A new exact analytical approach for free vibration of Reissner–Mindlin functionally graded rectangular plates

International Journal of Mechanical Sciences 53 (2011) 11–22

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International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci

A new exact analytical approach for free vibration of Reissner–Mindlin functionally graded rectangular plates Sh. Hosseini-Hashemi n, M. Fadaee, S.R. Atashipour School of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran 16846-13114, Iran

a r t i c l e in f o

abstract

Article history: Received 15 July 2010 Received in revised form 3 October 2010 Accepted 11 October 2010 Available online 14 October 2010

An exact closed-form procedure is presented for free vibration analysis of moderately thick rectangular plates having two opposite edges simply supported (i.e. Le´vy-type rectangular plates) based on the Reissner–Mindlin plate theory. The material properties change continuously through the thickness of the plate, which can vary according to a power law distribution of the volume fraction of the constituents. By introducing some new potential and auxiliary functions, the displacement fields are analytically obtained for this plate configuration. Several comparison studies with analytical and numerical techniques reported in literature are carried out to establish the high accuracy and reliability of the solutions. Comprehensive benchmark results for natural frequencies of the functionally graded (FG) rectangular plates with six different combinations of boundary conditions (i.e. SSSS–SSSC–SCSC–SCSF–SSSF–SFSF) are tabulated in dimensionless form for various values of aspect ratios, thickness to length ratios and the power law index. Due to the inherent features of the present exact closed-form solution, the present results will be a useful benchmark for evaluating the accuracy of other analytical and numerical methods, which will be developed by researchers in the future. & 2010 Elsevier Ltd. All rights reserved.

Keywords: Exact closed form solution Free vibration Functionally graded rectangular plates Reissner–Mindlin plate theory

1. Introduction Functionally graded materials (FGMs) are a new class of composite structures that is of great interest for engineering design and manufacture. These kinds of materials possess desirable properties for specific applications, particularly for aircrafts, space vehicles, optical, biomechanical, electronic, chemical, mechanical, shipbuilding and other engineering structures under stress concentration, high thermal and residual stresses. FGMs are heterogeneous composite materials, in which the material properties vary continuously from one interface to the other. This is achieved by gradually varying the volume fraction of the constituent materials. FG plate problems deal with two main concepts: the modeling of the plate (plate theories) and the procedure of solution. Most commonly used plate theories can be classified into three main categories: thin plate theory [1] (e.g. Kirchhoff theory or CPT), moderately thick plate theory [2] (e.g. first-order shear deformation plate theory of Mindlin or FSDT) and thick plate theory [3–5] (e.g. third-order shear deformation plate theory of Reddy or TSDT, higher-order shear deformation plate theory or HSDT and threedimensional (3-D) elasticity theory). The governing equation of

n

Corresponding author. Tel.: + 98 912 459 7032; fax: + 98 2177 240 488. E-mail addresses: [email protected] (Sh. Hosseini-Hashemi), [email protected] (M. Fadaee), [email protected] (S.R. Atashipour). 0020-7403/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2010.10.002

aforementioned plate theories must be solved through three types of solution: numerical methods [6–20] (e.g., Ritz energy method, finite element method, differential quadrature method, Galerkin method), semi-analytical methods [21–29] (e.g. power series method) or exact analytical methods [30–32] (e.g. state-space method). It is true that all researchers would like to present exact solutions for their FG plate problems on the basis of 3-D elasticity solution in which no assumptions are made. However, because of the mathematical and computational complexities, exact solutions for vibratory characteristics of FG plates are available only for simple cases (i.e., an elastic plate with either simply supported boundary conditions based on the 3-D elasticity theory or different boundary conditions based on simplified theories such as the CPT). According to a comprehensive survey of literature, it is found that a wide range of researches has been carried out on free vibration of the FG plates that the most of them used numerical solution methods. Free vibration of FG simply supported and clamped rectangular thin plates was considered by Abrate [6] using the CPT. Also, Abrate [7] analyzed free vibration, buckling and static deflections of FG square, circular and skew plates with different combinations of boundary conditions on the basis of the CPT, FSDT and TSDT. Qian et al. [8,9] carried out an analysis of free and forced vibrations of both homogeneous and FG thick plates with the higher-order shear and normal deformable plate theory using meshless local Petrov–Galerkin method. Free vibration analysis of FG simply supported square plates was studied by Pradyumna and Bandyopadhyay [10] using a higher order finite

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element formulation. Natural frequencies of FG square plates, with different boundary conditions at the edges, are obtained by Ferreira et al. [11] using the collocation method with multiquadric radial basis functions along with the FSDT and third-order shear deformation plate theory (TSDT). The static and dynamic thermo-elastic analysis of functionally graded plates (FGPs) are investigated by Praveen and Reddy [12] using the finite element method. Reddy [13] presented a theoretical formulation and finite element models for FGPs based on the third-order shear deformation theory having the thermomechanical coupling, time dependency and von Karman-type geometric nonlinearity of the plates. Free vibration of annular FG plates with variable thickness and different combinations of boundary conditions is investigated by Efraim and Eisenberger [14] based on the firstorder shear deformation theory and exact element method to derive the stiffness matrix. A new method of finite elements for analyzing Reissner–Mindlin FG plates based on the variational formulation was presented by Croce and Venini [15]. Cheng and Batra [16] have considered relations between deflection of a simply supported functionally graded polygonal plate given by a first order shear deformation theory and a third order shear deformation theory with an equivalent homogeneous Kirchhoff plate. The buckling and steady state vibrations of a simply supported functionally graded polygonal plate are carried out by Cheng and Batra [17] based on Reddy’s plate theory. Recently, Zhao et al. [18] have presented a free vibration analysis for FG square and skew plates with different boundary conditions using the elementfree kp-Ritz method on the basis of the FSDT. A finite element formulation for flexure of a symmetrically laminated plate based on a higher-order displacement model and a three-dimensional state of stress and strain is presented by Pandya and Kant [19]. The present higher-order theory incorporates linear variation of transverse normal strains and parabolic variation of transverse shear strains through the plate thickness, and as a result it does not require shear correction coefficients. Kant and Mallikarjuna [20] presented a refined higher-order theory for free vibration analysis of unsymmetrically laminated multilayered plates. The theory accounts for parabolic distribution of the transverse shear strains through the thickness of the plate and rotary inertia effects. A simple C0 finite element formulation is presented and the ninenoded Lagrangian element is chosen with seven degrees of freedom per node. Due to the coupling between in-plane and out-of-plane displacement components, the acquiring of an exact closed-form solution is more complicated. On the contrary, for isotropic plates, the coupling is vanished and presentation of an exact solution can be simpler. There are several published papers related to exact solution for free vibration of isotropic thin and thick plates [21,22], whereas, very little work is available on the analytical solution for free vibration of the FG plates. Woo et al. [23] provided an analytical solution for the nonlinear free vibration behavior of FG square thin plates using the von-Karman theory. Based on the CPT and an analytical method, Yang and Shen [24] presented free vibration and transient response of initially stressed FG rectangular thin plates subjected to impulsive lateral loads, resting on Pasternak elastic foundation. Using a semi-analytical approach, Yang and Shen [25] studied the dynamic response of FGPs subjected to impulsive lateral loads combined with initial in-plane actions in a thermal environment. Zhong and Yu [26] employed a state-space approach to analyze free and forced vibrations of an FG piezoelectric rectangular thick plate simply supported at its edges. The free vibration of FG plates with different combinations of boundary conditions are studied by Roque et al. [27] who utilized the multiquadric radial basis function method and the HSDT. Matsunaga [28] presented natural frequencies and buckling stresses of simply supported plates made of functionally graded

