Optimal rigid inclusion shapes in elastic bodies with cracks

Shape derivative of the energy functional in a problem for a thin rigid inclusion in an elastic body

“…Finally, by using asymptotic formulas (15), it is possible to rewrite down conditions (19) and (20) as follows:…”

“…Let us note the following papers: [4,8,[14][15][16]19,35,36] presenting the investigation of different models of elastic bodies with rigid inclusions and cracks with both linear and nonlinear boundary conditions on the crack faces.…”

Choosing an optimal shape of thin rigid inclusions in elastic bodies

“…Vtorushin [6] and Lazarev [7] proved the existence of extreme crack shapes for the model of a three-dimensional elastic body and a Timoshenko-type plate; moreover, the cost functional coincides with the energy functional derivative in terms of the crack perturbance parameter in [6] and characterizes the norm of the difference between the displacement field resulting from solving the equilibrium problem and the function given in the chosen subdomain in [7]. The influence of the shape of volume and thin rigid inclusions on the possible propagation of a crack with the use of the Griffith fracture criterion was analyzed in [8,9]. The traditional way of optimizing elastic bodies was considered in [10,11].…”

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

“…This approach is characterized by inequality type boundary conditions at the crack faces . 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 .…”