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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 Infinitesimal strain theory ： ウィキペディア英語版 Infinitesimal strain theory In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimally smaller) than any relevant dimension of the body; so that its geometry and the constitutive properties of the material (such as density and stiffness) at each point of space can be assumed to be unchanged by the deformation. With this assumption, the equations of continuum mechanics are considerably simplified. This approach may also be called small deformation theory, small displacement theory, or small displacement-gradient theory. It is contrasted with the finite strain theory where the opposite assumption is made. The infinitesimal strain theory is commonly adopted in civil and mechanical engineering for the stress analysis of structures built from relatively stiff elastic materials like concrete and steel, since a common goal in the design of such structures is to minimize their deformation under typical loads. ==Infinitesimal strain tensor== For ''infinitesimal deformations'' of a continuum body, in which the displacements and the displacement gradients are small compared to unity, i.e., $\backslash |\backslash mathbf\; u\backslash |\; \backslash ll\; 1\; \backslash ,\backslash !$ and $\backslash |\backslash nabla\; \backslash mathbf\; u\backslash |\; \backslash ll\; 1\; \backslash ,\backslash !$, it is possible to perform a ''geometric linearisation'' of the Lagrangian finite strain tensor $\backslash mathbf\; E\backslash ,\backslash !$, and the Eulerian finite strain tensor $\backslash mathbf\; e\backslash ,\backslash !$. In such a linearisation, the non-linear or second-order terms of the finite strain tensor are neglected. Thus we have :$\backslash mathbf\; E\; =\backslash frac\backslash left(\backslash nabla\_\backslash mathbf\; u\; +\; (\backslash nabla\_\backslash mathbf\; u)^T\; +\; (\backslash nabla\_\backslash mathbf\; u)^T\backslash nabla\_\backslash mathbf\; u\backslash right)\backslash approx\; \backslash frac\backslash left(\backslash nabla\_\backslash mathbf\; u\; +\; (\backslash nabla\_\backslash mathbf\; u)^T\backslash right)\backslash ,\backslash !$ or :$E\_=\backslash frac\backslash left(\backslash frac+\backslash frac+\backslash frac\backslash frac\backslash right)\backslash approx\; \backslash frac\backslash left(\backslash frac+\backslash frac\backslash right)\backslash ,\backslash !$ and :$\backslash mathbf\; e\; =\backslash frac\backslash left(\backslash nabla\_\backslash mathbf\; u\; +\; (\backslash nabla\_\backslash mathbf\; u)^T\; -\; \backslash nabla\_\backslash mathbf\; u(\backslash nabla\_\backslash mathbf\; u)^T\backslash right)\backslash approx\; \backslash frac\backslash left(\backslash nabla\_\backslash mathbf\; u\; +\; (\backslash nabla\_\backslash mathbf\; u)^T\backslash right)\backslash ,\backslash !$ or :$e\_=\backslash frac\backslash left(\backslash frac\; +\backslash frac-\backslash frac\backslash frac\backslash right)\backslash approx\; \backslash frac\backslash left(\backslash frac\; +\backslash frac\backslash right)\backslash ,\backslash !$ This linearisation implies that the Lagrangian description and the Eulerian description are approximately the same as there is little difference in the material and spatial coordinates of a given material point in the continuum. Therefore, the material displacement gradient components and the spatial displacement gradient components are approximately equal. Thus we have :$\backslash mathbf\; E\; \backslash approx\; \backslash mathbf\; e\; \backslash approx\; \backslash boldsymbol\; \backslash varepsilon\; =\; \backslash frac\backslash left((\backslash nabla\backslash mathbf\; u)^T\; +\; \backslash nabla\backslash mathbf\; u\backslash right)\; \backslash qquad$ or $\backslash qquad\; E\_\backslash approx\; e\_\backslash approx\backslash varepsilon\_=\backslash frac\backslash left(u\_+u\_\backslash right)\backslash ,\backslash !$ where $\backslash varepsilon\_\backslash ,\backslash !$ are the components of the ''infinitesimal strain tensor'' $\backslash boldsymbol\; \backslash varepsilon\backslash ,\backslash !$, also called ''Cauchy's strain tensor'', ''linear strain tensor'', or ''small strain tensor''. :$\backslash begin$ \varepsilon_ &= \frac\left(u_+u_\right) \\ &= \left(& \varepsilon_ & \varepsilon_ \\ \varepsilon_ & \varepsilon_ & \varepsilon_ \\ \varepsilon_ & \varepsilon_ & \varepsilon_ \\ \end\right ) \\ &= \left( \frac & \frac \left(\frac+\frac\right) & \frac \left(\frac+\frac\right) \\ \frac \left(\frac+\frac\right) & \frac & \frac \left(\frac+\frac\right) \\ \frac \left(\frac+\frac\right) & \frac \left(\frac+\frac\right) & \frac \\ \end\right ) \end or using different notation: : \varepsilon_ & \varepsilon_ & \varepsilon_ \\ \varepsilon_ & \varepsilon_ & \varepsilon_ \\ \end\right ) = \left( \frac & \frac \left(\frac+\frac\right) & \frac \left(\frac+\frac\right) \\ \frac \left(\frac+\frac\right) & \frac & \frac \left(\frac+\frac\right) \\ \frac \left(\frac+\frac\right) & \frac \left(\frac+\frac\right) & \frac \\ \end\right ) \,\! Furthermore, since the deformation gradient can be expressed as $\backslash boldsymbol\; =\; \backslash boldsymbol\backslash mathbf\; +\; \backslash boldsymbol$ where $\backslash boldsymbol$ is the second-order identity tensor, we have :$\backslash boldsymbol\backslash varepsilon=\backslash frac\backslash left(\backslash boldsymbol^T+\backslash boldsymbol\backslash right)-\backslash boldsymbol\backslash ,\backslash !$ Also, from the general expression for the Lagrangian and Eulerian finite strain tensors we have :$$ \begin \mathbf E_& =\frac(\mathbf U^-\boldsymbol) = \frac(- \boldsymbol ) \approx \frac(+ \boldsymbol\}^m - \boldsymbol )\approx \boldsymbol\\ \mathbf e_& =\frac(\mathbf V^-\boldsymbol)= \frac(- \boldsymbol )\approx \boldsymbol \end 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「Infinitesimal strain theory」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.