A whole process damage constitutive model for layered sandstone under uniaxial compression based on Logistic function

Bedding structural planes significantly influence the mechanical properties and stability of engineering rock masses.This study conducts uniaxial compression tests on layered sandstone with various bedding angles(0°,15°,30°,45°,60°,75°and 90°)to explore the impact of bedding angle on the deformational mechanical response,failure mode,and damage evolution processes of rocks.It develops a damage model based on the Logistic equation derived from the modulus’s degradation considering the combined effect of the sandstone bedding dip angle and load.This model is employed to study the damage accumulation state and its evolution within the layered rock mass.This research also introduces a piecewise constitutive model that considers the initial compaction characteristics to simulate the whole deformation process of layered sandstone under uniaxial compression.The results revealed that as the bedding angle increases from 0°to 90°,the uniaxial compressive strength and elastic modulus of layered sandstone significantly decrease,slightly increase,and then decline again.The corresponding failure modes transition from splitting tensile failure to slipping shear failure and back to splitting tensile failure.As indicated by the modulus’s degradation,the damage characteristics can be categorized into four stages:initial no damage,damage initiation,damage acceleration,and damage deceleration termination.The theoretical damage model based on the Logistic equation effectively simulates and predicts the entire damage evolution process.Moreover,the theoretical constitutive model curves closely align with the actual stress−strain curves of layered sandstone under uniaxial compression.The introduced constitutive model is concise,with fewer parameters,a straightforward parameter determination process,and a clear physical interpretation.This study offers valuable insights into the theory of layered rock mechanics and holds implications for ensuring the safety of rock engineering.

Microscopic cracking behaviors of rocks under uniaxial compression with microscopic multiphase heterogeneity by deep learning

Cracking behaviors of rocks significantly affect the safety and stability of the explorations of underground space and deep resources.To understand deeply the microscopic cracking process and mechanical property of rocks,X-ray micro-computed tomography(X-μCT)is applied to capture the rock microstructures.The digital color difference UNet(DCD-UNet)-based deep learning algorithm with 3D reconstruction is proposed to reconstruct the multiphase heterogeneity microstructure models of rocks.The microscopic cracking and mechanical properties are studied based on the proposed microstructure-based peridynamic model.Results show that the DCD-UNet algorithm is more effective to recognize and to represent the microscopic multiphase heterogeneity of rocks.As damage characteristic index of multiphase rocks increases,transgranular cracks in the same grain phase,transgranular and intergranular cracks of pore-grain phase,intergranular and secondary transgranular cracks and transgranular crack between different grains propagate.The ultimate microscopic failure modes of rocks are mainly controlled by the transgranular cracks-based T1-shear,T3-shear,T1-tension,T2-tension and T3-tension failures,and the intergranular cracks-based T1-tension,T1-shear and T3-shear failures under uniaxial compression.

Experimental study on 3D internal penny-shaped crack propagation in brittle materials under uniaxial compression

Fractures are widely present in geomaterials of civil engineering and deep underground engineering.Given that geomaterials are usually brittle,the fractures can significantly affect the evaluation of underground engineering construction safety and the early warning of rock failure.However,the crack initiation and propagation in brittle materials under composite loading remain unknown so far.In this study,a three-dimensional internal laser-engraved cracking technique was applied to produce internal cracks without causing damage to the surfaces.The uniaxial compression tests were performed on a brittle material with internal cracks to investigate the propagation of these internal cracks at different dip angles under compression and shear.The test results show that the wing crack propagation mainly occurs in the specimen with an inclined internal crack,which is a mixed-ModeⅠ–Ⅱ–Ⅲfracture;in contrast,ModeⅠfracture is present in the specimen with a vertical internal crack.The fractography characteristics of ModeⅢfracture display a lance-like pattern.The fracture mechanism in the brittle material under compression is that the internal wing cracks propagate to the ends of the whole sample and cause the final failure.The initial deflection angle of the wing crack is determined by the participation ratio of stress intensity factors KII to KI at the tip of the internal crack.

