 Advances in Technology Innovation, vol. 3, no. 3, 2018, pp. 109 - 117 Effects of Unsteady Aerodynamic Pressure Load in the Thermal Environment of FGM Plates Chih-Chiang Hong* Department of Mechanical Engineering, Hsiuping University of Science and Technology, Taichung, Taiwan, ROC. Received 13 August 2017; received in revised form 14 October 2017; accepted 20 October 2017 Abstract The effects of unsteady aerodynamic pressure load with varied shear correction coefficient on the functionally graded material (FGM) plates are investigated. Thermal vibrat ion is studied by using the first -order shear deformation theory (FSDT) and the generalized differential quadrature (GDQ) method. Usually, in the FGM analyses, the computed and varied values of shear correction coefficient are the function of the total thickness of plates, FGM power law index, and environment temperature. The effects of environment temperature and FGM power law index on the thermal stress and center deflection of airflow over the upper surface of FGM plates are obtained and investigated. In addition, the effects , with and without the fluid flow over the upper surface of FGM plates, on the center deflection and normal stress are also investigated. Keywords: aerodynamic pressure, varied shear correction coefficient, FGM, thermal vibration, GDQ 1. Introduction There are some investigations of aerodynamic pressure load on the functionally graded material (FGM) p lates and shells. In 2015, Fazelzadeh et al. [1] studied the effects of volume fraction, aspect ratio and non-dimensional in-plane forces on the nanocomposite FGM plates under the action of supersonic aerodynamic pressure. In 2015, Lee and Kim [2] investigated the structure characteristics of FGM panels under supersonic aerodynamic force. In 2013, Rafiee et al. [3] calculated and studied the nonlinear vib ration of piezoelectric FGM shells under combined electrical, thermal, mechanical and aerodynamic loading. In 2012, Ghadimi et al. [4] investigated and studied the thermal flutter characteristics of cantilever FGM plates under supersonic aerodynamic loads. In 2012, Prakash et al. [5] computed and investigated the large amplitude flexural v ibration characteristics of FGM plates with supersonic airflow by using the finite element method (FEM). In 2008, Sohn and Kim [6] presented and examined the thermal buckling and flutter characteristics of FGM panels under aero -thermal loads. In 2007, Fazelzadeh and Hosseini [7] studied and investigated a turbo-machinery FGM rotating blades beam under supersonic aero-thermo-elastic loading. In 2007, Wu et al. [8] calcu lated and analyzed the dynamic stability of FGM plates subjected to aero-thermo-mechanical loads by using the moving least squares differential quadrature method. In 2007, Navazi and Haddadpour [9] investigated the aero-thermo-elastic stability margins of FGM plates and panels in the supersonic flow. In 2006, Prakash and Ganapathi [10] investigated supersonic flutter behavior of FGM plates and flat panels by using the FEM . There are some computational investigations of generalized differential quadrature (GDQ) in the composited FGM plates and shells. In 2017, Hong [11] investigated the effects of varied shear correction on flutter value of the center deflection and the thermal v ibration of FGM shells in an unsteady supersonic flow. In 2015, Tornabene et al. [12] presented a survey of strong formulat ion FEM focused on the numerical investigation of differential quadrature method. In 2014, Hong [13] studied the thermal v ibration and transient response of