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# Saint-Venant's compatibility condition

In the mathematical theory of elasticity the strain ε is related to a displacement field U by

$\epsilon_{ij} = \frac{1}{2} \left( \frac{\partial U_i}{\partial x_j} + \frac{\partial U_i}{\partial x_j} \right)$

Saint-Venant 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 Saint-Venants tensor[1] defined by

$W_{ijkl} = \frac{\partial^2 \epsilon_{ij}}{\partial x_k \partial x_l} + \frac{\partial^2 \epsilon_{kl}}{\partial x_i \partial x_j} - \frac{\partial^2 \epsilon_{il}}{\partial x_j \partial x_k} -\frac{\partial^2 \epsilon_{jk}}{\partial x_i \partial x_l}$

Due to the symmetry conditions Wijkl = Wklij = − Wjikl = Wijlk there are only six (in the three dimensional case) distinct components of W. These six equations are not independent as verified by for example

$\frac{\frac{\partial^2 \epsilon_{22}}{\partial x_3^2} + \frac{\partial^2 \epsilon_{33}}{\partial x_2^2} - 2 \frac{\partial^2 \epsilon_{23}}{\partial x_2 \partial x_3}}{\partial x_1} = \frac{\frac{\partial^2 \epsilon_{22}}{\partial x_1 \partial x_3} - \frac{\partial}{\partial x_2} \left ( \frac{\partial \epsilon_{23}}{\partial x_1} - \frac{\partial \epsilon_{13}}{\partial x_2} + \frac{\partial \epsilon_{12}}{\partial x_3}\right)}{\partial x_2} + \frac{\frac{\partial^2 \epsilon_{33}}{\partial x_1 \partial x_2} - \frac{\partial}{\partial x_3} \left ( \frac{\partial \epsilon_{23}}{\partial x_1} + \frac{\partial \epsilon_{13}}{\partial x_2} - \frac{\partial \epsilon_{12}}{\partial x_3}\right)}{\partial x_3}$

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 skew-symmetric tensor fields (differential forms). The result can be generalized to higher rank symmetric tensor fields[3]. 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

$(dF)_{i_1... i_k i_{k+1}} = F_{(i_1... i_k,i_{k+1})}$

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 Saint-Venant tensor W of a symmetric rank-k tensor field U is defined by

$W_{i_1..i_k j_1...j_k}=V_{(i_1..i_k)(j_1...j_k)}$

with

$V_{i_1..i_k j_1...j_k} = \sum\limits_{p=1}^{k} (-1)^p {m \choose p} U_{i_1..i_{k-p}j_1...j_p,j_{p+1}...j_k i_{k-p+1}...i_k }$

On a simply connected domain in Euclidean space W = 0 implies that U = dF for some rank k-1 symmetric tensor field F.

## References

1. ^ N.I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity. Leyden: Noordhoff Intern. Publ., 1975.
2. ^ C Amrouche, PG Ciarlet, L Gratie, S Kesavan, On Saint Venant's compatibility conditions and Poincaré's lemma, C. R. Acad. Sci. Paris, Ser. I, 342 (2006), 887-891. doi:10.1016/j.crma.2006.03.026
3. ^ V.A. Sharafutdinov, Integral Geometry of Tensor Fields, VSP 1994,ISBN 906764165X. Chapter 2.