Saint-Venant's Compatibility Condition - Rank 2 Tensor Fields

Rank 2 Tensor Fields

The integrability condition takes the form of the vanishing of the Saint-Venant's tensor 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 there are only six (in the three dimensional case) distinct components of For example all components can be deduced from 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 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, where 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 is the permutation symbol. The operator maps symmetric tensor fields to symmetric tensor fields. The vanishing of the Saint Venant's tensor W(T) is equivalent to . This illustrates more clearly the six independent components of W(T). The divergence of a tensor field satisfies . 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 is thus the integrability condition for local existence of in the Euclidean case.

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