Method of Numerical Modeling for Constitutive Relations of Clay

In order to study the method of numerical modeling for constitutive relations of clay, on the basis of the principle of interaction between plastic volumetric strain and plastic generalized shear strain, the two constitutive functionals that include the function of stress path were used as the basic framework of the constitutive model, which are able to demonstrate the dependence of stress path. The two partial differential cross terms appear in the expression of stress-strain increment relation, which are used to demonstrate the interaction between plastic volumetric strain and plastic generalized shear strain. The elasoplastic constitutive models of clay under two kinds of stress paths, CTC and TC, have been constracted using the triaxtal test results. The three basic characteristics of deformation of soils, pressure sensitivity, dilatancy, and dependence of stress path, are well explained using these two models. Using visualization, the three-dimensional surfaces of shear and volume strains in the whole stress field under stress paths of CTC and TC are given. In addition, the two families of shear and volmetric yield Ioei under CTC and TC paths are plotted respectively. By comparing the results of deformation under these two stress paths, it has been found that, there are obvious differences in the strain peaks, the shapes of strain surfaces, and the trends of variation of volumetric yield loci, however both families of shear yield loci are similar. These results demonstrate that the influences of stress path on the constitutive relations of clay are considerably large and not negligible. The nmericul modeling method that can sufficiently reflect the dependence of stress path is superior to the traditional one.

STUDY OF CONSTITUTIVE RELATIONS FOR PSEUDO-ELASTICITY AND SHAPE MEMORY BEHAVIOR

Constitutive relations are given for the description of the deformation behavior of shape memory materials. The deformation is the superposition of the elastic, the thermal and the phase transformation deformation caused by the transformation from one to the other among the high temperature phase, the low temperature phase and the stress induced phase. The phase transformation is controlled by the driving force, i.e., the Gibbs energy difference between the phases.

CRITERION FOR JUDGING SATISFACTION OF CONSISTENCY CONDITIONS IN RATE-DEPENDENT CONSTITUTIVE RELATIONS

Based on irreversible thermodynamics, the criterion for judging the satisfaction of consistency conditions in rate-dependent constitutive relationship is deduced by introducing four basic hypotheses. Formulas for solving internal variables are given. It makes the rate-dependent model applicable no matter whether the consistency conditions can be satisfied or not.

Effect of hydrostatic stress on yield function and plastic constitutive relations

A new yield function taking effect of hydrostatic stress into account is presented through establishing and solving the functional equation satisfied by the yield function, and its characteristic is simple in form and strong in generality. In order to reveal its availability, a comparison is made between the results obtained with it and the experimental results of grey cast iron has be done, both seem to be in good agreement. At the same time, taking the yield function obtained here as a potential function, a new associative plastic constitutive equation taking effect of hydrostatic stress into account is built, and the plastic volume change ratio of plastic deformation is given.

Artificial neural network model of constitutive relations for shock-prestrained copper

Data from the deformation on Split Hopkinson Bar were used for constructing an artificial neural network model. When putting the thermodynamic parameters of the metals into the trained network model, the corresponding yielding stress can be predicted. The results show that the systematic error is small when the objective function is 0.5 , the number of the nodes in the hidden layer is 6 and the learning rate is about 0.1 , and the accuracy of the rate error is less than 3%. [

Neural Network Model for the Constitutive Relations of Soil

The soil constitutive relation is one of the important issues in soil mechanics. It is very difficult to establish mathematical models because of the complexity of soil mechanical behavior. We propose a new method of neural network analysis for establishing soil constitutive models. Based on triaxial experiments of sand in the laboratory, the nonlinear constitutive models of sand expressed by the neural network were set up. In comparison with Duncan\|Chang's model, the neural network method for sand modeling has been proved to be more convenient, accurate and it has a strong fault\|tolerance function.

Numerical simulation of hypersonic thermochemical nonequilibrium flows using nonlinear coupled constitutive relations

To predict aeroheating performance of hypersonic vehicles accurately in thermochemical nonequilibrium flows accompanied by rarefaction effect,a Nonlinear Coupled Constitutive Relations(NCCR)model coupled with Gupta’s chemical models and Park’s two-temperature model is firstly proposed in this paper.Three typical cases are intensively investigated for further validation,including hypersonic flows over a two-dimensional cylinder,a RAM-C II flight vehicle and a type HTV-2 flight vehicle.The results predicted by NCCR solution,such as heat flux coefficient and electron number densities,are in better agreement with those of direct simulation Monte Carlo or flight data than Navier-Stokes equations,especially in the extremely nonequilibrium regions,which indicates the potential of the newly-developed solution to capture both thermochemical and rarefied nonequilibrium effects.The comparisons between the present solver and NCCR model without a two-temperature model are also conducted to demonstrate the significance of vibrational energy source term in the accurate simulation of high-Mach flows.

