On implicit constitutive relations for materials with fading memory

Abstract

Starting with an implicit constitutive relation between the stress and relative deformation gradient histories to describe the response of a fluid, and using the assumption of fading memory, we show that both rate type and differential type fluid representations can be obtained, as approximations in retarded motions, that have the same form as the Maxwell, Oldroyd-B, Rivlin–Ericksen, and other popular fluid models. This result provides further evidence that the recently proposed implicit constitutive framework provides a very robust and general methodology to describe fluid response. Thus, fluids defined through an implicit constitutive relation between the stress and relative deformation gradient histories can be seen as an appropriate generalization of the notion of simple fluids and rate type fluids.

Introduction

The notion of a simple fluid (see Noll [1]), has played a central role in the theory of constitutive relations for describing the response of fluids. In the case of an incompressible and isotropic simple fluid, the general constitutive relation takes, after appealing to the principle of frame-indifference (see Noll [1] for details), the form

$$\mathbb{T}=-\pi \mathbb{I}+{\mathfrak{F}}_{s=0}^{+\infty }({\mathbb{C}}_t(t-s))\text{,}$$

where $\mathbb{T}$ is the Cauchy stress tensor, ${\mathbb{C}}_t(t-s)$ is the history of the relative right Cauchy–Green tensor, π is the indeterminate part of the stress due to the constraint of incompressibility and ${\mathfrak{F}}_{s=0}^{+\infty }$ is a functional¹ acting on the history of the relative Cauchy–Green tensor.

Constitutive relation (1.1) was introduced with a view towards defining a sufficiently general notion of a fluid that would include many popular fluid models as special instances. Although many incompressible fluid models such as power law-type models (see for example Carreau [3]) and differential type models (Rivlin and Ericksen [4]) are special instances of this general constitutive relation, the constitutive relation for a simple fluid is not of sufficient generality to include many rate type constitutive relations for fluids.

For example, some viscoelastic fluid models—rate type models that cannot be rewritten in an integral form—do not fit within the framework of a simple fluid. This has been observed already in the early papers dealing with the notion of simple materials (see for example the comments by Noll [1]). Widely used examples of such models are the model² introduced by Oldroyd [7]

$$\mathbb{T}=-\pi \mathbb{I}+\mathbb{S}\text{,}$$

$$\mathbb{S}+{\lambda }_1\stackrel{▿}{\mathbb{S}}+\frac{{\lambda }_3}{2}(\mathbb{DS}+\mathbb{SD})+\frac{{\lambda }_5}{2}\left(\mathrm{Tr}\mathbb{S}\right)\mathbb{D}+\frac{{\lambda }_6}{2}\left(\mathbb{S}:\mathbb{D}\right)\mathbb{I}=-\mu \left(\mathbb{D}+{\lambda }_2\stackrel{▿}{\mathbb{D}}+{\lambda }_4{\mathbb{D}}^2+\frac{{\lambda }_7}{2}(\mathbb{D}:\mathbb{D})\mathbb{I}\right)\text{,}$$

or the model

$$\mathbb{T}=-\pi \mathbb{I}+\mathbb{S}$$

$$Y\mathbb{S}+\lambda \stackrel{▿}{\mathbb{S}}+\frac{\lambda \xi }{2}(\mathbb{DS}+\mathbb{SD})=-\mu \mathbb{D}$$

$$Y={\mathrm{e}}^{-\epsilon \frac{\lambda }{\mu }\mathrm{Tr}\mathbb{S}}$$

introduced by Phan Thien [8]. Moreover, even simple models such as

$$\mathbb{T}=-\pi \mathbb{I}+2\mu (\pi \text{,}\mathbb{D})\mathbb{D}\text{,}$$

also do not fit into the framework (1.1) (see for example the discussion by Rajagopal [9], Rajagopal and Srinivasa [10]). Note that none of these models are recondite models fashioned by academics solely for the purpose of illustrating interesting constitutive features; these models are frequently used in engineering applications, for example in polymer science, geomechanics and lubrication theory.

Naturally the question arises as to whether there exists a general framework that allows one to unite rate type models such as (1.2a), (1.2b) or (1.3a), (1.3b), (1.3c) and models that fit into the framework of simple fluids (1.1). As we show below the answer is positive. The key step in establishing the answer is to do away with the usual procedure of defining the Cauchy stress in terms of the symmetric part of the velocity gradient

$$\mathbb{T}=-\pi \mathbb{I}+f(\mathbb{D})\text{,}$$

which is the underlying assumption that leads to a generalization of the type (1.1), and to replace (1.1) by an implicit relation between the histories of the stress and the deformation.

