Fingers Constitutive Equation

As any second rank tensor, the Cauchy strain tensor possesses three invariants. These are expressions in terms of the tensor components Cii which remain invariant under all rotations of the coordinate system. The three invariants of the Cauchy strain tensor are given by the following expressions 

There are different choices for the invariants since any combination gives new invariant expressions but the most common is the one cited here. As the free energy density depends on the local strain only and, being a scalar quantity, must be invariant under all rotations of the coordinate system, one can readily assume for the free energy density a functional dependence 

We now have the ingredients to formulate Finger s constitutive equation. It relates the Cauchy strain tensor to the stress tensor in the form 

C denotes the inverse of the Cauchy tensor, 1 is the unit tensor. c i,co,ci describe functions of the three invariants, and these are directly related to the free energy density. The relations are 

Finger derived this equation on the basis of general arguments. As we see, it provides us with a powerful tool Once one succeeds in determining the strain dependence of the free energy of a body, the stresses produced in all kinds of deformations can be predicted. 

There is an alternative form of Finger s equation, which gives us a choice and is, indeed, to be preferred when dealing with rubbers. We introduce the Finger strain tensor B, being defined as the reciprocal of the Cauchy strain tensor 

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A general single integral constitutive equation results if the Boltzmann superposition principle is applied to a nonrspecified tensor functional, of the macroscopic strain, represented by the Finger... 

The substitution of C by B implies, as the main point, that one is now choosing the invariants of B as independent variables, rather than those associated with C. The second form of Finger s constitutive equation is... 

The same modification is necessary when dealing in general with incompressible hyperelastic bodies. Introducing the additional term in Finger s constitutive equation, Eq. (7.63), and regarding the absence of Ills we obtain... 

Due to the unknown hydrostatic pressure, p, the individual normal stresses an are indeterminate. However, the normal stress differences are well-defined. Let us consider the difference between Gzz and Gxx- Insertion of the Finger strain tensor associated with uniaxial deformations, Eq. (7.80), in the constitutive equation (7.74) yields... 

Length, area, and volume change can also be expressed in terms of the invariants of B or C (see eqs. 1.4.45-1.4.47). Note that the Cauchy tensor operates on unit vectors that are defined in the past state. In the next section we will see that the Cauchy tensor is not as useful as the Finger tensor for describing the stress response at large strain for an elastic solid. But first we illustrate each tensor in Example 1.4.2. This example is particularly important because we will use the results direcfiy in the next section with our neo-Hookean constitutive equation. 

Since it is often simpler to write the Finger deformation tensor, and since it only differs fiom the strain tensor by unity, we usually write constitutive equations in terms of B. 

Whether to use the first or the second form of Finger s constitutive equation is just a matter of convenience, depending on the expression obtained for the free energy density in terms of the one or the other set of invariants. For the system under discussion, a body of rubbery material, the choice is clear The free energy density of an ideal rubber is most simply expressed when using the invariants of the Finger strain tensor. Equation (9.22), giving the result of the statistical mechanical treatment of the fixed junction model, exactly corresponds to... 

We have shown here that the Cauchy-Green and Finger tensors are not equivalent measures of finite strain, which is a very important fact to remember in the formulation of constitutive equations, as is discussed in Chapter 3. 

A primitive model of nonlinear behavior can be obtained by simply replacing the infinitesimal strain tensor in Eq. 10.3 by a tensor that can describe finite strain. However, there is no unique way to do this, because there are a number of tensors that can describe the configuration of a material element at one time relative to that at another time. In this book we will make use of the Finger and Cauchy tensors, B and C, respectively, which have been found to be most useful in describing nonlinear viscoelasticity. We note that the Finger tensor is the inverse of the Cauchy tensor, i.e., B = C. A strain tensor that appears in constitutive equations derived from tube models is the Doi-Edwards tensor Q, which is defined below and used in Chapter 11. The definitions of these tensors and their components for shear and uniaxial extension are given in Appendix B. 

Here h(/i, 12), the damping function , is a function of the invariants of the Finger strain tensor given in equations (31) and (32) the damping function is determined by requiring the constitutive equation to describe shear and elongational flow data. Extensive comparisons with experimental data show that this rather simple empiricism is extremely useful. Equation (47) gives a value of zero for the second normal stress coefficient. 

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