Anisotropic time-dependent behaviors of shale under direct shearing and associated empirical creep models

Understanding the anisotropic creep behaviors of shale under direct shearing is a challenging issue.In this context,we conducted shear-creep and steady-creep tests on shale with five bedding orientations (i.e.0°,30°,45°,60°,and 90°),under multiple levels of direct shearing for the first time.The results show that the anisotropic creep of shale exhibits a significant stress-dependent behavior.Under a low shear stress,the creep compliance of shale increases linearly with the logarithm of time at all bedding orientations,and the increase depends on the bedding orientation and creep time.Under high shear stress conditions,the creep compliance of shale is minimal when the bedding orientation is 0°,and the steady-creep rate of shale increases significantly with increasing bedding orientations of 30°,45°,60°,and 90°.The stress-strain values corresponding to the inception of the accelerated creep stage show an increasing and then decreasing trend with the bedding orientation.A semilogarithmic model that could reflect the stress dependence of the steady-creep rate while considering the hardening and damage process is proposed.The model minimizes the deviation of the calculated steady-state creep rate from the observed value and reveals the behavior of the bedding orientation's influence on the steady-creep rate.The applicability of the five classical empirical creep models is quantitatively evaluated.It shows that the logarithmic model can well explain the experimental creep strain and creep rate,and it can accurately predict long-term shear creep deformation.Based on an improved logarithmic model,the variations in creep parameters with shear stress and bedding orientations are discussed.With abovementioned findings,a mathematical method for constructing an anisotropic shear creep model of shale is proposed,which can characterize the nonlinear dependence of the anisotropic shear creep behavior of shale on the bedding orientation.

Development and Application of a Power Law Constitutive Model for Eddy Current Dampers

Eddy current dampers (ECDs) have emerged as highly desirable solutions for vibration control due to theirexceptional damping performance and durability. However, the existing constitutive models present challenges tothe widespread implementation of ECD technology, and there is limited availability of finite element analysis (FEA)software capable of accurately modeling the behavior of ECDs. This study addresses these issues by developing anewconstitutivemodel that is both easily understandable and user-friendly for FEAsoftware. By utilizing numericalresults obtained from electromagnetic FEA, a novel power law constitutive model is proposed to capture thenonlinear behavior of ECDs. The effectiveness of the power law constitutive model is validated throughmechanicalproperty tests and numerical seismic analysis. Furthermore, a detailed description of the application process ofthe power law constitutive model in ANSYS FEA software is provided. To facilitate the preliminary design ofECDs, an analytical derivation of energy dissipation and parameter optimization for ECDs under harmonicmotionis performed. The results demonstrate that the power law constitutive model serves as a viable alternative forconducting dynamic analysis using FEA and optimizing parameters for ECDs.

A creep model for ultra-deep salt rock considering thermal-mechanical damage under triaxial stress conditions

To investigate the specific creep behavior of ultra-deep buried salt during oil and gas exploitation,a set of triaxial creep experiments was conducted at elevated temperatures with constant axial pressure and unloading confining pressure conditions.Experimental results show that the salt sample deforms more significantly with the increase of applied temperature and deviatoric loading.The accelerated creep phase is not occurring until the applied temperature reaches 130℃,and higher temperature is beneficial to the occurrence of accelerated creep.To describe the specific creep behavior,a novel three-dimensional(3D)creep constitutive model is developed that incorporates the thermal and mechanical variables into mechanical elements.Subsequently,the standard particle swarm optimization(SPSO)method is adopted to fit the experimental data,and the sensibility of key model parameters is analyzed to further illustrate the model function.As a result,the model can accurately predict the creep behavior of salt under the coupled thermo-mechanical effect in deep-buried condition.Based on the research results,the creep mechanical behavior of wellbore shrinkage is predicted in deep drilling projects crossing salt layer,which has practical implications for deep rock mechanics problems.

A macro-mesoscopic constitutive model for porous and cracked rock under true triaxial conditions

The complex mechanical and damage mechanisms of rocks are intricately tied to their diverse mineral compositions and the formation of pores and cracks under external loads.Numerous rock tests reveal a complex interplay between the closure of porous defects and the propagation of induced cracks,presenting challenges in accurately representing their mechanical properties,especially under true triaxial stress conditions.This paper proposes a conceptualization of rock at the mesoscopic level as a two-phase composite,consisting of a bonded medium matrix and frictional medium inclusions.The bonded medium is characterized as a mesoscopic elastic material,encompassing various minerals surrounding porous defects.Its mechanical properties are determined using the mixed multi-inclusion method.Transformation of the bonded medium into the frictional medium occurs through crack extension,with its elastoplastic properties defined by the DruckerePrager yield criterion,accounting for hardening,softening,and extension.MorieTanaka and Eshelby’s equivalent inclusion methods are applied to the bonded and frictional media,respectively.The macroscopic mechanical properties of the rock are derived from these mesoscopic media.Consequently,a True Triaxial Macro-Mesoscopic(TTMM)constitutive model is developed.This model effectively captures the competitive effect and accurately describes the stress-deformation characteristics of granite.Utilizing the TTMM model,the strains resulting from porous defect closure and induced crack extension are differentiated,enabling quantitative determination of the associated damage evolution.

