Geometrically non‐linear enhanced strain mixed methods and the method of incompatible modes

TLDR

The paper introduces a class of assumed‑strain mixed finite element methods that generalize the classical incompatible‑mode formulation to fully nonlinear solid‑mechanics problems. The method uses a local multiplicative decomposition of the deformation gradient into conforming and enhanced parts within a three‑field variational formulation, and extends to nonlinear enhanced‑strain interpolations for axisymmetric problems. Simulations in 2‑D, 3‑D, and axisymmetric elasticity and elastoplasticity show the method performs well, particularly for problems with deformation localization.

Abstract

Abstract A class of ‘assumed strain’ mixed finite element methods for fully non‐linear problems in solid mechanics is presented which, when restricted to geometrically linear problems, encompasses the classical method of incompatible modes as a particular case. The method relies crucially on a local multiplicative decomposition of the deformation gradient into a conforming and an enhanced part , formulated in the context of a three‐field variational formulation. The resulting class of mixed methods provides a possible extension to the non‐linear regime of well‐known incompatible mode formulations. In addition, this class of methods includes non‐linear generalizations of recently proposed enhanced strain interpolations for axisymmetric problems which cannot be interpreted as incompatible modes elements. The good performance of the proposed methodology is illustrated in a number of simulations including 2‐D, 3‐D and axisymmetric finite deformation problems in elasticity and elastoplasticity. Remarkably, these methods appear to be specially well suited for problems involving localization of the deformation, as illustrated in several numerical examples.