MECHANICAL AND FAILURE PROPERTIES OF RIGID POLYURETHANE FOAM UNDER TENSION MUHAMMAD RIDHA

Transcription

1 MECHANICAL AND FAILURE PROPERTIES OF RIGID POLYURETHANE FOAM UNDER TENSION MUHAMMAD RIDHA NATIONAL UNIVERSITY OF SINGAPORE 007

2 MECHANICAL AND FAILURE PROPERTIES OF RIGID POLYURETHANE FOAM UNDER TENSION MUHAMMAD RIDHA (S.T., ITB) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT ON MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE

3 Acknowledgement In the name of Allah, the Mot Graciou, the Mot Merciful. All praie and thank be to Allah who ha given me the knowledge and trength to finih thi reearch. I would like to expre my incere gratitude to Profeor Victor Shim Phyau Wui for hi guidance, uperviion and upport during the coure of my reearch. I would alo like to thank Mr. Joe Low and Mr. Alvin Goh for their technical upport in undertaking thi tudy. My pecial thank to my friend and colleague in the Impact Mechanic Laboratory of the National Univerity of Singapore for their help and dicuion on variou reearch iue, a well a for making my tay in NUS enjoyable. I am grateful to the National Univerity of Singapore for providing me a Reearch Scholarhip to purue a Ph. D., and to NUS taff who have helped me in one way or another. I would alo like to expre my incere gratitude to my parent who ha upported me through all my effort and encouraged me to purue higher education; alo my wife for her undertanding, patience and upport during the completion of my tudy at the National Univerity of Singapore. Muhammad Ridha i

4 Table of content Acknowledgement...i Table of content...ii Summary...v Lit of figure... viii Lit of table...xix Lit of ymbol...xx Chapter Introduction.... Propertie of olid foam and it application.... Studie on mechanical behaviour.... Objective...4 Chapter Literature review...6. Microtructure of polymer foam...6. Baic mechanical propertie of olid foam Compreion Tenion...8. Factor influencing mechanical propertie of olid foam Studie on mechanical propertie of olid foam Experimental tudie Cell model Contitutive model... Chapter Rigid Polyurethane Foam...5. Fabrication of rigid polyurethane foam...5. Quai-tatic Tenile tet...6. Dynamic tenile tet...0 ii

5 .4 Micro CT imaging of rigid polyurethane foam cell....5 Microcopic obervation of cell trut Microcopic obervation of deformation and failure of polyurethane foam 8.6. Tenile repone Compreive repone Mechanical propertie of olid polyurethane Summary...54 Chapter 4 Analytical Model of Idealized Cell Rhombic dodecahedron cell model Relative denity Mechanical propertie in the z-direction Mechanical propertie in the y-direction Correction for rigid trut egment Tetrakaidecahedron cell model Relative denity Mechanical propertie in the z-direction Mechanical propertie in the y-direction Correction for rigid trut egment Contant C, C and C Reult and dicuion Cell geometry and parametric tudie Comparion between model and actual foam Summary...5 Chapter 5 Finite Element Model...9 iii

6 5. Modelling of cell Reult and dicuion Repone to tenile loading Influence of cell wall membrane on crack propagation Repone to tenile loading after modification Influence of randomne in cell geometric aniotropy and hape Summary...7 Chapter 6 Concluion and Recommendation for future work Concluion Recommendation for future work...79 Lit of Reference...8 Appendix A: SPHB experiment data proceing procedure...88 Appendix B: Figure and Table...90 iv

7 Summary Solid foam have certain propertie that cannot be elicited from many homogeneou olid; thee include a low tiffne, low thermal conductivity, high compreibility at a contant load and adjutability of trength, tiffne and denity. Thee propertie have made olid foam ueful for variou application, uch a cuhioning, thermal inulation, impact aborption and in lightweight tructure. The employment of olid foam for load-bearing application ha motivated tudie into their mechanical propertie and thi ha involved experiment a well a theoretical modelling. However, many apect of foam behaviour till remain to be fully undertood. Thi invetigation i directed at identifying the mechanical propertie of aniotropic rigid polyurethane foam and it repone to tenile loading, a well a developing a implified cell model that can decribe it behaviour. The invetigation encompae experimental tet, viual obervation of foam cell and their deformation and development of an idealized cell model. Three rigid polyurethane foam of different denity are fabricated and ubjected to tenion in variou direction. Quai-tatic tenile tet are performed on an Intron univeral teting machine, while dynamic tenion i applied uing a plit Hopkinon bar arrangement. The reult how that the tiffne and tenile trength increae with denity, but decreae with angle between the line of load application and the foam rie direction. Dynamic tenile tet data indicate that for the rate of deformation impoed, the foam i not rate enitive in term of the tiffne and trength. Obervation are made uing micro-ct canning and optical microcopy to examine the internal tructure of the rigid polyurethane and it behaviour under compreive and tenile load. Micro-CT image of cell in the foam indicate that the v

8 cell exhibit a good degree of reemblance with an elongated tetrakaidecahedron. Image of the cell trut how that their cro-ection are imilar to that of a Plateau border [], while microcopic examination of rigid polyurethane foam ample under tenile and compreive loading how that cell trut are both bent and axially deformed, with bending being the main deformation mechanim. The image alo reveal that trut egment immediately adjoining the cell vertice do not flex during deformation becaue they have a larger cro-ection there and are contrained by the greater thickne of the cell wall membrane in that vicinity. With regard to fracture, the image how that fracture in foam occur by crack propagation through trut and membrane perpendicular to the direction of loading. Idealized foam cell model baed on elongated rhombic dodecahedron and elongated tetrakaidecahedron cell are propoed and analyed to determine their load and deformation propertie elatic tiffne, Poion ratio, and tenile trength. A parametric tudy carried out by varying the value of tructural parameter indicate that: The elatic tiffne and trength of foam are not influenced by cell ize; they are governed by denity, geometric aniotropy of the cell, hape of the cell and their trut, a well a the length of the rigid trut egment. Foam trength and tiffne increae with denity but decreae with angle between the loading and foam rie direction. The aniotropic tiffne and trength ratio increae with greater aniotropy in cell geometry. The Poion ratio are primarily determined by the geometric aniotropy of the cell. vi

