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.
Modellers of viscoelastic fluids (especially those involved in the processsing of polymeric fluids) are frequently called upon to develop new thermomechanical models for these materials that are simple and computationally implementable. It is natural to consider the class of rate-type models as a means of carrying out this task owing to their relative simplicity (since they typically involve only current values of the stress, velocity gradient and their rates) when compared with integral models, wherein one has to keep track of the history of kinematical quantities. In other words, we are considering models of the class 1.1 where σ is the stress and L is the velocity gradient, and the superposed dot denotes material time differentiation. Such implicit models are extremely versatile and allow the material moduli to depend on the invariants of the stress and the symmetric part of the velocity gradient, as well as mixed invariants of the stress and the symmetric part of the velocity gradient. Thus, in such materials, the material moduli can depend on the mean normal stress (for a certain sub-class of incompressible materials, this implies the material moduli can depend on the pressure).
There is thus a vast literature on the subject of such rate-type models. However, owing to the fact that these models are typically completely phenomenological and are usually restricted to purely mechanical theories, there is currently no systematic way to generate new models in this class that can be guaranteed to satisfy the laws of thermodynamics. In fact, in the literature, special models of the type (1.1) (referred to as ‘hypoelastic materials’) are considered to be generally incompatible with thermodynamical criteria (see for example Truesdell & Noll 1994, ch. V and Simo & Hughes (1998)1 ch. 7), since they are thought not to admit a stored energy function. For isotropic materials, Bernstein (1960) has developed conditions under which such models can be considered to be thermodynamically acceptable. More recently, Xiao et al. (1999) have found explicit conditions under which the linear hypoelastic model can be considered as being thermodynamically acceptable.
Nor is there a way to systematically extend these models to include temperature dependence. Moreover, modelling flow-induced changes in anisotropy cannot be easily accomplished (unless one chooses to appeal to various director-based theories where one is immediately faced with problems concerning boundary conditions for the directors). There is thus a need to develop a thermodynamical framework for such models.
At the outset, it is necessary to mention that previous theories, which have been put into place to develop rate-type theories that are thermodynamically compatible (see Rajagopal & Srinivasa 2000), complement what is being developed here. The procedure developed by Rajagopal & Srinivasa (2000) using a Helmholtz potential cannot be used to develop all the classes of models that can be developed within the framework which is advocated here. Put more clearly, one cannot use a formulation of Rajagopal & Srinivasa (2000), use a Legendre transformation and arrive at the class of models that can be generated using the approach advocated in this paper. By the same token, one cannot arrive at all the models that can be obtained using the framework in Rajagopal & Srinivasa (2000) by using the procedure advanced in this paper. Thus, the two approaches go hand in hand towards developing a rich class of rate-type fluid models.
The aim of this note is to develop a framework for the modelling of a class of rate-type viscoelastic models (both isotropic and anisotropic) in a relatively straightforward way without the introduction of additional variables, such as elastic strains or conformation tensors, from a thermodynamical point of view.
It is worth emphasizing that the reason for rejecting models of the type (1.1) by Simo & Hughes (1998) are rooted in the fact that they do not reduce to the classical Helmholtz potential and deformation-based elasticity models that are referred to as Green elastic materials. However, as has been shown in a recent paper by Rajagopal & Srinivasa (2007), Green elastic models are only a small subclass of thermodynamically acceptable elasticity models. A different (and non-equivalent) class of elastic materials can be obtained by starting with the Gibbs potential.
We seek to develop a thermodynamical basis for models of the form (1.1) by starting with a Gibbs potential as a function of the stress, using the energy-dissipation equation and imposing the requirement that the rate of mechanical dissipation be non-negative. By strengthening this requirement and demanding that the rate of dissipation be the maximum possible (subject to the constraints imposed), we show how the constitutive equations are obtainable from two scalar functions—the Gibbs potential and the rate of dissipation function. We show how the classical rate-type models such as the Giesekus, Phan-Thien–Tanner and the White–Metzner models (among many others) can be derived from this approach. We emphasize that the procedure is not restricted to merely recovering these models, it is now possible to generate a host of new models that could prove to be useful.
The advantages of this approach are that it provides a scheme for generating new models that automatically satisfy the requirements of the second law and allows for the extension of these models to thermoviscoelastic materials in a straightforward way—all without the need for additional strain measures or any explicit introduction of new microstructural variables. Furthermore, the approach is fully Eulerian and does not use any notion of a special ‘reference state’ (other than the current state that is being used as a reference state), but instead uses kinematical and kinetic quantities measured ‘here and now’. The importance of such an approach cannot be overemphasized—especially with regard to computational implementation.
