Inverse design of mechanical metamaterial achieving a prescribedconstitutive curve

Besides exhibiting excellent capabilities such as energy absorption,phase-transforming metamaterials offer a vast design space for achieving nonlinear constitutive relations.This is facilitated by switching between different patterns under deformation.However,the related inverse design problem is quite challenging,due to the lack of appropriate mathematical formulation and the convergence issue in the post-buckling analysis of intermediate designs.In this work,periodic unit cells are explicitly described by the moving morphable voids method and effectively analyzed by eliminating the degrees of freedom in void regions.Furthermore,by exploring the Pareto frontiers between error and cost,an inverse design formulation is proposed for unit cells.This formulation aims to achieve a prescribed constitutive curve and is validated through numerical examples and experimental results.The design approach presented here can be extended to the inverse design of other types of mechanical metamaterials with prescribed nonlinear effective properties.

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.

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 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.

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.

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.

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.

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.

Experiment and constitutive modelling of creep deformation in the frozen silt-concrete interface

To ensure the long-term safety and stability of bridge pile foundations in permafrost regions,it is necessary to investigate the rheological effects on the pile tip and pile side bearing capacities.The creep characteristics of the pile-frozen soil interface are critical for determining the long-term stability of permafrost pile foundations.This study utilized a self-developed large stress-controlled shear apparatus to investigate the shear creep characteristics of the frozen silt-concrete interface,and examined the influence of freezing temperatures(−1,−2,and−5°C),contact surface roughness(0,0.60,0.75,and 1.15 mm),normal stress(50,100,and 150 kPa),and shear stress on the creep characteristics of the contact surface.By incorporating the contact surface’s creep behavior and development trends,we established a creep constitutive model for the frozen silt-concrete interface based on the Nishihara model,introducing nonlinear elements and a damage factor.The results revealed significant creep effects on the frozen silt-concrete interface under constant load,with creep displacement at approximately 2-15 times the instantaneous displacement and a failure creep displacement ranging from 6 to 8 mm.Under different experimental conditions,the creep characteristics of the frozen silt-concrete interface varied.A larger roughness,lower freezing temperatures,and higher normal stresses resulted in a longer sample attenuation creep time,a lower steady-state creep rate,higher long-term creep strength,and stronger creep stability.Building upon the Nishihara model,we considered the influence of shear stress and time on the viscoelastic viscosity coefficient and introduced a damage factor to the viscoplasticity.The improved model effectively described the entire creep process of the frozen silt-concrete interface.The results provide theoretical support for the interaction between pile and soil in permafrost regions.

Constitutive model of viscoelastic dynamic damage for the material of gas obturator in modular-charge howitzer

In order to investigate the mechanical response behavior of the gas obturator of the breech mechanism,made of polychloroprene rubber(PCR), uniaxial compression experiments were carried out by using a universal testing machine and a split Hopkinson pressure bar(SHPB), obtaining stress-strain responses at different temperatures and strain rates. The results revealed that, in comparison to other polymers, the gas obturator material exhibited inconspicuous strain softening and hardening effects;meanwhile, the mechanical response was more affected by the strain rate than by temperature. Subsequently, a succinct viscoelastic damage constitutive model was developed based on the ZWT model, including ten undetermined parameters, formulated with incorporating three parallel components to capture the viscoelastic response at high strain rate and further enhanced by integrating a three-parameter Weibull function to describe the damage. Compared to the ZWT model, the modified model could effectively describe the mechanical response behavior of the gas obturator material at high strain rates. This research laid a theoretical foundation for further investigation into the influence of chamber sealing issues on artillery firing.

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.

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.

A statistical damage-based constitutive model for shearing of rock joints in brittle drop mode

Some rock joints exhibit significant brittleness,characterized by a sharp decrease in shear stress upon reaching the peak strength.However,existing models often fail to accurately represent this behavior and are encumbered by numerous parameters lacking clear mechanical significance.This study presents a new statistical damage constitutive model rooted in both damage mechanics and statistics,containing only three model parameters.The proposed model encompasses all stages of joint shearing,including the compaction stage,linear stage,plastic yielding stage,drop stage,strain softening stage,and residual strength stage.To derive the analytical expression of the constitutive model,three boundary conditions are introduced.Experimental data from both natural and artificial rock joints is utilized to validate the model,resulting in average absolute relative errors ranging from 3%to 8%.Moreover,a comparative analysis with established models illustrates that the proposed model captures stress drop and post-peak strain softening more effectively,with model parameters possessing clearer mechanical interpretations.Furthermore,parameter analysis is conducted to investigate the impacts of model parameters on the curves and unveil the relationship between these parameters and the mechanical properties of rock joints.Importantly,the proposed model is straightforward in form,and all model parameters can be obtained from direct shear tests,thus facilitating the utilization in numerical simulations.

Constitutive Model for Rubberlike Materials

The isotropic continuum stored energy (CSE) functional, fully combined with the Poisson function, has been applied to constitutively model compressible as well as incompressible rubberlike materials. The isotropic CSE constitutive model fits uniaxial tension test and predicts unfitted pure shear and equibiaxial tension tests of incompressible S4035A thermoplastic elastomer (TPE). Furthermore, the isotropic CSE model along with the Poisson function fits uniaxial tension test along with the kinematic relation test and predicts unfitted pure shear and equibiaxial tension tests along with the corresponding kinematic relation tests of a compressible synthetic rubber. The detailed procedures for uniquely identifying constitutive parameters are emphasized, the experimental characterization results are discussed, and the principles for constitutive models are summarized and augmented.

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.

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.

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.

Effect of pre-deformation on aging creep of Al-Li-S4 alloy and its constitutive modeling

A set of uniaxial tensile creep tests at different pre-deformations, aging temperatures and stress levels were carried out for Al-Li-S4 alloy, and the creep behavior and the effects of pre-deformation on mechanical properties and microstructures were determined under basic thermodynamics conditions of aging forming. The results show that pre-deformation shortens the time of primary creep and raises the second steady-state creep rate. Then, the total creep strain is greater, but in the range of test parameters it is still smaller than that without pre-deformation. In addition, transmission electron microscopy（TEM） observation shows that pre-deformation promotes the formation of T1 phase and θ′ phase and makes them distribute more dispersively, while inhibits the generation of δ′ phase, which leads to the improvement of mechanical properties of the alloy. A unified constitutive model reflecting the effects of aging mechanism, stress levels and different pre-deformations was established. The fitting results agree with the experimental data well.

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).