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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 J integral ： ウィキペディア英語版 J integral The J-integral represents a way to calculate the strain energy release rate, or work (energy) per unit fracture surface area, in a material.〔(Van Vliet, Krystyn J. (2006); "3.032 Mechanical Behavior of Materials" )〕 The theoretical concept of J-integral was developed in 1967 by Cherepanov〔G. P. Cherepanov, '' The propagation of cracks in a continuous medium'', Journal of Applied Mathematics and Mechanics, 31(3), 1967, pp. 503-512.〕 and in 1968 by Jim Rice〔J. R. Rice, ''A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks'', Journal of Applied Mechanics, 35, 1968, pp. 379-386.〕 independently, who showed that an energetic contour path integral (called ''J'') was independent of the path around a crack. Later, experimental methods were developed, which allowed measurement of critical fracture properties using laboratory-scale specimens for materials in which sample sizes are too small and for which the assumptions of Linear Elastic Fracture Mechanics (LEFM) do not hold,〔Lee, R. F., & Donovan, J. A. (1987). J-integral and crack opening displacement as crack initiation criteria in natural rubber in pure shear and tensile specimens. Rubber chemistry and technology, 60(4), 674-688. ()〕 and to infer a critical value of fracture energy ''J''Ic. The quantity ''J''Ic defines the point at which large-scale plastic yielding during propagation takes place under mode one loading.〔〔Meyers and Chawla (1999): "Mechanical Behavior of Materials," 445-448.〕 The J-integral is equal to the strain energy release rate for a crack in a body subjected to monotonic loading.〔Yoda, M., 1980, ''The J-integral fracture toughness for Mode II'', Int. J. of Fracture, 16(4), pp. R175-R178.〕 This is generally true, under quasistatic conditions, only for linear elastic materials. For materials that experience small-scale yielding at the crack tip, ''J'' can be used to compute the energy release rate under special circumstances such as monotonic loading in mode III (antiplane shear). The strain energy release rate can also be computed from ''J'' for pure power-law hardening plastic materials that undergo small-scale yielding at the crack tip. The quantity ''J'' is not path-independent for monotonic mode I and mode II loading of elastic-plastic materials, so only a contour very close to the crack tip gives the energy release rate. Also, Rice showed that ''J'' is path-independent in plastic materials when there is no non-proportional loading. Unloading is a special case of this, but non-proportional plastic loading also invalidates the path-independence. Such non-proportional loading is the reason for the path-dependence for the in-plane loading modes on elastic-plastic materials. == Two-dimensional J-integral == The two-dimensional J-integral was originally defined as〔 (see Figure 1 for an illustration) :$$ J := \int_\Gamma \left(W~dx_2 - \mathbf\cdot\cfrac~ds\right) = \int_\Gamma \left(W~dx_2 - t_i\,\cfrac~ds\right) where ''W''(''x''1,''x''2) is the strain energy density, ''x''1,''x''2 are the coordinate directions, t=n.''σ'' is the surface traction vector, n is the normal to the curve Γ, ''σ'' is the Cauchy stress tensor, and u is the displacement vector. The strain energy density is given by :$$ W = \int_0^\epsilon \boldsymbol:d\boldsymbol ~;~~ \boldsymbol = \tfrac\left() ~. The J-Integral around a crack tip is frequently expressed in a more general form (and in index notation) as :$$ J_i := \lim_ \int_ \left(W n_i - n_j\sigma_~\cfrac\right) d\Gamma where $J\_i$ is the component of the J-integral for crack opening in the $x\_i$ direction and $\backslash epsilon$ is a small region around the crack tip. Using Green's theorem we can show that this integral is zero when the boundary $\backslash Gamma$ is closed and encloses a region that contains no singularities and is simply connected. If the faces of the crack do not have any surface tractions on them then the J-integral is also path independent. Rice also showed that the value of the J-integral represents the energy release rate for planar crack growth. The J-integral was developed because of the difficulties involved in computing the stress close to a crack in a nonlinear elastic or elastic-plastic material. Rice showed that if monotonic loading was assumed (without any plastic unloading) then the J-integral could be used to compute the energy release rate of plastic materials too. :~\cfrac\right) d\Gamma We can write this as :$$ J_1 = \int_ \left(W \delta_ - \sigma_~\cfrac\right)n_j d\Gamma From Green's theorem (or the two-dimensional divergence theorem) we have :$$ \int_ f_j~n_j~d\Gamma = \int_A \cfrac~dA Using this result we can express $J\_1$ as :$$ \begin J_1 & = \int_ \cfrac\left(W \delta_ - \sigma_~\cfrac\right) dA \\ & = \int_A \left() \qquad \implies \qquad \epsilon_ = \tfrac\left(\cfrac + \cfrac\right) ~. Therefore, :$$ \sigma_\cfrac = \tfrac\left(\sigma_\cfrac + \sigma_\cfrac\right) From the balance of angular momentum we have $\backslash sigma\_\; =\; \backslash sigma\_$. Hence, :$$ \sigma_\cfrac = \sigma_\cfrac The J-integral may then be written as :$$ J_1 = \int_A \left(ウィキペディア（Wikipedia）』 ■ウィキペディアで「J integral」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.