Effects of porosity on free vibration and nonlinear dynamic response of multi layered functionally graded materials subjected to blast loads

VIETNAM NATIONAL UNIVERSITY, HANOIVIETNAM JAPAN UNIVERSITY DO THI THU HA EFFECTS OF POROSITY ON FREE VIBRATION AND NONLINEAR DYNAMIC RESPONSE OF MULTI-LAYERED FUNCTIONALLY GRADED MATERIA

VIETNAM NATIONAL UNIVERSITY, HANOI

VIETNAM JAPAN UNIVERSITY

DO THI THU HA

EFFECTS OF POROSITY ON FREE

VIBRATION AND NONLINEAR DYNAMIC

RESPONSE OF MULTI-LAYERED

FUNCTIONALLY GRADED MATERIALS

SUBJECTED TO BLAST LOAD

MASTER’S THESIS

VIETNAM NATIONAL UNIVERSITY, HANOI

VIETNAM JAPAN UNIVERSITY

DO THI THU HA

EFFECTS OF POROSITY ON FREE

VIBRATION AND NONLINEAR DYNAMIC

RESPONSE OF MULTI-LAYERED

FUNCTIONALLY GRADED MATERIALS

SUBJECTED TO BLAST LOAD

MAJOR: INFRASTRUCTURE ENGINEERING

CODE: 8900201.04QTD

RESEARCH SUPERVISOR:

Dr TRAN QUOC QUAN

Ha Noi, 2020

First of all, I would like to express my sincere appreciation to my supervisor ,

Dr Tran Quoc Quan who has guided and created favorable conditions and regularlyencouraged me to complete this thesis Thank you for all your thorough andsupportive instructions, your courtesy and your enthusiasm Without your dedicatedguidance, I absolutely have not conducted this research well

Secondly, I would like to express my great thankfulness to Master’sInfrastructure Engineering Program for their wonderful supports, especiallyProf.Sci Nguyen Dinh Duc, Prof Kato, Prof Nagayama, Dr Phan Le Binh, Dr.Nguyen Tien Dung and Mr Bui Hoang Tan Their encouragement and assistancehas facilitated me a lot during 2 years studying in the VietNam – Japan University Ialso want to give my special thanks to all lecturers and staffs at The University ofTokyo for their warmly welcome and supports me in the internship time at Japan

Thirdly, I would like to thank all the members at the Advanced Materials andStructural Laboratory, University of Technology- VNU, especially for Mr Vu DinhQuang, Mr Vu Minh Anh, Mr Pham Dinh Nguyen spending their precious time topoint out for me which theories and methodology should I use and give me advices

to improve my thesis

Finally, there are my family and my friends, who always stay by my side,motivate and encourage me from the beginning until the end of my studying

TABLE OF CONTENTS

ACKNOWLEDGEMENT I LIST OF TABLES III LIST OF FIGURES IV NOMENCLATURES AND ABBREVIATIONS V ABSTRACT VI

CHAPTER 1: INTRODUCTION 1

1.1 Overview 1

1.1.1 Composite material – Functionally Graded Materials 1

1.1.2 FGM classification 2

1.1.3 Blast load 5

1.2 Research objectives 6

1.3 The layout of the thesis 6

CHAPTER 2: LITERATURE REVIEW 8

2.1 Structures 8

2.2 Porosity 9

2.3 Blast load 10

CHAPTER 3: METHODOLOGY 12

3.1 Configurations of analyzed models 12

3.2 Methodology 17

3.3 Theoretical formulation 18

3.4 Solution procedure 24

3.5 Vibration analysis 25

3.5.1 Dynamic response problem 25

3.5.2 Natural frequency 27

CHAPTER 4: NUMERICAL RESULTS AND DISCUSSION 28

4.1 Validation of the present results 28

4.2 Natural frequency 30

4.3 Dynamic response 33

CHAPTER 5: CONCLUSIONS 40

APPENDIX 41

II

Table 4.2 Comparison of natural fundamental frequency parameters  of simply square

FGM plates with other theories ( h/ b 0.1 ) .29 Table 4.3.

