A MICROMECHANICS METHOD TO PREDICT THE FRACTURE TOUGHNESS OF CELLULAR MATERIALS

Transcription

1 A MICROMECHANICS METHOD TO PREDICT THE FRACTURE TOUGHNESS OF CELLULAR MATERIALS By SUKJOO CHOI A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 00

2 Copyright 00 by Sukjoo Choi

3 Thi diertation i dedicated to my parent, Sunggu Choi and Jinil Yang

4 ACKNOWLEDGEMENTS I am very grateful to Dr. Bhavani V. Sankar for providing me the opportunity to complete my M.S. tudie under hi exceptional guidance and financial upport. He i not only my academic advior but alo a great influence in my life. Throughout thi reearch, I have greatly appreciated hi conitent encouragement, patience and poitive attitude. Alo, I would like to thank to Dr. Chen-Chi Hu, graduate coordinator, for hi advice. Becaue of great help and advice, tudying here wa a delightful experience. Many thank hould go to my colleague, Donald Myer, Nicoleta Apetre and Huadong Zhu. Moreover, I would like to thank my chool enior, Jongyoon Ok, Chungoo Ha and Kiloo Mok, who provided me invaluable academic feedback, encouragement and companionhip. I would like to thank my girlfriend, Sunghin Kang, who allowed me to devote myelf to tudying. I would like to expre my deepet appreciation to my parent for continuou upport and love. I am alo thankful to God for giving me the opportunity to extend my education at the Univerity of Florida. iv

5 TABLE OF CONTENTS page ACKNOWLEDGEMENTS... iv LIST OF TABLES... vii LIST OF FIGURES... viii ABSTRACT... xi CHAPTER INTRODUCTION... ELASTIC CONSTANTS OF THE FOAM...4 Formulation of Micro-Mechanic Propertie... 4 Finite Element Verification of Analytical Model... 7 FINITE ELEMENT METHODS OF FRACTURE TOUGHNESS... Boundary Diplacement at a Crack Tip... Model Setup... Convergence Analyi for Mode I... Mode I Fracture Toughne of Open Cell Foam... 7 Mode II Fracture Toughne of Open Cell Foam... 9 Fracture Toughne of Mixed Mode on Open Cell Foam... 0 Formulation of Fracture Toughne for Mode I and Mode II... Analytical Model for Mode I Fracture Toughne... Analytical Model for Mode II Fracture Toughne... 5 Mode I and Mode II Fracture Toughne with Angled Crack... 6 Prediction of the Maximum Strength under Tenile Loading... 7 Prediction of the Maximum Strength under Shear Loading... 9 Mode I Fracture Toughne of Inclined Crack... 4 FRACTURE TOUGHNESS OF CARBON FOAM...4 Material Propertie of Carbon Foam... 4 Mode I Fracture Toughne Tet (4-Pt. Bending Tet)... 4 Finite Element Analyi of Fracture Toughne... 8 v

6 Unit Cell of Carbon Foam Solid Model... 8 Micromechanic Analyi for Young Modulu and Shear Modulu... 9 Unit Cell of Carbon Foam Beam Model... 4 Fracture Toughne Etimation of the Solid Model Fracture Toughne of the Beam Model RESULTS AND DISCUSSION...47 Elatic Contant of Open-cell Foam Numerical Analyi of Fracture Toughne Fracture Toughne with an Angle Crack Fracture Toughne of Carbon Foam APPENDIX A ANALYTICAL METHOD TO ESTIMATE THE SOLIDITY OF SOLID MODEL..50 B CRACK TIP DISPLACEMENT FIELDS FOR ORTHOTROPIC MATERIAL...5 REFERENCES...55 BIOGRAPHICAL SKETCH...56 vi

7 LIST OF TABLES Table page. Material propertie of the Zoltex carbon fiber.... Foam model propertie for convergence analyi Mechanical propertie of carbon foam Denitie of variou form of carbon Fracture toughne with pecimen propertie of carbon foam Contant of fracture toughne curve Reult of fracture toughne vii

8 LIST OF FIGURES Figure page. SEM image of low (left) and high (right) denity carbon foam.... The packing of polyhedra to fill pace: tetrakaidecahedra.... Open rectangular cell model.... Applied tre for a open-cell foam Deformed hape of trut on open-cell foam Stre ditribution of tenile loading on open-cell foam Shear applied on open-cell foam Stre ditribution of hear loading on open-cell foam Open-cell foam model with a crack.... Variation of fracture toughne with variou model ize Axial tre variation from a crack tip for variou model ize Log-log plot of axial tre along a crack tip for variou model ize Fracture toughne in variou crack length Stre ditribution of Mode I fracture (c00 µm, h0 µm) Stre ditribution of Mode II fracture (c00µm,h0µm) Numerical reult of fracture toughne for Mode II Stre ditribution on the mixed mode fracture (c00µm, h0µm).... Mixed mode fracture toughne of contant length of cell edge.... Mixed mode fracture toughne of contant cell wall thickne... viii

9 . Crack tip force and moment in the actual foam and correponding crack tip tree in the idealized homogeneou continuum....4 Variation of α (l/c) with relative denity for Mode I Variation of α ( l/c) with relative denity for Mode II Force in a ligament under combined loading Variation of maximum tenile tre y a a function of angle between the xy coordinate ytem and the principal material direction Variation of maximum hear tre (hear trength) a a function of angle between the xy coordinate ytem and the principal material direction....9 Stre ditribution with variou crack angle for Mode I fracture....0 Mode I fracture toughne for variou crack angle with repect to the principal material direction.... Stre ditribution for variou angled crack under Mode II fracture.... Mode II fracture toughne for variou crack angle with repect to the principal material direction SENB pecimen geometry Four-point bending tet etup on a material teting machine Load-diplacement curve of four-point bending tet on carbon foam Unit cell of olid model Boundary condition on the unit-cell urface and maximum principal tre ditribution when the unit-cell i tretched in the y-direction Boundary condition on the unit-cell urface and maximum principal tre ditribution when the unit-cell i tretched in the y-direction Elatic modulu E a a function of relative denity (olidity) for E.6 GPa Shear modulu G a a function of relative denity (olidity) for E.6 GPa Unit cell of beam model... 4 ix

