Nonlocal thermo-mechanical vibration analysis of functionally graded nanobeams in thermal environment

Author's Accepted Manuscript

Nonlocal thermo-mechanical vibration analysis of functionally graded nanobeams in thermal environment Farzad Ebrahimi, Erfan Salari

www.elsevier.com/locate/actaastro

PII: DOI: Reference:

S0094-5765(15)00131-9 http://dx.doi.org/10.1016/j.actaastro.2015.03.031 AA5399

To appear in:

Acta Astronautica

Received date: 21 January 2015 Revised date: 23 March 2015 Accepted date: 27 March 2015 Cite this article as: Farzad Ebrahimi, Erfan Salari, Nonlocal thermo-mechanical vibration analysis of functionally graded nanobeams in thermal environment, Acta Astronautica, http://dx.doi.org/10.1016/j.actaastro.2015.03.031 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 galley proof before it is published in its final citable 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.

Nonlocal thermo-mechanical vibration analysis of functionally graded nanobeams in thermal environment Farzad Ebrahimi*,Erfan Salari Department of Mechanical Engineering, Faculty of Engineering, Imam Khomeini International University, Qazvin, Iran * Corresponding author email: [email protected] (F. Ebrahimi).

Abstract In this paper, the thermal effect on free vibration characteristics of functionally graded (FG) size-dependent nanobeams subjected to various types of thermal loading is investigated by presenting a Navier type solution and employing a semi analytical differential transform method (DTM) for the first time. Two kinds of thermal loading, namely, linear temperature rise and nonlinear temperature rise are studied. Material properties of FG nanobeam are supposed to vary continuously along the thickness according to the powerlaw form and the material properties are assumed to be temperature-dependent. 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 temperature-dependent FG nanobeams in detail. It is explicitly shown that the vibration behaviour 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: Thermal effect, Free vibration, Functionally graded nanobeam, Eringen’s nonlocal elasticity, DT method. 1. Introduction 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.1, 2 Presenting novel properties, FGMs have also attracted intensive research interests, which were mainly focused on their static, dynamic and vibration characteristics of FG structures.3, 4 Since functionally graded structures are most commonly used in high temperature environment, in the last decade, beams and plates made of FGMs have found wide applications as structural elements in modern industries such as aeronautics/astronautics 1

manufacturing industry, mechanical and engine combustion chamber, nuclear engineering and reactors. Also, FGMs will allow a tailored fit to a special purpose, i.e., with the static deflection not exceeding a specific level, or the buckling load not being less than a pre-specified level, or the natural frequency either exceeding or to being less than a pre-specified frequency. Thus, it is important to investigate and understand behaviour of the functionally graded materials due to difference between the making temperatures and working temperature of structures, for appropriate design. Moreover, structural components 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.5 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 applications6-8 such as sensors, actuators, 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. Since conducting experiments at the nanoscale is a daunting task, and atomistic modeling is restricted to small-scale 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 Eringen9 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.10 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.11 proposed a version of nonlocal elasticity theory which is employed to develop a nonlocal Euler beam model. Wang and Liew12 carried out the static analysis of micro- and nano-structures based on nonlocal continuum mechanics using Euler-Bernoulli beam theory and Timoshenko beam theory. Aydogdu13 proposed a generalized nonlocal beam theory to study bending, buckling, and free vibration of nanobeams based on Eringen model using different beam theories. Phadikar and Pradhan14 reported finite element formulations for nonlocal elastic Euler–Bernoulli beam and Kirchhoff plate. Pradhan and Murmu15 investigated the flapwise bending– vibration characteristics of a rotating nanocantilever by using Differential quadrature method (DQM). They 2

noticed that small-scale effects play a significant role in the vibration response of a rotating nanocantilever. Civalek et al.16 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 Demir17 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. Micro-electro-mechanical Systems, or MEMS, is a technology in its most general form can be defined as miniaturized mechanical and electro- mechanical elements (i.e., devices and structures) that are made using the techniques of microfabrication. Furthermore, with the development of the material technology, FGMs have also been employed in MEMS/NEMS. 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. 18, 19 Recently, the application of FGMs has been widely extended in micro- and nano-scale devices and systems such as thin films20, atomic force microscopes (AFMs)21, shape memory alloy, micro sensors, micro piezoactuator and nano-motors22, 23 which may experience vibration behavior. The bending deformations in the FG nanobeams are obtained by prescribing relatively high temperature environments through the thickness of the beams, which are relevant for actuator applications, such as high-precision controls in atomic force microscope and laser mirror alignment, and active vibration controls. In addition, all of the micro sensors and nano-motors are exposed to temperature changes. Nanobeams are the core structures widely used in MEMS, NEMS and AFMS with the order of microns or sub-microns, and their properties are closely related to their microstructures. On the other hands, FG nanobeams subjected to thermal environment are important structural members and hence, their thermo-mechanical vibration characteristic is of great interest for engineering design and accurate prediction of the mechanical response. Moreover, advanced nanocomposite materials propose numerous superior properties with respect to their constitutive elements, like high strength and stiffness. As a result, there will be many potential applications for nanocomposite materials, particularly in engineering and military aids. For example, a nanoscale layer of a ceramic material can be bonded to the surface of a metallic nanostructure to form a thermal barrier nanocoating in high-temperature applications. However, a sudden variation of material properties across the interface between the disparate materials of the nanocomposite can bring about large interlaminar stresses. Such extreme stresses would result in plastic deformation and cracking. Sequentially, such phenomena would hardly endanger the service life as well as the stability of the nanocomposite structure. An alternative strategy to overcome such a deficiency may be to exploit FG nanobeams. 3

Also, FG nanobeams are innovative nanocomposite materials that are designed to have a smooth variation of material properties across particular dimensions. This can be attained by fabricating the nanocomposite material to have a gradually spatial variation of the relative volume fractions as well as microstructure of its constitutive materials. Actually, the choice of material phases is affected by functional performance necessities. For instant, in a nanoscale metal/ceramic FGM structure, the metal-rich side is typically located in regions, where the toughness should to be high. In contrast, the ceramic-rich side is commonly placed in regions with high temperature gradients. The expected advantages of miniaturized FGMs to nanocomposites are: magnification the interface bond strength through introducing a continuous gradation of composition, elimination of deleterious stress concentrations at the intersection of interfaces, reduction of the peak thermal stresses, and enhancement fracture toughness of brittle ceramics by implementing a metallic phase. Due to the light weight but high stiffness of these small scale structures, they are ideal choice to be applied in space. For instance, through reducing weight, improving functionality and increasing durability, nanotechnology has changes radically the way NASA performs missions in aeronautics and space. From this point of view, aerospace structures often experience severe loading conditions. During their operations, life aerospace structures are subjected to both aerodynamic loads, which depend on aerodynamic pressure distributions and viscous forces, and aerothermal effects, which take into account surface heating-rate and inner temperature distributions. The rise of temperature due to aerodynamic heating strongly decreases the structural load-bearing capacity affecting the buckling phenomena. It becomes mandatory for design engineers to carry out their thermal vibration analysis to prevent failures. Thermo-mechanical vibration is a phenomenon that should be checked for the stability of structural design problems. Consequently, thermal vibration analysis of beam structures are common in structural mechanics. Thus, establishing an accurate model of FG nanobeams is a key issue for successful NEMS design. Fallah and Aghdam24 used Euler-Bernoulli beam theory to investigate thermos-mechanical buckling and nonlinear vibration analysis of functionally graded beams on nonlinear elastic foundation. The nonlinear static response of FGM beams under in-plane thermal loading is studied by Ma and Lee25. Asghari et al.26, 27 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. The dynamic characteristics of FG beam with power law material graduation in the axial or the transversal directions was examined by Alshorbagy et al.28. Ke and Wang29 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.30 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 of 4