Moreover, the Gibbs potential allows us to use the stress as a primitive instead of a kinematical quantity, such as some measure of strain. We feel that this is appropriate from a philosophical standpoint as the traction (and consequently the stress) is the cause, the kinematics (motion or deformation) being the effect.
(a) Current modelling approaches
Current macroscopic continuum-based approaches for modelling viscoelastic materials fall into three broad categories: (i) phenomenological models that provide a relation between a suitable rate of stress as a function of the stress (and possibly higher rates of stress) and velocity gradient and higher rates of the velocity gradient,2 (ii) phenomenological models based on the extension of elasticity (such as the Kage–Bernstein–Kearsley–Zapas (K–BKZ) models; see Bernstein et al. 1963) by writing the stress as a functional of the history of the relative deformation gradient, and (iii) models based on the introduction of new variables such as the conformation tensor or the instantaneous elastic strain (e.g. Leonov 1992). From the point of view of the nature of the constitutive equations, they can also be classified as rate-type models and integral models. While many of these models can be cast in multiple equivalent forms (for example, the classical Maxwell model can be expressed either as a rate-type model or an integral model), in general, these classes of models are not equivalent to one another.
Of these classes of models, the ones that are most easily amenable to computations belong to category (i) since they do not involve additional variables, nor do they result in integrodifferential equations. However, the development of rate-type models of category (i) has been somewhat unsatisfactory since, in general, a rigorous thermodynamical approach is not in place for their development. Most such models are developed from generalizations of purely mechanical spring-dashpot analogues.
A thermodynamic basis has been provided for the K–BKZ model (see Fong & Simmons 1972; Bernstein & Fong 1993; Rao & Rajagopal 2007), however, this basis is not applicable to the class of models considered here. Rajagopal & Srinivasa (2000, 2004) have developed a thermodynamical approach for a class of models based on a choice for the Helmholtz potential and a rate of dissipation, with the assumption that the natural configuration associated with the current configuration of a body could be accessed by an instantaneous elastic unloading. The results include those models that could be obtained by the introduction of conformation tensors. Rao & Rajagopal (2007) have shown how the K–BKZ model can be recast into a proper thermodynamic framework and have shown the close connection between the thermodynamical modelling of materials with multiple natural configurations and the K–BKZ type models.
Following the work of Rajagopal & Srinivasa (2000, 2004), Karra & Rajagopal (2009) developed rate-type constitutive equations for fluid models that did not possess instantaneous elasticity, but they used the notion of a ‘natural configuration’.
The question that naturally arises is the following: is it possible to develop a consistent thermodynamical approach for the models of category (i) that uses only macroscopic variables, such as the stress and velocity gradient, without invoking any additional microstructurally interpretable variables? We show in this paper that it is indeed possible to do this.
A salient aspect of the approach presented here is that we develop the theory directly from a Gibbs potential that is a function of the stress, thus avoiding the need to introduce strain-like variables completely. Furthermore, we show that we can obtain properly invariant stress rates either by adding terms that are workless—which we refer to as ‘gyroscopic terms’ (see Rajagopal & Srinivasa 2008)—or by using a suitably defined ‘rotated stress’ as the starting point. The latter approach allows for properly invariant stress rates to arise naturally in the theory and also shows how anisotropic materials can be modelled in a relatively straightforward way. Finally, we show how the use of the maximum rate of entropy production criterion substantially simplifies the constitutive theory and reduces it to the specification of two scalar functions.
(b) Main results
The central results of the theory are the following.
A whole new class of thermodynamically consistent viscoelastic fluid models can be obtained by starting with a Gibbs potential, which is a function of the Cauchy stress. These models are not equivalent to models that start with a Helmholtz potential (as a function of a strain-like variable)—a point that cannot be overemphasized.
(i) Isotropic materials
For compressible, isotropic viscoelastic materials, we assume the existence of two functions: (i) the Gibbs potential G, which is a function of the invariants of s:=σ/ϱ, and (ii) a rate of dissipation function f, which is stipulated to be non-negative and is an isotropic function of s,ϱ and D. In terms of these two functions G and f, the governing constitutive equations are of the form 1.2 where D and W are the symmetric and antisymmetric parts of the velocity gradient.
(ii) Anisotropic materials
For compressible anisotropic materials, we assume that the Gibbs potential G and the rate of dissipation f are arbitrary functions of the rotated stress tensor . Then, the governing equations are 1.3 where .
It will be demonstrated that the above set of equations automatically satisfies all the requirements of the second law of thermodynamics. Indeed, among all constitutive relations that have the same Gibbs potential and the same rate of dissipation function, this constitutive relation gives the largest rate of dissipation.
