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

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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 1 INTRODUCTION ................................................................................ 1 1.1 Background ....................................................................................................... 1 1.2 Research objectives ........................................................................................... 2 1.3 Structure of the thesis ........................................................................................ 2 CHAPTER 2 LITERATURE REVIEW ..................................................................... 4 CHAPTER 3 MODELING & METHODOLOGY ..................................................... 7 3.1 Material properties of sandwich plate ............................................................... 7 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 4 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 5 CONCLUSIONS................................................................................. 35 5.1 Conclusions ..................................................................................................... 35 APPENDIX ............................................................................................................... 36 LIST OF PUBLICATIONS ...................................................................................... 38 REFERENCES .......................................................................................................... 39 II LIST OF TABLES Table 4.1. Comparison  = a 2 12 (1 − v 2 ) / Ec ( 2h ) ( a / b = 1, Table of 2 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 ............................................................ 1 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 2 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 1 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 1 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. 2 ➢ Chapter 2: Literature review Introduction of articles related to research issues. Since then, explains why this study is necessary. ➢ Chapter 3: Methodology The material and model properties of the structure are presented. The method used as well as how to solve the problem are discussed. ➢ Chapter 4: Numerical results and discussion Check the reliability of the method through comparison with other authors considered. The results are expressed and discussed as the geometric transformations, temperature and mechanical load through the deflection amplitude - time curves and natural frequencies. ➢ Chapter 5: Conclusions Summarize the results achieved and provide further direction for the study. 3 CHAPTER 2 LITERATURE REVIEW Nowadays, sandwich structures are widely in many fields of life such as medical, electronics, energy, aerospace engineering, industry automotive and construction of civil with advantages such as high rigidity and lightweight Karlsson et al. [10]. A sandwich structure consists of two thin face sheet and core layer of low strength but thick brings high bending stiffness. The face sheet is often used with sheet metal and fiber-reinforced polymers, while the core is usually made of honeycomb or polymer foam. Recently, the advanced of carbon nanotubes such as superior strength and stiffness has been of interest to many scientists. Carbon nanotubes Liew et al. [15] are a potential candidate for sandwich structures with the replacement of the face sheet with nanocomposite material which is reinforced with carbon nanotubes that improves the bearing capacity. The sandwich structure carbon nanotube-reinforced composite face sheets are investigated by a number of authors. Wang et al. [37], Natarajan et al. [21] studied vibration and bending of sandwich plates with nanotube-reinforced composite face sheets. By using Extended High order Sandwich Panel Theory, the bending analysis of sandwich beam Salami [9] also presented. Di Sciuva et al. [7] investigated additionally buckling of sandwich plates adopted Refined zigzag theory with Rits method. The vibration of thermally postbuckled also studied by Shen et al. [28]. The dynamic instability analysis using shear flexible QUAD-8 serendipity element under periodic load is present by Sankar et al. [25]. Based on mesh-free method, Moradi-Dastjerdi et al [19]. studied static analysis of functionally grade nanocomposite sandwich plate reinforced by defected CNT. Safaei et al. [24] also using mesh-free method to investigate the influence of loading frequency on dynamic behavior of structure. Mehar et al. [18] researched thermoelastic nonlinear frequency analysis of CNT reinforced functional graded sandwich structure. Sobhy et al. [31] studied the effect of the magnetic field on thermomechanical buckling and vibration of viscoelastic sandwich nanobeams with CNT reinforced face sheets. 