SHEAR CORRECTION FACTOR FOR FGM PLATE BASED ON THICKNESS SHEAR VIBRATION. Cao Hangyu. Master of Science in Civil Engineering

Transcription

1 SHEAR CORRECTION FACTOR FOR FGM PLATE BASED ON THICKNESS SHEAR VIBRATION by Cao Hangyu Master of Science in Civil Engineering December 2012 Faculty of Science and Technology University of Macau

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3 SHEAR CORRECTION FACTOR FOR FGM PLATE BASED ON THICKNESS SHEAR VIBRATION by Cao Hangyu A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Civil Engineering Faculty of Science and Technology University of Macau 2012 Approved by Supervisor Date iii

4 In presenting this thesis in partial fulfillment of the requirements for a Master's degree at the University of Macau, I agree that the Library and the Faculty of Science and Technology shall make its copies freely available for inspection. However, reproduction of this thesis for any purposes or by any means shall not be allowed without my written permission. Authorization is sought by contacting the author at Address: Pak lok Garden V 2B Telephone: Fax: chyhunan@qq.com Signature Date iv

5 University of Macau Abstract SHEAR CORRECTION FACTOR FOR FGM PLATE BASED ON THICKNESS SHEAR VIBRATION by Cao Hangyu Thesis Supervisor: Prof. Iu Vai Pan Functionally graded materials have continuous variation of material properties from one surface to another, which is different from the stepped or discontinuous composite material. The gradation of properties in FGM reduces the thermal stresses, residual stresses, and stress concentrations found in traditional composites. Due to the high quality of the FGM, it meets the requirement of the growing new technologies especially in aerospace, microelectronics, biomechanical, nuclear engineering and so on. The first order shear deformation theory (FSDT) is a high efficient and convenient method for analyzing the moderate thick plates. But the shear correction factor of plate is a necessary parameter in FSDT to cater the non-uniformed shear strain in transverse section. The shear correction factor of homogenous plate is not suitable for the FGM plate, as the variation of material properties is not considered in the theory. This thesis gives a convenient and effective method to obtain the shear correction factor for FGM plates based on thickness shear vibration. Thickness shear vibration is an in-plane vibration of infinite plates characterized by shear deformation. The frequencies of thickness shear vibration of an infinite FGM plate according to three-dimensional theory of elasticity is determined by Galerkin method. The shear correction factor can then be obtained by matching the fundamental frequencies of thickness shear vibration obtained by three-dimensional theory of elasticity and by FSDT. Parametric studies of the shear correction factor with the variation of material properties, such as the mass density ratio, the Young s modulus ratio and the material index are discussed. Both the FGM plates and sandwich FGM plates are considered. The results show that the shear correction factor for FGM plates with the material index larger than 1 v

6 is small than π 2 /12 which is the shear correction factor for homogeneous plates. The comparisons of the shear correction factor from presented method and from the other method are also presented. The relative difference of shear correction factor is increased when the Young s modulus ratio R y increase. It is observed that the Young s modulus ratios R y have bigger influence on the shear correction factor in the thickness shear vibration method. The relative difference of shear correction factor is decreased when the mass density ratio R m increase, it means that the relative difference of shear correction factor will be very small when the proper value of R m has been used. Using the present shear correction factor and the first-order shear deformation theory, the free vibration frequency of simply-supported rectangular FGM plates is determined by the Navier s solution method. Comparison studies are given for the frequency parameters obtained by different shear correct factors and by different numerical model. Key words: FGM plate, shear correction factor, thickness shear vibration, first order shear deformation theory. vi

7 TABLE OF CONTENT TABLE OF CONTENT... I LIST OF FIGURE... III LIST OF TABLE... V NOMENCLATURE... VII ACKNOWLEDGMENTS... IX 1. INTRODUCTION FUNCTIONALLY GRADED MATERIAL LITERATURE REVIEW Two-Dimensional Plate Theories... 2 Classical plate theory (CPT)... 2 First order shear deformation theory (FSDT)... 3 Third and Higher order shear deformation plate theory (HSDT) Three-dimensional plate theory OUTLINE OF THE THESIS MATERIAL GRADIENT OF FGM PLATES INTRODUCTION MATERIAL GRADIENT OF FGM PLATES THICKNESS SHEAR VIBRATION INTRODUCTION THREE DIMENSIONAL ELASTIC ANALYSIS OF THICKNESS SHEAR VIBRATION FOR HOMOGENOUS PLATE THREE DIMENSIONAL ELASTIC ANALYSIS OF THICKNESS SHEAR VIBRATION FOR FGM PLATE Governing Equations Solution by the Galerkin Method... 18

8 3.4. THICKNESS SHEAR VIBRATION BASED ON FSDT SHEAR CORRECTION FACTOR SHEAR CORRECTION FACTOR OF FGM PLATE INTRODUCTION PARAMETRIC STUDY COMPARISON SUMMARY SHEAR CORRECTION FACTOR OF FGM SANDWICH PLATE INTRODUCTION MATERIAL PROPERTIES OF FGM SANDWICH PLATES PARAMETRICAL STUDY COMPARISON SUMMARY FREE VIBRATION OF SIMPLY SUPPORTED RECTANGULAR FGM PLATES INTRODUCTION GEOMETRICAL CONFIGURATION AND MATERIAL PROPERTIES GOVERNING EQUATIONS BOUNDARY CONDITIONS NAVIER S SOLUTION NUMERICAL RESULTS AND COMPARISONS SUMMARY CONCLUSION REFERENCE ii

9 LIST OF FIGURE Fig. 2.1 variation of metal material s volume fraction with different p Fig. 3.1 Cartesian coordinate system Fig. 3.2 Thickness shear vibration modes of infinite homogenous plate Fig. 3.3 Thickness shear vibration modes of infinite FGM plate Fig. 4.1 Variation of k according to Ry( Rm = 1) Fig. 4.2 Variation of k versus p(rm = 1) Fig. 4.3 Variation of k versus Ry( Rm = 2) Fig. 4.4 Variation of k versus p( Rm = 2) Fig. 4.5 Variation of k versus Ry( Rm = 3) Fig. 4.6 Variation of k versus p( Rm = 3) Fig. 5.1 Variation of volume fraction in sandwich FGM plate with different p Fig. 5.2 Variation of k versus Ry( Rm = 1, Rt = 4) Fig. 5.3 Variation of k versus p(rm = 1, Rt = 4) Fig. 5.4 Variation of k versus Ry( Rm = 2, Rt = 4) Fig. 5.5 Variation of k versus p( Rm = 2, Rt = 4) Fig. 5.6 Variation of k versus Ry( Rm = 3, Rt = 4) Fig. 5.7 Variation of k versus p( Rm = 3, Rt = 4) Fig. 5.8 Variation of k versus Rt( Rm = 1, Ry = 6) Fig. 5.9 Variation of k versus p( Rm = 1, Ry = 6) Fig Variation of k versus Rt(Rm = 2, Ry = 6) iii

10 Fig Variation of k versus p( Rm = 2, Ry = 6) Fig Variation of k versus Rt( Rm = 3, Ry = 6) Fig Variation of k versus p( Rm = 3, Ry = 6) Fig. 6.1 Geometry of a rectangular FGM plate and coordinate system iv

