  We study some differential complexes in continuum mechanics that involve both
symmetric and non-symmetric second-order tensors. In particular, we show that
the tensorial analogue of the standard grad-curl-div complex can simultaneously
describe the kinematics and the kinetics of motions of a continuum. The
relation between this complex and the de Rham complex allows one to readily
derive the necessary and sufficient conditions for the compatibility of the
displacement gradient and the existence of stress functions on non-contractible
bodies. We also derive the local compatibility equations in terms of the Green
deformation tensor for motions of $2$D and $3$D bodies, and shells in curved
ambient spaces with constant curvatures.
