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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 stress tensor ： ウィキペディア英語版 Cauchy stress tensor In continuum mechanics, the Cauchy stress tensor $\backslash boldsymbol\backslash sigma\backslash ,\backslash !$, true stress tensor,〔 or simply called the stress tensor, named after Augustin-Louis Cauchy, is a second order tensor of type (1,1) (that is, a linear map), with nine components $\backslash sigma\_\backslash ,\backslash !$ that completely define the state of stress at a point inside a material in the deformed placement or configuration. The tensor relates a unit-length direction vector n to the stress vector T(n) across an imaginary surface perpendicular to n: :$\backslash mathbf^=\; \backslash mathbf\; n\; \backslash cdot\backslash boldsymbol\backslash quad\; \backslash text\; \backslash quad\; T\_j^=\; \backslash sigma\_n\_i.\backslash ,\backslash !$ where, :$\backslash boldsymbol=$ \left(_ & \sigma _ & \sigma _ \\ \sigma _ & \sigma _ & \sigma _ \\ \sigma _ & \sigma _ & \sigma _ \\ \end}\right ) \equiv \left(_ & \sigma _ & \sigma _ \\ \sigma _ & \sigma _ & \sigma _ \\ \sigma _ & \sigma _ & \sigma _ \\ \end}\right ) \equiv \left(_x & \tau _ & \tau _ \\ \tau _ & \sigma _y & \tau _ \\ \tau _ & \tau _ & \sigma _z \\ \end}\right ) \,\! The Cauchy stress tensor obeys the tensor transformation law under a change in the system of coordinates. A graphical representation of this transformation law is the Mohr's circle for stress. The Cauchy stress tensor is used for stress analysis of material bodies experiencing small deformations: It is a central concept in the linear theory of elasticity. For large deformations, also called finite deformations, other measures of stress are required, such as the Piola–Kirchhoff stress tensor, the Biot stress tensor, and the Kirchhoff stress tensor. According to the principle of conservation of linear momentum, if the continuum body is in static equilibrium it can be demonstrated that the components of the Cauchy stress tensor in every material point in the body satisfy the equilibrium equations (Cauchy's equations of motion for zero acceleration). At the same time, according to the principle of conservation of angular momentum, equilibrium requires that the summation of moments with respect to an arbitrary point is zero, which leads to the conclusion that the stress tensor is symmetric, thus having only six independent stress components, instead of the original nine. There are certain invariants associated with the stress tensor, whose values do not depend upon the coordinate system chosen, or the area element upon which the stress tensor operates. These are the three eigenvalues of the stress tensor, which are called the principal stresses. == Euler-Cauchy stress principle - stress vector == (詳細はNewton and Euler, the motion of a material body is produced by the action of externally applied forces which are assumed to be of two kinds: surface forces $\backslash mathbf\; F$ and body forces $\backslash mathbf\; b$.〔Smith & Truesdell p.97〕 Thus, the total force $\backslash mathcal\; F$ applied to a body or to a portion of the body can be expressed as: :$\backslash mathcal\; F\; =\; \backslash mathbf\; b\; +\; \backslash mathbf\; F$ Only surface forces will be discussed in this article as they are relevant to the Cauchy stress tensor. When the body is subjected to external surface forces or ''contact forces'' $\backslash mathbf\; F\backslash ,\backslash !$, following Euler's equations of motion, internal contact forces and moments are transmitted from point to point in the body, and from one segment to the other through the dividing surface $S\backslash ,\backslash !$, due to the mechanical contact of one portion of the continuum onto the other (Figure 2.1a and 2.1b). On an element of area $\backslash Delta\; S\backslash ,\backslash !$ containing $P\backslash ,\backslash !$, with normal vector $\backslash mathbf\; n$, the force distribution is equipollent to a contact force $\backslash Delta\; \backslash mathbf\; F\backslash ,\backslash !$ and surface moment $\backslash Delta\; \backslash mathbf\; M\backslash ,\backslash !$. In particular, the contact force is given by :$\backslash Delta\backslash mathbf\; F=\; \backslash mathbf\; T^\backslash ,\backslash Delta\; S$ where $\backslash mathbf\; T^$ is the ''mean surface traction''. Cauchy’s stress principle asserts〔 that as $\backslash Delta\; S\backslash ,\backslash !$ becomes very small and tends to zero the ratio $\backslash Delta\; \backslash mathbf\; F/\backslash Delta\; S\backslash ,\backslash !$ becomes $d\backslash mathbf\; F/dS\backslash ,\backslash !$ and the couple stress vector $\backslash Delta\; \backslash mathbf\; M\backslash ,\backslash !$ vanishes. In specific fields of continuum mechanics the couple stress is assumed not to vanish; however, classical branches of continuum mechanics address non-polar materials which do not consider couple stresses and body moments. The resultant vector $d\backslash mathbf\; F/dS\backslash ,\backslash !$ is defined as the ''surface traction'',〔 also called ''stress vector'',〔 ''traction'',〔 or ''traction vector''.〔 given by $\backslash mathbf^=T\_i^\backslash mathbf\_i\backslash ,\backslash !$ at the point $P\backslash ,\backslash !$ associated with a plane with a normal vector $\backslash mathbf\; n\backslash ,\backslash !$: :$T^\_i=\; \backslash lim\_\; \backslash frac\; =\; .$ This equation means that the stress vector depends on its location in the body and the orientation of the plane on which it is acting. This implies that the balancing action of internal contact forces generates a ''contact force density'' or ''Cauchy traction field'' 〔 $\backslash mathbf\; T(\backslash mathbf\; n,\; \backslash mathbf\; x,\; t)$ that represents a distribution of internal contact forces throughout the volume of the body in a particular configuration of the body at a given time $t\backslash ,\backslash !$. It is not a vector field because it depends not only on the position $\backslash mathbf\; x$ of a particular material point, but also on the local orientation of the surface element as defined by its normal vector $\backslash mathbf\; n$.〔Lubliner〕 Depending on the orientation of the plane under consideration, the stress vector may not necessarily be perpendicular to that plane, ''i.e.'' parallel to $\backslash mathbf\; n\backslash ,\backslash !$, and can be resolved into two components (Figure 2.1c): * one normal to the plane, called ''normal stress'' :$\backslash mathbf\; \backslash frac\; =\; \backslash frac,$ :where $dF\_\backslash mathrm\; n\backslash ,\backslash !$ is the normal component of the force $d\backslash mathbf\; F\backslash ,\backslash !$ to the differential area $dS\backslash ,\backslash !$ * and the other parallel to this plane, called the ''shear stress'' :$\backslash mathbf\; \backslash tau=\; \backslash lim\_\; \backslash frac\; =\; \backslash frac,$ :where $dF\_\backslash mathrm\; s\backslash ,\backslash !$ is the tangential component of the force $d\backslash mathbf\; F\backslash ,\backslash !$ to the differential surface area $dS\backslash ,\backslash !$. The shear stress can be further decomposed into two mutually perpendicular vectors. 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 stress tensor」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.