Equivalence of formulations for problems in elasticity theory in terms of stresses

The problem of finding the number of independent compatibility equations in strains in n-dimensional Euclidean space is discussed. The number in question coincides with the number of independent Beltrami-Michell compatibility equations in terms of stresses used when formulating the problem in elasticity theory and also with the number of components of the incompatibility tensor and of the Riemann-Christoffel tensor. For n = 3, two counterexamples are presented that show the impossibility to transfer three “diagonal” or three “nondiagonal” Beltrami-Michell equations from the domain of an elastic solid to its boundary. In this case, the formulation of the problem becomes nonequivalent to either classical or new formulating of the problem in terms of stresses.

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