Free vibration analysis of four parameter functionally graded plates accounting transverse shear mode

Volume 36 Number 2 2 2014 Vietnam Journal of Mechanics, VAST, Vol. 36, No. 2 (2014), pp. 145 – 160 FREE VIBRATION ANALYSIS OF FOUR PARAMETER FUNCTIONALLY GRADED PLATES ACCOUNTING TRANSVERSE SHEAR MODE Gulshan Taj M. N. A.1,∗, Anupam Chakrabarti1, Mohammad Talha2 1Indian Institute of Technology Roorkee, India 2International Institute for Aerospace Engineering and Management, Jain University, India ∗E-mail: gulshantaj19@yahoo.co.in Received September 24, 2013 Abstract. In the present investigation, free vibration analysis of functionally graded material (FGM) plate is performed incorporating higher order shear deformation the- ory in conjunction with C0 based finite element formulation. The cubic component of thickness term is incorporated in in-plane fields and constant variation of thickness is assumed for transverse component. The theory incorporates the realistic parabolic vari- ation of transverse shear stresses thus eliminates the use of shear correction factor. Alu- minium/Zirconia plate is considered for the analysis and the effective properties are assumed to have smooth and gradual variation in the thickness direction and remain constant in in-plane direction. The spatial variation of properties pertaining to homoge- neous and FGM plate is estimated by means of power law, which is described by the four parameters. With respect to dynamic analysis, it is vital for an analyst to know whether the top of the plate is ceramic or metal rich, and inversely bottom of the plate is ceramic or metal rich. This phenomenon can be described by choosing the appropriate values of the parameters appears in the power law. In the study, prominence has been given to study the influence of power law parameters on frequency response of FGM plates so as to accomplish different combination of FGM profiles. Thin and moderately thick FGM plates are analyzed to generate the frequency values of the FGM plate. The imperative conclusions presented regarding the choice of parameter in the power law could be useful for designer to arrive for particular material profile of FGM plate. Keywords : Functionally graded plate, four parameter law, higher order shear deformation theory, free vibration analysis. 1. INTRODUCTION In view to eliminate the various shortcomings like, delamination, huge stiffness jump and stress discontinuity across the layer interface proffer by conventional composite mate- rials, a new class of materials are in demand by the research community to arrive for opti- mum and accurate design of structural elements. With regard to this, advanced composite materials with inhomogenous anatomy are discovered by Japan scientists and introduced 146 Gulshan Taj M. N. A., Anupam Chakrabarti, Mohammad Talha to the engineering society to serve under different operating conditions. Typically, FGMs are manufactured by using powder metallurgy techniques, where the two distinct materials are combined to extract their individual superior properties. For the purpose, metal and ceramics are usually united to form FGM structures, which are decidedly applicable in thermal environments e.g., nuclear structures, space shuttles and automotive industries. Moreover, tailoring of FGM material is probable to suit the various practical demands, by virtue of their smooth and gradual variation of properties in the preferred direction. Due to the aforementioned benefits, their vibration response becomes crucial for the safe and optimum design of structure under consideration. A wide range of publications are recorded in the scientific literature for dynamic analysis of FGM plate and shells concerning various numerical and analytical tools. This is evident from the review article by Jha et al. [1] where the studies related to thermo- elastic static and vibration are discussed briefly based on the various literature data exists since 1998. They noticed that most of the theories developed so far for the analysis of FGM structures considers the transverse shear deformation and the obtained 2D results are usually validated with 3D elasticity solutions. In spite of large number of vibration studies available for FGM plate and shells, only the studies related to the current topic is discussed for the sake of brevity of the presentation. Yang and Shen [2] analyzed the effect of thermal field on free and forced vibration analysis of functionally graded plates. In the study Reddy’s higher order shear deformation plate theory is combined with Galerkin technique. They observed that the vibration response of homogeneous plate do not show any intermediate sense, and this tendency is particular when material properties are tem- perature dependent. Further the authors [3] extended their work to examine the free vibration and stability analysis of FGM cylindrical shell panels under thermo-mechanical environment. Reddy’s higher order theory, Galerkin technique and Blotin’s method are applied to study the response of the FGM shell panels. Qian et al. [4] presented the static and dynamic analysis of functionally graded plates incorporating higher order shear and normal deformable plate theory. Isvandzibaei and Moarrefzadeh [5] performed the free vibration analysis of FGM shells and influence of different material and geometric param- eters on frequency characteristics of shell are discussed in the investigation. Free vibration characteristics of functionally graded cylindrical shell using Reddy’s higher order shear deformation theory is performed