Extraction and identifcation of spectrum characteristics of coal and rock hydraulic fracturing and uniaxial compression signals

Microseismic(MS)events generated during coal and rock hydraulic fracturing(HF)include wet events caused by fracturing fuid injection,in addition to dry events caused by stress perturbations.The mixture of these two events makes efective fracturing MS events pickup difcult.This study is based on physical experiments of diferent coal and rock HF and uniaxial compression.The diferences of waveform characteristic parameters of various coal and rock ruptures were analyzed using the Hilbert–Huang transform,leading to some useful conclusions.The phase characteristics of the acoustic emission(AE)energy difered signifcantly and responded well to the pumping pressure curve.The AE waveforms of HF exhibit similar energy and frequency distribution characteristics after Empirical mode decomposition.The main frequency bands for coal,sandstone,and shale samples are 100–300 kHz,while the mudstone sample is in the range of 50–150 kHz.The decay ratios for coal,sandstone,shale and mudstone samples are 0.78,0.83,0.67 and 0.85,respectively.When compared to the uniaxial compression test,the main frequency bands of HF were reduced for coal,sandstone and mudstone samples,whereas shale remained essentially unchanged.The duration,instantaneous energy,and total energy of the HF waveform are smaller than those of uniaxial compression,while the decay ratio is greater,especially for the mudstone samples.The waveform characteristic parameters,trained using the multilayer perceptron neural network,can efectively identify HF and uniaxial compression events with an accuracy of 96%.

A comparative study for determining rock joint normal stiffness with destructive uniaxial compression and nondestructive ultrasonic wave testing

Rock joints are one of the vital discontinuities in a natural rock mass.How to accurately and conveniently determine joint normal stiffness is therefore significant in rock mechanics.Here,first,seven existing methods for determining joint normal stiffness were introduced and reviewed,among which MethodⅠ(the indirect measurement method),MethodⅡ(the direct determination method),MethodⅢ(the across-joint strain gauge measurement method)and MethodⅣ(the deformation measuring ring method)are via destructive uniaxial compression testing,while MethodⅤ(the best fitting method),MethodⅥ(the rapid evaluation method)and MethodⅦ(the effective modulus method)are through wave propagation principles and nondestructive ultrasonic testing.Subsequently,laboratory tests of intact and jointed sandstone specimens were conducted following the testing requirements and pro-cedures of those seven methods.A comparison among those methods was then performed.The results show that Method I,i.e.the benchmark method,is reliable and stable.MethodⅡhas a conceptual drawback,and its accuracy is acceptable at only very low stress levels.Relative errors in the results from MethodⅢare very large.With MethodⅣ,the testing results are sufficiently accurate despite the strict testing environment and complicated testing procedures.The results from MethodⅤare greatly unstable and significantly dependent on the natural frequency of the transducers.The joint normal stiffness determined with MethodⅥis stable and accurate,although data processing is complex.MethodⅦcould be adopted to determine the joint normal stiffness corresponding to the rock elastic deformation phase only.Consequently,it is suggested that MethodsⅠ,ⅣandⅥshould be adopted for the mea-surement of joint normal stiffness.The findings could be helpful in selecting an appropriate method to determine joint normal stiffness and,hence,to better solve discontinuous rock mass problems.

Fracture evolution around pre-existing cylindrical cavities in brittle rocks under uniaxial compression

The development of fracture around pre-existing cylindrical cavities in brittle rocks was examined using physical models and acoustic emission technique. The experimental results indicate that when granite blocks containing one pre-existing cylindrical cavity are loaded in uniaxial compression condition, the profiles of cracks around the cavity can be characterized by tensile cracking （splitting parallel to the axial compression direction） at the roof-floor, compressive crack at two side walls, and remote or secondary cracks at the perimeter of the cavity. Moreover, fracture around cavity is size-dependent. In granite blocks containing pre-existing half-length cylindrical cavities, compressive stress concentration is found to initiate at the two sidewalls and induce shear crack propagation and coalescence. In granite blocks containing multiple parallel cylindrical cavities, the adjacent cylindrical cavities can influence each other and the eventual failure mode is determined by the interaction of tensile, compressive and shear stresses. Experimental results show that both tensile and compressive stresses play an important role in fracture evolution process around cavities in brittle rocks.

