Four-variable refined plate theory for forced-vibration analysis of sigmoid functionally graded plates on elastic foundation

Author’s Accepted Manuscript Four-variable refined plate theory for forcedvibration analysis of sigmoid functionally graded plates on elastic foundation Woo-Young Jung, Sung-Cheon Han, Weon-Tae Park www.elsevier.com/locate/ijmecsci

PII: DOI: Reference:

S0020-7403(16)00072-2 http://dx.doi.org/10.1016/j.ijmecsci.2016.03.001 MS3241

To appear in: International Journal of Mechanical Sciences Received date: 6 October 2015 Revised date: 21 February 2016 Accepted date: 1 March 2016 Cite this article as: Woo-Young Jung, Sung-Cheon Han and Weon-Tae Park, Four-variable refined plate theory for forced-vibration analysis of sigmoid functionally graded plates on elastic foundation, International Journal of Mechanical Sciences, http://dx.doi.org/10.1016/j.ijmecsci.2016.03.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Four-variable refined plate theory for forced-vibration analysis of sigmoid functionally graded plates on elastic foundation

Woo-Young Jung a , Sung-Cheon Han b , Weon-Tae Park c,* a

Department of Civil Engineering, Gangneung-Wonju National University, 7 Jukheon, Gangneung 210-702, Republic of Korea

b

Department of Civil & Railroad Engineering, Daewon University College, 599 Shinwol, Jecheon 390-702, Republic of Korea c

Division of Construction and Environmental Engineering, Kongju National University, 275 Budai, Cheonan 330-717, Republic of Korea

Abstract A refined higher-order shear deformation theory is developed for the analysis of free and forced vibration of sigmoid functionally graded materials (S-FGM) plates. The theory, proposed in this paper, considers the parabolic distribution of the transverse shear stress, and satisfies the condition that requires the transverse shear stress to be zero on the upper and lower surfaces of the plate, without the shear correction factor. Unlike the conventional higher-order shear deformation theory, the refined higher-order shear deformation theory, even though it uses only four unknown variables, shares strong similarities with classical plate theory (CPT) in many aspects such as boundary conditions, equation of motion, and stress-resultant expressions. The material properties of the plate are assumed to vary according to the two power law distributions of the volume fractions of the constituents. The equations of motion are derived from Hamilton’s principle. The solutions for a simply supported plate are derived, and a comparative analysis is carried out by comparing the results obtained with firstorder shear deformation theory and another, higher-order shear deformation theory. The results of the comparative analysis with the proposed theory provide accurate and relevant results for free-vibration problems of Functionally Graded Materials (FGM) plates. Analytical solutions for the forcedvibration problems are presented so as to reveal the effects of the power law index, length, aspect ratio, loading time interval, elastic medium parameters and side-to-thickness ratio of the plate on the dynamic response. Keywords: Four-variable Refined Shear Deformation Theory, Sigmoid Functionally Graded Materials (S-FGM), Free and Forced Vibration Analysis * Corresponding author. Tel.: +82-(0)41-550-0294; Fax: +82-(0)41-850-8630. E-mail: [email protected] (W.-T. Park)

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1. Introduction Functionally Graded Material (FGM) is a type of composite material in which the properties continuously vary from the surface of one side to the surface of the other side, thus removing the stress concentration phenomenon, which is a characteristic phenomenon observed in laminated composite materials. FGM is widely used in various structures in the civil engineering, aerospace, shipbuilding and machinery sectors. Given the widespread use of engineering structures incorporating FGM, many computational models have been developed to estimate its structural behavior. Reddy and Chin [1] performed a thermomechanical analysis of FGM cylinders and plates. Birman and Bird [2] presented a system of FGM-structural modeling. Akbarzadeh et al. [3, 4] discussed the dynamic analysis and mechanical behavior of FGM plates under thermomechanical and mechanical loading, respectively. Kiani et al. [5] studied FGM a doubly-curved panel on an elastic foundation. Taj and Chakrabarti [6] investigated FGM skew plates under static and dynamic loadings. Tornabene et al. [79] explored the free vibrations and stress recovery of doubly-curved FGM shells. Fantuzzi et al. [10] presented solutions for the free vibrations of cracked four-parameter FGM plates of arbitrary shape. Such computational models have been developed according to displacement-based theories based on the principle of virtual work, as well as displacement-stress-based theories based on Reissner’s mixed variational theorem. In general, these theories can be classified into three major types: classical plate theory (CPT), first-order shear deformation theory (FSDT), and higher-order shear deformation theory (HSDT). CPT provides accurate results for thin plates but neglects the effect of transverse shear stress [11-14]. Therefore, in cases of moderately thick plates, a small deflection and a greater frequency of vibration and buckling load are computed. FSDT considers the shear deformation effect but requires a shear correction factor in order to satisfy the condition that requires the transverse shear stress to be zero on the upper and lower surfaces of the plate [15-21]. Although FSDT provides relatively accurate results in cases of thin and moderately thick plates, it has constraints in determining an accurate shear correction factor. Thus, it is not a suitable method. To avoid the use of shear correction factors, HSDTs, which assume second-order, third-order, and higher-order variations of in-plane displacements through plate thickness, have been developed by Reddy [22-23], Touratier [24], Karama et al. [25], Zenkour [26], Xiao et al. [27], Pradyumna and Bandyopadhyay [28], Talha and Singh [29], Meiche et al. [30], and Mantari et al. [31]. Some of the well-known HSDTs among those above noted are: Reddy’s theory [22,23], the sinusoidal shear deformation theory [24,26], the exponential shear deformation theory [25], and the hyperbolic shear deformation theory [30]. Recently, Tornabene et al. [32-37] discussed free vibrations of composite structures based on the HSDTs. Even though HSDTs using five unknown variables can provide sufficiently accurate results in the analysis of a thin or a thick plate, they use equations of motion that are complex compared with those of FSDT and CPT. Therefore, the development of a simple HSDT is required.

