Vibration energy flow analysis of periodic nanoplate structures under thermal load using fourth-order strain gradient theory

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 single-layer 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.

Appendices

Appendix A

$\mathbf{w }_{\mathbf{e }}$ and $\mathbf{B }_{\mathbf{T }}$ are written as

$$\begin{aligned} \mathbf{w }_{\mathbf{e }}= & {} \left[ {{\begin{array}{cccccccccccc} {w_{1}} &{} \quad {\theta _{x_{1}}} &{} \quad {\theta _{y_{1}}} &{} \quad {w_{2}} &{} \quad {\theta _{x_{2}}} &{} \quad {\theta _{y_{2}}} &{} \quad {w_{3}} &{} \quad {\theta _{x_{3}}} &{} \quad {\theta _{y_{3}}} &{} \quad {w_{4}} &{} \quad {\theta _{x_{4}}} &{} \quad {\theta _{y_{4}}} \\ \end{array}}} \right] ^{\mathrm{T}}, \end{aligned}$$

(A.1)

$$\begin{aligned} \mathbf{B }_{\mathbf{T }}= & {} \left[ {{\begin{array}{cccccccccccc} 0 &{} \quad 1 &{} \quad 0 &{} \quad {2x} &{} \quad y &{} \quad 0 &{} \quad {3x^{2}} &{} \quad {2xy} &{} \quad {y^{2}} &{} \quad 0 &{} \quad {3x^{2}y} &{} \quad {y^{3}} \\ 0 &{} \quad 0 &{} \quad 1 &{} \quad 0 &{} \quad x &{} \quad {2y} &{} \quad 0 &{} \quad {x^{2}} &{} \quad {2xy} &{} \quad {3y^{2}} &{} \quad {x^{3}} &{} \quad {3xy^{2}} \\ \end{array}}} \right] \mathbf{A }^{-1}, \end{aligned}$$

(A.2)

where each node of four-node plate elements has three degrees of freedom $w_{i}$, $\theta _{x_{i}} =\partial w_{i} /\partial y$, and $\theta _{y_{i}} =-\partial w_{i} /\partial x$. $\mathbf{A }$ represents the shape function’s evaluations at the element nodes.

Appendix B

$\mathbf{B }$ is written as

$$\begin{aligned} \mathbf{B }=\left[ {{\begin{array}{cccccccccccc} 0 &{} \quad 0 &{} \quad 0 &{} \quad 2 &{} \quad 0 &{} \quad 0 &{} \quad {6x} &{} \quad {2y} &{} \quad 0 &{} \quad 0 &{} \quad {6xy} &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 2 &{} \quad 0 &{} \quad 0 &{} \quad {2x} &{} \quad {6y} &{} \quad 0 &{} \quad {6xy} \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 2 &{} \quad 0 &{} \quad 0 &{} \quad {4x} &{} \quad {4y} &{} \quad 0 &{} \quad {6x^{2}} &{} \quad {6y^{2}} \\ \end{array}}} \right] \mathbf{A }^{-1}. \end{aligned}$$

(B.1)

Appendix C

$\mathbf{S }$ is written as

$$\begin{aligned} \mathbf{S }=\left[ {{\begin{array}{cccccccccccc} 1 &{} \quad x &{} \quad y &{} \quad {x^{2}} &{} \quad {xy} &{} \quad {y^{2}} &{} \quad {x^{3}} &{} \quad {x^{2}y} &{} \quad {xy^{2}} &{} \quad {y^{3}} &{} \quad {x^{3}y} &{} \quad {xy^{3}} \\ \end{array}}} \right] \mathbf{A }^{-1}. \end{aligned}$$

(C.1)