Nonlinear dynamic response and vibration of sandwich plates with nanotube reinforced composite face sheets and FG porous core in thermal environments

VIETNAM NATIONAL UNIVERSITY, HANOI VIETNAM JAPAN UNIVERSITY NGO DINH DAT NONLINEAR DYNAMIC RESPONSE AND VIBRATION OF SANDWICH PLATES WITH FG POROUS HOMOGENEOUS CORE AND NANOTUBE-REINFORCED COMPOSITE FACE SHEETS INTEGRATED WITH PIEZOELECTRIC LAYERS IN THERMAL ENVIROMENTS MASTER’S THESIS Ha Noi, 2020 VIETNAM NATIONAL UNIVERSITY, HANOI VIETNAM JAPAN UNIVERSITY NGO DINH DAT NONLINEAR DYNAMIC RESPONSE AND VIBRATION OF SANDWICH PLATES WITH FG POROUS HOMOGENEOUS CORE AND NANOTUBE-REINFORCED COMPOSITE FACE SHEETS INTEGRATED WITH PIEZOELECTRIC LAYERS IN THERMAL ENVIROMENTS MAJOR: INFRASTRUCTURE ENGINEERING CODE: 8900201.04QTD RESEARCH SUPERVISOR: Prof Dr Sci NGUYEN DINH DUC Ha Noi, 2020 ACKNOWLEDGEMENT First of all, I would like to express my deep gratitude to the instructor, Professor Nguyen Dinh Duc, who devotedly guided, helped, created all favorable conditions and regularly encouraged me to complete this thesis I would like to express my deepest thanks to Professor Kato, Professor Dao Nhu Mai, Professor Nagayama, Dr Phan Le Binh and Dr Nguyen Tien Dung from the Infrastructure Engineering Program for always caring and helping, supporting and giving useful advice during the time I study and complete the thesis In addition, I feel very happy because of the enthusiastic support from the program assistant Bui Hoang Tan who assisted in studying at Vietnam Japan University In particular, I would like to express my gratitude to Dr Tran Quoc Quan, Master Vu Minh Anh for giving me valuable suggestions and advice to help me complete my thesis during meetings outside the lecture hall I would like to thank everyone at VJU, my classmate for creating unforgettable memories Finally, I would like to thank my family, my girlfriend Dang Thu Trang, who is always with me at difficult time who encourage and help me I TABLE OF CONTENTS ACKNOWLEDGEMENT LIST OF TABLES LIST OF FIGURES LIST OF ABBREVIATIONS ABSTRACT CHAPTER INTRODUCTION 1.1 Background 1.2 Research objectives 1.3 Structure of the thesis CHAPTER LITERATURE REVIEW CHAPTER MODELING & METHODOLOGY 3.1 Material properties of sandwich plate 3.2 Modeling of sandwich plate 3.3 Methodology 3.4 Basic Equation 3.5 Nonlinear vibration analysis 3.5.1 Nonlinear dynamic response 3.5.2 Natural frequencies CHAPTER RESULTS AND DISCUSSION 4.1 Validation analysis 4.2 Natural frequencies 4.3 Nonlinear dynamic response 4.3.1 The influence of geometric parameters 4.3.2 The influence of initial imperfection 4.3.3 The influence of temperature increment 4.3.4 The influence of mechanical load 4.3.5 The influence of elastic foundation 4.3.6 The influence of type of porosity distribution CHAPTER CONCLUSIONS 5.1 Conclusions APPENDIX LIST OF PUBLICATIONS REFERENCES II LIST OF TABLES Table = a2 12 − v / E (a / b = 1, 2h / a = 0.005) Table + v / Ec = 2h (a / b = 2, 2h / a =1/ 20,1/ 12) Table 4.3 The influence of porosity coefficient e0 , ratio width-to-length a / b and volume fraction VCNT* on natural frequencies of the sandwich plate with b / h = 20, T = 100 K , hc / hf = 5, hc / hp = 10, Table 4.4 The influence of type of porosity distribution, elastic foundation and temperature increment T on natural frequencies of the sandwich plate III LIST OF FIGURES Figure 1.1 Application of Advanced material Figure 3.1 Simulation model of the sandwich plate 10 Figure 4.1 Influence of ratio width-to-length a / b on the nonlinear dynamic response of the sandwich plate 28 Figure 4.2 Influence of ratio length-to-thickness b / h on the nonlinear dynamic response of the sandwich