Size-dependent vibration analysis of an axially moving sandwich beam with MR core and axially FGM faces layers in yawed supersonic airflow

European Journal of Mechanics / A Solids 77 (2019) 103792

Contents lists available at ScienceDirect

European Journal of Mechanics / A Solids journal homepage: www.elsevier.com/locate/ejmsol

Size-dependent vibration analysis of an axially moving sandwich beam with MR core and axially FGM faces layers in yawed supersonic airflow

T

A. Ghorbanpour Arani∗, T. Soleymani Department of Mechanical Engineering, Faculty of Engineering, University of Kashan, Kashan, Iran

ARTICLE INFO

ABSTRACT

Keywords: Supersonic airflow MR sandwich structures Axially moving structures Size-dependence First strain gradient theory FGM

The aim of this article is to study the influences of aerodynamic pressure and axially moving behavior on the size-dependent vibration of a sandwich structure. Here, the core of sandwich structure is a magnetorheological (MR) fluid and face layers are made of functionally graded material (FGM). In order to obtain the aerodynamic pressure due to supersonic flow over upper face of structure, the linear piston theory is considered. The displacement field of sandwich structure is written according to layerwise theory and the size-dependent strain energy is obtained based on modified first strain gradient theory (MFSGT). The Hamilton's principle is applied to derive the governing equations of motion. In order to solve the partial differential equations, the Galerkin method is applied. To validate the presented formulation and solution method, the obtained results are compared with the available results in the literature, which shows a good agreement. The first set of results investigate the first five natural frequencies of MR sandwich beam based on the different MR materials. The variations of frequency and corresponding loss factor are plotted against aerodynamic pressure, length scale parameters, axially speed, intensity of external magnetic field, yaw angle, and power-law index, and the resulting trends in the plots are discussed in detail. Also, the parameters of critical aerodynamic pressure and critical axially speed in different conditions are tabulated.

1. Introduction The use of sandwich structures in aerospace applications has a long history, and through the last three decades, there have been a great number of researches in the area of smart sandwich structures. In addition, the smart structures play a key role in the recent literature to control aeroelastic phenomena such as flutter and divergence. These phenomena may profoundly impress the performance of the structures due to fatigue of thin walled components or supporting structures, excessive noise levels in vehicle compartments near the fluttering panel or functional failure of the equipment attached to the structure (Asgari and Kouchakzadeh, 2016). A typical smart sandwich structure consists of the outer facings and the adaptive core embedded between them (Birman and Kardomateas, 2018). The electrorheological (ER) and magnetorheological materials are a good choice for the core of smart sandwich structure due to the rapid response time to external stimuli (Eshaghi et al., 2015). The ER fluid can go from the consistency of a liquid to that of a gel, and back, with response times on the order of milliseconds (Khanicheh et al., 2008). The behavior of MR fluid in ambient condition without magnetic

field is such as a fluid, while by increasing the intensity of magnetic field are changed to a quasi-solid. As a comparison between ER and MR fluids, the pre yield properties exhibited by MR indicated a key advantage over those observed for ER fluid (Weiss et al., 1994). In addition, MR elastomer is a rubber-like material that under the actuation of the magnetic field display more rapid response than MR fluids (Navazi et al., 2017). The main point of these material is change in shear modulus due to change in external electric or magnetic field. Therefore, the total stiffness of these materials is dependent on external field. Moreover, the top and bottom layers of sandwich structures are investigated by many researchers. Generally, these layers undertake the strength role in adaptive sandwich structures. The effect of nanocomposite facesheets in sandwich plate with ER core was studied by Ghorbanpour et al. (Ghorbanpour Arani et al., 2017). They found that the stability of structure increase with increasing volume fraction of reinforcement phase in nanocomposite facesheets. Allahverdizadeh et al. (2013) analyzed the vibration behavior of sandwich beams with ER core and functionally graded material (FGM) facesheets, and it was reported that resonant frequencies and amplitude of peak values have been decreased by increasing FGM volume fraction index at constant

Corresponding author. E-mail addresses: [email protected], [email protected], [email protected] (A. Ghorbanpour Arani), [email protected] (T. Soleymani). ∗

https://doi.org/10.1016/j.euromechsol.2019.05.007 Received 3 March 2019; Received in revised form 19 April 2019; Accepted 18 May 2019 Available online 23 May 2019 0997-7538/ © 2019 Published by Elsevier Masson SAS.

European Journal of Mechanics / A Solids 77 (2019) 103792

A. Ghorbanpour Arani and T. Soleymani

electric field. In Ref. (Ansari et al., 2018), the authors investigated vibration of sandwich annular plate with functionally graded carbon nanotube-reinforced composite facesheets and homogeneous core. They claim that the effect of carbon nanotubes distribution model on the vibration of sandwich plate become more remarkable when the face sheet-to-core thickness ratio increases. There are numerous publications that dealt with investigation of size effect in mechanical structures (Vardoulakis et al., 1998; Chasiotis and Knauss, 2003; Poole et al., 1996; McFarland and Colton, 2005; Chong and Lam, 1999; Fleck et al., 1994). When the size of structures approaches the order of Micro and Nano, the behavior of structure becomes dependent on the size (Jafari and Ezzati, 2017). Classical theory (CT) of continuum mechanics is not capable to predict the size effect on the mechanical behaviors (Hosseini and Bahaadini, 2016). Up to now, generally, two methods are presented to consider the influence of small size scale in the continuum mechanics. One of them is estimation of mechanical properties such as elasticity modulus, shear modulus, and Poisson's ratio in small scale (Spanos and Kontsos, 2008; Shokrieh and Rafiee, 2010a, 2010b), and the other is derivation of size dependent differential equation of motion. There exist several theories in continuum mechanics about derivation of size dependent governing equations for instance integral and differential form of Eringen's nonlocal theory (Eringen, 2002), couple stress theory (Mindlin, 1962), first strain gradient theory (Mindlin and Eshel, 1968), second strain gradient theory (Mindlin, 1965), Modified couple stress theory (MCST) (Yang et al., 2002), MFSGT (Lam et al., 2003). These theories were frequently applied on various problems. The study of boundary conditions relevant to strain gradient theory is conducted by Jafari and Ezzati (2017). They draw our attention to the vital role of non-classical boundary conditions in the beam structure. The free vibration of nanorods was studied by Li et al. (2015). They assumed hard and soft constraints for boundary conditions of nanorods. Hosseini and Bahaadini (2016) presented an article about size dependent stability of a micro pipe conveying fluid. They concluded that bifurcation critical speeds and natural frequencies predicted by MFSGT are higher than CT, respectively. In Ref. (Jalaei and Ghorbanpour Arani, 2018), the authors studied static and dynamic responses of embedded double-layered graphene sheets under longitudinal magnetic field based on differential form of Eringen theory. In their paper, it is claimed that by increasing the size scale parameter, the dimensionless deflection of nanoplate increase. The Size-dependent behavior of functionally graded sandwich plate were studied based on MCST in Ref. (Trinh et al., 2017). A mathematical model of an axially moving nanoscale beam with time-dependent speed was presented in Ref. (Liu et al., 2016). The authors of this paper investigated the dynamic response and stability of structure. The vibration and stability of a piezoelectric nanoplate with time-dependent axial speed under thermo-electro-mechanical loading were studied in Ref. (Li et al., 2017a). In this article, the Kelvin-Voigt viscoelastic model was considered to obtain the constitutive relations of structure. From aeroelasticity aspect, since February 1949, the study of supersonic airflow over body of structures has been growing, because of ignition of a V-2 rocket (McNamara and Friedmann, 2006). Van Dyke (Van Dyke, 1951) and Lighthill (1953) presented their theories about two-dimensional flow at high Mach number in 1950 and 1952, respectively. Due to simplicity and accuracy, the piston theory presented by Lighthill has been frequently used and developed by researchers (Asgari and Kouchakzadeh, 2016; Brouwer and McNamara, 2019; Meijer and Dala, 2018; Dowell and Bliss, 2013; Ghoman and Azzouz, 2012; Ashley., 1956; Goldman and Dowell, 2014, 2015; Bahaadini and Saidi, 2019; Nezami and Gholami, 2019). The basic concept of this theory is based on the unsteady waves motion and isentropic relations. These subjects are well described in Ref. (Anderson, 1982). Although a great number of researchers paid attention to the small scale effect on the vibration behavior, there is still considerable ambiguity with regard to the size-dependent vibration of sandwich structures in supersonic airflow. The main goal of present article is to fill