Energy evolution mechanism and failure criteria of jointed surrounding rock under uniaxial compression

The object of this article is to investigate the energy evolution mechanism and failure criteria of cross-jointed samples containing an opening during deformation and failure based on the uniaxial compression test and rock energy principle.The results show that the energy evolution characteristics of the samples correspond to a typical progressive damage mode.The peak total energy,peak elastic energy,and total input energy of the samples all first decrease and then increase with an increase of half of the included angle,reaching their minimum values when this angle is 45°,while the dissipated energy generally increases with this angle.The existence of the opening and cross joints can obviously weaken the energy storage capacity of the rock,and the change in the included angle of the cross joint has a great influence on the elastic energy ratio of the sample before the peak stress,which leads to some differences in the distribution laws of the input energy.The continuous change and the subsequent sharp change in the rate of change in the energy consumption ratio can be used as the criteria of the crack initiation and propagation and the unstable failure of the sample,respectively.

Experimental study on cracking behaviour of moulded gypsum containing two non-parallel overlapping flaws under uniaxial compression

Failure of rock mass that is subjected to compressive loads occurs from initiation, propagation, and linkage of new cracks from preexisting fissures. Our research investigates the cracking behaviour and coalescence process in a brittle material with two non-parallel overlapping flaws using a high-speed camera. The coalescence tensile crack and tensile wing cracks were the first cracks to occur from the preexisting flaws. The initiation stresses of the primary cracks at the two tips of each flaw were simultaneous and decreased with reduced flaw inclination angle. The following types of coalescence cracks were identified between the flaws: primary tensile coalescence crack, tensile crack linkage, shear crack linkage, mixed tensile-shear crack, and indirect crack coalescence. Coalescence through tensile linkage occurred mostly at pre-peak stress. In contrast, coalescence through shear or mixed tensile-shear cracks occurred at higher stress. Overall, this study indicates that the geometry of preexisting flaws affect crack initiation and coalescence behaviour.

Failure characteristics and its influencing factors of rock-like material with multi-fissures under uniaxial compression

The compression test on rock-like specimens with prefabricated closed multi-fissures made by pulling out the embedded metal inserts in the precured period was done on the servo control uniaxial loading instrument. The influence of fissure inclination angle and distribution density on the failure characteristics of fissure bodies was researched. It was found that, the fissure inclination angle was the major influencing factor on the failure modes of fissure bodies. The different developmental states of micro-cracks would appear on specimens under different fissure inclination angles. However, the influence of fissure distribution density on the failure mode of fissure bodies was achieved by influencing the transfixion pattern of fissures. It was shown by the sliding crack model that, the effective shear, which drove the relative sliding of the fissure, was a function of fissure inclination angle and friction coefficient of the fissure surface. The strain-softening model of fissure bodies was established based on the mechanical parameters that were obtained by the test of rock-like materials under the same experimental condition. And the reliability of experimental results was identified by using this model.

Stress-strain-acoustic responses in failure process of coal rock with different height to diameter ratios under uniaxial compression

Residual coal pillars play an important role in mining the adjacent coal seam safely,managing the gobs and maintaining the stability of abandoned coal mines.The height to diameter ratio(H/D)affects the stability of residual coal pillars.In this study,uniaxial compressive tests of coal specimens with five H/D(2.0,1.5,1.0,0.8 and 0.6)were performed,and the stress,strain and acoustic emission(AE)were monitored.Results show that the uniaxial compressive strength(UCS)and peak strain increase with H/D decreasing.An empirical equation is proposed to calculate the UCS based on the H/D.The AE activities during coal failure process can be separated into four periods.The span of quiet period and rapid decline period shorten with H/D decreasing.The smaller the H/D is,the more complicated the failure characteristics of coal will be.The failure form of coal with H/D of 2.0,1.5,and 1.0 is primarily shear failure,while splitting failure along the axial direction is the mainly mode when H/D is 0.8 or 0.6.The initiation,expansion,aggregation and connection of micro-cracks can be reflected by the real-time spatial evolution of AE event points.