The elegance of the approach presented here is best exemplified when dealing with incompressible materials, we simply stipulate that trD=0 and that G is a function only of the deviatoric part τ of s. Note that there is no restriction on f. Under these assumptions, for isotropic materials, the response becomes 1.4 However, care must be taken to ensure that all the derivatives are restricted to the five-dimensional deviatoric tensor space. This model allows for a thermodynamically consistent way to introduce pressure-dependent response functions for incompressible materials without invoking workless constraints, unlike traditional approaches.
(c) Advantages of the current approach
— The current approach can serve as a framework for developing many new models of viscoelastic materials simply by choosing different forms for G and f while, at the same time, ensuring that they meet all thermodynamic requirements.
— It is relatively straightforward to extend the theory to model anisotropic media by simply requiring G or f or both to be anisotropic functions of a suitably chosen rotated stress tensor.
— It is possible to develop models wherein the viscosity is pressure-dependent (particularly for high-pressure applications such as in elastohydrodynamic lubrication and other thin-film applications) by simply ensuring that f is a function of the pressure. This is possible because, in this formulation, incompressibility is a constitutive assumption and not a constraint, thus allowing wider latitude in the functional dependencies.
— It is possible to incorporate thermal effects by assuming that G is a function of the temperature. In this case, the entropy equation will become 1.5 where r is the specific body supply of heat (radiant heating), q the heat flux vector and f the rate of dissipation.
Consider a viscoelastic body that occupies a configuration kt at the current instant t. The position of any particle at the current instant is denoted by x and its velocity by v. The mass density of the material is denoted by ϱ and the Cauchy stress at the point x at time t is denoted by T. The governing balance equations for mass, momentum and angular momentum (in the absence of body forces) are given in local form by 2.1 and 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 denotes the transpose. Furthermore, we will state the balance of energy in the form 2.3 where ϵ is the internal energy per unit mass, r the specific body supply of heat, q the heat-flux vector and D the symmetric part of the velocity gradient.
Two broad approaches have generally been followed3 in the development of rate-type constitutive theories for viscoelastic fluids. The first is a purely phenomenological approach that consists of writing a constitutive equation for a suitably chosen rate of stress. Another alternative is to introduce the notion of a conformation tensor c which is a symmetric positive definite tensor that is meant to represent the conformation of the macromolecules that form the viscoelastic fluid. The stress is then written as a function of the conformation tensor, and an evolution equation is provided for the latter. This approach has some microscopic underpinnings and is quite versatile (see Leonov (1992) for a detailed discussion of models of this type). Furthermore, it is possible to introduce some aspects of thermodynamics into this model by requiring that the stress is related to the derivative of the Helmholtz potential with respect to the conformation tensor.
Recently, Rajagopal & Srinivasa (2000) introduced a fully thermodynamical framework for modelling viscoelastic fluids based on the notion of an instantly unloaded or natural configuration (similar in spirit to the notion of natural configurations that is used in plasticity). They postulated two scalar functions—the Helmholtz potential, which is a function of the deformation from the unloaded to the current configuration, and the rate of dissipation function, which is a function of a suitably chosen rate of the change of the current natural configuration. These functions are related by the energy-dissipation identity, which states that 2.4 where D is the symmetric part of the velocity gradient, and ψ and ξ are, respectively, the Helmholtz potential and rate of dissipation (both measured per unit mass). For later use (and especially for the purposes of carrying out Legendre transformations between the Gibbs and Helmholtz potentials), it is convenient to divide equation (2.4) by the density ϱ and rewrite it as 2.5 where s:=T/ϱ is the ‘Kirchhoff stress tensor’.
Rajagopal & Srinivasa (2000) stipulated that the natural configuration evolves in such a way as to maximize the rate of dissipation function, subject to the requirement that equation (2.5) be met. They found that
— the stress is the derivative of the Helmholtz potential, i.e. 2.6 where Be is the Finger tensor associated with the elastic stretch and α1 is related to the derivatives of the Helmholtz potential with respect to the invariants of Be and
— the evolution of the natural configuration is obtained from an implicit equation of the form 2.7 where and .
The derivation of these equations and the functional forms of λ1 and λ2 are discussed in detail by Rajagopal & Srinivasa (2000). Suffice to say that the entire structure of the constitutive equations is derivable from just two scalar functions, and—if the tensor Be is identified with the conformation tensor—the results can be extended to conformation-tensor-type theories also. Many commonly used models, such as the generalized Maxwell model and the Oldroyd-B model, and Burgers’ fluid model can be obtained within such a framework.