such a gap. In addition, the axially moving behavior for sandwich structure is considered. Then, the size-dependent equations of motion for an axially moving sandwich beam with MR core and axially FGM face layers subjected to supersonic flow are derived based on the Hamilton's principle along with MFSGT. The Galerkin method is applied as the solution procedure. The validation of paper is performed by a comparison of the first five natural frequencies with those mentioned in the literature. Then, a comparison between obtained natural frequencies based on different formulations of shear modulus of MR fluid is presented. Moreover, the effects of aerodynamic pressure, magnetic field, power-law index, and yaw angle on the first natural frequencies and corresponding loss factors of the sandwich beam are studied. In addition, the influences of various parameters on the critical aerodynamic pressure and critical speed of axially moving are investigated. Finally, the variation of first and second natural frequencies with aerodynamic pressure for different values of small scale parameter are plotted. 2. Problem statement An axially moving sandwich beam with a constant speed C, length L and width b under supersonic airflow with yaw angle is considered as shown in Fig. 1. The sandwich structure is made of an elastic base layer of thickness hb , a MR core of thickness hf , and a constraining elastic layer of thickness hc . The effective properties such as young modulus, shear modulus, and material density for base and constrained layers are assumed to vary in axial direction based on following distribution function (Li et al., 2017b).

E (x ) = (ER

EL)

(x ) = (

x L

R

s

+ EL L)

x L

G (x ) = (GR s

+

L

GL)

x L

s

+ GL (1)

where R and L indices indicate material properties at the right and left ends of the face layers. Also, non-negative parameter s is power-law index that indicates the material variation profile through the length of base and constrained layers. For the sake of simplification, the following assumptions are considered (Asgari and Kouchakzadeh, 2016; Navazi et al., 2017; Yeh and Shih, 2006; Nayak et al., 2013; Howson and Zare, 2005):

• The normal stress in the MR core is negligible due to small elastic modulus in comparison to that of the elastic layers. • The elastic layers are modeled based on the Euler-Bernoulli beam • •

theory. This means that the shear deformations are equal to zero because it is considered that the cross section of these layers is perpendicular to the bending line. There is not slippage and delamination between layers during deformations due to a perfect bonding. The transverse deflections for all three layers are same as together because of incompressibility condition for MR core.

3. Basic relations The relations of displacement field for elastic layers are assumed based on the Euler-Bernoulli theory as follows (Asgari and Kouchakzadeh, 2016):

ub (x , z b, t ) = ub (x , t )

zb

uc (x , z c , t ) = uc (x , t )

zc

w (x , t ) x w (x , t ) x

w b (x , z b, t ) = w c (x , z c , t ) = w (x , t )

(2)

The b and c indices are pertinent to base and constrained layers, respectively. u and w are the in-plane and transverse displacements of the mid-plane in the x and z directions. 2

European Journal of Mechanics / A Solids 77 (2019) 103792

A. Ghorbanpour Arani and T. Soleymani

Fig. 1. A schematic Statement of Problem.

Based on the linear strain-displacement assumption, strain components may be expressed as ub

b x c x

=

x uc x

ub x

=

zb

uc x

zc

u f (x , z f , t ) =

2w

b x c x

=

Eb Ec

b x c x

=

( E (

Eb c

uc x

(3)

x2

xz

zb zc

2w

x2 2w

x2

) )

uc = x

Ab Eb

ub x

xz

(7)

=

w uf w u uc + = + b x z x hf

=

D w hf x

hb + h c w 2hf x

(M + 1) u hf

(8)

=G

xz

(9)

where G is complex shear modulus that is depend on the intensity of applied magnetic field B. 4. Differential equations of motion

(5)

Here, Hamilton's principle is applied to derive the equations of motion. This method is expressed as below

(6)

Mub

hb + hc w 2h f x

where D = hf + The constitutive equation for MR core is presented as

(4)

Integrating the above equation with respect to x , results in

uc =

uc hf

hb + hc 2

where Ec and Eb denotes the Young's modulus of elastic layers. In according to a simplifier assumption mentioned in Refs. (Asgari and Kouchakzadeh, 2016; Rajamohan et al., 2009), it is considered that the longitudinal forces in elastic layers are equal in magnitude and opposite in direction. Thus, below relation can be written:

Ac Ec

ub

The shear strain of MR core can be expressed as

x2 2w

Constitutive relations of the elastic layers are defined as ub x

zf

t2

(U

Ab c11b . Ac C11c

where M = For the sake of simplicity, hereafter u is used instead of uc . The relationship between in plan displacement for core and displacement field pertinent to elastic layers based on continuity condition is derived as follow

t1

T + W ) dt = 0

(10)

where T, U, and W indicate kinetic energy, potential energy, and external work, respectively. Furthermore, is the first variation and t1 and t2 are the limits of integration time. 3

European Journal of Mechanics / A Solids 77 (2019) 103792

A. Ghorbanpour Arani and T. Soleymani

4.1. Kinetic energy The kinetic energy of a structure is the energy that it possesses due to its motion. Here, there is an axially moving structure with constant speed C in the direction of the x axis. To obtain the velocity vector of axially moving structure, the material derivative of displacement vector should be considered (Thomas Mase and Mase, 2019; Ghayesh et al., 2013; Arani et al., 2016; Sui et al., 2015). The kinetic energy for each layer can be defined as

( (C + ( (C +

Tb =

1 2 b

ub (x , z b, t ) t

+C

Tc =

1 2 c

u c (x , z c , t ) t

+C

(C +

1 2 f

Tf =

1 2 f

Vf +

1

Ub =

uf (x , zf , t )

+C

t

II f

( ) xz

2

t

) +( ) +( ) +(

ub (x , z b, t ) 2 x u c (x , z c , t ) 2 x

w (x , z, t ) t w (x , z , t ) t

uf (x , zf , t ) 2 x

w (x , z, t ) t

+C +C +C

w (x , z , t ) 2 x

b yyx

b yyx

+

b yyz

b zxz

+

b zzx

b zzx

+

b xxx

b xxx

+

b zzz

2

c yxy

c yxy

+

c yyx

c yyx

+

c yyz

c zxz

+

c zzx

c zzx

+

c xxx

c xxx

+

c zzz

2

ijk

=

=

ki, j

+

jk ( mm, i

+

mi, m )

1 ( 2 ipq qj, p

+

jpq qi, p )

[ ij

+

+ mij

ij )