Parameter calibration of the tensile-shear interactive damage constitutive model for sandstone failure

The tensile-shear interactive damage(TSID)model is a novel and powerful constitutive model for rock-like materials.This study proposes a methodology to calibrate the TSID model parameters to simulate sandstone.The basic parameters of sandstone are determined through a series of static and dynamic tests,including uniaxial compression,Brazilian disc,triaxial compression under varying confining pressures,hydrostatic compression,and dynamic compression and tensile tests with a split Hopkinson pressure bar.Based on the sandstone test results from this study and previous research,a step-by-step procedure for parameter calibration is outlined,which accounts for the categories of the strength surface,equation of state(EOS),strain rate effect,and damage.The calibrated parameters are verified through numerical tests that correspond to the experimental loading conditions.Consistency between numerical results and experimental data indicates the precision and reliability of the calibrated parameters.The methodology presented in this study is scientifically sound,straightforward,and essential for improving the TSID model.Furthermore,it has the potential to contribute to other rock constitutive models,particularly new user-defined models.

An improved strain-softening constitutive model of granite considering the effect of crack deformation

This paper presents an improved strain-softening constitutive model considering the effect of crack deformation based on the triaxial cyclic loading and unloading test results.The improved model assumes that total strain is a combination of plastic,elastic,and crack strains.The constitutive relationship between the crack strain and the stress was further derived.The evolutions of mechanical parameters,i.e.strength parameters,dilation angle,unloading elastic modulus,and deformation parameters of crack,with the plastic strain and confining pressure were studied.With the increase in plastic strain,the cohesion,friction angle,dilation angle,and crack Poisson's ratio initially increase and subsequently decrease,and the unloading elastic modulus and the crack elastic modulus nonlinearly decrease.The increasing confining pressure enhances the strength and unloading elastic modulus,and decreases the dilation angle and Poisson's ratio of the crack.The theoretical triaxial compressive stress-strain curves were compared with the experimental results,and they present a good agreement with each other.The improved constitutive model can well reflect the nonlinear mechanical behavior of granite.

Fatigue properties and damage constitutive model of salt rock based on CT scanning

To investigate the macroscopic fatigue properties and the mesoscopic pore evolution characteristics of salt rock under cyclic loading,fatigue tests under different upper-limit stresses were carried out on salt rock,and the mesoscopic pore structures of salt rock before and after fatigue tests and under different cycle numbers were measured using CT scanning instrument.Based on the test results,the effects of the cycle number and the upper-limit stress on the evolution of cracks,pore morphology,pore number,pore volume,pore size,plane porosity,and volume porosity of salt rock were analyzed.The failure path of salt rock specimens under cyclic loading was analyzed using the distribution law of plane porosity.The damage variable of salt rock under cyclic loading was defined on basis of the variation of volume porosity with cycle number.In order to describe the fatigue deformation behavior of salt rock under cyclic loading,the nonlinear Burgers damage constitutive model was further established.The results show that the model established can better reflect the whole development process of fatigue deformation of salt rock under cyclic loading.

Nonlinear constitutive models of rock structural plane and their applications

Structural planes play an important role in controlling the stability of rock engineering,and the influence of structural planes should be considered in the design and construction process of rock engineering.In this paper,mechanical properties,constitutive theory,and numerical application of structural plane are studied by a combination method of laboratory tests,theoretical derivation,and program development.The test results reveal the change laws of various mechanical parameters under different roughness and normal stress.At the pre-peak stage,a non-stationary model of shear stiffness is established,and threedimensional empirical prediction models for initial shear stiffness and residual stage roughness are proposed.The nonlinear constitutive models are established based on elasto-plastic mechanics,and the algorithms of the models are developed based on the return mapping algorithm.According to a large number of statistical analysis results,empirical prediction models are proposed for model parameters expressed by structural plane characteristic parameters.Finally,the discrete element method(DEM)is chosen to embed the constitutive models for practical application.The running programs of the constitutive models have been compiled into the discrete element model library.The comparison results between the proposed model and the Mohr-Coulomb slip model show that the proposed model can better describe nonlinear changes at different stages,and the predicted shear strength,peak strain and shear stiffness are closer to the test results.The research results of the paper are conducive to the accurate evaluation of structural plane in rock engineering.