There is, however, one issue that remains—both the conformation-tensor theories and the changing natural-configuration theory are anchored around a specific ‘ansatz’ for the elastic response of the material—in the case of the conformation tensor approach, it is that the elastic response depends upon a specific measure of the conformation of the macromolecules, while the changing natural-configuration approach depends upon the possibility of a specific process of unloading. They also necessitate the introduction of a new variable4 whose evolution has to be provided. A phenomenological model, writing everything in terms of the stress, the velocity gradient and their rates does not require additional variables, but it does not provide a thermodynamical framework.
The question that naturally arises is whether it is possible to completely obviate the need for introducing any new variable (other than as a mathematical convenience) and yet retain a fully thermodynamical viewpoint? We will show that such an approach is possible and that it is simple and direct in its approach.
We should however bear in mind that, if by natural configurations we mean configurations which are attained on the removal of external stimuli under a certain class of process, such configurations yet exist, but it may not be necessary to invoke the concept to build a certain class of models.
(a) The equation of state: Gibbs formulation
We begin our analysis by noting that there is absolutely no reason to start with the Helmholtz potential as 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.). The main point to note is that our primary interest is in avoiding the introduction of a new strain-like variable. In order to do this, we simply switch to a Gibbs-potential-based approach since the Gibbs potential is written as a function of the stress T in contrast to the Helmholtz potential, which is a function of strain.
For finite deformations of isotropic elastic materials, if the stored energy function is a convex function of the logarithmic strain, then the Helmholtz and Gibbs formulations are connected by a Legendre transformation and hence are equivalent. For these materials, we know that 2.8 where V is the right stretch tensor and is the tensor logarithm of V. It is well known that equations of the form (2.8)2 (if invertible) imply the existence of a function G(s) such that 2.9
(b) Helmholtz and Gibbs formulations are not always equivalent
The results (2.8) and (2.9) seem to suggest that these two formulations are equivalent in all cases. But this is not so. In the case of nonlinear materials, it is not always possible to convert Helmholtz potential formulations to Gibbs-potential formulations because of certain invertibility conditions that cannot be met. For example, consider a very simple model of an elastic material that has a ‘limiting strain’. This is a commonly used idea in the modelling of viscoelastic bead-spring models (which can be represented by a combination of a spring and an inextensible cable in parallel as shown below) such as the finitely extensible nonlinearly elastic models.
The response of a model such as the one shown in figure 1 cannot be defined by means of a Helmholtz potential at all. On the other hand, a Gibbs-potential formulation can be easily implemented as follows: 2.10 where s is the stress, a and k are constants and we have used the Macaulay bracket notation . A routine calculation gives 2.11
Once again, we reiterate that there is no corresponding Helmholtz potential formulation. Similarly, there are Helmholtz potential formulations that have no equivalent Gibbs-potential formulations. The point to recognize is that both approaches together increase the repertoire of the modeller. The above example proves that the Gibbs-potential formulation can be considered as an independent starting point for many nonlinear materials, entirely independent of whether a corresponding Helmholtz potential formulation exists. The situation is acute for finitely deforming materials. Consider a material whose constitutive equation is given by 2.12 where S=σ/ϱ, D is the symmetric part of the velocity gradient, R is the rotation matrix associated with the polar decomposition of the deformation gradient and is a constant fourth-order tensor. It is well known that there is no Helmholtz potential (as a function of the deformation gradient) from which this constitutive equation can be derived. However, the above model can be shown to be perfectly feasible as a non-dissipative material based on the Gibbs-potential approach. To see this, we simply set (using the notation ) 2.13 Then, upon using equations (2.9)1 and (2.5), we see that the constitutive relation 2.14 represents a non-dissipative material (which is not Green elastic).
These results place many hypoelastic models on a sound thermodynamical footing contrary to claims that such models do not have a proper thermodynamic basis (e.g. Simo & Hughes 1998, §7.3, p. 269 and p. 271). Thus, the Gibbs-potential viewpoint provides new insight into such models and is not simply a restatement of the results that are derivable from a Helmholtz potential model.
