Thermal buckling and free vibration analysis of size dependent Timoshenko FG nanobeams in thermal environments

Accepted Manuscript Thermal buckling and free vibration analysis of size dependent Timoshenko FG nanobeams in thermal environments Farzad Ebrahimi, Erfan Salari PII: DOI: Reference:

S0263-8223(15)00195-6 http://dx.doi.org/10.1016/j.compstruct.2015.03.023 COST 6292

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Composite Structures

Please cite this article as: Ebrahimi, F., Salari, E., Thermal buckling and free vibration analysis of size dependent Timoshenko FG nanobeams in thermal environments, Composite Structures (2015), doi: http://dx.doi.org/10.1016/ j.compstruct.2015.03.023

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Thermal buckling and free vibration analysis of size dependent Timoshenko FG nanobeams in thermal environments Farzad Ebrahimi1, Erfan Salari1 1

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 buckling and free vibration characteristics of functionally graded (FG) size-dependent Timoshenko nanobeams subjected to an in-plane thermal loading are investigated by presenting a Navier type solution for the first time. Material properties of FG nanobeam are supposed to be temperature-dependent and 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 based on Timoshenko beam theory through Hamilton’s principle and they are solved applying analytical solution. According to the numerical results, it is revealed that the proposed modeling can provide accurate frequency results of the FG nanobeams as compared to 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, beam thickness and mode number on the critical buckling temperature and normalized natural frequencies of the temperature-dependent FG nanobeams in detail. It is explicitly shown that the thermal buckling and vibration behaviour of a FG nanobeams is significantly influenced by these effects. Keywords: Thermal buckling, Timoshenko beam theory, Vibration, Functionally graded nanobeam, Nonlocal elasticity theory. 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 (Ebrahimi & Rastgoo, 2008a, b). Presenting novel properties, FGMs have also 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, transistors, probes, and resonators in NEMSs. For instance, in MEMS/NEMS; nanostructures have been used in many areas including communications, machinery, information technology, 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 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). 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 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. Phadikar and Pradhan (2010) reported finite element formulations for nonlocal elastic Euler–Bernoulli beam and Kirchhoff plate. 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.

Furthuremore, with the development of the material technology, FGMs have also been employed in MEMS/NEMS (Witvrouw, 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. Fallah and Aghdam (2012) used EulerBernoulli beam theory to investigate thermo-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 Lee (2012). Asghari et al. (2010, 2011) studied the free vibration of the FGM Euler–Bernoulli microbeams, which has been extended to consider a sizedependent 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. (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 of lower slenderness ratios. 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. 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). Using nonlocal Timoshenko and Euler– Bernoulli beam theory, Simsek and Yurtcu (2013) investigated bending and buckling of FG nanobeam by analytical method. More recently, vibration behaviour of simply supported Timoshenko FG nanobeams were investigated by Rahmani and Pedram (2014). 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. 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, thermal buckling and vibration characteristics of temperature dependent FG nanobeams considering the effect of linear temperature distribution is analyzed. An analytical method called Navier solution is employed for vibration and thermal buckling analysis of size-dependent FG nanobeams for the first time. 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 Timoshenko 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 Navier type method 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, beam thickness and small scale on critical buckling temperature and 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 , shear modulus G f and mass density 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 as (Simsek and Yurtcu, 2013):

Pf  PcV c  P mV m

(1)

Where Pm , Pc ,V m and V c are the material properties and the volume fractions of the metal and the ceramic constituents related by:

V c V m  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 (  ), thermal expansion (  ) and Poisson’s ratio ( ) can be expressed as follows: p

z 1    Em h 2

E (z )   E c  E m  

p

z 1  (z )   c   m       m h 2 p

z 1  ( z )   c   m       m h 2

(3)

p

 (z )   c  m      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 as (Touloukian , 1967) :

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 Timoshenko 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  (x , t )

(5.a)

u z (x , z , t )  w (x , t )

(5.b)

Where t is time,  is the total bending rotation of the cross-section, u and w are displacement components of the mid-plane along x and

z directions, respectively. Therefore, according to the

Timoshenko beam theory, the nonzero strains are obtained as:

u  z x x w   x

 xx 

(6)

 xz

(7)

Where  xx and  xy are the normal strain and shear 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 extremum (Tauchert, 1974): t

  (U T 0

V ) dt  0

(8)

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:

 U    ij   ij dV  ( xx   xx   xz   xz ) dV v

v

(9)