10 4.0 Maximum principal tre ditribution of olid and beam model for a unit K I at the crack tip Variation of fracture toughne of carbon foam with relative denity A. Unit cell of olid model B. Polar coordinate oriented from crack tip x

11 Abtract of Thei Preented to the Graduate School of the Univerity of Florida in Partial Fulfillment of the Requirement for the Degree of Mater of Science A MICROMECHANICS METHOD TO PREDICT THE FRACTURE TOUGHNESS OF CELLULAR MATERIALS By Sukjoo Choi December 00 Chairman: Dr. Bhavani V. Sankar Major Department: Mechanical and Aeropace Engineering Cellular material, uch a carbon foam, are ideal core material for andwich compoite in many application becaue of their thermal reitance, low denity, impact damage tolerance and cot effectivene. Moreover, carbon foam i a good material for heat exchanger and thermal protection ytem. The Mode I, Mode II and mixed mode fracture toughne of a cellular medium i predicted by imulating the crack propagation uing a finite element model. For the ake of implicity, the cellular medium i conidered a a rectangular open cell tructure, and the trut are aumed to be beam of quare cro ection. On the macro-cale, the cellular olid i conidered a an orthotropic material. A crack parallel to one of the principal material direction i aumed to exit in the olid and a mall region urrounding the crack tip i modeled uing beam finite element. Diplacement are impoed on the boundary of the region uch that the crack tip i ubjected to a given tre intenity factor for orthotropic material. The trut are aumed to fail in a brittle xi

12 manner when the maximum tenile tre become equal to the trength of the trut material, and the correponding tre intenity factor i taken a the fracture toughne of the cellular medium. Baed on the finite element reult a emi-empirical formula i alo derived to predict the Mode I and Mode II fracture toughne of cellular olid a a function of relative denity. Reult are alo preented for mixed mode fracture, and a imple mixed mode fracture criterion i derived for cellular olid. It i found that the fracture toughne i a trong function of the relative denity, but the cell ize alo ha a ignificant effect. Crack inclined to the principal material direction were alo conidered, and the Mode I and Mode II fracture toughne wa calculated a a function of the crack inclination angle. Fracture toughne of carbon foam wa etimated by experimental analyi and alo finite element method. Mode I fracture toughne of open cell carbon foam wa meaured uing ingle edge notched four-point bend pecimen. A micromechanical model wa developed auming a rectangular prim a the unit-cell. The cell wall were modeled uing three-dimenional olid element. Aforementioned finite element micromechanical analyi wa performed to predict the fracture. The micromechanical imulation were ued to tudy the variation of fracture toughne a a function of relative denity of the foam. The good agreement between the finite element and experimental reult for fracture toughne indicate that micro-mechanic can be an effective tool to tudy crack propagation in cellular olid. xii

13 CHAPTER INTRODUCTION Cellular material are made up of a net work of beam or plate tructure leaving an open pace or cell in between. Cellular material, e.g., carbon and polymeric foam, offer everal advantage of thermal reitance, durability, low denity, impact damage tolerance and cot effectivene. They have great potential a core material for andwich contruction, heat exchanger and thermal protection ytem in the military and commercial aeropace tructure. An excellent treatie on the tructure and propertie of cellular olid ha been written by Gibon and Ahby []. While analytical method of thermal and mechanical propertie of carbon foam are well documented, reearch on fracture behavior of variou foam i till in it infancy. Gibon and Ahby [] have preented approximate formula for Mode I fracture toughne of cellular olid in term of their relative denity and tenile trength. Thee are limited to crack parallel to the principal material direction. Alo, fracture behavior under mixed mode wa not tudied. A SEM wa ued to capture the image of carbon foam a hown in Figure.. In microcopic obervation, the open-cell foam are irregularly ized and paced. For highdenity carbon foam (00 to 800 kg/m ), the length of cell edge i in the range of,000 µm to,000 µm. For low-denity carbon foam (60 to 00 kg/m ), the cell length i in the range of 00 µm to 600 µm.

14 Figure. SEM image of low (left) and high (right) denity carbon foam The geometry of the foam i categorized by tetrakaidecahedra cell containing 4 face, 6 edge and 4 vertice a depicted in Figure.. Figure. The packing of polyhedra to fill pace: tetrakaidecahedra In thi tudy, finite element model are ued to predict the fracture toughne of ome cellular media under mixed mode condition. Baed on the microcopic obervation of carbon foam in Figure., both crack parallel to principal material direction and angled crack are conidered. For the ake of implicity, the cellular medium i conidered a a rectangular open cell tructure, and the trut are aumed to be beam of quare cro ection of a ide h a hown in Figure..

15 Figure. Open rectangular cell model On the macro-cale level, the cellular olid i conidered a an orthotropic material. A crack parallel to one of the principal material direction i aumed to exit in the olid and a mall region urrounding the crack tip i modeled uing beam finite element. For carbon foam, material propertie of cellular tructure are referred from pure carbon. Therefore, the Zoltex Pane 0MF High Purity Hilled carbon fiber (Table.) i choen becaue of the high percentage of carbon component weight (99.5%). Table. Material propertie of the Zoltex carbon fiber Denity 750 kg/m Modulu of Elaticity 07 Gpa Poion Ratio 0.7 Ultimate Tenile Strength 600 Mpa

16 CHAPTER ELASTIC CONSTANTS OF THE FOAM In order to etimate the Young modulu E in the principal material direction we ubject the foam to a uniform macro-tre in the direction (ee Figure -) where upercript of decribe the macromechanic propertie and the ubcript of i micromechanic propertie. Formulation of Micro-Mechanic Propertie The total tenile force acting on a trut i given by F c, where c i the length of the unit cell. The micro-tre (actual tre) in the trut can be derived a F c (.) h h Figure. Applied tre for a open-cell foam 4