Motivated by this, we assume that, for the viscoelastic materials we are developing, there is a Gibbs potential (or complementary energy potential) G per unit mass, which is a function of the Kirchhoff stress tensor s and the temperature θ. Invariance under Galilean transformations (which are transformations of the form x*=Qx+ct, where Q is a constant proper orthogonal tensor and c is a constant vector), immediately gives 2.15 where, without loss of generality, we can choose I1,I2 and I3 to be the moment invariants of s, defined through I1=tr(s), I2=1/2(tr(s)2) and I3=1/3(tr(s3)). Using these rather than the principal invariants allows for ease of differentiation so that explicit expressions of second derivatives can be displayed in a relatively simple manner. For future reference, we note that 2.16 where , and and are defined by 2.17
Given the Gibbs potential, we stipulate that the Helmholtz potential ψ, the internal energy ϵ and the entropy η (all per unit mass) are given, respectively, by 2.18
By substituting for ϵ in terms of G (from equation (2.182)) in the energy equation (2.3), we obtain—after some simplification and cancellation of terms—the following equation for the temperature: 2.19 where 2.20 The last of these equations, namely equation (2.20), is the most important and is the counterpart of the mechanical dissipation equation that is usually written in terms of the Helmholtz potential. Here, we obtain an expression that is directly in terms of the Gibbs potential.
Now, if we define the ‘elastic compliance tensor’ by 2.21 we get a remarkably simple form for the dissipation equation (2.20), namely that 2.22
We have hence arrived at a remarkably simple representation (2.19) for the heat equation as well for the rate of dissipation (2.22), both of which could be applicable to any material. Specifically, the terms inside the bracket on the right-hand side of equation (2.22) require close scrutiny. In words, one could say that they represent the difference between the symmetric part of the velocity gradient (a measure of the rate of deformation) and the elastic compliance times the rate of stress (which can be construed as ‘the rate of elastic deformation’). The constitutive framework is complete if we specify constitutive equations for in terms of other quantities in such a way that ensures ξ is non-negative. We turn to this aspect next.
3. Constitutive equations for isotropic viscoelastic materials
(a) Frame indifference and ‘gyroscopic terms’
It is tempting at this stage to believe that elastic response is obtained by setting since that would ensure that the rate of dissipation on the right-hand side of equation (2.22) vanishes identically. But this would be premature. To see the problem, note that while the symmetric part of the velocity gradient D and the elastic compliance are both objective tensors,5 is not an objective tensor.6
To overcome this difficulty, we will first propose a surprisingly simple way to make the quantities objective without affecting (2.22), and then provide a justification for this method.
We first observe that the Gibbs potential G is an isotropic function of s. It can then be shown by a simple calculation that the following result holds: 3.1 where αi, (i=1,2,3) are scalar functions of the invariants of s and are related to the second derivatives of G with respect to the invariants Ii. The exact expression for the αi is immaterial, but the fact that s commutes with B is important.
Now let A be any tensor and consider an expression of the form This can be rewritten (using the symmetry of ) as 3.2 where we have used the result (3.1). Thus, for any tensor A, we have 3.3
Therefore, the term is always orthogonal to the stress so that its inner product with the stress (which is the stress power) is always zero. In view of its similarity with ‘gyroscopic forces’ in dynamics, we will refer to terms of this form as ‘gyroscopic terms’.
Motivated by this observation, and in order to make sure that the stress rate is objective, we add equation (3.3) to equation (2.22), with A set to W=−WT, where W is the antisymmetric part of the velocity gradient (i.e. the spin tensor). This addition implies that the dissipation equation is now of the form 3.4 Now, the term is properly objective,7 being nothing but the Jaumann derivative of s. With this modification, we now see that the term within the square brackets on the left-hand side of equation (3.4) is objective. Thus, gyroscopic terms in the dissipation equation, while leaving the equation unaltered, are essential in enforcing objectivity of the stress rate.
One cannot dismiss this addition of terms to the energy-dissipation equations to achieve proper frame indifference as a conventional idea. To our knowledge, this idea was first explored by Rajagopal & Srinivasa (2008), who showed that such terms are necessary to obtain properly frame-indifferent thermodynamically consistent constitutive equations for fluids of the differential type.
(b) Properly invariant constitutive equations
Now that we have dealt with the issues of objectivity, we are in a position to make constitutive assumptions for the stress rate. In order to do so, it is more convenient to introduce the quantity Dp through 3.5 so that equation (3.4) can be written in compact form as 3.6 Note that the tensor Dp is objective.
It is now possible to propose thermodynamically consistent constitutive equations for the material by simply stipulating that Dp=f(s,D), where f is any isotropic tensor valued function of s and D that satisfies s⋅f(s,D)≥0.
With such an assumption, we now have a class of thermodynamically consistent constitutive equations of the form 3.7 where the notation is the Jaumann derivative of the Kirchhoff stress. In the above equation, is the second derivative of the Gibbs potential with respect to s (given by equation (2.16)) and s⋅f(s,D)≥0 is the rate of dissipation.
