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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Saint-Venant's compatibility condition ： ウィキペディア英語版 Saint-Venant's compatibility condition In the mathematical theory of elasticity the strain $\backslash varepsilon$ is related to a displacement field $\backslash \; u$ by :$\backslash varepsilon\_\; =\; \backslash frac\; \backslash left(\; \backslash frac\; +\; \backslash frac\; \backslash right)$ where $1\backslash le\; i,j\; \backslash le\; 3$. 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 on spaces of dimension $n\backslash ge\; 2$ ==Rank 2 tensor fields== For a symmetric rank 2 tensor field $F$ in n-dimensional Euclidean space the integrability condition takes the form of the vanishing of the Saint-Venant's tensor $W(F)$ 〔N.I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity. Leyden: Noordhoff Intern. Publ., 1975.〕 defined by :$W\_\; =\; \backslash frac\; +$ \frac - \frac -\frac 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〔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 〕 The Saint-Venant tensor $W$ is closely related to the Riemann curvature tensor $R\_$. Indeed the first variation $R$ about the Euclidean metric with a perturbation in the metric $F$ is precisely $W$.〔Philippe G. Ciarlet , Cristinel Mardare , Ming Shen, Recovery of a displacement field from its linearized strain tensor field in curvilinear coordinates, C. R. Acad. Sci. Paris, Ser. I 344 (2007) 535–540〕 Consequently the number of independent components of $W$ is the same as $R$〔 D. V. Georgiyecskii and B. Ye. Pobedrya,The number of independent compatibility equations in the mechanics of deformable solids, Journal of Applied Mathematicsand Mechanics,68 (2004)941-946〕 specifically $\backslash frac$ for dimension n.〔Weisstein, Eric W. Riemann Tensor. From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/RiemannTensor.html〕 Specifically for $n=2$, $W$ has only one independent component where as for $n=3$ there are six. In its simplest form of course the components of $F$ must be assumed twice continuously differentiable, but more recent work〔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. 〕 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 a reduced form of $W$ the Kröner tensor 〔 :$$ K_} = \epsilon_\epsilon_F_ where $\backslash epsilon$ is the permutation symbol. For $n=3$, $K$is a symmetric rank 2 tensor field. The vanishing of $K$ is equivalent to the vanishing of $W$ and this also shows that there are six independent components for the important case of three dimensions. While this still involves two derivatives rather than the one in the Poincaré lemma, it is possible to reduce to a problem involving first derivatives by introducing more variables and it has been shown that the resulting 'elasticity complex' is equivalent to the de Rham complex.〔M Eastwood, A complex from linear elasticity, Rendiconti del circolo mathematico di Palermo, Ser II Suppl 63 (2000), pp23-29〕 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. :$T\_=(\backslash mathcal\; L\_U\; g)\_\; =\; U\_+U\_$ where indices 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. As noted above this coincides with the vanishing of the linearization of the Riemann curvature tensor about the Euclidean metric. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Saint-Venant's compatibility condition」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.