Nonlinear Anisotropic Viscoelasticity

In this paper, we revisit the mathematical foundations of nonlinear viscoelasticity. We study the underlying geometry of viscoelastic deformations, and in particular, the intermediate configuration. Starting from the direct multiplicative decomposition of the deformation gradient $\mathbf{F}=\mathbf{F_e}\mathbf{F_v}\,$, into elastic and viscous distortions $\mathbf{F_e}$ and $\mathbf{F_v}\,$, respectively, we point out that $\mathbf{F_v}$ can be either a material tensor ($\mathbf{F_e}$ is a two-point tensor) or a two-point tensor ($\mathbf{F_e}$ is a spatial tensor). We show, based on physical grounds, that the second choice is unacceptable. It is assumed that the free energy density is the sum of an equilibrium and a non-equilibrium part. The symmetry transformations and their action on the total, elastic, and viscous deformation gradients are carefully discussed. Following a two-potential approach, the governing equations of nonlinear viscoelasticity are derived using the Lagrange--d'Alembert principle. We discuss the constitutive and kinetic equations for compressible and incompressible isotropic, transversely isotropic, orthotropic, and monoclinic viscoelastic solids. We finally semi-analytically study creep and relaxation in three examples of universal deformations.