Saint-Venant's Principle

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

${\epsilon }_{ij}=\frac{1}{2}(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i})$

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 $W(\epsilon )$ defined by

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

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

$\begin{array}{rl}\frac{\partial }{\partial x_1}& (\frac{{\partial }^2{\epsilon }_{22}}{\partial x_3^2}+\frac{{\partial }^2{\epsilon }_{33}}{\partial x_2^2}-2\frac{{\partial }^2{\epsilon }_{23}}{\partial x_2\partial x_3})-\frac{\partial }{\partial x_2}[\frac{{\partial }^2{\epsilon }_{22}}{\partial x_1\partial x_3}-\frac{\partial }{\partial x_2}(\frac{\partial {\epsilon }_{23}}{\partial x_1}-\frac{\partial {\epsilon }_{13}}{\partial x_2}+\frac{\partial {\epsilon }_{12}}{\partial x_3})]\\ & -\frac{\partial }{\partial x_3}[\frac{{\partial }^2{\epsilon }_{33}}{\partial x_1\partial x_2}-\frac{\partial }{\partial x_3}(\frac{\partial {\epsilon }_{23}}{\partial x_1}+\frac{\partial {\epsilon }_{13}}{\partial x_2}-\frac{\partial {\epsilon }_{12}}{\partial x_3})]=0\end{array}$

and there are two further relations obtained by cyclic permutation.

In its simplest form of course the components of $\epsilon$ 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 Poincaré's lemma can be understood more clearly using the operator $\mathrm{\nabla }\times \mathrm{\nabla }\times \mathbf{A}$, where $\mathbf{A}$ is a symmetric tensor field. The matrix curl of a symmetric rank 2 tensor field T is defined by

$(\mathrm{\nabla }\times T{)}_{ij}={ϵ}_{ilk}{\partial }_l{T}_{jk}$

where $ϵ$ is the permutation symbol. The operator $\mathrm{\nabla }\times \mathrm{\nabla }\times$ maps symmetric tensor fields to symmetric tensor fields. The vanishing of the Saint Venant's tensor W(T) is equivalent to $\mathrm{\nabla }\times \mathrm{\nabla }\times \mathbf{T}=\mathbf{0}$. This illustrates more clearly the six independent components of W(T). The divergence of a tensor field $(\mathrm{\nabla }\cdot T{)}_i={\partial }_j{T}_{ij}$ satisfies $\mathrm{\nabla }\cdot \mathrm{\nabla }\times \mathrm{\nabla }\times \mathbf{T}=\mathbf{0}$. 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. 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

$V_{i_1..i_k{j}_1...j_k}=\sum _{p=0}^k(-1{)}^p(\genfrac{}{}{0ex}{}{k}{p})T_{i_1..i_{k-p}{j}_1...j_p,j_{p+1}...j_k{i}_{k-p+1}...i_k}$

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)