materials (FGMs) based on 2D higher-order approximate plate theory. Bhangale and Ganesan [29] have investigated static analysis of simply supported FG and layered magneto-electroelastic plates. Very recently, Hosseini-Hashemi et al. [30] presented an exact closed-form frequency equation for free vibration analysis of circular and annular moderately thick FG plates based on the Mindlin’s first-order shear deformation plate theory. Based on the Reddy’s third-order shear deformation plate theory, exact closed-form solutions in explicit forms are presented by Hosseini-Hashemi et al. [31] for transverse vibration analysis of rectangular thick plates having two opposite edges hard simply supported (i.e., Le´vy-type rectangular plates). In addition, Hosseini-Hashemi et al. [32] presented a new exact closed-form procedure to solve free vibration analysis of functionally graded rectangular thick plates based on the Reddy’s third-order shear deformation plate theory while the plate has two opposite edges simply supported. Based on their proposed solution, five governing complicated partial differential equations of motion were exactly solved by introducing the auxiliary and potential functions and using the method of separation of variables. Hosseini-Hashemi et al. [33] presented an analytical method for free vibration analysis of moderately thick rectangular plates which are composed of functionally graded materials (FGMs) and supported by either Winkler or Pasternak elastic foundations. A new formula for the shear correction factors, used in the Mindlin plate theory, is obtained for FG plates. Using a semi-analytical approach, the exact 3-D solution for the thermo-elastic deformation of functionally graded simply supported plates is considered by Vel and Batra [34,35]. Moreover, Vel and Batra [36] described an excellent investigation on the analytical solution for free and forced vibrations of FG simply supported square plates based on the 3D elasticity solution. Based on a higher-order refined theory, Kant and Swaminathan [37] obtained the natural frequencies of simply supported composite and sandwich plates, analytically. The theoretical model incorporates laminate deformations which account for the effects of transverse shear deformation, transverse normal strain/stress and a nonlinear variation of in-plane displacements with respect to the thickness coordinate. Thus modeling the warping of transverse cross-sections more accurately and eliminating the need for shear correction coefficients. Bidirectional flexure analysis of functionally graded (FG) plate integrated with piezoelectric fiber reinforced composites (PFRC) is presented by Shiyekar et al [38]. A higher order shear and normal deformation theory is used to analyze such hybrid or smart FG plate subjected to electromechanical loading. The effective material properties at a point were estimated by either the Mori–Tanaka [39,40] or the selfconsistent schemes [41]. No exact closed-form solution is available in the literature for the free vibration analysis of rectangular FG Mindlin plates. The main objective of this paper is to present an exact close-form solution for free vibration analysis of moderately thick rectangular FG plates based on the first-order shear deformation theory. The material properties are assumed to be graded through the thickness in accordance with a power-law distribution. Hamiltonian principle is used to extract the equations of dynamic equilibrium and natural boundary conditions of the plate. Introducing potential functions and using the separation variables method, the exactclosed form solutions are obtained for both in-plane and out-ofplane displacements. To demonstrate the efficiency and accuracy of obtained results, the natural frequencies are compared with existing data available from other analytical and numerical techniques. The effect of the plate parameters such as aspect ratios, thickness to length ratios and the power law index on the natural frequencies of FG rectangular plates is considered for six combinations of classical boundary conditions, namely SSSS, SCSS, SCSC, SSSF, SFSC and SFSF.

Sh. Hosseini-Hashemi et al. / International Journal of Mechanical Sciences 53 (2011) 11–22

2. Problem formulation 2.1. Geometrical configuration Consider a flat, and moderately thick Functionally Graded rectangular plate of length a, width b and uniform thickness h,

13

as depicted in Fig. 1. The plate has two opposite edges simply supported along x2 axis (i.e. along the edges x1 ¼ 0 and x1 ¼a) while the other two edges may be free, simply supported or clamped. The Cartesian coordinate system (x1,x2,x3) is considered to extract mathematical formulations when x1 and x2 axes are located in the mid-plane of the plate. 2.2. Material properties Functionally graded materials (FGMs) are composite materials, the mechanical properties of which vary continuously due to gradually changing the volume fraction of the constituent materials, usually in the thickness direction. In this study, the properties of the plate are assumed to vary through the thickness of the plate with a power-law distribution of the volume fractions of the two materials in between the two surfaces. In fact, the top surface (x3 ¼ h=2) of the plate is ceramic-rich whereas the bottom surface (x3 ¼ h=2) is metal-rich. Poisson’s ratio n is assumed to be constant and is taken as 0.3 throughout the analyses. Young’s modulus and mass density are assumed to vary continuously through the plate thickness as Eðx3 Þ ¼ ðEc Em ÞVf ðx3 Þ þ Em , rðx3 Þ ¼ ðrc rm ÞVf ðx3 Þ þ rm ,

ð1Þ

in which the subscripts m and c represent the metallic and ceramic constituents, respectively, and the volume fraction Vf may be given by   x3 1 p þ ð2Þ Vf ðx3 Þ ¼ 2 h Fig. 1. Geometry of a Le´vy-type rectangular FG plate and coordinates.

Table 1 Material properties of the used FG plate. Material

Aluminum (Al) Alumina (Al2O3) Zirconia (ZrO2)

Properties E (GPa)

n

r (kg/m3)

70 380 200

0.3 0.3 0.3

2702 3800 5700

where p is the power law index and takes only positive values. Typical values for metal and ceramics used in the FG plate are listed in Table 1. In order to clarify behavior of Eqs. (1) and (2), the variation of Young’s modulus E in the thickness direction x3 for the Al/Al2O3 rectangular plate with various values of power law index p is shown in Fig. 2. For p ¼0 and N, the plate is fully ceramic and metallic, respectively; whereas the composition of metal and ceramic is linear for p ¼1. It is also observed from Fig. 2 that the Young’s modulus of the FG plate quickly approaches ceramic’s one for p o1 especially within 0:5 rx3 =h r 0. For p 41, the FG plate is made from a mixture in which the metal is used more than the ceramics.

Fig. 2. Variation of Young’s modulus through the dimensionless thickness of Al/Al2O3 plate.

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2.3. The FSDT assumptions

inertias Ii (i¼1,2,3) are defined by

According to the FSDT, in which the in-plane displacements are expanded as linear functions of the plate thickness and the transverse deflection is constant through the plate thickness, the displacement field is used as follows [2]:

ðI1 ,I2 ,I3 Þ ¼

u1 ðx1 ,x2 ,x3 ,tÞ ¼ uðx1 ,x2 ,tÞ þ x3 c1 ðx1 ,x2 ,tÞ

ð3aÞ

u2 ðx1 ,x2 ,x3 ,tÞ ¼ vðx1 ,x2 ,tÞ þx3 c2 ðx1 ,x2 ,tÞ

ð3bÞ

u3 ðx1 ,x2 ,x3 ,tÞ ¼ wðx1 ,x2 ,tÞ

 1 u þ uj,i , 2 i,j

2.4. Equations of motion Herein, Hamilton’s principle is used to drive equations of motion based on the FSDT. The principle can be stated as follows: 0 Z t Z þh Z b Z a 2 @ ðs11 de11 þ s22 de22 þ2s12 de12 0

!