Time-domain dynamic constitutive model suitable for mucky soil site seismic response

Soil nonlinear behavior displays noticeable effects on the site seismic response.This study proposes a new functional expression of the skeleton curve to replace the hyperbolic skeleton curve.By integrating shear modulus and combining the dynamic skeleton curve and the damping degradation coefficient,the constitutive equation of the logarithmic dynamic skeleton can be obtained,which considers the damping effect in a soil dynamics problem.Based on the finite difference method and the multi-transmitting boundary condition,a 1D site seismic response analysis program called Soilresp1D has been developed herein and used to analyze the time-domain seismic response in three types of sites.At the same time,this study also provides numerical simulation results based on the hyperbolic constitutive model and the equivalent linear method.The results verify the rationality of the new soil dynamic constitutive model.It can analyze the mucky soil site nonlinear seismic response,reflecting the deformation characteristics and damping effect of the silty soil.The hysteresis loop area is more extensive,and the residual strain is evident.

A thermodynamics-based three-scale constitutive model for partially saturated granular materials

A three-scale constitutive model for unsaturated granular materials based on thermodynamic theory is presented.The three-scale yield locus,derived from the explicit yield criterion for solid matrix,is developed from a series of discrete interparticle contact planes.The three-scale yield locus is sensitive to porosity changes;therefore,it is reinterpreted as a corresponding constitutive model without phenomenological parameters.Furthermore,a water retention curve is proposed based on special pore morphology and experimental observations.The features of the partially saturated granular materials are well captured by the model.Under wetting and isotropic compression,volumetric compaction occurs,and the degree of saturation increases.Moreover,the higher the matric suction,the greater the strength,and the smaller the volumetric compaction.Compared with the phenomenological Barcelona basic model,the proposed three-scale constitutive model has fewer parameters;virtually all parameters have clear physical meanings.

Damage constitutive model of lunar soil simulant geopolymer under impact loading

Lunar base construction is a crucial component of the lunar exploration program,and considering the dynamic characteristics of lunar soil is important for moon construction.Therefore,investigating the dynamic properties of lunar soil by establishing a constitutive relationship is critical for providing a theoretical basis for its damage evolution.In this paper,a split Hopkinson pressure bar(SHPB)device was used to perform three sets of impact tests under different pressures on a lunar soil simulant geopolymer(LSSG)with sodium silicate(Na_(2)SiO_(3))contents of 1%,3%,5%and 7%.The dynamic stressestrain curves,failure modes,and energy variation rules of LSSG under different pressures were obtained.The equation was modified based on the ZWT viscoelastic constitutive model and was combined with the damage variable.The damage element obeys the Weibull distribution and the constitutive equation that can describe the mechanical properties of LSSG under dynamic loading was obtained.The results demonstrate that the dynamic compressive strength of LSSG has a marked strain-rate strengthening effect.Na_(2)SiO_(3) has both strengthening and deterioration effects on the dynamic compressive strength of LSSG.As Na_(2)SiO_(3) grows,the dynamic compressive strength of LSSG first increases and then decreases.At a fixed air pressure,5%Na_(2)SiO_(3) had the largest dynamic compressive strength,the largest incident energy,the smallest absorbed energy,and the lightest damage.The ZWT equation was modified according to the stress response properties of LSSG and the range of the SHPB strain rate to obtain the constitutive equation of the LSSG,and the model’s correctness was confirmed.

Triaxial creep damage characteristics of sandstone under high crustal stress and its constitutive model for engineering application

The creep characteristics of rock under high crustal stress are of important influence on the long‐term stability of deep rock engineering.To study the creep characteristics and engineering application of sandstone under high crustal stress,this study constructed nonlinear creep damage(NCD)constitutive mode based on the triaxial graded loading‒unloading creep test of sandstone in the Yuezhishan Tunnel.A numerical NCD constitutive model and a breakable lining(BL)model were developed based on FLAC3D and then applied to the stability analysis of the Yuezhishan Tunnel.Based on the creep test results of sandstone,a power function of creep rate and stress level was constructed,by which the long‐term strength was solved.The results show that the long‐term strength of the red sandstone based on the related function of the steady‐state creep rate and stress level is close to the measured stress value in engineering.The NCD model considering damage factors reflects the instantaneous and viscoelastic plasticity deformation characteristics of the red sandstone.The numerical NCD constitutive model and the BL model can reflect surrounding rock deformation characteristics and lining failure characteristics in practical engineering.The research results provide theoretical references for long‐term stability analysis of rock engineering and the deformation control of surrounding rock under high crustal stress.