Terfenol-D FGM plates by using the GDQ method and considering the first -order * Corresponding author. E-mail address: cchong@mail.hust.edu.tw Advances in Technology Innovation, vol. 3, no. 3, 2018, pp. 109 - 117 Copyright © TAETI 110 shear deformat ion theory (FSDT) model and the varied modified shear correct ion factor effects. In 2014, Hong [14] investigated the rapid heating induced vibration of Terfenol-D FGM circular cylindrical shells by using the GDQ method and without considering the effects of shear deformation. In 2013, Hong [15] presented the thermal vib ration of Terfenol-D FGM shells by using the GDQ method and also without considering the effects of shear deformat ion. In 2012, Hong [16] studied the thermal vibrat ion of rapid heating for Terfenol-D FGM plates by using the GDQ method and considering the FSDT effects. In 2009, Tornabene and Viola [17] presented the free vibration analysis of FGM panels and shells by using the GDQ method and considering the FSDT model. It is interesting to study and investigate the thermal stresses and center deflection of GDQ computations by considering the FSDT and the varied effects of the shear correct ion coefficient of airflow over the upper surface of FGM plates with four edges in simply supported boundary conditions. Environment temperature and FGM power law index two parametric effects on the thermal stress and center deflection of airflow over the upper surface of FGM plates are also obtained and investigated. The effects of with and without the fluid flow over the upper surface of FGM plates on the center deflection and normal stress are also calculated. Fig. 1 Fluid flow over the upper surface of two-material FGM plates 2. Formulation For fluid flow over the upper surface of two-material FGM plates is shown in Fig. 1 with thickness h1 of FGM material 1 and thickness h2 of FGM material 2. The material properties are considered in the most dominated property Young’s modulus Efgm of FGM with standard variation fo rm of power law index Rn, the others of material properties are assumed in the simple average form [18]. The properties of indiv idual constituent material of FGM are functions of enviro nment temperature T. The time-dependent, linear FSDT equations of displacements u, vand w\ of FGM plates are assumed in the following [19]. ),,(),,(0 tyxztyxuu x (1) ),,(),,(0 tyxztyxvv y ),,( tyxww  where u 0 and v 0 are d isplacements in the x and y axes direction, respectively, w is t ransverse displacement in the z axis direction of the middle-plane of plates,  x and  y are the shear rotations, t is time. The normal stresses (σ𝑥 and σ𝑦 ) and the shear stresses (σ𝑥𝑦 , σ𝑦𝑧 andσ𝑥𝑧) in the FGM plate under temperature d ifference ∆𝑇 for the k th layer can be obtained as follows [20-21]. )()( 662616 262212 161211 )( k xyxy yy xx kk xy y x T T T QQQ QQQ QQQ                                         (2) )()(5545 4544 )( kxz yz kkxz yz QQ QQ                        Advances in Technology Innovation, vol. 3, no. 3, 2018, pp. 109 - 117 Copyright © TAETI 111 where α𝑥 and α𝑦 are the coefficients of thermal expansion, α𝑥𝑦 is the coefficient of thermal shear, �̅�𝑖𝑗 is the stiffness of FGM plates. ε𝑥, ε𝑦 and ε𝑥𝑦 are in-p lane strains, not negligible ε𝑦𝑧 and ε𝑥𝑧 are shear strains. The temperature difference between the FGM plate and curing area is given by the following equation. ),,(),,( 1*0 tyxT h z tyxTT  (3) in which T0 and T1 are temperature parameters in functions of x, y, and t, h * is the total thickness of plates. The dynamic equilibrium d ifferential equations of flu id flow over the upper surface of FGM plates can be represented and obtained in matrix fo rm [15, 21]. In the matrix elements, there are some coefficients (Aij, Bij, Dij), (i,j = 1,2,6), Ai*j*,(i * , j * = 4, 5) with partial derivatives of displacements and shear rotations . The external loads are subjected to f1,..., f5 with partial derivatives of thermal loads (𝑁, 𝑀), mechanical loads (p1, p2, q) and inertia terms (𝜌, H, I). In which (Aij, Bij, Dij), Ai*j*, f1,..., f5 and (𝜌, H, I) are in the following expressions. 