Closed-Form Coordinate-Free Decompositions of the Two-Dimensional Strain and Stress for Modeling Tension–Compression Dissymmetry

The modeling of the different mechanical behaviors of brittle and quasi-brittle materials in tension and compression leads to partitioning of the strain (or stress) tensor into a positive part and a negative part. In this study, applying a recently proposed general method to the two-dimensional (2D) strain and stress tensors, closed-form coordinate-free expressions are obtained for their decompositions which are orthogonal in the sense of an inner product where the forth-order elastic stiffness or compliance acts as a metric. The orthogonal decompositions are given analytically and explicitly for all possible 2D elastic symmetries, i.e., isotropic, orthotropic, square, and totally anisotropic elastic materials. These results can be directly used, for example, in developing phase field methods for modeling and simulating the fracture of isotropic and anisotropic brittle and quasi-brittle materials.