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f95acfc437147dad901d29945d4782a6a528c672 23713 F20101110_AACALU wang_j_Page_19.jpg b243df800e78b246d2c4e503e350e0cb 842327256527a0b91af3136b8af3cbc525bf73ae 45959 F20101110_AACAMK wang_j_Page_41.jpg 70e49cd651c85cc27166b87a5ec064df 83b1d802db7e433aa70b63f3cff74c1dad15ec3d 48181 F20101110_AACALV wang_j_Page_20.jpg ebe4a518dc71cddbf30ba2dd0ca3de04 0290355489396e02ae3d60abe87100c62d9242ee 34399 F20101110_AACAML wang_j_Page_44.jpg 2195af4a71a0f95ab4032b8824f617bf 884dc0df7af372e9eaf0674a5097600c1096c91d 23506 F20101110_AACALW wang_j_Page_21.jpg 706b2530155e8b2df899fdb0e1087a3c 0bd4139ee38690bb2b7712262ee0c076cc958f41 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 Nonuniform Strut T h ic k n e ss ......................................... ......... .......... ...... .......................... 3 6 Fracture Toughness of Functionally Graded Foam with Nonuniform 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 31 M material properties of struts .................................................... ......................................40 32 L ist of load cases .............. ....................................................................... .......40 33 Axial forces for Unit cell: c1=c2=c3=200 [tm, h=20 [tm; Load: remote traction ...............40 34 The ratio y for cell size: c2=c3=200 [am, h=20 [am, and cl varies ...................................40 35 The ratio y for cell size: c1=c2=c3=200 jam, h varies............................... ............... 40 36 M ode I fracture toughness(x 10 Pam05) ..................................................... ........... 40 37 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 38 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 39 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 42 Convergence study of fracture toughness................................. ...............59 43 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 11 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 21 An edgecracked plate: Hheight; acrack length; Wwidth. ...........................................26 22 F inite elem ent m odels ...... .. .......... .... ............ .. ............................ ....................26 23 Crack tip in microm echanical model. ........................................ .......................... 26 24 Stress field near cracktip .......................................................................... ................... 27 31 Micro and Macrostresses in an opencell foam............................................................42 32 Cell deform ation by cell strut bending........................................................... .... .......... 42 33 Example of graded foam with 50 strips and the discrete elastic modulus compared w ith the actual m odulus ......................... ......... .. .. ..... .. ............43 34 Five types of loading ...................................................... .......... .. ............ 43 35 Cracktip forces and moments and corresponding crack tip stresses in the idealized hom ogeneou s continuum ......................................................................... ....................44 36 Mode I fracture toughness as a function of relative density ............................................44 3 7 Cubic foam with c=1.8mm, Gu=3.5805 MPa........... .......................................... 45 38 The bendingtensile ratio convergence test for cell length over strut thickness c/h=10....45 39 Results of homogeneous foams under the remote bending load................... ............46 310 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. ........................................47 311 Fracture toughness under different loads for the foam (c=200 am, h=20 am) ..................48 312 Fracture toughness under different loads for the foam (c=50 am, h=5 am)......................48 313 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 ) ........................................ ....................49 314 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 ) ........................................ ....................49 315 Com prison of Eq(3.22) w ith Eq.(3.28) ............................................................ .......... 50 316 C om prison of three form s .................................................................. ....... ................. 50 317 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) ......................................................51 318 Plate size: W=0.1m, HIW=1; Graded foam: h0o=30am; a=2x104; c=c =c2=c3=200 tm ................... ............................. .. ............. ...5. 51 319 Fracture toughness under remote bending load. (Plate size: W=0.1m, HIW=1; Graded foam : co=200 am; 3= 0.15023 am; h=20am) ...................................... ............... 52 320 Fracture toughness under remote bending load. (Plate size: W=0.1m, HIW=1; Graded foam : co=200 am; 3= 0.15023 am; h=20am) ...................................... ............... 52 321 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.............................................. 53 322 Plate size: W=0.1m, HIW=1; Graded foam: co=200 am; 3= 0.15023 am; h=20am ........53 41 A tetrakaidecahedral unit cell and the cross section of a strut........................... ........62 42 A structure w ith 27 (3x3x3) cells .................................... .... ..... ........................ 62 43 Convergence study of Y oung's m odulus................................... ...................... .. .......... 63 44 Deformation of a micromechanical mode.................................... ........................ 63 45 Convergence study of Mode I fracture toughness ............................................................64 46 Mode I fracture toughness vs. strut thickness...........................................................64 47 M ode I fracture toughness vs. relative density ...................................... ............... 65 48 Normalized Mode I fracture toughness vs. relative density ...........................................65 49 Mode II fracture toughness vs. strut thickness.............................................................66 410 Mode II fracture toughness vs. relative density ...........................................................66 411 Normalized Mode II fracture toughness vs. relative density ..........................................67 412 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 Bl 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 macromechanical 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 macromechanical model can be evaluated using either the Jcontour integral or the stresses in the singularitydominated zone. The other approach is to directly apply displacements based on the Kfield 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 cracktip 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 energyabsorption. 