11 LIST OF TABLE Table 3.1 Frequency parameters of FGM plate for thickness shear vibration Table 4.1 Shear correction factor k (Rm = 1) Table 4.2 Shear correction factor k (Rm = 2) Table 4.3 Shear correction factor k ( Rm = 3) Table 4.4 Comparison of shear correction factor k (Rm = 1, 2) Table 5.1 Shear correction factor k (Rm = 1, Rt = 4) Table 5.2 Shear correction factor k (Rm = 2, Rt = 4) Table 5.3 Shear correction factor k (Rm = 3, Rt = 4) Table 5.4 Shear correction factor k ( Rm = 1, Ry = 6) Table 5.5 Shear correction factor k (Rm = 2, Ry = 6) Table 5.6 Shear correction factor k (Rm = 3, Ry = 6) Table 5.7 Comparison of Shear correction factor k ( Rm = 1, 2 and Rt = 4) Table 5.8 Comparison of shear correction factor k ( Rm = 1, 2 and Ry = 6) Table 6.1 Material properties of the used FGM plate Table 6.2 k for Al/ Al 2 O 3 with different material index Table 6.3 Boundary conditions Table 6.4 Comparison of frequency parameter β (η = 1, τ = 0.01) Table 6.5 Comparison of frequency parameter β ( η = 1, τ = 0.1) Table 6.6 Comparison of frequency parameter β ( η = 1, τ = 0.2) Table 6.7 Comparison of frequency parameter β ( η = 2, τ = 0.01) v

12 Table 6.8 Comparison of frequency parameter β ( η = 2, τ = 0.1) Table 6.9 Comparison of frequency parameter β ( η = 2, τ = 0.2) Table 6.10 Comparisons of frequency parameter β for square plate vi

13 Nomenclature a b g h k p t u i u, v, w u, v, w x i A, B, D A, B, D E G G I 1, I 2, I 3 I 1, I 2, I 3 N 1, N 2, N 12 N 1, N 2, N 12 M 1, M 2, M 12 M 1, M 2, M 12 K, M, U P, P t, P b Q 1, Q 2 Q 1, Q 2 R y R m The length of plate The width of plate Power law function The thickness of plate Shear correction factor Material index Time displacement Mid-plane displacement Non-dimensional mid-plane displacement Rectangular coordinate axis Coefficient related to material properties Non-dimensional coefficient related to material properties Young s modulus Shear modulus Non-dimensional shear modulus The inertia moment Non-dimensional inertia moment The force The non-dimensional force The bending moment The non-dimensional bending moment Coefficient matrix Material properties The shear force The non-dimensional shear force Young s modulus ratio Mass density ratio vii

14 R t U ij, V ij, W ij, M ij, N ij Z Z ε ij σ ij ν ρ ρ ω ψ 1, ψ 2 ψ 1, ψ 2 β λ η τ the layer ratios Constant coefficient Shape functions Non-dimensional shape functions Strains stress Poisson s Ratio Mass density Non-dimensional mass density Frequency of plate Rotation Non-dimensional rotation Frequency parameter Frequency parameter The ratio of length and width The ratio of length and thickness Indices i, j 1,2,3, viii

15 ACKNOWLEDGMENTS I would like to express my gratitude to all those who gave me the possibility to complete this thesis and express my sincerest thanks to my supervisor, Prof. Iu Vai Pan, for his guidance, valuable suggestion, comments and encouragement thought to this thesis. I wish to thank all the teachers and classmates for their concern and help. Finally, I would like to express my deep thanks and gratitude to my parents for their support and encouragement. Cao Hangyu December 2012 ix

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17 1. Introduction 1.1. Functionally Graded material Functionally graded material is a two-component composite material, in which the material properties vary continuously from one interface to the other. The FGM concept originated in Japan at 1984 during the space plane project, in the form of a proposed thermal barrier material capable of withstanding a surface temperature of 2000 K and a temperature gradient of 1000 K across a cross section <10 mm. Since 1984, FGM thin films have been comprehensively researched, and are almost a commercial reality. In contrast, traditional composites are homogenous mixtures, and they therefore involve a compromise between the desirable properties of the component materials. With the compromise, the traditional composites have disadvantages such as deboning problems at the interface and thermal mismatch that usually results in residual stress. Since the material properties of FGM vary continuously, the need for compromise is eliminated. The advantages of both components can be fully utilized. For example, the toughness of a metal can be mated with the refractoriness of a ceramic, without any compromise in the toughness of the metal side or the refractoriness of the ceramic side. Due to the high quality of the FGM, it meets the requirement of the growing new technologies especially in aerospace, microelectronics, biomechanical, nuclear engineering and so on. So the FGM is suitable for various applications, such as thermal coatings of barrier for ceramic engines, gas turbines, nuclear fusion and so on Literature review Because of wide application, many studies have been carried out to analyze the behaviors and understand the mechanics and mechanism of FGM structures. As a kind of structure components and mechanical element, plates and shells are widely studied by many researchers. Many published papers are focused on the problem of static and dynamic problems of FGM plates and shells.

18 FGM plate problems deal with two main questions: the model of the plate (plate theories) and the solution methodologies. Most commonly used plate model can be classified into two types: 1. Two dimensional elastic plate theories, such as the classical Kirchhoff thin plate theory (CPT), the first order shear deformation plate theory or the Mindlin theory (FSDT), the third order shear deformation theory (TSDT) and the higher order shear deformation plate theory (HSDT), 2. Three dimensional elastic plate theory. The available solution methodologies to solve the aforementioned plate theories can be classified as three types: 1. Numerical method, such as Ritz energy method, Galerkin method, finite element method, 2. Semi-analytical methods, such as power series method, 3.The analytical method Two-Dimensional Plate Theories Classical plate theory (CPT) When the plate is thin, the classical plate theory (CPT) is used to analyze FGM plate problem. A typical derivation of the classical linear equations for flexure of a thin plate was given in the standard text by Timoshenko and Woinowsky-Krieger [1]. This theory is established based on three assumptions. 1. The mid-plane is a neutral plane. 2. Straight lines normal to the mid-surface remain straight lines and normal to the midsurface after deformation. 3. The thickness of the plate does not change during a deformation. It means that the vertical strain of the plate is ignored. The research on the free vibration analysis of thin rectangular plates with a pair of opposite edges simply supported, exact solutions were presented by Leissa [2] for all possible combinations of classical boundary conditions along the other edges. Free vibration of FGM simply-supported and clamped rectangular thin plates was considered by Abrate [3] using the CPT. Woo et al. [4] provided an analytical solution for the nonlinear free vibration behavior of FGM square thin plates using the von Karman theory. For the problem of force vibration, Chi and Chung [5], [6] developed the analytical solution for simply supported FGM plates subjected to mechanical loads. A finite 2

19 1.Introduction element formulation based on the CPT was studied by He et al. [7] to control the shape and vibration of the FGM plate with integrated piezoelectric sensors and actuators. Yang and Shen [8] gives the solution of the dynamic response of initially stressed FGM rectangular thin plates based on the CPT. In practice, the CPT is not used in the thick plate and high modes of the thin plate, as the transverse shear deformation and the effects of rotary inertia cannot be ignored. First order shear deformation theory (FSDT) When the plate is moderate thick, the first order shear deformation theory is used to analyze FGM plate problem due to the high efficiency and simplicity. The first order shear deformation theory is an extension of classical plate theory that the shear deformations through the thickness of the plate are considered. A typical derivation of the first order deformation theory for thick plate was given in the standard text by Mindlin [17]. The assumptions in the FSDT are: 1. The mid-plane is a neutral plane. 2. Straight lines normal to the mid-surface remain straight lines but not necessary normal to the mid-surface after deformation. 3. The thickness of the plate does not change during a deformation. It means that the vertical strain of the plate is ignored. Comparing the assumptions, the FSDT extends the assumptions of CPT by releasing the restriction on the angle of shear deformations. The transverse shear strain is assumed to be constant through the thickness of the plate and a shear correction factor is introduced to cater the non-uniform shear strain. A research on the free vibration, buckling, and static deflections of FGM square, circular, and skew plates with different combinations of boundary conditions was carried out by Abrate [9] on the basis of the CPT, FSDT, and third order shear deformation theory (TSDT). Free vibration of annular FGM plates with variable thickness and different combinations of boundary conditions were investigated by Efraimand and Eisenberger [10] based on the first-order shear deformation theory and exact element method to derive the stiffness matrix. Zhao et al. [11] presented a free vibration analysis for FGM 3

