Purpose

The functionally graded circular nanoplate is a commonly seen component in the nano-electromechanical system. It is indispensable to examine free vibration behaviors of such an axisymmetric nanostructure subjected to uniformly distributed loads in the radial direction. Although the vibration engineering and technology have been fully studied at the macro-scale, there are still many unsolved problems at the micro-scale. The present research aims to promote the theoretical characterization of vibration behaviors at micro/nano-scale and further provide a basis for the development of vibration testing technologies.

Methods

Using the nonlocal strain gradient approach and Mindlin plate theory, we develop the theoretical model describing the transverse free vibration of the axisymmetric functionally graded circular nanoplate. First, by considering the nonlocal strain gradient constitutive relation, we derive the equation of motion via Hamilton’s principle in polar coordinate system. Subsequently, the differential quadrature method is employed to solve the equation of motion numerically.

Results

The natural frequencies of circular nanoplates decrease with an increase of a radial compression while increase with an increase of a radial tension. The first-mode natural frequency reduces to zero under a certain radial compression, bringing about the dynamical instability. The natural frequencies are sensitive to the radial compression, and the clamped boundary constraint is more resistant to external loads than the simply supported one. An increase in the nonlocal parameter results in lower natural frequencies, while an increase in the strain gradient characteristic parameter results in higher ones.

Conclusions

It is demonstrated that the strain gradient characteristic parameter has a threshold in the present model for functionally graded circular nanoplates. In the circumstance of a lower nonlocal parameter than the strain gradient characteristic parameter, the circular nanoplate shows hardening behaviors. In the circumstance of a greater nonlocal parameter, the circular nanoplate shows softening behaviors. When the two internal characteristic parameters are equal, the stiffness of nanoplates remains unchanged and degenerates into its classical counterpart.

中文翻译：

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具有径向载荷的轴对称功能梯度圆形纳米板的横向自由振动

目的

功能分级的圆形纳米板是纳米机电系统中的常见组件。检查这样的轴对称纳米结构在径向上受到均匀分布的载荷的自由振动行为是必不可少的。尽管在宏观上已经对振动工程和技术进行了充分的研究，但是在微观上仍然存在许多未解决的问题。本研究旨在促进微观/纳米尺度上振动行为的理论表征，并为振动测试技术的发展提供基础。

方法

使用非局部应变梯度方法和Mindlin板理论，我们开发了描述轴对称功能梯度圆形纳米板的横向自由振动的理论模型。首先，通过考虑非局部应变梯度本构关系，利用汉密尔顿原理在极坐标系中推导了运动方程。随后，采用微分求积法对运动方程进行数值求解。

结果

圆形纳米板的固有频率随着径向压缩的增加而降低，而随着径向张力的增加而增加。在一定的径向压缩下，第一模固有频率降低到零，从而导致动态不稳定。固有频率对径向压缩敏感，并且受约束的边界约束比简单支撑的约束更能抵抗外部载荷。非局部参数的增加导致较低的固有频率，而应变梯度特征参数的增加导致较高的固有频率。

结论

结果表明，在本模型中，梯度梯度特征参数对于功能梯度圆纳米板具有阈值。在非局部参数低于应变梯度特征参数的情况下，圆形纳米板表现出硬化行为。在较大的非局部参数的情况下，圆形纳米板显示出软化行为。当两个内部特征参数相等时，纳米板的刚度将保持不变，并退化为其经典对应物。