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I present a derivation of the static equations of a granular mechanical interpretation of Cosserat continuum based on a continuum formulated in the intra-granular fields. I assume granular materials with three-dimensional, non-spherical, and deformable grains, and interactions given by traction acting on finite contact areas. Surface traction is decomposed into a mean and a fluctuating part. These account for forces and contact moments. This decomposition leads to a split of the Cauchy stress tensor into two tensors, one of them corresponding to the stress tensor of the Cosserat continuum. Macroscopic variables are obtained by averaging over representative volume. The macroscopic Cauchy stress tensor is shown to be symmetric even in non-equilibrium. The stress tensor of the Cosserat continuum becomes asymmetric when the sum of the contact moments acting on the boundary of the representative volume is different from zero.

One of the key tasks for mathematical representation of a constitutive model is calibration of its parameters. An inverse computational–experimental solution with the data modelling approach was employed here to estimate the parameters of a hyperbolic sine type constitutive equation. The solution utilizes a multi-layer constitutive model. The computational component of the inverse solution includes a rigid viscoplastic finite element code based on the thermo-mechanical coupling. The hot torsion test data comprises the experimental component of the inverse solution. Determination of the primary constitutive parameters (PCPs) pertinent to a 303 Austenitic stainless steel is presented here as an example. In order to facilitate the calibration of the sub-models, a procedure to provide an initial guess vector for the secondary constitutive parameters (SCPs) is given.

Constitutive equations of non-isothermal polymer melt are presented by the analysis of entropic free energy contribution of the macromolecular chains, which are treated as elastic dumbbell models. With describing non-isothermal dumbbell spring, as the function of temperature, the non-linear elastic coefficient expression causes the appearance of temperature gradient in stress constitutive equations. Following the constitutive equation of Hookean dumbbell model, non-isothermal stress constitutive equations of FENE and FENE-P models are derived. In deriving process of constitutive equations, the second moment approximation is used to closure FENE model. Using the non-isothermal constitutive equations, numerical simulations of polymer flow through shear cavity and planar contraction cavity are presented. And the distributions of correlative stress functions and the effects of different temperatures on stress functions are discussed. The present results are shown to explore the non-isothermal constitutive equations of elastic dumbbell models, and to search more accurately describing way of non-isothermal polymer melt.

The determination of the equivalent (homogenized) constitutive tensor is one of the most important steps in multiscale models as well as in the classical homogenization theory. In this article, a procedure for determining the homogenized instantaneous (tangent) constitutive tensor of elastic materials containing growing cracks is proposed. The primary purpose of this procedure is its use in two-way coupled multiscale finite element algorithms that can model crack formation and propagation at the local microstructure. The procedure is basically developed by relating the local displacement field to the global strain tensor at each location and using first-order homogenization techniques. The finite element formulation is developed and some example problems are presented in order to verify and demonstrate the model capabilities.

A methodology is devised to utilize the statistical mechanical entropy of an isolated, constrained atomistic system to define constitutive response functions for the dissipative driving-force and energetic fields in continuum thermomechanics. A thermodynamic model of dislocation mechanics is discussed as an example. Primary outcomes are constitutive relations for the back-stress tensor and the Cauchy stress tensor in terms of the elastic distortion, mass density, polar dislocation density, and the scalar statistical density.

The present investigation has been conducted in order to develop a constitutive description for a low C-Mn steel employed in structural applications, when this material is deformed under hot-working conditions. The set of equations that encompass the constitutive description of the steel are based on a simplified form of the Mechanical Threshold Stress Model. Accordingly, it has been shown that the flow stress of the material can be considered to arise from two different contributions: solid solution and precipitation hardening, which do not evolve in the course of plastic deformation and work-hardening, the only evolving component of the flow stress. The evolution of work-hardening has been introduced in the model by means of the work-hardening law proposed by Estrin and Mecking, in differential form. Also, the temperature and strain rate dependence of the contributions of solid solution and precipitation hardening to the flow stress, as well as that of the saturation stress in the work-hardening law, are described by means of the model earlier advanced by Kocks. The constitutive formulation encompasses seven material parameters which have been determined employing the experimental data corresponding to the strain interval where the material only exhibits positive work-hardening. It is able to predict accurately both work-hardening and work-softening transients when changes in deformation conditions occur. Furthermore, it can be easily implemented in any advanced computational tool employed for the analysis of hot-working processes of this material.

This paper describes experiments and the development of constitutive equations to predict the steady-state response of filled rubber under cyclic loading. An MTS servo-hydraulic machine was used to obtain the dynamic hysteresis curves for a filled rubber compound in uniaxial tension-compression. The material tests were performed with mean strains from −0.1 to 0.1, strain amplitudes ranging from 0.02 to 0.1, and strain rates between 0.01 and 10 s−1. Temporary material set, the Payne effect and rate-dependence were observed from the experimental results. A hyper-viscoelastic constitutive model was developed to characterize the dynamic response of the rubber. A cornerstone of this constitutive modeling was to devise a scheme to evaluate material set and a finite strain, non-linear viscoelastic law from the test data. Predictions of the dynamic hysteresis curves using the proposed constitutive equation were found to be in good agreement with the uniaxial test results.

