# Saint-Venant's compatibility condition

In the mathematical theory of elasticity the strain $\varepsilon$ is related to a displacement field $\ u$ by

$\varepsilon_{ij} = \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right)$

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.

## Rank 2 tensor fields

The integrability condition takes the form of the vanishing of the Saint-Venant's tensor $W(\varepsilon)$ [1] defined by

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

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 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]

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 $W$ For example all components can be deduced from $W_{ijkl}$ the indices ijkl=2323, 2331, 1223, 1313, 1312 and 1212. The six components in such minimal sets are not independent as functions as they satisfy partial differential equations such as

\begin{align} \frac{\partial}{\partial x_1}& \left( \frac{\partial^2 \varepsilon_{22}}{\partial x_3^2} + \frac{\partial^2 \varepsilon_{33}}{\partial x_2^2} - 2 \frac{\partial^2 \varepsilon_{23}}{\partial x_2 \partial x_3}\right) - \frac{\partial}{\partial x_2}\left[ \frac{\partial^2 \varepsilon_{22}}{\partial x_1 \partial x_3} - \frac{\partial}{\partial x_2} \left ( \frac{\partial \varepsilon_{23}}{\partial x_1} - \frac{\partial \varepsilon_{13}}{\partial x_2} + \frac{\partial \varepsilon_{12}}{\partial x_3}\right) \right] \\ & - \frac{\partial}{\partial x_3}\left[ \frac{\partial^2 \varepsilon_{33}}{\partial x_1 \partial x_2} - \frac{\partial}{\partial x_3} \left ( \frac{\partial \varepsilon_{23}}{\partial x_1} + \frac{\partial \varepsilon_{13}}{\partial x_2} - \frac{\partial \varepsilon_{12}}{\partial x_3}\right)\right]=0 \end{align}

and there are two further relations obtained by cyclic permutation.

In its simplest form of course the components of $\varepsilon$ 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 the operator $\nabla \times \nabla \times \mathbf{A}$, where $\mathbf{A}$ is a symmetric tensor field. The matrix curl[2] of a symmetric rank 2 tensor field T is defined by

$(\nabla \times T)_{ij}= \epsilon_{ilk} \partial_l T_{jk}$

where $\ \epsilon$ is the permutation symbol. The operator $\nabla\times\nabla\times$ maps symmetric tensor fields to symmetric tensor fields.[2] The vanishing of the Saint Venant's tensor W(T) is equivalent to $\nabla\times\nabla\times \mathbf{T}=\mathbf{0}$. This illustrates more clearly the six independent components of W(T). The divergence of a tensor field $(\nabla \cdot T)_i = \partial_j T_{ij}$ satisfies $\nabla\cdot\nabla\times\nabla\times\mathbf{T}=\mathbf{0}$. This exactly the three first order differential equations satisfied by the components of W(T) mentioned above.

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.

$T_{ij}=(\mathcal L_U g)_{ij} = U_{i;j}+U_{j;i}$

where indices following a semicolon indicate covariant differentiation. The vanishing of $W(T)$ is thus the integrability condition for local existence of $U$ in the Euclidean case.

## Generalization to higher rank tensors

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.[4] 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 $T$ is defined by

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

with

$V_{i_1..i_k j_1...j_k} = \sum\limits_{p=0}^{k} (-1)^p {k \choose p} T_{i_1..i_{k-p}j_1...j_p,j_{p+1}...j_k i_{k-p+1}...i_k }$

On a simply connected domain in Euclidean space $W=0$ implies that $T = dF$ 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. ^ a b c d 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. ^ Giuseppe Geymonat, Francoise Krasucki, Hodge decomposition for symmetric matrix fields and the elasticity complex in Lipschitz domains,COMMUNICATIONS ON PURE AND APPLIED ANALYSIS, Volume 8, Number 1, January 2009, pp. 295–309 doi:10.3934/cpaa.2009.8.295
4. ^ V.A. Sharafutdinov, Integral Geometry of Tensor Fields, VSP 1994,ISBN 90-6764-165-0. Chapter 2.on-line version