## Abstract

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.

## 1. Introduction

Within the context of continuum mechanics, the notion of an elastic body has been interpreted in several ways, and it has been tacitly assumed that these different interpretations are not contradictory. For instance, the notion that an elastic body is one that is incapable of dissipation is thought of as being consistent, and in fact, implying that the stress in the body can be derived in terms of a stored energy. This is true only if several additional assumptions are made, which invariably are left unstated. The main thesis of this paper is that the class of solid bodies that are incapable of dissipation is infinitely richer than the class of solid bodies for which there exists a stored energy from which the stress can be derived, which is also incapable of dissipation. That is, there are solid bodies that are incapable of dissipation for which the stress is not a derivative of the stored energy with respect to the deformation gradient. Thus, when we have to describe the response of bodies that are incapable of dissipation, we have the possibility of a far more general class of models to pick from. In fact, it is easy to show that there are perfectly valid physical models that are incapable of dissipation for which we cannot express the stress as the derivative of the stored energy.

Of course, we now recognize that the ability to retain shape cannot be the characteristic by which means we define an elastic body. We know that an ideal gas falls within the class of classical elastic bodies as the stress in such a body is completely determined by the deformation gradient. Thus, we have to seek a better definition for an elastic body.

In this short paper, we show that the definition of an elastic solid body as being a body that is incapable of dissipation, while the response is the same regardless of the rate of deformation (i.e. the response is rate independent), includes a much more general class of bodies than those whose stress is derivable from a stored energy or those that are capable of retaining shape.

## 2. Preliminaries

Consider a body , a typical material point which is at a position * X* in a reference configuration

*k*

_{0}and position

*in the present configuration*

**x***k*

_{t}at time

*t*. The motion of the body is a smooth, invertible mapping(2.1)

The deformation gradient and the Cauchy–Green strain tensor for the motion are defined through(2.2)where the notation (.)^{t} stands for the transpose of a tensor.

We say that a body is Cauchy elastic body when the Cauchy stress in the body is completely determined by the deformation gradient (e.g. Truesdell & Noll 1992). We express this, in the case of a homogeneous body, as(2.3)i.e. the stress is prescribed as a function of a kinematic measure.

It is interesting to note that in thermodynamic treatments of gases, there is no preference given to kinematical or stress measures, i.e. at times the volume is expressed in terms of pressure and at other instances the pressure is expressed in terms of volume. The classical theory of linearized elasticity also shares this feature, i.e. we can express the strain in terms of stress or the stress in terms of strain. Depending on the situation, one can use one form or the other. In marked contrast, in theories that have been developed for nonlinear continua other than gases, stress is expressed in terms of kinematical quantities, and in general, one cannot invert this relationship to express an appropriate strain measure in terms of stress.

Even when one stays within the realms of classical nonlinear theories, one cannot always express the stress explicitly in terms of kinematical variables. This becomes clear when one considers incompressible fluids, whose Cauchy stress is related to the symmetric part of the velocity gradient in the following manner (see Hron *et al*. 2001):(2.4)where by virtue of the constraint of incompressibility ; and *μ* is a function of *p* and ||^{2}. This relationship is an *implicit* relationship between the stress and the symmetric part of the velocity gradient. Models of the form (2.4) are very useful in describing the response of fluids that are subject to a very high range of pressures, and the rationale for such models can be traced to the seminal work of Stokes (1845) and subsequent investigations by numerous eminent physicists (see Andrade 1930; Bridgman 1931). A detailed discussion of the implicit constitutive models can be found in Rajagopal (2003).

Of course, the constitutive equation (2.4) refers to a dissipative material. The question arises as to whether such implicit constitutive theories can be posited for *non-dissipative* materials.

Here, if one uses the conventional definition of Green elastic materials, such implicit constitutive theories are not possible. However, if one considers non-dissipative materials that are *not Green elastic*, then as we show here, it is possible to postulate such implicit constitutive equations. Such models are relevant when considering the purely mechanical behaviour of hysteretic materials, i.e. one seeks to answer the following question: is it possible for a material to be non-dissipative yet exhibit a response akin to that of a hysteretic material, with multivalued relations between stress and strain?