9 A comparion between the cell model with cell in actual foam indicate that the tetrakaidecahedron ha a greater geometric reemblance with cell in actual foam compared to the rhombic dodecahedron. Moreover, good correlation between the tetrakaidecahedron cell model and actual foam in term of elatic tiffne wa oberved. Finite element imulation are undertaken to examine the behaviour of foam baed on the tetrakaidecahedron cell model for cae that were not amenable to analytical olution i.e. tenile loading in variou direction and nonlinearity in cell trut material propertie. The imulation how that although thin membrane in foam do not have much effect on the tiffne, they affect the fracture propertie by influencing the direction of crack propagation. A comparion between foam propertie predicted by the model and thoe of actual foam how that they correlate reaonably well in term of tiffne and the aniotropy ratio for tenile trength. FEM imulation are alo performed to examine the influence of variation in cell geometry on the mechanical propertie. The reult how that the variation incorporated do not have much effect on the overall tiffne, but decreae the predicted tenile trength. In eence, thi tudy provide greater inight into the mechanical propertie of rigid polyurethane foam and the mechanim governing it deformation and failure. The propoed idealized cell model alo contitute ueful approache to account for pecific propertie of foam. vii

10 Lit of figure Fig.. (a) Cloe cell foam and (b) open cell foam...6 Fig.. Stre-train relationhip for foam under compreion...7 Fig.. Stre-train curve of foam under tenion []...9 Fig..4 Cubic cell model propoed by Gibon et al. [], Triantafillou et al. [8], Gibon and Ahby [, 0], Maiti et al. [], Huber and Gibon [6]...7 Fig..5 Tetrakaidecahedral foam cell model...0 Fig..6 Voronoi teellation cell model [4]... Fig..7 Cloed cell Gauian random field model [4]... Fig..8 Comparion of yield urface baed on everal model for foam [49]... Fig.. Dog-bone haped pecimen...6 Fig.. Foam pecimen attached to acrylic block...7 Fig.. Typical tre-train curve...7 Fig..4 Stiffne...8 Fig..5 Tenile trength...8 Fig..6 Strength and tiffne aniotropy ratio...0 Fig..7 Split Hopkinon bar arrangement... Fig..8 Typical tre-train curve... Fig..9 Stiffne... Fig..0 Tenile trength... Fig.. -D image of cell tructure...4 Fig.. Elongated tetrakaidecahedron cell model...5 Fig.. Cro-ection of cell trut in rigid polyurethane foam (foam B; 9.5kg m )...6 viii

11 Fig..4 Plateau border...8 Fig..5 Size meaurement...8 Fig..6 Foam pecimen loaded uing crew driven jig...9 Fig..7 Micrograph of fracture propagation for tenion along the foam rie direction...40 Fig..8 Micrograph of cell deformation for tenion along the foam rie direction.4 Fig..9 Micrograph of fracture propagation for tenion along the tranvere direction...4 Fig..0 Micrograph of cell deformation for tenion along the tranvere direction4 Fig.. Micrograph of fracture for tenion along the 45 o to the foam rie direction...4 Fig.. Micrograph of cell deformation for tenion along the 45 o to the foam rie direction...4 Fig.. Micrograph of cell deformation for compreion along the foam rie direction...44 Fig..4 Micrograph of cell deformation for compreion along the tranvere direction...45 Fig..5 Micrograph of cell deformation for compreion in the 45 o to the foam rie direction...45 Fig..6 Thick membrane at trut interconnection...46 Fig..7 Meaurement of rigid trut egment...46 Fig..8 Compreion pecimen...47 Fig..9 Tenion pecimen...48 Fig..0 Three point bending tet...48 Fig.. Compreion tre-train curve for Specimen...49 ix

12 Fig.. Compreion tre-train curve for Specimen...49 Fig.. Load-diplacement curve for three-point bending tet of Specimen...5 Fig..4 Load-diplacement curve for three-point bending tet of Specimen...5 Fig..5 Load-diplacement curve for three-point bending tet of Specimen...5 Fig..6 Three-point bending tet and it finite element model...5 Fig..7 Stre-train curve from tenion tet...5 Fig..8 Determination of yield trength...5 Fig. 4. Elongated rhombic dodecahedron cell...58 Fig. 4. Elongated FCC tructure made from rhombic dodecahedron cell...58 Fig. 4. Repeating unit for the analyi of an elongated rhombic dodecahedron cell loaded in the z-direction...59 Fig. 4.4 Three-dimenional view of repeating unit in the analyi of an elongated rhombic dodecahedron cell loaded in the z-direction...60 Fig. 4.5 Two-dimenional view of repeating unit in the analyi of an elongated rhombic dodecahedron cell loaded in the z-direction...60 Fig. 4.6 Strut OC...6 Fig. 4.7 Deformation of trut OC in plane OBCD...6 Fig. 4.8 Bending moment ditribution along trut OC...65 Fig. 4.9 Repeating unit for the analyi of an elongated rhombic dodecahedron cell loaded in the y-direction...67 Fig. 4.0 Three-dimenional view of repeating unit for analyi of an elongated rhombic dodecahedron cell loaded in y-direction...67 Fig. 4. Two-dimenional view of repeating unit for analyi of an elongated rhombic dodecahedron cell loaded in the y-direction...68 Fig. 4. Strut OC...69 x

