A thermodynamic frame work for rate type fluid models

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Abstract

In this paper, we develop a thermodynamic approach for modeling a class of viscoelastic fluids based on the notion of an `evolving natural configuration'. The material has a family of elastic (or non-dissipative) responses governed by a stored energy function that is parametrized by the `natural configurations'. Changes in the current natural configuration result in dissipative behavior that is determined by a rate of dissipation function. Specifically, we assume that the material possesses an infinity of possible natural (or stress-free) configurations. The way in which the current natural configuration changes is determined by a `maximum rate of dissipation' criterion subject to the constraint that the difference between the stress power and the rate of change of the stored energy is equal to the rate of dissipation. By choosing different forms for the stored energy function ψ and the rate of dissipation function ξ, a whole plethora of energetically consistent rate type models can be developed. We show that the choice of a neo-Hookean type stored energy function and a rate of dissipation function that is quadratic, leads to a Maxwell-like fluid response. By using this procedure with a different choice for the rate of dissipation, we also derive a model that is similar to the Oldroyd-B model. We also discuss several limiting cases, including the limit of small elastic strains, but arbitrarily large total strains, which leads to the classical upper convected Maxwell model as well as the Oldroyd-B model.

Introduction

In his celebrated paper on the dynamical theory of gases in 1866, Maxwell [1] discussed a mathematical model to describe the elastic and viscous behavior exhibited by gases. He recognized the notion of stress-relaxation and introduced the concept of `time of relaxation' for such materials. His model is based on the superposition of elastic and viscous response that can be thought of in mechanical analogy as a linear spring and a linearly viscous dashpot in series, with the strains in the spring and dashpot being additive, though Maxwell himself does not mention this explicitly. The one-dimensional model due to Maxwell leads to a linear differential equation for the stress. A little later, Boltzmann [2] also developed a linear one-dimensional model for describing the viscoelastic response of materials. Over five decades later, Jeffreys [3] developed another linear one-dimensional model for the viscoelastic response of a suspension of elastic spheres in a linearly viscous fluid.

Following the seminal work of Maxwell, Boltzmann, and Jeffreys, numerous models have been developed to describe the viscoelastic response of materials based on mechanical analogs. These models can be categorized as materials of the rate type. Early studies in the development of models in viscoelasticity were restricted to one-dimensional models. Furthermore, they were based on the mechanistic models of springs and dashpots that represent the co-existence of the two at a material point which in turn supposes an additive elastic and viscous response at each material point. Banding together such springs and dashpots in series and parallel leads to one-dimensional models of all manners and forms. All such models lead to rate type models (or equivalent integral models) with the possibility of a spectrum of relaxation times for the material (see e.g., Bland [4]).

Oldroyd [5] was the first to develop a systematic framework for the rheological behavior of rate type viscoelastic fluids. This original and remarkable study, recognized the restrictions imposed by frame invariance, introduced convective derivatives of the appropriate physical quantities to obtain properly frame invariant constitutive relations, incorporated the idea that the current state of stress in a body can depend on the history of deformation of the body, and even provided explicit formulae for computing the evolution of the material symmetry due to deformation (with regard to physical constants that appear in the constitutive equations) — all within the context of a fully three-dimensional frame work.

Since then, a plethora of frame indifferent rate type models have been suggested based on generalizations of one-dimensional models. As numerous frame-invariant time derivatives can be introduced, e.g., upper and lower convected derivatives, Jaumann derivative, Oldroyd derivative etc. (see Truesdell and Noll [6]), it is possible to generalize a one-dimensional model in more than one way. The introduction of the frame-invariant time derivatives leads to non-linear models. Thus, while the one-dimensional Maxwell model is linear, the frame-invariant three-dimensional model is non-linear.

Here, we discuss a thermodynamical framework that leads to a rational methodology for developing constitutive relations for the viscoelastic response of materials. The framework that is used, provides a unified basis to study a wide class of material response such as traditional plasticity 7, 8, twinning [9], solid to solid phase transition [10], multi-network theory [11], and includes, as special subcases, classical elasticity, the classical linearly viscous fluid and viscoelasticity1. Moreover, this general framework also allows one to develop models for anisotropic liquids.

There are numerous works in inelasticity and viscoelasticity in which internal variables (based on analogies with classical small deformation plasticity) are introduced. The papers by Leonov [12] and Mattos [13] are good representative examples of such approaches. However, our philosophy and framework are quite distinct, though there are certain shared features. We refer the readers to our papers 7, 8 for a detailed discussion of these issues.

The central feature of the general framework is that the body possesses numerous `natural configurations'.2 The response of the material is `elastic' from these natural configurations, and the rate of dissipation determines how these natural configurations evolve. The viscoelastic response is thus determined by a stored energy function that characterizes the elastic response from the `natural configuration' and a rate of dissipation function that describes the rate of dissipation due to the viscous effects. One way of describing the evolution of the `natural configurations' is by requiring that the rate of dissipation function is maximized. This is by no means a fundamental principle and there are other possible prescriptions depending upon the class of allowable processes. By changing the forms of the stored energy and the rate of dissipation functions, we can develop models that can describe various types of response with or without the requirement of maximization of the rate of dissipation. Here, we choose to require the maximization of the rate of dissipation.