Some simple boundary value problems are studied, for a new class of elastic materials, wherein deformations are expressed as non-linear functions of the stresses. Problems involving ‘homogeneous’ stress distributions and one-dimensional stress distributions are considered. For such problems, deformations are calculated corresponding to the assumed stress distributions. In some of the situations, it is found that non-unique solutions are possible. Interestingly, non-monotonic response of the deformation is possible corresponding to monotonic increase in loading. For a subclass of models, the strain–stress relationship leads to a pronounced strain-gradient concentration domain in the body in that the strains increase tremendously with the stress for small range of the stress (or put differently, the gradient of the strain with respect to the stress is very large in a narrow domain), and they remain practically constant as the stress increases further. Most importantly, we find that for a large subclass of the models considered, the strain remains bounded as the stresses become arbitrarily large, an impossibility in the case of the classical linearized elastic model. This last result has relevance to important problems in which singularities in stresses develop, such as fracture mechanics and other problems involving the application of concentrated loads.

A wave-like dynamic solution of elastic materials is studied. It is shown that if the material is not hyperelastic, the amplitude of the wave can grow exponentially. As a result, the kinetic energy in the material would increase indefinitely even when the applied forces do zero work on the material.

The constitutive equation of the material is an essential ingredient of any structural calculation. In this article, a phenomenological constitutive model is established to describe the dynamic deformation behavior of 30Cr2Ni4MoV steel in wide strain rate, strain, and temperature ranges. Also, the mathematical models to predict peak stress and corresponding strain were obtained. The stress–strain values predicted by the developed model well agree with experimental results, which confirmed that the developed constitutive equation gives an accurate and precise estimate for the flow stress of 30Cr2Ni4MoV steel.

The ever increasing number of shape memory alloy applications has motivated the development of appropriate constitutive models taking into account large rotations and moderate or finite strains. Up to now proposed finite-strain constitutive models usually contain an asymmetric tensor in the definition of the limit (yield) function. To this end, in the present work, we propose an improved alternative constitutive model in which all quantities are symmetric. To conserve the volume during inelastic deformation, an exponential mapping is used to arrive at the time-discrete form of the evolution equation. Such a symmetric model simplifies the constitutive relations and as a result of less nonlinearity in the equations to be solved, numerical efficiency increases. Implementing the proposed constitutive model with in a user-defined subroutine UMAT in the nonlinear finite element software ABAQUS/Standard, we solve different boundary value problems. Comparing the solution CPU times for symmetric and asymmetric cases, we show the effectiveness of the proposed constitutive model as well as of the solution algorithm. The presented procedure can also be used for other finite-strain constitutive models in plasticity and shape memory alloy constitutive modeling.

Since the constitutive information is one of the most important aspects of material deformation analysis, here a new constitutive model is proposed that can investigate the behavior of material during intense deformation better than existent models. The model that is completely based on physical mechanisms can predict all stages of flow stress evolution and also can elucidate the effects of strain and strain rate on flow stress evolution of material during intense plastic deformation. Here as an application, implementation of the constitutive model in finite element method (FEM) is used to compare two methods of sever plastic deformation (SPD) processes of copper sheet; repetitive corrugation and straightening (RCS) and constrained groove pressing (CGP). The modeling results are in good agreement with the experimental data and show that the hardness uniformity and its magnitude for RCSed sheet are higher than that for CGPed sheet. However, the prominence of these processes in strain uniformity depends on pass number.

An important theory on the dynamics of complex interfaces is the Doi and Ohta theory where the interfacial contribution to the Cauchy stress tensor is determined from an interfacial conformation tensor. For a uniform deformation field in the Eulerian framework, Doi and Ohta adopted a decoupling approximation to reduce a fourth-order tensor into two second-order tensors and derived a differential equation governing the evolution of the interfacial conformation tensor. In this paper, a different formulation is presented for establishing the Cauchy stress tensor based on a path-independent interfacial energy function. By differentiating this interfacial energy function against a Lagrangian strain tensor, a nearly closed-form solution for the stress tensor was determined, involving only elementary algebraic and matrix operations. From this process, the stress-conformation relation proposed by Doi and Ohta is also confirmed from a thermodynamic perspective. The testing cases with uniaxial elongation and simple shear further showed improved fitting to the analytical or exact solutions.