plate 29 Figure 4.3 Influence of volume fraction VCNT* on the nonlinear dynamic response of the sandwich plate 29 Figure 4.4 Influence of porosity coefficient e0 on the nonlinear dynamic response of the sandwich plate 30 Figure 4.5 Influence of initial imperfection W0 on the nonlinear dynamic response of the sandwich plate 30 Figure 4.6 Influence of temperature increment T on the nonlinear dynamic response of the sandwich plate 31 Figure 4.7 Influence of the magnitude Q0 of the external excitation on the nonlinear dynamic response of the sandwich plate 32 Figure 4.8 Influence of the Winkler foundation k1 on the nonlinear dynamic response of the sandwich plate 33 Figure 4.9 Influence of the Pasternak foundation k2 on the nonlinear dynamic response of the sandwich plate 33 Figure 4.10 Influence of the type of porosity distribution on the nonlinear dynamic response of the sandwich plate 34 IV LIST OF ABBREVIATIONS hc Functional graded Single-walled carbon nanotubes Carbon nanotube Functional graded- carbon nanotube-reinforced composite GygaPascal Thickness of FG porous homogeneous core hf Thickness of FG-CNTRC face sheet hp Thickness of piezoelectric layer FG SWCNTs CNT FG-CNTRC GPa V ABSTRACT Abstract: This thesis analytical solutions for the nonlinear dynamic response and vibration of sandwich plates with FG porous homogeneous core and nanotubereinforced composite face sheets integrated with piezoelectric layers in thermal environment Assuming that the characteristics of the plate depend on temperature and change consistent with the power functions of the plate thickness Motion and compatibility equations are used to base on the Reddy’s higher-order shear deformation plate theory and consider the influence of initial geometric imperfection and the thermal stress in the plate Besides, the Galerkin method and Runge – Kutta method are used to give clear expressions for nonlinear dynamic response and natural frequencies of the sandwich plate The influences of geometrical parameters, type of porosity distribution, initial imperfection, elastic foundation and temperature increment on the nonlinear dynamic response and vibration of thick sandwich plate are demonstrated in detail The results are reviewed with other authors in possible cases to check the reliability of the approach used Keywords: Nonlinear dynamic response, sandwich plate, FG porous, thermal environment, the Reddy’s higher order shear deformation theory VI CHAPTER INTRODUCTION 1.1 Background In all industries, materials are the most important factor to create certain products and details Materials determine the design, construction and cost of the product Metallic and non-metallic materials are materials commonly used in many industrial fields Recently with the development of science and technology has created a number of new materials such as composite materials, nanocomposite materials, sandwich materials Figure 1.1 Application of Advanced material In the world, sandwich materials are widely used in many fields of medical, electronics, energy, aerospace engineering, industry automotive and construction of civil, … (figure 1.1) Due to the outstanding characteristics of this material like light weight, heat resistance, energy dissipation reduction and superior vibrational damping, Especially, it is impossible not to mention the porous material It is lightweight cellular materials inspired by nature Wood, bones and sea sponges are some well-known examples of these types of structures Foams and other highly porous materials with a cellular structure are known to have many interesting combinations of physical and mechanical properties, such as high stiffness