Here again, there are subtle and important issues in modelling that need to be taken into account. Consider the traditional procedure for developing constitutive equations for an incompressible Newtonian viscous fluid using the Clausius–Duhem inequality. For ease of illustration, let us ignore the thermal variables and consider even the purely mechanical case. We would then begin with the inequality 4.1 Now, the standard procedure of imposing constraints, as described in Truesdell (1977, p. 176 eqn (IV.7-1)) is to require that 4.2 where is a function of the history FT of the deformation gradient and N is the constraint stress that does no work in any motion of the body. Note that in this development, the determinate part is independent of N. Thus, the classical approach to constraints disallows the viscosity of an incompressible viscous fluid to depend upon the pressure. The main point to note is that this is not a thermodynamical requirement, but simply an artefact of the structure of the constitutive equations. On the other hand, if we had simply assumed a constitutive equation of the form D=f(T), there is no restriction on T and we can easily construct constitutive functions where the viscosity depends upon the pressure. It is a non-trivial matter to enforce this for viscoelastic fluids in a manner that is consistent with thermodynamics.
However, modelling incompressible materials is a very simple affair within the Gibbs-potential formalism. The main point to observe with the Gibbs-potential formulation is that it is a framework for developing constitutive equations for kinematical variables, such as the symmetric part of the velocity gradient D, in terms of the ‘external stimuli’ such as the stress tensor. Thus, within this framework, incompressibility of a material is treated as a constitutive assumption rather than as a constraint.
This point cannot be overemphasized since, from the earliest days of the development of constitutive theories for materials, much attention has been paid to the notion of constraints, their worklessness,8 the determination of constraint responses, etc. All of the problems and questions considering constraints arise from the fact that we have almost always assumed that constitutive equations are to be represented as being of the form that the stress is a functional of the history of the deformation gradient.
In actuality, for ideal gases and for linearized elastic materials, the kinematical variables do not necessarily have such a role. In both these well-developed areas, we have always switched the roles of kinematical and kinetic variables to suit our convenience. If we choose the stresses as the primary variables, then all these questions regarding constraints simply disappear—the constraint simply turns into a constitutive equation!
To model incompressible materials, let us assume that trD=0 and that the Gibbs potential GI (where the subscript ‘I’ stands for incompressible) is a function only of the deviatoric part τ of s. In view of this and the incompressibility conditions, the reduced energy-dissipation equation becomes 4.3 where is a symmetric fourth order that maps the space of deviatoric tensors onto themselves. Specifically, if we assume that G is a function of the moment invariants J2=tr(τ2) and J3=tr(τ3), then a routine calculation leads to 4.4 where we have used the notation G,i for the derivatives of G with respect to the invariant Ji, the tensors and are defined as 4.5 and is the deviatoric part of τ2.
As before, by adding the ‘gyroscopic terms’, we can obtain the following frame-indifferent form for the reduced dissipation equation 4.6 with standing for the Jaumann derivative of τ.
We can now stipulate (as with the previous compressible example) that the evolution equation is given by 4.7 where f can be any suitably chosen isotropic function of the deviatoric stress τ, the pressure (or mean normal stress p) and the symmetric part of the velocity gradient D. Note that in this approach, there is absolutely no need to stipulate that the constitutive equations be independent of the pressure (as is commonly assumed for incompressible materials). Note that since the left-hand side of the above equation is traceless, we must also require f to be traceless.
Let us begin by assuming that G is only a function of J2=τ⋅τ/2. Then, . The left-hand side of equation (4.7) thus becomes 5.1
Many commonly used models for rate-type fluids belong to this class. For example, if we set GI=−aJ2 and f=bτ+cτ2, then the constitutive equation reduces to 5.2 If we divide this by b/ϱ:=(1/2)η, and set λ=a/b and c=d/(bϱ), we obtain9 the Giesekus model.
A different class of models can be obtained by setting f=τY (p), where Y (p) is a suitably chosen positive function of the mean normal stress. For this assumption, we obtain 5.3 Models of this kind were proposed on purely phenomenological grounds by Phan-Thien & Tanner.
Yet another family of models can be obtained by choosing f=τ/Y (I2(D)), where I2(D):=D⋅D. In this case, we obtain 5.4 resulting in the Metzner–White–Denn model. Note that in all these models, the Gibbs potential is of an extremely simple form.
Furthermore, it is important to note that the rate of dissipation is not D⋅D, but is given by f(τ,p,D)⋅τ. It is this quantity that has to be non-negative for thermodynamical consistency.
