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首页> 外文期刊>Applied Mathematical Modelling >Coupled longitudinal-transverse-rotational free vibration of post-buckled functionally graded first-order shear deformable micro- and nano-beams based on the Mindlin's strain gradient theory
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Coupled longitudinal-transverse-rotational free vibration of post-buckled functionally graded first-order shear deformable micro- and nano-beams based on the Mindlin's strain gradient theory

机译:基于Mindlin应变梯度理论的后屈曲功能梯度分级一阶剪切可变形微束和纳米束的纵向横向旋转自由振动耦合

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摘要

Presented herein is a comprehensive study on the size-dependent coupled longitudinal-transverse-rotational free vibration behavior of post-buckled functionally graded (FG) micro- and nano-beams based on the most general Mindlin's strain gradient theory. The current model enables us to incorporate size effects via introducing material length scale parameters and is developed in the framework of the first-order shear deformable beam model and the von Karman geometric nonlinearity. The FG micro- and nano-beams, whose volume fraction is expressed by using a power law function, are assumed to be made of a mixture of metals and ceramics. By using Hamilton's principle, the nonlinear governing equations and associated boundary conditions are derived for FG micro- and nano-beams in the postbuckling domain. Afterwards, the governing equations and boundary conditions are discretized using the generalized differential quadrature (GDQ) method in conjunction with a direct approach without linearization, before solving numerically by Newton's method. The effects of length scale parameter, length-to-thickness ratio, material gradient index and boundary conditions on the postbuckling path and frequency of FG micro-and nano-beams are carefully investigated. Finally, numerical results obtained from both the modified strain gradient theory (MSGT) and modified couple stress theory (MCST) are compared.
机译:本文介绍了基于最普遍的Mindlin应变梯度理论的后屈曲功能梯度(FG)微束和纳米束的尺寸相关的纵向-横向-旋转自由振动行为的综合研究。当前模型使我们能够通过引入材料长度比例参数来合并尺寸效应,并且是在一阶剪切可变形梁模型和von Karman几何非线性的框架中开发的。假设FG微束和纳米束的体积分数是通过使用幂律函数表示的,它是由金属和陶瓷的混合物制成的。利用汉密尔顿原理,推导了屈曲后域中FG微束和纳米束的非线性控制方程和相关边界条件。然后,在通过牛顿法进行数值求解之前,使用广义微分正交(GDQ)方法结合不带线性化的直接方法离散化控制方程和边界条件。仔细研究了长度尺度参数,长度厚度比,材料梯度指数和边界条件对FG微束和纳米束的后屈曲路径和频率的影响。最后,比较了从修正应变梯度理论(MSGT)和修正偶应力理论(MCST)获得的数值结果。

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