On the forced mechanics of doubly-curved nanoshell

doi:10.1016/j.ijengsci.2021.103538

International Journal of Engineering Science

Volume 168, 1 November 2021, 103538

https://doi.org/10.1016/j.ijengsci.2021.103538 Get rights and content

Highlights

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Both softening and hardening mechanisms of nanostructure are examined separately.
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Forced dynamic of FGM doubly-curved nano-size shell panel is investigated.
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A cosine function is utilized to capture the porosity distribution.
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An analytical solution is developed to solve the dynamical problem of the nanoshell.

Abstract

This paper is determined to study the forced vibration response of doubly-curved shells including different shape panels. A power-full higher-order shear-deformation theory in curvilinear coordinate is developed to model the doubly-curved nano-size shell. Furthermore, a general nonlocal strain gradient theory is employed in order to catch up with both phenomena of small-scale behaves. The nanoshell is made of advanced composite materials whose effective material properties vary continuously through the z-axis. After estimating the effective material properties utilizing a modified power-law, a virtual work of Hamilton statement is applied over the theories to obtain both governing equations as well as boundary conditions. Afterwards, an analytical technique based upon double Fourier series is exploited to satisfy conditions in edges. The numerical examples are presented to reveal the effect of the power-law index, porosity coefficient, elastic medium, aspect and length-to-thickness ratios and small-scale parameters, highlighted by loading time interval, on the dynamic response (i.e., transverse deflection and stresses) of spherical- elliptical, hyperbolic as well as cylindrical nano-panels.

Introduction

The interest in developing nanotechnology has substantially increased recently due to large marketplace of micro/nano-electromechanical systems (MEMS/NEMS) in commercial volumes (e.g. applicable in engineering inertial/pressure sensors, optical/communications devices, drug delivery, and inkjet printer heads (Natelson, 2015)). Mechanically analyzing of these devices have been conducted in a variety of ways; while the direct schemes of experimental and molecular dynamic simulation are thinkable, the other schemes, continuum mechanics, can be developed with the advantage of controlling challenges, time, and effort of investigating these tiny structures. The field of continuum mechanics for micro/nanostructures is followed the assumption of theme being continuous, rather than formed of discrete atomic units. A valuable level of accuracy has been observed via developing multiple non-classical continuum theories (see some of those in (Aminipour, Janghorban, & Li, 2020; Barretta & de Sciarra, 2018; Barretta & de Sciarra, 2019; Farajpour, Ghayesh, & Farokhi, 2018; Farajpour, Ghayesh, & Farokhi, 2019; Ghayesh, Farajpour, & Farokhi, 2019; Ghayesh, Farokhi, & Farajpour, 2019; Gholipour & Ghayesh, 2020; Karami, Shahsavari, & Li, 2018; Kiarasi, Babaei, Dimitri, & Tornabene, 2021; Lin et al., 2013; Nejad & Hadi, 2016; Nejad, Hadi, & Rastgoo, 2016; Sahmani & Fattahi, 2017; Shafiei & She, 2018; She, Liu, & Karami, 2021; Shen, Shen, & Zhang, 2010; Soltani, Atoufi, Mohri, Dimitri, & Tornabene, 2021; Thai, Vo, Nguyen, & Kim, 2017)) to predict size-dependent mechanical behavior of micro/nano continuous structures. The mechanics of continuous structures like beams, plates, and shells on the basis of size-dependent continuum theories are often in relation with concepts of elastic moduli, stress, and strain.

