Note_Fem_层合板单元格式

推导基于first order shear deformation theory 的层合板单元.给出刚度矩阵、质量矩阵.

假设位移场:

\[\begin{aligned} {{u ( x, y, z )}} & {{} {{} {}=u_{0} ( x, y )+z \theta_{x} ( x, y )}} \\ {{v ( x, y, z )}} & {{} {{} {}=v_{0} ( x, y )+z \theta_{y} ( x, y )}} \\ {{w ( x, y, z )}} & {{} {{}=w_{0} ( x, y )}} \\ \end{aligned} \]

因此,应变和应力表达式为:

其中:

\[ \left\{\begin{matrix} {{\epsilon_{x, x}^{m}}} \\ {{\epsilon_{y y}^{m}}} \\ {{\gamma_{x y}^{m}}} \\ \end{matrix} \right\}=\left\{\begin{matrix} {{\frac{\partial u_{0}} {\partial x}}} \\ {{\frac{\partial v_{0}} {\partial y}}} \\ {{\frac{\partial u_{0}} {\partial y}+\frac{\partial v_{0}} {\partial x}}} \\ \end{matrix} \right\} \]

\[ \left\{\begin{matrix} {{\epsilon_{x x}^{f}}} \\ {{\epsilon_{y y}^{f}}} \\ {{\gamma_{x y}^{f}}} \\ \end{matrix} \right\}=\left\{\begin{matrix} {{\frac{\partial\theta_{x}} {\partial x}}} \\ {{\frac{\partial\theta_{y}} {\partial y}}} \\ {{\frac{\partial\theta_{x}} {\partial y}+\frac{\partial\theta_{y}} {\partial x}}} \\ \end{matrix} \right\} \]

\[ \left\{\begin{aligned} {{\gamma_{x z}^{( 0 )}}} \\ {{\gamma_{y z}^{( 0 )}}} \\ \end{aligned} \right\}=\left\{\begin{aligned} {{\frac{\partial w_{0}} {\partial x}+\theta_{x}}} \\ {{\frac{\partial w_{0}} {\partial y}+\theta_{y}}} \\ \end{aligned} \right\} \]

应变-位移矩阵B是由三部分(膜+弯曲+剪切)组成的;