Junction problem for elastic and rigid inclusions in elastic bodies

Abstract

An equilibrium problem for an elastic body is considered. It is assumed that the body has a thin elastic inclusion and a thin rigid inclusion. We analyze a junction problem assuming that the inclusions have a joint point. Different equivalent problem formulations are discussed, and existence of solutions is proved. A set of junction conditions is found. We investigate a convergence to infinity and to zero of a rigidity parameter of the elastic inclusion. A delamination of the elastic inclusion is also investigated. In this case, inequality‐type boundary conditions are imposed at the crack faces to prevent a mutual penetration between crack faces. Copyright © 2015 John Wiley & Sons, Ltd.

Optimal control of the radius of a rigid circular inclusion in inhomogeneous two-dimensional bodies with cracks

“…It is well known that imposing of linear boundary conditions on the crack may lead to physical inconsistency of mathematical models since mutual penetration of the crack faces may happen [18,24]. In recent years, a crack theory with non-penetration conditions has been under active study [25,26,27,28,29,30]. This approach to solving crack problems is characterized by inequality type boundary conditions at the crack faces, is indeed what we employ in the present paper.…”

“…This approach to solving crack problems is characterized by inequality type boundary conditions at the crack faces, is indeed what we employ in the present paper. Within this approach, various problems for bodies with rigid inclusions has been successfully formulated and investigated using variational methods, see for example [9,25,27,31,32,33,34]. In contrast to a previous study of an optimal control problem for a two-dimensional elastic body with a rigid delaminated inclusion, as considered in [31], we suppose that crack curve touches the inclusion's boundary only at the crack's tip.…”

Optimal location of a thin rigid inclusion for a problem describing equilibrium of a composite Timoshenko plate with a crack

“…Among the studies in this field let us note the recently developed nonlinear models describing the equilibrium state of composite solids having cracks on the interfacial boundary of inclusions. For the mentioned models, the carrier matrix is assumed to be elastic, whereas the inclusion is either absolutely rigid (for instance, see [8-10, 13-18, 20, 23]) or elastic and described by other constitutive relations (in this case, we have the so-called junction problems [26][27][28][29][30][31]). The optimal control problems for the shape of the bodies with with cracks and rigid inclusions are studied, for example, in [10-12, 15, 17, 18, 21, 22].…”

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

“…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. [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.…”