Scalar, vectorial and tensorial damage parameters from the mesoscopic background

TL;DR

This paper introduces scalar, vectorial, and tensorial damage parameters derived from a mesoscopic crack distribution, linking them to free energy density and demonstrating their reduction in specific geometries.

Contribution

It defines new damage parameters based on crack distributions and relates them to free energy, providing a unified mesoscopic damage framework.

Findings

Damage parameters are derived from crack distributions.

Free energy density is expressed as a function of damage parameters.

Reductions to uniaxial and specific geometries are demonstrated.

Abstract

In the mesoscopic theory a distribution of different crack sizes and crack orientations is introduced. A scalar damage parameter, a second order damage tensor and a vectorial damage parameter are defined in terms of this distribution function. As an example of a constitutive quantity the free energy density is given as a function of the damage tensor. This equation is reduced in the uniaxial case to a function of the damage vector and in case of a special geometry to a function of the scalar damage parameter.