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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 yield surface ： ウィキペディア英語版 yield surface A yield surface is a five-dimensional surface in the six-dimensional space of stresses. The yield surface is usually convex and the state of stress of ''inside'' the yield surface is elastic. When the stress state lies on the surface the material is said to have reached its yield point and the material is said to have become plastic. Further deformation of the material causes the stress state to remain on the yield surface, even though the shape and size of the surface may change as the plastic deformation evolves. This is because stress states that lie outside the yield surface are non-permissible in rate-independent plasticity, though not in some models of viscoplasticity.〔Simo, J. C. and Hughes, T,. J. R., (1998), Computational Inelasticity, Spinger.〕 The yield surface is usually expressed in terms of (and visualized in) a three-dimensional principal stress space ($\backslash sigma\_1,\; \backslash sigma\_2\; ,\; \backslash sigma\_3$), a two- or three-dimensional space spanned by stress invariants ($I\_1,\; J\_2,\; J\_3$) or a version of the three-dimensional Haigh–Westergaard stress space. Thus we may write the equation of the yield surface (that is, the yield function) in the forms: *$f(\backslash sigma\_1,\backslash sigma\_2,\backslash sigma\_3)\; =\; 0\; \backslash ,$ where $\backslash sigma\_i$ are the principal stresses. *$f(I\_1,\; J\_2,\; J\_3)\; =\; 0\; \backslash ,$ where $I\_1$ is the first principal invariant of the Cauchy stress and $J\_2,\; J\_3$ are the second and third principal invariants of the deviatoric part of the Cauchy stress. *$f(p,\; q,\; r)\; =\; 0\; \backslash ,$ where $p,\; q$ are scaled versions of $I\_1$ and $J\_2$ and $r$ is a function of $J\_2,\; J\_3$. *$f(\backslash xi,\backslash rho,\backslash theta)\; =\; 0\; \backslash ,$ where $\backslash xi,\backslash rho$ are scaled versions of $I\_1$ and $J\_2$, and $\backslash theta$ is the Lode angle. == Invariants used to describe yield surfaces == The first principal invariant ($I\_1$) of the Cauchy stress ($\backslash boldsymbol$), and the second and third principal invariants ($J\_2,\; J\_3$) of the ''deviatoric'' part ($\backslash boldsymbol$) of the Cauchy stress are defined as: : :$$ \begin I_1 & = \text(\boldsymbol) = \sigma_1 + \sigma_2 + \sigma_3 \\ J_2 & = \tfrac \boldsymbol:\boldsymbol = \tfrac\left() \\ J_3 & = \det(\boldsymbol) = \tfrac (\boldsymbol\cdot\boldsymbol):\boldsymbol = s_1 s_2 s_3 \end where ($\backslash sigma\_1,\; \backslash sigma\_2\; ,\; \backslash sigma\_3$) are the principal values of $\backslash boldsymbol$, ($s\_1,\; s\_2,\; s\_3$) are the principal values of $\backslash boldsymbol$, and :$$ \boldsymbol = \boldsymbol-\tfrac\,\boldsymbol where $\backslash boldsymbol$ is the identity matrix. A related set of quantities, ($p,\; q,\; r\backslash ,$), are usually used to describe yield surfaces for cohesive frictional materials such as rocks, soils, and ceramics. These are defined as :$$ p = \tfrac~I_1 ~:~~ q = \sqrt = \sigma_\mathrm ~;~~ r = 3\left(\tfrac\,J_3\right)^ where $\backslash sigma\_\backslash mathrm$ is the equivalent stress. However, the possibility of negative values of $J\_3$ and the resulting imaginary $r$ makes the use of these quantities problematic in practice. Another related set of widely used invariants is ($\backslash xi,\; \backslash rho,\; \backslash theta\backslash ,$) which describe a cylindrical coordinate system (the Haigh–Westergaard coordinates). These are defined as: :$$ \xi = \tfrac~p ~;~~ \rho = \sqrt = \sqrt}~q ~;~~ \cos(3\theta) = \left(\tfrac\right)^3 = \tfrac~\cfrac \sigma_1 \\ \sigma_2 \\ \sigma_3 \end = \tfrac \xi \\ \xi \\ \xi \end + \sqrt}~\rho~\begin \cos\theta \\ \cos\left(\theta-\tfrac\right) \\ \cos\left(\theta+\tfrac\right) \end = \tfrac \xi \\ \xi \\ \xi \end + \sqrt}~\rho~\begin \cos\theta \\ -\sin\left(\tfrac-\theta\right) \\ -\sin\left(\tfrac+\theta\right) \end \,. A different definition of the Lode angle can also be found in the literature:〔Chakrabarty, J., 2006, ''Theory of Plasticity: Third edition'', Elsevier, Amsterdam.〕 :$$ \sin(3\theta) = ~\tfrac~\cfrac \sigma_1 \\ \sigma_2 \\ \sigma_3 \end = \tfrac \xi \\ \xi \\ \xi \end + \tfrac \cos\theta + \tfrac \,. 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「yield surface」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.