On the mathematical content of Volterra's principle in the boundary value problem of viscoelasticity: PMM vol. 35, no. 5, 1971, pp. 869–878

Abstract

A direct proof of Volterra's principle is given by reducing the boundary value problem of homogeneous viscoelasticity to solving the corresponding elastic problem and some operator equations. The conditions of applicability of the symbolic method are formulated as equivalence conditions between the realization of the viscoelastic operator function which arises from the elastic problem and the solutions of the operator equations. We note that the second procedure is more general and can be used in the problems of viscoelasticity whose solutions cannot be constructed by Volterra's principle. Volterra's principle [1 – 5] is widely used for the construction of the solution of boundary value problems in linear homogeneous viscoelasticity. The basis for its applicability is the independence of the operations with respect to the coordinates and time in the complete fundamental system of quasi-static equations of a viscoelastic body. As a result, the problem is divided into solving the corresponding boundary value problem of the elastic body and the determination of the operator functions. The latter are obtained from the elastic solution through the formal replacement of the mechanical moduli by the viscoelastic operators. However, the separation of the space and time operations in the equations of viscoelasticity is, by itself, not a sufficient criterion for the applicability of the operator-symbolic method, if only because the boundary conditions are not taken into account. In connection with this, an investigation of the problem of the rational application of Volterra's principle is required. An attempt for the mathematical foundation of Volterra's principle is contained in [6]. In the case of time-independent viscoelastic properties the identity between the first and second form of the correspondence principle has been established by the methods of operational calculus [7]. The symbolic method is justified by the con struction of an isomorphism between the sets of functions of the viscoelasticity operators and the functions of a complex variable. The conditions of applicability of Volterra's principle are determined by the possibility of performing a Laplace transform in the equations and the boundary conditions of the viscoelasticity problem.