Bending,Free Vibration and Buck ling Analysis of Functionally Graded Plates via

Zhibo YangXuefeng ChenYong Xie

1 Introduction

Over the past few decades,the science and technology mainly focus on homogeneous materials such as metal,alloy,ceramic and polymer,etc.However,the tradi-tional homogeneous materials are face with the challenge of ultrahigh temperature resistance with the rapid development of aerospace industry.Functionally graded materials(FGM)are new advanced composites firstly proposed as heat-shielding structural materials in space applications by Japanese material scientists in 1984[Koizumi(1997)].Generally,FGM are comprised of ceramic and metal with material properties varying smoothly and continuously throughout one surface to another.The smooth and continuity of FGM properties are able to effectively reduce the influence of interface and eliminate high interfacial stress.These excellent characteristics make FGM be widely used in the areas of aircraft,space vehicle,nuclear,mechanical,optical,chemical and biomechanical and other engineering structures.Due to the increasing applications of FGM in engineering structures,the theoretical research of FGM structures has attracted considerable researchers’attention.

Functionally graded(FG)plate structures,as one of basic structures in engineering areas,play a very important role in engineering practical and theoretical analysis[Akgoz and Civalek(2013a)].Hence,various plate theories are proposed for problems of FG plates.The classical plate theory(CPT),also named Kirchhoff plate theory,neglects the transverse shear deformation and rotary inertia terms.Shen[Yang and Shen(2001)]dealt with the dynamic response of initially stressed thin FG plates subjected to impulsive lateral loads based on CPT.The elastic foundation was considered in their research.Abrate[Abrate(2008)]adopted CPT to investigate free vibration of FG plates with simply supported and clamped boundary conditions.Eslami[Shariat et al.(2005)]investigated the buckling analysis of FG plates based on CPT.Saidi[Mohammadi et al.(2010);Baferani et al.(2011)]presented an analytical method for free vibration and buckling analysis of thin FG plates based on CPT.It is observed that the CPT gives satisfactory solutions for thin plates.

However,the effect of transverse shear deformation becomes remarkable with the increasing thickness of plates.For this reason,the first order shear deformation plate theory(FSDT)also named Mindlin plate theory was proposed for moderately thick plates by Mindlin[Mindlin(1951);Mindlin et al.(1955);Civalek and Acar(2007)].The FSDT can be considered as an extension of the Timoshenko theory to beam.Unlike Kirchhoff plate theory,this plate theory takes the effect of transverse shear deformations and rotational inertia into account which greatly improves the calculation accuracy of moderately thick plate(length/thickness＜10 or 20).Although high efficiency and simplicity of FSDT,shear correction factors are required to correct variation of transverse shear stress and shear strain through the thickness[Akgoz and Civalek(2013b)].Then various higher order shear deformation plate theories(HSDTs)are proposed for the problems of thick FG plates.These HSDTs also give satisfactory solutions for thick plates without requiring shear correction factors.

Kitipornchai[Yang et al.(2005)]studied the buckling analysis of FG plates resting on elastic foundations based on FSDT.Liew[Zhao et al.(2009b);Zhao et al.(2009a)]employed FSDT to investigate the free vibration,mechanical and thermal buckling analysis of FG plates using element-freekp-Ritz method.Then a local Kriging meshless method was proposed for free vibration of FG plates based on FSDT by Liew[Zhu and Liew(2011)].A group of satisfactory solutions for square,skew,quadrilateral plates and plates with holes are given in their literatures.An analytical solution based on FSDT was proposed for free vibration of moderately thick FG plates without or resting elastic foundations by Hashemi[Hosseini-Hashem i et al.(2010);Hosseini-Hashemi et al.(2011a)].The benchmark solutions of FG plates with SSSS,SSSC,SCSC,SCSF,SSSF,SFSF are reported in their literatures.Singha[Singha et al.(2011)]formulated a four-node high precision plate bending element to analyze the deflections and stresses of FG plates subjected to a sinusoidal or uniformly distributed loads.Choi[Thai and Choi(2013)]developed a simple FSDT for the bending and free vibration analysis of FG plates.Zenkour[Zenkour(2005b);Zenkour(2005a)]proposed sinusoidal shear deformation plate theory(SSDT)for bending,buckling and free vibration analysis of FG plates.After then Tounsi[Ameur et al.(2011);Merdaci et al.(2011);Tounsi et al.(2013)]and Thai[Thai and Vo(2013)]developed SSDT for similar problems of FG plates.Wu[Wu and Li(2010)]employed third order shear deformation theory(TSDT)to investigate the static behaviors of FG plates.A new exact closed-form procedure based on Reddy’s TSDT was proposed for free vibration of FG plates by Hashemi[Hosseini-Hashemi et al.(2011b)].The buckling analysis of FG plates are studied by Eslami[Shariat and Eslami(2007)]and Cheng[Cheng and Batra(2000)]based on TSDT.Liew[Liew et al.(2003)]employed Reddy’s high order shear deformation plate theory(HSDT)to study the postbuckling response of piezoelectric FG plates.The thermal,mechanical and electrical buckling are considered in their study.An analytical solution based on HSDT was proposed for nonlinear vibration and dynam ic response of FG plates in thermal environments by Shen[Huang and Shen(2004)].Ferreira and his co-workers employed SSDT[Neves et al.(2012)],TSDT[Ferreira et al.(2007);Ferreira et al.(2005)]and HSDT[Neves et al.(2013)]for static,free vibrations and buckling analysis of isotropic and sandwich FG plates.These literatures pointed that the effect of thickness stretching showed significance in thicker plates.Natural frequencies and buckling loads of FG plates were obtained by Matsunaga[Matsunaga(2008)]and Tzou[Chen et al.(2009)]using 2D HSDT and HSDT,respectively.Saidi[Bodaghi and Saidi(2010)]just considered the buckling problems of FG plates in the framework of HSDT.However,it is the first time to consider the buckling analysis of FG plates with various bound-ary conditions.The FG plates resting on a Winkler-Pasternak elastic foundation were considered by Tounsi[Benyoucef et al.(2010)]and Atmane[Atmane et al.(2010)].They employed HSDT for static response and free vibration of FG plates,respectively.Singh[Talha and Singh(2010)]employed HSDT for static response and free vibration of FG plate.The same work has also been done by Natarajan[Natarajan and Manickam(2012)].They constructed a C08-noded quadrilateral plates element with 13 degrees of freedom per node for the problems of FG plates.A higher order shear and normal deformable plates theory(HOSNDPT)has been proposed for static analysis of FG plates by Xiao[Gilhooley et al.(2007)]and Batra[Qian et al.(2004)].Recently,Dong[Dong et al.(2014a)]developed a simple locking-alleviated 3D 8-node mixed-collocation C0finite element for FG plates and shells based on[Dong et al.(2014b)].The widely-available theories of elasticity were employed for modeling FG structures without using HSDT in their works which solved the over-integration problem for FGM structures.