combined with very low specific gravity or high gas permeability combined with high thermal conductivity Among artificial cell materials, polymer foams are currently the most important with wide applications in most areas of technology Less known is that even metals and alloys can be manufactured in the form of cellular or foam materials, and these materials have such interesting properties that exciting new applications are expected in the near future 1.2 Research objectives The research objective of this thesis is to research nonlinear dynamic response and vibration of sandwich plate subjected to thermo-mechanical load combination Hence, to solve the problem, this thesis will set out the objectives should be achieved as below: Investigations on nonlinear dynamic response and vibration of sandwich plates subjected to thermo-mechanical load combination The natural frequency and the deflection – time curves of sandwich plate structures are determined In numerical results, the effects of the geometrical parameters, types of distribution of porosity, temperature increment, imperfections and elastic foundation on the nonlinear dynamic response and vibration of the sandwich plate will be studied 1.3 Structure of the thesis This thesis provides a detailed explanation of the nonlinear dynamic response and vibration of sandwich plate structure using analytical method In order to better understand the solution method as well as give an appropriate result, the thesis is presented in the following structure: ➢ Chapter 1: Introduction Highlights the role and importance of the material, especially the advanced material for industrial fields The background and research objective are introduced x 10-3 amplitude, W(m) Non-unifo =0.1, (m, Deflection -1 CNT -2 k1=0, k2= Figure 4.4 Influence of porosity coefficient e0 on the nonlinear dynamic x 10 -3 Non -5 * V Deflection amplitude, W(m) response of the sandwich plate C -10 e0 k1 Figure 4.5 Influence of initial imperfection W0 on the nonlinear dynamic response of the sandwich plate 30 4.3.2 The influence of initial imperfection Figure 4.5 describes the influence of initial imperfection W0 on the nonlinear dynamic response of the sandwich plate Three cares of the initial imperfection W0 = 0, 0.001 m, 0.005m are presented It is indicated that the amplitude of deflection changes significantly with the growth of initial imperfection 4.3.3 The influence of temperature increment Subsequently, the change of temperature environment on the nonlinear dynamic response of the sandwich porous plate with reinforced by nanotube composite face sheets are depicted in figure 4.6 with the increase of temperature from to 100 degrees Kelvin, we can easily see that the temperature greatly effects Deflection amplitude, W(m) load resistance of the material x 10 -3 1.5 0.5 -0.5 -1 -1.50 Figure 4.6 Influence of temperature increment T on the nonlinear dynamic response of the sandwich plate 31 4.3.4 The influence of mechanical load Continuously, in figure 4.7, the influence of the magnitudes Q0 of external force excitation on the dynamic response of sandwich porous plate are demonstrated In the care of Q0 = (15, 20, 25) kPa , when magnitude Q0 rises obviously the amplitude of vibration of sandwich plate will increase Deflection amplitude, W(m) x 10-3 Non-uni V* CNT -1 e0=0.2 k =0, k Figure 4.7 Influence of the magnitude Q0 of the external excitation on the nonlinear dynamic response of the sandwich plate 4.3.5 The influence of elastic foundation Next, the influence of the Winkler and Pasternak foundation on the nonlinear dynamic response of the sandwich plates with FG porous homogeneous core and carbon