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 opencell and closedcell 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 opencell 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 opencell foams. In a homogeneous continuum the neartip 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 cracktip 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 Jintegral. For homogeneous materials, the Jintegral is path independent, which allow us to get accurate J along a path away from the crack tip. Rice[ 1] introduced the pathindependent J integral for elastic solids under isothermal conditions. A general form of the J integral, suitable for elastic or elasticplastic 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 Jintegral. 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 Jintegral and is suitable for elastic, thermal elastic, and plastic materials. Gu et al. [15] applied this domain integral to evaluate the cracktip 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 elasticbrittle, 2D 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[1719] provided analytical solutions of some fracture problems 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 temperature distribution, and elastic or plastic cracktip singular fields of nonhomogeous materials are the same as those of homogeneous materials. Gu and Asaro[20] analytically studied a semiinfinite 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 neartip 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 monodispersed 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 lowdensity 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 bucklingtype 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 opencell foams with the unit cell shown as in Figure 11. 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 edgedcrack 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 unitcells 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 11. 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 globallocal 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 macroscale. 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 edgecracked 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 edgecracked plate. The edgecracked plate is shown in Figure 21. The plate is comprised of microstructure with the unit cell shown as Figure 11. Due to the symmetry of the geometry and loads, only one half of the plate is analyzed (Figure 22). A multiscale modeling approach consisting of three different length scales is used. Three models (Figure 22C) are used and they are: macro model, macro submodel, and micro model. The macro submodel and micro model are attached to the macro model. The boundary condition (BC) of the macro submodel is obtained from the macro model results and the BC of the micro model is obtained from the macro submodel. In other worlds, the displacements of the nodes on the boundary of the macro submodel 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 cracktip field. The material properties of this model are calculated through homogenization (see equations in Chapter 3). Due to the stress singularity near the cracktip, more elements are needed in this area to obtain accurate cracktip fields. The macro submodel plays such a role that allows us to increase the number of elements near the cracktip. The micro model is used to calculate the maximum tensile stresses in the unbroken strut ahead of the crack tip. Figure 23 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) 2r (2.3) r>O where a22 (r,0) is the stress in the ydirection near the crack tip, and is a function of r and 0 (see Figure 24). The stress intensity factors can also be calculated from the Jintegral 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 xy, yz, and zx planes. In this study we use domain integral in ABAQUS software to calculate the energy release rate. In the case of graded foams the Jintegral is not pathindependent. This is because the graded foam is an inhomogeneous material in macroscale. 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 cracktip stress. SIF calculated by Eq (2.3) expected to agree well with that based on Jintegral. Approach 2 Sih et al. [31] determined the Kfield in the vicinity of a crack tip in homogeneous orthotropic materials. We can directly apply displacements based on the Kfield 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 21. An edgecracked plate: Hheight; acrack length; Wwidth. Macrmodel A C77777 r ir * I T V0 nodEl Macro M icro subrmd del 'Micro model Figure 22. Finite element models: A) An edgedcrack 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 23. Crack tip in micromechanical model: Left crack tip in microstructure; right actual foam with resultant force and bending moment. Crack 1r Figure 24. Stress field near cracktip 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 xy, yz, and zx planes (G12*, G23*, G32*), and Poisson's ratios in xy, yz, zx planes (v12*, v23*, V31*). We choose the same carbon foam as Choi studied in [9].The material properties of microstructure are listed in Table 31. The Young's modulus iny direction could be derived as depicted Figure 31. 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 32. Because of symmetry, there is no curvature at the halflength 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 31, we can see that the strain in the xdirection 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 nonuniform strut thickness and with nonuniform 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 33A shows. As long as the regions are small enough, the gradient material properties of foam can be approximated by constant material properties; Figure 33B 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 34) 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 32, are studied. An Analytical Model for Fracture Toughness Maiti, Ashby and Gibson[2] used a KI field to calculate the cracktip 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 35), 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 33 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 36. 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 38 shows that the ratio converges to 0.409 as the number of beam elements increases. Table 34 and Table 35 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 34 and Table 35 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 21) with different aspect ratios: 1, 2 and 8 using ABAQUS. Figure 39a shows that the Jintegral increases as crack size increases. Figure 39c shows the maximum tensile stress in the first unbroken strut ahead of the crack tip. Figure 39b 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 36. The relative errors of fracture toughness are shown in Figure 39d, 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 Pam0o. The results in Table 36 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, 1.121.39a+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 37 and Table 38 show the variance of the ratio becomes less as the cell size decreases. Also by comparing the data in Table 39 with those in Table 37 we observe that the ratio varies for different crack sizes. Figure 310 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 311 and Figure 312 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 2Remote traction and Case 4 Bending are almost the same. Both cases correspond to remote traction. The results of Case 2 Surface traction and Case 5Thermal 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 311 and Figure 312, 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 Nonuniform 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 313 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 314, 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 315 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. Kc = 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 316 shows this conclusion. The stress gradient effects on the fracture toughness are shown in Figure 317 and Figure 318. The fracture toughness is normalized with the analytical one (Eq.