by Setareh and Isvandzibaei [6]. Static and free vibration analysis of functionally graded material plate considering variation of transverse displace- ment field is investigated by Talha and Singh [7]. They used variational approach to derive the fundamental equations and considered traction free boundary conditions on the top and bottom faces of the plate to solve for the unknown polynomial terms. Abrate [8] carried out the static, buckling and free vibration analysis of functionally graded plates and pointed out that, the natural frequencies of functionally graded plates are always proportional to those of homogeneous isotropic plates. Uymaz and Aydogdu [9] carried out vibration analysis of FGM plate and they used Chebyshev polynomials to express dis- placement fields along with Ritz method. Ebrahimi and Rastgo [10] investigated the free vibration behavior of functionally graded circular plates integrated with two uniformly distributed actuator layers made of piezoelectric material based on classical plate theory. In course of time meshless based method attained popularity due to the absence of mesh Free vibration analysis of four parameter functionally graded plates accounting . . . 147 technique. In this connection, global collocation method in conjunction with the first and the third-order shear deformation plate theories are used to analyze free vibrations of functionally graded plates by Ferreira et al. [11]. Other studies include vibration analysis of FGM structure under linear, non linear, dynamic and electrical field [12–17]. All the research works so far discussed employs either power law or exponential law to derive the effective properties of ceramic and metal as well. In recent times, Tornabene and his asso- ciates [18, 19] incorporated the power law type equation modeled with four parameters for the calculation of material properties. In the research, the author established that different material profiles (metal at top and ceramic at bottom, both top and bottom ceramic, top and bottom are occupied by 50 percent ceramic and 50 percent metal and perhaps other combination also) are possible, by means of appropriate selection of variables. Generalized differential quadrature technique is employed to decompose the governing equations of the plate. A vast number of examples are presented along with the different mode shapes for various types of FGM structures. The kinematic relations are based on the first order shear deformation theory which assumes the linear variation of transverse displacement through the thickness. It was noticed that some practical design requirements often demands combination of FGM material profiles to meet certain design criteria. Such a demand can be fulfilled by use of suitable general power law distribution reported in the literature that comprises of four parameters to describe the material profile along the thickness direction. The studies so far performed on this topic are based on first order shear deformation theory. For the realistic analysis, it is important to incorporate the actual transverse stress profile in the displacement field. Such an analysis will predict the accurate global response of the plate under loading conditions, in exact sense. To fill this gap, an attempt has been exerted by the authors to study the free vibration analysis of FGM plates using higher order shear deformation theory. A four parameter power law reported in the literature is adopted to estimate the volume fraction of ceramic and metal constituents. Further, the effect of parameters exists in the power law distribution has been studied by performing different numerical examples. Thin and moderately thick plates with different boundary conditions are incorporated in the developed MATLAB (R2011b) code. Conclusions regarding choice of various parameters exist in the power law, type of boundary conditions and thickness ratio could serve as crucial date for researchers involved in FGM analysis. Section 2 elabo- rates the assumed kinematics field incorporating transverse deformation mode along with the constitutive relationship of FGM material. In Section 3, various numerical examples performed are briefed and to finish, the important key points with respect to the free vibration analysis of four parameter FGM plate are arranged in conclusion part. 2. MATHEMATICAL FORMULATION AND MATERIAL PROPERTIES 2.1. Four parameter power law and constitutive relationship In general FGM are characterized by their gradual and continuous variation of material properties along the chosen direction (usually thickness direction), hence it is prime factor to capture the accurate particle size distribution in spatial direction. To incorporate this phenomena many methods were addressed and subsequently employed 148 Gulshan Taj M. N. A., Anupam Chakrabarti, Mohammad Talha in the literature by various researchers. Self consistent scheme [20], Mori-Tanaka scheme [21], power law [22], exponential law [23] and sigmoid function [24] are few to cite. Each method has is own superiorities to define the material distribution in a FGM plate. Most of the researchers prefer the power law function to estimate the given material properties [7, 11, 13]. In recent times, Tornabene and his associates [18, 19] established a simple four-parameter power-law and it is given by FGM-I(a, b, c, p) : Vc = ( 1− a ( 0.5 + z h ) + b ( 0.5 + z h )c)p (1a) FGM-II(a, b, c, p) : Vc = ( 1− a ( 0.5− z h ) + b ( 0.5− z h )c)p (1b) It is to be noted that, unlike the conventional distribution, the present power law