  This work concerns the continuum basis and numerical formulation for
deformable materials with viscous dissipative mechanisms. We derive a
viscohyperelastic modeling framework based on fundamental thermomechanical
principles. Since most large deformation problems exhibit the isochoric
property, our modeling work is constructed based on the Gibbs free energy in
order to develop a continuum theory using the pressure-primitive variables,
which is known to be well-behaved in the incompressible limit. With a general
theory presented, we focus on a family of free energies that leads to the
so-called finite deformation linear model. Our derivation elucidates the origin
of the evolution equations of that model, which was originally proposed
heuristically. In our derivation, the thermodynamic inconsistency is clarified
and rectified. We then discuss the relaxation property of the non-equilibrium
stress in the thermodynamic equilibrium limit and its implication on the form
of free energy. A modified version of the identical polymer chain model is then
proposed, with a special case being the model proposed by G. Holzapfel and J.
Simo. Based on the consistent modeling framework, a provably energy stable
numerical scheme is constructed for incompressible viscohyperelasticity using
inf-sup stable elements. In particular, we adopt a suite of smooth
generalization of the Taylor-Hood element based on Non-Uniform Rational
B-Splines (NURBS) for spatial discretization. The temporal discretization is
performed via the generalized-alpha scheme. We present a suite of numerical
results to corroborate the proposed numerical properties, including the
nonlinear stability, robustness under large deformation, and the stress
accuracy resolved by the higher-order elements.
