The correspondence principle of linear viscoelasticity theory for mixed boundary value problems involving time-dependent boundary regions

Abstract

1. Introduction. The classical method of .solving boundary value problems in the linear quasi-static theory of viscoelasticity is to apply an integral transform (with respect to time) to the time-dependent field equations and boundary conditions. The trans-formed field equations then have the same form as the field equations of elasticity theory and if a solution to these, which is compatible with the transformed boundary conditions, can be found then the solution to the original problem is reduced to transform inversion. This method of solving viscoelastic stress analysis problems is referred to as the "correspondence principle". The correspondence principle is clearly applicable whenever the type of boundary condition prescribed is the same at all points of the boundary. For mixed boundary value problems (i.e., problems for which different field quantities are prescribed over separate parts of the boundary) the method is still applicable provided the regions over which different types of boundary conditions are given do not vary with time. (We are, of course, assuming that the region occupied by the body does not vary with time.) There remain those viscoelastic mixed boundary value problems where the regions, over which different types of boundary conditions are given, do vary with time. Particular examples are indentation and crack propagation problems. For problems of this type there will be points of the boundary at which only partial histories of some field quantities will be prescribed. When this is the case the transforms of these quantities are not directly obtainable and the classical correspondence principle is not applicable. Following a statement of the fundamental field equations of linear