Nonlinear Vibration Analysis of Microscale Functionally Graded Timoshenko Beams using the Most General form of Strain Gradient Elasticity

2013 ◽  
Vol 30 (2) ◽  
pp. 161-172 ◽  
Author(s):  
R. Ansari ◽  
M. Faghih Shojaei ◽  
V. Mohammadi ◽  
R. Gholami ◽  
H. Rouhi
Keyword(s):  
Boundary Conditions ◽  
Nonlinear Vibration ◽  
Strain Gradient ◽  
Gradient Elasticity ◽  
Functionally Graded ◽  
Small Scale ◽  
Length Scale ◽  
Beam Model ◽  
Strain Gradient Elasticity ◽  
Governing Equations

ABSTRACTBased on the Timoshenko beam model, the nonlinear vibration of microbeams made of functionally graded (FG) materials is investigated under different boundary conditions. To consider small scale effects, the model is developed based on the most general form of strain gradient elasticity. The nonlinear governing equations and boundary conditions are derived via Hamilton's principle and then discretized using the generalized differential quadrature technique. A pseudo-Galerkin approach is used to reduce the set of discretized governing equations into a time-varying set of ordinary differential equations of Duffing-type. The harmonic balance method in conjunction with the Newton-Raphson method is also applied so as to solve the problem in time domain. The effects of boundary conditions, length scale parameters, material gradient index and geometrical parameters are studied. It is found that the importance of the small length scale is affected by the type of boundary conditions and vibration mode. Also, it is revealed that the classical theory tends to underestimate the vibration amplitude and linear frequency of FG microbeams.

scholarly journals Influence of flexoelectric, small-scale, surface and residual stress on the nonlinear vibration of sigmoid, exponential and power-law FG Timoshenko nano-beams

2018 ◽  
Vol 38 (1) ◽  
pp. 122-142 ◽  
Author(s):  
Mohammad Arefi ◽  
Mahmoud Pourjamshidian ◽  
Ali Ghorbanpour Arani ◽  
Timon Rabczuk
Keyword(s):  
Nonlinear Vibration ◽  
Elasticity Theory ◽  
Strain Gradient ◽  
Scale Effects ◽  
Nonlocal Parameter ◽  
Functionally Graded ◽  
Numerical Results ◽  
Small Scale ◽  
Beam Model ◽  
Surface Stresses

This research deals with the nonlinear vibration of the functionally graded nano-beams based on the nonlocal elasticity theory considering surface and flexoelectric effects. The flexoelectric functionally graded nano-beam is resting on nonlinear Pasternak foundation. Cubic nonlinearity is assumed for foundation. It is assumed that the material properties of the nano-beam change continuously along the thickness direction according to different patterns of material distribution. In order to include coupling of strain gradients and electrical polarizations in equation of motion, the nonlocal, nonclassical nano-beam model containing flexoelectric effect is employed. In addition, the effects of surface elasticity, di-electricity, and piezoelectricity as well as bulk flexoelectricity are accounted in constitutive relations. The governing equations of motion are derived using Hamilton principle based on first shear deformation beam theory and the nonlocal strain gradient elasticity theory considering residual surface stresses. The differential quadrature method is used to calculate nonlinear natural frequency of flexoelectric functionally graded nano-beam as well as nonlinear vibrational mode shape. After validation of the present numerical results with those results available in literature, full numerical results are presented to investigate the influence of important parameters such as flexoelectric coefficients of the surface and bulk, residual surface stresses, nonlocal parameter, length scale effects (strain gradient parameter), cubic nonlinear Winkler and shear coefficients, power gradient index of functionally graded material, and geometric dimensions on the nonlinear vibration behaviors of flexoelectric functionally graded nano-beam. The numerical results indicate that, considering the flexoelectricity leads to the decrease of the bending stiffness of the flexoelectric functionally graded nano-beams.

scholarly journals Stability analysis of a rotationally restrained microbar embedded in an elastic matrix using strain gradient elasticity

