A shape-topological control problem for nonlinear crack - defect interaction: the anti-plane variational model

TL;DR

This paper develops a variational framework for shape-topological control of crack-defect interactions in heterogeneous media, using asymptotic analysis and stress intensity factors to predict crack growth behavior.

Contribution

It introduces a novel approach combining singular perturbation theory and topological sensitivity analysis for nonlinear crack-defect interaction modeling.

Findings

Topological sensitivity of strain energy release rate derived

Stress intensity factors analyzed for micro-defects

Control strategies for crack growth proposed

Abstract

We consider the shape-topological control of a singularly perturbed variational inequality. The geometry-dependent state problem that we address in this paper concerns a heterogeneous medium with a micro-object (defect) and a macro-object (crack) modeled in 2d. The corresponding nonlinear optimization problem subject to inequality constraints at the crack is considered within a general variational framework. For the reason of asymptotic analysis, singular perturbation theory is applied resulting in the topological sensitivity of an objective function representing the release rate of the strain energy. In the vicinity of the nonlinear crack, the anti-plane strain energy release rate is expressed by means of the mode-III stress intensity factor, that is examined with respect to small defects like micro-cracks, holes, and inclusions of varying stiffness. The result of shape-topological control is useful either for arrests or rise of crack growth.