1 ( 15 ij mm, k

jk , i )

+

ki ( mm, j

i

ijk

= 2L12 G

dV

f zzz

+

f xxz

+ m xyf

f xxz f xy

+

c zzz

f xzx

+ m yxf

+

c yzy

c + m xy

f xzx f yx

+

+

f zxx

b + m yx

c xzx c yzy c xy

b zyy

c xzx

+

+

c zyy

c + m yx

f zxx

+

f zyy

b zyy

+

b yx

dVb

c zxx c zyy

+

c yx

dVc

f zyy

b xzz

c zxx

+

b xyy

+

+

c xzz

f yyz

b xzz

+

b zxz

+

c zxz

c xyy c xzz

f yyz

+

f yzy

dVf

+

+

mk, m )

=

ij

P w (x , t ) dA =

P w (x , t ) b dx

(18)

w(x, t) cos( ) x

w(x, t) t

µ

(19)

U2 M

2

1

µ=

U (M 2 (M 2

2) 1)3

(20)

where U , M , and are the free stream velocity, Mach number, and air density. The assumptions of piston theory are that the flow over body is a calorically perfect gas, isentropic, and parallel to the body surface. Also, the motion of the panel is negligible as compared with the motion of air flow (Lighthill, 1953; Ghoman and Azzouz, 2012; Ashley., 1956). In accordance with the Hamilton's principle, differential equations of motion are obtained as

1 15

mj, m )]

mij = 2L22 G

(17)

where is the yaw angle which is shown in Fig. 1, furthermore and µ are the aerodynamic pressure and damping parameter, respectively. These parameters are expressed as

(13)

(14)

ijk

f zzz

P=

where ij and ij are the kronecker delta and permutation tensor, respectively. Also, the parameters of pi , ijk , mij are the hyper stress tensors that are conjugate terms for corresponding tensors i , ijk , and ij , respectively (Lam et al., 2003). The hyper stresses are defined as follows:

pi = 2L02 G

+

xz ) xz

c yyz

c xxz

b xy

+

b zxx

In accordance with the linear piston theory (Cheng et al., 2003; Hasheminejad et al., 2013), the aerodynamic pressure can be written as

mm, i 1 ( 3 ij, k

(G

c xxz

+

+

b + m xy

b yzy

b zxx

+

The external work due to the aerodynamic pressure ( P ) can be expressed as

where ij and ij denote the tensor of classic stress and strain, respectively, which were introduced in previous section. Also, i , ijk , and ij are the dilatation gradient vector, deviatoric stretch gradient tensor, and symmetric rotation gradient tensor that are defined as below:

=

c z

b yzy

b xzx

4.3. External work

Accordance with the MFSGT, the stored strain energy for linear isotropic material is written as (Lam et al., 2003):

i

+ pzc

b zzz

+

b xzx

+

U = Ub + Uc + Uf

4.2. Potential energy

ijk ijk

c x

c xyy

W=

+ pi i +

+ pxc

b yyz

b xxz

The non-zero components of i , ijk , and ij are presented in Appendix A. Thus, the total stored strain energy can be written as

(12)

T = Tb + Tc + Tf

ij ij

c c x) x

(Ec

b xxz

+

(16)

dx

where parameter of is corresponding mass densities and V refers to volume of each layer which are shown by indices of b, c, and f. Also, IIf is the second momentum of area pertinent to core. The total of kinetic energy is expressed as

(

b z

+

f yzy

(11)

1 U= 2

+ pzb

b yxy

1

w (x , z, t ) 2 x

b x

b yxy

Uf =

c

+ pxb

+

1

b

b b x) x

(Eb

b xyy

Uc =

) ) dV ) ) dV ) d

w (x , z, t ) 2 x

2

L

[ (K1 w

+ K2 u ) w

+ (K3 w + K 4 u + K5 w + K 6 u + K7)

0

w + (K 8 w + K 9 u + K10 w + K11 u + K12 w¨ + K13 w + K14 u¨)

(15)

w + (K15 w + K16 w¨ + ( L

where G represent shear modulus and L0 , L1 , and L2 are independent higher-order length scale parameters. Here, it should be added when L0 and L1 are equal to zero, the constitutive relation reduces to MCST, besides CT is accordance with that all three higher-order parameters are equal to zero. The stored strain energies for each layer based on non-zero components of i , ijk , and ij are obtained as

[ (K2 w

bcos ( ) w

µbw )) w ] dx = 0

+ K17 u ) u + (K 4 w + K18 u + K19 w + K20 u + K21)

0

u + (K22 u + K 9 w + K23 w + K24 u + K25 u¨ + K14 w¨ ) u] dx = 0 (21) where K1 to K25 are presented in Appendix B. 5. Method of solution Here, the Galerkin method, as a weighted residual technique, is used 4

European Journal of Mechanics / A Solids 77 (2019) 103792

A. Ghorbanpour Arani and T. Soleymani

Table 1 Comparison of the first five natural frequencies. Natural Frequency (HZ) Paper

Mode Number

Present analysis Howson and Zare. Ref. (Howson and Zare, 2005) Nayak et al. Ref. (Nayak et al., 2013) Asgari and Kouchakzadeh. Ref. (Asgari and Kouchakzadeh, 2016) Rao et al. Ref. (Kameswara Rao et al., 2001)

1

2

3

4

5

57.2419 57.1358 57.146 57.139 57.068

219.858 219.585 223.919 219.585 218.569

465.427 465.172 465.932 465.171 460.925

768.162 768.177 772.133 768.163 757.642

1106.24 1106.68 1111.1 1106.626 1086.955

to solve the equations of motion. Based on this method, the solutions of governing differential equations are assumed to be approximated by a linear combination of trial functions. Also, in order to separate the variables to the temporal and spatial terms, the transverse and longitudinal displacements are assumed as

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

sandwich structure can be obtained based on the Eigen value ( ) of [A] as follows:

=

6. Validation, results, and discussion

nw wr (x ) qwr (t ) 1 nu u r (x ) qur (t ) 1

(22)

In this section, based on solution of governing equations, a parametric study of vibration response of MR sandwich beam is presented. It should be noted that because the aerodynamic damping term (µ ) always stabilizes the instability of structure (Asgari and Kouchakzadeh, 2016), in this research the effect of aerodynamic pressure ( ) of supersonic flow is only considered ( µ = 0 ). Also, in order to simplicity, it is assumed that the size scale parameters are equal together (L0 = L1 = L2 = Ls ) . To study the accuracy of this research, Table 1 presents a comparison of obtained natural frequencies of sandwich beam letting = = s = C = Ls = 0 with those reported in the literature. The geometric and material properties of MR sandwich beam for results mentioned in Table 1 are considered as

where wr (x ) and ur (x ) are the transverse and longitudinal mode shapes in natural vibration respectively. In addition, qwr (t ) and qur (t ) indicate the generalized coordinates. The number of modes for transverse and longitudinal vibration are identified by n w and nu . Here, it should be noted that the assumption of harmonic vibration and simply-supported boundary conditions are considered. By substituting these assumptions as vibrational response in the equations of motion, a set of coupled ordinary-differential equations is obtained as (23)

[M ]{q¨ (t )} + [C ]{q (t )} + [K ]{q (t )} = 0

where dot notation is referred to derivative with respect to time and the below relations show the vector of generalized coordinates as well as its derivatives.

q¨w1 (t )

q¨ (t ) =

q¨wnw (t ) q¨u1 (t )

qw1 (t )

