Pure plate bending in couple stress theories

In this paper, we examine the pure bending of plates within the framework of modified couple stress theory (M-CST) and consistent couple stress theory (C-CST). In this development, it is demonstrated that M-CST does not describe pure bending of a plate properly. Particularly, M-CST predicts no couple-stresses and no size effect for the pure bending of the plate into a spherical shell. This contradicts our expectation that couple stress theory should predict some size effect for such a deformation pattern. Therefore, this result clearly demonstrates another inconsistency of indeterminate symmetric modified couple stress theory (M-CST), which is based on considering the symmetric torsion tensor as the curvature tensor. On the other hand, the fully determinate skew-symmetric consistent couple stress theory (C-CST) predicts results for pure plate bending that tend to agree with mechanics intuition and experimental evidence. Particularly, C-CST predicts couple-stresses and size effects for the pure bending of the plate into a spherical shell, which represents an additional illustration of its consistency.