Thermo-mechanical vibration analysis of nonlocal temperature-dependent FG nanobeams with various boundary conditions

Accepted Manuscript Thermo-mechanical vibration analysis of nonlocal temperature-dependent FG nanobeams with various boundary conditions Farzad Ebrahimi, Erfan Salari, Majid Ghadiri PII:

S1359-8368(15)00199-7

DOI:

10.1016/j.compositesb.2015.03.068

Reference:

JCOMB 3502

To appear in:

Composites Part B

Received Date: 29 December 2014 Revised Date:

19 March 2015

Accepted Date: 21 March 2015

Please cite this article as: Ebrahimi F, Salari E, Ghadiri M, Thermo-mechanical vibration analysis of nonlocal temperature-dependent FG nanobeams with various boundary conditions, Composites Part B (2015), doi: 10.1016/j.compositesb.2015.03.068. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

Thermo-mechanical vibration analysis of nonlocal temperaturedependent FG nanobeams with various boundary conditions Farzad Ebrahimi*, Erfan Salari, Majid Ghadiri 1

Department of Mechanical Engineering, Faculty of Engineering, Imam Khomeini International University, Qazvin, Iran

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* Corresponding author email: [email protected] (F. Ebrahimi).

Abstract

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In this paper, the thermal effect on free vibration characteristics of functionally graded (FG) size-dependent nanobeams subjected to an in-plane thermal loading are investigated by presenting a Navier type solution and employing a semi analytical differential transform method (DTM) for the first time. Material properties of FG nanobeam are supposed to vary continuously along the thickness according to the power-law form. The small scale effect is taken into consideration based on nonlocal elasticity theory of Eringen. The nonlocal equations of motion are derived through Hamilton’s principle and they are solved applying DTM. According to the numerical results, it is revealed that the proposed modeling and semi analytical approach can provide accurate frequency results of the FG nanobeams as compared to analytical results and also some cases in the literature. The detailed mathematical derivations are presented and numerical investigations are performed while the emphasis is placed on investigating the effect of the several parameters such as thermal effect, material distribution profile, small scale effects, mode number and boundary conditions on the normalized natural frequencies of the temperaturedependent FG nanobeams in detail. It is explicitly shown that the vibration behavior of a FG nanobeams is significantly influenced by these effects. Numerical results are presented to serve as benchmarks for future analyses of FG nanobeams.

Keywords: A. Nano-structures; B. Thermomechanical; B. Vibration; C. Analytical modelling; C. Numerical analysis

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Introduction

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Functionally graded materials (FGMs) are composite materials with inhomogeneous micromechanical structure. They are generally composed of two different parts such as ceramic with excellent characteristics in heat and corrosive resistances and metal with toughness. The material properties of FGMs change smoothly between two surfaces and the advantages of this combination lead to novel structures which can withstand in large mechanical loadings under high temperature environments (Ebrahimi & Rastgoo, 2008a, b). Presenting novel properties, FGMs have attracted intensive research interests, which were mainly focused on their static, dynamic and vibration characteristics of FG structures (Ebrahimi et al. 2009a, b). Moreover, structural elements such as beams, plates, and membranes in micro or nanolength scale are commonly used as components in micro/nano electromechanical systems (MEMS/NEMS). Therefore understanding the mechanical and physical properties of nanostructures is necessary for its practical applications. Nanoscale engineering materials have attracted great interest in modern science and technology after the invention of carbon nanotubes (CNTs) by Iijima, (1991). They have significant mechanical, thermal and electrical performances that are superior to the conventional structural materials. In recent years, nanobeams and CNTs hold a wide variety of potential applications (Zhang et al., 2004; Wang, 2005; Wang and Varadan, 2006) such as sensors, actuators,

ACCEPTED MANUSCRIPT transistors, probes, and resonators in NEMSs. For instance, in MEMS/NEMS; nanostructures have been used in many areas including communications, machinery, information technology and biotechnology technologies.

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Since conducting experiments at the nanoscale is a daunting task, and atomistic modeling is restricted to smallscale systems owing to computer resource limitations, continuum mechanics offers an easy and useful tool for the analysis of CNTs. However the classical continuum models need to be extended to consider the nanoscale effects and this can be achieved through the nonlocal elasticity theory proposed by Eringen (Eringen, 1972) which consider the size-dependent effect. According to this theory, the stress state at a reference point is considered as a function of strain states of all points in the body. This nonlocal theory is proved to be in accordance with atomic model of lattice dynamics and with experimental observations on phonon dispersion (Eringen, 1983).

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Moreover, in recent years the application of nonlocal elasticity theory, in micro and nanomaterials has received a considerable attention within the nanotechnology community. Peddieson et al. (2003) proposed a version of nonlocal elasticity theory which is employed to develop a nonlocal Benoulli/Euler beam model. Wang and Liew (2007) carried out the static analysis of micro- and nano-structures based on nonlocal continuum mechanics using Euler-Bernoulli beam theory and Timoshenko beam theory. Aydogdu (2009) proposed a generalized nonlocal beam theory to study bending, buckling, and free vibration of nanobeams based on Eringen model using different beam theories. Pradhan and Murmu (2010) investigated the flapwise bending–vibration characteristics of a rotating nanocantilever by using Differential quadrature method (DQM). They noticed that small-scale effects play a significant role in the vibration response of a rotating nanocantilever. Civalek et al. (2010) presented a formulation of the equations of motion and bending of Euler–Bernoulli beam using the nonlocal elasticity theory for cantilever microtubules. The method of differential quadrature has been used for numerical modeling. Civalek and Demir (2011) developed a nonlocal beam model for the bending analysis of microtubules based on the Euler–Bernoulli beam theory. The size effect is taken into consideration using the Eringen’s nonlocal elasticity theory. According to the literature it can be observed that the material properties at the nano-scale are size dependent and consequently the small length scale effect should be considered for an enhanced modeling of the mechanical behavior of nanomaterials. Alizada and Sofiyev (2011) used continuum approach of continuum mechanics to develop the modified Young’s moduli in both directions for the two-dimensional single crystal body with a square lattice. They concluded that the influences of scale effects and vacancies on the Young’s moduli are considerable. In another study, they investigated the mechanics of deformation and the stability of an elastic beam with a nanocoating under an axial compression (Alizada and Sofiyev, 2011). In this work effects of the characteristics of a nanocoating on the values of the dimensionless critical axial load parameter have been studied.

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Furthermore, with the development of the material technology, FGMs have also been employed in MEMS/NEMS (Witvrouw and Mehta, 2005; Lee et al., 2006). Because of high sensitivity of MEMS/ NEMS to external stimulations, understanding mechanical properties and vibration behavior of them are of significant importance to the design and manufacture of FG MEMS/NEMS. Thus, establishing an accurate model of FG nanobeams is a key issue for successful NEMS design. Asghari et al. (2010, 2011) studied the free vibration of the FGM Euler–Bernoulli microbeams, which has been extended to consider a size-dependent Timoshenko beam based on the modified couple stress theory. Fallah and Aghdam (2012) used Euler-Bernoulli beam theory to investigate thermo mechanical buckling and nonlinear vibration analysis of functionally graded beams on nonlinear elastic foundation. The dynamic characteristics of FG beam with power law material graduation in the axial or the transversal directions was examined by Alshorbagy et al. (2011). Ke and Wang (2011) exploited the size effect on dynamic stability of functionally graded Timoshenko microbeams. The free vibration analysis of FG microbeams was presented by Ansari et al. (2011) based on the strain gradient Timoshenko beam theory. They also concluded that the value of gradient index plays an important role in the vibrational response of the FG microbeams for lower slenderness ratios. In another study, Sahmani et al. (2013) presented dynamic stability response of higher-order shear deformable cylindrical FG microshells based on modified couple stress theory.

