjected to constant amplitude cyclic tension are examined, namely a part-circular crack (a0/b0=1), part-elliptical crack (a0/b0=0.4) and straight crack (a0/b0=0). A strong dependence on the initial crack shape is observed. It is clear that for the initial straight shape (a0/b0 |
= 0), the crack grows much more rapidly in depth direction than along the free surface whilst for the part-circular shape (a0/b0 |
= 1) the growing is more balanced along the whole crack front. However, this relevant effect of the initial crack configuration existing in the early stage is gradually weakened as the crack extends and consequently the crack fronts become similar. The amount of crack growth needed to achieve this part of propagation also depends on the initial crack configuration and therefore shapes closer to the preferred propagation path reaches it faster than the others.The numerical predictions were compared with the literature results for validation. a plots a/b against a/D for different initial crack configurations (a0/b0=0, a0/b0=0.2, a0/b0=0.4, a0/b0=0.6, a0/b0=0.8, a0/b0=1) with the same length (a0/D=0.1). As can be seen, the trajectory drawn by the crack strongly depends on the initial crack aspect ratio. Nevertheless, this high dependence on initial crack shape gradually weakens, as the crack propagates, leading the crack shape to preferred propagation paths. Additionally, it is possible to observe a good agreement between the numerical results presented here and those found in the scientific literature a. In the early propagation period, no relevant changes are distinguished. However, as the crack grows, different propagation paths are clearly observed. On the other hand, after a certain time, the propagation paths cross-over (a/D |
≈ 0.48) following from then on divergent trajectories. The cross-over point depends on several variables such as the value of m, loading type and initial crack shape a (a0/b0 |
= 0 and a0/D |
= 0.1), the three curves tend to follow an unequivocal path for values of a/D greater than 0.2. In that sense, all experimental crack shapes used in this study were selected, as indicated in b shows the variation of Kmin/Kmax along the crack front during the crack growth for a surface crack of a round bar subjected to constant amplitude cyclic tension. Different initial crack configurations (a0/b0 |
= 0, a0/b0 |
= 1) with the same dimensionless crack length (a0/D |
= 0.1) were considered. This ratio (Kmin/Kmax) is interesting to characterise the SIF variations along the crack front with the crack growth. It increases suddenly at the early stage and then goes down slightly to values about 0.85–0.88. After that, the Kmin/Kmax ratio rises up slightly towards values close to unity (Kmin/Kmax |
≈ 0.98). Additionally, at the early stage, the gradient of ratio Kmin/Kmax is less intense for the initial part-circular crack shape (a0/b0 |
= 1) than for the initial straight crack shape (a0/b0 |
= 0). This explains the more significant shape changes observed in the latter case (c). Moreover, the results predicted in this study are in excellent agreement with those of Lin and Smith The first step encompassed the determination of the m constant. A set of numerical simulations was performed using values of m contained in the interval 2.6–3.6 (da/dN [mm/cycle], ΔK [MPa m0.5]). The crack shape 1 (a/D |
= 0.253), exhibited in b, was defined as the initial crack configuration. Each numerical simulation was interrupted when the first node of crack front (node 1 of d) reached the length of crack shape 2 (a/D |
= 0.418) of b. Then, for each case, the values of the accumulated difference (ad) between the numerical and experimental crack shapes were computed using Eq. . This dependent parameter has proven to be very sensitive to any shape change therefore its use is recommended for this purpose. In corner crack specimens, a different global parameter able to characterise asymmetrical crack shapes has been used a plots the accumulated difference (ad) against the Paris law exponent. A well-defined tendency emerges from the results. This is clear evidence of the suitability of the selected dependent parameter. The resulting data were fitted by the least square method to a second order polynomial function with a relatively high square correlation (r2 |
= 0.995). In theory, the correct value of m can be found by minimising the value of ad. So, it means that the derivative of ad (ad′) must be equal to zero: ad′(m)=0⇔m=3.102 (da/dN [mm/cycle], ΔK [MPa m0.5]). The error involved in the prediction of m, in relation to the experimental values, is of 4.73% which is clearly acceptable in this context. Note that due to the relatively low stress ratio (R |