Specimen aspect ratio and progressive field strain development of sandstone under uniaxial compression by three-dimensional digital image correlation

The complete stress-strain characteristics of sandstone specimens were investigated in a series of quasistatic monotonic uniaxial compression tests.Strain patterns development during pre-and post-peak behaviours in specimens with different aspect ratios was also examined.Peak stress,post-peak portion of stress-strain,brittleness,characteristics of progressive localisation and field strain patterns development were affected at different extents by specimen aspect ratio.Strain patterns of the rocks were obtained by applying three-dimensional(3D) digital image correlation(DIC) technique.Unlike conventional strain measurement using strain gauges attached to specimen,3D DIC allowed not only measuring large strains,but more importantly,mapping the development of field strain throughout the compression test,i.e.in pre-and post-peak regimes.Field strain development in the surface of rock specimen suggests that strain starts localising progressively and develops at a lower rate in pre-peak regime.However,in post-peak regime,strains increase at different rates as local deformations take place at different extents in the vicinity and outside the localised zone.The extent of localised strains together with the rate of strain localisation is associated with the increase in rate of strength degradation.Strain localisation and local inelastic unloading outside the localised zone both feature post-peak regime.

Cracking process of rock mass models under uniaxial compression

Anisotropic strength and deformability of the rock mass with non-persistent joints are governed by cracking process of the rock bridges. The dependence of cracking process of jointed rock masses on the two important geometrical parameters, joint orientation and joint persistence, was studied systematically by carrying out a series of uniaxial compression tests on gypsum specimens with regularly arranged multiple parallel pre-existing joints. According to crack position, mechanism and temporal sequence, seven types of crack initiations and sixteen types of crack coalescences, were identified. It was observed that both tensile cracks and shear cracks can emanate from the pre-existing joints as well as the matrix. Vertical joints were included and coplanar tensile cracks initiation and coalescence were observed accordingly. For specimen with joint inclination angle ,8=75~, it was found that collinear joints can be linked not only by coplanar shear cracks but also by mixed tensile-shear cracks, and that a pair of them can form a small rotation block. Seven failure modes, including axial cleavage, crushing, crushing and rotation of new blocks, stepped failure, stepped failure and rotation of new blocks, shear failure along a single plane and shear failure along multiple planes, were observed. These modes shift gradually in accordance with the combined variation of joint orientation and joint persistence. It is concluded that cracking process and failure modes are more strongly affected by joint orientation than by joint persistence, especially when joint inclination angle is larger than 45~. Finally, variations of macroscopic mechanical behaviors with the two geometrical parameters, such as patterns of the complete axial stress-axial strain curves, peak strength and elastic modulus, are summarized and their mechanisms are successfully explained according to their different cracking process.

Shape ratio effects on the mechanical characteristics of rectangular prism rocks and isolated pillars under uniaxial compression

Isolated pillars in underground mines are subjected to uniaxial stress,and the load bearing cross-section of pillars is commonly rectangularly shaped.In addition,the uniaxial compression test(UCT)is widely used for determining the basic mechanical properties of rocks and revealing the mechanism of isolated pillar disasters under unidimensional stress.The shape effects of rock mechanical properties under uniaxial compression are mainly quantitatively reflected in the specific shape ratios of rocks.Therefore,it is necessary to study the detailed shape ratio effects on the mechanical properties of rectangular prism rock specimens and isolated pillars under uniaxial compressive stress.In this study,granite,marble and sandstone rectangular prism specimens with various height to width ratios(r)and width to thickness ratios(u)were prepared and tested.The study results show that r and u have a great influence on the bearing ability of rocks,and thin or high rocks have lower uniaxial compressive strength.Reducing the level of r can enhance the u effect on the strength of rocks,and increasing the level of u can enhance the r effect on the strength of rocks.The lateral strain on the thickness side of the rock specimen is larger than that on the width side,which implies that crack growth occurs easily on the thickness side.Considering r and u,a novel strength prediction model of isolated pillars was proposed based on the testing results,and the prediction model was used for the safety assessment of 179 isolated pillars in the Xianglu Mountain Tungsten Mine.

