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

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 M

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

Ha Noi, 2020

Hanoi, 2019

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

I

ACKNOWLEDGEMENT

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 regularly encouraged me to complete this thesis Thank you for all your thorough and supportive instructions, your courtesy and your enthusiasm Without your dedicated guidance, I absolutely have not conducted this research well

Secondly, I would like to express my great thankfulness to Master’s Infrastructure Engineering Program for their wonderful supports, especially Prof.Sci Nguyen Dinh Duc, Prof Kato, Prof Nagayama, Dr Phan Le Binh, Dr Nguyen Tien Dung and Mr Bui Hoang Tan Their encouragement and assistance has facilitated

me a lot during 2 years studying in the VietNam – Japan University I also want to give my special thanks to all lecturers and staffs at The University of Tokyo 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 and Structural Laboratory, University of Technology- VNU, especially for Mr Vu Dinh Quang, Mr Vu Minh Anh, Mr Pham Dinh Nguyen spending their precious time to point 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

II

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

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

IV

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 son nonlinear response of the FGM sandwich plate with porosity I under blast load 39

V

NOMENCLATURES AND ABBREVIATIONS

VI

ABSTRACT

The effects of porosity ratio on free vibration and nonlinear dynamic response

of FGM sandwich plates with two FGM face-sheets and a homogeneous core as ceramic resting on elastic foundations subjected to blast load are investigated in this thesis by implementing the third-order shear deformation theory Two types of porosity are proposed, namely evenly distributed porosity and unevenly 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-order Runge-Kutta method illustrating the significant effects of porous fractions, geometrical parameters, the elastic foundation, blast loads on the nonlinear dynamic response of FGM sandwich plates

Key words: Porosity, Functionally graded sandwich plate, Blast loading, The

third-order shear deformation theory

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1.1 Overview

1.1.1 Composite material – Functionally Graded Materials

Composite material is a material composed of two or more different types of component materials in order to achieve superior properties such as light weight, high stiffness 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 are extensively 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 contact stresses 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 ceramic and 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 the functional materials avoid the common disadvantages in composite types such as the detachment between layers material, fibers breakage and high stress in the surface, which can 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 coefficient K and the very low coefficient of thermal expansion

 , the ceramic composition makes the material highly variable with high hardness and very good heat resistance While the metal components make the modified materials more flexible, more durable and overcome the cracks that may occur due to the brittleness of ceramic materials when subjected to high temperature (Table 1.1)

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]:

V V : the volume fractions of metal and ceramic, respectively

N : the volume distribution (0N )

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 to thickness of structure and conforming to the Sigmoid-law function as:

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 to thickness of structure and conforming to the exponential-law function as:

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1, ln ,

 

b t

E is the elastic modulus of structure on the bottom z h/ 2

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

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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 extended Sigmoid distribution law as follows:

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 in terrorist activities, explosions The damage from such events cannot be determined, not just economically because many of these ones are symbolic and important heritage, significant architectures and the spirit of the times Nowadays, considerable efforts in architecture and structural engineering in recent years are often

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focused towards optimal design and economic efficiency in construction It is essential to guarantee the safe and secure protection of important infrastructure for the present and future

Explosion loads usually act in a very short time (usually in milliseconds) but transmit very high pressure pulses ( 1 3

10  10 kPa) As a result, damage to structural systems can take many forms, such as damage to the outer surface and structural frame of a building; collapse of walls and bearing columns; blow debris of concrete, glass windows and furniture; and damaging safety systems Most existing buildings are not designed to withstand such extreme dynamic loads, so a comprehensive understanding of the explosion phenomena and the dynamic response of structures is required to be essential for the scientific basis improving the design and material improvement in a feasible manner, in order to improve explosion resistance and ensure the safety of structures Therefore, the investigation of the effects of explosive loading on structures should be focused

1.2 Research objectives

The research objective of this thesis is to investigate the effects of porosity on vibration and nonlinear dynamic response of multi-layered FGM subjected to blast load

 Studies the effects of porosity to FGM sandwich plates and comparison with different 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 on elastic foundation subjected to blast load In numerical results, the effects of the material properties, 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:

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 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 from their 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 on elastic 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

et al [11] basing on a semi-analytical approach Hoang Van Tung [12] are analyzed nonlinear bending and post buckling behavior of FGM sandwich plates under thermomechanical loading by using the first order shear deformation theory The effect of time constant, temperature, mid radius to thickness ratio and time on transient thermo-elastic behavior of sandwich plate with the core as FGM are taken into consideration by Alibeigloo [13] In his analysis, the sandwich plate’s time dependent response 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- and large-amplitude vibration of compressive and thermal post-buckling sandwich plates with FGM face sheets under uniform and non-uniform temperature fields Behzad Mohammadzadeh [15] combined higher-order shear deformation with Hamilton’s principle to analyze nonlinear dynamic responses of sandwich plates with FGM faces on elastic foundation subjected to blast loads Basing on a new four-variable shear deformation plate theory, Mohammed Sobhy

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

2.2 Porosity

The effects of porosities generated during actual manufacturing process to the vibration characteristics of FGM structures have been studied by several author s However, the number of researches in terms of the mechanical behaviors of porous materials is still limited The most recent investigations on structures with porosity are listed in the following

Ashraf M.Zenkour [19] used a quasi-3D shear deformation theory to investigate the bending responses of porous functionally graded single-layered and multi-layered thick rectangular plates By taking Galerkin Vlasov's method into account in thermo-mechanical analysis of sandwich S-FGM plate with three different types of porosity for diverse boundary conditions, Singha and co-author [20] obtained the approach for bending and stress under the thermal environment They deduced that the deflection and stress escalate significantly for even porosity distribution (P-1) to bring into comparison with uneven symmetric (P-2) or uneven non-symmetric (P-3) porosity distribution; and the effects of temperature on transverse shear stresses of the multi-layered plates Mojahedin [21] employed higher order shear deformation theory to investigate the buckling of functionally graded 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 Ni substrates from Y2O3 stabilized ZrO2 (YSZ) and NiCoCrAlY powders In their

