A Shape-Topological Control Problem for Nonlinear Crack-Defect Interaction: The Antiplane Variational Model

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.1991 Mathematics Subject Classification. 35B25, 49J40, 49Q10, 74G70.

Contacting crack faces within the context of bodies exhibiting limiting strains

“…As δ tends to zero, we can derive the existence theorem below. The proof of Theorem 3 follows the proof given in [17] based on the properties (10)- (12) for the specific case F(σ) = Ψ 2 (∥σ∥)σ.…”

“…The uniform bound in (10) together with (2) leads to ∥ε∥ ≤ M 1 implying limiting small strain for small M 1 . The bounds in (11) describe the monotone and Lipschitz continuous properties of F. After integration over Ω c , property (12) provides σ with the following lower bound in the L 1 -norm:…”

“…The principal challenge here is finding singular solutions at the crack tip [8,9] and obtaining a formula for the energy release rate [10][11][12], which is relevant to brittle as well as quasi-brittle materials fracturing [13]. Numerical methods suitable for this class of nonlinear crack problems were developed in [14].…”

Optimal location of a rigid inclusion in equilibrium problems for inhomogeneous two‐dimensional bodies with a crack

“…From a mathematical point of view an analysis of cracked composite structures is significantly more complicated because of the presence of nonregular boundary components . In this regard, the influence of mechanical and geometric properties of inclusions on crack‐tip sensitivities is a challenging mathematical problem . It is well known that imposing linear boundary conditions on a crack may lead to physical inconsistency of mathematical models since a mutual penetration of the crack faces may happen…”

“…In the last years, within the framework of crack models subject to nonpenetration (contact) conditions, a number of papers have been published, concerning shape optimization problems for delaminated rigid inclusions, see, for example . For a heterogeneous two dimensional body with a micro‐object (defect) and a macro‐object (crack), the antiplane strain energy release rate is expressed by means of the mode‐III stress intensity factor that is examined with respect to small defects such as microcracks, holes, and inclusions . The ability of velocity methods to describe changes of topology by creating defects like holes is investigated in [].…”