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

Abstract

In this paper, a novel numerical technique, namely, shifted Chebyshev polynomials based Rayleigh‐Ritz method has been employed to analyze the buckling characteristics of the nanobeam. The main advantage of the shifted Chebyshev polynomials is that, due to the orthogonality of the polynomials, ill‐conditioning of the system is being avoided with higher number of terms in the approximation. The nanobeam is exposed to both hygroscopic and thermal environments while being subjected to a longitudinal magnetic field. Further, the beam is modeled with Winkler‐Pasternak elastic foundation and nonlocal Euler–Bernoulli beam theory. The governing equation of motion of the proposed model has been derived using the Hamilton's principle, and critical buckling loads for Hinged‐Hinged (HH), Clamped‐Hinged (CH), and Clamped‐Clamped (CC) boundary conditions have been computed. The proposed model is validated against the existing model in special cases, exhibiting excellent agreement. Then a convergence study is performed to ensure the correctness and effectiveness of the method. Furthermore, a comprehensive parametric study has been carried out to determine the impact of various parameters such as the small scale parameter, Winkler modulus, shear modulus, magnetic field intensity parameter, hygroscopic parameter, and temperature parameter.

Winkler‐Pasternak foundation effect on the buckling loads of arbitrarily rigid or restrained supported nonlocal beams made of different FGM and porosity distributions

“…Euler-Bernoulli beam theory is considered in this study. In the context of Euler-Bernoulli beam theory, the non-zero strain expression is written as follows [28,[34][35][36]:…”

“…Euler‐Bernoulli beam theory is considered in this study. In the context of Euler‐Bernoulli beam theory, the non‐zero strain expression is written as follows [28, 34–36]: $$$$\begin{equation}{\epsilon }_{{x}_1{x}_1} = - {x}_3u{\left( {{x}_1} \right)}^{^{\prime\prime}}\end{equation}$$$$ Here, u ( x 1 ) denotes the transverse displacement. $${x}_1\ $$ is used to indicate the length direction of the nanobeam.…”

“…There have been several excellent properties of nano/micro-sized structures such as large surface area, high sensitivity and enhanced mechanical stability [18][19][20]. Consequently, several researches have been performed to study the stability, dynamic and static characteristics of these structures in recent decades [21][22][23][24][25][26][27][28][29][30][31][32][33][34].…”

The Application of the Modified Lindstedt–Poincaré Method to Solve the Nonlinear Vibration Problem of Exponentially Graded Laminated Plates on Elastic Foundations

“…Song et al [27] proposed an analytical method for linear vibration analysis of arbitrarily shaped non-homogeneous orthotropic plates of variable thickness resting on the Winkler-Pasternak foundation. Melaibari et al [28,29] examined the static response of 2D functionally graded porous plates resting on elastic foundations using midplane and neutral surfaces with movable constraints and differential quadrature method. Jena et al [30,31] presented the free vibration of functionally graded beams and the stability of nanobeams resting on the Winkler-Pasternak elastic foundation using a novel numerical approach.…”