20 square and skew plates with different boundary conditions using the element-free kp-ritz method on the basis of the FSDT. An exact closed-form procedure was presented for free vibration analysis of moderately thick rectangular plates having two opposite edges simply supported based on the FSDT by Hosseini-Hashemi et al [12]. For the force vibration of FGM plates, a wide range of problems in FGM cylinders and plates including FGM thermo-mechanical coupling effects were discussed by Reddy and Chin [13], in which finite element solutions were given for simply supported rectangular plates by using FSDT. Praveen and Reddy [14] investigated the nonlinear static and dynamic responses of FGM plates using a plate finite element based on FSDT that accounts for the transverse shear strains, rotary inertia and moderately large rotations in the von Karman sense. Liew et al. [15] analyzed the active vibration control of plate subjected to a thermal gradient based on first-order shear deformation theory. The shear correction factor of homogenous or FGM plate was determined by some researchers. In practice, the shear correction factor is typically taken to be 5/6 from Reissner [16] for homogeneous plates which is based on the parabolic variation of the shear stress distribution. Mindlin [17] also pointed out that shear correction factor is π 2 /12 by thickness shear vibration of homogeneous plate. The dynamic equations of orthotropic laminated plate was derived by Chow [18] using Timoshenko beam theory and a one dimensional procedure was adapted to obtain the shear correction factor by equating the shear strain energy from Timoshenko beam theory to the shear strain energy obtained from equilibrium conditions. Whitney [19] extended the procedure of Chow [18] to derive an expression for the shear correction factor of a cross-ply laminated plate. Also, Whitney [20] developed a procedure for accurately calculating the mechanical behavior of thick laminated composite and sandwich plates, including the effects of transverse shear deformation in the analysis. In his work, an expression for the shear correction factor was derived by considering cylindrical bending about the length and breadth of the plate, and discrete values of shear correction factors were presented for symmetric angleply, anti-symmetric angle-ply, and symmetric cross-ply laminated plates and sandwich plates. An expression for the shear correction factor of cross-ply laminated beams was derived by Bert [21] though an analysis of composite beams. Dharmarajan and Mccutchen [22] extended Cowper's [23] formulation for orthotropic beams and presented 4

21 1.Introduction expressions of shear coefficient for rectangular and tubular cross-sections. A predictorcorrector approach to obtain shear correction factors was proposed by Noor and Peters [24], who obtained the shear correction factors for laminated cylinders. Using an energy equivalence principle, a general expression for the shear correction factor of laminated rectangular beams with arbitrary lay-up configurations was derived by Madabhusi- Raman and Davalos [25]. These values are also no longer appropriate for functionally graded material analysis due to the continuously elastic properties changes in plates. Considering the varied material properties of the FGM plate, Hosseini-Hashemi et al [26] makes the frequency parameter of the analytic solution based on FSDT assumption identical to that acquired by finite element method to obtain the shear correction factor. Nguyen et al [27] identified the shear correction factor though an energy equivalence method. The first order shear deformation theory is the simplest one that accounts for the transverse shear strains. However it requires a shear correction factor to cater for the nonuniform transverse shear strain. Third and Higher order shear deformation plate theory (HSDT) Second and higher orders shear deformation plate theory use higher order polynomials in the expansion of the displacement components through the thickness of the plate. The restriction on warping of the cross section is relaxed and HSDT requires no shear correction factor due to the higher order polynomials is used. However, some additional unknowns are added so that the theory is complicated. A typical derivation of the higher order deformation theory for plates was given in the standard text by Reddy [28]. The assumption of HSDT is shown on the following: 1. The mid-plane is a neutral plane. 2. Straight lines normal to the mid-surface become curve dependent on the order of polynomials and not necessary normal to the mid-surface after deformation. 3. The thickness of the plate does not change during a deformation. It means that the vertical strain of the plate is ignored. 5

22 A review paper that summarized a higher order theory for functionally graded composites which explicitly coupled the microstructural details with material s macrostructures was presented by Pindera et al [29]. Natural frequencies and buckling stress of FGM shallow shells [30], FGM plates [31] and FGM circular cylindrical shells [32] were analyzed by Matsunaga based on higher order shear deformation theory. The free vibration characteristics of functionally graded elliptical cylindrical shells were analyzed by Patel et al [33] using finite element formulated based on the higher order shear deformation theory. A buckling analysis of FGM circular plate under various types of thermal loads has been carried out by Najafizadeh and Heydari [34] based on the third order shear deformation plate theory. The thermal buckling analysis of simply supported rectangular FGM plates under four different types of thermal loadings has been investigated by Javaheri and Eslami [35] on the basis of third order shear deformation theory. A two dimensional global higher order deformation theory was presented by Matsunaga [36] for thermal buckling analysis of simply supported rectangle FGM plate and the critical temperature prediction. The nonlinear deflection analysis of the large deflection behavior of a simply supported elastic rectangular FGM plate subjected to a pressure loading was done by Khabbaz et al [37] using a higher order shear deformation theory. In this research, the sinusoidal deflection is assumed and the solutions are achieved by minimizing the total potential energy. The nonlinear analysis of laminated composite and sigmoid through thickness FGM anisotropic structures was done by Han et al [38] using a higher order shear deformation theory based on finite element solutions. The nonlinear behavior of laminated plate in a general state of non-uniform initial stress was studied by Chen et al [39] at large vibration amplitudes based on higher order shear deformation theory Three-dimensional plate theory Plates are actually three dimensional structures which one dimension is relatively much smaller. With some simply assumptions, the governing equations of two dimensional theory, such as classic plate theory, first order shear deformation theory and higher order deformation theory are established. But when the plate is thick, the errors will increase with the increase of the thickness of plate due to the assumptions considered in the two 6

23 1.Introduction dimensional theories. Without any simply assumptions, the three dimensional plate theories only provide more accurate results and bring the physical insight of displacements along the thickness of plate. However, solving the governing equations of three dimensional theories is very complicated and has huge calculations. Many researches have been made for 3-D vibration analysis of rectangular plates with general boundary conditions in recently. Free vibration of FGM sandwich rectangular plates with simply supported and clamped edges was studied by Li et al [40] based on the three dimensional linear theory of elastic. The three displacements of the plates were expanded by a series of Chebyshev polynomials multiplied by appropriate functions to satisfy the essential boundary conditions. The free vibration of cantilevered and completely free isosceles triangular plates was studied by Cheung and Zhou [41] based on exact three-dimensional elasticity theory. An accurate and efficient solution procedure based on the three-dimensional elasticity theory for the free vibration analysis of thick laminated annular sector plates with simply supported radial edges and arbitrary boundary conditions on their circular edges was presented by Malekzadeh [42]. The free vibration analysis of thick FGM plates supported on two-parameter elastic foundation was solved by Malekzadeh [43] with a semi-analytical approach composed of differential quadrature method (DQM) and series solution based on the three-dimensional elasticity theory. A meshless collocation (MC) and an element-free Galerkin (EFG) method, using the differential reproducing kernel (DRK) interpolation, were developed by Wu and Chiu [44] for the quasi-three-dimensional free vibration analysis of simply supported, multilayered composite and FGM plates. For the problem in the thermal condition, free vibration of functionally graded material rectangular plates with simply supported and clamped edges in the thermal environment was studied by Li et al [45] based on the three-dimensional linear theory of elasticity. A generalized three dimensional high-order global local theory that satisfies all the kinematic and transverse stress continuity conditions at the interfaces of the layers, was proposed by Shariyat [46] to investigate dynamic buckling of imperfect sandwich plates subjected to thermo-mechanical loads. Some numerical methods also developed to obtain the static mechanical behavior of FGM plate. The Reissner mixed variational theorem (RMVT) and principle of virtual displacements (PVD) based on finite layer methods (FLMs) were developed by Wu and 7

