Saint-Venant's Principle

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

${\begin{aligned}{\frac {\partial }{\partial x_{1}}}&\left({\frac {\partial ^{2}\varepsilon _{22}}{\partial x_{3}^{2}}}+{\frac {\partial ^{2}\varepsilon _{33}}{\partial x_{2}^{2}}}-2{\frac {\partial ^{2}\varepsilon _{23}}{\partial x_{2}\partial x_{3}}}\right)-{\frac {\partial }{\partial x_{2}}}\left[{\frac {\partial ^{2}\varepsilon _{22}}{\partial x_{1}\partial x_{3}}}-{\frac {\partial }{\partial x_{2}}}\left({\frac {\partial \varepsilon _{23}}{\partial x_{1}}}-{\frac {\partial \varepsilon _{13}}{\partial x_{2}}}+{\frac {\partial \varepsilon _{12}}{\partial x_{3}}}\right)\right]\\&-{\frac {\partial }{\partial x_{3}}}\left[{\frac {\partial ^{2}\varepsilon _{33}}{\partial x_{1}\partial x_{2}}}-{\frac {\partial }{\partial x_{3}}}\left({\frac {\partial \varepsilon _{23}}{\partial x_{1}}}+{\frac {\partial \varepsilon _{13}}{\partial x_{2}}}-{\frac {\partial \varepsilon _{12}}{\partial x_{3}}}\right)\right]=0\end{aligned}}$

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 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 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-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

$V_{i_{1}..i_{k}j_{1}...j_{k}}=\sum \limits _{p=0}^{k}(-1)^{p}{k \choose p}T_{i_{1}..i_{k-p}j_{1}...j_{p},j_{p+1}...j_{k}i_{k-p+1}...i_{k}}$

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)