Although various numerical methods have already been proposed for bending,free vibration and buckling analysis of FG plates,the analytical solutions for problems of FG plates are rarely reported.In addition,traditional numerical methods,such as finite element method,often require more computing grids due to the continuous variation of material properties in FGM.According to the literatures mentioned before,literatures concerning bending problems,free vibration problems and buckling problems of FG plates at the same time are very few.Their attentions mainly focus on one or two of these problems.Hence,it is very meaningful and useful to propose an accurate and effective numerical method for deriving exact and comprehensive closed form solutions of FG plates.

Nowadays,High performance numerical computer methods,such as boundary element method(BEM)[Hall(1994)],meshless local Petrov-Galerkin(MLPG)method[Atluri and Zhu(1998)],local boundary integral equation(LBIE)method[Atluri et al.(2000)],the discontinuous Galerkin method[Hartmann and Houston(2002)]and hybrid/mixed finite element method[Dong and Atluri(2011)],play a crucial role for numerical simulation problems.The wavelet finite element method(WFEM)is another powerful analysis tool developed in recently[Chen and Wu(1995);Chen and Wu(1996);Zhong and Xiang(2011)].The WFEM employs a series of scaling functions as approximating functions.Compared with other wavelets,B-spline wavelet on the interval(BSWI)basis has the excellent characteristics of compact support,smoothness and symmetry in addition to the multiresolution analysis.Moreover,BSWI element,as a type of generalized spline finite element method,inherits the superiority of spline for structural analysis.Above all,it has explicit expressions so the coefficient integration and differentiation can be calculated conveniently.Hence,the WFEM employing BSWI as approximat-ing functions is the best choice for solving bending,free vibration and buckling problems of FG plates.In previous studies about BSWI elements,the main investigations are focused on the elements with uniform density and Young’s modulus on the out of plane direction,which induces the decoupling of the displacements between in plane and out of plane[Xiang et al.(2006);Jiawei et al.(2008);Yang et al.(2013)].Following previous work[Zuo et al.(in press)],the WFEM is extended to solve the FG plate problems in this paper.

Due to the excellent characteristics of BSWI,this paper adopts BSWI to investigate the bending,free vibration and buckling analysis of FG plates.The outline of this paper is organized as follows.The introduction of FGM is briefly presented in section 2.The formulation of FGM BSWI plate element and the equations of bending,free vibration and buckling of FG plate are derived in section 3.Various numerical examples and comparisons are provided to demonstrate the accuracy and efficiency of the constructed FGM BSWI element for FG plates in section 4.

2 Functionally graded material

A flat and moderately thick rectangular FG plate of lengtha,widthband thicknesshis considered and depicted in Fig.1.The Cartesian coordinate system is de fined on the neutral surface of plate wherex-axis is taken along the length direction,y-axis in the width direction andz-axis in the thickness direction.

Figure 1:Geometry and coordinates of rectangular FG plate.

Generally,the FG plate is always made of ceramic and metal and the material properties of FG plate,such as Young’s modulusE,mass densityρand Poisson’s ratioµ,are assumed to vary continuously throughout the thickness of plate according to the power law distribution of volume fraction of constituents.According to the rule

wherePm,Pc,VmandVcare defined as the material properties and volume fractions of metal and ceramic,respectively.The volume fractions of two constituent materials are assumed as

In this paper,the volume fraction for a plate with referential surface at its neutral surface can be written as of mixture,the effective material propertiesPcan be expressed as

wherenis anon-negative exponent named volume fraction index.The volume fraction indexndictates the material variation pro file thickness of FG plate.Then the effective material properties of FG plate which consists of two constituent materials can be expressed as

where the subscripts c and m represent ceramic and metal,respectively.The material properties of metal and ceramic used in FG plates are listed in Table 1.