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Employing modified couple stress theory the nonlinear free vibration of FG microbeams based on von-Karman geometric nonlinearity was presented by Ke et al. (2012). It was revealed that both the linear and nonlinear frequencies increase significantly when the thickness of the FGM microbeam was comparable to the material length scale parameter. Hosseini-Hashemi and Nazemnezhad (2013) investigated nonlinear free vibration of simply supported Euler-Bernoulli FG nanobeams with considering surface effects and balance condition between the FG nanobeam bulk and its surfaces. The multiple scales method was used as an analytical solution for the nonlinear governing equation. Asgharifard Sharabiani and Haeri Yazdi (2013) studied surface effects on nonlinear free vibration of functionally graded nanobeam within the framework of Euler-Bernoulli beam model on the basis of von Karman geometric nonlinearity. Recently, Eltaher et al. (2012, 2013a) presented a finite element formulation for free vibration analysis of FG nanobeams based on nonlocal Euler beam theory. They also exploited the size-dependent static-buckling behavior of functionally graded nanobeams on the basis of the nonlocal continuum model (Eltaher et al., 2013b). More recently, using nonlocal Timoshenko and Euler– Bernoulli beam theory, Simsek and Yurtcu (2013) investigated bending and buckling of FG nanobeam by analytical method. Material properties are assumed to be temperature independent in these works and the thermal environment effects were not considered. Furthermore, the common use of FGMs in high temperature environment leads to considerable changes in material properties. For example, Young’s modulus usually decreases when temperature increases in FGMs. To predict the behavior of FGMs under high temperature more accurately, it is necessary to consider the temperature dependency on material properties.

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It is found that most of the previous studies on vibration analysis of FG nanobeams have been conducted based on the ignorance of the thermal environment effects. As a result, these studies cannot be utilized in order to thoroughly study the FG nanobeams under investigation. Therefore, there is strong scientific need to understand the vibration behavior of FG nanobeams in considering the effect of temperature changes. Motivated by this fact, in this study, vibration characteristics of temperature dependent FG nanobeams considering the effect of thermal environment is analyzed. A new semi-analytical method called differential transformation method (DTM) is employed for vibration analysis of size-dependent FG nanobeams with three combinations of boundary conditions for the first time. The superiority of the DTM is its simplicity and good precision and depends on Taylor series expansion while it takes less time to solve polynomial series. It is different from the traditional high order Taylor’s series method, which requires symbolic competition of the necessary derivatives of the data functions. The Taylor series method is computationally taken long time for large orders. With this method, it is possible to obtain highly accurate results or exact solutions for differential equations. It is assumed that material properties of the beam, vary continuously through the beam thickness according to power-law form and are temperature dependent. Nonlocal Euler–Bernoulli beam model and Eringen’s nonlocal elasticity theory are employed. Governing equations and boundary conditions for the free vibration of a nonlocal FG beam have been derived via Hamilton’s principle. These equations are solved using DTM and numerical solutions are obtained. The detailed mathematical derivations are presented while the emphasis is placed on investigating the effect of several parameters such as thermal effects, constituent volume fractions, mode number, slenderness ratios, boundary conditions and small scale on vibration characteristics of FG nanobeams. Comparisons with analytical solotions and the results from the existing literature are provided for two-constituents metal–ceramic nanobeams and the good agreement between the results of this article and those available in literature validated the presented approach. Numerical results are presented to serve as benchmarks for the application and the design of nanoelectronic and nano-drive devices, nano-oscillators, and nanosensors, in which nanobeams act as basic elements. They can also be useful as valuable sources for validating other approaches and approximate methods.

2.Theory and formulation

2.1. Nonlocal powerpower-law FG nanobeam equations

ACCEPTED MANUSCRIPT Consider a FG nanobeam of length L, width b and uniform thickness h in the unstressed reference configuration. The coordinate system for FG nanobeam is shown in Figure 1. The nanobeam is made of elastic and isotropic functionally graded material with properties varying smoothly in the z thickness direction only. The effective material properties of the FG beam such as Young’s modulus  , shear modulus  and mass density ! are assumed to vary continuously in the thickness direction (z-axis direction) according to a power function of the volume fractions of the constituents. Therefore the average value can be used for calculation. According to the rule of mixture, the effective material properties, " , can be expressed as (Simsek and Yurtcu, 2013):

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" # "$ %$ & "' %'

(1)

Where "' , "$ , %' and %$ are the material properties and the volume fractions of the metal and the ceramic constituents related by: (2.a)

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%$ & %' # 1

The volume fraction of the ceramic constituent of the beam is assumed to be given by:

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) 1 , -). # -$ / ' . ( & + & ' * 2 ) 1 , 0-). # -0$ / 0' . ( & + & 0' * 2 ) 1 , !-). # -!$ / !' . ( & + & !' * 2

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) 1 , (2.b) %$ # ( & + * 2 Here p is the non-negative variable parameter (power-law exponent) which determines the material distribution through the thickness of the beam and z is the distance from the mid-plane of the FG beam. The FG beam becomes a fully ceramic beam when p is set to be zero. Therefore, from Eqs. (1)–(2), the effective material properties of the FG nanobeam can be expressed as follows:

(3)

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To predict the behavior of FGMs under high temperature more accurately, it is necessary to consider the temperature dependency on material properties. The nonlinear equation of thermo-elastic material properties in function of temperature T(K) can be expressed as (Touloukian , 1967) : " # "1 -"23 4 23 & 1 & "3 4 & "5 4 5 & "6 4 6 .

(4)

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Where 4 # 41 & ∆4 and 41 # 300 : (ambient or free stress temperature), ∆4 is the temperature change, "1 , "23 , "3 , "5 and "6 are the temperature dependent coefficients which can be seen in the table of materials properties (Table 1) for Al2O3 and SUS304. The bottom surface (z= -h/2) of FG nanobeam is pure metal (SUS304), whereas the top surface (z= h/2) is pure ceramics(Al2O3). 2.2. Kinematic relations

The equations of motion are derived based on the Euler–Bernoulli beam theory according to which the displacement field at any point of the beam can be written as: => ->, ?, @. # =->, @. / ?

AB->, @. A>

(5a)

ACCEPTED MANUSCRIPT =? ->, ?, @. # B->, @.

(5b)

Where t is time, u and w are the axial and transverse displacement of the mid-plane along x and z directions, respectively. Therefore, according to Euler–Bernoulli beam theory, every elements of strain tensor vanish except normal strain in the x-direction. Thus, the only nonzero normal strain is: AF->,@. , A>

E1 #

GH I-D,J. GD H

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1 1 CDD # CDD / ?E 1, CDD #

(6)

1 and E 1 are the extensional strain and bending strain respectively. Based on the Hamilton’s Where CDD principle, which states that the motion of an elastic structure during the time interval K3 L K L K5 is such that the time integral of the total dynamics potential is extremum (Tauchert, 1974) : J

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M N-O / 4 & %.PK # 0 1

(7)

NO # M QRS NCRS P% # M -QDD NCDD .P% T

T

Substituting Eq.(6) into Eq.(8) yields: X

1 . NO # M -U-NCDD / V-NE 1 ..PW 1

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Here U is strain energy, T is kinetic energy and V is work done by external forces. The virtual strain energy can be calculated as: (8)

(9)

U # M QDD PY, Z

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In which N, M are the axial force and bending moment respectively. These stress resultants used in Eq. (9) are defined as: V # M QDD )PY Z

(10)

The kinetic energy for Euler-Bernoulli beam can be written as: (11)

X \] \N] \b \Nb \] \ 5 Nb \N] \ 5 b \ 5 b \ 5 Nb N4 # M `a1 ( & + / a3 [ & c PW _ & a5 \K \K \K \K \K \K\W \K \K\W \K\W \K\W 1

(12)

-a1 , a3 , a5 . # M !-), 4.-1, ), ) 5 .PY

(13)

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1 X \]D 5 \]^ 5 4 # M M !-), 4. [( + &. _ PYPW 2 1 Z \K \K

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Also the virtual kinetic energy can be expressed as:

Where (a1 , a3 , a5 ) are the mass moment of inertias, defined as follows: Z

For a typical FG nanobeam which has been in high temperature environment for a long period of time, it is assumed that the temperature can be distributed uniformly across its thickness so that the case of uniform temperature rise is taken into consideration. In this investigation, initial uniform temperature

ACCEPTED MANUSCRIPT (41 # 300 :), which is a stress free state, changes to final temperature with ∆4. Hence, the first variation of the work done corresponding to temperature change can be written in the form: X

N% # M U d 1

\b \Nb PW \W \W

(14)

e/5

Ud # M

-), 4.0-), 4.∆4P)

2e/5

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Where U d is the thermal resultant and can be written as:

Under the following boundary conditions: N = 0 or u=0 at x = 0 and x=L Gi GD

GH F

Gj I

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\U \5] \6b # a1 5 / a3 \W \K \W\K 5 \5V \5b \5b \6] \hb d / U # a & a / a 1 3 5 \W 5 \W 5 \K 5 \W\K 5 \W 5 \K 5

/ a3 GJ H & a5 GDGJ H # 0 or w=0 at x= 0 and x=L

M = 0 or

GI GD

# 0 at x = 0 and x=L

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By Substituting Eqs. (9),(12) and (14) into Eq.(7) and setting the coefficients of N], Nb and the following Euler–Lagrange equation can be obtained:

gGI GD

(15)

to zero,

(16.a) (16.b)

(17.a) (17.b) (17.c)

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2.3. The nonlocal elasticity model for FG nanobeam

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Based on Eringen nonlocal elasticity model (Eringen & Edelen, 1972), the stress at a reference point x in a body is considered as a function of strains of all points in the near region. This assumption is agreement with experimental observations of atomic theory and lattice dynamics in phonon scattering in which for a homogeneous and isotropic elastic solid the nonlocal stress-tensor components QRS at any point x in the body can be expressed as: QRS -W. # M 0-|W l / W|, m.KRS -W l .PΩ-W l .