24 Li [47] for the three-dimensional analysis of simply-supported, multilayered composite and functionally graded material (FGM) plates subjected to mechanical loads. A version of meshless local Petrov Galerkin (MLPG) method was developed by Vaghefi et al [48] to obtain three-dimensional static solutions for thick FGM plates. A three-dimensional elastic solution was developed by Sburlati and Bardella [49] for FGM thick circular plate subject to axisymmetric conditions Outline of the Thesis Due to the high quality of the FGM, it meets the requirement of the growing new technologies especially in aerospace, microelectronics, biomechanical, nuclear engineering and so on. The first order shear deformation theory (FSDT) is a high efficiency and convenient method for analyzing the moderate thick plate. But the shear correction factor of plate is a necessary parameter in FSDT to cater the non-uniformed shear strain in transverse section. The shear correction factor of homogenous plate is not suitable for the FGM plate, as the variation of materiel properties is not considerate. This thesis gives a convenient and effective method to obtain the shear correction factor of FGM plate by thickness shear vibration. A brief summary of the research on the composite plates and shells, which based on the two dimensional elastic and three dimensional elastic theory, are given in the literature review on chapter 1. The material gradient of FGM plate is introduced on the Chapter 2. Thickness shear vibration is an in-plane vibration of infinite plates characterized by shear deformation. It is a kind of in-plane free vibration, the displacement only occurred in the horizontal direction, which means no vertical displacement. The detail assumptions and governing equations of infinite homogeneous plate on thickness shear vibration are introduced in Chapter 3. By establishing a three dimensional FGM plate model and a first order deformation FGM plate model for the thickness shear vibration of the same FGM plate, the shear correction factor can be obtained by matching the fundamental frequency of two model equally. The governing equation of the three dimensional FGM plate is solved by the Galerkin method. 8

25 1.Introduction In Chapters 4 and 5, Parametric studies of the shear correction factor with the variable of material properties, such as the mass density ratio, the Young s modulus ratio the layer ratios and the material index are discussed. Both the FGM plate and sandwich FGM plate are considerate. The presented shear correction factors of FGM plate and sandwich FGM plate are compared with the shear correction factors obtained by the energy method of Nguyen et al [27] and the difference of shear correction factors obtained from these two methods are discussed. In chapter 6, by substituting the presented shear correction factor into the FSDT model, the frequency parameter of FGM plate can be obtained by solving the governing equation. The numerical application of the FSDT model is performed with the four simply supported edges rectangle FGM plate. Herein, the Hamilton s principal is used to derive the equations of motion. Then the governing equations are solved by the Navier s solution. Some comparison studies are given for the frequency parameters of four simply support edges (SSSS) Al/Al 2 O 3 plate with different material indexes (p), thickness to length ratio (τ) and width to length ratio (η), to clarify the effect of the presented shear correction factors. Finally, conclusions of this thesis are given in the Chapter 7. 9

26 2. Material Gradient of FGM Plates 2.1. Introduction Due to the special manufacturing process, the FGMs are macroscopically homogenous in spite of microscopically inhomogeneous. They have continuous variation of material properties from one surface to another, which is different from the stepped or discontinuous composite material. Usually the variation of the material properties is in the thickness direction only and the volume fraction obeys a power law function. In this chapter, the material properties of FGM plate are given Material Gradient of FGM Plates The volume of two different materials in FGM varies continuously from top surface to the bottom surface, which results in continuous and gradual changes in the properties of the material. These variations of material properties are in the thickness direction and a power law function along the thickness direction is used to describe it. The power law function is assumed as follows: g(x 3 ) = X p (2.1) where p is the material index of FGM plate (p 0), X 3 is the non-dimensional thickness parameter. Then the typical material properties (P), such as the Young s modulus (E), the mass density ( ρ ), etc, are varied though the thickness according to the following expression: P(X 3 ) = P t (P t P b ) g(x 3 ) (2.2)

27 1.Introduction Fig. 2.1 variation of metal material s volume fraction with different p where P(X 3 ) is the material property of FGM in particular thickness, P t and P b are the material properties at the top and bottom surfaces of the FGM plate, respectively. In this thesis, the ceramic surface is on the bottom and the metal surface is on the top. The material index p is a very important variable parameter for the FGM plate, as it is directly determined the material distribution of the FGM plate. When p > 1, the FGM plate is ceramic rich which the ceramic properties is more distinct. In contrary, when p < 1, the FGM plate is metal rich which the metal properties is more distinct. The material index can be used to design the FGM with different property for the application purposes. Fig. 2.1 shows the variation of metal material s volume fraction with different material index p = 0.2, 0.5,1,2,5. 11

28 3. Thickness Shear Vibration 3.1. Introduction Thickness shear vibration is a kind of in-plane free vibration where the displacement only occurred in the horizontal direction, which means no vertical displacement. Firstly, the thickness shear vibration of infinite homogenous plate is analyzed by three dimensional theory. Then by establishing a three dimensional FGM plate model and a first order deformation FGM plate model for the thickness shear vibration of the same FGM plate, the shear correction factor can be obtained by matching the fundamental frequency of two models equally. The governing equation of the three dimensional FGM plate is solved by the Galerkin method. Fig. 3.1 Cartesian coordinate system

29 3.Thickness Shear Vibration 3.2. Three Dimensional Elastic Analysis of Thickness Shear Vibration for Homogenous Plate Thickness shear vibration is an in-plane vibration of infinite plates characterized by shear deformation. Fig. 3.1 shows the Cartesian coordinate system. With the infinite plate assumptions, the displacements in three directions are independent of x 1 and x 2. By taking the mid-plane at x 3 = 0 and assuming the displacement variables (u 1, u 2 and u 3 ) vary harmonically with respect to the time variable t, the displacement field of infinite plate is used as follows: u 1 (x 3, t) = Z(x 3 )e iωt u 2 (x 3, t) = 0 u 3 (x 3, t) = 0 (3.1) where u 1, u 2 and u 3 are the displacements in the x 1, x 2 and x 3 direction, respectively, Z(x 3 ) is the shape function of plate along the x 3 direction, ω is the thickness shear vibration frequency of the plate, t is the time. In the elastic deformation of the plate, the strain-displacement equation is as following: ε ij = 1 2 u i,j + u j,i ( i, j = 1,2,3) (3.2) where ε 11, ε 22 and ε 33 are the normal strain in the x 1, x 2 and x 3 direction, respectively, ε 12, ε 13 and ε 23 are the shear strains. Based on the Hooke s law, the stress and strain has a linear relationship in the elastic deformation. So the constitutive equations of the plate are: σ ij = E 1 + ν ε ij + νeδ ij(ε 11 + ε 22 + ε 33 ) (1 + ν)(1 + 2ν) (3.3) where E is the Young s modulus, ν is the Poisson ratio, δ ij = 1, when i = j and δ ij = 0, when i j. Based on the equations of equilibrium, the equations of motion are given as follows: σ ij,j = ρ 2 u i t 2 (3.4) 13

30 where ρ is the mass density of the plate. By substituting the equation (3.1), (3.2) and (3.3) into the equation (3.4), the governing equation of thickness shear vibration for infinite plate is obtained as GZ + ρω 2 Z = 0 (3.5) where G = E 2(1 + ν), divide by G on both side: Z + φ 2 Z = 0 (3.6) Where φ 2 = ρω 2 G. By solving the ordinary differential equation (3.6), the homogenous solution is written as: Z = c 1 sin φx 3 + c 2 cos φx 3 (3.7) where c 1 and c 2 are constants. As the top and bottom face of the plate are free surfaces, there is no traction and hence the following boundary condition can be written as: σ 13, σ 23 = 0; when x 3 = ± h 2 (3.8) where h is the thickness of the plate. By substituting equations (3.2) and (3.3) into (3.8), the boundary condition can be rewritten as: Z = 0; when x 3 = ± h 2. (3.9) By substituting the boundary condition (3.9) into the solution of the ordinary differential equation (3.7), the two equations of boundary condition can be obtained: c 1 sin φh 2 + c 2 cos φh 2 = 0 c 1 sin φh 2 + c 2 cos φh 2 = 0 (3.10) For the non-trivial solution, the coefficient of equations (3.10) must be linear dependent. Then the determinant of the coefficient of equations (3.10) on the below must be zero. 14

