Approximation methods in viscoelasticity theory

According to Volterra’s principle, the solution of the quasistatic problem in linear elasticity theory for an isotropic medium can be obtained from a solution of the corresponding problem of elasticity theory by substituting an integral operator with respect to time for the Poisson ratio and the subsequent interpretation of the constitutive operator function. The method of approximation uses an expansion of constitutive functions into a sum of rational operators. An evaluation is made for the accuracy of the solution, which is related to the quality of approximation. For an anisotropic medium, the constitutive function depends on several integral operators. In this case, a special method of approximation of the constitutive function is suggested which uses the introduction of “canonical” operators. A priori and a posteriori evaluations of the solution are given. Generalizations to nonhomogeneous (composite) media and nonlinear cases are indicated.