(18)

KRS # rRSpq Cpq

(19)

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Where KRS -W l . are the components of the classical local stress tensor at point x which are related to the components of the linear strain tensor Cpq by the conventional constitutive relations for a Hookean material, i.e:

The meaning of Eq. (18) is that the nonlocal stress at point x is the weighted average of the local stress of all points in the neighborhood of x, the size of which is related to the nonlocal kernel 0-|W l / W|, m.. Here |W l / W| is the Euclidean distance and m is a constant given by:

ACCEPTED MANUSCRIPT m#

s1 t u

(20)

Which represents the ratio between a characteristic internal length, a (such as lattice parameter, C–C bond length and granular distance) and a characteristic external one, l (e.g. crack length, wavelength) trough an

adjusting constant, s1 , dependent on each material. The magnitude of s1 is determined experimentally or approximated by matching the dispersion curves of plane waves with those of atomic lattice dynamics.

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According to for a class of physically admissible kernel 0-|W l / W|, m. it is possible to represent the

integral constitutive relations given by Eq. (18) in an equivalent differential form as: -1 / -s1 t.5 v 5 .QpqwJxy

(21)

\ 5 Q-W. # C-W. \W 5

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Q-W. / -e1 a.5

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Where v 5 is the Laplacian operator. Thus, the scale length e1 a takes into account the size effect on the response of nanostructures. For an elastic material in the one dimensional case, the nonlocal constitutive relations may be simplified as : (22)

Where σ and ε are the nonlocal stress and strain, respectively. E is the Young’s modulus. For Euler– Bernoulli nonlocal FG beam, Eq. (22) can be rewritten as: \ 5 QDD (23) # -).CDD \W 5 where (z # -s1 t.5) is the nonlocal parameter. Integrating Eq.(23) over the beam’s cross-section area, the force-strain and the moment-strain relations for the nonlocal FG beam can be obtained as follows:

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QDD / z

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\5U \] \5b # Y / { DD DD \W 5 \W \W 5 5 \ V \] \5b V/z # { / r DD DD \W 5 \W \W 5 In which the cross-sectional rigidities are defined as follows: U/z

-YDD , {DD , rDD . # M -), 4.-1, ), ) 5 .PY

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Z

(24) (25)

(26)

The explicit relation of the nonlocal normal force can be derived by substituting for the second derivative of N from Eq. (16.a) into Eq. (24) as follows: U # YDD

\] \5b \6] \hb / {DD 5 & z-a1 / a . 3 \W \W \W\K 5 \W 5 \K 5

(27)

Also the explicit relation of the nonlocal bending moment can be derived by substituting for the second derivative of M from Eq. (16.b) into Eq. (25) as follows: V # {DD

\] \5b \5b \6] \hb \5b d / rDD 5 & z-a1 5 & a3 / a & U . 5 \W \W \K \W\K 5 \W 5 \K 5 \W 5

(28)

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The nonlocal governing equations of FG nanobeam in terms of the displacement can be derived by substituting for N and M from Eqs. (27) and (28) , respectively, into Eq.(16) as follows: \5] \6b \h] \|b \5] \6b / { & z / a / a & a #0 [a _ DD 1 3 1 3 \W 5 \W 6 \K 5 \W 5 \K 5 \W 6 \K 5 \K 5 \W \6] \hb \5b \h] \|] \}b \hb \5b {DD 6 / rDD h / U d 5 & z [a1 5 5 & a3 5 6 / a5 5 h & U d h _ / a1 5 \W \W \W \K \W \K \W \K \W \W \K \6] \hb / a3 5 & a5 5 5 # 0 \K \W \K \W

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YDD

3. Solution method

(30)

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3.1. Analytical Solution

(29)

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€ ]-W, K. # ~ O cos- W. s R„… J ‚ w3 ƒ

€ b-W, K. # ~ † sin- W. s R„… J ‚ w3

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Here, based on the Navier type method, an analytical solution of the governing equations for free vibration of a simply supported FG nanobeam is presented. The displacement functions are expressed as product of undetermined coefficients and known trigonometric functions to satisfy the governing equations and the conditions at x = 0, L. The following displacement fields are assumed to be of the form:

(31)

(32)

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Where (O , † ) are the unknown Fourier coefficients to be determined for each n value. Boundary conditions for simply supported beam are as Eq. (33): \] -‚. # 0 ]-0. # 0, \W \5b \5b (33) -0. -‚. # 0 b-0. # b-‚. # 0, # \W 5 \W 5

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Substituting Eqs. (31) and (32) into Eqs. (29) and (30) respectively, leads to following relations: € € € € € ‡/YDD - .5 & a1 -1 & z- .5 .ˆ5 ‰ O & ‡{DD - .6 / a3 -- . & z- .6 .ˆ5 ‰ † # 0 ‚ ‚ ‚ ‚ ‚ € 6 € € 6 5 ‡{DD - . / a3 -- . & z- . .ˆ ‰ O ‚ ‚ ‚ € h € 5 € 5 € 5 € & (/rDD - . & U d ‡ ‰ -1 & z ‡ ‰ . & a1 -1 & z ‡ ‰ .ˆ5 & a5 -- .5 ‚ ‚ ‚ ‚ ‚ € h 5 & z- . .ˆ + † # 0 ‚

(34)

(35)

By setting the determinant of the coefficient matrix of the above equations, a quadratic polynomial for ˆ5 can be obtained. By setting this polynomial to zero, we can find ˆ . 3. 2. Implementation of differential transform method

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‹-E. #

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The differential transforms method provides an analytical solution procedure in the form of polynomials to solve ordinary and partial differential equations with small calculation errors and ability to solve nonlinear equations with boundary conditions value problems. Using DTM technique, the ordinary and partial differential equations can be transformed into algebraic equations, from which a closed-form series solution can be obtained easily. In this method, certain transformation rules are applied to both the governing differential equations of motion and the boundary conditions of the system in order to transform them into a set of algebraic equations. The solution of these algebraic equations gives the desired results of the problem. In this method, differential transformation of kth derivative function y(x) and differential inverse transformation of Y(k) are respectively defined as follows (Abdel-Halim Hassan, 2002):  -W. p! ŽD x 3

Žx

ƒ

-W. # ~ W p ‹-E.

Dw1

(37)

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1

(36)

-W. # ∑ƒ pw1 p! ŽD x -W. Dx

Žx

Dw1

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In which y(x) is the original function and Y(k) is the transformed function. Consequently from equations (36,37) we obtain: (38)

Equation (38) reveals that the concept of the differential transformation is derived from Taylor’s series expansion. In real applications the function y(x) in equation (38) can be written in a finite form as: p -W. # ∑’ pw1 W ‹-E.

p ∑ƒ “3 W ‹-E.

