A commutative-symmetrical multiplicative decomposition of left and right stretch tensors. (July 2018) - Heiduschke, Klaus
Abstract: Symmetric positive definite left or right stretch tensors are decomposed multiplicatively into Eulerian or Lagrangean triple tensor products of symmetrizing rotations in the middle between two symmetric positive definite partial stretches. The proper orthogonal rotation tensors in the middle are determined from the symmetry conditions of the whole triple tensor products. The substitutions of the symmetrizing rotation tensors yield two commutative-symmetrical partial-stretch tensor products, which are (isotropic tensor) functions of the partial-stretch tensors of either proper Eulerian type defined with respect to a present configuration or proper Lagrangean type defined with respect to a reference configuration and which are equal to the symmetric total stretch tensors. Commutative-symmetrical partial-stretch tensor products do not rely on intermediate (stress-free) configurations. The eigenbase vector orientations of their proper Eulerian or proper Lagrangean multiplicative-elastic stretch tensors are well-defined, which is essential in order to model constitutive equations properly. Finite material orthotropy can be modeled simultaneously for both constituents and without the interference of their deformation-induced anisotropies when the partial-stretch tensors of the Lagrangean commutative-symmetrical products are defined with respect to the same reference configuration of orthotropy. The commutative-symmetrical partial-stretch tensor products are applicable to the constitutive modeling of finite anisotropy, and they constitute a novel approach to the kinematics of multiplicatively coupled total and partial stretch tensors.