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 I LIST OF TABLES III LIST OF FIGURES IV LIST OF ABBREVIATIONS V ABSTRACT VI 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 10 3.3 Methodology 11 3.4 Basic Equation 11 3.5 Nonlinear vibration analysis 20 3.5.1 Nonlinear dynamic response 21 3.5.2 Natural frequencies 23 CHAPTER RESULTS AND DISCUSSION 24 4.1 Validation analysis 24 4.2 Natural frequencies 25 4.3 Nonlinear dynamic response 27 4.3.1 The influence of geometric parameters 28 4.3.2 The influence of initial imperfection 31 4.3.3 The influence of temperature increment 31 4.3.4 The influence of mechanical load 32 4.3.5 The influence of elastic foundation 32 4.3.6 The influence of type of porosity distribution 34 CHAPTER CONCLUSIONS 35 5.1 Conclusions 35 APPENDIX 36 LIST OF PUBLICATIONS 38 REFERENCES 39 II LIST OF TABLES Table 4.1 Comparison  = a 12 (1 − v ) / Ec ( 2h ) ( a / b = 1, Table of the with non-dimensional that reported in natural Refs frequencies [1,3,12] for 2h / a = 0.005) 24 4.2 Comparison  = 2h 2 (1 + v ) / Ec with of the that non-dimensional reported in natural Refs frequencies [1,32] for ( a / b = 2, 2h / a = 1/ 20,1/12 ) 25 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 = 100K , hc / hf = 5, hc / hp = 10, ( m, n ) = (1,1) , ( k1, k2 ) = ( 0,0 ) 26 Table 4.4 The influence of type of porosity distribution, elastic foundation and temperature increment T on natural frequencies of the sandwich plate with a / b = 1, b / h = 20, e0 = 0.2, hc / hf = 5, hc / hp = 10,VCNT = 0.12, ( m, n ) = (1,1) 27 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 k 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 FG SWCNTs CNT FG-CNTRC 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 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 Table 4.4 The influence of type of porosity distribution, elastic foundation and temperature increment T on natural frequencies of the sandwich plate with a / b = 1, b / h = 20, e0 = 0.2, hc / hf = 5, hc / hp = 10,VCNT = 0.12, ( m, n ) = (1,1) T ( k1 , k2 ) Non-uniform Non-uniform symmetric distribution 50 100 Uniform asymmetric distribution distribution ( 0, ) 1129.95 1117.74 1115.60 ( 0.1,0 ) 1341.22 1330.95 1329.34 ( 0.1,0.02) 1964.67 1957.68 1957.11 ( 0, ) 1003.75 990.36 988.69 ( 0.1,0 ) 1236.76 1225.92 1224.79 ( 0.1,0.02) 1894.91 1887.85 1887.65 ( 0, ) 841.78 826.22 825.23 ( 0.1,0 ) 1109.36 1097.60 1097.09 ( 0.1,0.02) 1814.32 1807.16 1807.41 4.3 Nonlinear dynamic response The effect of geometric parameters, type of porosity distribution, initial imperfection, elastic foundation and temperature increment on the nonlinear dynamic response of the sandwich plate is demonstrated and discussed in this chapter The results are achieved through the use of equations (47) 27 4.3.1 The influence of geometric parameters Firstly, the influence of geometric parameters on the nonlinear dynamic response of the sandwich plate with FG porous homogeneous core and carbon nanotubes-reinforced composite face sheets integrated with piezoelectric layer are