  We propose a new pressure-based structure-preserving (SP) and quasi
asymptotic preserving (AP) staggered semi-implicit finite volume scheme for the
unified first order hyperbolic formulation of continuum mechanics. The unified
model is based on the theory of symmetric-hyperbolic and thermodynamically
compatible (SHTC) systems and includes the description of elastic and
elasto-plastic solids in the nonlinear large-strain regime as well as viscous
and inviscid heat-conducting fluids, which correspond to the stiff relaxation
limit of the model. In the absence of relaxation source terms, the homogeneous
PDE system is endowed with two stationary linear differential constraints
(involutions), which require the curl of distortion field and the curl of the
thermal impulse to be zero for all times. In the stiff relaxation limit, the
unified model tends asymptotically to the compressible Navier-Stokes equations.
The new structure-preserving scheme presented in this paper can be proven to be
exactly curl-free for the homogeneous part of the PDE system, i.e. in the
absence of relaxation source terms. We furthermore prove that the scheme is
quasi asymptotic preserving in the stiff relaxation limit, in the sense that
the numerical scheme reduces to a consistent second-order accurate
discretization of the compressible Navier-Stokes equations when the relaxation
times tend to zero. Last but not least, the proposed scheme is suitable for the
simulation of all Mach number flows thanks to its conservative formulation and
the implicit discretization of the pressure terms.
