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.

Distinct behavior of electronic structure under uniaxial strain in BaFe_(2)As_(2)

We report a study of the electronic structure of BaFe_(2)As_(2) under uniaxial strains using angle-resolved photoemission spectroscopy and transport measurements. Two electron bands at the MY point, with an energy splitting of 50 meV in the strain-free sample, shift downward and merge into each other under a large uniaxial strain, while three hole bands at theГ point shift downward together. However, we also observed an enhancement of the resistance anisotropy under uniaxial strains by electrical transport measurements, implying that the applied strains strengthen the electronic nematic order in BaFe_(2)As_(2). These observations suggest that the splitting of these two electron bands at the MY point is not caused by the nematic order in BaFe_(2)As_(2).

Predicting uniaxial compressive strength of tuff after accelerated freeze-thaw testing: Comparative analysis of regression models and artificial neural networks

Ignimbrites have been widely used as building materials in many historical and touristic structures in the Kayseri region of Türkiye. Their diverse colours and textures make them a popular choice for modern construction as well. However, ignimbrites are particularly vulnerable to atmospheric conditions, such as freeze-thaw cycles, due to their high porosity, which is a result of their formation process. When water enters the pores of the ignimbrites, it can freeze during cold weather. As the water freezes and expands, it generates internal stress within the stone, causing micro-cracks to develop. Over time, repeated freeze-thaw (F-T) cycles lead to the growth of these micro-cracks into larger cracks, compromising the structural integrity of the ignimbrites and eventually making them unsuitable for use as building materials. The determination of the long-term F-T performance of ignimbrites can be established after long F-T experimental processes. Determining the long-term F-T performance of ignimbrites typically requires extensive experimental testing over prolonged freeze-thaw cycles. To streamline this process, developing accurate predictive equations becomes crucial. In this study, such equations were formulated using classical regression analyses and artificial neural networks (ANN) based on data obtained from these experiments, allowing for the prediction of the F-T performance of ignimbrites and other similar building stones without the need for lengthy testing. In this study, uniaxial compressive strength, ultrasonic propagation velocity, apparent porosity and mass loss of ignimbrites after long-term F-T were determined. Following the F-T cycles, the disintegration rate was evaluated using decay function approaches, while uniaxial compressive strength (UCS) values were predicted with minimal input parameters through both regression and ANN analyses. The ANN and regression models created for this purpose were first started with a single input value and then developed with two and three combinations. The predictive performance of the models was assessed by comparing them to regression models using the coefficient of determination (R2) as the evaluation criterion. As a result of the study, higher R2 values (0.87) were obtained in models built with artificial neural network. The results of the study indicate that ANN usage can produce results close to experimental outcomes in predicting the long-term F-T performance of ignimbrite samples.

Uniaxial Compressive Strength Prediction for Rock Material in Deep Mine Using Boosting-Based Machine Learning Methods and Optimization Algorithms

Traditional laboratory tests for measuring rock uniaxial compressive strength(UCS)are tedious and timeconsuming.There is a pressing need for more effective methods to determine rock UCS,especially in deep mining environments under high in-situ stress.Thus,this study aims to develop an advanced model for predicting the UCS of rockmaterial in deepmining environments by combining three boosting-basedmachine learning methods with four optimization algorithms.For this purpose,the Lead-Zinc mine in Southwest China is considered as the case study.Rock density,P-wave velocity,and point load strength index are used as input variables,and UCS is regarded as the output.Subsequently,twelve hybrid predictive models are obtained.Root mean square error(RMSE),mean absolute error(MAE),coefficient of determination(R2),and the proportion of the mean absolute percentage error less than 20%(A-20)are selected as the evaluation metrics.Experimental results showed that the hybridmodel consisting of the extreme gradient boostingmethod and the artificial bee colony algorithm(XGBoost-ABC)achieved satisfactory results on the training dataset and exhibited the best generalization performance on the testing dataset.The values of R2,A-20,RMSE,and MAE on the training dataset are 0.98,1.0,3.11 MPa,and 2.23MPa,respectively.The highest values of R2 and A-20(0.93 and 0.96),and the smallest RMSE and MAE values of 4.78 MPa and 3.76MPa,are observed on the testing dataset.The proposed hybrid model can be considered a reliable and effective method for predicting rock UCS in deep mines.

