Some issues concerning a version of the theory of thin solids based on expansions in a system of Chebyshev polynomials of the second kind

Abstract

We consider various forms of equations of motion and heat influx for deformable solids as well as various forms of Hooke’s law and Fourier’s heat conduction law under the nonclassical parametrization [1–5] of the domain occupied by a thin solid, where the transverse coordinate ranges in the interval [0, 1]. We write out several characteristics inherent in this parametrization. We use the above-mentioned equations and laws to derive the corresponding equations and laws, as well as statements of problems, for thin bodies in moments with respect to Chebyshev polynomials of the second kind. Here the interval [0, 1] is used as the orthogonality interval for the systems of Chebyshev polynomials. For this interval, we write out the basic recursion relations and, in turn, use them to obtain several additional recursion relations, which play an important role in constructing other versions of the theory of thin solids. In particular, we use the recursion relations to obtain the moments of the first and second derivatives of a scalar function, of rank one and two tensors and their components, and of some differential operators of these variables. Moreover, we give the statements of coupled and uncoupled dynamic problems in moments of the (r, N)th approximation in moment thermomechanics of thin deformable solids. We also state the nonstable temperature problem in moments of the (r, N)th approximation.