þ 2s13 de13 þ2s23 de23 Þ dx1 dx2 dx3 dt



d 2

Z 0

t

0 @

Z

h þ2

h 2

Z 0

bZ 0

a

ð8Þ

h=2

ðs11 , s22 , s12 Þdx3

ð9aÞ

h=2

ðM1 ,M2 ,M12 Þ ¼

Z

h=2

ðs11 , s22 , s12 Þx3 dx3

ð9bÞ

h=2

ðQ1 ,Q2 Þ ¼ K 2

Z

h=2

ðs13 , s23 Þdx3

ð9cÞ

h=2

Based on the FSDT, the following boundary conditions are imposed for an edge parallel to, for example, x1-axis:

N2 ¼ 0

ð5Þ

0

Z

rðzÞð1,x3 ,x23 Þdx3

or

u¼0

ð10aÞ

ð4Þ

where a comma followed by 1, 2 or 3 denotes the differentiation with respect to x1, x2 and x3 coordinates.

h 2

h 2

ðN1 ,N2 ,N12 Þ ¼

N12 ¼ 0 i,j ¼ 1,2,3

Based on Hooke’s law, the stress–displacement relations are defined as 08 9 1 u,1 þ nv,2 > > 9 8 > > 9 8 > > c1,1 þ nc2,2 >C B> > > > s11 > B> > > > v,2 þ nu,1 > > > >C > > > > B> C > > > > > > > 1n > >s > c2,2 þ nc1,1 > > > > > > > B C > > > > > = =C = < < < 22 > B ðu þ v Þ ,2 ,1 EðzÞ B C 1n 2 s12 ¼ þ x3 B C > >C > 1n2 B> > 1n > 2 ðc1,2 þ c2,1 Þ > > > > > > > > > > > > B C s ð c þw Þ > > > > > > 23 > ,2 > 2 > > > > B C 0 > > > > > 2 > > > > : B> C > ; > : s13 ; > > @> 1n A > 0 > ; : ðc1 þw,1 Þ > 2

0

h þ2

ð3cÞ

where u and v are the mid-plane displacements and c1, c2 denote the rotational displacements about the x2 and x1 axes at the middle surface of the plate, respectively, and t is the time. It should be pointed out that the normal transverse stress s33 is assumed to be zero. The strain–displacement relations are given as

eij ¼

Z

1

rðzÞðu_ 21 þ u_ 22 þ u_ 23 Þ dx1 dx2 dx3 Adt ¼ 0 ð6Þ

Substituting Eqs. (3) and (4) into Eq. (6) and collecting the coefficients of du, dv, dw, dc1, dc2, the following equations of motion are obtained:

or

M12 ¼ 0

v¼0

or

c1 ¼ 0

ð10bÞ ð10cÞ

M2 ¼ 0

or

c2 ¼ 0

ð10dÞ

Q2 ¼ 0

or

w¼0

ð10eÞ

For generality and convenience, the following non-dimensional terms are introduced: x1 x2 b h , X2 ¼ , Z¼ , t¼ a a a a u v w ~ ¼c , c ~ ¼c ~ ¼ , c u~ ¼ , v~ ¼ , w 1 1 2 2 a a a rffiffiffiffiffi I1 I2 I3 a2 rc , I2 ¼ I1 ¼ , I3 ¼ , b¼o rc a h Ec rc a2 rc a3

X1 ¼

ð11Þ

where o is the natural frequency of the plate. Substituting Eqs. (5) into Eqs. (9a)–(9c). The non-dimensional stress resultants are expressed as ~ Þ ~ þ nc N~ 1 ¼ Aðu~ ,1 þ nv~ ,2 Þ þ Bðc 1,1 2,2

ð12aÞ

~ þc ~ Þ N~ 2 ¼ Aðnu~ ,1 þ v~ ,2 Þ þ Bðnc 1,1 2,2

ð12bÞ

i 1n h ~ þc ~ Þ Aðu~ ,2 þ v~ ,1 Þ þ Bðc N~ 12 ¼ 1,2 2,1 2

ð12cÞ

~ Þ ~ þ nc ~ 1 ¼ Bðu~ ,1 þ nv~ ,2 Þ þDðc M 1,1 2,2

ð12dÞ

~ þc ~ Þ ~ 2 ¼ Bðnu~ ,1 þ v~ ,2 Þ þDðnc M 1,1 2,2

ð12eÞ

h i ~ þc ~ Þ ~ 12 ¼ 1n Bðu~ ,2 þ v~ ,1 Þ þDðc M 1,2 2,1 2

ð12fÞ

du :

@N1 @N12 € þ ¼ I1 u€ þ I2 c 1 @x1 @x2

ð7aÞ

1n 2 ~ ~ ,1 Þ K Aðc 1 þ w Q~ 1 ¼ 2

ð12gÞ

dv :

@N2 @N12 € þ ¼ I1 v€ þ I2 c 2 @x2 @x1

ð7bÞ

1n 2 ~ ~ ,2 Þ K Aðc 2 þ w Q~ 2 ¼ 2

ð12hÞ

@Q1 @Q2 € þ ¼ I1 w @x1 @x2

ð7cÞ

dw :

where Z ðA,B,DÞ ¼

h=2

Eðx3 Þ ð1,x3 ,x23 Þdx3 1n2 B D B¼ D¼ Ec a2 Ec a3 h=2

dc1 :

@M1 @M12 € þ Q1 ¼ I2 u€ þ I3 c 1 @x1 @x2

ð7dÞ

dc2 :

@M2 @M12 € þ Q2 ¼ I2 v€ þI3 c 2 @x2 @x1

ð7eÞ

where dot-overscript convention indicates the differentiation with respect to the time variable t. The stress resultants Ni, Mi, Qi and the

A¼

A Ec a

ð13Þ

K2 denotes the transverse shear correction coefficient which is introduced in FSDT in order to improve the transverse shear rigidities of the plate and is taken as K2 ¼5/6 for the isotropic material. For free

Sh. Hosseini-Hashemi et al. / International Journal of Mechanical Sciences 53 (2011) 11–22

15

vibration, we assume that the three coordinates c1, c2 and w vary harmonically with respect to the time variable t as follows:

1n 2 1þn ~ ,1 r f3 þ f2,1 ¼ k3 f1 þk4 f3 þk5 w 2 2

ð19cÞ

c~ 1 ðX1 ,X2 ,tÞ ¼ c1 ðx1 ,x2 Þeiot , c~ 2 ðX1 ,X2 ,tÞ ¼ c2 ðx1 ,x2 Þeiot

1n 2 1þn ~ ,2 r f4 þ f2,2 ¼ k3 f2 þk4 f4 þk5 w 2 2

ð19dÞ

~ þk8 w ~ k6 f1 þ k7 f2 ¼ k5 r2 w

ð19eÞ

~ 1 ,X2 ,tÞ ¼ wðX

wðx1 ,x2 Þ iot e a

ð14Þ

pffiffiffiffiffiffiffi where i ¼ 1. Since the present paper deals with the vibration of FG rectangular plates, the in-plane forces Ni (i¼1,2,12) and moments Mi(i¼1,2,12) are coupled due to existence of B term. Then, Eqs. (7a)–(7e) should be simultaneously solved. As a result, by substituting Eqs. (12a)–(12h) into Eqs. (7a)–(7e), the equations of motion are given in dimensionless form as follows:     1n 2 1þn 1n 2 ~ 1þn ~ ~ Þ ðu~ ,1 þ v~ ,2 Þ,1 þB ðc 1,1 þ c A r u~ þ r c1 þ 2,2 ,1 2 2 2 2 2 ~ ~ 2 t2 b2 c ¼ I1 t2 b uI 1

ð15aÞ

    1n 2 1þn 1n 2 ~ 1þ n ~ ~ Þ ðu~ ,1 þ v~ ,2 Þ,2 þ B ðc 1,1 þ c A r v~ þ r c2 þ 2,2 ,2 2 2 2 2 2 ~ ~ 2 t 2 b2 c ¼ I1 t2 b vI 2

