翻訳と辞書 Words near each other ・ "O" Is for Outlaw ・ "O"-Jung.Ban.Hap. ・ "Ode-to-Napoleon" hexachord ・ "Oh Yeah!" Live ・ "Our Contemporary" regional art exhibition (Leningrad, 1975) ・ "P" Is for Peril ・ "Pimpernel" Smith ・ "Polish death camp" controversy ・ "Pro knigi" ("About books") ・ "Prosopa" Greek Television Awards ・ "Pussy Cats" Starring the Walkmen ・ "Q" Is for Quarry ・ "R" Is for Ricochet ・ "R" The King (2016 film) ・ "Rags" Ragland ・ ! (album) ・ ! (disambiguation) ・ !! ・ !!! ・ !!! (album) ・ !!Destroy-Oh-Boy!! ・ !Action Pact! ・ !Arriba! La Pachanga ・ !Hero ・ !Hero (album) ・ !Kung language ・ !Oka Tokat ・ !PAUS3 ・ !T.O.O.H.! ・ !Women Art Revolution Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク objective stress rates ： ウィキペディア英語版 objective stress rates In continuum mechanics, objective stress rates are time derivatives of stress that do not depend on the frame of reference.〔 Many constitutive equations are designed in the form of a relation between a stress-rate and a strain-rate (or the rate of deformation tensor). The mechanical response of a material should not depend on the frame of reference. In other words, material constitutive equations should be frame indifferent (objective). If the stress and strain measures are material quantities then objectivity is automatically satisfied. However, if the quantities are spatial, then the objectivity of the stress-rate is not guaranteed even if the strain-rate is objective. There are numerous objective stress rates in continuum mechanics - all of which can be shown to be special forms of Lie derivatives. Some of the widely used objective stress rates are: # the Truesdell rate of the Cauchy stress tensor, # the Green-Naghdi rate of the Cauchy stress, and # the Jaumann rate of the Cauchy stress. The adjacent figure shows the performance of various objective rates in a pure shear test where the material model is hypoelastic with constant elastic moduli. The ratio of the shear stress to the displacement is plotted as a function of time. The same moduli are used with the three objective stress rates. Clearly there are spurious oscillations observed for the Jaumann stress rate. This is not because one rate is better than another but because it is a misuse of material models to use the same constants with different objective rates. For this reason, a recent trend has been to avoid objective stress rates altogether where possible. == Non-objectivity of the time derivative of Cauchy stress == Under rigid body rotations ($\backslash boldsymbol$), the Cauchy stress tensor $\backslash boldsymbol$ transforms as :$$ \boldsymbol_r = \boldsymbol\cdot\boldsymbol\cdot\boldsymbol^T ~;~~ \boldsymbol\cdot\boldsymbol^T = \boldsymbol is a spatial quantity and the transformation follows the rules of tensor transformations, $\backslash boldsymbol$ is objective. However, :$$ \cfrac(\boldsymbol_r) = \dot}\cdot\boldsymbol\cdot\boldsymbol^T + \boldsymbol\cdot\dot^T + \boldsymbol\cdot\boldsymbol\cdot\dot\cdot\dot^T \,. Therefore the stress rate is not objective unless the rate of rotation is zero, i.e. $\backslash boldsymbol$ is constant. For a physical understanding of the above, consider the situation shown in Figure 1. In the figure the components of the Cauchy (or true) stress tensor are denoted by the symbols $S\_$. This tensor, which describes the forces on a small material element imagined to be cut out from the material as currently deformed, is not objective at large deformations because it varies with rigid body rotations of the material. The material points must be characterized by their initial Lagrangian coordinates $x\_i$. Consequently, it is necessary to introduce the so-called objective stress rate $\backslash overset\_$, or the corresponding increment $\backslash Delta\; S\_\; =\; \backslash overset\_\; \backslash Delta\; t$. The objectivity is necessary for $\backslash overset\_$ to be functionally related to the element deformation. It means that $\backslash overset\_$ must be invariant with respect to coordinate transformations, particularly the rigid-body rotations, and must characterize the state of the same material element as it deforms. The objective stress rate can be derived in two ways: * by tensorial coordinate transformations,〔 which is the standard way in finite element textbooks〔 * variationally, from strain energy density in the material expressed in terms of the strain tensor (which is objective by definition)〔〔 While the former way is instructive and provides useful geometric insight, the latter way is mathematically shorter and has the additional advantage of automatically ensuring energy conservation, i.e., guaranteeing that the second-order work of the stress increment tensor on the strain increment tensor be correct (work conjugacy requirement). 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「objective stress rates」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.