Substituting Eqs. (6) and (7) into Eq. (9) yields: L

 U   (N ( 0

u  w )  M ( )  Q (   )) dx x x x

(10)

In which N is the axial force, M is the bending moment and Q is the shear force. These stress resultants used in Eq. (10) are defined as:

N    xx dA , M    xx z dA , Q   K s  xz dA A

A

A

(11)

The kinetic energy for Timoshenko beam can be written as:

T 

u u 1 L  (z ,T ) ( ( x )2  ( z ) 2 ) dA dx   2 0 A t t

(12)

Also the virtual kinetic energy can be expressed as: L u  u w w   u u      T   I 0 (  )  I1(  )I2 dx 0 t t t t t t t t   t t

(13)

Where (I 0 , I 1 , I 2 ) are the mass moment of inertias, defined as follows:

(I 0 , I 1 , I 2 )    (z ,T )(1, z , z 2 ) dA A

(14)

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 linearly across its thickness so that the case of linear

temperature rise is taken into consideration. Hence, the first variation of the work done corresponding to temperature change can be written in the form: L

V   N T 0

w  w dx x x

(15)

T

Where N is thermal resultant can be expressed as:

NT 

h /2

 h /2

E (z ,T )  (z ,T )(T T 0 ) dz

(16)

Where T 0 is the reference temperature. By Substituting Eqs. (10),(13) and (15) into Eq. (8) and setting the coefficients of  u , w and  to zero, the following Euler–Lagrange equation can be obtained:

N  2u  2  I 0 2  I1 2 x t t 2 Q  w  2w N T  I 0 x x 2 t 2 M  2u  2 Q  I 1 2  I 2 2 x t t

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

Under the following boundary conditions:

N  0 or u  0 at x  0 and x  L Q  0 or w  0 at x  0 and x  L M  0 or   0 at x  0 and x  L

(18.a) (18.b) (18.c)

2.3. The nonlocal elasticity model for FG nanobeam 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  ij at any point x in the body can be expressed as:

 ij (x )    ( x   x , ) t ij (x ) d  (x ) 

(19)

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

t ij  C ijkl  kl

(20)

The meaning of Eq. (19) 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 ( x   x , ) . Here x   x is the Euclidean distance and  is a constant given by:

  e0 a l

(21)

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, e 0 , dependent on each material. The magnitude of e 0 is determined experimentally or approximated by matching the dispersion curves of plane waves with those of atomic lattice dynamics. Thus, in the present study, a conservative estimate of the nonlocal parameter ( e 0a ) is considered to be in the range of 0-2 (nm) (Pradhan and Murmu, 2010). According to for a class of physically admissible kernel ( x   x , ) it is possible to represent the integral constitutive relations given by Eq. (19) in an equivalent differential form as:

(1  (e0a)2 2 ) kl  t kl

(22)

Where  2 is the Laplacian operator. Thus, the scale length e 0a 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:

 2 xx  E  xx x 2 2 2   xz  (e 0 a )  G  xz x 2

 xx  (e 0a )2

(23)

 xz

(24)

Where  and  are the nonlocal stress and strain, respectively. E is the Young’s modulus, G  E / 2(1   ) is the shear modulus (where is the poisson’s ratio). For Timoshenko nonlocal FG beam, Eqs. (23) and (24) can be rewritten as:

 xx  xz

 2 xx   E (z ) xx x 2  2 xz   G (z ) xz x 2

(25) (26)

Where (   (e 0a )2 ). Integrating Eqs. (25) and (26) over the beam’s cross-section area, the force-strain and the moment-strain of the nonlocal Timoshenko FG beam theory can be obtained as follows:

2N u  N   A xx  B xx 2 x x x 2 M u  M   B xx  D xx 2 x x x 2 Q w Q   C xz (  ) 2 x x

(27) (28) (29)

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

(A xx , B xx , D xx )   E (z ,T ) (1, z , z 2 ) dA

(30)

C xz  K s  G (z ) dA

(31)

A

A

Where K s  5 / 6 is the shear correction factor. The explicit relation of the nonlocal normal force can be derived by substituting for the second derivative of N from Eq. (17.a) into Eq. (27) as follows:

N  A xx

u   3u  3  B xx   (I 0  I1 ) x x x t 2 x t 2

(32)

Also the explicit relation of the nonlocal bending moment can be derived by substituting for the second derivative of M from Eq. (17.c) into Eq. (28) as follows:

M  B xx

2 u   2w  3u  3 T  w  D xx   (I 0 2  I 1  I  N ) 2 x x t x t 2 x t 2 x 2

(33)

By substituting for the second derivative of Q from Eq. (17.b) into Eq. (29), the following expression for the nonlocal shear force is derived:

Q  C xz (

3 w  3w T w   )   (I 0  N ) x x t 2 x 3

(34)

The nonlocal governing equations of Timoshenko FG nanobeam in terms of the displacement can be dreived by substituting for N, M and Q from Eqs. (32)-(34), respectively, into Eq. (17) as follows:

  2u  2  4u  4   2u  2 A xx 2  B xx 2    I 0 2 2  I 1 2 2   I 0 2  I 1 2  0 x x t x  t t  t x 2  2w   4w  4w  2w T  w C xz ( 2  )   (N T  I )  N  I 0 0 0 x x x 4 t 2x 2 x 2 t 2  2u  2 w  4u  4  2u  2 B xx 2  D xx 2  C xz (   )   (I 1 2 2  I 2 2 2 )  I 1 2  I 2 2  0 x x x t x t x t t 2.4. Linear temperature distribution

(35.a) (35.b) (35.c)

Assume a FG nanobeam where the temperature of the top surface (ceramic-rich) is T c and it is considered to vary linearly along the thickness from T c to the bottom surface (Metal-rich) temperature T m . Therefore, the temperature rise as a function of thickness is obtained as (Kiani and Eslami, 2013):

1 z  T  T m  T    2 h 

(36)

The T in Eq. (36) could be defined T Tc T m . 3. Solution procedures Here, based on the Navier type method, an analytical solution of the governing equations for free vibration and thermal buckling 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:

n x ) e i n t L n 1  n w (x , t )  W n sin ( x ) e i n t L n 1  n  (x , t )  n cos ( x ) e i n t L n 1 

u (x , t )  U n cos (

(37) (38) (39)

Where (U n ,W n , n ) are the unknown Fourier coefficients to be determined for each n value. Boundary conditions for simply supported beam are as Eq. (40) :

u (L )  0 x   w (0)  w (L )  0 , (0)  (L )  0 x x u (0)  0 ,

(40)

Substituting Eqs. (37)-(39) into Eqs. (35.a)-(35.c) respectively, leads to Eqs.(41)-(43):

n 2 n n n )  I0 (1   ( ) 2 )n 2 )U n  (B xx ( ) 2 +I1 (1   ( ) 2 )n 2 ) n  0 L L L L n n n n n (C xz ( )2  I0 (1   ( ) 2 )n 2  N T ( ) 2 (1   ( ) 2 ))W n  C xz ( ) n  0 L L L L L

(A xx (

n n (B xx ( ) 2  I1 (1   ( ) 2 )n 2 )U n L L n 2 n n (D xx ( )  C xz  I 2 (1   ( ) 2 )n 2 )) n  C xz ( )W n  0 L L L

(41)

(42)

(43)

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 ]  T [K T ])   [M ] W n   0    n



2



(44)

Where [K ] and [K T ] are stiffness matrix and the coefficient matrix of temperature change, respectively, and [M ] is the mass matrix. By setting this polynomial to zero, we can find natural frequencies n and critical buckling temperature Tcr . 4. Numerical results and discussions Through this section, the effect of temperature change, FG distribution, nonlocality effect and beam thickness on the natural frequencies and critical buckling temperature of the FG nanobeam will be figured out. 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 nm, b (width) = 1 nm and h (thickness) is variable. It is assumed that the temperature increase in metal surface to reference themperature T 0 of the FG nanobeam is T m T 0  5K (Kiani and Eslami, 2013). Relation described in equation (45) are performed in order to calculate the non-dimensional natural frequencies:

ˆ  ωL2 ρc A / EIc

(45)

Where I  b h / 12 is the moment of inertia of the cross section of the beam. To evaluate accuracy of the 3

natural frequencies predicted by the present method, the non-dimensional natural frequencies of simply supported FG nanobeam with various nonlocal parameters previously analyzed by Navier method are reexamined. Tables 2 compares the results of the present study and the results presented by Rahmani and Pedram (2014) which has been obtained by analytical method for FG nanobeam with different nonlocal parameters (varying from 0 to 4 (nm)2 ). The reliability of the presented method and procedure for FG nanobeam may be concluded from Table 2; where the results are in an excellent agreement as values of non-dimensional fundamental frequency are consistent with presented analytical solution. After extensive validation of the present formulation for S-S FG nanobeams, the effects of different parameters such as beam thickness, nonlocality parameter and gradient index on the thermal buckling of FG nanobeam are investigated.

