Symmetrisation and Hyperbolicity of First-Order Conservation Laws in Large Strain ViscoelasticitySTICITY
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This paper will present a first-order hyperbolic framework with relaxation (or dissipation) terms for large strain viscoelastic solids. The framework is based on a Maxwell-type viscoelastic model [1, 2] and integrates linear momentum conservation, geometric conservation laws, and evolution equations for internal variables. The viscous relaxation is formulated for both compressible and incompressible behaviour. First, we will propose a polyconvex strain energy function that is jointly convex with respect to the deformation measures and internal variables. Second, we will introduce a generalised convex entropy function to symmetrise the hyperbolic system in terms of dual conjugate (entropy) variables. Third, we will 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 will 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 will be validated through numerical examples and benchmarked against the in-house Updated Reference Lagragian SPH [3, 4] and vertex-centred finite volume algorithms [5], demonstrating stability, accuracy, and consistent energy dissipation for both compressible and incompressible viscous behaviour.
