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

VIETNAM NATIONAL UNIVERSITY, HANOIVIETNAM JAPAN UNIVERSITY NGO DINH DAT NONLINEAR DYNAMIC RESPONSE AND VIBRATION OF SANDWICH PLATES WITH FG POROUS HOMOGENEOUS CORE AND NANOTUBE-REINFORCE

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

First of all, I would like to express my deep gratitude to the instructor, ProfessorNguyen Dinh Duc, who devotedly guided, helped, created all favorable conditionsand regularly encouraged me to complete this thesis

I would like to express my deepest thanks to Professor Kato, Professor Dao NhuMai, Professor Nagayama, Dr Phan Le Binh and Dr Nguyen Tien Dung from theInfrastructure Engineering Program for always caring and helping, supporting andgiving useful advice during the time I study and complete the thesis In addition, Ifeel very happy because of the enthusiastic support from the program assistant BuiHoang 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 atVJU, my classmate for creating unforgettable memories Finally, I would like tothank my family, my girlfriend Dang Thu Trang, who is always with me at difficulttime who encourage and help me

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TABLE OF CONTENTS

ACKNOWLEDGEMENT

LIST OF TABLES

LIST OF FIGURES

LIST OF ABBREVIATIONS

ABSTRACT

CHAPTER 1 INTRODUCTION

1.1 Background

1.2 Research objectives

1.3 Structure of the thesis

CHAPTER 2 LITERATURE REVIEW

CHAPTER 3 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 4 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 5 CONCLUSIONS

5.1 Conclusions

APPENDIX

LIST OF PUBLICATIONS

REFERENCES

II

volume fraction V CNT* 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

III

LIST OF FIGURES

Figure 1.1 Application of Advanced material 1Figure 3.1 Simulation model of the sandwich plate 10Figure 4.1 Influence of ratio width-to-length a / b on the nonlinear dynamicresponse of the sandwich plate 28

response of the sandwich plate 29Figure 4.3 Influence of volume fraction V CNT on the nonlinear dynamic response of*the sandwich plate 29Figure 4.4 Influence of porosity coefficient e0 on the nonlinear dynamic response ofthe sandwich plate 30

of the sandwich plate 30Figure 4.6 Influence of temperature increment T on the nonlinear dynamicresponse of the sandwich plate 31Figure 4.7 Influence of the magnitude Q0 of the external excitation on the nonlineardynamic response of the sandwich plate 32Figure 4.8 Influence of the Winkler foundation k1 on the nonlinear dynamic

response of the sandwich plate 33Figure 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

Functional graded- carbon nanotube-reinforced composite

GygaPascalThickness of FG porous homogeneous core

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Abstract: This thesis analytical solutions for the nonlinear dynamic response and

vibration of sandwich plates with FG porous homogeneous core and reinforced composite face sheets integrated with piezoelectric layers in thermalenvironment Assuming that the characteristics of the plate depend on temperature andchange consistent with the power functions of the plate thickness Motion andcompatibility equations are used to base on the Reddy’s higher-order shear deformationplate theory and consider the influence of initial geometric imperfection and thethermal stress in the plate Besides, the Galerkin method and Runge – Kutta method areused to give clear expressions for nonlinear dynamic response and natural frequencies

nanotube-of the sandwich plate The influences nanotube-of geometrical parameters, type nanotube-of porositydistribution, initial imperfection, elastic foundation and temperature increment on thenonlinear dynamic response and vibration of thick sandwich plate are demonstrated indetail The results are reviewed with other authors in possible cases to check thereliability 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 certainproducts and details Materials determine the design, construction and cost of theproduct Metallic and non-metallic materials are materials commonly used in manyindustrial fields Recently with the development of science and technology hascreated a number of new materials such as composite materials, nanocompositematerials, 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 ofcivil, … (figure 1.1) Due to the outstanding characteristics of this material like lightweight, heat resistance, energy dissipation reduction and superior vibrationaldamping, Especially, it is impossible not to mention the porous material It islightweight cellular materials inspired by nature Wood, bones and sea sponges aresome well-known examples of these types of structures Foams and other highly

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porous materials with a cellular structure are known to have many interestingcombinations of physical and mechanical properties, such as high stiffnesscombined with very low specific gravity or high gas permeability combined withhigh thermal conductivity Among artificial cell materials, polymer foams arecurrently 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 ofcellular or foam materials, and these materials have such interesting properties thatexciting new applications are expected in the near future.

1.2 Research objectives

The research objective of this thesis is to research nonlinear dynamicresponse and vibration of sandwich plate subjected to thermo-mechanical loadcombination Hence, to solve the problem, this thesis will set out the objectivesshould be achieved as below:

Investigations on nonlinear dynamic response and vibration of sandwichplates subjected to thermo-mechanical load combination The natural frequency andthe deflection – time curves of sandwich plate structures are determined Innumerical results, the effects of the geometrical parameters, types of distribution ofporosity, temperature increment, imperfections and elastic foundation on thenonlinear 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 responseand vibration of sandwich plate structure using analytical method In order to betterunderstand the solution method as well as give an appropriate result, the thesis ispresented in the following structure:

Highlights the role and importance of the material, especially the advancedmaterial for industrial fields The background and research objective are introduced

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➢ Chapter 2: Literature review

Introduction of articles related to research issues Since then, explains why this study is necessary

The material and model properties of the structure are presented The method used as well as how to solve the problem are discussed

Check the reliability of the method through comparison with other authorsconsidered The results are expressed and discussed as the geometrictransformations, temperature and mechanical load through the deflection amplitude

