scholarly journals A Nonlocal Strain Gradient Approach for Out-of-Plane Vibration of Axially Moving Functionally Graded Nanoplates in a Hygrothermal Environment

2021 ◽  
Vol 2021 ◽  
pp. 1-19
Author(s):  
Chengxiu Zhu ◽  
Jianwei Yan ◽  
Pingyuan Wang ◽  
Cheng Li
Keyword(s):  
Scale Parameter ◽  
Nonlocal Parameter ◽  
Functionally Graded ◽  
Characteristic Scale ◽  
Vibration Frequencies ◽  
Material Characteristic ◽  
Moisture Concentration ◽  
Axially Moving ◽  
Nonlocal Strain Gradient ◽  
Axial Speed

Vibration analyses on axially moving functionally graded nanoplates exposed to hygrothermal environments are presented. The theoretical model of the nanoplate is described via the Kirchhoff plate theory in conjunction with the concept of the physical neutral layer. By employing the nonlocal strain gradient theory, the governing equation of motion is derived based on Hamilton’s principle. The composite beam function method, as well as the complex modal approach, is utilized to obtain the vibration frequencies of axially moving functionally graded nanoplates. Some benchmark results related to the effects of temperature changing, moisture concentration, axial speed, aspect ratio, nonlocal parameter, and the material characteristic scale parameter on the stiffness of axially moving functionally graded nanoplates are obtained. The results reveal that with increasing the nonlocal parameter, gradient index, temperature changing, moisture concentration, and axial speed, the vibration frequencies decrease. The frequencies increase while increasing the material characteristic scale parameter and aspect ratio. Moreover, there is an interaction between the nonlocal parameter and material characteristic scale parameter, influencing and restricting each other.

scholarly journals Free Vibration of Axially Moving Functionally Graded Nanoplates Based on the Nonlocal Strain Gradient Theory

2020 ◽  
Vol 25 (4) ◽  
pp. 587-596
Author(s):  
Cheng Li ◽  
P.Y. Wang ◽  
Q.Y. Luo ◽  
S. Li
Keyword(s):  
Strain Gradient ◽  
Natural Frequencies ◽  
Nonlocal Parameter ◽  
Functionally Graded ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Characteristic Scale ◽  
Bending Rigidity ◽  
Material Characteristic ◽  
Nonlocal Strain Gradient

Using the nonlocal strain gradient theory, we explore vibration behaviours of initially bidirectional tensioned functionally graded nanoplates with axial speed. The governing equation of motion can be obtained based on the differential type of nonlocal strain gradient constitutive relation, which characters the dynamics of nanostructures containing kinematic relation. The simply supported boundary constraints on four sides are considered and subsequently the numerical results are determined. It shows that natural frequencies of axially moving nanoplates decrease when increasing the axial speed and the nonlocal parameter. Hence the nonlocal and kinematic factors cause the natural frequencies to decrease or, weaken the equivalent bending rigidity. On the other hand, natural frequencies increase with an increase in the axial tension and material characteristic scale. Hence the strain gradient and tensile stress factors cause the natural frequencies to increase or, strengthen the equivalent bending rigidity. In addition, the natural frequencies get higher with a larger aspect ratio of the functionally graded nanoplate. The larger one between the nonlocal parameter and the material characteristic scale plays a dominant role in the softening and stiffening mechanisms of the nonlocal strain gradient effect. In case of the same magnitude of the nonlocal parameter and the material characteristic scale, the softening and hardening phenomena disappear. The equivalent bending rigidity neither increases nor decreases in such a situation, and its value degenerates to the classical one.

scholarly journals A nonlocal strain gradient shell model incorporating surface effects for vibration analysis of functionally graded cylindrical nanoshells

2019 ◽  
Vol 40 (12) ◽  
pp. 1695-1722 ◽  
Author(s):  
Lu Lu ◽  
Li Zhu ◽  
Xingming Guo ◽  
Jianzhong Zhao ◽  
Guanzhong Liu
Keyword(s):  
Strain Gradient ◽  
Scale Parameter ◽  
Nonlocal Parameter ◽  
Functionally Graded ◽  
Length Scale ◽  
Proposed Model ◽  
Material Length Scale Parameter ◽  
Material Length Scale ◽  
Material Length ◽  
Nonlocal Strain Gradient