Hence, in order to consider implicit constitutive equations, we discuss the response of a body that is incapable of dissipation, whose stress cannot be expressed as a function of the deformation gradient. Models such as this are routinely used in molecular dynamics simulations of polymer models, wherein the molecules are represented by sequences of nonlinear springs and dumbells. The dumbells have *finite extensibility*, i.e. the stretch of the springs cannot exceed a critical amount. Such models are referred to as finitely extensible nonlinear elastic models (e.g. Bird *et al*. 1987). The main point here is that since the stretch of the springs is restricted whereas the force on the spring is not, *it is impossible in these models to write the force as a function of the stretch*. A typical approach is to use an exponential or some such function, whose tangential stiffness tends to infinity, i.e. to achieve this in the limit as both the stress and the tangent modulus go to infinity at a finite strain. In this paper, we present an alternative means to model these materials, where the maximum strain is achievable at finite strains. While implicit models have been investigated in fluid mechanics, corresponding developments in purely elastic materials have not taken place, since such models fall outside the class of Cauchy or Green elastic materials.

To elaborate, consider a body whose mechanical analogue is represented in figure 1*a*. The body is essentially a spring in parallel with an inextensible string. In the unloaded state, the string is longer than the spring, and hence it is slack. As the body is loaded (figure 1*b*), the initial response is that of the spring alone until the string becomes taut. The total response of the body is shown in figure 1*c*. In the example cited previously, we note that there is a one-sided constraint, namely that the extension of the spring cannot exceed a critical value. We note that the force cannot be expressed as a function of the stretch beyond a point A, although the body is non-dissipative. Past the point A, we have to describe the body by stating that the magnitude of the strain is constant. This example as well as equation (2.4) suggests that we might not want to start with the assumption that the stress is a function of the deformation gradient, but rather look for implicit relations between the two of the form(2.5)

Taking frame indifference into account and in view of the fact that any frame indifferent constitutive equation of the form (2.5) can be recast in terms of *invariant* quantities, we note that equation (2.5) can be reduced to(2.6)where the symmetric Piola–Kirchhoff stress is defined by(2.7)and the symbols (.)^{−1} and (.)^{−t} stand for the inverse and inverse transpose of the tensor, respectively.

## 3. Implicit, elastic constitutive equations

Let us consider the implicit constitutive equation (2.6) in more detail. It follows from equation (2.6) that(3.1)where and are fourth-order tensors. Motivated by this, let us consider the following generalization of equation (3.1), namely:(3.2)where and are fourth-order tensors. All implicit relations of the form (2.6), if they are sufficiently smooth, lead to the relation (3.2), but equation (3.2) is more general in the sense that not all equations of the form (3.2) are integrable. In this paper, we shall look at constitutive equations of the form (3.2). It is important to note that if (,) is invertible, it is possible to pre-multiply equation (3.2) by ^{−1} and rewrite equation (3.2) as a rate-type equation of the form(3.3)

Truesdell (1955) introduced a class of materials that he called ‘hypoelastic’ through the constitutive relation(3.4)where is a fourth-order tensor that depends on the Cauchy stress ** T**; and

**and**

*D***, the symmetric and skew part of the velocity gradient, respectively. Bernstein (1960) carefully assessed the work of Truesdell (1955) and recognized that certain additional requirements ought to be made to make the model physically reasonable. Bernstein (1960) remarks**

*W*By a hypo-elastic material we shall mean the assignment of a set of equations (3.1) or (3.5) and a corresponding equivalence class of stress-configuration classes.(His eqn (3.1) refers to our equation (3.4).) Bernstein realized that such rate-type models referred to a class of constitutive relations, as do models defined through equation (3.2). Here, the class of models is larger than the class of hypoelastic solids. Constitutive models of the form (3.2) have been considered previously (e.g. Green & Naghdi 1973); however, the context within which they used this form of constitutive relations is totally different from that considered here. Green & Naghdi (1973) were interested in describing the inelastic response of bodies and the implication of the models of the form (3.2), with additional conditions, leading to an explanation for the notion of yield conditions. They did not consider the possibility that models of the form (3.2) can lead to non-dissipative or purely elastic behaviour (without being integrable) or the fact that they lead to models for elastic bodies that are not hyperelastic in the classical sense.