distribu- tion is described by the three parameters a, b, c and power law variable p in the expression. Such a representation would enable the designer to opt for different material profiles such as ceramic at top and metal at bottom, similarly, metal at top and ceramic at bottom and many others. The power law exponent p in the formula assumes the value between zero and infinity to represent different cases of FGM plates. For instance, the value of p = zero, represents homogenous case of ceramic plate, while in other hand the value of p = infinity resembles the FGM plate occupied by metal segment. The user can vary the range of power law exponent in between zero and infinity to get the plate with gradation properties. Different material combinations of FGM distributions are probable by Eqs. (1a) and (1b), and the type of material distribution depends on choice of the parameters a, b and c. However, the difference of frequency parameters obtained by FGM-I and FGM- II distributions are insignificant and shows deviation after third decimal point (refer the work [18]). Hence in the present work, FGM-I distribution is incorporated to solve the free vibration problem of Al/Zro2 plate. Several representations of volume fraction of ceramic are exhibited in Fig. 1. Classical FGM can be achieved by means of choosing a = 1 and b = 0. Other profiles are obtained by suitably assuming the values of the three parameters a, b and c in the power law formula. -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.2 0.4 0.6 0.8 1.0 Volume fraction of ceramic N o n -d im e n si o n a l d e p th (z /h ) p=0.2 p=0.4 p=0.5 p=0.8 p=1 p=1.5 p=2.00 p=2.5 p=2.8 p=3 p=4 p=5 p=10 p=20 p=30 p=50 a=1,b=0 (a) -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.2 0.4 0.6 0.8 1.0 Volume fraction of ceramic N o n -d im e n si o n a l d e p th (z /h ) p=0.2 p=0.4 p=0.5 p=0.8 p=1 p=1.5 p=2.5 p=2.8 p=3 p=4 p=5 p=10 p=20 p=30 p=50 a=b=1, c=2 (b) -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.2 0.4 0.6 0.8 1.0 Volume fraction of ceramic N o n -d im e n s io n a l d e p th (z /h ) p=0.2 p=0.4 p=0.5 p=0.8 p=1 p=1.5 p=2.00 p=2.5 p=2.8 p=3 p=4 p=5 p=10 p=20 p=30 p=50 a=1,b=0.5, c=2 (c) Fig. 1. Different profiles of FGM for several ranges of power law parameters (a) a = 1, b = 0 (classical profile) (b) a = b = 1, c = 2.0 (symmetric profile) (c) a = 1, b = 0.5, c = 2.0 The constitutive relationship of functionally graded plate assuming plane stress condition (σzz = 0) may be written as, Free vibration analysis of four parameter functionally graded plates accounting . . . 149   σxx σyy σyz σxz σxy   =   Q11 Q12 0 0 0 Q21 Q22 0 0 0 0 0 Q33 0 0 0 0 0 Q44 0 0 0 0 0 Q55     εxx εyy γyz γxz γxy   (2) Q11 = Q22 = E(z) 1− γ2 , Q12 = Q21 = γE(z) 1− γ2 , Q33 = Q44 = Q55 = E(z) 2(1 + γ) where stiffness co-efficient (Qij) contains the terms Young’s modulus (E) and Poisson’s ratio (γ), in which E alone is the function of depth since the properties are assumed as temperature in-dependent. Utilizing the available expression from Eqs. (1a) and (1b), one can arrive at the material properties of the FGM plate in the following manner. E(z) = Eb + (Et −Eb) ( 0.5 + z h )p ρ(z) = ρb + (ρt − ρb) ( 0.5 + z h )p (3) The subscripts ‘b’ and ‘t’ represent the bottom and top portion of the FGM plate which are usually represented by metal and ceramic, respectively. In the present study, Young’s modulus (E) and density (ρ) and treated as position dependent and Poisson’s ratio (ν) is assumed to be constant of 0.3. Throughout the analysis, the functional relationship of Vc + Vm = 1.0 between the ceramic and metal has to be maintained. 2.2. Displacement function A Reddy’s higher order theory [25] has been implemented in the present study which accounts for the parabolic distribution of transverse shear stresses in the plate. In the theory, the in-plane displacement fields (u and v) are expanded as cubic functions of the thickness coordinate (z), while the transverse displacement (w) variable has been assumed to be constant through the thickness. Any other choice of displacement field would either not satisfy the stress-free boundary conditions or lead to a theory that would involve more dependent unknowns than those in the first-order shear deformation theory [25]. Since the theory assumes the accurate profile of the transverse stress components, the use of shear correction factor could be avoided efficiently. According to Reddy’s higher order shear deformation theory [25], the in-plane displacements (u and v) and transverse displacement (w) are expressed in terms of corresponding displacements at the mid surface (u0, v0 and w0) by the following expression. u(x, y, z) = u0(x, y) + zθx(x, y) + z 2ξx(x, y) + z 3ζx(x, y) v(x, y, z) = v0(x, y) + zθy(x, y) + z 2ξy(x, y) + z 3ζy(x, y) w(x, y) = w0(x, y) (4) where the parameters u0, v0 and w0 are the displacements of points which are in the mid- surface (i.e., reference surface) of the plate and θx, θy are the bending rotations about the y and x axes respectively. ξx, ξy, ζx and ζy are higher order terms appears in Taylor’s series 150 Gulshan Taj M. N. A., Anupam Chakrabarti, Mohammad Talha expansion and are solved by the condition of zero transverse shear stains (γxz(x, y,±h/2) = γyz(x, y,±h/2) = 0) at the top and bottom of the plate. After incorporating the necessary boundary conditions the final displacement field turns into following form. u(x, y, z) = u0(x, y) + zθx − 4z3 3h2 ( θx + ∂w ∂x ) v(x, y, z) = v0(x, y) + zθy − 4z3 3h2 ( θy + ∂w ∂y ) w(x, y) = w0(x, y) (5) It is important to mention here that the above