A higher-order mathematical modeling for dynamic behavior of protein microtubule shell structures including shear deformation and small-scale effects

This work presents the buckling investigation of embedded orthotropic nanoplates by using a new hyperbolic plate theory and nonlocal small-scale effects. The main advantage of this theory is that, in addition to including the shear deformation effect, the displacement field is modeled with only three unknowns and three governing equation as the case of the classical plate theory (CPT) and which is even less than the first order shear deformation theory (FSDT) and higher-order shear deformation theory (HSDT). A shear correction factor is, therefore, not required. Nonlocal differential constitutive relations of Eringen is employed to investigate effects of small scale on buckling of the rectangular nanoplate. The elastic foundation is modeled as two-parameter Pasternak foundation. The equations of motion of the nonlocal theories are derived and solved via Navier's procedure for all edges simply supported boundary conditions. The proposed theory is compared with other plate theories. Analytical solutions for buckling loads are obtained for single-layered graphene sheets with isotropic and orthotropic properties. The results presented in this study may provide useful guidance for design of orthotropic graphene based nanodevices that make use of the buckling properties of orthotropic nanoplates. Verification studies show that the proposed theory is not only accurate and simple in solving the buckling nanoplates, but also comparable with the other higher-order shear deformation theories which contain more number of unknowns. Keywords: Buckling; orthotropic nanoplates; a simple 3-unknown theory; nonlocal elasticity theory; Pasternak’s foundations. * Corresponding author; [email protected]

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Bending and free vibration analyses of functionally graded material nanoplates via a novel nonlocal single variable shear deformation plate theory

Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science ◽

10.1177/0954406220964522 ◽

2020 ◽

pp. 095440622096452

Author(s):

Le Kha Hoa ◽

Pham Van Vinh ◽

Nguyen Dinh Duc ◽

Nguyen Thoi Trung ◽

Le Truong Son ◽

...

Keyword(s):

Shear Deformation ◽

Functionally Graded Material ◽

Numerical Solutions ◽

Scale Effects ◽

Plate Thickness ◽

Shear Deformation Theory ◽

Functionally Graded ◽

Small Scale ◽

Parameter Study ◽

Graded Material

A novel nonlocal shear deformation theory is established to investigate functionally graded nanoplates. The significant benefit of this theory is that it consists of only one unknown variable in its displacement formula and governing differential equation, but it can take into account both the quadratic distribution of the shear strains and stresses through the plate thickness as well as the small-scale effects on nanostructures. The numerical solutions of simply supported rectangular functionally graded material nanoplates are carried out by applying the Navier procedure. To indicate the accuracy and convergence of this theory, the present solutions have been compared with other published results. Furthermore, a deep parameter study is also carried out to exhibit the influence of some parameters on the response of the functionally graded material nanoplates.

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Nonlocal large-amplitude vibration of embedded higher-order shear deformable multiferroic composite rectangular nanoplates with different edge conditions

Journal of Intelligent Material Systems and Structures ◽

10.1177/1045389x17721377 ◽

2017 ◽

Vol 29 (5) ◽

pp. 944-968 ◽

Cited By ~ 8

Author(s):

R Gholami ◽

R Ansari ◽

Y Gholami

Keyword(s):

Boundary Conditions ◽

Shear Deformation ◽

Large Amplitude ◽

Higher Order ◽

Small Scale ◽

Nonlinear Frequency ◽

Amplitude Vibration ◽

Large Amplitude Vibration ◽

Multiferroic Composite ◽

Shear Deformable

Based on the nonlocal elasticity theory, a unified nonlocal, nonlinear, higher-order shear deformable nanoplate model is developed to investigate the size-dependent, large-amplitude, nonlinear vibration of multiferroic composite rectangular nanoplates with different boundary conditions resting on an elastic foundation. By considering a unified displacement vector and using von Kármán’s strain tensor, the strain–displacement components are obtained. Using coupled nonlocal constitutive relations, the coupled ferroelastic, ferroelectric, ferromagnetic, and thermal properties of multiferroic composite materials and small-scale effect are taken into account. The electric and magnetic potential distributions in the nanoplate are calculated via Maxwell’s electromagnetic equations. Furthermore, Hamilton’s principle is utilized to obtain the mathematical formulation associated with the coupled governing equations of motions and boundary conditions. The developed model enables us to consider the effects of rotary inertia and transverse shear deformation without using any shear correction factor. Also, it can be degenerated to the models based on the Kirchhoff and existing shear deformation plate theories. To solve the large-amplitude vibration problem, an efficient multistep numerical solution approach is utilized. Effects of various important parameters such as the type of the plate theory, and parameters of nonlocality and coupled fields on the nonlinear frequency response are investigated.