In Table 3 critical buckling temperature of the simply supported FG nanobeams are presented for various values of the gradient index ( p =0,0.2,0.5,1,2,5), nonlocal parameters (  =0,1,2,3,4 (nm)2 ) and three different values of beam thickness ( h =0.15, 0.25, 0.35 nm) based on analytical Navier solution method. It can be concluded from the results of the table that an increase in nonlocal scale parameter and gradient index gives rise to a decrement in the critical buckling temperature. In addition, it is seen that the Tcr increase by increasing beam thickness ( h ) and it can be stated that nonlocality parameter has a notable effect on the critical buckling temperature. Variations of the critical buckling temperature Tcr of the simply supported FG nanobeams with respect to beam thickness for different values of gradient indexes and nonlocal parameters are depicted in Figs. 2 and 3, respectively. Observing these two figures, it is easily deduced for a S-S FG nanobeam that, an increase in beam thickness parameter gives rise to a increase in the critical buckling temperature for all gradient indexes. In addition, it is deduced that the buckling temperature decreases by increasing nonlocality parameters. The critical buckling temperature versus the gradient index of S-S FG nanobeam for different values of nonlocality parameters and beam thickness is illustrated in figures 4 and 5, respectively. It can be observed from these figures that with an increase of the nonlocal parameter, critical buckling temperature decreases. It is seen from figures 4 and 5 that the beam thickness shows an increasing effect on critical temperature of FG nanobeam for all values of gradient indexes. Also in Figs. 4 and 5 it is noticed that, the buckling temperature reduce with high rate where the power exponent in range from 0 to 2 than that where power exponent in range between 2 and 10. In order to investigate the vibration characteristics of the FG nanobeam under linear temperature rise, the first three non-dimensional fundamental frequencies of simply-supported FG nanobeam is presented in Table 4, which figures out the effect of nonlocal parameter (varying from 0 to 4 (nm)2 ), gradient index (varying from 0 to 5) and three different values of linear temperature changes ( T =10, 30, 60 (K)) for h =0.5 nm on the natural frequency characteristics of FG nanobeam. First of all, when the two parameters vanish (  =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 this table. As seen in Table 4, by fixing the nonlocal parameter and varying the material distribution parameter results decreasing in the fundamental frequencies, due to increasing in ceramics phase constituent, and hence, stiffness of the beam. However, the increasing of nonlocal parameter causes the decreasing in fundamental frequency, at a constant material graduation

index. In addition, it is seen that the first three dimensionless natural frequencies decrease by increasing temperature change and it can be stated that temperature change has a significant effect on the dimensionless natural frequencies, especially for lower mode numbers. 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. 6, 7 and 8, respectively. 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. 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 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. 9 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 ( h =0.2 nm). Figure 10 demonstrate the variation of the first dimensionless natural frequency of FG nanobeam respect to temperature changes with varying of the beam thickness and nonlocality parameter at p = 0.5. It is shown that, by increasing the nonlocal parameter, the critical buckling temperature decreases at a constant material distribution. Finally, the fundamental frequency parameter as a function of power law indexes and temperature rise is presented in Figure 11 for the FG nanobeam with S–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 12 ( h =0.5 nm). Similarly, it is revealed that for a S-S FG nanobeam increasing temperature change, power law index, and nonlocal parameter, leads to decrease in natural frequency.

5. Conclusions Thermal buckling and vibrational behavior of the temperature-dependent FG nanobeams subjected to linear temperature distribution through the thickness direction are investigated on the basis of nonlocal elasticity theory in conjunction with Navier analytical method. Eringen’s theory of nonlocal elasticity together with Timoshenko 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 in thermal environment are derived by implementing Hamilton’s principle. 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, linear temperature change and beam thickness on critical buckling temperature and fundamental frequencies of FG nanobeams are investigated. It is concluded that various factors such as nonlocal parameter, gradient index, temperature-dependent material properties, thermal environment and beam thickness play important roles in dynamic behavior of FG nanobeams. It is illustrated that presence of nonlocality leads to reduction in natural frequency and buckling temperature. 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 linear temperature rise, 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 critical buckling temperature increases with the increase in beam thickness. References       

