An Analysis Of Buckling Behavior Of Multi-Cracked Functionally Graded Material Plates_TrinhDucTruong_2018_00051000345

An Analysis Of Buckling Behavior Of Multi-Cracked Functionally Graded Material Plates_TrinhDucTruong_2018_00051000345

VIETNAM NATIONAL UNIVERSITY, HANOI

V IETN A M JAPAN U NIV ERSITY

TR IN H DUC TRU O N G

AN ANALYSIS OF BUCKLING BEHAVIOR

OF MULTI-CRACKED FUNCTIONALLY

GRADED MATERIAL PLATES

M AJOR: INFRA STR UC TƯ RE E N G IN EE R IN G

R ESEA R C H SU PER V ISO R S:

Prof Dr Sci N G U Y E N D IN H DUC

Dr N G U YEN TIEN DUNG

ĐẠI HỌC QUỐC GIA HÀ NỘI ~ TRUNG TẰM THÔNG TIN THI r \'/K-M

Hanoi, 2018

A C K N O W L E D G E M E N T

First and foremost, I would like to express my heartfelt gratitude to my advisor, Proíessor Nguyen Dinh Duc, who accepted me with open arms as a member of the advanced materials and structures laboratory I was privileged and fortunate to be under his supervision I appreciate all his continuous support, great contributions of time, ideas, and marvelous guidance with the immense knowledge and thorough understanding This thesis cannot be accomplished without his great motivation since the íírst days I began working in the laboratory I will always remember his encouragement as well as his endurance for the countless íailures and mistakes, and guide me with the enormous patience and kindness

I would like to extend my gratitude to all the professors from the program ofInfrastructure Engineering: Proĩessor Hironori Kato, Professor Dao Nhu Mai, Professor Luong Xuan Binh, Doctor Phan Le Binh, Doctor Nguyen Tien Dung for their useíìil orientations and their willing advices during doing this research amidst their busy schedules This thesis cannot be fiilfĩlled without theừ valuable comments and great advices Importantly, I honestly appreciate to all the supports from Vietnam Japan ưniversity Indispensably, I would offer my most sincere thanks to Doctor Do Van Thom, Doctor Doan Hong Duc, Lecturer Pham Minh Phuc, not only for the invaluable help they provided in the developraent o f my research works, but also for their thoughtful discussions during the laboratory meetings, and supports throughout the most diffícult times of doing the thesis I also want give a thousand thanks to all the VJU friends for the happy time, companionship and encouragement Last but not least,

I would give my thankíiilness to my grandparents, parents, brother the support andbeloved attention, especiaỉly the understanding of my beloved wife

TABLE OF CONTENTS

ACKNOWLEDGEMENT I LIST OF FIGURES IV LIST OF TABLES VI NOMENCLATƯRES VII ABSTRACT IX

CHAPTER 1: INTRODƯCTĨON 1

1.1 Background and m otivation 1

1.2 Research objectives 5

1.3 The research scope and organization of contents 5

CHAPTER 2: LITERATURE REVIEW 8

2.1 Literature review on instability of FGM structures 8

2.2 Literature review on instability of FGM structure with fracture 9

CHAPTER 3: METHODOLOGY 13

3.1 Modelling o f íunctionally graded material plate structures 13

3.1.1 Symmetric íbrmula 13

3.1.2 Exponential method 14

3.1.3 Power law distribution approach 15

3.2 Constitutive íormulations for plate theory 17

3.2.1 Classic plate theory 17

3.2.2 The fìrst order shear deíormation theory 17

3.2.3 The new third order shear deformation theory 18

3.3 Phase íĩeld method in fracture mechanics 25

3.4 FreeFem++ numerical software 30

CHAPTER 4: RESƯLTS AND DISCƯSSION 31

4.1 Introduction 31

4.2 Veriíĩcation test and reliability level of the model 32

4.3 Analysis on critical load in buckling with two-equal-crack analysis 40

4.4 Analysis on critical load in buckling with three-equal-crack analysis 43

4.5 Analysis on critical load in buckling with three concentrated cracks at the center o f plate 47

4.6 Analysis on critical load in buckling in different boundary condition 51

4.7 Analysis on critical load in buckling with the change o f the size of plate 53

CHAPTER 5: CONCLUSIONS AND FƯRTHER W ORKS 55

5.1 Summary and conclusions 55

5.2 Limitation of the study and future w orks 57

PUBLIC ATIONS 58

REFERENCES 58

LIST OF FIGURES

Figure 1.1 Classiíĩcation method o f FGM [24] 2Figure 3.1 Geometry o f symmetric functionally graded material plate 13Figure 3.2 Geometry o f functionally graded material plate 15Figure 4.1 The reílnement o f mesh shape in triangle for square plate having a centercrack (4650 elements) 33Figure 4.2 The square geometry o f a Mindlin plate with a center crack having inclinedangle in Seiíĩ analysis 34Figure 4.3 The square geometry o f the FGM plate made b y A l/Z r02 with center crack