31 3.Thickness Shear Vibration sin φh 2 sin φh 2 cos φh 2 cos φh = 0 (3.11) 2 By solving the determinant (3.11), the thickness shear vibration frequency parameter of the plate φ is obtained as: φ = nπ h, n = 0,1 (3.12) By replacing the frequency parameter φ with ω. The thickness shear vibration frequency of the infinite homogenous plate is: ω = nπ h G ρ (3.13) And the solution of the ordinary differential equation (3.6) is: Z = c 1 sin nπx 3 h eiωt, n = 1,3,5 Z = c 2 cos nπx 3 h eiωt, n = 2,4,6 (3.14) The thickness shear vibration modes of infinite homogenous plate are shown in the Fig Due to the material properties are homogenous, when the mode number n is odd, the curvature of mode shape is anti-symmetric with respect to z; when the mode number n is even, the curvature of mode shape is symmetric with respect to x. 15

32 Fig. 3.2 Thickness shear vibration modes of infinite homogenous plate 3.3. Three Dimensional Elastic Analysis of Thickness Shear Vibration for FGM Plate Governing Equations Thickness shear vibration is an in-plane vibration of infinite plates characterized by shear deformation. With the infinite plate assumptions, the displacements in three directions are independent of x 1 and x 2. By taking the mid-plane at x 3 = 0 and assuming the displacement variables ( u 1, u 2 and u 3 ) vary harmonically with respect to the time variable t, the displacement field of infinite plate is used as follows: 16

33 3.Thickness Shear Vibration u 1 (x 3, t) = Z(x 3 )e iωt u 2 (x 3, t) = 0 u 3 (x 3, t) = 0 (3.15) FGM plate is isotropic but the material properties are varied with the thickness parameter. Based on the Hooke s law, the stress and strain has a linear relationship in the elastic deformation. So the constitutive equations of the plate are: σ ij = E(x 3) 1 + ν ε ij + νe(x 3)δ ij (ε 11 + ε 22 + ε 33 ) (1 + ν)(1 + 2ν) (3.16) where E(x 3 ) is the Young s modulus, ν is the Poisson ratio, δ ij = 1, when i = j and δ ij = 0, when i j. The equations of motion are given as follows: σ ij,j = ρ(x 3 ) 2 u i t 2 (3.17) where ρ(x 3 ) is the mass density in a given thickness position. By substituting the equation (3.15), (3.2) and (3.16) into the equation of motion (3.17), the governing equation of thickness shear vibration is obtained as GZ + G Z + ρ(x 3 )ω 2 Z = 0 (3.18) where G(x 3 ) = E(x 3 ) 2(1 + ν).for generality and convenience, the following nondimensional terms are introduced: X 3 = 2x 3 h, G(X 3) = G(x 3) E c, ρ(x 3 ) = ρ(x 3) ρ c, Z(X 3 ) = Z(x 3 ), λ 2 = h2 4 ρ c E c ω 2 (3.19) where λ is the frequency parameter, ρ c is the mass density of ceramic material and E c is the Young s modulus of ceramic material. By substituting the non-dimensional terms into the equation (3.18), the governing equation of thickness shear vibration is expressed as: G Z + G Z + λ 2 ρ Z = 0 (3.20) 17

34 As the top and bottom face are free surface, there is no traction and hence the following boundary condition can be written as: σ 13, σ 23 = 0; when x 3 = ± h 2 (3.21) By substituting (3.2) and (3.16) into (3.21), the boundary condition can be rewritten as: Z = 0; when X 3 = ± Solution by the Galerkin Method An approximate solution satisfying the boundary conditions is assumed by the following: Z (X 3 ) = a k φ k (X 3 ) (3.22) k=1 where φ k (X 3 ) = sin kπx 3 2 when k is odd, or cos kπx 3 2 when k is even. The Galerkin method is used to solve the equation (3.20). For practical purpose, a truncated series with n terms is used instead: n Z (X 3 ) = a k φ k (X 3 ) (3.23) k=1 The orthogonal conditions of the Galerkin method is L(Z ) φ k (z)dv = 0, (k = 1,2 n) (3.24) v where n L(Z ) = a k k2 π 2 k=1( k=2) n 4 G sin kπx kπ 2 G cos kπx ρ λ2 sin kπx a k k2 π 2 k=2( k=2) 4 G cos kπx 3 2 kπ 2 G sin kπx ρ λ2 cos kπx 3 2 (3.25) By substituting (3.25) into (3.24), a set of linear equations for the unknown a i is obtained. B ij a i = 0 1 i, j n (3.26) 18

35 3.Thickness Shear Vibration where B ij 1 i2 π 2 4 G sin iπx iπ 2 G cos iπx ρ λ2 sin iπx i2 π 2 4 G sin iπx iπ 2 G cos iπx ρ λ2 sin iπx = 1 i2 π 2 4 G cos iπx 3 2 iπ 2 G sin iπx ρ λ2 cos iπx i2 π 2 4 G cos iπx 3 2 iπ 2 G sin iπx ρ λ2 cos iπx sin jπx 3 2 dx 3, when i is odd and j is odd cos jπx 3 2 dx 3, when i is odd and j is even sin jπx 3 2 dx 3, when i is even and j is odd cos jπx 3 2 dx 3, when i is even and j is even For the non-trivial solutions, the coefficient of equations (3.26) must be linear dependent. So B nn = 0 (3.27) By solving the equation (3.27), the thickness shear vibration frequency parameters λ of the FGM plate are obtained. The frequency parameters of FGM plate for thickness shear vibration are shown on Table 3.1 with different number of terms used in (3.23). The FGM plate is Al/Al 2 O 3 plate, which the material properties are shown on the Table 6.1. From the frequency parameter values in Table 3.1, it can be found that the convergent results of the fundamental frequency parameter can be obtained, when n is larger than 5. The first four thickness shear vibration modes of infinite FGM plate are show on Fig Table 3.1 Frequency parameters of FGM plate for thickness shear vibration n Mode NO

36 Fig. 3.3 Thickness shear vibration modes of infinite FGM plate Due to the material properties are varied with the thickness, the modes shape are neither symmetric nor anti-symmetric with respect to z, which is different with the thickness shear vibration modes of homogenous plate (Fig. 3.2) Thickness Shear Vibration Based On FSDT In the first order shear deformation theory (FSDT) assumption for the FGM plate, the inplane displacements u 1 and u 2 are expanded as linear functions of the plate thickness parameter (x 3 ). Moreover the vertical displacement u 3 is independent on x 3 due to the negligence of the vertical strain ε 33. By taking the mid-plane at x 3 = 0 and assuming the displacement variables (u, v, w, ψ 1 and ψ 2 ) vary harmonically with respect to the time variable t, the displacement field of infinite plate is used as follows: u 1 (x 1, x 2, x 3, t) = u(x 1, x 2 )e iωt + x 3 ψ 1 (x 1, x 2 )e iωt u 2 (x 1, x 2, x 3, t) = v(x 1, x 2 )e iωt + x 3 ψ 2 (x 1, x 2 )e iωt u 3 (x 1, x 2, x 3, t) = w(x 1, x 2 )e iωt (3.28) 20

