Saint-Venant's compatibility condition

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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)

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.

[edit] Rank 2 tensor fields

The integrability condition takes the form of the vanishing of the Saint-Venant's tensor[1] 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}

Due to the symmetry conditions Wijkl = Wklij = − Wjikl = Wijlk there are only six (in the three dimensional case) distinct components of W. These six equations are not independent as verified by for example

\frac{\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}}{\partial x_1} = \frac{\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)}{\partial x_2} + \frac{\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)}{\partial x_3}

and there are two further relations obtained by cyclic permutation. However, in practise the six equations are preferred. In its simplest form of course the components of \varepsilon must be assumed twice continuously differentiable, but more recent work[2] proves the result in a much more general case.

In differential geometry the symmetrised 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 indicies 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.

[edit] Generalization to higher rank tensors

Saint-Vanants compatibility condition can be thought of as an analogue, for symmetric tensor fields, of Poincare's lemma for skew-symmetric tensor fields (differential forms). The result can be generalized to higher rank symmetric tensor fields.[3] 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 U 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=1}^{k} (-1)^p {m \choose p} U_{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 U = dF for some rank k-1 symmetric tensor field F.

[edit] References

  1. ^ N.I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity. Leyden: Noordhoff Intern. Publ., 1975.
  2. ^ C Amrouche, 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
  3. ^ V.A. Sharafutdinov, Integral Geometry of Tensor Fields, VSP 1994,ISBN 906764165X. Chapter 2.on-line version