,

q (t ) =

q¨unu (t )

qwnw (t ) qu1 (t )

Eb = Ec = 68.8 Gpa, f

qw1 (t )

,

q (t ) =

M11 M12 , M21 M22

C=

C11 C12 , C21 C22

K=

K11 K12 K21 K22

(24)

(25)

By definition of state vectors q = x1, q = x2 , the state space form of governing equation are derived as:

[M ] x2 + [C ] x2 + [K ] x1 = 0 x1 = x2

[0] [M ] 1 [K ]

,

hf = 12.7 mm ,

b = 25.4 mm ,

b

=

c

= 2700

kg m3

,

kg

f

b = 25.4 mm ,

= 3500 m3 L = 393.7 mm

The results, stated in Table 3, show that the predicted natural frequencies of MR sandwich beam when the shear modulus of MR core is based on the presented relation in Ref. (Chen et al., 2007), are more than the others. Also, it is obvious when the shear modulus of MR core of sandwich beam is based on the presented relation in Ref. (Yalcintas and Dai, 1999), there is a wide change of natural frequency by increasing magnetic field, for example the variation of first frequency from 19.1615 in B = 0 to 30.7609 in B = 0.2. The reported results in Table 3 show that the presented MR material in Ref. (Chen et al., 2007) has a good sensitivity to external magnetic field and this is a key point to choose the smart material. As a result, this kind of MR material has

(27)

[I ] [M ] 1 [C ]

kg m3

m3

hb = hc = hf = 0.7353 mm , (26)

where

[A] =

= 2680

kg

Eb = Ec = 70 Gpa,

By rewriting above equations

x1 x = [A] x1 2 x2

c

As shown in Table 1, no noteworthy differences are found between obtained natural frequencies and those mentioned in the literature. The first set of results investigate the difference between the first five natural frequencies of sandwich beam with the different relations of shear modulus of MR core. There are different kind of MR material that are investigated by researchers. Table 2 presents the three different relations of shear modulus of MR fluid reported in the literature (Rajamohan et al., 2009; Chen et al., 2007; Yalcintas and Dai, 1999). The results of this comparison are presented in Table 3 and the geometric and material properties are considered based on Ref. (Yeh and Shih, 2006) as

qu1 (t ) qunu (t )

=

b

L = 914.4 mm

qwnw (t )

qunu (t )

= 32.8

G = 82.68 Mpa ,

hb = hc = 0.4572 mm ,

The M, C, and K are the structural mass, damping, and stiffness matrices, respectively which are defined as

M=

(29)

(1 + i )

(28)

The natural frequency ( ) and corresponding loss factor ( ) of 5

European Journal of Mechanics / A Solids 77 (2019) 103792

A. Ghorbanpour Arani and T. Soleymani

Table 2 Different relations of Shear modulus (Mega Pascal) of MR core.

Table 4 Geometrical and physical parameters of the MRE sandwich beam.

Ref. (Yalcintas and Dai, 1999) G = 0.6125 + i (0.006375) under zero magnetic field G = 1250 B + i (13.75 B) with magnetic induction The unit of B is Oersted (Oe) Ref. (Rajamohan et al., 2009) G = G1 + i G2

G1 =

G = G (1 + i G )

6.9395 B 6

9.1077 B5 + 71.797 B 4

G = 5.3485 B6 17.787 B5 + 22.148 B 4 The unit of B is Tesla(T)

Notation and value

Parameter Length Width Thickness of base, constrained, and MR core

Notation and value L = 39.37 mm b = 2.54 mm hb = hc = hf = 73.53 µm

Base and Constrained layer Young's modulus at the right side Base and Constrained layer Young's modulus at the left side Base and Constrained layer shear modulus at the right side Base and Constrained layer shear modulus at the left side Base and Constrained layer density at the right side

3.3691 B2 + 4997.5 B + 0.873*106

G2 = 0.9 B2 + 812.4 B + 0.1855*106 The unit of B is Gauss(G) Ref. (Chen et al., 2007)

G =

Parameter

93.422 B3 + 38.788 B 2 + 2.43 B + 2.7006 12.185 B3 + 2.3522 B2 + 0.1526 B + 0.228

Base and Constrained layer density at the left side

Table 3 Comparison of the first five natural frequencies of sandwich beam ( = s = C = Ls = 0 ). Magnetic Field(T)

B=0

Natural Frequency (HZ)

1

2 3 4

B = 0.2

5 1

2 3 4

B = 0.4

5 1

2 3 4

B = 0.6

5 1

2 3 4

5

MR layer density

The formulation of shear modulus of MR core according to Ref. (Yalcintas and Dai, 1999)

Ref. (Rajamohan et al., 2009)

Ref. (Chen et al., 2007)

19.1615 52.3483 98.3755 159.739 237.406 30.7609 123.039 276.821 492.084 768.798 30.7609 123.039 276.821 492.085 768.8 30.7609 123.039 276.821 492.086 768.801

21.0923 57.7182 105.862 168.434 246.816 21.0981 57.7363 105.889 168.465 246.851 21.1039 57.7545 105.915 168.497 246.885 21.1097 57.7726 105.942 168.529 246.92

26.152 79.145 142.125 215.452 301.351 27.526 88.0228 160.557 242.236 334.759 28.2973 94.1685 175.1 265.081 364.746 28.4885 95.8948 179.576 272.532 374.912

( cr ) MFSGT > (

cr ) MCST

EbR = EcR = 70 Gpa

EbL = EcL = 116 Gpa

GbR = GcR = 25 Gpa GbL = GcL = 43 Gpa bR

=

cR

= 2700

bL

=

cL

= 4500

f

> (

cr )CT

= 3500

kg m3 kg m3

kg m3

(30)

For different values of magnetic flux density, the effect of dynamic pressure on the first natural frequency of the MR sandwich beam and corresponding loss factor based on MFSGT are illustrated in Fig. 3. It is observed that with increasing magnetic flux density the natural frequency increases while loss factors decreases. It is due to increasing the shear modulus of MR core that cause to increase the effective stiffness of structure. In addition, Table 5 expresses the numerical values of the calculated critical aerodynamic pressure in different magnetic field and small scale parameters. Here, it is obvious that by increasing magnetic field and small scale parameter the critical aerodynamic pressure increases. With a more detailed analysis, it can be seen that by increasing small scale parameter the effect of increasing magnetic field on rise of critical aerodynamic pressure decreases. The influence of axially velocity on the first natural frequency of the MR sandwich structure and corresponding loss factor based on CT, MCST, and MFSGT are shown in Fig. 4. It is observed that by increasing axially speed the natural frequency decays and corresponding loss factors increase. Thus, the axially moving has a softening effect on the structure. Furthermore, the type of instability due to axially moving is divergence and critical axially velocity (Ccr ) follows bellow relation:

(Ccr ) MFSGT > (Ccr ) MCST > (Ccr )CT high privilege in order to satisfy the demands of researchers. Also, the results mentioned in the highlighted part of Table 3 are in a good agreement with the stated results in Ref. (Yeh and Shih, 2006). Hereafter the results present for a sandwich beam with MR core based on the stated formulation in Ref. (Chen et al., 2007) taking the geometric and material properties in accordance with Table 4. The effect of aerodynamic pressure on the first natural frequency of the MR sandwich beam and corresponding loss factor based on CT, MCST, and MFSGT are presented in Fig. 2. The small scale parameter is considered 17.6 µm (Hosseini and Bahaadini, 2016). It is observed that the natural frequency increases with increasing aerodynamic pressure while the loss factor decays. When the sign of loss factor is changed from positive to negative or vice versa, the stability condition changes. When corresponding frequency at the point of changed sign of loss factor becomes zero, static instability (divergence) is happened, while at dynamic instability (flutter), corresponding frequency is not zero (Asgari and Kouchakzadeh, 2016; Hodges and Pierce, 2011). The parameters of aerodynamic pressure and axially moving speed at start the instability are critical parameters. As shown in Fig. 2, the type of instability due to aerodynamic pressure is flutter. In addition, the critical aerodynamic pressure ( cr ) obeys bellow relation:

(31)

The obtained result from Fig. 4 are logical, because it can be observed from Eq. (11) that the kinetic energy increase by increasing the axially velocity. So, the stiffness of structure decrease. It means that the axially moving structure tends to static instability. The effect of axially moving velocity on the first natural frequency and corresponding loss factor of the MR sandwich structure for different values of magnetic field are shown in Fig. 5. As can be seen, by increasing magnetic field natural frequency increases and loss factor decreases, so it is demonstrated that with rise of the magnetic field divergence due to axial traveling tacks place later. Table 6 illustrates the value of critical axial velocity for different values of magnetic field and small scale parameters. It is apparent from this table that the parameter of critical axial velocity increases by increasing magnetic field and small scale parameter. Moreover, by increasing the small scale parameter the effect of increasing magnetic field on rise of the critical axial velocity decreases. The effect of axial velocity on the critical aerodynamic pressure for different values of small scale parameters is investigated in Table 7. As expected, critical aerodynamic pressure decreases with increasing axial velocity. Furthermore, by increasing small scale parameter the influence of rise of the axial velocity on increasing of critical aerodynamic pressure increases. 6

European Journal of Mechanics / A Solids 77 (2019) 103792

A. Ghorbanpour Arani and T. Soleymani

Fig. 2. Effect of aerodynamic pressure on the first natural frequency and loss factor of MR sandwich beam based on CT, MCST (L2 = 17.6 µm) , and MFSGT (Ls = 17.6 µm) (B = 0.3, = s = C = 0 ).

Fig. 3. Effect of aerodynamic pressure and magnetic fields on the first natural frequency and loss factor of MR sandwich beam (Ls = 17.6 µm,

Fig. 6 illustrates the effect of different yaw angle and aerodynamic pressure on the first natural frequency and loss factor. As can be seen, with rise of the yaw angle first natural frequency decreases and the loss factors increases. In addition, the values of critical aerodynamic pressure in different values of yaw angle and small scale parameter are tabulated in Table 8. The observations detailed clearly indicate that by increasing yaw angle the critical aerodynamic pressure increases thus the situation of structure from stability aspect gets better. Also, by increasing small scale parameter the effect of rise of yaw angle on the aerodynamic pressure increases. The trend of presented results in this

= s = C = 0)

paper about the effect of yaw angle on the vibration and stability of structure has already been reported in Refs. (Cheng et al., 2003; Hasheminejad et al., 2013; Kordes et al., 1962). Here, should be mentioned that, in some papers, there are a contrary trend in the effect of yaw angle on the vibration behavior and stability of structure. This is for the boundary conditions and geometric properties of structure. Fig. 7 shows the variation of the first natural frequency and loss factor along with the aerodynamic pressure for different values of power-law index. Based on the plotted results, increasing the power-law index increases the first natural frequency while reduces the loss factor.

Table 5 Comparison of the critical aerodynamic pressure based on CT and MFSG T( = s = C = 0) . Magnetic field (T)

B=0 B = 0.3 B = 0.6

(

cr ) MFSGT

× 10 4

(

Ls = 10 µm

Ls = 15 µm

Ls = 20 µm

Ls = 25 µm

Ls = 30 µm

5.5 6.05 6.15

5.85 6.3 6.45

6.3 6.7 6.8

6.9 7.2 7.3

7.65 7.8 7.9

7

cr )CT

5.25 5.8 5.95

× 10 4

European Journal of Mechanics / A Solids 77 (2019) 103792

A. Ghorbanpour Arani and T. Soleymani

Fig. 4. Effect of axially velocity (C) on the first natural frequency and loss factor of MR sandwich beam based on CT, MCST (L2 = 17.6 µm) , and MFSGT = = s = 0 ). (Ls = 17.6 µm) (B = 0.3,

Fig. 5. Effect of axially velocity (C) and magnetic fields on the first natural frequency and loss factor of MR sandwich beam (Ls = 17.6 µm,

The critical aerodynamic pressure of MR sandwich structure in different values of power-law index and small scale parameter are stated in Table 9. Consistent with obtained results, the rise in powerlow index cause to increase the critical aerodynamic pressure. Also by increasing small scale parameter the effect of increasing of power-low index on the rise of critical aerodynamic pressure increases. In order to study the influence of aerodynamic pressure on the first and second natural frequencies for different values of small scale parameter Fig. 8 is plotted. It is obvious when the aerodynamic pressure Table 6 Comparison of the critical axial velocity based on CT and MFSGT ( = s = Magnetic field (T)

B=0 B = 0.3 B = 0.6

=

= s = 0 ).

increases, the first natural frequency increases, and there is not tangible variation for the second natural frequency at first and then it arises. Finally, both modes approach each other but do not merge due to damping property of MR layer. This result is also reported by Asgari et al. in Ref. (Asgari and Kouchakzadeh, 2016). Furthermore, it is clear that increasing the small scale parameter leads to an increase in the first and second natural frequencies. As mention previously, this is for the stiffening effect of increasing the small scale parameters at the stable position of structure. The reported results in Refs. (Li et al., 2015; Trinh

= 0) . (Ccr )CT

(Ccr ) MFSGT

Ls = 10 µm

Ls = 15 µm

Ls = 20 µm

Ls = 25 µm

Ls = 30 µm

25.2 27.2 27.7

25.8 27.7 28.2

26.7 28.5 28.9

27.8 29.4 29.8

29.1 30.5 30.9

8

24.7 26.7 27.3

European Journal of Mechanics / A Solids 77 (2019) 103792

A. Ghorbanpour Arani and T. Soleymani

Table 7 Comparison of the critical aerodynamic pressure with different values of axial velocity based on CT and MFSGT (B = 0.3, s = Axial velocity

(

C=0 C=5 C = 10

cr ) MFSGT

×

= 0) .

10 4

(

Ls = 10 µm

Ls = 15 µm

Ls = 20 µm

Ls = 25 µm

Ls = 30 µm

6.05 1.85 1.15

6.3 1.9 1.15

6.7 1.95 1.2

7.2 2.05 1.25

7.8 2.2 1.3

cr )CT

× 10 4

5.8 1.8 1.1

Fig. 6. Effect of aerodynamic pressure and yaw angle on the first natural frequency and loss factor of MR sandwich beam (B = 0.3, Ls = 17.6 µm, s = C = 0 ). Table 8 Comparison of the critical aerodynamic pressure with yaw angle based on CT and MFSGT (B = 0.3, C = s = 0) . Yaw angle

=0

= = =

8 4 3 8

(

cr ) MFSGT

× 10 4

(

cr )CT

Ls = 10 µm

Ls = 15 µm

Ls = 20 µm

Ls = 25 µm

Ls = 30 µm

6.05 6.55

6.3 6.85

6.7 7.25

7.2 7.8

7.8 8.45

5.8 6.3

8.55

8.95

9.5

10.2

11.05

8.25

15.8

16.5

17.5

18.85

20.45

15.2

Fig. 7. Effect of aerodynamic pressure and power-law index on the first natural frequency and loss factor of MR sandwich beam (B = 0.3, Ls = 17.6 µm, 9

× 10 4

= C = 0 ).

