  We develop a unified continuum modeling framework for viscous fluids and
hyperelastic solids using the Gibbs free energy as the thermodynamic potential.
This framework naturally leads to a pressure primitive variable formulation for
the continuum body, which is well-behaved in both compressible and
incompressible regimes. Our derivation also provides a rational justification
of the isochoric-volumetric additive split of free energies in nonlinear
continuum mechanics. The variational multiscale analysis is performed for the
continuum model to construct a foundation for numerical discretization. We
first consider the continuum body instantiated as a hyperelastic material and
develop a variational multiscale formulation for the hyper-elastodynamic
problem. The generalized-alpha method is applied for temporal discretization. A
segregated algorithm for the nonlinear solver is designed and carefully
analyzed. Second, we apply the new formulation to construct a novel unified
formulation for fluid-solid coupled problems. The variational multiscale
formulation is utilized for spatial discretization in both fluid and solid
subdomains. The generalized-alpha method is applied for the whole continuum
body, and optimal high-frequency dissipation is achieved in both fluid and
solid subproblems. A new predictor multi-corrector algorithm is developed based
on the segregated algorithm to attain a good balance between robustness and
efficiency. The efficacy of the new formulations is examined in several
benchmark problems. The results indicate that the proposed modeling and
numerical methodologies constitute a promising technology for biomedical and
engineering applications, particularly those necessitating incompressible
models.