4 On the other hand, a porous core in sandwich structures is capable of withstanding the transverse normal and shear loads as well as superior energy dissipation, not only thermal and acoustic insulation but also vibration damping due to the novel properties of porous materials. Li et al. [14] devoted to considering the energy-absorption performance of porous materials in sandwich composites. The paper demonstrate sandwich models can be abilities to prevent perforation subjected to up 7 km/s projectile hypervelocity impact loading. Talebitooti et al. [35] investigated the effect of nature of porous material on diffuse field acoustic transmission of the sandwich aerospace composite doubly curved shell. Qiao et al. [23] studied the sound insulation of a periodically rib-stiffened double panel with porous core by using space harmonic series and Biot theory. Moreover, a graded porosity leads to the continuous variation in material properties so reducing stress concentration often encountered in conventional sandwich structures. Chen et al. [5] employed the Ritz method to derive the nonlinear free vibration of shear deformation of sandwich beam with FG porous core. Li et al. [13] explored nonlinear vibration and dynamic buckling of sandwich functionally graded porous plates with reinforced graphene platelet. Chen et al. [6] analyzed bucking and bending of a novel FG porous plate. The results of research showed FG porosity were suggested could remove the mismatch stresses and effective buckling and bending significantly. In addition, based on the properties of piezoelectric materials, sensors are made such as ultrasonic transceiver sensors in machines (detecting defects in metal and concrete), which has made their applications more popular over the past decade (Liu et al. [16], Tao et al. [36]). Shuyu et al. [30] proposed a study of vibration properties for piezoelectric sandwich ultrasonic transducers. Masmoudi et al. [17] investigated mechanical behavior and health monitoring by acoustic emission of sandwich composite integrated by piezoelectric implant. The acoustic emission technique can be found damage in materials through transient ultrasonic detection. Belouettar et al. [4] adopted the Harmonic balance method to study active 5 control of nonlinear vibration of sandwich beam with piezoelectric face sheets. Azrar et al. [2] studied nonlinear vibration of imperfect sandwich piezoelectric beams. Thus, the sandwich materials with FG porous core and nanocomposite-reinforced face sheets and integrated with piezoelectric layers can be considered as new advanced material. Moradi-Dastjerdi et al. [20] employed the Reddy’s third order and meshfree method to study stability analysis of multifunctional smart sandwich plates with graphene nanocomposite and porous layers integrated with piezoelectric layers. By using generalized differential quadrature method, Setoodeh et al. [26] examined vibrational behavior of doubly curved smart sandwich shells with FG porous core and FG carbon nanotube-reinforced composite face sheets. Form the above literature reviews, we can see that although there are few authors studied 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 composite face sheets and FG porous homogeneous core is not studied so far. In this study, the governing 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 for nonlinear dynamic response and vibration of the sandwich plate. The influences of geometrical parameters, type of porosity distribution, initial imperfection, elastic foundation and temperature increment on the nonlinear dynamic of thick sandwich plate are demonstrated in detail. The results are reviewed with other authors very good agreement to verify of the approach used. 6 CHAPTER 3 MODELING & METHODOLOGY 3.1 Material properties of sandwich plate In this thesis, sandwich plate with three types of porous core are studied, in c which Young’s modulus of core layer E c , mass density of core layer  and thermal c expansion coefficients of core layer  are summed to be, respectively Chen et al. [5] E c = E1 1 − e0  ( z )   c = 1 1 − em  ( z )  (1)  c = 1 1 − em  ( z )  where  cos ( z / hc )   ( z ) = cos ( z / 2hc +  / 4 )    Non-uniform symmetric distribution Non-uniform asymmetric distribution (2) Uniform distribution in which e0 ( 0  e0  1) shows the coefficient of porosity and 1 12 2  em = 1 − 1 − e 0 ,  = −  1 − e 0 − + 1 e0 e0     2 (3) The homogeneous core layer made by titanium alloy, referred to as Ti-6Al4V. Except for Poisson’s ratio, the material properties are supposed to express as a nonlinear function of temperature Chen et al. [5]  c = 0.29, c = 4429kg / m3  c = 7.5788 (1 + 6.638 10−4 T + 3.147 10−6 T 2 ) 10−6 / K , (4) Ec = 122.56(1 − 4.586 10−4 T ) GPa, Recently, the FG-CNTRC material is made of Poly (methyl methacrylate), referred to as PMMA, reinforced by (10,10) SWCNTs. The effective Young’s and shear modulus of the FG-CNTRC material are determined as Shen [27] 7 E11 = 1VCNT E11CNT + Vm Em , 2 E22 3 G12 = VCNT Vm + , CNT Em E22 = VCNT Vm + , G12CNT Gm (5) where E11CNT , E11CNT , G12CNT are Young’s and shear modulus of the CNT; Em, Gm are mechanical properties of the matrix. VCNT and Vm are the volume fractions of the CNT and the matrix, respectively and i (i = 1,3) are the CNT efficiency parameters. In this thesis, it is assumed that the volume fractions of the CNTs have linear variations through the thickness layer as *  2 ( ( z − hc / 2 ) / h f )VCNT  VCNT ( z ) =   * 2 ( ( −hc / 2 − z ) / h f )VCNT Top face Bottom face (6) wCNT , / m ) − ( CNT / m ) wCNT (7) Vm ( z) = 1 − VCNT ( z), where * VCNT = wCNT + ( CNT in which wCNT is the mass fraction of CNTs, CNT and  m are the densities of CNT and matrix, respectively. Except Poisson’s ratio, the material properties of the matrix are assumed to express as a nonlinear function of temperature by Shen [27]  m = 0.34,  m = 45 (1 + 0.0005T ) 10−6 / K , Em = (3.52 − 0.0034T ) GPa, (8) in which T = T0 + T , T0 is the room temperature and T is the increase in temperature. 