With these preliminaries out of our way, we now consider the constitutive response of the material. In the following, we shall find it more convenient to work with a fixed Cartesian basis *e*_{A}(*A*=1,2,3) for the reference configuration and use component forms for all the tensors using capital letters for all subscripts that range from 1 to 3. Thus, *S*_{AB} are the Cartesian components of the tensor with respect to this basis, with similar notations holding for the other tensors.

With this notation, the constitutive equation (3.2) can be written as(3.5)where _{ABCD}(,) and _{ABCD}(,) are constitutively specified components of fourth-order tensors and possess the symmetries _{ABCD}=_{BACD}=_{ABDC} and _{ABCD}=_{BACD}=_{ABDC}, and the superposed dot signifies time differentiation.

The constitutive equation (3.5) is a generalization of a large class of hypoelastic materials and includes—as special cases—Cauchy elastic materials, Green or hyperelastic materials, hypoelastic materials of the class considered by Truesdell & Noll (1992) as well as a large variety of constitutive equations used to describe the response of inelastic materials such as clay, sandstone, etc. A more subtle issue is that the constitutive equation (3.5) includes commonly used kinematic constraints such as incompressibility as well. In other words, in this framework, kinematical constraints are treated in the same manner as any other constitutive specification. For example, the constraint of incompressibility becomes(3.6)and it can be clearly incorporated as one of the rate equations in (3.5),1.

### (a) Reduced energy-dissipation equation and hyperelasticity

Before we turn to an analysis within a full three-dimensional setting, we consider a one-dimensional example that addresses the crux of the matter and shows the richness that implicit theories accord.

Let us consider an implicit relationship within a one-dimensional, small strain setting for a stored energy *ψ* per unit volume as a function of both the stress *σ* and the strain *ϵ*, i.e.(3.7)

On requiring that the stress power be equal to the rate of change of the total stored energy, we obtain(3.8)Solving for the , we obtain(3.9)We note that equation (3.9) is of the form (3.2) and is completely determined by the form of the strain energy function. However, the body will not dissipate in any process.

To illustrate the kind of stress–strain response engendered by equation (3.9), let us consider a general quadratic form for the strain energy,(3.10)

A routine calculation reveals that the constitutive equation (3.9) reduces to(3.11)

Different values of the constants *a*_{1}−*a*_{5} give rise to widely different responses. For example, if we set *a*_{1}=0.01, *a*_{2}=−0.1, *a*_{3}=1.1, *a*_{4}=0.1 and *a*_{5}=−0.1, then we get a behaviour of a material which is very soft under uniaxial extension, but very stiff under uniaxial compression as shown in figure 2. Such a response is akin to that of a tightly coiled tension spring that can be extended but cannot be compressed. Moreover, the stress–strain relation in this case is not invertible.

On the other hand, if we set *a*_{1}=−*a*_{2}=1, *a*_{3}=*a*_{4}=*a*_{5}=0, then we get a response in the form of a spiral, as seen in figure 3. It is important to bear in mind that both the above responses are non-dissipative.

With the above motivation, we note that in the three-dimensional context, for purely mechanical theories, a version of the second law of thermodynamics can be stated in the form that the difference between the stress power and the stored energy (which is equal to the rate of dissipation, i.e. the rate at which work is converted into heat) be non-negative, with the vanishing of the difference being one of the defining characteristics of elastic materials. In other words, for elastic materials, we assume that(3.12)where *W* is the stored energy per unit reference volume. For Green elastic materials, it is typical to assume that(3.13)and demand that equation (3.12) be met for all possible values of and deduce that(3.14)Such materials are called ‘hyperelastic materials’. It is evident that the assumption is a special case of the general rate-type equation (3.5), as can be seen by differentiating the first of the two equations defined in (3.13). In view of this, in §4, we now seek to answer the following question:Given a constitutive equation of the form 12 and a strain energy function

The answer to this question leads to a class of materials, whose response functions are considerably more general than equation (3.14).*W*(,), what are the conditions on the functions _{ABCD} and _{ABCD}, such that ?

