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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 Strain rate tensor ： ウィキペディア英語版 Strain rate tensor In continuum mechanics, the strain rate tensor is a physical quantity that describes the rate of change of the deformation of a material in the neighborhood of a certain point, at a certain moment of time. It can be defined as the derivative of the strain tensor with respect to time, or as the symmetric component of the gradient (derivative with respect to position) of the flow velocity. The strain rate tensor is a purely kinematic concept that describes the macroscopic motion of the material. Therefore, it does not depend on the nature of the material, or on the forces and stresses that may be acting on it; and it applies to any continuous medium, whether solid, liquid or gas. On the other hand, for any fluid except superfluids, any gradual change in its deformation (i.e. a non-zero strain rate tensor) gives rise to viscous forces in its interior, due to friction between adjacent fluid elements, that tend to oppose that change. At any point in the fluid, these stresses can be described by a viscous stress tensor that is, almost always, completely determined by the strain rate tensor and by certain intrinsic properties of the fluid at that point. Viscous stress also occur in solids, in addition to the elastic stress observed in static deformation; when it is too large to be ignored, the material is said to be viscoelastic. ==Definition== Consider a material body, solid or fluid, that is flowing and/or moving in space. Let $v$ be the velocity field within the body; that is, a smooth function from $\backslash mathbb^3\backslash times\backslash mathbb$ such that $v(p,t)$ is the macroscopic velocity of the material that is passing through the point $p$ at time $t$. The velocity $v(p+r,t)$ at a point displaced from $p$ by a small vector $r$ can be written as a Taylor series: :$v(p+r,t)\; =\; v(p,t)+(\backslash nabla\; v)(p,t)(r)+\backslash text,$ where $\backslash nabla\; v$ the gradient of the velocity field, understood as a linear map that takes a displacement vector $r$ to the corresponding change in the velocity. In an arbitrary reference frame, $\backslash nabla\; v$ is related to the Jacobian matrix of the field, namely in 3 dimensions it is the 3×3 matrix :$(\backslash nabla\; v)^T\; =$ \begin \displaystyle & \displaystyle & \displaystyle\\ \displaystyle & \displaystyle & \displaystyle\\ \displaystyle & \displaystyle & \displaystyle \end = J . where $v\_i$ is the component of $v$ parallel to axis $i$ and $\backslash partial\_j\; f$ denotes the partial derivative of a function $f$ with respect to the space coordinate $x\_j$. Note that $J$ is a function of $p$ and $t$. In this coordinate system, the Taylor approximation for the velocity near $p$ is :$v\_i(p\; +\; r,t)\; =\; v\_i(p,t)\; +\; \backslash sum\_j\; J\_(p,t)\; r\_j\; =\; v\_i(p,t)\; +\; \backslash sum\_j\; \backslash partial\_j\; v\_i(p,t)\; r\_j;$ or simply $v(p\; +\; r,t)\; =\; v(p,t)\; +\; J(p,t)\; r$, if $v$ and $r$ are viewed as 3×1 matrices. 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「Strain rate 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.