lower slenderness ratios. In another study, vibration and buckling analysis of functionally graded nanoplates subjected to linear thermal loading based on surface elasticity theory and classical plate theory is studied by Ansari et al.31. Employing modified couple stress theory the nonlinear free vibration of FG microbeams based on von-Karman geometric nonlinearity was presented by Ke et al.32 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. Recently, Eltaher et al.33, 34 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.35 More recently, using nonlocal Timoshenko and Euler–Bernoulli beam theory, Simsek and Yurtcu36 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. 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. Also, these structures are often subjected to severe thermal environments during manufacturing and working, and thus the thermal effects become a primary design factor in specific cases. It is well known that the temperature rise leads to the reduction in the stiffness of the microbeams due to softening of the materials as well as the development of thermal stresses due to the thermal expansion. These effects in turn cause a quite significant change in the static and dynamic behaviors of the nanobeams. 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 linear and nonlinear temperature rises are analyzed. The lateral deformations in the FG nanobeams are due to applications of temperature rise through the thickness of the beams. Temperature change is one of the basic actuation parameters that can tune the system directly. Thermal actuation is known for its capability in producing a linear displacement with respect to a heating power. This mechanism is derived using a FG nanobeam including variable thermal expansion coefficient along its thickness. The usually used thermal boundary conditions for FG beams analyses in the literature are the prescribed temperature at the top and bottom surfaces of these structural elements. Also, it is assumed that the temperature rise occurs in the thickness direction only and one dimensional temperature field is assumed to be constant in the xy plane of the FG 5

nanobeam. 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 different thermal effects, constituent volume fractions, mode number, slenderness ratios, boundary conditions and small scale on vibration characteristics of FG nanobeams. Comparisons with analytical solutions 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 power-law FG nanobeam equations based 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 E f , thermal conductivity κ f , thermal expansion α f and mass density ρ f are assumed to vary continuously in the thickness direction (z-axis direction) according to a power function of the volume fractions of the constituents. According to the rule of mixture, the effective material properties, Pf , can be expressed as36:

Pf = PcV c + PmV m

(1)

6

Where Pm , Pc , Vm and Vc are the material properties and the volume fractions of the metal and the ceramic constituents related by:

Vc + Vm = 1

(2.a)

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

z 1 Vc = ( + ) P h 2

(2.b)

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 such as Young’s modulus ( E ), mass density ( ρ ), thermal expansion ( α ) and thermal conductivity ( κ ) can be expressed as follows: p

 z 1 +  + Em h 2

E ( z ) = ( Ec − Em ) 

p

 z 1 ρ ( z ) = ( ρc − ρm )  +  + ρm h 2

(3)

p

 z 1 α ( z ) = (α c − α m )  +  + α m h 2 p

κ ( z ) = (κ c − κ m )  +  + κ m h 2 z

1





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 as37: P = P0 ( P−1 T −1 + 1 + P1 T + P2 T 2 + P3 T 3 )

(4)

Where P0 , P−1 , P1 , P2 and P3 are the temperature dependent coefficients which can be seen in the table of materials properties (Table 1) for Si3N4 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( Si3N4 ).

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:

u x ( x, z , t ) = u ( x, t ) − z

∂w( x, t ) ∂x

(5.a)

7

u z ( x, z , t ) = w( x, t )

(5.b)

Where t is time, u and w are displacement components 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 strain is:

ε xx

∂u ( x, t ) ∂ 2 w( x, t ) 0 = ε − zk , ε = , k = ∂x ∂x 2 0 xx

0 x

0 xx

(6)

0 Where ε xx and k 0 are the extensional strain and bending strain respectively. Based on the Hamilton’s

principle, which states that the motion of an elastic structure during the time interval t1 < t < t 2 is such that the time integral of the total dynamics potential is extremum38: t

∫ δ (Π 0

S

− Π K + Π V ) dt = 0

(7)

Here Π S is strain energy, Π K is kinetic energy and Π V is work done by external forces. The virtual strain energy can be calculated as:

δ Π S = ∫ σ ijδ ε ij dV = ∫ (σ xxδ ε xx ) dV v

(8)

v

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

δ Π S = ∫ ( N (δε xx0 ) − M (δ k 0 )) dx

(9)

0

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

N = ∫ σ xx dA, M = ∫ σ xx z dA A

(10)

A

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

ΠK =

∂u 1 L ∂u ρ ( z ) ( ( x )2 + ( z ) 2 ) dA dx ∫ ∫ 2 0 A ∂t ∂t

(11)

Also the virtual kinetic energy can be expressed as: L ∂u ∂δ u ∂w ∂δ w ∂u ∂ 2δ w ∂δ u ∂ 2 w ∂ 2 w ∂ 2δ w  + ) − I1 ( + ) + I2 δ Π K = ∫  I0 (  dx 0 ∂t ∂t ∂t ∂t ∂x ∂t ∂t ∂x ∂t ∂x ∂t ∂x   ∂t ∂t

(12)

Where ( I 0 , I1 , I 2 ) are the mass moment of inertias, defined as follows:

( I 0 , I1 , I 2 ) = ∫ ρ ( z )(1, z, z 2 ) dA

(13)

A

For a typical FG nanobeam which has been in temperature environment for a long period of time, it is assumed that the temperature can be distributed across the thickness so that the two case of linear

8

temperature rise and nonlinear temperature rise (heat conduction) are taken into consideration. Hence, the first variation of the work done corresponding to temperature change can be written in the form: L

δ ΠV = ∫ NT 0

∂w ∂δ w dx ∂x ∂x

(14)

Where N T is thermal resultant can be expressed as:

NT = ∫

h /2

− h /2

E ( z ) α ( z ) (T − T0 ) dz

(15)

Where T0 is the reference temperature. By Substituting Eqs. (9),(12) and (14) into Eq. (7) and setting the coefficients of δ u , δ w and

δ ∂w ∂x

to zero, the following Euler–Lagrange equation can be obtained:

∂N ∂ 2u ∂3 w = I 0 2 − I1 ∂x ∂t ∂x∂t 2 2 ∂2M ∂2 w ∂ 3u ∂4 w T ∂ w − N = I + I − I 0 1 2 ∂x 2 ∂x 2 ∂t 2 ∂x∂t 2 ∂x 2 ∂t 2

(16.a) (16.b)

Under the following boundary conditions:

N = 0 or u = 0 at x = 0 and x = L ∂M ∂ 2u ∂3 w − I1 2 + I 2 = 0 or w = 0 at x = 0 and x = L ∂x ∂t ∂x∂t 2 ∂w M = 0 or = 0 at x = 0 and x = L ∂x

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

2.3. The nonlocal elasticity model for FG nanobeam Based on Eringen nonlocal elasticity model39, 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 σ ij at any point x in the body can be expressed as:

σ ij ( x) = ∫ α ( x′ − x ,τ ) tij ( x′) d Ω ( x′)

(18)

Ω

Where tij ( x′) are the components of the classical local stress tensor at point x which are related to the components of the linear strain tensor ε kl by the conventional constitutive relations for a Hookean material, i.e:

tij = Cijklε kl

(19)

9

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 α ( x′ − x ,τ ) . Here

x′ − x is the Euclidean distance and τ is a constant given by:

τ = e0 a

(20)

l

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, e0 , dependent on each material. The magnitude of e0 is determined experimentally or approximated by matching the dispersion curves of plane waves with those of atomic lattice dynamics. According to nonlocal theory for a class of physically admissible kernel α ( x′ − x ,τ ) it is possible to represent the integral constitutive relations given by Eq. (18) in an equivalent differential form as: (1 − (e0a) 2 ∇ 2 )σ kl = tkl

(21)

Where ∇ 2 is the Laplacian operator. Thus, the scale length e0 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 :

σ ( x) − (e0a)2 ∂ σ (2x) = Eε ( x) 2

∂x

(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:

σ xx − µ

∂ 2σ xx = E ( z ) ε xx ∂x 2

(23)

Where ( µ = (e0 a ) 2 ). 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:

∂2 N ∂u ∂2w = A − B xx xx ∂x 2 ∂x ∂x 2 ∂2M ∂u ∂2w M −µ = B − C xx xx ∂x 2 ∂x ∂x 2 N −µ

(24) (25)

In which the cross-sectional rigidities are defined as follows:

( Axx , Bxx , Cxx ) = ∫ E ( z ) (1, z , z 2 ) dA A

(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:

10

∂u ∂2 w ∂ 3u ∂4 w − Bxx 2 + µ ( I 0 − I1 2 2 ) N = Axx ∂x ∂x ∂x∂t 2 ∂x ∂t

(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:

M = Bxx

2 ∂u ∂2w ∂2w ∂ 3u ∂4w T ∂ w − C xx 2 + µ ( I 0 2 + I1 − + I N ) 2 ∂x ∂x ∂t ∂x∂t 2 ∂x 2∂t 2 ∂x 2

(28)

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:

Axx

 ∂ 2u ∂3w ∂ 4u ∂5 w  ∂ 2u ∂3w B I I I I − + µ − − + =0  0 2 2 1 2 3 0 2 xx 1 ∂x 2 ∂x 3 ∂t ∂x  ∂t ∂t 2∂x  ∂t ∂x

2 4 ∂ 3u ∂4w ∂4w ∂ 5u ∂6w T ∂ w T ∂ w Bxx 3 − Cxx 4 − N + µ ( I 0 2 2 + I1 2 3 − I 2 2 4 + N ) ∂x ∂x ∂x 2 ∂t ∂x ∂t ∂x ∂t ∂x ∂x 4 ∂2w ∂ 3u ∂4w − I 0 2 − I1 2 + I 2 2 2 = 0 ∂t ∂t ∂x ∂t ∂x

(29)

(30)

3. Solution method 3.1. Analytical Solution 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: ∞

nπ iω t x) e n L n =1 ∞ nπ iω t w( x, t ) = ∑ Wn sin ( x) e n L n =1

u ( x, t ) = ∑ U n cos (

(31) (32)

Where ( U n , Wn ) are the unknown Fourier coefficients to be determined for each n value. Boundary conditions for simply supported beam are as Eq. (33) :

∂u ( L) = 0 ∂x ∂2w ∂2w w(0) = w( L) = 0 , (0) = ( L) = 0 ∂x 2 ∂x 2

u (0) = 0 ,

(33)

Substituting Eqs. (31) and (32) into Eqs. (29) and (30) respectively, leads to following relations:

( − Axx (

nπ 2 nπ nπ nπ nπ ) + I 0 (1 + µ ( ) 2 )ωn 2 ) U n + ( Bxx ( )3 − I1 ( ( ) + µ ( )3 ) ωn 2 ) Wn = 0 L L L L L 11

(34)

nπ 3 nπ nπ nπ nπ nπ ) − I1 ( ( ) + µ ( )3 )ωn 2 ) U n + (− Cxx ( ) 4 + N T ( )2 (1 + µ ( ) 2 ) L L L L L L nπ 2 2 nπ 2 nπ 4 2 + I0 (1 + µ ( ) )ωn + I2 ( ( ) + µ ( ) )ωn ) Wn = 0 L L L

( Bxx (

(35)

By setting the determinant of the coefficient matrix of the above equations, the analytical solutions can be obtained from the following equations:

U n 

{[ K ] − ω [M ]} W  = 0 2



n

(36)



Where [ K ] and [ M ] are stiffness and mass matrix, respectively. By setting the determinant of the above equations, a quadratic polynomial for ωn 2 can be obtained. By setting this polynomial to zero, we can find natural frequencies ωn .

3. 2. Implementation of differential transform method 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 follows40:

Y (k) =

 1  dk  k y ( x ) k!  dx  x =0

(37)

∞

y ( x ) = ∑x k Y(k)

(38)

0

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

y(x) =

 xk  dk ∑  k y ( x ) k = 0 k!  dx  x =0 ∞

(39)

Equation (39) reveals that the concept of the differential transformation is derived from Taylor’s series expansion. In real applications the function y(x) in equation (39) can be written in a finite form as:

12

N

y ( x ) = ∑x k Y(k)

(40)

k =0

In this calculations y ( x ) =

∞

∑x Y(k) is small enough to be neglected, and N is determined by the k

n +1

convergence of the eigenvalues. From the definitions of DTM in Equations (37)-(39), 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. A sinusoidal variation of u ( x, t ) and w( x, t ) with a circular natural frequency ω is assumed and the functions are approximated as:

u ( x, t ) = u ( x) eiω t

(41a)

w( x, t ) = w( x) eiω t

(41b)

Substituting Eqs. (41a) and (41b) into Eqs. (29) and (30), equations of motion can be rewritten as follows:

Axx

2 3  ∂ 2u ∂3w ∂w 2 ∂ u 2 ∂ w − B + µ − I ω + I ω + I 0 ω 2u − I1 ω 2 =0  0 xx 1 2 3 2 3  ∂x ∂x ∂x ∂x  ∂x 

2 2 3 4 4 ∂ 3u ∂4w T ∂ w 2 ∂ w 2 ∂ u 2 ∂ w T ∂ w − C − N + µ ( − I ω − I ω + I ω + N ) xx 0 1 2 ∂x3 ∂x 4 ∂x 2 ∂x 2 ∂x 3 ∂x 4 ∂x 4 2 2 2 ∂u 2 ∂ w + I 0 ω w + I1 ω − I2 ω =0 ∂x ∂x 2

(42a)

Bxx

(42b)

