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ドゥーガルの変位関数

Dougall's displacement potential

 1904年J. Dougallが導いた変位関数である.\(\left( {x,y,z} \right)\)座標で\[\begin{array}{l} 2G{u_x} = \frac{{\partial {\varphi _0}}}{{\partial x}} + x\frac{{\partial {\varphi _4}}}{{\partial z}} + 2\frac{{\partial {\lambda _3}}}{{\partial y}}\\ 2G{u_y} = \frac{{\partial {\varphi _0}}}{{\partial y}} + y\frac{{\partial {\varphi _4}}}{{\partial z}} - 2\frac{{\partial {\lambda _3}}}{{\partial x}}\\ 2G{u_z} = \frac{{\partial {\varphi _0}}}{{\partial z}} - x\frac{{\partial {\varphi _4}}}{{\partial x}} - y\frac{{\partial {\varphi _4}}}{{\partial y}} - 4\left( {1 - \nu } \right){\varphi _4} \end{array}\]ここで\[\begin{array}{l} {\nabla ^2}{\varphi _0} = {\nabla ^2}{\varphi _4} = {\nabla ^2}{\lambda _3} = 0\\ {\nabla ^2} = \frac{{{\partial ^2}}}{{\partial {x^2}}} + \frac{{{\partial ^2}}}{{\partial {y^2}}} + \frac{{{\partial ^2}}}{{\partial {z^2}}} \end{array}\]Gは横弾性係数,νはポアソン比である.