A technique to determine the axisymmetric elastoplastic state of thin shells with allowance for the third invariant of the stress deviator is developed. The technique is based on the theory of thin shells that takes into account transverse shear and torsional strains. Plastic equations that relate the components of the stress tensor in Eulerian coordinates with the linear components of the finite-strain tensor are used as constitutive equations. The nonlinear scalar functions in the constitutive equations are found from base tests on tubular specimens under proportional loading for different stress modes. The boundary-value problem is solved by numerically integrating a system of ordinary differential equations

For an isotropic material, we construct the exact local equation for the influence of the stress tensor on the tensor of magnetic permeability. It is shown that, in the general case, this influence is determined by the magnetostriction properties of the material and, to a lesser degree, by the dependence of its elasticity on the applied magnetic field. The principal axes of the stress tensor are also the principal axes of the tensor of permeability. However, the spherical part of the tensor of permeability also depends on the orientation of the principal axes of stresses relative to the magnetic field. Under the assumption of weak magnetic dependence of the elastic properties, we deduce simplified expressions for the triaxial and plane stressed states.

We develop a systematic method for the derivation of closed form and thermodynamically consistent constitutive equations of complex fluids from microscopic models. The method builds upon our recent work {[}Ilg et al. Phys. Rev. E 79 011802 (2009)] on thermodynamically guided simulations within a consistent coarse graining scheme. These simulations are powerful at low to intermediate flow rates and have the considerable advantage that they do not require flow adapted boundary conditions i.e. operate at arbitrary homogeneous flows. The new method for deriving constitutive equations is illustrated for low molecular polymer melts subjected to imposed homogeneous flow fields. The differential constitutive equation we obtain for this model system is a simple nonlinear equation of change for the conformation tensor from which the stress tensor is readily obtained. The proposed constitutive model shows shear thinning (shear viscosity exhibiting fractional power laws in the range 0.40 to 0.86 the corresponding range for the first viscometric function is 1.20 to 1.43) stress overshoots normal stress coefficients and elongational viscosities in agreement with reference results. The constitutive equation can be interpreted as a molecularly derived modified Giesekus model with conformation dependent coefficients. (C) 2011 The Society of Rheology. {[}DOI: 10.1122/1.3523485]}}

It is well known that Shuttleworth's equation shows the relation between surface stress and surface tension of the solid. But surface stress is a kind of virtual concept defined to the mathematical surface without thickness, so that the physical meanings of surface stress can not be clear. In this paper, we tried to evaluate surface stress and surface tension of homogeneous isotropic elastic body by using primitive constitutive equation derived from continuum mechanics and thermodynamics of surface. From the results, it was shown that surface stress was the stress caused in the elastic bulk which had infinity small thickness and that surface tension was the function of strain in this elastic bulk.

Objectivity and compatibility with thermodynamics of evolution equations are examined in connection with the modelling of viscoelastic solids. The purpose of the paper is to show that the evolution equation for the stress is eventually obtained by means of a tensorial internal variable within the framework of the reference configuration. The non-simple character is realized by gradients of the internal variable. The thermodynamic analysis is developed by investigating the entropy inequality in the reference configuration and allowing for a non-zero extra-entropy flux. It follows that the evolution for the Cauchy stress tensor involves the Oldroyd derivative, irrespective of the form of the non-local terms.

The deformation work rate can be expressed by the time rate of pair functional potentials which describe the energy of materials in terms of atomic bonds and atom embedding interactions. According to Cauchy-Born rule, the relations between the microscopic deformations of atomic bonds and electron gas and macroscopic deformation are established. Further, atomic bonds are grouped according to their directions, and atomic bonds in the same direction are simplified as a spring-bundle component. Atom embedding interactions in unit reference volume are simplified as a cubage component. Consequently, a material model composed of spring-bundle components and a cubage component is established. Since the essence of damage is the decrease and loss of atomic bonding forces, the damage effect can be reflected by the response functions of these two kinds of components. Formulating the mechanical responses of two kinds of components, the corresponding elasto-damage constitutive equations are derived. Considering that slip is the main plastic deformation mechanism of polycrystalline metals, the slip systems of crystal are extended to polycrystalline, and the slip components are proposed to describe the plastic deformation. Based on the decomposition of deformation gradient and combining the plastic response with the elasto-damage one, the elasto-plastic damage constitutive equations are derived. As a result, a material model formulated with spring-bundle components, a cubage component and slip components is established. Different from phenomenological constitutive theories, the mechanical property of materials depends on the property of components rather than that directly obtained on the representative volume element. The effect of finite deformation is taken into account in this model. Parameter calibration procedure and the basic characteristics of this model are discussed.