1n 2 ~ ~ ~ 3 t 2 b2 c ~ ,1 Þ ¼ I2 t2 b2 uI K Aðc 1 þ w 1 2

1n 2 ~ ~ ~ ,2 Þ ¼ I2 t2 b2 vI ~ 3 t2 b2 c K Aðc 2 þ w 2 2

ð15eÞ

2

2ðB 2ADÞ 2 2

k4 ¼ 

2

K A ð1nÞ þ 2ðBI2 AI3 Þb t2 2

1n 2 K A, k5 ¼ 2

2ðB 2ADÞ K 2 ABð1nÞ k6 ¼ 2 2ðB ADÞ

2

k7 ¼ 

K 2 A ð1nÞ 2

,

2ðB ADÞ

2

k8 ¼ I1 b t2

ð20Þ

After differentiating Eqs. (19a) and (19b) with respect to x1 and x2, respectively, two obtained equations should be added together. Thus, we have

r2 f1 ¼ k1 f1 þ k2 f2

ð16Þ

ð21Þ

Similarly, using of Eqs. (19c) and (19d) leads to ð22Þ

~ In order to solve Eqs. (15a)–(15e), it is necessary to obtain w, firstly. To this end, we introduce f1 from Eq. (19e) into Eqs. (21) and ~ is obtained by eliminating f2 (22). The transverse deflection w between two obtained equations. After simplifying and collecting terms, we reach the following equation:

r6 w~ þ a1 r4 w~ þa2 r2 w~ þ a3 w~ ¼ 0

2

@ @ r ¼ 2þ 2 @X1 @X2

ðBI AI2 Þ 2 2 b2 t2 , k2 ¼  21 b t 2 B AD B AD 2 K 2 ABð1nÞ þ 2ðDI2 BI3 Þb t2

r f2 ¼ k3 f1 þ k4 f2 þk5 r2 w~

2

2

ðDI1 BI2 Þ

2

where r denotes the Laplace operator, given by 2

k3 ¼

ð15dÞ

    1n 2 1þn 1n 2 ~ 1þn ~ ~ Þ ðu~ ,1 þ v~ ,2 Þ,2 þD ðc 1,1 þ c B r v~ þ r c2 þ 2,2 ,2 2 2 2 2 

k1 ¼

ð15bÞ

1n 2 ~ ~ þ r2 wÞ ~ ¼ I1 t2 b2 w ~ K Aðc 1,1 þ c ð15cÞ 2,2 2     1n 2 1þn 1n 2 ~ 1þn ~ ~ Þ ðu~ ,1 þ v~ ,2 Þ,1 þ D ðc 1,1 þ c B r u~ þ r c1 þ 2,2 ,1 2 2 2 2 

where

ð23Þ

where a1 ¼ k7 k1 k4 

2.5. Exact solution procedure

k8 k5

ð24aÞ

Herein, we introduce the auxiliary functions f1, f2, f3, f4 and f1, f2 as follows:

a2 ¼

k2 k5 ðk6 k3 Þ þ k4 k8 þk1 ðk4 k5 k5 k7 þk8 Þ k5

ð24bÞ

~ f1 ¼ Au~ þ Bc 1

ð17aÞ

a3 ¼

ðk2 k3 k1 k4 Þk8 k5

ð24cÞ

~ f2 ¼ Av~ þBc 2

ð17bÞ

~ f3 ¼ Bu~ þDc 1

ð17cÞ

~ f4 ¼ Bv~ þ Dc 2

ð17dÞ

f1 ¼ f1,1 þ f2,2

ð18aÞ

f2 ¼ f3,1 þ f4,2

ð18bÞ

Also, from Eqs. (19e), (21) and (22), the introduced functions f1 and f2 can be given by ~ þe2 r2 w ~ þ e3 w ~ f1 ¼ e1 r4 w

ð25aÞ

~ þe5 r2 w ~ þ e6 w ~ f2 ¼ e4 r4 w

ð25bÞ

where

By using Eqs. (17a)–(17d) and (18a)–(18b) in Eqs. (15a)–(15e), the equation of motions can be rewritten as follows: 1n 2 1þn r f1 þ f1,1 ¼ k1 f1 þ k2 f3 2 2

ð19aÞ

1n 2 1þn r f2 þ f1,2 ¼ k1 f2 þ k2 f4 2 2

ð19bÞ

e1 ¼

k5 k7 k2 k26 k7 ðk1 k6 k4 k6 þ k3 k7 Þ

ð26aÞ

e2 ¼

k2 k5 k6 þ k7 ðk4 k5 k5 k7 þ k8 Þ k2 k26 þk7 ðk1 k6 k4 k6 þk3 k7 Þ

ð26bÞ

e3 ¼

k8 ðk2 k6 þk4 k7 Þ k2 k26 k7 ðk1 k6 k4 k6 þ k3 k7 Þ

ð26cÞ

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Sh. Hosseini-Hashemi et al. / International Journal of Mechanical Sciences 53 (2011) 11–22

k5 k6 k2 k26 þk7 ðk1 k6 k4 k6 þ k3 k7 Þ

ð26dÞ

e5 ¼

k1 k5 k6 þ k3 k5 k7 þk6 ðk5 k7 þ k8 Þ k2 k26 k7 ðk1 k6 k4 k6 þ k3 k7 Þ

ð26eÞ

e6 ¼

k8 ðk1 k6 þ k3 k7 Þ k2 k26 þk7 ðk1 k6 k4 k6 þ k3 k7 Þ

ð26fÞ

e4 ¼

Eq. (23) can be solved as ~ ¼ W1 þ W2 þ W3 w

ð27Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g1  g21 4g2

a25 ¼

The coefficients Ci ði ¼ 1,2,3,4,5,6,7,8Þ are 8 ð1þ nÞdi 2Ci þ 5 k4 2k5 ð1nÞCi þ 5 a2i > > > , > > 2k3 > > > < 2 2k4 þð1nÞai Ci ¼  > 2k3 > > > > gi3 > > > : a4 g a2 þ g i5

1 i5

r2 W1 þ a21 W1 ¼ 0

ð28aÞ

di ¼ e4 a4i þe5 a2i þ e6 ,

r2 W2 þ a22 W2 ¼ 0

ð28bÞ

gi ¼

2

2 3 W3

r W3 þ a and a 3

given as follows: i ¼ 1,2,3 i ¼ 4,5

ð36Þ

i ¼ 6,7,8

2

where

where

2 1,

ð35bÞ

2

2 2,

¼0

i ¼ 1,2,3

ðdi2 ð1 þ nÞ2k5 Þð2k1 þ ð1nÞa2i Þ2di2 k3 ð1 þ nÞ ð1nÞ2

ð28cÞ

ð37aÞ ,

i ¼ 3,4,5 ð37bÞ

2 3

a a are the roots of the following third-order equation: 2

y þ a1 y þa2 y þ a3 ¼ 0

ð29Þ

Then, it can be written as follows: 1 2 1 ð223 ða21 3a2 Þ2a1 A þ 23 A2 Þ 6A  pffiffiffi pffiffiffi  1 2 1 2i23 ði þ 3Þða21 3a2 Þ4a1 A23 ð1 þi 3ÞA2 a22 ¼ 12A

a21 ¼

 pffiffiffi pffiffiffi  1 2 1 2i23 ði þ 3Þða21 3a2 Þ4a1 A þ i23 ði þ 3ÞA2 12A pffiffiffiffiffiffiffi where i ¼ 1 and

a23 ¼

By virtue of the separation variables method, one set of solutions for Eqs. (28a)–(28c) and Eqs. (32a)–(32b) can be written as