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The purpose of this study is to develop an improved higher-order shear deformation theory for the analysis of the free and forced vibrations of an S-FGM plate. This study is based on the assumption that the transverse and in-plane displacements are each divided into a bending component and a shear component. Unlike the conventional higher-order shear deformation theory, the refined higher-order shear deformation theory proposed in this study, even though it uses only four unknown variables, has strong similarities with CPT in many aspects such as boundary conditions, equation of motion and stress-resultant expressions. The material properties of the plate are assumed to vary according to the power law distribution of the volume fractions of the constituents. The equations of motion are derived from Hamilton’s principle. The results of the analysis of a simply supported plate are obtained. To verify the accuracy of the results of the analysis of the free vibration of a Functionally Graded (FG) plate, various numerical results are presented. In fact, the concept of partitioning transverse displacements into the bending and shear components was first proposed by Huffington [38] and later adopted by Krishna Murty [39], Senthilnathan et al. [40], Han et al. [41], and Shimpi [42]. The more recent theories developed by Merdaci et al. [43], Tounsi et al. [44], Thai and Vo [45] and Thai and Choi [46] are similar to those employed in the present study. However, the scopes of those studies were limited to the problems of bending and eigenvalues. In the current investigation, S-FGM plates are developed to account for the effects of the power law index, length of plate, aspect ratio, loading time interval, elastic medium parameters and plate side-to-thickness ratio. Analytical solutions to the free- and forced-vibration problems of a simply supported plate are herein presented to reveal the effects of various parameters on the natural frequency and dynamic response.

2. Theoretical formulations 2.1 Higher-order plate theory A displacement at an arbitrary point on the plate can be expressed by Eq. (1),

u1 ( x, y, z, t )  u ( x, y, t )  z

w( x, y, t )  ( z ) x ( x, y, t ) , x

u2 ( x, y, z, t )  v( x, y, t )  z

w( x, y, t )  ( z ) y ( x, y, t ) , y

(1)

u3 ( x, y, z, t )  w( x, y, t )

where u1 , u2 , u3 are displacements in the x, y, z directions, u, v and w are displacements on the mid-surface,  x and  y are rotations caused by bending on the yz and xz planes respectively, and

( z ) denotes a shape function determining changes in the transverse shear strain and stress distribution along the thickness. If ( z )  0 , the displacement field of CPT can be easily obtained.

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Or, by assuming ( z )  z , the displacement field of the first-order shear deformation theory (FSDT) can be obtained. The different shape functions derived by researchers are given in Table 1.

Table 1. Different shape functions Model Ambartsumian [47]

Third-order theory with Polynomial

Reissner [48]

Third-order theory with Polynomial

Reddy [23] Touratier [24]

( z )

Function

z  h2 z 2   2  4 3  5z  4 z 2  1 4  3h 2   4z2  z 1  2   3h  h z  sin     h 

Third-order theory with Polynomial Sinusoidal

Karama et al. [25]

Exponential

Mantari et al. [31]

Trigonometric

ze2( z / h ) mh tan mz  m sec2 ( ) z, m  0 2 2

It can be observed that Mantari, using the trigonometric HSDT, obtained results very similar to Reddy’s and Touratier’s HSDTs. Therefore, in the present study, for the analysis of an S-FGM plate, the simple shape function proposed by Reddy was used. According to Reddy's theory, the displacement field can be expressed by Eq. (8),

 4z 2  w( x, y, t ) u1 ( x, y, z, t )  u ( x, y, t )  z  z 1  2  x ( x, y, t ) , x  3h 

 4z2  w( x, y, t ) u2 ( x, y, z, t )  v( x, y, t )  z  z 1  2  y ( x, y, t ) , y  3h 

(8)

u3 ( x, y, z, t )  w( x, y, t ) where h indicates the thickness of the plate, and  x and  y are defined as

 x ( x, y, t )   x ( x, y, t ) 

w( x, y, t ) w( x, y, t ) ,  y ( x, y, t )   y ( x, y, t )  . x y

(9)

The above-noted u, v, w and  x ,  y have the same physical meanings in FSDT, each denoting the

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displacement and rotation in the transverse normal stress on plane ( z  0 ):

u  u1 ( x, y,0, t ), v  u2 ( x, y,0, t ), w  u3 ( x, y,0, t ) ,

x 

u1 u ( x, y, 0, t ),  y  2 ( x, y, 0, t ) . z z

(10)

Displacement field Eq. (8) can be expressed by  x and  y using Eq. (9):

u1 ( x, y, z, t )  u ( x, y, t )  z x ( x, y, t ) 

4z3  w( x, y, t )   ( x, y, t )  , 2  x 3h  x 

u1 ( x, y, z, t )  v( x, y, t )  z y ( x, y, t ) 

4z3  w( x, y, t )   ( x, y, t )  , 2  y 3h  y 

(11)

u3 ( x, y, z, t )  w( x, y, t ) . 2.2 Refined plate theory for functionally graded (FG) plates Unlike other shear deformation theories that use five unknown variables, the version employed in this study uses only four. This proposed theory does not require any shear correction factor, and provides a transverse shear stress changing parabolically along the thickness of the plate while satisfying its free surface conditions. 2.2.1 Assumptions of refined plate theory [38 – 46]

(a) The displacements are small in comparison with the plate thickness and, therefore, the strains involved are infinitesimal. (b) The transverse displacement u3 includes two components of bending wb , and shear ws . These components are functions only of coordinates x, y :

u3 ( x, y, t )  w( x, y, t )  wb ( x, y, t )  ws ( x, y, t )

(12)

(c) The transverse normal stress  z is negligible in comparison with in-plane stresses  x and

y . (d) The displacements u1 in the x -direction and u2 in the y -direction consist of extension, bending and shear components:

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u1, u2  u, v  ub , vb   us , vs  .

