In the mathematical theory of elasticity, Saint-Venant's compatibility condition defines the relationship between the strain
Barré de Saint-Venant derived the compatibility condition for an arbitrary symmetric second rank tensor field to be of this form, this has now been generalized to higher rank symmetric tensor fields on spaces of dimension
For a symmetric rank 2 tensor field
) the integrability condition takes the form of the vanishing of the Saint-Venant's tensor
[1] defined by The result that, on a simply connected domain W=0 implies that strain is the symmetric derivative of some vector field, was first described by Barré de Saint-Venant in 1864 and proved rigorously by Beltrami in 1886.
[2] For non-simply connected domains there are finite dimensional spaces of symmetric tensors with vanishing Saint-Venant's tensor that are not the symmetric derivative of a vector field.
The situation is analogous to de Rham cohomology[3] The Saint-Venant tensor
is closely related to the Riemann curvature tensor
must be assumed twice continuously differentiable, but more recent work[2] proves the result in a much more general case.
The relation between Saint-Venant's compatibility condition and Poincaré's lemma can be understood more clearly using a reduced form of
is a symmetric rank 2 tensor field.
and this also shows that there are six independent components for the important case of three dimensions.
While this still involves two derivatives rather than the one in the Poincaré lemma, it is possible to reduce to a problem involving first derivatives by introducing more variables and it has been shown that the resulting 'elasticity complex' is equivalent to the de Rham complex.
[7] In differential geometry the symmetrized derivative of a vector field appears also as the Lie derivative of the metric tensor g with respect to the vector field.
is thus the integrability condition for local existence of
As noted above this coincides with the vanishing of the linearization of the Riemann curvature tensor about the Euclidean metric.
Saint-Venant's compatibility condition can be thought of as an analogue, for symmetric tensor fields, of Poincaré's lemma for skew-symmetric tensor fields (differential forms).
The result can be generalized to higher rank symmetric tensor fields.
[8] Let F be a symmetric rank-k tensor field on an open set in n-dimensional 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.
of a symmetric rank-k tensor field
is defined by with On a simply connected domain in Euclidean space