(3.22)). Similar to the homogeneous foam, the fracture toughness of Case 2Remote traction and Case 4Bending are almost the same. The results of Case 3Surface traction and Case 5Thermal 1 show similar trends as the crack size increases. Fracture Toughness of Functionally Graded Foam with Nonuniform Cell Length We obtain the same conclusion for the nonuniform 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 319 and Figure 320 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 321 and Figure 322. The fracture toughness is normalized with the analytical one (Eq.(3.22)). Similar to the homogeneous foam, the fracture toughness of Case 2Remote traction and Case 4Bending are almost the same. The results of Case 3Surface traction and Case 5Thermal 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 cracktip 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 micromacro combined method. The effect of stress gradients in the vicinity of the cracktip 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 micromacromechanics 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 31. 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 32. 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 33. 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 35. 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 36. Mode I fracture toughness(x10 Pam5) 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 34. ci (am) y 70 4.50 4.53 4.55 Table 37. 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 38. 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 39. 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 Ft F Figure 31. Micro and Macrostresses in an opencell foam F F m /* F F .4 Figure 32. 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 33. Example of graded foam with 50 strips and the discrete elastic modulus compared with the actual modulus rmax lw A B Figure 34. Five types of loading I 1 1 il I I1 L 7 oK Jde&PJ Hoinoge~neoue WueriatI Figure 35. Cracktip 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 36. 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 38. The bendingtensile ratio convergence test for cell length over strut thickness c/h=10 12000 HIW=1 10000 eHIW=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 39. Results of homogeneous foams under the remote bending load: A) Jintegral; 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 103 Figure 310. 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 311. 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 312. 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 105 Figure 313. 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 eIncreasing density Decreasing density *Analytical 0o01 0.02 0.03 0.04 Relative density eIncreasing density  Decreasing density 0.015 Analytical 0.05 0.06 0.02 0.03 0.04 Relative density Figure 314. 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 x0.2965p*/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 315. 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 II I SRelatie error  e0.3849(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 316. 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 317. Kic is normalized with the analytical value. (Plate size: W=0.lm, H/W=1; Graded foam: h0=30 am; a=2x104; 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 318. Plate size: W=0.lm, HIW=1; c=c1=c2=c3=200jam Graded foam: ho=30am; a=2x104; 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 104 Figure 319. 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 eIncreasing 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 320. 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 321. 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 322. 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 14sided 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 41). 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 11514 (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 41). Using FEM, we verified the above equation for the Young's modulus. By applying a compressive load (Figure 42), 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 43 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 44 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 onequarter 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 42 and Table 43 and also shown in Figure 45. Parametric Study In the parametric study, the two parameters, L and D (see Figure 41), 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 46. 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 47 and Figure 48. We use power law for deriving an empirical relation as: Kc = cp C2 (4.1) f a =ap0 (Nondimensional 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 48, 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 49, 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 410 and Figure 411. We use power law for deriving an empirical relation as: K1c = clpC2 (4.5) n = ap2' (Nondimensional 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 411, 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 412 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 plainstrain 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 42. 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 43. 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, (MPamm0 5) 1 0.0600 1.654x103 9.09x 102 1 0.1000 4.593 x10 3.92x101 1 0.1875 1.615x102 2.23 1 0.2308 2.446x 102 3.87 1 0.2727 3.416x102 5.98 1 0.3000 4.134x 102 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, (MPamm0 5) 2 0.1200 1.654x103 0.128 2 0.2000 4.593x103 0.554 2 0.3750 1.615x102 3.15 2 0.4615 2.446x 102 5.48 2 0.5455 3.416x102 8.46 2 0.6000 4.134x102 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 7.12x101 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 (Mpamm0 5) 1 0.0600 1.654x103 3.37x102 1 0.1000 4.593 x 10 1.62x101 1 0.1875 1.615x102 1.10 1 0.2308 2.446x 102 2.03 1 0.2727 3.416x102 3.32 1 0.3000 4.134x 102 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 (MPamm0 5) 2 0.1200 1.654x10.3 4.77x 102 2 0.2000 4.593x10.3 0.230 2 0.3750 1.615x102 1.55 2 0.4615 2.446x 102 2.87 2 0.5455 3.416x102 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 (MPamm 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 (MPammo 5) Standard Deviation (MPammo 5) Kic (MPamm 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 41. At a.,,''" '[" y x t tetrakaidecahedral unit cell and the cross section of a strut 4 6 6 4 8 0 z x Figure 42. 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 43. Convergence study of Young's modulus .af M 7y Sr U R:MM Dos %. IinA L. v Y y Figure 44. 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 45. 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 46. Mode I fracture toughness vs. strut thickness 12 O L=1 interpl 10 L=2 ^ interp2 2  0 / Figure 47. 