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

摘要

We consider the shape-topological control of a singularly perturbedvariational inequality. The geometry-dependent state problem that we address inthis paper concerns a heterogeneous medium with a micro-object (defect) and amacro-object (crack) modeled in 2d. The corresponding nonlinear optimization problem subject to inequalityconstraints at the crack is considered within a general variational framework.For the reason of asymptotic analysis, singular perturbation theory is appliedresulting in the topological sensitivity of an objective function representingthe 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-IIIstress intensity factor, that is examined with respect to small defects likemicro-cracks, holes, and inclusions of varying stiffness. The result ofshape-topological control is useful either for arrests or rise of crack growth.