Saint-Venant's compatibility condition

In the mathematical theory of elasticity, Saint-Venant's compatibility condition defines the relationship between the strain ε {\displaystyle \varepsilon } and a displacement field u {\displaystyle \ u} by ϵ i j = 1 2 ( ∂ u i ∂ x j + ∂ u j ∂ x i ) {\displaystyle \epsilon _{ij}={\frac {1}{2}}\left({\frac {\partial u_{i}}{\partial x_{j}}}+{\frac {\partial u_{j}}{\partial x_{i}}}\right)} where 1 ≤ i , j ≤ 3 {\displaystyle 1\leq i,j\leq 3} . 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 n ≥ 2 {\displaystyle n\geq 2} == Rank 2 tensor fields == For a symmetric rank 2 tensor field F {\displaystyle F} in n-dimensional Euclidean space ( n ≥ 2 {\displaystyle n\geq 2} ) the integrability condition takes the form of the vanishing of the Saint-Venant's tensor W ( F ) {\displaystyle W(F)} defined by W i j k l = ∂ 2 F i j ∂ x k ∂ x l + ∂ 2 F k l ∂ x i ∂ x j − ∂ 2 F i l ∂ x j ∂ x k − ∂ 2 F j k ∂ x i ∂ x l {\displaystyle W_{ijkl}={\frac {\partial ^{2}F_{ij}}{\partial x_{k}\partial x_{l}}}+{\frac {\partial ^{2}F_{kl}}{\partial x_{i}\partial x_{j}}}-{\frac {\partial ^{2}F_{il}}{\partial x_{j}\partial x_{k}}}-{\frac {\partial ^{2}F_{jk}}{\partial x_{i}\partial x_{l}}}} 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 1860 and proved rigorously by Beltrami in 1886.

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