Deviatoric couple stress theory and its application to simple shear and pure bending problems

Classical couple stress theory is indeterminate since the number of independent basic equations is inconsistent with that of field variables and the corresponding differential equation is not closed. The purpose of this paper is to remedy this gap and it is proven that the spherical part of the couple stress tensor vanishes when neglecting torsional deformation. With the vanishing trace of the couple stress tensor as a premise, the deviatoric (or traceless) couple stress theory (DCST) is considered. Besides basic equations, the governing equation along with appropriate boundary conditions is given for a three-dimensional problem. Two special cases of plane problems and anti-plane problems are also provided. A simple shear problem is considered to show the advantage of the DCST. An elastic layer with a clamped surface under uniform shear loading on the other surface is solved. Exact solution of the in-plane and anti-plane shear problems of an elastic strip is determined and however, the former has no solution if classical elasticity is used. The results indicate that there exists a boundary layer near the clamped surface of the strip or a great stress gradient occurs near the clamped surface when the characteristic length is sufficiently small. An annulus subjected to anti-plane shear loading on the inner edge and fixed on the outer edge and the pure bending of a 3D bar with rectangular cross-section are analyzed to illustrate the size-dependent effect. Modified couple stress theory and consistent couple stress theory can be reduced as two extreme cases of the present theory.