= 0.1) used, crack closure can exist which can affect the crack shape and therefore the m prediction. In order to avoid such a phenomenon, higher stress ratios can be adopted.The second step comprised the determination of the C constant. The value of m previously predicted was fixed. Then, a new set of numerical simulations was performed using values of C within the range of 7 × 10−9 to 2.4 × 10−9 (da/dN [mm/cycle], ΔK [MPa m0.5]). These simulations were computed using a maximum crack growth increment equal to D/2000. In a similar manner, the crack shape 1 (a/D |
= 0.253) of b was considered as the initial crack configuration. Each numerical simulation was interrupted when the first node of crack front (node 1 of d) reached the length of crack shape 2 (a/D |
= 0.418) of b. The number of loading cycles between these two prescribed crack shapes, computed using Eq. , was the result of the numerical simulation. The determination of C could also be done by integrating the Paris law (Eq. ). However, this approach requires a SIF-based solution which, in general, is computed with a certain level of error. Due to this fact and in order to take advantage from the numerical procedure developed here, the implementation of a SIF-based solution was not followed. b compares the number of loading cycles predicted with the number of cycles obtained experimentally. An exponential function was fitted to the data by employing the least square method (r2 |
= 0.990). The C constant was found by equalising the fitted function to the experimental number of cycles, i.e. solving the following equation: N(C)=46523⇔C=1.8489×10-9 (da/dN [mm/cycle], ΔK [MPa m0.5]). Numerical and experimental results are in good agreement. Note that the difference between both values is less than 2.87%. Naturally, this error magnitude can be considered acceptable.In order to evaluate the robustness of the proposed technique, the procedure was carefully repeated for other combinations of experimental crack shapes exhibited in b. Two different situations, corresponding to the combinations 1 to B (1–B) and A to 2 (A–2), were studied. For each of them, as described previously, the accumulated difference was computed through Eq. for various values of m. From the values of ad, a second order polynomial function was fitted. Then, the exponent of the Paris law was achieved by minimising this function. Next, with the predicted value of m, new numerical simulations were performed for different C constants aiming at obtaining the numerical number of cycles between both crack shapes. These data were fitted to an exponential function again. Equalising this function to the corresponding experimental number of cycles presented in , the C constant was calculated. Finally, the predicted constants were compared with the experimental values. presents the achieved C and m constants as well as the errors relatively to the experimental results. Regardless of the combination analysed, the error magnitude is similar. For example, in the combination 1–B, no significant effects were introduced due to the use of a different first visible crack shape; in the combination B–2, the influence of a different second visible crack shape was not relevant. These results reinforce the robustness of the proposed approach. The means of the C and m values (C |
= 1.8629 × 10−9 and m |
= 3.101) were considered the final predictions of the Paris law constants. Notwithstanding, taking into account the consistency of the predictions presented, this assumption is not expected to have a strong influence on the following analysis.The final numerical values of C and m were used to carry out an entire simulation from the initial straight crack shape used in the experimental tests. Then, the numerical and experimental crack shapes were compared. This comparison was done through the difference di, schematised in , assuming that both crack shapes were superimposed at the symmetry line of the cross-section of specimen. For each crack front, the evolution of di/ri with θi was computed, being ri the experimental radius. The results calculated are presented in for crack fronts X, 1, A, B, 2 and Y of b. As can be seen, these values have well-defined limits that vary between −1% and 5%. Besides, although some exceptions are observed, it is possible to distinguish a dominant tendency for the curves in which the differences increase progressively towards the surface. These results demonstrate that the reverse engineering technique proposed here is able to obtain the Paris law constants from materials in the form of round bars.In the present paper, a reverse engineering technique capable of determining both Paris law constants from the analysis of fracture surfaces of fatigued round bars was proposed. The technique consists of three main tasks. Firstly, experimental work aiming at obtaining at least two visible crack shapes and the number of loading cycles between them is performed. Secondly, a three-dimensional fatigue crack growth technique able to predict the crack shape evolution and the fatigue life is employed. Thirdly, a set of numerical simulations is carried out by using different values of m. The m constant is found by minimising the shape differences between the experimental and numerical crack fronts. Then, using the calculated m value, new numerical simulations are done for different C values. The correct C value is found by equalising both the numerical and the experimental numbers of loading cycles.This technique was successfully applied to a 12 mm-diameter and 190 mm-long circular cross-section specimen made of S45 steel subjected to tension loading. Experimental work was developed at constant amplitude ratio (R |
