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Fracture Toughness of Cellular Materials Using Finite Element Based Micromechanics

Permanent Link: http://ufdc.ufl.edu/UFE0021682/00001

Material Information

Title: Fracture Toughness of Cellular Materials Using Finite Element Based Micromechanics
Physical Description: 1 online resource (94 p.)
Language: english
Creator: Wang, Junqiang
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: cellular, finite, foam, fracture, functionally, micromechanics, orthotropic, rectangular, tetrakaidecahedron
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: A finite element method based micromechanical analysis is used to understand the fracture behavior of homogeneous and functionally graded foams. Both rectangular prism and tetrakaidecahedral unit cells are studied. Two approaches of predicting fracture toughness of foams and other cellular materials are used in this study. In one approach, the finite element analysis uses a micromechanical model in conjunction with a macro-mechanical model in order to relate the stress intensity factor to the stresses in the struts of the foam. The stress intensity factor at the crack tip of the macro-mechanical model can be evaluated using either the J-contour integral or the stresses in the singularity-dominated zone. The other approach is to directly apply displacements based on the K-field on the boundary of the micromechanical model. Using the first approach, the mode I fracture toughness is evaluated for various crack positions and length. Both homogeneous foam and graded foam are studied to investigate the effect of stress gradients in the vicinity of the crack-tip on the fracture toughness. Various types of loading such as remotely applied displacements, remote traction are studied. Preliminary results of this study show that the stress gradient has slight effects on the fracture toughness. However, since the effects are relatively small, KIc can be defined as a material property. Then the relationship between the fracture toughness of the graded foam and the local density at the crack tip is studied. The second approach is easy to apply in predicting the fracture toughness of homogeneous foam. By using this approach, convergence study of a micromechanical model is conducted. Also, an analytical model for the mode I fracture toughness of foams with rectangular prism cells is introduced. The mode I and mode II fracture toughness of homogeneous foam consisting of tetrakaidecahedral unit cells are predicted. A parametric study is performed to understand the effect of the geometric parameters of the unit cell and tensile strength of the foam ligament and also dislocation imperfection in the foam.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Junqiang Wang.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Sankar, Bhavani V.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021682:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021682/00001

Material Information

Title: Fracture Toughness of Cellular Materials Using Finite Element Based Micromechanics
Physical Description: 1 online resource (94 p.)
Language: english
Creator: Wang, Junqiang
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: cellular, finite, foam, fracture, functionally, micromechanics, orthotropic, rectangular, tetrakaidecahedron
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: A finite element method based micromechanical analysis is used to understand the fracture behavior of homogeneous and functionally graded foams. Both rectangular prism and tetrakaidecahedral unit cells are studied. Two approaches of predicting fracture toughness of foams and other cellular materials are used in this study. In one approach, the finite element analysis uses a micromechanical model in conjunction with a macro-mechanical model in order to relate the stress intensity factor to the stresses in the struts of the foam. The stress intensity factor at the crack tip of the macro-mechanical model can be evaluated using either the J-contour integral or the stresses in the singularity-dominated zone. The other approach is to directly apply displacements based on the K-field on the boundary of the micromechanical model. Using the first approach, the mode I fracture toughness is evaluated for various crack positions and length. Both homogeneous foam and graded foam are studied to investigate the effect of stress gradients in the vicinity of the crack-tip on the fracture toughness. Various types of loading such as remotely applied displacements, remote traction are studied. Preliminary results of this study show that the stress gradient has slight effects on the fracture toughness. However, since the effects are relatively small, KIc can be defined as a material property. Then the relationship between the fracture toughness of the graded foam and the local density at the crack tip is studied. The second approach is easy to apply in predicting the fracture toughness of homogeneous foam. By using this approach, convergence study of a micromechanical model is conducted. Also, an analytical model for the mode I fracture toughness of foams with rectangular prism cells is introduced. The mode I and mode II fracture toughness of homogeneous foam consisting of tetrakaidecahedral unit cells are predicted. A parametric study is performed to understand the effect of the geometric parameters of the unit cell and tensile strength of the foam ligament and also dislocation imperfection in the foam.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Junqiang Wang.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Sankar, Bhavani V.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021682:00001


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FRACTURE TOUGHNESS OF CELLULAR MATERIALS USING FINITE ELEMENT
BASED MICROMECHANICS






















By

JUNQIANG WANG


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2007


































2007 Junqiang Wang

































To my parents, Shiming Wang and Yuezhen Jing, and my wife, Baoning Zhang









ACKNOWLEDGMENTS

First, I thank my advisor, Dr. Bhavani Sankar for his support, guidance and dedication. It

is very fortunate for me to work with him. I sincerely thank my committee members, Dr. John

Mecholsky, Jr., Dr. Jacob Chung, and Dr. Ashok Kumar, for participating and evaluating my

research work.

It's our tradition that parents devote all their lives to their children. I thank my father,

Shiming Wang, and my mother, Yuezhen Jing, for their support and love. I thank my wife,

Baoning Zhang, for all of her support throughout my graduate study. I thank my son, Alex

Wang, for reminding me why I am working on this dissertation.









TABLE OF CONTENTS

page

A CK N O W LED G M EN T S ................................................................. ........... ............. .....

LIST O F TA BLE S ......... .... ........................................................................... 7

LIST OF FIGURES .................................. .. ..... ..... ................. .9

A B S T R A C T ......... ....................... ............................................................ 12

CHAPTER

1 INTRODUCTION ............... .......................................................... 14

B ack g rou n d ................... ...................1...................4..........
L iteratu re R ev iew .............................................................................15
Fracture T toughness .......................... ...................... .. .... ............. .... .. ... 15
Functionally G raded Foam ......................................................................... ............... 16
T etrakaidecahedral F oam ........................................................................ .................. 17
O bje ctiv e s ................... ...................1...................8..........
S c o p e ..........................................................................1 8

2 APPROACHES FOR PREDICTING FRACTURE TOUGHNESS ......................................22

A p p ro a c h 1 .............................................................................. 2 2
A p p ro ach 2 .......................... ...................................... ................ 2 4
C om prison of the Tw o A approaches ......................................................................... ...... 25

3 HOMOGENEOUS AND FUNCTIONALLY GRADED FOAMS .......................................28

H om ogeneous M material Properties .............................................................. .....................28
M material M odels for G raded Foam s.......................................................................... ....... 30
L o adin g C cases .................. ..... ..... .. ...................................................... .. 1
An Analytical M odel for Fracture Toughness............................................... .................. 31
Results and Discussion ..................................... ................. ........ .... 34
B ending Loading Case .......................... .............. ................. .... ....... 34
Stress in M icrostructure.................. ...... .... ...... .. ..... .... ............ 35
Stress Gradient Effects on Fracture Toughness of Homogeneous Foam ........................35
Fracture Toughness of Functionally Graded Foam with Non-uniform Strut
T h ic k n e ss ......................................... ......... .......... ...... .......................... 3 6
Fracture Toughness of Functionally Graded Foam with Non-uniform Cell Length.......37
C o n c lu sio n ................... .......................................................... ................ 3 9

4 FRACTURE TOUGHNESS PREDICTION OF A TETRAKAIDECAHEDRAL FOAM ...54

F E M M odel of a U nit C ell ............................................................................. ...................54









Elastic M oduli of H om ogeneous Foam .................................................................................54
Fracture Toughness.......... ...... ........... .. .... .... ...............55
Param etric Study ...................................... .. ......... ........ .... 56
M ode I fracture toughness.............................................. .............................. 56
M ode II fracture toughness ................................................. ............... 57
Progressive Fracture and Crack Propagation ...................................... ............... 57
Sum m ary and C onclu sion ............................................................................. .................... 58

5 IM PERFECTION EFFECTS ........................................................... .. ............... 68

H om ogeneous M material Properties .............................................................. .....................68
Fracture Toughness.......... ...... ........... .. .... .... ...............69
Sum m ary and C onclu sion ............................................................................. ....................70

6 PLASTIC DEFORMATION NEAR CRACK TIP ..................................... .................77

Elastic D reform action A analysis ........................................................... .. ............... 77
P plastic D reform action A naly sis ....................................................................... ....................78

7 CONCLUDING REMARKS AND SUGGESTED FUTURE WORK..............................84

APPENDIX

A CRACK TIP DISPLACEMENT FIELDS FOR ORTHOTROPIC MATERIALS ................86

B FRA M E ELEM EN T IN A BA QU S ............................................................. .....................88

L IST O F R E F E R E N C E S ...................................................................................... ...................9 1

B IO G R A PH IC A L SK E T C H .............................................................................. .....................94









LIST OF TABLES


Table page

3-1 M material properties of struts .................................................... ......................................40

3-2 L ist of load cases .............. ....................................................................... .......40

3-3 Axial forces for Unit cell: c1=c2=c3=200 [tm, h=20 [tm; Load: remote traction ...............40

3-4 The ratio y for cell size: c2=c3=200 [am, h=20 [am, and cl varies ...................................40

3-5 The ratio y for cell size: c1=c2=c3=200 jam, h varies............................... ............... 40

3-6 M ode I fracture toughness(x 10 Pam05) ..................................................... ........... 40

3-7 Tip stress at first unbroken strut normalized with total tip stress for the case with cell:
c=200 am, h=20 utm and a/W=0.5 .................................. .....................................41

3-8 Stress in the first unbroken strut normalized by total crack tip stress for the case with
cell: c=50 um h=5 um and a/W =0.5 ........................................... .......................... 41

3-9 Stress in the first unbroken strut normalized by total crack tip stress for the case with
cell: c=200 m h=20 m and a/W =0.1 .................................................. .....................41

4- 1 M material properties of struts ............. ...................... ......... ...................................... 59

4-2 Convergence study of fracture toughness................................. ...............59

4-3 Convergence study of fracture toughness................................. ...............59

4- 4 Fracture toughness for strut length L=1mm by using 40x 12 cells ..................................59

4- 5 Fracture toughness for strut length L=2mm by using 40x 12 cells ................... ................59

4- 6 Interpolation parameters for Mode I ............. .. ......................................... 59

4- 7 Mode II racture toughness for strut length L=1mm by using 30x 17 cells.......................60

4- 8 Mode II racture toughness for strut length L=2mm by using 30x 17 cells.........................60

4- 9 Interpolation param eters for M ode II.......................................... ........................... 60

4- 10 Maximum stress in the struts ahead of crack tip and model fracture toughness...............61

5- 1 Equivalent m material properties .............. ............................................................. 71

5- 2 N um erical results w ith respect to R ...................... ......... ........................ ............... 72









6 1 L o a d c criteria ...................... .. .............. .. .....................................................8 0

B -l Cross section of fram e elem ent ............................................................... .......................89





















































8









LIST OF FIGURES


Figure page

1-1 Microstructure of a cellular medium with rectangular unit cells: unit cell with cell
len g th s c l, c2 an d c3 ...................................................................... 2 0

1- 2 Micrograph of an AFRL carbon foam ................. ....... .......................... 20

1- 3 Three tetrakaidecahedral cells with strut length I and thickness t in a BCC lattice...........21

2-1 An edge-cracked plate: H-height; a-crack length; W-width. ...........................................26

2-2 F inite elem ent m odels ...... .. .......... .... ............ .. ............................ ....................26

2-3 Crack tip in microm echanical model. ........................................ .......................... 26

2-4 Stress field near crack-tip .......................................................................... ................... 27

3-1 Micro- and Macro-stresses in an open-cell foam............................................................42

3-2 Cell deform ation by cell strut bending........................................................... .... .......... 42

3-3 Example of graded foam with 50 strips and the discrete elastic modulus compared
w ith the actual m odulus ......................... ......... .. .. ..... .. ............43

3-4 Five types of loading ...................................................... .......... .. ............ 43

3-5 Crack-tip forces and moments and corresponding crack tip stresses in the idealized
hom ogeneou s continuum ......................................................................... ....................44

3-6 Mode I fracture toughness as a function of relative density ............................................44

3- 7 Cubic foam with c=1.8mm, Gu=3.5805 MPa........... .......................................... 45

3-8 The bending-tensile ratio convergence test for cell length over strut thickness c/h=10....45

3-9 Results of homogeneous foams under the remote bending load................... ............46

3-10 Normalized stress in the struts ahead of the crack tip for remote prescribed
displacement load case for the cell with h=5 am, c=50 am. ........................................47

3-11 Fracture toughness under different loads for the foam (c=200 am, h=20 am) ..................48

3-12 Fracture toughness under different loads for the foam (c=50 am, h=5 am)......................48

3-13 Fracture toughness under remote bending load. (Plate size: W=0.1, HIW=1; Graded
foam : ho=10 m ; a=2 m ; c=c =c2=c3=200 m ) ........................................ ....................49









3-14 Fracture toughness under remote bending load. (Plate size: W=0.1, H/W=1; Graded
foam : ho=10 am ; a=2 m ; c= =c =c 3=200 m ) ........................................ ....................49

3-15 Com prison of Eq(3.22) w ith Eq.(3.28) ............................................................ .......... 50

3-16 C om prison of three form s .................................................................. ....... ................. 50

3-17 Kic is normalized with the analytical value. (Plate size: W=0. m, HIW=1; Graded
foam: h0=30 am; a=-2 x104; c=C1=C2=C3=200am) ......................................................51

3-18 Plate size: W=0.1m, HIW=1; Graded foam: h0o=30am; a=-2x10-4;
c=c =c2=c3=200 tm ................... ............................. .. ............. ...5. 51

3-19 Fracture toughness under remote bending load. (Plate size: W=0.1m, HIW=1; Graded
foam : co=200 am; 3= -0.15023 am; h=20am) ...................................... ............... 52

3-20 Fracture toughness under remote bending load. (Plate size: W=0.1m, HIW=1; Graded
foam : co=200 am; 3= -0.15023 am; h=20am) ...................................... ............... 52

3-21 KI, is normalized with the analytical value; Plate size: W=0. m, HIW=1; Graded
foam: co=200 am; 3= -0.15023 am; h=20jam.............................................. 53

3-22 Plate size: W=0.1m, HIW=1; Graded foam: co=200 am; 3= -0.15023 am; h=20am ........53

4-1 A tetrakaidecahedral unit cell and the cross section of a strut........................... ........62

4-2 A structure w ith 27 (3x3x3) cells .................................... .... ..... ........................ 62

4-3 Convergence study of Y oung's m odulus................................... ...................... .. .......... 63

4-4 Deformation of a micromechanical mode.................................... ........................ 63

4-5 Convergence study of Mode I fracture toughness ............................................................64

4-6 Mode I fracture toughness vs. strut thickness...........................................................64

4-7 M ode I fracture toughness vs. relative density ...................................... ............... 65

4-8 Normalized Mode I fracture toughness vs. relative density ...........................................65

4-9 Mode II fracture toughness vs. strut thickness.............................................................66

4-10 Mode II fracture toughness vs. relative density ...........................................................66

4-11 Normalized Mode II fracture toughness vs. relative density ..........................................67

4-12 C rack develop ent history. ...................................................................... ...................67

5- 1 D dislocation of a vertex ...................... ........ ................. ......... 73









5- 2 Relative density as a function of the dislocation distance Ra of a vertex......................73

5- 3 Effective m oduli vs. R a ............................................ ... .... ........ ......... 74

5- 4 Equivalent Poisson's ratio versus R..................................................... ...............74

5- 5 Ra=0.50: left whole finite element model; right scaled structure near the crack tip....75

5- 6 Fracture toughness versus R .................................. .......................................... 75

5- 7 An example of the structure near the crack tip with only one imperfect cell (Ra=0.5)
ahead of the crack tip .............................. ...... ............... ........ .......... 76

5- 8 Fracture toughness versus Ra for foams with one imperfect cell ahead of the crack tip ...76

6- 1 P ip e cro ss section ..................................................................... 80

6- 2 Contour of axial elastic strain in struts ........................................ ......................... 81

6- 3 Data points generated for the perfect plastic model ............. ........................................ 81

6- 4 Stress intensity factor K1 vs. elastic strain in the strut......................................................82

6- 5 Stress intensity factor K1 vs. plastic strain in the strut .....................................................82

6- 6 Stress intensity factor Kj vs. total strain in the strut ............................................... 83

B-l Forces and moments on a frame element in space .......................................................90









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

FRACTURE TOUGHNESS OF CELLULAR MATERIALS USING FINITE ELEMENT
BASED MICROMECHANICS

By

Junqiang Wang

December 2007

Chair: Bhavani V. Sankar
Major: Mechanical Engineering

A finite element method based micromechanical analysis is used to understand the fracture

behavior of homogeneous and functionally graded foams. Both rectangular prism and

tetrakaidecahedral unit cells are studied. Two approaches of predicting fracture toughness of

foams and other cellular materials are used in this study. In one approach, the finite element

analysis uses a micromechanical model in conjunction with a macro-mechanical model in order

to relate the stress intensity factor to the stresses in the struts of the foam. The stress intensity

factor at the crack tip of the macro-mechanical model can be evaluated using either the J-contour

integral or the stresses in the singularity-dominated zone. The other approach is to directly apply

displacements based on the K-field on the boundary of the micromechanical model.

Using the first approach, the mode I fracture toughness is evaluated for various crack

positions and length. Both homogeneous foam and graded foam are studied to investigate the

effect of stress gradients in the vicinity of the crack-tip on the fracture toughness. Various types

of loading such as remotely applied displacements, remote traction are studied. Preliminary

results of this study show that the stress gradient has slight effects on the fracture toughness.

However, since the effects are relatively small, KI, can be defined as a material property. Then









the relationship between the fracture toughness of the graded foam and the local density at the

crack tip is studied.

The second approach is easy to apply in predicting the fracture toughness of homogeneous

foam. By using this approach, convergence study of a micromechanical model is conducted.

Also, an analytical model for the mode I fracture toughness of foams with rectangular prism cells

is introduced. The mode I and mode II fracture toughness of homogeneous foam consisting of

tetrakaidecahedral unit cells are predicted. A parametric study is performed to understand the

effect of the geometric parameters of the unit cell and tensile strength of the foam ligament and

also dislocation imperfection in the foam.









CHAPTER 1
INTRODUCTION

Background

Cellular materials are made up of a network of beam or plate like structures. There are a

number of cellular materials that occur in nature, such as honeycombs, wood, bone, and cork.

Cellular materials can offer high thermal resistance, low density, and high energy-absorption.

Foams are a class of cellular solids, generally made by dispersing gas into a liquid material and

then cooling it to solidify. Foams are categorized as open-cell and closed-cell foams. According

to the materials made into foams, foams are also categorized as polymeric, metallic, and ceramic

foams, e.g., carbon foams. Due to rapid developments in material science and manufacturing

techniques, a wide variety of foams have been developed and used in automobiles, aircraft, and

space vehicles. A special example is the thermal protection system (TPS) of space vehicles, e.g.,

Space Shuttle.

Traditional TPS cannot bear loads as they are designed for very low thermal conductivity,

and are easy to damage, which increases the risk of flight. For instance, a disassembled tile of old

TPS caused the tragedy of the Shuttle Columbia in 2003 [1]. NASA has started the study of novel

TPS concepts for the Crew Exploration Vehicle, which is essentially a replacement for the Space

Shuttle. An Integral Thermal Protection System (ITPS) concept is a new idea in which the load-

bearing function and insulation are combined into a single structure. This new concept can be

achieved by using foams as core of the sandwich structures since foams can be tailored to obtain

optimum performance. Under such conditions foams are subjected to various mechanical loads

and extreme heat loads.

Thus there exists an urgent need for the study on fracture toughness and other material

properties of foams.









Literature Review


Fracture Toughness

The most important parameter of a cellular material is the relative density p /ps where p is

the density of the cellular material or foam and ps the solid density, which is the density of the of

the strut or ligament material. The relative density is a measure of solidity, and most of the

material properties depend on the relative density. Analytical methods for determining the

mechanical and thermal properties of cellular solids are well documented. However, research on

fracture behavior of foams is still at its infancy. Maiti, Ashby, and Gibson[2] found that Mode I

fracture toughness Kic is proportional to (p*/ps)3 for open cell and to (p*/ps)2 for closed cell

foams. Huang and Gibson [3, 4] studied several open-cell foams with short crack and further

confirmed the above conclusion. Brezny and Green [5, 6] experimentally verified the factors that

determined the fracture toughness in the theoretical model. Gibson and Ashby [7] summarized

the formulations for Mode I fracture toughness. Recently, Choi and Sankar [8, 9], and Lee [10]

presented new results on fracture toughness of open-cell foams.

In a homogeneous continuum the near-tip stress and displacement fields uniquely depend

on the stress intensity factor (SIF). It is important to obtain accurate SIF value, which could be

calculated from crack-tip stresses. However, it is difficult to obtain accurate stress fields by using

FEM because of the existence of the singularity. In order to improve the accuracy, more

elements are needed near the crack tip, which causes more computational cost. Another way to

calculate SIF is based on the relation between the SIF and the J-integral. For homogeneous

materials, the J-integral is path independent, which allow us to get accurate J along a path away

from the crack tip.









Rice[ 1] introduced the path-independent J integral for elastic solids under isothermal

conditions. A general form of the J integral, suitable for elastic or elastic-plastic thermal crack

problems, is defined by Aoki et al.[12]. Jin [13] used this integral to solve thermal fracture

problems of inhomogeneous materials. However, this form is not a standard J-integral. Shih et al.

[14] provided a domain integral of J, and it has been proved to be more efficient and more

accurate than the direct calculation of the J-integral and is suitable for elastic, thermal elastic,

and plastic materials. Gu et al. [15] applied this domain integral to evaluate the crack-tip field in

inhomogeneous materials, such as functionally graded materials (FGM). The commercial

software ABAQUS also uses this domain integral method to calculate the Jintegral.

Another approach to investigate the fracture toughness is applying displacement boundary

conditions corresponding to a given SIF. Choi and Sankar[8, 9] first used this method to study

the fracture toughness of some carbon foams. Most recently Fleck and Qiu[16] have used this

method to study the damage tolerance of elastic-brittle, 2-D isotropic lattices.

Functionally Graded Foam

One should distinguish functionally graded foam (FGF) from functionally graded materials

(FGM). FGMs are a combination of two materials, e.g., a mixture of metals or ceramics, to

create a desired composite. However in our study, we assume the material properties of the solid

material are isotropic and only the cell size or the strut thickness varies along one direction in the

cellular medium. However, both FGF and FGM have thermal and mechanical inhomogeneities,

and the computational methods used to analyze FGMs are suitable for FGFs also. Some of the

results and conclusions on the behavior of FGMs also apply to FGFs.









There are a large amount of analytical studies available on FGMs. Erdogan and his co-

works[17-19] provided analytical solutions of some fracture problems for FGM. They found the

square root singularity of crack-tip stress is the same as that in a homogeneous material. Jin and

Noda[13] showed that temperature distribution, and elastic or plastic crack-tip singular fields of

nonhomogeous materials are the same as those of homogeneous materials. Gu and Asaro[20]

analytically studied a semi-infinite crack of a FGM. They concluded that material gradients do

not affect the order of the singularity and the angular function, but do affect stress intensity

factors (SIF). The near-tip stresses have the same form as that for a homogeneous material and

the propagation direction is the direction of maximum energy release rate. Sankar[21] derived an

elasticity solution for functionally graded beams with the conclusion that the stress

concentrations occur in short or thick beams. They are less than that in homogeneous beams,

when the softer side of FG beam is loaded and the reverse is true when the stiffer side loaded.

