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An exploratory discussion is presented on the application of egg-shaped function in elasto-plastic constitutive analysis for soft clay. Two main tasks of the paper are: 1) to propose a complete yield criterion based on egg-shaped function and supplement its definition in the deviatoric section, and then a yield criterion suitable for 3D stress conditions is obtained; 2) to elaborate its numerical implementation based on the drained triaxial tests. During the above discussion, a non-associated flow rule is proposed, in which the stress-dilatancy relationship in most classical theory is replaced by a linear dependence between the stress state parameter η and the rotation angle η of the plastic potential surface. Thereafter, isotropic and kinematic hardening behavior is considered by employing the hardening parameter H, which can be expressed as the function of plastic work Wp. Finally, comparisons between numerical results and test data on Taizhou soft clay are made to verify the effectiveness of the proposed model.

This paper considers conservation and balance laws and the constitutive theories for non-classical viscous fluent continua without memory, in which internal rotation rates due to the velocity gradient tensor are incorporated in the thermodynamic framework. The constitutive theories for the deviatoric part of the symmetric Cauchy stress tensor and the Cauchy moment tensor are derived based on integrity. The constitutive theories for the Cauchy moment tensor are considered when the balance of moments of moments 1) is not a balance law and 2) is a balance law. The constitutive theory for heat vector based on integrity is also considered. Restrictions on the material coefficients in the constitutive theories for the stress tensor, moment tensor, and heat vector are established using the conditions resulting from the entropy inequality, keeping in mind that the constitutive theories derived here based on integrity are in fact nonlinear constitutive theories. It is shown that in the case of the simplest linear constitutive theory for stress tensor used predominantly for compressible viscous fluids, Stokes' hypothesis or Stokes' assumption has no thermodynamic basis, hence may be viewed incorrect. Thermodynamically consistent derivations of the restrictions on various material coefficients are presented for non-classical as well as classical theories that are applicable to nonlinear constitutive theories, which are inevitable if the constitutive theories are derived based on integrity.

Nonlocal elastic models have attracted an increasing amount of attention in the past years, due to the promising feature of providing a viable simulation for scale effects in nano-structures and especially in nano-beams designed for use as actuators or sensors. In adapting Eringen ’s nonlocal model of elasticity to flexure of nano-beams, the bending field is expressed as convolution between the elastic curvature field and an averaging kernel. Basic difficulties are involved in this approach due to conflicting requirements imposed on the bending field by equilibrium on one hand and by constitutive conditions on the other one. In the newly proposed constitutive theory the bending field is placed in the proper position of input variable, giving to the elastic curvature field the role of output of the constitutive law, evaluated by convolution between the bending field and an averaging kernel. Conflicting restrictions on the bending field are thus eliminated and existence and uniqueness of the solution are assured under any data. Equivalence between integral and differential constitutive relations is proven to hold for nonlocal laws with a special kernel, under constitutive boundary conditions stemming naturally from the integral relation. When compared with the local limit, a stiffer elastic response is evaluated by the stress-driven nonlocal model, due to normalisation of the kernel. The theory provides an effective methodology for investigating small scale effects in nanobeams, by well-posed problems.

Material anisotropy is widely recognized as the fundamental reason that causes non-coaxiality between principal directions of stress and plastic strain increment. In order to model the non-coaxiality of soils from the view of cross-anisotropy, the anisotropic transformed stress method is introduced. By replacing the ordinary stress with the anisotropic transformed stress, and adopting a normal flow rule in the transformed stress space, non-coaxiality can be reflected simply within the framework of conventional elastoplastic constitutive theory. As an example, the unified hardening model is extended to be cross-anisotropic by this method, and then used to predict the non-coaxiality.

