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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 Voigt notation ： ウィキペディア英語版 Voigt notation In mathematics, Voigt notation or Voigt form in multilinear algebra is a way to represent a symmetric tensor by reducing its order. There are a few variants and associated names for this idea: Mandel notation, Mandel–Voigt notation and Nye notation are others found. Kelvin notation is a revival by Helbig (1994) of old ideas of Lord Kelvin. The differences here lie in certain weights attached to the selected entries of the tensor. Nomenclature may vary according to what is traditional in the field of application. For example, a 2×2 symmetric tensor ''X'' has only three distinct elements, the two on the diagonal and the other being off-diagonal. Thus it can be expressed as the vector :$\backslash langle\; x\_,\; x\_,\; x\_\backslash rangle$. As another example: The stress tensor (in matrix notation) is given as :$\backslash boldsymbol=$ \left( \sigma_ & \sigma_ & \sigma_ \\ \sigma_ & \sigma_ & \sigma_ \\ \sigma_ & \sigma_ & \sigma_ \end}\right ). In Voigt notation it is simplified to a 6-dimensional vector: :$\backslash tilde\backslash sigma=\; (\backslash sigma\_,\; \backslash sigma\_,\; \backslash sigma\_,$ \sigma_,\sigma_,\sigma_) \equiv (\sigma_1, \sigma_2, \sigma_3, \sigma_4, \sigma_5, \sigma_6). The strain tensor, similar in nature to the stress tensor -- both are symmetric second-order tensors --, is given in matrix form as :$\backslash boldsymbol=$ \left( \epsilon_ & \epsilon_ & \epsilon_ \\ \epsilon_ & \epsilon_ & \epsilon_ \\ \epsilon_ & \epsilon_ & \epsilon_ \end}\right ). Its representation in Voigt notation is :$\backslash tilde\backslash epsilon=\; (\backslash epsilon\_,\; \backslash epsilon\_,\; \backslash epsilon\_,$ \gamma_,\gamma_,\gamma_) \equiv (\epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4, \epsilon_5, \epsilon_6), where $\backslash gamma\_=2\backslash epsilon\_$, $\backslash gamma\_=2\backslash epsilon\_$, and $\backslash gamma\_=2\backslash epsilon\_$ are engineering shear strains. The benefit of using different representations for stress and strain is that the scalar invariance :$\backslash boldsymbol\backslash cdot\backslash boldsymbol\; =\; \backslash sigma\_\backslash epsilon\_\; =\; \backslash tilde\backslash sigma\; \backslash cdot\; \backslash tilde\backslash epsilon$ is preserved. Likewise, a three-dimensional symmetric fourth-order tensor can be reduced to a 6×6 matrix. == Mnemonic rule == A simple mnemonic rule for memorizing Voigt notation is as follows: * Write down the second order tensor in matrix form (in the example, the stress tensor) * Strike out the diagonal * Continue on the third column * Go back to the first element along the first row. Voigt indexes are numbered consecutively from the starting point to the end (in the example, the numbers in blue). File:Voigt notation Mnemonic rule.png 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「Voigt notation」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.