in Tiantang study 38Figure 4.4 Geometry model of functionally graded material square plate with twoequal cracks 40Figure 4.5 The analysis o f critical load in buckling varying o f material gradient indexand the crack’s length (Case o f 2-equal-crack) 40Figure 4.6 The analysis o f critical load in buckling when changing o f material gradientindex and distance among cracks (Case o f 2-equal-crack) 41Figure 4.7 Analysis on critical buckling load when changing o f crack length anddistance between cracks (Case o f 2-equal-crack) 42Figure 4.8 First five buckling modes of fully simple supported (SSSS) plate (2 equal

cracks, L - t ì = 0.2m, h = 0.002m, c / H = 0.5, d ì H = 0 3 ) 43

Figure 4.9 Geometry model o f ủinctionally graded material square plate with 3 equalcracks 44Figure 4.10 The analysis o f critical load in buckling varying o f gradient fraction indexand the length o f cracks (Case of 3-equal-cracks) 44Figure 4.11 The analysis o f critical load in buckling when changing the materialgradient index and distance among cracks (Case of 3-equal-crack) 45Figure 4.12 Analysis on critical buckling load when changing o f crack length anddistance among cracks (Case of 3-equal-crack) 46Figure 4.13 The ílrst buckling shaped modes of plate in simple-free boundary

condition (SFSF) (3 equal cracks,L = H = 0.2m,h = 0.002m,c / H = 0 5 ,d /H = 0.25)

46Figure 4.14 Geometry model o f functionally graded material square plate with 3 concentrated cracks at the center of p late 47

Figure 4.15 The analysis of critical load in buckling varying material gradient indexand the crack’s length (Case of 3 concentrated cracks at the plate center) 47Figure 4.16 The íĩrst buckling shaped modes of plate in simple-free boundary

condition (SFSF) (3 equal cracks, L = H = 0.2m, h - 0.002m, d H - 0.25, ớr:=45 ) 48

Figure 4.17 The íìrst buckling shaped modes o f plate in simple-free boundary

condition (SFSF) (3 equal cracks, L = H = 0.2m, h — 0.002m, c / H = 0.25, a =45 ) „50

Figure 4.18 Geometry model o f íìinctionally graded material rectangular plate with 2equal cracks 53

F igure 4.19 The analysis o f critical load in buckling with different ratio of plate’s sizesand the variation o f material gradient index 53

F igure 4.20 The ílrst buckling shaped modes o f plate in íìilly clamped boundary

condition (CCCC) (2 equal cracks,L = H = 0.2m, h - 0.002w, c / H = 0 4 ) 54

LIST OF TABLES

Table 4.1 Material properties of analyzed FGM in the study 32Table 4.2 The comparison table on criticaỉ load o f buckling of a square plate having the thickness ratio as h/H=0.01 and a crack at the center of plate as c/H=0.2 with theboundary condition o f fully simply supported 34Table 4.3 The comparison table on critical load o f buckling of a square plate having the thickness ratio as h/H=0.01 with the fully simply supported boundary condition(No crack effect) 34Table 4.4 Comparison o f critical buckling load o f square plate with exact and ref solutions having thickness ratio h/H=0.01 with fully simply supported boundarycondition (No crack effect) 34Table 4.5 The buckling comparison in critical load o f a Mindlin plate considered theeffect o f the crack length and crack angle 35Table 4.6 CBTR comparison of the square A l/Z r02 plate with the consideration ofcrack angle and volume ữaction exponent 39Table 4.7 A computation o f critical buckỉing load with changing crack length and

inclined cracking angle in the case o f three-crack concentrated at the Central geometry

: 49Table 4.8 Analysis on critical load o f buckling o f FGM square plate considered the

different boundary conditions (2 equal cracks, h í H = 0.01 , c / H = 0 5 , d / H = 0.25)

and varying the material gradient index 51Table 4.9 Analysis on critical load o f buckling o f FGM plate considered the difference

o f boundary conditions (3 equal cracks, h l H = 0.01 , c / H = 0.5 , d / H = 0.25 ) and

changing the material gradient index 51Table 4.10 Analysis on critical load of buckling of FGM plate with three concentrated

cracks at the center of plate considered the varying boundary conditions ( h ỉ H = 0.01,

c / H = 0.5, Ếí=45) with changing volume fraction index 52

N O M E N C L A T U R E S A ND A B B R E V IA T IO N S

H , L The height and length dimension of the FGM plate

Vnwlal, Vceramic The volume íraction o f metal and ceramic material

n The gradient volume íraction index (Material gradient index)