form of displacement components rep- resented by Eqs. (5) may invite the problem of C1 formulation, due to the existence of first order derivatives of transverse components appears in the in-plane field. To overcome the difficulties associated with the C1 formulation, in the present work, the derivatives of transverse displacement in in-plane displacement fields are replaced by the separate field variables, thus ensuring the C0 formulation i.e., γx = ( θx + ∂w ∂x ) and γy = ( θy + ∂w ∂y ) . In practice, C0 elements are preferred rather than C1 elements and for further details re- garding the C1 and C0 formulation the reader can refer any standard text book of finite element method [26]. A nine noded Lagrangian element modeled in the study is depicted in Fig. 2. Hence the displacement vector corresponding to each node can be represented as {X} = {u, v, w, θx, θy, γx, γy}. Each node has seven nodal unknowns and thus a total of sixty three unknowns are estimated at element level. 1 2 3 8 9 4 ξ 7 6 5 h Fig. 2. Lagrangian isoparametric element The derivations regarding strain-displacement relations and equilibrium equations are briefly discussed by the authors elsewhere [27–29], and not discussed here for the sake of space management. The governing equation for free vibration analysis is given by,( [K]− ω2 [M ] ) {X} = {0} (6) where [M ], [K] and ω are mass matrix, stiffness matrix and frequency parameter derived at element level. The detailed expressions for mass [M ] and stiffness matrix [K] are given below. [M ] = ∫∫ [C]T [L] [C] dxdy where [L] = ∫ ρ [F ]T [F ] dz (7) Free vibration analysis of four parameter functionally graded plates accounting . . . 151 where matrix [C] and [F ] represent the shape functions and thickness co- ordinate terms, respectively, and given in Appendix A. [K] = ∫∫ [B]T [D] [B] dxdy (8) where [B] is the strain-displacement matrix and [D] represent the rigidity matrix depends on material constitutive properties in Eq. (2). The corresponding strain-displacement re- lation is given in Appendix B. The right hand side zero of the Eq. (6) represents the problem of free vibration analysis. The mass and stiffness matrices formed at element level are assembled to get stiffness matrix at global domain. This can be achieved by taking the contribution of all the plate elements. The skyline storage scheme is used to store the elements in global stiffness matrix. A standard eigen value algorithm is utilized to extract the mode shapes of the FGM plate. The detailed steps involved in the algorithm are given in Appendix C. 3. NUMERICAL RESULTS AND DISCUSSION A computer code is developed in MATLAB environment based on the above formu- lation that accounts for the realistic parabolic variation of transverse stresses through the thickness. For the purpose of generating results, Aluminium/Zirconia plate is considered for all the numerical examples performed, unless otherwise specified. It is to be noted that, the bottom of the plate is enriched with Aluminium, while the top of the plate is made of Zirconia. The material properties of the FGM plate are: E = 168 GPa, ρ = 5700 kg/m3 for Zirconia (ceramic), and E = 70 GPa, ρ = 2707 kg/m3 for Aluminium (metal). Since the effect of Poisson’s ratio on deflection is insignificant [26], a constant value of 0.3 is assumed for both the materials. A square plate (a = b = 1 m) is considered with different boundary conditions (simply supported, clamped and simply supported-clamped) to tabulate the results. To study the influence of parameters a, b, c and p that appears in Eq. (1a), thin (h = 0.01) and moderately thick (h = 0.1) plates are assumed in the numerical part. The present formulation has been validated for number of analyses with respect to rectangular and skew FGM plates and reported by authors in their earlier works [27–29]. In the present study, the prominence has been exerted to study the influence of parameters a, b, c and p on frequency of FGM plate. Such a topic could able the designer to choose the appropriate value of power law parameters to solve for real time applications. The values of first six natural frequencies for simply supported thin and moderately thick FGM plate is furnished in Tabs. 1 and 2, respectively. Three types of power law profiles are considered for each case for example, classic, symmetric and asymmetric. The exact values of power law parameters (a, b and c) chosen for each case are furnished in Fig. 1. The value of power law exponents ranges from p (0 < p < 20). The case of p = 0, represents the homogenous case of ceramic plate. In both Tab. 1 and Tab. 2, it is manifested that the elevation of power law exponent from homogeneous to FGM plate increases the frequency of FGM plate. This trend is observed as common phenomenon in all the three profiles (classical, symmetric and asymmetric). The reason attributed is that the increase in metal content corresponds to low stiffness thus reducing the frequency as the power law exponent rises. Further for all the three profiles considered, the value of 152 Gulshan Taj M. N. A., Anupam Chakrabarti, Mohammad Talha p = 0, produces the same frequency, due to the isotropic property of plate. In Tab. 1, the symmetric profile exhibits high frequency value followed by asymmetric and classical profiles, when the power law exponent ranges from 0 to 1. Beyond the linear range (p > 1), the classical FGM plates produces higher frequency, thus ensuring the high stiffness of the plate under consideration. The low values of frequencies are recorded for thin plate compared to moderately