Tetrakaidecahedral Foam

It has been accepted for a long time that tetrakaidecahedron, packed in the BCC structure,

satisfies the minimum surface energy for mono-dispersed bubbles [22]. Only in 1994 a little

better example with smaller surface energy was found by Wearire and Phelan [23]. The

tetrakaidecahedral foams have held the interest of researchers for decades. Microcellular

graphitic carbon foams was first developed at the US Air Force Research Laboratory in the

1990s [24]. The repeating unit cells of this foam can be approximated by a regular

tetrakaidecahedron[25]. Micromechanical models have been used to predict mechanical

properties such as Young's modulus, bulk modulus, yield surface, etc. Warren and Kraynik [26]

studied the linear elastic behavior of a low-density Kelvin foam. Zhu [27] provided an analytical

solution of the elastic moduli. Li and Gao et al. [25, 28] developed some micromechanics models

to analyze the homogeneous material properties and simulate the macroscopic mechanical









behavior under compressive loading. Laroussi et al.[29] studied the compressive response of

foams with periodic tetrakaidecahedral cells. A failure surface is defined in macroscopic stress

space by the onset of the first buckling-type instability encountered along proportional load

paths. Ridha et al. [30] obtained a fracture model for rigid polyurethane foam based on the first

tensile failure of any strut in the cell. However, fracture toughness prediction of

tetrakaidecahedral foam is a new field, and there is no published work available in this topic.

Objectives

In this research, we plan to study open-cell foams with the unit cell shown as in Figure 1-1.

Since this is one of the simplest unit cells, it is easy to model and expected to be helpful in

understanding the fracture behavior of cellular solids. Our focus is the effect of stress gradients

on the fracture toughness. Both homogeneous and graded foams are investigated. We calculate

homogeneous material properties based on the cell geometry and its material properties. And

then the fracture behaviors of an edged-crack specimen with the homogeneous material

properties under different mechanical or thermal loadings are studied. A commercial FEM

software ABAQUS is used for FEM calculations and the input files of FEM are generated by

MATLAB.

Since the unit-cells of many foams such as the carbon foam in Figure 1- 2 could be well

approximated by tetrakaidecahedrons (Figure 1- 3), we shall do further study on the foam made

of this unit cell.

Scope

This research reviews some background information on cellular materials\foams including

fracture toughness determination, functionally graded foam analysis, and tetrakaidecahedral

foam study. We discuss two approaches to determine fracture toughness of foams which are used

in our study, and describe the finite element analysis of homogeneous and functionally graded









foams under different types of loading. We develop an analytical model for fracture toughness

and use it to compare the FEM results. We provide parametric study of fracture toughness of

tetrakaidecahedral foams, and analyze dislocation imperfection effects on material properties

such as elastic modulus and fracture toughness. We briefly discuss some plastic deformation in

the struts near crack tip in ductile foams. Concluding remarks and future work are included.

























Figure 1-1. Microstructure of a cellular medium with rectangular unit cells: unit cell with cell
lengths cl, c2 and c3.


Figure 1- 2. Micrograph of an AFRL carbon foam[24]
























Figure 1- 3. Three tetrakaidecahedral cells with strut length I and thickness t in a BCC lattice









CHAPTER 2
APPROACHES FOR PREDICTING FRACTURE TOUGHNESS

Our approach is a global-local approach wherein the microstructure is modeled in detail

near the crack tip (inner region), and boundary conditions are applied at far away points (outer

region) according to continuum fracture mechanics. The foam in the outer region is modeled as a

homogeneous orthotropic material. We also use two crack propagation criteria, one at the micro-

scale and one at the macro-scale. For brittle foams, once we know the stress intensity factor at

macroscale and the corresponding maximum tensile stress (microscale) in the struts ahead of the

crack, we can calculate the fracture toughness of the foam by the following equation:

K o-,P K
K, or Kc = (2.1)


where Ki is Mode I stress intensity factor, KIc is Mode I fracture toughness, au the tensile

strength of struts or the foam ligaments, and r, the maximum tensile stress in the first unbroken

strut ahead of the crack tip.

There are two approaches of predicting the fracture toughness of foams used in this study.

Approach 1

As an example, we study an edge-cracked plate and impose the displacements around the

outer region surrounding the crack. The maximum tensile stress in the microstructure is obtained

from a local model of the inner region. The stress intensity factor is obtained from the macro

model of the edge-cracked plate.

The edge-cracked plate is shown in Figure 2-1. The plate is comprised of microstructure

with the unit cell shown as Figure 1-1. Due to the symmetry of the geometry and loads, only one-

half of the plate is analyzed (Figure 2-2). A multi-scale modeling approach consisting of three

different length scales is used. Three models (Figure 2-2C) are used and they are: macro model,









macro sub-model, and micro model. The macro sub-model and micro model are attached to the

macro model. The boundary condition (BC) of the macro sub-model is obtained from the macro

model results and the BC of the micro model is obtained from the macro sub-model. In other

worlds, the displacements of the nodes on the boundary of the macro sub-model and the micro

model are the same as those values at the same position of the macro model and macro sub-

model, respectively. The values are automatically obtained by ABAQUS.

In the macro model, namely a model in macro scale, different loads are applied to

investigate the crack-tip field. The material properties of this model are calculated through

homogenization (see equations in Chapter 3). Due to the stress singularity near the crack-tip,

more elements are needed in this area to obtain accurate crack-tip fields. The macro sub-model

plays such a role that allows us to increase the number of elements near the crack-tip.

The micro model is used to calculate the maximum tensile stresses in the unbroken strut

ahead of the crack tip. Figure 2-3 shows the resultant force and bending moments in the strut of

rectangular foam. The maximum tensile stress is given by


M" F 6M, F,
tip bend + ten Ip 2.)
t AP h h

Mode I stress intensity factor (KI) can be determined by:

K, = lim U22 (r, 0) 2-r (2.3)
r->O

where a22 (r,0) is the stress in the y-direction near the crack tip, and is a function of r and 0

(see Figure 2-4).

The stress intensity factors can also be calculated from the J-integral or energy release rate.

Sih and Liebowbitz [31] presented such a relation for orthotropic materials










G = 2) K! K i + 2all 2 (2.4)
2 all 2a

1 1 1
where a, = E, a22 33 E, and in present case a2 a23 = a3 = 0, and
12 E 3

1 1 1 ,
a44 = a = a66 E *, E2*, E3* are Young's moduli in x, y and z directions
G23 G13 G12

respectively, and G12*, G23*, G32* are shear moduli in x-y, y-z, and z-x planes.

In this study we use domain integral in ABAQUS software to calculate the energy release

rate. In the case of graded foams the J-integral is not path-independent. This is because the

graded foam is an inhomogeneous material in macro-scale. Hence, we will use an extrapolation

technique to calculate the energy release rate in graded foams. The stress near the crack tip is

underestimated in the macro model and the macro submodel can capture the square root

singularity of the crack-tip stress. SIF calculated by Eq (2.3) expected to agree well with that

based on J-integral.

Approach 2

Sih et al. [31] determined the K-field in the vicinity of a crack tip in homogeneous

orthotropic materials. We can directly apply displacements based on the K-field on the boundary

of the microstructure.

The displacement fields near the crack tip for Mode I:


u = K,2r Re I I sp2 (cos + S2 sin 8)/2s -s (cosO+s sin O)1/2


u2 =K 2r Re 1 slq2 C(cos 8+ss2 sin 8) 2 S2q Ccos8 + s ins 80 2


The displacement fields near the crack tip for Mode II:










u =KI 2 Re 1 2 (coso + sin 0)12 (cos 8 + Sin 8)lJ2
\ x [s.-S, sI
(2.6)
u2 =K 2r Re 1 q2(cos +2 sin 0) / q (cos 0 + sin 8)1/}


The parameters p, q and s are dependent on material elastic constants and they are given in

Appendix A.

After we find the maximum tensile stress in the struts near the crack tip, we can use Eq

(2.1) to obtain the fracture toughness of the foam.

Comparison of the Two Approaches

Approach 2 is easier to use since only a micromechanical model is involved. Hence, this

approach is good for convergence tests. However this approach is related to a stress intensity

factor for homogeneous foams so that it cannot be used to predict the fracture toughness of

functionally graded foams. This simple expression for stress intensity factor hinders the use of

the approach in the stress gradient effects analysis.

On the other hand, Approach 1 requires a macro model except for the micromechanical

model. The stress intensity factor is needed to be determined from the macro model. So there is

much more effort involved in preparing the finite element models and calculations.

Since both approaches have advantages and disadvantages, the selection of the right

approach depends on the needs of the research task.












I -k-


a lH






Figure 2-1. An edge-cracked plate: H-height; a-crack length; W-width.


Macr-model






A


C77777

r ir


-* I T V0
nodEl

Macro
M icro subrmd del
'Micro model


Figure 2-2. Finite element models: A) An edged-crack plate under remote prescribed
displacement; B) Microstructure of the plate; C) Half model of the plate




I i I i IL

Crak tp Crack tip
Crack tip Mj_ r

Mthpl_ 1 Actual Foam
Flip

Figure 2-3. Crack tip in micromechanical model: Left- crack tip in microstructure; right- actual
foam with resultant force and bending moment.














Crack 1r


Figure 2-4. Stress field near crack-tip









CHAPTER 3
HOMOGENEOUS AND FUNCTIONALLY GRADED FOAMS

Homogeneous Material Properties

At first some notation should be specified. Symbols with denote properties belonging to

macrostructure or foam; symbols with a subscript s are of the strut/ligament material.

The material of the foam is orthotropic and so nine independent parameters are required to

be determined. These nine parameters are Young's moduli in x, y and z directions (E *, E2*, E3 ),

shear moduli in x-y, y-z, and z-x planes (G12*, G23*, G32*), and Poisson's ratios in x-y, y-z, z-x

planes (v12*, v23*, V31*). We choose the same carbon foam as Choi studied in [9].The material

properties of microstructure are listed in Table 3-1.

The Young's modulus iny direction could be derived as depicted Figure 3-1. When the

foam is loaded in the y direction, equilibrium requires the force in the unit area equal to that in

the strut.

o h2 = c1c3 s' Eh2 E= *E2c1C3 (3.1)

where h is the strut thickness, c2 and c3 are the cell length iny and z directions.

Since the strains ,, E* of micro and macro structure in y direction are equal. We have

h2
E =E,- (3.2)
CIC3

Similarly we can obtain:

S h2 h2
E =E E=E3 (3.3)
c2C3 CC 2

The derivation of shear modulus G12* is illustrated in Figure 3-2. Because of symmetry,

there is no curvature at the half-length of the strut. And thus we can use a half beam to solve for

62.










F (C2
2 -= (3.4)
3EJI

Again equilibrium requires:

F = rc3c1 (3.5)

Substitute Eq.(3.5) into Eq.(3.4), we obtain 62 as


g2 = 3 (3.6)
24EJ

In the same manner, 61 is


31 = C23C1 (3.7)
24EI

The shear strain is given as

71 23, 232 rcc2c3 (cl +C (3.8)
c1 c2 12EJ

And the shear modulus G12* can be derived as


G12 1E (3.9)
71 ccc3 (c2 +c)

Substitute the moment of inertia I = h /2


h4
G1 CC C +C E, (3.10)
cic2C3 (C1 +C2)

The shear moduli in the other two planes can be obtained by cyclic permutation as

h4
G23= E (3.11)


,* h4
G3 1 C23 E (3.12)
c, c2C3 (C3 + C, )









So far we have derived the shear and Young's moduli of the foam, the three undefined

parameters are Poisson's ratios. Based on Figure 3-1, we can see that the strain in the x-direction

is negligible and thus the Poisson's ratio v2 is approximately zero. Finally we conclude that

v12 = V3 = V3, = 0 (3.13)

The relative density p /ps is an important parameter of foam, which is a measure of

solidity. Based on the cell's geometry, the relative density can be expressed as

p* V (c +c +c )h2 -2h3
S V 1 (3.14)
Ps C1 C C3,

When cell length is much larger than strut thickness, the h3 term can be neglected.

Furthermore, when c1=C2=C3=c, the relative density is 3(h/c)2.

Material Models for Graded Foams

Two types of functionally graded foams are studied independently, namely, foams with

non-uniform strut thickness and with non-uniform cell length. They are defined respectively by

h(x)= ho +ax (3.15)

c =c+/7 (3.16)

where ac and P are constants, and h0 is the strut thickness at left edge of the foam. In the first kind

of foam, strut thickness varies in the x direction and cell length is constant, and the reverse for

the second kind of foam.

The orthotropic linear elastic material model is applied for the homogeneous foam. But

more effort is needed for graded foams. The material properties of graded foams vary along x-

direction since strut thickness or cell length varies in the direction. Instead of using graded

elements as Santare[32], we divided the foam into small regions with constant material

properties in each strip as Figure 3-3A shows. As long as the regions are small enough, the









gradient material properties of foam can be approximated by constant material properties; Figure

3-3B is an example for Young's modulus.

Loading Cases

Our main objective is to investigate stress gradient effects on fracture behavior. Since

different loads provide various stress gradient, we compare the results of the foams subjected to

five types of loading (Figure 3-4) including: A. Prescribed remote displacement; B. Remote

traction; C. Crack surface traction; D. Remote bending; and E. Thermal loads. In total, six cases

of the five types of loading, listed in Table 3-2, are studied.

An Analytical Model for Fracture Toughness

Maiti, Ashby and Gibson[2] used a KI field to calculate the crack-tip stress (Eq.(2.2)) of

homogeneous foam. The force and the bending moment in the strut were obtained by integration.

They assumed the bending stress in Eq.(2.2) is dominant and they ignored the tensile stress part.

However, we find that in some cases the tensile stress is greater than the bending stress. The ratio

of bending stress over the tensile stress is a constant, 0.415 in the present case, and thus neither

could be negligible. For the foam with a simple cubic cell (C1=c2=c3=c, Figure 3-5), Choi and

Sankar[9] introduced an effective length = ac, instead of using the actual cell length, as shown

in Eq. (3.18) and Eq.(3.19).


22=- (3.17)


F=cl u22dr =c(J s dr (3.18)


K 3.19
M = c \ rdr (3.19)
fO r/2)rr









However, there is no reason to let the effective lengths in Eq.(3.18) and Eq.(3.19) to be

equal. If the cell size is much smaller than the crack size, the homogeneous stress field represents

the stress field of microstructure accurately. Then the homogeneous stress must be balanced by

the tensile stress in the strut and thus we can get good results by setting / = c in Eq.(3.18). More

generally, in the case that the cell lengths in the three coordinate directions are not equal, this

equation is rewritten as


F = c3 c i dr = Ki C3 (3.20)


Table 3-3 gives an example that Eq.(3.20) is a good approximation of the axial force in the

first unbroken strut.

If KI, is a material property of the foam, KI, is a constant. And thus based on Eq.(2.1), the

ratio Kl/p must be a constant, which means o-, = CKI where C is a constant. And therefore,

the ratio -bend/ten is a constant as a result of Eq.(2.2) and Eq.(3.20). For convenience, denote

the ratio as y.

7 = bend /ten = (6M)/(Fh) (3.21)

Substituting Eq.(2.2), Eq.(3.21), and Eq.(3.20) into Eq.(2.1), we obtain

K/Iu KIu KIou r 1 h2
K c = K-= cl = h (3.22)
Ic bend + ten ten (1+ ) JKI cc3 2 1+ c
h2 (l+y)
h2

The relative density p*/ps can be related to the cell lengths and strut thickness with

Eq.(3.14). And then we plot KI, versus relative density in Figure 3-6. It shows that the above

equation agrees very well with Choi's results (Choi 2005: Fig. 13. and Eq. 19) for homogeneous

foam. We also can see that the relative density alone cannot determine KIc, and KI, also strongly

depends on cell size and shape. Figure 3- 7 shows the comparison of current model with Choi's









results in [8] and Gibson and Ashby[7]. Our current model is almost the same as Choi's result

and it give a little smaller fracture toughness. The relative error between our model and the

experimental results is 3%.

More Discussion on the Ratio of Bending Stress over the Tensile Stress. The reason

that the ratio is a constant lies in that the displacement fields in the vicinity of a crack tip in a

homogeneous orthotropic material depend on the stress intensity factor as discussed in Eq. 15 of

Choi 2005. The displacements of the boundary nodes in the micro model are equal to the

displacements at the same place of the homogeneous material, if there are enough cells near the

crack tip. Thus the ratio must be a constant. Figure 3-8 shows that the ratio converges to 0.409 as

the number of beam elements increases.

Table 3-4 and Table 3-5 show that the ratio varies for different foams with different unit

cells. These results are obtained from microstructures with more than 40000 beam elements.

The ratios vary a little. For the sake of simplification, a constant ratio y =0.409 is used.

The error between fracture toughness by using a fixed ratio and by using the ratio listed in Table

3-4 and Table 3-5 could be determined by following procedure. Using a Taylor series expansion,

we can rewrite Eq.(3.22) in terms of y and 7 as

SA A A (3.23)
l+y l+y+y- y (+) Y-?] (1+7) 1+/
11+


S t v t (3.24)
1+a

The absolute value of relative error is









A A


A
-. -i^ {(3.25)

(1+ /)

Corresponding to the largest ratio 0.427 and the smallest ratio 0.383 listed in the tables, the

absolute value of relative errors are 1.28% and 1.85% respectively, which gives us confidence to

use a fixed ratio in Eq.(3.22).

Results and Discussion

Bending Loading Case

We studied different loading cases; here we only show some detailed results of bending

loading to illustrate some conclusions.

We investigate plates (Figure 2-1) with different aspect ratios: 1, 2 and 8 using ABAQUS.

Figure 3-9a shows that the J-integral increases as crack size increases. Figure 3-9c shows the

maximum tensile stress in the first unbroken strut ahead of the crack tip. Figure 3-9b gives the

stress intensity factor calculated based on Eq.(2.4). Also, we compare the stress intensity with the

analytical solution for H->oo by Eq.(3.26) [33]. The FEM results agree well with the analytical

solution. Finally, the fracture toughness is calculated by Eq.(2.1) and listed in Table 3-6. The

relative errors of fracture toughness are shown in Figure 3-9d, where the true value is evaluated

by the mean value of fracture toughness of Case H/W=8. The analytical solution by Eq.(3.22) is

4.55x 105 Pa-m0o. The results in Table 3-6 show the aspect ratio has very little effect on the

fracture toughness. In other words, the plate size does not change the fracture toughness of the

foam.

a a2 a4
K, c, 1.12-1.39a+7.3 a213 a3+14a4 (3.26)
W ( 3 w









Stress in Microstructure

As mentioned in section 4, the ratio y, of maximum bending stress and tensile stress in the

first unbroken strut ahead of the crack tip is a constant when the cell size is small. Table 3-7 and

Table 3-8 show the variance of the ratio becomes less as the cell size decreases. Also by

comparing the data in Table 3-9 with those in Table 3-7 we observe that the ratio varies for

different crack sizes.

Figure 3-10 is an example of the total stress, bending stress and tensile stress in the struts

ahead of the crack tip. The tensile stress is continuously distributed in the struts ahead of the

crack tip. But the bending stress is discontinuous, especially for the first three struts. This

indicates it is difficult to derive an analytical form for the bending stress in the first strut.



Stress Gradient Effects on Fracture Toughness of Homogeneous Foam

Figure 3-11 and Figure 3-12 show the fracture toughness calculated using Eq.(2.1) under

different loads. Since the fracture toughness of remote displacement loading is almost constant,

the fracture toughness is normalized with the mean value of the fracture toughness of Case 1-

remote displacement loading. The fracture toughness of Case 2-Remote traction and Case 4-

Bending are almost the same. Both cases correspond to remote traction. The results of Case 2-

Surface traction and Case 5-Thermal 1 show similar trends as the crack size increases. The case

of Thermal 1 is involved with a negative stress intensity factor. There is a contact pressure

occurring in the crack surface. This is similar to a crack surface traction loading.

Comparing with Figure 3-11 and Figure 3-12, we can conclude that cell size does not

change much of the distribution trends. But the relative difference of fracture toughness for

foams with small cells is smaller than that of foams with large cells. Also we can see that the









stress ratio presented in previous section is not a constant. As a result of variable ratio y, the

fracture toughness varies a little.

Fracture Toughness of Functionally Graded Foam with Non-uniform Strut Thickness

We get the same conclusion as Lee[10] got for the remote displacement load case the

fracture toughness of graded foam is the same as that of homogeneous foam with the same cell

shape and size at the crack tip. This can also be explained by the analytical Eq.(3.22). The

fracture toughness depends on shape, size and material of the cell.

The conclusion is illustrated by the remote bending case. Figure 3-13 shows the fracture

toughness of the foam under remote bending load with ho=10pm and ca=2 am. The 'increasing h'

means that the crack propagates into higher density region, and vice versa. The fracture

toughness is very close to the analytical result for homogeneous foam. Only the fracture

toughness for 'increasing h' case is a slightly greater than the analytical one on the right side of

the figure. This phenomenon is similar to the homogeneous problem discussed in a previous

section: the fracture toughness is greater than the mean value for the small crack size.

The strut thickness h is related to the relative density by Eq.(3.14). The fracture toughness

is plotted with respect to the relative density in Figure 3-14, where fracture toughness is

calculated using Eq.(3.22). It shows that the fracture toughness linearly depends on the relative

density. As mentioned previously, when cell length is much greater than strut thickness and

c1=c2=c3=c, the relative density is 3(h/c)2. Then the fracture toughness can be written as shown

in Eq.(3.27). This equation illustrates that KI, linearly depends on the relative density (p* / p).


7 1 h2 1 \TC 1 p (3.
Kic = + o/-- -y -c 3 (3.27)
V2 l+ / c, 3 V2 l+ 7ps









By substituting / =0.409 into Eq.(3.27), the dimensionless fracture toughness takes the

following simple form as shown in Eq.(3.28). However, Figure 3-15 shows that the relative error

increases dramatically near zero relative density and approaches to 10% when the relative

density is 0.05. And thus, this simple form does not work well.

K--c = l P = 0.2965 (3.28)
,c 3 2 1 + p p

Based on Eq.(3.14), h2 can be derived as

2 3 2 2 2 2
h-- 1+- (3.29)
p 3c 2h P 3 2h p, 3 3c p, 3 33
3c

And then the fracture toughness is

1 c 1 p ( 2 p p p
K 1c 0-" c 21 =0.2965 ,P 1+ 0.3849 (3.30)
3 l+2 1+ p, 3 3p p, p,

Comparing this form with the simple form Eq.(3.28), the relative error should equal to

0.3849 p*/p, and Figure 3-16 shows this conclusion.

The stress gradient effects on the fracture toughness are shown in Figure 3-17 and Figure

3-18. The fracture toughness is normalized with the analytical one (Eq.(3.22)). Similar to the

homogeneous foam, the fracture toughness of Case 2-Remote traction and Case 4-Bending are

almost the same. The results of Case 3-Surface traction and Case 5-Thermal 1 show similar

trends as the crack size increases.