13 Fig. 4. Deformation of trut OC in plane OGCH...69 Fig. 4.4 Elongated tetrakaidecahedral cell...78 Fig. 4.5 Elongated BCC tructure made from tetrakaidecahedron cell...79 Fig. 4.6 Repeating unit for the analyi of an elongated tetrakaidecahedron cell loaded in the z-direction...80 Fig. 4.7 Three-dimenional view of repeating unit in the analyi of an elongated tetrakaidecahedron cell loaded in the z-direction...8 Fig. 4.8 Two-dimenional view of repeating unit for the analyi of an elongated tetrakaidecahedron cell loaded in the z-direction...8 Fig. 4.9 Deformation of trut OB...8 Fig. 4.0 Repeating unit for the analyi of an elongated tetrakaidecahedron cell loaded in the y-direction...86 Fig. 4. Three-dimenional view of repeating unit for the analyi of an elongated tetrakaidecahedron cell loaded in the y-direction...87 Fig. 4. Two-dimenional view of repeating unit ued for the analyi of elongated tetrakaidecahedron cell loaded in the y-direction...87 Fig. 4. Deformation of trut OS...88 Fig. 4.4 Deformation of trut OH...90 Fig. 4.5 Plateau border...98 Fig. 4.6 Elongated rhombic dodecahedron and tetrakaidecahedron cell...99 Fig. 4.7 Actual foam cell...00 Fig. 4.8 Variation of foam tiffne with relative denity baed on an iotropic rhombic dodecahedron cell model...0 Fig. 4.9 Variation of foam tiffne with relative denity baed on an iotropic tetrakaidecahedron model...04 xi

14 Fig. 4.0 Variation of foam tiffne with relative denity baed on an aniotropic rhombic dodecahedron cell model...06 Fig. 4. Variation of foam tiffne with relative denity baed on an aniotropic tetrakaidecahedron cell model...06 Fig. 4. Variation of foam tiffne with cell aniotropy baed on a rhombic dodecahedron cell model...07 Fig. 4. Variation of foam tiffne with cell aniotropy baed on a tetrakaidecahedron cell model...07 Fig. 4.4 Variation of aniotropy in foam tiffne with cell aniotropy baed on a rhombic dodecahedron cell model...08 Fig. 4.5 Variation of aniotropy in foam tiffne with cell aniotropy baed on a tetrakaidecahedron cell model...09 Fig. 4.6 Variation of aniotropy in foam tiffne with relative denity baed on a rhombic dodecahedron cell model...09 Fig. 4.7 Variation of aniotropy in foam tiffne with relative denity baed on a tetrakaidecahedron cell model...0 Fig. 4.8 Variation of foam tenile trength with relative denity baed on a rhombic dodecahedron cell model... Fig. 4.9 Variation of foam tenile trength with relative denity baed on a tetrakaidecahedron cell model... Fig Variation of foam tenile trength with relative denity baed on a rhombic dodecahedron cell model...5 Fig. 4.4 Variation of foam tenile trength with relative denity baed on a rhombic dodecahedron cell model...6 xii

15 Fig. 4.4 Variation of foam tenile trength with cell aniotropy baed on a rhombic dodecahedron cell model...7 Fig. 4.4 Variation of foam tenile trength with cell aniotropy baed on a tetrakaidecahedron cell model...7 Fig Variation of foam aniotropy in tenile trength with cell aniotropy baed on a rhombic dodecahedron cell model...8 Fig Variation of foam aniotropy in tenile trength with cell aniotropy baed on a tetrakaidecahedron cell model...9 Fig Variation of foam tenile trength aniotropy with relative denity baed on a rhombic dodecahedron cell model...9 Fig Variation of foam tenile trength aniotropy with relative denity baed on a tetrakaidecahedron cell model...0 Fig Open celled cubic model (GAZT) loaded in the tranvere direction... Fig Variation of Poion' ratio with cell geometric aniotropy ratio for a rhombic dodecahedron cell model...5 Fig Variation of Poion' ratio with cell geometric aniotropy ratio for a tetrakaidecahedron cell model...5 Fig. 4.5 Influence of cell aniotropy on υ zy ( υ zx )...7 Fig. 4.5 Influence of cell aniotropy on υ yx and υ yz for tetrakaidecahedron cell..8 Fig. 4.5 Influence of cell aniotropy on υ yx and υ yz for rhombic dodecahedron cell...9 Fig Influence of axial elongation and flexure of trut on Poion' ratio... Fig Variation of Poion' ratio with relative denity for a rhombic dodecahedron cell model ( tan )... xiii