17 5 where h i the cro ectional dimenion of the quare trut. It hould be noted that both the macro-train ε and micro-train ε are aumed to be equal. The micro-train i given by ε (.) E Subtituting for from Eq. (.) into Eq. (.) we obtain c ε h c ε E E h E (.) where E i elatic modulu of the trut material. The Young modulu of the foam can be obtained from Eq. (.) a E ε c h E h c E (.4) Figure. Deformed hape of trut on open-cell foam To calculate the hear modulu G, the unit-cell i ubjected to a tate of uniform hear a hown in Figure.. The deformed hape of the trut i hown in thin line. Due to anti-ymmetry the curvature of the deformed beam at the center of the trut mut be

18 6 equal to zero and hence the bending moment at the center of the trut mut alo be equal to zero []. Thi fact can be ued to find a relation between F and M a MFc/. Now half-of the trut, either OA or OB, can be conidered a a cantilever beam ubjected to a tip force F. The tranvere deflection i given by c F δ (.5) E I where I, the moment of inertia of the trut cro ection i given by 4 h I. The macrocopic hear tre τ i related to the hear force F a F c The macrocopic hear train γ can be calculated a The hear modulu of the foam τ (.6) δ 4δ γ (.7) c c G i defined a the ratio of the macrocopic hear tre τ and macro-hear train γ. Then from Eq. (.6) and Eq. (.7) we obtain G τ γ E h c 4 4 (.8) From the expreion for E Young modulu can be derived a and G a relationhip between the hear modulu and the E G h E c c (.9) 4 h h E 4 c

19 7 Due to the ymmetry of the tructure it i obviou that E E E and G G G. Further, ince we are uing beam theory to model the trut, all the Poion ratio referred to the principal material direction, ν, ν, and ν are equal to zero. The relative denity ρ /ρ, where ρ i the olid denity or the denity of the trut material, i a meaure of olidity of the cellular material. The denity of the foam can be obtained form the ma m and volume V of the unit-cell a hown below: ρ ρ m V ρ ρ ( h c h ) c ρ ( h c h ) c h c h c (.0) If the apect ratio of the trut h/c <<, the relative denity can be approximated a ρ ρ h h h c c c (.) Finite Element Verification of Analytical Model A portion near the crack tip of the foam wa modeled uing finite element. Each trut wa modeled a an Euler-Bernoulli beam. Each beam element ha integration point. Although we could have one unit-cell with periodic boundary condition by Marrey and Sankar [], a larger portion of the cellular medium wa modeled a the computational cot i not very high for thi cae. The model hown in Figure. conit of cell. The cell length c of the unit-cell i 00 µm and the trut i aumed to have a quare cro ection with a ide equal to 0 µm. A uniform diplacement (δ 0.0 m) i applied at node on the top edge of the model. The total force in the direction i computed from the FE reult. The average tenile tre (macro-tre ) can be obtained by

20 8 F y (.) L c where the Σ ign denote ummation of all nodal force and L i the width of the foam conidered in the FE model (ee Figure.). The Young modulu Figure. Stre ditribution of tenile loading on open-cell foam E of a foam can be determined by the tre-train definition a For the cae conidered the FE model gave a value Fy F L c y E ε (.) δ cδ L E.09 GPa and the analytical model Eq. (.4) yielded a value of.07 GPa. The difference of reult i %.

21 9 Figure.4 Shear applied on open-cell foam Figure.5 Stre ditribution of hear loading on open-cell foam To etimate the hear modulu by numerical analyi, a contant horizontal diplacement u i applied to all the node on the topide of the foam a hown in Figure.4. The hear tre τ can be obtained by the um of reaction force F divided by the area of the top urface:

22 0 F x τ (.4) Lc where L i the length of the foam conidered in the FE model and c i the unit-cell dimenion. The hear train γ can be calculated a Thu, hear modulu G u γ (.5) L of the foam can be etimated a G τ γ F L c u L x u F c x (.6) The deformed hape of the cellular olid under hear i hown in Figure.5. From the FE model the hear modulu of the foam wa etimated a 0.5 MPa, wherea the aforementioned analytical model Eq. (.8) yield 0.5 MPa. The difference between the two reult i about %. The FE model i lightly compliant becaue of lack of contraint on the ide and applying periodic boundary condition would have yielded value cloer to the analytical olution.

23 CHAPTER FINITE ELEMENT METHODS OF FRACTURE TOUGHNESS In thi ection, we decribe a finite element baed micromechanic model for etimating the fracture toughne of the cellular olid. The crack i aumed to be parallel to one of the principal material axe, and Mode I, Mode II and mixed mode fracture are conidered. To determine the fracture toughne a mall region around the crack tip i modeled uing beam element. A contant mode mixity K I /K II i aumed to be applied. The boundary of the cellular olid i ubjected to diplacement (u and u ) boundary condition correponding to an arbitrary value of K I (or K II ). The rotational degree of freedom at each node of the beam element on the boundary of the olid i left a unknown and no couple are applied at thee node. The calculation of boundary diplacement for a given tre intenity factor i decribed in the next ection. Boundary Diplacement at a Crack Tip The diplacement component in the vicinity of a crack tip in a homogenou orthotropic material are derived in Appendix B and they are a follow: Diplacement field for Mode I: u u x y K K I I r Re π r Re π [ p ( coθ + inθ ) ( + ) ] p coθ inθ [ q ( coθ + inθ ) ( + ) ] q coθ inθ (.) Diplacement field for Mode II:

24 u u x y K K II II r Re π r Re π [ p ( coθ + inθ ) ( + ) ] p coθ inθ [ q ( coθ + inθ ) ( + ) ] q coθ inθ (.) In deriving the above expreion, the crack i aumed to be parallel to the x axi, and r-θ i the polar coordinate ytem oriented at the crack tip. The complex parameter p, q, and depend on the elatic contant of the homogeneou orthotropic material a decribed in the Appendix B. Model Setup The commercial FE program ABAQUS i ued to conduct numerical analyi. A FE model of open cell foam i taken around the mall portion of the crack tip in Figure.. The boundary condition of mall portion i determined by uing the olution for boundary diplacement in the previou ection. A crack in the FE model i created by removing beam element. Figure. Open-cell foam model with a crack To reduce the modeling cot, a FORTRAN code wa written to generate nodal and element propertie with uer pecified unit-cell configuration. The code calculate