(39)

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In this calculations -W. # is small enough to be neglected, and N is determined by the convergence of the eigenvalues. From the definitions of DTM in Equations (36)- (38), the fundamental theorems of differential transforms method can be performed that are listed in Table 2 while Table 3 presents the differential transformation of conventional boundary conditions. According to the basic transformation operations introduced in Table 2, the transformed form of the governing equations (29,30) around W1 = 0 may be obtained as:

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YDD -E & 1.-E & 2.O”E & 2• / {DD -E & 1.-E & 2.-E & 3.†”E & 3• / a1 ˆ5 -/O”E• & z-E & 1.-E & 2.O”E & 2•. / a3 ˆ5 -/z-E & 1.-E & 2.-E & 3.†”E & 3• & -E & 1.†”E & 1•. # 0

(40)

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{DD -E & 1.-E & 2.-E & 3.O”E & 3• / rDD -E & 1.-E & 2.-E & 3.-E & 4.†”E & 4• / U d -E & 1.-E & 2.†”E & 2• / a1 ˆ5 -/†”E• & z-E & 1.-E & 2.†”E & 2•. / a3 ˆ5 -/-E & 1.O”E & 1• & z-E & 1.-E & 2.-E & 3.O”E & 3•. / a5 ˆ5 -/z-E & 1.-E & 2.-E & 3.-E & 4.†”E & 4• & -E & 1.-E & 2.†”E & 2•. (41) & zU d -E & 1.-E & 2.-E & 3.-E & 4.†”E & 4• # 0 Where O”E• and †”E• are the transformed functions of u, w respectively. Additionally, the differential transform method is applied to various boundary conditions by using the theorems introduced in Table 3 and the following transformed boundary conditions are obtained. Simply supported–Simply supported: †”0• # 0, †”2• # 0, O”0• # 0

ACCEPTED MANUSCRIPT ƒ

ƒ

ƒ

pw1

pw1

pw1

~ †”E• # 0, ~ E-E / 1.†”E• # 0, ~ E O”E• # 0

ƒ

ƒ

pw1

pw1

pw1

~ †”E• # 0, ~ E †”E• # 0, ~ O”E• # 0

Clamped–Simply supported: †”0• # 0, †”1• # 0, O”0• # 0 ƒ

ƒ

ƒ

pw1

pw1

pw1

~ †”E• # 0, ~ E-E / 1.†”E• # 0, ~ E O”E• # 0

(42.b)

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ƒ

(42.c)

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Clamped–Clamped: †”0• # 0, †”1• # 0, O”0• # 0

(42.a)

V33 -ˆ. V35 -ˆ. V36 -ˆ. —V53 -ˆ. V55 -ˆ. V56 -ˆ.˜ ”r• # 0 V63 -ˆ. V65 -ˆ. V66 -ˆ.

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By using Eqs. (40) and (41) together with the transformed boundary conditions one arrives at the following eigenvalue problem:

(43.a)

Where ”r• correspond to the missing boundary conditions at x = 0. For the non-trivial solutions of Eq. (43.a), it is necessary that the determinant of the coefficient matrix set equal to zero:

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V33 -ˆ. V35 -ˆ. V36 -ˆ. V ™ 53 -ˆ. V55 -ˆ. V56 -ˆ.™ # 0 V63 -ˆ. V65 -ˆ. V66 -ˆ.

(43.b)

Solution of Eq. (43.b) is simply a polynomial root finding problem. In the present study, the Newton– Raphson method is used to solve the governing equation of the non-dimensional natural frequencies.

following equation: -.

/ ˆR

-23.

šLC

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šˆR

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Solving equation (43.b), the ith estimated eigenvalue for nth iteration (ˆ # ˆR ) may be obtained and the total number of iterations is related to the accuracy of calculations which can be determined by the -.

(44)

In this study C =0.0001 considered in procedure of finding eigenvalues which results in 4 digit precision in estimated eigenvalues. Further a Matlab program has been developed according to DTM rule stated above, in order to find eigenvalues. As mentioned before, DT method implies an iterative procedure to obtain the highorder Taylor series solution of differential equations. The Taylor series method requires a long computational time for large orders, whereas one advantage of employing DTM in solving differential equations is a fast convergence rate and a small calculation error.

4. Numerical results and discussions

ACCEPTED MANUSCRIPT Through this section, a numerical testing of the procedure as well as parametric studies are performed in order to establish the validity and usefulness of the DTM approach. The effect of temperature, FG distribution, nonlocality effect and thickness ratios on the natural frequencies of the FG nanobeam will be figured out. The functionally graded nanobeam is composed of steel and ceramic where its properties are given in Table 1. The bottom surface of the beam is pure steel, whereas the top surface of the beam is

RI PT

pure alumina. The beam geometry has the following dimensions: L (length) = 10,000 nm, b (width) = 1000 nm and h (thickness) = 100 nm. Relation described in equation (45) are performed in order to calculate the non-dimensional natural frequencies: ˆ › # ˆ‚5 œ!$ Y/a$

(45)

SC

Where a # *6 /12 is the moment of inertia of the cross section of the beam and A is the area of cross

section. Table 4 shows the convergence study of DTM for first three frequencies of nanobeam with various gradient indexes. It is found that in DTM after a certain number of iterations (k) eigenvalues

M AN U

converged to a value with good precision. From results of Table 4, high convergence rate of the method may be easily observed and it may be deduced that the third natural frequency of S-S FG nanobeam with p =1 converged after 35 iterations with 4 digit precision while the first and secound natural frequencies converged after 17 and 27 iterations respectively.

To evaluate accuracy of the natural frequencies predicted by the present method, the non-dimensional natural frequencies of simply supported FG nanobeam with various nonlocal parameters previously

TE D

analyzed by finite element method are reexamined. Table 5 compares the results of the present study and the results presented by Eltaher et al. (2012) which has been obtained by finite element method for FG nanobeam with different nonlocal parameters (varying from 0 to 5). The reliability of the presented method and procedure for FG nanobeam may be concluded from Table 4;

EP

where the results are in an excellent agreement as values of non-dimensional fundamental frequency are consistent with presented analytical solution up to 4 digits. After extensive validation of the present formulation for S-S FG nanobeams, the effects of different

AC C

parameters such as temperature change, nonlocality, gradient index and boundary conditions on free vibration behavior of FG nanobeam are investigated. Three different combinations of clamped (C), and simply-supported (S) conditions x=0; L are considered in the following: simply-supported/simplysupported, simply-supported/clamped, and clamped/clamped. Each combination will be shortly indicated by a two-letter notation corresponding to edge conditions at x =0 and x=L. For example, SC stands for a FG beam with simply-supported edge at x=0 and clamped edge at x=L. In Tables 6, 7 and 8, first three dimensionless natural frequencies of the simply supported FG nanobeams are presented for various values of the gradient index (p =0, 0.2, 1, 5), nonlocal parameters (z =0,1,2,3,4), three different values of temperature changes (∆4 =0, 20, 40 K) for L/h=20 based on both DTM and analytical Navier solution

ACCEPTED MANUSCRIPT method. First of all, when the nonlocal parameters and gradient index vanish (z ∗ 10235= 0 and p=0) the classical isotropic beam theory is rendered. Furthermore, the effects of temperature change, nonlocal parameter and gradient indexes on the dimensionless frequencies are presented in these tables. It can be concluded from the results of the tables that an increase in nonlocal scale parameter gives rise to

RI PT

a decrement in the first three dimensionless natural frequencies. It can also be seen that the dimensionless natural frequencies predicted by DTM are in close agreement with those evaluated by analytical solution.

In addition, it is seen that the first three dimensionless natural frequencies decrease by increasing

SC

temperature change and it can be stated that temperature change has a significant effect on the dimensionless natural frequencies, especially for lower mode numbers.

M AN U

On the other hand, it is revealed that the dimensionless natural frequencies decrease with an increase in material gradient index due to increasing in stiffness of the beam because of increasing in ceramic’s phase constituent and this trend can be seen for higher mode frequencies either.

Variations of the first three dimensionless natural frequencies of the simply supported FG nanobeams with respect to temperature changes for different values of gradient indexes and nonlocal parameters are depicted in Figs. 2, 3 and 4, respectively.

TE D

It is seen from the figures that the fundamental frequency of FG nanobeam decreases with the increase of temperature until it approaches to the critical buckling temperature. This is due to the reduction in total stiffness of the beam, since geometrical stiffness decreases when temperature rises.

EP

Frequency reaches to zero at the critical temperature point. The increase in temperature yields in higher frequency after the branching point.