6. Rationale for adding the ‘gyroscopic terms’ and extension to anisotropic materials
Having shown that the approach presented here leads to reasonable (or at least widely accepted) classes of models, we will now turn to the question of why do we need to add terms of the form to the reduced energy-dissipation equation. Before we discuss this, we hasten to add that we need not have added these terms separately to the constitutive equation. We could have simply stipulated that 6.1 Said differently, we can ‘absorb’ the ‘gyroscopic terms’ into the constitutive equation for the rate of stress itself. We did not do this in order to highlight its necessity in the constitutive formulation.
To recognize the origin of these gyroscopic terms, we begin by letting Z(t) be any rotation tensor that is objective, i.e. if we consider two motions χ and χ* that differ by a rigid-body rotation Q, then the value of Z between the two motions is related by 6.2 Examples of such objective rotation tensors include the body-rotation tensor R obtained by the polar decomposition of the deformation gradient and the ‘spatial rotation tensor’ defined by 6.3
We know that the stress tensor s is objective in the sense that if motions differ by a rigid-body rotation, then the corresponding values of s are related by 6.4 Now consider the rotated stress tensor . It is a trivial matter to verify that: (i) is an invariant tensor, i.e. changes of frame have no effect on , and (ii) the moment invariants of are the same as that of s.
Thus, we can assume that the Gibbs potential G is actually a function of and invoke material symmetry (e.g. isotropy) rather than objectivity to reduce this to a function of the invariants Ii of . This seems to be a more rational way of obtaining isotropic response in contrast to the assumption that objectivity forces G to be a function of the invariants. With this simple modification, it is now possible to model materials with anisotropic response with relative ease.
Based on this modified assumption (i.e. that G is now a function of ), one can easily show that the dissipation equation becomes 6.5 Substituting for in terms of s and grouping terms, we obtain 6.6 where is given by 6.7
We can simplify the expression for the rate of dissipation further if we note that 6.8 Using the above result, a routine calculation reveals that 6.9 Note that the term inside the brackets on the right-hand side of the above equation is automatically a proper objective derivative of the stress s. Now substituting the expression for into the left-hand side of equation (6.6), we obtain 6.10 where is a generalized objective corotational derivative and 6.11 We thus obtain a properly invariant derivative of the stress without adding gyroscopic terms to the dissipation equation. But now we note that we can have anisotropic response for the material since there is no restriction imposed by frame indifference on the dependence of G on . This approach to anisotropic materials circumvents a major stumbling block in the description of anisotropy with respect to a given configuration, and allows for a simple and natural way for introducing anisotropic response.
It is a straightforward exercise to show that when G is a function only of the invariants of , then the result above will reduce to the form (3.7).
By choosing different rotated stress tensors, we can get a plethora of models. For example, choosing Z to be the spatial rotation whose rate is related to the spin tensor, we will get the Jaumann derivative of the stress. On the other hand, if we choose Z to be the body rotation, then we will get the Green–McInnis–Naghdi derivative.
Note that by using the rotated stress tensor, not only have we eliminated the need to add ad hoc ‘gyroscopic terms’—they are obtained naturally as the result of differentiating the rotated stress tensors—but we have also been able to introduce anisotropic response in a natural way.
7. The maximum rate of dissipation hypothesis and its consequences
Until now, we have focused all our attention on the form of the Gibbs potential, while only requiring that the constitutive function f be such that f⋅s≥0. This is somewhat unsatisfactory in the sense that we have to prescribe a tensor-valued function for the constitutive equation. We can considerably reduce the complexity of prescribing constitutive equations by requiring not only that ξ=f⋅s be non-negative, but requiring further that the rate of dissipation should be maximized.
Rajagopal & Srinivasa (2004) have developed a method for obtaining constitutive equations for dissipative processes by hypothesizing that if the body is in a particular state , then the manner in which the state evolves is so as to maximize the rate of dissipation among all possible values of consistent with the energy equation and any other additional constraints on state evolution that are imposed on the system.
While this idea has been extensively explored by Rajagopal & Srinivasa (2004) in a variety of contexts, these ideas have always been considered within the context of a Helmholtz potential formulation and using strain-like variables. In view of the discussion of the non-equivalence between the Helmholtz and Gibbs formulations for finite deformations, we now consider the implications of this criterion in the context of the Gibbs-potential formulation.
