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.

Theoretical investigation on axial cyclic performance of monopile in sands using interface constitutive models

Cyclic loads generated by environmental factors,such as winds,waves,and trains,will likely lead to performance degradation in pile foundations,resulting in issues like permanent displacement accumulation and bearing capacity attenuation.This paper presents a semi-analytical solution for predicting the axial cyclic behavior of piles in sands.The solution relies on two enhanced nonlinear load-transfer models considering stress-strain hysteresis and cyclic degradation in the pile-soil interaction.Model parameters are calibrated through cyclic shear tests of the sand-steel interface and laboratory geotechnical testing of sands.A novel aspect involves the meticulous formulation of the shaft loadtransfer function using an interface constitutive model,which inherently inherits the interface model’s advantages,such as capturing hysteresis,hardening,degradation,and particle breakage.The semi-analytical solution is computed numerically using the matrix displacement method,and the calculated values are validated through model tests performed on non-displacement and displacement piles in sands.The results demonstrate that the predicted values show excellent agreement with the measured values for both the static and cyclic responses of piles in sands.The displacement pile response,including factors such as bearing capacity,mobilized shaft resistance,and convergence rate of permanent settlement,exhibit improvements compared to non-displacement piles attributed to the soil squeezing effect.This methodology presents an innovative analytical framework,allowing for integrating cyclic interface models into the theoretical investigation of pile responses.

Novel damage constitutive models and new quantitative identification method for stress thresholds of rocks under uniaxial compression

Four key stress thresholds exist in the compression process of rocks,i.e.,crack closure stress(σ_(cc)),crack initiation stress(σ_(ci)),crack damage stress(σ_(cd))and compressive strength(σ_(c)).The quantitative identifications of the first three stress thresholds are of great significance for characterizing the microcrack growth and damage evolution of rocks under compression.In this paper,a new method based on damage constitutive model is proposed to quantitatively measure the stress thresholds of rocks.Firstly,two different damage constitutive models were constructed based on acoustic emission(AE)counts and Weibull distribution function considering the compaction stages of the rock and the bearing capacity of the damage element.Then,the accumulative AE counts method(ACLM),AE count rate method(CRM)and constitutive model method(CMM)were introduced to determine the stress thresholds of rocks.Finally,the stress thresholds of 9 different rocks were identified by ACLM,CRM,and CMM.The results show that the theoretical stress−strain curves obtained from the two damage constitutive models are in good agreement with that of the experimental data,and the differences between the two damage constitutive models mainly come from the evolutionary differences of the damage variables.The results of the stress thresholds identified by the CMM are in good agreement with those identified by the AE methods,i.e.,ACLM and CRM.Therefore,the proposed CMM can be used to determine the stress thresholds of rocks.

Flow stress prediction of Hastelloy C-276 alloy using modified Zerilli−Armstrong,Johnson−Cook and Arrhenius-type constitutive models

To better understand the hot deformation behaviors of Hastelloy C-276 alloy under elevated temperatures,hot tensile tests were carried out in the temperature range of 1223−1423 K and the strain rate range of 0.01−10 s^−1,respectively.Based on the modified Zerilli−Armstrong,modified Johnson-Cook,and strain-compensated Arrheniustype models,three constitutive equations were established to describe the high-temperature flow stress of this alloy.Meanwhile,the predictability of the obtained models was evaluated by the calculation of correlation coefficients(r)and absolute errors(Δ),where the values of r for the modified Zerilli−Armstrong,Johnson−Cook,and Arrhenius-type constitutive models were computed to be 0.935,0.968 and 0.984,and the values ofΔwere calculated to be 13.4%,10.5%and 6.7%,respectively.Moreover,the experimental and predicted flow stresses were compared in the strain range of 0.1−0.5,the results further indicated that the obtained modified Arrhenius-type model possessed better predictability on hot flow behavior of Hastelloy C-276.

