Abstract

This article deals with introducing a closed-form solution for considering the vibration of macro-micro-nano plates and shells made of unit material with functionally graded porosity (FGP). Classic continuum theory beside modified nonlocal strain gradient theory are used at macro and micro/nano scales, respectively. The governing equations are obtained in the frame work of third order plate and shell theories. In the solution procedure, a differential operator is defined and by using it, five equations of motion are converted to the five algebraic equations. Then, by satisfying the equations of boundary conditions and achieving the nonzero solution, the frequencies are extracted.

摘要

本文介绍了一种封闭形式的解决方案，用于考虑由具有功能梯度孔隙率 (FGP) 的单元材料制成的宏观-微纳米板和壳的振动。除了修正的非局部应变梯度理论之外，经典的连续统理论分别用于宏观和微/纳米尺度。控制方程是在三阶板壳理论框架下获得的。在求解过程中，定义了一个微分算子，并通过使用它，将五个运动方程转换为五个代数方程。然后，通过满足边界条件方程并获得非零解，提取频率。