nanotube-reinforced composite face sheets integrated piezoelectric layers are investigated respectively in figures 4.8 and 4.9 From the results, we can see that the amplitude of deflection reduces with the appearance of elastic foundation 32 Deflection amplitude, W(m) -0 -1 Figure 4.8 Influence of the Winkler foundation k1 on the nonlinear dynamic response of the sandwich plate W(m) x 10-4 No Deflection amplitude, V * C -1 e0= -2 Vp Figure 4.9 Influence of the Pasternak foundation k2 on the nonlinear dynamic response of the sandwich plate 33 4.3.6 The influence of type of porosity distribution Eventually, the influence of type of porosity distribution on the nonlinear dynamic response of sandwich plate are considered in figure 4.10 Three type of porosity distribution: uniform distribution, non-uniform symmetric distribution and non-uniform asymmetric distribution are researched As can be seen from the result, the amplitude of nonlinear dynamic response lowest with non-uniform symmetric distribution On the other hand, the comparison between porosity distribution types then non-uniform symmetric distribution gives the best load capacity x 10-3 Deflection amplitude, W(m) Non-uniform symmetric distribution Non-uniform asymmetric distribution Uniform distribution V*CNT=0.12, =0.1, (m,n)=(1,1), q=20sin600t kPa -1 e0=0.2, k1=0, k2=0, hc/hf=5, hc/hp=10, W0=0 -20 Time, t(s) Figure 4.10 Influence of the type of porosity distribution on the nonlinear dynamic response of the sandwich plate 34 CHAPTER CONCLUSIONS 5.1 Conclusions Based on analytical method, nonlinear dynamic response and vibration of sandwich plates with carbon nanotube-reinforced composite face sheets and FG porous homogeneous core integrated with piezoelectric layers under thermomechanic loads on elastic foundation are researched The effect of geometrical parameters, elastic foundation, thermal environment and type of porosity distribution on natural frequency is also considered The method of us was evaluated and verified by comparison with previous study From the results obtained, we can draw some important conclusions: - Among three type of porosity distribution, the non-uniform symmetric distribution has highest natural frequency and the amplitude of deflection is lowest - The initial imperfection and elastic foundation have a dramatic influence on the nonlinear dynamic response and vibration of sandwich plate, whereas the porosity has slight effect on the deflection amplitude of sandwich plate - The thermo-mechanical loads have influence in a negative way on the nonlinear dynamic response and vibration of sandwich plates with nanotubereinforced composite face sheets and FG porous homogeneous core 35 APPENDIX − h / 2−h (A , Bij , Dij , Eij , Fij , H ij − )= ij c Qij (1, z , z f , z , z , z ) dz h f − hc / 2−hp −hc / + Qij (1, z , − h f − hc / hc / 2+hf Qij (1, z , z , z , z , z ) dz + hc / ( Akl , Dkl , Fkl ) = hc / + h f Qij (1, z , z ) dz + + hc / (1, , )=−h / 2−h c f − h f − hc / 2−hp −h / + (Q11f c 11 T + Q12f T 22 h/2 )(1, z , z )dz + (Q11c c − h f − hc / T + Q12c 22 T )(1, z , z )dz −hc / h + 11 f +hc / (Q11f T + Q12f 11 T 22 )(1, z , z )dz + h f (Q11p 11 T + Q12p T + Q22c 22 T (Q12p 11 T + Q22p + hc / 2+hp 22 T )(1, z, z )dz, hc / (2, , )=−h c / 2−h f − h f − hc / 2−hp −h / + (Q12f c 11 T + Q22f 22 T h/2 )(1, z , z ) dz + c − h f − hc / 11 )(1, z , z )dz −hc / h + (Q12c f +hc / (Q12f 11 T + Q22f 22 T )(1, z , z )dz + h hc / f + hc / 2+hp 22 T )(1, z , z )dz, hc / 2+h f O11 = A44 − 6c1 D44 + 9c12 F44 , X 12 = A55 − 6c1 D55 + 9c12 F55 , O13 = − c12 ( E11 j15 + E12 j25 + H 11 ), O15 = −c12 ( E12 j16 + E 22 j 26 + H22 ), O14 = − c12 (4 E66 j33 + H 66 + E11 