Abstract
We consider equilibrium problems for a cracked composite plate with a thin cylindrical rigid inclusion. Deformation of an elastic matrix is described by the Timoshenko model. The plate is assumed to have a through crack that does not touch the rigid inclusion. In order to describe mutual nonpenetration of the crack faces we impose a boundary condition in the form of inequality on the crack curve. For a family of appropriate variational problems, we analyze the dependence of their solutions on the location of the rigid inclusion. We formulate an optimal control problem with a cost functional defined by an arbitrary continuous functional on the solution space, while the location parameter of inclusion is chosen as the control parameter. The existence of a solution to the optimal control problem and a continuous dependence of the solutions in a suitable Sobolev space with respect to the location parameter are proved.
Highlights
Mathematical models and methods related to analysis of the mechanical properties and behavior of composite materials are sought to provide optimal engineering properties of composites [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31]
While studying the problems of the theory of cracks, it is appropriate to take into account the possibility of contact mechanical interaction between the crack faces, which, in particular, leads to the need to use of boundary conditions of the Signorini type
These conditions have the form of inequalities and describe the nonpenetration of the opposite faces of the crack
Summary
Mathematical models and methods related to analysis of the mechanical properties and behavior of composite materials are sought to provide optimal engineering properties of composites [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31]. The above properties of the energy functional Π (χ ), the bilinear form B(Ωγ , ·, ·), and the set Kt allow one to establish the existence of a unique solution ξt = (Ut, ut, φt) ∈ Kt for problem (6); see [34].
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
