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

作者：Roman Poya, Rogelio Ortigosa, Antonio J. Gil, Theodore Kim, Javier Bonet · 发表于：Computer Methods in Applied Mechanics and Engineering · 年份：2024 · DOI：10.1016/j.cma.2024.117701 · 被引用次数：6 · 研究领域：Elasticity and Material Modeling、Advanced Numerical Analysis Techniques、Advanced Numerical Methods in Computational Mathematics

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 cond...