Saint-Venant compatibility condition

Remarks

Necessity is a direct computation: substituting $\varepsilon = \tfrac12 (\nabla u + (\nabla u)^T)$, all third derivatives of $u$ cancel. For sufficiency, one first builds the infinitesimal rotation field $\omega$ from $\varepsilon$ by a line integral. The condition says that the integrand is curl-free, and simple connectivity makes the integral independent of the path. Then $u$ is recovered from $\varepsilon + \omega$ by a second line integral. Of the $81$ equations only $6$ are independent; they say that the double curl of $\varepsilon$ vanishes. Every field $\varepsilon$ whose entries are affine in $X$ is compatible. The field with $\varepsilon_{11} = X_2^2$ and all other entries $0$ is not: with $i = j = 1$ and $k = l = 2$ the left side is $\partial_2^2 \varepsilon_{11} = 2 \neq 0$. Without simple connectivity the condition is only locally sufficient; in a hollow cylinder a compatible strain field may correspond to a multivalued displacement, which models a dislocation.

Going deeper

For a student

In linear elasticity one may solve for stresses first; the compatibility equations then ensure that the corresponding strains come from a displacement. Written in terms of stresses through Hooke's law, they become the Beltrami–Michell equations. In two dimensions only one equation remains, $\partial_2^2 \varepsilon_{11} + \partial_1^2 \varepsilon_{22} = 2 \partial_1 \partial_2 \varepsilon_{12}$.