W1 ¼ A1 sinhðm1 X2 Þ þ A2 coshðm1 X2 Þ sinðx1 X1 Þ

ð38aÞ þ B1 sinhðm1 X2 Þ þB2 coshðm1 X2 Þ cosðx1 X1 Þ

ð30aÞ

ð30bÞ

ð30cÞ

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi13 A ¼ 2a31 þ 9a1 a2 27a3 þ 4ða21 3a2 Þ3 þ ð2a31 9a1 a2 þ 27a3 Þ2

Substituting Eqs. (25a), (25b) and (27) into Eqs. (19a)–(19d), the general solutions to Eqs. (19a)–(19d) in terms of the five dimensionless potentials W1, W2, W3, W4 and W5 may be expressed as



W2 ¼ A3 sinhðm2 X2 Þ þ A4 coshðm2 X2 Þ sinðx2 X1 Þ

þ B3 sinhðm2 X2 Þ þB4 coshðm2 X2 Þ cosðx2 X1 Þ

ð38bÞ



W3 ¼ A5 sinðm3 X2 Þ þ A6 cosðm3 X2 Þ sinðx3 X1 Þ

þ B5 sinðm3 X2 Þ þ B6 cosðm3 X2 Þ cosðx3 X1 Þ

ð38cÞ



W4 ¼ A7 sinhðm4 X2 Þ þ A8 coshðm4 X2 Þ cosðx4 X1 Þ

þ B7 sinhðm4 X2 Þ þB8 coshðm4 X2 Þ sinðx4 X1 Þ

ð38dÞ



W5 ¼ A9 sinhðm5 X2 Þ þ A10 coshðm5 X2 Þ cosðx5 X1 Þ

þ B9 sinhðm5 X2 Þ þB10 coshðm5 X2 Þ sinðx5 X1 Þ

ð38eÞ

f1 ¼ C1 W1,1 þC2 W2,1 þ C3 W3,1 þC4 W4,2 þ C5 W5,2

ð31aÞ

f2 ¼ C1 W1,2 þC2 W2,2 þ C3 W3,2 C4 W4,1 C5 W5,1

ð31bÞ

f3 ¼ C6 W1,1 þC7 W2,1 þ C8 W3,1 þW4,2 þ W5,2

ð31cÞ

a21 ¼ m21 þ x21 x21 40 m21 o 0

ð39aÞ

f4 ¼ C6 W1,2 þC7 W2,2 þ C8 W3,2 W4,1 W5,1

ð31dÞ

a22 ¼ m22 þ x22 x22 40 m22 o 0

ð39bÞ

where

where

r2 W4 þ a24 W4 ¼ 0

ð32aÞ

a23 ¼ m23 þ x23 x23 40 m23 4 0

ð39cÞ

r2 W5 þ a25 W5 ¼ 0

ð32bÞ

a24 ¼ m24 þ x24 x24 40 m24 o 0

ð39dÞ

a25 ¼ m25 þ x25 x25 40 m25 o 0

ð39eÞ

and a24 , a25 are the roots of the following quadratic equation: r 2 þ g1 r þ g2 ¼ 0

ð33Þ

where

g1 ¼ g2 ¼

2ðk1 þ k4 Þ 1 þ n

ð34aÞ

4k2 k3 þ 4k1 k4 ð1 þ nÞ

2

Clearly by solving Eq. (33) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g1 þ g21 4g2 a24 ¼ 2

Finally, the exact closed form displacement field of the plate according to Mindlin’s theory, are obtained by substituting Eqs. (31a)–(31d) into Eqs. (17a)–(17d) u~ ¼

Df1 þ Bf3 2

c~ 1 ¼

ð35aÞ

ð40aÞ

B AD

ð34bÞ

v~ ¼

Bf1 Af3 2

ð40bÞ

B AD

Df2 þ Bf4 2

B AD

ð40cÞ

Sh. Hosseini-Hashemi et al. / International Journal of Mechanical Sciences 53 (2011) 11–22

c~ 2 ¼

Bf2 Af4

ð40dÞ

2

B AD

Based on the FSDT, the classical boundary conditions may be obtained for an edge parallel to, for example, X1-axis as the following non-dimensional equations:

N~ 2 ¼ 0

~ ¼0 w

~ 2 ¼0 c~ 1 ¼ 0 M

ð41aÞ

 Clamped u~ ¼ 0

v~ ¼ 0

~ ¼0 w

c~ 1 ¼ 0 c~ 2 ¼ 0

ð41bÞ

 Free N~ 12 ¼ 0

2 terms pffiffiffiffiffiffiffiffiffiffiffiffiof the non-dimensional frequency parameter b ¼ oa 2 rc =Ec =h. For all calculations here, shear correction factor K and Poisson’s ratio n have been taken as 5/6 and 0.3, respectively. For the brevity, SCSF denotes that the edges x1 ¼ 0, x2 ¼0, x1 ¼a and x2 ¼b are simply supported, clamped, simply supported and free, respectively.

3.1. Comparison studies

 Simply Support u~ ¼ 0

17

N~ 2 ¼ 0

~ 12 ¼ 0 M

~ 2¼0 M

Q~ 2 ¼ 0

ð41cÞ

By changing subscripts 1 and 2 in Eqs. (41a)–(41c), the different boundary conditions are obtained for the edges X1 ¼0 and 1. 2.6. Le´vy-type solution In this section we will develop the Le´vy-type solutions corresponding to the cases where two opposite edges are simply supported and the remaining edges of the plate can have any boundary conditions. According to Fig. 1, the boundary conditions of plate at X1 ¼0 and 1 are simply supported, then Eqs. (38a)–(38e) may be written as

ð42aÞ W1 ¼ A1 sinhðm1 X2 Þ þ A2 coshðm1 X2 Þ sinðx1 X1 Þ

W2 ¼ A3 sinhðm2 X2 Þ þ A4 coshðm2 X2 Þ sinðx2 X1 Þ

ð42bÞ



W3 ¼ A5 sinðm3 X2 Þ þ A6 cosðm3 X2 Þ sinðx3 X1 Þ

ð42cÞ



W4 ¼ A7 sinhðm4 X2 Þ þ A8 coshðm4 X2 Þ cosðx4 X1 Þ

ð42dÞ



W5 ¼ A9 sinhðm5 X2 Þ þ A10 coshðm5 X2 Þ cosðx5 X1 Þ

ð42eÞ

where

x1 ¼ x2 ¼ x3 ¼ x4 ¼ x5 ¼ mp, m ¼ 1,2,3,:::::

ð43Þ

Substituting Eqs. (31a)–(31d) into three appropriate boundary conditions (i.e., Eqs. (41a)–(41c)) along the edges X2 ¼0 and Z leads to a coefficient matrix. For a nontrivial solution, the determinant of the coefficient matrix must be set to zero for each m. Solving the eigenvalue equations yields the frequency parameters b. For more information on this solution procedure, the reader is encouraged to read the most relevant earlier papers [30–32].