thick plates as expected, considering different values of power law exponent. The observations regarding the influence of chosen profile on frequency extracted from Tab. 1 are analogous for Tab. 2 also. Since the symmetric profile of FGM plate ensures the ceramic segment at top and bottom of the plate (refer Fig. 1(a)) having high stiffness, shows better performance compared with other two cases of FGM profiles. Table 1. Natural frequencies of four-parameter FGM plate for first six modes (simply supported, h = 0.1) p 0 0.2 0.4 1 5 10 20 Classic 324.1771 323.3509 322.4659 319.5509 287.9179 200.7168 64.20281 810.0952 808.0331 805.8241 798.5465 719.5333 501.6757 160.5009 1295.472 1292.178 1288.65 1277.022 1150.724 802.4174 256.7668 1619.64 1615.526 1611.117 1596.589 1438.734 1003.339 321.1017 Symmetric 324.1771 324.9004 325.5331 326.7571 282.8979 138.2552 23.62688 810.0952 811.9001 813.4788 816.5307 706.9492 345.567 59.06632 1295.472 1298.354 1300.875 1305.744 1130.537 552.74 94.49552 1619.64 1623.24 1626.388 1632.467 1413.444 691.1558 118.1739 Asymmetric 324.1771 323.6979 323.145 321.0901 287.5848 187.9573 51.54532 810.0952 808.8992 807.5188 802.3878 718.6918 469.7836 128.8586 1295.472 1293.561 1291.356 1283.157 1149.364 751.4057 206.1462 1619.64 1617.253 1614.498 1604.253 1437.022 939.5536 257.798 Table 2. Natural frequencies of four-parameter FGM plate for first six modes (simply supported, h = 0.01) p 0 0.2 0.4 1 5 10 20 Classic 3132.852 3125.462 3117.48 3090.867 2793.455 1963.022 635.5957 7481.315 7465.368 7447.947 7388.901 6703.212 4757.477 1565.769 10577.5 10575.27 10574.15 10577.09 10339.33 7397.019 2469.472 10577.5 10575.27 10574.15 10577.09 10755.76 9086.254 3059.584 Symmetric 3132.852 3139.264 3144.812 3155.066 2735.686 1354.249 234.2482 7481.315 7494.977 7506.619 7526.66 6537.607 3287.879 578.2702 10577.5 10580.63 10584.62 10601.7 10049.83 5118.174 913.7849 10577.5 10580.63 10584.62 10601.7 10853.49 6290.415 1133.513 Asymmetric 3132.852 3128.547 3123.524 3104.584 2788.14 1838.044 510.3354 7481.315 7471.966 7460.891 7418.333 6684.272 4453.685 1257.335 10577.5 10576.15 10575.84 10580.93 10302.35 6922.942 1983.185 10577.5 10576.15 10575.84 10580.93 10773.04 8502.296 2457.191 Free vibration analysis of four parameter functionally graded plates accounting . . . 153 Table 3. Natural frequencies of four-parameter FGM plate for first six modes (clamped, h = 0.1) p 0 0.2 0.4 1 5 10 20 Classic 5351.332 5340.513 5328.595 5287.73 4800.356 3412.43 1129.664 10228.3 10210.51 10190.54 10120.19 9231.368 6643.842 2248.666 14361.95 14339.35 14313.63 14221.47 13013.22 9442.17 3246.746 16951.14 16926.57 16898.25 16795.22 15401.39 11234.82 3903.099 Symmetric 5351.332 5360.513 5368.242 5380.771 4667.175 2355.196 417.5748 10228.3 10243.06 10255.12 10271.63 8927.615 4593.57 833.7104 14361.95 14380.41 14395.17 14412.54 12546.16 6535.852 1206.544 16951.14 16970.9 16986.37 17001.59 14812.36 7780.121 1452.583 Asymmetric 5351.332 5344.962 5337.332 5307.609 4783.676 3192.332 907.0319 10228.3 10217.71 10204.7 10152.5 9188.311 6212.726 1805.685 14361.95 14348.39 14331.45 14262.16 12943.47 8827.05 2607.327 16951.14 16936.28 16917.44 16839.1 15310.54 10499.89 3134.476 Table 4. Natural frequencies of four-parameter FGM plate for first six modes (clamped, h = 0.01) p 0 0.2 0.4 1 5 10 20 Classic 590.5254 589.0234 587.4141 582.1116 524.5195 365.718 117.0151 1203.612 1200.557 1197.284 1186.493 1069.19 745.6393 238.6523 1773.148 1768.657 1763.843 1747.97 1575.276 1098.792 351.7923 2157.098 2151.64 2145.789 2126.494 1916.483 1336.935 428.1054 Symmetric 590.5254 591.84 592.9895 595.2106 515.3173 251.9106 43.06357 1203.612 1206.285 1208.622 1213.131 1050.323 513.6197 87.83163 1773.148 1777.077 1780.51 1787.127 1547.327 756.9039 129.4756 2157.098 2161.872 2166.043 2174.078 1882.394 920.9689 157.5654 Asymmetric 590.5254 589.6541 588.6485 584.9095 523.9001 342.465 93.94575 1203.612 1201.84 1199.794 1192.183 1067.903 698.2254 191.6026 1773.148 1770.543 1767.533 1756.334 1573.345 1028.915 282.4378 2157.098 2153.931 2150.273 2136.658 1914.115 1251.913 343.7065 The free vibration results of Aluminium/Zirconia thin and moderately thick plate with clamped boundary is furnished in Tabs. 3 and 4, respectively. Because of the high bending nature of the clamped boundary, the higher values of frequency are reported for both the cases. The observations regarding the profile type on natural frequencies drawn from Tabs. 1 and 2 holds true for Tabs. 3 and 4 also, except the frequency values are higher for the later case. In Tabs. 5 and 6, the simply supported-clamped FGM plates are considered to generate the frequency values. Intermediate values of frequency are recorded, since two of the edges correspond to simply-supported boundary, thereby reducing the total stiffness of the plate. Once again, symmetric profile is turned to be a better choice compared with classical and asymmetric profiles, by virtue of high stiffness at top and bottom of the plate. Further to show the influence of each parameter on frequency, three examples are illustrated. In all the cases, one of the parameter is varied, whilst remaining 154 Gulshan Taj M. N. A., Anupam Chakrabarti, Mohammad Talha two parameters are treated as constant. The value of power law exponent is varied from 0 to 100. Table 5. Natural frequencies of four-parameter FGM plate for first six modes (simply supported- clamped, h = 0.1) p 0 0.2 0.4 1 5 10 20 Classic 4156.973 4147.89 4137.986 4104.511 3718.128 2628.357 860.5396 8792.393 8775.471 8756.735 8692.041 7908.192 5654.128 1887.878 12051.01 