Cauchy strain tensor invariants

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... 

In Equation (1.28) function M(t - r ) is the time-dependent memory function of linear viscoelasticity, non-dimensional scalars 4>i and 4>2 and are the functions of the first invariant of Q(t - f ) and F, t t ), which are, respectively, the right Cauchy Green tensor and its inverse (called the Finger strain tensor) (Mitsoulis, 1990). The memory function is usually expressed as... 

Here we describe the strain history with the Finger strain tensor C 1(t t ) as proposed by Lodge [55] in his rubber-like liquid theory. This equation was found to describe the stress in deforming polymer melts as long as the strains are small (second strain invariant below about 3 [56] ). The permanent contribution GcC 1 (r t0) has to be added for a linear viscoelastic solid only. C 1(t t0) is the strain between the stress free state t0 and the instantaneous state t. Other strain measures or a combination of strain tensors, as discussed in detail by Larson [57], might also be appropriate and will be considered in future studies. A combination of Finger C 1(t t ) and Cauchy C(t /. ) strain tensors is known to express the finite second normal stress difference in shear, for instance. 

Any second-order tensor has a number of invariants associated with it. One such is the trace of the tensor, equal to the sum of its diagonal terms, applicable to any strain tensor. We define the first invariant h as the trace of the Cauchy-Green strain measure tr(C) ... 

In the large strain situation, we can split the deviatoric and volumetric terms 9] by redefining the deformation gradient tensor as F = Then, the right Cauchy-Green deformation tensor invariants become... 

The general and detailed constitutive relations of E.H. Lee s elastic-plastic theory at finite strain have been derived by Lubarda and Lee [5]. In this work, let the specid constitutive relations which are employed in the general purpose finite element program be listed as follows. First, the Helmholtz free energy density, E, as a function of the invariants of the elastic Cauchy-Green tensor, c/y, may be expressed as... 

The K-BKZ Theory Model. The K-BKZ model was developed in the early 1960s by two independent groups. Bernstein, Kearsley, and Zapas (70) of the National Bureau of Standards (now the National Institute of Standards and Technology) first presented the model in 1962 and published it in 1963. Kaye (71), in Cranfield, U.K., published the model in 1962, without the extensive derivations and background thermodynamics associated with the BKZ papers (82,107). Regardless of this, only the final form of the constitutive equation is of concern here. Similar to the idea of finite elasticity theory, the K-BKZ model postulates the existence of a strain potential function U Ii, I2, t). This is similar to the strain energy density function, but it depends on time and, now, the invariants are those of the relative left Cauchy-Green deformation tensor The relevant constitutive equation is... 

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. 

Similar to the idea of finite elasticity theory, the K-BKZ model postulates the existence of a strain potential function U(Ji, I2, i). This is similar to the strain energy density function, but it depends on time and, now, the invariants are those of the relative left Cauchy-Green deformation tensor The relevant constitutive equation is... 

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