New mixed formulation and mesh dependency of finite elements based on the consistent couple stress theory

TL;DR

This paper introduces a finite element formulation based on the consistent couple stress theory, addressing mesh dependency issues and providing a reliable approach for modeling structures with translational and rotational degrees of freedom.

Contribution

It develops a six-field variational finite element formulation incorporating couple stress theory with a simple, efficient solution method applicable to elastic and inelastic materials.

Findings

Objective rotation definition via couple stress theory

Mesh dependency issues in softening/fracture problems

Reliable modeling of structures with translational and rotational DOFs

Abstract

This work presents a general finite element formulation based on a six--field variational principle that incorporates the consistent couple stress theory. A simple, efficient and local iteration free solving procedure that covers both elastic and inelastic materials is derived to minimise computation cost. With proper interpolations, membrane elements of various nodes are proposed as the examples. The implemented finite elements are used to conduct numerical experiments to investigate the performance of the in-plane drilling degrees of freedom introduced by the consistent couple stress theory. The mesh dependency issue is also studied with both elastic and inelastic materials. It is shown that the consistent couple stress theory provides an objective definition of rotation compared with the Cauchy theory but additional regularisation (or other techniques) is required to overcome mesh/size dependency in softening or fracture related problems. In the case of hardening continuum problems and/or large characteristic lengths, the proposed formulation and elements offer a more reliable approach to model structures with both translational and rotational degrees of freedom.