Symmetrisation and hyperbolicity of first-order conservation laws in large strain compressible viscoelasticity using the smoothed particle hydrodynamics method

This paper presents a first-order hyperbolic framework with relaxation (or dissipation) terms for large strain viscoelastic solids. The framework is based on a compressible Maxwell-type viscoelastic model and integrates linear momentum conservation, geometric conservation laws, and evolution equations for internal variables. First, we propose a polyconvex strain energy function that is jointly convex with respect to the deformation measures and internal variables. Second, we introduce a generalised convex entropy function to symmetrise the hyperbolic system in terms of dual conjugate (entropy) variables. Third, we demonstrate that the system is hyperbolic (i.e., real wave speeds) under all deformation states, and that the relaxation terms correctly capture viscoelastic dissipation. Fourth, we present an upwinding Smoothed Particle Hydrodynamics (SPH) scheme that enforces the second law of thermodynamics semi-discretely and uses the time rate of the generalised convex entropy to monitor internal dissipation and stabilise the simulation. Finally, the proposed framework is validated through numerical examples and benchmarked against an in-house Updated Reference Lagragian SPH algorithm, demonstrating stability, accuracy, and consistent energy dissipation.