Dynamic modelling of CNTs under multiple moving nanoparticles based on non-local strain gradient theory

A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation

In this paper, a numerical procedure is proposed for analyzing the effects of length scale parameter, external electric field, angular speed and nonlocal parameter on the free vibration of a functionally graded piezoelectric cylindrical nanoshell. Nonlocal strain gradient theory (NSGT) is employed to study Eringen’s size-dependent effect and the length scale parameter. This new proposed method can be considered as a combination of Eringen’s nonlocal model and classical strain gradient theory. The obtained results show that this model can be used reliably for small-scale systems. The effects of boundary conditions, applied voltage, nonlocal parameter, rotational speed and length scale parameter on natural frequencies are presented. Compared to other elasticity theories, NSGT achieves the highest natural frequency and critical rotational speed and also a wider stability region. Doubling and tripling the length scale increases the natural frequency by approximately 1.8 and 2.6 times, respectively; while doubling and tripling the nonlocal parameter value reduces the natural frequency by approximately 1.2 and 1.4 times, respectively. Therefore, the natural frequency is more sensitive to the length scale parameter than the nonlocal parameter. Finally, it was shown that the critical angular speed goes up by increasing the length scale parameter, applied voltage, or nonlocal parameter.

New observations on transverse dynamics of microtubules based on nonlocal strain gradient theory

Size-dependent vibration analysis of nanobeams based on the nonlocal strain gradient theory

The purpose of this research is to study the propagation of surface waves in transversely isotropic piezoelastic medium based on nonlocal strain gradient theory. A characteristics equation for the existence of surface waves is discussed. This equation could be easily reduced to the ones of the gradient strain theory, nonlocal theory, and classical theory. It has also been concluded that there exist cut-off frequency for the wave propagating in size-dependent materials based on the nonlocal strain gradient theory. The dispersion equation which surface wave speed satisfies is derived from the free traction condition on the surface of half-space with consideration of electrically open circuit conditions. The effect of the nonlocal parameter, the strain gradient parameter on the existence of surface waves as well as the Rayleigh wave propagation is illustrated through some numerical examples.

Calibration of nonlocal strain gradient shell model for vibration analysis of a CNT conveying viscous fluid using molecular dynamics simulation

The main objective of this investigation is to study the size-dependent dynamic pull-in instability of nanobeams based on the nonlocal strain gradient theory (NLSGT) and Euler–Bernoulli beam model. To this end, the partial differential equation is obtained based on the NLSGT considering the electrostatic, fringing field, and intermolecular nonlinear forces. Then, the Galerkin method and the homotopy analysis method (HAM) were employed to solve the nonlinear governing equation. To validate the proposed results, the non-dimensional natural frequency and pull-in voltage are compared with the previously published results. Likewise, the analytical results of the HAM are compared with those obtained based on the Runge–Kutta numerical method. Besides, the impacts of the NLSGT, strain gradient theory, nonlocal theory, and classical theory on the dynamic behavior of nanobeams are investigated in the same situation. The pull-in voltage is also presented and the effects of electrostatic forces, fringing field, and initial gap are discussed in detail for different boundary conditions.

Strain gradient and nonlocal influences have important roles in the mechanical characteristics of structures as they are scaled down to nanoscopic dimensions. In this research, an attempt is made to capture both effects on the static bending response of nanoscopic Bernoulli–Euler beams by means of a comprehensive model. For this purpose, the strain gradient theory of Mindlin is combined with the integral (original) form of Eringen’s nonlocal theory. Also, the integral nonlocal formulation is written on the basis of both strain-driven and stress-driven versions of nonlocal theory. This mixed nonlocal strain gradient formulation is capable of reducing to the size-independent and combination of integral nonlocal strain gradient family theories by employing simple substitutions. Through constructing finite difference-based differential and integral matrix operators, numerical solution approaches are developed to obtain the deflection of nanobeams under various end conditions. In addition, for the solution of governing equation of strain-driven nonlocal strain gradient theory, an efficient technique is presented that is applied to the variational statement of the problem in a direct approach. Selected numerical results are provided to explore the simultaneous influences of strain gradient terms and nonlocality based on different theories on the static bending response of beams. It is revealed that the developed size-dependent model has the ability of considering strain gradient and nonlocal influences in the most general way.

In this paper, non-linear free vibration analysis of nano-beam has been studied. The non-local strain gradient theory and curvature tensor are used to show the size effect. The length scale parameter expresses the effect of strain gradient tensor in the non-local strain gradient theory. However, the aim of this article is to show the simultaneous effect of curvature and strain gradient tensors in non-linear vibration of functionally graded porous nano-beams. The effect of curvature tensor is demonstrated with the curvature tensor dependent parameter. Considering non-linear Von Kármán strains and Euler–Bernoulli beam theory, the governing vibrational equation of FG porous nano-beams are derived using Hamilton’s principle in the presence of strain gradient and curvature tensors simultaneously. The non-linear differential equation is extracted by using Galerkin’s method and the non-linear natural frequency of nano-beam is obtained according to Hamiltonian approach. Results represent the simultaneous effects of the length scale and curvature tensor dependent parameters on dimensionless non-linear natural frequencies. Also effects of different parameters such as non-local parameter, length scale parameter, porosity volume index, and power-law index are discussed in the presence and absence of the curvature tensor dependent parameter. Also, the beginning points of stiffness-hardening and stiffness-softening of nano-beam are always constant values in the non-local strain gradient theory, whereas considering the curvature tensor changes the beginning points of stiffness-hardening and stiffness-softening. The results are also compared with previous researches for validation.

Free vibration of self-powered nanoribbons subjected to thermal-mechanical-electrical fields based on a nonlocal strain gradient theory

Wave propagation in viscoelastic thin cylindrical nanoshell resting on a visco-Pasternak foundation based on nonlocal strain gradient theory

A unified size-dependent plate model based on nonlocal strain gradient theory including surface effects

Buckling analysis of orthotropic nanoplates based on nonlocal strain gradient theory using the higher-order finite strip method (H-FSM)

This paper has the objective of studying the propagation of surface waves in a transversely isotropic medium based on nonlocal strain gradient theory (NSGT). A characteristics equation for the existence of surface waves is discussed. This equation could be easily reduced to the ones of the gradient strain theory, nonlocal theory and classical theory. It has also been concluded that there exist escape frequency and cut-off frequency for the wave propagating in size-dependent materials based on the NSGT. Dispersion equation for the propagation of Rayleigh-type waves at the free surface has been derived. The effect of the nonlocal parameter, the strain gradient parameter on the Rayleigh wave propagation is considered through some numerical examples.

A nonlocal strain gradient isogeometric model for free vibration analysis of magneto-electro-elastic functionally graded nanoplates