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-REINFO

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

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

I

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

II

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

( )

2h 2 1 v /E c

(a b/ =2, 2 /h a=1/ 20,1/12) 25 Table 4.3 The influence of porosity coefficient e0, ratio width-to-length a b/ and volume fraction *

0

/ 1, / 20, 0.2, / 5, / 10, CNT 0.12, , 1,1

a b= b h= e = hc hf = hc hp= V = m n = 27

IV

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 excitationon 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

VI

ABSTRACT

Abstract: This thesis analytical solutions for the 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

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

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

2

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

3

➢ 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

4

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

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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

6

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 mesh-free 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 nanocomposite- reinforced 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

7

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 which Young’s modulus of core layer E , mass density of core layer c c and thermal expansion coefficients of core layer care summed to be, respectively Chen et al [5]

( ) ( ) ( )

c c

In this thesis, it is assumed that the volume fractions of the CNTs have linear variations through the thickness layer as

w V

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:

10

11 11 11

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

Figure 3.1 Simulation model of the sandwich plate

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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]

0

1 2 1

2

xz y

yz

y xy

c w

12

2 2

2 2

y w

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 = and 0 c1 =4 / (3 ).h2

The Hooke’s law for the FG porous homogeneous core

( )

( ) ( )

C

C C

xz

xz C C

xz f f

and we presume G13 =G12 and G23 =1.2G12

The Hooke’s law for the piezoelectric layers

( )

( ) ( )

15 44

E e

Q

E e

Q Q

We have the transverse electric field component E z is nonzero due to the use

of panel type piezoelectric material by

in which the detail of coefficients (A ij, B ij, D ij, E ij, F ij, H ij),ij 11,12, 22, 66; =

(A kl,D kl, F kl),kl=44,55;m,m=1, 2,3, 4,5, 6. may be found in Appendix

The nonlinear motion equation of the sandwich plate can be determined

(Nosier et al [22], Reddy [34])

in which  is the viscous damping coefficient, q is an external pressure uniformly

distributed on the surface of the plate and

( ) 1, , , , , ( ) 1, , , , ,

( ) 1, , ,

hc c hc