Numerical methods for treating problems of viscoelastic isotropic solid deformation

Abstract

Mathematical models for treating problems of linear viscoelasticity involving hereditary constitutive relations for compressible solids are discussed, and their discretization using finite element methods in space together with quadrature rules in time to treat the hereditary integrals is described. The range of applicability of this type of formulation is reviewed in the context of geometric and constitutive linearity/nonlinearity, and the limitations imposed by the availability of physical data are discussed. One of the above models is a Volterra integral equation of the second kind. In this, when the kernel is separable, an established technique due to Goursat (1933) can be exploited to reformulate the problem as a system of ordinary differential equations. This approach will be described. For the special case of a linear viscoelastic, isotropic, homogeneous, synchronous (constant Poisson's ratio) solid this method results in a complete decoupling of the space and time dependence. In this case the problem can be solved at each time level by solving first a problem of linear elasticity and then a system of ordinary differential equations for each point in the spatial mesh at which the viscoelastic displacements are required. The advantages, disadvantages and limitations offered by this, and various other schemes for solving problems of viscoelasticity as outlined below, are discussed.