2019 ◽  
Vol 6 (1) ◽  
pp. 1-10 ◽  
Author(s):  
Mustafa Özgür Yayli
Keyword(s):  
Boundary Conditions ◽  
Strain Gradient ◽  
Scale Parameter ◽  
Gradient Elasticity ◽  
Buckling Load ◽  
Elastic Matrix ◽  
Small Scale ◽  
Strain Gradient Elasticity ◽  
Critical Buckling Load ◽  
Fourier Sine Series

AbstractThe buckling of rotationally restrained microbars embedded in an elastic matrix is studied within the framework of strain gradient elasticity theory. The elastic matrix is modeled in this study as Winkler’s one-parameter elastic matrix. Fourier sine series with a Fourier coefficient is used for describing the deflection of the microbar. An eigenvalue problem is obtained for buckling modes with the aid of implementing Stokes’ transformation to force boundary conditions. This mathematical model bridges the gap between rigid and the restrained boundary conditions. The influences of rotational restraints, small scale parameter and surrounding elastic matrix on the critical buckling load are discussed and compared with those available in the literature. It is concluded from analytical results that the critical buckling load of microbar is dependent upon rotational restraints, surrounding elastic matrix and the material scale parameter. Similarly, the dependencies of the critical buckling load on material scale parameter, surrounding elastic medium and rotational restraints are significant.

scholarly journals Study on the small-scale effect on wave propagation characteristics of fluid-filled carbon nanotubes based on nonlocal fluid theory

2019 ◽  
Vol 11 (1) ◽  
pp. 168781401882332
Author(s):  
Yang Yang ◽  
Wuhuai Yan ◽  
Jinrui Wang
Keyword(s):  
Wave Propagation ◽  
Fluid Flow ◽  
Carbon Nanotube ◽  
Scale Effect ◽  
Strain Gradient ◽  
Small Scale ◽  
Beam Model ◽  
Governing Equations ◽  
Mode Wave ◽  
Fluid Theory

In this article, Timoshenko’s beam model is established to investigate the wave propagation behaviors for a fluid-conveying carbon nanotube when employing the nonlocal stress–strain gradient coupled theory and nonlocal fluid theory. The governing equations of motion for the carbon nanotube are derived. The small-scale influences induced by the nanotube are simulated by nonlocal and strain gradient effects, and the scale effect induced by fluid flow is first investigated applying nonlocal fluid theory. Numerical results obtained by solving the governing equations indicate that the nonlocal effect induced by the nanotube leads to wave damping and a decrease in stiffness, while the strain gradient effect contributes to wave promotion and an enhancement in stiffness. The scale effect caused by the inner fluid only leads to a decay for a high-mode wave since there is no influence from fluid flow on the low-mode wave. The numerical solution is validated by comparing with Monte Carlo simulation and interval analysis method.

On the size-dependent nonlinear dynamics of viscoelastic/flexoelectric nanobeams

2020 ◽  
pp. 107754632095222
Author(s):  
Rasoul Bagheri ◽  
Yaghoub Tadi Beni
Keyword(s):  
Boundary Conditions ◽  
Viscoelastic Medium ◽  
Medium Effect ◽  
Small Scale ◽  
Beam Model ◽  
Governing Equations ◽  
Size Dependent ◽  
Flexoelectric Effect ◽  
Nonlinear Forced Vibration ◽  
Euler Bernoulli Beam

In this study, size-dependent nonlinear forced vibration of viscoelastic/flexoelectric nanobeams has been investigated. By calculating enthalpy and kinetic energy and using Hamilton’s principle, the coupled governing equations of viscoelastic/flexoelectric nanobeams are derived along with dependent electrical and mechanical boundary conditions. Furthermore, to take the effects of the small scale into account, the nonclassical theory of continuous medium has been used and the Euler–Bernoulli beam model has been adopted to model the nanobeams. Finally, the governing equations are solved using numerical methods for distributed loaded and clamped–clamped boundary conditions. By comparing the results, it is determined that the parameters of the size effect and the viscoelastic medium effect can increase the vibrational frequency of the nanobeams. Also, the results show that the frequency of nanobeams outside of the viscoelastic medium strongly depends on the size-dependent parameters, and the increase in the length and thickness of the nanobeam decreases the frequency. The results also show that with the increasing flexoelectric effect, the amplitude of the nonlinear oscillation increases.

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