16 Fig Variation of Poion' ratio with relative denity for a tetrakaidecahedron cell model ( tan )... Fig Stiffne of actual foam and that baed on a rhombic dodecahedron cell model... Fig Stiffne of actual foam and that baed on a tetrakaidecahedron cell model...4 Fig Normalized tiffne of actual foam and that baed on a rhombic dodecahedron cell model...4 Fig Normalized tiffne of actual foam and that baed on a tetrakaidecahedron cell model...5 Fig. 5. Elongated tetrakaidecahedron cell packed together in an elongated BCC lattice...40 Fig. 5. Element a tetrakaidecahedral cell model...4 Fig. 5. Star hape for beam cro ection...4 Fig. 5.4 Localied area of weakne in a finite element model...4 Fig. 5.5 Loading condition in the finite element model...4 Fig. 5.6 Stre-train curve for foam B ( 9.5kg m ; geometric aniotropy ratio )...44 Fig. 5.7 Crack pattern for tenion in the cell elongation/rie direction...44 Fig. 5.8 Crack pattern for tenion in the tranvere direction...45 Fig. 5.9 Cell model loaded in the tranvere (y) direction...49 Fig. 5.0 Cell model loaded in the rie (z) direction...50 Fig. 5. Single cell loaded in the cell elongation (foam rie) direction...50 Fig. 5. Single cell loaded in the tranvere direction...5 Fig. 5. Strut in a tetrakaidecahedron cell...5 xiv

17 Fig. 5.4 Crack propagation for loading in the 0 o, 45 o, 60 o, and 8.5 o direction...5 Fig. 5.5 Single cell loaded 0 o to the cell elongation (foam rie) direction...5 Fig. 5.6 Single cell loaded 45 o to the cell elongation (foam rie) direction...5 Fig. 5.7 Single cell loaded 60 o to the cell elongation (foam rie) direction...5 Fig. 5.8 Single cell loaded 8.5 o to the cell elongation (foam rie) direction...5 Fig. 5.9 FEM imulation reult for foam A (.kg m ; geometric aniotropy ratio.5)...56 Fig. 5.0 FEM imulation reult for foam B ( 9.5kg m ; geometric aniotropy ratio )...57 Fig. 5. FEM imulation reult for foam C ( 5.kg m ; geometric aniotropy ratio.7)...58 Fig. 5. Stre-train curve for foam A (.kg m ; geometric aniotropy ratio.5)...59 Fig. 5. Stre-train curve for foam B ( 9.5kg m ; geometric aniotropy ratio )...59 Fig. 5.4 Stre-train curve for foam C ( 5.kg m ; geometric aniotropy ratio.7)...60 Fig. 5.5 Stiffne of foam A (.kg m ; geometric aniotropy ratio.5).60 Fig. 5.6 Stiffne of foam B ( 9.5kg m ; geometric aniotropy ratio )...6 Fig. 5.7 Stiffne of foam C ( 5.kg m ; geometric aniotropy ratio.7).6 Fig. 5.8 Comparrion between tiffne predicted by FEM and analytical model..6 Fig. 5.9 Tenile trength for foam A (.kg m ; geometric aniotropy ratio.5)...6 xv

18 Fig. 5.0 Tenile trength for foam B ( 9.5kg m ; geometric aniotropy ratio )...64 Fig. 5. Tenile trength for foam C ( 5.kg m ; geometric aniotropy ratio.7)...64 Fig. 5. Normalized tenile trength for foam A (.kg m ; geometric aniotropy ratio.5)...65 Fig. 5. Normalized tenile trength for foam B ( 9.5kg m ; geometric aniotropy ratio )...65 Fig. 5.4 Normalized tenile trength for foam C ( 5.kg m ; geometric aniotropy ratio.7)...66 Fig. 5.5 Model with random variation in cell geometric aniotropy ratio...68 Fig. 5.6 Model with random variation in cell vertex location...69 Fig. 5.7 Random cell model for loading in the rie and tranvere direction...70 Fig. 5.8 Stre-train curve for uniform and random cell model for loading in the rie direction...7 Fig. 5.9 Stre-train curve for uniform and random cell model for loading in the tranvere direction...7 Fig Elatic tiffne of uniform and random cell model...7 Fig. 5.4 Tenile trength of uniform and random cell model...7 Fig. A. Split Hopkinon bar arrangement...88 Fig. A. SPHB pecimen with two reference point along the centre-line...89 Fig. A. Example of train-time data and application of linear regreion...89 Fig. B. Stre-train curve for loading in the rie direction (foam A.kg m ; geometric aniotropy ratio.5)...90 xvi

19 Fig. B. Stre-train curve for loading 0 o to the rie direction (foam A.kg m ; geometric aniotropy ratio.5)...90 Fig. B. Stre-train curve for loading 45 o to the rie direction (foam A.kg m ; geometric aniotropy ratio.5)...9 Fig. B.4 Stre-train curve for loading 60 o to the rie direction (foam A.kg m ; geometric aniotropy ratio.5)...9 Fig. B.5 Stre-train curve for loading in the tranvere direction (foam A.kg m ; geometric aniotropy ratio.5)...9 Fig. B.6 Stre-train curve for loading in the rie direction (foam B 9.5kg m ; geometric aniotropy ratio )...9 Fig. B.7 Stre-train curve for loading 0 o to the rie direction (foam B 9.5kg m ; geometric aniotropy ratio )...9 Fig. B.8 Stre-train curve for loading 45 o to the rie direction (foam B 9.5kg m ; geometric aniotropy ratio )...9 Fig. B.9 Stre-train curve for loading 60 o to the rie direction (foam B 9.5kg m ; geometric aniotropy ratio )...94 Fig. B.0 Stre-train curve for loading in tranvere direction (foam B 9.5kg m ; geometric aniotropy ratio )...94 Fig. B. Stre-train curve for loading in the rie direction (foam C 5.kg m ; geometric aniotropy ratio.7)...95 Fig. B. Stre-train curve for loading 0 o to the rie direction (foam C 5.kg m ; geometric aniotropy ratio.7)...95 xvii