= 0.1) to obtain visible crack shapes and the number of cycles between them. Marking was produced by changing the stress ratio. A three-dimensional finite element model was created to predict the crack shape evolution and the fatigue life by using the FCG software Lynx. The numerical procedure was optimised and the results of crack shape evolution and stress intensity factors were successfully compared with values published by other authors. Additionally, as reported in the literature, it was demonstrated that the initial crack configuration affects significantly the early propagation period while the subsequent propagation is influenced by the Paris law exponent.Three predictions of C and m from three different combinations of experimental crack shapes were carried out. The values found were clearly convergent. The maximum errors in C and m, relatively to the experimental constants, were 4.82% and 2.87%, respectively. Although not perfect, both errors are entirely acceptable in this context. The proposed values of C and m were calculated by means of three predictions obtained from different crack shapes, being C |
= 1.8629 × 10−9 and m |
= 3.101, respectively. The differences between the numerical and the experimental results evidence accurate predictions in C and m and therefore the reverse engineering technique proposed here is very promising for the determination of the Paris law constants of materials in the form of round bars.A comprehensive review on effect of process parameters and heat treatment on tensile strength of additively manufactured Inconel-625Additive manufacturing (AM) is a novel deposition technique to fabricate a 3D complex component. In AM, fabricating complex geometries is easy to fabricate due to layer-by-layer deposition and provides many advantages over conventional processes. AM technique is used in many industries like aerospace, automobile, marine etc. In the Selective laser melting (SLM) process, the laser is used as a heat source to fuse the material. The mechanical and microstructural properties of SLM samples are good due to high cooling rate. The Inconel 625 superalloy has superior properties in strength, good fatigue, and high creep resistance. Inconel is known for a high-temperature application in the field of aerospace and automobile. Tensile strength is one of the important factors to evaluate the quality of the build part. This comprehensive review focuses on ultimate tensile strength, elongation, yield strength, process parameter and heat treatment conditions. The effect of process parameters such as laser power, scanning speed and orientation on tensile strength is covered. It also includes effect of different heat treatment such as solution annealing, direct aging, thermal exposure and solution treatment on tensile strength, yield strength and elongation. This study would be helpful to optimum process parameters and post-processing techniques for improving tensile strength.Additive manufacturing (AM) is a trending technology in manufacturing industries shows the schematic diagram of the laser powder bed fusion technique. Powder bed fusion technique consists of material in the form of powder on a bed, then with the help of rollers, the material is swiped and fused by laser and this process again repeat with several layers up to the final height. This technique is also known as selective laser melting (SLM), direct laser metal sintering (DMLS) In this process, various process parameters contribute to building a 3D product like Laser power, layer thickness, scanning speed, and hatch space It gives the heat input according to the selection of the process parameter. The effect of the process parameter on building a part is important. If the process parameter is not optimum, then there are chances of formation of cracks, porosity, balling effect and distortion In laser powder-based fusion technique, fine-grain were formed due to high Cooling rate, which leads to better mechanical properties than the conventionally fabricated part. Tensile strength is an important factor for mechanical properties to check whether the product is good enough or not. Tensile strength varies by two major things, process parameter and post-heat treatment. In AM process, tensile strength varies with laser power, hatch space, layer