One important observation within the range of temperature before the critical temperature, it is seen that

AC C

the FG nanobeams with lower value of gradient index (higher percentage of ceramic phase) usually provide larger values of the frequency results. However, this behavior is opposite in the range of temperature beyond the critical temperature. It is also observable that the branching point of the FG nanobeam is postponed by consideration of the lower gradient indexes due to the fact that the lower gradient indexes result in the increase of stiffness of the structure. Fig. 5 displays the variations of the first dimensionless natural frequency of the simply supported FG nanobeam with respect to temperature change for different values of nonlocal parameters and gradient indexes (L/h=50).

ACCEPTED MANUSCRIPT The effect of material graduation on the first dimensionless frequency of simply supported FG nanobeam with different temperatures and nonlocality parameters for (L/h =20) has been shown in Figures 6 and 7. Temperature and material graduation effects on first dimensionless frequency (fundamental frequency) of a C-S FG nanobeam with different nonlocality parameters are presented in Table 9.

RI PT

The fundamental frequency parameter as a function of power law indexes temperature rise is presented in Figure 8 for the FG nanobeam with C–S boundary condition. Similarly, the variation of the first dimensionless frequency with material graduation and nonlocality parameters for different temperature rises are depicted in Figure 9.

SC

Observing these two figures, it is easily deduced for a C-S FG nanobeam that, an increase in nonlocal scale parameter gives rise to a decrease in the first dimensionless natural frequency for all gradient indexes.

M AN U

In addition, it is deduced that the fundamental frequency decreases by increasing temperature changes and it can be stated that temperature change has a significant effect on the fundamental frequency of the graded C-S nanobeam.

Besides, for higher values of material property gradient index, graded nanobeam becomes stiffer and thus, the dimensionless fundamental frequency decreases while the gradient index increases which causes the

TE D

reduction in the value of the elasticity modulus.

Comparing the frequency values for FG nanobeams with various boundary conditions presented in Tables 9 and 10 reveals that for a prescribed temperature rise and gradient index the greatest frequency at prebuckling region, is obtained for the FG beam with C–C boundary conditions followed with S-C and S-S,

EP

respectively. The last analysis deals with the temperature change and material graduation effect on vibration behavior of C-C FG nanobeam with different nonlocality parameters. Table 10 contains the results of different cases with and without temperature rise effects for different material inhomogeneity

AC C

and nonlocal parameters which influence the first three frequencies of C-C functionally graded nanobeam. Also the variation of the first dimensionless frequency of C-C FG nanobeam with material graduation and nonlocality parameters for different temperature rises are depicted in Figures 10 and 11. Similar to two previously discussed edge conditions, it is revealed that for a C-C FG nanobeam increasing temperature change, power law index, and nonlocal parameter, leads to decrease in natural frequency. Also, as it is expected, a beam with stiffer edges; i.e. C-C and C-S, respectively shows higher natural frequencies.

5. Conclusions

ACCEPTED MANUSCRIPT In this paper, thermo-mechanical vibrational behavior of the temperature-dependent FG nanobeams based on Euler–Bernoulli beam theory and Eringen’s nonlocal constitutive equations is investigated. The governing differential equations and related boundary conditions are derived by implementing Hamilton’s principle. By using the differential transformation technique, the governing partial differential equation is reduced to algebraic equations. Accuracy of the results is examined using available date in the literature. Finally, through some numerical examples, the effect of different parameters are investigated. It is illustrated that FG nanobeam

RI PT

model produces smaller natural frequency than the classical (local) beam model. Therefore, the small scale effects should be considered in the analysis of mechanical behavior of nanostructures. It is observed that the fundamental frequency decreases with the increase in temperature and tends to the minimum point closing to zero at the critical temperature. This decrease in frequency with thermal load is attributed to the fact that the

SC

thermally induced compressive stress weakens the beam stiffness. However, after the critical temperature region, the fundamental frequency increases with the increment of temperature. Also, it is concluded that under temperature rise, with the increase in the gradient index value leads to the decrease in frequency. Moreover it

M AN U

is revealed that under the temperature rise, the greatest frequency at pre-buckling region is obtained for the FG nanobeam with C–C boundary conditions followed with S-C and S-S conditions, respectively.

References

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Abdel-Halim Hassan, I. H. (2002). On solving some eigenvalue problems by using a differential transformation. Applied Mathematics and Computation, 127(1): 1-22. Alshorbagy A.E., Eltaher M.A., Mahmoud F.F., Free vibration characteristics of a functionally graded beam by finite element method, Applied Mathematical Modelling 35 (2011) 412–425. Alizada, A. N., & Sofiyev, A. H. (2011). Modified Young’s moduli of nano-materials taking into account the scale effects and vacancies. Meccanica, 46(5), 915-920. Alizada, A.N., Sofiyev, A.H, On the mechanics of deformation and stability of the beam with a nanocoating. J. Reinforced Plastics Compos. 30, 1583–1595 (2011) Ansari R, Gholami R, Sahmani S. Free vibration analysis of size-dependent functionally graded microbeams based on the strain gradient Timoshenko beam theory. Compos Struct 2011;94:221–8. Asghari M., M.T. Ahmadian, M.H. Kahrobaiyan, M. Rahaeifard, On the size-dependent behavior of functionally graded micro-beams, Materials and Design 31 (2010) 2324–2329. Asghari, M., Rahaeifard, M., Kahrobaiyan, M.H., Ahmadian, M.T., The modified couple stress functionally graded Timoshenko beam formulation, Mater. Des. 32 (2011) 1435–1443. Aydogdu, Metin. "A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration." Physica E: Low-dimensional Systems and Nanostructures 41.9 (2009): 1651-1655. Chen, C. O. K., Ju, S. P. (2004). Application of differential transformation to transient advective–dispersive transport equation. Applied Mathematics and Computation, 155(1): 25-38. Civalek O., Demir C., Akgoz B., Free vibration and bending analyses of cantilever microtubules based on nonlocal continuum model, Mathematical and Computational Applications 15 (2) (2010) 289–298. Civalek, Ömer, and Çiğdem Demir. "Bending analysis of microtubules using nonlocal Euler–Bernoulli beam theory." Applied Mathematical Modelling 35.5 (2011): 2053-2067. Ebrahimi, F., and A. Rastgoo. "Free vibration analysis of smart annular FGM plates integrated with piezoelectric layers." Smart Materials and Structures 17.1 (2008a): 015044. Ebrahimi, F, and A. Rastgoo. "An analytical study on the free vibration of smart circular thin FGM plate based on classical plate theory." Thin-Walled Structures 46.12 (2008b): 1402-1408. Ebrahimi, F., A. Rastgoo, and A. A. Atai. "A theoretical analysis of smart moderately thick shear deformable annular functionally graded plate." European Journal of Mechanics-A/Solids 28.5 (2009a): 962-973.

