三次元でねじりのない軸対称弾性問題でよく用いられる．Michellの変位関数Φは重調和関数であり，円柱座標系\(\left( {r,\theta ,z} \right)\)での変位成分は\[\begin{array}{l} 2G{u_r} = - \frac{{{\partial ^2}\it {\Phi} }}{{\partial r\partial z}},\;{u_\theta } = 0,\\ 2G{u_z} = 2\left( {1 - \nu } \right){\nabla ^2}\it {\Phi} - \frac{{{\partial ^\rm{2}}\it {\Phi} }}{{\partial {z^\rm{2}}}} \end{array}\]ここで\[\begin{array}{l} {\nabla ^4}\it{\Phi} = \rm{0}\\ {\nabla ^2} = \frac{{{\partial ^2}}}{{\partial {r^2}}} + \frac{1}{r}\frac{\partial }{{\partial r}} + \frac{{{\partial ^2}}}{{\partial {z^2}}} \end{array}\]Gは横弾性係数，νはポアソン比である．【応力関数】