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 48. 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 49. 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 410. Mode II fracture toughness vs. relative density x 103 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 411. Normalized Mode II fracture toughness vs. relative density 1 4 5 12 7 6 15 14 8 11 9 10 Figure 412. 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 * Nonuniform 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 inplane dislocation, which means a vertex is within a circle of radius R in the xy 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 cracktip 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 planestrain 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  I2 2= 1, 01= 12 = 0 AL L 02 o 0 (calculated based on resultant force) Lo E2=02 ^2 12 2 =0, 12 =0, o,O 0 & o2 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) (MPamm05) 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 31). 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 (MPammo.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 Nmm and Ftp=36.7165 N. Hence, ML F = + + =15,287 MPa I A And Kic = o= 61.23 MPamm.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 MPamm05 2 1+Y7 fcc The numerical result of fracture toughness agrees well the analytical solution. Plastic Deformation Analysis To include elasticplastic response, we need to specify the nodal forces N, M1, M2, and T directly as functions of their conjugate plastic deformation variables. For elasticperfectly 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 31 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 31. 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 elasticperfectly 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 MPamm05. 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_elasticK1004Model_. OOObclL_D30_OOO.odb AEAQUS/STANDARD Version 6.55 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 globallocal 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 microscale and one at the macroscale, 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 micromacro 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 micromacromechanics 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 plainstrain 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 opencell foams used in cooling system wherein hot air/fluid flows through the foam. As energyabsorption function, foams are under compression and closedcell foams are often used. Hence, fracture behavior under compression and research on closedcell foams are good future topics. APPENDIX A CRACK TIP DISPLACEMENT FIELDS FOR ORTHOTROPIC MATERIALS The stressstrain 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 stressstrain relation can be transformed from the principal 12 coordinate system to the xy 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 xy 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 12 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 B1. 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 B1. Since those values are needed to calculate stress in the struts, Table B1 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 elasticplastic 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 B1. 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 Bl. 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. 2. Maiti, S.K., M.F. Ashby, and L.J. Gibson, FractureToughness ofBrittle Cellular Solids. Scripta Metallurgica, 1984. 18(3): p. 213217. 3. Huang, J.S. and L.J. Gibson, FractureToughness ofBrittle Foams. Acta Metallurgica Et Materialia, 1991. 39(7): p. 16271636. 4. Huang, J.S. and L.J. Gibson, FractureToughness ofBrittle Honeycombs. Acta Metallurgica Et Materialia, 1991. 39(7): p. 16171626. 5. Brezny, R., D.J. Green, and C.Q. Dam, Evaluation of Strut Su.engil in OpenCell Ceramics. Journal of the American Ceramic Society, 1989. 72(6): p. 885889. 6. Brezny, R. and D.J. Green, The Effect of CellSize on the MechanicalBehavior of Cellular Materials. Acta Metallurgica Et Materialia, 1990. 38(12): p. 25172526. 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. 21012116. 9. Choi, S. and B.V. Sankar, A micromechanical method to predict the fracture toughness of cellular materials. International Journal of Solids and Structures, 2005. 42(56): p. 1797 1817. 10. Lee, S., J. Wang, and B.V. Sankar, A micromechanical modelfor predicting the fracture toughness offunctionally graded foams. International Journal of Solids and Structures, 2007. 44: p. 40534067. 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. 379&. 12. Aoki, S., K. Kishimoto, and M. Sakata, EnergyRelease Rate in ElasticPlastic Fracture Problems. Journal of Applied MechanicsTransactions of the Asme, 1981. 48(4): p. 825 829. 13. Jin, Z.H. and N. Noda, CrackTip Singular Fields in Nonhomogeneous Materials. Journal of Applied MechanicsTransactions of the Asme, 1994. 61(3): p. 738740. 14. Shih, C.F., B. Moran, and T. Nakamura, EnergyRelease Rate Along a 3Dimensional Crack Front in a Thermally Stressed Body. International Journal of Fracture, 1986. 30(2): p. 79102. 15. Gu, P., M. Dao, and R.J. Asaro, A simplified method for calculating the cracktip field of functionally graded materials using the domain integral. Journal of Applied Mechanics Transactions of the Asme, 1999. 66(1): p. 101108. 16. Fleck, N.A. and X. Qiu, the damage tolerance of elasticbrittle, two dimensional isotropic lattices. Journal of the Mechanics and Physics of Solids, 2007. 55(3): p. 562 588. 17. Delale, F. and F. Erdogan, The Crack Problem for a NonHomogeneous Plane. Journal of Applied MechanicsTransactions of the Asme, 1983. 50(3): p. 609614. 18. Delale, F. and F. Erdogan, Interface Crack in a Nonhomogeneous Elastic Medium. International Journal of Engineering Science, 1988. 26(6): p. 559568. 19. Erdogan, F., FractureMechanics ofFunctionally Graded Materials. Composites Engineering, 1995. 5(7): p. 753770. 20. Gu, P. and R.J. Asaro, Cracks in functionally graded materials. International Journal of Solids and Structures, 1997. 34(1): p. 117. 21. Sankar, B.V., An elasticity solution for functionally graded beams. Composites Science and Technology, 2001. 61(5): p. 689696. 22. Thompson, W., On the division of space i/ ith minimum partitional area. Philosophical Magazine, 1887. 24(151): p. 503. 23. Weaire, D. and R. Phelan, A Counterexample to Kelvin Conjecture on MinimalSurfaces. Philosophical Magazine Letters, 1994. 69(2): p. 107110. 24. Hall, R.B. and J.W. Hager, Performance limitsfor stiffnesscritical graphiticfoam structures .1. Comparisons i/ ith highmodulus foams, refractory alloys and graphite epoxy composites. Journal of Composite Materials, 1996. 30(17): p. 19221937. 25. Li, K., X.L. Gao, and A.K. Roy, Micromechanics model for ith eelimeni,\iiiul opencell foams using a tetrakaidecahedral unit cell and Castigliano's second theorem. Composites Science and Technology, 2003. 63(12): p. 17691781. 26. Warren, W.E. and A.M. Kraynik, Linear elastic behavior of a lowdensity Kelvin foam ii ith open cells. Journal of Applied MechanicsTransactions of the Asme, 1997. 64(4): p. 787794. 