On a new approach to identifying inhomogeneous mechanical properties of elastic bodies

A new approach to solving the problem of identifying the variable characteristics of an inhomogeneous elastic isotropic body is presented. The most common formulations of problems on determining variable mechanical characteristics (the Lamé parameters and density are functions of coordinates) are presented. The inverse problem of identifying properties, due to its significant nonlinearity, is usually solved iteratively, with each iteration requiring the solution of a direct problem for some initial approximation and a system of the Fredholm integral equations of the first kind with smooth kernels to find corrections. This approach, in turn, requires specifying the displacement field in the area in which the loading occurs. An approach is proposed on the basis of which it is possible to carry out reconstruction by obtaining additional information about the displacement field in an area other than the loading area in a narrower search space. An example of such a reconstruction is presented in the problem of longitudinal vibrations of an inhomogeneous rod, where the amplitude-frequency response is specified at the internal point of the rod, and the loading is implemented at the end. The results of computational experiments on the reconstruction of the elasticity modulus and density in the form of two functions of the longitudinal coordinate are presented.