Fracture Toughness of Functionally Graded Foam with Non-uniform Cell Length

We obtain the same conclusion for the non-uniform cell length case: the fracture toughness

of graded foam equals to the fracture toughness of homogeneous foams with the same cell as that









of the graded foam at the crack tip. However the fracture toughness does not linearly depend on

the relative density for this case. cl can be derived from Eq.(3.14) as

(C2 + C3)h2 2h3
c1 -= (3.31)
c2c3 ( h


and thus the fracture toughness is related to the relative density as follows:


-= 1 h2 U C + C 2hc (3.32)
2KIC l+7Y- cz 2+7 c c c +c -2h


which shows the nonlinear relationship between the fracture toughness and the relative density.

The dimensionless fracture toughness can be derived as Eq.(3.33), which shows the

dimensionless fracture toughness depends not only on the relative density but also on the

geometry of the foam. However, the dimensionless fracture toughness linearly depends on the

relative density of the foam since c2, c3 and h are constant.


C2c3 -h2
hh

K 1 h _2 1 h2 r 1 1 ((3.33)
C, 2 1+7 cc, 2 1+y c3 (c + c3)h 2h3 +2 1+ c 2h 1p c C3
C2 C2

Figure 3-19 and Figure 3-20 show that the fracture toughness of the remote bending case

depends on cell length or relative density.

The stress gradient effects on the fracture toughness are shown in Figure 3-21 and Figure

3-22. The fracture toughness is normalized with the analytical one (Eq.(3.22)). Similar to the

homogeneous foam, the fracture toughness of Case 2-Remote traction and Case 4-Bending are

almost the same. The results of Case 3-Surface traction and Case 5-Thermal 1 show similar

trends as the crack size increases.









Conclusion

Through this study, we find that the fracture toughness of the foam could be predicted by

the strength of the strut or ligament material and the shape and size of the cells that constitute the

foam. The crack-tip singular fields of the graded foam, as a nonhomogeneous material, are the

same as those of homogeneous foam. Different loading cases are studied by using a micro-macro

combined method. The effect of stress gradients in the vicinity of the crack-tip on the fracture

toughness is studied. Our results lead to the following conclusions:

* Except for remote displacement loading cases, the fracture toughness of the homogeneous
foam decreases as the crack size increases.

* The aspect ratio of the plate does not have much effect on the fracture toughness.

* As the cell size become smaller, the fracture toughness of the homogeneous foam under
different types of loads becomes uniform;

* Since the relative differences of the fracture toughness of the homogeneous foam under
different loads are within +5%, the fracture toughness can be treated as a material property;

* The analytical model matches well with numerical results for both homogeneous and
graded foams; the fracture toughness of the analytical model agrees with that determined
by the combined micro-macro-mechanics method;

* The fracture toughness of graded foam equals to the fracture toughness of homogeneous
foam with the same cell as that of the graded foam at the crack tip;

* The fracture toughness does not simply depend on the relative density. It depends on both
the material and the shape and size of the cell.










Table 3-1. Material properties of struts
Density, p, 1750 Kg / m3
Elastic Modulus, E, 207 GPa
Poisson's ratio, v, 0.17
Ultimate Tensile Strength, G,, 3600 MPa


Table 3-2. List of load cases.
Case # Description
1. A Remote displacement: vo=7 x 10' m
2. B Remote traction: Go=5 x 107 Pa

3. C Crack surface traction: G=5 x 107 Pa
4. D Max remote bending stress: Gmax= 1x 106 Pa
5 E Temperature change: AT(x) = 75000x2 C


Temperature change: AT(x)


100x2 C


Table 3-3. Axial forces for Unit cell: ci=c2=c3=200 [tm, h:
a/W 0.1 0.2 0.3 0.4
Force(FEM) (N) 33.20 66.66 106.2 154.7
Analytical Force (N) 33.50 67.37 107.4 156.5
Relative error* 0.90 1.05 1.09 1.13
* Relative error=(Analytical FEM)/Analytical x 100%


=20 [tm; Load: remote traction
0.5 0.6 0.7
217.7 307.2 452.9
220.3 310.9 458.4
1.19 1.19 1.19


The ratio y for cell size: c2
100 150
0.383 0.399


=c3=200 am, h=20
200 250
0.409 0.41'


pm, and c, varies
300
5 0.420


Table 3-5. The ratio y for cell size: c1=c2=c3=200 [tm, h varies
h ([m) 10 20 30 40
7 0.396 0.409 0.419 0.427


Table 3-6. Mode I fracture toughness(x10 Pa-m5)
a/Wx100% 10 20 30 40
HIW 1 4.56 4.54 4.53 4.52
2 4.57 4.56 4.55 4.54
8 4.58 4.57 4.56 4.56


50
4.51
4.54
4.55


60
4.51
4.53
4.55


Table 3-4.
ci (am)
y


70
4.50
4.53
4.55












Table 3-7. Tip stress at first unbroken strut normalized with total tip stress for the case with cell:


c=200 tm, h=20 pm and a/W=0.5
Load case # 1
Normalized bending stress 0.2797
Normalized tensile stress 0.7203
Ratio y 0.388


Table 3-8. Stress in the first unbroken strut normalized
cell: c=50 pm, h=5 pm and a/W=0.5
Load case # 1 2 3
Normalized bending stress 0.2875 0.2934 0.2944
Normalized tensile stress 0.7125 0.7066 0.7056
Ratio y 0.404 0.415 0.417


Table 3-9. Stress in the first unbroken strut normalized
cell: c=200 am, h=20 pm and alW=O.1
Load case # 1 2 3
Normalized bending stress 0.2801 0.2900 0.3025
Normalized tensile stress 0.7199 0.7100 0.6975
Ratio y 0.389 0.408 0.434


by total crack tip stress for the case with


0.2938
0.7062
0.416


0.2919
0.7081
0.412


0.2957
0.7043
0.420


by total crack tip stress for the case with


0.2908
0.7092
0.410


0.2150
0.7850
0.274


6
0.3009
0.6991
0.430


2
0.2981
0.7019
0.425


3
0.3000
0.7000
0.429


4
0.2998
0.7002
0.429


0.2936
0.7064
0.416


0.3085
0.6915
0.446

















-I F7 F-t F



Figure 3-1. Micro- and Macro-stresses in an open-cell foam


F





F m-


/--* F



F .4-


Figure 3-2. Cell deformation by cell strut bending: A) the undeformed cell and deformed cell;
B) the loads, moments in a strut; C) The loads and moment in a half strut























IIIIIIIIIJ I| ILL IUI II IIII 1111 Il llll Il[[II I 0I 001 002 003 004 005 006 007 008 009 01
x (m)

A B

Figure 3-3. Example of graded foam with 50 strips and the discrete elastic modulus compared
with the actual modulus


rmax


lw


A B


Figure 3-4. Five types of loading











I 1 1 il I I1 L


7 oK


Jde&PJ Hoinoge~neoue WueriatI


Figure 3-5. Crack-tip forces and moments and corresponding crack tip stresses in the idealized
homogeneous continuum. (Refer to (Choi 2005) Fig. 19)


x 106


- Choi: constant h
e constant h
Choi: constant c
Constant c


0.02 0.04 0.06 0.08
relative density


0.1 0.12 0.14 0.16


Figure 3-6. Mode I fracture toughness as a function of relative density












x 10"
Current model
-- Gibson and- Iiii 1 j...i..
3 Choi's beam :l00 -------- --------
Chos's solid ..."liii ... i
Experimental results





S------- -------- ---------------- ---- -------- ------ --------


j------------ -------- --- -------------i------- ----------I-----------


----- --- ---- ------. ----- -- ------- ------- ------------

iiii---------------


006 0.08 0.1 0.12 0.14 0.16 0.18
Relative density


Figure 3- 7. Cubic foam with c=1.8mm, ou=3.5805 MPa


0 2 0 22


p --
I I I


0.4



0.38-



0.36



0.34



0.32



0.3


0 0.5 1 1.5 2 2.5 3 3.5 4
Number of elements x 104



Figure 3-8. The bending-tensile ratio convergence test for cell length over strut thickness c/h=10















12000-
HIW=1
10000 -e-HIW=2
H/W=8


0.2 0.4
a/W


4x109
14


1 H/W=8
10


0.2 0.4
a/W


0.6 0.8


x106
--Analytical h/w=c
-- H/W=1


x 103


S0
0 ----




/ -5 -
N




15
0.6 0.8 0


0.2 0.4
a/W


Figure 3-9. Results of homogeneous foams under the remote bending load: A) J-integral;
B).Stress intensity factor; C).Tip stress; D) Relative difference in fracture toughness


SH/W=1
-e- H/W=2
-- H/W=8


0.6 0.8











Normalized stress of micro model ahead of crack tip


Total stress
- Bending stress
Tensile stress


0.8



0.6



0.4


0.2



n *


r(m)


10
x 10-3


Figure 3-10. Normalized stress in the struts ahead of the crack tip for remote prescribed
displacement load case for the cell with h=5 [tm, c=50 [tm.





























0.98


0.94
0.1


-- 1.Displ.
S2.Remote traction
3.Surface traction
4.Bending
5.Thermall
6.Thermal2


0.2 0.3 0.4 0.5 0.6 0.7 0.8
a/W


Figure 3-11. Fracture toughness under different loads for the foam (c=200 pm, h=20 pm)






1.025
S 1. Displ.
1.02 e 2. Remote traction
3.Surface traction
1 \ -- 4. Bending
1.015
5.Thermall
6.Thermal2
o 1.01


N 1.005

z 1
Z 1 --- ~ --- -


0.995


0.985
0.1


0.2 0.3 0.4 0.5 0.6 0.7 0.8


Figure 3-12. Fracture toughness under different loads for the foam (c=50 pm, h=5 pm)


r
x_


=c~












x 105
10


SIncreasing h
Decreasing h
- Analytical


a


1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3
h(m) x 10-5


Figure 3-13. Fracture toughness under remote bending load. (Plate size: W=0.1, H/W=1; Graded
foam: ho=10 pm; a=2 pm; c=c1=C2=c3=200[im)


10 5
x 10


-e-Increasing density
---Decreasing density
-*-Analytical


0o01 0.02 0.03 0.04
Relative density


-e-Increasing density
-- Decreasing density
0.015 --Analytical


0.05 0.06


0.02 0.03 0.04
Relative density


Figure 3-14. Fracture toughness under remote bending load. (Plate size: W=0.1, H/W=1; Graded
foam: ho=10 pm; a=2 pm; c=c1=C2=c3=200j[m)


E 8



t-
0)
0)
o 4

-2

LL
m3


0.05 0.06














---Analytical
-x-0.2965p*/p


8
0

6





2
4
._>

of 2

n


0 0.01 0.02 0.03 0.04 0.05
Relative density (p*/ps)


Figure 3-15. Comparison of Eq(3.22) with Eq.(3.28)


0.02 0.04
Relative density (p*/ps)


10

8
0
6




2 6

0)
-01


I- -I-I I


S--Relatie error
---- -e-0.3849(p*/ps) 5
-- -- Relative error 1

I I I I-


0 0.01 0.02 0.03 0.04 0.05
Relative density (p*/ps)


Figure 3-16. Comparison of three forms: A) Eq(3.22): analytical, and Eq.(3.30): modified; B)
Relative error is between Eq.(3.22)and Eq.(3.28); Relative error 1 is between
Eq.(3.22)and Eq.(3.30).


--------------




































0.92
0.1


-0 1.Displ.
S2.Remote traction
3.Surface traction
4.Bending
5.Thermall
6.Thermal2
Analytical


0.2 0.3 0.4 0.5 0.6 0.7 0.8
a/W


Figure 3-17. Kic is normalized with the analytical value. (Plate size: W=0.lm, H/W=1; Graded
foam: h0=30 am; a=-2x10-4; c=ci=c2=c3=200[tm)


x 105
10
-- 1. Displ.
9c\ e 2.Remote traction
3.Surface traction
4. Bending
c^ 5.Thermall
E 6.Thermal2
7 Analytical


0.2 0.3 0.4 0.5 0.6 0.7 0.8
a/W


Figure 3-18. Plate size: W=0.lm, HIW=1;
c=c1=c2=c3=200jam


Graded foam: ho=30am; a=-2x10-4;


1.08

1.06 -


.0



E 1
.8
z
0.98

















e Increasing c
- Decreasing c
- Analytical
- Homogeneous


II


1.2 1.3 1.4 1.5 1.6
cl(m)


1.7 1.8 1.9 2
x 10-4


Figure 3-19. Fracture toughness under remote bending load. (Plate size: W=0.lm, H/W=1;
Graded foam: co=200 pm; 3= -0.15023 pm; h=20pm)


-Increasing c
-Decreasing c
-Analytical
-Homogeneous


0.03 0.035
Relative density


0.016
-e-Increasing c
0.015 ---Decreasing c

0.014 Analytical
"- Homogeneous
j 0.013

_ 0.012

0.011


0.04 0.045


0.0 8o0
0.025


0.03 0.035
Relative density


0.04 0.045


Figure 3-20. Fracture toughness under remote bending load. (Plate size: W=0.1m, HIW=1;
Graded foam: co=200 pm; 3= -0.15023 pm; h=20pm)


x 105
6.2


6


5.8

E
S5.6
0)
(U)
- 5.4
0
2 5.2

LL


4.6
1.1


6.5105
-e

E
6 -

S5.5
0

LI..
4 5
LL
U-

0J25














-0 1. Displ.
S2. Remote traction
S3.Surface traction
4. Bending
5.Thermall
6.Thermal2
Analytical








---


0.2 0.3 0.4 0.5 0.6 0.7 0.8
a/W


Figure 3-21. KI, is normalized with the analytical value; Plate
foam: co=200 am; 3= -0.15023 am; h=20[m


size: W=0.1m, H/W=1; Graded


-* 1. Displ.
S2. Remote traction
3.Surface traction
4. Bending
5.Thermall
6.Thermal2
Analytical


4.4 1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
a/W



Figure 3-22. Plate size: W=0. m, HIW=1; Graded foam: co=200 jam; B= -0.15023 jam; h=20jam


0.95
0.1


x 105


S


n.
~-Q"









CHAPTER 4
FRACTURE TOUGHNESS PREDICTION OF A TETRAKAIDECAHEDRAL FOAM

FEM Model of a Unit Cell

The tetrakaidecahedral unit cell that we propose to study is a 14-sided polyhedron with six

square and eight hexagonal faces. It is more precisely called truncated octahedron, since it is

created by truncating the corners of an octahedron [34]. From a different viewpoint, it can be

generated by truncating the covers of a cube [27]. All the edges of the cell are of equal length L

and cross sectional area A.

The tetrakaidecahedral foam has a BCC lattice. The axes of the BCC lattice are parallel to

the axes of the cube. Due to the symmetry of the structure, the Young's moduli of the foam in

the lattice vector directions are equal:

E0o0 = E010 = E1o (4.1)

Each strut of the cell is treated as a beam element. In our study, the cross section of the

struts is assumed to be an equilateral triangle with side length D (Figure 4-1). A reticulated

vitreous carbon (RVC) foam will be studied and the material properties of the RVC are listed in

Table 4- 1.

Elastic Moduli of Homogeneous Foam

Zhu [27] obtained analytical expressions for the Young's modulus and Poisson's ratio

based on the symmetry of the microstructure:


Eio0 0.726E p2 (4.2)
1+1.09p


S0.5 1-1514 (4.3)
1+2 1.514p)









where p is the relative density, which is related to the side length D and strut length L as shown

below:

p = 0.4593(D/L)2 (44)


In our study, the cross section of struts is an equilateral triangle with side length D (Figure

4-1). Using FEM, we verified the above equation for the Young's modulus. By applying a

compressive load (Figure 4-2), we calculated the nominal strain from the change in height of the

structure and the original height, and the nominal stress is obtained by total resultant forces in y-

direction per unit area. Figure 4-3 shows that the Young's modulus converges to the analytical

solution as the number of cells increases. In the following sections, the homogeneous material

properties of the foam will be calculated with the above equations unless specified otherwise.

Fracture Toughness

We study the fracture toughness of plane strain problems for tetrakaidecahedral foams.

Approach 2 is used to obtain fracture toughness, in other words, by imposing the displacements

of K field on the boundary to micromechanical model, we can obtain the maximum tensile stress

near the crack tip from the ABAQUS results. Figure 4-4 gives an example of the deformation of

the micromechanical model. However, in order to reduce the cost of computation and storage,

we take advantage of the symmetry and model only one-quarter of the cellular medium.

Two convergence tests are conducted: Case 1 in which the cell number is increased

gradually in both x and y directions; Case 2 in which the cell number in x direction is increased

and that in y direction is kept constant. The results are listed in Table 4-2 and Table 4-3 and also

shown in Figure 4-5.









Parametric Study

In the parametric study, the two parameters, L and D (see Figure 4-1), are varied to study

their effects on the fracture toughness. At first, the detail results for Mode 1 fracture toughness

will be presented. And the results of Mode 2 fracture toughness are also included.

Mode I fracture toughness

At first, we fixed the strut length at L=1 mm and varied the strut thickness. The effect of

strut thickness on Mode I fracture toughness is shown in Table 4- 4. The procedures were

repeated for L=2 mm and the corresponding results are given in Table 4- 5.

The results presented in Table 4- 4 and Table 4- 5 are also plotted in Figure 4-6. In general,

we can conclude that the fracture toughness decreases as L increases for a given strut thickness

D. For the same L, the fracture toughness increases as D increases.

The relationship between fracture toughness and relative density is shown in Figure 4-7

and Figure 4-8. We use power law for deriving an empirical relation as:

Kc = cp C2 (4.1)


f- a =ap0 (Non-dimensional form) (4.2)


The coefficients in the above relations are listed in the Table 4- 6. Base on results

presented in Table 4- 6 and Figure 4-8, the relative differences of coefficients al and a2 for L=l

mm and L=2 mm are less than 1% and the two curves for two cases collapse into one curve.

Hence, we can conclude that the dimensionless fracture toughness of tetrakaidecahedral foam

mainly depends on its relative density. The dimensionless fracture toughness increases as relative

density increases.









Mode II fracture toughness

Mode II fracture toughness is obtained by Approach 2, that is, by imposing the

displacements of K1 field on the boundary to micromechanical model, we can obtain the

maximum tensile stress near the crack tip from the ABAQUS results. At first, fixing the strut

length at L=l mm and varying the strut thickness, we obtain Mode II fracture toughness as

shown in Table 4- 7. And then we choose L=2 mm and follow the same procedure to obtain

fracture toughness listed in Table 4- 8. The results presented in Table 4- 7 and Table 4- 8 are

plotted in Figure 4-9, and we can conclude that the fracture toughness decreases as L increases

for the foam with the same strut thickness D.

The relationship between fracture toughness and relative density is shown in Figure 4-10

and Figure 4-11. We use power law for deriving an empirical relation as:

K1c = clpC2 (4.5)


-n = ap2' (Non-dimensional form) (4.6)


The coefficients in the above relations are listed in Table 4- 9. Base on results presented

in Table 4- 9 and Figure 4-11, we can conclude that the normalized fracture toughness of

tetrakaidecahedral foam mainly depends on its relative density.

Progressive Fracture and Crack Propagation

So far the fracture toughness we have presented is calculated based on the maximum stress

in one strut near the crack tip. In this section we will study progressive fracture by continuously

loading the plate and failing a series of struts.

In this study we assume that the crack is sufficiently long compared to the cell dimension

L, and hence the crack propagation is considered under Mode I loading condition. After a strut

fails, the failed strut is removed, and displacements corresponding to an arbitrary KI are applied









along the boundary of the model. In the present study we used K1=0.01. The stresses in the struts

in the vicinity of the crack tip are calculated. From the maximum stress, the stress intensity factor

K1c that will cause a strut to break is calculated using the relation This procedure repeated until

several struts fail in the vicinity of the crack tip. Figure 4-12 depicts the sequence in which the

struts break in the FE model. It is interesting to see that the crack does not propagate in a self-

similar manner (horizontally). Instead there are two kink cracks occurring in 450 and -450

directions. Maximum stresses in the strut at each stage for K1=0.01 (MPammo ) and

corresponding fracture toughness are listed in Table 4- 10. One can note that the fracture

toughness slightly increases as the kinked crack grows.

Summary and Conclusion

A finite element based method developed by Choi and Sankar has been used to study the

fracture toughness of tetrakaidecahedral foam. We obtain the plain-strain fracture toughness of

the foam by relating the fracture toughness to the tensile strength of the cell struts. Also, we have

studied the effects of various geometric parameters that describe the cell. The fracture toughness

decreases as strut length L increases for the foam with the same strut thickness D. For the same

strut length, as D increases the fracture toughness increases. However, the dimensionless fracture

toughness only depends on the relative density.












Table 4- 1. Material properties of struts
Density, p, 1650 Kg / m3
Elastic Modulus, E, 23.42GPa
Poisson's ratio, Vs 0.33
Ultimate Tensile Strength, o, 689.5 MPa


Table 4-2. Convergence study of fracture toughness
Number of Cell 10x5 16x8 20x10 25x12 32x16
50 128 200 300 512
Kc, 0.405 0.399 0.397 0.395 0.393


Table 4-3. Convergence study of fracture toughness
Number of Cell 10x10 20x10 30x10 40x10 50x10
100 200 300 400 500
Kc, 0.405 0.397 0.394 0.393 0.392


Table 4- 4. Fracture toughness for strut length L=1 mm by using 40x 12 cell
L (mm) D (mm) Relative density KI, (MPa-mm0 5)


1 0.0600 1.654x10-3 9.09x 10-2
1 0.1000 4.593 x10- 3.92x10-1
1 0.1875 1.615x10-2 2.23
1 0.2308 2.446x 10-2 3.87
1 0.2727 3.416x10-2 5.98
1 0.3000 4.134x 10-2 7.63


Table 4- 5. Fracture toughness for strut length L=2 mm by using 40x 12 cells
L (mm) D (mm) Relative density KI, (MPa-mm0 5)
2 0.1200 1.654x10-3 0.128
2 0.2000 4.593x10-3 0.554
2 0.3750 1.615x10-2 3.15
2 0.4615 2.446x 102 5.48
2 0.5455 3.416x10-2 8.46
2 0.6000 4.134x10-2 10.8





Table 4- 6. Interpolation parameters for Mode I
Cl C2 a, a2
L=I 494 1.31 7.17x10- 1.31
L=2 694 1.31 7.12x10-1 1.31


s












Table 4- 7. Mode II fracture toughness for strut length L=I mm by using 30x 17 cells
L (mm) D (mm) Relative density KIIn (Mpa-mm0 5)
1 0.0600 1.654x103 3.37x10-2
1 0.1000 4.593 x 10 1.62x10-1
1 0.1875 1.615x10-2 1.10
1 0.2308 2.446x 10-2 2.03
1 0.2727 3.416x10-2 3.32
1 0.3000 4.134x 10-2 4.40





Table 4- 8. Mode II fracture toughness for strut length L=2 mm by using 30x 17 cells
L (mm) D (mm) Relative density KIc (MPa-mm0 5)
2 0.1200 1.654x10.3 4.77x 10-2
2 0.2000 4.593x10.3 0.230
2 0.3750 1.615x10-2 1.55
2 0.4615 2.446x 10-2 2.87
2 0.5455 3.416x10-2 4.70
2 0.6000 4.134x 102 6.22




Table 4- 9. Interpolation parameters for Mode II
C1 C2 a, a2
L=I 486 1.48 0.704 1.48
L=2 687 1.48 0.704 1.48










Table 4- 10. Maximum stress in the struts ahead of crack tip and Mode I fracture toughness for
kinked cracks


Sequence of analysis


Maximum stress (MPa)
for KI=0.01 (MPa-mm 5)
22.0
32.4
25.1
29.9
19.4
22.4
21.2
15.8
23.9
19.1
21.2
17.1


13 15.1
14 17.9
15 17.6
Mean Fracture Toughness (MPa-mmo 5)
Standard Deviation (MPa-mmo 5)


Kic (MPa-mm 5)


0.313
0.213
0.275
0.231
0.355
0.308
0.325
0.435
0.289
0.361
0.326
0.403
0.457
0.384
0.391
0.338
0.070 (21%)













15
1 5 ..