European Journal of Mechanics / A Solids 77 (2019) 103792

A. Ghorbanpour Arani and T. Soleymani

Table 9 Comparison of the critical aerodynamic pressure with power-law index based on CT and MFSGT (B = 0.3, C = Power-law index

(

s=0 s = 0.5 s=1 s=5

cr ) MFSGT

×

= 0) .

10 4

(

Ls = 10 µm

Ls = 15 µm

Ls = 20 µm

Ls = 25 µm

Ls = 30 µm

6.05 7.05 7.7 9.1

6.3 7.4 8.1 9.6

6.7 7.9 8.65 10.35

7.2 8.55 9.4 11.3

7.8 9.35 10.3 12.45

cr )CT

× 10 4

5.8 6.8 7.35 8.7

Fig. 8. Effect of aerodynamic pressure on the first and second natural frequencies of MR sandwich beam based on different values of small scale parameter (B = 0.3) ( = C = s = 0 ).

et al., 2017; Shen and Li, 2017) verify the obtained results about the effect of small scale parameter on the natural frequencies in Fig. 8.

Some of the main conclusions are summarized as.

• The critical aerodynamic pressure of MR sandwich beam follows

7. Conclusion

from bellow relation:

(

A size-dependent aeroelastic evaluation of the axially moving sandwich beam has been presented in this article. The sandwich beam has three layer that the face layers are FGM and core is MR material. The piston theory, as a supersonic theory, and layerwise theory, as structural theory, were used to develop the mathematical relations of problem. The size-dependent governing equations have been derived based on the Hamilton's principal and MFSGT. Partial differential equations of motion have been converted to a set of ordinary differential equations by using Galerkin method. In order to validate the obtained results, a comparison of the first five natural frequencies with those found in the literature have been presented. First, the influence of different relations of shear modulus for MR core on the first five natural frequencies has been investigated. Next, the effects of different parameters on the natural frequency, the corresponding loss factor, critical aerodynamic pressure, and critical axial velocity have been studied.

=

b z

=

b xxz

=

2

x2

u (x , t ) 2

x2 b xzx

zb

3

x3

b zxx

=

cr ) MCST

> (

cr )CT

from bellow relation:

(Ccr ) MFSGT > (Ccr ) MCST > (Ccr )CT

• The type of instability due to aerodynamic pressure and axially moving is flutter and divergence, respectively. • By increasing the small scale parameter, the effect of increasing

magnetic field on the critical aerodynamic pressure and axial velocity decreases.

w (x , t )

w (x , t )

=

> (

• There is an increase in frequency and a decay in loss factor with increasing magnetic field. • The critical axially moving speed of MR sandwich beam follows

Appendix A b x

cr ) MFSGT

4 2 w (x , t ) 15 x 2

10

European Journal of Mechanics / A Solids 77 (2019) 103792

A. Ghorbanpour Arani and T. Soleymani

b xyy

=

b yxy

=

b yyx

=

b yyz

=

b yzy

=

b zyy

=

b xxx

=

2 2 u (x , t ) 5 x2

b zzz

=

1 2 w (x , t ) 5 x2

b xy

=

c x

=

c z

=

b yx

2

x2 2

x2

u (x , t )

c zxx

=

c xyy

=

c yxy

=

c yyx

=

c yyz

=

c yzy

=

c zyy

=

c xxx

=

c zzz

=

c xy

=

c yx

=

f xzx

=

f yzy

=

1 5

=

3

x3

w (x , t )

x3

w (x , t )

3

zc

x3

w (x , t )

4 2 w (x , t ) 15 x 2

=

f xy

1 2 1 u (x , t ) + z b 5 x2 5

=

w (x , t )

c xzx

f zzz

3

2 zb 5

=

f zyy

b zzx

=

1 2 w (x , t ) 15 x 2

c xxz

f xxz

b zxz

=

1 2 w (x , t ) 2 x2

=

M

b xzz

2

2 M 5

x2

c xzz

c zxz

=

c zzx

=

=

2

1 M 5

x2

u (x , t ) +

3

1 zc 5

x3

w (x , t )

1 2 w (x , t ) 15 x 2

2 zc 5

u (x , t )

3

x3

w (x , t )

1 2 w (x , t ) 5 x2

f yx

1 2 w (x , t ) 2 x2

=

=

f zxx

=

f yyz

D

(

2

x2

4 = 15

=

w (x , t )

=

D

(

2

x2

w (x , t )

D

(

2

x2

w (x , t ) hf

2

x2

(

4 (M + 1) x u (x , t ) 15 hf

)+

) + 1 (M + 1) ( 5

(

)

hf

1 15

hf

1 4

D

w (x , t ) hf

x

(

)

1 (M + 1) x u (x , t ) 15 hf

u (x , t )

)

)

hf

) + 1 (M + 1) ( 4

x

u (x , t )

)

hf

Appendix B

K1 = 2Gb L 0 2IIb + K2 = K3 =

2Gb L 0 2Ib cC

2II c

+ 2A c

4 4 Gb L12IIb + 2Gc L0 2IIc + Gc L12 IIc 5 5 4 4 Gb L12Ib + 2MGc L 0 2Ic + MGc L12Ic 5 5 bC

2II b

2IIf

2 2 2 2 1 C f hb 1 C f hc 4 D 2G L12 1 D 2G L 2 2 4 1 + + 2Af + + 2Ab Gb L12 + Gb L 22 + Gb L0 2 8 hf 2 8 hf 2 15 hf 2 8 hf 2 15 2

