Character of couple and couple-stress in continuum mechanics

TL;DR

This paper critically examines the concept of couples and couple-stresses in continuum mechanics, revealing issues with indeterminacy and establishing the skew-symmetric nature of the couple-stress tensor based on physical and mathematical considerations.

Contribution

It clarifies the physical meaning of couples in continuum mechanics and demonstrates the skew-symmetric property of the couple-stress tensor, resolving indeterminacy issues.

Findings

Couples in continuum mechanics must specify the line of action for accuracy.

The couple-stress tensor is inherently skew-symmetric.

Distribution of body couples is unsupported in a consistent continuum model.

Abstract

In this paper, the concept of moment and couple in mechanics is examined from a fundamental perspective. It turns out that although representing a couple by its moment vector is very useful in rigid body mechanics and strength of materials, it has been very misleading in continuum mechanics. To specify the effect of a concentrated couple in continuum mechanics, not only the couple moment, but also the line of action of its constituent parallel opposite forces must be specified. However, in the governing equations of equilibrium or motion of a continuum, only moment of body couple, moment of couple-tractions, and moment of couple-stresses appear without specifying the line of action of any couple density forces. This results in non-uniqueness of the state of stresses and deformation in the continuum, which has shown itself in the indeterminacy of the couple-stress tensor. Nevertheless, the physical state of stress and deformation in the continuum is unique and determinate. Therefore, this imposes some restrictions on the form of body couple, couple-traction and couple-stresses. Here, the uniqueness of interactions in the continuum is used to establish that the continuum does not support a distribution of body couple, and a distribution of surface twisting couple-traction with normal moment. From this the mechanism of action of the couple-traction as a double layer of shear force-tractions, and the skew-symmetric character of the couple-stress moment tensor in continuum mechanics are also established.