(13a)

The bending components are assumed to be similar to the displacements given by CPT:



w ( x, y, t ) , ub , vb    z  x ,  y    z  b 

x

wb ( x, y, t )  . y 

(13b)

The shear components give rise to the parabolic variations of shear strains, and the shear stresses are zero at the top and bottom surfaces of the plate. These can be given as

 ws ( x, y, t ) ws ( x, y, t )  , . x y  

us , vs    f ( z) 

(13c)

2.2.2 Displacement field of refined plate theory Now, if Eqs. (12) and (13) are substituted into Eq. (11), the new displacement field

u1 ( x, y, z, t )  u ( x, y, t )  z

wb ( x, y, t ) w ( x, y, t )  f (z) s x x

u2 ( x, y, z, t )  v( x, y, t )  z

wb ( x, y, t ) w ( x, y, t )  f (z) s y y

(14)

u3 ( x, y, z, t )  wb ( x, y, t )  ws ( x, y, t )



can be obtained, where f ( z )  z  ( z ) and  ( z )  z 1 



4z2  . 3h2 

Considering this new displacement field of Eq. (14), the liner strain rate can be derived, in Eq. (15), as

 xx(1)   xx(3)   xx   xx(0)   (1)   (3)     (0)   yy    yy   z  yy   f ( z )  yy  ,    (0)   (1)   (3)   xy   xy   xy   xy 

 u     xx(0)   x   (0)   v   yy    ,  y  (0)     xy   u v      y x 

  2 wb   2  x  (1)  xx   2  (1)    wb     ,  yy   2   (1)   y   xy    2 wb  2   xy 

(15a)

  2 ws   2  x  (3)  xx   2  (3)    ws     ,  yy   2   (3)   y   xy    2 ws  2   xy 

(15b)

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 zz(0)   zz   (0)     xz   g ( z )  xz  ,    (0)   yz   yz 

where g ( z ) 

   0   zz(0)     (0)   ws   xz    ,  x  (0)     yz   ws   y   

(15c)

d ( z) (0) (0) (0) and  xx ,  yy ,  xy  are the membrane strains,  xx(1) ,  yy(1) ,  xy(1)  are the dz

(3) (3) (3) flexural (bending) strains, and  xx ,  yy ,  xy  are the higher-order strains.

2.3 Functionally graded materials (FGM) The functions and volume fractions of two FGMs are shown in Table 2, where p is the power law index.

Table 2. Functions and volume fractions of two FGMs FGM P-FGM

Function

H ( z)

Power law function

V f ( z ) H1  1  V f ( z )  H 2

Sigmoid function

 z h/2    h 

for  h / 2  z  h / 2

V f1 ( z ) H1  1  V f1 ( z )  H 2 , S-FGM

V f ( z) p

for 0  z  h / 2

1 h/2 z  V ( z)  1    , 2 h/2 

for  h / 2  z  0

1 h/2 z  V ( z)    2 h/2 

V f2 ( z ) H1  1  V f2 ( z )  H 2 ,

p

1 f

p

2 f

By utilizing the rule of mixture, the material properties of the FGMs can be calculated as

E( z),  ( z)  H ( z) E1, 1  1  H ( z) E2 , 2  , where subscripts 1 and 2 denote the two types of materials. Stress can be obtained from plane-stress-reduced constitutive relations as

(16)

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1   xx       yy  E ( z )  0  xy   2    1   0   xz    yz  0 



0

1

0 1  0 2

0 0 0

0

0

1  2

0

0

0

0  0   xx    0   yy      xy  , 0   xz      yz  1   2 

(18)

where E ( z ) and  are the Young’s modulus and the Poisson’s ratio of the FG plate, respectively. As shown in Zimmerman [49] and Akbarzadeh et al. [50], micromechanics models significantly affect FGM results;, therefore, application of the simple rule of mixture is questionable and indeed impractical. In fact, the simple rule of mixture is valid only for metallic FGM with very close Poisson's ratios, whereas the concept of FGM is valid not only for metallic composites but others as well, and the method given in Akbarzadeh et al. [50,51] can be used for a non-constant Poisson's ratio. In this study, for simplicity, the Poisson’s ratio is assumed to be constant [52].

2.4 Elastic medium models A sigmoid functionally graded materials (S-FGM) plate embedded in an elastic medium is considered (Fig. 1). Chemical bonds are assumed to be formed between the plate and the medium. A Pasternak-type foundation model [53] polymer matrix accounts for both normal pressure and the transverse shear deformation of the surrounding elastic medium. When the shear effects are neglected, the model reduces to the Winkler-type foundation [54]. The normal pressure or Winkler elastic foundation parameter is approximated as a series of closely spaced, mutually independent, vertical linear elastic springs. Here the foundation modulus is assumed to be equivalent to the stiffness of the springs. Tornabene [55], Ceruti and Tornabene [56], and Tornabene et al. [57,58] studied the WinklerPasternak foundation effect on static and dynamic analyses of FGM and laminated doubly-curved shells. The normal pressure and the incompressible layer that resists transverse shear deformation are represented by the Winkler and Pasternak elastic medium models, respectively. These loadings are expressed as

qWinkler  kW u3  kW  wb  ws 

(19)

qPasternak  kW u3  kG2u3  kW  wb  ws   kG2  wb  ws 

(20)

where 2   2 / x 2   2 / y 2 , kW and kG denote the Winkler modulus and the shear modulus of the surrounding elastic medium, respectively. Now, consider the origin at one corner of the plate. The x coordinate of the axis is taken along the length of the plate, the y coordinate along the width, and the

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z coordinate along the thickness (Fig. 1). z

a

b

S-FGM plate E( z) ,  ( z)

h

x

y

Winkler modulus ( kW ) + shear modulus of surrounding elastic medium ( kG ) Fig. 1 S-FGM plate on Pasternak’s elastic medium

2.5 Equations of motion The strain energy of the plate is expressed by Eq. (21) as

U  

  h/2

A h / 2

 xx   yy yy   xy xy   xz xz   yz yz  dzdxdy

xx

(21)

where  is a variational operator. Eqs. (15) and (18) can be substituted into Eq. (21), and, after integrating with respect to thickness, Eq. (21) can be expressed as Eq. (22),





 U     M xx(i ) xx  M yy(i ) yy  M xy(i ) xy   M xz xz  M yz yz  dxdy A 2

 i 0



(22)

where

M

(0) ij

M

xz

, M ij(1) , M ij(2)   

h/2

h / 2

, M yz   

h/2

h / 2

 ij (1, z, f ( z )) dz,

 ij ( g ( z )) dz,

ij  xx, yy, xy  ,

ij  xz, yz  .