0.5 -

-0.5 -
-1
-1.5








Figure 4-1. At


a-.,,''" '["


y x t


tetrakaidecahedral unit cell and the cross section of a strut


4 6
6 4
8 0
z x

Figure 4-2. A structure with 27 (3x3x3) cells


2


/\
D



















80
0
S4


3.5
E

S3


o 2.5


2


1.5



0 200 400 600 800 1000 1200 1400 1600 1800
Number of cells



Figure 4-3. Convergence study of Young's modulus












.af -M
7y Sr U









R:MM












Dos %. IinA L. v Y

y


Figure 4-4. Deformation of a micromechanical mode















0.408

0.406


10x5 0xl10


S0.404
0
E
E
6 0.402
2I
2 0.4

2 0.398


S0.396
L_


0.394


100 150 200 250 300 350 400 450
Number of unit cells


Figure 4-5. Convergence study of Mode I fracture toughness



12


_ 10
Ip
0

E
E



v,
6



4
L)
2




0
0


0.1 0.2 0.3 0.4 0.5 0.6 0.7
D (mm)


Figure 4-6. Mode I fracture toughness vs. strut thickness















12
O L=1
interpl
10- L=2
^ interp2







2 -







0 /




Figure 4-7. Mode I fracture toughness vs. relative density






0.012

o L=1
0.01 -interpl

0i nterp2
0.008

0_ /
-?
0.006


0.004


0.002


0
0 0.01 0.02 0.03 0.04 0.05
P*/Ps


Figure 4-8. Normalized Mode I fracture toughness vs. relative density
















eL= 1
6 L=2

d /
E
E 5-
a_

(1)
a4-
0










0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7





D(mm)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
D(mm)


Figure 4-9.


6
E
E 5
m
n


o4



0



2
0)
0


Mode II fracture toughness vs. strut thickness


interpl
L=2
interp2


0 0.005 0.01


0.015 0.02 0.025

P*/Ps


0.03 0.035 0.04 0.045


Figure 4-10. Mode II fracture toughness vs. relative density











x 10-3


L=1
interpl
L=2
interp2


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045
p*/ps


Figure 4-11. Normalized Mode II fracture toughness vs. relative density


1 4


5 12
7 6


15 14
8 11
9 10


Figure 4-12. Crack development history: first broken strut is labeled 1 and the last broken strut is
15.









CHAPTER 5
IMPERFECTION EFFECTS

So far only idealized foams are studied. In reality there are always imperfections in foams:

* Dislocation of a vertex which connects several struts

* Non-uniform strut thickness or material properties

* Voids in the microstructure

* Inclusion in the microstructure

We shall study the first kind of imperfection effects on foams consisting of

tetrakaidecahedral unit cells. Generally for the dislocation imperfection, a vertex is assumed to

be somewhere within a sphere with radius R, the center of which corresponds to its perfect

position. However, due to computer and software limitations, we will assume only in-plane

dislocation, which means a vertex is within a circle of radius R in the x-y plane with the center of

the circle at the perfect position. As Figure 5- 1 shows, O is the perfect position of a vertex, and

O' is the actual position. R' and a are uniformly distributed in [0, R] and [0, 2nr] respectively. We

introduce a new parameter Ra.

R
S= Re[0,0.5] (5.1)
L

Homogeneous Material Properties

When we study imperfection problems, the microstructure is no longer symmetric. Hence,

in this section we study the whole model instead of the half model. We conducted four

simulations for each Ra value, which means that there are four finite element models randomly

generated and analyzed. The relative density of the foam is calculated and plotted in Figure 5- 2.

The results show that the relative density increases as Ra increases. Since the tetrakaidecahedral

unit cell has almost minimum surface area, as Ra increase, the total length of struts increases and

thus relative density becomes larger. In order to obtain the nodal displacements on the boundary









of the microstructure, we need to calculate the equivalent material properties E1 E2, 12 and Gc2

in Eq (A. 1).

Symmetric displacement conditions are still applied in the nodes on the front and back

surfaces. Detailed schemes to obtain these equivalent material properties are shown in Table 5-

1. For a given Ra, the inner nodes are randomly generated within a circle with radium R=RaL and

the center located at the perfect position, but the nodes on the boundary are located in the perfect

position so that it is easy to apply BC. In order to observe the random parameter Ra effect at each

value of Ra we conduct four FEM analyses, where four finite element models are randomly

generated.

Figure 5- 3 shows that the elastic modulus increases as Ra increases. This result confirms

the general conclusion that higher the relative density larger its modulus. The Poisson's ratio

decreases as Ra increases (Figure 5- 4). Generally, when Ra increases the deviation from the

mean value of these material properties also increase.

Fracture Toughness

Once we obtain the homogeneous material properties using a finite element model, we use the
same structure and break the elements at the crack (see Figure 5- 5). The fracture
toughness analysis is similar to that in Chapter 4. The only difference is that we use
finite element model to calculate the equivalent material properties instead of the
analytical solution. In previous section, we found that the deviation of material
properties is small and negligible. However, the deviation of fracture toughness is
large as shown in Figure 5- 6. All these results are also listed in
Table 5- 2.

We also studied the foams with only one imperfection cell ahead of crack-tip as Figure 5- 7

shows. The overall material properties were assumed not to be affected by the imperfection. We

conducted three simulations for each R value. Figure 5- 8 shows that the imperfection of this

single cell has significant effect on the fracture toughness.









Summary and Conclusion

In this chapter, the dislocation imperfection effects are studied. Equivalent homogeneous

material properties are obtained by finite element analysis. The fracture toughness of

tetrakaidecahedral foams is analyzed with the same approach as in Chapter 4. We obtain the

plane-strain fracture toughness of the foam by relating the fracture toughness to the tensile

strength of the cell struts. We find that dislocation distance Ra affects the elastic modulus

slightly. But it has a huge effect on the fracture toughness. The deviation of the fracture

toughness increases as the dislocation distance Ra increases. The results of foams with one

imperfect cell ahead of crack tip also confirm this conclusion.










Table 5- 1. Equivalent material properties
Material properties Deformation and boundary condition Equations
* AL
Ei -= 2 = r12 = 0,
L
o, 0 (calculated based on resultant
force)





2* 2
o kL AL --



I-2 2= 1, 01= 12 = 0
AL L
0-2 o 0 (calculated based on resultant
force)

Lo E2=0-2
^2




12 2 =0, 12 =0,
o,O 0 & o-2 0 (calculated based on
resultant force)

22 2


L oo




GC2 Applying Periodic BC:
u x X = 0, vx1 -v, = 0,
u, u = AL, v v = 0.
AL
Shear strain: 12 = -
y1 y0
X- - +- x- Strain energy density:
1 _1 2 2


2U
(2 =( 12)2











Table 5- 2. Numerical results with respect to Ra


Ra Relative density E1
(MPa) (


0 0.004663
0.05 0.004666
0.05 0.004666
0.05 0.004666
0.05 0.004666
0.1 0.004673
0.1 0.004673
0.1 0.004673
0.1 0.004674
0.2 0.004704
0.2 0.004702
0.2 0.004703
0.2 0.004703
0.3 0.004753
0.3 0.004751
0.3 0.004757
0.3 0.004752
0.4 0.004826
0.4 0.004824
0.4 0.004821
0.4 0.004822
0.5 0.004917
0.5 0.004909
0.5 0.004914
0.5 0.004917


0.4664
0.4671
0.4666
0.4676
0.4662
0.4700
0.4686
0.4692
0.4687
0.4793
0.4765
0.4794
0.4758
0.4903
0.4882
0.4883
0.4888
0.4961
0.4957
0.5044
0.5039
0.4999
0.5087
0.5005
0.4948


0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.5
0.5
0.5
0.5
0.5
0.5


E2 G2
MPa) (MPa)
664 0.1160
673 0.1163
667 0.1164
676 0.1163
662 0.1165
703 0.1173
696 0.1175
695 0.1176
706 0.1175
806 0.1213
757 0.1216
801 0.1213
775 0.1224
901 0.1284
902 0.1288
900 0.1296
833 0.1294
969 0.1394
998 0.1367
103 0.1364
141 0.1363
033 0.1478
142 0.1456
109 0.1454
043 0.1475


v12 Max tip stress Kic
(MPa) (MPa-mm05)
0.9807 21.11 0.327
0.9787 19.59 0.352
0.9786 20.22 0.341
0.9785 21.18 0.326
0.9786 21.03 0.328
0.9720 25.85 0.267
0.9728 22.39 0.308
0.9722 23.71 0.291
0.9737 20.81 0.331
0.9448 23.88 0.289
0.9451 18.08 0.381
0.9449 26.63 0.259
0.9470 26.93 0.256
0.8940 30.38 0.227
0.8992 34.24 0.201
0.8908 29.77 0.232
0.8922 23.09 0.299
0.8214 17.78 0.388
0.8257 26.01 0.265
0.8250 35.24 0.196
0.8338 22.10 0.312
0.7290 26.44 0.261
0.7279 33.53 0.206
0.7319 22.50 0.306
0.7259 27.85 0.248

































Figure 5- 1. Dislocation of a vertex




x103
4.95 ----- -----------------------
p p s i
4.9- an\ale of p/p -


>,4.85 -


o 4.8


S4.75 ----- ---- --- -- --- ---------


4.7 ---------


4651
0 0.1 0.2 0.3 0.4 0.5

Figure 5- 2. Relative density as a function of the dislocation distance R of a vertex.
Figure 5- 2. Relative density as a function of the dislocation distance Ra of a vertex.













S E1 o G12
0.51 E2 2 0.15 -. Meanvalueof G

o 0.5 -Meanvalueof E
o -Mean value of E 0
E0.49- E
D 0.13----- ---- ----
0.48 -
0 0 A012
E 0.4 ---- -- ------
, 0 0----- -----------
I
0.46 0.11
0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5
Ra Ra
A B
Figure 5- 3 Effective moduli vs. R,: A) Equivalent elastic modulus; B). Equivalent shear
modulus








1 1

0.95 Meanvalueof v12
() \


0.9

U)
0


0.85
o

o
0.8


0.75


0.7
0


Figure 5- 4.


0.1 0.2 0.3 0.4 0.5
Ra

Equivalent Poisson's ratio versus Ra






























Figure 5- 5. Ra=0.50: left
tip


whole finite element model; right scaled structure near the crack


>o 0.35 --------------------- ----
OE *

S0.3 ------- --- ,------ ------^- -


0.25 -




SMean value of o

0 0.1 0.2 0.3 0.4 0.5
Ra


Figure 5- 6. Fracture toughness versus Ra


- T



























Figure 5- 7. An example of the structure near the crack tip with only one imperfect cell (Ra=0.5)
ahead of the crack tip








0.4

0.38 Meanalueof -- ---


E 0.36 -- --



0.34 -


LI
S0.3


0.1 0.2 0.3 0.4 0.5


Figure 5- 8. Fracture toughness versus Ra for foams with one imperfect cell ahead of the crack
tip


0.26
0









CHAPTER 6
PLASTIC DEFORMATION NEAR CRACK TIP

We have studied brittle foams in previous chapters. Now we will have a glimpse of

fracture behavior of ductile materials, which means plastic deformation will occur when the

principal stress is greater than the yield stress of the material. In order to use ABAQUS's

capability, pipe cross section (Figure 6- 1) is chosen for the struts of cells in the rectangular

prism foam. The outer radius is r and the thickness is t. The cross section area is A and the

moment of inertia is I (Eq.(6.1) and Eq.(6.2)).

A= 2r(r -0.5t)t (6.1)


I= '(r4 (r _4 (6.2)


Elastic Deformation Analysis

Before studying plastic deformation, we first follow the same procedures in Chapter 3 to

analyze fracture toughness of brittle material (Table 3-1). Here we use Approach 2 to obtain the

fracture toughness.

Similar to derivation of Eq.(3.2), we obtained elastic homogeneous material properties:

A A A
El* = E E E,- E3= E, (6.3)
C2C3 C1C3 C1C2

The shear modulus G12* takes the same form as Eq.(3.9) and the Poisson's ratio are given

as Eq.(3.13).

The maximum tensile stress of the first unbroken strut ahead of the crack tip is calculated

as the resultant force and bending moments on the strut are obtained.

AMpr Fp
p = -bend + ten A+ (6.4)
I A









Case 1: input parameters are KI=500 (MPa-mmo.5), cell size c1=c2=C3=0.2mm, cross section

r=c1/5 and t=r/6.

The elastic strain contour (Figure 6- 2) shows the strains in the struts near crack tip are

much larger than those in other struts as expected. We obtain Mtp=0.122691 N-mm and

Ftp=36.7165 N. Hence,

ML F
= +- +-- =15,287 MPa
I A

And


Kic = o-= 61.23 MPa-mm.5
Otip


The ratio of bending stress over the tensile stress y is


S= -bend ten = 0.2964
I A

The analytical solution of fracture toughness is


Kic = or- I = 60 MPa-mm05
2 1+Y7 fcc

The numerical result of fracture toughness agrees well the analytical solution.


Plastic Deformation Analysis

To include elastic-plastic response, we need to specify the nodal forces N, M1, M2, and T

directly as functions of their conjugate plastic deformation variables. For elastic-perfectly plastic

deformation, for each of the above nodal forces we need to provide the value at which plastic

deformation sets in (denoted by Fo) and the force at which the section becomes fully plastic (F1).









These are given in the form of a graph depicted Figure 6- 3. In this figure P1 is the plastic

deformation per unit length corresponding to F1 and P2 is an arbitrarily big value.

The ultimate stress in Table 3-1 is taken as the yield stress for the elastic -plastic

deformation. The other properties of the foam are the same as for Case 1 in Table 3-1. The forces

Fo and Fi are calculated for each mode, extension, flexure and torsion, using mechanics of

materials formulas, and are listed in Table 6- 1.

ABAQUS assumes the displacement and rotation increments can be decomposed into

elastic and plastic parts. Plastic strain will occur when the strain is larger than the yield strain

given by cy/Es=3600/(207 x103)=0.01739.

For this study we used the full micromechanical model. The displacements corresponding

to a given KI are applied along the boundary of the model and they were increased incrementally

starting from KI=0. The strain in the crack tip strut is monitored for each increment. Figure 6- 4

shows the elastic strain vs. KI plot. It shows the KI corresponding to the onset of plastic strain in

the strut. This value will be approximately equal to the KI, obtained for brittle foams with the

rupture strength equal to the yield stress. The elastic strain does not increase beyond the onset of

yielding as we are using elastic-perfectly plastic model. However, the plastic strain increases as

KI increases (Figure 6- 5).

If maximum strain criterion is used, then the fracture toughness can be determined based

on the curve in Figure 6- 6 in which the KI is plotted against the total strain in the crack tip strut.

For example, if strain to failure of the strut material is 0.2, then KIc=207 MPa-mm05.










Table 6- 1. Forces and deformations for onset of yielding and fully plastic conditions
Fo F; P1
*Plastic axial 5.5292 N 5.8057 N 0.04348
*Plastic M1 0.09292 MPa 0.12544 MPa 0.5031 Rad
*Plastic M2 0.09292 MPa 0.12544 MPa 0.5031 Rad
*Plastic Torque 0.01014 MPa 0.13685 MPa 0.6421 Rad


Figure 6- 1. Pipe cross section




































2
L DB: propaOS3007_elasti-cK1004Model_. OOObclL_D30_OOO.odb AEAQUS/STANDARD Version 6.5-5 Thu Aug 30 14:54
Step: acro compression
Increment 334: Step Time = 1.000
Primary Var: SEE, SEE1
Deformed Var: U Deformation Scale Factor: +5.000e+00

Figure 6- 2. Contour of axial elastic strain in struts


0 P1 P2 qP'

L

Figure 6- 3. Data points generated for the perfect plastic model










81













300



250
d
E
E
A 200
O_


S150

, 1
-
C 100
()
()
0)
_T


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Elastic strain

Figure 6- 4. Stress intensity factor K1 vs. elastic strain in the crack tip strut





300



250

E
E
I 200



150



E 100
lOO /


01
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Plastic strain

Figure 6- 5. Stress intensity factor Kj vs. plastic strain in the crack tip strut













300



250

d
E
E
A 200
n


150

/


100 -
) /
U)


50



0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Total strain



Figure 6- 6. Stress intensity factor KI vs. total strain in the crack tip strut







































83









CHAPTER 7
CONCLUDING REMARKS AND SUGGESTED FUTURE WORK

In this dissertation, we have studied two types of foams: foams with rectangular prism unit

cells, including homogeneous foams and functionally graded foams, and tetrakaidecahedral

foams. The geometry of first one is simple and easy to model which provide a means to

understand fracture behavior of foams. The other one is close to reality as some carbon foams

can be approximated to be tetrakaidecahedral foams. Our approach to study the fracture

toughness of foams is a global-local approach wherein the microstructure was modeled in detail

near the crack tip (inner region), and boundary conditions are applied at far away points (outer

region) according to continuum fracture mechanics. Two crack propagation criteria, one at the

micro-scale and one at the macro-scale, are used. The fracture toughness of brittle foam is

calculated based on the stress intensity factor and the corresponding maximum tensile stress in

the struts ahead of the crack.

We have studied stress gradient effects on the homogeneous and graded foams with

rectangular prism unit cells. The fracture toughness of the foam could be predicted by the

strength of the strut or ligament material and the shape and size of the cells that constitute the

foam. An analytical model of fracture toughness was derived. Different loading cases were

studied by using a micro-macro combined method. Fracture toughness of the homogeneous foam

decreases as the crack size increases except for remote displacement loading cases. The aspect

ratio of the plate does not have much effect on the fracture toughness. As the cell size becomes

smaller, the fracture toughness of the homogeneous foams under different types of loads

becomes uniform. Since the relative differences of the fracture toughness of the homogeneous

foam under different loads are within +5%, the fracture toughness can be treated as a material

property. The fracture toughness of the analytical model agrees with that determined by the









combined micro-macro-mechanics method. It is found that the fracture toughness of graded foam

equals to the fracture toughness of homogeneous foam with the same cell as that of the graded

foam at the crack tip. The fracture toughness does not simply depend on the relative density. It

also depends on both the material and the shape and size of the cell.

Approach 2 has been used to study the fracture toughness of tetrakaidecahedral foam. We

obtain the plain-strain fracture toughness of the foam by relating the fracture toughness to the

tensile strength of the cell struts. Also, we have studied the effects of various geometric

parameters that describe the cell. The fracture toughness decreases as strut length L increases for

the foam with the same strut thickness D. For the same strut length, as D increases the fracture

toughness increases. However, the dimensionless fracture toughness only depends on the relative

density. In the study of the dislocation imperfection effects, we find that dislocation distance Ra

has no significant effect on the elastic modulus. But it has a huge effect on fracture toughness.

The deviation of the fracture toughness increase as the dislocation distance Ra increases.

Finally, we have taken the first step to study plastic deformation near crack tip.

However, there are some supplement study and new areas needed to be studied. We only

have one experimental result for homogeneous foam for comparing our results. Hence,

experimental study could be an area of future study. Since foams other than brittle foams are

widely used, large deformation of foams will be an interesting topic. There are also some

research needs in open-cell foams used in cooling system wherein hot air/fluid flows through the

foam. As energy-absorption function, foams are under compression and closed-cell foams are

often used. Hence, fracture behavior under compression and research on closed-cell foams are

good future topics.








APPENDIX A
CRACK TIP DISPLACEMENT FIELDS FOR ORTHOTROPIC MATERIALS

The stress-strain relation in the principal direction for plane stress problem is given as:

R 1 1/E* -v21/E2 0 (a
2 = [Slo{a = -v12/E* 1/E2 0 2 (A.1)
712 0 0 1/G*2 _r2

The stress-strain relation can be transformed from the principal 1-2 coordinate system to

the x-y coordinate system by using the transformation matrix [T]:


-2 y=[T r.y (A.2)


where the transformation matrix is defined as:

cos2 o sin2 0 2 cos 0 sin 0
[T]= sin2 0 cos2 0 -2 cos 0sin 0 (A.3)
-cos 0sin cos 0 sin 0 cos2 0 sin2 0

The compliance matrix[S] in the x-y plane is

s11 S12 S16
S]= I2 S22 S =[T [S][T] (A.4)
LS16 g26 S66J

The characteristic equation of the orthotropic material is given by Sih and Liebowitz(1968)

S,, 4 2S,16U3 +(2S12 + S66) I2 2261 + S22 = 0 (A.5)

There are four roots of the characteristic equation. We denote sl and s2 as the two unequal

roots with positive conjugate values:

S1 = A/ = al + i/'f s2 = u=2 a2 i82 (A.6)

The constantspj and qj (j=1,2) are related sl and s2 as bellows









P, = S,112 + S12 S16S1, P2 S 11s21 + S12 S16s2 (A.7)

S12S1 + S22 S26S1 S12S + S22 S26 2
q= q2=
Sl S2

The displacement field in the vicinity of crack tip is a function of the orthotropic material

parameters pi,, q2, q2, 2, and s2 as shown in Eq(2.5).

For plane strain problem, the strain and stress relation is

0 C,, C12 0 E
2 =[] 21 C22 0 o c2 (A.8)
12 0 0 C66 712


where C11 E 2 3 C2 E2 + 2 V13V23

I E E3 V21

C22 D E2 3 123 C21 = C12, C66 =G12,


D E1E2E3 V23 E2 v12E 2E3 2v12v13v23 E 2 123E2E
D EE, v2 C E 2 V EE 2V12V13V23E2E3 VI3E2E3
E, E2E3

And then the compliance matrix [S ] in the 1-2 coordinates is the inverse matrix of [Q ]:


[S,]= [Q,]1 (A.8)

In order to obtain the displacement field near the crack tip, simply replace [S] in Eq.(A.4)

with [S,] and then the solution takes the same form as plane stress problems.