11 p y N x N f xyx        (4) 22 p y N x N f yxy        22 2 3 ** hzhz t w M U x w M U qf              y M x M f xyx      4 y M x M f yxy      5 , dzzTQQQMN xyy h h xxx ),1()(),( 1612 2 2 11 * *     (5) dzzTQQQMN xyyx h hyy ),1()(),( 2622 2 2 12 * *    dzzTQQQMN xyyx h hxyxy ),1()(),( 6626 2 2 16 * *    * * 22 2 ( , , ) (1, , ) , ( , 1,2,6) h ij ij ij ijh A B D Q z z dz i j    * ** * * * * *2 2 , ( , 4,5) h hi j i j A k Q dz i j   (6) dzzzIH h h ),,1(),,( 22 2 0 * *  (7) in which 𝑘𝛼 is the shear correction coefficient, 𝜌0 is the density of ply. The values of 𝑘𝛼 are usually functions of h * , T, and Rn. q is the aerodynamic pressure load for the unsteady, in viscid fluid flow over the upper surface of FGM plate with free stream density 𝜌∞ , velocity 𝑈∞ , and Mach number 𝑀∞ . Advances in Technology Innovation, vol. 3, no. 3, 2018, pp. 109 - 117 Copyright © TAETI 112 The simple forms of �̅� 𝑖𝑗 and �̅�𝑖∗𝑗∗for FGM plates were introduced by Shen [22] in 2007 and used to calculate normal, shear stresses and Aij. The modified shear correction factor 𝑘𝛼 can be derived and obtained directly from the total strain energy principle derivation as follows for the FGM plates [13] by considering the varied value effect of 𝑘𝛼 on the plates. * 1 FGMZSV k h FGMZIV   (8) in which the expressions of FGMZSV and FGMZIV are functions of h * , Rn, the Poisson’s ratios vfgm, the Young’s modulus of the FGM constituent materials E1 and E2 of the FGM plates, respectively. The dynamic GDQ d iscrete equations in matrix notation can be derived and obtained for the dynamic equilibrium differential equations by considering four sides simply supported, fluid flow over the upper surface of FGM plates. The GDQ method was presented by Shu and Richards in 1990 [13, 23-25]. 3. Computational results To study and obtain the GDQ results of varied shear correct ion coefficient calcu lations with plates layers in the stacking sequence (0°/0°), under four sides simply supported boundary condition, no in-plane distributed forces (p1 = p2 = 0) and under the external aerodynamic pressure load (q) of airflow over the upper surface of FGM plates with 𝜌∞ = 0.00000678𝑙𝑏/𝑖𝑛3, 𝑈∞ = 23304𝑖𝑛 /𝑠 and 𝑀∞ = 2 at altitude 50,000ft. The following coordinates xi and yi for the grid points numbers N and M of FGM plates are used 1 0.5 [1 cos( )] , 1, 2, ..., 1 i i x a i N N       (9) 1 0.5 [1 cos( )] , 1, 2, ..., 1 j j y b j M M       The displacement and temperature of thermal vibrat ions are used in time sinusoidal form as follows for a simple case study. )sin()],(),([ 0 tyxzyxuu mnx  (10) )sin()],(),([ 0 tyxzyxvv mny  )sin(),( tyxww mn * * 2 2 2 ( , ) sin( ) ( , ) cos( ) mn mn mnz h z h U Uw x y q t w x y t M x M                 (11) 0 1* [ ( , ) ( , )]sin( ) z T T x y T x y t h    (12) And with the temperature parameter in the following simple vibration 0),(0 yxT (13) )/sin()/sin(),( 11 byaxTyxT  in which 𝜔𝑚𝑛 is the natural frequency in mode shape numbers m and n of the plates, γ is the frequency of applied heat flux, �̅�1 is the amplitude of temperature. Advances in Technology Innovation, vol. 