A soft solid is said to be initially stressed if it is subjected to a state of internal stress in its unloaded reference configuration. In physical terms, its stored elastic energy may not vanish in the absence of an elastic deformation, being also dependent on the spatial distribution of the underlying material inhomogeneities. Developing a sound mathematical framework to model initially stressed solids in nonlinear elasticity is key for many applications in engineering and biology. This work investigates the links between the existence of elastic minimizers and the constitutive restrictions for initially stressed materials subjected to finite deformations. In particular, we consider a subclass of constitutive responses in which the strain energy density is taken as a scalar-valued function of both the deformation gradient and the initial stress tensor. The main advantage of this approach is that the initial stress tensor belongs to the group of divergence-free symmetric tensors satisfying the boundary conditions in any given reference configuration. However, it is still unclear which physical restrictions must be imposed for the well-posedness of this elastic problem. Assuming that the constitutive response depends on the choice of the reference configuration only through the initial stress tensor, under given conditions we prove the local existence of a relaxed state given by an implicit tensor function of the initial stress distribution. This tensor function is generally not unique, and can be transformed according to the symmetry group of the material at fixed initial stresses. These results allow one to extend Ball's existence theorem of elastic minimizers for the proposed constitutive choice of initially stressed materials. This article is part of the theme issue ‘Rivlin's legacy in continuum mechanics and applied mathematics’.

Stress power-law fluids are a special sub-class of fluids defined through implicit constitutive relations, wherein the symmetric part of the velocity gradient depends on a power-law of the stress (see Eq. (2.2)), and were introduced recently to describe the non-Newtonian response of fluid bodies. Such fluids are counterparts to the classical power-law fluids wherein the stress is given in terms of a power-law for the symmetric part of the velocity gradient. Stress power-law fluids can describe phenomena that cannot be described by classical power-law fluids (see [1]). In this paper, first a new exact solution for a variant of Stokes' first problem for stress power-law fluids, when the exponent n=0 (Navier-Stokes fluid), is obtained. Such an exact solution for the stress is in terms of a convolution integral, for which we establish bounds. We then compute the convolution integral using Gauss-Kronrod quadrature by ensuring that its value always lies within the bounds. Using the validated quadrature, we can accurately evaluate the exact solution and we the exact solution it to validate the numerical scheme employed in solving the governing equations for stress-power law fluids with arbitrary exponent n. Finally, for stress power-law fluids wherein the exponent n

In Bertram (Continuum Mech Thermodyn. doi: 10.1007/s00161-014-0387-0 , 2015), a mechanical framework for finite gradient elasticity and plasticity has been given. In the present paper, this is extended to thermodynamics. The mechanical theory is only briefly repeated here. A format for a rather general constitutive theory including all thermodynamic fields is given in a Euclidian invariant setting. The plasticity theory is rate-independent and unconstrained. The Clausius–Duhem inequality is exploited to find necessary and sufficient conditions for thermodynamic consistency. The residual dissipation inequality restricts the flow and hardening rules in combination with the yield criterion.

Continuum dislocation dynamics (CDD) is a single crystal strain gradient plasticity theory based exclusively on the evolution of the dislocation state. Recently, we derived a constitutive theory for the average dislocation velocity in CDD in a phase field-type description for an infinite domain. In the current work, so-called rational thermodynamics is employed to obtain thermodynamically consistent boundary conditions for the dislocation density variables of CDD. We find that rational thermodynamics reproduces the bulk constitutive equations as obtained from irreversible thermodynamics. The boundary conditions we find display strong parallels to the microscopic traction conditions derived by Gurtin and Needleman (M.E. Gurtin and A. Needleman, J. Mech. Phys. Solids 53 (2005) 1–31) for strain gradient theories based on the Kroner–Nye tensor.

