Size dependent nonlinear free vibration of an axially functionally graded (AFG) microbeam using He’s variational method

Accepted Manuscript Size Dependent Nonlinear Free Vibration of an Axially Functionally Graded (AFG) Microbeam Using He’s Variational Method Mesut Şimşek PII: DOI: Reference:

S0263-8223(15)00364-5 http://dx.doi.org/10.1016/j.compstruct.2015.05.004 COST 6417

To appear in:

Composite Structures

Please cite this article as: Şimşek, M., Size Dependent Nonlinear Free Vibration of an Axially Functionally Graded (AFG) Microbeam Using He’s Variational Method, Composite Structures (2015), doi: http://dx.doi.org/10.1016/ j.compstruct.2015.05.004

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Size Dependent Nonlinear Free Vibration of an Axially Functionally Graded (AFG) Microbeam Using 1

Mesut

Yildiz Technical University, Faculty of Civil Engineering, Department of Civil Engineering, -Istanbul, TURKEY

Abstract

Nonlinear free vibration of axially functionally graded (AFG) Euler- Bernoulli microbeams with immovable ends is studied by using the modified couple stress theory. The nonlinearity of the problem stems from the von-

-displacement relationships.

Elasticity modulus and mass density of the microbeam vary continuously in the axial direction according to a simple power-law form. The nonlinear governing partial differential equation and the associated boundary conditions are derived by

. By duced to

a nonlinear ordinary differential equation. approximate closed form solution of the nonlinear ordinary governing equation. Pinnedpinned and clamped-clamped boundary conditions are considered. The influences of the length scale parameters, material variation, vibration amplitude, and boundary conditions on vibration responses are examined in detail.

Keywords: Vibration; Microbeam; Functionally Graded Material; Modified Couple Stress Theory; Variational Method

1

Corresponding Author: Tel: +902123835146, Fax: +902123835102 E-mail addresses: [email protected], [email protected]

1

1.Introduction

Functionally graded materials (FGMs) are a new generation of advanced composites where the volume fractions of material constituents vary gradually along certain direction. In general, one face of FGM structure is ceramic-rich which can resist severe thermal loading, and the other face is metal-rich that has excellent structural strength. Functionally graded material (FGMs) were proposed as thermal barrier materials during a space project in Japan in 1984. Nowadays, FGMs have been encountered in various engineering applications such as aircrafts, space vehicles, automobile, defense industries, electronics, and biomedical sector. Due to gradually varying the material properties in FGMs, interface disbanding, cracking and residual stresses can be prevented, and therefore structural integrity can be maintained to a desirable level. However, FGMs need advanced manufacturing techniques, including powder metallurgy, chemical vapor deposition, centrifugal casting. Because of the wide applications of FGMs structures,

2

have received increasing attention in modeling beamtype structures, and ] has recently investigated the dynamic behavior of AFG Euler-Bernoulli beam subjected to a moving harmonic load. A nonlocal continuum model for free vibration AFG tapered nanorod was proposed

]. Shahba et al. [35] used the differential transform method (DTM) to

scrutinize free vibration and stability of AFG tapered Euler-Bernoulli beam. Huang et al. [36] proposed a new approach by introducing an auxiliary function for investigating free vibration of AFG Timoshenko beams with non-uniform cross-section. Akgöz and Civalek [37] performed free vibration analysis of a non-uniform AFG microbeam based on the modified couple stress theory. Akgöz and Civalek [38] studied longitudinal vibration of AFG microbars within the framework of the strain gradient theory.

