Buckling Behavior of Nanobeams Placed in Electromagnetic Field Using Shifted Chebyshev Polynomials-Based Rayleigh-Ritz Method

Abstract

In the present investigation, the buckling behavior of Euler–Bernoulli nanobeam, which is placed in an electro-magnetic field, is investigated in the framework of Eringen’s nonlocal theory. Critical buckling load for all the classical boundary conditions such as “Pined–Pined (P-P), Clamped–Pined (C-P), Clamped–Clamped (C-C), and Clamped-Free (C-F)” are obtained using shifted Chebyshev polynomials-based Rayleigh-Ritz method. The main advantage of the shifted Chebyshev polynomials is that it does not make the system ill-conditioning with the higher number of terms in the approximation due to the orthogonality of the functions. Validation and convergence studies of the model have been carried out for different cases. Also, a closed-form solution has been obtained for the “Pined–Pined (P-P)” boundary condition using Navier’s technique, and the numerical results obtained for the “Pined–Pined (P-P)” boundary condition are validated with a closed-form solution. Further, the effects of various scaling parameters on the critical buckling load have been explored, and new results are presented as Figures and Tables. Finally, buckling mode shapes are also plotted to show the sensitiveness of the critical buckling load.

A novel numerical approach for the stability of nanobeam exposed to hygro‐thermo‐magnetic environment embedded in elastic foundation

“…The Rayleigh-Ritz method using shifted Chebyshev polynomials as shape functions have been used to transform Equation ( 25) into the generalized eigenvalue problem given in Equation (29). The generalized eigenvalue problem has been solved to find critical buckling loads and 2nd buckling loads for HH, CH, and CC boundary conditions.…”

“…Chebyshev polynomials of the first kind (𝐶 𝑛 (𝑋)) are a sequence of orthogonal polynomials with 𝑋 ∈ [−1 1], and few terms of the sequence are defined as [29] 𝐶 0 (𝑋) = 1…”

“…This is accomplished by transforming 𝑋 ↦ 2𝑋 − 1, and there exists a one-to-one correspondence between [0 1] and [− 1 1]. As a result, the first few terms of first-order shifted Chebyshev polynomials (𝐶 * 𝑛 (𝑋)) can be expressed as [29], (where 𝐶 * 𝑛 (𝑋) = 𝐶 𝑛 (2𝑋 − 1))…”

“…Chebyshev polynomials of the first kind $$( {{C_n}( X )} )$$ are a sequence of orthogonal polynomials with $$X \in [ { - 1\,\,\,1} ]$$, and few terms of the sequence are defined as [29] $$\begin{equation} \def\eqcellsep{&}\begin{array}{l} {C_0}(X) = 1\\[3pt] {C_1}(X) = X\\[3pt] {C_n}(X) = 2X{C_{n - 1}}(X) - {C_{n - 2}}(X),\quad n = 2,\,3,\, \ldots \end{array} \end{equation}$$…”

“…In the pioneering work of Samaniego et al [21], Deep Neural Networks was explored as an option for approximation to investigate mechanical problems and analyze the energetic format of the PDE. Other related works of nanostructures considering the magnetic field and the hygrothermal environment can be seen in [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38].…”