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.1991 Mathematics Subject Classification. 35B25, 49J40, 49Q10, 74G70.
A major drawback of the study of cracks within the context of the linearized theory of elasticity is the inconsistency that one obtains with regard to the strain at a crack tip, namely it becoming infinite. In this paper we consider the problem within the context of an elastic body that exhibits limiting small strain wherein we are not faced with such an inconsistency. We introduce the concept of a non-smooth viscosity solution which is described by generalized variational inequalities and coincides with the weak solution in the smooth case. The well-posedness is proved by the construction of an approximation problem using elliptic regularization and penalization techniques.
The ability of velocity methods to describe changes of topology by creating defects like holes is investigated. For the shape optimization energy-type objective functions are considered, which depend on the geometry by means of state variables. The state system is represented by abstract, quadratic, constrained minimization problems stated over domains with defects. The velocity method provides the shape derivative of the objective function due to finite variations of a defect. Sufficient conditions are derived which allow us to pass the shape derivative to the limit with respect to diminishing defect, thus, to obtain the "topological derivative" of the objective function due to a topology change. An illustrative example is presented for a circular hole bored at the tip of a crack.
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