According to the basic transformation operations introduced in Table 2, the transformed form of the governing equations (42a) and (42b) around x0 = 0 may be obtained as:

Axx ( k + 1)( k + 2 ) U [k + 2] − Bxx ( k + 1)( k + 2 )( k + 3) W [k + 3] − I 0ω 2 ( −U [k ] + µ ( k + 1)( k + 2 ) U [k + 2] − I1ω 2 (− µ ( k + 1)( k + 2 )( k + 3) W [k + 3] + ( k + 1) W [k + 1]) = 0

(43)

Bxx ( k + 1)( k + 2 )( k + 3 ) U [k + 3] − Cxx ( k + 1)( k + 2 )( k + 3)( k + 4 ) W [k + 4] − N T (k + 1)(k + 2) W [k + 2] − I 0ω 2 ( −W [k ] + µ ( k + 1)( k + 2 ) W [k + 2]) − I1ω 2 ( − ( k + 1) U [k + 1] + µ ( k + 1)( k + 2 )( k + 3) U [k + 3] − I 2ω 2 ( − µ ( k + 1)( k + 2 )( k + 3 )( k + 4 ) W [k + 4] + ( k + 1)( k + 2 ) W [k + 2]) + µ N T (k + 1)(k + 2)(k + 3)(k + 4) W [k + 4] = 0

(44)

Where U [k ] and W [ k ] 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:

W [ 0] = 0, W [ 2] = 0, U [ 0] = 0 ∞

∞

∞

k =0

k =0

k =0

∑ W [k ] = 0, ∑ k (k − 1) W [k ] = 0, ∑ k U [k ] = 0 13

(45.a)

Clamped–Clamped:

W [ 0] = 0, W [1] = 0, U [ 0] = 0 ∞

∞

∞

k =0

k =0

k =0

∑ W [k ] = 0, ∑ k W [k ] = 0, ∑ U [k ] = 0

(45.b)

Clamped–Simply supported:

W [ 0] = 0, W [1] = 0, U [ 0] = 0 ∞

∞

∞

k =0

k =0

k =0

∑ W [k ] = 0, ∑ k (k − 1) W [k ] = 0, ∑ k U [k ] = 0

(45.c)

By using Eqs. (43) and (44) together with the transformed boundary conditions one arrives at the following eigenvalue problem:

 M 11 (ω ) M 12 (ω ) M 13 (ω )   M (ω ) M (ω ) M (ω )  C = 0 22 23  21 [ ]  M 31 (ω ) M 32 (ω ) M 33 (ω ) 

(46)

Where [C ] correspond to the missing boundary conditions at x = 0. For the non-trivial solutions of Eq. (46), it is necessary that the determinant of the coefficient matrix set equal to zero:

M 11 (ω )

M 12 (ω )

M 13 (ω )

M 21 (ω ) M 22 (ω ) M 23 (ω ) = 0 M 31 (ω ) M 32 (ω ) M 33 (ω )

(47)

Solution of Eq. (47) 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. Solving equation (47), the ith estimated eigenvalue for nth iteration ( ω = ω (i n ) ) may be obtained and the total number of iterations is related to the accuracy of calculations which can be determined by the following equation:

ωi( n ) − ωi( n −1) < ε

(48)

In this study ε=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 high-order 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.

14

4. Linear and nonlinear temperature rises 4.1. Linear temperature rise (LTR) Assume a FG nanobeam where the temperature of the top surface (ceramic-rich) is Tc and it is considered to vary linearly along the thickness from Tc to the bottom surface (Metal-rich) temperature Tm . Therefore, the temperature rise as a function of thickness is obtained as41:   T = Tm + ( ∆T )  1 + z  2 h

(49)

The ∆T in Eq. (49) could be defined ∆T = Tc − Tm .

4.2. Nonlinear temperature rise (NLTR) In this case, it is assumed that the nonlinear temperature distribution varies across the thickness of the FGM beam and no heat generation source exists within the FG nanobeam.The steady-state onedimensional heat conduction equation with the known temperature boundary conditions on bottom and top surfaces of the FG nanobeam can be obtained by solving the following equation42, 43 : −

d  dT  κ ( z) =0 dz  dz 

h  h T   = Tc , T  −  = Tm 2  2

(50)

The solution of Eq. (50) subjected to the boundary conditions can be solved by the following equation:

1

z

T = Tm + (∆T )

∫− h2 κ ( z ) dz h 2 −h 2

∫

1 dz κ ( z)

(51)

Where ∆T = Tc − Tm .

5. Numerical results and discussions 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. To demonstrate the nonlocal parameter effects on the thermo-mechanical vibration of FG nanobeams, variations of the natural frequencies versus temperature change, power law index, and thickness ratios of the FG nanobeam for different boundary conditions, are presented in this section. The functionally graded nanobeam is composed of Steel ( SUS304 ) and Silicon nitride ( Si3N4 ) 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 pure Silicon nitride. The beam geometry has the following dimensions: L (length) = 10,000 nm, b (width) = 1000 nm and h(thickness) = 100 nm. It is assumed that the temperature increase in metal surface to reference themperature T0 of the FG nanobeam is Tm − T0 = 5K .41

15

Relation described in equation (52) are performed in order to calculate the non-dimensional natural frequencies:

ωˆ = ωL2 ρ c A / EIc

(52)

Where I = b h 3 / 12 is the moment of inertia of the cross section of the beam. Table 4 shows the convergence study of DT method for first dimensionless natural frequency of FG nanobeam with various gradient indexes and boundary conditions. It is found that in DT method after a certain number of iterations eigenvalues converged to a value with good precision, so the number of iterations is important in DT method convergence. From results of Table 4, high convergence rate of the method may be easily observed and it may be deduced that the natural frequency of S-S FG nanobeam converged after 15 iterations with 4 digit precision while the first natural frequency of C-S FG nanobeam and C-C nanobeam converged after 19 and 21 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 and gradient indexes previously analyzed by finite element method are reexamined. Table 5 compares the results of the present study and the results presented by Eltaher et al.33 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; 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 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 at 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, S-C stands for a FG beam with simply-supported edge at x=0 and clamped edge at x=L. In Tables 6, 7 and 8, a comparison between first dimensionless natural frequency of the simply supported FG nanobeams are presented for various values of the gradient index ( p =0.1, 0.2, 0.5, 1), nonlocal parameters ( µ =0, 1, 2, 3, 4) and three different values of temperature changes subjected to linear (LTR) and nonlinear (NLTR) temperature rises ( ∆T =20, 40, 80 K ) for L / h = 20 based on both DTM and analytical Navier solution method. It can be concluded from the results of the tables that an increase in nonlocal scale parameter