u, V, vv The displacements in the X, y and z directions

u0, v0, w0 The displacements in the X, y and z directions

Px,Py The transverse normal rotation o f the y axis and X axis

D h, Ds The bending and shear stiffiiess matrix

N,M, P,Q , R The normal, shear forces, bending and higher-order moments

Gc The rate of energy release in Griffíth’s theory

kc{x,y) The nearest distance from an arbitrary point (x,y) to the line /

r ( ỏ ) The energy o f the crack region

kc The non-dimensionalized value of the critical buckling load

A BSTR A C T

The research íocuses mainly on the investigations o f the buckling behavior of the plate structure vvhich is made by functionally graded material (FGM) included the effect of multiple fractures as cracks by numerical simulation based on íinite element method The numerical computation for crackũig problems is developed by a new approach in ữacture mechanics named phase fíeld theory The constitutive íòrmulation o f FGM plate is derived from a new plate theory in third order shear deíormation (TSDT), while the analyzed FGM plate is combined from metal and ceramic materials The material properties are graded in the thickness direction o f the FGM plate according to a simple power law distribution In the discussion part, the analysis is implemented in order to gain the well understanding about how the critical buckling load is affected when having the change on the characteristics of cracks such as the size, the declination angle or the shape, also the effects of different boundary conditions and so on to the stability o f plate are discussed Then, the buckling behavior considered by buckling mode shape will be presented by some visual coníìgurations It is observed from the calculated results that the cracks have a great impact to the instability o f FGM plate, not only the considerations involved to the crack parameters such as the length, the numbers o f cracks and crack shape, but also the ratio o f plate size and boundary conditions effects sigĩiiílcantly to the buckling behavior of FGM plate

Key words: Buckling analysis, new TSDT, phase-field method, multi-cracks, FGM plate

C H A PT E R 1: IN T R O D U C T IO N

1.1 Background and motỉvation

Advanced materials has been researching and developing for long times, by combining the different mixtures o f metals and non-metals, these new advanced material kinds called composites could make the strengths frora the material compositions for speciíĩc íunctional requirements Composites could be considered as the most advanced mode

of materials, composites are made from two or more material components with the complete difference in material properties compared to the individual materials Although composite materials could provide many advantages such as greater strength with stiffness ratio, increasing the wear resistance, corrosion, and fatigue, high reliability over the pure individual material However, at the hard working conditions, the normal composites as the laminated composite consisting several individual layers, the material properties between the layers will be changed suddenly, the sharp change

of material properties along the interíace o f composites could result in failure as the delamination betvveen material layers, and the structure could be suffered the premature failures, these problems are the great drawbacks o f composite This leads to the great demanding o f eliminating the drawbacks o f conventional composites, and develops advanced material by modifíed a new composite form called íunctionally graded materials (FGM) These materials have ability to replaces the sharp interface with a material gradient index which gives a smooth transition o f material properties from one material to the other

In FGM, the material properties controlled of two or more than two material compositions could be varied gradually and continuously over the volume íraction obeying a speciíĩc rule or íunction of the in-plane geometric dimensions such as along the thickness o f structures Due to the volume fraction o f material properties in FGM

which is varied intentionally, it is reasonable to design, control and optimize the characteristics of the new materials to enhance the períormance of some target engineering applications Although there are many types of FGM in reality, Bhavar et

aì [5] has proposed a method to classify the types o f FGM based on two criteria such

that the structure o f material and the size of FGM structure

Figure 1.1 ClassịỊìcation method ofF G M [24]

As shown in íĩgure 1.1, the continuous FGM type belonged to the FGM structure based group, this type will use a continuous gradient to express the material properties from one material to the other; while in the discontinuous FGM type, the gradient material index is set from layer to layer On the other hand, based on the size o f material structure, FGM is classiíĩed to be thin and bulk FGM structure The FGM having a relatively thin section in the interface like a separate suríace o f the structure is the thin FGM, while the bulk FGM concludes a complete volume of materials Each types of FGM structure requires high innovative technology in manufacturing processes, some

of them now are only theoretically, but several techniques are already available and applicable in reality to produce the íunctionally graded material such as physical or chemical vapor deposition to manufacture thin FGMs only, powder metallurgy technology is used to produce bulk discontinuous FGM structure On the other side,

centrifugal method is applied for manuíầcture the continuously structure bulk FGM, ceramic-metal FGM plate is one of the FGM types which is manutầctured by this method Finally, solid free form fabrication or additive manufacturing technology is the most complicated method for FGM manufacture, this method is also known as 3D printing method, and generally applied for very complicated FGM bulk components This study does not concentrate on the methods o f manufacturing process o f FGM structure.

Though manuíacturing technology in FGM has many difficulties and requires very modem innovative technologies, several types o f the FGMs have been produced successíully based on the appropriate manufacturing processes, among them, ceramic- metal FGM seems to be one the o f most popular types which is used widely in various engineering application, it has been being applicable in the very difficult working conditions For instance, the applications o f FGM are focused on aerospace, nuclear industry or electronics sectors The FGM turbine blade was made by metal-ceramic FGM to achieve the thermal resistance at high temperatures, as well as own extremely high strength and toughness Furthermore, the properties o f ceramics also help diminish the interface defects [8] Another problem in aircraít solved by ceramic-metal FGM is that the shuttles normally used ceramic tiles to protect the high temperature from heat generation, however, the ceramic tiles are likely to crack because o f the diíĩerences in thermal expansion coefficient, using the ceramic-metal FGM tiles will provide better characteristics in both thermal protection and load carrying ability The research in this thesis will focus on the analysis o f ceramic-metal FGM type in plate structure for the consideration o f buckling phenomenon

