Generalised tangent stabilised nonlinear elasticity: An automated framework for controlling material and geometric instabilities

Tangent stabilised large strain isotropic elasticity was recently proposed by Poya et al. (2023) wherein by working directly with principal stretches the entire eigenstructure of constitutive and geometric/initial stiffness terms were found in closed-form, giving fresh insights into exact convexity conditions of highly non-convex functions in discrete settings. Consequently, owing to these newly found tangent eigenvalues an analytic tangent stabilisation was proposed (for common non-convex strain energies that exhibit material and/or geometric instabilities) bypassing incumbent numerical approaches routinely used in nonlinear finite element analysis. This formulation appears to be extremely robust for quasi-static simulation of complex deformations even with no load increments and time stepping while still capturing instabilities (similar to dynamic analysis) automatically in ways that are infeasible for path-following techniques in practice. In this work, we generalise the notion of tangent stabilised elasticity to virtually all known invariant formulations of nonlinear elasticity. We show that, closed-form eigen-decomposition of tangents is easily available irrespective of invariant formulation or integrity basis. In particular, we work out closed-form tangent eigensystems for isotropic Total Lagrangian deformation gradient ( )-based and right Cauchy–Green ( )-based as well as Updated Lagrangian left Cauchy–Green ( )-based formulations and present their exact convexity conditions postulated in terms of their corresponding tangent and geometric stiffness eigenvalues. In addition, we introduce the notion of geometrically stabilised polyconvex large strain elasticity for models that are materially stable but exhibit geometric instabilities for whom we construct their geometric stiffness in a spectrally-decomposed form analytically. We further extend this framework to the case of transverse isotropy where once again, closed-form tangent eigensystems are found for common transversely isotropic invariants. In this context, we augment the recent work on mixed variational formulations in principal stretches for deformable and rigid bodies, by presenting a mixed variational formulation for models with arbitrarily directed inextensible fibres. Since, tangent stabilisation unleashes an unparallelled capability for extreme deformations new numerical techniques are required to guarantee element-inversion-safe analysis. To this end, we propose a discretisation-aware load-stepping together with a line search scheme for a robust industry-grade implementation of tangent stabilised elasticity over general polyhedral meshes. Extensive comparisons with path-following techniques provide conclusive evidence that utilising tangent stabilised elasticity can offer both faster and automated results.