(23)

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The first variation of the additional strain energy induced by the elastic medium can be expressed as



 u3  u3 u3  u3      dxdy . y y    x x

 U EM    kW u3 u3  kG  A



(24)

The kinetic energy is given by Eq. (25),

K  



h/2

A h / 2

 u1  u1 u2  u2 u3  u3     dzdxdy t t t t   t t



(25)

where  is the mass density.





Vertical load q z and in-plane loads N x , N y , N xy can be expressed by Eq. (26) as



 V    qz  u3 dxdy    N x A

A



u3  u3 u  u3 u  u3   Ny 3  2 N xy 3  dxdy . x x y y x y 

(26)

Hamilton's principle is applied to derive the equation of motion [59] as

  K   (U  U T

0

EM

)   V  dt  0

(27)

which can be obtained by substituting Eqs. (22), (24), (25) and (26) into Eq. (27) and arranging the coefficients based on displacement after integrating by parts:

u :

v:

(0) M xx(0) M xy w w   m0u  m1 b  m2 s x y x x

M xy(0) x



M yy(0) y

 m0v  m1

wb w  m2 s y y

(28a)

(28b)

(1) (1) (1)   M xx(1) M xy    M xy M yy   wb :       q x  x y  y  x y  z

kW  wb  ws   kG2  wb  ws   H (u3 )

 u v   m0  wb  ws   m1     m3 2 wb  m4 2 ws  x y 

(28c)

11

(2) (2) (2)   M xx(2) M xy    M xy M yy  M xz M yz  ws :      qz    x  x y  y  x y  x y

kW  wb  ws   kP2  wb  ws   H (u3 )

 u v   m0  wb  ws   m2     m4 2 wb  m5 2 ws  x y  where H (u3 ) 

(28d)

  wb  ws       wb  ws    wb  ws       wb  ws   N xy  Ny  Nx    N xy , x  x y x y  y  

the superposed dots indicating differentiation with respect to time, and mi being the mass moment of inertia.

 m0 , m1 , m2 , m3 , m4 , m5    h / 2  1, z, h/2

f ( z ), z 2 , zf ( z ), f 2 ( z )  dz

(29)

The boundary conditions are in the following explicit forms:

(1) Free edge

M xx(0)  M xy(0)  M xz  M xx(1)  M xx(2)  0,

at x  0, a ,

M xy(0)  M yy(0)  M yz  M yy(1)  M yy(2)  0,

at y  0, b .

(30a)

(2) Simply supported edge

M xx(0)  v  wb  ws  M xx(1)  M xx(2)  0,

at x  0, a ,

M yy(0)  u  wb  ws  M yy(1)  M yy(2)  0,

at y  0, b .

(30b)

(3) Clamped edge

u  v  wb  ws 

wb ws   0, x x

at x  0, a ,

u  v  wb  ws 

wb ws   0, y y

at y  0, b .

(30c)

3. Closed-form solutions An analysis of a simply supported square plate of length a and width b under transverse load q z is carried out. The expansions of displacements based on the Navier method are assumed as in the following Eq. (30),

12

   imn t   U mn 1e  m 1 n 1  u  x, y , t      i t     Vmn  2 e mn  v  x, y, t    m 1 n 1      wb  x, y, t    Wbmn  3eimnt  ws  x, y, t       m 1 n 1    W  eimnt   m 1 n 1 smn 3

     ,      

(30)

where 1  cos  x sin y , 2  sin  x cos y , 3  sin  x sin y , and  

m n , ,  mn b a

is the frequency of vibration. The transverse load q z is expanded in double-Fourier sine series as 



qz  x, y, t    Qmn 3 . m 1 n 1

The equations of motion can be obtained in matrix form by substituting Eq. (30) into Eqs. (28a)-(28d),

K   M  Q , where

(31)

  U mn ,Vmn ,Wbmn ,Wsmn  , K  is

Q  0, 0, Qmn , Qmn 

T

the stiffness matrix,  M  is the mass matrix, and

is the load vector. For the forced vibration analysis, the Newmark time

integration method is adopted. A parametric study of an FG plate is conducted, considering two types of load conditions, in order to study the transient response of the plate. The load applied to the FG plate is applied to the entire surface, and therefore as a function of time, as

Qmn  q0 P(t )

(39)

where

P(t )  1, 0  t : step loading,

(40a)

1, 0  t  t1  P(t )    : pulse loading, t  t1  0,

(40b)

where, t1  50 ms ( ms = 10 s ). -3

4. Results and discussion In this section, the results of various numerical analyses of the free and forced vibration of a simply supported S-FGM plate are presented for verification. An S-FGM plate made of aluminum (as

13

metal) and alumina (as ceramic) is considered. The properties of the two materials are given in Table 3. Table 3. Material properties of FGM plate Properties Elastic modulus

: E (GPa )

Poisson’s ratio

:

Mass density

:  (kg/m3 )

Metal Aluminum 70

Zirconia 200

Ceramic Alumina 380

0.3

0.3

0.3

2702

5700

3800

The results verification was by comparison with the various existing FGM plate analysis theories. These theories are presented in Table 4.

Table 4. FGM plate analysis theories Theory CPT FSDT TSDT1 HSDT TSDT2 SSDT Present

(Classical plate theory) (First-order shear deformation theory) (Third-order shear deformation theory) (Hyperbolic shear deformation theory) (Third-order shear deformation theory) (Sinusoidal shear deformation theory) (New third-order shear deformation theory)

Unknowns 3 5 5 5 4 4 4

In all of the examples, FSDT used a shear correction factor of 5/6, and all of the theories were evaluated by rotary inertias. The Poisson’s ratio of the plate was assumed to be constant along the thickness (value: 0.3). The analytical results were compared using a non-dimensional form of equation

ww

E2 h3 a2 ,    1 q0 a 4 h

2 E2

, ˆ   h

1 E1

,  

a2 h

1 E1

.