16

gives rise to a decrement in the first dimensionless natural frequency, at a constant material graduation index. This results indicates that the effect of nonlocal parameter soften the FG nanobeam. It can also be seen that the dimensionless natural frequencies predicted by DT method are in close agreement with those evaluated by analytical solution. In addition, it is seen that the first dimensionless natural frequencies decrease by increasing temperature change (both LTR and NLTR) and it can be stated that temperature change has a significant effect on the dimensionless natural frequencies, especially for lower mode numbers. On the other hand, it can be concluded that the dimensionless natural frequencies decrease with an increase in material gradient index due to increasing in ceramics phase constituent, and hence, stiffness of the beam and this trend can be seen for higher modes either. Figs. 2, 3 and 4 demonstrate the variation of first three dimensionless natural frequencies with changing of the nonlocality parameter at constant slenderness ratio ( L / h = 50 ) of FG nanobeam with simplysupported edge conditions and different material distribution under nonlinear temperature rise. It is observed from the results of figures 2-4 that the dimensionless natural frequencies of FG nanobeam decrease with the increase of temperature until it approaches to the critical buckling temperature. Also it is seen that, small scale parameter, has a softening effect on nonlocal FG beam at pre-buckling region and a rise in small scale increases this effect. Variations of the first three dimensionless natural frequencies of the simply supported FG nanobeams with respect to linear temperature changes for different values of gradient indexes and nonlocal parameters are depicted in Figs. 5, 6 and 7, respectively. As known, thermal stress due to the temperature rise in micro/nano-beams produces compressive axial force which can lead to buckling the beams if its value increases over the critical value. If a high external pressure is applied to the FG nanobeam structure, the high stresses occurred in the structure will affect its integrity and the structure is susceptible to failure. From this point of view, it is seen from the figures that the dimensionless frequencies of FG nanobeam decreases with the increase of temperature until it reaches to zero at the critical temperature point. This is due to the reduction in total stiffness of the beam, since geometrical stiffness decreases when temperature rises. One important observation within the range of temperature before the critical temperature, it is seen that 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.

17

The effect of material graduation on the first dimensionless frequency of simply supported FG nanobeam with different linear temperature distributions and nonlocality parameters for ( L / h = 20 ) has been shown in Figure 8. It is seen from this figure that frequencies predicted by the the nonlocal solution is smaller than the those of the local frequency due to the small scale effects. For this reason, the size effect has a major role on vibration behavior of nonlocal FG beams. Temperature and material graduation effects on first dimensionless frequency (fundamental frequency) of a C-C FG nanobeam under the both temperature rises with different nonlocality parameters are presented in Table 9. It can be concluded from the results of the table that increasing the nonlocality parameter yields the reduction in dimensionless frequencies for every material graduation and temperature change, which highlights the significance of the nonlocal effect. The fundamental frequency parameter as a function of power law indexes for various linear temperature rises and nonlocality parameters is presented in Figure 9 for the FG nanobeam with C–C boundary condition. Observing this figure, it is easily deduced for a C-C FG nanobeam that, an increase in nonlocal scale parameter gives rise to a decrease in the first dimensionless natural frequency for all gradient indexes. In addition, it is deduced that the fundamental frequency decreases by increasing temperature changes and it can be stated that linear temperature change has a significant effect on the fundamental frequency of the graded C-C 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 reduction in the value of the elasticity modulus. Comparing the frequency values for FG nanobems 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 C-S and S-S, respectively. The last analysis deals with the linear and nonlinear temperature rises and material graduation effect on vibration behaviour of C-S FG nanobeam with different nonlocality parameters. Table 10 contain the results of different cases including the temperature rise effect for different material inhomogeneity and nonlocal parameters which influence the first dimensionless natural frequency of C-S functionally graded microstructure-dependent nanobeam. Also the variation of the first dimensionless frequency of C-S FG nanobeam with material graduation and nonlocality parameters for different linear temperature rises are depicted in Figure 10. Similar to two previously discussed edge conditions, it is revealed that for a C-S FG nanobeam increasing temperature change, power law index, and nonlocal

18

parameter, leads to decrease in natural frequency. Also, as it is expected, a beam with stiffer edges; i.e. CC and C-S, respectively shows higher natural frequencies.

6. Conclusions Thermo-mechanical vibrational behavior of the size-dependent FG nanobeams subjected to linear and nonlinear temperature rises through the thickness direction with various boundary conditions is investigated on the basis of nonlocal elasticity theory in conjunction with differential transform method. Eringen’s theory of nonlocal elasticity together with Euler–Bernoulli beam theory are used to model the nanobeam. Thermo-mechanical properties of the FG nanobeams are assumed to be functions of both temperature and thickness. The governing differential equations and related boundary conditions with both thermal distributions are derived by implementing Hamilton’s principle. Implementing 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 parametric study and numerical examples, the effect of different parameters are investigated. The effects of small scale parameter, material property gradient index, mode number, different temperature change and boundary conditions on fundamental frequencies of FG nanobeams are investigated. It is concluded that various factors such as nonlocal parameter, gradient index, temperature-dependent material properties, thermal effect and boundary conditions play important roles in dynamic behavior of FG nanobeams. The FG nanobeam model produces smaller natural frequency than the classical 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 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 the both temperature rises, with the increase in the gradient index value leads to the decrease in frequency; however, the trend is reversed after the pre-buckling stage is passed. Moreover it is revealed that under the linear and nonlinear temperature rises, the greatest frequency at pre-buckling region is obtained for the FG beam with C–C boundary conditions followed with S-C and S-S, respectively.

19

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

22. 23.

Ebrahimi, F., and A. Rastgoo. "Free vibration analysis of smart annular FGM plates integrated with piezoelectric layers." Smart Materials and Structures 17.1 (2008): 015044. Ebrahimi, Farzad, and Abbas Rastgo. "An analytical study on the free vibration of smart circular thin FGM plate based on classical plate theory." Thin-Walled Structures 46.12 (2008): 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 (2009): 962973. Ebrahimi, Farzad, Mohammad Hassan Naei, 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 (2009): 2107-2124. Iijima, Sumio. "Helical microtubules of graphitic carbon." nature 354.6348 (1991): 56-58. Zhang, Y. Q., G. R. Liu, and J. S. Wang. "Small-scale effects on buckling of multiwalled carbon nanotubes under axial compression." Physical review B 70.20 (2004): 205430. Wang, Q. "Wave propagation in carbon nanotubes via nonlocal continuum mechanics." Journal of Applied Physics 98.12 (2005): 124301. Wang, Q., and V. K. Varadan. "Vibration of carbon nanotubes studied using nonlocal continuum mechanics." Smart Materials and Structures 15.2 (2006): 659. Eringen, A. Cemal. "Nonlocal polar elastic continua." International Journal of Engineering Science 10.1 (1972): 1-16. Eringen, A. Cemal. "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves." Journal of Applied Physics 54.9 (1983): 4703-4710. 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. Wang, Q., and K. M. Liew. "Application of nonlocal continuum mechanics to static analysis of micro-and nano-structures." Physics Letters A 363.3 (2007): 236-242. 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. Phadikar, J. K., and S. C. Pradhan. "Variational formulation and finite element analysis for nonlocal elastic nanobeams and nanoplates." Computational Materials Science 49.3 (2010): 492-499. 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): 1944-1949. Civalek, Ömer, Cigdem Demir, and Bekir Akgöz. "Free vibration and bending analyses of cantilever microtubules based on nonlocal continuum model." Math. Comput. Appl 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. Witvrouw, Ann, and A. Mehta. "The use of functionally graded poly-SiGe layers for MEMS applications." Materials Science Forum. Vol. 492. 2005. Lee, Z., C. Ophus, L. M. Fischer, N. Nelson-Fitzpatrick, K. L. Westra, S. Evoy, V. Radmilovic, U. Dahmen, and D. Mitlin. "Metallic NEMS components fabricated from nanocomposite Al–Mo films." Nanotechnology 17, no. 12 (2006): 3063. Lü, C. F., Lim, C. W., & Chen, W. Q. (2009). Size-dependent elastic behavior of FGM ultra-thin films based on generalized refined theory. International Journal of Solids and Structures, 46(5), 1176-1185. Rahaeifard, M., Kahrobaiyan, M. H., & Ahmadian, M. T. (2009, January). Sensitivity analysis of atomic force microscope cantilever made of functionally graded materials. In ASME 2009 international design engineering technical conferences and computers and information in engineering conference (pp. 539-544). American Society of Mechanical Engineers. Carbonari, R. C., Silva, E. C., & Paulino, G. H. (2009). Multi‐actuated functionally graded piezoelectric micro‐tools design: A multiphysics topology optimization approach. International Journal for Numerical Methods in Engineering, 77(3), 301-336. Lun, F. Y., Zhang, P., Gao, F. B., & Jia, H. G. (2006). Design and fabrication of micro optomechanical vibration sensor. Microfabrication Technology, 120(1), 61-64.