The effects of porosity ratio on natural frequency of FGM sandwich plates

31

Table 4.4 Influences of temperature increment, elastic foundations and the volume

fraction index on natural frequencies of the FGM sandwich plate with porosity I 32

LIST OF FIGURES

Fig 1.1 The distribution types of FGM sandwich material 5

Fig 3.1 FGM sandwich plate resting on elastic foundation 12

Fig 3.2 FGM-ceramic- FGM model 13

Fig 3.3 Porosity – I: evenly distributed, Porosity – II: unevenly distributed 14

Fig 3.4 Blast pressure function 17

Fig 4.1 Influences of power law index N on the nonlinear dynamic response of the FGM sandwich plates with porosity I 33

Fig 4.2 Influences of power law index N on nonlinear dynamic response of the FGM sandwich plates with porosity II 33

Fig 4.3 Influences of porous ratio on nonlinear dynamic response of the FGM sandwich plates with porosity I 34

Fig 4.4 Influences of type of porosity on nonlinear dynamic response 35

Fig 4.5 Influences of a/b ratio on nonlinear dynamic response of the FGM sandwich plates with porosity I 36

Fig 4.6 Influences of a/h ratio on nonlinear dynamic response of the FGM sandwich plates with porosity I 37

Fig 4.7 Influences of Pasternak foundation on nonlinear dynamic response of the FGM sandwich plates with porosity I 37

Fig 4.8 Influences of Winkler foundation on nonlinear dynamic response of the FGM sandwich plates with porosity I 38

Fig 4.9 Effect of parameter characterizing the duration of the blast pulse T s on nonlinear response of the FGM sandwich plate with porosity I under blast load 39

NOMENCLATURES AND ABBREVIATIONS

Functionally Graded MaterialThe length of plate

The width of plateThe thickness of plateThe thickness of the core layerThe thickness of FGM face-sheetThe Winker foundation

The Pasternak foundationGigaPascal =10 9 PascalNumbers of half waves in x,y direction

The effects of porosity ratio on free vibration and nonlinear dynamicresponse of FGM sandwich plates with two FGM face-sheets and a homogeneouscore as ceramic resting on elastic foundations subjected to blast load areinvestigated in this thesis by implementing the third-order shear deformation theory.Two types of porosity are proposed, namely evenly distributed porosity andunevenly distributed porosity Assumption that the material properties of multi-layered FGM plate to be changed in the thickness direction accord with a simple-power law distribution with regard to the volume proportion of the components.This study obtains numerical results by using the Galerkin method and fourth-orderRunge-Kutta method illustrating the significant effects of porous fractions,geometrical parameters, the elastic foundation, blast loads on the nonlinear dynamicresponse of FGM sandwich plates

Key words: Porosity, Functionally graded sandwich plate, Blast loading, The third-order shear deformation theory.

VI

CHAPTER 1: INTRODUCTION

1.1 Overview

1.1.1 Composite material – Functionally Graded Materials

Composite material is a material composed of two or more different types ofcomponent materials in order to achieve superior properties such as light weight, highstiffness and strength, ability of heat resistance and chemical corrosion resistance,

good soundproofing, thus it plays a crucial role in advanced industries in the world that areextensively applied across wide range of fields such as: aviation, aerospace, mechanics,construction, automotive [1] [2] However, this material has a defect as a sudden change

of material properties at the junction between the layers is likely to generate large contactstresses at this surface One of the solutions to overcome this

disadvantage of layered composite material is to use Functionally Graded Material(FGM) which is a material made up of two main component materials as ceramicand metal, in which the volume ratio of each component varies smoothly and

continuously from one side to the other according to the thickness of the structure so thefunctional materials avoid the common disadvantages in composite types such as thedetachment between layers material, fibers breakage and high stress in the surface, whichcan cause material destruction and reduce the efficiency of the structure, especially in heat-resistant structures Due to the high modulus of elasticity E , the thermal conduction

very good heat resistance While the metal components make the modified materials moreflexible, more durable and overcome the cracks that may occur due to the brittleness of ceramicmaterials when subjected to high temperature (Table 1.1)

Table 1.1 Properties of component materials of FGM material [3]

PropertiesMaterial

Depending on the power law of the volume ratio of component materials, we

can classify different types of FGM Each of these FGM materials is characterized

by different mechanical and physical properties by a function that determines the

material properties (effective properties), and the value of the function varies with

thickness Mathematical functions of material properties used to classify materials

[4] Specifically, there are three main types of FGM

A power-law distribution P-FGM: is a type of material having a volume

fractions of ceramic and metal components which is assumed to vary according to

thickness of structure and conforming to the power-law function [5, 6]:

: the volume fractions of metal and ceramic, respectively

: the volume distribution (0N )

(1.1)

2

The effective properties P eff of the P- FGMs are established using the

modified mixed rules as follows [7]:

in which Pr denotes a specific property of the material such as elastic modulus E,thermal expansion coefficient  or density  , thermal conduction K

Sigmoid-law distribution S-FGM: is a type of material having a volume

fractions of ceramic and metal components which is assumed to vary according tothickness of structure and conforming to the Sigmoid-law function as:

The effective properties

mixed rules as follows:

An exponential-law distribution E-FGM: is a type of material having a volume

fractions of ceramic and metal components which is assumed to vary according tothickness of structure and conforming to the exponential-law function as:

E  z   AeB  z h / 2, (1.5)where

3

FGM sandwich material: The multi-layered sandwich structure is a

particularly important type of structure in the aerospace industry as well as in a

number of other industries such as ships, automobiles, construction Sandwich

structure consists of 3 main layers: core layer and two face-sheets The core layer is

made of lightweight material, low hardness between two face -sheets made of very

high hardness material The great advantage of sandwich structure is that it

increases the stiffness and bending resistance of the structure while ensuring a small

volume, because the core layer is made of light material that can be made with a

large thickness that will have an effect to transfer the two face-sheets away from the

neutral axis

To avoid the phenomenon of flaking between the layers as well as the

phenomenon of stress caused as with conventional multi-layer structures, it was

thought that FGM sandwich material with ceramic or metal core layer and two face

- sheets made of FGM material

4

a Sandwich FGM- Metal-FGM b Sandwich FGM- Metal-FGM

Fig.1.1 The distribution types of FGM sandwich material

The effective properties of this materials vary according to the extendedSigmoid distribution law as follows:

(1.7)

1.1.3 Blast load

Blast load: In recent years, the safety of important buildings and infrastructure

around the globe has become more fragile by extreme dynamic loads due to the increase interrorist activities, explosions The damage from such events cannot be determined, notjust economically because many of these ones are symbolic and important heritage,significant architectures and the spirit of the times Nowadays, considerable efforts in

5

focused towards optimal design and economic efficiency in construction It isessential to guarantee the safe and secure protection of important infrastructure forthe present and future.

Explosion loads usually act in a very short time (usually in milliseconds) buttransmit very high pressure pulses (10 1 10 3 kPa) As a result, damage to structuralsystems can take many forms, such as damage to the outer surface and structuralframe of a building; collapse of walls and bearing columns; blow debris of concrete,glass windows and furniture; and damaging safety systems Most existing buildingsare not designed to withstand such extreme dynamic loads, so a comprehensiveunderstanding of the explosion phenomena and the dynamic response of structures

is required to be essential for the scientific basis improving the design and materialimprovement in a feasible manner, in order to improve explosion resistance andensure the safety of structures Therefore, the investigation of the effects ofexplosive loading on structures should be focused

1.2 Research objectives

The research objective of this thesis is to investigate the effects of porosity onvibration and nonlinear dynamic response of multi-layered FGM subjected to blastload

cases: FGM plates without porosity, porous -I FGM plates and porous-II FGM plates

 Investigations on nonlinear dynamic analysis on the structure in FGM plates onelastic foundation subjected to blast load In numerical results, the effects of the materialproperties, geometrical parameters, blast load… on the nonlinear dynamic response will

be analyzed

1.3 The layout of the thesis

The thesis includes an introduction, five chapters, conclusions, references and appendices The main contents of the chapters involve:

 Chapter 1: Introduction

The thesis presents an overview of FGM materials Porosities are also

mentioned in this chapter

 Chapter 2: Literature review

Chapter 2 presents some studies which have been reported to this thesis’s field

In those publications, I also pointed out their main outstanding results obtained fromtheir research as well as those research’s limitation

 Chapter 3: Methodology

Chapter 3 introduces the analytical method by using high-order shear

deformation to approach and solve problems…

 Chapter 4: Numerical results and discussion

The numerical results are presented in this chapter for a FGM sandwich plate onelastic foundation in terms of natural frequencies, effects of geometrical parameters,materials properties on nonlinear dynamic response

 Chapter 5: Conclusions

Chapter 5 summarize the main results obtained from this thesis

7

CHAPTER 2: LITERATURE REVIEW

Because of their remarkable properties, in recent years, sandwich FGM structurehas been attracted a lot of attention of scientists Among those, Zhaobo Chen et.al [8]presented the free vibration of the functionally graded material sandwich doubly-curvedshallow shells under simply supported conditions according to a new shear deformationtheory with stretching effects The wave propagation of FGM sandwich plates withporosities putting on viscoelastic foundation was studied by Chen Liang and Yan QingWang [9] based on a quasi-3D trigonometric shear deformation theory Singh and his co-authors [10] used a semi-analytical approach to analyze thermo-mechanical of poroussandwich S-FGM plate for different boundary conditions using Galerkin Vlasov's method.Furthermore, the nonlinear vibration of imperfect sandwich plates with FGM face sheetsalso investigated by Kitipornchaii et al [11] basing on a semi-analytical approach HoangVan Tung [12] are analyzed nonlinear bending and post buckling behavior of FGMsandwich plates under thermomechanical loading by using the first order shear deformationtheory The effect of time constant, temperature, mid radius to thickness ratio and time ontransient thermo-elastic behavior of sandwich plate with the core as FGM are taken intoconsideration by Alibeigloo [13] In his analysis, the sandwich plate’s time dependentresponse is built from generalized coupled thermo-elasticity when applying the Lord-Shulman expression Moreover, Xia and Shen [14] introduced an analytical using higher-order shear deformation and a general von Kármán-type function to obtain small- andlarge-amplitude vibration of compressive and thermal post-buckling sandwich plates withFGM face sheets under uniform and non-uniform temperature fields BehzadMohammadzadeh [15] combined higher-order shear deformation with Hamilton’s principle