Constitutive modelling of fabric effect on sand liquefaction

Sand liquefaction under static and dynamic loading can cause failure of embankments,slopes,bridges and other important infrastructure.Sand liquefaction in the seabed can also cause submarine landslides and tsunamis.Fabric anisotropy related to the internal soil structure such as particle orientation,force network and void space is found to have profound influence on sand liquefaction.A constitutive model accounting for the effect of anisotropy on sand liquefaction is proposed.Evolution of fabric anisotropy during loading is considered according to the anisotropic critical state theory for sand.The model has been validated by extensive test results on Toyoura sand with different initial densities and stress states.The effect of sample preparation method on sand liquefaction is qualitatively analysed.The model has been used to investigate the response of a sand ground under earthquake loading.It is shown that sand with horizontal bedding plane has the highest resistance to liquefaction when the sand deposit is anisotropic,which is consistent with the centrifuge test results.The initial degree of fabric anisotropy has a more significant influence on the liquefaction resistance.Sand with more anisotropic fabric that can be caused by previous loading history or compaction methods has lower liquefaction resistance.

Study on hot deformation behavior of homogenized Mg-8.5Gd-4.5Y-0.8Zn-0.4Zr alloy using a combination of strain-compensated Arrhenius constitutive model and finite element simulation method

Isothermal hot compression experiments were conducted on homogenized Mg-8.5Gd-4.5Y-0.8Zn-0.4Zr alloy to investigate hot deformation behavior at the temperature range of 673-773 K and the strain rate range of 0.001-1 s^(-1)by using a Gleeble-1500D thermo mechanical simulator.Metallographic characterization on samples deformed to true strain of 0.70 illustrates the occurrence of flow localization and/or microcrack at deformation conditions of 673 K/0.01 s^(-1),673 K/1 s^(-1)and 698 K/1 s^(-1),indicating that these three deformation conditions should be excluded during hot working of homogenized Mg-8.5Gd-4.5Y-0.8Zn-0.4Zr alloy.Based on the measured true stress-strain data,the strain-compensated Arrhenius constitutive model was constructed and then incorporated into UHARD subroutine of ABAQUS software to study hot deformation process of homogenized Mg-8.5Gd-4.5Y-0.8Zn-0.4Zr alloy.By comparison with measured force-displacement curves,the predicted results can describe well the rheological behavior of homogenized Mg-8.5Gd-4.5Y-0.8Zn-0.4Zr alloy,verifying the validity of finite element simulation of hot compression process with this complicated constitutive model.Numerical results demonstrate that the distribution of values of material parameters(α,n,Q and ln A)within deformed sample is inhomogeneous.This issue is directly correlated to the uneven distribution of equivalent plastic strain due to the friction effect.Moreover,at a given temperature the increase of strain rate would result in the decrease of equivalent plastic strain within the central region of deformed sample,which hinders the occurrence of dynamic recrystallization(DRX).

A rate-dependent constitutive model for saturated frozen soil considering local breakage mechanism

A rate-dependent constitutive model for saturated frozen soil is vital in frozen soil mechanics,especially when simultaneously describing the nonlinearity,dilatancy and strain-softening characteristics.The distribution of the non-uniform strain rate of saturated frozen soil at the meso-scale due to the local icecementation breakage is described by a newly binary-medium-based homogenization equation.Based on the field-equation-based approach of the meso-mechanics theory,the interaction expression of the strain rate at macro-and meso-scale is derived,which can give the strain rate concentration tensor at different crushed degrees.With the thermodynamics and empirical assumption,a breakage ratio in the rate-dependent form is determined.This overcomes the limitations of the existing binary-medium-based models that are difficult to simulate rate-dependent mechanical response.Based on these assumptions,a newly binary-medium-based rate-dependent model is proposed considering both the ice bond breakage and material composition characteristics of saturated frozen soil.The proposed constitutive model has been validated by the test results on frozen soils with different temperatures and strain rates.

