Application of the Higher-Order Hamilton Approach to the Nonlinear Free Vibrations Analysis of Porous FG Nano-Beams in a Hygrothermal Environment Based on a Local/Nonlocal Stress Gradient Model of Elasticity

Abstract

Nonlinear transverse free vibrations of porous functionally-graded (FG) Bernoulli–Euler nanobeams in hygrothermal environments through the local/nonlocal stress gradient theory of elasticity were studied. By using the Galerkin method, the governing equations were reduced to a nonlinear ordinary differential equation. The closed form analytical solution of the nonlinear natural flexural frequency was then established using the higher-order Hamiltonian approach to nonlinear oscillators. A numerical investigation was developed to analyze the influence of different parameters both on the thermo-elastic material properties and the structural response, such as material gradient index, porosity volume fraction, nonlocal parameter, gradient length parameter, mixture parameter, and the amplitude of the nonlinear oscillator on the nonlinear flexural vibrations of metal–ceramic FG porous Bernoulli–Euler nano-beams.

Numerical Study on the Buckling Behavior of FG Porous Spherical Caps Reinforced by Graphene Platelets

“…[ 51 , 52 ] to treat the buckling response of isotropic and orthotropic shallow spherical caps, whose problem was solved analytically by means of Chebychev series [ 51 ], or numerically according to classical finite elements [ 52 ]. At the present state, however, there is a general lack of works from the literature focusing on the dynamic buckling of GPL-reinforced porous nanocomposite spherical shells, whose aspects are explored here according to the 3D elasticity basics and Green deformation nonlinearities, rather than common shell theories and Von-Karman nonlinearities, as proposed in [ 53 ].…”

Integral nonlocal stress gradient elasticity of functionally graded porous Timoshenko nanobeam with symmetrical or anti‐symmetrical condition

“…Darban et al [35] studied the influence of Pasternak elastic foundation on the buckling behavior of Euler-Bernoulli nanobeam. Penna et al [36] utilized the Galerkin method to study the influence of hygrothermal environment on the nonlinear free vibration of FG porous Bernoulli-Euler nanobeams based on nonlocal stress gradient model. Caporale et al [37] proposed a mathematical formulation for nonlocal strain and stress gradient models with loading discontinuity.…”

“…Penna et al. [36] utilized the Galerkin method to study the influence of hygrothermal environment on the nonlinear free vibration of FG porous Bernoulli–Euler nanobeams based on nonlocal stress gradient model. Caporale et al.…”