37 3.Thickness Shear Vibration where u, v and w are the mid-plane displacements in the x 1, x 2 and x 3 direction respectively, ψ 1, ψ 2 are the rotational displacements about the x 1 and x 2 axes at the middle surface of the plate, respectively. Herein, Hamilton s principle is used to derive equations of motion based on the FSDT. The principle can be stated as follows: t h 2 b a (σ 11 δε 11 + σ 22 δε σ 12 δε σ 13 δε σ 23 δε 23 ) dx 1 dx 2 dx 3 dt 0 h δ 2 0 t h 2 b a ρ(x 3)(u 1 2 +u 2 2 +u 3 2 ) dx 1 dx 2 dx 3 h dt = 0 (3.29) Where a, b and h is the length, width and thickness of the plate. By substituting equation (3.2), (3.15) and (3.28) into the equation (3.29) and collecting the coefficients of δu, δv, δw, δψ 1 and δψ 2, the equation of motions are obtained: δu: N 1 + N 12 = I x 1 x 1 u + I 2 ψ 1 2 δv: N 2 + N 12 = I x 2 x 1 v + I 2 ψ 2 1 δw: Q 1 x 1 + Q 2 x 2 = I 1 w δψ 1 : M 1 + M 12 Q x 1 x 1 = I 2 u + I 3 ψ 1 2 δψ 2 : M 2 + M 12 Q x 2 x 2 = I 2 v + I 3 ψ 2 (3.30) 1 Where the stress resultants N i, Q i, M i and the inertias I i (i = 1,2,3) are defined by h 2 (I 1, I 2, I 3 ) = ρ(x 3 )(1, x 3, x 3 2 )dx 3 h 2 h 2 (N 1, N 2, N 12 ) = (σ 11, σ 22, σ 12 )dx 3 h 2 21

38 h 2 (M 1, M 2, M 12 ) = (σ 11, σ 22, σ 12 )x 3 dx 3 h 2 h 2 (Q 1, Q 2 ) = (σ 13, σ 23 )dx 3 h 2 (3.31) By substituting the equation (3.2) and (3.15) into the stress resultants Q i : h 2 Q i = G(x 3 )ε i3 dx 3 h 2 (3.32) where ε i3 = w x i + ψ i. The shear strain ε i3 is a constant in the x 3 direction based on FSDT assumptions which is different with the real conditions, a shear correction factor k is necessary to cater for the non-uniform shear strain. h 2 (Q 1, Q 2 ) = k 2 (σ 13, σ 23 )dx 3 h 2 For generality and convenience, the following non-dimensional terms are introduced: X 1 = x 1 a, X 2 = x 2 a, η = b a, τ = h a u = u a, v = v a, w = w a, ψ 1 = ψ 1, ψ 2 = ψ 2 I 1 = I 1 ρ c a, I 2 = I 2 ρ c a 2, I 3 = I 3 a2, β = ω ρ c a3 h ρ c (3.33) E c By substituting the non-dimensional terms (3.33) into the equation stress resultants (3.31), the non-dimensional stress resultants are expressed as N 1 = A u,1 + νv,2 + B ψ 1,1 + νψ 2,2 N 2 = A νu,1 + v,2 + B νψ 1,1 + ψ 2,2 22

39 3.Thickness Shear Vibration N 12 = 1 ν 2 A u,2 + v,1 + B ψ 1,2 + ψ 2,1 M 1 = B u,1 + νv,2 + D ψ 1,1 + νψ 2,2 M 2 = B νu,1 + v,2 + D νψ 1,1 + ψ 2,2 M 12 = 1 ν 2 B u,2 + v,1 + D ψ 1,2 + ψ 2,1 Q 1 = 1 ν 2 k2 A ψ 1 + w,1 Q 2 = 1 ν 2 k2 A ψ 2 + w,2 (3.34) where h 2 (A, B, D) = E(x 3) 1 ν 2 (1, x 3, x 2 3 )dx 3 h 2, A = A E c a, B = B E c a 2, D = D E c a 3 k 2 denotes the shear correction factor. By substituting the non-dimensional stress resultants (3.34), (3.2) and (3.3) into the equations (3.30), then the equations of motion are expressed in dimensionless form as follows: A 1 υ 2 2 u υ 2 u,1 + v,2,1 + B 1 υ 2 2 ψ υ 2 ψ 1,1 + ψ 2,2,1 = I 1τ 2 β 2 u I 2τ 2 β 2 ψ 1 A 1 υ 2 2 v υ 2 u,1 + v,2,2 + B 1 υ 2 2 ψ υ 2 ψ 1,1 + ψ 2,2,2 = I 1τ 2 β 2 v I 2τ 2 β 2 ψ 2 1 υ 2 k2 A ψ 1,1 + ψ 2,2 + 2 w = I 1τ 2 β 2 w 23

40 B 1 υ 2 2 u υ 2 u,1 + v,2,1 + D 1 υ 2 2 ψ υ 2 ψ 1,1 + ψ 2,2,1 1 υ 2 k2 A ψ 1 + w,1 = I 2τ 2 β 2 u I 3τ 2 β 2 ψ 1 B 1 υ 2 2 v υ 2 u,1 + v,2,2 + D 1 υ 2 2 ψ υ 2 ψ 1,1 + ψ 2,2,2 1 υ 2 k2 A ψ 2 + w,2 = I 2τ 2 β 2 v I 3τ 2 β 2 ψ 2 (3.35) For the purpose of obtaining the fundamental shear vibration frequency of the FGM plate based on FSDT, ψ 1 = 1, w = 0, u = 0, v = 0 and ψ 2 = 0 are set in the equation (3.35) [17]. Then the fundamental shear vibration frequency of the infinite FGM plate based on FSDT is shown on the below: β 2 = (1 ν)k2 A 2I 3τ 2 (3.36) 3.5. Shear Correction Factor In section 3.3 and 3.4, the fundamental thickness shear vibration frequency parameters of FGM plate based on three-dimensional plate model (λ) and the FSDT plate model (β) are obtained, respectively. By rewriting the frequency parameter in (3.19), the frequency of infinite plate in the three-dimensional plate model can be written as: ω E = 4λ2 h 2 E c ρ c (3.37) At the same time, by rewriting the frequency parameter in (3.33), the frequency of infinite plate in the FSDT plate model can be written as: ω F = β h a 2 E c ρ c (3.38) By substituting the equation (3.36) into (3.38): 24

41 3.Thickness Shear Vibration ω F = (1 ν)k2 A 2I 3 (3.39) Then the shear correction factors can be determined by matching these two frequency ω E and ω F equally. k 2 = 8λ 2 I 3 E c (1 ν)ah 2 ρ c (3.40) 25

42 4. Shear Correction Factor of FGM Plate 4.1. Introduction The thickness shear vibration of the infinite homogenous and FGM plate has been discussed in the Chapter 3 based on three-dimensional theory and the first order shear deformation theory respectively. The procedure of obtaining the shear correction factors of FGM plate though the thickness shear vibration is also shown on the chapter 3. In this chapter, the present shear correction factors are compared with the shear correction factors obtained by the energy method of Nguyen et al [27] and the difference of shear correction factors obtained from these two methods are discussed. Parametric studies of the shear correction factor with different modulus ratios, the mass density ratios and the material index for FGM plates are presented. The material properties of the FGM plates vary continuously in the thickness directions except the Poisson s ratio which is assumed to be a constant of 0.3. The volume fraction of FGM plate is assumed to obey a powerlaw function along the thickness direction Parametric study For the homogenous plate, the shear correction factor is identified by many researches. Reissner [16] has given a value of 5/6 by assuming a parabolic variation of the shear stress. Moreover Mindlin [17] has given a value of π 2 /12 by the thickness shear vibration. But it will lead a different value for the FGM plates due to the material properties continuously change in the thickness direction. With the thickness shear vibration method shown on the chapter 3, the shear correction factors of different FGM plate are obtained. In the following illustrations, R y = E c E m is the Young s modulus ratio of metal material and ceramic material and on the range between 0 and 20. R m = ρ c ρ m is the mass density ration of metal material and ceramic material and on the range between 1 and 3. p is the material index and on the range between 0 and 20. The values of the shear correction factors for every couple (p, R y ) is given in the Table 4.1, Table 4.2 and

