Structure of anisotropic bi-dimensional elasticity tensors in micromorphic and related higher-order media

Abstract

This paper addresses the classification of constitutive tensors in bi-dimensional micromorphic elasticity, a higher-order continuum theory that extends Cauchy elasticity by introducing an additional degree of freedom in the form of a microdeformation tensor. We systematically investigate the impact of material symmetries on the forms of the constitutive elasticity tensors, reducing the number of independent parameters in the micromorphic energy. The analysis includes the first-order, second-order, and coupling elastic energies, with results presented in terms of symmetry classes and explicit matrix forms. Additionally, analogous results are derived for related higher-order theories, including the microdilatation, micropolar, microstretch, and microstrain theories.