Multidimensional rank-one convexification of incremental damage models at finite strains

TL;DR

This paper introduces a computationally feasible rank-one convexification method for incremental damage models at finite strains, improving numerical stability and mesh independence in simulations of damage evolution.

Contribution

It develops novel algorithms for rank-one relaxation of nonconvex stress potentials, enabling efficient and stable simulations of damage models in multiple dimensions.

Findings

The relaxed model captures softening effects accurately.

Solutions are mesh independent.

The approach prevents non-existence of minimizers.

Abstract

This paper presents computationally feasible rank-one relaxation algorithms for the efficient simulation of a time-incremental damage model with nonconvex incremental stress potentials in multiple spatial dimensions. While the standard model suffers from numerical issues due to the lack of convexity, the relaxation by rank-one convexification prevents non-existence of minimizers and mesh dependence of the solutions of finite element discretizations. By the combination, modification and parallelization of the underlying convexification algorithms, the novel approach becomes computationally feasible. A descent method and a Newton scheme enhanced by step-size control prevent stability issues related to local minima in the energy landscape and the computation of derivatives. Numerical techniques for the construction of continuous derivatives of the approximated rank-one convex envelope are discussed. A series of numerical experiments demonstrates the ability of the computationally relaxed model to capture softening effects and the mesh independence of the computed approximations. An interpretation in terms of microstructural damage evolution is given, based on the rank-one lamination process.