The paper presents rate constitutive theories for finite deformation of homogeneous, isotropic, compressible, and incompressible thermoviscoelastic solids without memory in Lagrangian description derived using the second law of thermodynamics expressed in terms of Gibbs potential Ψ. To ensure thermodynamic equilibrium during evolution, the rate constitutive theories must be derived using entropy inequality [as other three conservation and balance laws are do not provide a mechanism for deriving constitutive theories for the deforming matter (Surana in Advanced mechanics of continuua. in preparation, 2014)]. The two forms of the entropy inequality in Ψ derived using conjugate pairs \({\mathbf{\sigma}^*}\), \({[\dot{J}]}\) : first Piola–Kirchhoff stress tensor and material derivative of the Jacobian of deformation and \({\mathbf{\sigma}^{[0]}}\), \({\dot{\mathbf{\varepsilon}}_{[0]}}\) ; second Piola–Kirchhoff stress tensor and material derivative of Green’s strain tensor are precisely equivalent as the conjugate pairs \({\mathbf{\sigma}^*}\), \({[\dot{J}]}\) and \({\mathbf{\sigma}^{[0]}}\), \({\dot{\mathbf{\varepsilon}}_{[0]}}\) are transformable from each other. In the present work, we use \({\mathbf{\sigma}^{[0]}}\), \({\dot{\mathbf{\varepsilon}}_{[0]}}\) as conjugate pair. Two possible choices of dependent variables in the constitutive theories: Ψ, η, \({\mathbf{\sigma}^{[0]}}\), \({\mathbf{q}}\) and Ψ, η, \({\mathbf{\varepsilon}_{[0]}}\), \({\mathbf{q}}\) (in which η is entropy density and \({\mathbf{q}}\) is heat vector) are explored based on conservation and balance laws. It is shown that the choice of Ψ, η, \({\mathbf{\varepsilon}_{[0]}}\), \({\mathbf{q}}\) is essential when the entropy inequality is expressed in terms of Ψ. The arguments of these dependent variables are decided based on desired physics. Viscoelastic behavior requires considerations of at least \({\mathbf{\varepsilon}_{[0]}}\) and \({\dot{\mathbf{\varepsilon}}_{[0]}}\) (or \({\mathbf{\varepsilon}_{[1]}}\)) in the constitutive theories. We generalize and consider strain rates \({\mathbf{\varepsilon}_{[i]}}\); i = 0, 1, …, n−1 as arguments of the dependent variables in the derivations of the ordered rate theories of up to orders n. At the onset, \({\mathbf{\sigma}^{[0]}}\), \({\mathbf{\varepsilon}_{[i]}}\) ; i = 0, 1, …, n−1, θ and \({\mathbf{g}}\) are considered as arguments of Ψ, η, \({\mathbf{\varepsilon}_{[n]}}\) and \({\mathbf{q}}\). When \({\dot{\Psi}}\) is substituted in the entropy inequality, the resulting conditions eliminate η as a dependent variable, reduce arguments of some of the dependent variables in the constitutive theory etc. but do not provide a mechanism to derive constitutive theories for \({\mathbf{\varepsilon}_{[i]}}\) and \({\mathbf{q}}\). The stress tensor \({\mathbf{\sigma}^{[0]}}\) is decomposed into equilibrium stress \({{}_e \mathbf{\sigma}^{[0]}}\) and deviatoric stress \({{}_d \mathbf{\sigma}^{[0]}}\). Upon substituting this in the entropy inequality, we finally arrive at the inequality that must be satisfied by \({{}_e \mathbf{\sigma}^{[0]}}\), \({{}_d \mathbf{\sigma}^{[0]}}\) and \({\mathbf{q}}\). Derivations of the constitutive theory for \({{}_e \mathbf{\sigma}^{[0]}}\) follow directly from \({{}_e \mathbf{\sigma}^{(0)}}\), equilibrium Cauchy stress tensor, and the constitutive theory for \({\mathbf{\varepsilon}_{[n]}}\) is derived using the theory of generators and invariants. Constitutive theories for the heat vector \({\mathbf{q}}\) of up to orders n that are consistent (in terms of the argument tensors) with the constitutive theories for \({\mathbf{\varepsilon}_{[n]}}\) are also derived. Many simplified forms of the rate theories of orders n are presented. Material coefficients are derived by considering Taylor series expansions of the coefficients in the linear combinations representing \({\mathbf{\varepsilon}_{[n]}}\) and \({\mathbf{q}}\) using the combined generators of the argument tensors about a known configuration \({\underline{\Omega}}\) in the combined invariants of the argument tensors and temperature. It is shown that the rate constitutive theories of order one (n = 1) when further simplified results in constitutive theories that resemble currently used theories but are in fact different. The solid materials characterized by these theories have mechanisms of elasticity and dissipation but have no memory, i.e., no relaxation behavior or rheology. Fourier heat conduction law is shown to be an over-simplified case of the rate theory of order one for \({\mathbf{q}}\). The paper establishes when there is equivalence between the constitutive theories derived here using Ψ and those presented in Surana et al. (Acta Mech 224(11):2785—2816, 2013), that are derived using Helmholtz free energy density Φ.