The effective Young’s modulus through the thickness of Al/Al2O3plate with various volume fraction indexes is shown in Fig.2.to clarify the behavior of FG plate.It is clearly that the bottom surface(z=-h/2)of FG plate is metal rich while the top surface(z=h/2)material of FG plate is ceramic rich.And the material properties vary continuously and smoothly from metal rich to ceramic rich with different volume fraction indexes.Moreover,the FG plate is fully ceramic(Al2O3)and metal(Al)forn=0 andn=∞,respectively.

3 Formulation of B-spline wavelet on the interval Mindlin plate element

3.1 Two-dimensional tensor product BSWI on the interval[0,1]

Themth order B-spline functions need to be construed on the interval[0,1]due to any one-dimensional functionf(x)on the interval[a,b]can be transformed tothe standard interval[0,1]by means of a linear mappingξ=(x-a)/(b-a).In order to meet at least one inner wavelet on the interval[0,1],the condition should be satisfied as

Table 1:Material properties of metal and ceramic used in FG plates.

Figure 2:The effective Young’modulus through the thickness of Al/Al2O3 plate with various volume fraction indexes.

wherejis the scale number of BSWI.

For anyjscalemth order BSWI,which can be written as BSWImj,the scaling functionscan be calculated by the following formula[Xiang et al.(2007)]

Letj0be the scale as the condition Eq.(5)is satisfied.Then for eachj＞j0,letl=0,the scaling functions can be obtacined through Eq.(6).Therefore,the scaling functions on the interval[0,1]can be written in vector form as

whereξandηdepict the normalizedxandycoordinates,respectively.

Tensor product is the best way to construct two-dimensional BSWI from one dimensional BSWI.It is assumed that two-dimensional tensor product of BSWI at scalejof L2(R2)can be constructed by multi-resolution approximation spaceand the scaling functions of two-dimensional BSWI are

where Φ1and Φ2are two different variations in scaling functions given by Eq.(7),the symbol⊗denotes the kronecker function.Some selected BSWI43scaling functions are presented in Fig.3 to clarify the BSWI shape function.

3.2 The theory formulation of Mindlin plate

According to Mindlin plate theory,normal to the undeformed middle plane of the plate remains straight,but not normal to the deformed middle surface.In Cartesian coordinate system,the assumed displacement field is defined as follows

whereu0,v0,w0are thex-direction,y-direction andz-direction displacements of the plate on the neutral plane,respectively,θxandθyare the rotations of a transverse normal about axisyandx,andtdenotes time.

Figure 3:(a-d)some typical two-dimensional BSW I43 scaling functions.

Based on the small deformations assumption,the bending strainsεxx,εyyandγxycan be written as

while the transverse shear strainsγxzandγyzcan be written as

According to the Mindlin plate theory,the total strain energy of Mindlin plate consists of two parts

where

whereεrepresents the bending strain which can be written asε=[εxx,εyy,γxy]Tandγrepresents the transverse shear strain which can be written asγ=[γxz,γyz]T.According to the Hooke’s law,the bending stressσand transverse shear stressτare obtained as

whereDbandDsare corresponding elasticity matrixes which can be de fined as

whereE(z)is the Young’s modulus varying continuously throughout the thickness direction,µis the Poisson’s ratio and the variablekis known as the shear correction factor.

The kinetic energy for Mindlin plate consists of two parts.One of them is related with translations and the other is related with rotations.Then the kinetic energy can be obtained as

in whichρ(z)denotes the mass density varying continuously throughout the thickness direction.Then the variational energy function can be de fined as the difference between the strain energy and the kinetic energy

3.3 The formulation of FGM BSWI plate elements

In Mindlin plate element,the displacements and rotations can be interpolated by tensor product of BSWI scaling functions,respectively.The displacement fields can been derived as follow

whereu,v,w,θx,θyare the displacement vectors in BSWI scaling space,respectively,andTis the BSWI element transform matrix.Tis obtained by the tensor product aswhich can be written as

Substitute the displacement field Eqs.(26)into Eqs.(12-16)and the results are

wheredcan be expressed asd=[uvwθxθy]T.

3.3.1Bending analysis of BSWI Mindlin plate

In this section,the bending analysis of Mindlin plate is formulated and implemented.According to Hamilton’s principle,the equation of motions for bending analysis of Mindlin plate can be expressed as

Substituting Eqs.(29-30)into Eq.31,the basic governing equation of bending problem is obtained as

whereis stiffness matrix,is force vector.

The stiffness matrixis defined by the summation of two parts

where

with

where the details of integration matrixes Γ can be found in Appendix.

where

The force vectorFcan be expressed as

wherelexandleyare the element lengths,respectively,andq(ξ,η)is uniform distributed load.

