An Incremental formulation for the linear analysis of viscoelastic beams: Relaxation differential approach using generalized variables

This paper is concerned with the development of a new incremental formulation in the time domain for linear, non-aging viscoelastic materials undergoing mechanical deformation. The formulation is derived from linear differential equations based on a discrete spectrum representation for the relaxation function. The incremental constitutive equations are then obtained by finite difference integration. Thus the difficulty of retaining the stress history in computer solutions is avoided. The influence of the whole past history on the behaviour at any time is given by a pseudo second order tensor. A complete general formulation of linear viscoelastic stress analysis is developed in terms of increments of midsurface strains and curvatures and corresponding stress resultants. The generality allowed by this approach has been established by finding incremental formulation through simple choices of the tensor relaxation components. This approach appears to open a wide horizon (to explore) of new incremental formulations according to particular relaxation components. Remarkable incremental constitutive laws, for which the above technique is applied, are given. This formulation is introduced in a finite element discretization in order to resolve complex boundary viscoelastic problems.