Analytical method of nonlinear coupled constitutive relations for rarefied non-equilibrium flows

It is well known that Navier-Stokes equations are not valid for those high-Knudsen and high-Mach flows, in which the local thermodynamically non-equilibrium effects are dominant. To extend the non-equilibrium describing the ability of macroscopic equations, Nonlinear Coupled Constitutive Relation(NCCR) model was developed from Eu’s generalized hydrodynamic equations to substitute linear Newton’s law of viscosity and Fourier’s law of heat conduction in conservation laws. In the NCCR model, how to solve the decomposed constitutive equations with reasonable computational cost is a key ingredient of this scheme. In this paper, an analytic method is proposed firstly. Compared to the iterative procedure in the conventional NCCR model, the analytic method not only obtains exact roots of the decomposed constitutive polynomials, but also preserves the nonlinear constitutive relations in the original framework of NCCR methods. Numerical tests to assess the efficiency and accuracy of the proposed method are conducted for argon shock structures, Couette flows, two-dimensional hypersonic flows over a cylinder and threedimensional supersonic flows over a three-dimensional sphere. These superior advantages of the current method are expected to render itself a powerful tool for simulating the hypersonic rarefied flows and microscale flows of high Knudsen number for engineering applications.

Constitutive relations for the shear band evolution in granular matter under large strain

A so-called ＂split-bottom ring shear cell＂ leads to wide shear bands under slow, quasi-static deformation. Unlike normal cylindrical Couette shear cells or rheometers, the bottom plate is split such that the outer part of it can move with the outer wall, while the other part （inner disk） is immobile. From discrete element simulations （DEM）, several continuum fields like the density, velocity, deformation gradient and stress are computed and evaluated with the goal to formulate objective constitutive relations for the powder flow behavior. From a single simulation, by applying time- and （local） space-averaging, a non-linear yield surface is obtained with peculiar stress dependence. The anisotropy is always smaller than the macroscopic friction coefficient. However, the lower bound of anisotropy increases with the strain rate, approaching the maximum according to a stretched exponential with a specific rate that is consistent with a shear path of about one particle diameter.

Modified Arrhenius-type Constitutive Model and Artificial Neural Network-based Model for Constitutive Relationship of 316LN Stainless Steel during Hot Deformation

Hot compression experiments of 316LN stainless steel were carried out on Gleeble-3500 thermo-simulator in deforma- tion temperature range of 1 223-1 423 K and strain rate range of 0.001-1 s 1. The flow behavior was investigated to evaluate the workability and optimize the hot forging process of 316LN stainless steel pipes. Constitutive relationship of 316LN stainless steel was comparatively studied by a modified Arrhenius-type analytical constitutive model considering the effect of strain and by an ar- tificial neural network model. The accuracy and effectiveness of two models were respectively quantified by the correlation coeffi- cient and absolute average relative error. The results show that both models have high reliabilities and could meet the requirements of engineering calculation. Compared with the analytical constitutive model, the artificial neural network model has a relatively higher predictability and is easier to work in cooperation with finite element analysis software.

A REVIEW ON RESEARCH ON DEVELOPMENT OF LIGAMENT CONSTITUTIVE RELATIONS ON MACRO, MESO, AND MICRO LEVELS

Ligaments are densely connective soft tissues capable of maintaining stability and function of knee joint.As an important factor,the constitutive relation of ligament would affect its biomechanics and further play an essential role in the research on ligament injury,healing and treatment.The objective of this paper is to provide an overview of the current research on ligament constitutive relations on the macro,meso,and micro levels as well as the anatomy and histological structure of ligament.Some studies of biomechanical behaviors during ligament injury and healing periods have also been investigated.Based on the research on ligament constitutive relation in the past three decades,a discussion of some research perspectives is also presented,such as a validated accurate measuring method of in situ strain in ligament,a new constitutive relation involving the distribution of ultra-structural properties,and a rational estimation of ligament injury and healing process by the change of its ultra-structural or histological characteristics.