3.3.2Free vibration analysis of BSWI Mindlin plate

In this section,the free vibration analysis of Mindlin plate is formulated and implemented.According to Hamilton’s principle,the equation of motions for free vibration analysis of Mindlin plate can be expressed as

Substituting Eqs.(29-30)into Eq.37,the basic governing equation of free vibration problem is obtained as

whereωis natural frequency andXis the mode shape of Mindlin plate.The stiffness matrixKhas been obtained in previous section.Similarly,the corresponding mass matrixMcan be obtained as

wherewith

3.3.3Buckling analysis of BSWI Mindlin plate

In this section,the buckling analysis of Mindlin plate is formulated and implemented.The buckling analysis of Mindlin plate involves the solution of eigenvalue problem

whereKis the stiffness matrix,KGis the geometric matrix,λis the critical load andXis the corresponding buckling mode shape of Mindlin plate.The critical loadλcan be obtained by solving Eq.40.The geometric stiffness matrixKGcan be written as[Hinton(1988)]

with

Substituting Eq.26 and Eq.42 into Eq.41,the geometric matrixKGformulation via BSWI can be obtained

4 Numerical examples and discussion

In this section,various numerical examples and comparisons are presented and discussed to validate the accuracy and reliability of the proposed BSWI method for bending,free vibration and buckling analysis of FG plates.The shear correct factorkis taken as 5/6 for all comparison studies.For convenience,the boundary conditions such as clamped supported,simply supported and free are indicated as C,S and F,respectively.The non-dimensional parameters used in this paper are de fined as follows

4.1 Bending problem

4.1.1Accuracy and efficient of BSWI method for bending analysis of FG plate

Example 1:The first comparison study is considered to verify the accuracy and efficient of proposed BSWI method for bending analysis of SSSS FG square plate(a/b=1).The FG plate is made of aluminum(Al)and zirconia(ZrO2),and the length-to-thickness ratio is taken asa/h=5 with volume fraction indexesn=0,0.5,1,2 and∞.The A l/ZrO2plate is subjected to uniform distributed load.Ferreira[Ferreira et al.(2005)]has given referential solutions for this problem based on the collocation multi-quadric radial basis functions by TSDT.Moreover,Singh[Talha and Singh(2010)]has conducted the convergence of this problem based on HSDT.By aid of one BSWI Mindlin plate element,the calculated non-dimensional deflectionsˆware in good agreement with results given by Singh(5×5)[Talha and Singh(2010)]mesh and also almost identical with results given by Ferreira[Ferreira et al.(2005)]shown in Table 2.This example demonstrates that the proposed BSWI method is highly accuracy and efficient for bending analysis of FG plate.It should be noted that the proposed method adopts just one BSWI element.Hence,one BSWI element is adopted in the follow ing examples if no explanation is given.

Table 2:The deflectionsˆw of SSSS A l/ZrO2 square plate with various volume fraction indexes(a/h=5).

4.1.2Effect of volume fraction indexes

Example 2:Another comparison study is employed to investigate a moderately thick Al/Al2O3square plate(a/h=10)with various volume fraction indexes.The A l/A l2O3is also subjected to uniform distributed load.The A l/A l2O3plate is fully simply supported(SSSS)and the volume fraction index varies from 0 to∞.The obtained non-dimensional deflectionstabulated in Table 3 are compared with those referential solutions given by Zenkour[Zenkour(2006)]based on the generalized shear deformation theory(GSDT),Thai[Thai and Choi(2013)]based on FSDT.It can be observed that the proposed method achieves an excellent agreement with Thai[Thai and Choi(2013)]for Al/Al2O3plates with all volume fraction indexes.For ceramic rich(n=0)and metal rich(n=∞),the results are almost identical.However,the other obtained results are slightly small compared with results given by Zenkour[Zenkour(2006)].These differences may be caused by shear correct factor used in FSDT while this factor is not needed in GSDT.The differences is so slight that the BSWI method still gives satisfactory solutions for A l/A l2O3plates subjected to uniform distributed load.It is also observed that the non-dimensional deflection increases as volume fraction index increases.The full ceramic plate(n=0)has the maximum bending stiffness and the bending stiffness reduces gradually as volume fraction index increases.The variation of non-dimensional stressesσxxandτxythrough the thickness of A l/A l2O3plate under uniform distributed load are presented in Fig.4.The stresses also vary continuously and smoothly throughout the thickness of A l/Al2O3plate which can effectively reduce the influence of interface and eliminate high interfacial stress.

Figure 4:The stresses σxx and τxy through the thickness of SSSS A l/A l2O3 square plate subjected to uniform distributed load(a/h=10).

Table 3:The deflections¯w of SSSS A l/Al2O3 square plate subjected to uniform distributed load(a/h=10).

4.1.3Effect of length-to-thickness ratios

Example 3:The next comparison study is employed to discuss a Al/Al2O3square plate subjected to sinusoidal distributed load(q=q0sin(πx/a)×sin(πy/b))with different length-to-thickness ratios.The length-to-thickness ratios are chosen asa/h=4,10 and 100 for thick plate,moderately thick plate and thin plate.The nondimensional deflections¯wwith volume fraction indexesn=1,4 and 10 are calculated and presented in Table 4.The similar problem has been investigated by Carrera[Carrera et al.(2008)]using unified formulation method(UFM),Brischetto[Carrera et al.(2011)]based on HSDT,and Ferreira[Neves et al.(2013)]based on 3D HSDT.These literatures also give solutions based on CPT and FSDT.Compared with other methods,the obtained results are satisfactory especially for thin plate(a/h=100).with the decreasing of length-to-thickness ratio,the plate becomes thicker while the differences among the Carrera[Carrera et al.(2008)],Brischetto[Carrera et al.(2011)]and Ferreira[Neves et al.(2013)]become obvious,especially for lager volume fraction index.Considering the higher terms,the HSDT method can obtain more accurate results than FSDT for thick plate.The differences between the HSDT and FSDT method become smaller since the effect of transverse shear deformation becomes weak with the increasing length-to-height ratio.Although the HSDT method gives better solutions for thick and thin plate,their equations are much more complicated than those of FSDT.And solutions in Table 4.show that the BSWI element formulated by FSDT could give satisfactory results and it is very effective to investigate behavior of FG plate.Fig.5.shows the non-dimensional deflections of plates alongy=b/2 with various volume fraction indexes.