3. Numerical results and discussion Due to the simplicity and clarification of the presented closed form solution procedure, computing the exact frequency parameters of free flexural vibration of Le´vy-type FG rectangular plates with different combinations of free (F), simply supported (S) and clamped (C) boundary conditions can be easily possible. In this section, results are given for various values of the aspect ratios, thickness to length ratios and the power law index. Also, the FG plates are composed of Al/Al2O3 or Al/ZrO2 that their material properties are shown in Table 1. All frequencies are expressed in

To clarify the reliability and the high accuracy of the aforementioned formulation, some comparison studies are given for the types of frequency parameters of SSSS square Al/Al2O3 and Al/ZrO2 FG plates (Z ¼1) with different thickness to length ratios and the power law index. According to a beneficial literature review, three examples are categorized and the present results are compared with those available in open literature. Example 1. The convergence and stability of the present exact procedure are shown in Tables 2 and 3. According to Eqs. (1) and (2), when The power law index p tends to zero or infinite (N), the plate is isotropic, composed of fully ceramic or metallic, respectively. Keeping in mind the concept of isotropic materials as a special case of FGMs, the first and second natural frequency parameters, as provided in Tables 2 and 3, are compared with the exact solutions of Hosseini-Hashemi et al [21] using a first-order deformation theory. It is found that the results show a trend of monotonic convergence trend, and that the solutions are identical to those given in the literature. It is evident that there is a very good agreement among the results confirming the high accuracy of the current analytical approach. Example 2. In Table 4, the natural frequency parameters of the SSSS square FG plates (Z ¼1) for different values of the thickness to length ratios (t ¼0.05, 0.1 and 0.2) are presented. Fundamental frequency and the lowest third frequency parameters are given when t ¼0.05 and t ¼0.1, 0.2, respectively. The power law index p are selected as 0, 0.5, 1, 4, 10 and N. The plates are made of a mixture of aluminum (Al) and alumina (Al2O3). It should be noted that the solutions reported by Matsunaga [28] were based on a higher-order deformation theory, whereas Zhao et al. [18] employed the FSDT by using the element-free kp-Ritz method and Hosseini-Hashemi et al. [33] obtained frequency parameters with the aid of an analytical method. For simplicityp inffiffiffiffiffiffiffiffiffiffiffiffi comparison, another frequency parameter is defined as b^ ¼ oh rc =Ec . As it is seen, when t ¼0.05, due to the low thickness effect, an excellent agreement exists between the present exact results with those obtained by the FSDT numerical results [18]. It is also seen that the present exact solution is reported the good agreement with those obtained by the HSDT [28] for the thicker FG square plates (t ¼0.1, 0.2) particularly at the higher modes of vibration. In addition, the discrepancy between the FSDT [18] and three other studies (i.e., the present exact solution, FSDT [33] and HSDT [28]) is also considerable. Example 3. In Table 5, the exact solution procedure is validated by comparing the evaluation of fundamental frequency parameters pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b~ ¼ oh rm =Em for a simply supported Al/ZrO2 square plates with those of two-dimensional higher-order theory [28], threedimensional theory by employing the power series method [36], finite element HSDT method [10], finite element FSDT method [10] and an analytical FSDT solution [33]. From Table 5, it is found that the present method is in close agreement with other methods in [10,28,33,36]. The difference between the present exact natural frequencies with those obtained by the 3-D method (Vel and Batra [36]) may be due to the estimation of material properties of FG plates. In Ref. [36], the functionally graded material properties at a point were expressed by the local volume fractions and the

18

Sh. Hosseini-Hashemi et al. / International Journal of Mechanical Sciences 53 (2011) 11–22

material properties of the phases using two methods: Mori–Tanaka [39,40] and the self-consistent scheme [41], whereas, in the present exact analysis, The properties of the plate are assumed to vary through the thickness of the plate with a power-law distribution of the volume fractions of the two materials between the two surfaces.

Table 4

pffiffiffiffiffiffiffiffiffiffiffiffi Comparison of the natural frequency parameter b^ ¼ oh rc =Ec for SSSS Al/Al2O3 square plates (Z ¼1).

t ¼ h/a (m,n)a Method

From Tables 2–5, following interesting concepts can be implied: (a) The good agreement between the present exact solution and the other methods may be attributed to the fact that the exact natural frequencies have insignificant sensibility with respect to the shear correction factor K2. That is, the appropriate and stable results are obtained by taking K2 ¼5/6. (b) The proposed exact solution reveals very good agreement between the present results and those obtained by the numerical HSDT solution of Matsunaga [28] using the method of power series expansion of displacement components. (c) When the FG plate convert to its corresponding isotropic plate (for p ¼0 and N), the present exact natural frequencies are equal to those obtained by the exact solutions of isotropic plate [21] using the first order shear deformation theory. (d) The difference between the present exact solution with those obtained by the analytical FSDT solutions [33] is due to vanishing the in-plane displacement components of FG plate

Table 2 Comparison study of frequency parameter b ¼ oa2 Material

Fully ceramic

Present

Presenta

4

10

N

0.05

(1,1)

Present FSDT [18] FSDT [33]

0.0148 0.0146 0.0148

0.0125 0.0124 0.0128

0.0113 0.0112 0.0115

0.0098 0.0097 0.0101

0.0094 0.0093 0.0096

– –

0.1

(1,1)

Present HSDT [28] FSDT [18] FSDT [33] Present HSDT [28] FSDT [18] Present HSDT [28] FSDT [18]

0.0577 0.0577 0.0568 0.0577 0.1376 0.1381 0.1354 0.2112 0.2121 0.2063

0.0490 0.0492 0.0482 0.0492 0.1173 0.1180 0.1154 0.1805 0.1819 0.1764

0.0442 0.0443 0.0435 0.0445 0.1059 0.1063 0.1042 0.1631 0.1640 0.1594

0.0382 0.0381 0.0376 0.0383 0.0911 0.0904 – 0.1397 0.1383 –

0.0366 0.0364 0.0359 0.0363 0.0867 0.0859 0.0850 0.1324 0.1306 0.1289

0.0293 0.0293 – 0.0294 0.0701 0.0701 – 0.1076 0.1077 –

Present HSDT [28] FSDT [18] FSDT [33] Present HSDT [28] Present HSDT [28]

0.2112 0.2121 0.2055 0.2112 0.4618 0.4658 0.6676 0.6753

0.1805 0.1819 0.1757 0.1806 0.3978 0.4040 0.5779 0.5891

0.1631 0.1640 0.1587 0.1650 0.3604 0.3644 0.5245 0.5444

0.1397 0.1383 0.1356 0.1371 0.3049 0.3000 0.4405 0.4362

0.1324 0.1306 0.1284 0.1304 0.2856 0.2790 0.4097 0.3981

0.1076 0.1077 – 0.1075 0.2352 0.2365 0.3399 0.3429

(1,2)

(2,2)

0.2

(1,1)

(1,2) (2,2)

a

m and n are wave numbers in direction x1 and x2, respectively.

First mode

Second mode

t ¼ h/a

t ¼h/a 0.2

0.05

0.1

0.2

p ¼ 10 p ¼ 10  4 p ¼ 10  5 p ¼ 10  6

5.9191 5.9212 5.9214 5.9214 5.9214

5.7728 5.7748 5.7751 5.7751 5.7751

5.2954 5.2972 5.2974 5.2974 5.2974

14.611 14.616 14.616 14.616 14.616

13.789 13.794 13.795 13.795 13.795

11.612 11.615 11.616 11.616 11.616

p ¼ 102 p ¼ 103 p ¼ 104 p ¼ 105

3.1833 3.0329 3.0158 3.0139 3.0139

3.1002 2.9575 2.9413 2.9394 2.9394

2.8315 2.7115 2.6978 2.6963 2.6963

7.8518 7.4858 7.4443 7.4396 7.4396

7.3875 7.0622 7.0254 7.0212 7.0212

6.1756 5.9416 5.9152 5.9123 5.9123

Shear correction factor K2 ¼ 0.86667.

Table 3 Comparison study of frequency parameter b ¼ oa2 Material

Fully ceramic

pffiffiffiffiffiffiffiffiffiffiffiffi rc =Ec =h for SCSF Al/Al2O3 square plates (Z ¼ 1).

Method

Presenta

Fully metallic

Presenta

Exact FSDT [21]a Shear correction factor K2 ¼ 0.86667.

