l-组合变形-应变状态理论

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1、2.应变状态分析Analysisofstrians正应变:PABxy(1)应变的定义（Difinitionofstrains)2.1一点应变状态概念Conceptofstrainstateatonepoint线应变以拉长为正，缩短为负；pab切应变:PABxypab角应变以直角的减小量为负，增加量为正。(2)应变状态矩阵(strainstatematrix)Ifthelinearstrainsalonganydirectionandtheshearingstrainsbetweenanytwovert

2、icaldirectionsatonepointareallknown,wesaythestateofstrainatthispointisdetermined.结构一点处应变情况的总和称为一点应变状态.如果一点处沿任何方向的线应变以及任意两个相互垂直方向间的切应变都已知,则说该点的应变状态是确定的.Likethestateofstressatonepoint,thestateofstraincanbedeterminedbyamatrix.Itiscalledstrainstatematrix.类似

3、一点应力状态的描述,一点应变状态可以用一个矩阵确定,该矩阵称为应变状态矩阵.xyzoStrainstatematrixStrainstatematrixisasymmetricalmatrixtoo.Thatis:应变状态矩阵(3)应变状态的分类（Formsofstrainstates)(a)三向（空间）应变状态Three-dimensionalorspacestateofstrainIfinmatrixEtheelementseveryrowandarrangearenotallequaltozer

4、o,thestateofstrainatonepointiscalledthree-dimensionalstateofstrain.(b)平面（双向）应变状态Two-dimensionalorplanestateofstrainThestateofstrainiscalledtwo-dimensionalorplanestateofstrain.(c)单向应变状态One-dimensionalstateofstrainThestateofstrainiscalledone-dimensionalst

5、ateofstrain.xOthersstrains=0RigidwallForthefreebodythereisnoone-dimensionalstraininengineering.PABxypabPxypqQPxypQqdxdyxy(1)斜方向上的应变考察倾斜微元线段的伸长率时，无须考虑可能存在的局部的刚体平移。也无须考虑可能存在的局部的刚体转动。Normalstrainalongtheslopedirections2.2平面（双向）应变状态分析Two-dimentionalanal

6、ysisofstrainx方向的线应变将影响倾斜微元线段的伸长率。y方向的线应变将影响倾斜微元线段的伸长率。角应变也将影响倾斜微元线段的伸长率。(a)公式推导应变与应力的运算是类似的。xyα(b)正应变(c.)切应变xyα从斜线起逆时针方向90°的增大量斜方向上的应变:拉为正压为负直角增大为正减小为负(2)主应变与应变主方向PrincipalstrainsandprincipalstraindirectionsAtonepointiftherearethreedirectionsverticaltoe

7、achotherandtheshearingstrainsbetweenthemareallequaltozero,thesedirectionsarecalledpricipalstraindirectionsandthenormalstrainsalongthesedirectionsarecalledprincipalstrains.PrincipalstraindirectionsPrincipalstrains(a)概念(Concepts)A123一点存在三个相互垂直的方向,这三个方向之间的

8、切应变均为零,则这三个方向称为该点的应变主方向,这三个方向的正应变称为该点的主应变.Fortheisotropicmaterials,theprincipalstressdirectionsandprincipalstraindirectionsarejustsame.Directions1,2,3aretheprincipalstraindirectionstoo.Principalelement123对线弹性各向同性材料来说,主应力方向和主应变方向是