We provide a thermodynamic basis for compressible fluids of a Korteweg type that are characterized by the presence of the dyadic product of the density gradients ∇ϱ ⊗ ∇ϱ in the constitutive equation for the Cauchy stress. Our approach does not need to introduce any new or non-standard concepts such as multipolarity or interstitial working and is based on prescribing the constitutive equations for two scalars: the entropy and the entropy production. In comparison with the Navier–Stokes–Fourier fluids we suppose that the entropy is not only a function of the internal energy and the density but also of the density gradient. The entropy production takes the same form as for a Navier–Stokes–Fourier fluid. For a Navier–Stokes–Fourier fluid one can express the entropy production equivalently in terms of either thermodynamic affinities or thermodynamic fluxes. Following the ideas of K.R. Rajagopal concerning the systematic development of implicit constitutive theory and primary role of thermodynamic fluxes (such as force) that are cause of effects in thermodynamic affinities (such as deformation) in considered processes, we further proceed with a constitutive equation for entropy production expressed in terms of thermodynamic fluxes. The constitutive equation for the Cauchy stress is then obtained by maximizing the form of the rate of entropy production with respect to thermodynamic fluxes keeping as the constraint the equation expressing the fact that the entropy production is the scalar product of thermodynamic fluxes and thermodynamic affinities. We also look at how the form of the constitutive equation changes if the material in question is incompressible or if the processes take place at constant temperature. In addition, we provide several specific examples for the form of the internal energy and make the link to models proposed earlier. Starting with fully implicit constitutive equation for the entropy production, we also outline how the methodology presented here can be extended to non-Newtonian fluid models containing the Korteweg tensor in a straightforward manner.

All the investigations on the mechanics and physics of electro-magneto-thermo-elastic materials are based on the set of constitutive equations in which the strain, electric displacement, and magnetic induction and entropy are expressed in terms of the stress, electric field, magnetic field and temperature. A concise formulation of relevant non-linear constitutive relations is presented in this paper. The electro-magneto-thermo-elastic materials show significant interaction among the elastic, electric, magnetic and thermal fields due to the coupled nature of constitutive equations. The thermodynamic Gibbs function of state describes stress, strain, internal energy, electric field, electric displacement, magnetic field, magnetic induction, temperature and entropy. The anisotropic non-linear problem of electro-magneto-thermo-elasticity has been derived on the basis of thermodynamic principle. The Gibbs function is expanded into Taylor series. Material constants have been analyzed from strain, electric displa...

It is of theoretical and engineering interest to establish a macro-mechanical constitutive model of the shape memory polymer (SMP), which includes the mechanical constitutive equation and the material parameter function, from the viewpoint of practical application. In this paper, a new three-dimensional macro-mechanical constitutive equation, which describes the mechanical behaviors associated with the shape memory effect of SMP, is developed based on solid mechanics and the viscoelasticity theorem. According to the results of the DMA test, a new material parameter function is established to express the relationship of the material parameters and temperature during the glass transition of SMP. The new macro-mechanical constitutive equation and material parameter function are used to numerically simulate the process producing the shape memory effect of SMP, which includes deforming at high temperature, stress freezing, unloading at low temperature and shape recovery. They are also used to investigate and analyze the influences of loading rate and temperature change rate on the thermo-mechanical behaviors of SMP. The numerical results and the comparisons with Zhou’s material parameter function and Tobushi’s mechanical constitutive equation illustrate that the proposed three-dimensional macro-mechanical constitutive model can effectively predict the thermo-mechanical behaviors of SMP under the state of complex stress.

The rate constitutive equations based on upper convected, lower convected, Jaumann, Truesdell and Green–Naghdi stress rates, etc. [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] have been used for the solid matter and polymeric liquids when the mathematical models are derived employing conservation laws in the Eulerian description. The rate constitutive equations provide relationship between convected time derivatives of the stress tensor and the convected time derivatives of strain tensor through the constitution of the matter. It is well known that if one assumes constant properties of the matter, then the use of various rate constitutive equations yield different responses [9] when the strains and strain rates are not infinitesimal (finite deformation) even though the rate constitutive equations are objective or frame invariant. This has been rationalized by the argument that in order for the different rate constitutive equations to produce the same response, the material tensor must be different for each case [9]. Ou...

A large deformation analysis has been given by using a modified Gao’s second constitutive relation. In certain circumstances, the new constitutive relation may be simplified to the Mooney-Rivlin and Neo-Hookean models. With regard to incompressible materials, when the influence of the second strain invariant is neglected, the modified model brings a unified form of Gao’s two constitutive models. The new constitutive relation has been utilized to resolve a problem of a hyperelastic membrane containing a hole or rigid inclusion. The basic governing equations of incompressible materials have been obtained and numerical solutions of equations illustrate how constitutive parameters affect the result of membranes’ deformation. These conclusions may provide helps to practical problems.