20

24. 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): 1523-1530. 25. Ma, L. S., and D. W. Lee. "Exact solutions for nonlinear static responses of a shear deformable FGM beam under an in-plane thermal loading." European Journal of Mechanics-A/Solids 31.1 (2012): 13-20. 26. Asghari, M., M. T. Ahmadian, M. H. Kahrobaiyan, and M. Rahaeifard. "On the size-dependent behavior of functionally graded micro-beams." Materials & Design 31, no. 5 (2010): 2324-2329. 27. Asghari, M., M. Rahaeifard, M. H. Kahrobaiyan, and M. T. Ahmadian. "The modified couple stress functionally graded Timoshenko beam formulation." Materials & Design 32, no. 3 (2011): 1435-1443. 28. Alshorbagy, Amal E., M. A. Eltaher, and F. F. Mahmoud. "Free vibration characteristics of a functionally graded beam by finite element method." Applied Mathematical Modelling 35.1 (2011): 412-425. 29. Ke, Liao-Liang, and Yue-Sheng Wang. "Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory." Composite Structures 93.2 (2011): 342-350. 30. Ansari, R., R. Gholami, and S. Sahmani. "Free vibration analysis of size-dependent functionally graded microbeams based on the strain gradient Timoshenko beam theory." Composite Structures 94.1 (2011): 221-228. 31. Ansari, R., Ashrafi, M. A., Pourashraf, T., & Sahmani, S. (2015). Vibration and buckling characteristics of functionally graded nanoplates subjected to thermal loading based on surface elasticity theory. Acta Astronautica. 32. Ke, Liao-Liang, Yue-Sheng Wang, Jie Yang, and Sritawat Kitipornchai. "Nonlinear free vibration of sizedependent functionally graded microbeams." International Journal of Engineering Science 50, no. 1 (2012): 256-267. 33. Eltaher, M. A., Samir A. Emam, and F. F. Mahmoud. "Free vibration analysis of functionally graded sizedependent nanobeams." Applied Mathematics and Computation 218.14 (2012): 7406-7420. 34. Eltaher, M. A., A. E. Alshorbagy, and F. F. Mahmoud. "Determination of neutral axis position and its effect on natural frequencies of functionally graded macro/nanobeams." Composite Structures 99 (2013): 193-201. 35. Eltaher, M. A., Samir A. Emam, and F. F. Mahmoud. "Static and stability analysis of nonlocal functionally graded nanobeams." Composite Structures 96 (2013): 82-88. 36. Şimşek, M., and H. H. Yurtcu. "Analytical solutions for bending and buckling of functionally graded nanobeams based on the nonlocal Timoshenko beam theory." Composite Structures 97 (2013): 378-386. 37. Thermophysical Properties Research Center. Thermophysical properties of high temperature solid materials. Vol. 1. Elements.-Pt. 1. Ed. Yeram Sarkis Touloukian. Macmillan, 1967. 38. Tauchert, Theodore R. Energy principles in structural mechanics. NY: McGraw-Hill, 1974. 39. Eringen, A. Cemal, and D. G. B. Edelen. "On nonlocal elasticity." International Journal of Engineering Science 10.3 (1972): 233-248. 40. Abdel-Halim Hassan, I. H. "On solving some eigenvalue problems by using a differential transformation." Applied Mathematics and Computation 127.1 (2002): 1-22. 41. Kiani, Y., and M. R. Eslami. "An exact solution for thermal buckling of annular FGM plates on an elastic medium." Composites Part B: Engineering 45.1 (2013): 101-110. 42. Fu, Yiming, Jianzhe Wang, and Yiqi Mao. "Nonlinear analysis of buckling, free vibration and dynamic stability for the piezoelectric functionally graded beams in thermal environment." Applied Mathematical Modelling 36.9 (2012): 4324-4340. 43. Zhang, Da-Guang. "Nonlinear bending analysis of FGM beams based on physical neutral surface and high order shear deformation theory." Composite Structures 100 (2013): 121-126. 44. Chen, Cha’o-Kuang, and Shin-Ping Ju. "Application of differential transformation to transient advective– dispersive transport equation." Applied Mathematics and Computation 155.1 (2004): 25-38.

21

Table 1. Temperature dependent coefficients of Young’s modulus, thermal expansion coefficient, mass density and thermal conductivity for Si3 N4 and SUS304 . Material

Properties

P0

P−1

P1

P2

P3

Si3 N4

E (Pa)

348.43e+9

0

-3.070e-4

2.160e-7

-8.946e-11

5.8723e-6

0

9.095e-4

0

0

ρ (Kg/m )

2370

0

0

0

0

κ (W/mK)

13.723

0

-1.032e-3

5.466e-7

-7.876e-11

E (Pa)

201.04e+9

0

3.079e-4

-6.534e-7

0

12.330e-6

0

8.086e-4

0

0

ρ (Kg/m )

8166

0

0

0

0

κ (W/mK)

15.379

0

-1.264e-3

2.092e-6

-7.223e-10

-1

α (K ) 3

SUS304

-1

α (K ) 3

Table 2. Some of the transformation rules of the one-dimensional DTM.44

Original function

Transformed function

f ( x ) = g ( x ) ± h( x )

F ( K ) = G ( K ) ± H (K )

f ( x ) = λ g ( x)

F ( K ) = λG ( K )

f ( x ) = g ( x ) h( x )

F (K ) =

K

∑G ( K − l ) H (l ) l =0

( k + n )! G ( K + n) =

d n g ( x) f ( x) = dx n

F (K )

f ( x ) = xn

1 F ( K ) = δ ( K − n) =  0

k!

k=n k≠n

Table 3. Transformed boundary conditions (B.C.) based on DTM.44 x =0

Original B.C.

x =L

Transformed B.C.