j16 + E12 j 26 + H 12 + E12 j15 + E 22 j25 ), O = c (E j − c E j + F − c H + E j − c E j ), 16 11 13 11 15 11 11 12 23 12 25 O17 = c1 (2 E66 j32 − 2c1 E66 j33 + F66 − 2c1 H 66 + c1 E12 j13 − c1 E12 j15 + F12 36 − c1 H 12 + E 22 j23 − c1 E 22 j25 ), O18 = c1 (E12 j14 − c1 E12 j16 + E 22 j24 − c1 E 22 j26 + F22 − c1 H 22 O = c (2 E j − 2c E j + F − 2c H + E j − c E j + E j − c E j ), 19 + 66 32 66 33 66 66 11 14 11 16 12 24 12 26 F12 − c1 H 12 ), O112 = c1 ( E12 j11 − E 22 j12 ), O110 = − c1 ( E11 j12 − E12 j21 ), O111 = − c1 (2 E66 j31 − E11 j11 + E12 j12 − E22 j21 ), O21 = − A44 + 6c1 D44 − 9c12 F44 , O = − c ( B j + F + B j − c E j − c H − c E j ), O = − c (B j + B j 22 11 15 11 12 25 11 15 11 12 25 23 11 16 12 26 + ), F12 + B66 j33 + F66 − c1 E66 j33 − 2c1 H 66 − c1 E11 j16 − c1 E12 j26 − c1 H12 O24 = B11 j13 − c1 B11 j15 + D11 − c1 F11 + B12 j23 − c1 B12 j25 − c1 E11 j13 + c12 E11 j15 − c1 F11 + c12 H11 − c1 E12 j23 + c12 E12 j25 , X 25 = B66 j32 − c1 B66 j33 + D66 − c1 F66 − c1 E66 j32 + c12 E66 j33 − c1 F66 H ,O =B j −c B j +B j −c B j +D −c F +B j −c B j +D 66 26 11 14 11 16 12 24 12 26 12 12 66 32 + c12 E66 j33 − c1 F66 + c12 H 66 − c1 E11 j14 + c12 E11 j16 − c1 E12 j24 + c12 E12 j26 B j +B j +c E j −c E j ,O =B j −B j −B j 11 12 12 21 11 12 12 21 28 11 11 66 31 − 12 12 66 − 66 33 +c c1 F66 − c1 E66 j32 −c F +c 2H ,O = − 12 12 27 c1 E11 j11 + c1 E12 j12 + c1 E66 j31 ,O31 = − A55 + 6c1 D55 − 9c12 F55 , O32 = − c1 (2 B66 j33 + 2F66 B j + F + B j − 2c E + O = −c (B j +B j +F −c E 12 15 12 22 25 j − 2c H − c E j − c H − c E j ), 66 33 12 16 66 12 15 22 34 12 22 25 j − c E j − c H ), O = B j − c B j + D − cF+Bj−cBj+D−cF+Bj−cBj−cEj+ c2Ej−cF 33 12 16 22 26 22 22 26 66 32 66 33 66 166 1213 11215 12 112 2223 12225 16632 6633 166 + j26 D66 − c1 F66 − c1 E66 j32 + c12 E66 j33 − c1 F66 + c12 H 66 ,O36 = B12 j14 − c1 B12 j16 + B22 j24 − c1 B22 + j31 D22 − c1 F22 − c1 E12 j14 + c12 E12 j16 − c1 E 22 j24 + c12 E 22 j26 − c1 F22 + c12 H 22 , O37 = −B66 −B j +B j +c E j +c E j −c E j ,O =B j −B j −c E j +c E j 12 12 22 21 66 31 12 12 22 21 38 12 11 22 12 12 11 22 12 l11 = − k1 − k ( m2 + n2 )+ O13 m4 + O14 m2 n2 + O15 n4 + O110 G1 m4 + O111G1 m2 n2 + O112 G1 n4 , l12 = −O11 +O m + O16 m + X 17 m n + O110 G2 m + O111G2 m n + O112 G2 n , l13 = −O12 n + O18 n +OG4 19 n=−O m n 110 −O 11 16 n = mn −OG ,l 21 −O G 27 m l = − (O + G O ) − m ,l 28 31 l =O −O 33 n 33 31 35 m 37 LIST OF PUBLICATIONS [1] Dat, N D., Khoa, N D., Nguyen, P D., & Duc, N D (2019) An analytical solution for nonlinear dynamic response and vibration of FG-CNT reinforced nanocomposite elliptical cylindrical shells resting on elastic foundations ZAMM Zeitschrift fuer Angewandte Mathematik Und Mechanik, doi:10.1002/zamm.201800238 (WILEY, SCIE, IF= 1.467) [2] Dat, N D., Quan, T Q., & Duc, N D (2019) Nonlinear thermal vibration of carbon nanotube polymer composite elliptical cylindrical shells International Journal of Mechanics and Materials in Design, doi:10.1007/s10999-019-09464-y (Springer, SCIE, IF=3.143) 38 REFERENCES [1] Askari Farsangi, M A., Saidi, A R., & Batra, R C (2013) Analytical solution for free vibrations of moderately thick hybrid piezoelectric laminated plates Journal of Sound and Vibration, 332(22), 5981–5998 [2] Azrar, L., Belouettar, S., & Wauer, J (2008) Nonlinear vibration analysis of actively loaded sandwich piezoelectric beams with geometric imperfections Computers & Structures, 86(23-24), 2182–2191 [3] Baferani, A H., Saidi, A R., & Jomehzadeh, E (2010) An exact solution for free vibration of thin functionally graded rectangular plates Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 225(3), 526–536 [4] Belouettar, S., Azrar, L., Daya, E