On the sensitivity and reconstruction of 2D prestress state in a thin plate

Despite of the relevance of studies devoted to the identification of fully 2D or 3D nonhomogeneous prestress fields in solids based on a number of surface measurements using a nondestructive approach, there is still a lack of publications on this topic in literature. In the present paper, we continue to develop the methodology for nondestructive acoustical identification of inhomogeneous 2D residual stress state in a plate, the foundations of which have been laid and published by the authors in their early works. In the framework of the linearized boundary-value problem, we formulate and investigate the direct and inverse problems of in-plane vibrations of a prestressed thin plate. The variational and weak formulations of the direct problem are presented. A new iterative-regularized scheme is proposed for solving the inverse problem of identification of 2D residual stress state in a plate using the measurements of displacements on some part of the boundary in a given frequency range. This technique is based on the projection and finite-element methods and leads to solving an ill-conditioned algebraic system at each iteration. It makes it possible to use the results of a series of vibration tests performed by applying different types of loading. Numerical results of reconstruction of some 2D inhomogeneous residual stress states in a rectangular plate were obtained and analyzed. Additionally, the finite-element analysis of sensitivity of the prestress parameters to sounding load types was carried to formulate recommendations on the choice of sounding parameters for the most advantageous reconstruction procedure.