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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 Föppl–von Kármán equations ： ウィキペディア英語版 Föppl–von Kármán equations The Föppl–von Kármán equations, named after August Föppl〔Föppl, A., "Vorlesungen über technische Mechanik", ''B.G. Teubner'', Bd. 5., p. 132, Leipzig, Germany (1907)〕 and Theodore von Kármán,〔von Kármán, T., "Festigkeitsproblem im Maschinenbau," ''Encyk. D. Math. Wiss.'' IV, 311–385 (1910)〕 are a set of nonlinear partial differential equations describing the large deflections of thin flat plates.〔E. Cerda and L. Mahadevan, 2003, "Geometry and Physics of Wrinkling" (Phys. Rev. Lett. 90, 074302 (2003) )〕 With applications ranging from the design of submarine hulls to the mechanical properties of cell wall,〔http://focus.aps.org/story/v27/st6〕 the equations are notoriously difficult to solve, and take the following form: 〔"Theory of Elasticity". L. D. Landau, E. M. Lifshitz, (3rd ed. ISBN 0-7506-2633-X)〕 :$$ \begin (1) \qquad & \frac\Delta^2 w-h\frac\left(\sigma_\frac\right)=P \\ (2) \qquad & \frac=0 \end where is the Young's modulus of the plate material (assumed homogeneous and isotropic), is the Poisson's ratio, is the thickness of the plate, is the out–of–plane deflection of the plate, is the external normal force per unit area of the plate, is the Cauchy stress tensor, and are indices that take values of 1 or 2. The 2-dimensional biharmonic operator is defined as〔The 2-dimensional Laplacian, , is defined as $\backslash Delta\; w\; :=\; \backslash frac\; =\; \backslash frac\; +\; \backslash frac$〕 :$$ \Delta^2 w := \frac\left(kinematic assumptions and the constitutive relations for the plate. Equations (2) are the two equations for the conservation of linear momentum in two dimensions where it is assumed that the out–of–plane stresses () are zero. == Validity of the Föppl–von Kármán equations == While the Föppl–von Kármán equations are of interest from a purely mathematical point of view, the physical validity of these equations is questionable.〔(von Karman plate equations http://imechanica.org/node/6618 Accessed Tue July 30 2013 14:20. )〕 Ciarlet states: ''The two-dimensional von Karman equations for plates, originally proposed by von Karman (), play a mythical role in applied mathematics. While they have been abundantly, and satisfactorily, studied from the mathematical standpoint, as regards notably various questions of existence, regularity, and bifurcation, of their solutions, their physical soundness has been often seriously questioned.'' Reasons include the facts that # the theory depends on an approximate geometry which is not clearly defined # a given variation of stress over a cross-section is assumed arbitrarily # a linear constitutive relation is used that does not correspond to a known relation between well defined measures of stress and strain # some components of strain are arbitrarily ignored # there is a confusion between reference and deformed configurations which makes the theory inapplicable to the large deformations for which it was apparently devised. Conditions under which these equations are actually applicable and will give reasonable results when solved are discussed in Ciarlet.〔 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「Föppl–von Kármán equations」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.