Plates are one o f the most important parts in engineering structures and applicable widely in many engineering íĩelds, from mechanical, civil to aeronautical engineering applications Generally, plate structures are subjected in different load such as

mechanical loads, or thermal loads with various differences of directions applying, so

a t íailure S ta te , the structures could be prone to fail at different modes It could be said

that one of the most important and common modes of failure in plate structure is buckling phenomenon Once plate member is loaded to an axial compression forces which is bigger than the critical load for that member, at the same time, the plate stores

a great compressive strain energy, before reach to the yield stress region, the structure will easily be failed suddenly, and buckling happens, the most dangerous point is that it happens un-expectable, the structure form will disproportionate greatly in deíormation behavior, obviously, changing form one confíguration shape to another can cause unexpected deformations and the structure will be unusable for its original designed

purpose since the loading capacity is decreased at this State The buckling behaviors of

plate structures are highly sensitive in failure modes which could be called as imperíection, for FGM plate, it is clearly that the buckling State becomes much more complicated Not only the change o f material properties, the variation of thickness, but also the loading conditions could affect the buckling phenomenon o f structures

When applying FGM in extremely hard working environments, there might be appeared of failures such cracks These deíects could affect much to the hardness, toughness and durability o f the plate These kinds o f defects indeed weaken signiíĩcantly the hardness, the toughness, and the durability o f the structure Once the structure is taken under compressive load, it will be rauch easier to make to cracked-

p late b e in in stab ility State rather than the p late do n ot b e aíTected b y cracks, ev en that

the considerations of the size, position, or number o f cracks are accounted into the instability analysis o f plate structures It is clearly that the buckling behavior in FGM plate included the consideration of cracking eíĩect which the crack is through the plate thickness will become much more seriously

1.2 Research objectives

The íìnal goal of this research is to investigate the buckỉing behavior of the ceramic- metal functionally graded material plate with the effect of multiple intemal cracks by numerical simulation based on the new shear deíbrmation plate theory in third order coupling with phase field method along the framework of íĩnite element method This

To consider the stability of plate, the calculated formula o f buckling load at critical point will be used to analyze the effect o f crack characteristics Thereíore, in order to have the fmal achievement, the following targets have been set as the upgrade level in this Master thesis:

❖ Modelling the functionally graded material plate by deriving the constitutive equations for plate using the very new shear deformation plate theory in third order which is proposed by Shi [47] while the model of FGM structure is chosen appropriately due to the complexity o f computation The buckling behavior with the effect of intemal crack could be íĩgured out by the total potential energy of the plate by fmding the crack shape and phase fĩeld variable

❖ Developing the constitutive model with the consideration o f several cracking characteristics as well as the boundary conditions, then analyzing and evaluating the buckling behaviors o f FGM plate through the critical buckling load

1.3 The research scope and organizatỉon of contents

This thesis is focusing on the investigation o f linear problem on the critical buckling load and its effects to the buckling behaviors o f the íìinctionally graded material plate with multiple cracking The numerical simulation for ceramic-metal FGM plate is proposed by the combination o f íinite element method along with the derived constitutive model based on the new third order shear deformation theory while the intemal fractures are modeled based on the phase íĩeld method The development of

modelling and discussing the buckling behaviors o f ceramic-metal PGM plate will be explained in detail in this thesis following structures as shown below:

> Chapter 1: Introduction

In this chapter, the research backgrounds, as well as the motivation of the research will be presented An introduction o f íìinctionally graded material, their types and engineering applications will be shortly described

> Chapter 2: Literature review

Several literature reviews that related to the topic are introduceđ in in chapter 2 They also show what previous research has done, and the limitation o f those studies The objective of this thesis is outlined Also, an explanation for the reason why it is important to investigate the critical buckling behaviors of FGM plate with multiple íractures will be shown in this section

> Chapter 3: Methodology

In this chapter, the method in development o f the numerical simulation model will

be described The model was based on the new shear deíòrmation theory in third order which is proposed by Shi [47] There is a comparison o f the íĩrst order shear deformation theory and the third order shear deformation theory in modelling numerically the structure o f plate The introduction o f phase íĩeld method with its advantages in modelling the multiple cracks, and the application of phase íìeld method into the constitutive equations in buckling problem will be presented

> Chapter 4: Numerical results and discussion

Chapter 4 shows the numerical results of the ceramic-metal functionally graded material with and without a single intemal crack to veriíy the constative models that will be used in the simulation, the compared results are taken from the references

[4,54] Results and discussions of simulations of the FGM plate with 2 equidistant cracks, 3 equidistant cracks as well as 3 cracks cutting each other at the Central of plate, and other considerations such as boundary conditions, plate ratios will be shovvn in this chapter Furthermore, several modes of buckling forms o f failure criterion o f multi-crack FGM plate will be also revealed in this chapter.

> Chapter 5: Conclusions

In this fínal chapter, some remarkable points about the buckling analysis of the multiple cracks in FGM plate are emphasized Also, some limitation during doing this research will be ílgured out, and then some commentaries for íìưther research are proposed