The elastoplastic deformation of an isotropic material is described using constitutive equations and allowing for the stress mode. The equations include two nonlinear functions that relate the first and second invariants of the stress and linear-strain tensors to the stress mode angle. It is proposed to use a linear rather than nonlinear relationship between the first invariants of the tensors. This simplification is validated by comparing calculated and experimental strains under loading with constant and variable stress mode angle

A thermoelasto-viscous material is defined by a set of constitutive laws in which the stress, entropy, heat flux and free energy are functions of the present configuration, temperature, temperature gradient and the rate of change of all three of these. Here these materials are presented within the framework of Walter Noll’s new theory of simple materials, so that the constitutive laws are specified without the use of a frame of reference. The Coleman-Noll procedure is carried out, and the symmetry group of the material is also discussed, both without using a frame of reference. It is then shown what form the constitutive laws of a fluid thermoelasto-viscous material take when a frame of reference is considered. Finally, the governing equations for these materials are explicitly obtained and discussed. The results in this paper may serve as a foundation for new and better mathematical models to deal with phenomena such as heat transfer, heat exchanges and thermophoresis.

In solid mechanics, the phenomenological description of processes that occur at microand nanolevel means that new material parameters modeling some characteristics of the object structure are introduced in the mathematical model and, first of all, in the constitutive relations. These parameters can be either of scalar or tensor nature.

The problem of the search of the invariants of an anisotropic elastic tensor representing the mechanical response of a complex elastic body in a two dimensional space is addressed, in particular for a tensor that does not possess all the tensor symmetries typical of classical elasticity. The invariants of the stiffness tensor are found and all the possible types of orthotropy are discussed.

Today, the identification of material model parameters is based more and more on full-field measurements. This article explains how an appropriate use of the constitutive equation gap method (CEGM) can help in this context. The CEGM is a well-known concept which, until now, has been used mainly for the verification of finite element simulations. This has led to many developments, especially concerning the techniques for constructing statically admissible stress fields. The originality of the present study resides in the application of these recent developments to the identification problem. The proposed CEGM is described in detail, then evaluated through the identification of heterogeneous isotropic elastic properties. The results obtained are systematically compared with those of the equilibrium gap method, which is a well-known technique for the resolution of such identification problems. We prove that the use of the enhanced CEGM significantly improves the quality of the results.

In the present paper, we consider basic relations of the mathematical theory of plasticity for the spatial state corresponding to the edge of the Coulomb-Tresca prism, which follow from the generalized associated flow law restricting the plastic flow freedom for the above states to the minimal possible extent. We found that the spatial relations of the theory of plasticity, formulated by A. Yu. Ishlinsky in 1946, can be derived from the above version of the theory of flow. We show that the A. Yu. Ishlinsky constitutive relations for states on the Coulomb-Tresca prism edge express the commutativity of the stress tensor and the tensor of plastic strain increments. We obtained one explicit form of the constitutive relation relating the stress tensor to the plastic strain increments for the stressed states corresponding to the Coulomb-Tresca prism edge.

This paper treats the numerical implementation of a finite strain, elasto-viscoelastic-viscoplastic constitutive model for semi-crystalline polymers, written in principal stretches. A parallel configuration of the three model elements is used that enables the decoupled algorithmic treatment of each response within a stress update numerical scheme. The numerical aspects associated with the use of principal stretch constitutive expressions in a tensor space numerical environment are initially developed for the general cases of any elastic or inelastic constitutive element. Included is the formulation of the closed-form, consistent tangential modulus tensor. The principal space algorithmic treatments of the elastic, viscoelastic and viscoplastic elements are then used as specific examples. Of particular importance is the development of a principal space, closest point projection return mapping algorithm for viscoplasticity including isotropic strain hardening. Preliminary numerical examples are presented to illustrate the versatility of the model.

We present a new algorithm for solving viscoelastic flows with a general constitutive equation. In our approach the hyperbolic constitutive equation is split such that the term for the convective transport of stress tensor is treated as a source. This allows the stress tensor at each grid point to be expressed mainly in terms of the velocity gradient tensor at the same point. Then, the set of six stress tensor components is found after inverting a six by six matrix at each grid point. Thus we call this algorithm the grid-by-grid inversion method. The convective transport of stress tensor in the constitutive equation, which has been treated as a source, is updated iteratively. The present algorithm can be combined with finite volume method, finite element method or the spectral methods. To corroborate the accuracy and robustness of the present algorithm we consider viscoelastic flow past a cylinder placed at the center between two plates, which has served as a benchmark problem. Also considered is the investigation of the pattern and strength of the secondary flows in the viscoelastic flows through a rectangular pipe. It is found that the present method yields accurate results even for large relaxation times.

We present a material frame formulation analogous to the spatial frame formulation developed by Hardy, whereby expressions for continuum mechanical variables such as stress and heat flux are derived from atomic-scale quantities intrinsic to molecular simulation. This formulation is ideally suited for developing an atomistic-to-continuum correspondence for solid mechanics problems. We derive expressions for the first Piola-Kirchhoff (P-K) stress tensor and the material frame heat flux vector directly from the momentum and energy balances using localization functions in a reference configuration. The resulting P-K stress tensor, unlike the Cauchy expression, has no explicit kinetic contribution. The referential heat flux vector likewise lacks the kinetic contribution appearing in its spatial frame counterpart. Using a proof for a special case and molecular dynamics simulations, we show that our P-K stress expression nonetheless represents a full measure of stress that is consistent with both the system virial and the Cauchy stress expression developed by Hardy. We also present an expanded formulation to define continuum variables from micromorphic continuum theory, which is suitable for the analysis of materials represented by directional bonding at the atomic scale.

