Construct a convex extension for the viscous case with volumetric deformation J greater than one

Construct a symmetric-hyperbolic or otherwise thermodynamically admissible formulation for the viscoelastic multi-material balance laws that provides a convex energy extension and a signed entropy-production source when the volumetric deformation satisfies J>1.

Background

The viscous formulation introduces the energy terms −Glog⁡J-G\log J and −G2log⁡∣A∣-\frac{G}{2}\log|A|. Joint convexity of these terms imposes incompatible restrictions on the exponent choices in the domain J>1J>1. Consequently, the paper establishes well-posedness only on restricted deformation domains and does not guarantee a formulation of the second principle of thermodynamics for J>1J>1.

The authors suggest doubling the variable GG as a possible polyconvex remedy, but do not develop or prove such a construction.

References

One idea to overcome the previous limit is to double the variable $G$ into two different variables, as in the polyconvexity approach. We let this for future studies.

— Multi-material flow with a new viscoelastic model  (2609.35172 - Boyaval, 28 Sep 2026) in Section 2, immediately after Proposition 2 (Proposition \ref{prop:viscous})

In particular, the present formulation of viscous effects cannot be guaranteed stable on the whole natural domain $J>0$ % one may look for another stored energy ! / ``entropy'' % (the dilatation/compression rate should not vary too much within subdomains) and (a formulation of) the second principle of thermodynamics cannot be guaranteed in the dilatation case $J>1$ (with respect to the reference state).

— Multi-material flow with a new viscoelastic model  (2609.35172 - Boyaval, 28 Sep 2026) in Section 3, Conclusion