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Differential Complexes in Continuum Mechanics

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.