27. Zhu, H.X., J.F. Knott, and N.J. Mills, Analysis of the elastic properties of opencell foams ii ith 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 threedimensional open cell foams using the matrix method for spatial frames. Composites Part BEngineering, 2005. 36(3): p. 249262. 29. Laroussi, M., K. Sab, and A. Alaoui, Foam mechanics: nonlinear response of an elastic 3Dperiodic microstructure. International Journal of Solids and Structures, 2002. 39(13 14): p. 35993623. 30. Ridha, M., V.P.W. Shim, and L.M. Yang, An elongated tetrakaidecahedral cell modelfor fracture in rigid polyurethane foam, in Fracture and Su eugil of Solids Vi, Pts 1 and 2. 2006. p. 4348. 31. Sih, G.C. and H. Liebowitz, Mathematical Theories ofBrittle Fracture. Fracture, ed. H. Liebowitz. Vol. 2: Mathematical Fundamentals. 1968, New York and London: Academic press. 32. Santare, M.H. and J. Lambros, Use of gradedfinite elements to model the behavior of nonhomogeneous materials. Journal of Applied MechanicsTransactions of the Asme, 2000. 67(4): p. 819822. 33. Hellan, K., Introduction to fracture mechanics. 1984: McGrawHill Book Company. 34. Weisstein, E.W., Truncated Octahedron, in MathWorldA Wolfram Web Resource: http://mathworld.wolfram. com/TruncatedOctahedron. html 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 Nonuniform Strut Thickness.....................................................................................................................36 Fracture Toughness of Functionally Graded Foam with Nonuniform 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 31 Material properties of struts.............................................................................................. .40 32 List of load cases......................................................................................................... .......40 33 Axial forces for Unit cell: c1= c2= c3=200 m, h =20 m; Load: remote traction................40 34 The ratio for cell size: c2= c3=200 m, h =20 m, and c1 varies......................................40 35 The ratio for cell size: c1= c2= c3=200 m, h varies..........................................................40 36 Mode I fracture toughness(5Pam0.5)............................................................................40 37 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 38 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 39 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 42 Convergence study of fracture toughness..........................................................................59 43 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 B1 Cross section of frame element..........................................................................................89 PAGE 9 9 LIST OF FIGURES Figure page 11 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 21 An edgecracked plate: H height; acrack length; W width..............................................26 22 Finite el ement models...................................................................................................... ..26 23 Crack tip in micr omechanical model.................................................................................26 24 Stress field near cracktip................................................................................................ ...27 31 Microand Macrostress es in an opencell foam...............................................................42 32 Cell deformation by cell strut bending...............................................................................42 33 Example of graded foam with 50 strips and the discrete elastic modulus compared with the actual modulus.....................................................................................................43 34 Five types of loading...................................................................................................... ....43 35 Cracktip forces and moments and corres ponding crack tip stress es in the idealized homogeneous continuum...................................................................................................44 36 Mode I fracture toughness as a function of relative density..............................................44 37 Cubic foam with c =1.8mm, u=3.5805 MPa.....................................................................45 38 The bendingtensile ratio convergence test for cell length over strut thickness c / h =10....45 39 Results of homogeneous foams under the remote bending load........................................46 310 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..............................................47 311 Fracture toughness under different loads for the foam ( c =200 m, h =20 m)..................48 312 Fracture toughness under different loads for the foam ( c =50 m, h =5 m)......................48 313 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 PAGE 10 10 314 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)...............................................................49 315 Comparison of Eq(3.22) with Eq.(3.28)............................................................................50 316 Comparison of three forms................................................................................................50 317 KIc is normalized with the analytical value. (Plate size: W =0.1m, H / W =1; Graded foam: h0=30 m; =24; c = c1= c2= c3=200 m)............................................................51 318 Plate size: W =0.1m, H / W =1; Graded foam: h0=30 m; =24; c = c1= c2= c3=200 m............................................................................................................51 319 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)..................................................................52 320 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)..................................................................52 321 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....................................................................53 322 Plate size: W =0.1m, H / W =1; Graded foam: c0=200 m; = 0.15023 m; h =20 m........53 41 A tetrakaidecahedral unit cell and the cross section of a strut...........................................62 42 A structure with 27 (3) cells......................................................................................62 43 Convergence study of Youngs modulus...........................................................................63 44 Deformation of a micromechanical mode..........................................................................63 45 Convergence study of Mode I fracture toughness.............................................................64 46 Mode I fracture toughness vs. strut thickness....................................................................64 47 Mode I fracture toughness vs. relative density..................................................................65 48 Normalized Mode I fracture toughness vs. relative density..............................................65 49 Mode II fracture toughness vs. strut thickness...................................................................66 410 Mode II fracture toughne ss vs. relative density.................................................................66 411 Normalized Mode II fracture toughness vs. relative density.............................................67 412 Crack development history................................................................................................67 51 Dislocation of a vertex................................................................................................... ....73 PAGE 11 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 B1 Forces and moments on a frame element in space.............................................................90 PAGE 12 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 macromechanical 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 macromechanical model can be evaluated using either the J contour integral or the stresses in the singularitydominated 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 cracktip 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 PAGE 13 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. PAGE 14 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 energyabsorption. 