thickness, scan speed, and energy density. In post-heat treatment tensile strength varies by direct aging (DA), stress relieve annealing (SRA), solution treatment (SOT). Solution annealing (SA), thermal exposure. Various other factors can change the tensile strength such as densification, the orientation of the part. All the conditions were shown in . LPBF had better mechanical properties than binder jet, like UTS, elongation and yield strength. They also investigated build direction to the mechanical properties they have found that get better properties in horizontal build part shows the relation between ultimate tensile strength to laser power. Observed from the figure increasing laser power up to certain value tensile strength also increases. However, by further increasing laser power it decreases the tensile strength. Up to 200 W laser power energy, heat input is enough to fuse powders properly. Further, increase a laser power up to 300 W it causes to overheat input and it leads to vaporized powders and forming a coarse grain. The trend of the UTS is increased and then decrease to increase the laser power. show effect of scanning speed on UTS. It depicts that UTS decreases with increase in scanning speed. The heat input decreases with increase in scanning speed and result in unfused, which creates porosity in part. Therefore, the value of UTS is reduced. They also build a component in vertical direction and found corresponding value of UTS 934 MPa, 966 MPa, 840 MPa and 749 MPa. Hence, in vertical direction properties were not good compared to horizontal. However, in horizontal and vertical direction maximum UTS observed was at 44 J/mm3 energy density and UTS decreases with increase in scanning speed.In laser powder bed fusion, part undergoes a thermal cycle of rapid heating and cooling. It generates anisotropy properties as well as and high amount of residual stress. This leads to deteriorating mechanical properties. To eliminate this problem, heat treatment required on as a build condition of additive manufactured part. Heat treatment can decrease the microstructure effect on directionally and generate isotropy properties all over the part, which leads to improve the mechanical properties. Effect of different heat treatments on tensile strength given in shows the effect of different heat treatment on UTS. It depicts that lowest UTS with HIP and comparatively more UTS in a Solution treatment. In a Recrystallization annealing 820 Mpa and maximum UTS get 910 MPa in stress relief annealing. Maximum elongation was observed in HIP condition where UTS got minimum and minimum elongation observed in stress relief annealing where UTS got minimum. As tensile strength decreases as elongation increases. shows the different heat treatment process to the tensile strength. It depicts that the maximum tensile strength obtains 1137 MPa at 600C and minimum strength obtain 977 MPa at 900C in thermal exposure condition. In the case of Solution heat treatment at a temperature of 700C it gives high UTS 1187 MPA, and the lowest UTS observed in a SOL at a temperature of 900C value of UTS 920 MPa. All SLM process parameters were the same for all heat treatment conditions, which gives a heat input of 90 J/mm3. In thermal exposures, as the temperature increases, tensile strength decreases and elongation increases. In solution treatment up to 700C, tensile strength decreases, then increases. Marchese et al. shows the different heat treatment to tensile strength. From the figure observed that In any case of heat treatment, the horizontal build direction gives a maximum UTS compared to the vertical direction. In both case of horizontal and vertical build direction, at stress relief condition gives the highest value and solution annealing gives the lowest value of UTS as shown in . The samples were prepared at a constant energy density of 90 j/mm3 then various heat treatment processes are used to improve tensile strength.In additive manufacturing, laser powder bed fusion technique has great potential for fabricating complex geometries. Current literature focuses on tensile strength, yield strength, elongation, energy density and different heat-treatment process. Tensile strength is an important parameter to finalize the product quality of the additively manufactured part. Researchers have performed a tensile strength with different process parameters, different build direction, different additive manufacturing processes, and different heat-treatment processes. As per the literature mentioned, UTS was observed higher in a horizontal direction compared to a vertical direction because in the horizontal direction, tensile stress is applied in parallel to the layer and a vertical direction it is perpendicular to the layer. As scanning speed increases, tensile strength decreases because of low heat input, leading to partial melting of metal powder. As laser power increases, tensile