On determining invariants of a three-dimensional elasticity tensor in anisotropic media

Abstract

Invariants of elasticity tensors embody the most important mechanical properties of materials, effectively generalizing and extending the traditional concept of "spring stiffness" to much more complex, multidimensional systems. Unlike simple springs, where the stiffness is characterized by a single constant, elasticity tensors allow us to describe complex interactions in materials with anisotropic properties, including materials with different types of symmetry. These invariants are a powerful tool for analyzing and classifying mechanical properties, since they allow us to consider both linear and nonlinear responses of materials to external influences. In this paper, we use the generalized Kelvin representation to parameterize the stress tensor, which significantly simplifies and makes more visual the process of determining the influence of various factors on the elasticity tensor. In this study, we calculate the invariants of the anisotropic elasticity tensor, taking into account the rotational symmetry defined by the SO(3) group. As a result of the analysis, we identified 18 independent invariants, including 5 first-order invariants, especially relevant for isotropic materials, and 13 higher-order invariants. These results highlight the importance and complexity of studying anisotropic materials and open up new perspectives for their mechanical interpretation and classification.