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Nomenclature

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•

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•

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•

Ebrahimi, F., Naei, M.H. and Abbas Rastgoo. "Geometrically nonlinear vibration analysis of piezoelectrically actuated FGM plate with an initial large deformation." Journal of mechanical science and technology 23.8 (2009b): 2107-2124. Eltaher M.A., Emam S.A., Mahmoud F.F., Free vibration analysis of functionally graded size-dependent nanobeams, Appl. Math. Comput. 218 (2012) 7406–7420. Eltaher M.A., Alshorbagy A.E., Mahmoud F.F., Determination of neutral axis position and its effect on natural frequencies of functionally graded macro/ nanobeams, Compos. Struct. 99 (2013a) 193–201. Eltaher M.A., Emam S.A., Mahmoud F.F., Static and stability analysis of nonlocal functionally graded nanobeams, Composite Structures 96 (2013b) 82–88 Eringen, A. C. (1972). Nonlocal polar elastic continua. International Journal of Engineering Science, 10(1): 116. Eringen, A. C. (1983). On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics,54(9): 4703-4710. Eringen AC, Edelen DGB. “Nonlocal elasticity”. International Journal of Engineering Sciences (1972); 10(3):233–48. Fallah, A., and M. M. Aghdam. "Thermo-mechanical buckling and nonlinear free vibration analysis of functionally graded beams on nonlinear elastic foundation." Composites Part B: Engineering 43.3 (2012): 15231530. Hosseini-Hashemi, S., & Nazemnezhad, R. (2013). An analytical study on the nonlinear free vibration of functionally graded nanobeams incorporating surface effects. Composites Part B: Engineering, 52, 199-206. Iijima, S. (1991). Helical microtubules of graphitic carbon. Nature, 354.6348: 56-58. Ke L.L., Wang Y.S., Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory, Composite Structures 93 (2011) 342–350. Ke LL, Wang YS, Yang J, Kitipornchai S. Nonlinear free vibration of size dependent functionally graded microbeams. Int J Eng Sci 2012; 50:256–67. Lee Z, Ophus C, Fischer LM, Nelson-Fitzpatrick N, Westra KL, Evoy S, et al. Metallic NEMS components fabricated from nanocomposite Al–Mo films. Nanotechnology 2006;17(12):3063–70. Peddieson, John, George R. Buchanan, and Richard P. McNitt. "Application of nonlocal continuum models to nanotechnology." International Journal of Engineering Science 41.3 (2003): 305-312. Pradhan, S. C., and T. Murmu. "Application of nonlocal elasticity and DQM in the flapwise bending vibration of a rotating nanocantilever." Physica E: Low-dimensional Systems and Nanostructures 42.7 (2010): 19441949. Sahmani, S., Ansari, R., Gholami, R., & Darvizeh, A. (2013). Dynamic stability analysis of functionally graded higher-order shear deformable microshells based on the modified couple stress elasticity theory. Composites Part B: Engineering, 51, 44-53. Sharabiani, P. A., & Yazdi, M. R. H. (2013). Nonlinear free vibrations of functionally graded nanobeams with surface effects. Composites Part B: Engineering, 45(1), 581-586. Simsek M., Yurtcu H.H., Analytical solutions for bending and buckling of functionally graded nanobeams based on the nonlocal Timoshenko beam theory, Compos. Struct. 97 (2013) 378–386. Tauchert T.R., Energy principles in structural mechanics, Library of Congress Cataloging in Publication Data, 1974. Touloukian Y S, Thermo-physical Properties Research Center. Thermo-physical properties of high temperature solid materials. Vol. 1. Elements.-Pt. 1.Macmillan, 1967. Wang, Q. (2005). Wave propagation in carbon nanotubes via nonlocal continuum mechanics. Journal of Applied Physics, 98(12): 124301. Wang, Q., and K. M. Liew. "Application of nonlocal continuum mechanics to static analysis of micro-and nanostructures." Physics Letters A 363.3 (2007): 236-242. Wang, Q., Varadan, V. K. (2006). Vibration of carbon nanotubes studied using nonlocal continuum mechanics. Smart materials and structures, 15(2): 659. Witvrouw A, Mehta A. The use of functionally graded poly-SiGe layers for MEMS applications. Funct Graded Mater VIII 2005;492–493:255–60. Zhang, Y. Q., Liu, G. R., Wang, J. S. (2004). Small-scale effects on buckling of multiwalled carbon nanotubes under axial compression. Physical review B, 70(20): 205430.

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•

ACCEPTED MANUSCRIPT

I

k ¦£ L M N §¨ p

©¡ ©ª ©£ , ©2¤ , ©¤ , ©¥ , ©«

t T

initial uniform temperature temperature change axial displacement strain energy unknown Fourier coefficients transformed functions work done by external forces transverse displacement x coordinate z coordinate normal strain extensional strain mass density thermal expansion coefficient nonlocal stress-tensor components natural frequency non-dimensional natural frequency nonlocal parameter

u O

O , † O”E•, †”E•

% w x z

RI PT

¢£, ¢¤, ¢¥

41 ∆4

CDD 1 CDD

! 0

QRS

ˆ ˆ › z

SC

E h

area of the cross section cross-sectional rigidities Young’s modulus thickness of the beam moment of inertia of the cross section mass moment of inertias number of iterations bending strain length of the beam bending moment axial force thermal resultant power-law exponent material properties of the ceramic constituent material properties of the metal constituent temperature dependent coefficients time kinetic energy

M AN U

Ÿ Ÿ>>,  >>, ¡>>

Tables:

Table 1. Temperature dependent coefficients of Young’s modulus, thermal expansion coefficient and mass density for Al2O3 and SUS304 (Touloukian, 1967). Material Properties "1 "23 "3 "5 "6 E (Pa) 349.55e+9 0 -3.853e-4 4.027e-7 1.673e-10 Al2O3 23 6.8269e-6 0 1.838e-4 0 0 0-K . 3800 0 0 0 0 !-Kg/m6 .

0-K 23 . !-Kg/m6 . E (Pa)

201.04e+9 12.330e-6 8166

0 0 0

TE D

SUS304

3.079e-4 8.086e-4 0

-6.534e-7 0 0

0 0 0

Table 2. Some of the transformation rules of the one-dimensional DTM (Chen and Ju, 2004) Transformed function

y-x. # λφ-x.

Y-k. # λΦ-k.

EP

Original function

y-x. # φ-x. ³ θ-x.

AC C

y-x. #

y-x. #

dφ dx

d5 φ dx 5

y-x. # φ-x.θ-x. y-x. # x ¸

Y-k. # Φ-k. ³ Θ-k.

Y-k. # -k & 1.Φ-k & 1.

Y-k. # -k & 1.-k & 2.Φ-k & 1. ¶

Y-k. # ~ Φ-l.Θ-k / l. ·w1

Y-k. # δ-k / m. # º

1 0

k#m k»0

Table 3. Transformed boundary conditions (B.C.) based on DTM (Chen and Ju, 2004) X=0

X=1

ACCEPTED MANUSCRIPT Original BC ƒ

~ F”k• # 0

f(0)=0

F[0]=0

f(1)=0

df -0. # 0 dx

F[1]=0

df -1. # 0 dx

d5 f -0. # 0 dx 5

¶w1 ƒ

~ kF”k• # 0

¶w1 ƒ

d5 f -1. # 0 dx 5

F[2]=0

d6 f -0. # 0 dx 6

Transformed BC

~ k-k / 1.F”k• # 0

¶w1 ƒ

d6 f -1. # 0 dx 6

F[3]=0

RI PT

Transformed BC

~ k-k / 1.-k / 2.F”k• # 0

¶w1

SC

Original BC

z # 4, ∆4 # 20-K., ¼ # 1

Table 4. Convergence study for the first three natural frequencies of S-S FG nanobeam with L/h=20,

5.5193

ˆ ›5 -

ˆ ›6

13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43

5.5051 5.5060 5.5060 5.5060 5.5060 5.5060 5.5060 5.5060 5.5060 5.5060 5.5060 5.5060 5.5060 5.5060 5.5060 5.5060

15.9965 17.2150 16.8807 16.9121 16.9090 16.9093 16.9093 16.9093 16.9093 16.9093 16.9093 16.9093 16.9093 16.9093 16.9093

28.4099 28.7696 28.7125 28.7186 28.7180 28.7180 28.7180 28.7180 28.7180 28.7180

5.5058

16.9063

28.7065

z 0 1 2 3 4 5

ˆ ›3

C-C

11.5246

ˆ ›5 -

ˆ ›6 -

ˆ ›3

8.3555

ˆ ›5 -

ˆ ›6

13.2222 12.5524 12.6131 12.6066 12.6071 12.6071 12.6071 12.6071 12.6071 12.6071 12.6071 12.6071 12.6071 12.6071 12.6071 12.6071

24.4847 25.3454 25.1435 25.1655 25.1631 25.1633 25.1633 25.1633 25.1633 25.1633 25.1633 25.1633

37.1449 37.5900 37.5107 37.5197 37.5187 37.5188 37.5188

8.7307 8.6636 8.6695 8.6691 8.6691 8.6691 8.6691 8.6691 8.6691 8.6691 8.6691 8.6691 8.6691 8.6691 8.6691 8.6691

20.7997 20.9576 20.9367 20.9387 20.9385 20.9386 20.9386 20.9386 20.9386 20.9386 20.9386 20.9386

33.5322 32.9669 33.0180 33.0119 33.0125 33.0125 33.0125 33.0125

-

-

-

-

-

-

TE D

-

AC C

Present Analytical

S-S

EP

Present DTM

k

M AN U

11

ˆ ›3

Method

C-S

-

Table 5. Comparison of the nondimensional fundamental fequency for a S-S FG nanobeam with various gradient indexes when L/h=20. p=0 p =0.2 p =5 FEM Eltaher et al. (2012)

Present DTM

Present Analytical

FEM Eltaher et al. (2012)

