Towards a phase-field model for thin structures: a coarse-grained constitutive law for brittle fracture of beams

TL;DR

This paper develops a new coarse-grained constitutive law for beams with non-symmetric cracks, capturing the coupling between axial and bending strains, to improve phase-field models for thin structures.

Contribution

It introduces a homogenized 1D elastic energy model that accounts for crack asymmetry and strain coupling, advancing phase-field modeling of brittle fracture in beams.

Findings

Derived a homogenized 1D elastic energy coupling axial and bending strains.

Identified how constitutive parameters depend on crack depth.

Provided a foundation for phase-field gradient models for thin structures.

Abstract

Damage gradient models approximate fracture mechanics using a modulation of the material stiffness. To this aim a single scalar field, the damage, is used to degrade as a whole the elastic energy. If applied to the structural models of beams and shells, where the elastic energy is the sum of the stretching and bending contributions, a similar approach is not able to capture some important features. For instance, the coupling between axial and bending strains induced by cracks non-symmetric with respect to the center line is completely missed. In this contribution, we deduce a constitutive law for a beam having a crack non-symmetric with respect to the center line. To achieve this, we perform an asymptotic coarse-grained procedure from a 2D problem, using a sharp interface model. We deduce a homogenized 1D elastic energy, coupling the axial and bending strains, the constitutive parameters of which depend in a different way on the crack depth and we state how precisely they do. This should pave the way to a rational development of a phase-field gradient model for thin structures.