4 1 Gc L12 + Gc L22 + Gc L0 2 + Eb IIb + Ec IIc 15 2

2 1 C hb hc II f 2 hf 2

f

11

European Journal of Mechanics / A Solids 77 (2019) 103792

A. Ghorbanpour Arani and T. Soleymani

K4 =

cC

2I M c

2 b C Ib

+

+ Ec Ic M

2 2 1 C f hb 1 C f hc + 4 hf 2 4 hf 2

K5 =

II f

K6 =

C c Ic M + C b Ib + II f

K7 =

1 C f hb If 2 hf

2

K9 =

G D2 hf 2

1 C f II f hb hc 2 hf 2

C 2 b Ab

C 2 c Ac

GD hf 2

Af

K11 = C b Ib + C c Ic M

II f

b IIb

K13 =

Ab

+

c IIc

bC

b Ib

K15 = Ab

bC

K16 = Ab

b

c Ic M

c

CAb

K21 =

C 2Ab

f

+

D fM 1 hb f 1 hc f 1 hb f M 1 hc f M + + + + 2 hf 2 2 hf 2 hf 2 2 hf 2 2 hf 2

C

f

b

+

hf 2 CIb

f

C 2M

+

b

b

f

hf 2

hf 2

+ CIc c M

+

C2

f

hf 2

C f M2

IIf

hf 2 f

+ C 2A c c M + I f

Af G M 2

+

1 Chb f 1 Chc f 1 Chb f M 1 Chc f M + + + 2 2 2 2 hf 2 hf 2 hf 2 hf 2

CAc c M 2

b

Af G

K24 = CAb

f

hf 2

CIc c M + IIf

b

K20 =

K23 =

f

2 2 Gb L12 + 2Ac Gc L 0 2M 2 + Gc L12 M 2 5 5

2 2 1CM 2 hf 2

2IIf

K22 =

2 1 Chc 4 hf 2

+

1 G L22 4 G L12 1 G L 2 2M 2 1 G L2 2 M 4 G L12 M 2 8 G L12M + + + + + 2 2 2 2 2 8 hf 15 hf 8 hf 4 hf 15 hf 15 hf 2

K18 = 2Af

K19 = CIb

f

C

f

D

IIf

+ Af

K17 = 2Ab Gb L0 2 +

2 1 Chb 4 hf 2

+ II f

1 Chb f 1 Chc f 1 Chb f M 1 Chc f M + + + 2 hf 2 2 hf 2 2 hf 2 2 hf 2

Af

+ A c c C + Af

+ Ac

f

2 2 2 1 hb hc f II f 1D f 1 hb f 1 hc f + 2 II f + + 2 2 2 2 hf 2 hf 8 hf 8 hf 2

+

Ac c C

+

C b IIb

C 2 f Af

1 Chb hc IIf 2 hf 2

K12 =

C c IIc

1 C f hc 1 C f hc M 1 C f hb 1 C f hb M + + + 2 2 2 2 hf 2 hf 2 hf 2 hf 2

K10 = C b IIb + C c IIc +

K14 =

1 MDG L 2 2 8 MDG L12 1 DG L 2 2 8 DG L12 + + + hf 2 hf 2 4 15 4 hf 2 15 hf 2

2 1 C f hc I f + C 2 c Ic + C 2 b Ib 2 hf

G DM hf 2

Af

Af

2 2 2 2 1 C f hb 1 C f hc 1 C f hb M 1 C f hc M + + + 2 2 2 2 hf 2 hf 2 hf 2 hf 2

+ II f

K 8 = Af

Eb Ib

C2

hf

+

2C f M

+

+

hf 2 f

C

f

hf 2

C 2M hf

2Af G M hf 2 IIf

+ CAc c M 2 + II f

1 Chb f 1 CMhb + 2 hf 2 2 hf 2

C f M2 hf 2

+

2C f M hf 2

f

+

+

C

1 Chc f 1 Chc f M + 2 hf 2 2 hf 2 f

hf 2

12

Ab

bC

2

Ac c C 2M 2 + Ab Eb + Ac Ec M 2

European Journal of Mechanics / A Solids 77 (2019) 103792

A. Ghorbanpour Arani and T. Soleymani

K25 = Ab

b

+ Ac c M 2 +

Where (A, I , II ) =

2IIf hf

f 2

+ 2II f

M2 hf 2

f

+

2M

f

hf 2

(1, Z , Z 2) dydz .

Appendix C. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.euromechsol.2019.05.007.

three-layered sandwich beams. J. Sound Vib. 282, 753–767. Jafari, A., Ezzati, M., 2017. Investigating the non-classical boundary conditions relevant to strain gradient theories. Phys. E Low-dimens. Syst. Nanostruct. 86, 88–102. Jalaei, M.H., Ghorbanpour Arani, A., 2018. Size-dependent static and dynamic responses of embedded double-layered graphene sheets under longitudinal magnetic field with arbitrary boundary conditions. Compos. Part B Eng. 142, 117–130. Kameswara Rao, M., Desai, Y.M., Chitnis, M.R., 2001. Free vibrations of laminated beams using mixed theory. Compos. Struct. 52, 149–160. Khanicheh, A., Mintzopoulos, D., Weinberg, B., Tzika, A.A., Mavroidis, C., 2008. Evaluation of electrorheological fluid dampers for applications at 3-T MRI environment. IEEE ASME Trans. Mechatron. 13, 286–294. Kordes, E.E., Noll, R.B., United, S., 1962. Theoretical Flutter Analysis of Flat Rectangular Panels in Uniform Coplanar Flow with Arbitrary Direction. National Aeronautics and Space Administration : [For sale by the Office of Technical Services, Dept. of Commerce], Washington, D.C. Lam, D.C.C., Yang, F., Chong, A.C.M., Wang, J., Tong, P., 2003. Experiments and Theory in Strain Gradient Elasticity. Li, C., Li, S., Yao, L., Zhu, Z., 2015. Nonlocal theoretical approaches and atomistic simulations for longitudinal free vibration of nanorods/nanotubes and verification of different nonlocal models. Appl. Math. Model. 39, 4570–4585. Li, C., Liu, J.J., Cheng, M., Fan, X.L., 2017a. Nonlocal vibrations and stabilities in parametric resonance of axially moving viscoelastic piezoelectric nanoplate subjected to thermo-electro-mechanical forces. Compos. Part B Eng. 116, 153–169. Li, X., Li, L., Hu, Y., Ding, Z., Deng, W., 2017b. Bending, buckling and vibration of axially functionally graded beams based on nonlocal strain gradient theory. Compos. Struct. 165, 250–265. Lighthill, M.J., 1953. Oscillating airfoils at high Mach number. J. Aeronaut. Sci. 20, 402–406. Liu, J., Li, C., Yang, C., Shen, J., Xie, F., 2016. Dynamical responses and stabilities of axially moving nanoscale beams with time-dependent velocity using a nonlocal stress gradient theory. J. Vib. Control 23, 3327–3344. McFarland, A.W., Colton, J.S., 2005. Role of material microstructure in plate stiffness with relevance to microcantilever sensors. J. Micromech. Microeng. 15, 1060–1067. McNamara, J., Friedmann, P., 2006. Aeroelastic and aerothermoelastic analysis of hypersonic vehicles: current status and future trends. In: 14th AIAA/AHI Space Planes and Hypersonic Systems and Technologies Conference. American Institute of Aeronautics and Astronautics. Meijer, M.-C., Dala, L., 2018. Role of higher-order terms in local piston theory. J. Aircr. 56, 388–391. Mindlin, R.D.T.H., 1962. Effects of couple-stresses in linear elasticity. Archive for rational mechanics and analysis. Arch. Ration. Mech. Anal. 11, 415–448. Mindlin, R.D., 1965. Second gradient of strain and surface-tension in linear elasticity. Int. J. Solids Struct. 1, 417–438. Mindlin, R.D., Eshel, N.N., 1968. On first strain-gradient theories in linear elasticity. Int. J. Solids Struct. 4, 109–124. Navazi, H.M., Bornassi, S., Haddadpour, H., 2017. Vibration analysis of a rotating magnetorheological tapered sandwich beam. Int. J. Mech. Sci. 122, 308–317. Nayak, B., Dwivedy, S.K., Murthy, K.S.R.K., 2013. Dynamic stability of magnetorheological elastomer based adaptive sandwich beam with conductive skins using FEM and the harmonic balance method. Int. J. Mech. Sci. 77, 205–216. Nezami, M., Gholami, B., 2019. Optimal locations of magnetorheological fluid pockets embedded in an elastically supported honeycomb sandwich beams for supersonic flutter suppression. Eur. J. Mech. A Solid. 74, 81–95. Poole, W.J., Ashby, M.F., Fleck, N.A., 1996. Micro-hardness of annealed and work-hardened copper polycrystals. Scripta Mater. 34, 559–564. Rajamohan, V., Sedaghati, R., Rakheja, S., 2009. Vibration Analysis of a Multi-Layer Beam Containing Magnetorheological Fluid. Shen, J.P., Li, C., 2017. A semi-continuum-based bending analysis for extreme-thin micro/nano-beams and new proposal for nonlocal differential constitution. Compos. Struct. 172, 210–220. Shokrieh, M.M., Rafiee, R., 2010a. Prediction of Young's modulus of graphene sheets and carbon nanotubes using nanoscale continuum mechanics approach. Mater. Des. 31, 790–795. Shokrieh, M., Rafiee, R., 2010b. Stochastic Multi-Scale Modeling of CNT/polymer Composites. Spanos, P.D., Kontsos, A., 2008. A multiscale Monte Carlo finite element method for determining mechanical properties of polymer nanocomposites. Probabilist. Eng.