25 boundary diplacement at correponding boundary nodal coordinate. After execution, it export an ABAQUS input file o that the ABAQUS can read the input directly. The maximum tenile tre in the trut i calculated from the finite element method. The FE analyi output axial force, bending moment and hear force at each node of the beam element, and the maximum tenile tre wa calculated uing a eparate program. Uually the maximum tre occur at the crack tip trut. From the reult the value of K I (or K II ) that will caue rupture of the trut i etimated, which then i taken a the fracture toughne of the cellular olid. Before we decribe an analytical model for etimating the fracture toughne, we will dicu the reult from the FE imulation. At firt, we will check the validity of applying continuum fracture mechanic. It wa found that a model coniting of cell (total 0,000 cell) gave a converged reult for fracture toughne and the ame model i ued throughout the tudy. The run time for thi model i about ½ minute in a.7 GHz Intel Pentium 4 computer. Convergence Analyi for Mode I The convergence analyi i performed to evaluate variation of fracture toughne with variou ize of foam model,.cm.cm (,600 cell), cm cm (0,000 cell), 4cm 4cm (40,000 cell) and 8cm 8cm (60,000 cell). The foam model propertie are hown in Table.. Table.. Foam model propertie for convergence analyi Length of cell edge 00 µm Cell wall thickne 0 µm Relative denity Tenile trength 600Mpa

26 4 A the model ize increae, fracture toughne become table at approximately Pa m / in Figure.. For a foam model containing from 0,000 to 60,000 cell, variation in fracture toughne i.7%. Therefore, the 0,000 cell model i choen for further numerical analyi to maintain output accuracy with le CPU time. For the 0,000 cell model, a computer with.7ghz Intel Pentium 4 take ½ minute to complete the job, but the 60,000 cell take hour more. Figure. Variation of fracture toughne with variou model ize The axial tre along the crack path from a crack tip i hown in Figure.. In the reult, maller model produce higher axial tre at the crack tip. Further from the crack tip, variation in axial tree become inignificant for all model ize.

27 5 Figure. Axial tre variation from a crack tip for variou model ize The log-log plot of maximum trut tre variation from the crack tip i hown in Figure., the curve i fitted to the equation 4.69 (.) r Uing the relation between micro and macro tree in Eq. (.), Eq..4 can be written a.05 (.4) π r indicating that the macro tre in tenile factor i equal to.05 which i approximately equal to the applied the tre intenity factor K I of unity.

28 6 Figure.4 Log-log plot of axial tre along a crack tip for variou model ize If the fracture toughne etimated by uing the preent method i truly a material(macrocopic) property, then it hould be independent of the crack length. Hence the crack length wa varied in the micromechanical method. The reult are hown in Figure.5. The predicted fracture toughne wa plotted a a function of percentage of crack length in the model. The reult clearly how that idea of modeling the foam a a homogenou material i quite acceptable.

29 7 Figure.5 Fracture toughne in variou crack length Mode I Fracture Toughne of Open Cell Foam Since the relative denity depend on the length of the trut c and a cell wall thickne h, the fracture imulation i conducted in two cae. In the firt cae, c i varied and h i kept contant. The econd cae, h varied while c i contant. The reult of the FE imulation are axial force, bending moment and hear force in each element, which are ued to calculate the maximum principal tre at the crack tip, and then the fracture toughne. A ample tre ditribution due to Mode I fracture i hown in Figure.6.

30 8 Figure.6 Stre ditribution of Mode I fracture (c00 µm, h0 µm) The variation of fracture toughne with relative denity i hown in Figure.7. The reult of fracture toughne in Figure.7 are fitted to a curve with a power of.045 for contant wall thickne and for contant length of trut. To obtain a higher fracture toughne, increaing cell wall thickne h i more effective while cell length c i contant. Figure.7 Mode I fracture toughne for two cae

31 9 Mode II Fracture Toughne of Open Cell Foam The analyi of Mode II fracture toughne i performed in ame manner a the Mode I fracture analyi in previou ection. Unlikely to Mode I fracture, the boundary diplacement are not ymmetric about the crack plane a hown in Figure.8. Figure.8 Stre ditribution of Mode II fracture (c00µm,h0µm) The reult of Mode II fracture toughne are fitted in a curve with power of.0654 for contant wall thickne h and. for the contant cell length c in Figure.9. Therefore, to obtain the higher fracture toughne, increaing cell wall thickne h i more effective than increaing cell length c for Mode II.

32 0 Figure.9 Numerical reult of fracture toughne for Mode II Fracture Toughne of Mixed Mode on Open Cell Foam For the mixed mode fracture analyi, the boundary diplacement for Mode I and Mode II are uperpoed for variou ratio of Mode I and Mode II in Figure.0. A ample deformation under mied mode i hown in Figure.0. Fracture toughne for mixed mode i hown for contant length of cell edge in Figure. and contant wall thickne in Figure.. The fracture toughne for mixed mode i inverely linearly proportional. K K I IC a K + K II IIC b (.5) where the contant a and b are. The combination of K I and K II for fracture are hown in Figure. for contant cell length and in Figure. for contant cell wall thickne.