A nonlinear creep model for surrounding rocks of tunnels based on kinetic energy theorem

The initiating condition for the accelerated creep of rocks has caused difficulty in analyzing the whole creep process.Moreover,the existing Nishihara model has evident shortcomings in describing the accelerated creep characteristics of the viscoplastic stage from the perspective of internal energy to analyze the mechanism of rock creep failure and determine the threshold of accelerated creep initiation.Based on the kinetic energy theorem,Perzyna viscoplastic theory,and the Nishihara model,a unified creep constitutive model that can describe the whole process of decaying creep,stable creep,and accelerated creep is established.Results reveal that the energy consumption and creep damage in the process of creep loading mainly come from the internal energy changes of geotechnical materials.The established creep model can not only describe the viscoelasticeplastic creep characteristics of rock,but also reflect the relationship between rock energy and creep deformation change.In addition,the research results provide a new method for determining the critical point of creep deformation and a new idea for studying the creep model and creep mechanical properties.

Creep behaviour and constitutive model of coal filled with gas

Coal exhibits different creep behaviours when filled with different amounts of gas. Creep tests of coal filled with 0 and 0.5 MPa gas were performed, and strain under different axial stress was compared.The three creep constitutive models which were analysed using the method fitting experimental data for determining which creep model can reflect the creep process of the test best. The results show that the deformation of coal filled with 0.5 MPa gas is more higher than that of coal filled with 0 MPa gas under the same axial stress. Gas plays a positive effect on the deformation of coal process and will accelerate creep process. And gas will reduce coal intensity and change coal creep properties.Compared with Nishihara Model and Extensional Nishihara Model, Burgers Model can reflect the three stages of creep process of coal filled with gas better. The research results can contribute to reveal coal and gas outburst mechanism.

Viscoelastoplastic constitutive model for creep deformation behavior of asphalt sand

A uniaxial viscoelastoplastic model that can describe whole creep behaviors of asphalt sand at different temperatures was presented.The model was composed of three submodels in series,which describe elastoplastic,viscoelastic and viscoplastic characteristics respectively.The constitutive equation was established for uniaxial loading condition,and the creep representation was also obtained.The constitutive parameters were determined by uniaxial compression tests under controlled-stress of 0.1 MPa with five different test temperatures of 20,40,45,50 and 60 ℃.Expressions of the model parameters in terms of temperatures were also given.The model gave prediction at various temperatures consistent with the experimental results,and can reflect the total deformation characterization of asphalt sands.

QUICK ASSESSMENT METHODOLOGY FOR RELIABILITY OF SOLDER JOINTS IN BALL GRID ARRAY (BGA) ASSEMBLY——PART Ⅰ: CREEP CONSTITUTIVE RELATION AND FATIGUE MODEL

In this study, a new unified creep constitutive relation and a mod- ified energy-based fatigue model have been established respectively to describe the creep flow and predict the fatigue life of Sn-Pb solders. It is found that the relation successfully elucidates the creep mechanism related to current constitutive relations. The model can be used to describe the temperature and frequency dependent low cycle fatigue behavior of the solder. The relation and the model are further employed in part Ⅱ to develop the numerical simulation approach for the long-term reliability assessment of the plastic ball grid array (BGA) assembly.

Constitutive model for Ya'an mudstone based on mesoscopic breakage mechanism

The slope stability of Ya’an expressway in Sichuan dominated by mudstone strata,is influenced greatly by both the mechanical properties and stressstrain relationship of mudstone.In this paper,the mechanical properties of the Ya’an mudstone samples under triaxial compression conditions were studied,based on an established constitutive model under the framework of breakage mechanics to simulate the mechanical properties of mudstone.Firstly,triaxial compression tests and SEM tests at the confining pressures of 0.5 MPa,1.0 MPa,and 2.0 MPa were carried out on the mudstone samples,and it was found that the mudstone sample undergoes strain softening and dilatancy followed by the volumetric compaction.Then,based on analysis on the breakage mechanism of the above test results,we idealized the mudstone sample as a binary medium material consisting of the bonded elements and frictional elements,of which the bonded elements are composed of solid matrix and pores,and the frictional elements are composed of broken aggregates.During the loading process,the cementation between clay minerals and non-clay minerals in the mudstone sample is first destroyed,leading to the formation of micro-cracks within the particle aggregate,that is,the bonded elements are gradually damaged during the loading process and gradually turned into the frictional elements,and the two jointly bear the external load.The bonded elements are composed of mudstone matrix and pores,which have the cementitious characteristics of mudstone,and the frictional elements are composed of the broken aggregate with the frictional characteristics of the broken particles.Based on the homogenization theory,the constitutive model for the mudstone is established,and the determining method for model parameters is also given.Finally,the results of the triaxial compression tests of the mudstone samples are predicted by the constitutive model proposed here,which can reflect the main mechanical properties of the mudstone samples.