A first order hyperbolic framework for large strain computational solid dynamics. Part II: Total Lagrangian compressible, nearly incompressible and truly incompressible elasticity

In Part I of this series, Bonet et al. (2015) introduced a new computational framework for the analysis of large strain isothermal fast solid dynamics, where a mixed set of Total Lagrangian conservation laws was presented in terms of the linear momentum and an extended set of strain measures, namely the deformation gradient, its co-factor and its Jacobian. The main aim of this paper is to expand this formulation to the case of nearly incompressible and truly incompressible materials. The paper is further enhanced with three key novelties. First, the use of polyconvex nearly incompressible strain energy functionals enables the definition of generalised convex entropy functions and associated entropy fluxes. Two variants of the same formulation can then be obtained, namely, conservation-based and entropy-based, depending on the unknowns of the system. Crucially, the study of the eigenvalue structure of the system is carried out in order to demonstrate its hyperbolicity and, thus, obtain the correct time step bounds for explicit time integrators. Second, the development of a stabilised Petrov-Galerkin framework is presented for both systems of hyperbolic equations, that is, when expressed in terms of either conservation or entropy variables. Third, an adapted fractional step method, built upon the work presented in Gil et al. (2014), is presented to extend the range of applications towards the incompressibility limit. Finally, a series of numerical examples are presented in order to assess the applicability and robustness of the proposed formulation. The overall scheme shows excellent behaviour in compressible, nearly incompressible and truly incompressible scenarios, yielding equal order of convergence for velocities and stresses.