In this paper, a deformation measure is introduced which leads to a class of strain measures in the Lagrangian and Eulerian descriptions. In order to develop a constitutive equation, a second-order constitutive relation based on these strain measures is considered for modeling the mechanical behavior of solids at finite deformation. For this purpose and performance evaluation of the proposed strains, a Hookean-type constitutive equation is considered and the uniaxial loading as well as simple shear and pure shear tests are examined. It is shown that the constitutive modeling based on the proposed strains give results which are in good agreement with the experimental data.

Due to the significant thermal-mechanical effects during hot spot formation in PBX explosives, a thermodynamic constitutive model has been constructed for HMX anisotropic single crystal subjected to dynamic impact loading. The crystal plasticity model based on dislocation dynamics theory was employed to describe the anisotropic plastic behavior along the preferential slip systems. A modified equation of state (EOS) was introduced into the constitutive equations through the decomposing stress tensor and the nonlinear elasticity for materials was taken into account. The one-dimensional strain impact simulations for HMX single crystal and quasi-bicrystal were performed respectively, in which the cohesive elements were inserted over the interface areas for the latter. The predicted particle velocities for the single crystal sample agreed well with the experimental results in the literature. Furthermore, the effects of crystal orientations, interface, misorientations on localized strain, stress and temperature distributions were predicted and discussed.

In the theory of Cosserat continuum, the faulting-related deformation of rocks is described using translational and rotational degrees of freedom, producing definitions for a symmetric macrostrain tensor and a skew-symmetric relative microrotation tensor. The macrostrain tensor describes the large-scale deformation of the region, whilst the relative microrotation tensor describes the difference between the large-scale regional rotation and local systematic microrotations of blocks between faults. Faults are activated when the resolved shear stress in the direction of movement exceeds frictional resistance for sliding, according to Amontons's Law of Friction. The direction of slip along the faults depends on the Cosserat strain tensor, which is defined as the sum of the macrostrain tensor and the relative microrotation tensor. We develop a constitutive relation for the faulting-related strain of rocks (cataclastic flow) based on the J-2 plasticity model for the Cosserat continuum, from which we derive the generally asymmetric stress tensor. We also develop the Cosserat stress–strain inverse method for fault-slip data analysis. We show that the geometry of fault systems is controlled by both the Cosserat strain tensor and the stress tensor, and present a field example of a fault system that conforms to the predictions of the Cosserat theory.

A new continuum theory of the constitutive equation of co-rotational derivative type is developed for anisotropic viscoelastic fluid—liquid crystalline (LC) polymers. A new concept of simple anisotropic fluid is introduced. On the basis of principles of anisotropic simple fluid, stress behaviour is described by velocity gradient tensor and spin tensor instead of the velocity gradient tensor in the classic Leslie—Ericksen continuum theory. Analyzing rheological nature of the fluid and using tensor analysis a general form of the constitutive equ- ation of co-rotational type is established for the fluid. A special term of high order in the equation is introduced by author to describe the sp- ecial change of the normal stress differences which is considered as a result of director tumbling by Larson et al. Analyzing the experimental results by Larson et al., a principle of Non- oscillatory normal stress is introduced which leads to simplification of the problem with relaxation times. The special behaviour of non- symmetry of the shear stress is predicted by using the present model for LC polymer liquids. Two shear stresses in shear flow of LC polymer liquids may lead to vortex and rotation flow, i.e. director tumbling in the flow. The first and second normal stress differences are calculated by the model special behaviour of which is in agree- ment with experiments. In the research, the com- putational symbolic manipulation such as computer software Maple is used. For the anisotropic viscoelastic fluid the constitutive equation theory is of important fundamental significance.

A general method of estimating constitutive equation of material from experiment of free oscillation is presented. The information on attenuation and frequency in the oscillation changing the environment temperature is utilized to estimate the linear visco-elastic constitutive equation with deformation process. Linear visco-elastic models of 2-element, 3-element or 4-element are adapted with the activation process in the dash-pot elements. Method of examining the profile in the attenuation versus temperature curve is proposed to obtain the elements constants using the Arrhenius plots, as an additional method to the ordinary Arrhenius plot for the frequency versus peak attenuation temperature relation. Deformation characteristic of solid material is generally expressed as elastic, plastic, viscous or their combinational form of constitutive equation. Most of materials exhibit more or less strain rate depen- dency in the stress-strain relationship and the strain rate dependency is expressed as viscous terms in the constitutive equation. Especially as the environmental temperature increases the rate dependency becomes more remarkable so that viscous effect becomes crucial at elevated temperature. This temper- ature dependency is often explained by the activation theory and has also been confirmed for various materials by experiments. Constitutive equation that expresses the activation process includes activation energy terms in the equation and the values of activation energy obtained from experiments lead to know the deformation mechanism of the material. The activation process is generally given by the equation