On the other hand, microbeams, which are novel structures, have been started to be used in various applications of micro-electro-mechanical Systems (MEMS), such as atomic force microscopes (AFMs) [39-41], microactuator [42], microsensor [43-45], and so on. It is known that since the thickness of microbeams is generally on the order of microns and sub-microns, the mechanical behavior and the deformation characteristics of microbeams are sizedependent [46-50], as verified with the experimental studies. Therefore, the size effect should be taken into account in the analysis of mechanical behavior of microstructures. Furthermore, the appropriate theories and mathematical models should be developed for the analysis of microstructures since it is not always possible to perform the controlled experiments in microscale. The classical continuum theories, which do not account for such size effects because of the lack of additional length scale parameter, are inadequate to evaluate the sizedependent responses of microstructures. Thus, there is a strong encouragement to develop the

3

size-dependent models to determine the microstructural size dependency of these structures. For this purpose, several non-classical higher-order continuum theories, such as the couple stress theory [46-49] that contains four material constants (two classical and two additional), strain gradient theory [51], nonlocal elasticity theory [52], have been developed for modeling small-sized structures. The most important difficulty in the non-classical higher-order continuum theories is the determining the micro-structural material length scale parameters. Hence, Yang et al [53] developed the modified couple stress theory with only one higherorder material parameter. In addition, the another advantage of the modified couple stress theory is due to the fact that the strain energy density function depends only on the strain tensor and the symmetric part of the curvature tensor. After the first study on the static analysis of a microbeam based on the modified couple stress theory by Park and Gao [54], many researches have been reported on the characteristics of static, buckling and vibration for microbeams [55-72].

The nonlinear effects on slender structural elements, which are extensively used in civil and mechanical engineering, have been studied for decades now. As is known, the vibration frequencies are independent of the vibration amplitude in the linear theory of vibrations. However, in many cases, the linear theory is inadequate to predict the vibration characteristics of structures if the amplitude of the vibration is large. Xia et al. [73] established a nonlinear Euler Bernoulli microbeam model based on the modified couple stress theory for static bending, post-buckling and free vibration analysis. Asghari et al. [74] investigated the sizedependent nonlinear behavior of Timoshenko microbeams based on the modified couple stress theory. Ke et al. [75] predicted nonlinear free vibration behavior of

FG

microbeams based on the modified couple stress theory and von-Kármán geometric nonlinearity. Nonlinear forced vibration of a microbeam by using the modified couple stress

4

theory and the strain gradient theory is examined by Ghayesh et al. [76, 77]. In a very recent paper

], nonlinear static and free vibration of homogeneous microbeams based

on a three-layered nonlinear elastic foundation is investigated within the framework of the modified couple stress theory and Euler-Bernoulli beam theory together with the von].

It should be noted that the above-mentioned studies on the nonlinear free vibration of microbeams is related to microbeams made of transversely functionally graded materials (FGMs). No literature is available for the large amplitude free vibration of axially functionally graded (AFG) microbeams. This paper is the first attempt to the nonlinear free vibration of AFG microbeams with different boundary conditions based on the modified couple stress A simple power-law function is employed

theory and von-

to predict the material properties of AFG microbeam vary along the length of the beam. The

principle.

ethod and

ional method, which is a novel solution technique

for the nonlinear differential equations, is conducted to determine the nonlinear vibration frequencies of the AFG microbeams with different boundary conditions. A detailed parametric study is carried out to highlight the influences of the length scale parameter, material variation, and end supports on the nonlinear free vibration characteristics of AFG microbeams.

2. The governing equations based on the modified couple stress theory

5

According to the modified couple stress theory, the strain energy in an isotropic linear elastic body can be expressed as

1 2

Us

m:

:

dV

U

V

in which

is the stress tensor,

stress tensor, and

1 2

ij

ij

m ij

ij

dV

(1)

i, j 1, 2, 3

V

is the strain tensor, m is the deviatoric part of the couple

is the symmetric curvature tensor. The following kinematic relations can

be written:

1 2

u

u

1 2

ij

1 u i, j 2

ij

1 2

T

T

u j, i

(2)

j, i

(3)

i, j

in which u is the displacement vector,

1 curl u 2

i

is the rotation vector that can be expressed as

1 eijk uk , j 2

(4)

