Abstract

In the current study, vibration analysis of functionally graded (FG) nano-sized beams resting on a elastic foundation is presented via a finite element method. The elastic foundation is simulated by using one-parameter Winkler type elastic foundation model. Euler-Bernoulli beam theory and Eringen’s nonlocal elasticity theory are utilized to model the functionally graded nano-sized beams with various boundary conditions such as simply supported at both ends (S-S), clamped-clamped (C-C) and clamped-simply supported (C-S). Material properties of functionally graded nanobeam vary across the thickness direction according to the power-law distribution. The vibration behaviors of functionally graded nanobeam composed of alumina (Al₂O₃) and steel are shown using nonlocal finite element formulation. The importance of this paper is the utilize of shape functions and the Eringen's nonlocal elasticity theory to set up the stiffness matrices and mass matrices of the functionally graded nano-sized beam resting on Winkler elastic foundation for free vibration analysis. Bending stiffness, foundation stiffness and mass matrices are obtained to realize the solution of vibration problem of the FG nanobeam. The influences of power-law exponent (k), dimensionless nonlocal parameters (e₀a/L), dimensionless Winkler foundation parameters (KW), mode numbers and boundary conditions on frequencies are investigated via several numerical examples and shown by a number of tables and figures.

摘要

在当前的研究中，通过有限元方法介绍了基于弹性地基的功能梯度 (FG) 纳米梁的振动分析。采用单参数Winkler型弹性地基模型模拟弹性地基。利用欧拉-伯努利梁理论和埃林根的非局部弹性理论对具有各种边界条件的功能梯度纳米级梁进行建模，例如两端简支（SS）、夹紧-夹紧（CC）和夹紧-简支（CS） . 功能梯度纳米束的材料特性根据幂律分布在厚度方向上变化。_{由氧化铝（Al 2} O ₃）组成的功能梯度纳米梁的振动行为) 和钢使用非局部有限元公式显示。本文的重要性在于利用形状函数和 Eringen 的非局部弹性理论建立了基于 Winkler 弹性基础的功能梯度纳米梁的刚度矩阵和质量矩阵，用于自由振动分析。获得弯曲刚度、基础刚度和质量矩阵，实现FG纳米梁振动问题的求解。幂律指数 (k)、无量纲非局部参数 (e ₀ a/L)、无量纲 Winkler 基础参数 (KW)、模式数和边界条件对频率的影响通过几个数值示例进行研究，并通过多个表格显示和数字。