The paper considers an approach to the development of a crystal plasticity model comprising constitutive equations, equations of evolution and closure equations. The approach is based on the hypothesis that there exists a finite set of internal tensor variables and physical mechanical parameters fully representative of the “here-and-now” state of material. This makes possible physical equations in the form of simple relations (tensor-algebraic or differential), while not discarding the loading history; its “carriers” are introduced internal variables. Consideration is given to a possible structure type of the constitutive model. The derivation of constitutive relations is exemplified with a 2D plastic strain problem for single crystals. Algorithms are presented for determination of active slip systems under force and kinematic loading. The behavioral peculiarities of crystals in lattice rotation are studied for different loading conditions. The evolution of the orientation distribution function for the crystallographic coordinate system of grains under kinematic loading is determined.

Observations are reported on three grades of isotactic polypropylene in uniaxial cyclic tensile tests with various maximum strains. Constitutive equations are derived for the viscoplastic behavior of a semicrystalline polymer. A characteristic feature of the model is that the rate-of-strain tensor for plastic flow is split into the sum of two components, one of which is proportional to the deviator of the stress tensor, while the other is proportional to the rate-of-strain tensor. The stress–strain relations involve four adjustable parameters that are found by fitting the experimental data. Numerical simulation demonstrates that the model correctly predicts the mechanical response of semicrystalline polymers in cyclic tests.

Authors have proposed constitutive equations for rock based on the theory of non-linear visco-elasticity. The constitutive equations are grouped into two types: compliance or irrecoverable-strain monotonically increases with stress. In earlier studies, it was reported that both types of constitutive equations are applicable at least to the post-failure region.In this study, applicability of the constitutive equations to the pre-failure region was examined. At first, parameters were estimated from the results of unloading-reloading and alternating loading-rate tests under uniaxial compression. The calculated results with the estimated parameters were compared with the experimental results under various loading conditions.If a constitutive equation was properly selected, the equation well simulated the various time-dependent behaviors in the pre-failure region: loading-rate dependency of Young's modulus, creep and generalized relaxation where a set of parameter values was used in not only pre-failure but also post-failure region.

We investigate regularity properties of the stress tensor near the boundary for models of elasto-plasticity in arbitrary dimension. Focusing on special geometries, namely on balls and infinite strips, we obtain L 2-estimates for the tangential derivatives of the stress tensor near the boundary. We indicate why these estimates may fail for more general domains. In addition, we establish L 2-estimates for (the trace of) the stress tensor on the boundary.

In granular mechanics, macroscopic approaches treat a granular material as an equivalent continuum at macro-scale, and study its constitutive relationship between macro-quantities, such as stresses and strains. On the other hand, microscopic approaches consider a granular material as an assembly of individual particles interacting with each other at micro-scale (i.e., particle-scale), and the physical quantities under study are forces and displacements. This paper aims at linking up the findings from these two scales and to establish the macro–micro relations in granular mechanics. Three aspects of the macro–micro relations are investigated. They are about the internal structure, the stress tensor and the strain tensor. The internal structure is described with geometrical systems at the particle scale. Micro-structural definitions of the stress and strain tensors are derived, which link the macro-stress tensor with the contact forces, and the macro-strain tensor with the relative displacements at contact. In addition to a brief review of the past research work on these topics, further generalizations are made in this paper. In particular, the two cell systems proposed by Li and Li (2009), namely the solid cell system and the void cell system, are introduced and used for the derivation of the macro-structural expressions. The stress expression is derived based on Newton’s second law of motion. The result is valid for both static and dynamic cases. The strain expression is derived based on the compatibility requirement. And the expression is valid for any tessellation subdividing the granular assembly into polyhedral elements. The homogenization for deriving a macroscopic constitutive relationship from microscopic behaviour is discussed. Attention is placed on the macroscopic quantification of the internal structure in terms of a second rank tensor, known as the fabric tensor. Existing definitions of the fabric tensors have been reviewed. The correlations among different fabric tensors and their relations with the stress–strain behaviours have been investigated.

The constitutive equations describing the elastoplastic deformation of an isotropic material and taking into account the stress mode are validated against available experimental data. We propose a method for the approximate determination of the base functions appearing in the constitutive equations and relating the first and second invariants of the stress tensors and the linear components of finite strains. The strain components obtained by this method are compared to experimental data

An Eulerian rate-independent constitutive model for isotropic materials undergoing finite elastoplastic deformation is formulated. Entirely fulfilling the multiplicative decomposition of the deformation gradient, a constitutive equation and the coupled elastoplastic spin of the objective corotational rate therein are explicitly derived. For the purely elastic deformation, the model degenerates into a hypoelastic-type equation with the Green–Naghdi rate. For the small elastic- and rigid-plastic deformations, the model converges to the widely-used additive model where the Jaumann rate is used. Finally, as an illustration, using a combined exponential isotropic-nonlinear kinematic hardening pattern, the finite simple shear deformation is analyzed and a comparison is made with the experimental findings in the literature.