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 opencell 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. PAGE 15 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 opencell 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 opencell foams. In a homogeneous continuum the neartip 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 cracktip 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. PAGE 16 16 Rice[11] introduced the pathindependent J integral for elastic solids under isothermal conditions. A general form of the J integral, suitable for elastic or elasticplastic 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 cracktip 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 asticbrittle, 2D 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. PAGE 17 17 There are a large amount of analytical st udies available on FGMs. Erdogan and his coworks[1719] 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 cracktip singular fields of nonhomogeous materials are the same as those of homogeneous materials. Gu and Asaro[20] analytically studied a se miinfinite 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 neartip 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 lowdensity 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 PAGE 18 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 bucklingtype 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 openc ell foams with the unit cell shown as in Figure 11. 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 edgedcr 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 unitcells 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 PAGE 19 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. PAGE 20 20 Figure 11. 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] PAGE 21 21 Figure 13. Three tetrakaidecahedral cells with strut length l and thickness t in a BCC lattice PAGE 22 22 CHAPTER 2 APPROACHES FOR PREDICTI NG FRACTURE TOUGHNESS Our approach is a globallocal 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 macroscale. 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 edgecracked 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 edgecracked plate. The edgecracked plate is shown in Figure 21. The plate is comprised of microstructure with the unit cell shown as Figur e 11. Due to the symmetry of th e geometry and loads, only onehalf of the plate is analyzed (Figure 22). A multiscale modeling approach consisting of three different length scales is used. Three models (Figure 22C) are used and they are: macro model, PAGE 23 23 macro submodel, and micro model. The macro s ubmodel and micro model are attached to the macro model. The boundary conditio n (BC) of the macro submodel is obtained from the macro model results and the BC of the micro model is obtained from the macro submodel. In other worlds, the displacements of th e nodes on the boundary of the macro submodel 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 cracktip field. The material properties of this model are calc ulated through homogenization (see equations in Chapter 3). Due to the stress singularity near the cracktip, more elements are needed in this area to obtai n accurate cracktip fields. The macro submodel plays such a role that allows us to increase the number of elements near the cracktip. The micro model is used to calculate the maximum tensile stresses in the unbroken strut ahead of the crack tip. Figure 23 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 24). 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 PAGE 24 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 xy yz and zx 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 pathindepe ndent. This is because the graded foam is an inhomogeneous material in macr oscale. 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 cracktip 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: PAGE 25 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. PAGE 26 26 Figure 21. An edgecracked plate: H height; acrack length; W width. Figure 22. Finite element models: A) An edgedcrack plate under remote prescribed displacement; B) Microstruc ture of the plate; C) Half model of the plate Figure 23. Crack tip in micromechanical model: Le ftcrack tip in microstructure; rightactual foam with resultant force and bending moment. PAGE 27 27 Figure 24. Stress fi eld near cracktip PAGE 28 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 xy yz and zx planes ( G12 *, G23 *, G32 *), and Poissons ratios in xy yz zx planes (12 *, 23 *, 31 *). We choose the same carbon foam as Choi studied in [9].The material properties of microstructure are listed in Table 31. The Youngs modulus in y direction could be derived as depicted Figure 31. 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 32. Because of symmetry, there is no curvature at the halflen gth of the strut. And thus we can use a half beam to solve for 2. PAGE 29 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) PAGE 30 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 31, we can see that th e strain in the xdirection 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 nonuniform strut thickness and w ith nonuniform 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 33A shows. As long as the regions are small enough, the PAGE 31 31 gradient material properties of foam can be approximated by constant material properties; Figure 33B 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 34) 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 32, 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 35), 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) PAGE 32 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 33 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 36. 