(41)

4.1 Static bending and free-vibration problems

The maximum non-dimensional deflections w and of the square P-FGM plates are given, respectively, in Tables 5 and 6 for a side-to-thickness ratio a / h and the power law index p . The results are compared with those of Akbarzadeh et al. [50] using TSDT. The material properties are given by

14

E1  380GPa, E2  69GPa, 1 =3,980 kg/m3 , 2 =2,710 kg/m3 .

(42)

As shown in Tables 5 and 6, there are small discrepancies between the present results and those given by Akbarzadeh et al. [50]. This is due to the fact that Akbarzadeh et al. [50] employed a non-constant effective Poisson’s ratio.

Table 5. Comparison of non-dimensional deflections w of square plate

a/h 5 10 100

Method TSDT1[50] Present TSDT1[50] Present TSDT1[50] Present

P-FGM Power law index ( p ) 0 0.5 1 9.9796 14.9113 19.1894 10.1060 15.2675 19.7643 8.8105 13.2870 17.1282 8.7149 13.3829 17.4095 8.4241 12.7502 16.4470 8.2546 12.7595 16.6304

2 24.7938 25.6989 21.9014 22.4525 20.9452 21.3779

10 34.9869 36.5023 29.0130 29.9808 27.0369 27.8202

Table 6. Comparison of non-dimensional fundamental frequencies 1 of square plate

a/h 5 10 100

Method TSDT1[50] Present TSDT1[50] Present TSDT1[50] Present

P-FGM Power law index ( 0 0.5 10.0885 8.7277 10.2271 8.7905 10.9548 9.4370 11.1723 9.5354 11.3080 9.7234 11.5630 9.8383

p) 1 7.9227 7.9533 8.5671 8.6165 8.8277 8.8860

2 7.1877 7.1943 7.8215 7.8340 8.0824 8.0978

10 6.3804 6.3839 7.1517 7.1426 7.4913 7.4753

In the second example, the non-dimensional fundamental frequencies ˆ of the square plate are obtained for the various thickness ratios a / h and the power law index p . These are listed in Table 7. The results are compared with those of Hosseini-Hashemi and others [20] using FSDT, with those of Thai and Vo [45] using SSDT, and with those of Hosseini-Hashemi et al. [60] using HSDT. It can be observed that the results of this study are in fact in good agreement with those earlier ones. In the third example, to prove the higher-order modes in the vibration analyses, the first four non-dimensional frequencies of the square plate ( b  2a ) are compared with the references, by changing the power law index from 1 to 10 and the thickness ratio from 5 to 20. The results are shown in Table 8. These results are next compared with the non-dimensional frequencies using TSDT1 [22] and with Hosseini-Hashemi et al. [20] using FSDT. It can be observed that the present theory and the results using TSDT1 [22] agree in both the thin- and thick-plate cases. It is also apparent that the present theory provides a more accurate natural frequency than does FSDT. The maximum error of

15

5

the present study with TSDT1 is 110 % in mode 4 ( a / h  5 ), whereas that of FSDT with TSDT1 is only 3.7% in mode 4 ( a / h  5 ). However, this difference is reduced in the lower-vibration modes. For example, the differences for FSDT and TSDT1, in modes 1 to 4, increase by 110

4

and 3.7%,

respectively. The results for the present theory almost agree with those for TSDT1. Table 7. Comparison of non-dimensional fundamental frequencies ˆ of square plate S-FGM Power law index ( p ) Method a/h Homogeneous 1 2 5 Plate FSDT[20] 0.2112 0.1631 HSDT[60] 0.2113 0.1631 5 TSDT1[22] 0.2113 0.1631 0.1554 0.1484 SSDT[45] 0.2113 0.1631 Present 0.2113 0.1631 0.1554 0.1484 FSDT[20] 0.0577 0.0442 HSDT[60] 0.0577 0.0442 10 TSDT1[22] 0.0577 0.0442 0.0419 0.0398 SSDT[45] 0.0577 0.0442 Present 0.0577 0.0442 0.0419 0.0398 FSDT[20] 0.0148 0.0113 HSDT[60] 0.0148 0.0113 20 TSDT1[22] 0.0148 0.0113 0.0107 0.0102 SSDT[45] 0.0148 0.0113 Present 0.0148 0.0113 0.0107 0.0102 * TSDT1[22]: Reddy’s higher-order shear deformation theory with 5DOF

10 0.1462 0.1462 0.0392 0.0392 0.0100 0.0100

Table 8. Comparison of first four non-dimensional frequencies  of rectangular plate ( b  2a )

a/h

Mode ( m, n ) 1 (1,1)

2 (1,2) 5 3 (1,3) 4 (2,1)

Method FSDT[20] TSDT1[22] SSDT[45] Present FSDT[20] TSDT1[22] SSDT[45] Present FSDT[20] TSDT1[22] SSDT[45] Present FSDT[20] TSDT1[22] SSDT[45]

S-FGM Power law index ( p ) Homogeneous 1 2 Plate 3.4409 2.6473 3.4412 2.6475 2.5162 3.4416 2.6478 3.4412 2.6475 2.5162 5.2802 4.0773 5.2813 4.0781 3.8850 5.2822 4.0787 5.2813 4.0781 3.8850 8.0710 6.2636 8.0749 6.2663 5.9889 8.0772 6.2678 8.0749 6.2663 5.9889 9.7416 7.8711 10.1164 7.8762 7.5437 10.1201 7.8787 -