## 4. Non-dissipative materials that are not hyperelastic

### (a) The twelve-dimensional state space

Before answering the question posed at the end of §3*a*, the first issue that needs to be addressed is the concept of a state space for such materials at each material point and the allowable values for the state variables. At the outset, the state variables would include the density field *ϱ*(** X**,

*t*), the stress field (

**,**

*X**t*) and the strain field (

**,**

*X**t*).

The allowable values of these fields are subject to certain constraints, such as the balance of mass, linear and angular momentum. We deal with each of these in turn. The referential form of the balance of mass implies that , where the function *ϱ*_{0}(** X**) is the (fixed) density field in the reference configuration. Thus, given the reference density

*ϱ*

_{0}, we note that the present density field

*ϱ*is completely determined by the strain field; hence, we can remove it from the list of state variables. Similarly, the local form of the balance of angular momentum implies that not all values of

**are possible, but only for those with**

*S***=**

*S*

*S*^{t}, i.e. the stress response should be symmetric. We are now left with the requirement for the balance of linear momentum. Here, we note that

*homogeneous deformations with the stress and strain fields being constant throughout the body are always possible for quasistatic processes*. Thus, we restrict our attention at first to homogeneous quasistatic motions. However, since we are interested only in local constitutive theories, namely those for which there exists a relationship only between the stress history and the strain history at a given material point, consideration of homogeneous motions is sufficient to establish the constitutive relationship. In classical elastic materials, a further simplification is possible since the stress is assumed to be a function of the deformation gradient, so that even the stress can be removed from the list of state variables, leaving the strain tensor as the sole independent state variable.

Here, however, since there is no explicit relationship between the stress and the strain, we shall convert the state variables , into column vectors and rewrite the constitutive equation (3.5) as well as the reduced energy-dissipation equation (3.12) as matrix equations2. To this end, we introduce an isomorphism between the set of all symmetric second-order tensors and elements of ^{6}. Specifically, the components of the symmetric Piola–Kirchhoff stress are written as a six-dimensional column vector of the form(4.1)with a similar notation for . The off-diagonal components are multiplied by in order for the Euclidean norm in ^{6} to coincide with the Frobenius norm for second-order tensors. The linear transformation from the space of symmetric second-order tensors and ^{6} and its inverse can be represented, respectively, by the matrix equations(4.2)In this and subsequent equations, we shall stipulate that Greek indices run from 1 to 6. In equation (4.2), *M*_{αAB} and *N*_{ABα} are all zero, except the following:(4.3)(4.4)(4.5)(4.6)

Using these definitions, we can rewrite the constitutive equation (3.5) as(4.7)where(4.8)Finally, writing the column vectors *S*_{α} and *E*_{α} as a single-column vector *Σ*_{i} in ^{12}, where(4.9)we can rewrite the entire constitutive equation in matrix notation as(4.10)where [** Π**] is a 6×12 matrix, whose components

*Π*_{αj}are given by(4.11)In matrix notation, the 6×12 matrix

**is obtained by juxtaposing the 6×6 matrices and side by side as(4.12)Thus, given that the material is in a state**

*Π**Σ*, the allowable ways in which the states can evolve are given by those values of that satisfy equation (4.10), i.e. the right null space

*N*(

**) of the matrix**

*Π***. We now stipulate that**

*Π***is of rank 6. Now, since the domain on which**

*Π***operates is a twelve-dimensional vector space, the null space must be of dimension 12−6=6.**

*Π*In other words, the constitutive equation (4.10) is such that at most 6 of the 12 components of can be assigned, the other six being determined by solving the six simultaneous linear equations represented by equation (4.10). ‘This is the counterpart of the standard assumption that the six components of can be chosen arbitrarily’. In the present approach, it is not known *a priori* which 6 of the 12 components of can be chosen arbitrarily, the choice being determined by the structure of the matrix ** Π** and could vary from state to state since

**itself depends upon**

*Π**Σ*.