First mode

Second mode

t ¼ h/a

t ¼ h/a

0.05

0.1

0.2

0.05

0.1

0.2

p ¼ 10  3 p ¼ 10  4 p ¼ 10  5 p ¼ 10  6

3.7958 3.7971 3.7973 3.7973 3.7973

3.7088 3.7101 3.7102 3.7102 3.7102

3.4464 3.4476 3.4476 3.4477 3.4477

9.7515 9.7550 9.7553 9.7554 9.7554

9.2185 9.2216 9.2220 9.2220 9.2220

7.8347 7.8372 7.8375 7.8369 7.8369

p ¼ 102 p ¼ 103 p ¼ 104 p ¼ 105

2.0414 1.9449 1.9340 1.9328 1.9328

1.9918 1.9001 1.8896 1.8885 1.8885

1.8447 1.7648 1.7558 1.7548 1.7548

5.2399 4.9963 4.9687 4.9654 4.9654

4.9390 4.7214 4.6967 4.6939 4.6939

4.1713 4.0089 3.9909 3.9889 3.9889

Exact FSDT [21]a

a

1

0.1

Exact FSDT [21]a a

0.5

0.05 3

Exact FSDT [21]a Fully metallic

0

pffiffiffiffiffiffiffiffiffiffiffiffi rc =Ec =h for SSSS Al/Al2O3 square plates (Z ¼1).

Method

a

Power law index (p)

Sh. Hosseini-Hashemi et al. / International Journal of Mechanical Sciences 53 (2011) 11–22

19

Table 5

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Comparison of fundamental frequency parameter b~ ¼ o h rm =Em for SSSS Al/ZrO2 square plates (Z ¼1). Method

p¼0

Present HSDT [28] 3-D [36] HSDT [10] FSDT [10] FSDT [33]

t ¼0.1

t ¼ 0.05

t ¼ 0.1

t ¼0.2

p¼ 2

p¼3

p ¼5

0.4618 0.4658 0.4658 0.4658 0.4619 0.4618

0.0577 0.0577 0.0577 0.0578 0.0577 0.0576

0.0158 0.0158 0.0153 0.0157 0.0162 0.0158

0.0619 0.0619 0.0596 0.0613 0.0633 0.0611

0.2276 0.2285 0.2192 0.2257 0.2323 0.2270

0.2264 0.2264 0.2197 0.2237 0.2325 0.2249

0.2276 0.2270 0.2211 0.2243 0.2334 0.2254

0.2291 0.2281 0.2225 0.2253 0.2334 0.2265

Table 6 First four natural frequency parameter b ¼ oa2

t¼

h a

t ¼ 0.2

p¼1

pffiffiffiffiffiffi t ¼ 1= 10

Mode no.

pffiffiffiffiffiffiffiffiffiffiffiffi rc =Ec =h for SSSS Al/Al2O3 plates (Z ¼2).

Power law index (p) 0

0.5

1

2

5

8

10

0.05

b1 b2 b3 b4

3.712311 5.919812 9.566813 12.45621

3.145611 5.017512 8.112113 10.56621

2.835211 4.522812 7.313213 9.526121

2.577711 4.111512 6.647113 8.657221

2.442511 3.893912 6.290313 8.187521

2.394811 3.817012 6.163913 8.020721

2.364211 3.768112 6.084313 7.916621

0.1

b1 b2 b3 b4

3.651811 5.769312 9.187613 11.83121

3.098311 4.899712 7.814513 10.07421

2.793711 4.419212 7.051213 9.092821

2.538611 4.014212 6.401513 8.251521

2.399811 3.788112 6.024713 7.750521

2.350411 3.707212 5.888713 7.568821

2.319711 3.658012 5.808613 7.463921

0.2

b1 b2 b3 b4

3.440911 5.280212 8.071013 9.741621

2.932211 4.512212 6.923113 8.692621

2.647311 4.077312 6.263613 7.871121

2.401711 3.695312 5.669513 7.118921

2.252811 3.449212 5.257913 6.574921

2.198511 3.358712 5.104513 5.906221

2.167711 3.309412 5.025313 5.751821

Table 7 First four natural frequency parameter b ¼ oa2

t¼

h a

Mode no.

pffiffiffiffiffiffiffiffiffiffiffiffi rc =Ec =h for SSSC Al/Al2O3 square plates (Z ¼1).

Power law index (p) 0

0.5 11

1 11

2 11

5

8

10

0.05

b1 b2 b3 b4

7.0526 15.24321 17.17512 24.95122

5.9810 12.93721 14.58812 21.20822

5.3926 11.66721 13.15912 19.13622

4.9019 10.60121 11.95612 17.38022

4.6382 10.01621 11.28312 16.37922

4.5443 9.806821 11.04012 16.01622

4.485411 9.677921 10.89312 15.80022

0.1

b1 b2 b3 b4

6.775111 14.25421 15.78112 22.42522

5.764911 12.15821 13.49112 19.20522

5.203911 10.98221 12.19912 17.37222

4.726111 9.961621 11.06312 15.74122

4.446211 9.330221 10.32512 14.64322

4.343911 9.098521 10.04812 14.04422

4.283911 8.968521 9.898012 13.67222

0.2

b1 b2 b3 b4

5.962511 11.78621 12.54312 17.19922

5.118811 10.16521 10.86512 14.93022

4.635611 9.216521 9.873912 13.57122

4.199611 8.331021 8.923912 12.24922

3.891611 7.656721 8.144212 11.14222

3.774611 7.401221 7.843812 10.71722

3.714611 7.276821 7.703112 10.52122

in Ref. [33]. In fact, as the present exact procedure provided, inplane displacement components u and v should be taken into account and are coupled with transverse displacement components w, c1 and c2.

3.2. Benchmark results In this section, the following new results for the free vibration analysis of Le´vy-type rectangular FG Mindlin plates are expressed as benchmark solutions for validating approximate two-dimensional theories and new computational techniques in the future.

11

11

11

In Tables 6–11, based on the present exact closed-form solutions, Numerical results have been performed for each of the six possible cases of boundary conditions (i.e., SSSS, SSSC, SCSC, SSSF, SCSF and SFSF). For simply supported boundary condition, the aspect ratio of plate is taken to be 2 and for other boundary conditions, the plate is assumed square (Z ¼1) while three different thickness to length ratios 0.05 (corresponding to thin plates), 0.1 and 0.2 (corresponding to moderately thick plates) are used. All of calculations are obtained for first four natural frequencies and seven values of the power law index p¼0, 0.5, 1, 2, 5, 8 and 10. In three combinations of classical boundary conditions, including SSSS, SSSC and SCSC, the FG plates are composed of Al/Al2O3

20

Sh. Hosseini-Hashemi et al. / International Journal of Mechanical Sciences 53 (2011) 11–22

Table 8 First four natural frequency parameter b ¼ oa2

t¼

h a

Mode no.

pffiffiffiffiffiffiffiffiffiffiffiffi rc =Ec =h for SCSC Al/Al2O3 square plates (Z ¼ 1).

Power law index (p) 0

0.5 11

1

2

7.2715 13.64221 17.05112 23.06022

6.5585 15.39021 12.30512 20.81622

5.9612 11.18121 13.98112 18.90422

5.6332 10.55521 13.16712 17.78522

5.5152 10.33021 12.86912 17.37622

5.442311 10.19321 12.69312 17.13622

0.1

b1 b2 b3 b4

8.070211 14.86221 17.91812 23.85022

6.884711 12.69321 15.37112 20.47522

6.222211 11.47221 13.92112 18.54322

5.649411 10.40521 12.62112 16.79922

5.293011 9.725521 11.71412 15.56622

5.159411 9.473421 11.36612 15.09822

5.084411 9.334821 11.18612 14.85822

0.2

b1 b2 b3 b4

6.766311 12.06021 13.50112 17.71822

5.840911 10.42021 11.75812 15.42322

5.303911 9.456121 10.71212 14.04022

4.803211 8.546621 9.675912 12.67022

4.412711 7.833121 8.754812 11.47322

4.260411 7.561021 8.395712 11.01122

4.186511 7.430721 8.234512 10.80222

h a

Mode no.