9.8797 9.4238 9.0257 8.6741 8.3607 8.0789

9.8594 9.4062 9.0102 8.6603 8.3483 8.0677

9.8594 9.4062 9.0102 8.6603 8.3483 8.0677

8.7200 8.3175 7.9661 7.6557 7.3791 7.1303

Present Present DTM Analytical

8.6859 8.2866 7.9377 7.6295 7.3546 7.1075

8.6858 8.2865 7.9376 7.6294 7.3545 7.1074

FEM Eltaher et al. (2012)

Present DTM

Present Analytical

6.0025 5.7256 5.4837 5.2702 5.0797 4.9086

5.9373 5.6643 5.4258 5.2152 5.0273 4.8583

5.9370 5.6641 5.4256 5.2149 5.0271 4.8581

ACCEPTED MANUSCRIPT

1

2

3

4

RI PT

0

SC

z

Table 6. Effect of material inhomogeneity on first three dimensionless frequencies of a S-S FG nanobeam with different nonlocal parameters ∆4 # 0-K. when L/h=20. p=0 p =0.2 p =1 p =5 ˆ ›R Present Present Present Present Present Present Present Present DTM Analytical DTM Analytical DTM Analytical DTM Analytical 9.8594 8.6846 8.6845 7.0641 7.0638 6.0498 6.0496 ½ # 1 9.8594 34.6276 34.6263 28.1677 28.1627 24.1294 24.1270 ½ # 2 39.3171 39.3171 77.5013 77.4947 63.0483 63.0229 54.0320 54.0198 ½ # 3 88.0158 88.0158 9.4062 8.2854 8.2853 6.7393 6.7390 5.7717 5.7715 ½ # 1 9.4062 29.3204 29.3193 23.8505 23.8463 20.4312 20.4292 ½ # 2 33.2911 33.2911 56.3998 56.3950 45.8820 45.8635 39.3205 39.3117 ½ # 3 64.0515 64.0515 9.0102 7.9365 7.9365 6.4556 6.4553 5.5287 5.5286 ½ # 1 9.0102 25.8850 25.8841 21.0561 21.0524 18.0373 18.0356 ½ # 2 29.3905 29.3905 46.5112 46.5073 37.8375 37.8223 32.4265 32.4192 ½ # 3 52.8213 52.8213 8.6603 7.6284 7.6283 6.2050 6.2047 5.3140 5.3139 ½ # 1 8.6603 23.4294 23.4285 19.0586 19.0552 16.3262 16.3246 ½ # 2 26.6023 26.6023 40.4841 40.4806 32.9343 32.9211 28.2245 28.2181 ½ # 3 45.9765 45.9765 8.3483 7.3535 7.3535 5.9814 5.9811 5.1226 5.1224 ½ # 1 8.3483 21.5618 21.5610 17.5394 17.5363 15.0248 15.0234 ½ # 2 24.4818 24.4818 36.3210 36.3179 29.5476 29.5357 25.3221 25.3164 ½ # 3 41.2486 41.2486

2

3

4

#1 #2 #3 #1 #2 #3 #1 #2 #3 #1 #2 #3 #1 #2 #3

Present Analytical 9.5065 38.9700 87.6713 9.0356 32.8804 63.5773 8.6226 28.9245 52.2453 8.2563 26.0866 45.3136 7.9284 23.9205 40.5083

TE D

1

½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½

Present DTM 9.5065 38.9700 87.6713 9.0356 32.8804 63.5773 8.6226 28.9245 52.2453 8.2563 26.0866 45.3136 7.9284 23.9205 40.5083

p =0.2 Present Present DTM Analytical 8.3093 8.3092 34.2596 34.2584 77.1364 77.1298 7.8910 7.8910 28.8848 28.8837 55.8972 55.8925 7.5240 7.5239 25.3906 25.3897 45.9005 45.8966 7.1982 7.1981 22.8820 22.8812 39.7809 39.7775 6.9062 6.9062 20.9657 20.9649 35.5355 35.5325

EP

0

ˆ ›R

∆4 # 20-K.

p=0

AC C

z

M AN U

Table 7. Effect of material inhomogeneity on first three dimensionless frequencies of a S-S FG nanobeam with different nonlocal parameters with temperature effect ∆4 # 20-K. when L/h=20 Present DTM 6.6664 27.7797 62.6639 6.3212 23.3911 45.3522 6.0178 20.5342 37.1933 5.7481 18.4803 32.1922 5.5060 16.9093 28.7180

p =1 Present Analytical 6.6661 27.7749 62.6387 6.3209 23.3869 45.3340 6.0175 20.5305 37.1784 5.7478 18.4771 32.1793 5.5058 16.9063 28.7065

Present DTM 5.6476 23.7386 53.6448 5.3486 19.9680 38.7867 5.0854 17.5110 31.7770 4.8512 15.7428 27.4759 4.6406 14.3887 24.4849

p =5 Present Analytical 5.6474 23.7362 53.6328 5.3484 19.9661 38.7780 5.0853 17.5093 31.7699 4.8510 15.7412 27.4697 4.6405 14.3873 24.4794

Table 8. Effect of material inhomogeneity on first three dimensionless frequencies of a S-S FG nanobeam with different nonlocal parameters with temperature effect ∆4 # 40-K. when L/h =20

z 0

1

2

½ ½ ½ ½ ½ ½ ½

ˆ ›R

#1 #2 #3 #1 #2 #3 #1

Present DTM 9.1374 38.6173 87.3231 8.6463 32.4617 63.0962 8.2138

p =0 Present Analytical 9.1374 38.6173 87.3231 8.6463 32.4617 63.0962 8.2138

∆4 # 40-K.

p =0.2 Present Present DTM Analytical 7.9105 7.9105 33.8804 33.8792 76.7593 76.7528 7.4701 7.4700 28.4349 28.4339 55.3796 55.3749 7.0813 7.0812

Present DTM 6.2335 27.3724 62.2554 5.8631 22.9080 44.7956 5.5348

p =1 Present Analytical 6.2332 27.3676 62.2303 5.8629 22.9040 44.7776 5.5346

Present DTM 5.2021 23.3240 53.2279 4.8761 19.4762 38.2199 4.5861

p =5 Present Analytical 5.2019 23.3217 53.2159 4.8759 19.4742 38.2113 4.5859

ACCEPTED MANUSCRIPT

3

4

½ ½ ½ ½ ½ ½ ½ ½

#2 #3 #1 #2 #3 #1 #2 #3

28.4477 51.6588 7.8284 25.5568 44.6361 7.4818 23.3415 39.7490

28.4477 51.6588 7.8284 25.5568 44.6361 7.4818 23.3415 39.7490

24.8782 45.2704 6.7342 22.3124 39.0532 6.4212 20.3429 34.7196

24.8773 45.2665 6.7341 22.3116 39.0499 6.4211 20.3421 34.7166

19.9836 36.5167 5.2404 17.8676 31.4105 4.9738 16.2382 27.8408

19.9801 36.5020 5.2402 17.8645 31.3979 4.9736 16.2353 27.8296

16.9498 31.0880 4.3250 15.1173 26.6791 4.0877 13.7025 23.5894

16.9481 31.0810 4.3249 15.1158 26.6731 4.0876 13.7012 23.5841

3

4

5 9.4494 30.5286 63.3806 8.9466 25.5252 45.4631 8.5151 22.3819 37.3246 8.1397 20.1759 32.4280 7.8093 18.5186 29.0691

0 15.1386 49.4456 102.9310 14.2806 41.1953 73.5707 13.5418 35.9942 60.1873 12.8970 32.3309 52.1059 12.3276 29.5687 46.5420

5 9.1542 30.1940 63.0319 8.6073 25.0812 44.9227 8.1351 21.8481 36.6413 7.7216 19.5636 31.6248 7.3554 17.8355 28.1598

0 14.8707 49.1442 102.6170 13.9720 40.7945 73.0839 13.1955 35.5114 59.5706 12.5152 31.7760 51.3795 11.9121 28.9483 45.7178

SC

0 15.3997 49.7431 103.2410 14.5803 41.5895 74.0509 13.8770 36.4674 60.7934 13.2651 32.8727 52.8175 12.7267 30.1723 47.3463

∆4 # 40-K.

M AN U

2

#1 #2 #3 #1 #2 #3 #1 #2 #3 #1 #2 #3 #1 #2 #3

∆4 # 20-K.

