Nonlinear Dynamic Behavior of Porous and Imperfect Bernoulli-Euler Functionally Graded Nanobeams Resting on Winkler Elastic Foundation

Abstract

Nonlinear free vibrations of functionally graded porous Bernoulli–Euler nano-beams resting on an elastic foundation through a stress-driven nonlocal elasticity model are studied taking into account von Kármán type nonlinearity and initial geometric imperfection. By using the Galerkin method, the governing equations are reduced to a nonlinear ordinary differential equation. The closed form analytical solution of the nonlinear natural flexural frequency is then established using the Hamiltonian approach to nonlinear oscillators. Several comparisons with existing models in the literature are performed to validate the accuracy and reliability of the proposed approach. Finally, a numerical investigation is developed in order to analyze the effects of the gradient index coefficient, porosity volume fraction, initial geometric imperfection, and the Winkler elastic foundation coefficient, on the nonlinear flexural vibrations of metal–ceramic FG porous Bernoulli–Euler nano-beams.

Stochastic thermo-elastic vibration characteristics of functionally graded porous nano-beams using first-order perturbation-based nonlocal finite element model

“…[61][62][63][64][65] Few researchers emphasized the influence of porosity on stress, electric field, bending, bifurcation buckling, linear, and nonlinear vibrations of nano-beams. [66][67][68][69][70][71][72] Recently, Rahmani et al 73 studied the vibration characteristics of porous nano-beams in rotational motion, and Rastehkenari et al 74 studied the nonlinear random vibrations of FGMs porous nano-beams. Zhao et al 75 studied the effects of porosity on the bending and free vibration analysis of axially graded flexoelectric nano-beams.…”

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

“…The main aim of this study is to help fill these gaps by proposing an application of the higher-order Hamilton approach [49][50][51][52][53][54][55][56][57] to the nonlinear free vibrations analysis of porous FG nano-beams in a hygro-thermal environment based on the L/NStressG model.…”

Hygro-Thermal Vibrations of Porous FG Nano-Beams Based on Local/Nonlocal Stress Gradient Theory of Elasticity

“…In the framework of nonlocal elasticity, two of the most notable purely nonlocal constitutive laws are surely the softening or Eringen’s strain-driven nonlocal integral model (StrainDM) [ 14 , 15 ], in which the total stress of a given point is a function of the strain at all other adjacent points of the continuum, and the more recently hardening or stress-driven nonlocal integral model (StressDM) developed by Romano and Barretta [ 16 ], in which the strain at any point is resulted from the stress of all points. As widely discussed in [ 17 , 18 , 19 ], the differential formulation of StrainDM is ill-posed and leads to the unexpected paradoxical results for some boundary and loading conditions, unlike the well-posed StressDM that provides a consistent approach for the analysis of nanostructures [ 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 ].…”