4.1.4Effect of boundary conditions

Figure 5:The deformations of Al/Al2O3 plates along y=b/2 with various volume fraction indexes(a)a/h=4(b)a/h=10(c)a/h=100.

Table 4:The deflections¯w of SSSS Al/Al2O3 square plate subjected to sinusoidal distributed load.

Example 4:In addition,a comparison study is presented to investigate a A l/A l2O3square thin plate(a/h=100)subjected to uniform ly distributed load with simply supported(SSSS)and clamped supported(CCCC).The non-dimensional deflectionsw∗are tabulated in Table 5 for different volume fraction indexes.The obtained results are compared with those solutions given by Singha[Singha et al.(2011)]based on FSDT.It is clearly that the solutions obtained by the proposed BSWI method get a great agreement with Singha’s[Singha et al.(2011)]solutions for all volume fraction indexes in SSSS and CCCC cases.It is also found that the non-dimensional deflection decreases as the constrain increase.The variations of non-dimensional deflectionsw∗of A l/A l2O3plates for different length-to-thickness ratios are illustrated in Fig.6.for SSSS and CCCC cases.In order to clarify the deflections of A l/A l2O3plates,the non-dimensional deflection maps of the whole Al/Al2O3plates withn=10 are shown in Fig.7.The solutions of the whole A l/A l2O3plate can be directly observed in Fig.7.

Figure 6:The deflections w∗of A l/A l2O3 plates for different length-to-thickness ratios(a)SSSS(b)CCCC.

Figure 7:The deflection maps of the whole Al/Al2O3 plates with n=10(a)SSSS(b)CCCC.

Table 5:The deflections w∗of SSSS and CCCC square Al/Al2O3 plate subjected to uniformly distributed load(a/h=100).

4.2 Free vibration problem

4.2.1Accuracy and efficient of BSWI method for free vibration analysis of FG plate

Example 5:Firstly,a comparison study is employed to verify the accuracy and efficient of the proposed BSWI method for free vibration analysis of A l/A l2O3square plate.The length-to-thickness ratios are taken asa/h=5 and 10 and volume fraction indexes are chosen asn=0,0.5,1,4,10.The fundamental nondimensional frequency parametersˆωobtained using the BSWI element are compared with element-freekp-Ritz method solutions of Liew[Zhao et al.(2009a)],the FSDT solutions of Thai[Thai and Choi(2013)]and Hashemi[Hosseini-Hashem i et al.(2010)]in Table 6 for SSSS case.Obviously,the solutions given by the proposed BSWI method are in excellent agreement with the referential solutions given by Liew[Zhao et al.(2009a)],Thai[Thai and Choi(2013)]and Hashemi[Hosseini-Hashem i et al.(2010)],especially for moderately thick ones(a/h=10).This comparison study shows that the accuracy and efficiency of the present method are valid and effective for free vibration analysis of FG plates.

Example 6:Another comparison study is considered to validate the accuracy and efficiency of proposed BSWI method for square FG plate with different boundary conditions(BCS).In this study,the length-to-thickness ratio of square A l/ZrO2plate is taken asa/h=5.The Al/ZrO2plate material properties are assumed to vary continuously throughout the thickness of plate with volume fraction indexn=5.The non-dimensional frequency parameters¯ωare calculated and shown in Table 7 for SSSS,CCCC and SCSC cases.The comparison results have also been given by Singh[Talha and Singh(2010)]based on HSDT and Aydogdu[Uymaz and Aydogdu(2007)]based on small strain linear elasticity theory.It is obviously that the BSWI method achieves a high accuracy compared with Aydogdu[Uymaz and Aydogdu(2007)].Due to excellent characteristics of BSWI,the proposed BSWI method with adopting just one BSWI element can get excellent results for various boundary conditions.

Table 6:The frequency parameters of SSSS square Al/Al2O3 plate.

Table 6:The frequency parameters of SSSS square Al/Al2O3 plate.

volume fraction indexes n a/h Method 0 0.5 1 4 10 Liew(9×9)0.2018 0.1726 0.1559 0.1332 0.1261 Liew(13×13)0.2045 0.1748 0.1579 0.1349 0.1277 Liew(17×17)0.2055 0.1757 0.1587 0.1356 0.1284 5 Thai 0.2112 0.1805 0.1631 0.1397 0.1324 Hashem i 0.2112 0.1806 0.1650 0.1371 0.1304 Present 0.2112 0.1804 0.1630 0.1396 0.1323 Liew(9×9)0.0561 0.0476 0.0430 0.0371 0.0355 Liew(13×13)0.0565 0.0480 0.0433 0.0374 0.0358 Liew(17×17)0.0567 0.0482 0.0435 0.0376 0.0359 10 Thai 0.0577 0.0490 0.0442 0.0382 0.0366 Hashemi 0.0578 0.0492 0.0445 0.0383 0.0363 Present 0.0577 0.0490 0.0442 0.0382 0.0366

Table 7:The frequency parameters of Al/ZrO2 square plate for various boundary conditions(a/h=5,n=5).

Table 7:The frequency parameters of Al/ZrO2 square plate for various boundary conditions(a/h=5,n=5).