APPENDIX B
FRAME ELEMENT IN ABAQUS

Space frame elements are used in the study of tetrakaidecahedral foam. Forces and

moments on a frame element in space are shown in Figure B-1. These forces and moments can

be output at three nodes, that is, two end nodes and the middle node.

Since equilateral triangle is not a default cross section in ABAQUS, general cross section

option is used in the frame element for tetrakaidecahedral foam. Area A, the moment of inertia I,

and 12, the polar moment of inertia J are required for input data. Those values could be

determined by the equations listed in Table B-1. Since those values are needed to calculate stress

in the struts, Table B-1 also gives equations for other types of cross section used in this study.

If we ignore the shear stress, stress at a point in the section at the middle of a strut is given

as

N Mxx M~x
oe = + +i (B.1)
A I, I,

For rectangular prism foam, since M2=0 this foam becomes


N MI N 6M1
ten -2+ = + 3 (B.2)
te" A I h h

In order to include elastic-plastic response, we need specify N, M1, M2, and T directly as

functions of their conjugate plastic deformation variables. The plasticity is lumped at the element

ends. There are no plastic strains as output in the frame element. Plastic displacements and

rotations in the element coordinate system are output for plastic deformation. ABAQUS assumes

the displacement and rotation increments can be decomposed into elastic and plastic parts. We

can obtain plastic deformation in axial direction by adding the plastic displacements on element

ends. Then the plastic strain is assumed to be the plastic deformation divided by the element









length. However, this simple method is an approximate method to evaluate the plastic

deformation near the crack tip.


Table B-1. Cross section of frame element
Cross section

Equilateral triangle


~/ D
-:D


Pipe


Square


2



---4---


Equations


A = D2
4


18 2

1 18


A2
J=,f
5s


27r (r

(4


;(r4
2


0.5t)t

(r t)4)



( r t)4
4)*



4s'


A=h2

h4
',



h4
1 12


J= h
6











I N N,



T ---N*-

M, 1
I


T
X /.,


3


iL L


Figure B-l. Forces and moments on a frame element in space.









LIST OF REFERENCES


1. Chen, Y.-R., Columbia .\lmnit Tragedy, in
http://www.csa.com/discoveryguides/shuttle/overview.php. 2003.

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5. Brezny, R., D.J. Green, and C.Q. Dam, Evaluation of Strut Su.engil in Open-Cell
Ceramics. Journal of the American Ceramic Society, 1989. 72(6): p. 885-889.

6. Brezny, R. and D.J. Green, The Effect of Cell-Size on the Mechanical-Behavior of
Cellular Materials. Acta Metallurgica Et Materialia, 1990. 38(12): p. 2517-2526.

7. Gibson, L.J. and M.F. Ashby, Cellular Solids: structure and properties. 2 ed. 1997:
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Materials, 2003. 37(23): p. 2101-2116.

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toughness offunctionally graded foams. International Journal of Solids and Structures,
2007. 44: p. 4053-4067.

11. Rice, J.R., A Path Independent Integral and Approximate Analysis of Strain
Concentration by Notches and Cracks. Journal of Applied Mechanics, 1968. 35(2): p.
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Problems. Journal of Applied Mechanics-Transactions of the Asme, 1981. 48(4): p. 825-
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14. Shih, C.F., B. Moran, and T. Nakamura, Energy-Release Rate Along a 3-Dimensional
Crack Front in a Thermally Stressed Body. International Journal of Fracture, 1986. 30(2):
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functionally graded materials using the domain integral. Journal of Applied Mechanics-
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BIOGRAPHICAL SKETCH

Junqiang Wang was born in China in 1973. He received his Bachelor of Engineering in

mechanical engineering from University of Science and Technology Beijing in 1996. He worked

for 2 years for Qinhuangdao Branch of Baotou Engineering and Research Corp. of Iron and Steel

Industry, China. He received his master's degree in the speciality of materials processing

engineering in Tsinghua University, China. He also got a Master of Science in mechanical

engineering at University of Florida. He is pursuing his doctoral degree at the Center for

Advanced Composites in the Department of Mechanical and Aerospace Engineering, University

of Florida.





PAGE 1

1 FRACTURE TOUGHNESS OF CELLULA R MATERIALS USING FINITE ELEMENT BASED MICROMECHANICS By JUNQIANG WANG A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2007

PAGE 2

2 2007 Junqiang Wang

PAGE 3

3 To my parents, Shiming Wang and Yuezh en Jing, and my wife, Baoning Zhang

PAGE 4

4 ACKNOWLEDGMENTS First, I thank my advisor, Dr. Bhavani Sanka r for his support, guida nce and dedication. It is very fortunate for me to work with him. I sincerely thank my committee members, Dr. John Mecholsky, Jr., Dr. Jacob Chung, and Dr. Ashok Ku mar, for participating and evaluating my research work. Its our tradition that parents devote all their lives to thei r children. I thank my father, Shiming Wang, and my mother, Yuezhen Jing, for their support and love. I thank my wife, Baoning Zhang, for all of her support throughout my graduate study. I thank my son, Alex Wang, for reminding me why I am working on this dissertation.

PAGE 5

5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........7 LIST OF FIGURES................................................................................................................ .........9 ABSTRACT....................................................................................................................... ............12 CHAPTER 1 INTRODUCTION..................................................................................................................14 Background..................................................................................................................... ........14 Literature Review.............................................................................................................. .....15 Fracture Toughness.........................................................................................................15 Functionally Graded Foam..............................................................................................16 Tetrakaidecahedral Foam................................................................................................17 Objectives..................................................................................................................... ..........18 Scope.......................................................................................................................... .............18 2 APPROACHES FOR PREDICTING FRACTURE TOUGHNESS......................................22 Approach 1..................................................................................................................... .........22 Approach 2..................................................................................................................... .........24 Comparison of the Two Approaches......................................................................................25 3 HOMOGENEOUS AND FUNCTI ONALLY GRADED FOAMS.......................................28 Homogeneous Material Properties.........................................................................................28 Material Models for Graded Foams........................................................................................30 Loading Cases.................................................................................................................. .......31 An Analytical Model for Fracture Toughness........................................................................31 Results and Discussion......................................................................................................... ..34 Bending Loading Case....................................................................................................34 Stress in Microstructure...................................................................................................35 Stress Gradient Effects on Fractur e Toughness of Homogeneous Foam........................35 Fracture Toughness of Functionally Grad ed Foam with Non-uniform Strut Thickness.....................................................................................................................36 Fracture Toughness of Functionally Graded Foam with Non-uniform Cell Length.......37 Conclusion..................................................................................................................... .........39 4 FRACTURE TOUGHNESS PREDICTION OF A TETRAKAIDECAHEDRAL FOAM...54 FEM Model of a Unit Cell......................................................................................................54

PAGE 6

6 Elastic Moduli of Homogeneous Foam..................................................................................54 Fracture Toughness............................................................................................................. ....55 Parametric Study.............................................................................................................56 Mode I fracture toughness........................................................................................56 Mode II fracture toughness......................................................................................57 Progressive Fracture and Crack Propagation..................................................................57 Summary and Conclusion.......................................................................................................58 5 IMPERFECTION EFFECTS..................................................................................................68 Homogeneous Material Properties.........................................................................................68 Fracture Toughness............................................................................................................. ....69 Summary and Conclusion.......................................................................................................70 6 PLASTIC DEFORMATION NEAR CRACK TIP................................................................77 Elastic Deformation Analysis.................................................................................................77 Plastic Deformation Analysis.................................................................................................78 7 CONCLUDING REMARKS AND SU GGESTED FUTURE WORK..................................84 APPENDIX A CRACK TIP DISPLACEMENT FIELDS FOR ORTHOTROPIC MATERIALS................86 B FRAME ELEMENT IN ABAQUS........................................................................................88 LIST OF REFERENCES............................................................................................................. ..91 BIOGRAPHICAL SKETCH.........................................................................................................94

PAGE 7

7 LIST OF TABLES Table page 3-1 Material properties of struts.............................................................................................. .40 3-2 List of load cases......................................................................................................... .......40 3-3 Axial forces for Unit cell: c1= c2= c3=200 m, h =20 m; Load: remote traction................40 3-4 The ratio for cell size: c2= c3=200 m, h =20 m, and c1 varies......................................40 3-5 The ratio for cell size: c1= c2= c3=200 m, h varies..........................................................40 3-6 Mode I fracture toughness(5Pam0.5)............................................................................40 3-7 Tip stress at first unbroken strut normalized with total tip stress for the case with cell: c =200 m, h =20 m and a / W =0.5.....................................................................................41 3-8 Stress in the first unbroken strut normalized by total crack tip stress for the case with cell: c =50 m, h =5 m and a / W =0.5.................................................................................41 3-9 Stress in the first unbroken strut normalized by total crack tip stress for the case with cell: c =200 m, h =20 m and a / W =0.1.............................................................................41 41 Material properties of struts............................................................................................. ..59 4-2 Convergence study of fracture toughness..........................................................................59 4-3 Convergence study of fracture toughness..........................................................................59 44 Fracture toughness for strut length L =1mm by using 40 cells....................................59 45 Fracture toughness for strut length L =2mm by using 40 cells....................................59 46 Interpolation parameters for Mode I..................................................................................59 47 Mode II racture toughness for strut length L =1mm by using 30 cells.........................60 48 Mode II racture toughness for strut length L =2mm by using 30 cells.........................60 49 Interpolation parameters for Mode II.................................................................................60 410 Maximum stress in the struts ahead of crack tip and mode l fracture toughness................61 51 Equivalent ma terial properties...........................................................................................71 52 Numerical results with respect to Ra..................................................................................72

PAGE 8

8 61 Load criteria............................................................................................................. ..........80 B-1 Cross section of frame element..........................................................................................89

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9 LIST OF FIGURES Figure page 1-1 Microstructure of a cellular medium w ith rectangular unit ce lls: unit cell with cell lengths c1, c2 and c3............................................................................................................20 12 Micrograph of an AFRL carbon foam...............................................................................20 13 Three tetrakaidecahedral cells with strut length l and thickness t in a BCC lattice...........21 2-1 An edge-cracked plate: H -height; a-crack length; W -width..............................................26 2-2 Finite el ement models...................................................................................................... ..26 2-3 Crack tip in micr omechanical model.................................................................................26 2-4 Stress field near crack-tip................................................................................................ ...27 3-1 Microand Macro-stress es in an open-cell foam...............................................................42 3-2 Cell deformation by cell strut bending...............................................................................42 3-3 Example of graded foam with 50 strips and the discrete elastic modulus compared with the actual modulus.....................................................................................................43 3-4 Five types of loading...................................................................................................... ....43 3-5 Crack-tip forces and moments and corres ponding crack tip stress es in the idealized homogeneous continuum...................................................................................................44 3-6 Mode I fracture toughness as a function of relative density..............................................44 37 Cubic foam with c =1.8mm, u=3.5805 MPa.....................................................................45 3-8 The bending-tensile ratio convergence test for cell length over strut thickness c / h =10....45 3-9 Results of homogeneous foams under the remote bending load........................................46 3-10 Normalized stress in the struts ahead of the crack tip for remote prescribed displacement load case for the cell with h =5 m, c =50 m..............................................47 3-11 Fracture toughness under different loads for the foam ( c =200 m, h =20 m)..................48 3-12 Fracture toughness under different loads for the foam ( c =50 m, h =5 m)......................48 3-13 Fracture toughness under remote bending load. (Plate size: W =0.1, H / W =1; Graded foam: h0=10 m; =2 m; c = c1= c2= c3=200 m)...............................................................49

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10 3-14 Fracture toughness under remote bending load. (Plate size: W =0.1, H / W =1; Graded foam: h0=10 m; =2 m; c = c1= c2= c3=200 m)...............................................................49 3-15 Comparison of Eq(3.22) with Eq.(3.28)............................................................................50 3-16 Comparison of three forms................................................................................................50 3-17 KIc is normalized with the analytical value. (Plate size: W =0.1m, H / W =1; Graded foam: h0=30 m; =-2-4; c = c1= c2= c3=200 m)............................................................51 3-18 Plate size: W =0.1m, H / W =1; Graded foam: h0=30 m; =-2-4; c = c1= c2= c3=200 m............................................................................................................51 3-19 Fracture toughness under remote bending load. (Plate size: W =0.1m, H / W =1; Graded foam: c0=200 m; = -0.15023 m; h =20 m)..................................................................52 3-20 Fracture toughness under remote bending load. (Plate size: W =0.1m, H / W =1; Graded foam: c0=200 m; = -0.15023 m; h =20 m)..................................................................52 3-21 KIc is normalized with the analytical value; Plate size: W =0.1m, H / W =1; Graded foam: c0=200 m; = -0.15023 m; h =20 m....................................................................53 3-22 Plate size: W =0.1m, H / W =1; Graded foam: c0=200 m; = -0.15023 m; h =20 m........53 4-1 A tetrakaidecahedral unit cell and the cross section of a strut...........................................62 4-2 A structure with 27 (3) cells......................................................................................62 4-3 Convergence study of Youngs modulus...........................................................................63 4-4 Deformation of a micromechanical mode..........................................................................63 4-5 Convergence study of Mode I fracture toughness.............................................................64 4-6 Mode I fracture toughness vs. strut thickness....................................................................64 4-7 Mode I fracture toughness vs. relative density..................................................................65 4-8 Normalized Mode I fracture toughness vs. relative density..............................................65 4-9 Mode II fracture toughness vs. strut thickness...................................................................66 4-10 Mode II fracture toughne ss vs. relative density.................................................................66 4-11 Normalized Mode II fracture toughness vs. relative density.............................................67 4-12 Crack development history................................................................................................67 51 Dislocation of a vertex................................................................................................... ....73

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11 52 Relative density as a func tion of the dislocation distance Ra of a vertex...........................73 53 Effective moduli vs. Ra......................................................................................................74 54 Equivalent Poissons ratio versus Ra..................................................................................74 55 Ra=0.50: left whole finite element model; right scaled structure near the crack tip....75 56 Fracture toughness versus Ra.............................................................................................75 57 An example of the structure near th e crack tip with only one imperfect cell ( Ra=0.5) ahead of the crack tip.........................................................................................................76 58 Fracture toughness versus Ra for foams with one imperfect cell ahead of the crack tip...76 61 Pipe cross section........................................................................................................ .......80 62 Contour of axial elastic strain in struts..............................................................................81 63 Data points generated fo r the perfect plastic model...........................................................81 64 Stress intensity factor KI vs. elastic strain in the strut........................................................82 65 Stress intensity factor KI vs. plastic strain in the strut.......................................................82 66 Stress intensity factor KI vs. total strain in the strut...........................................................83 B-1 Forces and moments on a frame element in space.............................................................90

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12 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy FRACTURE TOUGHNESS OF CELLULA R MATERIALS USING FINITE ELEMENT BASED MICROMECHANICS By Junqiang Wang December 2007 Chair: Bhavani V. Sankar Major: Mechanical Engineering A finite element method based micromechanical analysis is used to understand the fracture behavior of homogeneous and functionally graded foams. Both rectangular prism and tetrakaidecahedral unit cells ar e studied. Two approaches of pr edicting fracture toughness of foams and other cellular materials are used in this study. In one approach, the finite element analysis uses a micromechanical model in conjunction with a macro-mechanical model in order to relate the stress intensity factor to the stresse s in the struts of the foam. The stress intensity factor at the crack tip of th e macro-mechanical model can be evaluated using either the J -contour integral or the stresses in the singularity-dominated zone. The other approach is to directly apply displacements based on the K -field on the boundary of the micromechanical model. Using the first approach, the mode I fractur e toughness is evaluated for various crack positions and length. Both homogeneous foam and graded foam are studied to investigate the effect of stress gradients in the vicinity of th e crack-tip on the fracture toughness. Various types of loading such as remotely applied displacem ents, remote traction ar e studied. Preliminary results of this study show that the stress gradie nt has slight effects on the fracture toughness. However, since the effects are relatively small, KIc can be defined as a material property. Then

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13 the relationship between the fract ure toughness of the graded foam and the local density at the crack tip is studied. The second approach is easy to apply in pr edicting the fracture toughness of homogeneous foam. By using this approach, convergence study of a micromechanical model is conducted. Also, an analytical model for the mode I fractur e toughness of foams with rectangular prism cells is introduced. The mode I and mode II fracture toughness of homogeneous foam consisting of tetrakaidecahedral unit cells ar e predicted. A parametric study is performed to understand the effect of the geometric parameters of the unit ce ll and tensile strength of the foam ligament and also dislocation imperfection in the foam.

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14 CHAPTER 1 INTRODUCTION Background Cellular materials are made up of a network of beam or plate like structures. There are a number of cellular materials that occur in nature, such as honeycombs, wood, bone, and cork. Cellular materials can offer high thermal resist ance, low density, and high energy-absorption. Foams are a class of cellular so lids, generally made by dispersing gas into a liquid material and then cooling it to solidify. Foams are categorized as open-cell and closed -cell foams. According to the materials made into foams, foams are also categorized as polymeric, metallic, and ceramic foams, e.g., carbon foams. Due to rapid developm ents in material science and manufacturing techniques, a wide variety of foam s have been developed and used in automobiles, aircrafts, and space vehicles. A special example is the thermal protection system (TPS) of space vehicles, e.g., Space Shuttle. Traditional TPS cannot bear loads as they are designed for very low thermal conductivity, and are easy to damage, which increases the risk of flight. For instance, a disassembled tile of old TPS caused the tragedy of the Shuttle Columbia in 2003[1]. NASA has st arted the study of novel TPS concepts for the Crew Exploration Vehicle, which is essentially a replacement for the Space Shuttle. An Integral Thermal Protection System (ITPS) concept is a new idea in which the loadbearing function and insulation ar e combined into a single struct ure. This new concept can be achieved by using foams as core of the sandwich st ructures since foams can be tailored to obtain optimum performance. Under such conditions foam s are subjected to various mechanical loads and extreme heat loads. Thus there exists an urgent need for th e study on fracture toughness and other material properties of foams.

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15 Literature Review Fracture Toughness The most important parameter of a cellular material is the relative density */ s where is the density of the cellular material or foam and s the solid density, which is the density of the of the strut or ligament material. The relative de nsity is a measure of so lidity, and most of the material properties depend on the relative dens ity. Analytical methods for determining the mechanical and thermal properties of cellular soli ds are well documented. However, research on fracture behavior of foams is still at its infanc y. Maiti, Ashby, and Gibson[2] found that Mode I fracture toughness KIc is proportional to ( */ s)3 for open cell and to ( */ s)2 for closed cell foams. Huang and Gibson [3, 4] studied several open-cell foams with short crack and further confirmed the above conclusion. Brezny and Green [5, 6] experimentally veri fied the factors that determined the fracture toughness in the theore tical model. Gibson and Ashby [7] summarized the formulations for Mode I fracture toughness. R ecently, Choi and Sankar [8, 9], and Lee [10] presented new results on fractur e toughness of open-cell foams. In a homogeneous continuum the near-tip stre ss and displacement fields uniquely depend on the stress intensity factor (SIF ). It is important to obtain accurate SIF value, which could be calculated from crack-tip stresses. However, it is di fficult to obtain accurate stress fields by using FEM because of the existence of the singular ity. In order to improve the accuracy, more elements are needed near the crack tip, which causes more computationa l cost. Another way to calculate SIF is based on the relation between the SIF and the J -integral. For homogeneous materials, the J -integral is path independent, which allow us to get accurate J along a path away from the crack tip.

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16 Rice[11] introduced the path-independent J integral for elastic solids under isothermal conditions. A general form of the J integral, suitable for elastic or elastic-plastic thermal crack problems, is defined by Aoki et al.[12]. Jin [13] used this integral to solve thermal fracture problems of inhomogeneous materials. However, this form is not a standard J -integral. Shih et al. [14] provided a domain integral of J and it has been proved to be more efficient and more accurate than the direct calculation of the J -integral and is suitable for elastic, thermal elastic, and plastic materials. Gu et al. [ 15] applied this domain integral to evaluate the crack-tip field in inhomogeneous materials, such as functionally graded materials (FGM). The commercial software ABAQUS also uses this doma in integral method to calculate the J integral. Another approach to investigate the fractur e toughness is applying displacement boundary conditions corresponding to a given SIF. Choi a nd Sankar[8, 9] first used this method to study the fracture toughness of some carbon foams. Most recently Fleck and Qiu[16] have used this method to study the damage tolerance of el astic-brittle, 2-D isotropic lattices. Functionally Graded Foam One should distinguish functionally graded foam (FGF) from functionally graded materials (FGM). FGMs are a combination of two materials, e.g., a mixture of metals or ceramics, to create a desired composite. However in our study, we assume the material properties of the solid material are isotropic and only the cell size or the strut thickness varies al ong one direction in the cellular medium. However, both FGF and FGM ha ve thermal and mechanical inhomogeneities, and the computational methods used to analyze FGMs are suitable for FGFs also. Some of the results and conclusions on the behavior of FGMs also apply to FGFs.

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17 There are a large amount of analytical st udies available on FGMs. Erdogan and his coworks[17-19] provided analytical solutions of some fracture pr oblems for FGM. They found the square root singularity of cracktip stress is the same as that in a homogeneous material. Jin and Noda[13] showed that temperatur e distribution, and elastic or plastic crack-tip singular fields of nonhomogeous materials are the same as those of homogeneous materials. Gu and Asaro[20] analytically studied a se mi-infinite crack of a FGM. They c oncluded that material gradients do not affect the order of the singularity and the angular function, but do affect stress intensity factors (SIF). The near-tip stresses have the sa me form as that for a homogeneous material and the propagation direction is the direction of ma ximum energy release rate Sankar[21] derived an elasticity solution for functionally graded beams with the conclu sion that the stress concentrations occur in short or thick beams. They are less than that in homogeneous beams, when the softer side of FG beam is loaded and th e reverse is true when th e stiffer side loaded. Tetrakaidecahedral Foam It has been accepted for a long time that te trakaidecahedron, packed in the BCC structure, satisfies the minimum surface energy for monodispersed bubbles [22]. Only in 1994 a little better example with smaller surface energy was found by Wear ire and Phelan [23]. The tetrakaidecahedral foams have held the intere st of researchers for decades. Microcellular graphitic carbon foams was first developed at the US Air For ce Research Laboratory in the 1990s [24]. The repeating unit cells of this foam can be approximated by a regular tetrakaidecahedron[25]. Micromech anical models have been used to predict mechanical properties such as Youngs modulus, bulk modulus, yield surface, etc. Warren and Kraynik [26] studied the linear elastic behavior of a low-density Kelvin foam. Zhu [27] provided an analytical solution of the elastic moduli. Li and Gao et al [25, 28] developed some micromechanics models to analyze the homogeneous material propertie s and simulate the macroscopic mechanical

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18 behavior under compressive loading. Laroussi et al.[29] studied the compressive response of foams with periodic tetrakaidecahedral cells. A failure surface is define d in macroscopic stress space by the onset of the first buckling-type in stability encountered along proportional load paths. Ridha et al. [30] obtained a fracture mode l for rigid polyurethane foam based on the first tensile failure of any stru t in the cell. However, fr acture toughness prediction of tetrakaidecahedral foam is a new field, and ther e is no published work available in this topic. Objectives In this research, we plan to study open-c ell foams with the unit cell shown as in Figure 1-1. Since this is one of the simplest unit cells, it is easy to model and expected to be helpful in understanding the fracture behavior of cellular solids. Our focus is the effect of stress gradients on the fracture toughness. Both homogeneous and gr aded foams are investigated. We calculate homogeneous material properties based on the ce ll geometry and its material properties. And then the fracture behaviors of an edged-cr ack specimen with the homogeneous material properties under different mechanical or ther mal loadings are studi ed. A commercial FEM software ABAQUS is used for FEM calculation s and the input files of FEM are generated by MATLAB. Since the unit-cells of many foam s such as the carbon foam in Figure 12 could be well approximated by tetrakaidecahedrons (Figure 13), we shall do further study on the foam made of this unit cell. Scope This research reviews some background info rmation on cellular materials\foams including fracture toughness determination, functionally graded foam anal ysis, and tetrakaidecahedral foam study. We discuss two approaches to dete rmine fracture toughness of foams which are used in our study, and describe the finite element analysis of homogeneous and functionally graded

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19 foams under different types of loading. We deve lop an analytical model for fracture toughness and use it to compare the FEM results. We pr ovide parametric study of fracture toughness of tetrakaidecahedral foams, and analyze dislocat ion imperfection effects on material properties such as elastic modulus and fr acture toughness. We briefly discu ss some plastic deformation in the struts near crack tip in ductile foams. C oncluding remarks and future work are included.

