Foundations of consistent couple stress theory

Abstract

In this paper, we examine the recently developed skew-symmetric couple stress theory and demonstrate its inner consistency, natural simplicity and fundamental connection to classical mechanics. This hopefully will help the scientific community to overcome any ambiguity and skepticism about this theory, especially the validity of the skew-symmetric character of the couple-stress tensor. We demonstrate that in a consistent continuum mechanics, the response of infinitesimal elements of matter at each point decomposes naturally into a rigid body portion, plus the relative translation and rotation of these elements at adjacent points of the continuum. This relative translation and rotation captures the deformation in terms of stretches and curvatures, respectively. As a result, the continuous displacement field and its corresponding rotation field are the primary variables, which remarkably is in complete alignment with rigid body mechanics, thus providing a unifying basis. For further clarification, we also examine the deviatoric symmetric couple stress theory that, in turn, provides more insight on the fundamental aspects of consistent continuum mechanics.