Constitutive Equations for Isotropic and Anisotropic Linear Viscoelastic Materials

Abstract

In case of isotropic material symmetry, the elastic-viscoelastic correspondence principle is well established to provide the solution of linear viscoelasticity from the coupled fictitious elastic problem by use of the inverse Laplace transformation (Alfrey–Hoff’s analogy). Aim of this chapter is to show useful enhancement of the Alfrey–Hoff’s analogy to a broader class of material anisotropy for which separation of the volumetric and the shape change effects from total viscoelastic deformation does not occur. Such extension requires use of the vector–matrix notation to description of the general constitutive response of anisotropic linear viscoelastic material (see Pobiedria Izd. Mosk. Univ., (1984) [10]). When implemented to the composite materials which exhibit linear viscoelastic response, the classically used homogenization techniques for averaged elastic matrix, can be implemented to viscoelastic work-regime for associated fictitious elastic Representative Unit Cell of composite material. Next, subsequent application of the inverse Laplace transformation (cf. Haasemann and Ulbricht Technische Mechanik, 30(1–3), 122–135 (2010)) is applied. In a similar fashion, the well-established upper and lower bounds for effective elastic matrices can also be extended to anisotropic linear viscoelastic composite materials. The Laplace transformation is also a convenient tool for creep analysis of anisotropic composites that requires, however, limitation to the narrower class of linear viscoelastic materials. In the space of transformed variable $s$, instead of time space $t$, the classical homogenization rules for fictitious elastic composite materials can be applied. For the above reasons in what follows, we shall confine ourselves to the linear viscoelastic materials, isotropic, or anisotropic.