5

10

2.3976 2.3976 3.7099 3.7099 5.7366 5.7366 7.2408 -

2.3611 2.3611 3.6561 3.6561 5.6589 5.6589 7.1474 -

16

Present 10.1164 7.8762 7.5437 FSDT[20] 3.6518 2.7937 TSDT1[22] 3.6518 2.7937 2.6456 1 (1,1) SSDT[45] 3.6519 2.7937 Present 3.6518 2.7937 2.6456 FSDT[20] 5.7693 4.4192 TSDT1[22] 5.7694 4.4192 4.18842 (1,2) SSDT[45] 5.7697 4.4194 Present 5.7694 4.4192 4.1884 10 FSDT[20] 9.1876 7.0512 TSDT1[22] 9.1880 7.0515 6.6913 3 (1,3) SSDT[45] 9.1887 7.0519 Present 9.1880 7.0515 6.6913 FSDT[20] 11.8310 9.0928 TSDT1[22] 11.8315 9.0933 8.6368 4 (2,1) SSDT[45] 11.8326 9.0940 Present 11.8315 9.0933 8.6368 FSDT[20] 3.7123 2.8352 TSDT1[22] 3.7123 2.8352 2.6822 1 (1,1) SSDT[45] 3.7123 2.8353 Present 3.7123 2.8352 2.6822 FSDT[20] 5.9198 4.5228 TSDT1[22] 5.9199 4.5228 4.2796 2 (1,2) SSDT[45] 5.9199 4.5228 Present 5.9199 4.5228 4.2796 20 FSDT[20] 9.5668 7.3132 TSDT1[22] 9.5669 7.3132 6.9224 3 (1,3) SSDT[45] 9.5671 7.3133 Present 9.5669 7.3132 6.9224 FSDT[20] 12.4560 9.5261 TSDT1[22] 12.4562 9.5261 9.0195 4 (2,1) SSDT[45] 12.4565 9.5263 Present 12.4563 9.5261 9.0195 * TSDT1[22]: Reddy’s higher-order shear deformation theory with 5DOF

7.2408 2.5124 2.5124 3.9804 3.9804 6.3664 6.3664 8.2244 8.2244 2.54472.5447 4.0610 4.0610 6.5710 6.5710 8.5638 8.5638

7.1474 2.4716 2.4716 3.9166 3.9166 6.2666 6.2666 8.0976 8.0976 2.5025 2.5025 3.9940 3.9940 6.4632 6.4632 8.4240 8.4240

In the fourth example, based on the present theory, comprehensive results for nondimensional frequency  of FGM plates with simple support are tabulated in Table 9 for future comparison. There, the aspect ratios a / b are taken to be 1, while four different values of thickness ratio, a / h =5, 10, 20, and 100, are examined. Additionally, four arbitrary values of power law index

p along with three combinations of foundation parameters kW , kP are considered. The results show that the Winkler and Pasternak foundation parameters have the effect of increasing the nondimensional frequency, and that that of the Pasternak parameter is stronger. The effect of the power law index on the non-dimensional frequency is very interesting. It is observed that if a plate is just rested on the Winkler foundation, the increase of the power law index decreases the non-dimensional frequency. The situation is different if the plate is rested on the Pasternak foundation, regardless of the

17

presence/absence of the Winkler foundation. In this case, the non-dimensional frequency is almost constant with respect to the variation of the power law index. [61,62]

The following non-dimensionalizations are used:

kW 

kW a 4 k a2 E2 h3 a2 , kP  P , D  ,    D D 12(1  2 ) h

2 E2

.

(43)

Table 9. Comparison of non-dimensional fundamental frequencies  of S-FGM plate on elastic foundation

kW

kP

a/h

5

10 0

0 20

100

5

10 0

100 20

100

5

100

0

10

20

Method TSDT1[22] TSDT2[62] Present TSDT1[22] TSDT2[62] Present TSDT1[22] TSDT2[62] Present TSDT1[22] TSDT2[62] Present TSDT1[22] TSDT2[62] Present TSDT1[22] TSDT2[62] Present TSDT1[22] TSDT2[62] Present TSDT1[22] TSDT2[62] Present TSDT1[22] TSDT2[62] Present TSDT1[22] TSDT2[62] Present TSDT1[22] TSDT2[62] Present

S-FGM Power law index ( p ) Homogeneous 1 2 Plate 10.3761 8.0122 7.6328 10.3761 8.0122 10.3761 8.0122 7.6328 11.3351 8.6824 8.2288 11.3351 8.6824 11.3351 8.6824 8.2288 11.6307 8.8859 8.4080 11.6307 8.8859 11.6307 8.8859 8.4080 11.7315 8.9549 8.4686 11.7315 8.9549 8.4686 15.1868 14.3818 14.1518 15.1867 14.3818 15.1868 14.3818 14.1518 15.9732 14.9401 14.6768 15.9732 14.9401 15.9732 14.9401 14.6768 16.2263 15.1177 14.8409 16.2263 15.1177 16.2263 15.1177 14.8409 16.3138 15.1789 14.8972 16.3138 15.1789 14.8972 10.6723 8.4517 8.0912 10.6723 8.4517 10.6723 8.4517 8.0912 11.6147 9.1035 8.6716 11.6147 9.1035 11.6147 9.1035 8.6716 11.9062 9.3025 8.8471 11.9062 9.3025 11.9062 9.3025 8.8471

5

10

7.2889 7.2889 7.8202 7.8202 7.9786 7.9786 8.0319 8.0319 13.9491 13.9491 14.4478 14.4478 14.6010 14.6010 14.6533 14.6533 7.7660 7.7660 8.2845 8.2845 8.4400 8.4400

7.1830 7.1830 7.6948 7.6948 7.8469 7.8469 7.8981 7.8981 13.8879 13.8879 14.3792 14.3792 14.5292 14.5292 14.5804 14.5804 7.6662 7.6662 8.1662 8.1662 8.3156 8.3156

18

TSDT1[22] 12.0058 9.3702 8.9066 TSDT2[62] Present 12.0058 9.3702 8.9066 FSDT[38] 15.3891 14.6299 TSDT1[12] 15.3904 14.6305 14.4031 5 TSDT2[62] 15.3904 14.6305 Present 15.3904 14.6305 14.4031 FSDT[61] 16.1727 15.1886 TSDT1[22] 16.1728 15.1887 14.9295 10 TSDT2[62] 16.1728 15.1887 Present 16.1728 15.1887 14.9295 100 100 FSDT[61] 16.4249 15.3663 TSDT1[22] 16.4249 15.3663 15.0940 20 TSDT2[62] 16.4249 15.3663 Present 16.4249 15.3663 15.0940 FSDT[61] 16.5121 15.4276 TSDT1[22] 16.5121 15.4276 15.1504 100 TSDT2[62] Present 16.5121 15.4276 15.1504 * TSDT1[22]: Reddy’s higher-order shear deformation theory with 5DOF 100

