Evolution of generalized couple-stress continuum theories: a critical analysis

TL;DR

This paper critically analyzes various generalized couple-stress continuum theories, emphasizing fundamental mathematical and mechanical consistency, and clarifies the proper degrees of freedom and tensor properties necessary for a valid size-dependent continuum mechanics framework.

Contribution

It provides a rigorous critique of existing theories, establishing key requirements for consistency and clarifying the roles of displacement, rotation, and tensor properties in couple-stress theories.

Findings

Couple stress theory must be based on displacement and macrorotation fields.

Normal component of surface moment traction must vanish in consistent theories.

Deviatoric curvature tensors can cause indeterminacy in couple stress tensors.

Abstract

In this paper, we examine different generalized couple-stress continuum mechanics theories, including couple stress, strain gradient and micropolar theories. First, we investigate the fundamental requirements in any consistent size-dependent couple stress continuum mechanics, for which satisfying basic rules of mathematics and mechanics are crucial to establish a consistent theory. As a result, we show that continuum couple stress theory must be based on the displacement field and its corresponding macrorotation field as degrees of freedom, while an extraneous artificial microrotation cannot be a true continuum mechanical concept. Furthermore, the idea of generalized force and independent generalized degrees of freedom show that the normal component of the surface moment traction vector must vanish. Then, with these requirements in mind, various existing couple stress theories are examined critically, and we find that certain deviatoric curvature tensors create indeterminacy in the spherical part of the couple stress tensor. We also examine micropolar and micromorphic theories from this same perspective.