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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 Neo-Hookean solid ： ウィキペディア英語版 Neo-Hookean solid A neo-Hookean solid〔Ogden, R. W. , 1998, Nonlinear Elastic Deformations, Dover.〕〔C. W. Macosko, 1994, Rheology: principles, measurement and applications, VCH Publishers, ISBN 1-56081-579-5.〕 is a hyperelastic material model, similar to Hooke's law, that can be used for predicting the nonlinear stress-strain behavior of materials undergoing large deformations. The model was proposed by Ronald Rivlin in 1948. In contrast to linear elastic materials, the stress-strain curve of a neo-Hookean material is not linear. Instead, the relationship between applied stress and strain is initially linear, but at a certain point the stress-strain curve will plateau. The neo-Hookean model does not account for the dissipative release of energy as heat while straining the material and perfect elasticity is assumed at all stages of deformation. The neo-Hookean model is based on the statistical thermodynamics of cross-linked polymer chains and is usable for plastics and rubber-like substances. Cross-linked polymers will act in a neo-Hookean manner because initially the polymer chains can move relative to each other when a stress is applied. However, at a certain point the polymer chains will be stretched to the maximum point that the covalent cross links will allow, and this will cause a dramatic increase in the elastic modulus of the material. The neo-Hookean material model does not predict that increase in modulus at large strains and is typically accurate only for strains less than 20%.〔Gent, A. N., ed., 2001, Engineering with rubber, Carl Hanser Verlag, Munich.〕 The model is also inadequate for biaxial states of stress and has been superseded by the Mooney-Rivlin model. The strain energy density function for an incompressible neo-Hookean material is :$$ W = C_ (I_1-3) \, where $C\_$ is a material constant, and $I\_1$ is the first invariant of the right Cauchy-Green deformation tensor, i.e., :$$ I_1 = \lambda_1^2 + \lambda_2^2 + \lambda_3^2~ where $\backslash lambda\_i$ are the principal stretches. For three-dimensional problems the compressible neo-Hookean material the strain energy density function is given by :$$ W = C_~(\bar_1 - 3) + D_1~(J-1)^2 ~;~~ J = \det(\boldsymbol) = \lambda_1\lambda_2\lambda_3 where $D\_1$ is a material constant, $\backslash bar\_1\; =\; J^\; I\_1$ is the first invariant of the isochoric part of the right Cauchy-Green deformation tensor, and $\backslash boldsymbol$ is the deformation gradient. It can be shown that in 2D, the strain energy density function is :$$ W = C_~(\bar_1 - 2) + D_1~(J-1)^2 ~; where, in this case, $\backslash bar\_1\; =J^\; I\_1$. Several alternative formulations exist for compressible neo-Hookean materials, for example 〔 :$$ W = C_~(I_1 - 3 - 2\ln J) + D_1~(\ln J)^2 For consistency with linear elasticity, :$$ C_ = \cfrac ~;~~ D_1 = \cfrac where $\backslash mu$ is the shear modulus and $\backslash kappa$ is the bulk modulus. == Cauchy stress in terms of deformation tensors == 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「Neo-Hookean solid」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.