Shape optimization of rigid inclusions for elastic plates with cracks

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

“…In particular, an analysis of appropriate mathematical models makes it possible to reveal the characteristics allowing one to create composites with a good margin of structural integrity. 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 .…”

“…An alternative approach to modelling crack problems that do not allow the opposite crack faces to penetrate each other has been elaborated since 1990s. 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 .…”

“…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 .…”

“…From a mathematical point of view an analysis of cracked composite structures is significantly more complicated because of the presence of nonregular boundary components. [1][2][3][4][5] In this regard, the influence of mechanical and geometric properties of inclusions on crack-tip sensitivities is a challenging mathematical problem. [6][7][8] 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.…”

“…[1,2,[10][11][12][13][14][15][16][17] 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. [2,[18][19][20][21][22] 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. [7] The ability of velocity methods to describe changes of topology by creating defects like holes is investigated in [23].…”