Gradient index 0.2 1 13.2874 10.7409 43.4941 35.3046 90.5771 73.6074 12.5245 10.1101 36.2098 29.3544 64.6915 52.5064 11.8672 9.5660 31.6145 25.5959 52.8840 42.8682 11.2931 9.0902 28.3755 22.9431 45.7490 37.0363 10.7857 8.6691 25.9314 20.9386 40.8330 33.0125

TE D

1

½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½

∆4 # 0-K.

Gradient index 0.2 1 13.5647 11.0336 43.8094 35.6368 90.9059 73.9537 12.8428 10.4464 36.6277 29.7951 65.2004 53.0427 12.2233 9.9425 32.1164 26.1254 53.5266 43.5459 11.6843 9.5041 28.9504 23.5501 46.5036 37.8326 11.2100 9.1183 26.5721 21.6154 41.6862 33.9135

EP

0

ˆ ›R

AC C

z

RI PT

Table 9. Temperature and material graduation effect on first three dimensionless frequency of a C-S FG nanobeam with different nonlocality parameters (L/h =20). Gradient index 0.2 1 12.9994 10.4310 43.1693 34.9547 90.2361 73.2359 12.1926 9.7527 35.7790 28.8917 64.1673 51.9421 11.4944 9.1640 31.0958 25.0389 52.2214 42.1567 10.8817 8.6456 27.7792 22.3024 44.9688 36.1984 10.3375 8.1838 25.2643 20.2208 39.9475 32.0608

5 8.8381 29.8377 62.6520 8.2422 24.6103 44.3480 7.7237 21.2808 35.9166 7.2659 18.9102 30.7707 6.8570 17.1025 27.1886

ACCEPTED MANUSCRIPT Table 10. Temperature and material graduation effect on first three dimensionless frequency of a C-C FG nanobeam with different nonlocality parameters (L/h=20).

4

1 15.7954 43.6841 85.4118 14.8139 35.8672 60.0724 13.9749 31.0388 48.8324 13.2469 27.6781 42.1183 12.6071 25.1633 37.5188

5 13.4954 37.3801 73.1565 12.6442 30.6638 51.4135 11.9159 26.5103 41.7552 11.2832 23.6162 35.9790 10.7266 21.4481 32.0175

RI PT

5 13.7110 37.6706 73.4724 12.9323 31.1057 51.9686 12.2699 27.0706 42.4808 11.6980 24.2769 36.8457 11.1980 22.1978 33.0082

SC

3

1 16.0094 43.9727 85.7255 15.0997 36.3058 60.6233 14.3260 31.5946 49.5520 13.6581 28.3332 42.9774 13.0741 25.9062 38.5003

Fig. 1. Geometry and coordinates of functionally graded nanobeam.

(a) z # 0

∆4 # 40-K.

Gradient index 0 0.2 21.9585 19.2695 60.8590 53.4998 119.1110 104.7670 20.5591 18.0124 49.8880 43.7860 83.6450 73.4549 19.3612 16.9348 43.1002 37.7680 67.8843 59.5212 18.3199 15.9967 38.3677 33.5661 58.4528 51.1699 17.4031 15.1695 34.8204 30.4115 51.9800 45.4288

M AN U

2

#1 #2 #3 #1 #2 #3 #1 #2 #3 #1 #2 #3 #1 #2 #3

∆4 # 20-K.

Gradient index 0 0.2 22.1532 19.4789 61.1205 53.7827 119.3950 105.0770 20.8197 18.2924 50.2864 44.2147 84.1447 73.9935 19.6820 17.2796 43.6064 38.3119 68.5388 60.2245 18.6966 16.4017 38.9659 34.2088 59.2362 52.0112 17.8322 15.6311 35.5005 31.1425 52.8773 46.3927

TE D

1

½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½

∆4 # 0-K.

Gradient index 0 0.2 22.3447 19.6819 61.3790 54.0567 119.6770 105.3750 21.0751 18.5634 50.6790 44.6308 84.6381 74.5162 19.9954 17.6122 44.1033 38.8389 69.1826 60.9070 19.0632 16.7909 39.5510 34.8295 60.0041 52.8254 18.2482 16.0729 36.1633 31.8459 53.7537 47.3222

EP

0

ˆ ›R

AC C

z

(b) z # 1

1 15.5703 43.3779 85.0712 14.5131 35.4064 59.4921 13.6045 30.4548 48.0769 12.8113 26.9878 41.2148 12.1097 24.3773 36.4830

5 13.2663 37.0678 72.8074 12.3379 30.1947 50.8221 11.5383 25.9156 40.9857 10.8385 22.9126 35.0581 10.2181 20.6459 30.9604

RI PT

ACCEPTED MANUSCRIPT

(d) z # 3

TE D

M AN U

SC

(c) z # 2

(e) z # 4

AC C

EP

Fig. 2. Variations of the first dimensionless natural frequency of the S-S FG nanobeam with respect to temperature change for different values of gradient indexes and nonlocal parameters (L/h=50).

(a) z # 0

(b) z # 1

RI PT

ACCEPTED MANUSCRIPT

(d) z # 3

TE D

M AN U

SC

(c) z # 2

(e) z # 4

AC C

EP

Fig. 3. Variations of the second dimensionless natural frequency of the S-S FG nanobeam with respect to temperature change for different values of gradient indexes and nonlocal parameters (L/h=50).

(a) z # 0

(b) z # 1

RI PT

ACCEPTED MANUSCRIPT

(d) z # 3

TE D

M AN U

SC

(c) z # 2

(e) z # 4

AC C

EP

Fig. 4. Variations of the third dimensionless natural frequency of the S-S FG nanobeam with respect to temperature change for different values of gradient indexes and nonlocal parameters (L/h=50).

SC

RI PT

ACCEPTED MANUSCRIPT

(b) p=0.2

EP

TE D

M AN U

(a) p=0

AC C

(c) p=1

(d) p=5

Fig. 5. Variations of the first dimensionless natural frequency of the S-S FG nanobeam with respect to temperature change for different values of nonlocal parameters and gradient indexes (L/h=50).

M AN U

(b) ∆4 # 20-K.

EP

TE D

(a) ∆4 # 0-K.

SC

RI PT

ACCEPTED MANUSCRIPT

AC C

(c)∆4 # 40-K.

(d) ∆4 # 60-K.

Fig. 6. The variation of the first dimensionless frequency of S-S FG nanobeam with material graduation and nonlocality parameters for different temperatures (L/h=20).

RI PT

ACCEPTED MANUSCRIPT

(b) z # 1

TE D

M AN U

SC

(a) z # 0

(d) z # 3

AC C

EP

(c) z # 2

(e) z # 4

Fig. 7. The variation of the first dimensionless frequency of S-S FG nanobeam with material graduation and temperatures for different nonlocality parameters (L/h=20).

M AN U

(b) ∆4 # 20-K.

EP

TE D

(a) ∆4 # 0-K.

SC

RI PT

ACCEPTED MANUSCRIPT

AC C

(c) ∆4 # 40-K.

(d) ∆4 # 60-K.

Fig. 8. The variation of the first dimensionless frequency of C-S FG nanobeam with material graduation and nonlocality parameters for different temperatures (L/h=20).

SC

RI PT

ACCEPTED MANUSCRIPT

(b) z # 1

TE D

M AN U

(a) z # 0

(d) z # 3

AC C

EP

(c) z # 2

(e) z # 4

Fig. 9. The variation of the first dimensionless frequency of C-S FG nanobeam with material graduation and temperatures for different nonlocality parameters (L/h=20).

M AN U

(b) ∆4 # 20-K.

EP

TE D

(a) ∆4 # 0-K.

SC

RI PT

ACCEPTED MANUSCRIPT

AC C

(c) ∆4 # 40-K.

(d) ∆4 # 60-K.

Fig. 10. The variation of the first dimensionless frequency of C-C FG nanobeam with material graduation and nonlocality parameters for different temperatures (L/h=20).

RI PT

ACCEPTED MANUSCRIPT

(b) z # 1

TE D

M AN U

SC

(a) z # 0

(d) z # 3

AC C

EP

(c) z # 2

(e) z # 4 Fig. 11. The variation of the first dimensionless frequency of C-C FG nanobeam with material graduation and temperatures for different nonlocality parameters (L/h=20).