Constitutive models and salt migration mechanisms of saline frozen soil and the-state-of-the-practice countermeasures in cold regions

A series of saline soil-related problems,including salt expansion and collapse,frost heave and thaw settlement,threaten the safety of the road traffic and the built infrastructure in cold regions.This article presents a comprehensive review of the physical and mechanical properties,salt migration mechanisms of saline soil in cold environment,and the countermeasures in practice.It is organized as follows:(1)The basic physical characteristics;(2)The strength criteria and constitutive models;(3)Water and salt migration characteristics and mechanisms;and(4)Countermeasures of frost heave and salt expansion.The review provides a holistic perspective for recent progress in the strength characteristics,mechanisms of frost heave and salt expansion,engineering countermeasures of saline soil in cold regions.Future research is proposed on issues such as the effects of salt erosion on concrete and salt corrosion of metal under the joint action of evaporation and freeze-thaw cycles.

Review on cyclic plasticity of magnesium alloys:Experiments and constitutive models

Fatigue analysis has always been a concern in the design and assessment of Mg alloy structure components subjected to cyclic loading,and research on the cyclic plasticity is fundamental to investigate the corresponding fatigue failure.Thus,this work reviews the progress in the cyclic plasticity of Mg alloys.First,the existing macroscopic and microscopic experimental results of Mg alloys are summarized.Then,corresponding macroscopic phenomenological constitutive models and crystal plasticity-based models are reviewed.Finally,some conclusions and recommended topics on the cyclic plasticity of Mg alloys are provided to boost the further development and application of Mg alloys.

A comparative study of 85 hyperelastic constitutive models for both unfilled rubber and highly filled rubber nanocomposite material

Nonlinear finite element analysis is widely used for structural optimization of the design and the reliability analysis of complex elastomeric components.However,high-precision numerical results cannot be achieved without reliable strain energy functions(SEFs)of the rubber or rubber nanocomposite material.Although hyperelastic constitutive models have been studied for nearly 80 years,selecting one that accurately describes rubber's mechanical response is still a challenge.This work reviews 85 isotropic SEFs based on both the phenomenological theory and the micromechanical network theory proposed from the 1940s to 2019.A fitting algorithm which can realize the automatic fitting optimization and determination of the parameters of all SEFs reviewed is developed.The ability of each SEF to reproduce the experimental data of both the unfilled and highly filled rubber nanocomposite is quantitatively assessed based on a new proposed evaluation index.The top 30 SEFs for the unfilled rubber and the top 14 SEFs for the highly filled rubber nanocomposite are presented in the ranking lists.Finally,some suggestions on how to select an appropriate hyperelastic constitutive model are given,and the perspective on the future progress of constitutive models is summarized.

Application of thermodynamics-based rate-dependent constitutive models of concrete in the seismic analysis of concrete dams

This paper discusses the seismic analysis of concrete dams with consideration of material nonlinearity. Based on a consistent rate-dependent model and two thermodynamics-based models, two thermodynamics-based rate-dependent constitutive models were developed with consideration of the influence of the strain rate. They can describe the dynamic behavior of concrete and be applied to nonlinear seismic analysis of concrete dams taking into account the rate sensitivity of concrete. With the two models, a nonlinear analysis of the seismic response of the Koyna Gravity Dam and the Dagangshan Arch Dam was conducted. The results were compared with those of a linear elastic model and two rate-independent thermodynamics-based constitutive models, and the influences of constitutive models and strain rate on the seismic response of concrete dams were discussed. It can be concluded from the analysis that, during seismic response, the tensile stress is the control stress in the design and seismic safety evaluation of concrete dams. In different models, the plastic strain and plastic strain rate of concrete dams show a similar distribution. When the influence of the strain rate is considered, the maximum plastic strain and plastic strain rate decrease.