20 Fig. B. Stre-train curve for loading 45 o to the rie direction (foam C 5.kg m ; geometric aniotropy ratio.7)...96 Fig. B.4 Stre-train curve for loading 60 o to the rie direction (foam C 5.kg m ; geometric aniotropy ratio.7)...96 Fig. B.5 Stre-train curve for loading in the tranvere direction (foam C 5.kg m ; geometric aniotropy ratio.7)...97 Fig. B.6 Stre-train curve for loading in the rie direction (foam B 9.5kg m ; geometric aniotropy ratio )...97 Fig. B.7 Stre-train curve for loading in the 45 o direction (foam B 9.5kg m ; geometric aniotropy ratio )...98 Fig. B.8 Stre-train curve for loading in tranvere direction (foam B 9.5kg m ; geometric aniotropy ratio )...98 Fig. B.9 Cro-ection of trut in rigid polyurethane foam A (.kg m ; geometric aniotropy ratio.5)...99 Fig. B.0 Cro-ection of trut in rigid polyurethane foam B ( 9.5kg m ; geometric aniotropy ratio )...00 Fig. B. Cro-ection of trut in rigid polyurethane foam C ( 5.kg m ; geometric aniotropy ratio.7)...0 xviii

21 Lit of table Table. Solid foam data...6 Table. Average dimenion of rigid polyurethane foam trut...7 Table. Stiffne from compreion tet...50 Table.4 Stiffne and yield trength from three point bending tet...5 Table.5 Mechanical propertie from tenile tet...54 Table 5. Value of parameter in finite element cell model...40 Table B. Strut dimenion...0 Table B. Dimenion of rigid egment in trut in foam B ( 9.5kg m ; geometric aniotropy ratio )...0 xix

22 Lit of ymbol A A y area of trut cro-ection area correponding to load in the y-direction A z A b area correponding to load in the z-direction area of bar cro-ection A area of pecimen cro-ection C C contant relating econd moment of area to the area of the trut cro-ection contant relating ditance from centroid to the extremitie and the area of the trut cro-ection C contant relating the length of the rigid trut egment to the ditance from centroid to the extremitie of the trut cro-ection C f contant relating the mechanical propertie to the denity of foam d E b length of rigid trut egment tiffne of bar E f overall tiffne of foam E tiffne of olid cell trut material E overall foam tiffne in the y-direction yy E zz F y overall foam tiffne in the z-direction load in the y-direction F z I L load in the z-direction econd moment of area of the trut cro-ection length of trut in the tetrakaidecahedron cell xx

23 Lˆ P f length of trut in the rhombic dodecahedron cell mechanical property of foam P mechanical property of olid material R R ditance from centroid to the extremitie of the trut cro-ection average ditance from the centroid to the extremitie of the trut croection r α β δ x radiu of a circle incribed within a Plateau border angle between a trut and the xy-plane angle between a trut and the xz-plane deformation in the x-direction δ y deformation in the y-direction δ z deformation in the z-direction platic train at fracture of olid material pl max normal train in the x-direction xx normal train in the y-direction yy normal train in the z-direction zz angle defining cell geometric aniotropy ratio overall denity of foam denity of olid material σ tenile trength of foam max f σ tenile trength of olid material in cell max σ normal tre in the y-direction yy xxi

24 σ tenile trength in the y-direction y max σ normal tre in the z-direction zz σ tenile trength in the z-direction z max υ Poion ratio of train in the x-direction ariing from normal tre in the yx y-direction υ Poion ratio of train in the z-direction ariing from normal tre in the yz y-direction υ Poion ratio of train in the x-direction ariing from normal tre in the zx z-direction υ Poion ratio of train in the y-direction ariing from normal tre in the zy z-direction u n diplacement in the normal direction u t diplacement in the tranvere direction x length of the flexible egment of an inclined trut projected onto the x axe flex X X fraction of trut that i flexible fraction of trut that i rigid y length of the flexible egment of an inclined trut projected onto the y axe flex z length of the flexible egment of an inclined trut projected onto the z axe flex xxii

25 Chapter Introduction A cellular material i defined a one which i made of an interconnected network of olid trut or plate which form edge and face of cell []. Cellular material can be natural occurring a well a man-made. They have been ued in many engineering application, e.g., andwich tructure, kinetic energy aborber, heat inulator, etc. Man-made cellular material generally come in two form olid foam with ome variation in cell geometry and tructure with regular cell uch a honeycomb. Solid foam are cellular material with a three-dimenional tructural arrangement, while honeycomb eentially poe a two-dimenional pattern. Solid foam made from metal or polymer have been ued in tructural application and kinetic energy aborption device, whereby they are ubject to tatic and dynamic load. Hence, the mechanical behaviour of foam under different rate of loading, a well a their failure propertie, mut be conidered in engineering deign that incorporate their uage. Although numerou invetigation on foam have been performed, their mechanical behaviour, epecially with regard to failure, i till not fully undertood. Thi motivate continued reearch with regard to thee apect.. Propertie of olid foam and it application Solid foam poe certain unique propertie that are different from thoe of homogeneou olid material. Some of thee propertie and how they facilitate application are: Relatively low tiffne Low tiffne foam made from elatic polymer are ueful in cuhioning application uch a bedding and eat.

