Couple stress theories: Theoretical underpinnings and practical aspects from a new energy perspective

In this paper, we examine theoretical and practical aspects of several versions of couple stress theory. This includes indeterminate Mindlin-Tiersten-Koiter couple stress theory (MTK-CST), indeterminate symmetric modified couple stress theory (M-CST) and determinate skew-symmetric consistent couple stress theory (C-CST). We observe that MTK-CST and M-CST not only suffer from inconsistencies, these theories also cannot describe properly several elementary deformations, such as pure torsion of a circular bar and pure bending of a plate. By using an energy method, we also demonstrate another aspect of the inconsistency of the indeterminate MTK-CST and M-CST for elastic solids. This is achieved by deriving the governing equilibrium equations for elastic bodies in MTK-CST, M-CST and C-CST. This development shows that the direct minimization of the total potential energy for MTK-CST and M-CST violates the divergence free compatibility condition of the rotation vector field. On the other hand, the direct minimization of the total potential energy for C-CST satisfies this compatibility condition automatically. This result demonstrates another aspect of the inner consistency of C-CST.