The subject of objective rates in continuum damage mechanics is under consideration. It is shown that by changing the definition of the objective rate in constitutive theory, one obtains different material behavior and different final fracture modes as causes of intrinsic microdamage evolution. Theoretical considerations and a numerical example include the comparison of two special cases of objective rates for which the symmetric part of the spatial velocity gradient defines a true rate of the assumed strain rates, namely Lie and Logarithmic. The problem is described in the framework of the recently proposed Perzyna-type viscoplasticity, including an anisotropic damage description; nevertheless, final conclusions are general.

The need for a proper geometric approach to constitutive theory in non-linear continuum mechanics (NLCM) is witnessed by lasting debates about basic questions concerning time-invariance, integrability, conservativeness and frame invariance. Our aim is to bring geometry to play a central role in theoretical and computational issues of NLCM. This demand is imposed by the present state of art, dominated by a mainly algebraic approach which, being a modified heritage of the linearized theory, is inadequate to manage concepts and methods in a non-linear framework. A proper definition of spatial and material fields and the statement of the ensuing covariance paradigm, provide a firm foundation to the theory of constitutive behavior in NLCM. The notion of constitutive frame invariance (CFI) is introduced as geometric correction to the formulation of material frame indifference (MFI). Standard models of constitutive behavior are critically discussed and compared with the ones consistent with the new approach. The outcome is a physically testable theory which eventually results in new effective computation tools for structural engineers.

The mathematical foundation of the traditional elastoplastic constitutive theory for geomaterials is presented from the mathematical point of view, that is, the expression of stress-strain relationship in principal stress/strain space being transformed to the expression in six-dimensional space. A new framework is then established according to the mathematical theory of vectors and tensors, which is applicable to establishing elastoplastic models both in strain space and in stress space. Traditional constitutive theories can be considered as its special cases. The framework also enables modification of traditional constitutive models.

A new type constitutive equation of the rate type T = h(T,D,e) for granular materials is introduced in the frame of Rational Mechanics. This new approach to constitutive modelling, called “barodesy”, is based on the asymptotic behaviour of sand. Barodesy can be seen as a further development of hypoplasticity. Despite its striking mathematical simplicity, it captures many aspects of sand behaviour. Barodesie als eine neue hypoplastische Stofftheorie auf der Grundlage des asymptotischen Verhaltens von Sand. Es wird eine konstitutive Gleichung vom Ratentyp T = h(T,D,e) fur granulare Stoffe im Rahmen der sog. Rational Mechanics eingefuhrt. Diese neue Betrachtung von Stoffmodellen, genannt “Barodesie”, beruht auf das asymptotische Verhalten von Sand und kann als eine Weiterentwicklung der Hypoplastizitat betrachtet werden. Trotz der auffalligen mathematischen Einfachheit, kann sie viele Aspekte des Verhaltens von Sand beschreiben.

Interconversion of viscoelastic material functions is a longstanding problem that has received attention since the 1950s. There is currently no accepted methodology for interconverting viscoelastic material functions due to the lack of stability and accuracy of the existing methods. This paper presents a new exact, analytical interconversion method for linearly viscoelastic material functions expressed as Prony series. The new algorithm relies on the equations of the thermodynamics of irreversible processes used for defining linearly viscoelastic constitutive theories. As a result, interconversion is made possible for unidimensional and tridimensional materials for arbitrary material symmetry. The algorithm has been tested over a broad range of cases and was found to deliver accurate interconversion in all cases. Based on its accuracy and stability, the authors believe that their algorithm provides a closure to the interconversion of linearly viscoelastic constitutive theories expressed with Prony series. © 2012 Springer Science+Business Media, B. V.

Constitutive laws for multi-component fluids (MCF) is one of the thorniest problems in science. Two questions explored here are: how to ensure that these relations reduce to accepted forms when all but one of the constituents vanishes, and what constraints does the Second Law impose on the dynamics of viscous fluids at different temperatures? The analysis suggests an alternative to the metaphysical principles for MCF proposed by Truesdell [1].