12048.47 12047.17 12050.42 11642.45 8391.26 2844.406 12893.85 12871.23 12845.91 12757.06 12251.44 10163.68 3476.406 Symmetric 4156.973 4164.77 4164.77 4183.034 3627.797 1813.751 317.6164 8792.393 8806.671 8806.671 8837.217 7678.918 3908.736 698.6065 12051.01 12054.58 12054.58 12078.47 11268.46 5807.666 1054.814 12893.85 12912.67 12912.67 12949.89 12360.48 7037.288 1290.834 Asymmetric 4156.973 4151.655 4145.37 4121.291 3708.067 2459.947 690.9498 8847.72 8837.613 8825.438 8777.686 7926.335 5319.885 1523.394 12051.01 12049.47 12049.1 12054.79 11589.97 7848.985 2284.268 12893.85 12880.4 12863.95 12798.19 12270.74 9504.534 2791.885 Table 6. Natural frequencies of four-parameter FGM plate for first six modes (simply supported-clamped, h = 0.01) p 0 0.2 0.4 1 5 10 20 Classic 444.1443 443.0135 441.802 437.8109 394.4827 275.0275 87.98444 993.2975 990.7727 988.0675 979.1529 882.3082 615.2365 196.8725 1522.186 1518.323 1514.183 1500.538 1352.21 943.0533 301.8472 1878.978 1874.214 1869.109 1852.276 1669.238 1164.262 372.7016 Symmetric 444.1443 445.1342 446.000 447.6739 387.5844 189.4415 32.37916 993.2975 995.5071 997.4393 1001.171 866.8104 423.7926 72.45336 1522.186 1525.566 1528.52 1534.222 1328.358 649.6201 111.0897 1878.978 1883.147 1886.79 1893.817 1639.73 802.0132 137.1686 Asymmetric 444.1443 443.4884 442.7314 439.9176 394.0218 257.5425 70.63843 993.2975 991.833 990.1423 983.856 881.2617 576.1202 158.0596 1522.186 1519.945 1517.357 1507.733 1350.583 883.0914 242.3392 1878.978 1876.215 1873.023 1861.15 1667.216 1090.233 299.2252 Fig. 3 depicts the free vibration results of FGM plate in which the parameters b and c are kept constant and the parameter a is varied from 0 to 1.2. In all the cases, a ceramic line is established which contributes high stiffness to the plate. A fast descending behavior of frequency is discerned as the plate turned from isotropic to FGM case. Because the increase in value of power law exponent tends to reduce the stiffness of the plate further for all cases. In some cases, the natural frequency of FGM plate exceeds the limit case of ceramic plate. For particular, in frequency mode 5 and 6, lower value of the parameter a (0.2 to ∼= 0.8) exceeds the maximum frequency of the plate. This is due to the choice of parameter b and c to decide the frequency value. In particular, the types of vibration Free vibration analysis of four parameter functionally graded plates accounting . . . 155 0 20 40 60 80 100 0 500 1000 1500 2000 2500 3000 3500 Fr eq ue nc y (m od e 1) Power law exponent a=0 a=0.2 a=0.4 a=0.6 a=0.8 a=1.0 a=1.2 ceramic line (a) 0 20 40 60 80 100 0 2000 4000 6000 8000 a=0 a=0.2 a=0.4 a=0.6 a=0.8 a=1.0 a=1.2 Fr eq ue nc y (m od e 2 & 3) Power law exponent ceramic line (b) 0 20 40 60 80 100 0 2000 4000 6000 8000 10000 12000 F re qu en cy (m od e 4) Power law exponent a=0 a=0.2 a=0.4 a=0.6 a=0.8 a=1.0 a=1.2 F re qu en cy (m od e 5) (c) 0 20 40 60 80 100 0 2000 4000 6000 8000 10000 12000 Power law exponent a=0 a=0.2 a=0.4 a=0.6 a=0.8 a=1.0 a=1.2 (d) 0 20 40 60 80 100 0 2000 4000 6000 8000 10000 12000 Fr eq u en cy (m o de 6 ) Power law exponent a=0 a=0.2 a=0.4 a=0.6 a=0.8 a=1.0 a=1.2 ceramic line (e) Fig. 3. First six natural frequencies of FGM plate (0 < a < 1.2) 0 10 20 30 40 50 60 0 500 1000 1500 2000 2500 3000 3500 Fr eq ue nc y (m od e 1) Power law exponent b=0 b=0.2 b=0.4 b=0.6 b=0.8 b=1.0 b=1.2 (a) 0 10 20 30 40 50 60 0 1000 2000 3000 4000 5000 6000 7000 8000 Fr eq ue nc y (m od e 2 & 3) Power law exponent b=0 b=0.2 b=0.4 b=0.6 b=0.8 b=1.0 b=1.2 (b) 0 10 20 30 40 50 60 0 2000 4000 6000 8000 10000 12000 Fr eq ue nc y (m od e 4) b=0 b=0.2 b=0.4 b=0.6 b=0.8 b=1.0 b=1.2 (c) 0 10 20 30 40 50 60 0 2000 4000 6000 8000 10000 12000 Fr eq ue nc y (m od e 5) Power law exponent b=0 b=0.2 b=0.4 b=0.6 b=0.8 b=1.0 b=1.2 (d) 0 20 40 60 80 100 0 2000 4000 6000 8000 10000 12000 Fr eq u en cy (m o de 6 ) Power law exponent a=0 a=0.2 a=0.4 a=0.6 a=0.8 a=1.0 a=1.2 ceramic line (e) Fig. 4. First six natural frequencies of FGM plate (0 < b < 1.2) 156 Gulshan Taj M. N. A., Anupam Chakrabarti, Mohammad Talha 0 20 40 60 80 100 0 500 1000 1500 2000 2500 3000 3500 F re q u e nc y (m od e 1 ) Power law exponent c=1 c=3 c= 5 c=7 c=9 c=11 ceramic line (a) 0 20 40 60 80 100 0 1000 2000 3000 4000 5000 6000 7000 8000 F re q u e nc y (m od e 2 & 3 ) Power law exponent c=1 c=3 c= 5 c=7 c=9 c=11 ceramic line (b) 0 20 40 60 80 100 0 2000 4000 6000 8000 10000 12000 F re q u e n cy (m o de 4 ) Power law exponent c=1 c=3 c= 5 c=7 c=9 c=11 (c) 0 20 40 60 80 100 0 2000 4000 6000 8000 10000 12000 F re q u e n cy (m o de 5 ) Power law exponent c=1 c=3 c= 5 c=7 c=9 c=11 (d) 0 20 40 60 80 100 0 2000 4000 6000 8000 10000 12000 F re q u e n cy (m o de 6 ) Power law exponent c=1 c=3 c= 5 c=7 c=9 c=11 ceramic line (e) Fig. 5. First six natural frequencies of FGM plate (1 < c < 11) mode that ensures this type of monotone decrease of frequency are torsional, bending and axisymmetric mode shapes. In Fig. 4, the parameter b is varied from 0 to 1.2 while other two parameters are kept constant. A convex type of descending behavior is discerned in all the type of frequency modes. For homogeneous case of plate, all the cases merge at same frequency value. Beyond certain value of power law exponent (say p ∼= 25), the frequency of plate considering different values of the parameter b establishes stable path. Hence it can be inferred that change in value of the parameter b has no significant effect beyond certain value of p. The first six mode shapes of FGM plate for several ranges of the power law parameter c is exhibited in