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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 Arruda–Boyce model ： ウィキペディア英語版 Arruda–Boyce model In continuum mechanics, an Arruda–Boyce model〔Arruda, E. M. and Boyce, M. C., 1993, A three-dimensional model for the large stretch behavior of rubber elastic materials,, J. Mech. Phys. Solids, 41(2), pp. 389–412.〕 is a hyperelastic constitutive model used to describe the mechanical behavior of rubber and other polymeric substances. This model is based on the statistical mechanics of a material with a cubic representative volume element containing eight chains along the diagonal directions. The material is assumed to be incompressible. The strain energy density function for the incompressible Arruda–Boyce model is given by〔Bergstrom, J. S. and Boyce, M. C., 2001, Deformation of Elastomeric Networks: Relation between Molecular Level Deformation and Classical Statistical Mechanics Models of Rubber Elasticity, Macromolecules, 34 (3), pp 614–626, .〕 :$$ W = Nk_B\theta\sqrt\left(- \sqrt\ln\left(\cfrac\right)\right ) where $n$ is the number of chain segments, $k\_B$ is the Boltzmann constant, $\backslash theta$ is the temperature in Kelvin, $N$ is the number of chains in the network of a cross-linked polymer, :$$ \lambda_} ~;~~ \beta = \mathcal^\left(\cfrac^(x) is the inverse Langevin function which can approximated by :$$ \mathcal^(x) = \begin 1.31\tan(1.59 x) + 0.91 x & \quad\mathrm~|x| < 0.841 \\ \tfrac & \quad\mathrm~ 0.841 \le |x| < 1 \end For small deformations the Arruda–Boyce model reduces to the Gaussian network based neo-Hookean solid model. It can be shown〔Horgan, C.O. and Saccomandi, G., 2002, A molecular-statistical basis for the Gent constitutive model of rubber elasticity, Journal of Elasticity, 68(1), pp. 167–176.〕 that the Gent model is a simple and accurate approximation of the Arruda–Boyce model. ==Alternative expressions for the Arruda–Boyce model== An alternative form of the Arruda–Boyce model, using the first five terms of the inverse Langevin function, is〔Hiermaier, S. J., 2008, Structures under Crash and Impact, Springer.〕 :$$ W = C_1\left(+ \tfrac(I_1^2 -9) + \tfrac(I_1^3-27) + \tfrac(I_1^4-81) + \tfrac(I_1^5-243)\right ) where $C\_1$ is a material constant. The quantity $N$ can also be interpreted as a measure of the limiting network stretch. If $\backslash lambda\_m$ is the stretch at which the polymer chain network becomes locked, we can express the Arruda–Boyce strain energy density as :$$ W = C_1\left(+ \tfrac(I_1^2 -9) + \tfrac(I_1^3-27) + \tfrac(I_1^4-81) + \tfrac(I_1^5-243)\right ) We may alternatively express the Arruda–Boyce model in the form :$$ W = C_1~\sum_^5 \alpha_i~\beta^~(I_1^i-3^i) where $\backslash beta\; :=\; \backslash tfrac\; =\; \backslash tfrac$ and $$ \alpha_1 := \tfrac ~;~~ \alpha_2 := \tfrac ~;~~ \alpha_3 := \tfrac ~;~~ \alpha_4 := \tfrac ~;~~ \alpha_5 := \tfrac. If the rubber is compressible, a dependence on $J=\backslash det(\backslash boldsymbol)$ can be introduced into the strain energy density; $\backslash boldsymbol$ being the deformation gradient. Several possibilities exist, among which the Kaliske–Rothert〔Kaliske, M. and Rothert, H., 1997, On the finite element implementation of rubber-like materials at finite strains, Engineering Computations, 14(2), pp. 216–232.〕 extension has been found to be reasonably accurate. With that extension, the Arruda-Boyce strain energy density function can be expressed as :$$ W = D_1\left(\tfrac - \ln J\right) + C_1~\sum_^5 \alpha_i~\beta^~(\overline_1^i-3^i) where $D\_1$ is a material constant and $\backslash overline\_1\; =\; \_1\; J^$ . For consistency with linear elasticity, we must have $D\_1\; =\; \backslash tfrac$ where $\backslash kappa$ is the bulk modulus. 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「Arruda–Boyce model」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.