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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 Lode Coordinates ： ウィキペディア英語版 Lode Coordinates Lode coordinates $(z,r,\backslash theta)$ or Haigh-Westergaard coordinates $(\backslash xi,\backslash rho,\backslash theta)$.〔Menetrey, P.H., Willam, K.J., 1995, ''Triaxial Failure Criterion for Concrete and Its Generalization'', ACI Structural Journal〕 are a set of tensor invariants that span the space of real, symmetric, second-order, 3-dimensional tensors and are isomorphic with respect to principal stress space. This right-handed orthogonal coordinate system is named in honor of the German scientist Dr. Walter Lode because of his seminal paper written in 1926 describing the effect of the middle principal stress on metal plasticity.〔Lode, W. (1926). '' Versuche über den Einfuss der mittleren Hauptspannung auf das Fliessen der Metalle Eisen Kupfer und Nickel''. Zeitung Phys., vol. 36, pp. 913–939.〕 Other examples of sets of tensor invariants are the set of principal stresses $(\backslash sigma\_1,\; \backslash sigma\_2,\; \backslash sigma\_3)$ or the set of mechanics invariants $(I\_1,\; J\_2,\; J\_3)$. The Lode coordinate system can be described as a cylindrical coordinate system within principal stress space with a coincident origin and the z-axis parallel to the vector $(\backslash sigma\_1,\backslash sigma\_2,\backslash sigma\_3)=(1,1,1)$. ==Mechanics Invariants== The Lode coordinates are most easily computed using the mechanics invariants. These invariants are a mixture of the invariants of the Cauchy stress tensor, $\backslash boldsymbol$, and the stress deviator, $\backslash boldsymbol$, and are given by〔Asaro, R.J., Lubarda, V.A., 2006, ''Mechanics of Solids and Materials'', Cambridge University Press〕 :$I\_1\; =\; \backslash mathrm(\backslash boldsymbol)$ :$J\_2\; =\; \backslash frac\backslash left(-\; \backslash frac\backslash text(\backslash boldsymbol)^2\backslash right)\; =\; \backslash frac\backslash mathrm\backslash left(\backslash boldsymbol\backslash cdot\backslash boldsymbol\backslash right)\; =\; \backslash frac\backslash lVert\; \backslash boldsymbol\; \backslash rVert^2$ :$J\_3\; =\; \backslash mathrm(\backslash boldsymbol)\; =\; \backslash frac\backslash mathrm\backslash left(\backslash boldsymbol\backslash cdot\backslash boldsymbol\backslash cdot\backslash boldsymbol\backslash right)$ which can be written equivalently in Einstein notation :$I\_1\; =\; \backslash sigma\_$ :$J\_2\; =\; \backslash frac\backslash left(-\; \backslash frac\backslash text(\backslash boldsymbol)^2\backslash right)\; =\; \backslash fracs\_s\_\; =\; \backslash fracs\_s\_$ :$J\_3\; =\; \backslash frac\backslash epsilon\_\backslash epsilon\_\backslash sigma\_\backslash sigma\_\backslash sigma\_\; =\; \backslash fracs\_s\_s\_$ where $\backslash epsilon$ is the Levi-Civita symbol (or permutation symbol) and the last two forms for $J\_2$ are equivalent because $\backslash boldsymbol$ is symmetric ($s\_=s\_$). The gradients of these invariants〔Brannon, R.M., 2009, ''KAYENTA: Theory and User's Guide'', Sandia National Laboratories, Albuquerque, New Mexico.〕 can be calculated by :$\backslash frac$ :$\backslash frac\; =\; \backslash boldsymbol\; -\; \backslash frac\backslash right)\}\backslash boldsymbol$ :$\backslash frac\; =\; \backslash boldsymbol\backslash cdot\backslash boldsymbol\; -\; \backslash frac\backslash boldsymbol$ where $\backslash boldsymbol$ is the 3x3 identity matrix and $\backslash boldsymbol$ is called the Hill 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「Lode Coordinates」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.