BCS Method SSSS CCCC SCSC Singh(3×3)1.4321 2.2669 2.0260 Singh(4×4)1.4222 2.1944 1.9287 Singh(5×5)1.4165 2.1540 1.8161 Aydogdu 1.4106 2.1447 1.8055 Present 1.4102 2.1429 1.7990

4.2.2Effect of the thickness-to-width ratios

Example 7:The influence of thickness-to-width ratios on the free vibration of Al/Al2O3plate is considered in this comparison.The aim of this example is to verify the obtained results with the SSDT solutions of Thai[Thai and Vo(2013)],the Reddy’s TSDT solutions of Hashemi[Hosseini-Hashemi et al.(2011b)]and HSDT solutions of Matsunaga[Matsunaga(2008)].The square A l/Al2O3plate is full simply support(SSSS)and the first three non-dimensional frequency parametersfora/h=5,10 and fundamental non-dimensional frequency parametersˆωfora/h=20 with volume fraction indexes varying from 0 to∞are tabulated in Table 8.It can be observed that the results obtained by present BSWI method are good in agreement with almost solutions of Thai[Thai and Vo(2013)],Hashemi[Hosseini-Hashem i et al.(2011b)]and Matsunaga[Matsunaga(2008)],especially for moderately plate(a/h=10)or thin plate(a/h=20).This example verifies the proposed BSWI method has an excellent accuracy for free vibration problem of A l/A l2O3plates.It is also found that the non-dimensional frequency parameter decreases as the volume fraction index increases.This phenomenon may be caused by larger volume fraction index leads to metal rich of A l/A l2O3plate which w ill result in decrease of stiffness.In order to verify the correctness of the proposed BSWI method,the fist six mode shapes of SSSS square Al/A l2O3plates witha/h=10 andn=10 are shown in Fig.8.It is observed that the solving mode shapes are very consistent with the real vibration of square plate which veri fies the correctness of the proposed BSWI method once more.

Example 8:The next example is considered for Al/Al2O3rectangular plate with different thickness-to-w idth ratios.The other corresponding parameters are the same with the example 7.The first four non-dimensional frequency parameters˜ωof A l/A l2O3rectangular plate(b/a=2)are calculated and shown in Table 9.The comparison solutions are given by Choi[Thai et al.(2013)]based on an efficient shear deformation theory and Hashemi[Hosseini-Hashemi et al.(2011a)]based on the Reissner-Mindlin plate theory(FSDT).A good agreement between the ref-erential solutions and results obtained by proposed BSWI method is observed for almost all the vibration mode shapes of Al/Al2O3plates.However,the differences between the results for the forth vibration modes of thick plate(a/h=5)increase slightly with the increase of volume fraction indexes.These differences may be caused by the fewer degrees of freedom used in BSWI for estimating frequencies.The fist six mode shapes of SSSS A l/A l2O3rectangular plates witha/h=10 andn=10 are shown in Fig.9.These mode shapes are also very consistent with the real vibration of rectangular plate which verifies the correctness of the proposed BSWI method once more.So the proposed BSWI method is not only suitable for solving the free vibration problem of A l/Al2O3square plate but also for those of Al/Al2O3rectangular plate.

Table 8:The frequency parameters of SSSS square A l/Al2O3 plate.

Table 8:The frequency parameters of SSSS square A l/Al2O3 plate.

volume fraction indexes n a/h (m,n)Method 0 0.5 1 4 10∞TSDT 0.2113 0.1807 0.1631 0.1378 0.1301 0.1076 HSDT 0.2121 0.1819 0.1640 0.1383 0.1306 0.1077(1,1)SSDT 0.2112 0.1805 0.1631 0.1397 0.1324 0.1076 Present 0.2112 0.1802 0.1625 0.1384 0.1315 0.1075 TSDT 0.4623 0.3989 0.3607 0.2980 0.2771 0.2355 HSDT 0.4658 0.4040 0.3644 0.3000 0.2790 0.2365(1,2)5 SSDT 0.4618 0.3978 0.3604 0.3049 0.2856 0.2352 Present 0.4618 0.3986 0.3625 0.3107 0.2865 0.2351 TSDT 0.6688 0.5803 0.5254 0.4284 0.3948 0.3404 HSDT 0.6753 0.5891 0.5444 0.4362 0.3981 0.3429(2,2)SSDT 0.6676 0.5779 0.5245 0.4405 0.4097 0.3399 Present 0.6676 0.5779 0.5248 0.4401 0.4090 0.3398 TSDT 0.0577 0.0490 0.0442 0.0381 0.0364 0.0293 HSDT 0.0578 0.0492 0.0443 0.0381 0.0364 0.0293(1,1)SSDT 0.0577 0.0490 0.0442 0.0382 0.0366 0.0293 Present 0.0577 0.0491 0.0443 0.0384 0.0367 0.0294 TSDT 0.1377 0.1174 0.1059 0.0903 0.0856 0.0701 HSDT 0.1381 0.1180 0.1063 0.0904 0.0859 0.0701(1,2)10 SSDT 0.1376 0.1173 0.1059 0.0911 0.0867 0.0701 Present 0.1376 0.1171 0.1055 0.0903 0.0864 0.0701 TSDT 0.2113 0.1807 0.1631 0.1378 0.1301 0.1076 HSDT 0.2121 0.1819 0.1640 0.1383 0.1306 0.1077(2,2)SSDT 0.2112 0.1805 0.1631 0.1397 0.1324 0.1076 Present 0.2112 0.1808 0.1638 0.1405 0.1327 0.1075 TSDT 0.0148 0.0125 0.0113 0.0098 0.0094-20(1,1)SSDT 0.0148 0.0125 0.0113 0.0098 0.0094-Present 0.01480.01260.01140.00990.00950.0075

Table 9:The frequency parameters ω∗of SSSS rectangular A l/A l2O3 plate(b/a=2).

