## Abstract

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’.

## 1. Introduction

The main thrust of our paper is to develop a class of implicit constitutive relations for materials that do not have the capability to dissipate mechanical power. These equations, while being rate independent, do not necessarily lead to a hyperelastic model.

Within the context of purely mechanical considerations, by elasticity1 one refers to either Cauchy elasticity or Green elasticity (hyperelasticity). In Cauchy elasticity, the stress is defined to be a function of the deformation gradient from a specific reference configuration, while in Green elasticity, one associates a stored energy function with the body in question, with the stored energy being a function of the deformation gradient. However, from a more general thermodynamic point of view, one views a body as being elastic if the body is incapable of dissipating energy, i.e. the body is incapable of converting working (rate at which work is done) into heat (energy in thermal form), and all the mechanical work is recoverable. The assumption that a Green-elastic body is incapable of dissipation implies that the Piola–Kirchhoff stress can be obtained as the derivative of the stored energy with respect to the deformation gradient. Recently, Rajagopal & Srinivasa (2007) have shown that there is a large class of bodies for which the stress (or stored energy) is not expressed explicitly as a function of the deformation gradient, but the stress and the deformation gradient are defined by an implicit relation, or the stored energy depends on both the stress and the deformation gradient, the bodies being incapable of dissipation in any process that they undergo. As is evident, such bodies are neither Cauchy elastic nor Green elastic. In order to illustrate the implications of their ideas, they considered a one-dimensional response wherein the stress and the strain are related by an implicit relation and showed that the locus for the stress and strain could be even spiral like, among a host of other intriguing possibilities. The study of Rajagopal & Srinivasa (2007) is by no means restricted to a one-dimensional response; within the context of a three-dimensional response, they obtain sufficient conditions to ensure that the rate of dissipation is zero in all possible motions of the body under consideration.

The paper by Rajagopal & Srinivasa (2007) seems to have merely scratched the surface with regard to the possibility of bodies that are incapable of dissipation, i.e. elastic bodies. Though their study greatly increased the class of non-dissipative bodies, they were yet looking at a special class of implicit relations. In this paper, we shall build upon the work of Rajagopal & Srinivasa (2007) and extend it to rate-type materials that are non-dissipative. As we shall be concerned with the response of rate-type materials, it is important to discuss the response of an important class of rate-type materials that were introduced by Truesdell (1955), namely hypoelastic materials. The name unfortunately suggests that it might pertain to a class of elastic bodies. Bernstein (1960) studied such materials and derived conditions under which such hypoelastic materials would be elastic. However, Olsen & Bernstein (1984) subsequently obtained conditions within which hypoelastic materials that satisfied the demands of the second law of thermodynamics could be non-elastic. Recently, Bernstein & Rajagopal (2007) have considered the thermodynamics of hypoelastic bodies by requiring that such bodies meet the second law (interpreted as the Clausius–Duhem inequality). They find that enforcing the second law underdetermines the material coefficients that appear in the definition of the rate-type constitutive equation that defines hypoelastic materials and allows for some leeway in specifying these coefficients. The study of Rajagopal & Srinivasa (2007) concerning the thermodynamics of non-dissipative materials is quite different from those of Bernstein & Rajagopal (2007) in that the class of materials considered by Rajagopal & Srinivasa (2007) is much more general. Moreover, even the thermodynamic approach is very different, as Rajagopal & Srinivasa (2007) adopt a framework that they have employed with a great deal of success to study both dissipative and non-dissipative materials, namely that the constitutive relations be such that the body undergoes processes wherein the rate of entropy production is maximal. More importantly, they define a rate of entropy production function and make assumptions concerning the forms of the specific Helmholtz potential and the rate of entropy production to arrive at the constitutive relation for the stress. The usual procedure in continuum thermodynamics allows the body to undergo arbitrary processes, an assumption that might violate the domain in which the constitutive relation is assumed to hold in the first place. In this paper, we use the thermodynamic approach used by Rajagopal & Srinivasa (2007) to study rate-type models. The approach here is fundamentally different from those based on hypoelasticity since we obtain *implicit constitutive relations* relating the symmetric part of the velocity gradient and suitable rates of stress, rather than explicit equations for the stress rates.

