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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 Hypoelastic material ： ウィキペディア英語版 Hypoelastic material In continuum mechanics, a hypoelastic material〔Truesdell (1963).〕 is an elastic material that has a constitutive model independent of finite strain measures except in the linearized case. Hypoelastic material models are distinct from hyperelastic material models (or standard elasticity models) in that, except under special circumstances, they cannot be derived from a strain energy density function. Hypoelastic materials are described by a relation of the form :$$ \dot:\dot is a stress measure, $\backslash mathsf$ is the material property tensor, $\backslash boldsymbol$ is the deformation gradient, and superposed dots indicate time derivatives. In hypoelasticity, the quantity $\backslash mathsf$ is a function of stress while, in hyperelasticity, the material tensor may also depend on strains and rotations. == Hypoelasticity and objective stress rates == In many practical problems of solid mechanics, it is sufficient to characterize material deformation by the small (or linearized) strain tensor :$$ \varepsilon_ = \frac 1 2 (u_ + u_) where $u\_i$ are the components of the displacements of continuum points, the subscripts refer to Cartesian coordinates $x\_i$ $(i=1,2,3)$, and the subscripts preceded by a comma denote partial derivatives (e.g., $u\_\; =\; \backslash partial\; u\_i\; /\backslash partial\; x\_j$). But there are also many problems where the finiteness of strain must be taken into account. These are of two kinds: # large nonlinear elastic deformations possessing a potential energy, $W(\backslash boldsymbol)$ (exhibited, e.g., by rubber), in which the stress tensor components are obtained as the partial derivatives of $W$ with respect to the finite strain tensor components; and # inelastic deformations possessing no potential, in which the stress-strain relation is defined incrementally. In the former kind, the total strain formulation described in the article on finite strain theory is appropriate. In the latter kind an incremental (or rate) formulation is necessary and must be used in every load or time step of a finite element computer program using updated Lagrangian procedure. The absence of a potential raises intricate questions due to the freedom in the choice of finite strain measure and characterization of the stress rate. For a sufficiently small loading step (or increment), one may use the deformation rate tensor (or velocity strain) :$$ d_ = \dot \varepsilon_ = \frac 1 2 (v_ + v_) or increment :$$ \Delta \varepsilon_ = \dot \varepsilon_ \Delta t = d_ \Delta t representing the linearized strain increment from the initial (stressed and deformed) state in the step. Here the superior dot represents the time derivative $\backslash partial\; /\backslash partial\; t$, $\backslash Delta$ denotes a small increment over the step, $t$ = time, and $v\_i\; =\; \backslash dot\; u\_i$ = material point velocity or displacement rate. However, it would not be objective to use the time derivative of the Cauchy (or true) stress $\backslash sigma\_$. This stress, which describes the forces on a small material element imagined to be cut out from the material as currently deformed, is not objective because it varies with rigid body rotations of the material. The material points must be characterized by their initial coordinates $X\_i$ (called Lagrangian) because different material particles are contained in the element that is cut out (at the same location) before and after the incremental deformation. Consequently, it is necessary to introduce the so-called objective stress rate $\backslash hat\; \backslash sigma\_$, or the corresponding increment $\backslash Delta\; \backslash sigma\_\; =\; \backslash hat\; \backslash sigma\_\; \backslash Delta\; t$. The objectivity is necessary for $\backslash hat\; \backslash sigma\_$ to be functionally related to the element deformation. It means that that $\backslash hat\; \backslash sigma\_$ must be invariant with respect to coordinate transformations (particularly rotations) and must characterize the state of the same material element as it deforms. 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「Hypoelastic material」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.