Figure 9:The first six mode shapes of SSSS rectangular Al/Al2O3 plates(a/h=10 and n=10).

4.2.3Effect of boundary conditions

Example 9:The boundary conditions also play a very important role in estimating the frequencies of A l/Al2O3plate.Hence,the following example is employed to verify the validity of proposed BSWI method for free vibration analysis of A l/A l2O3square plate with different boundary conditions.The fundamental nondimensional frequency parametersω∗for thickness-to-width ratiosa/h=5,10 with SSSS,CCCC,CFFF,SCSC boundary conditions are calculated and presented in Table 10.The identical problem has been discussed by Liew[Zhu and Liew(2011);Zhao et al.(2009a)]using the local Kriging meshless method and element-freekp-Ritz method,and Hashemi[Hosseini-Hashemi et al.(2011a)]using FSDT.Compared with those referential solutions,the proposed BSWI method gives satisfactory solutions for different boundary conditions.The variations of non-dimensional frequency parametersω∗with volume fraction indexes are illustrated in Fig.10.for SSSS,CCCC,CFFF,SCSC boundary conditions.The constraints of boundary conditions affect stiffness.So the non-dimensional frequency parameter is higher in CCCC cases than other cases.This comparison shows that the formulated FGM BSWI element is very suitable for analyzing free vibration of FG plate with various boundary conditions.

4.3 Buckling problem

4.3.1Accuracy and stability of BSWI method for buckling analysis of FG plate

Figure 10:The frequency parameters ω∗of square Al/Al2O3 plates with volume fraction index(a)a/h=5(b)a/h=10.

Example 10:The first comparison study is employed to verify the accuracy and stability of BSWI method for buckling analysis of FG plate.Since the exact buckling solutions of FG plate are not available in literatures,a homogenous material plate is used here for verification.The homogenous material plate represents a special FG plate with volume fraction indexn=0 or∞.The thin plate(a/h=100)subjected to uniaxial compression with SSSS and CCCC cases is considered in this example.The exact buckling factorshave been given by Timoshenko[Timoshenko and Gere(2012)]and other referential results have also been given by Cheung[Cheung et al.(2000)],Aliabadi[Purbolaksono and Aliabadi(2005)],Liew[Liew and Chen(2004)],Civalek[Ersoy et al.(2009)]and Xuan[Nguyen-Xuan et al.(2010)].The obtained results listed in Table 11 has an excellent agreement with the exact solution given by Timoshenko[Timoshenko and Gere(2012)]in SSSS case compared with other methods.Although Cheung’s method seems to show a better agreement with exact solutions for CCCC case,the proposed BSWI method also still performances well.This comparison study validates the accuracy of the proposed BSWI method for buckling analysis of FG plate.

4.3.2A FG square plate subjected to uniaxial compression

Example 11:The buckling factorsof FG square plate subjected to uniaxial compression with SSSS boundary condition are considered in this comparison.The FG plate consists of SUS304 and Si3N4.The buckling factors are calculated and presented in Table 12.And the present results are compared with the referential solutions given by Tzou[Chen et al.(2009)]based on HSDT with thickness-to-w idth ratiosa/h=10,20,100.Compared with the referential solutions,the present method obtains excellent solutions for different thickness-to-width ratios.Fig.11.presents the contour plots of the first six buckling mode shapes of square SUS304/Si3N4plate subjected to uniaxial compression onxaxis(a/h=100,n=10).Obviously,more wrinkles may arise onxaxis with the increase of mode shape.The compression load is mainly subjected onxaxial,so the plates are most likely to lose stability inxaxial direction.

Table 10:The frequency parameters ω∗of square Al/Al2O3 plates with different boundary conditions.

Table 11:The buckling factors of square FG plate subjected to uniaxial compression(a/h=100,n=0).

Table 11:The buckling factors of square FG plate subjected to uniaxial compression(a/h=100,n=0).

BCS Cheung[54]FEM[55]Liew[56]DSC[57]DSG3[58]ESDSG3[58]Exact[53]Present SSSS 4.002 4.011 4.017 4.011 4.1590 4.0170 4.00 3.9982 CCCC 10.075 10.392 10.308 10.310 11.0446 10.2106 10.07 10.1842

Figure 11:The contour plots of the first six buckling mode shapes of square SUS304/Si3N4 plate subjected to uniaxial compression on x axis(a/h=100,n=10).

Table 12:The buckling factors of SSSS square SUS304/Si3N4 plate subjected to uniaxial compression.

