====== ミッチェルの変位関数 ======
==== Michell's displacement function ====

{{tag>..c07}}

　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．【[[07:1001447|応力関数]]】


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