AbstractIn this paper, a novel size-dependent functionally graded (FG) cylindrical shell model is developed based on the nonlocal strain gradient theory in conjunction with the Gurtin-Murdoch surface elasticity theory. The new model containing a nonlocal parameter, a material length scale parameter, and several surface elastic constants can capture three typical types of size effects simultaneously, which are the nonlocal stress effect, the strain gradient effect, and the surface energy effects. With the help of Hamilton’s principle and first-order shear deformation theory, the non-classical governing equations and related boundary conditions are derived. By using the proposed model, the free vibration problem of FG cylindrical nanoshells with material properties varying continuously through the thickness according to a power-law distribution is analytically solved, and the closed-form solutions for natural frequencies under various boundary conditions are obtained. After verifying the reliability of the proposed model and analytical method by comparing the degenerated results with those available in the literature, the influences of nonlocal parameter, material length scale parameter, power-law index, radius-to-thickness ratio, length-to-radius ratio, and surface effects on the vibration characteristic of functionally graded cylindrical nanoshells are examined in detail.

A nonlocal strain gradient theory for rotating thermo-mechanical characteristics on magnetically actuated viscoelastic functionally graded nanoshell

2020 ◽  
Vol 31 (12) ◽  
pp. 1511-1523
Author(s):  
Mohammad Mahinzare ◽  
Hossein Akhavan ◽  
Majid Ghadiri
Keyword(s):  
Boundary Conditions ◽  
Strain Gradient ◽  
Equations Of Motion ◽  
Nonlocal Parameter ◽  
Functionally Graded ◽  
Smart Structure ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Nonlocal Strain Gradient Theory ◽  
Nonlocal Strain Gradient

In this article, a first-order shear deformable model is expanded based on the nonlocal strain gradient theory to vibration analysis of smart nanostructures under different boundary conditions. The governing equations of motion of rotating magneto-viscoelastic functionally graded cylindrical nanoshell in the magnetic field and corresponding boundary conditions are obtained using Hamilton’s principle. To discretize the equations of motion, the generalized differential quadrature method is applied. The aim of this work is to investigate the effects of the temperature changes, nonlocal parameter, material length scale, viscoelastic coefficient, various boundary conditions, and the rotational speed of this smart structure on natural frequencies of rotating cylindrical nanoshell made of magneto-viscoelastic functionally graded material.

Studies on the dynamic stability of an axially moving nanobeam based on the nonlocal strain gradient theory

2018 ◽  
Vol 32 (16) ◽  
pp. 1850167 ◽  
Author(s):  
Jing Wang ◽  
Huoming Shen ◽  
Bo Zhang ◽  
Juan Liu
Keyword(s):  
Scale Effect ◽  
Strain Gradient ◽  
Nonlocal Parameter ◽  
Excitation Frequency ◽  
Instability Region ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Nonlocal Strain Gradient Theory ◽  
Axially Moving ◽  
Nonlocal Strain Gradient

In this paper, we studied the parametric resonance issue of an axially moving viscoelastic nanobeam with varying velocity. Based on the nonlocal strain gradient theory, we established the transversal vibration equation of the axially moving nanobeam and the corresponding boundary condition. By applying the average method, we obtained a set of self-governing ordinary differential equations when the excitation frequency of the moving parameters is twice the intrinsic frequency or near the sum of certain second-order intrinsic frequencies. On the plane of parametric excitation frequency and excitation amplitude, we can obtain the instability region generated by the resonance, and through numerical simulation, we analyze the influence of the scale effect and system parameters on the instability region. The results indicate that the viscoelastic damping decreases the resonance instability region, and the average velocity and stiffness make the instability region move to the left- and right-hand sides. Meanwhile, the scale effect of the system is obvious. The nonlocal parameter exhibits not only the stiffness softening effect but also the damping weakening effect, while the material characteristic length parameter exhibits the stiffness hardening effect and damping reinforcement effect.