Table 12:The buckling factors of SSSS square SUS304/Si3N4 plate subjected to uniaxial compression.

volume fraction indexes n a/h Method 0 0.1 0.5 1 2 5 10∞HSDT 0.5148 0.4957 0.4525 0.4299 0.4115 0.3907 0.3753 0.3426 10 Present 0.5140 0.4942 0.4484 0.4247 0.4079 0.3918 0.3784 0.3454 HSDT 0.5396 0.5194 0.4743 0.4512 0.4329 0.4118 0.3955 0.3600 20 Present 0.5396 0.5188 0.4709 0.4465 0.4296 0.4134 0.3994 0.3640 HSDT 0.5482 0.5276 0.4818 0.4585 0.4403 0.4192 0.4026 0.3660 100 Present 0.5485 0.5274 0.4788 0.4546 0.4374 0.4209 0.4067 0.3702

Example 12:The thin A l/SiC plate(a/h=100)with SCSC,SSSC,SSSS,SCSF,SSSF,SFSF cases are employed to verify the accuracy and applicability of proposed BSWI method for buckling analysis of A l/SiC square plate subjected to uniaxial compression.Table 13 presents the comparisons of BSWI method with Levytype solutions based on HSDT.It can be seen that the obtained results are found to be in excellent agreement with Levy-type solutions given by Saidi[Bodaghi and Saidi(2010)]for different boundary conditions.The variations of non-dimensional buckling factorsλ∗of Al/SiC plates subjected to uniaxial compression with volume fraction indexes are illustrated in Fig.12.for SCSC,SSSC,SSSS,SCSF,SSSF,SFSF cases.It can be observed that the critical buckling factor decreases as the volume fraction index increases.However,the influence of volume fraction index becomes weaker as the volume fraction index increases.This is due to the fact that the larger volume fraction index may lead to metal rich.The constraint of boundary conditions directly affects the buckling load of A l/SiC plates.Thus,the boundary conditions play a very important role for buckling analysis of FG plates.It is meaningful to investigate the buckling analysis of FG plates with different boundary conditions which can be used to guide the practical engineering design.

Figure 12:The buckling factors λ∗of SiC plates subjected uniaxial compression with volume fraction indexes(a/h=100).

4.3.3A FG square plate subjected to biaxial compression

Example 13:Another comparison of SSSS square A l/A l2O3plate subjected to biaxial compression is considered.Table 14 lists the buckling factorsλ∗of A l/A l2O3plate with thickness-to-w idth ratiosa/h=10,20,100.The obtained results are compared with similar solutions given by Choi[Thai et al.(2013)]based on the re fined theory of Shimpi.It is observed that the present BSWI method gives satisfactory solutions compared with referential solutions given by Choi[Thai et al.(2013)].These results get closer with the increase ofa/h.The difference may be caused by the influence of transverse shear deformation.The first six buckling modes of SSSS square Al/Al2O3plate subjected to biaxial compression witha/h=100 andn=10 are identical with vibration mode shapes of SSSS square A l/A l2O3.Obviously,the wrinkles may arise onxaxis oryaxis with the increase of mode shape.The compression load is mainly subjected onxaxial andyaxial at the same time,so the plates may lose stability inxor y axial direction with the same probability.Example 14:Table 15 presents the buckling factorsλ∗of a square Al/SiC FG plate(a/h=100)subjected to biaxial compression for SCSC,SSSC,SSSS,SCSF,SSSF,SFSF cases.Saidi[Bodaghi and Saidi(2010)]has also given the similar solutions for this problem.The present solutions show a good approximation with the results given by Saidi[Bodaghi and Saidi(2010)].The variations of non-dimensional buckling factorsλ∗of SiC plates subjected to biaxial compression with volume fraction indexes are illustrated in Fig.13.for SCSC,SSSC,SSSS,SCSF,SSSF,SFSF cases.The similar phenomenon has been explained in example 12.The buckling factors subjected to biaxial compression are smaller than those subjected to uniaxial compression.

Table 13:The buckling factors λ∗of square Al/SiC plate subjected to uniaxial compression with different boundary conditions(a/h=100).

Table 14:The buckling factors λ∗of SSSS square Al/Al2O3 plate subjected to biaxial compression.

Figure 13:The buckling factors λ∗of A l/SiC plates subjected to biaxial compression with volume fraction indexes(a/h=100).

Table 15:The buckling factors λ∗of square Al/SiC plate subjected to biaxial compression with different boundary conditions(a/h=100).

5 Conclusion

The objective of this paper is to present an accurate and effective numerical method for the comprehensive study of bending,free vibration and buckling analysis of FG plates.For this purpose,a wavelet finite element,which employs scaling functions of two-dimensional tensor product BSWI as shape functions,is proposed for theoretical analysis of FG plates.The governing motion equations are derived by using the Mindlin plate theory and Hamilton’s principle.Then two-dimensional FGM BSWI element is formulated for bending,free vibration and buckling analysis of FG plates.Different numerical examples concerning various length-to-thickness ratios,volume fraction indexes,aspect ratios and boundary conditions are provided to validate the accuracy,efficiency and the reliability of FGM BSWI element compared with available analytical and numerical solutions in literatures.Satisfactory solutions for bending,free vibration and buckling analysis of FG plates can be achieved using fewer degrees of freedoms.These excellent solutions can be attributed to the excellent characteristics of BSWI.This paper reveals that the proposed wavelet based BSWI finite element method is an efficient numerical tool for the bending,free vibration and buckling problems of FGM structures.What’s more,the proposed wavelet-based BSWI finite element method w ill be promising to be an effective and accurate tool for analyses of FGM structures.

Acknowledgement:This work was supported by the National Natural Science Foundation of China(Nos.51405369 and 51421004),the China Postdoctoral Science Foundation(No.2014M 560766),and the Fundamental Research Funds for the Central Universities(No.xjj2014107).

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