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A Novel Refined Plate Theory for Free Vibration Analyses of Single-Layered Graphene Sheets Lying on Winkler-Pasternak Elastic Foundations

Journal of Nano Research ◽

10.4028/www.scientific.net/jnanor.58.151 ◽

2019 ◽

Vol 58 ◽

pp. 151-164 ◽

Cited By ~ 2

Author(s):

Fatima Boukhatem ◽

Aicha Bessaim ◽

Abdelhakim Kaci ◽

Abderrahmane Mouffoki ◽

Mohammed Sid Ahmed Houari ◽

...

Keyword(s):

Free Vibration ◽

Shear Deformation ◽

Elastic Foundation ◽

Displacement Field ◽

Scale Effects ◽

Plate Theory ◽

Small Scale ◽

Graphene Sheets ◽

Refined Plate Theory ◽

Foundation Parameters

In this article, the analyses of free vibration of nanoplates, such as single-layered graphene sheets (SLGS), lying on an elastic medium is evaluated and analyzed via a novel refined plate theory mathematical model including small-scale effects. The noteworthy feature of theory is that the displacement field is modelled with only four unknowns, which is even less than the other shear deformation theories. The present one has a new displacement field which introduces undetermined integral variables, the shear stress free condition on the top and bottom surfaces of the plate is respected and consequently, it is unnecessary to use shear correction factors. The theory involves four unknown variables, as against five in case of other higher order theories and first-order shear deformation theory. By using Hamilton’s principle, the nonlocal governing equations are obtained and they are solved via Navier solution method. The influences played by transversal shear deformation, plate aspect ratio, side-to-thickness ratio, nonlocal parameter, and elastic foundation parameters are all examined. From this work, it can be observed that the small-scale effects and elastic foundation parameters are significant for the natural frequency.

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Dynamic Behavior of Multi-Layered Viscoelastic Nanobeam System Embedded in a Viscoelastic Medium with a Moving Nanoparticle

Journal of Mechanics ◽

10.1017/jmech.2016.91 ◽

2016 ◽

Vol 33 (5) ◽

pp. 559-575 ◽

Cited By ~ 25

Author(s):

Sh. Hosseini Hashemi ◽

H. Bakhshi Khaniki

Keyword(s):

Dynamic Behavior ◽

Viscoelastic Medium ◽

Scale Effects ◽

Nonlocal Parameter ◽

Nonlocal Theory ◽

Function Technique ◽

Small Scale ◽

Transform Method ◽

Damping Parameter ◽

Moving Nanoparticle

AbstractIn this paper, dynamic behavior of multi-layered viscoelastic nanobeams resting on a viscoelastic medium with a moving nanoparticle is studied. Eringens nonlocal theory is used to model the small scale effects. Layers are coupled by Kelvin-Voigt viscoelastic medium model. Hamilton's principle, eigen-function technique and the Laplace transform method are employed to solve the governing differential equations. Analytical solutions for transverse displacements of double-layered is presented for both viscoelastic nanobeams embedded in a viscoelastic medium and without it while numerical solution is achieved for higher layered nanobeams. The influences of the nonlocal parameter, stiffness and damping parameter of medium, internal damping parameter and number of layers are studied while the nanoparticle passes through. Presented results can be useful in analysing and designing nanocars, nanotruck moving on surfaces, racing nanocars etc.

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Nonlocal Elasticity Theory for Transient Analysis of Higher-Order Shear Deformable Nanoscale Plates

Journal of Nanomaterials ◽

10.1155/2014/208393 ◽

2014 ◽

Vol 2014 ◽

pp. 1-8 ◽

Cited By ~ 7

Author(s):

Woo-Young Jung ◽

Sung-Cheon Han

Keyword(s):

Dynamic Response ◽

Shear Deformation ◽

Nonlocal Elasticity ◽

Deformation Theory ◽

Transient Analysis ◽

Shear Deformation Theory ◽

Nonlocal Theory ◽

Higher Order ◽

Small Scale ◽

Nano Scale

The small scale effect on the transient analysis of nanoscale plates is studied. The elastic theory of the nano-scale plate is reformulated using Eringen’s nonlocal differential constitutive relations and higher-order shear deformation theory (HSDT). The equations of motion of the nonlocal theories are derived for the nano-scale plates. The Eringen’s nonlocal elasticity of Eringen has ability to capture the small scale effects and the higher-order shear deformation theory has ability to capture the quadratic variation of shear strain and consequently shear stress through the plate thickness. The solutions of transient dynamic analysis of nano-scale plate are presented using these theories to illustrate the effect of nonlocal theory on dynamic response of the nano-scale plates. On the basis of those numerical results, the relations between nonlocal and local theory are investigated and discussed, as are the nonlocal parameter, aspect ratio, side-to-thickness ratio, nano-scale plate size, and time step effects on the dynamic response. In order to validate the present solutions, the reference solutions are employed and examined. The results of nano-scale plates using the nonlocal theory can be used as a benchmark test for the transient analysis.

