A.R. Srinivasa

Mechanics of the inelastic behavior of materials—part 1, theoretical underpinnings

Abstract This is the first of a two-part paper that is concerned with the modeling of the behavior of inelastic materials from a continuum viewpoint, taking into account changes in the elastic response and material symmetry that occur due to changes in the microstructure of the material. The first part discusses some of the fundamental issues that must be addressed when modeling the elastic response of these materials. In particular, we discuss in detail the far reaching effects of the notion of materials with families of elastic response functions with corresponding natural configurations that was introduced by Wineman and Rajagopal (1990, Archives of Mechanics, 42, 53–75) and Rajagopal and Wineman (1992, Int. J. Plasticity 8, 385–395) for the study of the inelastic behavior of polymeric materials and later generalized and extended to the study of deformation twinning of polycrystals by Rajagopal and Srinivasa (1995 Int. J. Plasticity 11(6), 653–678, 1997, 13(1/2) 1–35). For these materials, a definition of material symmetry is introduced, that makes it possible to discuss the concept of “evolving material symmetry”.

ON THE OBERBECK-BOUSSINESQ APPROXIMATION

This paper deals with a derivation (using a perturbation technique) of an approximation, due to Oberbeck8,9 and Boussinesq,1 to describe the thermal response of linearly viscous fluids that are mechanically incompressible but thermally compressible. The present approach uses a nondimensionalization suggested by Chandrasekhar2 and utilizing the ratio of two characteristic velocities as a measure of smallness, systematically derives the Oberbeck-Boussinesq approximation as a third-order perturbation. In the present approach, the material is subjected to the constraint that the volume change is determined solely by the temperature change in the body and uses a novel approach in deriving the thermodynamical restrictions. Consequently, it is free from the additional assumptions usually added on in earlier works in order to obtain the correct equations.

Mechanics of the inelastic behavior of materials. Part II: inelastic response

Abstract This is the second of a two part paper dealing with the inelastic response of materials. Part I (see Rajagopal and Srinivasa, 1998 International Journal of Plasticity 14 , 945–967) dealt with the structure of the constitutive equations for the elastic response of a material with multiple natural configurations. We now focus attention on the evolution of the natural configurations. We introduce two functions-the Helmholtz potential and the rate of dissipation function—representing the rate of conversion of mechanical work into heat. Motivated by, and generalizing the work of Ziegler (1963) in Progress in Solid Mechanics , Vol. 4, North-Holland, Amsterdam/New York; (1983) An Introduction to Thermodynamics, North-Holland, Amsterdam/New York) we then assume that the evolution of the natural configurations occurs in such a way that the rate of dissipation is maximized. This maximization is subject to the constraint that the rate of dissipation is equal to the difference between the rate of mechanical working and the rate of increase of the Helmholtz potential per unit volume. This then allows us to derive the constitutive equations for the stress response and the evolution of the natural configurations from these two scalar functions. Of course, the maximum rate of dissipation criterion that is stated here is only an assumption that holds for a certain class of materials under consideration. Our quest is to see whether such an assumption gives reasonable results. In the process, we hope to gain insight into the nature of such materials. We demonstrate that the resulting constitutive equations allow for response with and without yielding behavior and obtain a generalization of the normality and convexity conditions. We also show that, in the limit of quasistatic deformations, if one considers materials that possess yielding behavior, then the constitutive equations reduce to those corresponding to the strain space formulation of the rate independent theory of plasticity (see e.g. Naghdi (1990) Journal of Applied Mathematics and Physics A345 , 425–458.). Moreover, in this limit, the maximum rate of dissipation criterion, as stated here, is equivalent to the work inequality of Naghdi and Trapp (1975) Quartely Journal of Mechanics and Applied Mathematics 28 , 25–46). The main results together with an illustrative example are presented.

On thermomechanical restrictions of continua

The central idea proposed here is that, in entropy–producing processes, a specific choice from among a competing class of constitutive functions can be made so that the state variables evolve in a way that maximizes the rate of entropy production. When attention is restricted to quadratic forms for the rate of entropy production, the assumption leads to results that are fully in keeping with linear phenomenological relations that satisfy the Onsager relations. In other words, the usual linear evolution laws such as Fouriers law of heat conduction, Ficks law, Darcys law, Newtons law of viscosity, etc., all corroborate this assumption. We clarify the difference between the maximum rate of entropy production criterion that characterizes choices among constitutive relations and the minimum entropy production theorem due to Onsager (1931) that characterizes steady states for special choices of the rate of entropy production. We then show that for other forms of entropy production that are not quadratic for which the Onsager relations and related theorems cannot be applied, we can use the procedure described here to obtain nonlinear laws. We demonstrate by means of an example that even yield–type phenomena can be accommodated within this framework, while they cannot within the framework of Onsager. We also discuss issues concerning constraints, especially in thermoelasticity within the context of our ideas.

On the response of non-dissipative solids

In addition to being incapable of dissipation, in any process that it is subject to, there are other tacit requirements that a classical elastic body has to meet. The class of solids that are incapable of dissipation is far richer than the class of bodies that is usually understood as being elastic. We show that, unlike the case of a classical elastic body, the stress in non-dissipative bodies is not necessarily derivable from a stored energy that depends only on the deformation gradient.