Memristor interpretations based on constitutive relations

The attractive memristor is interpreted based on its constitutive relation.The memory property of the memristor is explained,along with the explanation on its three fingerprints：（1） Pinched hysteresis loop;（2） Hysteresis lobe area decreases as frequency increases;（3） Pinched hysteresis loop shrinks to a single-valued function at infinite frequency.Where the magnetic flux is in Strukov＇s memristor is also introduced.Resistive elements including the memristor are taken as an example to argue that the constitutive relation determines the electrical property of a circuit element and diagram method is used to distinguish different elements in the resistive element series.

BIAXIAL LOADING OF NEO-CLASSICAL LIQUID CRYSTAL ELASTOMERS:CONSTITUTIVE RELATIONS AND SOFT BEHAVIOR

The thermo-order-mechanical behaviors of liquid crystal elastomers （LCEs） under biaxial loading are studied in this paper. Inverse method for nonlinear elastic problems is utilized by imposing biaxial stretching to thin rectangular samples. Neo-classical elastic energy is used together with the Landau-de Gennes nematic free energy. Under plane stress assumptions, the constitutive equations are derived. Due to the possible reorientations of the liquid crystal molecules induced by the imposed biaxial loading, the in-plane nonlinear stress-strain relations can have different expressions depending on which loading axis will have the largest effective principal strain. And the free energy is a multi-well non-convex potential function. As shown by some typical loading paths, the LCE samples will exhibit an anisotropic nonlinear elastic behavior, as long as the loading has not induced a reorientation of the liquid crystal molecules. When this did occur, jumps of stresses could take place for dead loadings due to the losing of stability.

Dynamic compressive behavior and constitutive relations of lanthanum metal

The dynamic compressive behavior and constitutive relations of Lanthanum（La） metal was determined by using the first compression in split Hopkinson pressure bar（SHPB） tests at different strain rates and temperatures.The constitutive relation of La metal determined in a certain range of strains was employed and adjusted in numerically simulating large deformations of La metal specimens generated by multi-compression in SHPB tests and recorded by a high-speed camera.The dynamic compressive behavior and constitutive relations of La metal under multiple SHPB tests loading was also revealed.The results of scanning electron microscope（SEM） investigation of the recovered La metal specimens for typical tests showed that there was a variety of deformation microstructures depending on strain rate,temperature and stress state.

Isogeometric Analysis of Hyperelastic Material Characteristics for Calcified Aortic Valve

This study explores the implementation of computed tomography(CT)reconstruction and simulation techniques for patient-specific valves,aiming to dissect the mechanical attributes of calcified valves within transcatheter heart valve replacement(TAVR)procedures.In order to facilitate this exploration,it derives pertinent formulas for 3D multi-material isogeometric hyperelastic analysis based on Hounsfield unit(HU)values,thereby unlocking foundational capabilities for isogeometric analysis in calcified aortic valves.A series of uniaxial and biaxial tensile tests is executed to obtain an accurate constitutive model for calcified active valves.To mitigate discretization errors,methodologies for reconstructing volumetric parametric models,integrating both geometric and material attributes,are introduced.Applying these analytical formulas,constitutive models,and precise analytical models to isogeometric analyses of calcified valves,the research ascertains their close alignment with experimental results through the close fit in displacement-stress curves,compellingly validating the accuracy and reliability of the method.This study presents a step-by-step approach to analyzing themechanical characteristics of patient-specific valves obtained fromCT images,holding significant clinical implications and assisting in the selection of treatment strategies and surgical intervention approaches in TAVR procedures.

A nonlinear constitutive model for magnetostrictive materials

A general nonlinear constitutive model is proposed for magnetostrictive materials, based on the important physical fact that a nonlinear part of the elastic strain produced by a pre-stress is related to the magnetic domain rotation or movement and is responsible for the change of the maximum magnetostrictive strain with the pre-stress. To avoid the complicity of determining the tensor function describing the nonlinear elastic strain part, this paper proposes a simplified model by means of linearizing the nonlinear function. For the convenience of engineering applications, the expressions of the 3-D （bulk）, 2-D （film） and 1-D （rod） models are, respectively, given for an isotropic material and their applicable ranges are also discussed. By comparison with the experimental data of a Terfenol-D rod, it is found that the proposed model can accurately predict the magnetostrictive strain curves in low, moderate and high magnetic field regions for various compressive pre-stress levels. The numerical simulation further illustrates that, for either magnetostrictive rods or thin films, the proposed model can effectively describe the effects of the pre-stress or residual stress on the magnetization and magnetostrictive strain curves, while none of the known models can capture all of them. Therefore, the proposed model enjoys higher precision and wider applicability than the previous models, especially in the region of the high field.