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Nonlocal Dynamic Stability of DWCNTs Containing Pulsating Viscous Fluid Including Surface Stress and Thermal Effects Based on Sinusoidal Higher Order Shear Deformation Shell Theory

Journal of Mechanics ◽

10.1017/jmech.2017.97 ◽

2017 ◽

Vol 34 (3) ◽

pp. 387-398 ◽

Cited By ~ 1

Author(s):

A. Mirafzal ◽

A. Fereidoon ◽

E. Andalib

Keyword(s):

Viscous Fluid ◽

Shear Deformation ◽

Dynamic Stability ◽

Surface Stress ◽

Shell Theory ◽

Fluid Velocity ◽

Higher Order ◽

Local Theory ◽

Small Scale ◽

Non Local

AbstractIn this study, the dynamic stability of double-walled carbon nanotubes (DWCNTs) conveying pulsating viscous fluid located on visco-Pasternak medium is investigated. The effects of small scale are considered using Eringen's non-local theory. The surrounding medium of DWCNT is anticipated as a visco-Pasternak foundation including the normal, shear, and damping forces. The van der Waals (vdWs) force is considered among the inner and outer layers of DWCNT. Three types of temperature changes, i.e. uniform, linear and sinusoidal temperature rise, through the thickness are studied. The equations of motion are obtained using higher order sinusoidal shear deformation shell theory (SSDT), in which the surface effects are included. Dynamical equations of DWCNT are extracted using the energy method and Hamilton's principle. Due to orthogonal conditions, the governing equations are solved based on Bolotin method. The effects of different parameters such as surface stress, non-local parameter, fluid velocity, Knudsen's number, fluid density, dimensions ratio and various temperature loadings on the dynamic stability regions of DWCNTs are discussed.

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On well-posedness of two-phase nonlocal integral models for higher-order refined shear deformation beams

Applied Mathematics and Mechanics ◽

10.1007/s10483-021-2750-8 ◽

2021 ◽

Author(s):

Pei Zhang ◽

Hai Qing

Keyword(s):

Shear Deformation ◽

Scale Effects ◽

Differential Quadrature Method ◽

Higher Order ◽

Functionally Graded ◽

Integral Model ◽

Well Posedness ◽

Curved Beams ◽

Two Phase ◽

Constitutive Requirements

AbstractDue to the conflict between equilibrium and constitutive requirements, Eringen’s strain-driven nonlocal integral model is not applicable to nanostructures of engineering interest. As an alternative, the stress-driven model has been recently developed. In this paper, for higher-order shear deformation beams, the ill-posed issue (i.e., excessive mandatory boundary conditions (BCs) cannot be met simultaneously) exists not only in strain-driven nonlocal models but also in stress-driven ones. The well-posedness of both the strain- and stress-driven two-phase nonlocal (TPN-StrainD and TPN-StressD) models is pertinently evidenced by formulating the static bending of curved beams made of functionally graded (FG) materials. The two-phase nonlocal integral constitutive relation is equivalent to a differential law equipped with two restriction conditions. By using the generalized differential quadrature method (GDQM), the coupling governing equations are solved numerically. The results show that the two-phase models can predict consistent scale-effects under different supported and loading conditions.

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Free Vibration Analysis of Simply Supported P-FGM Nanoplate Using a Nonlocal Four Variables Shear Deformation Plate Theory

Strojnícky casopis – Journal of Mechanical Engineering ◽

10.2478/scjme-2019-0039 ◽

2019 ◽

Vol 69 (4) ◽

pp. 9-24 ◽

Cited By ~ 1

Author(s):

Chikh Abdelbaki

Keyword(s):

Free Vibration ◽

Shear Deformation ◽

Power Law ◽

Material Properties ◽

Scale Effects ◽

Plate Theory ◽

Functionally Graded ◽

Small Scale ◽

Simply Supported ◽

Shear Deformation Plate Theory

AbstractThis paper shows an analysis of the free vibration of functionally graded simply supported nanoplate. The nonlocal four variables shear deformation plate theory is used to predict the free vibration frequencies of functionally graded nanoplate simply supported using non-local elasticity theory with the introduction of small-scale effects. The effect of the material properties, thickness-length ratio, aspect ratio, the exponent of the power law, the vibration mode is presented, the current solutions are compared to those obtained by other researchers. Equilibrium equations are obtained using the virtual displacements principle. P-FGM Power law is used to have a distribution of material properties that vary across the thickness. The results are in good agreement with those of the literature.

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