8 For the CNT, the material properties of (10,10) SWCNTs which are highly dependent to temperature. The Poisson’s ratio of SWCNTs is chosen to be constant 12CNT = 0.175. The temperature-dependent material properties for (10, 10) SWCNTs with 300  T  1000 by Shen et al. [29] E11CNT = ( 6.18387 − 0.00286T + 4.22867 10 −6 T 2 − 2.2724 10 −9 T 3 ) (TPa) CNT E22 = ( 7.75348 − 0.00358T + 5.30057 10 −6 T 2 − 2.84868 10 −9 T 3 ) (TPa) E11CNT = (1.80126 − 7.7845 10−4 T − 1.1279 10 −6 T 2 + 4.93484 10 −9 T 3 ) (TPa) 11CNT = ( −1.12148 + 0.02289T − 2.88155 10−5 T 2 + 1.13253 10 −8 T 3 ) (10 −6 / K )  22CNT = ( 5.43874 − 9.95498 10 −4 T + 3.13525 10 −7 T 2 − 3.36332 10 −12 T 3 ) (10 −6 / K ) The CNT efficiency parameters i (i = 1,3) used in equation (5) are estimated by matching Young’s modulus E11 , E22 and the shear modulus G12 of FG-CNTRC material obtained by the rule of mixtures extended to molecular simulation results (Kwon et al. [11]). For three different volume fractions of CNTs, these parameters are as: * * with VCNT = 0.12 (12%) then 1 = 0.137,2 = 1.022,3 = 0.715 ; with VCNT = 0.17 (17%) then 1 = 0.142,2 = 1.626,3 = 1.138 and 1 = 0.141,2 = 1.585,3 = 1.109 corresponds * to VCNT = 0.28 (28%) . The effective Poisson’s ratio depends weakly on temperature change and position and is given as * CNT 12 = VCNT v12 + Vm m , (9) where  12CNT and  m are Poisson’s ratio of the CNT and the matrix, respectively. The thermal expansion coefficients in the longitudinal and transverse directions of the CNTRCs are expressed by 9 VCNT E11CNT 11CNT + Vm Em m 11 = , VCNT E11CNT + Vm Em  22 = (1 +  CNT 12 )VCNT  CNT 22 (10) + (1 +  m )Vm m −  1211 , with 11CNT ,  22CNT and  m are the thermal expansion coefficients of the CNT and the matrix, respectively. 3.2 Modeling of sandwich plate Consider sandwich plate with total thickness h , length a, and width b is placed in the spatial coordinate system ( x, y, z ) be illustrated in the figure 3.1a. In this figure can see that sandwich plate is consisted by 5 layers: two layers piezoelectric, two layers CNTs and one layer FG porous homogeneous core. Besides, in the figure 3.1, z direction is attached in the thickness direction of the sandwich plate while ( x, y ) plane will be attached to the middle face of sandwich plate. a) 2D model b) 3D model Figure 3.1. Simulation model of the sandwich plate 10 3.3 Methodology In order to obtain the proposed purpose, an analytical method is used. I assume that the deflection of structures is relatively large, the material is elastic and the structural damage does not occur. Depending on the form of structures, the problems are posed in terms of stress and deflection functions. Basic equations will be established taking into account the influences of geometric nonlinearity and initial imperfection. Specifically, the Reddy’s higher-order shear deformation plate theory is used for thick sandwich plate structures. Then these equations are solved by combination of the Galerkin method and the Runge-Kutta method. I also use popular software to calculate such as Matlab, Maple, etc. Some numerical results are given and compared with one of other authors to verify the accuracy of the research. 3.4 Basic Equation The higher-order shear deformation plate theory is used to set up basic equations and determine nonlinear dynamic response and vibration of the sandwich plate. The deformed components of the sandwich plate are at a point away the mid – plane at the distance z are defined as Reddy [33]   x    x0   k x1   k x3  0 2    0  1  3  3   xz   xz  2  k xz    y  =   y  + z  k y  + z  k y  ,   =  0  + z  k 2  ,  yz   xy   xy0   k xy1   k xy3   yz   yz          where  u 1  w  2  +      x 2  x    0 x   2   0   v 1  w    = +  y   y 2  y   ,    xy0       u + v + w w   y x x y     w  + x     x ,   = −3c1  w    + y   y  0 xz 0 yz 11 (11)  x       x 1 3     kx   kx        1  3 y k = , k = − c   1  y  y  y 1 3  k    k xy       xy        y  x+   x  y y   y  x  2 w + 2  x x  w  (12)  +  x  x   k xz2   y  2 w , +  ,  2  = −3c1  w k y y 2  yz    + y    2  y  y  w + +2  y xy  with u, v, w are displacement components parallel to the coordinates ( x, y, z ) , respectively. Also, x , y are respectively the rotations of the transverse normal about the y and x axes at z = 0 and c1 = 4 / (3h2 ). The Hooke’s law for the FG porous homogeneous core  xx  Q11C    C  yy  Q12  xy  =  0     xz   0  yz    C  0 Q12C 0 0 Q22C 0 0 C 66 0 Q 0 C 44 0 0 Q 0 0 0  (  ) − 11T  0   xx C   (  yy ) −  22 T  0  C    xy )C 0  ( ,   0  ( xz )C   Q55C   ( yz )C   (13a) in which Q11C = Q22C = EC 1 − ( vC ) , Q12C = 2 vC E C 1 − ( vC ) , Q66C = Q44C = Q55C = 2 EC 2 (1 + vC ) (13b) The Hooke’s law for FG-CNTRC face sheets  xx  Q11f    f  yy  Q12  xy  =  0     xz   0  yz    f  0 f 12 f 22 Q 0 0 Q 0 0 0 Q66f 0 0 0 Q44f 0 0 0  (  xx ) f − 11T    0     −   T  ( ) 22 0   yy f   0  ( xy ) f  ,   0   ( xz ) f f   Q55    ( yz ) f   12 (14a)
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