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SaintVenant's compatibility conditionIn the mathematical theory of elasticity the strain ε is related to a displacement field U by Additional recommended knowledgeSaintVenant derived the compatibility condition for an arbitrary symmetric second rank tensor field to be of this form. The integrability condition takes the form of the vanishing of the SaintVenants tensor^{[1]} defined by Due to the symmetry conditions W_{ijkl} = W_{klij} = − W_{jikl} = W_{ijlk} there are only six (in the three dimensional case) distinct components of W. These six equations are not independent as verified by for example and there are two further relations obtained by cyclic permutation. However, in practise the six equations are preferred. In its simplest form of course the components of ε must be assumed twice continuously differentiable, but more recent work ^{[2]} proves the result in a much more general case. This can be thought of as an analogue, for symmetric tensor fields, of Poincare's lemma for skewsymmetric tensor fields (differential forms). The result can be generalized to higher rank symmetric tensor fields^{[3]}. Let F be a symmetric rankk tensor field on an open set in ndimensional Euclidean space, then the symmetric derivative is the rank k+1 tensor field defined by where we use the classical notation that indices following a comma indicate differentiation and groups of indices enclosed in brackets indicate symmetrization over those indices. The SaintVenant tensor W of a symmetric rankk tensor field U is defined by with On a simply connected domain in Euclidean space W = 0 implies that U = dF for some rank k1 symmetric tensor field F. References

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "SaintVenant's_compatibility_condition". A list of authors is available in Wikipedia. 