Rivlin-Ericksen tensor

Rivlin-Ericksen tensor of order n, for a viscoelastic liquid or solid in homogeneous deformation, is the nth time derivative of the Cauchy strain tensor at reference time, t. Note 1 For an inhomogeneous deformation the material derivatives have to be used. 

It is of interest to note that the relative Cauchy-Green tensor C,(r ) can be expressed in terms of the Rivlin-Ericksen tensors A by (Coleman 1962 Rivlin and Ericksen 1955)... 

Here, we present constitutive equations that do not fall within the categories of differential- and integral-type, presented in the preceding sections. For the reason of historical importance, we summarize the constitutive equations of Rivlin and Ericksen (1955) and Coleman and Noll (1959, 1960, 1961a). Because the stress tensor a in the constitutive equations is expressed in terms of the Rivlin-Ericksen tensor A. ,( = 1,2,...) (see Chapter 2), we shall refer them to as rate-type constitutive... 

For a slow flow, E(y) defined by Eq. (3.73) can be expressed as a series of the Rivlin-Ericksen tensors (Coleman and Noll 1961b) ... 

The term rate-type is used for the reason that the Rivlin-Ericksen tensors for n>2 contain the time derivative term, although such classification may not be justified from a rigorous viewpoint. 

Ericksen tensors in this way leads to the so-called Rivlin-Ericksen (RE) fluid. Part of the complexity of the RE constitutive equation (say, staying with the case of A[ and A2 dependency, only) is due to keeping the full representation of the stress tensor as an isotropic tensor-valued function in A and A2. It is possible to simplify this relation by considering an alternative approach. Now one essentially looks at the problem as a perturbation expansion for slow flows. Thus, at rest, the stress tensor is given by the isotropic hydrostatic pressure only. The first order correction includes an additional term proportional to /4, which gives us the Newtonian fluid. At second order, we would include the square terms only, viz. 

Rivlin and Ericksen [109] showed that a scalar-valued function of a symmetric tensor G is isotropic if and only if it is expressible as a function of Ii(G), 12(G) and 13(G). 

The Cayley-Hamilton theorem has been widely used in the formulation of constitutive equations (Rivlin and Ericksen 1955). As an example, let the stress tensor a be expressed as a function of powers of the rate-of-deformation tensor d ... 

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