Here eijk is the permutation symbol. For an isotropic linear elastic material, the constitutive relations can be formulated as shown below

tr

I 2

m 2l 2 where

ij

mij

2 l2

and

kk

ij

2

(5)

ij

(6)

ij

l is the material length scale parameter,

Kronecker delta. The von-

ij

is the

-displacement relationship based on 6

assumptions of large transverse displacements, moderate rotations and small strains for a straight Euler-Bernoulli beam are given by

u x, t xx

w x, t

1 2

x

2

2

z

x

w x, t

(7)

x2

where u and w are the axial and the transverse displacement of any point on the neutral axis, and

xx

is the longitudinal strain. The explicit expressions of the components of the rotation

vector are derived by using Eq. (4) as

w x, t , x

y

x

0

z

(8)

Substitution of Eq. (8) into Eq. (4) leads to the non-zero components of the symmetric curvature tensor

2

1 2

xy

yx

N x

f x

w x, t x2

,

xx

yy

zz

xz

yz

(9)

0

2

2

M x2

x A

2

Y x2

x

N

u t2

w x

(10) 2

q x

x A

w t2

(11)

At the same time, the related boundary conditions can be acquired at x

0 and x

L as

follows

either

N

0

or

u 0

(12) 7

either

M x

Y x

either

M Y

w x

N

w x

or

0

or

0

(13)

w 0

(14)

0

x is the mass density of the microbeam, A is the area of the cross-section, q is

where

distributed transverse load, and f is the distributed axial load. The stress resultants (N, M, Y) defined in the above equations are given by

N

xx

dA

E x A

A

u x

1 2

w x

2

(15)

2

M

z

xx dA

w x2

E x I

A

(16) 2

Y

mxy dA A

2

x l2

xy dA

A

x Al 2

w x2

(17)

where N is the normal force, M is the bending moment, Y is a new stress resultant due to the couple stress, E x

x

section.

is the elasticity modulus, I is the inertia moment of the cross-

E x / 2(1

( x)) is known as shear modulus, where

x

ratio. As it is seen from Fig. 1, the material properties of AFG microbeam (i.e., elasticity modulus, mass density) changes continuously in the axial direction according to the rule of mixture as

[Figure 1 about here]

E x

EL

ER 1 x / L

k

(18)

ER

8

x

L

where

L

R

and

1 x/L

R

k

(19)

R

are the effective material properties of the beam at the left and the right

end of the microbeam, k is the volume fraction exponent (also known as power-law exponent or gradient index). It is clear from Eqs. (18-19) that when x

L, E

x

ER ,

R

0, E

EL ,

L

and when

. It should be noted that the material length scale parameter l and the

assumed to be constant in this study.

Making use of Eqs. (10-11) and (15-17), the governing equations are expressed in terms of displacements as

E x A 2

u E x A x x 2

x

2

w x

2

2

f x

2

x Al 2

E x I

w x2

x

N

x A

u

(20)

t2 2

w x

q x

x A

w t2

(21)

As seen from above the equations, the governing equations are coupled with respect to the displacements u and w . Thus, in order to reduce the equations to a single equation in terms of w , the in-plane inertia can be neglected [80, 81]. Further, considering that the distributed axial load f is zero for free vibration, Eq. (20) can be rewritten as

E x A 2

u E x A x x

w x

2

0

(22)

Integration of Eq. (22) with respect to x gives

u x

C E x A

1 2

w x

2

(23)

9

where C is an integration constant to be calculated with respect to the boundary conditions. It is assumed that the microbeam has immovable supports in the axial direction. Thus, the boundary conditions related to the axial motion are expressed as

u 0, t

u L, t

0

(24)

Integrating both side of Eq. (23) from 0 to L together with the above boundary conditions, one obtains

L

u L, t

u 0, t 0

L

C 1 dx E x A 20

A 2

2

w dx 0 x

C

2

L

0 L

0

w dx x dx E x

(25)