After developing the general theory in 2 Kinematics, 3 Constitutive assumptionsSection 4.1, we first formulate in Section 4.2, a model which reflects a neo-Hookean elastic response from the current natural configuration and a dissipative response that is quadratic in the stretching tensor associated with the current natural configuration (see , ). Such a choice for the stored energy and rate of dissipation function, and the requirement that the rate of dissipation be maximized, leads to a model that is very similar to a Maxwell model.

In Section 5we discuss several limiting cases of the above model to illustrate some of its features. Linearization of the proposed model without restrictions on the total strain, under the assumption of small elastic strain, leads to the classical upper convected Maxwell model.

Section 6is devoted to the study of the steady plane Poiseuille problem for the above model. In order to bring out the efficacy of the present framework, we develop two other examples in Section 7by choosing different forms for the rate of dissipation function. The first example leads to a variant of the Maxwell-like model while the second leads to a Oldroyd-B type model which, under conditions of small elastic strains, reduces to the Oldroyd-B model. We end with some remarks concerning the modeling of anisotropic fluids.

One of the principal features of the present approach is that, since we are not generalizing from any one-dimensional model, there is no need to choose a priori a particular frame invariant rate. Instead, the form of the constitutive equations (including the rates that appear) is dictated by the choice of two scalar functions, namely the stored energy function and the rate of dissipation function.

Section snippets

Kinematics

As we mentioned earlier, we are interested in modeling viscoelastic fluids which exhibit instantaneous elastic response. Let κR denote the reference configuration and κt denote the current configuration of the viscoelastic fluid of interest (see Fig. 1). The motion χκR of the fluid is a mapping that at time t assigns to each position in a reference configuration a corresponding position in the current configuration, i.e.,x≔χκR(XκR,t)The deformation gradient FκR is defined throughFκR∂χκRXκR.

Constitutive assumptions

Let W denote the stored energy and ξ the rate of dissipation associated with the material. In an isothermal process, the rate of dissipation ξ is given by (see [6])ξ=T·DẆ,where T·D is usually referred to as the stress power.

We shall make constitutive assumptions for the stored energy W, and rate of dissipation ξ for the viscoelastic fluid. We assume thatW=Ŵ(I,II),andξ=ξ̂Bκp(t),Dκp(t),whereI=TrBκp(t),II=TrB2κp(t).We shall also stipulate that ξ̂,0)=0.

We see that the above assumptions are

General results

On differentiating Eq. (21)with respect to time we obtainẆ=α011Bκp(t)·Ḃκp(t),where α0=(∂Ŵ/∂I) and α1=2(∂Ŵ/∂II). Next, on substituting Eq. (14)into Eq. (27)and using , , we obtainẆ=2α0Bκp(t)1B2κp(t)·DDκp(t).

It follows from , thatT−2α0Bκp(t)1B2κp(t)·D+2α0Bκp(t)1B2κp(t)·Dκp(t)=ξ̂Bκp(t),Dκp(t).Motivated by the fact that the second term on the left-hand side and the term on the right-hand side of the above equation are independent of D and since only isochoric motions are possible, we

Some limiting cases

It is generally accepted that viscoelastic materials show response that is intermediate between that of classical elastic solids and viscous fluids. In the current approach, this idea is made explicit by the use of an elastic response that is like that of a neo-Hookean solid and a dissipative response that is akin to a Newtonian fluid. This idea will be made even more transparent if we consider certain limiting cases. In particular, we shall show that the above representation reduces to that of

Plane Poiseuille flow

Returning once more to the `full' model (, , ), we illustrate the effect of the `stretch dependent viscosity' by considering the problem of plane Poiseuille flow between parallel plates, separated by a distance 2h (see Fig. 2).

Other fluid models

To illustrate the kinds of models that one can generate by this procedure, we shall consider two cases. In both the examples that follow, we shall retain the strain energy function (33). The first example results in a `variant' of the Maxwell model. The second example is one where we illustrate a derivation using a `mixture-like' approach within the current framework, so that the stress response is composed of two terms – the expression for the stress in the Maxwell-like model developed in

Remarks on anisotropic fluids

According to the classification of Noll [19], a simple fluid by definition is an isotropic material. While some isotropic simple fluids can be cast into the framework presented here, the structure allows us to consider more general non-simple fluids, namely fluids that are anisotropic. In the present framework, this anisotropy can be brought into play in two ways, by making the instantaneous elastic response anisotropic or the dissipative response anisotropic. The former is achieved by

Acknowledgements

The authors thank one of the reviewers for bringing to their attention the relationship of this approach to theories based on the conformation tensors. The authors acknowledge the support of the National Science Foundation in the performance of the research. One of us (ARS) would also like to acknowledge the support of the NIST ATP.

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