Proof of the Saint-Venant compatibility equations

Question

In the Mechanics of deformable solids, and specifically in the Theory of Elasticity, the following mathematical relation is known:
$$ 2 cdot frac{partial^2 epsilon_{12}}{partial x_{1} partial x_{2}} = frac{partial^2 epsilon_{11}}{partial x_{2}^2} + frac{partial^2 epsilon_{22}}{partial x_{1}^2}$$
which makes it possible to establish a connection between tangential or shear deformation and normal deformations.
I have tried to locate in several books on Elasticity Theory a mathematical demonstration of the above equation, but the search has not been satisfactory, since I have not found anything in any book. On the other hand, I have tried to prove it without going to books, but to be honest I don't know how to proceed.

nicoguaro · Answer

Suppose that you have a strain field and you want to obtain the displacements. You could directly integrate the strain-displacement relations
$$frac{partial u_j}{partial x_i} + frac{partial u_i}{partial x_j} = 2epsilon_{ij}, .$$
This is a system of six equations and three unknowns. Theoretically, you only need three equations to determine the displacement fields. You can't pick the strain components $epsilon_{ij}$ arbitrarily.
To have a unique solution for $u_i$ you need some constraints on the strains. If you differentiate the above equation an interchange some of the indices you obtain
$$frac{partial^2 epsilon_{ij}}{partial x_k partial x_l}
+ frac{partial^2 epsilon_{kl}}{partial x_i partial x_j}
- frac{partial^2 epsilon_{il}}{partial x_j partial x_l}
- frac{partial^2 epsilon_{jl}}{partial x_i partial x_k}  = 0, .$$
From there you have 81 equations, but some of them are trivial. From those only six are nontrivial, and one of those is the one you present.

Proof of the Saint-Venant compatibility equations

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