The main reasons for the consideration of such constitutive equations are manifold and have both a philosophical and pragmatic basis: (i) at a philosophical level, we would like to extend and generalize the notion of elasticity to include classes of materials that were previously thought to be inelastic and (ii) at a pragmatic level, we would like to develop a fully Eulerian theory for non-dissipative rate-independent materials that does not use any notion of a ‘reference state’, but instead uses kinematical and kinetic quantities measured ‘here and now’. The importance of such an approach cannot be overemphasized. For instance, in the study of the mechanics of biological matter one has to take into account the fact that cells are born and die, there is constant turnover of the matter and thus a Lagrangian point of view is not feasible or meaningful. Furthermore, the material that is born is not created in a stress-free state. For the study of most biological bodies, not only is an Eulerian approach convenient, but it is also the only one that ought to be used. In certain problems, where the turnover is negligible, as in a mature organ, one might be able to ignore the fact that a body is not a fixed set of particles (even in mature organs there is turnover, but it can be ignored without serious consequences). It is important to consider a different point of view that, in many ways, may be more suitable for certain purposes (such as the finite-element implementation of large deformation contact problems in Abaqus, to provide a concrete practical example) and, as will become clear, the Eulerian approach presented here shows one such point of view. Thus, from both the philosophical and practical standpoint, the development of a rate-type constitutive equation for non-dissipative materials is highly desirable.

## 2. Preliminaries

Consider a body that at current instant *t* occupies a configuration *k*_{t}. The position of any particle at the current instant is denoted by ** x** and its velocity by

**. The mass density of the material is denoted by**

*v**ϱ*and the Cauchy stress at the point

**at time**

*x**t*is denoted by

**. The governing balance equations for mass, momentum and angular momentum (in the absence of body forces), are given in local form by(2.1)(2.2)where the superposed dot is the material time derivative that, in terms of spatial coordinates, is given by and the superscript notation (.)**

*T*^{T}denotes the transpose.

For the development of classical Cauchy-elastic or Green-elastic (hyperelastic) materials, it is necessary to introduce a *reference state* from which appropriate measures of strain and stress are used. In the case of Green-elastic materials, the stress is obtained as the transpose of the derivative of the strain energy function with respect to the deformation gradient.

The point of departure here from the classical theories is to note that, thermodynamically speaking there is absolutely no necessity to consider the strain energy function, and hence its generalization, the Helmholtz potential is the primary state function. Indeed, even a casual perusal of most texts on thermodynamics quickly reveals that there are many other potentials that have equal, if not greater, claim to primacy (the Gibbs potential, the enthalpy, etc.). Furthermore, from the point of view of statistical mechanics, the Gibbs potential (or more precisely the grand canonical ensemble) appears as the potential that has the widest possible application (including for open systems, chemically reacting systems, etc.). Keeping this in mind, it seems appropriate to start the consideration of the thermodynamics of non-dissipative materials within the context of the Gibbs potential. Thus, we begin by assuming a form for the Gibbs function *Φ* per unit mass is of the form(2.3)as the starting point for our analysis of elastic materials.

Invariance under Galilean transformations immediately gives(2.4)where I, II and III are the invariants of ** T**, defined through I=tr

*, and . Thus, if we start with the Gibbs function only depending upon the stress, we will obtain an extremely restrictive constitutive equation that only allows for isotropic response.*

**T**### (a) Modelling of anisotropic materials

In order to eliminate such a restriction that only allows models for isotropic bodies, and to enable us to consider a much wider class of materials, we introduce three vectors *a*_{i}, *i*=1, 2, 3, which represent three special directions in the current configuration. For example, for the case of a single crystal, they would be the lattice vectors of the crystal, whereas for a rolled material, they would be the rolling, transverse and thickness direction. We do not yet stipulate that they are directions related to material properties, however, leaving such stipulations to specific constitutive assumptions. We only assume that, as they are vectors defined in the current configuration of the material, they will transform to *a*_{i} under Galilean transformations. Furthermore, if we define a tensor by(2.5)where *e*_{i} are the fixed basis vectors of the laboratory frame, we see that transforms to under Galilean transformations. Note that has characteristics that are similar to the deformation gradient ** F** (which requires one to use a reference configuration) and it will be identical to it

