Galerkin-based mechanical integrators for finite elastodynamics formulated in principal stretches - Pitfalls and remedies

In the present paper we deal with integrators relying on Finite Elements in time for general hyperelastic models which guarantee the conservation of the momentum maps as well as of the total energy within the calculation accuracy. In the isotropic case, eigenvalue-based constitutive laws are, on the one hand, very significant to model the behaviour of various materials. On the other hand, it will be shown that the efficient algorithmic treatment of principal stretches based on a perturbation approach requires, especially within the context of conserving schemes, advanced techniques to circumvent numerical problems. Accordingly, we propose an adequate solution strategy to avoid potential pitfalls which are related to numerical artefacts of the applied non-standard quadrature rule. Finally, the excellent numerical and mechanical performance of the resulting Galerkin-based time-stepping schemes will be demonstrated by means of representative numerical examples.