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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 Cauchy elastic material ： ウィキペディア英語版 Cauchy elastic material In physics, a Cauchy-elastic material is one in which the stress at each point is determined only by the current state of deformation with respect to an arbitrary reference configuration.〔 A Cauchy elastic material is also called a simple elastic material. It follows from this definition that the stress in a Cauchy-elastic material does not depend on the path of deformation or the history of deformation, or on the time taken to achieve that deformation or the rate at which the state of deformation is reached. The definition also implies that the constitutive equations are spatially local; that is, the stress is only affected by the state of deformation in an infinitesimal neighborhood of the point in question, without regard for the deformation or motion of the rest of the material. It also implies that body forces (such as gravity), and inertial forces cannot affect the properties of the material. Finally, a Cauchy elastic material must satisfy the requirements of material objectivity. Cauchy-elastic materials are mathematical abstractions, and no real material fits this definition perfectly. However, many elastic materials of practical interest, such as steel, plastic, wood and concrete, can often be assumed to be Cauchy-elastic for the purposes of stress analysis. == Mathematical definition == Formally, a material is said to be Cauchy-elastic if the Cauchy stress tensor $\backslash boldsymbol$ is a function of the strain tensor (deformation gradient) $\backslash boldsymbol$ alone: :$\backslash \; \backslash boldsymbol\; =\; \backslash mathcal(\backslash boldsymbol)$ This definition assumes that the effect of temperature can be ignored, and the body is homogeneous. This is the constitutive equation for a Cauchy-elastic material. Note that the function $\backslash mathcal$ depends on the choice of reference configuration. Typically, the reference configuration is taken as the relaxed (zero-stress) configuration, but need not be. Frame indifference requires that the constitutive relation $\backslash mathcal$ should not change when the location of the observer changes. Therefore the constitutive equation for another arbitrary observer can be written $\backslash boldsymbol^$ * = \mathcal(\boldsymbol^ *) . Knowing that the Cauchy stress tensor $\backslash sigma$ and the deformation gradient $F$ are objective quantities, one can write: :$$ \begin & \boldsymbol^ * &=& \mathcal(\boldsymbol^ *) \\ \Rightarrow & \boldsymbol\cdot\boldsymbol\cdot\boldsymbol^T &=& \mathcal(\boldsymbol\cdot\boldsymbol) \\ \Rightarrow & \boldsymbol\cdot\mathcal(\boldsymbol)\cdot\boldsymbol^T &=& \mathcal(\boldsymbol\cdot\boldsymbol) \end where $\backslash boldsymbol$ is a proper orthogonal tensor. The above is a condition that the constitutive law $\backslash mathcal$ has to respect to make sure that the response of the material will be independent of the observer. Similar conditions can be derived for constitutive laws relating the deformation gradient to the first or second Piola-Kirchhoff stress tensor. 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「Cauchy elastic material」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.