Cauchy tensor

For uniaxial deformation the Finger and Cauchy tensors read... 

The Cauchy tensor is a device for describing the change in length at any point of the material body. Equation (5.29) indicates that in performing such a function the Cauchy tensor operates on the unit vector u at some time t in the past. However, the stress tensor is always measured with respect to the present form of the material body or the deformed state. Thus, there is the need to find a deformation tensor that operates on the unit vector at the present time. This can be done by using the inverse of E, E to express dX in terms of dX, namely. 

In Chapter 5, we defined the deformation gradient tensor E, Cauchy tensor C, and Finger tensor B, respectively, as... 

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... 

In this case inversion of the Cauchy tensor, which is necessary when deriving the Finger tensor, is trivial. For non-orthogonal deformations this is more complicated, and here one can make use of a direct expression for the components of B which reads... 

Next, using the relative Cauchy tensor, we take d/dr and then set r = leading to... 

The particle paths allow one to calculate the relative Cauchy tensor which leads to the stress tensor... 

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. 

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. 

Wagner s Eq. 10.10 is special case of a more general integral model of Rivlin and Sawyer, which is, in turn, a generalization called the K-BKZ model [7]. In the K-BKZ model, the strain is described by a linear combination of the Finger and Cauchy tensors, and as a result it is possible to accommodate a non-zero second normal stress difference. ... 

Another special case of the Rivlin-Sawyer model that is a generalization of Eq. 10.10 was proposed by Wagner and Demarmels [12] who added a dependency on the Cauchy tensor to Eq. 10.10 to provide for a non-zero second normal stress difference and a better fit to data for planar extension, which is defined in Section 10.9. Their model is shown as Eq. 10.11. 

Other possibilities exist to solve the frame invariant problem Cauchy-Maxwell equation uses the Cauchy tensor, C, which is also independent of the system of reference, the Lodge rubber-like liquid model uses the Finger tensor but contrarily to the Lodge model, it uses a generalized memory function ... 

Example 10.4 Obtain expressions for the deformation gradient and Cauchy tensors for the shear deformation illustrated in Figure 10.9. Here, the only nonzero velocity component is Uj, and it equals yx2, where y is the constant shear rate. 

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