The Character of Couples and Couple Stresses in Continuum Mechanics

Abstract

In this paper, the concepts of moments and couples in mechanics are examined from a fundamental perspective. Representing a couple by its moment vector is very useful in rigid body mechanics, where the states of internal stresses and deformation are not studied. This is because only the moment of couples appears in the governing equation of moment equilibrium. On the other hand, when considering the state of internal stresses and deformation in continuum mechanics, not only the moment of couples but also the line of action of their constituent parallel opposite forces must be specified. In defining a well-posed problem for a continuum, including the governing equations of moment equilibrium or motion, boundary conditions, and constitutive relations, only the moment of couples (e.g., body couples, couple tractions, couple stresses) appear without specifying the line of action of the constituent parallel forces. Nevertheless, the physical state of stress and deformation in the continuum must be unique and determinate. Therefore, this physical requirement imposes some restrictions on the form of body couples, couple tractions, 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 couples or a distribution of surface twisting couple tractions with normal moments. Furthermore, the mechanism of action of the couple traction as a double layer of shear force tractions is established, along with the skew-symmetric character of the couple stress moment tensor.