C H A PT E R 2: LIT E R A TU R E R EV IE W

2.1 Literature review on instabiỉỉty of FGM structures

There are huge numbers o f studies in buckling of plate structures In Yang study [60],

an analysis on buckling is done for FGM plate setting up on the elastic foundation which is modeled based on the formulation o f Pastemak In the research o f Thai [51], the principle o f Hamilton is applied to derive the constitutive equations for buckling behaviors of the plate which is set on Pastemak foundation After that, Thai and Kim [50], with the same condition for plate resting on the foundation o f Pastemak, an investigation for buckling of FGM plate are presented as a closed-form solution Numerical method is also used comprehensively in the study o f Praveen and Reddy [36], on the basement o f the shear deíbrmation plate theory in íírst order (FSDT), the mechanical behaviors o f nonlinear statics and dynamics on the ceramic-metal FGM plate are analyzed by finite element method under a steady temperature and dynamics transverse ỉoad conditions In the study o f Duc and Cong [13], the nonlinear post- buckling is investigated analytically for the imperíect eccentric thin FGM plate which

is stiíTened under the temperature load, the plate is rested on elastic foundation which

is modelled by Pastemak type using the Galerkin method Especially, Reddy [38] has studied the buckling phenomenon of FGM plate with the boundary condition as simple supported using the shear deíòrmation plate theory in higher order This study has avoided the consideration of the zero transverse shear components at the interface o f plate

In ceramic-metal FGM, the structure combines both material properties o f material components which are here ceramics and metals, they are inhomogeneous, FGM ’s material properties change smoothly and continuously in one or many directions The excellent characteristics o f ceramic material have the capability to against the high

temperature, scratch and corrosion whereas the good features at the toughness, the durability and hardiness of metal material could attract the energy and the plasíic deílection It can be said that the ceramic-metal FGM have extensively abilities to against extreme high temperature gradient, maintain the structure integrity, and reject the interface problems, as well as thermal stress concentration [59] Yu et al [61] used

a moditìed Mori-Tanaka and a self-consistent method to study the material properties

by modelling the FGMs, Mirzavand et al [26] studied the thermal buckling o f FGM plate that combined with piezoelectric actuators bonded in interface or in the research

of Nemat-Alla [31], a 2D-FGM was developed to achieve a better reduction o f the thermal stress Thom et al [53] investigated the buckling and bending behavior o f a bi- directional FGM plate using a new plate theory with no shear correction factor Additionally, a new solution for thermal buckling o f FGM plate was proposed by Shariat and Eslami [45] in the closed-form that they have used the shear deformation plate theory in fìrst order to investigate a thick plate, being similar in using the first order plate theory, Golmakani et al [15] has studied the deformation in circular mode

o f FGM plate with the applying load as thermal loading which the properties are depended on the temperature Besides that, the critical load o f buckling with the effect

o f temperature rise o f a FGM in circular form which is formulated according to the third order plate theory in shear deíbrmation has analyzed by Najafizadeh and Haydari [29] Then in Talha et al [49] research, the behavior and vibration in static State o f a plate made by FGM have been presented with using high order plate theory while the boundary conditions are varied

2.2 Literature review on ỉnstabilỉty of FGM structure with fracture

Though plates made by FGM provide many advantages with higher strength, toughness, vvorking in high temperature, and FGM has been applying in engineering applications for decades, the unđerstanding of buckling behaviors of FGM with the considerations

of other factors such as defects, fracture is still not researched comprehensively, even though íracture also has an importantly significant effect to the instability of the plate structure, for FGM plate, the effect of fracture such as crack is becoming more unpredictable to the buckling behavior of the structure Furthermore, treating the crack problems in numerically computing problem always meets many difficulties Thereíore, study about the instability o f FGM plate accounted the effects o f numerous intemal cracks is dispensable in real engineering applications [14, 17] Noda et al [33] also considered a single surface crack on FGM plate to analyze the thermal shock effect, Nemat-Alla et al [32] included the edge crack effect in the model o f semi-infinite FGM plate with a bi-direction coeffícient o f thermal expansion when applying the thermal loading in 2-dimention The topic is not only about the material structure, but also the methods and solutions to deal with the diíTiculties o f interaal cracks As a remarkable methođ in mechanics o f fracture, one of the most popular methods used to analyze the íracture as cracks is extended íĩnite element method (XFEM), as an example, Dolbow et al [9] has used XFEM to analyze a combination o f the deformation modes o f the plate obeying Mindlin-Reissner theory, and then íĩgured out the serious effect of the stress concentration in plates having crack Continuously, Shaterzadeh [46] made the investigation on a composite plate having a circular cutout with consideration many different characteristic factors such as boundary conditions, crack size, and material gradient index by applying FEM to calculate the criticai load

o f buckling included the effect o f temperature rise Besides that, by using XFEM and SFEM, Baiz and Natarajan [4] have attempted a linear analysis on buckling for plate included effects of ữacture based on the Mindlin-Reissner plate theory Similarly by using the application o f XFEM and íĩrst order shear deformation plate theory, Baiz and Natarajan [30] also investigated the effect o f cracking geometry to the buckling behavior in FGM plate As a noticeable study, Tiantang [54] already clariíĩed the effect

o f thermal buckling of the cracked FGM in plate structure using extended isogeometric method.