Now imagine a new class of inhomogeneous material obtaining the effective elastic moduli which owns high thermic and corrosion resistance in addition to high strength and further toughness, known as functionally graded materials (FGMs). Classical micromechanics approaches including Voigt-Reuss bounds methods, Mori-Tanaka scheme, Hashin-Shtrikman bounds model, Tamura model, and cubic local representative volume elements and a variety of other models have been developed to calculate the multi-physics of effective elastic properties of perfect FGMs (Akbarzadeh, Abedini, & Chen, 2015; Karami, Shahsavari, Janghorban, & Li, 2019). As its name implies, FGM made up of at least two distinct material phases in which the material properties are varied continuously along the demanding direction/s supporting with other points in designing, processing and engineering applications in Ref. (Miyamoto, Kaysser, Rabin, Kawasaki, & Ford, 2013). It should not be surprising that some material imperfections are caused during the process of manufacturing FGMs. One example may be that the creation of some internal voids (or porosities) inside the original of FGMs during their sintering process because of the difference in solidification temperature of the distinct material phases. Since that the porosity affects tensile strength as well as Young's modulus of FGMs in addition to assist FGMs behave more effective in terms of thermal insulation, absorbing the shocks (Miyamoto et al., 2013), and reducing the overall weight, there are some applications of those in the implants and human tissue. Hence, the role of porosity has been taken into account recently using some self-consistent methods (Faleh, Ahmed, & Fenjan, 2018; Karami & Janghorban, 2019; Karami, Shahsavari, & Janghorban, 2019; Liu, Yang, Gao, Wu, & Li, 2019; Shahsavari, Shahsavari, Li, & Karami, 2018; Shahverdi & Barati, 2017; She, Ren, Yuan, & Xiao, 2018; Xu, Karami, & Shahsavari, 2021). Using these methods, mechanical behavior of imperfect FGMs containing different shapes and size of the porosity inside the spatial position of the bulk material have been studied in the open literature (Barati, 2017; Shafiei, Mousavi, & Ghadiri, 2016). The symmetric/asymmetric/uniform porosity distribution patterns were developed to study vibration of porous curved and straight beams by Zhao et al. (2019). Power/Sigmoid distribution patterns of porosity were developed by Rezaiee-Pajand, Rajabzadeh-Safaei, & Masoodi (2020) to evaluate mechanical behavior of curved FGM beams. They showed the necessity of using different patterns of estimating porosities inside FGMs, when the obtained results compared with experimental data.

Relaxing the stresses would be that the another consequence of being porosities inside FGMs (Miyamoto et al., 2013). Static and dynamic response of micro/nano-sized structures made up of FGMs have been examined for beam/plate/shell in Refs. (Batra & Nie, 2018; Chaht et al., 2015; Dastjerdi, Akgöz, & Civalek, 2020; Jalaei & Civalek, 2019; She, Yuan, Karami, Ren, & Xiao, 2019). Between structural elements, shells are applicable three-dimensional (3D) structures due of their curvature. It has reported that FGM nanoshell type biological materials associated with a large thickness stretching, can feature very large deformations, as in most cases found in arteries under internal/external pressures (Carpenter, Gholipour, Ghayesh, Zander, & Psaltis, 2020; Ghayesh & Farajpour, 2019). Moreover, it is supposed to consider an elastic foundation is particularly advantageous in FGM curved panel to control their possible deformations under external loads (Alimoradzadeh, Salehi, & Esfarjani, 2019; Dinh Duc, Quang, Nguyen, & Chien, 2018; Hadi, Ovesy, Shakhesi, & Fazilati, 2017; She, Yuan, & Ren, 2017; Zhang & Liu, 2020). While, various hypotheses of elastic foundation to consider the interaction between foundation and shells, which are applicable in MEMS (Maluf & Williams, 2004), have been introduced, in this article, we consider a Pasternak elastic foundation containing spring and shear layers.

Having said that, due to application of FGM nanoshell biological arteries as well as composite implants and tissues, we defined this research study to find forced vibration response of such structures. While structural mechanics of curved/straight-panels respect to the type of load and analysis may be grouped into static and dynamic analysis with valuable applications (Dehrouyeh-Semnani, Nikkhah-Bahrami, & Yazdi, 2017; Farokhi & Ghayesh, 2018; Ghayesh & Farajpour, 2018; Huang, 2012; Karami, Janghorban, & Rabczuk, 2020; Qatu, Asadi, & Wang, 2012; Qatu, Sullivan, & Wang, 2010; Safaei, Moradi-Dastjerdi, Qin, & Chu, 2019), we should focus on understanding different dynamic responses of shell structures subjected to an external load, not just its displacement or deformation. Finding transverse deflection, normal and shear stresses, as well as other mechanical resultants, which are time-dependent, in a curved nano-size panel subjected to an external mechanical load is the main motivation of the current work.

In the present work, based on a cosine model of considering porosity reported in Ref (Kim, Żur, & Reddy, 2019), a curved nanoshell made of FGMs whose effective material properties varies continuously through the z-axis is modeled. For the mathematical modeling of time-dependent response of doubly-curved nanoshell, the assumptions of an efficient second-order shear-deformation shell theory in curvilinear coordinate is developed with considering a general nonlocal strain gradient model inside derivations using Hamilton statement due to the importance of size-dependency. The derived governing equations clearly reveals the effects of several parameters inside spherical- elliptical, hyperbolic as well as cylindrical nano-panels made of FGMs. An analytical solution based upon double Fourier series is utilized to satisfy simply boundary conditions in edges. Particularly, for the first time, a such model is developed to find time-dependent dynamic deflection, stresses, and moments of curved panel made of nano-sized FGMs.