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20 Figure 1-1. Microstructure of a cellular medium with rectangul ar unit cells: unit cell with cell lengths c1, c2 and c3. Figure 12. Micrograph of an AFRL carbon foam[24]

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21 Figure 13. Three tetrakaidecahedral cells with strut length l and thickness t in a BCC lattice

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22 CHAPTER 2 APPROACHES FOR PREDICTI NG FRACTURE TOUGHNESS Our approach is a global-local approach wher ein the microstructure is modeled in detail near the crack tip (inner region), and boundary cond itions are applied at fa r away points (outer region) according to continuum fracture mechanics. The foam in the outer region is modeled as a homogeneous orthotropic material We also use two crack propaga tion criteria, one at the microscale and one at the macro-scale. For brittle fo ams, once we know the stre ss intensity factor at macroscale and the corresponding maximum tensile stre ss (microscale) in the struts ahead of the crack, we can calculate the fracture toughness of the foam by the following equation: tip I ICuK K or I ICu tipK K (2.1) where KI is Mode I stress intensity factor, KIc is Mode I fracture toughness, u the tensile strength of struts or the foam ligaments, and tip the maximum tensile stress in the first unbroken strut ahead of the crack tip. There are two approaches of predicting the fr acture toughness of foams used in this study. Approach 1 As an example, we study an edge-cracked plate and impose the displacements around the outer region surrounding the crack. The maximum tens ile stress in the micros tructure is obtained from a local model of the inner region. The stre ss intensity factor is obtained from the macro model of the edge-cracked plate. The edge-cracked plate is shown in Figure 2-1. The plate is comprised of microstructure with the unit cell shown as Figur e 1-1. Due to the symmetry of th e geometry and loads, only onehalf of the plate is analyzed (Figure 2-2). A multi-scale modeling approach consisting of three different length scales is used. Three models (Figure 2-2C) are used and they are: macro model,

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23 macro sub-model, and micro model. The macro s ub-model and micro model are attached to the macro model. The boundary conditio n (BC) of the macro sub-model is obtained from the macro model results and the BC of the micro model is obtained from the macro sub-model. In other worlds, the displacements of th e nodes on the boundary of the macro sub-model and the micro model are the same as those values at the same position of the macro model and macro submodel, respectively. The values are au tomatically obtained by ABAQUS. In the macro model, namely a model in m acro scale, different loads are applied to investigate the crack-tip field. The material properties of this model are calc ulated through homogenization (see equations in Chapter 3). Due to the stress singularity near the crack-tip, more elements are needed in this area to obtai n accurate crack-tip fields. The macro sub-model plays such a role that allows us to increase the number of elements near the crack-tip. The micro model is used to calculate the maximum tensile stresses in the unbroken strut ahead of the crack tip. Figure 2-3 shows the resultant force and bending moments in the strut of rectangular foam. The maximum tensile stress is given by 326 2tip tip tiptiptip tipbendten tiptiph M FMF I Ahh (2.2) Mode I stress intensity factor ( KI) can be determined by: 22 0lim,02I rKrr (2.3) where 22, r is the stress in the y -direction near the crack tip, and is a function of r and (see Figure 2-4). The stress intensity factors can also be calculated from the J -integral or energy release rate. Sih and Liebowbitz [31] pr esented such a relation for orthotropic materials

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24 1 1 1 2 2 2 2 1266 112222 11112 22Iaa aaa GK aa (2.4) where 11 11 a E 22 21 a E 33 31 a E and in present case 1223310 aaa and 44 231 a G 55 131 a G 66 121 a G E1 *, E2 *, E3 are Youngs moduli in x y and z directions respectively, and G12 *, G23 *, G32 are shear moduli in x-y y-z and z-x planes. In this study we use domain in tegral in ABAQUS software to calculate the energy release rate. In the case of graded foams the J -integral is not path-indepe ndent. This is because the graded foam is an inhomogeneous material in macr o-scale. Hence, we will use an extrapolation technique to calculate the energy release rate in graded foams. The stress near the crack tip is underestimated in the macro model and the m acro submodel can capture the square root singularity of the crack-tip st ress. SIF calculated by Eq (2.3) expected to agree well with that based on J -integral. Approach 2 Sih et al. [31] determined the K -field in the vicinity of a crack tip in homogeneous orthotropic materials. We can direc tly apply displacements based on the K -field on the boundary of the microstructure. The displacement fields near the crack tip for Mode I: 1212 1122211 12 1212 2122211 1221 Recossincossin 21 RecossincossinI Ir uKspssps ss r uKsqssqs ss (2.5) The displacement fields near the crack tip for Mode II:

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25 1212 12211 12 1212 22211 1221 Recossincossin 21 RecossincossinII IIr uKpsps ss r uKqsqs ss (2.6) The parameters p q and s are dependent on material elastic constants and they are given in Appendix A. After we find the maximum tensile stress in th e struts near the crack tip, we can use Eq (2.1) to obtain the frac ture toughness of the foam. Comparison of the Two Approaches Approach 2 is easier to use since only a mi cromechanical model is involved. Hence, this approach is good for convergence tests. However th is approach is related to a stress intensity factor for homogeneous foams so that it cannot be used to predict the fracture toughness of functionally graded foams. This simple expression for stress intensity factor hinders the use of the approach in the stress gradient effects analysis. On the other hand, Approach 1 requires a macro model except for the micromechanical model. The stress intensity factor is needed to be determined from the macro model. So there is much more effort involved in preparing th e finite element models and calculations. Since both approaches have advantages and disadvantages, the selection of the right approach depends on the needs of the research task.

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26 Figure 2-1. An edge-cracked plate: H -height; a-crack length; W -width. Figure 2-2. Finite element models: A) An edged-crack plate under remote prescribed displacement; B) Microstruc ture of the plate; C) Half model of the plate Figure 2-3. Crack tip in micromechanical model: Le ftcrack tip in microstructure; rightactual foam with resultant force and bending moment.

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27 Figure 2-4. Stress fi eld near crack-tip

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28 CHAPTER 3 HOMOGENEOUS AND FUNCTIONALLY GRADED FOAMS Homogeneous Material Properties At first some notation should be specified. Symbols with denote properties belonging to macrostructure or foam; symbols with a subscr ipt s are of the strut/ligament material. The material of the foam is orthotropic and so nine independent parameters are required to be determined. These nine parameters are Youngs moduli in x y and z directions ( E1 *, E2 *, E3 *), shear moduli in x-y y-z and z-x planes ( G12 *, G23 *, G32 *), and Poissons ratios in x-y y-z z-x planes (12 *, 23 *, 31 *). We choose the same carbon foam as Choi studied in [9].The material properties of microstructure are listed in Table 3-1. The Youngs modulus in y direction could be derived as depicted Figure 3-1. When the foam is loaded in the y direction, equilibrium requi res the force in the unit area equal to that in the strut. 2* 13 shcc2** 213 ssEhEcc (3.1) where h is the strut thickness, c2 and c3 are the cell length in y and z directions. Since the strains s of micro and macro structure in y direction are equal. We have 2 2 13 sh EE cc (3.2) Similarly we can obtain: 2 1 23 sh EE cc 2 3 12 sh EE cc (3.3) The derivation of shear modulus G12 is illustrated in Figure 3-2. Because of symmetry, there is no curvature at the half-len gth of the strut. And thus we can use a half beam to solve for 2.

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29 3 2 22 3sc F EI (3.4) Again equilibrium requires: 31Fcc (3.5) Substitute Eq.(3.5) into Eq.(3.4), we obtain 2 as 3 312 224sccc EI (3.6) In the same manner, 1 is 3 231 124sccc EI (3.7) The shear strain is given as 12312 12 12 1222 12sccccc ccEI (3.8) And the shear modulus G12 can be derived as 12 121231212sEI G ccccc (3.9) Substitute the moment of inertia 412h I 4 12 12312 s h GE ccccc (3.10) The shear moduli in the other two planes can be obtained by cyclic permutation as 4 23 12323 s h GE ccccc (3.11) 4 31 12331 s h GE ccccc (3.12)

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30 So far we have derived the shear and Y oungs moduli of the foam, the three undefined parameters are Poissons ratios. Based on Figure 3-1, we can see that th e strain in the x-direction is negligible and thus the Poissons ratio 12 is approximately zero. Fi nally we conclude that *** 1223310v (3.13) The relative density */s is an important parameter of foam, which is a measure of solidity. Based on the cells geometry, the relative density can be expressed as 23 123 1232s sccchh V Vccc (3.14) When cell length is much larger than strut thickness, the h3 term can be neglected. Furthermore, when c1=c2=c3=c, the relative density is 23(/)hc. Material Models for Graded Foams Two types of functionally graded foams are studied independently, namely, foams with non-uniform strut thickness and w ith non-uniform cell length. They are defined respectively by 0hxhx (3.15) 1 11 iicc (3.16) where and are constants, and 0his the strut thickness at left edge of the foam. In the first kind of foam, strut thickness varies in the x direction and cell length is constant, and the reverse for the second kind of foam. The orthotropic linear elastic material m odel is applied for the homogeneous foam. But more effort is needed for graded foams. The material properties of graded foams vary along xdirection since strut thic kness or cell length varies in the di rection. Instead of using graded elements as Santare[32], we divided the foam into small regions with constant material properties in each strip as Figure 3-3A shows. As long as the regions are small enough, the

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31 gradient material properties of foam can be approximated by constant material properties; Figure 3-3B is an example for Youngs modulus. Loading Cases Our main objective is to investigate stress gradient effects on fr acture behavior. Since different loads provide various st ress gradient, we compare the results of the foams subjected to five types of loading (Figure 3-4) including: A. Prescribed remote displacement; B. Remote traction; C. Crack surface traction; D. Remote bending; and E. Thermal loads. In total, six cases of the five types of loading, listed in Table 3-2, are studied. An Analytical Model for Fracture Toughness Maiti, Ashby and Gibson[2] used a KI field to calculate the crack -tip stress (Eq.(2.2)) of homogeneous foam. The force and the bending moment in the strut were obtained by integration. They assumed the bending stress in Eq.(2.2) is do minant and they ignored the tensile stress part. However, we find that in some cases the tensile stress is greater than the bending stress. The ratio of bending stress over the tensile stress is a constant, 0.415 in the present case, a nd thus neither could be negligible. For the foam with a simple cubic cell (c1=c2=c3=c, Figure 3-5), Choi and Sankar[9] introduced an effective length lc instead of using the act ual cell length, as shown in Eq. (3.18) and Eq.(3.19). 222IK r (3.17) 22 002ll IK Fcdrcdr r (3.18) 02l IK M crdr r (3.19)

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32 However, there is no reason to let the effective lengths in Eq.(3.18) and Eq.(3.19) to be equal. If the cell size is much sm aller than the crack size, the ho mogeneous stress field represents the stress field of microstructure accurately. Th en the homogeneous stress must be balanced by the tensile stress in the strut and thus we can get good results by setting lc in Eq.(3.18). More generally, in the case that the cell lengths in the three coordinate directions are not equal, this equation is rewritten as 1313 02 2c I IK FcdrKcc r (3.20) Table 3-3 gives an example that Eq.(3.20) is a good approximation of the axial force in the first unbroken strut. If KIc is a material property of the foam, KIc is a constant. And thus based on Eq.(2.1), the ratio ItipK must be a constant, which means tipICK where C is a constant. And therefore, the ratio bendten is a constant as a resu lt of Eq.(2.2) and Eq.(3.20). For convenience, denote the ratio as (6)/()bendten M Fh (3.21) Substituting Eq.(2.2), Eq.(3.21), and Eq.(3.20) into Eq.(2.1), we obtain 2 2 13 13 21 121 1IuIuIu Ic u bendtenten IKKK h K cc Kcc h (3.22) The relative density */s can be related to the cell le ngths and strut thickness with Eq.(3.14). And then we plot KIc versus relative density in Figure 3-6. It shows that the above equation agrees very well with Chois result s (Choi 2005: Fig. 13. and Eq. 19) for homogeneous foam. We also can see that the rela tive density alone cannot determine KIc, and KIc also strongly depends on cell size and shape. Figure 37 shows the comparison of current model with Chois

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33 results in [8] and Gibson and Ashby[7]. Our current model is almost the same as Chois result and it give a little smaller fracture toughness. The relative error between our model and the experimental results is 3%. More Discussion on the Ratio of Bending Stress over the Tensile Stress. The reason that the ratio is a constant lies in that the displacement fields in the vicinity of a crack tip in a homogeneous orthotropic material depend on the stress intensity fact or as discussed in Eq. 15 of Choi 2005. The displacements of the boundary n odes in the micro mode l are equal to the displacements at the same place of the homogeneous material, if there ar e enough cells near the crack tip. Thus the ratio must be a constant. Figure 3-8 shows that the ra tio converges to 0.409 as the number of beam elements increases. Table 3-4 and Table 3-5 show that the ratio varies for different foams with different unit cells. These results are obtained from microstr uctures with more than 40000 beam elements. The ratios vary a little. For the sake of simplification, a constant ratio =0.409 is used. The error between fracture toughness by using a fi xed ratio and by using the ratio listed in Table 3-4 and Table 3-5 could be determined by following procedure. Using a Taylor series expansion, we can rewrite Eq.(3.22) in terms of and as 1 11(1)1 (1)1 1ICAAAA K (3.23) 1ICA K (3.24) The absolute value of relative error is

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34 (1)1 1 (1)IcIc IcAA KK A K (3.25) Corresponding to the largest ratio 0.427 and the smallest ratio 0.383 liste d in the tables, the absolute value of relative errors are 1.28% and 1.85% respectively, which gives us confidence to use a fixed ratio in Eq.(3.22). Results and Discussion Bending Loading Case We studied different loading cases; here we only show some detailed results of bending loading to illustrate some conclusions. We investigate plates (Figure 2-1) with different aspect ra tios: 1, 2 and 8 using ABAQUS. Figure 3-9a shows that the J-integral increases as crack size increases. Figure 3-9c shows the maximum tensile stress in the first un broken strut ahead of the crack tip. Figure 3-9b gives the stress intensity factor calculated based on Eq.(2.4) Also, we compare the stress intensity with the analytical solution for H by Eq.(3.26) [33]. The FEM results ag ree well with the analytical solution. Finally, the fracture toughness is calculated by Eq.(2.1) and listed in Table 3-6. The relative errors of fract ure toughness are shown in Figure 3-9d, where the tr ue value is evaluated by the mean value of fracture toughness of Case H / W =8. The analytical solution by Eq.(3.22) is 4.555 Pam0.5. The results in Table 3-6 show the aspect ratio has very little effect on the fracture toughness. In other words, the plate si ze does not change the fracture toughness of the foam. 234 2341.121.397.31314Iaaaa Ka wwww (3.26)

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35 Stress in Microstructure As mentioned in se ction 4, the ratio of maximum bending stress and tensile stress in the first unbroken strut ahead of th e crack tip is a constant when the cell size is small. Table 3-7 and Table 3-8 show the variance of the ratio b ecomes less as the cell size decreases. Also by comparing the data in Table 3-9 with those in Table 3-7 we observe that the ratio varies for different crack sizes. Figure 3-10 is an example of the total stress, bending stress and tensil e stress in the struts ahead of the crack tip. The tensile stress is con tinuously distributed in th e struts ahead of the crack tip. But the bending stress is discontinuous, especially fo r the first three struts. This indicates it is difficult to derive an analytical form for the bendi ng stress in the first strut. Stress Gradient Effects on Fractu re Toughness of Homogeneous Foam Figure 3-11 and Figure 3-12 show the fracture toughne ss calculated using Eq.(2.1) under different loads. Since the fracture toughness of re mote displacement loading is almost constant, the fracture toughness is normalized with the mean value of the fracture toughness of Case 1remote displacement loading. The fracture tough ness of Case 2-Remote traction and Case 4Bending are almost the same. Both cases corresp ond to remote traction. The results of Case 2Surface traction and Case 5-Thermal 1 show simila r trends as the crack size increases. The case of Thermal 1 is involved with a negative stress intensity factor. There is a contact pressure occurring in the crack surface. This is similar to a cr ack surface traction loading. Comparing with Figure 3-11 and Figure 3-12, we can conclude that cell size does not change much of the distribution trends. But th e relative difference of fracture toughness for foams with small cells is smaller than that of foams with large cells. Also we can see that the

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36 stress ratio presented in previous section is not a constant. As a resu lt of variable ratio the fracture toughness varies a little. Fracture Toughness of Functionally Graded Foam with Non-uniform Strut Thickness We get the same conclusion as Lee[10] got for the remote displacement load case the fracture toughness of graded foam is the same as that of homogeneous foam with the same cell shape and size at the crack tip. This can al so be explained by the analytical Eq.(3.22). The fracture toughness depends on shape, size and material of the cell. The conclusion is illustrated by the remote bending case. Figure 3-13 shows the fracture toughness of the foam under remote bending load with h0=10 m and =2 m. The increasing h means that the crack propagates into highe r density region, and vi ce versa. The fracture toughness is very close to the analytical resu lt for homogeneous foam. Only the fracture toughness for increasing h case is a slightly grea ter than the analytical one on the right side of the figure. This phenomenon is similar to the homogeneous problem discussed in a previous section: the fracture toughness is greater than the mean value for the small crack size. The strut thickness h is related to the relative density by Eq.(3.14). The fracture toughness is plotted with respect to the relative density in Figure 3-14, where fracture toughness is calculated using Eq.(3.22). It shows that the fracture toughness linearly depends on the relative density. As mentioned previously, when cell leng th is much greater than strut thickness and c1= c2= c3= c the relative density is 23(/) hc Then the fracture toughness can be written as shown in Eq.(3.27). This equation illustrates that KIc linearly depends on the relative density*/ s 2* 13111 21321ICuu s hc K cc (3.27)

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37 By substituting =0.409 into Eq.(3.27), the dimensionless fracture toughness takes the following simple form as shown in Eq.(3.28). However, Figure 3-15 shows that the relative error increases dramatically near zero relative dens ity and approaches to 10% when the relative density is 0.05. And thus, this simple form does not work well. **11 0.2965 321IC s s uK c (3.28) Based on Eq.(3.14), h2 can be derived as *3*2*2*2* 222 11 2 3233333 31 3sssssccchc h h chc c (3.29) And then the fracture toughness is ****112 10.296510.3849 32133ICuu ssssc Kc (3.30) Comparing this form with the simple form Eq.(3.28), the relative error should equal to *0.3849 s and Figure 3-16 shows this conclusion. The stress gradient effects on th e fracture toughness are shown in Figure 3-17 and Figure 3-18. The fracture toughness is norma lized with the analytical one (Eq.(3.22)). Similar to the homogeneous foam, the fracture toughness of Ca se 2-Remote traction and Case 4-Bending are almost the same. The results of Case 3-Surf ace traction and Case 5-Thermal 1 show similar trends as the crack size increases. Fracture Toughness of Functionally Graded Foam with Non-uniform Cell Length We obtain the same conclusion for the non-unifo rm cell length case: the fracture toughness of graded foam equals to the fracture toughness of homogeneous foams with the same cell as that

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38 of the graded foam at the crack tip. Howeve r the fracture toughness doe s not linearly depend on the relative density for this case. c1 can be derived from Eq.(3.14) as 23 23 1 2 23()2scchh c cch (3.31) and thus the fracture toughness is relate d to the relative density as follows: 2*22 2 2 3323 1311 21212ICuu sc hhh K cccch cc (3.32) which shows the nonlinear relationship between th e fracture toughness and the relative density. The dimensionless fracture t oughness can be derived as Eq.(3.33), which shows the dimensionless fracture toughness depends not on ly on the relative density but also on the geometry of the foam. However, the dimens ionless fracture toughness linearly depends on the relative density of the foam since c2, c3 and h are constant. 2 23 22*2 23 3 1332323 1 221111 2 2121()221 1ICs s ucch K hhh c h ccccchhcc c cc (3.33) Figure 3-19 and Figure 3-20 show that the fracture toughness of the remote bending case depends on cell length or relative density. The stress gradient effects on th e fracture toughness are shown in Figure 3-21 and Figure 3-22. The fracture toughness is norma lized with the analytical one (Eq.(3.22)). Similar to the homogeneous foam, the fracture toughness of Ca se 2-Remote traction and Case 4-Bending are almost the same. The results of Case 3-Surf ace traction and Case 5-Thermal 1 show similar trends as the crack size increases.

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39 Conclusion Through this study, we find that the fracture toughness of the foam could be predicted by the strength of the strut or ligament material and the shape and size of the cells that constitute the foam. The crack-tip singular fields of the grad ed foam, as a nonhomogeneous material, are the same as those of homogeneous foam. Different lo ading cases are studied by using a micro-macro combined method. The effect of stress gradients in the vicinity of the crack-tip on the fracture toughness is studied. Our results l ead to the following conclusions: Except for remote displacement loading cas es, the fracture toughne ss of the homogeneous foam decreases as the crack size increases. The aspect ratio of the pl ate does not have much eff ect on the fracture toughness. As the cell size become smaller, the fract ure toughness of the homogeneous foam under different types of loads becomes uniform; Since the relative differences of the fractur e toughness of the homogeneous foam under different loads are within %, the fracture t oughness can be treated as a material property; The analytical model matches well with numerical results for both homogeneous and graded foams; the fracture toughness of the anal ytical model agrees with that determined by the combined micro-macro-mechanics method; The fracture toughness of graded foam equals to the fracture toughness of homogeneous foam with the same cell as that of the graded foam at the crack tip; The fracture toughness does not simply depend on the relative densit y. It depends on both the material and the shape and size of the cell.