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Tables: Table 1. Temperature dependent coefficients of Young’s modulus, thermal expansion coefficient, mass density and Poisson’s ratio for Si3N4 and SUS304 . Material

Properties

P0

P1

P1

P2

P3

Si3N4

E (Pa)

348.43e+9

0

-3.070e-4

2.160e-7

-8.946e-11

5.8723e-6

0

9.095e-4

0

0

 (Kg/m )

2370 0.24

0 0

0 0

0 0

0 0

E (Pa)

201.04e+9

0

3.079e-4

-6.534e-7

0

12.330e-6

0

8.086e-4

0

0

 (Kg/m )

8166

0

0

0

0



0.3262

0

-2.002e-4

3.797e-7

0

 (K ) -1

3

 SUS304

 (K ) -1

3

 (nm)2

0 1 2 3 4

Table 2. Comparison of the nondimensional fundamental frequency for a S-S FG nanobeam with various gradient indexes when L =10 nm, h =0.2 nm. p =0 p =0.2 p =1 p =5 Rahmani and Present Rahmani and Present Rahmani and Rahmani and Present Present Pedram(2014) Analytical Pedram(2014) Analytical Pedram(2014) Pedram(2014) Analytical 9.8631 9.86315733 8.6895 8.68954599 6.9917 6.99174004 5.9389 5.93894397 9.4097 9.40973040 8.2901 8.29007206 6.6703 6.67031728 5.6659 5.66592012 9.0136 9.01358936 7.9411 7.94106762 6.3895 6.38950303 5.4274 5.42739007 8.6636 8.66360601 7.6327 7.63272858 6.1414 6.14140878 5.2166 5.21665314 8.3515 8.35146095 7.3577 7.35772548 5.9201 5.92013713 5.0287 5.02869994 Table 3. Material graduation and beam thickness effect on the critical buckling temperature nanobeam with different nonlocality parameters ( L =10 nm). Gradient index  (nm)2 h (nm) 0 0.2 0.5 1 2 0 0.15 39.4887 32.9070 27.7259 23.8047 20.9463 0.25 127.3340 109.7510 95.5739 84.6229 76.4715 0.35 258.7810 224.7380 197.1000 175.6260 159.5460

Tcr [K ] of a S-S FG

5 18.5146 69.4307 145.6000

1

0.15 0.25 0.35

35.0431 114.9980 234.6360

29.0183 98.9592 203.6170

24.2924 86.0456 178.4520

20.7268 76.0818 158.9100

18.1362 68.6735 144.2860

15.9375 62.2798 131.6070

2

0.15 0.25 0.35

31.3304 104.6950 214.4720

25.7706 89.9465 185.9780

21.4249 78.0881 162.8770

18.1563 68.9486 144.9500

15.7893 62.1610 131.5410

13.7853 56.3077 119.9200

3

0.15

28.1831

23.0175

18.9941

15.9774

13.7998

11.9608

4

0.25 0.35

95.9606 197.3790

82.3065 171.0250

71.3424 149.6750

62.9019 133.1150

56.6404 120.7370

51.2452 110.0140

0.15 0.25 0.35

25.4813 88.4628 182.7040

20.6541 75.7477 158.1890

16.9073 65.5514 138.3410

14.1067 57.7108 122.9560

12.0919 51.9010 111.4620

10.3946 46.8992 101.5090

Table 4. Temperature and material graduation effect on first three dimensionless frequency of a S-S FG nanobeam with different nonlocality parameters ( L =10 nm, h =0.5 nm).  (nm)2

ˆ i

0

T  10[K ]