Recently in mechanics o f ữacture, phase íĩelđ method as a very new numerical method have got many concentrations from researchers and experts because of its great advantages in modelling in both static and dynamic analysis, calculating and representing the cracking geometry with high accuracy [2, 6, 7, 22] For example, as a great remark in strong advantage points o f phase íĩeld method, a numericaỉ simulation

o f íĩnite strain plates in bending problems having the effect o f íracture which is modelling by phase fĩeld method was studied by Areias and Rabczuk [3]; or Amiri [1] expressed the advantages in modeling the intemal crack by phase íĩeld approach as compared to XFEM, that a crack model for thin Shell has been built and not required tracking of the crack paths Also, Phuc et al [35] has used phase fíeld method to express the effect of crack in buckling o f plate having a linear change in thickness While classical FEM or extended FEM treat the multiple crack problera discretely, handle the crack one by one independently, and the discontinuities are not accounted into the displacement field as well as geometrically expressed in the model, in phase fĩeld method, the displacement ííeld is govemed continuously by partial diíĩerential equations, and the discontinuities such cracks could be modelled with various complex geometries and their evolution based on a diffiisive crack approach [3] There might have been several studies in buckling problem looking into normal materials of which material properties are constant; however, analyses in buckling problems for FGM plate with multiple cracks in which the material properties vary according to the thickness of materials are still very few Thus, study and understand thoroughly the buckling behavior in FGM plate with multiple cracks using phase field method is useful and necessary Over more, once the constitutive formulated is succeed, the framework o f numerical method has been developed not only for íìinctionally graded material but also other materials In this research, a new plate theory named the new

shear deíbrmation plate theory in third order to modelling the ceramic-metal FGM plate coupling with phase fíeỉd method in treating the effect of intemal cracks will be used to formulate the constitutive equation and simulate the buckling behavior, and then the buckling behavior o f multi-cracked FGM pỉate will be explored signiíĩcantly

to have a well understand about the effect of cracks to the instability o f the plate

C H A PT E R 3: M E TH O D O L O G Y

3.1 Modelling of ĩunctỉonally graded materỉal plate structures

The most important characteristic point in modeling íiinctionally graded material (FGM) plate is how describes the material properties that are actually the combination

o f two independent materials, which is changed smoothly along the thickness from one

to another interíace, as mentioned in the introduction section, the composition materials in this research are metal and ceramic material There have been several theories proposed for modelling o f FGM plate structures However, based on the structure and the complexity level o f computation, there are three method to model the material properties ofFG M íbrplate structure

Figure 3.1 Geometry o f sỵmmetric/unctionally graded materỉal plate

The symmetric method is applied in the case that the FGM plate structure is symmetrical only, as shown geometrically in íígure 3.1 In order to make the FGM plate model be easier for íbrmulating from the constitutive equations, the xy-plane is set in the middle plane, and the positive direction o f z-axis is upward from the xy-plane The volume fractions are assumed to change through the plate thickness, and considered as a distribution obeying the sigmoid law

h/2

ĩ

h/2

Vme ,Ả Z ) = V ceramĩc.(z) = i - r , ( z )v ' metal V / (1)

Where n is the volume íraction index with 0 < n < 00, h is the thickness of the plate, and z is the coordinate defíning the thickness of plate w ith- h / 2 < z < h / 2 A parameter

Pris used to deíĩne the material properties o f the FGM structure according to the linear rule of combination as

The prparameter stands for the two most important material properties that are the

elastic modulus E and Poisson ratio V Substituting Eq (1) into (2), the speciíĩc

expressions o f material properties could be obtained as

It is clearly to observe that the top suríiace and the bottom surface of the symmetric FGM plate are rich o f ceramic, oppositely, tíie middle suríầce is metal rich

Ceramic surĩace

h/2

Mid-plane

h/2

Fỉgure 3.2 Geometrv o f ịunctionally graded material plate

The material properties o f FGM structure are modeled as a distribution following the exponential law, using the volume ratios o f constituent for modifying the smooth changing through the thickness of plate as the íormula (1)

In which A - E , , B = —ln

V top

Eu is stand for the elastic modulus o f the top surface ( z = - h / 2 ) o f the FGM plate,

while describes the elastic modulus o f the bottom surface ( z = h / 2) of the FGM plate, h is used to detĩne the thickness o f the pỉate.

3.1.3 Power law distribution approach

This method is applied for the same FGM structure as previous part described in íĩgure3.2 However, the distribution in power law will give great advantage as avoiding the

natural logarithm mathematical constant e , then decreases much the level of

computation, but still provide the accuracy o f the needed calculation In this thesis, the research will apply this power law distribution approach to model the material characteristics o f FGM plate The model is still supposing that the top surface is íìilly ceramic while the bottom suríiace is totally composed by metallic material The material properties of FGM plate vvhich is varying following the thickness o f the plate

( 4 )

are computed with the volume ratio index ( n ) Hence, the volume íractions of the

metal as well as the ceramic material are obeyed the power law rule [12, 39, 54]

:(z) = - Ị- —-z 1

h 2

The n is considered as the non-negative variable which stands for the volume ratio

gradient index to deíĩne the volume of each material property component z is used to

determine the thickness o f plate, - h / 2 < z < h / 2 Then to deíĩne the effectiveness P ro f

material properties o f the FGM structure, a formula o f Pr is íòrmed by the linear rule

of combination as follows

Where Prparameter denotes a material properties, here two most important material

properties that are the elastic modulus E and Poisson ratio V Substituting Eq (6) into

(7), the material properties could be expressed in speciíĩc as

^2 z + h Ỵ

pw ( z) = p r^ + (pr™ c- pr^ (1/)