Strength weakening effect of high static pre-stressed granite subjected to low-frequency dynamic disturbance under uniaxial compression

This study aimed to elucidate the strength weakening effect of high static pre-stressed rocks subjected to low-frequency disturbances under uniaxial compression.Based on the uniaxial compressive strength(UCS)of granite under static loading,70%,80%,and 90%of UCS were selected as the initial high static pre-stress(σ_(p)),and then the pre-stressed rock specimens were disturbed by sinusoidal stress with amplitudes of 30%,20%,and 10%of UCS under low-frequency frequencies(f)of 1,2,5,and 10 Hz,respectively.The results show that the rockburst failure of pre-stressed granite is caused by low-frequency disturbance,and the failure strength is much lower than UCS.When theσp or f is constant,the specimen strength gradually decreases as the f or σ_(p) increases.The experimental study illustrates the influence mechanism of the strength weakening effect of high static pre-stress rocks under low-frequency dynamic disturbance,that is,high static pre-stress is the premise and leading factor of rock strength weakening,while low-frequency dynamic disturbance induces rock failure and affects the strength weakening degree.

Unified analytical stressstrain curve for quasibrittle geomaterial in uniaxial tension, direct shear and uniaxial compression

Considering strain localization in the form of a narrow band initiated just at peak stress, three analytical expressions for stressstrain curves of quasibrittle geomaterial (such as rock and concrete) in uniaxial tension, direct shear and uniaxial compression were presented, respectively. The three derived stressstrain curves were generalized as a unified formula. Beyond the onset of strain localization, a linear strain-softening constitutive relation for localized band was assigned. The size of the band was controlled by internal or characteristic length according to gradient-dependent plasticity. Elastic strain within the entire specimen was assumed to be uniform and decreased with the increase of plastic strain in localized band. Total strain of the specimen was decomposed into elastic and plastic parts. Plastic strain of the specimen was the average value of plastic strains in localized band over the entire specimen. For different heights, the predicted softening branches of the relative stressstrain curves in uniaxial compression are consistent with the previously experimental results for normal concrete specimens. The present expressions for the post-peak stressdeformation curves in uniaxial tension and direct shear agree with the previously numerical results based on gradient-dependent plasticity.

Influence of infilling stiffness on mechanical and fracturing responses of hollow cylindrical sandstone under uniaxial compression tests

Hollow cylindrical sandstone specimens filled with Al,Pb and polymethyl methacrylate(PMMA),as well as hollow and solid specimens were tested under monotonic unconfined compression.The discrepancies in the elastic modulus,unconfined compressive strength and failure pattern of the specimens were studied and then illustrated.The interaction stress threshold and localized failure stress threshold were identified by the strain gauges on the rock and filling rod.The results indicated that unobvious changes in the strength and elastic modulus were found between the solid and hollow specimens,while for the hollow specimens with infillings,the strength decreases with increasing the stiffness of the infilling material.The filling material with a higher stiffness leads to a high hoop stress,and hence a stronger interfacial force.The specimens coupled with filling rod are mainly fractured with tensile cracks,while the solid and hollow specimens are typically split into blocky fragments with dominated shear fractures.Finally,the equivalent inner pressure in the opening was theoretically derived.The findings suggested in the experiments can be well explained using the theoretical thick-walled cylinder model.