References Allahverdizadeh, A., Mahjoob, M.J., Eshraghi, I., Nasrollahzadeh, N., 2013. On the vibration behavior of functionally graded electrorheological sandwich beams. Int. J. Mech. Sci. 70, 130–139. Anderson, J.D., 1982. Modern Compressible Flow, with Historical Perspective, second ed. McGraw-Hill. Ansari, R., Torabi, J., Hasrati, E., 2018. Axisymmetric nonlinear vibration analysis of sandwich annular plates with FG-CNTRC face sheets based on the higher-order shear deformation plate theory. Aero. Sci. Technol. 77, 306–319. Arani, A.G., Haghparast, E., BabaAkbar Zarei, H., 2016. Nonlocal vibration of axially moving graphene sheet resting on orthotropic visco-Pasternak foundation under longitudinal magnetic field. Phys. B Condens. Matter 495, 35–49. Asgari, M., Kouchakzadeh, M.A., 2016. Aeroelastic characteristics of magneto-rheological fluid sandwich beams in supersonic airflow. Compos. Struct. 143, 93–102. Ashley, H.Z.G., 1956. Piston theory-A new aerodynamic tool for the aeroelastician. J. Aeronaut. Sci. 23, 1109–1118. Bahaadini, R., Saidi, A.R., 2019. Aerothermoelastic flutter analysis of pre-twisted thinwalled rotating blades reinforced with functionally graded carbon nanotubes. Eur. J. Mech. A Solid. 75, 285–306. Birman, V., Kardomateas, G.A., 2018. Review of current trends in research and applications of sandwich structures. Compos. Part B Eng. 142, 221–240. Brouwer, K.R., McNamara, J.J., 2019. Enriched piston theory for expedient aeroelastic loads prediction in the presence of shock impingements. AIAA J. 57, 1288–1302. Chasiotis, I., Knauss, W.G., 2003. The mechanical strength of polysilicon films: Part 2. Size effects associated with elliptical and circular perforations. J. Mech. Phys. Solids 51, 1551–1572. Chen, L., Ae, G., Ae, J., Ae, Y., Ae, D., Li, W.-H., 2007. Investigation on Magnetorheological Elastomers Based on Natural Rubber. Cheng, G., Lee, Y.Y., Mei, C., 2003. Flow angle, temperature, and aerodynamic damping on supersonic panel flutter stability boundary. J. Aircr. 40, 248–255. Chong, A.C.M., Lam, D.C.C., 1999. Strain gradient plasticity effect in indentation hardness of polymers. J. Mater. Res. 14, 4103–4110. Dowell, E.H., Bliss, D.B., 2013. New look at unsteady supersonic potential flow aerodynamics and piston theory. AIAA J. 51, 2278–2281. Eringen, A.C., 2002. Nonlocal Continuum Field Theories. Eshaghi, M., Sedaghati, R., Rakheja, S., 2015. Dynamic characteristics and control of magnetorheological/electrorheological sandwich structures: a state-of-the-art review. J. Intell. Mater. Syst. Struct. 27, 2003–2037. Fleck, N.A., Muller, G.M., Ashby, M.F., Hutchinson, J.W., 1994. Strain gradient plasticity: theory and experiment. Acta Metall. Mater. 42, 475–487. Ghayesh, M.H., Amabili, M., Païdoussis, M.P., 2013. Nonlinear dynamics of axially moving plates. J. Sound Vib. 332, 391–406. Ghoman, S.S., Azzouz, M.S., 2012. Supersonic aerothermoelastic nonlinear flutter study of curved panels: frequency domain. J. Aircr. 49, 1075–1090. Ghorbanpour Arani, A., Jamali, S.A., BabaAkbar Zarei, H., 2017. Differential quadrature method for vibration analysis of electro-rheological sandwich plate with CNT reinforced nanocomposite facesheets subjected to electric field. Compos. Struct. 180, 211–220. Goldman, B.D., Dowell, E., 2014. In: In-Flight Aeroelastic Stability of the Thermal Protection System on the NASA HIAD, Part I: Linear Theory. 55th AIAA/ASME/ ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference. American Institute of Aeronautics and Astronautics. Goldman, B.D., Dowell, E., 2015. In-flight aeroelastic stability of the thermal protection system on the NASA HIAD, Part II: nonlinear theory and extended aerodynamics. In: 56th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference. American Institute of Aeronautics and Astronautics. Hasheminejad, S.M., Nezami, M., Panah, M.E.A., 2013. Flutter suppression of an elastically supported plate with electro-rheological fluid core under yawed supersonic flows. Int. J. Struct. Stab. Dyn. 13, 1250073. Hodges, D.H., Pierce, G.A., 2011. Introduction to Structural Dynamics and Aeroelasticity, 2 ed. Cambridge University Press, Cambridge. Hosseini, M., Bahaadini, R., 2016. Size dependent stability analysis of cantilever micropipes conveying fluid based on modified strain gradient theory. Int. J. Eng. Sci. 101, 1–13. Howson, W.P., Zare, A., 2005. Exact dynamic stiffness matrix for flexural vibration of

13

European Journal of Mechanics / A Solids 77 (2019) 103792

A. Ghorbanpour Arani and T. Soleymani Mech. 23, 456–470. Sui, S., Chen, L., Li, C., Liu, X., 2015. Transverse vibration of axially moving functionally graded materials based on timoshenko beam theory %. J. Math. Probl. Eng. 2015, 9. Thomas Mase, G.,E., Mase, G., 2019. Continuum Mechanics for Engineers/G.T. Mase, G.E. Mase. Trinh, L.C., Vo, T.P., Thai, H.-T., Mantari, J.L., 2017. Size-dependent behaviour of functionally graded sandwich microplates under mechanical and thermal loads. Compos. Part B Eng. 124, 218–241. Van Dyke, M.D., 1951. First- and second-order theory of supersonic flow past bodies of revolution. J. Aeronaut. Sci. 18, 161–178. Vardoulakis, I., Kourkoulis, S.K., Exadaktylos, G., 1998. Bending of marble with intrinsic

length scales: a gradient theory with surface energy and size effects. J. Phys. 4 (8), 399–406. Weiss, K.D., Carlson, J.D., Nixon, D.A., 1994. Viscoelastic properties of magneto- and electro-rheological fluids. J. Intell. Mater. Syst. Struct. 5, 772–775. Yalcintas, M., Dai, H., 1999. Magnetorheological and electrorheological materials in adaptive structures and their performance comparison. Smart Mater. Struct. 8, 560–573. Yang, F., Chong, A.C.M., Lam, D.C.C., Tong, P., 2002. Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct. 39, 2731–2743. Yeh, Z.-F., Shih, Y.-S., 2006. Dynamic stability of a sandwich beam with magnetorheological core. Mech. Based Des. Struct. Mach. 34, 181–200.

14