基于一种新修正偶应力理论建立了微尺度平面正交各向异性功能梯度梁模型.模型中包含两个材料尺度参数,因此能够分别描述在两个正交方向上由尺度效应带来的不同大小弯曲刚度增强.基于最小势能原理推导了平衡方程和边界条件,并以自由端受集中载荷作用的悬臂梁为例给出了弯曲问题的解析解.该梁模型的控制方程以及解的形式和经典梁模型是一致的,只是在刚度项中增加了一项和尺度效应有关的项.算例结果表明:采用本文模型所预测的梁挠度总是小于经典理论的结果,即捕捉到了尺度效应.尺度效应会随着梁几何尺寸的减小而增大,并在梁的几何尺寸远大于尺度参数时逐渐消失.
Based on the new couple stress theory,a new model of plane orthotropic functionally graded beam mod el was proposed.The two material scale parameters included in the model enables present model to describe different bending stiffness enhancements in two orthogonal directions caused by scale effect.The equilibrium equations and related boundary conditions were derived using the principle of minimum potential energy.Analytical solution of bending problem was developed for a bending cantilever beam with concentrate force applied on the free end.The governing equations and solutions of present model have similar form with classical beam model,except one additional term related to scale effect in rigidity.Numerical results indicate that deflection of the beam predicted by present model is smaller than that of the classical one,which means the scale effect can be captured.The scale effect will increase with the decrease of the geometrical size of the beam,but diminish little by little when the geometrical size is much larger than the material scale parameter.
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