Analytical solutions to the precession relaxation of magnetization with uniaxial anisotropy

Based on the Landau-Lifshitz-Gilbert(LLG)equation,the precession relaxation of magnetization is studied when the external field H is parallel to the uniaxial anisotropic field H_(k).The evolution of three-component magnetization is solved analytically under the condition of H=nH_(k)(n=3,1 and 0).It is found that with an increase of H or a decrease of the initial polar angle of magnetization,the relaxation time decreases and the angular frequency of magnetization increases.For comparison,the analytical solution for H_(k)=0 is also given.When the magnetization becomes stable,the angular frequency is proportional to the total effective field acting on the magnetization.The analytical solutions are not only conducive to the understanding of the precession relaxation of magnetization,but also can be used as a standard model to test the numerical calculation of LLG equation.

Mechanical behavior of rock under uniaxial tension:Insights from energy storage and dissipation

Many rock engineering projects show that the growth of tensile cracks is often an important cause of engineering disasters,and the mechanical behavior of rocks is essentially the transmission,storage,dissipation and release of energy.To investigate the tensile behavior of rock from the perspective of energy,uniaxial tension tests(UTTs)and uniaxial compression tests(UCTs)were carried out on three typical rocks(granite,sandstone and marble).Different unloading points were set before the peak stress to separate elastic energy and dissipated energy.The input energy density ut,elastic energy density ue,and dissipated energy density ud at each unloading point were calculated by integrating stress-strain curves.The results show that there is a strong linear relationship between the three energy parameters and the square of the unloading stress in UCT,but this linear relationship is weaker in UTT.The ue and ud increase linearly with the increase in ut in UCT and UTT.Based on the phenomenon that ue and ud increase linearly with ut,the applicability of W_(et)^(p) index in UTT was proved and the relative energy storage capacity and absolute energy distribution characteristics of three rocks in UCT and UTT were evaluated.The tensile behavior of marble and sandstone in UTT can be divided into two stages vaguely according to the energy distribution,but granite is not the case.In addition,based on dissipated energy,the damage evolution of three types of rocks in UCT and UTT was discussed.This study provides some new insights for understanding the tensile behavior of rock.

Determination of Material Parameters of EVA Foam under Uniaxial Compressive Testing Using Hyperelastic Models

The objective of this research was to determine the mechanical parameter from EVA foam and also investigate its behavior by using Blatz-Ko,Neo-Hookean,Mooney model and experimental test.The physical characteristic of EVA foam was also evaluated by scanning electron microscopy(SEM).The results show that Blatz-Ko and Neo-Hookean model can fit the curve at 5%and 8%strain,respectively.The Mooney model can fit the curve at 50%strain.The modulus of rigidity evaluated from Mooney model is 0.0814±0.0027 MPa.The structure of EVA foam from SEM image shows that EVA structure is a closed cell with homogeneous porous structure.From the result,it is found that Mooney model can adjust the data better than other models.This model can be applied for mechanical response prediction of EVA foam and also for reference value in engineering application.

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.

Experiment and simulation of creep performance of basalt fibre asphalt mortar under uniaxial compressive loadings

The creep performance of basalt fibre（BF）reinforced in asphalt mortar under uniaxial compressive loadings is investigated. The samples of basalt fibre asphalt mortar（BFAM） with different BF mass fractions（0. 1%,0. 2%, and 0. 5%） and without BF in asphalt mixture are prepared, and then submitted for the compressive strength test and corresponding creep test at a high in-service temperature.Besides, numerical simulations in finite element ABAQUS software were conducted to model the compressive creep test of mortar materials, where the internal structure of the fibre mortar was assumed to be a two-component composite material model such as fibre and mortar matrix. Finally, the influence factors of rheological behaviors of BFAM are further analyzed. Results indicate that compared to the control sample, the compressive strength of BFAM samples has a significant increase, and the creep and residual deformation are decreased. However, it also shows that the excessive fibre, i.e. with the BF content of 0. 5%, is unfavorable to the high-temperature stability of the mortar. Based on the analysis results, the prediction equations of parameters of the Burgers constitutive model for BFAM are proposed by considering the fibre factors.

Investigating the mechanical and acoustic emission characteristics of brittle failure around a circular opening under uniaxial loading

The size of underground openings in rock masses in metal mines is critical to the performance of the openings. In this study, the mechanical and acoustic emission (AE) characteristics of brittle rock-like specimens containing a circular opening with different ratios of opening diameter to sample size λ (λ = 0.1, 0.13, 0.17, 0.2, and 0.23) were investigated under uniaxial compression with AE monitoring. The results indicate that the opening size strongly affected the peak strength and the elastic modulus. Crack initiation first started from the upper surface of the specimens, not from the periphery of the openings. Tensile and shear cracks coexisted on the roof and floor of the specimens, whereas tensile cracks were dominant on the two sides. The fracture mode of samples with openings was partially affected by the relative size of the pillars and openings. The AE response curves (in terms of counts, cumulative energy, cumulative counts, and b-value) show that brittle failure was mainly a progressive process. Moreover, the AE information corresponded well with microcrack evolution in the samples and thus can be used to predict sample failure.