A formulation of elasticity theory is presented, for which a response function is taken for the deformation gradient in terms of the stress. The issue of the response function being multi-valued is discussed. The objectivity, the theory of hyperelasticity, internally constrained materials, and material symmetry are studied.

A theoretical framework for the hierarchical multiscale modeling of inelastic response of heterogeneous materials is presented. Within this multiscale framework, the second gradient is used as a nonlocal kinematic link between the response of a material point at the coarse scale and the response of a neighborhood of material points at the fine scale. Kinematic consistency between these scales results in specific requirements for constraints on the fluctuation field. The wryness tensor serves as a second-order measure of strain. The nature of the second-order strain induces anti-symmetry in the first-order stress at the coarse scale. The multiscale internal state variable (ISV) constitutive theory is couched in the coarse scale intermediate configuration, from which an important new concept in scale transitions emerges, namely scale invariance of dissipation. Finally, a strategy for developing meaningful kinematic ISVs and the proper free energy functions and evolution kinetics is presented.

In the present study a variational principle is proposed for the revised Goodman–Cowin theory with internal length for cohesionless granular materials (Fang et al. in Continuum Mech Thermodyn in press). The balance equations of the internal variables employed in the theory in equilibrium states, the equilibrium expressions of the constitutive variables and the corresponding natural boundary conditions are derived by use of the proposed variational principle for both cases of compressible and incompressible grains. It is demonstrated that the derived results coincide with those obtained by use of the thermodynamic analysis. The current work serves as a supplementary variational verification of the constitutive theory proposed in Fang et al. (in Continuum Mech Thermodyn in press).

Based on the general 5D plastic strain vector space, we previously developed an integral elasto-plastic constitutive theory in which the stress is expressed as a vectorial functional of intrinsic geometry of plastic strain trajectory in a non-Euclidean space of plastic strains. This paper presents a further discussion on the geometric descriptions in this theory and gives a practical model with loading path dependence for non-proportional cyclic plasticity. In the model, the hardening variables and the constitutive functional are assumed to be simply related to the accumulation of plastic strain component along the normal of stress trajectory and the geometric parameters of loading path. The simulations by the model are compared with the experiments in the references and a good agreement is obtained.

The constitutive theory of micromorphic thermoplasticity has been formulated in Lagrangian form. The generalized Lagrangian strains, strain rates, temperature, temperature rate and temperature gradients are considered as the independent constitutive variables. Internal variables, including generalized Lagrangian plastic strains, are incorporated. The axioms of objectivity and equipresence are strictly followed. The entropy inequality has been fully utilized to guide the derivations of the constitutive relations. No restrictive assumption has been made to the magnitude of any independent constitutive variables. Finite element formulations for micromorphic thermoplasticity are developed based on the balance laws and the constitutive theory.

Recently some important new findings in the porous media theory have been worked out and partly published, which has led to a widely consistent theory. Thus, the time seems to be ripe to give a survey of the current state of the porous media theory.

A constitutive theory for large elastic‐plastic deformations is presented by employing F=F p F e decomposition of the total deformation gradient. A duality in constitutive formulation based on this and the well-known Lee’s decomposition F=FeFp is established for isotropic polycrystalline and single crystal plasticity. # 1999 Published by Elsevier Science Ltd. All rights reserved.

Based on the crystallography theory of martensitic transformation and Hill-Rice’s internal variable constitutive theory, a generalized micromechanics constitutive model is established to describe the thermoelastic martensitic transformation and reorientation of single crystal. This model can describe the macroscopic constitutive behavior due to the microstructure changes of forward transformation, reverse transformation and reorientation in single crystal under complex thermodynamic loading condition. The theoretical predictions agree well with the available experiment.