26 Low thermal conductivity Non metallic foam are ueful for thermal inulation, which are employed in application ranging from lagging of indutrial pipe to encapulating frozen food. High compreibility at a contant load Foam can be compreed to a relatively high train under an approximately contant load. Thi make them very ueful for impact aborption becaue they can diipate ignificant kinetic energy while limiting the magnitude of the force tranferred to more fragile component that they hield. Hence, they are ued in the packaging of electronic product and in car bumper. Adjutable trength, tiffne and weight The trength, tiffne and weight of foam depend on their denity, which can be varied. Hence, the mechanical propertie of foam can be controlled, making them attractive in tructural application requiring particular trength or tiffne to weight ratio e.g., compoite tructure ued in aircraft. The ue of foam in kinetic energy aborption and tructural application, whereby they are ubjected to tatic and dynamic loading, motivate the need to tudy their mechanical propertie. Variou approache have been employed and thee are briefly dicued in the following ection.. Studie on mechanical behaviour The mechanical behaviour of foam ha been tudied and analyed uing everal approache. Thee include experimentation, analyi baed on implified cell model and development of contitutive relationhip. Experiment involving threedimenional mechanical tet have been performed by everal reearcher [-0]. Thee yielded reult in the form of empirical failure criteria of the foam. The failure criteria do not eem to agree with each other, and different reearcher have defined

27 their failure criteria baed on different type of tre. Experimental invetigation have contributed ignificantly to undertanding the mechanical behaviour of foam; however, they have not yielded much information on the micromechanic involved in the deformation and failure of foam material. A number of reearcher [, 8, -46] have propoed implified cell model for foam. They ued thee to predict mechanical propertie uch a tiffne, Poion ratio, failure criteria, etc. Thi approach i ueful in decribing ome of the micromechanic involved in the deformation and failure of foam. However, mot model involve ome empirical contant that need to be determined from experiment and hence they do not give direct relationhip between the propertie of the cell e.g. the overall foam denity, the mechanical propertie of the material the trut and membrane are made from, cell geometry and trut cro-ection with the overall mechanical propertie of actual foam. Some of the model propoed are alo not realitic becaue of everal reaon e.g. the model cannot fill pace in three dimenion (i.e. they cannot be arranged to form large cell aemblie) and the cell do not reemble thoe in actual foam. Moreover, mot of thee cell model are iotropic and hence cannot be ued to decribe aniotropic foam behaviour, which often reult from the manufacturing proce. Contitutive model for foam have been developed and analyed by everal reearcher [, 47-5]. Some of them have alo been implemented in finite element code uch a ABAQUS and LS-DYNA. Thee model eem to differ from one another becaue they are developed baed on different type of tre and train e.g. Miller [50] ued the von Mie tre and mean tre with a hardening model defined by a function of the volumetric and platic train; Dehpande and Fleck [] on the other hand, ued their own definition of equivalent tre and train baed on a von

28 Mie criterion combined with a volumetric energy criterion. Hanen et al. [49], who have examined everal contitutive model and compared them with experiment on aluminium foam, uggeted that thee model are inaccurate becaue they do not conider local and global fracture for hear and tenion.. Objective The extenive ue of foam in many engineering application, epecially for kinetic energy aborption and in advanced tructure, ha motivated the tudy of their mechanical behaviour. Although many uch invetigation have been undertaken, variou apect of the mechanical behaviour of foam have yet to be fully undertood, epecially with regard to it repone and failure under tenion. Reearcher have propoed contitutive model baed on experimental reult, cell model analyi, or combination of both. However, thee model do not eem to agree with one another particularly with regard to the type of tre and train ued to define the contitutive relationhip. Moreover, a tudy by Hanenn et al. [49] how that ome of the model which have been implemented in finite element package cannot repreent actual foam, mainly becaue the model do not include fracture criteria for tenion and hear. Simplified cell model have alo been propoed, analyed and compared with actual foam; however, they do not eem to be able to fully repreent and explain the mechanical behaviour oberved. Moreover, the implified cell model have limitation, uch a the dependence on empirical contant, lack of geometrical realim, and current applicability only to iotropic foam. Experimental tudie and development of idealized cell model for olid foam have their particular ditinctive advantage. Experimental tudie provide good inight into the mechanical behaviour of foam but they are not able to explain the micromechanic behind it behaviour. On the other hand, idealized cell model are 4

29 able to explain the micromechanic. Thu, a tudy that combine both i expected to give a fuller inight into foam behaviour. Conequently, thi tudy aim to provide an undertanding of everal apect that appear to be lacking in information; thee include: the mechanical propertie of rigid polyurethane foam under tatic and dynamic tenion microcopic feature of the rigid polyurethane foam, uch a the ize and geometry of contituent cell and cell trut, a well a tiffne and tenile trength of the trut micromechanic of the deformation and fracture of cell within foam ubjected to tenion, a revealed by microcopic obervation development of a implified cell geometry that can model the behaviour of rigid polyurethane foam under tenion and which directly relate the overall mechanical propertie of rigid polyurethane foam with the mechanical behaviour of the contituent cell Thi tudy combine experimental teting, viual obervation of cell deformation, and development of an idealized cell model. The information generated will help facilitate future development of contitutive model for foam. The focu include an undertanding of how rigid polyurethane repond to tenion and the development of an idealized cell model. It i enviaged that the reult of thi tudy and the cell model propoed can be applied to invetigation of other type of foam and loading. The following chapter provide an overview of the baic mechanical propertie of foam, apect that influence their behaviour and other tudie that have been conducted on the propertie of foam. 5