V 2 h

(8)

Or the Young’s elastic modulus E , the Poisson’s ratio V will become

Eplate(z ) ~ Emetal+ (Eceramic Emetal)ị^

v plale( z ) = y metal + {Vceramic - ^metaỉ) ) ( 1 °)

According to the Eq (8), it is evident that the upper surface and the lower suríace of the FGM plate are rich of ceramic and metal material, respectively For more

convenient in developing the constitutive equation, from now on, the subscripts m and

c will be used in the related equations which stand for the metal and ceramic

constituents, respectively

3.2 Constitutive íormulations for plate theory

3.2.1 Classic plaíe theory

The classic plate theory proposed by Kirchhoff [19, 20] is very well known in engineering mechanics which is basically based on the Kirchhoff hypothesis theory, the theory assumes that the straight lines which is perpendicular to the under-formed mid-plane will keep being straight and be perpendicular to the deíormed plane, and do not undertake the stretch of the thickness o f plate In another word, in the kinematic assumptions, the transverse shear and transverse normal strains are assumed to 0 Hence, the displacement íĩeld is as follows

3.2.2 The first order shear deformation theory

Shear deformation ứieory has been developed with including the transverse shear deíbrmation, and this theory named as the shear deíòrmation theory in íĩrst order

17

ĐAI HỌC QUỐC GIA HÀ NÒI TRUNG TẦM THÔNG TIN THI r VIĨ-M

developed by Mindlin [25] could be considered as the simplest plate theory included the transverse shear strain components in the model, it can be said that the Mindlin plate theory is a generalized model of Kirchhoff plate theory In this theory, the assumption is set through the thickness o f the plate for the linear distribution of in- plane displacement, due to this base, the constitutive equations are derived Hence, the shear stress in the plane suríace cannot be 0, but these transverse shear stresses will become a constant across the thickness o f plate Nonetheless, this leads to a problem of using the shear coưection factors, and the problem is becoming very complicated because o f the dependence o f shear coưection factor with how to choose a shear correction factor accurately for the model.

The displacement fíeld o f the íĩrst order shear deíbrmation theory is described as follows

u = u0(x, y) + zj3x(x ,y )

w = w ữ(x,y)

With u , V and w are the displacement components in the X, y and z axes o f the

Cartesian coordinate system M0,v0 and vv0are the unknown displacement o f middle

plane, while /3 X and Py stand for the transverse normal rotation o f the y axis and X axis,

respectively

3.2.3 The new third order shear deformation theory

It can be seen that the constraint o f the classical plate theory is the neglect o f the transverse shear strain components whereas the limitation in the íĩrst order shear deformation theory is the dependence o f the shear correction element To avoid the use

o f the shear correction factor, a development o f plate theory is compelled into the higher order shear deíbrmation theory or an equivalent plate theory to have appropriate

distribution such as the realistic varying of transverse shear deíormations as well as

stresses across the thickness o f the plate, and in fact, these such high order shear

deformation theories have been shown that they could be applicable with high effectiveness to the composite material plates

Murthy [28] has improved and developed the plate theory into the third order polynomials which can describe the in-plane deíormation through the thickness with no use o f shear correction factors Nonetheless, this theory o f Murthy is based on the equilibrium equations from the classic plate theory which is differently conílict with the kinematic law o f displacements In order to improve this problem, Reddy [40, 41] has developed the consistent equilibrium equations by a variational technique, the numerical analysis in the accuracy o f several high order plate theory studied comprehensively later on by Rohwer [42] has shown that the plate theory proposed by Reddy is one of the bests for high order shear deformation, until now, the high order plate theory o f Reddy is still considered as the most popular high order shear deformation theory for the composite structure plate analysis Nevertheless, Shi [47] has pointed out that the boundary conditions at every edge o f the plate in the high order plate theory o f Reddy are inconsistent with the 101*1 order differential equations Therefore, it is necessary to develop an improved new higher order shear deíòrmation theory that is based on a rigorous and accurate constitutive kinematics, and has a variational consistent diíĩerential equations combined with a system o f boundary conditions being rationally steady with the differential equations

Based on the works on the development o f kinematic relations with high order displacement theory from the elastic theory o f Voyiadjis and Shi [58], Shi [47] has proposed a new third order plate theory o f shear deformation for plate structure This theory recently gets many concentrations from researchers due to its advantages as pointed out by Shi that by included the appropriate constitutive relations for the

equivalent rigidity computation o f composite material plate, it is simply to applicable

to analyze the composite plate with various layers [41], the nonlinear investigation can

be easily achieve geometrically by combining the Von Karman nonlinear strain components [40], additionally, the theory coulđ be covered for dynamic problems of composite material by applying the principle o f Hamilton [40], and the shear locking eíĩect could be avoided in the analysis o f the ílexible plate problem with a corresponding assumption of strain approach [48] Thereíòre, it could say that this new shear deformation plate theory in third order proposed by Shi [47] provides a higher order numerical method for plate with high accuracy and reliability compared to other high order shear deíbrmation theories due to a more rigorous form of the kinematic equations of displacement which is formed based on the elasticity theory, not írom the hypothesis o f displacements, and accounts both a consistent system o f differential equations and also boundary conditions In this research, the new third order shear deíòrmation theory of Shi is applied for the static analysis o f íìinctionally graded material plate