When materials such as Armco iron, titanium etc., are subject to impact it can be observed that two basic inelastic processes take place – slip and deformation twinning. Of these processes, inelasticity associated with the slip mechanism has received considerable attention. For example, Zerilli and Armstrong (1988) modeled the Taylor impact test for a variety of materials using traditional plasticity theories. They found that there was a significant discrepancy between the theoretical and experimental results for some materials. They attributed this to the fact that they had neglected deformation twinning in their models. Subsequent metallurgical studies have indicated that twinning had indeed taken place in these materials. In this study, we focus on the inelastic processes solely due to deformation twinning (i.e., neglecting slip). We model these processes using the approach of Rajagopal and Srinivasa (1995, 1997) and Srinivasa et al., (1997), the results of which are briefly summarized in section 2.1. In order to better understand the twinning process, we study the Taylor impact test for a 2-D slab under the assumption that only deformation twinning takes place and solve the governing dynamical equations by using the finite element method. The results show that the twinned zone is concentrated near the point of impact and indeed it contributes significantly to the overall permanent shape change due to the impact.

In this second part of a three-part paper we derive constitutive theory for a multiphase, multicomponent, three-scale, swelling system which includes interfaces. In Part I, the governing field equations and the definitions of all mesoscopic and macroscopic variables therein were defined in terms of microscopic variables. In this paper, we choose the independent variables and derive constitutive restrictions for two cases of a dual-porosity multiple-component swelling media: one which assumes no interfacial effects, and one which includes interfacial effects. For each case, the entropy inequality is fully derived using a Lagrange multiplier technique. Novel definitions for macroscopic pressures and chemical potentials are given, and generalized Darcy's and Fick's laws are presented. Although these models are developed with a clay soil in mind, the results hold for any medium which assumes the same set (or subset of) independent variables as constitutive unknowns, e.g. swelling biopolymers.

In part I of this two-part paper, a new theoretical framework was presented to describe the response of electro-elastic bodies. The constitutive theory that was developed consists of two implicit constitutive relations: one that relates the stress, stretch and the electric field, and the other that relates the stress, the electric field and the electric displacement field. In part II, several boundary value problems are studied within the context of such a construct. The governing equations allow for nonlinear coupling between the electric and stress fields. We consider boundary value problems wherein both homogeneous and inhomogeneous deformations are considered, with the body subject to an electric field. First, the extension and the shear of an electro-elastic slab subject to an electric field are studied. This is followed by a study of the problem of a thin circular plate and a long cylindrical tube, both subject to an inhomogeneous deformation and an electric field. In all the boundary value problems considered, the relationships between the stress and the linearized strain are nonlinear, in addition to the nonlinear relation to the electric field. It is emphasized that the theories that are currently available are incapable of modelling such nonlinear relations.

As illustrated variously by wetting and drying scanning curves, flow in unsaturated porous media is inherently nonlocal. This nonlocality is also manifest in hysteresis in the classical Darcy conductivity. It is the authors' belief that most current theories of unsaturated/saturated flow are often inadequate, as they do not account for spatial nonlocality and memory. Here we provide a fundamental theory in which nonlocality of the flow constitutive theory is a natural consequence of force balances. The results are derived from general principles in statistical physics and under appropriate limiting conditions, the classical Darcy's Law is recovered for saturated flow. A notable departure in this theory from other nonlocal flow theories is that a classical Darcy type equation on a small scale need not exist.

In the preceding chapter, we discussed the mechanical modeling of the damaged state of materials by means of internal variables, and called them damage variables. The present chapter is concerned with the thermodynamic constitutive theory with internal variables, which furnishes a consistent basis to formulate the mechanical behavior of damaged materials. In Section 3.1, the fundamental principles and the basic laws of the non-equilibrium continuum thermodynamics are presented as the foundations for the succeeding discussions. In Section 3.2, the notion and the procedure of the thermodynamic constitutive theory with internal variables will be described in detail. It is shown that the inelastic constitutive equations and the evolution equations for internal variables are formulated as a set of generalized normality rule defined by a dissipation potential function and a common multiplier.

A continuum form of design sensitivity analysis is developed for the history dependent non linear structural problems. In particular, a rearranged form of the non-yield-surface endochronic constitutive theory is used which has a unifying feature of describing the plastic, viscoplastic and viscoelatic behavior in one simple form. Both the total Lagrangian and the updated Lagrangian formulations are developed.