After some mathematical efforts, the axial normal force N is obtained as follows

A 2K

N

2

L

L

w dx , K x

0

0

dx E x

(26)

The governing equation is derived in terms of w by substituting Eq. (26) into Eq. (21) as follows

2

x

2

2

E x I

x Al

2

w x2

A 2K

2

L

w dx x

0

2

2

w x2

x A

w t2

q x

(27)

For the general purpose, the following non-dimensional parameters can be defined

x

x ,w L

w , Eratio r

EL , ER

L ratio R

,

l , t h

10

t

AL4 , ER I R

(28)

where r

I / A is the radius of gyration of the cross-section. Considering Eq. (28) and

omitting the distributed load q for free vibration analysis, Eq. (27) can be written in the nondimensional form as

2

x

2

w x2

D x

2

1 2K

2

1

2

w dx x

0

w F x x2

2

w

t2

0

(29)

where

D x

Eratio 1 1 x

F x

ratio

1

k

1 1

6 1

2

(30)

(31)

1

dx

K 0

1 1 x

k

Eratio 1 1 x

k

(32)

1

Based on the Galerkin method, displacement function w x , t

w x, t

Q t

Here Q t

can be represented as follows

x

(33)

is the unknown time-dependent coefficient to be determined and

x is the test

function which must satisfy the kinematic boundary conditions. In this study, the exact mode shapes of pinned-pinned (PP) and clamped-clamped (CC) homogeneous beams are chosen for

x as shown below [82]

For pinned-pinned (PP) microbeam:

x

sin

x ,

(34)

11

For clamped-clamped (CC) microbeam: x

cosh

x

cos

cosh sinh

x

cos sin

sinh

x

sin

x

,

4.7300

(35)

x is substituted into Eq. (29), then the both sides of the resulting equation

If the solution are multiplied by

1

0

2

x

2

x , and this equation is integrated over the domain 0, 1 , one obtains

2

D x

w x2

dx

1 2K

1

0

2

w dx x

1

0

2

w dx x2

1

2

F x

w

t2

0

dx

(36)

0

The second derivative in the first term of the governing equation causes some mathematical difficulties. Therefore, in order to remove the second derivative, one has to apply the partial

x

integration to this term. However, it is necessary that the function

must satisfy the

homogenous form of all specified boundary condition. The integration by parts two times together with the homogeneous boundary conditions yields

1

2

D x 0

w 2 dx x2 x2

1 2K

1

0

2

w dx x

1

0

2

w dx x2

1

2

F x 0

w

t2

dx

0

(37)

After some mathematical arrangements, Eq. (37) can be written in a compact form as

(38) Here

is the second derivative of Q with respect to time. The coefficients S1 , S 2 can be

expressed as

12

1

D x S1

dx

0 1

, F x

2

S2

1 2K

1

0 1

dx

0

1 2

dx

dx 0

F x

2

(39) dx

0

where the prime denotes the derivative with respect to the axial coordinate x . It should be noted that the microbeam has the following initial conditions

(40)

wmax 0.5 / r is the dimensionless maximum vibration amplitude of AFG

where microbeam.

3. Approximate a

In general, finding the exact solution of nonlinear differential equations is impossible in most cases. Therefore, nonlinear differential equations are generally solved with the help of approximate methods. More recently, He [79] has proposed a novel variational method to obtain analytical solution for nonlinear differential equations. By using the semi-inverse

(41)

where T is the period of the nonlinear oscillator. The approximate solution is given as [79]

Q t

cos t

(42)

13

Here

is the natural frequency of the nonlinear oscillator. When the approximate solution in

Eq. (42) is substituted into Eq. (44), one has the following equation with the transformation of t

J

/2

1

,

1 2

0

2

2

S1 2

sin 2

2

S2 4

cos 2

Accordin

4

cos 4

d

], the stationary conditions dJ / d

(43)