33 Figure.0 Stre ditribution on the mixed mode fracture (c00µm, h0µm) Figure. Mixed mode fracture toughne of contant length of cell edge

34 Figure. Mixed mode fracture toughne of contant cell wall thickne Formulation of Fracture Toughne for Mode I and Mode II Analytical Model for Mode I Fracture Toughne Figure. Crack tip force and moment in the actual foam and correponding crack tip tree in the idealized homogeneou continuum In order to derive an analytical model for fracture toughne, the tre intenity factor of the homogeneou model hould be related to the actual tree in the crack tip ligament of the foam. Thi can be obtained by auming that the internal force and bending moment in the crack tip trut are caued by a portion of the crack tip tre field

35 ahead of the crack tip in the homogeneou model. Let u define a non-dimenional factor α that decribe the effective length l a follow: l α c (.9) The ideal tre ditribution y ahead of crack tip i decribed a follow: y K I πr (.0) where r i the ditance from the crack tip. The axial force in the crack tip ligament i obtained by integrating the tenile tre y over the effective length F l l c ydr K I c (.) π 0 The bending moment M i given by M l l / / K I c K I c l l c r ydr rdr K I c (.) π π 0 0 π Auming fracture occur when the maximum bending tre equal the tenile trength of the ligament material, a relationhip between the tenile trength and fracture toughne can be derived a h M / / My 6M 6 l l K I c K I c (.) I h h h π π h By ubtituting for l in term of α from Eq. (.9), the fracture toughne can be related to the tenile trength a / c K / I α c (.4) π h Then the fracture toughne of the cellular material can be derived a

36 4 K π h π h h u u (.5) / 5 / / α c α c c I The non-dimenional ditance α can be found from the FE reult for fracture toughne K Ic and the tenile trength of the ligament material a follow: / u π h α 5 / (.6) K I c The reult for α a a function relative denity are plotted in Fig..4. It may be noted that α increae with relative denity and the variation can be accurately repreented by a power law. Interetingly both contant wall thickne cae and the contant pacing cae we have tudied o far fit accurately into a ingle power law. The effective crack tip ditance i about 0% of the cell pacing for low-denity foam and about 05 for highdenity foam.

37 5 Figure.4 Variation of α (l/c) with relative denity for Mode I Analytical Model for Mode II Fracture Toughne The effective crack tip ditance for Mode II can be derived following tep imilar to that in the preceding ection for Mode I. The hear tre (τ xy ) ditribution ahead of the crack tip i given by: τ xy K II (.7) πr The total hear force F over the effective ditance l can be derived a F l l c τ xydr K II c (.8) π 0 The maximum bending moment in the crack tip element i the product of force F and the length c/: M l c K II c (.9) π The maximum bending tre i derived a h M My 6M l c K II (.0) I h h π h An expreion for fracture toughne K II in term of trut tenile trength, trut dimenion and the effective ditance can be derived a K II u h π (.) c l Since, fracture toughne can be expreed in term of the non-dimenional ditance α: K u h π uh π (.) 5 c αc c α II

38 6 From (.) an expreion for α can be derived a 6 uh π π u h α (.) 5 5 K c 8 K II c II The contant α can be evaluated uing the FE reult of fracture toughne. The value of α a a function of relative denity i plotted in Fig..5. Again, it can be noted that a power law decription i adequate for Mode II alo. Further the value of α depend only on the relative denity and not on individual cell or trut dimenion. The effective ditance eem to be lightly le for Mode II. It i about 5% of the trut pacing for low denity and about % for high denity foam. Figure.5 Variation of α ( l/c) with relative denity for Mode II Mode I and Mode II Fracture Toughne with Angled Crack So far our attention ha been focued on crack parallel to the principal material direction. The next tep will be to tudy crack inclined at an angle to the principal direction. Before that, an undertanding of how the material will fail under combined

39 7 loading will be beneficial. In the following we decribe methodologie to predict the failure of the cellular medium under combined loading. The reult will then be ued to predict the tenile and hear trength in an arbitrary coordinate ytem. Prediction of the Maximum Strength under Tenile Loading When the cellular medium i ubjected to combined loading under plane tre condition, the ligament in a typical repreentative volume element will be ubjected to axial force, hear force and bending moment a hown in Fig..6. Figure.6 Force in a ligament under combined loading From the equation (.), axial tre can be obtained a h (.5) axial c The relation between maximum bending moment and the hear force i given by M c c τ c F τ c (.6) The maximum tre due to the bending moment can be derived a τ c h M y c bending τ 4 (.7) I h h

40 8 It hould be noted that we have aumed that only y component of the macro-tre exit. The ligament tree in Eq..5 and Eq..7 can be expreed in term of macro-tree referred to the x-y coordinate ytem a: in y co τ y θ θ inθ coθ y (.8a) (.8b) (.8c) Then the micro-tre u under combined loading can be derived a axial c y in θ (.9) h bending c τ y inθ coθ (.0) h The micro-tre u can be derived a c c c c u y in θ ± y inθ coθ in ± in co y θ θ θ (.) h h h h The maximum tre that can be applied to the cellular medium i obtained by equating the maximum micro-tre to the trength of the ligament material. The ratio of the macro-tre and the micro-tre for ligament L (refer to Fig..6) can be derived a y u in c c θ ± inθ coθ h h (.) In the ame manner, the ratio of the macro-tre and the micro-tre for ligamnet L can be derived a

41 9 y u co c c θ ± inθ coθ h h (.) Eq. (.) and Eq. (.) are ued to calculate the maximum tre y that ca be applied to the cellular medium before any ligament failure. Then maximum tre under tenile loading i a function of the orientation θ a hown in Figure.7. The trength variation i ymmetric about θ45 o. Figure.7 Variation of maximum tenile tre y a a function of angle between the xy coordinate ytem and the principal material direction Prediction of the Maximum Strength under Shear Loading The calculation of hear trength at an arbitrary orientation i very imilar to that decribed for tenile trength in the preceding ection. The tranformation of macrotree between the global xy coordinate and the principal material coordinate are given by:

42 0 τ in θ xy τ in θ xy τ τ co θ xy (.4a) (.4b) (.5c) It may be noted that only hear component of the macro-tre (τ xy ) i preent. Subtituting Eq. (.4) into the Eq. (.5) and (.7) we obtain c τ xy in c axial θ (.6) h h The micro-tree can be derived a c τ xy co c bending τ θ (.7) h h c c c c u τ xy in θ ± τ xy co θ τ xy in θ ± co θ (.8) h h h h The ratio of the macro-hear tre and the micro-tre in ligament L can be derived a τ xy u c c in θ ± co θ h h (.9) In the ame manner, the ratio of the applied hear tre and the maximum tre in ligament L can be derived a τ xy u c c in θ ± co θ h h (.40) The macrocopic hear trength (maximum hear tre) i plotted a a function of orientation θ in Figure 5.. The maximum tre i ymmetric about θ45 o.