Vibration characteristics and stable region of a parabolic FGM thin-walled beam with axial and spinning motion

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Baichuan Lin ◽  
Bo Chen ◽  
Yinghui Li ◽  
Jie Yang
Keyword(s):  
Natural Frequencies ◽  
Angular Speed ◽  
Vibration Characteristics ◽  
Functionally Graded ◽  
Stable Region ◽  
Graded Material ◽  
Axially Moving ◽  
Fgm Beam ◽  
Spinning Speed ◽  
Axial Speed

Abstract This paper focuses on the vibration characteristics of the parabolic functionally graded material (FGM) beam considering the axially moving and spinning motion. Based on the Hamilton’s principle, the governing equation of the beam is derived. Then, the Galerkin’s method is employed to solve the equation. The combined influence of axial speed, spinning speed, and geometric parameters on natural frequencies of the beam are investigated. What’s more, the axially moving and spinning motion can lead to a critical axial speed and critical spinning angular speed, respectively. These two critical speeds and stable region affected by different parameters are also discussed.

scholarly journals An analytical study of vibration in functionally graded piezoelectric nanoplates: nonlocal strain gradient theory

2019 ◽  
Vol 40 (12) ◽  
pp. 1723-1740 ◽  
Author(s):  
Z. Sharifi ◽  
R. Khordad ◽  
A. Gharaati ◽  
G. Forozani
Keyword(s):  
Strain Gradient ◽  
Analytical Study ◽  
Natural Frequencies ◽  
Nonlocal Parameter ◽  
Functionally Graded ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Nonlocal Strain Gradient Theory ◽  
Functionally Graded Piezoelectric ◽  
Nonlocal Strain Gradient

AbstractIn this paper, we analytically study vibration of functionally graded piezoelectric (FGP) nanoplates based on the nonlocal strain gradient theory. The top and bottom surfaces of the nanoplate are made of PZT-5H and PZT-4, respectively. We employ Hamilton’s principle and derive the governing differential equations. Then, we use Navier’s solution to obtain the natural frequencies of the FGP nanoplate. In the first step, we compare our results with the obtained results for the piezoelectric nanoplates in the previous studies. In the second step, we neglect the piezoelectric effect and compare our results with those obtained for the functionally graded (FG) nanoplates. Finally, the effects of the FG power index, the nonlocal parameter, the aspect ratio, and the side-tothickness ratio, and the nanoplate shape on natural frequencies are investigated.

Porosity-dependent vibration and dynamic stability of compositionally gradient nanofilms using nonlocal strain gradient theory

2017 ◽  
Vol 232 (17) ◽  
pp. 3144-3155 ◽  
Author(s):  
Mohammad Reza Barati
Keyword(s):  
Dynamic Stability ◽  
Strain Gradient ◽  
Functionally Graded ◽  
Strain Gradient Theory ◽  
Load Parameter ◽  
Gradient Theory ◽  
Vibration Frequencies ◽  
Stability Regions ◽  
Nonlocal Strain Gradient Theory ◽  
Nonlocal Strain Gradient

Porosity-dependent free vibration and dynamic stability of functionally graded nanofilms are studied according to the nonlocal strain gradient theory. Two-scale coefficients are considered to incorporate both nonlocality and strain gradient impacts. The nanofilm is subjected to in-plane hygro-thermal and harmonic mechanical loads. Uniform dispersion of porosities is considered according to a power-law model for functionally graded materials. Galerkin's approach is employed to obtain the vibration frequencies as well as stability regions. One can see that stability regions and vibration frequencies of a functionally graded nanofilm are significantly affected by static load parameter, dynamic load parameter, porosities, moisture change, temperature change, and elastic substrate nonlocal strain gradient coefficients.

scholarly journals On the Vibrations and Stability of Moving Viscoelastic Axially Functionally Graded Nanobeams

Materials ◽  
2020 ◽  
Vol 13 (7) ◽  
pp. 1707 ◽  
Author(s):  
Ali Shariati ◽  
Dong won Jung ◽  
Hamid Mohammad-Sedighi ◽  
Krzysztof Kamil Żur ◽  
Mostafa Habibi ◽  
...  
Keyword(s):  
Critical Velocity ◽  
Natural Frequencies ◽  
Nonlocal Parameter ◽  
Dynamical Behavior ◽  
Functionally Graded ◽  
Fine Tuning ◽  
Dynamical Response ◽  
Power Law Distribution ◽  
Axially Moving ◽  
The Stability