0 should be

satisfied as follows

dJ d

1

/2 2

sin 2

S1 cos 2

S2

3

cos4

d

(44)

0

0

After some mathematical process, the natural frequency (

) is found as

/2

S1 cos 2 2

S2

2

cos4

d

0

(45)

/2

sin

2

d

0

The nonlinear natural frequency (

NL

S1

3 S2 4

NL

) can be obtained by taking the above integral as

2

(46)

Consequently, the following approximate solution ( Q t ) can be found by using Eq. (44) as follows

Q t

cos

S1

3 S2 4

2

(47)

t

4. Numerical Results 14

In the numerical results, in order to examine the effects of the length scale parameters, material variation, vibration amplitude, and boundary conditions, some numerical results are presented for the nonlinear free vibration properties of AFG microbeams based on the modified couple stress theory. In order to obtain the general results, the ratio of nonlinear frequency to linear frequency of homogeneous microbeam, which is called the nonlinear frequency ratio, is computed for PP and CC AFG microbeam. Thus, the nonlinear frequency ratio can be written as

NL

(48)

ratio LO

where

LO

is the linear frequency of homogeneous beam basis on the classical elasticity

theory. It is very important to check the reliability of the present formulation and the analytical solution before the extensive numerical presentations. In the comparison study, the frequency ratio of PP and CC homogeneous beam by using the classical beam theory is computed and compared with those of previous studies of Azrar et al. [83], Fallah and Aghdam [84], and Pirbodaghi et al. [85]. Good agreement is observed. However, these results 78, 86].

are not presented here again since they are presented

[Table 1 about here] [Table 2 about here]

In Tables 1 and 2, the nonlinear frequency ratio of PP and CC microbeams is presented for different values of the ratio of elasticity modulus ( Eratio

0.5, 1, 2 ), dimensionless scale

parameter ( l / h 0, 0.25, 0.50, 0.75, 1 ) and volume fraction exponent ( k

0, 0.3, 1, 3, 10 ).

In these calculations, the vibration amplitude and the mass density ratio are kept constant as 15

1 and

ratio

1 , respectively. It can be observed from the results of the tables that an

increase in the dimensionless scale parameter ( l / h ) gives rise to an increment in the nonlinear frequency ratio (

ratio

NL

/

LO

). This is due to the fact that the microbeam

becomes stiffer by incorporating the couple stress in the new beam model. Especially, the size effect is very strong when the length scale parameter is equal to the microbeam length, namely l / h 1. It is also seen that the nonlinear frequency ratio increases as the ratio of elasticity modulus ( Eratio ) increases for a fixed value of the volume fraction index (or gradient index). At the same time, the tables reveal that an increase in the gradient index yields an increase or a decrease in the nonlinear frequency ratio for the cases that Eratio

Eratio

0.5 and

2 , respectively. The reason of the above two findings can be explained as follows:

As remembered from the non-dimensionalization process (see Eq.(30)), the dimensionless elasticity modulus is defined as E x

Eratio 1 1 x

k

1 . It is apparent from this

equation that the elasticity modulus at the left side of the microbeam (when x

E 0 is E 1

EL

0 ) is

Eratio , and the elasticity modulus at the right side of the microbeam (when x 1 )

ER 1. Moreover, it should be noted that if the gradient index is zero ( k

microbeam is homogeneous with the constant elasticity modulus, E x the other hand, when the gradient index goes from zero to infinity ( k elasticity modulus changes from E x

Eratio to E x

0 ), the

Eratio : constant . On k

), the

1 . Finally, for k

, the

0

microbeam is also made of homogeneous material with the constant elasticity modulus,

E x

1: constant . Based on the above comments, for the case that Eratio

modulus of the microbeam at the supports is EL

0.5 and ER

0.5 , the elasticity

1 , respectively. In this case,

when the gradient index ( k ) is increased, the elasticity modulus of the microbeam is also 16

increases ( E

0.5

E 1). In other words, the microbeam becomes stiffer with the increase

of the gradient index. As a result of this, the stiffer microbeam gives the larger nonlinear frequency and then larger nonlinear frequency ratio. In contrast, if the elasticity ratio is taken as Eratio