studied Figure 4.1 and figure 4.2 shows the influence of ratio width-to-length a / b and length-to-thickness b / h on the nonlinear dynamic response of the sandwich plate, respectively It is clear that the increasing of ratio width-to-length or length-tothickness will almost increase deflection amplitude -3 Deflection amplitude, W(m) x 10 a/b=0.5 a/b=1 a/b=1.5 Non-uniform symmetric distribution V*CNT=0.12,  =0.1, (m,n)=(1,1), q=20sin600t kPa -2 e =0.2, T=100 K, b/h=20, Vp=200 V, k1=0, k2=0, hc /hf=5, hc /hp=10, W0=0 0.01 0.02 Time, t(s) 0.03 0.04 Figure 4.1 Influence of ratio width-to-length a / b on the nonlinear dynamic response of the sandwich plate * The effect of volume fraction VCNT on dynamic response of the sandwich plate are illustrated in figure 4.3 Obviously, when the volume fraction increases from 12% to 28%, the amplitude of vibration decrease This is due to reinforcement of carbon nanotubes leads to increased stiffness of the sandwich plate Figure 4.4 indicates the effect of porosity e0 on the deflection amplitude-time curve of the sandwich plate As can be observed, the deflection amplitude tended to decrease as we increased the porosity 28 -3 Deflection amplitude, W(m) x 10 b/h=15 b/h=20 b/h=25 Non-uniform symmetric distribution * VCNT=0.12, =0.1, (m,n)=(1,1), q=20sin600t kPa -1 -2 e0=0.2, T=100 K, a/b=1, Vp=200 V, k1=0, k2=0, hc /hf=5, hc /hp=10, W0=0 -3 0.01 0.02 Time, t(s) 0.03 0.04 Figure 4.2 Influence of ratio length-to-thickness b / h on the nonlinear dynamic response of the sandwich plate -3 Deflection amplitude, W(m) x 10 1.5 V*CNT=0.12 V*CNT=0.17 V*CNT=0.28 Non-uniform symmetric distribution  =0.1, (m,n)=(1,1), q=20sin600t kPa 0.5 -0.5 -1 -1.5 e0=0.2, T=100 K, a/b=1, b/h=20, Vp=200 V, k1=0, k2=0, hc/hf=5, hc/hp=10, W0=0 0.01 0.02 0.03 Time, t(s) * 0.04 0.05 Figure 4.3 Influence of volume fraction VCNT on the nonlinear dynamic response of the sandwich plate 29 -3 Deflection amplitude, W(m) x 10 e0=0 e0=0.1 e0=0.2 Non-uniform symmetric distribution  =0.1, (m,n)=(1,1), q=20sin600t kPa -1 -2 V*CNT=0.12, T=100 K, a/b=1, b/h=20, Vp=200 V, k1=0, k2=0, hc /hf=5, hc /hp=10, W0=0 0.005 0.01 0.015 0.02 Time, t(s) 0.025 0.03 Figure 4.4 Influence of porosity coefficient e0 on the nonlinear dynamic response of the sandwich plate Deflection amplitude, W(m) -3 x 10 W0=0 W0=0.001 m W0=0.005 m Non-uniform symmetric distribution -5 V*CNT=0.12, =0.1, (m,n)=(1,1), q=20sin600t kPa -10 e0=0.2, T=100 K, a/b=1, b/h=20, Vp=200 V, k1=0, k2=0, hc /hf=5, hc /hp=10 0.01 0.02 Time, t(s) 0.03 0.04 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, 0.001m, 0.005 m 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 load resistance of the material Deflection amplitude, W(m) -3 x 10 T=0 T=50 K T=100 K 1.5 Non-uniform symmetric distribution * VCNT=0.12, =0.1, (m,n)=(1,1), q=20sin600t kPa 0.5 -0.5 -1 -1.5 e0=0.2, a/b=1, b/h=20, Vp=200 V, k1=0, k2=0, hc /hf=5, hc /hp=10, W0=0 0.01 0.02 Time, t(s) 0.03 0.04 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 -3 Deflection amplitude, W(m) x 10 Q0=15 kPa