Original B.C.

Transformed B.C. ∞

f (0) = 0

F [0] = 0

f ( L) = 0

∑F[k ] = 0 k =0

df (0) =0 dx

F [1] = 0

2

df ( L) =0 dx 2

d f (0) =0 dx 2

F [2] = 0

d 3 f (0) =0 dx 3

F [3] = 0

d f ( L) =0 dx 2 d 3 f ( L) =0 dx 3

22

∞

∑k F[k ] = 0 k =0

∞

∑k ( k − 1) F[k ] = 0 k =0

∞

∑k ( k − 1)( k − 2) F[k ] = 0 k =0

Table 4. Convergence study for the first dimensionless natural frequency of FG nanobeam under nonlinear temperature rise with L / h = 20 , µ = 1 , ∆T = 40[ K ] . k

S-S p =0.2

11 13 15 17 19 21 23 25 Analytical

7.1952 7.1772 7.1783 7.1782 7.1783 7.1783 7.1783 7.1783 7.1781

p =1

C-C p =0.2

p =0.5

5.2713 5.2577 5.2585 5.2585 5.2585 5.2585 5.2585 5.2585 5.2582

16.0989 17.0802 16.7525 16.7802 16.7769 16.7771 16.7771 16.7771 -

13.6270 14.4747 14.1944 14.2184 14.2155 14.2157 14.2157 14.2157 -

p =0.5 6.0618 6.0464 6.0473 6.0473 6.0473 6.0473 6.0473 6.0473 6.0470

C-S p =0.2

p =1 11.9070 12.6621 12.4146 12.4360 12.4335 12.4337 12.4337 12.4337 -

p =0.5

11.2480 11.5228 11.4554 11.4596 11.4592 11.4592 11.4592 11.4592 -

p =1

9.5068 9.7473 9.6899 9.6935 9.6932 9.6932 9.6932 9.6932 -

8.2944 8.5112 8.4607 8.4639 8.4636 8.4636 8.4636 8.4636 -

Table 5. Comparison of the nondimensional fundamental fequency for a S-S FG nanobeam with various gradient indexes when L / h = 20 . p =0.1 p =0.5 p =1

µ FEM 0 1 2 3 4 5

Eltaher et al.33 9.2129 8.7879 8.4166 8.0887 7.7964 7.5336

Present DTM 9.1887 8.7663 8.3972 8.0712 7.7804 7.5189

Present Analytical 9.1887 8.7663 8.3972 8.0712 7.7804 7.5189

FEM Present Eltaher et al.33 DTM 7.8061 7.7380 7.4458 7.3823 7.1312 7.0715 6.8533 6.7969 6.6057 6.5520 6.3830 6.3318

Present Analytical 7.7377 7.3820 7.0712 6.7966 6.5518 6.3316

FEM Eltaher et al.33 7.0904 6.7631 6.4774 6.2251 6.0001 5.7979

Present DTM 6.9889 6.6676 6.3869 6.1389 5.9177 5.7189

Present Analytical 6.9885 6.6672 6.3865 6.1386 5.9174 5.7185

Table 6. Effect of material inhomogeneity on first dimensionless natural frequency of a S-S FG nanobeam with different nonlocal parameters under various temperature rises when L / h = 20 , ∆T = 20[ K ] . p =0.1 p =0.2 p =0.5 p =1

µ

Load type

0

LTR NLTR

Present DTM 8.4634 8.4675

Present Analytical 8.4633 8.4674

Present DTM 7.7347 7.7382

Present Analytical 7.7346 7.7381

Present DTM 6.5415 6.5437

Present Analytical 6.5412 6.5435

Present DTM 5.7114 5.7124

Present Analytical 5.7110 5.7121

1

LTR NLTR

8.0488 8.0532

8.0487 8.0532

7.3542 7.3581

7.3541 7.3580

6.2166 6.2195

6.2164 6.2192

5.4251 5.4269

5.4248 5.4266

2

LTR NLTR

7.6854 7.6902

7.6853 7.6901

7.0206 7.0250

7.0205 7.0249

5.9316 5.9351

5.9314 5.9349

5.1740 5.1764

5.1737 5.1761

3

LTR NLTR

7.3633 7.3685

7.3633 7.3684

6.7248 6.7296

6.7248 6.7295

5.6789 5.6829

5.6787 5.6827

4.9511 4.9541

4.9508 4.9539

4

LTR NLTR

7.0751 7.0806

7.0751 7.0806

6.4602 6.4654

6.4601 6.4653

5.4526 5.4571

5.4524 5.4569

4.7514 4.7551

4.7512 4.7548

23

Table 7. Effect of material inhomogeneity on first dimensionless natural frequency of a S-S FG nanobeam with different nonlocal parameters under various temperature rises when L / h = 20 , ∆T = 40[ K ] . p =0.1 p =0.2 p =0.5 p =1

µ

Load type

0

LTR NLTR

Present DTM 8.2781 8.2911

Present Analytical 8.2780 8.2910

Present DTM 7.5559 7.5674

Present Analytical 7.5558 7.5673

Present DTM 6.3717 6.3803

Present Analytical 6.3715 6.3800

Present DTM 5.5469 5.5527

Present Analytical 5.5466 5.5523

1

LTR NLTR

7.8535 7.8676

7.8535 7.8675

7.1656 7.1783

7.1655 7.1781

6.0372 6.0473

6.0370 6.0470

5.2511 5.2585

5.2508 5.2582

2

LTR NLTR

7.4805 7.4955

7.4805 7.4955

6.8225 6.8363

6.8224 6.8362

5.7430 5.7544

5.7428 5.7542

4.9907 4.9996

4.9904 4.9993

3

LTR NLTR

7.1491 7.1651

7.1491 7.1650

6.5176 6.5325

6.5175 6.5324

5.4812 5.4940

5.4810 5.4937

4.7588 4.7691

4.7585 4.7688

4

LTR NLTR

6.8518 6.8687

6.8518 6.8687

6.2440 6.2599

6.2439 6.2598

5.2461 5.2601

5.2459 5.2599

4.5503 4.5620

4.5500 4.5617

Table 8. Effect of material inhomogeneity on first dimensionless natural frequency of a S-S FG nanobeam with different nonlocal parameters under various temperature rises when L / h = 20 , ∆T = 80[ K ] . p =0.1 p =0.2 p =0.5 p =1

µ

Load type

0

LTR NLTR

Present DTM 7.8795 7.9265

Present Analytical 7.8794 7.9265

Present DTM 7.1711 7.2136

Present Analytical 7.1710 7.2135

Present DTM 6.0063 6.0402

Present Analytical 6.0061 6.0400

Present DTM 5.1927 5.2186

Present Analytical 5.1925 5.2183

1

LTR NLTR

7.4319 7.4824

7.4318 7.4824

6.7581 6.8043

6.7580 6.9042

5.6493 5.6873

5.6491 5.6871

4.8743 4.9044

4.8740 4.9041

2

LTR NLTR

7.0363 7.0902

7.0363 7.0902

6.3927 6.4425

6.3926 6.4425

5.3329 5.3749

5.3327 5.3747

4.5915 4.6257

4.5912 4.6254

3

LTR NLTR

6.6826 6.7399

6.6826 6.7399

6.0658 6.1192

6.0658 6.1191

5.0492 5.0951

5.0490 5.0949

4.3373 4.3756

4.3371 4.3753

4

LTR NLTR

6.3634 6.4239

6.3633 6.4239

5.7704 5.8273

5.7704 5.8272

4.7922 4.8419

4.7920 4.8418

4.1066 4.1488

4.1063 4.1486

Table 9. Temperature and material graduation effect on first dimensionless natural frequency of a C-C FG nanobeam with different nonlocality parameters and thermal loading ( L / h = 20 ). ∆T = 20[ K ]