Section snippets

Second-order shear deformation theory

A field of displacement at an arbitrary point of the shell can be evaluated as below: $\begin{matrix} u_{1} (α, β, z, t) = u (α, β, t) + z ϕ_{1} (α, β, t) + z^{2} ϕ_{2} (α, β, t) \\ u_{2} (α, β, z, t) = v (α, β, t) + z ψ_{1} (α, β, t) + z^{2} ψ_{2} (α, β, t) \\ u_{3} (α, β, z, t) = w (α, β, t) \end{matrix}$ where the middle surface displacement components of the shell are indicated, respectively, by u, v and w in the x-, y- and z- axes; ϕ₁ and ψ₁ are the rotations for the y- and x- axes, respectively; ϕ₂ and ψ₂ are the variables of the second-order terms. All components of displacement are the function of a

Closed-form solutions

This section is proposed to solve the partial differential equations which are obtained through the general nonlocal strain gradient second-order shear deformation theory for forced dynamic problems in the previous section. Adopting the double-Fourier series, the simply-supported boundary conditions in edges have been satisfied. The conditions are: $\begin{matrix} v = w = ψ_{1} = ψ_{2} = N_{α α} = M_{α α} = L_{α β} = 0 at α = 0, a \\ u = w = ϕ_{1} = ϕ_{2} = N_{β β} = M_{β β} = L_{α β} = 0 at β = 0, b \end{matrix}$

Through the mentioned boundary conditions, the following expressions are given as below, ${\begin{matrix} u \end{matrix}$

Results and discussion

In the current section, an analytical analysis is employed to study the effects of material composition, porosity coefficient as well as size-dependency on the time-dependent transverse deflection and normal/-shear stresses of doubly-curved nanoshells with different curves (i.e., spherical (R_α = R_β = R), hyperbolic (R_α = R, R_β = -R), elliptical (R_α = R, R_β = 2R), and cylindrical (R_α = R, R_β = →∞) panels). The current work supports a nanoshell made of Ti-6Al-4 V/Al₂O₃ with Young's modulus E_c

Concluding remarks

The second-order shear deformation theory is developed in curvilinear coordinate for the forced dynamics of nano-sized curved panels. The general nonlocal strain gradient theory is adopted in order to consider both softening and hardening mechanisms of nanostructures separately. While there are many works which have been done on the forced mechanics of simple structures such as beam and plates, the aim of the work is to study the forced mechanics of much more complex structures (doubly-curved

Declaration of Competing Interest

None.

References (61)

A. Akbarzadeh et al.
Effect of micromechanical models on structural responses of functionally graded plates
Composite Structures
(2015)
M.R. Barati
On wave propagation in nanoporous materials
International Journal of Engineering Science
(2017)
R. Barretta et al.
Constitutive boundary conditions for nonlocal strain gradient elastic nano-beams
International Journal of Engineering Science
(2018)
R. Barretta et al.
Variational nonlocal gradient elasticity for nano-beams
International Journal of Engineering Science
(2019)
R. Batra et al.
Torsional deformations and material tailoring of orthotropic bi-directional FGM hollow truncated conical cylinders with curved lateral surfaces
International Journal of Engineering Science
(2018)
H.J. Carpenter et al.
A review on the biomechanics of coronary arteries
International Journal of Engineering Science
(2020)
S. Dastjerdi et al.
On the effect of viscoelasticity on behavior of gyroscopes
International Journal of Engineering Science
(2020)
A.M. Dehrouyeh-Semnani et al.
On nonlinear vibrations of micropipes conveying fluid
International Journal of Engineering Science
(2017)
N.M. Faleh et al.
On vibrations of porous FG nanoshells
International Journal of Engineering Science
(2018)
A. Farajpour et al.
A review on the mechanics of nanostructures
International Journal of Engineering Science
(2018)

M. Rezaiee-Pajand et al.

An efficient curved beam element for thermo-mechanical nonlinear analysis of functionally graded porous beams, in: structures

(2020)

B. Safaei et al.

Frequency-dependent forced vibration analysis of nanocomposite sandwich plate under thermo-mechanical loads

Composites Part B: Engineering

(2019)

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