Fig. 5. The value of c = 1, establishes the ceramic line corresponds to high stiffness of the plate. For further value of c (c = 3, 7, 9 and 11), a steep tendency of frequency value is noticed. Exceeding the power law exponent beyond 40, shows stable point for all the cases of c considered, except for the case c = 1. Further in mode 5, the value of the parameter c corresponds to 3, exceeds the limit case of ceramic plate due to the choice of the other parameters a and b. This behavior depends on the type of vibration mode and value of the parameter c. 4. CONCLUSIONS An efficient C0 based finite element formulation is presented for free vibration re- sponse of four-parameter functionally graded plates in conjunction with higher order shear REFERENCES 157 deformation theory. Four parameter based power law function is utilized in order to esti- mate the volume fraction of ceramic and metal components. The four parameters exists in the power law expression describes the various material profile of functionally graded plate along the thickness direction. To perform the numerical examples various combina- tion of parameters are considered. Natural frequencies of free vibration of FGM plate are presented in the form of tables and figures. Classical, symmetric and asymmetric profiles are generated by the suitable assump- tion of value of power law parameters. It was noticed symmetric profiles exhibits maximum frequency value for certain value of power law exponent (p > 1) and this tendency is irre- spective of the plate thickness and boundary condition. Variation of single parameter in a power function leads to fall-off in frequency parameter when the power law exponent rises. Due to the choice of other two parameters in the power function, for certain types of modes, the plate with gradation properties records frequency greater than homogenous ceramic plate. For a designer it is vital to acquire the knowledge about the material dis- tribution of plate (either ceramic or metal) at the top and bottom to meet the practical demands. Henceforth, the free vibration of FGM plate based on four-parameter power law could serve as key topic from dynamic point of view. REFERENCES [1] D. Jha, T. Kant, and R. Singh. A critical review of recent research on functionally graded plates. Composite Structures, 96, (2013), pp. 833–849. [2] J. Yang and H.-S. Shen. Vibration characteristics and transient response of shear- deformable functionally graded plates in thermal environments. Journal of Sound and Vibration, 255, (3), (2002), pp. 579–602. [3] J. Yang and H.-S. Shen. Free vibration and parametric resonance of shear deformable functionally graded cylindrical panels. Journal of Sound and Vibration, 261, (5), (2003), pp. 871–893. [4] L. Qian, R. Batra, and L. Chen. Static and dynamic deformations of thick functionally graded elastic plates by using higher-order shear and normal deformable plate theory and meshless local Petrov-Galerkin method. Composites Part B: Engineering, 35, (6), (2004), pp. 685–697. [5] M. R. Isvandzibaei and A. Moarrefzadeh. Vibration two type functionally graded cylindrical shell. Int. J. Multidisciplinary Sciences and Engineering, 2, (8), (2011), pp. 7–11. [6] M. Setareh and M. R. Isvandzibaei. A finite element formulation for analysis of functionally graded plates and shells. J. Basic and Applied Scientific Research, 1, (2011), pp. 1236–1243. [7] M. Talha and B. Singh. Static response and free vibration analysis of FGM plates using higher order shear deformation theory. Applied Mathematical Modelling, 34, (12), (2010), pp. 3991–4011. [8] S. Abrate. Free vibration, buckling, and static deflections of functionally graded plates. Composites Science and Technology, 66, (14), (2006), pp. 2383–2394. 158 REFERENCES [9] B. Uymaz, M. Aydogdu, and S. Filiz. Vibration analyses of FGM plates with in- plane material inhomogeneity by Ritz method. Composite Structures, 94, (4), (2012), pp. 1398–1405. [10] F. Ebrahimi and A. Rastgo. Free vibration analysis of smart FGM plates. Interna- tional Journal of Mechanical Systems Science & Engineering, 2, (2008), pp. 94–98. [11] A. Ferreira, R. Batra, C. Roque, L. Qian, and R. Jorge. Natural frequencies of functionally graded plates by a meshless method. Composite Structures, 75, (1), (2006), pp. 593–600. [12] J. Woo and S. Meguid. Nonlinear analysis of functionally graded plates and shallow shells. International Journal of Solids and structures, 38, (42), (2001), pp. 7409–7421. [13] J. S. Kumar, B. S. Reddy, C. E. Reddy, and K. Reddy. Higher order theory for free vibration analysis of functionally graded material plates. Journal of Engineering & Applied Sciences, 6, (10), (2011), pp. 105–111. [14] S. M. Hasheminejad and A. Rafsanjani. Three-dimensional vibration analysis of thick FGM plate strips under moving line loads. Mechanics of Advanced Materials and Structures, 16, (6), (2009), pp. 417–428. [15] G. Praveen and J. Reddy. Nonlinear transient thermoelastic analysis of functionally graded ceramic-metal plates. International Journal of Solids and Structures, 35, (33), (1998), pp. 4457–4476. [16] H. Matsunaga. Free vibration and stability of functionally graded plates according to a 2-D higher-order deformation theory. Composite structures, 82, (4), (2008), pp. 499–512. [17] C.-S. Chen, C.