43 Figure.8 Variation of maximum hear tre (hear trength) a a function of angle between the xy coordinate ytem and the principal material direction Mode I Fracture Toughne of Inclined Crack The procedure for predicting the fracture toughne of angled crack (crack inclined at an angle to the material principal direction) are very imilar to that decribed in the preceding ection. The only change i in the material elatic contant which have to be tranformed from the material principal direction to the global xy coordinate ytem. The tre field under Mode I fracture for variou crack angle are hown in Figure.9.

44 Figure.9 Stre ditribution with variou crack angle for Mode I fracture The variation of Mode I fracture toughne with crack orientation i hown in Fig.0 for variou relative denitie. It may be noted that the variation of Mode I fracture toughne with θ i very imilar to that of tenile trength hown in Fig..7 with the reult being ymmetric about θ45 o. Figure.0 Mode I fracture toughne for variou crack angle with repect to the principal material direction

45 Figure. Stre ditribution for variou angled crack under Mode II fracture The tre filed for variou angled crack under Mode II fracture i hown in Figure.. The variation of Mode II fracture toughne i hown in Figure.. Again one can note the imilaritie in the variation of K IIc and hear trength preented in Fig..8 Figure. Mode II fracture toughne for variou crack angle with repect to the principal material direction

46 CHAPTER 4 FRACTURE TOUGHNESS OF CARBON FOAM Material Propertie of Carbon Foam From the SEM image of low-denity carbon foam hown in Figure., one can note that the cell are irregularly ized and paced. The cell ize i meaured to be in the range of to mm. Mechanical propertie of the carbon foam a reported by Touchtone Reearch Laboratory, Inc., manufacturer of the foam, are hown in Table 4.. The olidity (relative denity) i determined by dividing, ρ, denity of the foam, by ρ, the denity of olid carbon that make the trut or the cell wall. The denitie of variou form of carbon are given in Table 4.. The olidity of the carbon foam ued in the preent tudy wa baed on the value of olid denity ρ,50 kg/m, and the olidity i determined a 0.. Table 4. Mechanical propertie of carbon foam Elatic Modulu (E).79 MPa Tenile trength ( u ).5805 MPa Denity (ρ ) 95. kg/m Table 4. Denitie of variou form of carbon Diamond (C Wt. %00),50 kg/m Graphite carbon fiber (C Wt. %00),50 kg/m Zoltec Pane 0MF carbon fiber (C Wt. % 99.5),750 kg/m Mode I Fracture Toughne Tet (4-Pt. Bending Tet) There are everal method available for meauring the fracture toughne of cellular material. Compact tenion tet (CT), Single edge notched bend tet (SENB) and 4

47 5 Double edge notched tenion tet (DENT) performed by Fowlke [6] are ome of the tet that are uitable for foam material. In the preent tudy, we choe the ingle edge notched four-point bend pecimen for meauring the fracture toughne of carbon foam. It wa thought that the four-point bending tet would yield more accurate and repeatable reult a the crack i in a region under contant bending moment and no tranvere hear force. Hence mall offet of the loading point with repect to the crack location will not ignificantly affect the reult. The pecimen dimenion are depicted in Figure 4.. The height of the pecimen wa about 50 mm and the crack length wa about 5 mm. Individual pecimen dimenion are given in Table 4.. A notch wa cut uing a diamond aw, and then a razor blade wa ued to harpen the crack tip. The crack length wa the ditance of the crack tip from the bottom urface edge of the beam. The tet were conducted under diplacement control in a material teting machine at the rate of 0.5 mm/min (Figure 4.). Load-deflection diagram are given in Figure 4.. It may be noted from the curve that the crack propagated intantaneouly and the pecimen failed in a brittle manner. The fracture load for variou pecimen are lited in Table 4.. The Mode I fracture toughne wa calculated from the load at failure uing the following formula []: 4 a a a a K I πa (4.) w w w w 4 where the maximum bending tre in the uncracked beam i determined by w M M y 6M (4.) I w Bw B

48 6 In Eq. 4., M i the contant bending moment in the central region, h i the height of the pecimen and B i the width. The bending moment M i given by MPd/, where d i ditance between one of the top loading point and the correponding bottom upport a hown in Figure 4.. The reult for fracture toughne are lited in Table 4.. For the carbon foam ample teted the average Mode I fracture toughne i found to be 0.7 MPa m / with a tandard deviation of 0.0 MPa m / (about 8%). Figure 4. SENB pecimen geometry Table 4. Fracture toughne with pecimen propertie of carbon foam Speci men Span L (m) Height h (m) Width B (m) Crack length a (m) Denity (Kg/m ) Fracture Load (N) K Ic (MPa m / ) IF IF IF IF

49 7 Figure 4. Four-point bending tet etup on a material teting machine Figure 4. Load-diplacement curve of four-point bending tet on carbon foam

50 8 Finite Element Analyi of Fracture Toughne Unit Cell of Carbon Foam Solid Model The firt tep in imulating the crack propagation in carbon foam i to idealize the microtructure of the foam. The unit-cell i aumed a a perfect cube of ide c in Figure 4.4. The foam model i created by placing a pherical void (bubble) at the center of the cube. By varying the radiu of the bubble R, foam in variou olidity can be modeled. A relation between the olidity and the R/c ratio can be derived in Appendix A: ρ ρ 4 + π 8 R + π 4 a R π a (4.) Figure 4.4 Unit cell of olid model The average dimenion of the unit-cell wa obtained from the SEM image of the cro ection of carbon foam. Then, the radiu of the pherical void can be determined from the olidity of the foam. The Pro/Engineering, modeling application, wa ued to model the unit-cell and calculate the olid volume. In the preent tudy, the unit-cell dimenion c i taken a.8 mm and the olidity a 0..

