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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 Size effect on structural strength ： ウィキペディア英語版 Size effect on structural strength According to the classical theories of elastic or plastic structures made from a material with non-random strength (''f''t), the nominal strength (''σ''N) of a structure is independent of the structure size (''D'') when geometrically similar structures are considered.〔The nominal strength of a structure (''σ''N) has units of stress and is related to the maximum load (''P''max) that the structure can support. For structures that can be approximated as two-dimensional, ''σ''N = ''P''max/''bD'' where ''b'' is the thickness of the two-dimensional structure. For three-dimensional structures, ''σ''N = ''P''max/''D''2. Any structure dimension can be chosen for ''D'' but it must be homologous for every size.〕 Any deviation from this property is called the size effect. For example, conventional strength of materials predicts that a large beam and a tiny beam will fail at the same stress if they are made of the same material. In the real world, because of size effects, a larger beam will fail at a lower stress than a smaller beam. The structural size effect concerns structures made of the same material, with the same microstructure. It must be distinguished from the size effect of material inhomogeneities, particularly the Hall-Petch effect, which describes how the material strength increases with decreasing grain size in polycrystalline metals. The size effect can have two causes: # statistical, due to material strength randomness, and # energetic (and non-statistical), due to energy release when a large crack or a large fracture process zone (FPZ) containing damaged material develops before the maximum load is reached. The limitations of elasticity theory are discussed in good textbooks on the topic. The same holds for plasticity theory. Modern computational models do not have these limitations and they predict structural strength correctly for any size. The scientists that develop new material models make sure that the results agree with the size effect laws. The engineers that design exceptionally large structures make sure that the calculations do not include a size effect mistake. ==Statistical Theory of Size Effect in Brittle Structures== The statistical size effect occurs for a broad class of brittle structures that follow the weakest-link model. This model means that macro-fracture initiation from one material element, or more precisely one representative volume element (RVE), causes the whole structure to fail, like the failure of one link in a chain (Fig. 1a). Since the material strength is random, the strength of the weakest material element in the structure (Fig. 1a) is likely to decrease with increasing structure size $D$ (as noted already by Mariotte in 1684). Denoting the failure probabilities of structure as $P\_f$ and of one RVE under stress $\backslash sigma\_k$ as $P\_1(\backslash sigma\_k)$, and noting that the survival probability of a chain is the joint probability of survival of all its $N$ links, one readily concludes that The key is the left tail of the distribution of $P\_1(\backslash sigma\_k)$. It was not successfully identified until Weibull in 1939 recognized that the tail is a power law. Denoting the tail exponent as $m$, one can then show that, if the structure is sufficiently larger than one RVE (i.e., if ), the failure probability of a structure as a function of $\backslash sigma\_N$ is Eq. 2 is the cumulative Weibull distribution with scale parameter $S\_0$ and shape parameter $m$; $\backslash Psi\; =\; \backslash int\_V\; \backslash left(\backslash right)^m\; \backslash ,\; \backslash mbox\; V(\backslash xi)$ = constant factor depending on the structure geometry, $V$ = structure volume; $\backslash xi$ = relative (size-independent) coordinate vectors, $\backslash hat\backslash sigma\; (\backslash xi)$ = dimensionless stress field (dependent on geometry), scaled so that the maximum stress be 1; $n\_d$ = number of spatial dimensions ($n\_d$ = 1, 2 or 3); $l\_0$ = material characteristic length representing the effective size of the RVE (typically about 3 inhomogeneity sizes). The RVE is here defined as the smallest material volume whose failure suffices to make the whole structure fail. From experience, the structure is sufficiently larger than one RVE if the equivalent number $N\_$ of RVEs in the structure is larger than about $10^4$ ; $N\_\; =(\; D\; /\; l\_0\; )^\backslash Psi$ = number of RVEs giving the same $P\_f$ if the stress field is homogeneous (always , and usually $N\_\; \backslash ll\; N$). For most normal-scale applications to metals and fine-grained ceramics, except for micrometer scale devices, the size is large enough for the Weibull theory to apply (but not for coarse-grained materials such as concrete). From Eq. 2 one can show that the mean strength and the coefficient of variation of strength are obtained as follows: (where $\backslash Gamma$ is the gamma function) The first equation shows that the size effect on the mean nominal strength is a power function of size $D$, regardless of structure geometry. Weibull parameter $m$ can be experimentally identified by two methods: 1) The values of $\backslash sigma\_N$ measured on many identical specimens are used to calculate the coefficient of variation of strength, and the value of $m$ then follows by solving Eq. (4); or 2) the values of $\backslash bar\_N$ are measured on geometrically similar specimens of several different sizes $D$ and the slope of their linear regression in the plot of $\backslash log\; \backslash bar\_N$ versus $\backslash log\; D$ gives $-1/m$. Method 1 must give the same result for different sizes, and method 2 the same as method 1. If not, the size effect is partly or totally non-Weibullian. Omission of testing for different sizes has often led to incorrect conclusions. Another check is that the histogram of the strengths of many identical specimens must be a straight line when plotted in the Weibull scale. A deviation to the right at high strength range means that $N\_$ is too small and the material quasibrittle. 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「Size effect on structural strength」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.