In this article, size-dependent vibrations and the stability of moving viscoelastic axially functionally graded (AFG) nanobeams were investigated numerically and analytically, aiming at the stability enhancement of translating nanosystems. Additionally, a parametric investigation is presented to elucidate the influence of various key factors such as axial gradation of the material, viscosity coefficient, and nonlocal parameter on the stability boundaries of the system. Material characteristics of the system vary smoothly along the axial direction based on a power-law distribution function. Laplace transformation in conjunction with the Galerkin discretization scheme was implemented to obtain the natural frequencies, dynamical configuration, divergence, and flutter instability thresholds of the system. Furthermore, the critical velocity of the system was evaluated analytically. Stability maps of the system were examined, and it can be concluded that the nonlocal effect in the system can be significantly dampened by fine-tuning of axial material distribution. It was demonstrated that AFG materials can profoundly enhance the stability and dynamical response of axially moving nanosystems in comparison to homogeneous materials. The results indicate that for low and high values of the nonlocal parameter, the power index plays an opposite role in the dynamical behavior of the system. Meanwhile, it was shown that the qualitative stability of axially moving nanobeams depends on the effect of viscoelastic properties in the system, while axial grading of material has a significant role in determining the critical velocity and natural frequencies of the system.

Geometrically Nonlinear Electromechanical Instability of FG Nanobeams by Nonlocal Strain Gradient Theory

2021 ◽  
pp. 2150051
Author(s):  
S. M. J. Hosseini ◽  
J. Torabi ◽  
R. Ansari ◽  
A. Zabihi
Keyword(s):  
Differential Equation ◽  
Strain Gradient ◽  
Electrostatic Force ◽  
Nonlocal Parameter ◽  
Functionally Graded ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Nonlocal Strain Gradient Theory ◽  
Electromechanical Instability ◽  
Nonlocal Strain Gradient

This paper is concerned with studying the size-dependent nonlinear dynamic pull-in instability and vibration of functionally graded Euler–Bernoulli nanobeams (FG-EBNs) with the von Kármán hypothesis based on the nonlocal strain gradient theory (NLSGT). To this end, the partial differential equation (PDE) is developed by Hamilton’s principle considering the intermolecular, fringing field and electrostatic nonlinear forces. Then, the Galerkin method (GM) is utilized to acquire the ordinary differential equation (ODE) and the results are obtained with the help of an analytical approach called the homotopy analysis method (HAM). To verify the outcome of this study, the nonlinear and linear frequencies obtained are compared with those of the literature. Consequently, the pull-in voltage of the FG nanobeam is obtained and the variations of nonlinear and linear frequencies are discussed in detail. Also, the effects of initial amplitude, electrostatic force, length scale, nonlocal parameter, material gradient index and boundary condition (BC) on the electromechanical behavior of FG-EBNs are analyzed with the results commented.

Buckling analysis of nonlocal strain gradient axially functionally graded nanobeams resting on variable elastic medium

2017 ◽  
Vol 232 (11) ◽  
pp. 2067-2078 ◽  
Author(s):  
Farzad Ebrahimi ◽  
Mohammad R Barati
Keyword(s):  
Elastic Medium ◽  
Power Law ◽  
Strain Gradient ◽  
Scale Parameter ◽  
Functionally Graded ◽  
Strain Gradient Theory ◽  
Solution Technique ◽  
Buckling Loads ◽  
Functionally Graded Nanobeam ◽  
Nonlocal Strain Gradient

This paper provides the first examination of buckling behavior of axially functionally graded nanobeams. A nonlocal strain gradient theory consisting of two scale parameter is employed for modeling of size-dependent behavior axially functionally graded nanobeam much accurately. This theory takes into account both nonlocal stress field and strain gradient effects on the response of nanostructures. A power-law model is used to describe the distribution of material properties along the axial direction. The axially functionally graded nanobeam is in contact with a non-uniform elastic medium which consists of a Winkler layer with variable stiffness and also a Pasternak layer with constant stiffness. Linear, parabolic and sinusoidal variations of Winkler foundation in longitudinal direction are considered. A Galerkin-based solution technique is implemented to solve the governing equation obtained from Hamilton’s principle. Buckling loads of functionally graded nanobeam are verified with those of previous papers. It is shown that buckling loads of axially functionally graded nanobeams are significantly influenced by power-law index, nonlocal parameter, length scale parameter, type of elastic foundation and boundary conditions.

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