3, no. 3, 2018, pp. 109 - 117 Copyright © TAETI 113 The SUS304 (stainless steel) for FGM material 1 and the Si3N4 (silicon nitride) for FGM material 2 are used in the numerical GDQ computations. Firstly, the dynamic convergence study of center deflection amplitude w (a/2, b/2) (unit mm) in airflow over the upper surface of FGM plates are obtained in Table 1 by considering the varied effects of shear correction coefficient and with h * = 1.2 mm, h1 = h2 = 0.6 mm, m = n = 1, Rn =1, 𝑘𝛼 = 0.149001, T = 100K, 1 100T K , t = 6s. The error accuracy is 7.2E-05 for the center deflection amplitude of a / b = 1, a / h * =10. The 17×17 grid point can be considered in the good convergence result and treated in the following GDQ computations of time responses for deflection and stress of FGM plates. It might be mentioned that the deformations of plates usually increase with the increasing a / h * for the cases of non thermal loads. However, in the Table1 shows the absolute values of center deflection for thin a / h * =100 are much smaller than that for thick a / h * =10 and 5 due to the phenomenon effect of thermal loads. In the FGM plates (𝐵𝑖𝑗 ≠ 0), varied values of 𝑘𝛼 are usually functions of h * , Rn and T. For a / h * =10, a / b = 1, h * from 0.12mm to 2.4 mm, h1 = h2, calculated values of 𝑘𝛼 under T =100K are shown in Table 2, used for the GDQ and shear calculations. For h * = 0.12mm, values of 𝑘𝛼 (from 0.109359 to 1.08902) are increasing with Rn (from 0.1 to 2). For h * = 1.2mm, values of 𝑘𝛼 (from 0.891024E-01 to 0.508881) are increasing with Rn (from 0.1 to 10). For h * = 2.4mm, values of 𝑘𝛼 (from 0.836925E-01 to 0.118029E-02) are small decreasing with Rn (from 0.1 to 10). Usually, the values of 𝑘𝛼 are dominantly in the inverse proportion to h * at a given values of Rn and T, e.g. values of 𝑘𝛼 (firstly 0.109359, then 0.891024E-01, finally 0.836925E-01) are decreasing with h * (from 0.12mm, then 1.2mm to 2.4mm) at Rn = 0.1 and T =100K. Table 1 Dynamic convergence of airflow over the upper surface of FGM plates a / h * GDQ method Deflection w (a/2, b/2) at t = 6s 𝑁 × 𝑀 a / b = 0.5 a / b = 1 a / b = 2 100 13 × 13 0.166916E-18 -0.864896E-16 -0.113379E-14 15 × 15 0.166916E-18 -0.864895E-16 -0.113379E-14 17 × 17 0.166914E-18 -0.864888E-16 -0.113378E-14 14 13 × 13 -0.270625E-14 -0.436330E-12 -0.948251E-10 15 × 15 -0.270623E-14 -0.436326E-12 -0.947482E-10 17 × 17 -0.270623E-14 -0.436324E-12 -0.944192E-10 10 13 × 13 -0.190895E-13 -0.318221E-11 -0.674404E-09 15 × 15 -0.190895E-13 -0.318221E-11 -0.673224E-09 17 × 17 -0.190893E-13 -0.318244E-11 -0.674404E-09 8 13 × 13 -0.703716E-13 -0.118106E-10 -0.323377E-08 15 × 15 -0.703718E-13 -0.118357E-10 -0.466268E-08 17 × 17 -0.703708E-13 -0.118450E-10 -0.595615E-08 5 13 × 13 -0.108713E-11 -0.170468E-09 0.259696E-06 15 × 15 -0.108830E-11 -0.167507E-09 0.271164E-06 17 × 17 -0.108713E-11 -0.174309E-09 0.278226E-06 Table 2 Varied shear correction coefficient 𝑘𝛼 vs. Rn under T =100K h* (mm) 𝑘𝛼 Rn = 0.1 Rn = 0.2 Rn = 0.5 Rn = 1 Rn = 2 Rn = 5 Rn = 10 0.12 0.109359 0.140739 0.288792 0.687883 1.08902 0.989219 0.878104 1.2 0.891024E-01 0.939102E-01 0.111874 0.149001 0.231364 0.415802 0.508881 2.4 0.836925E-01 0.828082E-01 0.815941E-01 0.796610E-01 0.632624E-01 0.219191E-01 0.118029E-02 Secondly, the amplitude of center deflection w (a/2, b/2) (unit mm) for the airflow over the upper surface of FGM plates is calculated. Fig. 2 shows the response values of center deflection amplitude w (a/2, b/2) (unit mm) versus time t in FGM plate for a / h * =10, 14 and th in a / h * = 100, respectively, a / b = 1, h * = 1.2 