Investigation on Uniaxial Tensile Instability of USIBOR1500 Steel Sheets at Elevated Temperature

In the sheet forming process with stretch, diffuse instability and localized instability usually occur in the process one after another. The necking rate has a great impact on the instability process and the forming limits of the rate sensitive material, such as USIBOR1500 steel sheets at elevated temperature. The available reports about this steel mainly focus on hot uniaxial tensile, martensitic transformation and forming process, but there are few investigations on instability behavior and forming limits. Based on Inoue Kachiro＇s flow stress constitutive model at elevated temperature, combined with Swift＇s diffuse instability rule and Hill＇s localized instability nile, the relation is theoretically deduced between the diffuse necking rate along with the localized necking rate and the exponent of strain rate sensitivity together with the strain rate in the case of the USIBOR1500 steel sheets at elevated temperature. According to the time-temperature characteristics of the hot stamping process, tensile tests of the steel sheets were carried out on Gleeble3800, and the stress-strain curves were obtained at different temperatures and different strain rates. Then the values of the exponent of strain rate sensitivity and the hardening exponent were obtained through fitting the curves by least squares. The tests also helped to provide the distribution laws of the major strain of the specimens and the uniform strains as well as the necking width. The uniform strains obtained from the tests are matched well with the theoretical calculations.

Effect of the layer orientation on mechanics and energy evolution characteristics of shales under uniaxial loading

The uniaxial compression tests were conducted on the cylindrical shale specimens with bedding plane inclined at 0° and 90° to the axial loading direction, respectively. Effect of the bedding orientation on the mechanical property and energy evolution characteristics of shales was revealed. The failure mechanisms of the specimens with layers in 0° orientation showed splitting failure along weak bedding, while the specimens with layers in 90° orientation were failed by shearing sliding. The values of compressive strength, elastic modulus and shear modulus of samples at 0° were higher than those of samples at 90°and there was little difference of Poisson's ratio between samples at 0° and 90°. The analysis of the stress–strain energy and acoustic emission(AE) energy indicated that the growth rate of absorbed energy density and elastic energy density at 0° was significantly faster than that at 90°, hence their final values at 0°were relatively larger than the latter. Moreover, higher energy release was observed for specimens at 0°.The energy release and rapid growth of energy dissipation also appeared more early at 0°. The stress ratio63% was a critical point of energy distribution at which differences started to arise between samples at 0°and 90°. These results indicated that the failure of shale at 0° was more violent and devastative than the failure of shale at 90°.

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.

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.

Uniaxial experimental study of the acoustic emission and deformation behavior of composite rock based on 3D digital image correlation(DIC)

In this paper, uniaxial compression tests were carried out on a series of composite rock specimens with different dip angles, which were made from two types of rock-like material with different strength. The acoustic emission technique was used to monitor the acoustic signal characteristics of composite rock specimens during the entire loading process. At the same time, an optical non-contact 3 D digital image correlation technique was used to study the evolution of axial strain field and the maximal strain field before and after the peak strength at different stress levels during the loading process. The effect of bedding plane inclination on the deformation and strength during uniaxial loading was analyzed. The methods of solving the elastic constants of hard and weak rock were described. The damage evolution process, deformation and failure mechanism, and failure mode during uniaxial loading were fully determined. The experimental results show that the θ = 0?–45?specimens had obvious plastic deformation during loading, and the brittleness of the θ = 60?–90?specimens gradually increased during the loading process. When the anisotropic angle θincreased from 0?to 90?, the peak strength, peak strain,and apparent elastic modulus all decreased initially and then increased. The failure mode of the composite rock specimen during uniaxial loading can be divided into three categories:tensile fracture across the discontinuities(θ = 0?–30?), slid-ing failure along the discontinuities(θ = 45?–75?), and tensile-split along the discontinuities(θ = 90?). The axial strain of the weak and hard rock layers in the composite rock specimen during the loading process was significantly different from that of the θ = 0?–45?specimens and was almost the same as that of the θ = 60?–90?specimens. As for the strain localization highlighted in the maximum principal strain field, the θ = 0?–30?specimens appeared in the rock matrix approximately parallel to the loading direction,while in the θ = 45?–90?specimens it appeared at the hard and weak rock layer interface.

Propagation of Airy Gaussian vortex beams in uniaxial crystals

The propagation dynamics of the Airy Gaussian vortex beams in uniaxial crystals orthogonal to the optical axis has been investigated analytically and numerically. The propagation expression of the beams has been obtained. The propagation features of the Airy Gaussian vortex beams are shown with changes of the distribution factor and the ratio of the extraordinary refractive index to the ordinary refractive index. The correlations between the ratio and the maximum intensity value during the propagation, and its appearing distance have been investigated.