to analyze nonlinear dynamic responses of sandwich plates with FGM faces on elasticfoundation subjected to blast loads Basing on a new four-variable shear deformation platetheory, Mohammed Sobhy

[16] evaluated the hydrothermal vibration and buckling of various types of FGMsandwich plates resting on elastic foundations exposed moisture condition, risingtemperature, Winkler–Pasternak foundation coefficients and power- law distributionindex Chien et al [17] used isogeometric approach to investigate static, free vibrationand buckling analysis of FGM isotropic and sandwich plates Tao Fu et al [18] adoptedthe space harmonic approach and virtual work principle to describe analytically soundloss when transmitting through two types of porous FGM sandwich structures.

2.2 Porosity

The effects of porosities generated during actual manufacturing process tothe vibration characteristics of FGM structures have been studied by several authors.However, the number of researches in terms of the mechanical behaviors of porousmaterials is still limited The most recent investigations on structures with porosityare listed in the following

Ashraf M.Zenkour [19] used a quasi-3D shear deformation theory to investigate thebending responses of porous functionally graded single-layered and multi-layered thickrectangular plates By taking Galerkin Vlasov's method into account in thermo-mechanicalanalysis of sandwich S-FGM plate with three different types of porosity for diverseboundary conditions, Singha and co-author [20] obtained the approach for bending andstress under the thermal environment They deduced that the deflection and stress escalatesignificantly for even porosity distribution (P-1) to bring into comparison with unevensymmetric (P-2) or uneven non-symmetric (P-3) porosity distribution; and the effects oftemperature on transverse shear stresses of the multi-layered plates Mojahedin [21]employed higher order shear deformation theory to investigate the buckling of functionallygraded porous circular plates Polat et al [22] utilized an atmospheric plasma spray system

to obtain functionally gradient coatings from five layers which were prepared on Nisubstrates from Y2O3 stabilized ZrO2 (YSZ) and NiCoCrAlY powders In their

9

research, they found that an escalation in porosity ratio of layers lead to the decrease ofresidual stresses Chen et al [23] employed Chebyshev-Ritz method to analyze bucklingand bending loads of a novel functionally graded porous plates Wang et al.

rectangular plates with two types of porosity, namely, even and uneven distributed porosity, andtransferring in thermal environment Based on a sinusoidal shear deformation theory incombination with the Rayleigh–Ritz method, Yuewe Wang et al [25] depicted the effects ofporosity, boundary conditions, and geometrical parameters on free vibration of the functionallygraded porous cylindrical shell An isogeometric finite element model and the nonlocal elasticitywere introduced by Phung-Van et al [26] to investigate the transient responses of functionallygraded nanoplates with porosity Small size effects, nonlocal parameters, and porositydistributions, volume index, the characteristics of dynamic load have considerably influenced onthe plate nonlinear transient deflections Cong et al [27] acquired closed-form expression inregard to critical bucking loads and post-buckling paths of a porous functionally graded plates onelastic foundations subjected to the coupling of mechanical and thermal loads by applyingReddy's higher-order shear deformation plate theory in conjunction Galerkin method Analyticalsolutions and numerical results revealed that porosity I (evenly distribution) behaves better thanporosity II (unevenly distribution) according to the static buckling investigations Chien et al

[28] adopted the first-order shear deformation theory taking the out-of-plane sheardeformation into account to calculate the fundamental frequencies and nonlinear dynamicresponses of porous functionally graded sandwich shells with double curvature under theinfluence of thermomechanical loads This study proved that porosities help the shellstructures stiffen to some extent

2.3 Blast load

In recent years, explosive loads and their impacts on the safety and efficiency

of building and structures have received considerable attention Tuan et al [29]presented the results of an empirical investigation conducted in Woomera, Southern