- time curves and natural frequencies

Summarize the results achieved and provide further direction for the study

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CHAPTER 2 LITERATURE REVIEW

Nowadays, sandwich structures are widely in many fields of life such asmedical, electronics, energy, aerospace engineering, industry automotive andconstruction 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 lowstrength but thick brings high bending stiffness The face sheet is often used withsheet metal and fiber-reinforced polymers, while the core is usually made ofhoneycomb or polymer foam Recently, the advanced of carbon nanotubes such assuperior strength and stiffness has been of interest to many scientists Carbonnanotubes Liew et al [15] are a potential candidate for sandwich structures with thereplacement of the face sheet with nanocomposite material which is reinforced withcarbon nanotubes that improves the bearing capacity The sandwich structure carbonnanotube-reinforced composite face sheets are investigated by a number of authors.Wang et al [37], Natarajan et al [21] studied vibration and bending of sandwichplates with nanotube-reinforced composite face sheets By using Extended Highorder Sandwich Panel Theory, the bending analysis of sandwich beam Salami [9]also presented Di Sciuva et al [7] investigated additionally buckling of sandwichplates adopted Refined zigzag theory with Rits method The vibration of thermallypostbuckled also studied by Shen et al [28] The dynamic instability analysis usingshear flexible QUAD-8 serendipity element under periodic load is present bySankar et al [25] Based on mesh-free method, Moradi-Dastjerdi et al

[19] studied static analysis of functionally grade nanocomposite sandwich platereinforced by defected CNT Safaei et al [24] also using mesh-free method toinvestigate the influence of loading frequency on dynamic behavior of structure.Mehar et al [18] researched thermoelastic nonlinear frequency analysis of CNTreinforced functional graded sandwich structure Sobhy et al [31] studied the effect

of the magnetic field on thermomechanical buckling and vibration of viscoelasticsandwich nanobeams with CNT reinforced face sheets

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On the other hand, a porous core in sandwich structures is capable ofwithstanding the transverse normal and shear loads as well as superior energydissipation, 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 theenergy-absorption performance of porous materials in sandwich composites Thepaper 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 acoustictransmission of the sandwich aerospace composite doubly curved shell Qiao et al.[23] studied the sound insulation of a periodically rib-stiffened double panel withporous core by using space harmonic series and Biot theory Moreover, a gradedporosity leads to the continuous variation in material properties so reducing stressconcentration often encountered in conventional sandwich structures Chen et al [5]employed the Ritz method to derive the nonlinear free vibration of sheardeformation of sandwich beam with FG porous core Li et al [13] explorednonlinear vibration and dynamic buckling of sandwich functionally graded porousplates with reinforced graphene platelet Chen et al

research showed FG porosity were suggested could remove the mismatch stressesand effective buckling and bending significantly

In addition, based on the properties of piezoelectric materials, sensors aremade such as ultrasonic transceiver sensors in machines (detecting defects in metaland 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 vibrationproperties for piezoelectric sandwich ultrasonic transducers Masmoudi et al [17]investigated mechanical behavior and health monitoring by acoustic emission ofsandwich composite integrated by piezoelectric implant The acoustic emissiontechnique can be found damage in materials through transient ultrasonic detection.Belouettar et al [4] adopted the Harmonic balance method to study active

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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-reinforcedface sheets and integrated with piezoelectric layers can be considered as newadvanced material Moradi-Dastjerdi et al [20] employed the Reddy’s third orderand mesh-free method to study stability analysis of multifunctional smart sandwichplates with graphene nanocomposite and porous layers integrated with piezoelectriclayers By using generalized differential quadrature method, Setoodeh et al [26]examined vibrational behavior of doubly curved smart sandwich shells with FGporous core and FG carbon nanotube-reinforced composite face sheets

Form the above literature reviews, we can see that although there are fewauthors studied smart sandwich material with FG porous core and nanocomposite-reinforced face sheets integrated with piezoelectric layers, the nonlinear dynamicresponse and vibration of sandwich plate with FG nanotube-reinforced compositeface sheets and FG porous homogeneous core is not studied so far In this study, thegoverning equations are used to base on the Reddy’s higher-order shear deformationplate theory and consider the influence of initial geometric imperfection and thethermal stress in the plate Besides, The Galerkin method and Runge – Kuttamethod are used for nonlinear dynamic response and vibration of the sandwichplate The influences of geometrical parameters, type of porosity distribution, initialimperfection, elastic foundation and temperature increment on the nonlineardynamic of thick sandwich plate are demonstrated in detail The results arereviewed with other authors very good agreement to verify of the approach used

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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

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

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

material obtained by the rule of mixtures extended to molecular simulation results

The thermal expansion coefficients in the longitudinal and transverse directions

of the CNTRCs are expressed by

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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 assumethat the deflection of structures is relatively large, the material is elastic and thestructural damage does not occur Depending on the form of structures, the problemsare posed in terms of stress and deflection functions Basic equations will beestablished taking into account the influences of geometric nonlinearity and initialimperfection Specifically, the Reddy’s higher-order shear deformation plate theory isused for thick sandwich plate structures Then these equations are solved bycombination of the Galerkin method and the Runge-Kutta method I also use popularsoftware to calculate such as Matlab, Maple, etc Some numerical results are given andcompared 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 basicequations and determine nonlinear dynamic response and vibration of the sandwichplate The deformed components of the sandwich plate are at a point away the mid –

The Hooke’s law for the FG porous homogeneous core

12

(13b)

(14a)where

layers as

31 31 11

of panel type piezoelectric material by

The force and moment resultants of sandwich plate are demonstrated by

From equation (19), the constitutive relations can write

The nonlinear motion equation of the sandwich plate can be determined

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

distributed on the surface of the plate and

By substituting equations (25a) and (25b) into equations (23c) – (23e) gives

L31 (w) + L32 (x) + L33(y)+ L34

where

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