Effect of freeze-thaw cycles on uniaxial mechanical properties of cohesive coarse-grained soils

Freeze-thaw cycles are closely related to the slope instability in high-altitude mountain regions. In this study, cohesive coarse-grained soils were collected from a high-altitude slope in the Qinghai–Tibet Plateau to study the effect of cyclic freeze-thaw on their uniaxial mechanical properties. The soil specimens were remolded with three dry densities and three moisture contents. Then, after performing a series of freeze-thaw tests in a closed system without water supply, the soil specimens were subjected to a uniaxial compression test. The results showed that the stress-strain curves of the tested soils mainly performed as strain-softening. The softening feature intensified with the increasing dry density but weakened with an increase in freeze-thaw cycles and moisture content. The uniaxial compressive strength, resilient modulus, residual strength and softening modulus decreased considerably with the increase of freeze-thaw cycles. After more than nine freeze-thaw cycles, these four parameters tended to be stable. These parameters increased with the increase of dry density and decreased with the increasing moisture content, except for the residual strength which did not exhibit any clear variation with an increase in moisture content. The residual strength, however, generally increased with an increase in dry density. The soil structural damage caused by frozen water expansion during the freeze-thaw is the major cause for the changes in mechanical behaviors of cohesive coarse-grained soils. With results in this study, the deterioration effect of freeze-thaw cycles on the mechanical properties of soils should be considered during the slope stability analysis in high-altitude mountain regions.

Evaluation of empirical estimation of uniaxial compressive strength of rock using measurements from index and physical tests

The uniaxial compressive strength(UCS) of rock is an important parameter required for design and analysis of rock structures,and rock mass classification.Uniaxial compression test is the direct method to obtain the UCS values.However,these tests are generally tedious,time-consuming,expensive,and sometimes impossible to perform due to difficult rock conditions.Therefore,several empirical equations have been developed to estimate the UCS from results of index and physical tests of rock.Nevertheless,numerous empirical models available in the literature often make it difficult for mining engineers to decide which empirical equation provides the most reliable estimate of UCS.This study evaluates estimation of UCS of rocks from several empirical equations.The study uses data of point load strength(Is(50)),Schmidt rebound hardness(SRH),block punch index(BPI),effective porosity(n) and density(ρ)as inputs to empirically estimate the UCS.The estimated UCS values from empirical equations are compared with experimentally obtained or measured UCS values,using statistical analyses.It shows that the reliability of UCS estimated from empirical equations depends on the quality of data used to develop the equations,type of input data used in the equations,and the quality of input data from index or physical tests.The results show that the point load strength(Is(50)) is the most reliable index for estimating UCS among the five types of tests evaluated.Because of type-specific nature of rock,restricting the use of empirical equations to the similar rock types for which they are developed is one of the measures to ensure satisfactory prediction performance of empirical equations.

Random field characterization of uniaxial compressive strength and elastic modulus for intact rocks

Rock properties exhibit spatial variabilities due to complex geological processes such as sedimentation,metamorphism, weathering, and tectogenesis. Although recognized as an important factor controlling the safety of geotechnical structures in rock engineering, the spatial variability of rock properties is rarely quantified. Hence, this study characterizes the autocorrelation structures and scales of fluctuation of two important parameters of intact rocks, i.e. uniaxial compressive strength(UCS) and elastic modulus(EM).UCS and EM data for sedimentary and igneous rocks are collected. The autocorrelation structures are selected using a Bayesian model class selection approach and the scales of fluctuation for these two parameters are estimated using a Bayesian updating method. The results show that the autocorrelation structures for UCS and EM could be best described by a single exponential autocorrelation function. The scales of fluctuation for UCS and EM respectively range from 0.3 m to 8.0 m and from 0.3 m to 8.4 m.These results serve as guidelines for selecting proper autocorrelation functions and autocorrelation distances for rock properties in reliability analyses and could also be used as prior information for quantifying the spatial variability of rock properties in a Bayesian framework.

A Uniaxial Optimal Perfectly Matched Layer Method for Time-harmonic Scattering Problems

We develop a uniaxial optimal perfectly matched layer （opt PML） method for solving the time-harmonic scattering problems by choosing a particular absorbing function with unbounded integral in a rectangular domain. With this choice, the solution of the optimal PML problem not only converges exponentially to the solution of the original scatting problem, but also is insensitive to the thickness of the PML layer for sufficiently small parameter ε0. Numerical experiments are included to illustrate the competitive behavior of the proposed optimal method.