Almansi strain tensor

The referential formulation is translated into an equivalent current spatial description in terms of the Cauchy stress tensor and Almansi strain tensor, which have components relative to the current spatial configuration. The spatial constitutive equations take a form similar to the referential equations, but the moduli and elastic limit functions depend on the deformation, showing effects that have misleadingly been called strain-induced hardening and anisotropy. Since the components of spatial tensors change with relative rigid rotation between the coordinate frame and the material, it is relatively difficult to construct specific constitutive functions to represent particular materials. 

The referential constitutive equations for an inelastic material may be set into spatial terms. Casey and Naghdi [14] did so for their special case of finite deformation rigid plasticity discussed by Casey [15], Using the spatial (Almansi) strain tensor e and the relationships of the Appendix, it is possible to do so for the full inelastic referential constitutive equations of Section 5.4.2. 

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

For a geometrically nonlinear analysis, the vector e contains components of the Almansi strain tensor. 

In order to consider the inelastic stress rate relation (5.111), some assumptions must be made about the properties of the set of internal state variables k. With the back stress discussed in Section 5.3 in mind, it will be assumed that k represents a single second-order tensor which is indifferent, i.e., it transforms under (A.50) like the Cauchy stress or the Almansi strain. Like the stress, k is not indifferent, but the Jaumann rate of k, defined in a manner analogous to (A.69), is. With these assumptions, precisely the same arguments... 

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