2 , the increasing the volume fraction exponent ( k ) causes a reverse effect on the

nonlinear frequency ratio. Besides, for a fixed value of the gradient index ( k ), the stiffness of the microbeam increases when the elasticity ratio ( Eratio ) is increased. Moreover, it can be emphasized that the gradient index ( k ) has no effect on the nonlinear frequency ratio of the microbeam for the case that Eratio

1 which means that the microbeam is homogeneous.

[Figure 2 about here] [Figure 3 about here]

Variation of the nonlinear frequency ratio with the dimensionless scale parameter for various values of the elasticity modulus ratio ( Eratio

1/ 3, 0.5, 1, 2, 3 ) is depicted in Fig. 2. One can

recognize from Fig. 2 that the frequency ratio is greatly affected by the dimensionless scale parameter. Fig. 3 shows the nonlinear frequency ratio as a function of the dimensionless vibration amplitude (

) for the same values of Eratio like in Fig. 1. Here, the scale parameter

is l / h 0.25 , and the gradient index is k 1 . It is clear that the nonlinear frequency ratio increase with the vibration amplitude, which indicates the hardening spring behavior. It can be noticed from Figs. 3a and 3b that PP microbeam is more sensitive to the effect of the large deflection than CC microbeam.

[Figure 4 about here] [Figure 5 about here]

17

Fig. 4 illustrates the effect of the elasticity ratio ( Eratio ) on the nonlinear vibrational characteristic for the some specific values of the material gradient index ( k

0, 0.3, 1, 3, 10 ).

Fig. 5 plots the nonlinear frequency ratio with the volume fraction index ( k ) for different values of Eratio . In these figures, the dimensionless scale parameter and vibration amplitude are taken as l / h 0.25 and

1 , respectively. As mentioned before, the nonlinear

frequency ratio increases depending on an increment in the elasticity modulus ratio. It can be found from Figs. 4, 5 that the gradient index plays opposite roles on the responses depending on whether Eratio

1 or Eratio

1, and as seen from Fig. 4, all curves pass through the same

point where Eratio

1. It is notable from Fig. 5 that for a specific value of Eratio , the nonlinear

frequency ratio approaches to a constant value after reaching a certain value of the gradient index.

5. Conclusions

In this study, large amplitude free vibration analysis of axially functionally graded (AFG) microbeam with immovable ends has been performed for the first time using Euler-Bernoulli beam theory in conjunction with the modified couple stress theory. The vonnonlinear strain-displacement relationship is considered to construct nonlinear model of the microbeam. It is assumed that the material properties change continuously in the axial direction according to a simple power-law form. The nonlinear governing partial differential equation and the associated boundary conditions are derived with the help of the principle. A finite degree-of-freedom system of nonlinear ordinary differential equations is obtained by using the Galerkin procedure.

s variational method is employed to obtain the

approximate closed form solution of the nonlinear ordinary governing equation. Pinnedpinned and clamped-clamped boundary conditions are considered. A detailed parametric

18

study is performed to determine the effects of the length scale parameters, material variation, vibration amplitude, and boundary conditions on nonlinear vibrational responses. Prominent observations from the numerical results can be summarized as follows:

Size-dependency on the nonlinear frequency ratio becomes significant when the thickness of AFG microbeam is comparable to the length scale parameter.

The vibration responses are greatly influenced by the material gradient index, and the effect of the material gradient index on the nonlinear frequency ratio changes according to the elasticity modulus ratio.

This indicates that by choosing a suitable gradient index, the material properties of the AFG microbeam can be tailored to meet the desired goals of optimizing the nonlinear frequency ratio the microbeam.

