Abstract
We consider a family of variational problems on the equilibrium of a composite Kirchhoff–Love plate containing two flat rectilinear rigid inclusions, which are connected in a hinged manner. It is assumed that both inclusions are delaminated from an elastic matrix, thus forming an interfacial crack between the inclusions and the surrounding elastic media. Displacement boundary conditions of an inequality type are set on the crack faces that ensure a mutual nonpenetration of opposite crack faces. The problems of the family depend on a parameter specifying the coordinate of a connection point of the inclusions. For the considered family of problems, we formulate a new inverse problem of finding unknown coordinates of a hinge joint point. The continuity of solutions of the problems on this parameter is proved. The solvability of this inverse problem has been established. Using a passage to the limit, a qualitative connection between the problems for plates with flat and bulk hinged inclusions is shown.
Highlights
The success of the modern industry is largely based on the widespread use of composite materials
We consider a family of variational problems on the equilibrium of a composite Kirchhoff–Love plate containing two flat rectilinear rigid inclusions, which are connected in a hinged manner
5 Conclusion We have considered the inverse problem (11), (12), which is motivated by applications to fracture mechanics
Summary
The success of the modern industry is largely based on the widespread use of composite materials. The presence of an inclusion or a crack means that there can be significant stress values in the vicinity of these inhomogeneities For this reason, the choice of one or another method for specifying the boundary conditions characterizing mechanical processes near a crack can affect the physical adequacy of an entire mathematical model as a whole. Equilibrium problems for composite bodies with elastic or rigid inclusions have been successfully formulated and investigated, see for example [14,15,16,17,18,19,20,21,22,23,24,25,26,27,28]. Under the assumption that the sum of lengths of the inclusions is constant, a solvability of this inverse problem is proved
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