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ひずみ-変位関係式

strain-displacement relations

 物体の変形が微小であるならば,直角座標系\(\left( {x,y,z} \right)\)の工学ひずみ成分\(\left( {{\varepsilon _x},{\varepsilon _y},{\varepsilon _z},{\gamma _{xy}},{\gamma _{yz}},{\gamma _{zx}}} \right)\)と変位成分\(\left( {u,v,w} \right)\)の間には\[\begin{array}{l} {\varepsilon _x} = \frac{{\partial u}}{{\partial x}},\;{\varepsilon _y} = \frac{{\partial v}}{{\partial y}},\;{\varepsilon _z} = \frac{{\partial w}}{{\partial z}}\\ {\gamma _{xy}} = {\gamma _{yx}} = \frac{{\partial u}}{{\partial y}} + \frac{{\partial v}}{{\partial x}}\\ {\gamma _{yz}} = {\gamma _{zy}} = \frac{{\partial w}}{{\partial y}} + \frac{{\partial v}}{{\partial z}}\\ {\gamma _{xz}} = {\gamma _{xz}} = \frac{{\partial u}}{{\partial z}} + \frac{{\partial w}}{{\partial x}} \end{array}\]の関係がある.なおテンソルひずみの場合,せん断ひずみ成分はそれぞれ\({\varepsilon _{xy}} = \frac{1}{2}{\gamma _{xy}}\),\({\varepsilon _{yz}} = \frac{1}{2}{\gamma _{yz}}\),\({\varepsilon _{zx}} = \frac{1}{2}{\gamma _{zx}}\)で与えられる.これをひずみ-変位関係式という.