11

10

8.5674 16.06521 20.05012 27.10022

t¼

11

8

b1 b2 b3 b4

First four natural frequency parameter b ¼ oa2

11

5

0.05

Table 9

11

11

pffiffiffiffiffiffiffiffiffiffiffiffi rc =Ec =h for SSSF Al/ZrO2 square plates (Z ¼1).

Power law index (p) 0

0.5

1

2

5

8

10

0.05

b1 b2 b3 b4

3.505811 8.240712 12.21021 17.32822

3.299511 7.759612 11.50021 16.32922

3.224411 7.583412 11.23921 15.96022

3.220811 7.572112 11.22121 15.92722

3.294311 7.738112 11.46221 16.25522

3.298311 7.745712 11.47221 16.26522

3.287111 7.719112 11.43221 16.20822

0.1

b1 b2 b3 b4

3.441711 7.914712 11.58621 16.11322

3.242411 7.468412 10.94421 15.24222

3.168911 7.299912 10.69821 14.90222

3.163111 7.277812 10.65921 14.83122

3.229711 7.411012 10.83521 15.04022

3.232111 7.411412 10.83021 15.02422

3.220911 7.385112 10.79021 14.96922

0.2

b1 b2 b3 b4

3.237411 7.006512 9.900221 13.18622

3.059811 6.648012 9.417821 12.57422

2.991211 6.500412 9.212521 12.30222

2.978911 6.454212 9.132821 12.17422

3.025211 6.512012 9.176321 12.18322

3.023111 6.496912 9.143521 12.12822

3.012111 6.472212 9.106821 12.07822

Table 10 First four natural frequency parameter b ¼ oa2

t¼

h a

Mode no.

pffiffiffiffiffiffiffiffiffiffiffiffi rc =Ec =h for SCSF Al/ZrO2 square plates (Z ¼1).

Power law index (p) 0

0.5

1

2

5

8

10

0.05

b1 b2 b3 b4

3.796211 9.748712 12.34621 18.37522

3.573211 9.183612 11.62921 17.32222

3.492011 8.975512 11.36521 16.93022

3.487711 8.959512 11.34521 16.89222

3.566511 9.149112 11.58821 17.23022

3.570611 9.156212 11.59821 17.23722

3.558411 9.124412 11.55821 17.17722

0.1

b1 b2 b3 b4

3.706811 9.202112 11.69121 16.88622

3.493611 8.695312 11.04521 15.98722

3.414611 8.500612 10.79721 15.63222

3.407411 8.467012 10.75621 15.54922

3.476611 8.601812 10.93121 15.74622

3.478611 8.596412 10.92621 15.72322

3.466411 8.565112 10.88621 15.66422

0.2

b1 b2 b3 b4

3.438311 7.793712 9.954121 13.53422

3.252811 7.417012 9.470921 12.92122

3.180411 7.255312 9.264721 12.64322

3.165111 7.189412 9.183321 12.50222

3.209211 7.218312 9.224221 12.48922

3.205511 7.191612 9.190321 12.42622

3.193611 7.162812 9.153321 12.37322

whereas the FG plates of SSSF, SCSF and SFSF boundary conditions are made of Al/ZrO2. Moreover, the corresponding mode shapes m and n, denoting the number of half-waves in the x1- and x2-direction, respectively, are presented for any of the frequency parameters b, listed in Tables 6–11. For example, superscript 21 means two half-waves in the x1-direction (m¼2) and one

half-wave in the x2-direction (n ¼1). Following points can be noticeable from Tables 6–11:

(a) Regardless of boundary constraints, power law index and mode numbers, frequency parameters b are considerably reduced by

Sh. Hosseini-Hashemi et al. / International Journal of Mechanical Sciences 53 (2011) 11–22

Table 11 First four natural frequency parameter b ¼ oa2

t¼

h a

Mode no.

21

pffiffiffiffiffiffiffiffiffiffiffiffi rc =Ec =h for SFSF Al/ZrO2 square plates (Z ¼1).

Power law index (p) 0

0.5

1

2

5

8

10

11

11

11

11

11

11

0.05

b1 b2 b3 b4

2.8977 4.798612 10.80913 11.56121

2.7268 4.517612 10.18213 10.88821

2.6648 4.415012 9.950513 10.64121

2.6620 4.409012 9.933013 10.62421

2.7233 4.507112 10.14513 10.85321

2.7267 4.512012 10.15413 10.86321

2.717411 4.496512 10.11913 10.82621

0.1

b1 b2 b3 b4

2.856911 4.657012 10.24613 11.00221

2.690611 4.390812 9.676413 10.39021

2.629611 4.291512 9.458213 10.15721

2.625311 4.281112 9.423313 10.12121

2.682011 4.365612 9.583013 10.29221

2.684411 4.367412 9.580613 10.28821

2.675111 4.352112 9.546413 10.25121

0.2

b1 b2 b3 b4

2.718411 4.265012 8.817113 9.462921

2.567211 4.037612 8.376913 8.998321

2.509611 3.947412 8.190213 8.802021

2.500611 3.926012 8.121713 8.728221

2.542911 3.976412 8.177013 8.775221

2.541911 3.971012 8.154913 8.745121

2.532711 3.956312 8.124113 8.710121

increasing the thickness to length ratio t from 0.05 to 0.2. The effect of the thickness to length ratio becomes more significant for higher-mode natural frequencies. Such behavior is due to the influence of rotary inertia and shear deformations. (b) For SSSS, SSSC and SCSC rectangular plates, as the power law index p enhances, frequency parameter b decreases, keeping all other parameters fixed. When the free boundary condition is applied, (i.e., SSSF, SCSF and SFSF), the frequency parameter b as a function of the power law index p, shows nonlinear behavior. (c) Frequency parameters b increase when higher restraining boundary is used at the other two edges of Le´vy-type rectangular plates. In other words, the lowest and highest values of frequency parameters correspond to SFSF and SCSC rectangular FG plates, respectively. This is due to the fact that higher constraints at the edges increase the flexural rigidity of the plate, leading to a higher frequency response. 4. Conclusions This paper presented exact closed-form solutions for free vibration analysis of Le´vy-type rectangular FG plates with different combinations of free, simply supported and clamped boundary conditions using first-order shear deformation theory. The properties of the plate were assumed to vary through the thickness of the plate with a power-law distribution of the volume fractions of the two materials in between the two surfaces. Based on the Reissner– Mindlin’s first-order shear deformation theory for FG plates, five highly coupled governing partial differential equations of motion for freely vibrating Le´vy-type rectangular FG plates were exactly solved by introducing the potential functions and using the method of separation of variables. Several comparison cases by those reported in the literature, for simply supported rectangular thin and moderately thick plate, were presented to demonstrate highly stability and accuracy of present exact procedure. It was observed that the proposed exact method yields an improved accuracy for vibration of FG plates over the existing analytical and numerical methods based on FSDT in literature. Exact frequency parameters of rectangular FG plates under different boundary conditions were reported in tabular form for a wide range of thickness to length ratios and the power law index. All analytical results presented here can be applied as a useful and reliable source by other research groups. References [1] Leissa AW. The free vibration of rectangular plates. Journal of Sound and Vibration 1973;31:257–93.

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