mm, h1= h2= 0.6mm, Rn = 1, 𝑘𝛼 = 0.117077, T =653K , 𝑇1=100K, starting time t = 0.001s and t =0.1s-3.0s with time step is 0.1s. The absolute value of center deflection amplitude is 7.04E-12mm occurs at t = 0.2s, the steady state value of center deflection is -4.54E-12mm for a / h * =10. The absolute value of center deflection amplitude is 1.31E-12mm occurs at t = 0.1s, the steady state value of center deflection is -6.21E-13mm for a / h * =14. The absolute values of center deflection for thin a / h * =100 are much smaller than that for a / h * =10 and 14. Advances in Technology Innovation, vol. 3, no. 3, 2018, pp. 109 - 117 Copyright © TAETI 114 (a) w (a/2, b/2)versus t for a / h * = 10 (b) w (a/2, b/2)versus t for a / h * = 14 (c) w (a/2, b/2) versus t for a / h * = 100 Fig. 2 w (a/2, b/2) versus t for a / h * = 10, 14 and 100 (a) x  versus z for a / h * = 10 (b) xz  versus z for a / h * = 10 (c) x  versus t for a / h * = 10 (d) x  versus t for a / h * = 14 Fig. 3 Stresses versus z and t for a / h * = 10, 14 and 100 (continued) -1.0E-11 -5.0E-12 0.0E+00 5.0E-12 0 0.5 1 1.5 2 2.5 3 w (a/2, b/2) t -2.5E-12 -1.3E-12 0.0E+00 1.3E-12 2.5E-12 0 0.5 1 1.5 2 2.5 3 w (a/2, b/2) t -2.5E-12 -1.3E-12 0.0E+00 1.3E-12 2.5E-12 0 0.5 1 1.5 2 2.5 3 w (a/2, b/2) t -1.2E-03 -8.0E-04 -4.0E-04 1.0E-19 4.0E-04 8.0E-04 -0.5 -0.25 0 0.25 0.5 𝜎𝑥 z / h* -1.6E-12 -1.2E-12 -8.0E-13 -4.0E-13 -0.5 -0.25 0 0.25 0.5 𝜎𝑥z z / h* -9.20E-04 -9.15E-04 -9.10E-04 -9.05E-04 -9.00E-04 0 0.5 1 1.5 2 2.5 3 𝜎𝑥 t -9.140E-04 -9.135E-04 -9.130E-04 -9.125E-04 -9.120E-04 0 0.5 1 1.5 2 2.5 3 𝜎𝑥 t Advances in Technology Innovation, vol. 3, no. 3, 2018, pp. 109 - 117 Copyright © TAETI 115 (e) x  versus t for a / h * = 100 Fig. 3 Stresses versus z and t for a / h * = 10, 14 and 100 Normal stress 𝜎𝑥 and shear stress 𝜎𝑥𝑧 are three-dimensional components and usually in functions of x, y, and z. Typically their values vary through the plate thickness for the airflow over the upper surface of FGM plates. Fig. 3(a) shows the normal stress 𝜎𝑥 (unit GPa) versus z and Fig. 3 (b) shows the shear stress 𝜎𝑥𝑧 (unit GPa) versus z at center position (x = a / 2, y = b / 2) of plates, respectively at t = 0.1s, a / h * =10 and a / b = 1. The absolute value (9.15E-04 GPa) of normal stress 𝜎𝑥 at z = 0.5h * is found in the much greater value than the value (1.3E-12 GPa) of shear stress 𝜎𝑥𝑧 at z = 0.5h * , thus the normal stress 𝜎𝑥 can be treated as the dominated stress for the airflow over the upper surface of FGM plates. Figs. 3(c)-3(e) shows the time responses of the dominated stresses 𝜎𝑥 (unit GPa) at the center position of outer surface z = 0.5h * as the analyses of deflection case in Fig. 2 for L / h * =10, 14 and thin L / h * =100, respectively. The maximum absolute value of stresses 𝜎𝑥 is 9.15E-04 GPa occurs constantly in the periods t =0.2s-3s for L / h * =10. Fig. 4 shows the center deflection amplitude ( / 2, / 2)w a b (unit mm) versus T for all different values Rn (from 0.1 to 10) of FGM plates calculated and varied values of k  , for the airflow over the upper surface of FGM plates * / 10L h  , / 1a b  , * h = 1.2 mm, 1 h = 2 h = 0.6 mm, 1 100T K , at 3t  s. The maximum value of center deflection amplitude is 1.22E-11mm occurs at 653T K for Rn = 10. The center deflection amplitude values are all small, decreasing versus T from 653T K to 1000T K , for Rn = 2, 5 and 10, they can withstand for higher temperature ( 1000T K ) of environment. The center deflection amplitude values are all small, increasing versus T from 653T K to 1000T K , for Rn = 0.1, 0.2 and 0.5. Fig. 4 w (a/2, b/2) versus T Fig. 