Vibration energy flow analysis of periodic nanoplate structures under thermal load using fourth-order strain gradient th
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O R I G I NA L PA P E R
Tao Chen · Haixia Chen · Liangmei Liu
Vibration energy flow analysis of periodic nanoplate structures under thermal load using fourth-order strain gradient theory
Received: 1 April 2020 / Revised: 24 May 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract A nonlocal Kirchhoff plate model with fourth-order strain gradient theory is firstly proposed to study variations with band gap frequencies and vibration energy distribution. The temperature rise is supposed to vary linearly through the thickness of the periodic nanoplate structures. The dynamic equations of finite periodic nanoplate structures under thermal load with the small-scale effect and the nonlinear membrane strain taken into consideration are derived based on the finite element method. The structural intensity approach is developed to predict the vibration energy flow of the periodic nanoplates. Effects of the temperature rise on band gaps and the structural intensity are divided into two parts. One is nonlinear effects of temperature rise, and the other is effects of the thermal load. In the numerical calculation, the natural frequencies of singlelayer graphene sheets computed by the nonlocal finite element method with fourth-order strain gradient agree well with analytical results, which validate the effectiveness of the present method. The influences of nonlocal parameters and thermal load on the band gap and structural intensity are considered, respectively. The boundary value has been achieved to determine the critical mechanical load or thermal load for analyzing in depth the effects of the mechanical load and thermal load on the structural intensity. The proposed method shows that the vibration energy flow pattern may be controlled by adjusting the magnitude of the mechanical or thermal load.
1 Introduction Nanoscale structures such as single-layer graphene or nanoplates have been widely used as basic elements in nanosensors, nanoresonators, and nanoelectromechanical systems, due to their good vibration characteristics [1–5]. At the same time, single-layer graphene also has been made as heat dissipation film in electronic devices because of the outstanding thermal properties and energy dissipation characteristics [6–8]. Moreover, periodic nanostructures have been investigated or fabricated to exhibit some interesting properties of nanoscale structures [9–16]. In order to improve their vibration or energy dissipation characteristics, it is necessary and important to assess the vibration energy flow in the periodic nanoplates under dynamic loads, such as thermal load and mechanical load. Generally, molecular dynamics simulations and nonlocal continuum mechanics are used to research the thermal dynamic characteristics of the nanoplates. Liu and Wang investigated the dynamics problems in the thermal vibration of single-layer graphene using molecular dynamics [17]. Nonlocal continuum mechanics theory can be widely employed to investigate the thermal dynamic characteristics of the nanoplates, due to th
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