Q0=20 kPa Q0=25 kPa Non-uniform symmetric distribution V*CNT=0.12, =0.1, (m,n)=(1,1), q=Q0sin600t -1 e0=0.2, T=100 K, a/b=1, b/h=20, Vp=200V, k1=0, k2=0, hc /hf=5, hc /hp=10, W0=0 0.01 0.02 Time, t(s) 0.03 0.04 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 -3 Deflection amplitude, W(m) x 10 k1=0 1.5 k1=0.1 GPa/m k1=0.3 GPa/m Non-uniform symmetric distribution =0.1, (m,n)=(1,1), q=20sin600t kPa 0.5 -0.5 -1 -1.5 e0=0.2, T=100 K, a/b=1, b/h=20, Vp=200 V, V*CNT=0.12, k2=0, hc /hf=5, hc /hp=10, W0=0 0.01 0.02 0.03 Time, t(s) 0.04 0.05 Figure 4.8 Influence of the Winkler foundation k1 on the nonlinear dynamic response of the sandwich plate -4 Deflection amplitude, W(m) x 10 k2=0 k2=0.02 GPa.m k2=0.04 GPa.m Non-uniform symmetric distribution V*CNT=0.12, =0.1, (m,n)=(1,1), q=20sin600t kPa -1 -2 e0=0.2, T=200K, a/b=1, b/h=20, k1=0.3 GPa/m, Vp=200 V, hc /hf=5, hc /hp=10, W0=0 0.01 0.02 Time, t(s) 0.03 0.04 Figure 4.9 Influence of the Pasternak foundation k 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 -3 Deflection amplitude, W(m) x 10 Non-uniform symmetric distribution Non-uniform asymmetric distribution Uniform distribution * VCNT=0.12, =0.1, (m,n)=(1,1), q=20sin600t kPa -1 -2 e0=0.2, T=100 K, a/b=1, b/h=20, Vp=200 V, k1=0, k2=0, hc /hf=5, hc /hp=10, W0=0 0.01 0.02 Time, t(s) 0.03 0.04 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 − hc /2 − h f (A , B , D , E , F , H ) = ij + ij − hc /2  ij ij ij  ij − h f − hc /2 − h p Qij (1, z , z , z , z , z ) dz + +  hc /2  − h f − hc /2 hc /2 + h f Qij (1, z , z , z , z , z ) dz − hc /2 Qij (1, z , z , z , z , z ) dz + hc /2 + h f + h p  ( Akl , Dkl , Fkl ) = − h f − hc /2   Qij (1, z , z ) dz + − hc /2 − h f ( 1 ,  ,  ) =   − h f − hc /2 + − h f − hc /2 hc /2 + h f + h p  (Q  p 11 − h f − hc /2 − h p + 11 h f + hc /2  (Q  f 11 11 T + Q  22 T )(1, z , z ) dz + f 12 − hc /2 − h f   (Q  f 12 11 − h f − hc /2 h f + hc /2  (Q  f 12 11 (Q  p 12 − h f − hc /2 − h p − hc /2 hc /2  Qij (1, z , z ) dz − hc /2 T + Q12p 22 T )(1, z , z ) dz ( Q11f 11T + Q12f  22T )(1, z, z ) dz + ( 2 , 4 , 6 ) = + hc /2 Qij (1, z , z ) dz , kl = 44,55, hc /2 +  Qij (1, z , z ) dz + hc /2 + h f hc /2 − hc /2 − hc /2 Qij (1, z , z ) dz + − h p − h f − hc /2 hc /2 + h f Qij (1, z , z , z , z , z ) dz , ij = 11,12, 22,66, hc /2 + h f hc /2 + Qij (1, z , z , z , z , z ) dz 11  (Q  c 11 11 − hc /2 T + Q12c  22 T )(1, z , z ) dz h f + hc /2 + h p  (Q  p 11 11 hc /2 + h f T + Q12p 22 T ) (1, z , z ) dz , T + Q22p  22 T )(1, z , z ) dz T + Q  22 T )(1, z , z f 22 hc /2 hc /2 ) dz +  (Q  T + Q22f  22 T )(1, z , z ) dz + c 12 11 − hc /2 T + Q22c  22 T )(1, z , z ) dz h f + hc /2 + h p  (Q  p 12 hc /2 + h f 11 T + Q22p  22 T )(1, z , z ) dz , O11 = A44 − 6c1D44 + 9c12 F44 , X 12 = A55 − 6c1D55 + 9c12 F55 , O13 = −c12 ( E11 j15 + E12 j25 + H11 ), O15 = −c12 ( E12 j16 + E22 j26 + H 22 ), O14 = −c12 (4 E66 j33 + H 66 + E11 j16 + E12 j26 + 2H12 + E12 j15 + E22 j25 ), O16 = c1 ( E11 j13 − c1E11 j15 + F11 − c1H11 + E12 j23 − c1 E12 j25 ), O17 = c1 (2 E66 j32 − 2c1E66 j33 + F66 − 2c1H 66 + c1E12 j13 − c1E12 j15 + F12 36 −c1H12 + E22 j23 − c1E22 j25 ), O18 = c1 ( E12 j14 − c1E12 j16 + E22 j24 − c1E22 j26 + F22 − c1H 22 ) , O19 = c1 (2 E66 j32 − 2c1E66 j33 + F66 − 2c1H 66 + E11 j14 − c1E11 j16 + E12 j24 − c1E12 j26 + F12 − c1H12 ), O112 = c1 ( E12 j11 − E22 j12 ), O110 = −c1 ( E11 j12 − E12 j21 ), O111 = −c1 (2 E66 j31 − E11 j11 + E12 j12 − E22 j21 ), O21 = − A44 + 6c1D44 − 9c12 F44 , O22 = −c1 ( B11 j15 + F11 + B12 j25 − c1E11 j15 − c1H11 − c1E12 j25 ), O23 = −c1 ( B 11 j16 + B12 j26 + F12 + B66 j33 + F66 − 2c1E66 j33 − 2c1H 66 − c1E11 j16 − c1E12 j26 − c1H12 ), O24 = B11 j13 − c1B11 j15 + D11 − c1F11 + B12 j23 − c1B12 j25 − c1E11 j13 + c12 E11 j15 − c1F11 + c12 H11 −c1E12 j23 + c12 E12 j25 , X 25 = B66 j32 − c1B66 j33 + D66 − c1F66 − c1E66 j32 + c12 E66 j33 − c1F66 +c12 H 66 , O26 = B11 j14 − c1B11 j16 + B12 j24 − c1B12 j26 + D12 − c1F12 + B66 j32 − c1B66 j33 + D66 −c1F66 − c1E66 j32 + c12 E66 j33 − c1F66 + c12 H 66 − c1E11 j14 + c12 E11 j16 − c1E12 j24 + c12 E12 j26 −c1F12 + c12 H12 , O27 = − B11 j12 + B12 j21 + c1E11 j12 − c1E12 j21 , O28 = B11 j11 − B12 j12 − B66 j31 −c1E11 j11 + c1E12 j12 + c1E66 j31 , O31 = − A55 + 6c1D55 − 9c12 F55 , O32 = −c1 (2 B66 j33 + F66 + B12 j15 + F12 + B22 j25 − 2c1 E66 j33 − 2c1 H 66 − c1 E12 j15 − c1H12 − c1E22 j25 ), O33 = −c1 ( B12 j16 + B22 j26 + F22 − c1E12 j16 − c1E22 j26 − c1H 22 ), O34 = B66 j32 − c1 B66 j33 + D66 −c1F66 + B12 j13 − c1B12 j15 + D12 − c1F12 + B22 j23 − c1B22 j25 − c1 E66 j32 + c12 E66 j33 − c1 F66 +c12 H 66 − c1E12 j13 + c12 E12 j15 − c1F12 + c12 H12 − c1E22 j23 + c12 E22 j25 , X 35 = B66 j32 − c1 B66 j33 + D66 − c1F66 − c1E66 j32 + c12 E66 j33 − c1F66 + c12 H 66 , O36 = B12 j14 − c1B12 j16 + B22 j24 − c1 B22 j26 + D22 − c1F22 − c1E12 j14 + c12 E12 j16 − c1E22 j24 + c12 E22 j26 − c1F22 + c12 H 22 , O37 = − B66 j31 − B12 j12 + B22 j21 + c1E66 j31 + c1E12 j12 − c1E22 j21 , O38 = B12 j11 − B22 j12 − c1 E12 j11 + c1 E22 j12 l11 = − k1 − k2 ( m2 +  n2 ) + O13m4 + O14m2 n2 + O15 n4 + O110G1m4 + O111G1m2 n2 + O112G1 n4 , l12 = −O11m + O16m3 + X 17 m n2 + O110G2 m4 + O111G2 m2 n2 + O112G2 n4 , l13 = −O12 n + O18 n3 32G2m n 32G3m n , l15 = , 3ab 3ab 32G1m n 8O   8O   4 4 n1 = −O11m2 − O12 n2 , n2 = , n3 = − 110 m n − 112 m n , n4 = − m − n , 3ab 3abj21 3abj11 16 j11 16 j21 +O19m2 n + O110G3m4 + O111G3m2 n2 + O112G3 n4 , l14 = n5 = 16 , l21 = −m3 (O22 + G1O27 ) − m n2 (O23 + G2O28 ), l22 = X 21O24 m2 − O25 n2 mn −O27G2 m3 − O28G2 m n2 , l23 = −O26 m n − O27G3m3 − O28G3m n2 , n6 = O21m , n7 = 8O27 n , 3abj21 l31 = − n3 (O33 + G1O38 ) − m2 n (O32 + G1O37 ), l32 = −O34 m n − O38G2 n3 − D37G2 m2 n , l33 = O31 − O35m2 − O36 n2 − O38G3 n3 − O37G3m2 n , n8 = O31 n , n9 = 37 8O38m 3abj11 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 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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. .. buckling and vibration of viscoelastic sandwich nanobeams with CNT reinforced face sheets On the other hand, a porous core in sandwich structures is capable of withstanding the transverse normal and