A selection of models for the variation in porosity in dry granular flows is investigated and compared on the basis of thermodynamic consistency to illustrate their performance and limitations in equilibrium situations. To this end, the thermodynamic analysis, based on the Muller–Liu entropy principle, is employed to deduce the ultimate constitutive equations at equilibrium. Results show that while all the models deliver appropriate equilibrium expressions of the Cauchy stress tensor for compressible grains, the model in which the variation in porosity is treated kinematically yields a spherical stress tensor for incompressible grains. Only the model in which the variation in porosity is modeled by a dynamic equation can give rise to a non-spherical stress tensor at equilibrium. The present study illuminates the validity and thermodynamic justification of the two modeling approaches for the porosity variation in dry granular matter.

To make constitutive modeling of materials more approaching reality, a new theory is proposed, in which a corresponding constitutive model can be constructed and characterized experimentally via two steps, one relates to the characterization of yielding behavior of material, and the second relates to the characterization of plastic flow of material deformation. The constitutive model involves two functions, yield function and plastic potential. A relationship between two functions is suggested, therefore, a corresponding plastic potential can be easily created after we have an appropriate yield function. To consider the non-isotropic hardening feature of strength differential in the constitutive model, the concept of equivalent hardening state is introduced, and then, multi-experimental flow stresses can be addressed in the model. When pressure sensitive materials are taken as an example in discussions, the Drucker–Prager yield function is employed to express the yielding behavior of material and a differently experimental characterization of the model is created as the corresponding plastic potential to describe the feature of plastic flow of material. This simple constitutive model can reproduce three sets of experimental results; including two flow-stresses and the volumetric plastic strain. The constitutive model can also well predict stress–strain relations with different pressures loaded on the material. Study shows that the feature of plastic flow is not that sensitive to the pressure loaded on the material when the yielding stress is.

A complete analysis was made for an isotropic hardening rate-type elastoplastic constitutive model with the logarithmic stress rate utilizing solid shafts torsion test in the large strain range. The deformation rate, the logarithmic spin, the Kirchhoff stress and the logarithmic stress rate of the Kirchhoff stress were obtained for the free-end solid shaft torsion test when considering Swift effect. Utilizing the results obtained from the solid shaft torsion test, the plastic rigidity function in the isotropic hardening elastoplastic constitutive model was determined at finite strain range. It was shown that the influence of Swift effect on finite strain constitutive model was related to the varying rate of axial deformation and the varying rate of radius deformation to shear strain, and the plastic rigidity function corresponding to the logarithmic stress rate was the same as that corresponding to the Jaumann stress rate when neglecting Swift effect. Solid shaft can achieve very large strain without buckling in torsion test. They can be used to accurately determine the large strain elastoplastic behavior [1-3]. Many researchers are interesting in the theoretical study and experimental study on large strain torsion deformation [1-7]. Jaumann objective stress rate is often adopted for study on constitutive models [1], but Nagtegaa J C [8] discovered that Jaumann stress rate may render oscillatory stress response for anisotropic kinematics hardening simple shear problem. After that, great interests focus on the issues of choosing an appropriate objective stress rate in rate type constitutive models. The expression of logarithmic stress rate with logarithmic spin can be found in Xiao H [9]. It had been proved that among all rate type elastoplastic constitutive models, only those with the newly discovered logarithmic stress rate fulfill the self-consistency criterion. In this paper, the parameter in large strain isotropic hardening elastoplastic constitutive model is determined with logarithmic rate of Kirchhoff stress utilizing solid shaft torsion test.

Using mixture theory we formulate the balance laws for unsaturated porous media composed of a double-porosity solid matrix infiltrated by liquid and gas. In this context, the term ‘double porosity’ pertains to the microstructural characteristic that allows the pore spaces in a continuum to be classified into two pore subspaces. We use the first law of thermodynamics to identify energy-conjugate variables and derive an expression for the ‘effective’, or constitutive, stress that is energy-conjugate to the rate of deformation of the solid matrix. The effective stress has the form σ ¯ = σ + B p ¯ 1 , where σ is the total Cauchy stress tensor, B is the Biot coefficient, and p ¯ is the mean fluid pressure weighted according to the local degrees of saturation and pore fractions. We identify other emerging energy-conjugate pairs relevant for constitutive modeling of double-porosity unsaturated continua, including the local suction versus degree of saturation pair and the pore volume fraction versus weighted pore pressure difference pair. Finally, we use the second law of thermodynamics to determine conditions for maximum plastic dissipation in the regime of inelastic deformation for the unsaturated two-porosity mixture.