The three dimentional (3D) displacement fíeỉd contains u , V and w at a speciílc point ( X , y , z ) in the plate could be expressed by 5 unknown variables as:

w ( x , y , z ) = wữ( x , y )

Where M0, v0 and w0 determine the displacements at the middle plane of the analyzed

plate according to the x , y a n d zaxes respectively Adđitionally, p and p, means the

transverse normal rotations around the y axis and JC asix, on the other hand, the

diíTerentiation is described as the comma in the goveming equations with respect to the

5

( 1 3 )

jcand y coordinates As the considering assumptions are very small strain, the strain-

displacement relationship can be described in equations as follows

Where Dh(z)is the bending stiffness matrix, and D (z) is the shear stiffness matrix

changing across the thickness of plate, respectively It is important to notice that equation (18) symbolizes £(0);£(I);c(3);ỵ (0);y (2) for the strain and shear components which are shorten from equations (14) o f displacements in the plate [10, 52, 56]

The strain can be reíormed to be:

The normal and shear forces, bending moments, as well as higher order moments can

be calculated in detail as bellows

Where ô the displacement vector and decides the mode shape of buckling behavior in the model.

3.3 Phase íleld method in íracture mechanics

Fracture problems such as cracking ha ve an indispensable role in engineering desigrt, analysis, and application, so it is very important to have a usetiiỉ tool to achieve a well understanding in fracture investigations The development o f numerical engineering theories along with the enhancing of Computer power, modelling through simulation method is considered as a great solution for analyzing the mechanical behaviors of engineering structures and its problems, including fracture mechanics Many ửacture problems have been modeiled by íĩnite element method based on various techniques to treating the cracks Among the techniques for handling the cracks, Griffith’s theory of elastic brittle fracture is one o f the most popular techniques that the model is based on the energy release rate, it means that the crack could happen and propagate only if the energy release rate reaches to the íracture field at the critical point Additionally, the approaches by fínite element method have been used extensively, most o f the models

using íĩnite element method are so fundamental with the Virtual crack closure

technique [21], or as a remarkable method in modelling fracture, the extended ílnite element raethod which is introduced by Moes [27] has also got many íocuses ííom researchers Nevertheless, these kinds o f method still deal with cracking problems in which the cracks are represented discretely and discontinuously, and generally require

an explicit and rigorous technique to track the crack path, or need to have applying the strategy o f taking average of re-meshing model, in another word, the discontinuities due to the cracking defects are not described well in the model

It can be said that the discontinuities o f sharp crack in modelling the íracture problem bring an extremely demanding, especially to the cases o f complex crack geometries Diíĩiisive crack modelling which is based on phase íĩeld method can overcome this

drawback Recently, a new approach named phase fíeld method got much attention because of its great advantages in treating fracture problems The phase íĩeld method is applied vvidely in engineering structures for a diverse physic model [16, 43] In the phase íĩeld method, the discontinuities like cracks, fractures will not be included in the displacement goveming relations, instead that, the displacement íĩeld is still modeled continuously by partial diíĩerential equation while the phase íĩeld method begins with sharp interfaces as a smaỉl region in crack boundary using a continuous variable named the phase íĩeld order variable, this variable is under control to differentiate inside the phases o f materiaỉ structures across a smooth evolution, then approximates the discontinuous suríace where crack is happened The mainly strong point of using the phase íield order parameter is that the evolution of cracking region obeys the solution

o f the coupling system o f partial differential equations, thereíore, it decreases much the complexity of building the tracking algorithm for cracking path Additionally, in the crack propagating problem, the stress intensity factor is not necessary to be included, and cracking path tracking is done automatically due to the smooth cracking ííeld variable in a fíxed mesh, this brings a better choice compared to the classic finite element method that require explicit method or implicit in the extended íĩnite element method to treat cracks [34] The phase íĩeld method naturally owns the abílity to deal with crack forming, crack propagation, crack merging with a simple implementation, especially through the partial differential equations, it could handle on various complicated cracking geometry

In the phase íield method, consider s to be the phase íĩeld order parameter, or cracking

field variable which is introduced into the partial diữerential equation to denote the

cracks in the model, parameter s will converge to 0 if the fĩeld model reaches to a

cracking region (totally damaged), and to 1 if the model is away from the discrete areas (unbroken material region) Hence, the fíeld order parameter sis controlled only on the

interval [0,1], and varied smoothly in this set If the variable s has the values between the interval(0,l), it means that the model is in the soíìening process The total potential energy now can be expressed as the partial diíĩerential equation included the phase íleld order variable as the 52 function, so the strain energy o f cracking region in the model will be converged to 0 The total potential energy of FGM plate with the in- plane stress could be íiormed as follows:

In the equation (33), Gc stands for the rate o f energy release in Griffith’s theory, or the

critical fracture energy density taking the value o f Gc = ( 5 ỉ 2 / 2 1 ) ( ơc)2(l/E) [6] is

the total surface analyzing area of the FGM plate, Vs is the spatial gradient, / is used

to control and adjust the width o f the cracking region and being a positive