Grain growth in AZ31 alloy after uniaxial compression

The grain growth morphology,kinetics and texture change after uniaxial compression at 430 ℃ of an extruded AZ31 alloy were studied.The samples were loaded following two routes insuring two initial textures of the samples with compression direction parallel and normal to the extrusion direction.For both initial textures,a stable grain size is attained upon isothermal annealing and the grain growth kinetics can be described by:dn= dRn+kt with an n value of around 15.The annealing texture with grown grains is a retained hot deformation texture without emerging or strengthening other components.Abnormal grain growth is not observed for annealing time up to 10 000 h at 450℃.

Strain field investigation of limestone specimen under uniaxial compression loads using particle image velocimetry

The particle image velocimetry (PIV) method was used to investigate the full-field displacements and strains of the limestone specimen under external loads from the video images captured during the laboratory tests.The original colorful video images and experimental data were obtained from the uniaxial compression test of a limestone.To eliminate perspective errors and lens distortion,the camera was placed normal to the rock specimen exposure.After converted into a readable format of frame images,these videos were transformed into the responding grayscale images,and the frame images were then extracted.The full-field displacement field was obtained by using the PIV technique,and interpolated in the sub-pixel locations.The displacement was measured in the plane of the image and inferred from two consecutive images.The local displacement vectors were calculated for small sub-windows of the images by means of cross-correlation.The video images were interrogated in a multi-pass way,starting off with 64×64 images,ending with 16×16 images after 6 iterations,and using 75% overlap of the sub-windows.In order to remove spurious vectors,the displacements were filtered using four filters:signal-to-noise ratio filter,peak height filter,global filter and local filter.The cubic interpolation was utilized if the displacements without a number were encountered.The full-field strain was then obtained using the local least square method from the discrete displacements.The strain change with time at different locations was also investigated.It is found that the normal strains are dependant on the locations and the crack distributions.Between 1.0 and 5.0 s prior to the specimen failure,normal strains increase rapidly at many locations,while a stable status appears at some locations.When the specimen is in a failure status,a large rotation occurs and it increases in the inverse direction.The strain concentration bands do not completely develop into the large cracks,and meso-cracks are not visible in some bands.The techniques presented here may improve the traditional measurement of the strain field,and may provide a lot of valuable information in investigating the deformation/failure mechanism of rock materials.

Deformation and failure characteristics of sandstone under uniaxial compression using distributed fiber optic strain sensing

This paper investigates the deformation and fracture propagation of sandstone specimen under uniaxial compression using the distributed fiber optic strain sensing(DFOSS)technology.It shows that the DFOSS-based circumferential strains are in agreement with the data monitored with the traditional strain gage.The DFOSS successfully scans the full-field view of axial and circumferential strains on the specimen surface.The spatiotemporal strain measurement based on DFOSS manifests crack closure and elastoplastic deformation,detects initialization of microcrack nucleation,and identifies strain localization within the specimen.The DFOSS well observes the effects of rock heterogeneity on rock deformation.The advantage of DFOSS-based strain acquisition includes the high spatiotemporal resolution of signals and the ability of full-surface strain scanning.The introduction to the DFOSS technology yields a better understanding of the rock damage process under uniaxial compression.

Statistical damage constitutive model for concrete materials under uniaxial compression

According to the damage mechanism of concrete material during the uniaxial compressive failure process,this paper further establishes the statistical damage constitutive model of concrete subjected to uniaxial compressive stress based on the statistical damage model under uniaxial tension. The damage evolution law in the direction subjected to pressure is confirmed by the tensile damage evolution process of lateral deformation due to the Poisson effect,and then the compressive stress-strain relationship is defined. The peak nominal stress state and the critical state occurring in the macro longitudinal distributed splitting cracks are distinguished. The whole loading process can be divided into the even damage phase and the local breakage phase. The concrete specimen is divided into the failure process zone and the resting unloading zone. The size effects during the local breakage phase under the uniaxial monotonic compressive process and the hysteretic phenomenon under the cyclic compressive loading process are analyzed. Finally,the comparison between theoretical results and experimental results preliminarily verifies the rationality and feasibility of understanding the failure mechanism of concrete through the statistical damage constitutional law.