*if we are to assume that the*

*a*_{i}

*are material line elements*. However, this is a specific constitutive assumption and we will postpone such assumptions until we actually require it. Here, we simply note that, if the

*a*_{i}are always linearly independent, the polar decomposition theorem can be applied to and so it can be resolved into a pure stretch and a pure rotation as(2.6)

With the introduction of , we now stipulate that the Gibbs potential for an anisotropic material is to be of the form(2.7)In view of the Galilean invariance of the Gibbs potential, the potential function reduces to(2.8)Now introducing the notation , we can write (2.8) as(2.9)For our present purposes (i.e. to discuss non-dissipative response within the context of purely mechanical processes), we shall make the following assumptions concerning the Gibbs potential *Φ*.

We shall assume that the Gibbs potential becomes zero when the stress

or alternatively*T**T*^{*}is zero, i.e. in any stress-free state.We shall also assume that, near the stress-free state, i.e. as the stress tends to zero, the Gibbs potential becomes a smooth convex function of the stress.2 Thus,

*Φ*as well as its first derivative (with respect to*T*^{*}) go to zero while its second derivative is positive definite at=0.*T*

With these conditions, it is not hard to see that the Gibbs potential can be written as(2.10)where the notation ‖.‖ stands for the Frobenius norm and is defined as . Also the function *a* is finite as the stress vanishes.

## 3. The Helmholtz potential and its time derivative

Given the Gibbs potential (2.9), it is relatively straightforward to define the Helmholtz potential *ψ* through the relation(3.1)By a routine use of the chain rule for differentiation, we can rewrite this as(3.2)

Furthermore, the material time derivative of *ψ* is given by(3.3)where we have used (2.10) as well as the conservation of mass in the form (2.1), and where ** D** is the symmetric part of the velocity gradient whose trace (denoted by tr(.)) is the divergence of

**. We note that, owing to the fact that transforms as under Galilean transformations, the term automatically involves a properly objective corotational rate of stress. To see this, we take the time derivative of**

*v*

*T*^{*}and get(3.4)where , is the spin tensor associated with the rate of change of and has to be specified by a constitutive equation. Thus, we will get different objective rates of stress depending upon the different constitutive choices we make regarding the evolution of the vectors

*a*_{i}or equivalently of the tensor . Specifically, if we assume that the vectors coincide with material line elements, then , where is the rotation tensor obtained by the polar decomposition of

**. Then,**

*F*

*T*^{*}becomes the rotated stress tensor and the terms in the bracket of (3.4) become the Green–McInnis–Naghdi stress rate.

A particularly simple form results, on the other hand, if we assume that *a*_{i} are always *orthonormal* and that the Gibbs potential is independent of the density *ϱ*. In this case, becomes an orthogonal tensor (so that and ** K**=

**) and the constitutive equation for**

*I**Φ*reduces to . In this case, equation (3.3) simplifies considerably to give(3.5)

## 4. Implicit constitutive equations

Having obtained expressions for the Helmholtz potential and its rate, we now invoke the reduced energy dissipation criterion in the form(4.1)as a requirement for all non-dissipative materials, where ** D** is the symmetric part of the velocity gradient. The traditional way to use this restriction is to postulate a form for the Helmholtz potential and demand that (4.1) be met for any choice of

**. However, as Rajagopal & Srinivasa (2007) have shown, a much richer class of models can be obtained by choosing implicit, rate-type constitutive equations and to demand that they be consistent with (4.1), with the right-hand side set to zero for all allowable values of the stress**

*D***. The general conditions for such constitutive equations to be possible were laid out in that paper. Here, we show how such equations may be obtained. To this end, we substitute the form (3.3) into (4.1) and consider constitutive equations for that will allow us to ensure the satisfaction of (4.1).**

*T*In view of (3.3), it is easy to see (by substituting (3.3) into the left-hand side of (4.1) and grouping terms) that a *sufficient condition* for the satisfaction of (4.1) is the condition(4.2)

We note several interesting features of the above equations.

The constitutive equation for

is truly implicit.*D*The resulting model is non-dissipative, irrespective of the choice of the constitutive equation for the rate of

and .*K*The first term on the right-hand side of (4.2) is the usual ‘tangent compliance tensor’ acting on the stress rate.