30 Chapter Literature review. Microtructure of polymer foam Solid foam comprie cell with olid material defining their edge, and membrane wall in ome cae (ee Fig..). Foam that have membrane cell wall are conidered cloed cell foam (ee Fig..(a)) while foam that do not have uch membrane are called open cell foam (ee Fig..(b)). Due to it tructure, cloed cell foam can have liquid or ga trapped inide it cell while open cell foam doe not. a b Fig.. (a) Cloe cell foam and (b) open cell foam Fig.. how that the dimenion and hape of cell in foam vary even within a mall area. Thi i becaue it i not poible to control the foaming proce to produce uniform cell. Thermoetting polymeric foam are made by introducing a gaing agent to a mixture of polymer rein and hardener. Thi reult in foaming proce whereby the mixture expand and rie a the reult of cell/bubble formation. Thi proce produce foam cell that are elongated in the foam rie direction. Other proceing parameter uch a gravity [49] can alo caue cell that have larger or maller dimenion in certain direction. Cell with larger or maller dimenion in one direction give rie to aniotropy in foam propertie, uch a a higher tiffne and trength in the direction of elongation. 6

31 . Baic mechanical propertie of olid foam.. Compreion The mechanical behaviour of olid foam under compreive loading i probably the primary property that ditinguihe it from non-cellular olid. Typical tretrain curve for olid foam made from three different kind of olid material elatomeric foam, elatic-platic foam and elatic-brittle foam are hown in Fig... They all have imilar characteritic i.e. linear elaticity at low tree, followed by an extended plateau terminating in a regime of denification, whereby the tre rie teeply. Thee characteritic are different from thoe of common olid material uch a metal, which normally do not have an extended tre-train plateau under compreion. a. elatomeric foam denification b. elatic-platic foam σ σ plateau linear elatic c. elatic-brittle foam σ Fig.. Stre-train relationhip for foam under compreion In all the three type of olid foam, initial linear elaticity arie primarily from the bending of cell trut, and in cloed cell foam, tretching of the membrane in the cell wall and change in fluid preure inide the cell []. On the other hand, 7

32 the mechanim correponding to the tre-train plateau are different for the three type of foam elatic buckling for elatomeric foam, formation of platic hinge in elatic-platic foam and brittle cruhing in elatic-brittle foam []. The long plateau in the compreive tre-train curve endow foam with a very high compreibility and enable them to exert a relatively contant tre up to a very high train. Thee two characteritic make foam an ideal material for cuhioning purpoe becaue the low and contant tre contribute to comfort and for crah protection (e.g. in helmet), becaue the foam i able to aborb kinetic energy while limiting the tre tranmitted to relatively low level... Tenion Typical tre-train curve for three kind of olid foam, i.e. elatomeric foam, elatic-platic foam and elatic-brittle foam are hown in Fig... At low train, all the foam exhibit linear elaticity, imilar to their compreive behaviour []. On the other hand, at higher train, different deformation mechanim occur in the three type of foam, cauing difference in the hape of their tre-train curve. An increae in the modulu i experienced by elatomeric foam becaue of cell trut/wall re-alignment, whereby the deformation mechanim change from bending to tenion in cell trut. On the other hand, platic yielding occur in elatic-platic foam, creating a hort plateau in the tre-train curve followed by a rapid increae of tre due to cell wall re-alignment. For brittle foam, their tre train curve do not how any non-linearity, a brittle fracture occur immediately at the end of the linear elaticity. [] 8

33 σ a. elatomeric foam σ b. elatic-platic foam σ c. elatic-brittle foam Fig.. Stre-train curve of foam under tenion [] Note that thee behaviour typical of foam are not all-encompaing, ince each foam ha it own ditinctive characteritic. Banhart and Baumeiter [5] aerted that the linear portion i not really elatic, a ome of the deformation i irreverible. The tenile tre-train curve obtained by Motz and Pippan [5] for a cloed-cell aluminium foam (an elatic-platic foam) how no rapid increae of tre after platic collape, which i expected according to Fig... Intead, fracture occur after platic yielding reulting in a tre-train curve which i imilar to that for olid aluminium.. Factor influencing mechanical propertie of olid foam The mechanical propertie of olid foam are influenced by everal factor the mechanical propertie of the olid material defining the cell edge (trut) and wall (membrane), cell tructure and propertie of fluid inide the cell. 9

34 Mechanical propertie of the olid material The mechanical propertie of olid foam, uch a tiffne, trength and vicoelaticity, depend largely on the mechanical propertie of the olid material in the cell edge and wall e.g. the tiffer and tronger the cell trut and wall material, the tiffer and the tronger the olid foam. Cell tructure of the foam The mechanical propertie of olid foam depend not only on the mechanical propertie of the olid material in the cell edge and wall, but alo on cell tructure. Thi i becaue how the cell trut and wall deform determine the overall mechanical behaviour of foam. When olid foam are loaded by compreion/tenion, the trut at the cell edge deform, and they undergo bending and tenion/compreion. Their compliance in bending i much higher than that in tenion/compreion; and hence, bending i the primary deformation mechanim [] and conequently, tiffne of olid foam i trongly influenced by the bending of cell edge. A highlighted in Section., cell in many foam are uually geometrically aniotropic, with a larger dimenion in one direction. Conequently, thi caue aniotropy in the mechanical propertie of foam. Uually, the foam i tiffer and tronger in the direction of cell elongation. Poion ratio i another mechanical property of olid foam that depend on cell tructure, with value ranging from -0.5 to a large poitive value []. Fluid inide the foam cell A mentioned in Section., olid foam can be claified into two type cloed celled and open celled. Due to the difference in their microtructure, thee two type of foam behave differently when loaded. Cloed cell foam have fluid trapped inide their cell and hence their mechanical propertie are influenced by the propertie of the fluid contained e.g. a fluid with low compreibility can tiffen the foam. On the other hand, fluid can flow freely 0