Much experimental research has proved that material constitutive theory built on the data of material uniaxial experiment can not accurately describe constitutive behavior of material under nonproportional cyclic loadings. This is attributable to the fact that material uniaxial experiments are by no means able to reveal some basic deformation behavioral characteristics of materials when subjected to nonproportional cyclic loading environments. This paper, by a series of experiments, finds that the curvature of deformation path is an appropriate phenomenological parameter to reflect influence of shape of deformation path on the response of materials. Based upon these experimental results, this paper develops a modified endochronic constitutive theory which can better describe the nonproportional cyclic constitutive behavior of materials.

In order to obtain a more fundamental basis for constitutive modeling, it seems intriguing to investigate whether thermodynamics may provide this basis. Thermodynamics deals with relations between heat and mechanical work, and it is based on two fundamental laws of nature, the first and second law of thermodynamics; these laws are accepted as axioms. It turns out to be possible to derive constitutive theories from thermodynamics and, therefore, this chapter discusses some fundamental results of thermodynamics. Moreover, it turns out that, with this thermodynamic basis, it is possible to combine various material behaviors in a straightforward manner. As an example, a constitutive theory that combines plasticity and damage might be established. Without the thermodynamic basis such a combined model is rather difficult to establish, but with the thermodynamic formulation the establishment turns out to be pretty straightforward. This chapter introduces various concepts, such as temperature, heat, entropy that enter thermodynamics.

In a thermodynamic framework which exploits the entropy inequality to obtain constitutive equations, it is common practice to assume charge neutrality and enforce this restriction using Lagrange multipliers. In this paper we show that the Lagrange multiplier used to enforce charge neutrality does not correspond to any known physical parameter, raising the question of whether charge neutrality can really be enforced.

The classical micropolar constitutive theory is reformulated within the framework of the microplane concept in order to obtain constitutive equations and models including available and, hopefully, more precise information of the complex microstructure of engineering materials like concrete and other composites. The main objective is the macroscopic modeling and description of anisotropic material response behavior by means of the well-developed microplane concept applied within a micropolar continuum setting. A thermodynamically consistent approach is considered to derive the so called micropolar microplane theory. The main assumption of the present proposal is the integral relation between the macroscopic and the microscopic free energy as advocated by Carol et al. [Int.J.Solids Struct. 38 (2001) 2921] and Kuhl et al. [Int.J.Solids Struct. 38 (2001) 2933] whereby the microplane laws are chosen such that the macroscopic Clausius–Duhem inequality is fully satisfied. This theoretical framework is considered to derive both elastic and elastoplastic micropolar microplane models. Fil: Etse, Jose Guillermo. Universidad Nacional de Tucumán. Facultad de Ciencias Exactas y Tecnología; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Tucumán; Argentina Fil: Nieto, Maria Marcela. Universidad Nacional de Santiago del Estero. Facultad de Ciencias Exactas y Tecnologías; Argentina Fil: Steinmann, Paul. University of Kaiserslautern; Alemania

These lecture notes deal with the approximation of conservation laws via viscosity or relaxation. The following topics are covered

In this chapter we give an exposition of three physically interesting recent pieces of work which deal with very rapid instabilities due to the nature of the partial differential equations changing from being hyperbolic to being elliptic.

The principles of invariance and symmetry have played an important role ever since the early days of science. This chapter presents what many believe is an appropriate spacetime invariance principle for the continuum thermodynamics: the principle of material frame indifference, also called the principle of objectivity. This principle establishes a link between the general thermodynamic assertions such as the equation of balance of energy and the boundedness of energy from below on the one hand and the specific continuum thermodynamic concepts on the other. Among the latter concepts will be the mass, its positiveness, the equations of balance of linear and angular momenta, and the concepts of internal and kinetic energies. Apart from this theoretical role, the principle of material frame indifference is also important in the constitutive theory where it restricts the form of the constitutive functions. This feature of the principle of material frame indifference will be explained in Part III; here we shall concentrate on the fundamental aspects.

A relatively simple set of unified elastic-viscoplastic constitutive equations was formulated by Bodner and Partom (B-P) [1–3] which has been shown to provide good representational capability for metals over a wide range of strain rates and temperatures. A material effect not included in the basic equations is the strain rate dependence of hardening. Also, the reference flow equation is taken to be that of Prandtl-Reuss which is not fully adequate under conditions of strong non-proportional loading. The present paper is concerned with describing and discussing modifications to the constitutive equations to account for those effects.

