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Differential complexes in continuum mechanics. (English) Zbl 1310.35224

Summary: 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 motion 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 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 2D and 3D bodies, and shells in curved ambient spaces with constant curvatures.

MSC:

35Q74 PDEs in connection with mechanics of deformable solids
58A12 de Rham theory in global analysis
74K25 Shells
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