# Micromechanics based framework with second-order damage tensors

## TL;DR

This paper introduces a micromechanics framework utilizing second-order damage tensors derived from harmonic factorization, enabling efficient modeling of damage in 3D structures with improved sensitivity analysis.

## Contribution

It develops a novel micromechanics approach based on second-order damage tensors using harmonic factorization, simplifying damage modeling in 3D structures.

## Key findings

- Harmonic factorization allows exact harmonic square representation in 2D and accessible directions in 3D.
- The framework models hydrostatic sensitivity effectively at high damage levels.
- Application example demonstrates the framework's capability in damage analysis.

## Abstract

The harmonic product of tensors---leading to the concept of harmonic factorization---has been defined in a previous work (Olive et al, 2017). In the practical case of 3D crack density measurements on thin or thick walled structures, this mathematical tool allows us to factorize the harmonic (irreducible) part of the fourth-order damage tensor as an harmonic square: an exact harmonic square in 2D, an harmonic square over the set of so-called mechanically accessible directions for measurements in the 3D case. The corresponding micro-mechanics framework based on second---instead of fourth---order damage tensors is derived. An illustrating example is provided showing how the proposed framework allows for the modeling of the so-called hydrostatic sensitivity up to high damage levels.