In the previous report the linearized stability results for the flow of granular materials down an inclined plane, modeled by a constitutive theory based on the kinetic theory approach were presented. In this report, the authors derive the governing equations for the flow of granular materials down an inclined plane, modeled by the constitutive theory proposed by Boyle and Massoudi (1990). The governing equations obtained will be solved numerically to obtain the basic solutions.

In the context of a strain-space formulation of finite plasticity, a constitutive theory is presented for rate-independent rigid-plastic materials. Both Eulerian and Lagrangian descriptions of the theory are discussed, and the nature of strain-hardening behavior is analyzed. Arbitrary objective rates are admitted. For special constitutive equations, a number of classical results, including the Hencky theorems, hold even at finite deformations.

The term intermediate configuration denotes a general concept, which can be applied to represent some types of inelastic material behavior. Basically, the 'idea of an intermediate configuration leads to a decomposition of the constitutive functional: It implies a rather simple constitutive equation of elastic or viscoelastic type, but this constitutive equation is related to a varying configuration, which depends on the history of the mechanical process in a more complicated way. The concept of an intermediate configuration is related to the question of a decomposition of the total strain into components corresponding to different physical qualities. These different components are called elastic and plastic parts or, more generally, recovered and unrecovered parts of the total strain. There are situations where the solution of engineering problems requires a consistent analysis of finite deformations. In these cases the question of decomposition together with the associated problem of a convenient formulation of constitutive assumptions has to be carefully considered. The history of the discussion about the decomposition problem and its main arguments within a rational foundation of the idea of an intermediate state will not be described here. The reader might be referred to papers of GREEN & NAGI-,IDI [1965, 1966, 1971], BESSELING [1966], LEE [1969, 1981], KLEIBER [1975], LEHMANN [1982] etc.; of course, this list should not be regarded as complete. In this paper some arguments are reviewed to understand the phenomenological description of finite plastic deformations within the constitutive theory of simple materials with memory in the sense of TRUESDELL ~ Note [1965]. Essentially, the intermediate state represents an accumulation of permanent memory effects, whereas the relation between the actual and the intermediate state is characterized by fading memory properties (see also HAUPT [1977]). In the following the intermediate configuration is understood as a noncompatible strain field, which arises as a result of a local unloading process. In other words, this intermediate state of strain is the asymptotic limit of the response of the material element to a given mechanical process, which is followed by a state of zero stress. Essentially, this is the idea of a natural state, which has been discussed by HOLSAPPLE [1973a, b, c] in the context o f the functional theory of simple materials with memory. In general, the local unloading process is an abstract idea, that means: it does not take place in reality. Therefore, as an alternative, it could be governed by a fictitious elasticity law: This version might be useful in view of a more general incorporation of

Equations which describe single phase fluid flow and transport through an elastic porous media are obtained by applying constitutive theory to a set of general multiphase mass, momentum, energy, and entropy equations. Linearization of these equations yields a set of equations solvable upon specification of the material coefficients which arise. Further restriction of the flow to small velocities proves that Darcy's law is a special case of the general momentum balance.

In the theory of constitutive relations in continuum mechanics, various axioms of locality are postulated. These axioms restrict the amount of information concerning the motion of the body that is needed in order to determine the stress at any material point.

The author has previously proposed an extension of classical plasticity theory which accounts for thermal effects and creep behavior and is applicable to transient temperature loading. This theory has been designed for use within large scale nonlinear finite element computer codes. Although the constitutive theory has been previously presented, the complete derivation was not reported due to its length. In this paper the author will present an abbreviated derivation of the uniaxial theory for those who have found the theory difficult to follow. In addition, some computational examples of the theory are presented in order to better clarify the capability of the model as well as details of its utilization.

Taylor stability in Couette flow of a thermoviscoelastic fluid between two rotating coaxial circular cylinders is investigated employing the constitutive theory of Eringen and Koh. The stability problem is reduced to a characteristic value problem which is solved to obtain stability criteria in terms of critical Values of Taylor numbers. The results are compared with those of the classical viscous fluids and viscoinelastic fluids and the stabilizing effects of the thermoviscoelastic nature of the fluid are discussed.