Rajagopal [9], [6] has gainfully exploited the notion of implicit constitutive relations and the fact that incompressibility implies that π, the indeterminate part of the stress due to incompressibility, coincides with the mean normal stress $p=-\frac{1}{3}\mathrm{Tr}\mathbb{T}$ to rewrite (1.4) as

$${\mathbb{T}}_{\delta }-2\mu \left(-\frac{1}{3}\mathrm{Tr}\mathbb{T}\text{,}\mathbb{D}\right)\mathbb{D}=\mathbb{0}\text{,}$$

where ${\mathbb{T}}_{\delta }{=}_{\mathrm{def}}\mathbb{T}-\frac{1}{3}\mathrm{Tr}\mathbb{T}$ denotes the traceless part of the Cauchy stress tensor. We note that (1.6) is of the form

$$f(\mathbb{T}\text{,}\mathbb{D})=0\text{.}$$

Based on this observation one can argue that the formula (1.7) is the right form of the general algebraic type constitutive model. Formulation of the constitutive relation in the form (1.7) is also convenient for expressing some standard models namely the models of the type

$$\mathbb{T}=-\pi \mathbb{I}+\mu (\mathbb{T})\mathbb{D}\text{,}$$

where $\mu (\mathbb{T})$ is given by formulae³ such as

$$\mu (\mathbb{T})={\mu }_{\infty }+({\mu }_0-{\mu }_{\infty }){\mathrm{e}}^{-\frac{|{\mathbb{T}}_{\delta }|}{{\tau }_0}}\text{,}$$

$$\mu (\mathbb{T})=\frac{A}{{\left(|{\mathbb{T}}_{\delta }{|}^2+{\tau }_0^2\right)}^{\frac{n-1}{2}}}\text{,}$$

$$\mu (\mathbb{T})=\frac{{\mu }_0}{1+\alpha |{\mathbb{T}}_{\delta }{|}^{n-1}}$$

(see for example Seely [11], Blatter [12] and Matsuhisa and Bird [13]).

Let us now write the constitutive relation in the form (1.7), and let us generalize it to the following implicit relation between the histories of the stress and relative stretch:

$${\mathfrak{H}}_{s=0}^{+\infty }(\mathbb{T}(t-s)\text{,}{\mathbb{C}}_t(t-s))=\mathbb{0}\text{.}$$

This is the same type of generalization as one that goes from (1.5) to (1.1).

If one uses (1.1) then simple “point” models—meaning models that replace the functional of the history of the Cauchy–Green tensor with a combination of functions of point values of the Cauchy–Green tensor and/or its derivatives—can be obtained by a suitable approximation of the functional relation (1.1). Such approximations have been studied for example by Coleman and Noll [2], Wang [14] and Chacon and Rivlin [15], and techniques have been developed that can be also used to approximate the functional (1.12). To find such an approximation is the aim of the present study. If we however want to obtain a frame indifferent approximation of (1.12), we cannot start directly with (1.12).

The approximation procedure will be carried out by appealing to the following steps. First, we write the constitutive relation as a relation between the stress and stretch history in a convected coordinate system, see Section 2.1. Then we approximate this constitutive relation using a variant of the technique introduced by Coleman and Noll [2] (see Section 2.2) that is based on the notions of materials with fading memory and retarded motion expansion. This will allow us to approximate the functional relation by a relation between functions of point values of the stress and stretch and their derivatives with respect to time. Then we apply (see Section 2.3) representation theorems for isotropic multilinear functions (see for example Truesdell and Noll [16]). This will significantly reduce the complexity of the approximation. Finally (see Section 2.4) we will—following Oldroyd [17]—transform the approximated relation to a fixed-in-space coordinate system and obtain an approximation of the implicit constitutive relation that will be frame indifferent. (It will include only objective time derivatives of tensor quantities.)

As a consequence of the above procedure, we will end up with the approximation that will include, as special instances, constitutive equations of type (1.2a), (1.2b), (1.3a), (1.3b), (1.3c) (see Section 2.5). The word “include” must be however interpreted carefully (see the thorough discussion at the end of Section 2.5). To indicate what type of results can be expected from the procedure describe above, let us now, without describing the necessary technical details, present a part of the main result. The result is the following. If we consider only slow processes, then the model

$$\mathbb{T}=-\pi \mathbb{I}+\mathbb{S}\text{,}$$

$$b_0(\mathrm{Tr}\mathbb{S})\mathbb{I}+b_1\mathbb{S}+2b_3\mathbb{D}+b_4(\mathrm{Tr}\stackrel{▿}{\mathbb{S}})\mathbb{I}+b_5\stackrel{▿}{\mathbb{S}}=\mathbb{0}\text{,}$$

where $\{b_i{\}}_{i=0}^5$ are constants, is a first order approximation of (1.12). (See (2.32a) for details.) In the body of the paper we also develop a second order approximation—see formulae (2.2a), (2.2b) and (2.32b).

When possible we try to follow the notation used in the original papers by Oldroyd [17] and Coleman and Noll [2]. In the part dealing with differential geometry we however tend to use the modern index-free notation for the tensors. For the reader’s convenience the meaning of some “nonstandard” symbols is summarized in Table 1.