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40 Table 3-1. Material properties of struts Density, s 1750 Kg / m3 Elastic Modulus, Es 207 GPa Poissons ratio, s 0.17 Ultimate Tensile Strength, u 3600 MPa Table 3-2. List of load cases. Case # Description 1. A Remote displacement: v0=7-5 m 2. B Remote traction: 0=57 Pa 3. C Crack surface traction: 0=5107 Pa 4. D Max remote bending stress: max=16 Pa 5 E Temperature change: 2()75000 Txx C 6 E Temperature change: 2()100 Txx C Table 3-3. Axial forces for Unit cell: c1= c2= c3=200 m, h =20 m; Load: remote traction a/W 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Force(FEM) (N) 33.20 66.66 106.2 154.7 217.7 307.2 452.9 Analytical Force (N) 33.50 67.37 107.4 156.5 220.3 310.9 458.4 Relative error* 0.90 1.05 1.09 1.13 1.19 1.19 1.19 Relative error=(Analytical FEM)/Analytical 100% Table 3-4. The ratio for cell size: c2= c3=200 m, h =20 m, and c1 varies c1 ( m) 100 150 200 250 300 0.383 0.399 0.409 0.415 0.420 Table 3-5. The ratio for cell size: c1= c2= c3=200 m, h varies h ( m) 10 20 30 40 0.396 0.409 0.419 0.427 Table 3-6. Mode I fracture toughness(105Pam0.5) a / W 100% 10 20 30 40 50 60 70 1 4.56 4.54 4.53 4.52 4.51 4.51 4.50 2 4.57 4.56 4.55 4.54 4.54 4.53 4.53 H / W 8 4.58 4.57 4.56 4.56 4.55 4.55 4.55

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41 Table 3-7. Tip stress at first unbroken strut normali zed with total tip stress for the case with cell: c =200 m, h =20 m and a / W =0.5 Load case # 1 2 3 4 5 6 Normalized bending stress 0.2797 0.2981 0.3000 0.29 98 0.2936 0.3085 Normalized tensile stress 0.72 03 0.7019 0.7000 0.7002 0.7064 0.6915 Ratio 0.388 0.425 0.429 0.429 0.416 0.446 Table 3-8. Stress in the first unbroken strut normalized by total cr ack tip stress for the case with cell: c =50 m, h =5 m and a / W =0.5 Load case # 1 2 3 4 5 6 Normalized bending stress 0.2875 0.2934 0.2944 0.29 38 0.2919 0.2957 Normalized tensile stress 0.71 25 0.7066 0.7056 0.7062 0.7081 0.7043 Ratio 0.404 0.415 0.417 0.416 0.412 0.420 Table 3-9. Stress in the first unbroken strut normalized by total cr ack tip stress for the case with cell: c =200 m, h =20 m and a / W =0.1 Load case # 1 2 3 4 5 6 Normalized bending stress 0.2801 0.2900 0.3025 0.29 08 0.2150 0.3009 Normalized tensile stress 0.71 99 0.7100 0.6975 0.7092 0.7850 0.6991 Ratio 0.389 0.408 0.434 0.410 0.274 0.430

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42 Figure 3-1. Microand Macro-st resses in an open-cell foam Figure 3-2. Cell deformation by cell strut bending: A) the undeformed cell and deformed cell; B) the loads, moments in a strut; C) The loads and moment in a half strut

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43 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 2 3 4 5 6 7 8 9 x 109 x (m)E (Pa) A B Figure 3-3. Example of graded foam with 50 st rips and the discrete elastic modulus compared with the actual modulus Figure 3-4. Five types of loading

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44 Figure 3-5. Crack-tip forces and moments and co rresponding crack tip stre sses in the idealized homogeneous continuum. (Ref er to (Choi 2005) Fig. 19) 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0 0.5 1 1.5 2 2.5 3 x 106 relative densityKI (Pa-m0.5) Choi: constant h constant h Choi: constant c Constant c Figure 3-6. Mode I fracture toughness as a function of relative density

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45 Figure 37. Cubic foam with c =1.8mm, u=3.5805 MPa 0 0.5 1 1.5 2 2.5 3 3.5 4 x 104 0.3 0.32 0.34 0.36 0.38 0.4 Number of elements(6M)/(Fh) Figure 3-8. The bending-tensile ratio convergence test for cel l length over strut thickness c / h =10

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46 0 0.2 0.4 0.6 0.8 0 2000 4000 6000 8000 10000 12000 a/WJ (N/m) H/W=1 H/W=2 H/W=8 0 0.2 0.4 0.6 0.8 0 0.5 1 1.5 2 x 106 a/WKI (Pam1/2) Analytical h/w= H/W=1 H/W=2 H/W=8 A B 0 0.2 0.4 0.6 0.8 0 2 4 6 8 10 12 14 x 109 a/Wtip (Pa) H/W=1 H/W=2 H/W=8 0 0.2 0.4 0.6 0.8 -15 -10 -5 0 5 x 10-3 a/WRelative error of KIc H/W=1 H/W=2 H/W=8 C D Figure 3-9. Results of homogeneous foams under the remote bending load: A) J -integral; B).Stress intensity factor; C).Tip stress; D) Relative differenc e in fracture toughness

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47 0 2 4 6 8 10 x 10-3 0 0.2 0.4 0.6 0.8 1 Normalized stress of micro model ahead of crack tip r (m)Normalized22 (Pa) Total stress Bending stress Tensile stress Figure 3-10. Normalized stress in the struts ahead of the crack tip for remote prescribed displacement load case for the cell with h =5 m, c =50 m.

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48 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.94 0.96 0.98 1 1.02 1.04 1.06 a/WNormalized KIc 1.Displ. 2.Remote traction 3.Surface traction 4.Bending 5.Thermal1 6.Thermal2 Figure 3-11. Fracture toughness under different loads for the foam ( c =200 m, h =20 m) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.985 0.99 0.995 1 1.005 1.01 1.015 1.02 1.025 a/WNormalized KIc 1.Displ. 2.Remote traction 3.Surface traction 4.Bending 5.Thermal1 6.Thermal2 Figure 3-12. Fracture toughness under different loads for the foam ( c =50 m, h =5 m)

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49 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 x 10-5 1 2 3 4 5 6 7 8 9 10 x 105 h(m)Fracture toughness(Pam1/2) Increasing h Decreasing h Analytical Figure 3-13. Fracture toughness under remote bending load. (Plate size: W =0.1, H / W =1; Graded foam: h0=10 m; =2 m; c = c1= c2= c3=200 m) 0.01 0.02 0.03 0.04 0.05 0.06 0 2 4 6 8 10 x 105 Relative densityFracture toughness(Pam1/2) Increasing density Decreasing density Analytical 0.01 0.02 0.03 0.04 0.05 0.06 0 0.005 0.01 0.015 0.02 Relative densityKIc/(u*c0.5) Increasing density Decreasing density Analytical Figure 3-14. Fracture toughness under remote bending load. (Plate size: W =0.1, H / W =1; Graded foam: h0=10 m; =2 m; c = c1= c2= c3=200 m)

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50 0 0.02 0.04 0.06 0 0.005 0.01 0.015 0.02 Relative density (*/s)KIc/(u*c0.5) Analytical 0.2965*/s 0 0.01 0.02 0.03 0.04 0.05 0.06 0 2 4 6 8 10 Relative density (*/s)Relative error (100%) Figure 3-15. Comparison of Eq(3.22) with Eq.(3.28) 0 0.02 0.04 0.06 0 0.005 0.01 0.015 0.02 Relative density (*/s)KIc/(u*c0.5) Analytical 0.2965*/s Modified 0 0.01 0.02 0.03 0.04 0.05 0.06 -2 0 2 4 6 8 10 Relative density (*/s)Relative error (100%) Relatie error 0.3849(*/s)0.5 Relative error 1 A B Figure 3-16. Comparison of three forms: A) Eq(3.22): analytical, and Eq.(3.30): modified; B) Relative error is between Eq.(3.22)and Eq.(3.28); Relative error 1 is between Eq.(3.22)and Eq.(3.30).

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51 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1 a/WNormalized KIc 1.Displ. 2.Remote traction 3.Surface traction 4.Bending 5.Thermal1 6.Thermal2 Analytical Figure 3-17. KIc is normalized with the analytical value. (Plate size: W =0.1m, H / W =1; Graded foam: h0=30 m; =-2-4; c = c1= c2= c3=200 m) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 2 3 4 5 6 7 8 9 10 x 105 a/WFracture toughness(Pam1/2) 1.Displ. 2.Remote traction 3.Surface traction 4.Bending 5.Thermal1 6.Thermal2 Analytical Figure 3-18. Plate size: W =0.1m, H / W =1; Graded foam: h0=30 m; =-2-4; c = c1= c2= c3=200 m

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52 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 x 10-4 4.6 4.8 5 5.2 5.4 5.6 5.8 6 6.2 x 105 c1(m)Fracture toughness(Pam1/2) Increasing c Decreasing c Analytical Homogeneous Figure 3-19. Fracture toughness under remote bending load. (Plate size: W =0.1m, H / W =1; Graded foam: c0=200 m; = -0.15023 m; h =20 m) 0.025 0.03 0.035 0.04 0.045 4.5 5 5.5 6 6.5 x 105 Relative densityFracture toughness(Pam1/2) Increasing c Decreasing c Analytical Homogeneous 0.025 0.03 0.035 0.04 0.045 0.009 0.01 0.011 0.012 0.013 0.014 0.015 0.016 Relative densityKIc/(u*L0.5) Increasing c Decreasing c Analytical Homogeneous Figure 3-20. Fracture toughness under remote bending load. (Plate size: W =0.1m, H / W =1; Graded foam: c0=200 m; = -0.15023 m; h =20 m)

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53 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.95 1 1.05 1.1 1.15 1.2 1.25 a/WNormalized KIc 1.Displ. 2.Remote traction 3.Surface traction 4.Bending 5.Thermal1 6.Thermal2 Analytical Figure 3-21. KIc is normalized with the analytical value; Plate size: W =0.1m, H / W =1; Graded foam: c0=200 m; = -0.15023 m; h =20 m 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 x 105 a/WNormalized KIc 1.Displ. 2.Remote traction 3.Surface traction 4.Bending 5.Thermal1 6.Thermal2 Analytical Figure 3-22. Plate size: W =0.1m, H / W =1; Graded foam: c0=200 m; = -0.15023 m; h =20 m

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54 CHAPTER 4 FRACTURE TOUGHNESS PREDICTION OF A TETRAKAIDECA HEDRAL FOAM FEM Model of a Unit Cell The tetrakaidecahedral unit cell that we propos e to study is a 14-sided polyhedron with six square and eight hexagonal faces. It is more pr ecisely called truncated octahedron, since it is created by truncating the corners of an octahedron [34]. From a different viewpoint, it can be generated by truncating the corner s of a cube [27]. All the edges of the cell are of equal length L and cross sectional area A The tetrakaidecahedral foam has a BCC lattice. The axes of the BCC lattice are parallel to the axes of the cube. Due to the symmetry of the structure, the Youngs moduli of the foam in the lattice vector di rections are equal: *** 001010100EEE (4.1) Each strut of the cell is treated as a beam el ement. In our study, the cross section of the struts is assumed to be an equ ilateral triangle with side length D (Figure 4-1). A reticulated vitreous carbon (RVC) foam will be studied and the material properties of the RVC are listed in Table 41. Elastic Moduli of Homogeneous Foam Zhu [27] obtained analytical expressions for the Youngs modulus and Poissons ratio based on the symmetry of the microstructure: 2 1000.726 11.09sE E (4.2) 1211.514 0.5 11.514 (4.3)

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55 where is the relative density, which is related to the side length D and strut length L as shown below: 20.4593(/) DL (4.4) In our study, the cross section of struts is an equilateral triangle with side length D (Figure 4-1). Using FEM, we verified the above e quation for the Youngs modulus. By applying a compressive load (Figure 4-2), we calculated th e nominal strain from the change in height of the structure and the original height, and the nominal stress is obtained by total resultant forces in y direction per unit area. Figure 4-3 shows that the Youngs modul us converges to the analytical solution as the number of cells increases. In the following sections, th e homogeneous material properties of the foam will be calculated with the above equations unless specified otherwise. Fracture Toughness We study the fracture toughness of plane strain problems for tetrakaidecahedral foams. Approach 2 is used to obtain fracture toughness, in other words, by imposing the displacements of KI field on the boundary to micromechanical mode l, we can obtain the maximum tensile stress near the crack tip from the ABAQUS results. Figure 4-4 gives an example of the deformation of the micromechanical model. However, in order to reduce the cost of computation and storage, we take advantage of the symmetry and model only one-quarter of the cellular medium. Two convergence tests are conducted: Case 1 in which the cell number is increased gradually in both x and y directions; Case 2 in which the cell number in x direction is increased and that in y direction is kept constant The results are listed in Table 4-2 and Table 4-3 and also shown in Figure 4-5.

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56 Parametric Study In the parametric study, the two parameters, L and D (see Figure 4-1), are varied to study their effects on the fracture toughness. At first, the detail results for Mode 1 fracture toughness will be presented. And the results of Mode 2 fracture toughness are also included. Mode I fracture toughness At first, we fixed the strut length at L =1 mm and varied the strut thickness. The effect of strut thickness on Mode I fr acture toughness is shown in Table 44. The procedures were repeated for L =2 mm and the corresponding results are given in Table 45. The results presented in Table 44 and Table 45 are also plotted in Figure 4-6. In general, we can conclude that the fracture toughness decreases as L increases for a given strut thickness D For the same L the fracture toughness increases as D increases. The relationship between fracture toughness and relative density is shown in Figure 4-7 and Figure 4-8. We use power law for deriving an empirical relation as: 21 c ICKc (4.1) 21 a IC uK a L (Non-dimensional form) (4.2) The coefficients in the above relations are listed in the Table 46. Base on results presented in Table 46 and Figure 4-8, the relative di fferences of coefficients a1 and a2 for L =1 mm and L =2 mm are less than 1% and the two curves for two cases collapse into one curve. Hence, we can conclude that the dimensionle ss fracture toughness of tetrakaidecahedral foam mainly depends on its relative density. The dimens ionless fracture toughness increases as relative density increases.

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57 Mode II fracture toughness Mode II fracture toughness is obtained by Approach 2, that is, by imposing the displacements of KII field on the boundary to micromech anical model, we can obtain the maximum tensile stress near the crack tip from the ABAQUS results. At first, fixing the strut length at L =1 mm and varying the strut thickness, we obtain Mode II fracture toughness as shown in Table 47. And then we choose L =2 mm and follow the same procedure to obtain fracture toughness listed in Table 48. The results presented in Table 47 and Table 48 are plotted in Figure 4-9, and we can conclude th at the fracture toughness decreases as L increases for the foam with the same strut thickness D The relationship between fracture toughness and relative density is shown in Figure 4-10 and Figure 4-11. We use power law for deriving an empirical relation as: 21 c IIcKc (4.5) 21 a IIc uK a L (Non-dimensional form) (4.6) The coefficients in the a bove relations are listed in Table 49. Base on results presented in Table 49 and Figure 4-11, we can conc lude that the normaliz ed fracture toughness of tetrakaidecahedral foam mainly depends on its relative density. Progressive Fracture and Crack Propagation So far the fracture toughness we have presen ted is calculated based on the maximum stress in one strut near the crack tip. In this section we will study pr ogressive fracture by continuously loading the plate and fai ling a series of struts. In this study we assume that the crack is sufficiently long compared to the cell dimension L, and hence the crack propagation is considered under Mode I loading condition. After a strut fails, the failed strut is removed, and di splacements corresponding to an arbitrary KI are applied

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58 along the boundary of the model. In the present study we used KI=0.01. The stresses in the struts in the vicinity of the crack tip are calculated. Fr om the maximum stress, the stress intensity factor KI c that will cause a strut to break is calculated using the relation This procedure repeated until several struts fail in the vi cinity of the crack tip. Figure 4-12 depicts the sequence in which the struts break in the FE model. It is interesting to see that the crack does not propagate in a selfsimilar manner (horizontall y). Instead there are two ki nk cracks occurring in 45 and directions. Maximum stresses in the strut at each stage for KI=0.01 (MPamm0.5) and corresponding fracture toughness are listed in Table 410. One can note that the fracture toughness slightly increases as the kinked crack grows. Summary and Conclusion A finite element based method developed by Choi and Sankar has been used to study the fracture toughness of tetrakaidecahedral foam. We obtain the plain-strain fracture toughness of the foam by relating the fracture toughness to the tensile strength of the cell struts. Also, we have studied the effects of various geometric paramete rs that describe the cell. The fracture toughness decreases as strut length L increases for the foam with the same strut thickness D For the same strut length, as D increases the fracture toughness increases. However, the dimensionless fracture toughness only depends on the relative density.

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59 Table 41. Material properties of struts Density, s 1650 Kg / m3 Elastic Modulus, Es 23.42 GPa Poissons ratio, s 0.33 Ultimate Tensile Strength, u 689.5 MPa Table 4-2. Convergence study of fracture toughness 10 16 20 25 32 Number of Cell 50 128 200 300 512 KIc 0.405 0.399 0.397 0.395 0.393 Table 4-3. Convergence study of fracture toughness 10 20 30 40 50 Number of Cell 100 200 300 400 500 KIc 0.405 0.397 0.394 0.393 0.392 Table 44. Fracture toughness for strut length L =1 mm by using 40 cells L (mm) D (mm) Relative density KIc (MPamm0.5) 1 0.0600 1.654-3 9.09-2 1 0.1000 4.593-3 3.92-1 1 0.1875 1.615-2 2.23 1 0.2308 2.446-2 3.87 1 0.2727 3.416-2 5.98 1 0.3000 4.134-2 7.63 Table 45. Fracture toughness for strut length L =2 mm by using 40 cells L (mm) D (mm) Relative density KIc (MPamm0.5) 2 0.1200 1.654-3 0.128 2 0.2000 4.593-3 0.554 2 0.3750 1.615-2 3.15 2 0.4615 2.446-2 5.48 2 0.5455 3.416-2 8.46 2 0.6000 4.134-2 10.8 Table 46. Interpolation parameters for Mode I c1 c2 a1 a2 L =1 494 1.31 7.17-1 1.31 L =2 694 1.31 7.12-1 1.31

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60 Table 47. Mode II frac ture toughness for strut length L =1 mm by using 30 cells L (mm) D (mm) Relative density KIIc (Mpamm0.5) 1 0.0600 1.654-3 3.37-2 1 0.1000 4.593-3 1.62-1 1 0.1875 1.615-2 1.10 1 0.2308 2.446-2 2.03 1 0.2727 3.416-2 3.32 1 0.3000 4.134-2 4.40 Table 48. Mode II frac ture toughness for strut length L =2 mm by using 30 cells L (mm) D (mm) Relative density KIIc (MPamm0.5) 2 0.1200 1.654-3 4.77-2 2 0.2000 4.593-3 0.230 2 0.3750 1.615-2 1.55 2 0.4615 2.446-2 2.87 2 0.5455 3.416-2 4.70 2 0.6000 4.134-2 6.22 Table 49. Interpolation parameters for Mode II c1 c2 a1 a2 L =1 486 1.48 0.704 1.48 L =2 687 1.48 0.704 1.48

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61 Table 410. Maximum stress in the struts ahea d of crack tip and Mode I fracture toughness for kinked cracks Sequence of analysis Maximum stress (MPa) for KI=0.01 (MPamm0.5) KIc (MPamm0.5) 1 22.0 0.313 2 32.4 0.213 3 25.1 0.275 4 29.9 0.231 5 19.4 0.355 6 22.4 0.308 7 21.2 0.325 8 15.8 0.435 9 23.9 0.289 10 19.1 0.361 11 21.2 0.326 12 17.1 0.403 13 15.1 0.457 14 17.9 0.384 15 17.6 0.391 Mean Fracture Toughness (MPamm0.5) 0.338 Standard Deviation (MPamm0.5) 0.070 (21%)

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62 Figure 4-1. A tetrakaide cahedral unit cell and the cross section of a strut -2 0 2 4 6 8 0 2 4 6 -10 -5 0 5 x z y Figure 4-2. A structure with 27 (3) cells

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63 0 200 400 600 800 1000 1200 1400 1600 1800 1 1.5 2 2.5 3 3.5 4 4.5 5 Number of cellsRelative error of Young's moudulus(x100%) Figure 4-3. Convergence study of Youngs modulus y Figure 4-4. Deformation of a micromechanical mode

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64 50 100 150 200 250 300 350 400 450 500 550 0.392 0.394 0.396 0.398 0.4 0.402 0.404 0.406 0.408 1010 2010 3010 4010 5010 Number of unit cellsFracture toughness(MPamm0.5)105 168 2010 2512 3216 Figure 4-5. Convergence study of Mode I fracture toughness 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 2 4 6 8 10 12 D (mm)Fracture toughness(MPa mm0.5) L=1 L=2 Figure 4-6. Mode I fracture toughness vs. strut thickness

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65 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0 2 4 6 8 10 12 */sFracture toughness(MPamm0.5) L=1 interp1 L=2 interp2 Figure 4-7. Mode I fracture toughness vs. relative density 0 0.01 0.02 0.03 0.04 0.05 0 0.002 0.004 0.006 0.008 0.01 0.012 */ sKIc /( u*L0.5) L=1 interp1 L=2 interp2 Figure 4-8. Normalized Mode I fr acture toughness vs. relative density

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66 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 1 2 3 4 5 6 7 D(mm)Mode II Fracture toughness(MPamm0.5) L=1 L=2 Figure 4-9. Mode II fractu re toughness vs. strut thickness 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0 1 2 3 4 5 6 7 */sMode II Fracture toughness(MPamm0.5) L=1 interp1 L=2 interp2 Figure 4-10. Mode II fracture toughness vs. relative density

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67 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0 1 2 3 4 5 6 7 x 10-3 */sKIIc/(u*L0.5) L=1 interp1 L=2 interp2 Figure 4-11. Normalized Mode II fr acture toughness vs. relative density Figure 4-12. Crack development history: first broken strut is labeled 1 and the last broken strut is 15.