51 9 The trength of the olid carbon in the foam can be eaily etimated from the tenile trength of the carbon foam, which i meaured experimentally. The relation between the foam tenile trength and the olid carbon trength i given by c u u (4.4) A min where A min i the minimum cro ectional area of the trut in the carbon foam normal to the principal material axi. It hould be noted that the tenile trength of the foam i for a direction parallel to one of the principal material axe. The area A min wa obtained from the modeling oftware and it wa equal to mm. Subtituting the dimenion of the unit.cell and the meaured carbon foam tenile trength, the trength of olid carbon wa etimated a 6 Mpa. The procedure for determining the Young modulu and tenile trength of the olid carbon i a follow. Micromechanic Analyi for Young Modulu and Shear Modulu The Young modulu E of olid carbon wa etimated by a trial and error method. An initial value for E i aumed and the elatic contant of the carbon foam are determined by uing the micromechanical method developed by Sankar et al. [4]. Then the value of E can be caled to match the micromechanical reult for E to the experimentally meaured E. The unit-cell wa modeled by uing 4-noded tetrahedral olid element. Due to ymmetry only a portion of the unit-cell wa modeled and periodic boundary condition were impoed uch that only one of the macro-train i non zero [4]. We aume a value for E and ν, and the force required to deform the unit cell are calculated from the nodal reaction. From the force the macro-tree can be computed. For example

52 40 Fx F y x, y (4.5) c c where ΣF x repreent the um of all nodal force on a face normal to the x axi. Similary ΣF x i the um of all the nodal force F y on the face normal to the y axi. The carbon foam i aumed a an orthotropic material and the macro-tree and train are ubtituted in the contitutive relation to obtain the compliance coefficient S ij. From the S matrix the elatic contant can be etimated uing the following relation: Ex ν E ν E ε x S ε y S ε z S xy x xz x ν E E ν E y yz y yx y S S S S S S ν zx Ez S ν zy S Ez S Ez x y z S S S S S S (4.6) The FE model contained approximately 00,000 olid tetrahedral element. A diplacement u y wa applied to the top urface of a unit cell (Figure 4.5). The contour plot of maximum principal tree i depicted in Figure 4.5.

53 4 Figure 4.5 Boundary condition on the unit-cell urface and maximum principal tre ditribution when the unit-cell i tretched in the y-direction For the ake of implicity no attempt wa made to etimate the Poion ratio ν uing the micro-mechanical method and it wa aumed to be equal to 0.7 baed on a previou analyi. Baed on the foam propertie given in Table 4. and the unit-cell ize c of.8mm, the Young modulu of olid carbon E wa etimated to be.6 GPa. Uing thi value for E the Young modulu of carbon foam of variou oliditie wa calculated uing the FE model and they are hown in Figure 4.7. Figure 4.6 Boundary condition on the unit-cell urface and maximum principal tre ditribution when the unit-cell i tretched in the y-direction

54 4 The hear modulu wa alo calculated uing the micromechanic analyi developed by Marrey and Sankar [4]. The variation of hear modulu a function of olidity i hown in Figure 4.8. Figure 4.7 Elatic modulu E a a function of relative denity (olidity) for E.6 GPa. Figure 4.8 Shear modulu G a a function of relative denity (olidity) for E.6 GPa.

55 4 Unit Cell of Carbon Foam Beam Model In addition to the olid model decribed in the preceding ection a imple lattice model wa alo attempted. In thi model the foam i aumed to be made up of trut arranged in a cubic lattice pattern. The length of the trut wa equal to the unit-cell dimenion c in Figure 4.9. The trut were aumed to be uniform quare cro ection beam and their dimenion were determined from the relative denity of the foam. Figure 4.9 Unit cell of beam model From the Eq. (.), the wall thickne h i determined to be mm for a olidity of 0. and c.8 mm. Since thi beam model i compoed a a implet hape, FE method are not neceary to determine the relation between the olid propertie and foam propertie. Analytical expreion for variou elatic propertie have been derived earlier ection. Baed on the Eq. (.) and Eq. (.4), the Young modulu E and the trength u are found to be.4 MPa and 69.5 MPa, repectively. The variation of E and G with the olidity for the beam model are hown, repectively, in Figure 4.7 and 4.8.

56 44 Fracture Toughne Etimation of the Solid Model In thi ection, we decribe a finite element baed micromechanic model to etimate the fracture toughne of the cellular olid. The crack i aumed to be parallel to one of the principal material axe. The crack i created by breaking the ligament of the unit cell (Figure 4.0). To determine the fracture toughne, a mall region of the foam around the crack tip i modeled uing finite element. Only Mode I fracture i conidered in the preent tudy. The boundary of the cellular olid i ubjected to diplacement boundary condition (u x and u y ) correponding to a unit K I., i.e., K I. The diplacement component in the vicinity of a crack tip in an orthotropic olid can be found in Appendix B. The maximum tenile tre in the unit-cell correponding to unit tre intenity factor i calculated from the FE model. In the cae of three-dimenional olid model the maximum tre i obtained a an output of the FE program. From the reult the value of K I that will caue rupture of the trut i etimated, which then i taken a the fracture toughne of the cellular olid. The olid model ued 4 cell with 5,000 olid tetrahedral element a hown in Figure 4-0. The maximum principal tre ditribution i hown in Figure 4.0. When unit K I wa applied to the crack tip, maximum principal tre wa found equal to,46 Pa. Then the fracture toughne i obtained from the trength of the olid carbon ( u 0.6 MPa) a K u IC 0. max 46 MPa m (4.)