Applying this maximum rate of dissipation hypothesis requires that we define a rate of dissipation function ξ as a non-negative scalar function of the state and its rate . In the present case, the state variables are the stress s (or ) and the velocity gradient L. Let us assume that the rate of dissipation depends upon these state variables and their rates only through Dp (that, from equation (3.5), is a function of s, and L). Then, satisfaction of the energy equation implies the state has to evolve in such a way that the following equation is met: 7.1 We thus have to maximize Dp subject to the condition (7.1). To this end, we introduce the auxiliary function 7.2 where the two constraints (satisfaction of the mechanical dissipation equation and the fact that only the deviatoric part of Dp is involved) are incorporated by Lagrange multipliers. By differentiating Φ with respect to Dp, we obtain that 7.3 where μ=(λ1−1)/λ1 and λ2 are obtained by satisfaction of the two constraints. By means of a routine calculation, we obtain that 7.4 Thus, in this method, we obtain an implicit constitutive equation for , as seen by combining (7.3) and (3.5). This form of the constitutive equation may not be particularly convenient for simulations of the response of these materials. It is, however, possible to obtain a more convenient equation in the following manner. We will first assume that tr ∂ξ/∂Dp is zero. Next, we will assume that ξ is only a function of Dp and possibly D. Finally, we will assume that equation (7.3), in principle, is invertible for Dp as a function of τ, i.e. that it is possible to write Dp=Dp(τ,D). Under these assumptions, we note that 7.5 Next, we consider the dissipation equation in the form 7.6 Note that in view of the fact that the equations (7.3) and (7.4) automatically satisfy (7.1), equation (7.6) is an identity and is true for all τ. If we differentiate with respect to τ and use equation (7.3), we will obtain that 7.7 where ϕ is given by 7.8 We have thus arrived at an important simplification.
The maximum rate of dissipation assumption implies that there exists a scalar dissipation potential ξ=f(τ,D) such that 7.9 In other words, we see that the function f(s,D), which was introduced earlier, is now given in terms of the derivative of a scalar dissipation function f. Thus, the entire structure of the theory can be obtained from two functions: the Gibbs potential and the dissipation function.
8. Concluding remarks
The thermodynamic framework that is presented here for viscoelastic fluid models is based on constitutive assumptions for just two scalar functions, the Gibbs potential (as a function of the stress) and the rate of dissipation (in general when thermal effects are included, the rate of entropy production) as a function of the stress and velocity gradient, together with the additional assumption that amongst all allowable processes, that which is attained maximizes the rate of dissipation. We obtain as a consequence of the framework the form for the rate of the stress tensor. Unlike approaches such as those of Leonov (1992), which need a choice for the reference configuration and elastic strains measured from such a configuration (the previous approach of Rajagopal & Srinivasa (2000) used kinematical quantities that require the specification of a reference configuration and the evolution of a natural configuration that corresponds to the configuration in the absence of external stimuli), or the introduction of additional tensors such as the ‘conformation tensor’, we are able to replicate models that are obtained by using such ideas.
The framework is fully Eulerian, with the Gibbs potential and the rate of dissipation depending on quantities measured with respect to the current configuration, and thus has special relevance to bodies such as biological bodies, where the particles of the body are not fixed owing to growth and atrophy, and hence the notion of elastic strains, etc. are ill defined, at best. The approach that is developed can also be used to generate models for anisotropic fluids without introducing directors; this has the great advantage of not having to worry about what boundary conditions ought to be prescribed with regard to the directors. The theory also leads naturally to the development of implicit models and can thus be used to describe fluids with pressure-dependent material functions
↵1 On p. 258, in the remarks following eqn (7.1.92), Simo & Hughes state ‘Given any rate constitutive eqn of the form (7.1.92), is there a stored energy function such that τ is given by (7.1.82)b? In general, the answer to this question is negative.’ It is clear that they have a Green elastic material in mind. We will show that these materials do indeed have a stored energy that can be written in terms of the stress.
↵2 Phenomenological models that are referred to as differential-type models do not exhibit stress relaxation and we shall not consider them here.
↵3 There are other types of models including integral-type models and those based on microstructural details that we are not addressing here.
↵4 For anisotropic materials, the introduction of additional variables is somewhat unavoidable since there must be some means for identifying the orientation of the material; however, the conformation tensor and the natural-configuration models need additional variables, even for the isotropic case.
↵5 The latter is an objective fourth-order tensor since it is the second derivative of an isotropic function of s with respect to s, which is objective.
↵6 However, it can be shown that is objective.
↵7 By adding other types of gyroscopic terms, we can get other objective derivatives of s.
↵8 Going back to the work of D’Alembert, Bernoulli and Lagrange, despite Gauss’s work that points to the contrary (see Rajagopal & Srinivasa (2005) and the references therein for a detailed discussion).
↵9 The division by ϱ is owing to the fact that we have defined τ=Tdev/ϱ. So the parameters will involve ϱ, which, for incompressible materials is constant.
- Received March 5, 2010.
- Accepted April 30, 2010.
- This journal is © 2010 The Royal Society