Optimal control of transverse crack length in the equilibrium problem of Timoshenko plate with two intersecting cracks

A mathematical model of the equilibrium of an elastic plate with two mutually intersecting cracks is considered. One of the cracks is described by a part of the plane perpendicular to the midplane of the plate, and the other is specified by a smooth curve in the midplane. The nonlinearity of the problem is due to the non-penetration conditions in the form of inequalities imposed on the curves corresponding to the cracks. An analysis is made of the dependence of solutions of a family of variational inequalities on a parameter characterizing the variation of the length of a rectilinear crack. Based on the described family of problems, an optimal control problem is formulated with a quality functional determined by the Griffiths formula, which characterizes the possibility of crack development along a given trajectory. In this case, the control is specified by a numerical parameter specifying the length of the rectilinear crack. The existence of a solution for the optimal control problem is proved, and a continuous dependence of solutions in the Sobolev space on a change in the crack length parameter is established.