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 $W(\varepsilon)$ 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. 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

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 proves the result in a much more general case.

The relation between Saint-Venant's compatibility condition and Poincare'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 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. 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-Vanant's compatibility condition 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. 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 $F$.

See also

• Compatibility (mechanics)