Two-dimensional dynamic boundary-value problems for curvilinear thermoviscoelastic bodies

A new numerical-analytical method is proposed and demonstrated using an example of dynamic problems of a thermoviscoelastic body. In the general formulation, the thermoviscoelastic problem is split into three simpler problems. In the first problem, boundary functions that should satisfy only boundary conditions are determined. The second problem with homogeneous boundary conditions and inhomogeneous initial conditions is reduced to an eigenvalue problem by introducing special $\xi$ variables and separating time. This problem is solved by organizing integral superpositions with respect to the angular parameter. A linear algebraic system is obtained as a result of satisfaction of the boundary conditions at points that partition the curvilinear boundary of the body into small segments. After the eigenfunctions and eigenvalues are determined, the third problem with homogeneous boundary and initial conditions is solved by spectral decomposition of unknown functions and inhomogeneous terms in a coupled system of ordinary differential equations.