COMPATIBILITY EQUATIONS AND STRESS FUNCTIONS IN ELASTICITY THEORY

Two formulations of problems in the theory of elasticity in stresses are considered. The first one is based on Papkovich’s compatibility equations. The second one is based on the Saint-Venant compatibility equations. It is shown that the Cesaro formulas in both formulations make it possible to introduce as a vector of indefinite Lagrange multipliers the vector of partial solutions of inhomogeneous equilibrium equations satisfying the Neumann vector problem. On the other hand, it is shown that the compatibility equations introduced as links between distortions (Papkovich compatibility) or deformations (Saint-Venant compatibility) allow one to introduce the corresponding tensors of indefinite Lagrange multipliers. It is shown that these tensors can be considered as functions of stresses. In the first setting, the stress function tensor of the second rank has nine components, since it is generally asymmetric. In the second formulation, the stress function tensor is symmetrical and has six components. In particular, the possibility of introducing three stress functions is also discussed.

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