On derivative of energy functional for elastic bodies with cracks and unilateral conditions
In this paper we consider elasticity equations in a domain having a cut (a crack) with unilateral boundary conditions considered at the crack faces. The boundary conditions provide a mutual nonpenetration between the crack faces, and the problem as a whole is nonlinear. Assuming that a general perturbation of the cut is given, we find the derivative of the energy functional with respect to the perturbation parameter. It is known that a calculation of the material derivative for similar problems has the difficulty of finding boundary conditions at the crack faces. We use a variational property of the solution, thus avoiding a direct calculation of the material derivative.There are many results related to the differentiation of the potential energy functional with respect to variable domains (see, e.g., [9, 4, 5, 16, 18, 17, 3]). The general theory of calculating material and shape derivatives in linear and nonlinear boundary value problems is developed in [6].Derivatives of energy functionals with respect to the crack length in classical linear elasticity can be found by different ways. It is well known that the classical approach to the crack problem is characterized by the equality-type boundary conditions considered at the crack faces [13, 4, 7, 14, 12, 15]. As for the analysis of solution dependence on the shape domain for a wide class of elastic problems, we refer the reader to [8].In the works [1,2] the appropriate technique of finding derivatives of the energy functional with respect to the crack length for unilateral boundary conditions is proposed, which can be used for a wide class of the unilateral problems. Qualitative properties of solutions (solution existence, solution regularity, dependence of solutions on parameters, etc.) in the crack problem for plates, shells, two- and three-dimensional bodies with unilateral conditions on the crack faces are analysed in [2] (see also [20, 19, 10, 11]).