8.4924 8.4924 14.1959 14.2029 14.2029 14.7019 14.7043 14.7043 14.8575 14.8582 14.8582 14.9107 14.9107 14.9107

8.3660 8.3660 14.1341 14.1425 14.1425 14.6340 14.6368 14.6368 14.7869 14.7876 14.7876 14.8390 14.8390 14.8390

4.2 Parametric studies of forced-vibration analysis The geometry and material data of the S-FGM plate are presented in Table 3. The values of

 and  in the Newmark integration scheme are both taken to be 0.5, which correspond to the constant-average acceleration method. The forced-vibration analysis results for the dynamic responses of an S-FGM plate under a distributed rectangular step load at loading time interval t  100s according to various power law index effects ( p  1, 2,5,10) are presented in Fig. 2. These results, as illustrated in Fig. 3’s graphical schematic, reflect the profound power law index effect for higher values.

19

Nondimensional Center Deflection

5

p = 0.0 p = 1.0 p = 2.0 p = 5.0 p = 10.0

4

Static Response (p = 10.0)

3

2

1

0 0

200

400

600

800

1000

Time Step

Fig. 2 Dynamic response of S-FGM plate under suddenly applied step load with variable power law index values

Fig. 3 3D graphical representation of dynamic response of S-FGM plate under suddenly applied step load with variable power law index values

20

Fig. 4 shows the variation of the dynamic response with respect to the length of the S-FGM plate, which varies from 5h to 20h . The power law index ( p) is assumed to be 10. As represented, the amplitude and frequency of the dynamic response decrease as the size of the S-FGM plate increases. 6

Nondimensinal Center Deflection

a = b = 5h a = b = 10h a = b = 20h

4

2

0 0

200

400

600

800

1000

Time Step

Fig. 4 Dynamic response of S-FGM plate under suddenly applied step load with variable plate length ( p  10)

The dynamic response due to the change of the aspect ratio of the S-FGM plate is plotted in Fig. 5. The power law index ( p) is again assumed to be 10. The aspect ratios are taken as a / b = 0.5, 0.75 and 1.0. It is shown that the amplitude and period of the dynamic response decreases as the aspect ratio increases. When, for example, the aspect ratio changes from 0.5 to 0.75, the maximum deflection is reduced by 35%; and when the aspect ratio changes from 0.75 to 1.0, the maximum deflection is reduced by 38%. Fig. 6’s 3D graphical schematic captures this effect of increasing aspect ratio ( a / b ).

21

Nondimensional Center Deflection

10

a/b = 0.50 a/b = 0.75 a/b = 1.0 8

6

4

2

0 0

200

400

600

800

1000

Time Step

Fig. 5 Dynamic response of S-FGM plate under suddenly applied step load with variable aspect ratio ( p  10)

Fig. 6 3D graphical representation of dynamic response of S-FGM plate under suddenly applied step load with variable aspect ratio ( p  10)

22

Fig. 7 plots the effect of loading time interval (t ) on the accuracy of the solutions for an SFGM plate under distributed step loading. Fig. 7 shows the non-dimensional center transverse deflection for four different loading time intervals. As is evident, the effect of a longer loading time interval is to reduce the amplitude and increase the period. For all of the loading time intervals below

500 s , the difference is not noticeable on the graphs. In all of the following examples, the loading time interval (t ) = 100 s is used. It represents the difference between the maximum deflection of loading time intervals 100s and 2000 s , which is about 8.5%. It is expected that an appropriate choice of loading time interval (t ) is required in order to obtain an accurate dynamic response.

5

t = 100 s t = 200 s

Nondimensional Center Deflection

t = 500 s t = 2,000 s

4

3

2

1

0 0

20

40

60

80

100

Time (ms)

Fig. 7 Dynamic response of plate under suddenly applied step load with variable t ( p  10)

To illustrate the effect of Winkler’s elastic medium parameter on the responses of an S-FGM plate, Fig. 8 plots the dynamic response with respect to that parameter for a simply supported S-FGM plate with kG  0 , p  10.0 , and a / h  10 . The inclusion of the elastic medium effect will increase the stiffness of the S-FGM plate and, consequently, lead to a reduction of dynamic response. Fig. 9’s 3D graphical schematic captures this effect of Winkler’s elastic medium parameter.

23

Nondimensional Center Deflecton

5

KW = 0

KW = 500

KW = 100

KW = 1,000

KW = 200 4

3

2

1

0 0

200

400

600

800

1000

Time Step

Fig. 8 Effect of elastic medium parameter ( kW ) on non-dimensional deflection of S-FGM plate ( a / h  10, p  10 )

Fig. 9 3D graphical representation of effect of elastic medium parameter ( kW ) on non-dimensional deflection of S-FGM plate ( a / h  10, p  10 )

24

Nondimensional Center Deflection

5

KG = 0

KG = 50

KG = 10

KG = 100

KG = 20 4

3

2

1

0 0

200

400

600

800

1000

Time Step

Fig. 10 Effect of elastic medium parameter ( kG ) on non-dimensional deflection of S-FGM plate ( a / h  10, p  10 , kW  0 )

Fig. 11 3D graphical representation of effect of elastic medium parameter ( kG ) on non-dimensional deflection of S-FGM plate ( a / h  10, p  10 , kW  0 )

25

To account for the effect of Pasternak’s elastic medium parameter on the responses of an SFGM plate, Fig. 10 plots the dynamic response with respect to the Pasternak’s elastic medium parameter for a simply supported S-FGM plate with kW  0 , p  10.0 and a / h  10 . The inclusion of the Pasternak’s elastic medium effect will increase the stiffness of the S-FGM plate and, consequently, lead to a reduction of dynamic response. The results show that Pasternak’s elastic medium parameter has a greater effect than Winkler’s elastic medium parameter in decreasing the dynamic response. In Fig. 11, the 3D graphical schematic captures this effect of Pasternak’s elastic medium parameter. The variation of the dynamic response of an S-FGM plate with respect to Winkler’s elastic medium effect and power law index for various Pasternak’s elastic medium effects are plotted in Fig. 12. As can be seen, the increasing value of Pasternak’s elastic medium parameter leads to a decrease in the magnitude of the dynamic response. In both Figs. 10 and 12, the results show that the dynamic response decreases with the increase of elastic medium parameters kW , kG and that Pasternak parameter kG has a greater effect in decreasing the dynamic response than does the Winkler parameter.

