Green strain tensor

The right Cauchy-Green strain tensor corresponding to this deformation gradient is thus expressed as... 

Another generalization uses referential (material) symmetric Piola-Kirchhoff stress and Green strain tensors in place of the stress and strain tensors used in the small deformation theory. These tensors have components relative to a fixed reference configuration, and the theory of Section 5.2 carries over intact when small deformation quantities are replaced by their referential counterparts. The referential formulation has the advantage that tensor components do not change with relative rotation between the coordinate frame and the material, and it is relatively easy to construct specific constitutive functions for specific materials, even when they are anisotropic. 

The pressure P0 represents the arbitrary additive contribution to the normal components of stress in an incompressible system, 8i is the Kronecker delta, C[ j 1(t t) is the inverse of the Cauchy-Green strain tensor for the configuration of material at t with respect to the configuration at the current time t [a description of the motion (221)], and M(t) is the junction age distribution or memory function of the fluid. 

Components of the Cauchy-Green strain tensor and its inverse, with the current configuration as the reference configuration. 

The Green strain tensor, e°a, and the Almansi strain tensor, ea, are given by... 

A typical choice to model compressible materials is to decompose the left Cauchy-Green strain tensor into a pure isochoric and a pure volumetric part [110, 111]... 

Here C is the right-Cauch-Green strain tensor and is the elastic strain... 

Here, C represents the history of the right-Green strain tensor up to time t, i.e.,... 

Hence, the right-Cauchy-Green strain tensor reads as C(t)... 

It can be easily shown that the Cauchy-Green strain tensor also transforms like this. 

In the above two equations, as well as in the rest of the equations in this section, subscripts 1, 2, and 3 indicate x, y, and z directions, respectively. The deformation tensor and its transpose can be combined to yield the right relative Cauchy-Green strain tensor, C, with components... 

The components of the Cauchy-Green strain tensor, C, are calculated from Eq. 6.85. The only nonzero components are... 

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