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research, they found that an escalation in porosity ratio of layers lead to the decrease

of residual stresses Chen et al [23] employed Chebyshev-Ritz method to analyze buckling and bending loads of a novel functionally graded porous plates Wang et al [24] focused on effects of parameters on the vibrations of functionally graded material rectangular plates with two types of porosity, namely, even and uneven distributed porosity, and transferring in thermal environment Based on a sinusoidal shear deformation theory in combination with the Rayleigh–Ritz method, Yuewe Wang et

al [25] depicted the effects of porosity, boundary conditions, and geometrical parameters on free vibration of the functionally graded porous cylindrical shell An isogeometric finite element model and the nonlocal elasticity were introduced by Phung-Van et al [26] to investigate the transient responses of functionally graded nanoplates with porosity Small size effects, nonlocal parameters, and porosity distributions, volume index, the characteristics of dynamic load have considerably influenced on the plate nonlinear transient deflections Cong et al [27] acquired closed-form expression in regard to critical bucking loads and post-buckling paths of

a porous functionally graded plates on elastic foundations subjected to the coupling

of mechanical and thermal loads by applying Reddy's higher-order shear deformation plate theory in conjunction Galerkin method Analytical solutions and numerical results revealed that porosity I (evenly distribution) behaves better than porosity 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 shear deformation into account to calculate the fundamental frequencies and nonlinear dynamic responses of porous functionally graded sandwich shells with double curvature under the influence of thermomechanical loads This study proved that porosities help the shell structures 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

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Australia, in May 2004 on the explosion-resistance of concrete-panel created by ultrahigh-strength concrete material A finite-element method was used to analyze concrete structures under blast and impact loading In the study conducted by Tin and co-authors [30] , they proposed using the explicit finite element software LS-DYNA

to induce stress wave propagation and the impacts on structural responses of precast concrete segmental columns subjected to simulated blast loads Balkan et al [31] examined the effects of sandwich stiffeners on the dynamic response of laminated composite plates under the non-uniform blast loading Moreover, the dynamic behavior of stiffened plates exposed to confined blast loads 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 to assess the effects of the degrees of confinement of the deformation to blast loads Asoylar et al [34] studied the transient stability analysis metal-fiber laminated composite plates under no-ideal explosion load by experiment and finite element methods In addition, Uybeyli and colleagues [35] used SiC reinforced functionally gradient material via powder metallurgy to investigate the impact of armor piercing projectile Bodaghi et al [36] studied non-linear active control of dynamic response of functionally graded beams with rectangular cross-section in thermal environments under blast loadings

Based on meticulous investigations in the available literature, it can be concluded that there are few free vibration and nonlinear dynamic behaviors of porous 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 This study has been implemented to meet the demand

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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

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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 The face-sheets and the core satisfy power-law distribution and the constituent volume fraction varying continuously along thickness direction The assumption of the metal and ceramic volume is written as:

2 ( ) |

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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:

The FG sandwich plate contains porosities in its structure, which can be dispersed evenly or unevenly along the plate thickness Two types of porosity are considered, namely evenly distribution (Porosity I) and unevenly distribution (Porosity II) as shown in Fig.3.3

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

The effective properties P z( , , ) T such as the elastic moduli E z( , , ) T , the mass density  ( , , )z T and the thermal expansion coefficient  ( , , )z T are defined

h z

h z

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Fig 3.4 Blast pressure function

where the "1.8" factor accounts for the effects of a hemispherical blast, Ps max is the maximum (or peak) static over-pressure, b is the parameter controlling the rate of wave amplitude decay and Ts is the parameter characterizing the duration of the blast pulse

3.2 Methodology

Natural frequencies and nonlinear dynamic responses of functionally graded sandwich plates with porosities under the influence of blast loadings are studied in this thesis The investigated FGM sandwich plate consists of two face-sheets and a core layer as FGM-ceramic-FGM which satisfy the continuity requirement of material properties The strain-displacement relations taking into account the Von Karman nonlinear terms and the higher order shear deformation theory deduce motion and geometric compatibility equations of the sandwich plate After introducing the Airy stress function, the number of primary variables diminishes from five to three This thesis utilizes the Bubnob-Galerkin procedure to solve the governing equation of the dynamic system The natural frequencies of the sandwich plate are analytically determined directly by taking the smallest value as solving eigenvalue problems Applying the fourth-order Runge-Kutta method obtains numerical results to illustrate the nonlinear dynamic behaviors of the porous multi-layered FGM plates exposed to the effects of diverse geometry configurations, the

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types of porosity, the volume fraction index

3.3 Theoretical formulation

Suppose that the FGM plate is subjected to blast loads The Reddy’s higher

order shear deformation theory (HSDT) in conjunction with the stress function are

used to establish the governing motion, compatibility equations and determine the

nonlinear dynamic response and vibration of the multi-layered FGM plate

The strain components at the distance from the mid – plane taking into

consideration von Karman nonlinear terms is the following [38] :

y x

where c14 / 3 ,h2  x, y are normal strains, xy is the in-plane shear strain, and  xz, yz

are the transverse shear deformations Also u v, , w are the displacement components

parallel to the coordinates ( , , )x y z , respectively, and  x, y are correspondingly the

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

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

presented by Hooke's law as follows:

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