On a class of non-dissipative materials that are not hyperelastic

The purpose of this brief note is to develop fully Eulerian, implicit constitutive equations for the mechanical response of a class of materials that do not dissipate mechanical work in any process. We show that such materials can be modelled by obtaining a form for the Helmholtz potential as a function of the current mass density, the Cauchy stress and certain other parameters that capture anisotropic response. The resulting constitutive equations are of the form , where and are functions of the state variables of the system. The class of materials that can be obtained from such a constitutive relation is considerably more general than conventional Green-elastic hyperelastic materials. Such response functions may be suitable for the modelling of biological tissue where, due to the constant remodelling that takes place, there may be no physical meaning to a ‘reference configuration’.

Diffusion of a fluid through an elastic solid undergoing large deformation

Abstract This paper is concerned with the modeling of slow diffusion of a fluid into a swelling solid undergoing large deformation. Both the stress in the solid as well as the diffusion rates are predicted. The approach presented here, based on the balance laws of a single continuum with mass diffusion, overcomes the difficulties inherent in the theory of mixtures in specifying boundary conditions. A “natural” boundary condition based upon the continuity of the chemical potential is derived by the use of a variational approach, based on maximizing the rate of dissipation. It is shown that, in the absence of inertial effects, the differential equations resulting from the use of mixture theory can be recast into a form that is identical to the equations obtained in our approach. The boundary value problem of the steady flow of a solvent through a gum rubber membrane is solved and the results show excellent agreement with the experimental data of Paul and Ebra-Lima (J. Appl. Polym. Sci. 14 (1970) 2201) for a variety of solvents.

Modeling anisotropic fluids within the framework of bodies with multiple natural configurations

Abstract This is a follow up of a paper on a thermodynamic framework for rate type models [J. Non-Newtonian Fluid Mech. 88 (2000) 207] published in this journal. The previous paper used the notion that certain materials have multiple natural configurations and that their response can be characterized as a class of elastic responses from an evolving set of natural configurations, and used this framework to model the behavior of a class of visoelastic fluids that are isotropic with regard to their viscous as well as their elastic response. Here, we extend the framework to the modeling of anisotropic fluids . Anisotropic fluids are invariably modeled within the framework of director theories, and such theories require boundary conditions for the directors for the resolution of boundary value problems. Here, we present an approach to the modeling of anisotropic fluids, which is not a director theory; no balance laws for directors are posited nor is there a notion of a director body force, director (or cosserat) stress or director kinetic energy. Thus, the present approach does not require specifying any additional boundary conditions other than that usually specified for viscous fluids, even for flows that involve spatially inhomogeneous fields. Moreover, the framework is based on sound thermodynamical footing, the evolution of the natural configurations being determined by the rate of dissipation of the material. To delineate the efficacy of the theory, we solve a problem associated with a shearing flow of the fluid in which we discuss the tumbling and alignment of certain vectors that represent the axes of anisotropy of the fluid and which may be associated with rod-like structures in the fluid.

The inelastic behavior of metals subject to loading reversal

Abstract One of the consequences of memory effects in the plastic deformation of metals is the Bauschinger effect (Civilingenieur 27 (1881) 289–348), which manifests itself as a difference in the values of the yield stress in tension and compression for a material that has undergone plastic deformation. The Bauschinger effect has been modeled with the kinematic hardening rules e.g., Ziegler (Quart. Appl. Math. 17 (1959) 55) and Chaboche (Int. J. Plasticity 2 (1986) 149). These models, though, are not able to reproduce the stress-strain response accurately at points of loading reversal: it has been observed (Acta Metall. 34 (1986) 1553; Mater. Sci. Engineering A113 (1989) 441) that, for some materials, the stress has a plateau after the loading is reversed. This is not reflected by the kinematic hardening rule nor by its modifications. In this paper we will develop a general three dimensional model that is able to reproduce the stress–strain response at loading reversals and can be applied also to more general changes of loading direction. The central idea of our model is to link the hardening behavior of the material to thermodynamical quantities such as the stored energy due to cold work and the rate of dissipation. The predictions of the theory show good agreement with the stress–strain curve and also with the manner in which the stored energy varies with the inelastic strain, as obtained from experiments (Progress in Materials Science (1973) Vol. 17. Pergamon, Oxford; Trans. Met. Soc. AIME 224 (1962) 719).

A Gibbs-potential-based formulation for obtaining the response functions for a class of viscoelastic materials

In this article, we demonstrate the use of a Gibbs-potential-based formulation as a means for developing a thermodynamically consistent model for a class of viscoelastic fluids of the rate type. Since one cannot always use a formulation based on a Helmholtz potential to model rate-type models, the formulation takes on added significance. The salient features of this approach are the following: — this approach provides a thermodynamical rationalization of many commonly used models that are developed on purely phenomenological grounds; furthermore, the study provides a framework for generating other classes of models and allows for a relatively straightforward means for the inclusion of thermal effects, — the approach provides a simple means for including anisotropic effects without the need for directors or other new internal variables, and — the approach does not use any additional variables (such as conformation tensors or elastic strains measured from stress free configurations) other than the current (or Cauchy) stress, the current mass density and the velocity gradient. We also show how the entire structure of the theory is obtained from just two scalar functions, the Gibbs potential and the rate of dissipation function.