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 PAGE 33 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 38 shows that the ra tio converges to 0.409 as the number of beam elements increases. Table 34 and Table 35 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 34 and Table 35 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 PAGE 34 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 21) with different aspect ra tios: 1, 2 and 8 using ABAQUS. Figure 39a shows that the Jintegral increases as crack size increases. Figure 39c shows the maximum tensile stress in the first un broken strut ahead of the crack tip. Figure 39b 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 36. The relative errors of fract ure toughness are shown in Figure 39d, 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 36 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) PAGE 35 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 37 and Table 38 show the variance of the ratio b ecomes less as the cell size decreases. Also by comparing the data in Table 39 with those in Table 37 we observe that the ratio varies for different crack sizes. Figure 310 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 311 and Figure 312 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 2Remote traction and Case 4Bending are almost the same. Both cases corresp ond to remote traction. The results of Case 2Surface traction and Case 5Thermal 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 311 and Figure 312, 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 PAGE 36 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 Nonuniform 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 313 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 314, 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) PAGE 37 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 315 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 316 shows this conclusion. The stress gradient effects on th e fracture toughness are shown in Figure 317 and Figure 318. The fracture toughness is norma lized with the analytical one (Eq.(3.22)). Similar to the homogeneous foam, the fracture toughness of Ca se 2Remote traction and Case 4Bending are almost the same. The results of Case 3Surf ace traction and Case 5Thermal 1 show similar trends as the crack size increases. Fracture Toughness of Functionally Graded Foam with Nonuniform Cell Length We obtain the same conclusion for the nonunifo rm cell length case: the fracture toughness of graded foam equals to the fracture toughness of homogeneous foams with the same cell as that PAGE 38 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 319 and Figure 320 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 321 and Figure 322. The fracture toughness is norma lized with the analytical one (Eq.(3.22)). Similar to the homogeneous foam, the fracture toughness of Ca se 2Remote traction and Case 4Bending are almost the same. The results of Case 3Surf ace traction and Case 5Thermal 1 show similar trends as the crack size increases. PAGE 39 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 cracktip 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 micromacro combined method. The effect of stress gradients in the vicinity of the cracktip 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 micromacromechanics 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. PAGE 40 40 Table 31. 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 32. List of load cases. Case # Description 1. A Remote displacement: v0=75 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 33. 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 34. 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 35. 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 36. 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 PAGE 41 41 Table 37. 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 38. 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 39. 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 PAGE 42 42 Figure 31. Microand Macrost resses in an opencell foam Figure 32. 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 PAGE 43 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 33. Example of graded foam with 50 st rips and the discrete elastic modulus compared with the actual modulus Figure 34. Five types of loading PAGE 44 44 Figure 35. Cracktip 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 (Pam0.5) Choi: constant h constant h Choi: constant c Constant c Figure 36. Mode I fracture toughness as a function of relative density PAGE 45 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 38. The bendingtensile ratio convergence test for cel l length over strut thickness c / h =10 PAGE 46 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 103 a/WRelative error of KIc H/W=1 H/W=2 H/W=8 C D Figure 39. 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 PAGE 47 47 0 2 4 6 8 10 x 103 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 310. 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. PAGE 48 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 311. 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 312. Fracture toughness under different loads for the foam ( c =50 m, h =5 m) PAGE 49 49 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 x 105 1 2 3 4 5 6 7 8 9 10 x 105 h(m)Fracture toughness(Pam1/2) Increasing h Decreasing h Analytical Figure 313. 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 314. 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) PAGE 50 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 315. 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 316. 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). PAGE 51 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 317. KIc is normalized with the analytical value. (Plate size: W =0.1m, H / W =1; Graded foam: h0=30 m; =24; 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 318. Plate size: W =0.1m, H / W =1; Graded foam: h0=30 m; =24; c = c1= c2= c3=200 m PAGE 52 52 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 x 104 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 319. 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 320. 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) PAGE 53 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 321. 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 322. Plate size: W =0.1m, H / W =1; Graded foam: c0=200 m; = 0.15023 m; h =20 m PAGE 54 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 14sided 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 41). 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) PAGE 55 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 41). Using FEM, we verified the above e quation for the Youngs modulus. By applying a compressive load (Figure 42), 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 43 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 44 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 onequarter 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 42 and Table 43 and also shown in Figure 45. PAGE 56 56 Parametric Study In the parametric study, the two parameters, L and D (see Figure 41), 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 46. 