Nondimensional Center Deflection

2

KW = 1,000, KG = 0

KW = 1,000, KG = 50

KW = 1,000, KG = 10

KW = 1,000, KG = 100

KW = 1,000, KG = 20 1.6

1.2

0.8

0.4

0 0

200

400

600

800

1000

Time Step

Fig. 12 Effect of elastic medium parameter ( kG ) on non-dimensional deflection of S-FGM plate ( a / h  10, p  10 , kW  1,000 )

26

The effects of the side-to-thickness ratio on the dimensionless deflection with elastic medium parameters is presented for a simply supported square plate with power law index, p  10 . The symbols wb and ws denote the bending and shear components of the transverse displacement, respectively. The double-coupled dynamic response with the two-parameter elastic medium is plotted in Figs. 13 and 14. The bending and shear components of the transverse displacement are represented. It is evident that the bending component is larger than the shear. The effect of the shear component of transverse displacement is significant when the side-to-thickness ratio is small, but is negligible when the side-to-thickness ratio is larger. Subsequently, using the same material, a forced-vibration analysis of center deflection under a rectangular pulse loading is carried out. A plot of the center deflection versus time, in Fig. 14, shows that after t1  50 ms , the amplitude under the pulse load varies between −0.3 and 0.3. This transient response is caused by the removal of loading. 0.7

Nondimensional Center Deflection

0.6

0.5

0.4

0.3

0.2

0.1

0

-0.1

Wb Ws

-0.2

Wb + Ws

KW = 1,000, KG = 100

-0.3 0

20

40

60

80

100

120

140

160

Time Step

Fig. 13 Dynamic response of S-FGM plate under suddenly applied step load ( a / h  5, p  10 , kW  1,000 , kG  100 )

180

200

27

0.8

Nondimensional Center Deflection

KW = 1,000, KG = 100 0.6

0.4

0.2

0

Wb

-0.2

Ws Wb + Ws -0.4 0

200

400

600

800

1000

Time Step

Fig. 14 Dynamic response of S-FGM plate under suddenly applied pulse load ( a / h  10, p  10 , kW  1,000 , kG  100 , t1  50ms )

5. Conclusions

A refined plate theory for analysis of the free and forced vibration of an S-FGM plate is developed. This refined higher-order shear deformation theory, unlike the conventional higher-order shear deformation theory, has strong similarities with classical plate theory (CPT) in many aspects such as boundary conditions, equation of motion and stress-resultant expressions, even though it uses only four unknown variables. The equations of motion are derived from Hamilton’s principle. The analytical solutions of a simply supported plate are obtained. The accuracy of the theory proposed in this study is verified through the analysis of the free vibration of the FGM plate. The practical utility of this study can be summarized as follows.

(1) According to all of the comparative analyses, the frequency results for only four unknown variables show good agreement with those of other shear deformation theories for four or five unknown variables. The theory requires no shear correction factor in free- and forced-vibration analysis of isotropic and FGM plates. (2) Because the CPT is applied as a special case in the present study, the finite element model using this theory will be free from shear locking.

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(3) The dynamic response computed by S-FGM elasticity is always larger than the homogeneous elasticity value. The stiffness of a plate by S-FGM elasticity is small compared with that by homogeneous elasticity. Correspondingly, the amplitude and period are large. (4) In the case of an S-FGM plate of side-to-thickness ratio larger than 10, the effect of shear deformation is very small. The effect of a larger time step is to reduce the amplitude and increase the period. It is expected, therefore, that in order to obtain an accurate dynamic response, an appropriate choice of loading time interval is required. (5) As the size of an S-FGM plate decreases, the effect of S-FGM elasticity becomes more significant, and the dynamic responses are increased. With aspect ratio increases of a / b =0.5 to 0.75 and 0.75 to 1.0, the amplitude and frequency of the dynamic response increase: from 0.5 to 0.75, the maximum deflection is reduced by 35%, and from 0.75 to 1.0, there is a 38% reduction.

In summary, the theory proposed in this paper is simple and effective in terms of computational time, since it uses only four unknown variables. The dynamic response of the S-FGM plate can serve as a benchmark for future S-FGM plate structural design guidelines. Further, it will be necessary to include, for the purposes of static and dynamic analysis, a theory of S-FGM shells and various micromechanical models with effective Poisson’s ratios.

Acknowledgements This work was supported by the National Research Foundation of Korea [NRF] grant funded by the Korea Government [MEST] (No. 2011-0028531).

Conflict of Interest The authors declare that there is no conflict of interests regarding the publication of this article.

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Highlights Editors-in-Chief, International Journal of Mechanical Sciences Marian Wiercigroch University of Aberdeen, Aberdeen, UK

Four-variable refined plate theory for forced-vibration analysis of sigmoid functionally graded plates on elastic foundation Highlights

1. A four-variable refined higher-order shear deformation theory is used. 2. An S-FGM composed of two power-law functions to define volume fraction is proposed.

3.

Fig. 6 3D graphical representation of dynamic response of S-FGM plate under suddenly applied step load with variable aspect ratio ( p  10)

34

4.

Fig. 9 3D graphical representation of effect of elastic medium parameter ( kW ) on non-dimensional deflection of S-FGM plate ( a / h  10, p  10 ) 5.

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Fig. 11 3D graphical representation of effect of elastic medium parameter ( kG ) on non-dimensional deflection of S-FGM plate ( a / h  10, p  10 , kW  0 )

Weon-Tae Park Professor Weon-Tae Park Division of Construction and Environmental Engineering, Kongju National University, 275 Budai, Cheonan, 330-717, Republic of Korea Tel) +82-41-550-0294 fax) +82-41-850-8630 E-mail) [email protected]