Saint-Venant's compatibility condition: Difference between revisions
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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 form]]s). The result can be generalized to higher rank [[symmetric tensor]] fields.<ref> |
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 form]]s). The result can be generalized to higher rank [[symmetric tensor]] fields.<ref> |
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V.A. Sharafutdinov, Integral Geometry of Tensor Fields, VSP 1994,ISBN 906764165X. Chapter 2.</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 |
V.A. Sharafutdinov, Integral Geometry of Tensor Fields, VSP 1994,ISBN 906764165X. 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 |
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:<math> (dF)_{i_1... i_k i_{k+1}} = F_{(i_1... i_k,i_{k+1})}</math> |
:<math> (dF)_{i_1... i_k i_{k+1}} = F_{(i_1... i_k,i_{k+1})}</math> |
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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>U</math> is defined by |
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>U</math> is defined by |
Revision as of 10:43, 29 March 2008
It has been suggested that this article be merged into Strain (materials science) and Talk:Strain (materials science)#Merger proposal. (Discuss) Proposed since February 2008. |
In the mathematical theory of elasticity the strain is related to a displacement field by
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[1] defined by
Due to the symmetry conditions there are only six (in the three dimensional case) distinct components of . These six equations are not independent as verified by for example
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 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.
where indicies following a semicolon indicate covariant differentiation. The vanishing of is thus the integrability condition for local existence of in the Euclidean case.
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
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 of a symmetric rank-k tensor field is defined by
with
On a simply connected domain in Euclidean space implies that for some rank k-1 symmetric tensor field .
References
- ^ N.I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity. Leyden: Noordhoff Intern. Publ., 1975.
- ^ 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
- ^ V.A. Sharafutdinov, Integral Geometry of Tensor Fields, VSP 1994,ISBN 906764165X. Chapter 2.on-line version