Australia, in May 2004 on the explosion-resistance of concrete-panel created by strength concrete material A finite-element method was used to analyze concrete structuresunder blast and impact loading In the study conducted by Tin and co-authors [30] , theyproposed using the explicit finite element software LS-DYNA to induce stress wavepropagation and the impacts on structural responses of precast concrete segmental columnssubjected to simulated blast loads Balkan et al [31] examined the effects of sandwichstiffeners on the dynamic response of laminated composite plates under the non-uniformblast loading Moreover, the dynamic behavior of stiffened plates exposed to confined blastloads are carried out by Zhao et al [32] through experimental and numerical studies.Geretto et al [33] analyzed a series of experiments of square monolithic steel plates toassess the effects of the degrees of confinement of the deformation to blast loads Asoylar

ultrahigh-et al [34] studied the transient stability analysis multrahigh-etal-fiber laminated composite platesunder no-ideal explosion load by experiment and finite element methods In addition,Uybeyli and colleagues [35] used SiC reinforced functionally gradient material via powdermetallurgy to investigate the impact of armor piercing projectile Bodaghi et al [36]studied non-linear active control of dynamic response of functionally graded beams withrectangular cross-section in thermal environments under blast loadings

Based on meticulous investigations in the available literature, it can beconcluded that there are few free vibration and nonlinear dynamic behaviors ofporous functionally graded sandwich plates resting on elastic foundations regardless

of the high demand for understanding In particular, literature review indicates lack

of investigations on effects of porosity on this structure exposed to blast loads Thisstudy has been implemented to meet the demand

11

CHAPTER 3: METHODOLOGY

3.1 Configurations of analyzed models

The geometry configuration of the rectangular FG sandwich plate with two FGM

face-sheets and the core as ceramic resting on elastic foundations under blast load

are as follows (Figs 3.1 and 3.2) The plate is referred to a Cartesian coordinate

system x, y , z , where xy is the mid-plane of the plate andzis the thickness coordinator,

: the length and width of the plate

, h f :thickness of the total plate, the core and the face-sheets

z

shear layer b

x

Fig 3.1 FGM sandwich plate resting on elastic foundation.

Fig 3.2 FGM-ceramic- FGM model.

The sandwich plate is composed of three elastic layers, namely: “Layer 1”,

“Layer 2”, “Layer 3” corresponding with FGM sheet, the core and FGM sheet The FGM face-sheets are made from a mixture of metal and ceramic Theface-sheets and the core satisfy power-law distribution and the constituent volumefraction varying continuously along thickness direction The assumption of themetal and ceramic volume is written as:

ratio of the ceramic is denoted as V ( z) c The volume fraction index

distribution of component materials in the structure and N [0,)

(3.2)

The volume

N defines the

13

In the case of N = 0, the face-sheet is made entirely of ceramic In the case of

N   , the face-sheet is made entirely of metal

The reaction–deflection relation of Pasternak foundation is defined as follows:

Fig 3.3 Porosity – I: evenly distributed, Porosity – II: unevenly distributed.

The effective properties P(z,,T) such as the elastic moduli

mass density  (z,  ,T) and the thermal expansion coefficient  (z, 

and the Poisson

The blast load q (t) is a short–term load and is a rapid release of stored energyfrom an explosion, a shock-wave disturbance or a supersonic projectile… Part ofthe energy is released as thermal radiation (flash); and part is coupled into the air asair blast and into the soil (ground) as ground shock, both as radially expandingshock waves It can be considered as the models in [37] and shown in Fig 3.4

16

Fig 3.4 Blast pressure function

where the "1.8" factor accounts for the effects of a hemispherical blast, Ps max is themaximum (or peak) static over-pressure, b is the parameter controlling the rate of

wave amplitude decay and T

is the parameter characterizing the duration of the blast

types of porosity, the volume fraction index.

3.3 Theoretical formulation

Suppose that the FGM plate is subjected to blast loads The Reddy’s higherorder shear deformation theory (HSDT) in conjunction with the stress function areused to establish the governing motion, compatibility equations and determine thenonlinear dynamic response and vibration of the multi-layered FGM plate

The strain components at the distance

consideration von Karman nonlinear terms is the

slope rotations in the x, zand y, z planes.

18

The compatibility equation for a multi-layered functionally graded are given

The thermal stress-strain relation of the functionally graded sandwich plate

are presented by Hooke's law as follows:

in which T is temperature rise from stress free

difference between two surfaces of the FGM plate

initial state or temperature

The force and moment resultants of the plate can be expressed in terms of

stress components across the plate thickness as:

The constitutive relations are established by introducing Eqs (3.4), (3.5) and (3.9)

into Eq (3.10) as follows:

 ,



) T

(3.11)