strength also increases, but up to certain laser power, the further increment in laser power reduces tensile strength. As laser power increases, heat input increases and makes a proper fusion of powders, but as laser power increases, it impacts as overheat input. It leads to coarse grain and vaporization of powders it produced porous structures. In the stress relief process, good UTS was observed compared to recrystallization and solution treatment post process. It works as to regain the ductility and improves tensile strength. The direct aging post-process has made a higher strength compared to all other post-processing. As the thermal exposure of any heat treatment process increases, it decreases the UTS and subsequently increases elongation. The future study can focus on density, hardness, microstructure in correlation with heat treatment and different conditions.The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.The extreme value distribution and dynamic reliability analysis of nonlinear structures with uncertain parametersA new approach for evaluation of the extreme value distribution and dynamic reliability assessment of nonlinear structures with uncertain parameters is proposed. The approach is established based on the thoughts of the newly developed probability density evolution method, which is capable of capturing the instantaneous probability density function (PDF) and its evolution of the responses of nonlinear stochastic structures. In the proposed method, a virtual stochastic process associated to the extreme value of the studied stochastic process is firstly constructed in such a way that the extreme value equals the value of the virtual stochastic process at a certain “instant of time”. The probability density evolution method is then employed to evaluate the instantaneous PDF of the virtual stochastic process. This will yield the PDF of the extreme value as a natural byproduct. After that, dynamic reliability could be evaluated from the extreme value distribution, instead of the level-crossing process theory. A simple integration in terms of the extreme value distribution over the safe domain will give the dynamic reliability, requiring neither the joint PDF of the response and its velocity, nor the assumptions on properties of the level-crossing events. Numerical algorithm is outlined. Two examples, of which one is to capture the extreme value distribution of the random sampling, the other is to evaluate the extreme value distribution and dynamic reliability of a nonlinear stochastic structure, are studied. Some features of the instantaneous PDF and its evolution, the extreme value distribution and the dynamic reliability are discussed. The investigations indicate that the proposed approach is of versatility, accuracy and efficiency.The extreme value distributions of random variables and stochastic processes are of paramount importance in engineering, particularly in reliability evaluation and engineering risk analysis Since pioneering researches of Fisher and Tippet For the first-passage problems, the reliability evaluation is intimately related to the extreme value distribution. Similar to the situation in the theory of the extreme value distribution, the level-crossing process theory is one of the most widely-used theories in dynamic reliability evaluation In recent years, a family of probability density evolution method (PDEM), which is capable of capturing the instantaneous PDF and its evolution of the response of structures involving random parameters, has been developed and used successfully in linear and nonlinear dynamical systems As mentioned above, a family of probability density evolution method (PDEM) for analysis of nonlinear stochastic dynamical systems, including multi-degree-of-freedom nonlinear structures involving random parameters and stochastic excitations, has been developed Without loss of generality, the state equation of a nonlinear stochastic dynamical system readswhere X=(X1,X2,…,Xnd)T is the state vector; nd the dimension of the state space; A=(A1,A1,…,And)T the dynamics operator vector; Θ=(Θ1,Θ2,…,ΘnΘ) the random parameter vector with known PDF pΘ(θ); θ=(θ1,θ2,⋯,θnΘ); nΘ is the number of the involved random parameters. It is noted that the random parameter vector Θ includes the stochastic parameters involved in the system properties and the external excitations. In the case that the excitation is a stochastic process, it could be transformed to a series composed of combination of a set of mutually independent random variables and deterministic time history functions with, say, the Karhunen–Loeve decomposition represents a stochastic dynamical system with randomness coming simultaneously from the initial conditions, the excitations and the system properties.The PDEM starts with the solution to Eq. itself. For a well-posed dynamics problem, the solution to the system exists, is unique and must be a function of Θ. It is convenient to assume the solution takes the formLikewise, the velocity of X also exists and is a function of Θ, and could be assumed to take the formwhere Xl,Hl,hl (l |
