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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 Michell solution ： ウィキペディア英語版 Michell solution The Michell solution is a general solution to the elasticity equations in polar coordinates ($r,\; \backslash theta\; \backslash ,$). The solution is such that the stress components are in the form of a Fourier series in $\backslash theta\; \backslash ,$. Michell showed that the general solution can be expressed in terms of an Airy stress function of the form :$$ \begin \varphi &= A_0~r^2 + B_0~r^2~\ln(r) + C_0~\ln(r) + D_0~\theta \\ & + \left(A_1~r + B_1~r^ + B_1^~r~\theta + C_1~r^3 + D_1~r~\ln(r)\right) \cos\theta \\ & + \left(E_1~r + F_1~r^ + F_1^~r~\theta + G_1~r^3 + H_1~r~\ln(r)\right) \sin\theta \\ & + \sum_^ \left(A_n~r^n + B_n~r^ + C_n~r^ + D_n~r^\right)\cos(n\theta) \\ & + \sum_^ \left(E_n~r^n + F_n~r^ + G_n~r^ + H_n~r^\right)\sin(n\theta) \end The terms $A\_1~r~\backslash cos\backslash theta\backslash ,$ and $E\_1~r~\backslash sin\backslash theta\backslash ,$ define a trivial null state of stress and are ignored. == Stress components == The stress components can be obtained by substituting the Michell solution into the equations for stress in terms of the Airy stress function (in cylindrical coordinates). A table of stress components is shown below.〔 J. R. Barber, 2002, ''Elasticity: 2nd Edition'', Kluwer Academic Publishers. 〕 \, ! $\backslash sigma\_\backslash ,$ |- | $r^2\backslash ,$ | $2$ | $0$ | $2$ |- | $r^2~\backslash ln\; r$ | $2~\backslash ln\; r\; +\; 1$ | $0$ | $2~\backslash ln\; r\; +\; 3$ |- | $\backslash ln\; r\backslash ,$ | $r^\backslash ,$ | $0$ | $-r^\backslash ,$ |- | $\backslash theta\backslash ,$ | $0$ | $r^\backslash ,$ | $0$ |- |- | $r^3~\backslash cos\backslash theta\; \backslash ,$ | $2~r~\backslash cos\backslash theta\; \backslash ,$ | $2~r~\backslash sin\backslash theta\; \backslash ,$ | $6~r~\backslash cos\backslash theta\; \backslash ,$ |- | $r\backslash theta~\backslash cos\backslash theta\; \backslash ,$ | $2~r^~\backslash sin\backslash theta\; \backslash ,$ | $0$ | $0$ |- | $r~\backslash ln\; r~\backslash cos\backslash theta\; \backslash ,$ | $r^~\backslash cos\backslash theta\; \backslash ,$ | $r^~\backslash sin\backslash theta\; \backslash ,$ | $r^~\backslash cos\backslash theta\; \backslash ,$ |- | $r^~\backslash cos\backslash theta\; \backslash ,$ | $-2~r^~\backslash cos\backslash theta\; \backslash ,$ | $-2~r^~\backslash sin\backslash theta\; \backslash ,$ | $2~r^~\backslash cos\backslash theta\; \backslash ,$ |- |- | $r^3~\backslash sin\backslash theta\; \backslash ,$ | $2~r~\backslash sin\backslash theta\; \backslash ,$ | $-2~r~\backslash cos\backslash theta\; \backslash ,$ | $6~r~\backslash sin\backslash theta\; \backslash ,$ |- | $r\backslash theta~\backslash sin\backslash theta\; \backslash ,$ | $2~r^~\backslash cos\backslash theta\; \backslash ,$ | $0$ | $0$ |- | $r~\backslash ln\; r~\backslash sin\backslash theta\; \backslash ,$ | $r^~\backslash sin\backslash theta\; \backslash ,$ | $-r^~\backslash cos\backslash theta\; \backslash ,$ | $r^~\backslash sin\backslash theta\; \backslash ,$ |- | $r^~\backslash sin\backslash theta\; \backslash ,$ | $-2~r^~\backslash sin\backslash theta\; \backslash ,$ | $2~r^~\backslash cos\backslash theta\; \backslash ,$ | $2~r^~\backslash sin\backslash theta\; \backslash ,$ |- |- | $r^~\backslash cos(n\backslash theta)\; \backslash ,$ | $-(n+1)(n-2)~r^n~\backslash cos(n\backslash theta)\; \backslash ,$ | $n(n+1)~r^n~\backslash sin(n\backslash theta)\; \backslash ,$ | $(n+1)(n+2)~r^n~\backslash cos(n\backslash theta\; \backslash ,$ |- | $r^~\backslash cos(n\backslash theta)\; \backslash ,$ | $-(n+2)(n-1)~r^~\backslash cos(n\backslash theta)\; \backslash ,$ | $-n(n-1)~r^~\backslash sin(n\backslash theta)\backslash ,$ | $(n-1)(n-2)~r^~\backslash cos(n\backslash theta)$ |- | $r^n~\backslash cos(n\backslash theta)\; \backslash ,$ | $-n(n-1)~r^~\backslash cos(n\backslash theta)\; \backslash ,$ | $n(n-1)~r^~\backslash sin(n\backslash theta)\; \backslash ,$ | $n(n-1)~r^~\backslash cos(n\backslash theta)\; \backslash ,$ |- | $r^~\backslash cos(n\backslash theta)\; \backslash ,$ | $-n(n+1)~r^~\backslash cos(n\backslash theta)\; \backslash ,$ | $-n(n+1)~r^~\backslash sin(n\backslash theta)\; \backslash ,$ | $n(n+1)~r^~\backslash cos(n\backslash theta)\; \backslash ,$ |- |- | $r^~\backslash sin(n\backslash theta)\; \backslash ,$ | $-(n+1)(n-2)~r^n~\backslash sin(n\backslash theta)\; \backslash ,$ | $-n(n+1)~r^n~\backslash cos(n\backslash theta)\; \backslash ,$ | $(n+1)(n+2)~r^n~\backslash sin(n\backslash theta\; \backslash ,$ |- | $r^~\backslash sin(n\backslash theta)\; \backslash ,$ | $-(n+2)(n-1)~r^~\backslash sin(n\backslash theta)\; \backslash ,$ | $n(n-1)~r^~\backslash cos(n\backslash theta)\backslash ,$ | $(n-1)(n-2)~r^~\backslash sin(n\backslash theta)$ |- | $r^n~\backslash sin(n\backslash theta)\; \backslash ,$ | $-n(n-1)~r^~\backslash sin(n\backslash theta)\; \backslash ,$ | $-n(n-1)~r^~\backslash cos(n\backslash theta)\; \backslash ,$ | $n(n-1)~r^~\backslash sin(n\backslash theta)\; \backslash ,$ |- | $r^~\backslash sin(n\backslash theta)\; \backslash ,$ | $-n(n+1)~r^~\backslash sin(n\backslash theta)\; \backslash ,$ | $n(n+1)~r^~\backslash cos(n\backslash theta)\; \backslash ,$ | $n(n+1)~r^~\backslash sin(n\backslash theta)\; \backslash ,$ |} 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「Michell solution」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. 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