将上式对z积分,得到 p az 2 (2'(=)+z()+v(二) 再对z积分,得到 =(0()+0(-)+|v(二)dz+8(E) 2 y(二)dz=(=) 即 (z)=x(=) 则 p=(E0(二)+20(-)+(=)+g(2) 16
16 ( '( ) '( ) ( )) 2 1 z z z z z z = + + 再对z积分，得到 ( ( ) ( ) ( )d ( )) 2 1 z z z z z z g z = + + + (z)dz (z) 令 = 即 (z) = '(z) ( ( ) ( ) ( ) ( )) 2 1 = z z + z z + z + g z 将上式对 z 积分，得到 则
Notice the biharmonic function on the left side of the above equation is a real function. It is obvious that the four terms on the right side must be conjugate two and two. The first two terms is conjugate, and the next two terms should be also conjugate Let g(三)=X(=) we obtain the famous gusa formula =[E()+20(二)+(2)+(= 2 it can be also written as =Re[z0(z)+(z) 17
17 Notice the biharmonic function on the left side of the above equation is a real function. It is obvious that the four terms on the right side must be conjugate two and two. The first two terms is conjugate, and the next two terms should be also conjugate: g(z) = (z) Let we obtain the famous gusa formula [ ( ) ( ) ( ) ( )] 2 1 = z z + z z + z + z it can be also written as = Re[z(z) + (z)]
注意上式左边的重调和函数中是实函数,可见该 式右边的四项一定是两两共轭,前两项已经是共 轭的,后两项也应是共轭的: 令 g(三)=X(=) 即得有名的古萨公式 O==E0 2140(-)+z0(2)+x(x)+x(2) (=) 也可以写成 o=re[zo(z)+x(z) 18
18 注意上式左边的重调和函数φ是实函数，可见该 式右边的四项一定是两两共轭，前两项已经是共 轭的，后两项也应是共轭的： g(z) = (z) 令 即得有名的古萨公式 [ ( ) ( ) ( ) ( )] 2 1 = z z + z z + z + z 也可以写成 = Re[z(z) + (z)]
So in plane problems when body force is constant, stress function can be represented by two analytical functions of complex variable z, (p(z) and x(z), named K-M function. So solving plane problems is just selecting K-M function properly and determining any constant in them according to boundary condition 19
19 So in plane problems when body force is constant, stress function φ can be represented by two analytical functions of complex variable z, (z) and (z), named K-M function. So solving plane problems is just selecting K-M function properly and determining any constant in them according to boundary condition
于是可见,在常量体力的平面问题中,应力函数 总可以用复变数z的两个解析函q(z)和x(z)来表示, 称为K-M函数。而求解各个具体的平面问题,可归结 为适当地选择这两个解析函数,并根据边界条件决定其 中的任意常数。 20
20 于是可见，在常量体力的平面问题中，应力函数φ 总可以用复变数z的两个解析函 (z)和(z)来表示， 称为K-M 函数。而求解各个具体的平面问题，可归结 为适当地选择这两个解析函数，并根据边界条件决定其 中的任意常数