  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
multiplicative decomposition of deformation gradient into elastic and viscous
parts $\mathbf{F}=\Fe\Fv\,$, we point out that $\Fv$ can be either a material
tensor ($\Fe$ is a two-point tensor) or a two-point tensor ($\Fe$ is a spatial
tensor). We show that based on physical grounds 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.