= 1,2,…,nd) are the lth component of X, H and h, respectively. For simplicity of writing, the subscript l will be omitted hereafter without inducing confusions.To make the physical sense clearer, employing X˙(θ,t) in place of h(θ,t) (Eq. It is easy to give the initial condition of Eq. ) is solved, the PDF of X(t) could then be evaluated bywhere ΩΘ is the distribution domain of Θ. are referred to as the generalized density evolution equation. Remarkably, it tells that even for a multi-dimensional dynamical system a one-dimensional partial differential equation governing the joint PDF exists. This was thought almost impossible in relevant previous investigations says that as long as the velocity of a physical quantity is known, the PDF of the quantity could be easily captured because varying of the PDF results from varying of the state of the quantity. This, as will be stressed later, makes it possible to obtain the PDF of arbitrary quantity of interest through constructing a virtual stochastic process.By the way, although the explicit expression of the velocity X˙(θ,t) needed in Eq. is usually unavailable, it has been shown that in the numerical algorithm the value of X˙(θ,t), which is obtainable through a deterministic dynamic analysis, instead of the expression itself, is essentially used. As a matter of fact, that is why the physical sense of Eq. The above outlined PDEM proves to be of accuracy and efficiency in dynamic response analysis of linear and nonlinear stochastic structures In general, the extreme value of a stochastic process is a random variable. As discussed in the section of introduction, how to get the extreme value distribution of a general stochastic process is still a difficult problem. Only some special results were achieved for some particular stochastic processes. In contrast, based on the above PDEM, the extreme value distribution could be evaluated through constructing a virtual stochastic process.Denote the extreme value of the response X(t) of the system For instance, if one considers the maximum absolute value of X(t),t |
∈ [0, |
T], Eq. , it is seen that the extreme value of X(t),t |
∈ [0, |
T] depends on Θ. Therefore, for convenience it could be assumed to take the formwhich means that the extreme value of X(t),t |
∈ [0, |
T] exists, is unique and a function of Θ and T.As discussed in the preceding section on physical sense of the generalized density evolution Eq. , if the PDF of arbitrary physical quantity is of interest, the information of the corresponding “velocity” associated with the quantity is essential and sufficient. This makes it possible to construct a stochastic process associated with the physical quantity under investigation and, simultaneously, the “velocity” of the constructed stochastic process is known. Consequently, through the PDEM the instantaneous PDF of the constructed stochastic process is obtainable which contains the probabilistic information of the physical quantity under investigation.According to the preceding thoughts, construct a virtual stochastic processwhere τ is somewhat like the time and is called as “virtual time”; Y(τ) is a “virtual stochastic process” whose randomness comes from the random vector Θ. on both sides with regard to τ will yieldwhere the overdot stands for differentiation with regards to τ., respectively. Therefore, the PDEM outlined in the above section could be employed here to obtain the PDF of Y(τ). Likewise, a generalized density evolution equation, i.e., the counterparts of Eq. where pYΘ(y,θ,τ) is the joint PDF of (Y(τ),Θ). it is seen that the extreme value Xext equals to the value of the virtual stochastic process Y(τ) at the instant of time τ |
= 1, i.e., one immediately gets the PDF of Xext byIt is seen that based on the PDEM the extreme value distribution of the response of a stochastic dynamical system is captured through constructing a virtual stochastic process, without inducing any additional approximation or assumption.Dynamic reliability of a nonlinear stochastic structure could be evaluated in a straightforward way through integration of the extreme value distribution. In general stochastic dynamical systems, the first passage reliability is defined aswhere Pr {·} denotes the probability of the random events; Ωs is the safe domain. In the case the symmetric double boundary is employed, Eq. in which xB is the value of the symmetric boundary.As discussed before, the accuracy of the widely-used level-crossing-process-based reliability theory could hardly be ensured. In contrast, viewed from the extreme value distribution dynamic reliability evaluation is straightforward and convenient.where Xext is the extreme value associated to the