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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 Lame's stress ellipsoid ： ウィキペディア英語版 Lame's stress ellipsoid Lame's stress ellipsoid (Figure coming) is an alternative to Mohr's circle for the graphical representation of the stress state at a point. The surface of the ellipsoid represents the locus of the endpoints of all stress vectors acting on all planes passing through a given point in the continuum body. In other words, the endpoints of all stress vectors at a given point in the continuum body lie on the stress ellipsoid surface, i.e., the radius-vector from the center of the ellipsoid, located at the material point in consideration, to a point on the surface of the ellipsoid is equal to the stress vector on some plane passing through the point. In two dimensions, the surface is represented by an ellipse (Figure coming). Once the equations of the ellipsoid is known, the magnitude of the stress vector can then be obtained for any plane passing through that point. To determine the equation of the stress ellipsoid we consider the coordinate axes $x\_1,\; x\_2,\; x\_3\backslash ,\backslash !$ taken in the directions of the principal axes, i.e., in a principal stress space. Thus, the coordinates of the stress vector $\backslash mathbf\; T^\backslash ,\backslash !$ on a plane with normal unit vector $\backslash mathbf\; n\backslash ,\backslash !$ passing through a given point $P\backslash ,\backslash !$ is represented by :$T\_1^=\backslash sigma\_1n\_1,\; \backslash qquad\; T\_2^=\backslash sigma\_2n\_2,\; \backslash qquad\; T\_3^=\backslash sigma\_3n\_3\backslash ,\backslash !$ And knowing that $\backslash mathbf\; n\backslash ,\backslash !$ is a unit vector we have :$n\_1^2+n\_2^2+n\_3^2=\backslash frac^2\}^2+\backslash frac^2\}^2+\backslash frac^2\}^2=1\backslash ,\backslash !$ which is the equation of an ellipsoid centered at the origin of the coordinate system, with the lengths of the semiaxes of the ellipsoid equal to the magnitudes of the principal stresses, i.e. the intercepts of the ellipsoid with the principal axes are $\backslash pm\backslash sigma\_1,\; \backslash pm\backslash sigma\_2,\; \backslash pm\backslash sigma\_3\backslash ,\backslash !$. * The first stress invariant $I\_1\backslash ,\backslash !$ is directly proportional to the sum of the principal radii of the ellipsoid. * The second stress invariant $I\_2\backslash ,\backslash !$ is directly proportional to the sum of the three principal areas of the ellipsoid. The tree principal areas are the ellipses on each principal plane. * The third stress invariant $I\_3\backslash ,\backslash !$ is directly proportional to the volume of the ellipsoid. * If two of the three principal stresses are numerically equal the stress ellipsoid becomes an ellipsoid of revolution.〔Timoshenko〕 Thus, two principal areas are ellipses and the third is a circle. * If all of the principal stresses are equal and of the same sign, the stress ellipsoid becomes a sphere and any three perpendicular directions can be taken as principal axes.〔 The stress ellipsoid by itself, however, does not indicate the plane on which the given traction vector acts. Only for the case where the stress vector lies along one of the principal directions it is possible to know the direction of the plane, as the principal stresses act perpendicular to their planes. To find the orientation of any other plane we used the ''stress-director surface''〔 or ''stress director quadric''〔Timoshenko〕 represented by the equation :$n\_1^2+n\_2^2+n\_3^2=\backslash frac^2\}+\backslash frac^2\}+\backslash frac^2\}=1\backslash ,\backslash !$ The stress represented by a radius-vector of the stress ellipsoid acts on a plane oriented parallel to the tangent plane to the stress-director surface at the point of its intersection with the radius-vector.〔 ==References== 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「Lame's stress ellipsoid」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.