43 4.Shear Correction Factor of FGM Plate Table 4.3, when R m = 1,2,3 respectively. Fig. 4.1, Fig. 4.3, and Fig. 4.5 present the variation profile of the shear correction factors according to the Young s modulus ratio R y, when R m = 1,2,3 respectively. When p > 1, the shear correction factor will decrease with the increase of R y. Fig. 4.2, Fig. 4.4 and Fig. 4.6 present the variation profile of the shear correction factors according to the material index p when R m = 1,2,3 respectively. When p > 1 and R y > 1, the shear correction factor will be smaller than π 2 /12, which is the shear correction factor of homogenous plate based on thickness shear vibration method. When p < 1, they may be higher than π 2 /12. And the shear correction factor will go back to π 2 /12, when p or 0, for fully ceramic or metal plate. The shear correction factors slightly increase with the increase of R m. The shear correction factor of FGM plate is within the region between 0.30 and 0.84, when R y, R m and p are on the range of [0, 20], [1,3] and [0,20], respectively Comparison The present shear correction factors obtained from thickness shear vibration of infinite FGM plate are compared with the shear correction factor obtained by shear energy equivalent method (EM) of Nguyen et al [27]. The relative difference between the shear correction factor obtained by the thickness shear vibration method and energy equivalent method according to the Young s modulus ratio R y and material index p when mass density ratio R m = 1,2 is shown in Table 4.4. The maximum value of relative difference can be up to 42.44% when p = 2, R y = 20. The relative difference of shear correction factor is increased when the Young s modulus ratio R y increases, it is easy to found that the Young s modulus ratios R y has bigger influence on the shear correction factor in the thickness shear vibration method. And the relative difference of shear correction factor is decreased when the mass density ratio R m increase, it means that the relative difference will be very small when the proper value of R m has been used. Along with the increase of p, the relative difference of shear correction factor is initially increased when p 2 and then decreased when p > 2. 27

44 4.4. Summary In this chapter, the shear correction factor of FGM plate is obtained by the thickness shear vibration of the plate. The relationship of the shear correction factor and variable parameters is also shown. The variable parameters, such as material index, the Young s modulus ratio and the mass density ratio have important influence on the shear correction factor. The comparison results show the difference of the shear correction factor between the present method and the energy equivalent method. It can be easily observed that the difference will be very small, when the proper R m is used. 28

45 4.Shear Correction Factor of FGM Plate Table 4.1 Shear correction factor k (R m = 1) p R y Table 4.2 Shear correction factor k (R m = 2) p R y

46 Table 4.3 Shear correction factor k ( R m = 3) p R y

47 4.Shear Correction Factor of FGM Plate Table 4.4 Comparison of shear correction factor k (R m = 1, 2) p R y [27] R m = difference 1.30% 1.30% 1.30% 1.30% 1.30% 1.30% 1.30% R m = difference 1.30% 1.30% 1.30% 1.30% 1.30% 1.30% 1.30% [27] R m = difference 1.30% 0.83% 0.72% 0.71% 0.71% 0.71% 0.72% R m = difference 2.07% 1.27% 0.96% 0.88% 0.83% 0.80% 0.77% [27] R m = difference 1.30% 1.79% 3.58% 4.70% 5.43% 5.92% 7.08% R m = difference 2.37% 1.30% 1.85% 2.43% 2.84% 3.13% 3.84% [27] R m = difference 1.30% 3.10% 7.72% 10.97% 13.24% 14.92% 19.26% R m = difference 2.03% 1.42% 3.93% 6.18% 7.85% 9.13% 12.53% [27] R m = difference 1.30% 4.72% 12.85% 19.55% 24.88% 29.22% 42.44% R m = difference 1.06% 1.58% 7.10% 12.65% 17.38% 21.38% 34.18% [27] R m = difference 1.30% 4.56% 10.30% 15.07% 19.09% 22.58% 35.35% R m = difference 0.43% 1.30% 5.56% 9.58% 13.17% 16.40% 28.90% [27] R m = difference 1.30% 4.07% 8.08% 11.20% 13.80% 16.03% 24.15% R m = difference 0.84% 1.20% 4.43% 7.10% 9.41% 11.46% 19.12% 31

48 Fig. 4.1 Variation of k according to R y ( R m = 1) Fig. 4.2 Variation of k versus p(r m = 1) 32

49 4.Shear Correction Factor of FGM Plate Fig. 4.3 Variation of k versus R y ( R m = 2) Fig. 4.4 Variation of k versus p( R m = 2) 33

50 Fig. 4.5 Variation of k versus R y ( R m = 3) Fig. 4.6 Variation of k versus p( R m = 3) 34

51 5. Shear Correction Factor of FGM Sandwich Plate 5.1. Introduction This chapter studies the shear correction factor of the sandwich FGM plate, which is composed of two FGM face sheets and a homogenous core. The shear correction factors of different sandwich FGM plate can be obtained by the thickness shear vibration method presented in Chapter 3. The present shear correction factors are compared with the shear correction factors obtained by the energy method of Nguyen et al [27] and the difference of shear correction factors obtained by these two methods are discussed. Parametric studies of the shear correction factor with different modulus ratios, the layer ratios, the mass density ratios and the material index for FGM plates are presented. The material properties of the FGM plates vary continuously in the thickness directions except the Poisson s ratio which is assumed to be a constant of 0.3. The volume fraction of FGM plate is assumed to obey a power-law function along the thickness direction Material Properties of FGM Sandwich Plates A sandwich FGM plate composes of three sub layers, two FGM sub layers and one homogenous sub layer. The volume of two different materials in FGM also varies continuously from top surface to the bottom surface, which results in continuous and gradual changes in the properties of the material. The vertical positions of the bottom face, two interfaces and top face are respectively denoted by h 1, h 2, h 3 and h 4.These variations of material properties are also in the thickness direction and a power law function is used to describe it. The power law function is assumed as follows:

52 X p 3 h 1, X h 2 h 3 [h 1, h 2 ] 1 g(x 3 ) = 1, X 3 [h 2, h 3 ] X p 3 h 3, X h 4 h 3 [h 3, h 4 ] 3 (5.1) where p is the material index(p 0), X 3 is the non-dimensional thickness parameter. Then the typical material properties (P), such as the Young s modulus (E), the mass density (ρ), etc, are varied though the thickness according to the following expression: P(X 3 ) = P t (P t P c ) g(x 3 ) (5.2) where P(X 3 ) is the material property of FGM in particular thickness, P t and P c are the material properties at the top and core surfaces of the FGM plate, respectively. Fig. 5.1 Variation of metal s volume fraction in sandwich FGM plate with different p In this thesis, the variation of material is according to the type of ceramic-metalceramic. The material index p is a very important variable parameter for the FGM plate, as it is directly determined the material distribution of the FGM face sheet in sandwich 36

53 5.Shear Correction Factor of FGM Sandwich Plate plate. When p > 1, the sandwich plate is ceramic rich in the FGM face sheet which the ceramic properties is more distinct. In contrary, when p < 1, the sandwich plate is metal rich in the FGM face sheet which the metal properties is more distinct. The material index can be used to design the FGM with different property for the application purposes Parametrical Study With the assumed material properties in the section 5.2, the thickness shear vibration method, which shows on the chapter 3, also can be used to obtain the shear correction factor of sandwich FGM plate. In the following illustrations, R t = (h 3 h 2 ) (h 2 h 1 ) is the thickness ratio of FGM face sheet and homogenous core. It is on the range between 1 and 40. The values of the shear correction factors for every couple (p, R y ) is given in the Table 5.1, Table 5.2 and Table 5.3, when R t = 4 and R m = 1, 2, 3 respectively. Fig. 5.2, Fig. 5.4 and Fig. 5.6 present the variation profile of the shear correction factors according to the Young s modulus ratio R y when R t = 4 and R m = 1, 2, 3 respectively. Along with the increase of R y, the shear correction factor will initially increase then decrease when p < 1. Fig. 5.3, Fig. 5.5 and Fig. 5.7 present the variation profile of the shear correction factors according to the material index p when R t = 4 and R m = 1,2,3 respectively. Fig. 5.9, Fig and Fig present the variation profile of the shear correction factors according to the material index p when R y = 6 and R m = 1,2,3 respectively. When p > 1 and R y > 1, the shear correction factor will be smaller than π 2 /12. When p < 1 and R m > 1, they may be higher than π 2 /12. And the shear correction factor will go back to π 2 /12, when p or 0, for fully ceramic or metal plate. Fig. 5.8, Fig and Fig present the variation profile of the shear correction factors according to the layer ratios R t when R y = 6 and R m = 1,2,3 respectively. The shear correction factor will increase when the thickness ratio R t increase. The shear correction factor of FGM sandwich plate is within the region between 0.12 and 0.89, when R y, R m, p and R t are on the range of [0, 20], [1,3], [0,20] and [1,40] respectively. 37

