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Nonlinear Anisotropic Viscoelasticity
In this work, we revisit the mathematical foundations of nonlinear viscoelasticity. We study the un-
derlying geometry of viscoelastic deformations, and in particular, the intermediate configuration. Starting from the direct multiplicative decomposition of the deformation gradient F = FeFv, into elastic and viscous distortions Fe and Fv, respectively, 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, 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, and the constitutive relations are found following the thermodynamics Coleman-Noll procedure. 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.
Keywords : Nonlinear viscoelasticity, multiplicative decomposition, intermediate configuration, anisotropic solids.