1 2 3

Gradient index 0 0.2 9.6461 7.7964 38.6688 31.3222 85.5816 69.3332

1 5.7717 23.2890 51.5787

1

1 2 3

9.1852 32.6807 62.1603

7.4215 26.4636 50.3428

2

1 2 3

8.7818 28.7973 51.1628

3

1 2 3

4

1 2 3

T  30[K ] 5 4.6925 19.0001 42.0838

Gradient index 0 0.2 9.4538 7.6211 38.4794 31.1579 85.3911 69.1801

1 5.6105 23.1492 51.4659

5.4907 19.6648 37.4287

4.4616 16.0353 30.5228

8.9831 32.4564 61.8977

7.2369 26.2655 50.1172

7.0932 23.3117 41.4229

5.2445 17.3124 30.7783

4.2593 14.1098 25.0862

8.5701 28.5425 50.8434

8.4247 26.0161 44.4465

6.8025 21.0537 35.9737

5.0264 15.6260 26.7130

4.0799 12.7288 21.7611

8.1056 23.8969 39.7984

6.5427 19.3325 32.2012

4.8313 14.3399 23.8969

3.9193 11.6750 19.4565

T  60[K ] 5 4.5363 18.8693 41.9855

Gradient index 0 0.2 9.1475 7.3420 38.1838 30.9016 85.0946 68.9412

1 5.3537 22.9319 51.2893

5 4.2875 18.6672 41.8321

5.3204 19.4913 37.2400

4.2964 15.8708 30.3473

8.6601 32.1054 61.4880

6.9419 25.9558 49.7655

5.0480 19.2211 36.9466

4.0317 15.6160 30.0761

6.8996 23.0843 41.1420

5.0654 17.1100 30.5340

4.0852 13.9166 24.8550

8.2310 28.1427 50.3438

6.5892 22.7281 40.7034

4.7777 16.7941 30.1539

3.8049 13.6164 24.4972

8.2038 25.7338 44.0785

6.6002 20.8001 35.6462

4.8388 15.3980 26.4225

3.8972 12.5101 21.4838

7.8488 25.2897 43.5014

6.2747 20.4017 35.1334

4.5360 15.0408 25.9694

3.6015 12.1689 21.0532

7.8758 23.5893 39.3870

6.3319 19.0549 31.8323

4.6354 14.0883 23.5657

3.7283 11.4329 19.1387

7.5053 23.1039 38.7400

5.9916 18.6175 31.2531

4.3177 13.6931 23.0477

3.4172 11.0537 18.6434

Figures:

Fig. 1. Geometry and coordinates of Timoshenko FG nanobeam.

(a) p  0

(b) p  0.2

(c) p  0.5

(d) p  1

(e) p  2

(f) p  5

Fig. 2. The variation of the critical buckling temperature of S-S FG nanobeam with beam thickness and nonlocality parameters for different material graduations.

(a)   0(nm)2

(b)   1(nm)2

(c)   2(nm)2

(d)   3(nm)2

(e)   4 (nm)2

Fig. 3. The variation of the critical buckling temperature of S-S FG nanobeam with beam thickness and material graduations for different nonlocality parameters.

(a)   0(nm)2

(b)   1(nm)2

(c)   2(nm)2

(d)   3(nm)2

(e)   4(nm)2 Fig. 4. The variation of the critical buckling temperature of S-S FG nanobeam with material graduations and beam thickness for different nonlocality parameters.

(a) h  0.15(nm)

(b) h  0.25(nm)

(c) h  0.35(nm)

(d) h  0.45(nm)

Fig. 5. The variation of the critical buckling temperature of S-S FG nanobeam with material graduations and nonlocality parameters for different beam thickness.

(a)   0(nm)2

(b)   1(nm)2

(d)   3(nm)2

(c)   2(nm)2

(e)   4(nm)2 Fig. 6. 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 ( h =0.2 nm).

(a)   0(nm)2

(b)   1(nm)2

(c)   2(nm)2

(d)   3(nm)2

(e)   4(nm)2

Fig. 7. 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 ( h =0.2 nm).

(a)   0(nm)2

(b)   1(nm)2

(c)   2(nm)2

(d)   3(nm)2

(e)   4(nm)2 Fig. 8. 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 ( h =0.2 nm).

(a) p  0

(b) p  0.2

(c) p  0.5

(d) p  1

(e) p  2

(f) p  5

Fig. 9. 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 ( h =0.2 nm).

(a)   0(nm)2

(b)   1(nm)2

(d)   3(nm)2

(c)   2(nm)2

(e)   4(nm)2 Fig. 10. 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 beam thickness ( p =0.5).

(a)   0(nm)2

(b)   1(nm)2

(c)   2(nm)2

(d)   3(nm)2

(e)   4(nm)2

Fig. 11. The variation of the first dimensionless frequency of S-S FG nanobeam with material graduation and temperatures for different nonlocality parameters ( h =0.5 nm).

(a) T  0[K ]

(b) T  20[K ]

(c) T  40[K ]

(d) T  60[K ]

(e) T  80[K ] Fig. 12. The variation of the first dimensionless frequency of S-S FG nanobeam with material graduation and nonlocality parameters for different temperatures ( h =0.5 nm).