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サンブナンの適合条件式

Saint Venant's compatibility equations

 微小変形に対するひずみの適合条件式であって,直角座標(x,y,z)に関しては次の六個の式によって与えられる.\[\begin{array}{l} \frac{{{\partial ^2}{\varepsilon _{xx}}}}{{\partial y\partial z}} = \frac{\partial }{{\partial x}}\left( { - \frac{{\partial {\varepsilon _{yz}}}}{{\partial x}} + \frac{{\partial {\varepsilon _{zx}}}}{{\partial y}} + \frac{{\partial {\varepsilon _{xy}}}}{{\partial z}}} \right),\\ 2\frac{{{\partial ^2}{\varepsilon _{yz}}}}{{\partial y\partial z}} = \frac{{{\partial ^2}{\varepsilon _{zz}}}}{{\partial {y^2}}} + \frac{{{\partial ^2}{\varepsilon _{yy}}}}{{\partial {z^2}}}\\ \frac{{{\partial ^2}{\varepsilon _{yy}}}}{{\partial z\partial x}} = \frac{\partial }{{\partial y}}\left( {\frac{{\partial {\varepsilon _{yz}}}}{{\partial x}} + \frac{{\partial {\varepsilon _{zx}}}}{{\partial y}} + \frac{{\partial {\varepsilon _{xy}}}}{{\partial z}}} \right),\\ 2\frac{{{\partial ^2}{\varepsilon _{zx}}}}{{\partial z\partial x}} = \frac{{{\partial ^2}{\varepsilon _{xx}}}}{{\partial {z^2}}} + \frac{{{\partial ^2}{\varepsilon _{zz}}}}{{\partial {x^2}}}\\ \frac{{{\partial ^2}{\varepsilon _{zz}}}}{{\partial x\partial y}} = \frac{\partial }{{\partial z}}\left( {\frac{{\partial {\varepsilon _{yz}}}}{{\partial x}} + \frac{{\partial {\varepsilon _{zx}}}}{{\partial y}} + \frac{{\partial {\varepsilon _{xy}}}}{{\partial z}}} \right),\\ 2\frac{{{\partial ^2}{\varepsilon _{xx}}}}{{\partial x\partial y}} = \frac{{{\partial ^2}{\varepsilon _{yy}}}}{{\partial {x^2}}} + \frac{{{\partial ^2}{\varepsilon _{zz}}}}{{\partial {y^2}}} \end{array}\]【適合条件