Junction problem for rigid and Timoshenko elastic inclusions in elastic bodies

Abstract

This paper concerns an equilibrium problem for a two-dimensional elastic body with a thin Timoshenko elastic inclusion and a thin rigid inclusion. It is assumed that the inclusions have a joint point and we analyze a junction problem for these inclusions. The existence of solutions is proved and the different equivalent formulations of the problem are discussed. In particular, the junction conditions at the joint point are found. A delamination of the elastic inclusion is also assumed. In this case, the inequality-type boundary conditions are imposed at the crack faces to prevent a mutual penetration between the crack faces. We investigate the convergence to infinity and zero of a rigidity parameter of the elastic inclusion. It is proved that in the limit, we obtain a rigid inclusion and a zero rigidity inclusion (a crack).

Junction problem for rigid and semirigid inclusions in elastic bodies

“…Sinceũ ∈ H 1 Γ (Ω γ ) 2 In fact, the problem formulation (60)-(64) can be simplified since, in fact, we have the inclusion γ which is a rigid one. Indeed, introduce a set…”

“…A junction problem for rigid and Timoshenko elastic inclusions incorporated into an elastic body is analyzed in [2].…”

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

“…It remains to show that̃satisfies the inequality [̃] ≥ 0 on . Bearing in mind the convergence (11), if necessary, we can once again extract a subsequence satisfying | →̃| a.e. on ± .…”

“…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.…”