The dynamic problem of thermoelectroelasticity for functionally graded layer

A general formulation of the coefficient inverse thermoelasticity problem for an inhomogeneous body is presented. The inverse problem is to determine the material characteristics of a finite inhomogeneous thermoelectric body as a function of the coordinates. A weak formulation is formulated. To solve the inverse problem on the basis of a weak formulation in Laplace transforms and the linearization method, operator equations are obtained that relate the functions sought and measured in the experiment. As an example, we consider the problem for a thermoelectroelastic rod. The direct problem of thermoelectroelasticity for a rod after the Laplace transform is solved on the basis of reduction to the system of Fredholm integral equations of the second kind, construction of solutions in the form of rational functions relative to transformants, and finding the originals on the basis of the theory of residues. To solve the inverse thermoelasticity problem, an iterative process is constructed, at each stage of which there are corrections of the reconstructed characteristics by solving the Fredholm integral equations of the first kind. The effect of changing the material characteristics on the additional information needed to solve the inverse problem is investigated. Computational experiments were carried out to reconstruct the laws of distribution of inhomogeneity in the classes of power and exponential functions, most often used in the modeling of functional gradient materials. Recommendations are given on the choice of the most informative time ranges for the extraction of additional information. A series of computational experiments showed that the error in restoring the dimensionless characteristics does not exceed 5%.