M., Laptev, V., & Potier-Ferry, M (2008) Active control of nonlinear vibration of sandwich piezoelectric beams: A simplified approach Computers & Structures, 86(3-5), 386–397 [5] Chen, D., Kitipornchai, S., & Yang, J (2016) Nonlinear free vibration of shear deformable sandwich beam with a functionally graded porous core Thin-Walled Structures, 107, 39–48 [6] Chen, D., Yang, J., & Kitipornchai, S (2019) Buckling and bending analyses of a novel functionally graded porous plate using Chebyshev-Ritz method Archives of Civil and Mechanical Engineering, 19(1), 157–170 [7] Di Sciuva, M., & Sorrenti, M (2019) Bending, free vibration and buckling of functionally graded carbon nanotube-reinforced sandwich plates, using the extended Refined Zigzag Theory Composite Structures, 227, 111324 [8] Duc, N D., & Cong, P H (2013) Nonlinear postbuckling of symmetric S-FGM plates resting on elastic foundations using higher order shear deformation plate theory in thermal environments Composite Structures, 100, 566–574 [9] Jedari Salami, S (2016) Extended high order sandwich panel theory for bending analysis of sandwich beams with carbon nanotube reinforced face sheets Physica E: Low-Dimensional Systems and Nanostructures, 76, 187–197 [10] Karlsson, K F., & TomasÅström, B (1997) Manufacturing and applications of structural sandwich components Composites Part A: Applied Science and Manufacturing, 28(2), 97–111 [11] Kwon, H., Bradbury, C R., & Leparoux, M (2010) Fabrication of Functionally Graded Carbon Nanotube-Reinforced Aluminum Matrix Composite Advanced Engineering Materials, 13(4), 325–329 [12]Leissa, A.W., Vibration of Plates, NASA Report SP-160, Washington, DC, 1969 39 [13] Li, Q., Wu, D., Chen, X., Liu, L., Yu, Y., & Gao, W (2018) Nonlinear vibration and dynamic buckling analyses of sandwich functionally graded porous plate with graphene platelet reinforcement resting on Winkler-Pasternak elastic foundation International Journal of Mechanical Sciences, 148, 596-610 [14] Li, Y., Li, J B., & Zhang, R (2004) Energy-absorption performance of porous materials in sandwich composites under hypervelocity impact loading Composite Structures, 64(1), 71–78 [15] Liew, K M., Lei, Z X., & Zhang, L W (2015) Mechanical analysis of functionally graded carbon nanotube reinforced composites: A review Composite Structures, 120, 90–97 [16] Liu, S W., Lye, S W., & Miao, J M (2012) Sandwich structured electrostatic/electrets parallel-plate power generator for low acceleration and low frequency vibration energy harvesting 2012 IEEE 25th International Conference on Micro Electro Mechanical Systems (MEMS) [17] Masmoudi, S., El Mahi, A., & El Guerjouma, R (2014) Mechanical behaviour and health monitoring by acoustic emission of sandwich composite integrated by piezoelectric implant Composites Part B: Engineering, 67, 76–83 [18] Mehar, K., Panda, S K., & Mahapatra, T R (2017) Thermoelastic nonlinear frequency analysis of CNT reinforced functionally graded sandwich structure European Journal of Mechanics - A/Solids, 65, 384–396 [19] Moradi-Dastjerdi, R., & Aghadavoudi, F (2018) Static analysis of functionally graded nanocomposite sandwich plates reinforced by defected CNT Composite Structures, 200, 839–848 [20] Moradi-Dastjerdi, R., & Behdinan, K (2019) Stability analysis of multifunctional smart sandwich plates with graphene nanocomposite and porous layers International Journal of Mechanical Sciences, 105283 [21] Natarajan, S., Haboussi, M., & Manickam, G (2014) Application of higherorder structural theory to bending and free vibration analysis of sandwich plates with CNT reinforced composite facesheets Composite Structures, 113, 197–207 [22] Nosier, A., & Reddy, J N (1991) A study of non-linear dynamic equations of higher-order shear deformation plate theories International Journal of Non-Linear Mechanics, 26(2), 233–249 [23] Qiao, H., He, Z., Jiang, W., & Peng, W (2019) Sound transmission of periodic composite structure lined with porous core: Rib-stiffened double panel case Journal of Sound and Vibration, 440, 256–276 [24] Safaei, B., Moradi-Dastjerdi, R., Qin, Z., & Chu, F (2019) Frequencydependent forced vibration analysis of nanocomposite sandwich plate under thermo-mechanical loads Composites Part B: Engineering, 161, 44-54 40 [25] Sankar, A., Natarajan, S., & Ganapathi, M (2016) Dynamic instability analysis of sandwich plates with CNT reinforced facesheets Composite Structures, 146, 187– 200 [26] Setoodeh, A R., Shojaee, M., & Malekzadeh, P (2019) Vibrational behavior of doubly curved smart sandwich shells with FG-CNTRC face sheets and FG porous core Composites Part B: Engineering, 165, 798–822 [27] Shen, H.-S (2009) Nonlinear bending of functionally graded carbon nanotube-reinforced composite plates in thermal environments Composite Structures, 91(1), 9–19 [28] Shen, H.-S., Wang, H., & Yang, D.-Q (2017) Vibration of thermally postbuckled sandwich plates with nanotube-reinforced composite face sheets resting on elastic foundations International Journal of Mechanical Sciences, 124-125, 253– 262 [29] Shen, H.-S., & Zhang, C.-L (2010) Thermal buckling and postbuckling behavior of functionally graded carbon nanotube-reinforced composite plates Materials & Design, 31(7), 3403–3411 [30] Shuyu, L., & Fucheng, Z (1994) Study of vibrational characteristics for piezoelectric sandwich ultrasonic transducers Ultrasonics, 32(1), 39–42 [31] Sobhy, M., & Zenkour, A M (2018) Magnetic field effect on thermomechanical buckling and vibration of viscoelastic sandwich nanobeams with CNT reinforced face sheets on a viscoelastic substrate Composites Part B: Engineering, 154, 492-506 [32] Srinivas, S., Joga Rao, C V., & Rao, A K (1970) An exact analysis for vibration of simply-supported homogeneous and laminated thick rectangular plates Journal of Sound and Vibration, 12(2), 187–199 [33] Reddy, J.N., 1984 A refined nonlinear theory of plates with transverse shear deformation International Journal of Solids and Structures, 20, 881–896 [34] Reddy, J.N., Mechanics of laminated composite plates and shells: Theory and analysis Boca Raton: CRC Press; 2004 [35] Talebitooti, R., & Zarastvand, M R (2018) The effect of nature of porous material on diffuse field acoustic transmission of the sandwich aerospace composite doubly curved shell Aerospace Science and Technology, 78, 157–170 [36] Tao, K., Miao, J., Lye, S W., & Hu, X (2015) Sandwich-structured twodimensional MEMS electret power generator for low-level ambient vibrational energy harvesting Sensors and Actuators A: Physical, 228, 95–103 [37] Wang, Z.-X., & Shen, H.-S (2012) Nonlinear vibration and bending of sandwich plates with nanotube-reinforced composite face sheets Composites Part B: Engineering, 43(2), 411–421 41 ... NGO DINH DAT NONLINEAR DYNAMIC RESPONSE AND VIBRATION OF SANDWICH PLATES WITH FG POROUS HOMOGENEOUS CORE AND NANOTUBE- REINFORCED COMPOSITE FACE SHEETS INTEGRATED WITH PIEZOELECTRIC LAYERS IN THERMAL. .. smart sandwich material with FG porous core and nanocompositereinforced face sheets integrated with piezoelectric layers, the nonlinear dynamic response and vibration of sandwich plate with FG nanotube- reinforced. .. 33 in which l1i (i = This is basic equation to determine nonlinear dynamic response and vibration of sandwich plates with FG porous homogeneous core and nanotube- reinforced composite face sheets