failure criterion. For instance, if the failure criterion in Eq. is used, Xext is essentially the extreme value defined in Eq. Since the extreme value distribution pXext(x) is captured in the above section in Eq. , it is quite easy to evaluate the reliability in Eq. In the case of the symmetric double boundary, the reliability readsIf the boundary xB is a random variable with the PDF pB(b), the dynamic reliability yieldswhere ΩB is the distribution domain of xB.From the above analysis, it is seen that viewed from the extreme value distribution, the problem of dynamic reliability evaluation is transformed to a simple integration problem. In contrast to the reliability theory based on the level-crossing process, the proposed method requires neither the joint PDF of the response and its velocity, nor the assumption on properties of the level-crossing events.To get the instantaneous PDF or other probabilistic indices of the response, a set of equations, consisting of the state Eq. , the generalized density evolution Eq. As is pointed out that, except for some special cases, a general closed form solution is usually unfeasible. Therefore, a numerical algorithm combining the solution of Eqs. is developed, where the solution of Eqs. gives the value of the coefficient in Eq. . The procedure includes the following steps:Select representative points θq(q |
= 1,2,…,Nsel) in the domain ΩΘ; Nsel is the total number of the selected points.Let Θ |
= |
θq. Solve the deterministic ordinary differential equation to evaluate the value of X˙(θq,tm), where tm |
= |
m |
· Δt (m |
= 0,1,2,…), Δt is the time step.Employ θq in place of θ, and X˙(θq,tm) in place of X˙(θ,t) in Eq. . Solve the partial differential equation with the finite difference method to obtain the discretized value pXΘ(xj,θq,tk), where xj |
= |
j |
· Δx (j |
= 0,±1,±2,…), Δx is the space step, tk=k·Δtˆ(k=0,1,2,…), Δtˆ is the time step in the finite difference method.For dynamic reliability evaluation, take the numerical integration in Eq. . In this case, in step (ii) the value of X˙(θq,tm) is obtained and then the extreme value in Eq. is easily captured; in step (iii) Eqs. is numerically integrated instead of Eq. The strategy of selecting points in step (i) is of paramount importance to the accuracy and efficiency of the PDEM. In the case that the number of the random parameters nΘ is small, for instance, nΘ |
= 1,2, the grid-type point set is feasible, which is exemplified in the present paper. In the case of nΘ |
⩾ 3, special strategies of selecting points, including the mapping-based dimension-reduction technique and the Number-Theoretical-Method-based algorithm, have been developed and are elaborated in other papers The step (ii) is a deterministic procedure of solving ordinary differential equation set, which is extensively studied is a conservative type partial differential equation, the numerical methods for which are well developed in computational fluid dynamics From the above steps, it is seen that a traditional deterministic dynamic response analysis process is embedded in the procedure. Additional finite difference method is then employed. This makes it convenient to use the existent commercial softwares.To verify the proposed method, two examples are presented. In example 1, the extreme value distribution (EVD) of random sampling is investigated where the closed form solution of the EVD is available. The EVD computed through the PDEM is compared with the closed form solution, indicating the accuracy of the proposed method. In example 2, the extreme value distribution of the response of nonlinear stochastic structures is computed. Dynamic reliabilities are evaluated and compared with the Monte Carlo simulation method (MCS) and other widely-used distribution. Some features of the EVD are discussed.Consider a random variable X with the PDF pX(x). Carrying out random sampling from the population, the samples (X1,X2,…,Xr) with the size r construct a set of mutually independent identically distributed random variables. Denote the extreme value of the samples asIn the mathematical statistics, the closed form solution of the PDF of Xext is available where PX(x) is the cumulated probability distribution functionObviously, in the case r |
= 1, pXmax(x)=pX(x) is the original PDF of the random variable.The EVD of the random sampling could also be computed by the proposed method. Note that Eq. , the procedure constructing a virtual stochastic process discussed in the preceding section could be used herein. In this case, Θ |
= (X1,X2,…,Xr). The step (ii) should be modified correspondingly such that the value of Xmax (θq) = max(θq) is evaluated where θq |
= (x1,q,x2,q,…, |
xr,q) (q |
= 1,2,…,Nsel), xj,q(j |
= 1,2,…,r) is the representative value of Xj. In step (iii), Eq. , where the coefficient W(θq,T) = |