-Y. Hsu, and G. J. Tzou. Vibration and stability of functionally graded plates based on a higher-order deformation theory. Journal of Reinforced Plastics and Composites, 28, (10), (2009), pp. 1215–1234. [18] F. Tornabene. Free vibration analysis of functionally graded conical, cylindrical shell and annular plate structures with a four-parameter power-law distribution. Computer Methods in Applied Mechanics and Engineering, 198, (37), (2009), pp. 2911–2935. [19] F. Tornabene and E. Viola. Static analysis of functionally graded doubly-curved shells and panels of revolution. Meccanica, 48, (4), (2013), pp. 901–930. [20] R. Hill. A self-consistent mechanics of composite materials. Journal of the Mechanics and Physics of Solids, 13, (4), (1965), pp. 213–222. [21] T. Mori and K. Tanaka. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta metallurgica, 21, (5), (1973), pp. 571–574. [22] S. Suresh, A. Mortensen, and S. Suresh. Fundamentals of functionally graded mate- rials. Institute of Materials London, (1998). [23] Z.-H. Jin and R. Batra. Stress intensity relaxation at the tip of an edge crack in a functionally graded material subjected to a thermal shock. Journal of Thermal Stresses, 19, (4), (1996), pp. 317–339. [24] Y. Chung and S. Chi. The residual stress of functionally graded materials. J Chin Inst Civil Hydraulic Eng, 13, (2001), pp. 1–9. [25] J. N. Reddy. A simple higher-order theory for laminated composite plates. Journal of Applied Mechanics, 51, (4), (1984), pp. 745–752. REFERENCES 159 [26] C. Krishnamoorthy. Finite element analysis: Theory and programming. TataMcGraw- Hill Education, (1994). [27] M. Gulshan Taj and A. Chakrabarti. Static and dynamic analysis of functionally graded skew plates. Journal of Engineering Mechanics, 139, (7), (2012), pp. 848– 857. [28] M. Gulshan Taj, A. Chakrabarti, and A. H. Sheikh. Analysis of functionally graded plates using higher order shear deformation theory. Applied Mathematical Modelling, 37, (18), (2013), pp. 8484–8494. [29] M. Gulshan Taj and A. Chakrabarti. Dynamic response of functionally graded skew shell panel. Latin American Journal of Solids and Structures, 10, (6), (2013), pp. 1243 –1266. APPENDIX A C(1, 1) = N1;C(1, 8) = N2;C(1, 15) = N3;C(1, 22) = N4;C(1, 29) = N5;C(1, 36) = N6;C(1, 43) = N7;C(1, 50) = N8;C(1, 57) = N9 and all other terms of the row one will be zero. Similarly, all other row values are obtained according to each degree of freedom. The detailed expression for shape functions for the assumed Lagrangian element is presented below. N1 = 1 4 (ξ − 1) (η − 1) ξη, N2 = 1 4 (ξ + 1) (η − 1) ξη, N3 = 1 4 (ξ + 1) (η + 1) ξη, N4 = 1 4 (ξ − 1) (η + 1) ξη, N5 = − 1 2 ( 1− ξ2 ) (1− η) η, N6 = − 1 2 (1 + ξ) ( η2 − 1 ) ξ, N7 = − 1 2 ( ξ2 − 1 ) (1 + η) η, N8 = − 1 2 (ξ − 1) ( η2 − 1 ) ξ, N9 = ( 1− ξ2 ) ( 1− η2 ) . (A.1) where ξ and η represents the natural co-ordinate system of the element (Fig. 2). [F ] =   1 0 0 z 0 −4z3 3h2 0 0 1 0 0 z 0 −4z 3 3h2 0 0 1 0 0 0 0   . (A.2) APPENDIX B By utilizing the displacement components given in Eqs. (5) the mechanical strain at a point can be represented as {εm} =   ∂u ∂x ∂v ∂y ∂u ∂y + ∂v ∂x ∂u ∂z + ∂w ∂x ∂v ∂z + ∂w ∂y   =   ∂u0 ∂x + z ∂θx ∂x − 4z3 3h2 ∂γx ∂x ∂v0 ∂y + z ∂θy ∂y − 4z3 3h2 ∂γy ∂y ∂u0 ∂y + z ∂θx ∂x − 4z3 3h2 ∂γx ∂x + ∂v0 ∂y + z ∂θy ∂y − 4z3 3h2 ∂γy ∂y ∂w0 ∂x + θx − 4z2 h2 γx ∂w0 ∂y + θy − 4z2 h2 γy   (B.1) 160 REFERENCES Mechanical strain in terms of total strain can be rewritten as {εm} = [H ] {ε} The strain matrix {ε} can be written in terms of nodal displacement vector {X} by means of strain-displacement matrix [B]. The components of matrix [B] involve derivatives of shape function terms and having the matrix order of 15× 63. APPENDIX C The stiffness matrix [K] in Eq. (6) is positive definite and can be decomposed into Cholesky factors as [K] = [L][T ]T (C.1) where [L] is the lower triangular matrix. Using Eq. (6), Eq. (C.1) is rewritten for the free vibration analysis as: {[L]−1[M ][L]−T}[L]T{X} = 1 ω2 [L]T{X} (C.2) The Eq. (C.2) represents standard eigen value problem and this has been solved to extract the eigen values and the eigen vectors. The term 1/ω2 appear in Eq. (C.2) is the eigen value. The eigen value corresponding to the lowest natural frequency is obtained using the simultaneous iteration technique. The detailed methodology is explained as follows: (i) Set a trial vector [U ] and ortho-normalize. (ii) Back substitute [L][X ] = [U ] (iii) Multiply [Y ] = [M ][X ] (iv) Forward substitute [L]T [V ] = [Y ] (v) Form [B] = [U ]T [V ] (vi) Construct [T ] so that tij =1 and tij = −2bij [bii − bij + s(bii − bij)2] , where s is the sign of (bii − bij) (vii) Multiply [W ] = [V ][T ] (viii) Perform Schmidt ortho-normalization to derive [U¯ ] (ix) Check tolerance [U ]− [U¯ ] (x) If not satisfactory, go to step (ii). VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY VIETNAM JOURNAL OF MECHANICS VOLUME 36, N. 2, 2014 CONTENTS Pages 1. Dao Huy Bich, Nguyen Dang Bich, A coupling successive approximation method for solving Duffing equation and its application. 77 2. Nguyen Thai Chung, Hoang Xuan Luong, Nguyen Thi Thanh Xuan, Dynamic stability analysis of laminated composite plate with piezoelectric layers. 95 3. Vu Le Huy, Shoji Kamiya, A direct evidence of fatigue damage growth inside silicon MEMS structures obtained with EBIC technique. 109 4. Nguyen Tien Khiem, Duong The Hung, Vu Thi An Ninh, Multiple crack identification in stepped beam by measurements of natural frequencies. 119 5. Nguyen Hong Son, Hoang Thi Bich Ngoc, Dinh Van Phong, Nguyen Manh Hung, Experiments and numerical calculation to determine aerodynamic char- acteristics of flows around 3D wings. 133 6. Gulshan Taj M. N. A., Anupam Chakrabarti, Mohammad Talha, Free vi- bration analysis of four parameter functionally graded plate accounting for realistic transverse shear mode. 145