Mathematical analysis of a solution method for finite-strain holonomic plasticity of Cosserat materials

Abstract

This article deals with the mathematical derivation and the validation over benchmark examples of a numerical method for the solution of a finite-strain holonomic (rate-independent) Cosserat plasticity problem for materials, possibly with microstructure. Two improvements are made in contrast to earlier approaches: First, the micro-rotations are parameterized with the help of an Euler-Rodrigues formula related to quaternions. Secondly, as main result, a novel two-pass preconditioning scheme for searching the energyminimizing solutions based on the limited memory Broyden-Fletcher-Goldstein-Shanno quasi-Newton method is proposed that consists of a predictor step and a corrector-iteration. After outlining the necessary adaptations to the model, numerical simulations compare the performance and efficiency of the new and the old algorithm. The proposed numerical model can be effectively employed for studying the mechanical response of complicated materials featuring large size effects.

Simple shear in nonlinear Cosserat micropolar elasticity: Existence of minimizers, numerical simulations, and occurrence of microstructure

“…Yet, numerical optimization favors (u, a 1 ) over (u, a 3 ), underlining that the energy landscape for geometrically nonlinear Cosserat materials is extremely complicated and emphasizing why it is so hard to numerically compute the correct global minimizers. In comparison with Equation (91), the theoretical infimal energy is E(u, a 1 ) = m 2 g 2 = 0:32, but (u, a 1 ) violates the consistent coupling condition (27).…”

“…For L c > 0 , various advanced numerical tools such as multigrid methods are available [27]. We will not discuss this here.…”

The Gamma limit of the simple shear problem in nonlinear Cosserat elasticity

“…The relaxed functionals E 0 and F 0 may also be of interest for numerical simulations. Using E 0 instead of the original Cosserat functional E given by (5) for simulations with a small but finite L c > 0 corresponds to a convexification or homogenization of the problem and may help apart from a very significant speed up to avoid some of the numerical problems encountered in [5,6,7].…”

Editorial

“…Blesgen and Amendola [1] present a mathematical derivation and validation of a numerical method for the solution of a finite-strain holonomic (rate-independent) Cosserat plasticity problem for materials. This works improved the present literature by considering parametrized micro-rotations and a novel twopass preconditioning scheme for searching the energyminimizing solutions based on the limited memory Broyden-Fletcher-Goldstein-Shanno quasi-Newton method.…”