Material symmetry rotation for oblique 2D-elasticity

Abstract

Definition of the material symmetry axes rotation in case of purely elastic finite-strain deformation is given. This rotation explains the rigid motion of the orthonormal vector frame, which coincides with the material symmetry axes in the initial configuration of the continuum body and uniquely corresponds to these axes in the actual configuration, although it does not consist with them due to the distortion of the angles between the actual material fibers. The reloaded configuration of the local material volume is found using the minimizing principle of the summary length of the material elements’ paths under releasing. The obtained exact expression for the material symmetry axes rotation includes the deformation gradient tensor, unit vectors and axial parameters of the initial material anisotropy axes. This solution allows obtaining a new variant of decomposing any elastic finite-strain motion onto deformational and rigid parts and introducing the material symmetry-based corotational rate. The latter is used for the formulation of the anisotropic ratetype elastic law in the actual configuration. For the tetragonal symmetry, the introduced material rotation tensor coincides with the proper orthogonal tensor of rotation from the polar decomposition of the deformation gradient tensor.