《弹性力学》（双语版） 第四章 平面问题的极坐标解答

POELSOUIONSFOR2 Angle of rotation of radial line segment PA, have (uo+ a= O Angle of rotation of hoop line segment PB, have B=-- Thus shear strain have: rre=a+B. ao If exists radial and loop displaces, from superposition method have: u, I Oup rr ae 1 au a y + r a8 a Such are geometric formulas in polar coordinates 21

21 If exists radial and loop displaces,from superposition method have：                   r u r u u r u r r u r u r r r r r           1 1 Such are geometric formulas in polar coordinates. Thus shear strain,have: Angle of rotation of hoop line segment PB,have ： r u r u r             r u     Angle of rotation of radial line segment PA,have ： r u dr dr u r u u              ( )

径向线段PA的转角为: ar dr)-ug a a= ar 环向线段PB的转角为: B= 可见剪应变为: fre a+B= ouo ug 如果同时存在径向和环向位移,则由叠加法得: rr ae Yre 1 our+ ra0 ar 这就是极坐标中的几何方程。 22

22 如果同时存在径向和环向位移，则由叠加法得：            ru r u u r u r r u ru r r r r r           1 1 这就是极坐标中的几何方程。 径向线段 PA 的转角为： ru drdr u ru u            ( ) 环向线段 PB的转角为： ru    可见剪应变为： ru ru r           

POtLSOUTIOFORP II Physical Equations (1) states of planar stress E 6 (oo-) E 2(1+) Yre Gre e (2) In planar strain's situation E Substitute 1-f and 1-u for E andu in above formula, respectively 8. E 6 8 E 6 2(1+4) Yre E

23                                    r r r r r E E E 2(1 ) ) 1 ( 1 ) 1 ( 1 2 2 (2) In planar strain’s situation: Substitute 2 and for E and in above formula,respectively. 1  E    1 II、Physical Equations (1) states of planar stress：                            r r r r r r G E E E 1 2(1 ) ( ) 1 ( ) 1

物理方程 (1)平面应力情况: E,=(0,-Hoe E EO.-Ho Yre =-T.o= 2(1+) G E (2)平面应变情况: 将上式中的E换为 E ,μ换为 E=<Al E E E y E

24 （2）平面应变情况：                             r r r r r EEE2(1 ) ) 1 ( 1 ) 1 ( 1 22 将上式中的 E 换为 2 ， 换为 。 1  E   1 二、物理方程                         r r r r r r G E EE1 2(1 ) ( ) 1 ( ) （ 1 1）平面应力情况：

84-3 Stress Functions and Consistent Equations in Polar coordinates To get stresses and consistent equations denoted by stress functions in polar coordinates, using relationship between polar and rectangular coordinates: r2=x2+y2, 0=arctan 2 x=rcos6, y=rsin g have Or x r s0, ay cOS SIn 0 y sin 80 x COS 6 x y 25

25 §4-3 Stress Functions and Consistent Equations in Polar Coordinat e s To get stresses and consistent equations denoted by stress functions in polar coordinates，using relationship between polar and rectangular coordinates：    cos , sin , arctan 2 2 2 x r y r x y r x y      have： r r x r r y y x r y y r r x x r             cos , sin cos , sin , 2 2          