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Volumetric strain : 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., \|\mathbf u\| \ll 1 \,\! and \|\nabla \mathbf u\| \ll 1 \,\!, it is possible to perform a ''geometric linearisation'' of the Lagrangian finite strain tensor \mathbf E\,\!, and the Eulerian finite strain tensor \mathbf e\,\!. In such a linearisation, the non-linear or second-order terms of the finite strain tensor are neglected. Thus we have
:\mathbf E =\frac\left(\nabla_\mathbf u + (\nabla_\mathbf u)^T + (\nabla_\mathbf u)^T\nabla_\mathbf u\right)\approx \frac\left(\nabla_\mathbf u + (\nabla_\mathbf u)^T\right)\,\!
or
:E_=\frac\left(\frac+\frac+\frac\frac\right)\approx \frac\left(\frac+\frac\right)\,\!
and
:\mathbf e =\frac\left(\nabla_\mathbf u + (\nabla_\mathbf u)^T - \nabla_\mathbf u(\nabla_\mathbf u)^T\right)\approx \frac\left(\nabla_\mathbf u + (\nabla_\mathbf u)^T\right)\,\!
or
:e_=\frac\left(\frac +\frac-\frac\frac\right)\approx \frac\left(\frac +\frac\right)\,\!
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
:\mathbf E \approx \mathbf e \approx \boldsymbol \varepsilon = \frac\left((\nabla\mathbf u)^T + \nabla\mathbf u\right) \qquad
or
\qquad E_\approx e_\approx\varepsilon_=\frac\left(u_+u_\right)\,\!
where \varepsilon_\,\! are the components of the ''infinitesimal strain tensor'' \boldsymbol \varepsilon\,\!, also called ''Cauchy's strain tensor'', ''linear strain tensor'', or ''small strain tensor''.
:\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:
:\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
) \,\!
Furthermore, since the deformation gradient can be expressed as \boldsymbol = \boldsymbol\mathbf + \boldsymbol where \boldsymbol is the second-order identity tensor, we have
:\boldsymbol\varepsilon=\frac\left(\boldsymbol^T+\boldsymbol\right)-\boldsymbol\,\!
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



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