### (b) The matrix formulation of the reduced energy-dissipation equation

Having rewritten the rate-type constitutive equation (3.5) in the matrix form (4.10), we now turn our attention to the stored energy function (or the Helmholtz potential) and assume the existence of a function *W*(,), such that equation (3.12) is satisfied. We first note that we have generalized the independent variables upon which *W* depends to include both the stress and the strain, so that we can write . We shall first rewrite equation (3.12) using the column vector *Σ* as follows:(4.13)where the row vector *Γ* is given by(4.14)

### (c) Non-dissipative response

We now consider the following counterpart of the condition defining the response of hyperelastic materials:What are the conditions on the matrix [

** Π**], such that every allowable process—i.e. possible values of that satisfy equation (4.10)—also satisfies equation (4.13)?

In other words, we need to find conditions on [** Π**], such that(4.15)When presented in this form, the answer to the above question is rather simple. ‘The row vector [

*Γ*], containing the derivatives of W, must be an element of the row space of

**, i.e.**

*Π**Γ*must be a linear combination of the rows of

**’.**

*Π*To see this, we augment the original 6×12 matrix [** Π**] by appending [

*Γ*] to the rows to get the 7×12 matrix , i.e.(4.16)Thus, the condition (4.15) can be rephrased to mean that(4.17)In other words, the null space of [

**] must coincide with that of the augmented matrix . This then implies that the row spaces of [**

*Π***] and must coincide. Hence the result that**

*Π**Γ*must be a linear combination of the rows of

**. Of course, it is a trivial result once the proper state space has been identified and all the constraints are written in terms of the elements of this state space; But the main point here is that**

*Π**rather than treating the reduced energy-dissipation equation*

*(3.12)*

*as a restriction on the constitutive variables, we use it as one among a set of six rate-type constitutive equations (which include constraint conditions)*. In other words, rather than separate the formulation into constitutive equations, restrictions on the variables and constraints, the formulation presented here treats each of them as simply one among the six constitutive equations.

## 5. Constitutive equations for non-dissipative behaviour

Summarizing the results obtained, in order for the rate-type constitutive equation (3.5) to represent a non-dissipative response, i.e. the stress power is equal to that rate of increase in the strain energy, we must have the following set of conditions:

the 6×12 matrix [

] formed by juxtaposing the 6×6 matrices and must be of rank 6, and*Π*the row vector

*Γ*defined by equation (4.14) must be a linear combination of the rows of [].*Π*

Given the rate-type equations and the strain energy function *W*, one easy way to modify the response to ensure that it is non-dissipative is to simply replace one of the rows with *Γ*, the row to be replaced being so chosen to ensure that resulting modified matrix ** Π** has rank 6.

On the other hand, given the strain energy function *W*, we can first find the vector *Γ* defined by equation (4.14) and choose five other (twelve-dimensional) vectors *Γ*_{i}, so that the six vectors are linearly independent. Then, the matrix ** Π** with these six vectors as rows will be guaranteed to satisfy equation (4.17) and the response will be non-dissipative.

Thus, there is a rich class of constitutive models that are non-dissipative, i.e. all processes of such bodies satisfy equation (4.13), the stored energy for such bodies being given by *W*=*W*(,) and the stress is not the derivative of *W* with respect to , i.e. these models can be chosen so that they are neither hyperelastic in the traditional sense nor hypoelastic according to the definition of Truesdell & Noll (1992).

## Footnotes

↵While the present approach deals with constraints that are at most linear in the rate of deformation, a general theory of constraints is beyond the scope of this paper and is discussed extensively by Rajagopal & Srinivasa (2005).

↵It is not

*necessary*to do so, but it is much more convenient. Furthermore, we wish to emphasize the fact that both the stress and the strain ought to be considered on an equal footing in the development that follows. Finally, when written in this way, the central result becomes evident.- Received May 10, 2006.
- Accepted July 4, 2006.

- © 2006 The Royal Society