《弹性力学》（双语版） 第五章 平面问题的复变函数

Chapter s compleyoyeriable Methods for Plane Elasticity

1 Elasticity

弹单性生力学

2

Chapter 5 Complex-variable Methods for Plane elasticity 85-1 Complex-variable representation of stress function D$5-2 Complex-variable representation of stress and displacement D85-3 Complex-variable representation of boundary condition >85-4 The single-valued condition of stress and displacement in multiply connected region 85-5 Situation of infinite multiply connected body 85-6 Problem of infinite plane including hole

3 §5-4 The single-valued condition of stress and displacement in multiply connected region §5-3 Complex-variable representation of boundary condition §5-2 Complex-variable representation of stress and displacement §5-1 Complex-variable representation of stress function §5-6 Problem of infinite plane including hole §5-5 Situation of infinite multiply connected body Chapter 5 Complex-variable Methods for Plane Elasticity

第五章平面问题的复变函数法 §5-1应力函数的复变函数表示 §5-2应力和位移的复变函数表示 §5-3边界条件的复变函数表示 §5-4多连通域内应力与位移的单值条件 §5-5无限大多连体的情形 §5-6含孔口的无限大板问题

4 §5-4 多连通域内应力与位移的单值条件 §5-3 边界条件的复变函数表示 §5-2 应力和位移的复变函数表示 §5-1 应力函数的复变函数表示 §5-6 含孔口的无限大板问题 §5-5 无限大多连体的情形 第五章 平面问题的复变函数法

Chapter 5 Complex-variable Methods for Plane Elasticity When solving plane problems by Cartesian coordinates or polar coordinates, the boundary of object is straight line or circular arc To other boundary, for example ellipse, hyperbola, non concentric circles and so on, we need use different curvilinear coordinates. Applying complex-variable can predigest these problems In this chapter, we just introduce the simple application of complex-variable in elasticity 5

5 Chapter 5 Complex-variable Methods for Plane Elasticity When solving plane problems by Cartesian coordinates or polar coordinates, the boundary of object is straight line or circular arc .To other boundary, for example ellipse, hyperbola, non￾concentric circles and so on, we need use different curvilinear coordinates. Applying complex-variable can predigest these problems. In this chapter, we just introduce the simple application of complex-variable in elasticity

第五章平面问题的复变函数法 直角坐标及极坐标求解平面问题,所涉及的物体 边界是直线或圆弧形。对于其他一些边界,例如椭 圆形、双曲形、非同心圆等就要用不同的曲线坐标。 应用复变函数可使该类问题得以简化。本章只限于 介绍复变函数方法在弹性力学中的简单应用

6 第五章 平面问题的复变函数法 直角坐标及极坐标求解平面问题，所涉及的物体 边界是直线或圆弧形。对于其他一些边界，例如椭 圆形、双曲形、非同心圆等就要用不同的曲线坐标。 应用复变函数可使该类问题得以简化。本章只限于 介绍复变函数方法在弹性力学中的简单应用

s5-1 Complex-variable representation of stress function In chapter 2, we have proved, in plane problems, there is a stress function that is biharmonic function of position coordinates, if body force is constant, i.e Vo=O Now introduce complex variable =x tiy and 2-x=iy to replace real variable x and y , noticing az_1 =1 ax Oy az =1 7

7 §5-1 Complex-variable representation of stress function In chapter 2,we have proved, in plane problems, there is a stress function φ that is biharmonic function of position coordinates, if body force is constant, i.e. 0 4   = 1, i 1, i = −   =   =   =   y z x z y z x z Now introduce complex variable z= x＋iy and z＝x－iy to replace real variable x and y. Noticing

§5-1应力函数的复变函数表示 在第二章中已经证明,在平面问题里,如 果体力是常量,就一定存在一个应力函数φ, 它是位置坐标的重调和函数,即 Vo=O 现在,引入复变数=x+i和z=x-以代替实 变数x和y。注意 az =1 ar 02_ z l

8 §5-1 应力函数的复变函数表示 在第二章中已经证明，在平面问题里，如 果体力是常量，就一定存在一个应力函数φ， 它是位置坐标的重调和函数，即 0 4   = 现在，引入复变数z= x＋iy和 z＝x－iy以代替实 变数x 和y。注意 1, i 1, i = −   =   =   =   y z x z y z x z

dextrane Methods for Pane bestial We find the transformation are ao 00 0z 000z O ox ae ax az az az ay az Oy Oz Oy az 0z ) + N2O0022 Ox Oy az Ox Oy dz furthermore (+=)y=-(-=) Ox 0z az Oz az 9

9 We find the transformation are          −   =     +     =     +   =     +     =           i( ) ( ) y z z z y z z y z x z z z x z z x z x y z x y z  =   −     =   +        i 2 , i 2 furthermore,     2 2 2 2 2 2 ( ) , ( ) x z z y z z  −   = −     +   =  

可以得到变换式 ao 00 0z 000z O ax a ax az az az Oy az Oy az ay az 0z ) + N2O0022 Ox Oy az Ox Oy dz 进而 (+=)y=-(-=) Ox 0z az Oz az 10

10 可以得到变换式          −   =     +     =     +   =     +     =           i( ) ( ) y z z z y z z y z x z z z x z z x z x y z x y z  =   −     =   +        i 2 , i 2 进而     2 2 2 2 2 2 ( ) , ( ) x z z y z z  −   = −     +   =  