54 5.4. Comparison The present shear correction factors obtained from thickness shear vibration method are compared with the shear correction factor obtained by energy equivalent method (EM) of Nguyen et al [27]. The relative difference of shear correction factor between the thickness shear vibration method (TSVM) and energy method (EM) according to the Young s modulus R y and material index p when R m = 1,2 and R t = 4 is shown on the Table 5.7. The maximum value of relative difference is 21.63% when p = 1, R y = 20. The relative difference of shear correction factor is increased when the Young s modulus ratio R y increase, it is easy to found that the Young s modulus ratios R y has bigger influence on the shear correction factor in the thickness shear vibration method. And the relative difference of shear correction factor is decreased when the mass density ratio R m increase, it means that the relative difference of shear correction factor will be very small when the proper value of R m has been used. Along with the increase of p, the relative difference of shear correction factor is initially increased when p 2 and then decreased when p > 2. The relative difference of shear correction factor between the thickness shear vibration method and energy method according to the thickness ratio R t and material index p when R m = 1,2 and R y = 6 is shown on the Table 5.8. The maximum value of relative difference is 17.69% when p = 1, R t = 1. The relative difference of shear correction factor is decreased when the layer s ratio of FGM face sheet and homogenous core R t increase Summary In this chapter, the shear correction factor of sandwich FGM plate is obtained by the thickness shear vibration. The relationship of the shear correction factor and variable parameters is also shown. The variable parameters, such as material index, the Young s modulus ratio, the mass density ratio and layer s ratio have important influence on the shear correction factor. The comparison results show the difference of the shear correction factor between the present method and the energy equivalent method. The relative difference of shear correction factor is increased when the Young s modulus ratio 38

55 5.Shear Correction Factor of FGM Sandwich Plate R y increases. It is obvious that the Young s modulus ratio R y has bigger influence on the shear correction factor in the thickness shear vibration method. And the relative difference of shear correction factor is decreased when the mass density ratio R m increase, it means that the relative difference of shear correction factor will be very small when the proper value of R m has been used. The relative difference of shear correction factor is decreased when the thickness ratio R t increases. 39

56 Table 5.1 Shear correction factor k (R m = 1, R t = 4) P R y Table 5.2 Shear correction factor k (R m = 2, R t = 4) P Ry

57 5.Shear Correction Factor of FGM Sandwich Plate Table 5.3 Shear correction factor k (R m = 3, R t = 4) P Ry Table 5.4 Shear correction factor k ( R m = 1, R y = 6) P Rt

58 Table 5.5 Shear correction factor k (R m = 2, R y = 6) P Rt Table 5.6 Shear correction factor k (R m = 3, R y = 6) P Ry

59 5.Shear Correction Factor of FGM Sandwich Plate Table 5.7 Comparison of Shear correction factor k ( R m = 1, 2 and R t = 4) p R y [27] R m = difference 1.30% 1.30% 1.30% 1.30% 1.30% 1.30% 1.30% R m = difference 1.30% 1.30% 1.30% 1.30% 1.30% 1.30% 1.30% [27] R m = difference 1.30% 3.92% 7.66% 10.29% 12.25% 13.79% 18.42% R m = difference 1.36% 1.22% 4.92% 7.52% 9.46% 11.00% 15.58% [27] R m = difference 1.30% 6.07% 11.39% 14.43% 16.43% 17.86% 21.50% R m = difference 3.37% 1.16% 6.28% 9.22% 11.14% 12.53% 16.00% [27] R m = difference 1.30% 6.91% 12.54% 15.50% 17.33% 18.59% 21.63% R m = difference 4.03% 1.14% 6.44% 9.24% 10.97% 12.15% 14.93% [27] R m = difference 1.30% 7.59% 13.19% 15.84% 17.39% 18.41% 20.65% R m = difference 4.39% 1.12% 6.20% 8.60% 9.96% 10.88% 12.80% [27] R m = difference 1.30% 7.71% 12.70% 14.80% 15.92% 16.61% 17.97% R m = difference 4.16% 1.02% 5.22% 6.97% 7.87% 8.41% 9.41% [27] R m = difference 1.30% 7.60% 12.30% 14.16% 15.16% 15.76% 16.86% R m = difference 3.99% 0.94% 4.79% 6.27% 7.02% 7.47% 8.21% 43

60 Table 5.8 Comparison of shear correction factor k ( R m = 1, 2 and R y = 6) p R t [27] R m = difference 1.30% 1.30% 1.30% 1.30% 1.30% 1.30% 1.30% R m = difference 1.30% 1.30% 1.30% 1.30% 1.30% 1.30% 1.30% [27] R m = difference 12.17% 11.52% 10.29% 9.35% 8.63% 8.05% 6.93% R m = difference 8.61% 8.22% 7.52% 6.98% 6.55% 6.19% 5.49% [27] R m = difference 16.81% 16.07% 14.43% 13.32% 12.44% 11.75% 10.48% R m = difference 10.42% 10.01% 9.22% 8.77% 8.40% 8.12% 7.59% [27] R m = difference 17.69% 17.10% 15.50% 14.44% 13.60% 12.94% 11.71% R m = difference 10.39% 9.98% 9.24% 8.91% 8.62% 8.42% 8.07% [27] R m = difference 17.23% 17.02% 15.84% 15.05% 14.37% 13.79% 12.71% R m = difference 9.67% 9.21% 8.60% 8.57% 8.43% 8.30% 8.20% [27] R m = difference 13.95% 14.78% 14.86% 14.63% 14.33% 13.98% 13.20% R m = difference 7.47% 7.06% 7.04% 7.42% 7.63% 7.68% 7.86% [27] R m = difference 12.12% 13.64% 14.31% 14.29% 14.16% 13.88% 13.22% R m = difference 6.18% 6.07% 6.43% 6.93% 7.30% 7.43% 7.69% 44

61 5.Shear Correction Factor of FGM Sandwich Plate Fig. 5.2 Variation of k versus R y ( R m = 1, R t = 4) Fig. 5.3 Variation of k versus p(r m = 1, R t = 4) 45

62 Fig. 5.4 Variation of k versus R y ( R m = 2, R t = 4) Fig. 5.5 Variation of k versus p( R m = 2, R t = 4) 46

63 5.Shear Correction Factor of FGM Sandwich Plate Fig. 5.6 Variation of k versus R y ( R m = 3, R t = 4) Fig. 5.7 Variation of k versus p( R m = 3, R t = 4) 47

64 Fig. 5.8 Variation of k versus R t ( R m = 1, R y = 6) Fig. 5.9 Variation of k versus p( R m = 1, R y = 6) 48

65 5.Shear Correction Factor of FGM Sandwich Plate Fig Variation of k versus R t (R m = 2, R y = 6) Fig Variation of k versus p( R m = 2, R y = 6) 49

66 Fig Variation of k versus R t ( R m = 3, R y = 6) Fig Variation of k versus p( R m = 3, R y = 6) 50

67 6. Free Vibration of Simply Supported Rectangular FGM Plates 6.1. Introduction In this chapter, shear correction factors obtained by the present method are used to the free vibration frequency of simply supported rectangular FGM plates based on the firstorder shear deformation theory. The governing equations for simply supported FGM plates made of Al/Al 2 O 3 are solved by the Navier s solution. Comparison studies are given for the frequency parameters obtained by different shear correct factors and by different numerical model. Fig. 6.1 Geometry of a rectangular FGM plate and coordinate system