INVESTIGATION ON GRADIENT-DEPENDENT NONLOCAL CONSTITUTIVE MODELS FOR ELASTO-PLASTICITY COUPLED WITH DAMAGE

Firstly, typical) gradient-dependent nonlocal inelastic models were briefly reviewed. Secondly, based on the principle of ‘gradient-dependent energy dissipation', a gradient-dependent constitutive model for plasticity coupled with isotropic damage was presented in the framework of continuum thermodynamics. Numerical scheme for calculation of Laplacian term of damage field with the numerical results obtained by FEM calculation was proposed. Equations have been presented on the basis of Taylor series for both 2-dimensional and 3-dimensional cases, respectively. Numerical results have indicated the validity of the proposed gradient-dependent model and corresponding numerical scheme.

Application and comparison of different elastoplastic constitutive models for springback simulation of aluminum sheet forming

Springback is considered to be one of the most important problems in aluminum sheet stamp forming, leading to deviation from the designed target shape and assembly defects. In this study, a springback simulation model based on the benchmark of a Jaguar Land Rover aluminum panel is established. We embed several elastoplastic constitutive models （ Barlat＇ s 89, Barlat＇ s YLD2000, Yoshida-Uemori （YU） ＋ Barlat＇ s 89, and YU ＋ Barlat＇ s YLD2000） in the finite element model,in order to discuss the influence of the constitutive model selection on springback prediction in aluminum sheet forming.

Review on constitutive models of road materials

Road material constitutive model is the essential condition of pavement structure design,calculation,and maintenance decision.Experts have conducted in-depth studies on the problems and proposed many meaningful constitutive models.However,these constitutive models have not been summarized and no uniform conclusion on the applicability of them is reported in the academic circle.The linear elastic model based on Hooke's law is still used in pavement structure design.This paper systematically reviewed the existing constitutive models that reflect the materials'mechanical properties and used for pavement structure calculation.These constitutive models were divided into three categories:linear,nonlinear,and damage models under repeated loads.The applicable conditions,parameter determination methods,advantages,and disadvantages of each model were introduced.Moreover,the model's feasibility in structural design and calculation was analyzed.Follow-up studies should focus on the parameter determination standard of viscoelastic constitutive models,the extension of the 1D constitutive model to 3D state,and the nonlinear constitutive models related to stress-strain state to drive the application of the constitutive models in pavement structure calculation and design.Furthermore,to evaluate the life cycle service performance of road materials and structures,establishing the pavement damage model from the structure perspective will be the future direction.

Geomaterial Constitutive Models in Toe-Shooting Method

For toe-shooting method, geomaterial constitutive models concerned are studied. Analysis shows that, although extensively applied in soil mechanics, due to its angular singularity of yielding surface, the Mohr-Coulomb model is not suitable for numerical simulations in large deformation; in this case the rock-fills may be regarded as the Drucker-Prager model and the seaooze as the Prandtl-Reuss model. By comparing experimental data with numerical results, the constitutive model of the seaooze is numerically verified. It shows that, in high strain rate stage forming the blasting crater, the seaooze behaves as ideal non-compressible fluid, while in low strain rate stage during which the reck-fills flow to the blasting crater, the viscosity of the seaooze is negligible.

IDENTIFICATION OF QUASI-STATIC DISPLACEMENT FOR CONSTITUTIVE MODELS OF VISCOELASTICITY

An explicit relation between constitutive parameters and quasi-static displacement of viscoelasticity is derived under a kind of boundary condition, and an iterative form of optimized identification is presented. Viscoelastic constitutive models are identified from a two order differential model, and effects of information errors on results of inverse analysis are discussed.

Comparative Study on Constitutive Models to Predict Flow Stress of Fe-Cr-Ni Preform Reinforced Al-Si-Cu-Ni-Mg Composite

The isothermal compression tests were carried out on Gleeble-3500 thermal-mechanical simulation machine in a temperature range of 298-473 K and strain rate range of 0.001-10 s^-1. The experimental results show that the flow stress data are negatively correlated with temperature for temperature softening, and the strain rates sensitivity of this composite increases with elevating temperature. Based on the experimental data, Johnson-Cook, modified Johnson-Cook and Arrhenius constitutive models were established. The accuracy of these three constitutive models was analyzed and compared. The results show that the values predicted by Johnson-Cook model could not agree well with the experimental values. The prediction accuracy of Arrhenius model is higher than that of Johnson-Cook model but lower than that of the Modified Johnson-Cook model.

