  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.