Xmax(θq). shows the extreme value distribution evaluated by the PDEM, compared with the closed form solutions. The original PDFs are, respectivelyThe uniform distribution over the interval [1, 2], i.e., pX (x) = |
u(x |
− 1) − |
u(x |
− 2), where u(·) is the Heaviside’s function.The exponential distribution, i.e., pX(x) = |
λ |
e−λxu(x) with λ |
= 1.The normal distribution N(4, 1) with the mean = 4 and the standard deviation = 1. it is seen that the EVDs computed by the PDEM accord perfect well with the closed form (analytical) solutions, indicating high accuracy of the proposed method. The computed EVD could capture the precise features of the exact EVD. For instance, in (a) the extreme values should distribute (but not uniformly for r≠1) over the interval [1, 2]. It is found that the computed EVD gives an interval close to it. Also seen in (b) is the extreme values being larger than zero, i.e., distributing over the positive parts, which meets the requirement of the closed form solution when the original random variable is exponentially distributed. By the way, this may mean that the PDEM could give the distributing interval as a byproduct with more information than that could be provided by the interval method (a) is subjected to stochastic ground motion in the shape of scaled El Centro record. Lumped masses of each story are listed in . The geometric sizes of the structure read: the height h1 |
= 4 m, h |
= 3 m; the section of the columns = 500 mm × 500 mm; the EI of the beams →∞. Rayleigh damping is employed, i.e., C |
= |
aM |
+ |
bK, where a |
= 0.01, b |
= 0.005, K is the initial stiffness matrix. The restoring force adopts the Bouc–Wen model where k is the initial stiffness, x the inter-story drift and z is the hysteretic component satisfyingin which the parameters take the value α |
= 0.01, A |
= 1.2, β |
= 1.4, γ |
= 0.2 and n |
= 1.In the analysis, the initial Young’s modulus and the peak ground acceleration (PGA) are taken as random parameters with the probabilistic information listed in . Two cases are studied where the coefficients of variation of the random parameters are different.As stated earlier in dynamic response analysis a deterministic time-integration is embedded and then a finite difference method is carried out to capture the instantaneous PDF of the response. In the deterministic time-integration, a one-step algorithm proposed by Chung and Lee (b) shows typical hysteretic loops when the model is used in a single-degree-of-freedom system with the mass = 0.5 × 105 |
kg, the damping and the stiffness identical to that of the structure in (a) and the excitation x¨g=1000sin10t. Pictured in (c) is a typical realization of hysteretic loops of the bottom story of the structure in The probabilistic information of the top displacement of the structure is shown in it is seen that the mean and the standard deviation computed by the PDEM accord perfectly with those by the MCS. The coefficient of variation (C.O.V.) of the random parameters in reaches 0.2 and 0.25 but the accuracy of the PDEM does not deteriorate. This is a great advantage in comparison to the previous proposed approaches, say, the random perturbation method depicts the instantaneous PDFs of the response and its evolution against time. In (a) are the typical PDFs at certain instants of time. It is found that the PDFs are quite different from widely-used regular distribution such as the normal distribution, the lognormal distribution, etc. It is common for them to have two or more peaks. Evolution of the PDF against time constructs a PDF surface like a mountain ((c) seems like water flowing in a river with many whirl pools. In fact, it is evolution of the state that leads to the probability flow in the time-state space. These phenomena and features of the probabilistic information is analogous to that discussed in the previous literatures of Li and Chen The extreme value distribution of the response over the time interval [0, 15] s is pictured in are the PDFs of the Rayleigh distribution, the normal distribution and the lognormal distribution with the same second-order statistics as the computed EVD. It is found from the figures that the computed EVD is different from the afore-mentioned widely-used regular distributions. This indicates that, although in some special cases the Rayleigh distribution is the approximate closed form solution, it might be imprudent to use the Rayleigh distribution as the adopted EVD in the case of general stochastic processes. pictures the cumulated probability distribution function (CDF) of the extreme values from the computed EVD in . They are compared with some of the points computed by the MCS. Perfect accordance between the PDEM and the MCS is seen in , demonstrating that the proposed method is of high accuracy.The EVD of a stochastic process is obviously related to the time interval under consideration. The EVD over different time intervals [0, |
T] with T |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.