57 45 Comparing Eq. (4-) with the reult of fracture toughne experiment (Table 4.), the difference i about 6%. One reaon for the difference could be the mall number of cell ued in the FE imulation. The FE model wa ued to tudy the variation of fracture toughne with relative denity and the reulting relationhip i hown in Figure 4.. Gibon and Ahby [] provide analytical reult for fracture toughne for open-cell foam a given below: K u IC πc 0 ρ.65 ρ (4.) where u i the tenile trength of the olid. Uing the above formula the fracture toughne for the carbon foam conidered in thi tudy can be obtained a 0.6 MPa m. The variation of K Ic according to Eq. (4-) i preented in Figure 4.. Figure 4.0 Maximum principal tre ditribution of olid and beam model for a unit K I at the crack tip Fracture Toughne of the Beam Model The procedure for imulating fracture uing the beam model i the ame a that for the FE olid model. The beam model of carbon foam (Figure 4.0) conited of

58 46 0,000 cell uing approximately 0,000 beam element. When uing the beam model, the rotational degree of freedom at each node of the beam element on the boundary of the olid i left a unknown and no couple are applied at thee node. The maximum principal tre ditribution in the beam model for K I i hown in Figure 4.0. The maximum principal tre for a unit K I wa found to be equal to 506 Pa. Therefore, the fracture toughne can be etimated a 6 u 69 0 K IC 0. 7 MPa 506 max m (4.4) The difference between the experimental reult and that from the beam model i only %. The beam model wa alo ued to tudy the variation of fracture toughne with the olidity and it i preented in Figure 4.. Figure 4. Variation of fracture toughne of carbon foam with relative denity

59 CHAPTER 5 RESULTS AND DISCUSSION Elatic Contant of Open-cell Foam Young modulu for carbon foam in the principal material direction i etimated to be.09 GPa from the FE micro-mechanic and.07 GPa baed on analytical olution. Shear modulu i predicted a 0.45 MPa from FE analyi and 0.50 MPa from analytical olution. Difference between analytical and numerical reult i 0.96% for elatic modulu and.0% for hear modulu. For open-cell foam model, analytical olution and FE olution for elatic contant how good agreement. Numerical Analyi of Fracture Toughne A Finite Element method baed micro-mechanic method wa developed to predict the fracture toughne of cellular material. A portion of material around the crack tip wa modeled uing finite element. Boundary diplacement calculated uing orthotropic fracture mechanic were applied to the FE model. The tre intenity factor correponding to the failure of the crack tip trut wa taken a the fracture toughne of the cellular medium. It ha been found that the fracture toughne i a trong function of relative denity of the foam, however it alo depend on the trut dimenion and pacing. Baed on the FE reult an analytical model wa developed to predict the fracture toughne. A imple empirical formula ha been derived for the effective length ahead of the crack tip that contribute to the crack tip ligament force and bending moment. For the type of cellular medium conidered in thi tudy the fracture toughne can be 47

60 48 expreed in the form of power law of the type where K i fracture toughne and R i relative denity. The contant a and b for Mode I and Mode II are preented in Table 5. Table 5. Contant of fracture toughne curve Contant length of cell edge with variou wall thickne Contant wall thickne with variou length of cell edge A b a b Mode I.96x x Mode II 6.95x x The effective length l ahead of the crack-tip i expree a lαc. The contant α wa evaluated analytically uing the reult for fracture toughne obtained uing the FE micro-mechanic model. It ha been found that α depend only on relative denity and i independent of cell length or trut cro ectional dimenion. Fracture Toughne with an Angle Crack Variation of trength under combined loading for open-cell foam wa evaluated analytically. The reult were ued to determine the tenile and hear trength when the load are applied at an angle to the principal material direction. FE micro-mechanic imulation were performed for inclined crack. Mode I fracture toughne i maximum at 0º and Mode II fracture toughne i maximum at 45º to the principal material direction. Fracture Toughne of Carbon Foam Four point bend tet were performed on SENB pecimen made of carbon foam, and their Mode I fracture toughne wa meaured. In addition to the experimental approach, a finite element baed micromechanic ha been developed to predict the fracture toughne. Two micro-mechanical model were developed to imulate Mode I fracture. Both model aumed a cube a the unit-cell of the foam. In the firt model olid finite element were ued to model the foam. The meaured denity of the carbon foam

61 49 wa ued in determining the void ize in the micromechanical model. Young modulu and tenile trength of the olid carbon were alo determined from the correponding value of the carbon foam meaured experimentally. A mall region urrounding the crack tip wa modeled uing finite element. The crack wa aumed parallel to one of the principal material direction. Boundary diplacement were calculated uing linear elatic fracture mechanic for orthotropic material. From the FE imulation the tre intenity factor K I that will caue the failure of the crack-tip element wa determined, and thi wa taken a the fracture toughne of the cellular material hown in Table 5.. The agreement between the tet reult and numerical reult are good indicating micromechanic can be a powerful tool in predicting the fracture behavior of foam and other cellular olid. Table 5. Reult of fracture toughne K ic ( MPa m / ) % Difference from Experiment Beam Model 0.7 Solid Model Gibon & Ahby 0.6 Experimental 0.

62 APPENDIX A ANALYTICAL METHOD TO ESTIMATE THE SOLIDITY OF SOLID MODEL Figure A. Unit cell of olid model To etimate the olidity of porou medium in the olid model, the volume of the pore (void) left inide the unit-cell need to determined. In order to do that, the volume of top potion V B in Fig. A i ubtracted from entire phere volume of a void. The volume V A i obtained a V A ψ R (A.) where Ψ i olid angle. The olid angle can be obtained a follow: da ψ R (A.) da π r dr coθ (A.) da π r dr coθ π r hdr dψ (A.4) h + r h + r ( h + r ) 50

63 5 Integrating both ide, ( ) + b b r h r hdr d 0 0 π ψ (A.5) R h h π ψ (A.6) By ubtitute Eq. (A6) into Eq. (A), V A i obtained a follow; [ h R R R h hr R V A π π ψ ] (A.7) The volume of the top portion V B can be obtained by ubtracting the volume of the cone from V A [ ] h h R R b h R R V B π π π π π + (A.8) The volume of the pore left inide the unit-cell i derived a V C, a a R R R V R V B c π π π π π + (A.9) Therefore, the olidity can be obtained from it definition a: a a R R R a a V a V a c c π π π π ρ ρ (A.0) For open cell model, the equation i only valid in the range where R i the radiu of the phere in Fig. A. After implifying the equation, the olidity can be expreed in term of the ratio R/a in the form of a polynomial: a R a R π π π ρ ρ (A.)