5 shows the dominated stresses x  (unit GPa) at the center position of outer surface * 0.5z h versus T for all different values Rn of FGM plates as the analyses of deflection case in Fig. 4. The absolute values of dominated stresses x  versus T are increasing (from 100T K to 653T K ) and then decreasing (from 653T K to 1000T K ) for Rn = 1, all decreasing (from 100T K to 1000T K ) for Rn = 10, all increasing (from 100T K to 1000T K ) for Rn = 0.1, 0.2, 0.5 and 2. -9.1250E-04 -9.1225E-04 -9.1200E-04 0 0.5 1 1.5 2 2.5 3 𝜎𝑥 t -1.5E-11 -1.0E-11 -5.0E-12 0.0E+00 5.0E-12 1.0E-11 1.5E-11 0 250 500 750 1000 Rn=10 Rn=5 Rn=2 Rn=1 Rn=0.5 Rn=0.2 Rn=0.1 w (a/2, b/2) T Advances in Technology Innovation, vol. 3, no. 3, 2018, pp. 109 - 117 Copyright © TAETI 116 Fig. 5 x  versus T The effects of with and without the fluid flow over the upper surface of FGM plates on the center deflection and normal stress for / 1a b  , * h = 1.2 mm, 1 h = 2 h = 0.6 mm, 1 n R  , k  = 0.117077, 653T K , 1 100T K are also considered as follows. The Fig. 6 (a) shows the response values of center deflection amplitude ( / 2, / 2)w a b (unit mm) versus time t for * / 10a h  with and without airflow over the upper surface of FGM plates. The absolute values of center deflection amplitude for without airflow (0.111079 mm) are much greater than that with airflow (5.26E-14 mm) at t = 0.001s for * / 10a h  . Fig. 6 (b) shows the normal stress x  (unit GPa) versus z at the center position ( / 2, / 2x a y b  ) of plates at t = 0.001s for * / 10a h  with and without airflow over the upper surface of FGM plates. The normal stresses are almost in the same values for with ( x  =-9.12E-04 GPa at * 0.5z h ) and without ( x  =-9.25E-04 GPa at * 0.5z h ) airflow cases. (a) w (a/2, b/2) versus t for a / h * = 10 (b) w (a/2, b/2) versus t for a / h * = 10 Fig. 6 w (a/2, b/2) versus t and x  versus z for a / h * = 10 with and without airflow 4. Conclusions In this study, the GDQ solutions have been obtained and investigated for the deflections and stresses in the thermal vibration of FGM plates by considering the varied effects of shear correction coefficient and the airflow over the upper surface of FGM plates. The GDQ results have shown that varied values of k  are usually functions of * h , Rn and T. The absolute value of center deflection amplitude is 7.04E-12 mm occurs at t = 0.2s for * / 10a h  at 653T K . The center deflect ion amplitude values are all small and decreasing versus T from 653T K to 1000T K , for Rn = 2, 5 and 10, the FGM plates also can withstand for higher temperature environment. References [1] S. A. Fazelzadeh, S. Pouresmaeeli, and E. Ghavanloo, “Aeroelastic characteristics of functionally graded carbon nanotube-reinforced composite plates under a supersonic flow,” Computer Methods in Applied Mechanics and Engineering vol. 285, pp. 714-729, March 2015. -1.1E-03 -1.0E-03 -9.0E-04 -8.0E-04 -7.0E-04 -6.0E-04 -5.0E-04 0 250 500 750 1000 Rn=10 Rn=5 Rn=2 Rn=1 Rn=0.5 Rn=0.2 Rn=0.1 𝜎𝑥 T -0.20 -0.15 -0.10 -0.05 0.00 0.05 0 0.5 1 1.5 2 2.5 3 without air flow with air flow w (a/2, b/2) t -1.2E-03 -8.0E-04 -4.0E-04 1.0E-19 4.0E-04 8.0E-04 -0.5 -0.25 0 0.25 0.5 without air flow with air flow 𝜎𝑥 z / h* Advances in Technology Innovation, vol. 3, no. 3, 2018, pp. 109 - 117 Copyright © TAETI 117 [2] C. Y. Lee and J. H. Kim, “Evaluation of homogenized effective p roperties for FGM panels in aero-thermal environments,” Composite Structures, vol. 120, pp. 442-450, February 2015. [3] M. Rafiee, M. Mohammadi, B. S. Aragh, and H. 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