The constitutive equation cannot necessarily be integrated to give explicit expressions for any specific measure of the strain in terms of the stress. However, the equation is entirely Eulerian in nature. If one specifies that is the ‘logarithmic rotation tensor’ (such as that used by Xiao

*et al*. 1999) then, one can show that equation (4.2) can be integrated to give an implicit equation for the logarithmic stretch tensor for isotropic materials. However, for anisotropic materials, since the right-hand side of equation (4.2) involves, it cannot be directly integrated, unless certain additional simplifying assumptions are made.*D*The constitutive equation is not complete unless (or its rate) and

(or its rate) are specified. Such a specification is tantamount to specifying evolution equations for or, equivalently,*K**a*_{i}. It is also pertinent to state that Xiao*et al*. (1999) were only concerned with the response of hyperelastic bodies that could be expressed in Eulerian form. Also, their work was restricted to a generalization of hypoelastic materials, while the class of material being considered in this paper is not obtained by merely generalizing hyperelasticity; the class considered here is of far wider scope.

## 5. The evolution of the vectors *a*_{i}

*a*

Several simple choices present themselves for the evolution of the vectors *a*_{i}. One obvious choice is to stipulate that *a*_{i} are material line elements, in which case, the evolution law becomes(5.1)With this choice, it is not difficult to show that(5.2)which, when substituted into (4.2), gives(5.3)which is an *implicit* equation of the form(5.4)

Furthermore, it can be shown that coincides with the rotation tensor , obtained by the polar decomposition of the deformation gradient, so that *T*^{*} is the rotated stress tensor and its rate is indeed the Green–McInnis–Naghdi rate.

Other, more general choices for are possible, giving rise to a wide range of non-dissipative behaviour.

## 6. Implicit theories and a generalization of Green elasticity

One of the most widely accepted notions in classical elasticity is that, for such materials, the work done in a closed cycle of deformation must be zero. Many implicit or rate-type constitutive theories do not satisfy this criterion. For the class of theories considered here, one has to define what is meant by a mechanical cycle with care. A proper statement of this requirement would be that the work done in a mechanical cycle of states, i.e. a cycle in which *both the stress and the deformation gradient return to their original values*, the work done must be zero.3 The models developed here satisfy this condition, as can be seen from the fact that the stress power is equal to the rate of change of the Helmholtz potential. Thus, the work done in a process is equal to the difference in the Helmholtz potential. Thus, if we consider a cycle in which ** F**, as well as

**, return to their original values, the difference in the Helmholtz potential vanishes and hence the work done is zero. This could be considered as a generalization of Green-elastic materials.**

*T*## Footnotes

↵We refer the reader to a recent paper by Rajagopal (2007), entitled ‘The elasticity of elasticity’ for a discussion of the various ways in which the term ‘elasticity’ has been used. It will be clear from the discussion there that the word has been used in senses far from its usage in mechanics today. Even Lord Kelvin (Thomson 1865) used the word ‘Elastic’ to describe frictional materials. In a paper entitled ‘The elasticity and viscosity of metals’, Kelvin states ‘Hence there is in elastic solid molecular friction which may be properly called the viscosity of solids, because, as being an internal resistance to change of shape depending on the rapidity of the change, it must be classed with fluid molecular friction, which by general consent is called viscosity of fluids’. Within the context of present day usage of the word ‘elasticity’ we would have to conclude that Kelvin is confusing a viscoelastic material with an elastic material, as a purely elastic material has no viscosity and that there is no internal friction associated with such materials.

↵Note that, unlike the case of classical linearized elasticity, no assumption is made regarding the order of magnitude of the strain; rather, the linearization is for small values of the stress. Hence, within the context of our general theory, it is possible for the strain to be small and the stress large or the stress to be small and the strain large or both the stress and the strain to be small. Thus, it becomes possible to separate out nonlinearity associated with the kinematics from the nonlinearity associated with the constitutive relation, i.e. within the context of the implicit theory developed here, the relation between the strain and the stress could be nonlinear, even though the strain is small (also see Rajagopal 2007).

↵In the classical theory of elasticity, as the stress is defined in terms of the deformation gradient, a cycle of deformations implies a cycle of stress.

- Received August 1, 2008.
- Accepted September 30, 2008.

- © 2008 The Royal Society