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 47 and Figure 48. We use power law for deriving an empirical relation as: 21 c ICKc (4.1) 21 a IC uK a L (Nondimensional form) (4.2) The coefficients in the above relations are listed in the Table 46. Base on results presented in Table 46 and Figure 48, 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. PAGE 57 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 49, 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 410 and Figure 411. We use power law for deriving an empirical relation as: 21 c IIcKc (4.5) 21 a IIc uK a L (Nondimensional form) (4.6) The coefficients in the a bove relations are listed in Table 49. Base on results presented in Table 49 and Figure 411, 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 PAGE 58 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 412 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 plainstrain 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. PAGE 59 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 42. 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 43. 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 1.6543 9.092 1 0.1000 4.5933 3.921 1 0.1875 1.6152 2.23 1 0.2308 2.4462 3.87 1 0.2727 3.4162 5.98 1 0.3000 4.1342 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 1.6543 0.128 2 0.2000 4.5933 0.554 2 0.3750 1.6152 3.15 2 0.4615 2.4462 5.48 2 0.5455 3.4162 8.46 2 0.6000 4.1342 10.8 Table 46. Interpolation parameters for Mode I c1 c2 a1 a2 L =1 494 1.31 7.171 1.31 L =2 694 1.31 7.121 1.31 PAGE 60 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 1.6543 3.372 1 0.1000 4.5933 1.621 1 0.1875 1.6152 1.10 1 0.2308 2.4462 2.03 1 0.2727 3.4162 3.32 1 0.3000 4.1342 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 1.6543 4.772 2 0.2000 4.5933 0.230 2 0.3750 1.6152 1.55 2 0.4615 2.4462 2.87 2 0.5455 3.4162 4.70 2 0.6000 4.1342 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 PAGE 61 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%) PAGE 62 62 Figure 41. 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 42. A structure with 27 (3) cells PAGE 63 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 43. Convergence study of Youngs modulus y Figure 44. Deformation of a micromechanical mode PAGE 64 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 45. 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 46. Mode I fracture toughness vs. strut thickness PAGE 65 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 47. 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 48. Normalized Mode I fr acture toughness vs. relative density PAGE 66 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 49. 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 410. Mode II fracture toughness vs. relative density PAGE 67 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 103 */sKIIc/(u*L0.5) L=1 interp1 L=2 interp2 Figure 411. Normalized Mode II fr acture toughness vs. relative density Figure 412. Crack development history: first broken strut is labeled 1 and the last broken strut is 15. PAGE 68 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 Nonuniform 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 inplane dislocation, which means a vertex is within a circle of radius R in the xy 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 PAGE 69 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 cracktip 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. PAGE 70 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 planestrain 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. PAGE 71 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 PAGE 72 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 PAGE 73 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 103 RaRelative density */ s Mean value of */ s Figure 52. Relative density as a function of the di slocation distance Ra of a vertex. PAGE 74 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 PAGE 75 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 PAGE 76 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 PAGE 77 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 31). 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) PAGE 78 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 elasticplastic response, we need to specify the nodal forces N, M1, M2, and T directly as functions of their conjugate plastic de formation variables. For elasticperfectly 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). PAGE 79 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 31 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 31. 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 elasticperfectly 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. PAGE 80 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 PAGE 81 81 Figure 62. Contour of axia l elastic strain in struts Figure 63. Data points generate d for the perfect plastic model PAGE 82 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 PAGE 83 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 PAGE 84 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 globallocal 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 microscale and one at the macroscale, 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 micromacro 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 PAGE 85 85 combined micromacromechanics 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 plainstrain 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 opencell foams used in coolin g system wherein hot air/fluid flows through the foam. As energyabsorption function, foams ar e under compression and closedcell foams are often used. Hence, fracture behavior under comp ression and research on closedcell foams are good future topics. PAGE 86 86 APPENDIX A CRACK TIP DISPLACEMENT FIELDS FOR ORTHOTROPIC MATERIALS The stressstrain relation in the principal di rection for plane stress problem is given as: ** 112121 ** 212122 12121210 10 001 EE SEE G (A.1) The stressstrain relation can be transformed from the principa l 12 coordinate system to the xy 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 xy 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 PAGE 87 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 12 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. PAGE 88 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 B1. 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 B1. Since those values are needed to calculate stress in the struts, Table B1 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 elasticplast 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 PAGE 89 89 length. However, this simple method is an approximate method to evaluate the plastic deformation near the crack tip. Table B1. 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 PAGE 90 90 Figure B1. Forces and moment s on a frame element in space. PAGE 91 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, FractureToughness of Br ittle Cellular Solids. Scripta Metallurgica, 1984. 18(3): p. 213217. 3. Huang, J.S. and L.J. Gibson, FractureToughness of Brittle Foams. Acta Metallurgica Et Materialia, 1991. 39(7): p. 16271636. 4. Huang, J.S. and L.J. Gibson, FractureToughness of Brittle Honeycombs. Acta Metallurgica Et Materialia, 1991. 39(7): p. 16171626. 5. Brezny, R., D.J. Green, and C.Q. 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Hellan, K., Introduction to fracture mechanics 1984: McGrawHill Book Company. 34. Weisstein, E.W., Truncated Octahedron in MathWorldA Wolfram Web Resource: http://mathworld.wolfram.com /TruncatedOctahedron.html PAGE 94 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. 