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68 CHAPTER 5 IMPERFECTION EFFECTS So far only idealized foams are studied. In real ity there are always imperfections in foams: Dislocation of a vertex wh ich connects several struts Non-uniform strut thickness or material properties Voids in the microstructure Inclusion in the microstructure We shall study the first kind of impe rfection effects on foams consisting of tetrakaidecahedral unit cells. Gene rally for the dislocation imperfec tion, a vertex is assumed to be somewhere within a sphere with radius R, the center of which corr esponds to its perfect position. However, due to computer and software limitations, we will assume only in-plane dislocation, which means a vertex is within a circle of radius R in the x-y plane with the center of the circle at the perfect position. As Figure 51 shows, O is the perfect position of a vertex, and O is the actual position. R and are uniformly distributed in [0, R ] and [0, 2 ] respectively. We introduce a new parameter Ra. ,0,0.5aaR RR L (5.1) Homogeneous Material Properties When we study imperfection problems, the micr ostructure is no longe r symmetric. Hence, in this section we study the whole model instead of the ha lf model. We conducted four simulations for each Ra value, which means that there are f our finite element models randomly generated and analyzed. The relative density of the foam is calculated and plotted in Figure 52. The results show that the relative density increases as Ra increases. Since the tetrakaidecahedral unit cell has almost mini mum surface area, as Ra increase, the total length of struts increases and thus relative density becomes larger. In orde r to obtain the nodal di splacements on the boundary

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69 of the microstructure, we need to calc ulate the equivalent material properties* 1E, 2E, 12 and 12G in Eq (A.1). Symmetric displacement conditions are still applied in the nodes on the front and back surfaces. Detailed schemes to obtain these eq uivalent material properties are shown in Table 51. For a given Ra, the inner nodes are randomly generate d within a circle with radium R = RaL and the center located at th e perfect position, but the nodes on the boundary are located in the perfect position so that it is easy to apply BC. In order to observe the random parameter Ra effect at each value of Ra we conduct four FEM analyses, where four finite element mo dels are randomly generated. Figure 53 shows that the el astic modulus increases as Ra increases. This result confirms the general conclusion that highe r the relative density larger its modulus. The Poissons ratio decreases as Ra increases (Figure 54). Generally, when Ra increases the deviation from the mean value of these material properties also increase. Fracture Toughness Once we obtain the homogeneous material propertie s using a finite element model, we use the same structure and break the elements at the crack (see Figure 55). The fracture toughness analysis is similar to that in Chap ter 4. The only difference is that we use finite element model to calculate the equiva lent material proper ties instead of the analytical solution. In previous section, we found that the de viation of material properties is small and negligible. Howeve r, the deviation of fracture toughness is large as shown in Figure 56. All these results are also listed in Table 52. We also studied the foams with only one imperfection cell ahead of crack-tip as Figure 57 shows. The overall material properties were assu med not to be affected by the imperfection. We conducted three simulations for each Ra value. Figure 58 shows that th e imperfection of this single cell has significant eff ect on the fracture toughness.

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70 Summary and Conclusion In this chapter, the dislocation imperfecti on effects are studied. Equivalent homogeneous material properties are obtained by finite element analysis. Th e fracture toughness of tetrakaidecahedral foams is analyzed with the same approach as in Chapter 4. We obtain the plane-strain fracture toughness of the foam by relating the frac ture toughness to the tensile strength of the cell struts. We find that dislocation distance Ra affects the elastic modulus slightly. But it has a huge effect on the fract ure toughness. The deviation of the fracture toughness increases as th e dislocation distance Ra increases. The results of foams with one imperfect cell ahead of crack tip also confirm this conclusion.

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71 Table 51. Equivalent material properties Material properties Deformation and boundary cond ition Equations 1E 1L L 2120 10 (calculated based on resultant force) 1 1 1E 2E 2L L 1120 20 (calculated based on resultant force) 2 2 2E 12 20 120 120&0 (calculated based on resultant force) 2 12 1 12G Applying Periodic BC: 100xxuu 100xxvv 10yyuuL 100yyvv Shear strain: 12 10L y y Strain energy density: 2 1212121211 22 UG 12 2 122U G

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72 Table 52. Numerical results with respect to Ra Ra Relative density 1E (MPa) 2E (MPa) 12G (MPa) 12 Max tip stress (MPa) Ic K (MPamm0.5) 0 0.004663 0.4664 0.4664 0.1160 0.9807 21.11 0.327 0.05 0.004666 0.4671 0.4673 0.1163 0.9787 19.59 0.352 0.05 0.004666 0.4666 0.4667 0.1164 0.9786 20.22 0.341 0.05 0.004666 0.4676 0.4676 0.1163 0.9785 21.18 0.326 0.05 0.004666 0.4662 0.4662 0.1165 0.9786 21.03 0.328 0.1 0.004673 0.4700 0.4703 0.1173 0.9720 25.85 0.267 0.1 0.004673 0.4686 0.4696 0.1175 0.9728 22.39 0.308 0.1 0.004673 0.4692 0.4695 0.1176 0.9722 23.71 0.291 0.1 0.004674 0.4687 0.4706 0.1175 0.9737 20.81 0.331 0.2 0.004704 0.4793 0.4806 0.1213 0.9448 23.88 0.289 0.2 0.004702 0.4765 0.4757 0.1216 0.9451 18.08 0.381 0.2 0.004703 0.4794 0.4801 0.1213 0.9449 26.63 0.259 0.2 0.004703 0.4758 0.4775 0.1224 0.9470 26.93 0.256 0.3 0.004753 0.4903 0.4901 0.1284 0.8940 30.38 0.227 0.3 0.004751 0.4882 0.4902 0.1288 0.8992 34.24 0.201 0.3 0.004757 0.4883 0.4900 0.1296 0.8908 29.77 0.232 0.3 0.004752 0.4888 0.4833 0.1294 0.8922 23.09 0.299 0.4 0.004826 0.4961 0.4969 0.1394 0.8214 17.78 0.388 0.4 0.004824 0.4957 0.4998 0.1367 0.8257 26.01 0.265 0.4 0.004821 0.5044 0.5103 0.1364 0.8250 35.24 0.196 0.4 0.004822 0.5039 0.5141 0.1363 0.8338 22.10 0.312 0.5 0.004917 0.4999 0.5033 0.1478 0.7290 26.44 0.261 0.5 0.004909 0.5087 0.5142 0.1456 0.7279 33.53 0.206 0.5 0.004914 0.5005 0.5109 0.1454 0.7319 22.50 0.306 0.5 0.004917 0.4948 0.5043 0.1475 0.7259 27.85 0.248

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73 Figure 51. Disloc ation of a vertex 0 0.1 0.2 0.3 0.4 0.5 4.65 4.7 4.75 4.8 4.85 4.9 4.95 x 10-3 RaRelative density */ s Mean value of */ s Figure 52. Relative density as a function of the di slocation distance Ra of a vertex.

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74 0 0.1 0.2 0.3 0.4 0.5 0.46 0.47 0.48 0.49 0.5 0.51 0.52 RaHomogeneous moulous (MPa) E1 E2 Mean value of E1 Mean value of E2 0 0.1 0.2 0.3 0.4 0.5 0.11 0.12 0.13 0.14 0.15 0.16 RaHomogeneous moulous (MPa) G12 Mean value of G12 A B Figure 53 Effective moduli vs. Ra: A) Equivalent elastic mo dulus; B). Equivalent shear modulus 0 0.1 0.2 0.3 0.4 0.5 0.7 0.75 0.8 0.85 0.9 0.95 1 RaPoisson s ratio 12 Mean value of 12 Figure 54. Equivalent Poissons ratio versus Ra

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75 Figure 55. Ra=0.50: left whole finite element model; right scaled structure near the crack tip 0 0.1 0.2 0.3 0.4 0.5 0.2 0.25 0.3 0.35 0.4 RaFracture toughness (MPa mm0.5) KIc Mean value of KIc Figure 56. Fracture toughness versus Ra

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76 Figure 57. An example of the structure ne ar the crack tip with only one imperfect cell (Ra=0.5) ahead of the crack tip 0 0.1 0.2 0.3 0.4 0.5 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4 RaFracture toughness (MPa mm0.5) KIc Mean value of KIc Figure 58. Fracture toughness versus Ra for foams with one imperf ect cell ahead of the crack tip

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77 CHAPTER 6 PLASTIC DEFORMATION NEAR CRACK TIP We have studied brittle foams in previous chapters. Now we will have a glimpse of fracture behavior of ductile materials, which means plastic deformation will occur when the principal stress is greater than the yield stre ss of the material. In order to use ABAQUSs capability, pipe cross section (Figure 61) is chosen for the st ruts of cells in the rectangular prism foam. The outer radius is r and the thickness is t. The cross section area is A and the moment of inertia is I (Eq.(6.1) and Eq.(6.2)). 20.5 Artt (6.1) 4 44 I rrt (6.2) Elastic Deformation Analysis Before studying plastic deformation, we first follow the same procedures in Chapter 3 to analyze fracture toughness of brittle material (Table 3-1). Here we use Approach 2 to obtain the fracture toughness. Similar to derivation of Eq.(3.2), we obtai ned elastic homogeneous material properties: 1 23sA EE cc 2 13sA EE cc 3 12sA EE cc (6.3) The shear modulus G12 takes the same form as Eq.(3.9) and the Poissons ratio are given as Eq.(3.13). The maximum tensile stress of the first unbroken strut ahead of the crack tip is calculated as the resultant force and bending moments on the strut are obtained. tiptip tipbendten M rF I A (6.4)

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78 Case 1: input parameters are KI=500 (MPamm0.5), cell size c1=c2=c3=0.2mm, cross section r=c1/5 and t=r/6. The elastic strain contour (Figure 62) shows the strains in the struts near crack tip are much larger than those in other struts as expected. We obtain Mtip=0.122691 Nmm and Ftip=36.7165 N. Hence, 15,287tiptip tipMrF IA MPa And 61.23Iu IC tipK K MPamm0.5 The ratio of bending stre ss over the tensile stress is /0.2964tiptip bendtenMrF IA The analytical solution of fracture toughness is 131 60 21ICuA K cc MPamm0.5 The numerical result of fracture toughness agrees well the analytical solution. Plastic Deformation Analysis To include elastic-plastic response, we need to specify the nodal forces N, M1, M2, and T directly as functions of their conjugate plastic de formation variables. For elastic-perfectly plastic deformation, for each of the above nodal forces we need to provide the value at which plastic deformation sets in (denoted by F0) and the force at which the section becomes fully plastic (F1).

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79 These are given in the form of a graph depicted Figure 63. In this figure P1 is the plastic deformation per unit length corresponding to F1 and P2 is an arbitrarily big value. The ultimate stress in Table 3-1 is taken as the yield stress for the elastic -plastic deformation. The other properties of the foam are the same as for Case 1 in Table 3-1. The forces F0 and F1 are calculated for each mode, extension, flexure and torsion, using mechanics of materials formulas, and are listed in Table 61. ABAQUS assumes the displacement and rotati on increments can be decomposed into elastic and plastic parts. Plastic strain will occur when the strain is larger than the yield strain given by Y/Es=3600/(2073)=0.01739. For this study we used the full micromechanical model. The displacements corresponding to a given KI are applied along the boundary of the model and they were increased incrementally starting from KI=0. The strain in the crack tip stru t is monitored for each increment. Figure 64 shows the elastic strain vs. KI plot. It shows the KI corresponding to the onse t of plastic strain in the strut. This value will be approximately equal to the KIc obtained for brittle foams with the rupture strength equal to the yiel d stress. The elastic strain does not increase beyond the onset of yielding as we are using elastic-perfectly plastic model. However, the plastic strain increases as KI increases (Figure 65). If maximum strain criterion is used, then the fracture toughness can be determined based on the curve in Figure 66 in which the KI is plotted against the total strain in the crack tip strut. For example, if strain to failure of the strut material is 0.2, then KIc=207 MPamm0.5.

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80 Table 61. Forces and defo rmations for onset of yieldi ng and fully plastic conditions F0 F1 P1 *Plastic axial 5.5292 N 5.8057 N 0.04348 *Plastic M1 0.09292 MPa 0.12544 MPa 0.5031 Rad *Plastic M2 0.09292 MPa 0.12544 MPa 0.5031 Rad *Plastic Torque 0.01014 MPa 0.13685 MPa 0.6421 Rad Figure 61. Pipe cross section

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81 Figure 62. Contour of axia l elastic strain in struts Figure 63. Data points generate d for the perfect plastic model

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82 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 50 100 150 200 250 300 Elastic strainStress intensity factor (MPamm0.5) Figure 64. Stress intensity factor KI vs. elastic strain in the crack tip strut 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 50 100 150 200 250 300 Plastic strainStress intensity factor (MPamm0.5) Figure 65. Stress intensity factor KI vs. plastic strain in the crack tip strut

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83 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 50 100 150 200 250 300 Total strainStress intensity factor (MPamm0.5) Figure 66. Stress intensity factor KI vs. total strain in the crack tip strut

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84 CHAPTER 7 CONCLUDING REMARKS AND SUGGESTED FUTURE WORK In this dissertation, we have studied two type s of foams: foams with rectangular prism unit cells, including homogeneous foams and functionally graded foams, and tetrakaidecahedral foams. The geometry of first one is simple and easy to model which provide a means to understand fracture behavior of foams. The other one is close to reality as some carbon foams can be approximated to be tetrakaidecahedral foams. Our approach to study the fracture toughness of foams is a global-local approach wher ein the microstructure was modeled in detail near the crack tip (inner region), and boundary cond itions are applied at fa r away points (outer region) according to continuum fracture mechanics. Two crack propagation criteria, one at the micro-scale and one at the macro-scale, are used. The fracture toughness of brittle foam is calculated based on the stress intensity factor and the corresponding maximum tensile stress in the struts ahead of the crack. We have studied stress gradient effects on the homogeneous and graded foams with rectangular prism unit cells. The fracture toughness of the foam could be predicted by the strength of the strut or ligament material and the shape and si ze of the cells that constitute the foam. An analytical model of fracture toughness was derived. Different loading cases were studied by using a micro-macro combined met hod. Fracture toughness of the homogeneous foam decreases as the crack size increases except for remote displacement loading cases. The aspect ratio of the plate does not have much effect on the fracture toughness. As the cell size becomes smaller, the fracture toughness of the homogene ous foams under different types of loads becomes uniform. Since the relative differences of the fracture toughness of the homogeneous foam under different loads are within %, the fracture toughness can be treated as a material property. The fracture toughness of the analytical model agrees with that determined by the

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85 combined micro-macro-mechanics method. It is found that the fracture toughness of graded foam equals to the fracture toughness of homogeneous foam with the same cell as that of the graded foam at the crack tip. The fr acture toughness does not simply de pend on the relative density. It also depends on both the material and the shape and size of the cell. Approach 2 has been used to study the fract ure toughness of tetrakai decahedral foam. We obtain the plain-strain fracture toughness of the foam by relating the fracture toughness to the tensile strength of the cell struts. Also, we have studied the effects of various geometric parameters that describe the cell. The fr acture toughness decreases as strut length L increases for the foam with the same strut thickness D. For the same strut length, as D increases the fracture toughness increases. However, the dimensionless fracture toughness only depends on the relative density. In the study of the disl ocation imperfection effects, we find that dislocation distance Ra has no significant effect on the elastic modulus But it has a huge eff ect on fracture toughness. The deviation of the fr acture toughness increase as the dislocation distance Ra increases. Finally, we have taken the fi rst step to study plastic de formation near crack tip. However, there are some supplement study and ne w areas needed to be studied. We only have one experimental result for homogeneous foam for comparing our results. Hence, experimental study could be an ar ea of future study. Since foams other than brittle foams are widely used, large deformation of foams will be an interesting topic. There are also some research needs in open-cell foams used in coolin g system wherein hot air/fluid flows through the foam. As energy-absorption function, foams ar e under compression and closed-cell foams are often used. Hence, fracture behavior under comp ression and research on closed-cell foams are good future topics.

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86 APPENDIX A CRACK TIP DISPLACEMENT FIELDS FOR ORTHOTROPIC MATERIALS The stress-strain relation in the principal di rection for plane stress problem is given as: ** 112121 ** 212122 12121210 10 001 EE SEE G (A.1) The stress-strain relation can be transformed from the principa l 1-2 coordinate system to the x-y coordinate system by using th e transformation matrix [T]: 1 2 12 x y x yT (A.2) where the transformation matrix is defined as: 22 22 22cossin2cossin sincos2cossin cossincossincossin T (A.3) The compliance matrix S in the x-y plane is 111216 122226 162666TSSS SSSSTST SSS (A.4) The characteristic equation of the orthotropic material is given by Sih and Liebowitz(1968) 432 1116126626222220 SSSSSS (A.5) There are four roots of the char acteristic equation. We denote s1 and s2 as the two unequal roots with positive conjugate values: 1111s i 2222s i (A.6) The constants pj and qj (j=1,2) are related s1 and s2 as bellows

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87 2 111112161 p SsSSs 2 2112112162 p SsSSs (A.7) 2 12122261 1 1SsSSs q s 2 12222262 2 2SsSSs q s The displacement field in the vi cinity of crack tip is a functi on of the orthotropic material parameters p1, p2, q1, q2, s1 and s2 as shown in Eq(2.5). For plane strain problem, the strain and stress relation is 111121 221222 1266120 0 00 CC QCC C (A.8) where 2 3 11123 21 1 E CE DE 3 122121323 21 E CE DE 2 3 22213 11 1 E CE DE 2112CC 6612CG 2222222 12323131223121323231323 1232 EEEEEEEEEEE D EEE And then the compliance matrix S in the 1-2 coordinates is the inverse matrix of Q : 1SQ (A.8) In order to obtain the displacement fi eld near the crack tip, simply replace Sin Eq.(A.4) with S and then the solution takes the same form as plane stress problems.

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88 APPENDIX B FRAME ELEMENT IN ABAQUS Space frame elements are used in the study of tetrakaidecahedral foam. Forces and moments on a frame element in space are shown in Figure B-1. These forces and moments can be output at three nodes, that is two end nodes and the middle node. Since equilateral triangle is not a default cross section in ABAQUS, general cross section option is used in the frame element for tetrakaidecahedral foam. Area A the moment of inertia I1 and I2, the polar moment of inertia J are required for input data Those values could be determined by the equations listed in Table B-1. Since those values are needed to calculate stress in the struts, Table B-1 also give s equations for other types of cr oss section used in this study. If we ignore the shear stress, stress at a point in the section at the middl e of a strut is given as 1221 12ten M xMx N AII (B.1) For rectangular prism foam, since M2=0 this foam becomes 1 1 23 16 2tenh M M NN AIhh (B.2) In order to include elastic-plast ic response, we need specify N M1, M2, and T directly as functions of their conjugate plasti c deformation variables. The plas ticity is lumped at the element ends. There are no plastic strain s as output in the frame elemen t. Plastic displacements and rotations in the element coordinate system are output for plastic deformation. ABAQUS assumes the displacement and rotation increments can be decomposed into elastic and plastic parts. We can obtain plastic deformation in axial directi on by adding the plastic displacements on element ends. Then the plastic strain is assumed to be the plastic deformati on divided by the element

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89 length. However, this simple method is an approximate method to evaluate the plastic deformation near the crack tip. Table B-1. Cross section of frame element Cross section Equations Equilateral triangle 23 4 AD 2 13 18 I A 12 I I 253 A J Pipe 20.5 Artt 4 4 14 I rrt 12 I I 4 42 Jrrt Square 2 A h 4 112 h I 12 I I 46 h J

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90 Figure B-1. Forces and moment s on a frame element in space.

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91 LIST OF REFERENCES 1. Chen, Y.-R., Columbia Shuttle Tragedy in http://www.csa.com/discoveryguides/shuttle/overview.php 2003. 2. Maiti, S.K., M.F. Ashby, and L.J. Gibson, Fracture-Toughness of Br ittle Cellular Solids. Scripta Metallurgica, 1984. 18(3): p. 213-217. 3. Huang, J.S. and L.J. Gibson, Fracture-Toughness of Brittle Foams. Acta Metallurgica Et Materialia, 1991. 39(7): p. 1627-1636. 4. Huang, J.S. and L.J. Gibson, Fracture-Toughness of Brittle Honeycombs. Acta Metallurgica Et Materialia, 1991. 39(7): p. 1617-1626. 5. Brezny, R., D.J. Green, and C.Q. Dam, Evaluation of Strut Strength in Open-Cell Ceramics. Journal of the American Ceramic Society, 1989. 72(6): p. 885-889. 6. Brezny, R. and D.J. Green, The Effect of Cell-Size on the Mechanical-Behavior of Cellular Materials. Acta Metallurgica Et Materialia, 1990. 38(12): p. 2517-2526. 7. Gibson, L.J. and M.F. Ashby, Cellular Solids: structure and properties 2 ed. 1997: Cambridge university press, U.K. 8. Choi, S. and B.V. Sankar, Fracture toughness of carbon foam. Journal of Composite Materials, 2003. 37(23): p. 2101-2116. 9. Choi, S. and B.V. Sankar, A micromechanical method to pred ict the fracture toughness of cellular materials. International Journal of Solids and Structures, 2005. 42(5-6): p. 17971817. 10. Lee, S., J. Wang, and B.V. Sankar, A micromechanical model for predicting the fracture toughness of functionally graded foams. International Journal of Solids and Structures, 2007. 44: p. 4053-4067. 11. Rice, J.R., A Path Independent Integral and A pproximate Analysis of Strain Concentration by Notches and Cracks. Journal of Applied Mechanics, 1968. 35(2): p. 379-&. 12. Aoki, S., K. Kishimoto, and M. Sakata, Energy-Release Rate in Elastic-Plastic Fracture Problems. Journal of Applied MechanicsTransactions of the Asme, 1981. 48(4): p. 825829. 13. Jin, Z.H. and N. Noda, Crack-Tip Singular Fields in Nonhomogeneous Materials. Journal of Applied Mechanics-Transactions of the Asme, 1994. 61(3): p. 738-740.

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93 27. Zhu, H.X., J.F. Knott, and N.J. Mills, Analysis of the elastic pr operties of open-cell foams with tetrakaidecahedral cells. Journal of the Mechanics and Physics of Solids, 1997. 45(3): p. 319-&. 28. Li, K., X.L. Gao, and A.K. Roy, Micromechanical modeling of three-dimensional opencell foams using the matrix method for spatial frames. Composites Part B-Engineering, 2005. 36(3): p. 249-262. 29. Laroussi, M., K. Sab, and A. Alaoui, Foam mechanics: nonlinear response of an elastic 3D-periodic microstructure. International Journal of Solids and Structures, 2002. 39(1314): p. 3599-3623. 30. Ridha, M., V.P.W. Shim, and L.M. Yang, An elongated tetrakaidecahedral cell model for fracture in rigid polyurethane foam in Fracture and Strength of Solids Vi, Pts 1 and 2 2006. p. 43-48. 31. Sih, G.C. and H. Liebowitz, Mathematical Theories of Brittle Fracture Fracture, ed. H. Liebowitz. Vol. 2: Mathematical Fundament als. 1968, New York and London: Academic press. 32. Santare, M.H. and J. Lambros, Use of graded finite elements to model the behavior of nonhomogeneous materials. Journal of Applied Mechanics-Transactions of the Asme, 2000. 67(4): p. 819-822. 33. Hellan, K., Introduction to fracture mechanics 1984: McGraw-Hill Book Company. 34. Weisstein, E.W., Truncated Octahedron in MathWorld--A Wolfram Web Resource: http://mathworld.wolfram.com /TruncatedOctahedron.html

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94 BIOGRAPHICAL SKETCH Junqiang Wang was born in China in 1973. He received his Bachelor of Engineering in mechanical engineering from University of Sc ience and Technology Beijing in 1996. He worked for 2 years for Qinhuangdao Branch of Baotou Engi neering and Research Corp. of Iron and Steel Industry, China. He received his masters de gree in the speciality of materials processing engineering in Tsinghua Universi ty, China. He also got a Master of Science in mechanical engineering at University of Florida. He is pursuing his doc toral degree at the Center for Advanced Composites in the Department of Mech anical and Aerospace Engineering, University of Florida.