The frequency ratio is independent of the gradient index for homogeneous microbeam.

An increase in the elasticity modulus ratio leads to a higher nonlinear frequency and higher nonlinear frequency ratio for all values of the gradient index.

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19

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29

FIGURE CAPTIONS

Figure 1 Geometry of an axially functionally graded (AFG) simply-supported microbeam with immovable supports. Figure 2 Variation of the nonlinear frequency ratio with the dimensionless scale parameter for various values of the elasticity modulus ratio, and for k 1 ,

1 , a) PP microbeam, b)

CC microbeam.

Figure 3 Variation of the nonlinear frequency ratio with the dimensionless amplitude for various values of the elasticity modulus ratio, and for l / h 0.25 ,

1 , a) PP microbeam, b)

CC microbeam.

Figure 4 Variation of the nonlinear frequency ratio with the elasticity modulus ratio for various values of the volume fraction exponent, and for l / h 0.25 ,

1 , a) PP microbeam,

b) CC microbeam.

Figure 5 Variation of the nonlinear frequency ratio with the volume fraction exponent for various values of the elasticity modulus ratio, and for l / h 0.25 , CC microbeam.

30

1 , a) PP microbeam, b)

Table 1 The nonlinear frequency ratio values for PP AFG microbeam for Eratio

l/h

k

0

k

0.3

0.5

0 0.25 0.50 0.75 1

0.77055 0.85905 1.08197 1.37543 1.70336

0.84469 0.94164 1.18586 1.50740 1.86671

0.94087 1.04957 1.32311 1.68296 2.08489

1

0 0.25 0.50 0.75 1

1.08972 1.21489 1.53014 1.94516 2.40891

1.08972 1.21489 1.53014 1.94516 2.40891

1.08972 1.21489 1.53014 1.94516 2.40891

2

0 0.25 0.50 0.75 1

1.54110 1.71811 2.16395 2.75087 3.40672

1.45842 1.62664 2.05011 2.60728 3.22969

1.33060 1.48431 1.87116 2.38007 2.94848

31

k 1

1,

Table 2 The nonlinear frequency ratio values for CC AFG microbeam for Eratio

l/h

k

0

k

0.3

0.5

0 0.25 0.50 0.75 1

0.746119 0.837212 1.06471 1.36189 1.69244

0.836589 0.93917 1.19520 1.52948 1.90116

0.912026 1.02378 1.30275 1.66700 2.07203

1

0 0.25 0.50 0.75 1

1.05517 1.18399 1.50573 1.92601 2.39348

1.05517 1.18399 1.50573 1.92601 2.39348

1.05517 1.18399 1.50573 1.92601 2.39348

2

0 0.25 0.50 0.75 1

1.49223 1.67442 2.12942 2.72379 3.38489

1.39113 1.56087 1.98482 2.53867 3.15472

1.28980 1.44785 1.84237 2.35750 2.93030

32

k 1

1,

FIGURES

Figure 1 Geometry of an axially functionally graded (AFG) simply-supported microbeam with immovable supports.

a)

b)

Figure 2 Variation of the nonlinear frequency ratio with the dimensionless scale parameter for various values of the elasticity modulus ratio, and for k 1 , microbeam.

33

1 , a) PP microbeam, b) CC

a)

b)

Figure 3 Variation of the nonlinear frequency ratio with the dimensionless amplitude for various values of the elasticity modulus ratio, and for l / h 0.25 ,

1 , a) PP microbeam, b) CC

microbeam.

a)

b)

Figure 4 Variation of the nonlinear frequency ratio with the elasticity modulus ratio for various values of the volume fraction exponent, and for l / h 0.25 , microbeam.

34

1 , a) PP microbeam, b) CC

a)

b)

Figure 5 Variation of the nonlinear frequency ratio with the volume fraction exponent for various values of the elasticity modulus ratio, and for l / h 0.25 , microbeam.

35

1 , a) PP microbeam, b) CC