On the direct and reverse multiplicative decompositions of deformation gradient in nonlinear anisotropic anelasticity.

In this paper we discuss nonlinear anisotropic anelasticity formulated based on the two multiplicative decompositions F = F e F a and F = F a F e . Using the Bilby–Kröner–Lee decomposition F = F e F a one can define a Riemannian material manifold (the natural configuration of an anelastic body) whose metric explicitly depends on the anelastic deformation F a. We call this the global material intermediate configuration. Deformation is a map from this Riemannian manifold to the flat ambient space. Using the reverse decomposition F = F a F e , the reference configuration is a (flat) submanifold of the Euclidean ambient space, while the global intermediate configuration is a Riemannian manifold whose metric explicitly depends on the elastic deformation F e. We call this the global spatial intermediate configuration. We show that the direct F = F e F a and reverse F = F a F e decompositions correspond to the same anelastic motion if and only if F e and F e are equal up to local isometries of the reference configuration. We discuss the constitutive equations of anisotropic anelastic solids in terms of both intermediate configurations. It is shown that the two descriptions of anelasticity are equivalent in the sense that the Cauchy stresses calculated using them are identical. We note that, unlike isotropic solids, for an anisotropic solid the material metric is not sufficient for describing the constitutive behavior of the solid; the energy function explicitly depends on F a (or F a) through the structural tensors. [ABSTRACT FROM AUTHOR]