Spectral decompositions in dynamical viscoelastic problems

Theoretical relations obtained from solutions of dynamic problems of viscoelasticity represent an effective framework for experimental identification of dynamic rheological properties of materials. For the construction of such relations, closed solutions of boundary value problems (i.e. written in the form of convergent series or integrals) are preferred, because they(unlike solutions obtained by numerical methods) allow strict error estimates. However, the construction of analytical solutions is associated with the following difficulties. 1. As usual, the hypothesis of proportionality is accepted for relaxation operators corresponding to the first and second Lamé moduli, which is equivalent to the hypothesis of a constant Poisson's ratio. This significantly reduces the generality of consideration. 2. Representation of solutions for three-dimensional problems in the form of expansions in eigenfunctions makes it necessary to taking into account the large eigenvalues which in the vast majority of problems can be found only numerically, as the roots of transcendental equations, thus, it is likely to skip closely spaced and multiple roots. 3. Constructed series converge slowly. In this paper we suggest ways to overcome these difficulties. Solutions of initial boundary value problems are presented in the form of spectral expansions, but in contrast to the classical method of Fourier decomposition they are expanded over biorthogonal system of eigenfunctions of mutually conjugate pencils of differential operators. This pencils define generalized Sturm-Liouville problem with a polynomial spectral parameter. This eliminates the hypothesis of proportionality relaxation operators. Effective relations for the terms of spectral(in particular normalization factors) coordinate functions and asymptotic formulas for the initial approximations of eigenvalues excluding their omission in calculations are obtained. Power related ranking of elements of the spectral decomposition is proposed which allows achievingthe required accuracy of calculations on the partial sums of a low order.