Mechanical properties and constitutive behaviors of as-cast 7050 aluminum alloy from room temperature to above the solidus temperature

The mechanical properties and constitutive behaviors of as-cast AA7050 in both the solid and semi-solid states were determined using the on-cooling and in situ solidification approaches, respectively. The results show that the strength in the solid state tends to increase with decreasing temperature. The strain rate plays an important role in the stress–strain behaviors at higher temperatures, whereas the influence becomes less pronounced and irregular when the temperature is less than 250°C. The experimental data were fitted to the extended Ludwik equation, which is suitable to describe the mechanical behavior of the materials in the as-cast state. In the semi-solid state, both the strength and ductility of the alloy are high near the solidus temperature and decrease drastically with decreasing solid fraction. As the solid fraction is less than 0.97, the maximum strength only slightly decreases, whereas the post-peak ductility begins to increase. The experimental data were fitted to the modified creep law, which is used to describe the mechanical behavior of semi-solid materials, to determine the equivalent parameter fGBWL, i.e., the fraction of grain boundaries covered by liquid phase.

A Nonlinear Magneto-Mechanical Coupled Constitutive Model for the Magnetostrictive Material Galfenol

In order to predict the performance of magnetostrictive smart material and pushits applications in engineering, it is necessary to build the constitutive relations for themagnetostrictive material Galfenol. For Galfenol rods under the action of the pre-stress andmagnetic field along the axial direction, the one-dimensional nonlinear magneto-mechanicalcoupling constitutive model is proposed based on the elastic Gibbs free energy, where theTaylor expansion of the elastic Gibbs free energy is made to obtain the polynomial forms. Andthen the constitutive relations are derived by replacing the polynomial forms with the propertranscendental functions based on the microscopic magneto-mechanical coupling mechanism.From the perspective of microscopic mechanism, the nonlinear strain related to magneticdomain rotation results in magnetostrictive strain changing with the pre-stress among theelastic strains induced by the pre-stress. By comparison, the predicted stress-strain,magnetostrictive strain, magnetic induction and magnetization curves agreed well withexperimental results under the different pre-stresses. The proposed model can describe notonly the influences of pre-stress on magnetostrictive strain and magnetization curves, butalso nonlinear magneto-mechanical coupling effect of magnetostrictive materialsystematically, such as the Young’s modulus varying with stress and magnetic field. In theproposed constitutive model, the key material constants are not chosen to obtain a good fitwith the experimental data, but aremeasured directly by experiments, such as the saturationmagnetization, saturation magnetostrictive coefficient, saturation Young’s modulus, linearmagnetic susceptibility and so on. In addition, the forms of the new constitutive relationsare simpler than the existing constitutive models. Therefore, this model could be appliedconveniently in the engineering applications.

Analysis of Bonding Properties of Corroded Reinforcement Concrete

In order to investigate the degradation of bonding properties between corroded steel bars and concrete,this study employs the half-beam method to conduct bond-slip tests between corroded steel bars and concrete after impressed-current accelerated corrosion of the steel bars in concrete.The effects of steel corrosion rate,steel bar diameter,steel bar strength grade,and concrete strength grade on the bonding properties between concrete and corroded steel bars were analyzed.The influence of different corrosion rates on specimens’bonding strength and bond-slip curves was determined,and a constitutive relationship for bond-slip between corroded steel bars and concrete was proposed.The results indicate that the ultimate bonding strength of corroded reinforced concrete specimens decreases with increasing corrosion rate.Additionally,an increase in corrosive crack width leads to a linear decrease in bonding strength.Evaluating the decline in adhesive properties through rust expansion crack width in engineering applications is feasible.Furthermore,a bond-slip constitutive relationship between corroded steel bars and concrete was established using relative bond stress and relative slip values,which aligned well with the experimental findings.

A GENERAL CONSTITUTIVE RELATION FOR FATIGUE CRACK GROWTH ANALYSIS OF METAL STRUCTURES

Crack growth rate curves are the fundamental material property for metal structures under fatigue loading. Although there are many crack growth rate curves available in the literature, few of them showed the capability to explain various special phenomena observed in tests. A modified constitutive relation recently proposed by McEvily and his co-workers showed very promising capability. This modified constitutive relation is further generalized by (1) introducing an unstable fracture condition; (2) defining a virtual strength to replace the yield stress; and (3) defining an overload and underload parameter. The performances of this general constitutive relation for fatigue crack growth is extensively studied and it is found that this general constitutive relation is able to explain various phenomena observed with particular strong capability on load sequence effect.