View source for Saint-Venant's Principle

In the mathematical theory of [[elasticity (mathematics)|elasticity]] the strain <math>\varepsilon</math> is related to a displacement field <math>\ u</math> by
:<math>\varepsilon_{ij} = \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right)</math>
[[Adhémar Jean Claude Barré de Saint-Venant|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 <math>W(\varepsilon)</math> <ref>N.I. Muskhelishvili,   Some Basic Problems of the Mathematical Theory of Elasticity. Leyden: Noordhoff Intern. Publ., 1975.</ref> defined by 
:<math>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}
</math>

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 [[Eugenio Beltrami|Beltrami]] in 1886.<ref name="Amrouche"/> 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]]<ref>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}}
</ref> 

Due to the symmetry conditions <math>W_{ijkl}=W_{klij}=-W_{jikl}=W_{ijlk}</math> there are only six (in the three dimensional case) distinct components of <math>W</math> For example all components can be deduced from <math>W_{ijkl}</math> 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
:<math>
 \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}
</math>
and there are two further relations obtained by cyclic permutation. 

In its simplest form of course the components of <math>\varepsilon</math> must be assumed twice continuously differentiable, but more recent work<ref name="Amrouche">C Amrouche, [[Philippe G. Ciarlet|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}}</ref> 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 <math>\nabla \times \nabla \times \mathbf{A}</math>, where <math>\mathbf{A}</math> is a symmetric tensor field. The matrix curl<ref name="Amrouche" /> of a symmetric rank 2 tensor  field ''T'' is defined by
:<math>
(\nabla \times T)_{ij}= \epsilon_{ilk} \partial_l T_{jk}
</math>
where <math>\ \epsilon</math> is the [[permutation symbol]]. The operator <math>\nabla\times\nabla\times</math> maps symmetric tensor fields to symmetric tensor fields.<ref name="Amrouche" /> The vanishing of the Saint Venant's tensor ''W(T)'' is equivalent to <math>\nabla\times\nabla\times \mathbf{T}=\mathbf{0}</math>. This illustrates more clearly the six independent components of ''W(T)''. The divergence of a tensor field <math> (\nabla \cdot T)_i = \partial_j T_{ij}</math> satisfies <math>\nabla\cdot\nabla\times\nabla\times\mathbf{T}=\mathbf{0}</math>. 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#Lie_derivative_of_tensor_fields|Lie derivative]] of the metric tensor ''g'' with respect to the vector field.
:<math> T_{ij}=(\mathcal L_U g)_{ij} = U_{i;j}+U_{j;i}
</math>
where indices following a semicolon indicate covariant differentiation. The vanishing of <math>W(T)</math> is thus the integrability condition for local existence of <math>U</math> 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 form]]s). The result can be generalized to higher rank [[symmetric tensor]] fields.<ref>
V.A. Sharafutdinov, Integral Geometry of Tensor Fields, VSP 1994,ISBN 90-6764-165-0. Chapter 2.[http://www.math.nsc.ru/~sharafutdinov/files/book.pdf on-line version]</ref> 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
:<math> (dF)_{i_1... i_k i_{k+1}} = F_{(i_1... i_k,i_{k+1})}</math>
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 <math>W</math> of a  symmetric rank-k tensor field <math>T</math> is defined by
:<math> W_{i_1..i_k j_1...j_k}=V_{(i_1..i_k)(j_1...j_k)}</math>
with
:<math> 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 } </math>
On a [[simply connected]] domain in Euclidean space <math>W=0</math> implies that <math> T = dF</math> for some rank k-1 symmetric tensor field <math>F</math>.

==References==
<references/>

== See also ==
* [[Compatibility (mechanics)]]

[[Category:Elasticity (physics)]]
[[Category:Tensors]]
[[Category:Partial differential equations]]