Modified constitutive models for Inconel 718 considering current density and temperature in electrically assisted forming process

Electrically Assisted Forming(EAF)technology has obvious advantages in material forming.To develop an effective constitutive model considering electrical effects,room temperature and electrically assisted quasi-static uniaxial tensile tests were conducted using ultrathin nickelbased superalloy plates with a thickness of 0.25 mm.The research focused on the two most widely recognized effects:the Joule thermal and the electric athermal effects.The mechanism of current action can be divided into two scenarios:one considering the Joule thermal effect only,and the other considering both effects simultaneously.Two basic constitutive models,namely the Modified-Hollomon model and the Johnson-Cook(J-C)model,were selected to be optimized through the classification of two different situations,and four optimized constitutive models were proposed.It was found that the J-C model with simultaneous consideration of the Joule thermal effect and electric athermal effect had the best prediction effect by comparing the results of these four models.Finally,the accuracy of the optimization model was verified by finite element simulation of the electrically assisted stretching optimization model.The results show that the constitutive model can effectively predict the temperature effect caused by the Joule heat effect and the athermal effect of current on the material.

Coupled elasto-plasticity damage constitutive models for concrete

The paper is to design and construct a coupled elasto-plasticity damage constitutive model for concrete.Based on the energy dissipation principle,the Hsieh-Ting-Chen four-parameter yield function is used.The model can reflect different strength characteristics of concrete in tension and compression,and reduce the limitation and lacuna of the traditional damage constitutive models for concrete.Furthermore,numerical test for concrete stress-strain relation under uniaxial tension and compression is given.Moreover,the damage process of concrete gravity dam is calculated and analyzed in seismic load.Compared with other damage constitutive models,the proposed model contains only one unknown parameter and the other parameters can be found in the Hsieh-Ting-Chen four-parameter yield function.The same damage evolution law,which is used for tension and compression,is good for determining stress-strain constitutive and damage characteristics in complex stress state.This coupled damage constitutive models can be applied in analyzing damage of concrete gravity dam and arch dam.

Evaluation on prediction abilities of constitutive models considering FEA application

Constitutive model plays an important role in the numerical simulations of metal forming. However, th~ influence of the models on the calculation is vague. Based on the stress-strain data of A1 7050 and Ti-6A1-4V alloy generated by isothermal compressive tests, the Johnson-Cook （JC） and Arrhenius-type （A-type） hyperbolic sine model were fitted to obtain the constants. Flow stresses directly calculated by the equations were compared with th~ experiment results, and rigid-plastic finite element analyses （FEA） utilizing these models were employed to simulate th~ same compression processes. The results show that A-type model has higher accuracy in the direct prediction of roy stress, even outside of the fit domain. The simulation results using A-type model also have higher agreement with th~ experiment; however, the suitability is affected by the referential parameters employed in the regression process, h terms of the overall deformation and strain distributions, there are slight differences among the simulation results usint these two models.

Constitutive Models for Compressive Deformation of AZ80Magnesium Alloy under Multiple Loading Directions and Strain Rates

Two constitutive models,the modified Johnson-Cook model and the logarithm linear relation model based on empirical approach and data analysis,were presented to illustrate compressive deformation of magnesium alloys AZ80 under multiple loading directions and strain rates.The results of stress-strain curve analysis and sensitivity index analysis suggested that the stress held large fluctuations in loading direction of 90°.Model testing signified that the logarithm linear relation model was more proper than the modified Johnson-Cook model in view of relative mean square error and correlation coefficients.Moreover,numerical simulation building on established models also indicated that the logarithm linear model is more precise than the modified Johnson-Cook model.

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.

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.