  We present a novel Finite Volume (FV) scheme on unstructured polygonal meshes
that is provably compliant with the Second Law of Thermodynamics and the
Geometric Conservation Law (GCL) at the same time. The governing equations are
provided by a subset of the class of symmetric and hyperbolic thermodynamically
compatible (SHTC) models. Our numerical method discretizes the equations for
the conservation of momentum, total energy, distortion tensor and thermal
impulse vector, hence accounting in one single unified mathematical formalism
for a wide range of physical phenomena in continuum mechanics. By means of two
conservative corrections directly embedded in the definition of the numerical
fluxes, the new schemes are proven to satisfy two extra conservation laws,
namely an entropy balance law and a geometric equation that links the
distortion tensor to the density evolution. As such, the classical mass
conservation equation can be discarded. Firstly, the GCL is derived at the
continuous level, and subsequently it is satisfied by introducing the new
concepts of general potential and generalized Gibbs relation. Once
compatibility of the GCL is ensured, thermodynamic compatibility is tackled in
the same manner, thus achieving the satisfaction of a local cell entropy
inequality. The two corrections are orthogonal, meaning that they can coexist
simultaneously without interfering with each other. The compatibility of the
new FV schemes holds true at the semi-discrete level, and time integration of
the governing PDE is carried out relying on Runge-Kutta schemes. A large suite
of test cases demonstrates the structure preserving properties of the schemes
at the discrete level as well.
