Three-dimensional strain and stress orthogonal decompositions via an elastic energy preserving transformation

Abstract

Motivated by the recent development of phase field methods for modeling and simulating the fracture of brittle and quasi-brittle materials, a general approach is proposed to decompose the infinitesimal strain tensor (or Cauchy stress tensor) into a positive part and a negative part which are orthogonal in the sense of an inner product where the elastic stiffness (or compliance) tensor acts as a metric. This approach is based on a strain (or stress) transformation preserving the elastic energy, and gives rise to a generalized Pythagorean theorem. It is valid not only for the isotropic elastic case but also for all the anisotropic ones. Decompositions of the strain (or stress) tensor are given in a coordinate-free way. The main results obtained are illustrated and detailed for the elastic isotropic case in an analytical explicit manner.