∆T = 40[ K ]

∆T = 80[ K ]

µ

Load type

0

LTR NLTR

0.1 19.6398 19.6390

0.2 17.9776 17.9742

0.5 15.2580 15.2501

1 13.3671 13.3558

0.1 19.5436 19.5449

0.2 17.8869 17.8833

0.5 15.1759 15.1635

1 13.2905 13.2715

0.1 19.3420 19.3552

0.2 17.6968 17.6999

0.5 15.0040 14.9886

1 13.1304 13.1011

1

LTR NLTR

18.4670 18.4674

16.9005 16.8987

14.3375 14.3316

12.5555 12.5465

18.3371 18.3417

16.7769 16.7771

14.2236 14.2157

12.4477 12.4337

18.0640 18.0874

16.5174 16.5311

13.9849 13.9810

12.2224 12.2047

2

LTR NLTR

17.4669 17.4685

15.9819 15.9815

13.5521 13.5480

11.8627 11.8558

17.3063 17.3139

15.8284 15.8320

13.4093 13.4054

11.7266 11.7169

16.9677 17.0003

15.5051 15.5285

13.1090 13.1155

11.4409 11.4339

3

LTR NLTR

16.6012 16.6038

15.1865 15.1874

12.8718 12.8693

11.2624 11.2573

16.4122 16.4225

15.0054 15.0120

12.7022 12.7020

11.0999 11.0942

16.0124 16.0537

14.6225 14.6549

12.3444 12.3605

10.7578 10.7605

4

LTR NLTR

15.8421 15.8457

14.4891 14.4911

12.2750 12.2740

10.7355 10.7321

15.6267 15.6396

14.2821 14.2916

12.0803 12.0836

10.5484 10.5464

15.1693 15.2188

13.8430 13.8840

11.6681 11.6934

10.1528 10.1647

Power-law Exponent

24

Table 10. Temperature and material graduation effect on first dimensionless natural frequency of a C-S FG nanobeam with different nonlocality parameters and thermal loading ( L / h = 20 ). ∆T = 20[ K ]

∆T = 40[ K ]

∆T = 80[ K ]

µ

Load type

0

LTR NLTR

0.1 13.4380 13.4395

0.2 12.2947 12.2948

0.5 10.4238 10.4211

1 9.1227 9.1178

0.1 13.3037 13.3105

0.2 12.1663 12.1700

0.5 10.3040 10.3020

1 9.0082 9.0018

0.1 13.0201 13.0483

0.2 11.8951 11.9162

0.5 10.0515 10.0594

1 8.7674 8.7648

1

LTR NLTR

12.6832 12.6855

11.6017 11.6027

9.8316 9.8302

8.6007 8.5972

12.5283 12.5371

11.4531 11.4592

9.6923 9.6932

8.4670 8.4636

12.2002 12.2347

11.1385 11.1663

9.3977 9.4130

8.1848 8.1897

2

LTR NLTR

12.0337 12.0367

11.0053 11.0071

9.3218 9.3215

8.1511 8.1489

11.8597 11.8704

10.8380 10.8462

9.1644 9.1678

7.9995 7.9990

11.4899 11.5304

10.4827 10.5168

8.8302 8.8523

7.6783 7.6902

3

LTR NLTR

11.4670 11.4707

10.4848 10.4874

8.8768 8.8775

7.7584 7.7574

11.2752 11.2877

10.3001 10.3102

8.7023 8.7082

7.5901 7.5921

10.8659 10.9121

9.9061 9.9462

8.3306 8.3591

7.2317 7.2501

4

LTR NLTR

10.9669 10.9712

10.0253 10.0287

8.4837 8.4854

7.4115 7.4115

10.7582 10.7723

9.8241 9.8361

8.2932 8.3013

7.2273 7.2317

10.3112 10.3629

9.3933 9.4390

7.8854 7.9201

6.8332 6.8577

Power-law Exponent

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

25

(a) p = 0

(b) p = 0.2

(c) p = 0.5

(d) p = 1

(e) p = 2

(f) p = 5

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

26

(a) p = 0

(b) p = 0.2

(c) p = 0.5

(d) p = 1

(e) p = 2

(f) p = 5

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

27

(a) p = 0

(b) p = 0.2

(c) p = 0.5

(d) p = 1

(e) p = 2

(f) p = 5

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

28

(a) µ = 0

(b) µ = 1

(c) µ = 2

(d) µ = 3

(e) µ = 4 Fig. 5. Variations of the first dimensionless natural frequency of the S-S FG nanobeam with respect to linear temperature change for different values of gradient indexes and nonlocal parameters ( L / h = 50 ).

29

(a) µ = 0

(b) µ = 1

(c) µ = 2

(d) µ = 3

(e) µ = 4 Fig. 6. Variations of the second dimensionless natural frequency of the S-S FG nanobeam with respect to linear temperature change for different values of gradient indexes and nonlocal parameters ( L / h = 50 ).

30

(a) µ = 0

(b) µ = 1

(c) µ = 2

(d) µ = 3

(e) µ = 4 Fig. 7. Variations of the third dimensionless natural frequency of the S-S FG nanobeam with respect to linear temperature change for different values of gradient indexes and nonlocal parameters ( L / h = 50 ).

31

(a) ∆T = 0

(b) ∆T = 20

(c) ∆T = 40

(d) ∆T = 60

(e) ∆T = 80 Fig. 8. The variation of the first dimensionless frequency of S-S FG nanobeam with material graduation and nonlocality parameters for different linear temperature rises ( L / h = 20 ).

32

(a) µ = 0

(b) µ = 1

(c) µ = 2

(d) µ = 3

(e) µ = 4 Fig. 9. The variation of the first dimensionless frequency of C-C FG nanobeam with material graduation and linear temperature rises for different nonlocality parameters ( L / h = 20 ).

33

(a) µ = 0

(b) µ = 1

(c) µ = 2

(d) µ = 3

(e) µ = 4 Fig. 10. The variation of the first dimensionless frequency of C-S FG nanobeam with material graduation and linear temperature rises for different nonlocality parameters ( L / h = 20 ).

34

Highlight • We study the thermo-mechanical vibration of FG nanobeams (FGNBs) under various B.C.s • Various types of thermal loadings are examined. • Numerical simulations show the effectiveness of analytical method and obtained results. • Several significant parameters on thermo-vibrational behavior of FGNBs are discussed. • Normalized frequencies of the temperature-dependent FGNBs are discussed in detail.

35