l-组合变形-应变状态理论课件

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。Likethestateofstressatonepoint,thestateofstraincanbedeterminedbyamatrix.Itiscalledstrainstatematrix.类似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-dimensionalorspacestateofstrainIfinmatrixEtheelementseveryrowandarrangearenotallequaltozero,thestateofstrainatonepointiscalledthree-dimensionalstateofstrain.(c)

单向应变状态One-dimensionalstateofstrainThestateofstrainiscalledone-dimensionalstateofstrain.xOthersstrains=0RigidwallForthefreebodythereisnoone-dimensionalstraininengineering.PABxypabPxypqQPxypQqdxdyxy(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-dimentionalanalysisofstrain

x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。

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(2)主应变与应变主方向PrincipalstrainsandprincipalstraindirectionsAtonepointiftherearethreedirectionsverticaltoeachotherandtheshearingstrainsbetweenthemareallequaltozero,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一致的.Directionzisoneofprincipalstraindirections,isaprincipalstrain.

Sothethreeprincipalstrainsfortheplanestateofstrainareasfollows

:0具有最大切应变的方位与主方向相差45°（3）最大切应变(Maximumshearingstrain)2.3

平面（双向）应力状态的应变分析Analysisofstrainsfortheplanestressstate在xy平面内的应变情况与平面应变状态一致平面应力状态xyFortheplanestressstate,theprincipalstrainsare:电阻应变仪应变片电桥电阻变化电压变化放大器