35 through open cell foam, but, thi doe not mean that the fluid doe not affect it mechanical propertie. At high train rate, the vicoity of the fluid flowing through the cell when foam i loaded can increae it tiffne, thu, introducing train rate enitivity..4 Studie on mechanical propertie of olid foam The extenive ue of foam in engineering application, epecially in kinetic energy aborption and compoite tructure, ha motivated invetigation into their mechanical behaviour. Different approache have been ued experimentation, development of implified cell model and formulation of continuum contitutive relationhip for numerical modelling; ome of thee are now dicued.4. Experimental tudie Reearcher, uch a Zalawky [9], McIntyre and Anderton [7], Zhang et al. [0], Triantafillou et al. [8], Doyoyo and Wierzbicki [5], Dehpande and Fleck [, 4] and Gdouto et al. [6], have performed experiment on olid foam. Thee include uniaxial tenion and compreion, hear, and multiaxial loading, with mot of thee effort are aimed at obtaining empirical failure criteria. Zalawky [9] carried out tet on thin walled tube of rigid polyurethane foam to imulate multiaxial loading by impoing internal preure together with axial tenion or compreion. Thee tet yielded an empirical failure criterion in form of a rectangular envelope with repect to axe defined by the two principal tree, uggeting that the foam follow a maximum principal tre failure criterion. A with Zalawky [9], Zhang et al. [0, 54] alo teted everal polymeric foam. Their experiment included uniaxial compreion, hear, and hydrotatic compreion to etablih a failure criterion and a contitutive model. Unlike the failure criterion by Zalawky [9] which i only

36 defined by the maximum principal tre, the failure criterion by Zhang et al. [0, 54] i quantified by the effective tre and hydrotatic preure. Triantafillou et al. [8] tudied polymeric foam and determined a failure envelope baed on the von Mie effective tre and mean tre. Thi failure envelope wa then compared with analytical failure criteria derived from a cubic open cell model developed by Gibon et al. []. Triantafillou et al. [8] uggeted that their model i able to decribe yield in open cell polyurethane and cloed cell polyethylene foam quite well, howing that the principal tre criterion propoed by Zalawky [9] i inadequate. Unlike Zalawky [9], Zhang et al. [0, 54] and Triantafillou et al. [8] who tudied polymeric foam, Doyoyo and Wierzbicki [5] examined metallic foam and performed biaxial tet on iotropic Alpora and aniotropic Hydro cloed cell aluminium foam. From thee tet, they alo propoed a failure criterion baed on mean tre and von Mie effective tre. A with Zalawky [9], Zhang et al. [0, 54] and Doyoyo and Wierzbicki [5], Gdouto et al. [6] alo tudied the mechanical behaviour of foam in order to propoe failure criteria. They performed uniaxial tenion, compreion, hear and biaxial loading tet on Divinycell foam and found that failure could be decribed by the Tai-Wu failure criterion [55] which i expreed a a econd-order polynomial equation with principal tree a the parameter. Again, thi failure criterion differ from thoe decribed earlier, a it i defined uing different tree. Dehpande and Fleck [, 4] performed multiaxial mechanical tet on PVC and aluminium foam, and compared their reult with a phenomenological yield urface they propoed []. Their criterion alo ue the von Mie-mean tre plane, imilar to the failure criterion propoed by Gibon et al. [], Triantafillou et al. [8] and Doyoyo and Wierzbicki [5].

37 The tudie dicued o far have concentrated on obtaining empirical failure criterion. McIntyre and Anderton [7] and McCullough et al. [56] choe to ue another approach, employing fracture mechanic and carried out fracture toughne tet on rigid polyurethane foam. They concluded that fracture in the foam they tudied could be characterized by G Ic, K Ic and a crack opening diplacement criterion. Although the foam ued in their tet wa aniotropic, they neglected thi factor in their tudy. McCullough et al. [56], invetigated aluminium foam and performed fracture toughne tet to get J-integral curve. They found that crack propagation in foam i interrupted by crack bridging by cell edge. Thi differentiate crack propagation in foam from that in olid material. Their approach of uing fracture mechanic to analye the propertie of foam eem to be omewhat peculiar becaue fracture mechanic i a continuum-baed approach while foam i not really a continuum material. The experimental tudie dicued have focued mainly on finding appropriate failure criteria. Some of thee criteria have alo been developed into a contitutive model and thi i dicued later. It i noted that reearcher have propoed failure criteria that differ one another becaue different type of tre are ued. Although thee tudie have provided inight into the mechanical behaviour of foam, they have not yielded information on the micromechanic governing foam behaviour. An undertanding of the underlying micromechanic i important in explaining the repone of foam, thu leading to improved failure criteria and contitutive model. Hence, ome reearcher have alo tried to decribe the micromechanic governing foam deformation by analyzing idealized cell model. Some of thee tudie are now dicued.

38 .4. Cell model Cell model for olid foam have been propoed and analyed both analytically and numerically to find generic relationhip between the overall mechanical propertie of olid foam and it microtructural characteritic, uch a cell hape and ize, denity and the mechanical propertie of the olid defining cell trut and wall. Thi approach i alo ueful for relating the overall deformation of olid foam to the tenion, bending and torion experienced by cell trut and wall and decribing how failure i governed by buckling, platic deformation and fracture in cell. Early cell model for foam by Gent and Thoma [8, 9], Lederman [9], Cunningham [5], Chritenen [4] and Kanakkanatt [7] ugget that elatic deformation of foam i caued mainly by the tretching of cell trut/wall which lead to a linear dependence of tiffne on foam denity. However, later development by Ko [8], Gibon et al. [], Triantafillou et al. [8], Gibon and Ahby [, 0], Huber and Gibon [6], Maiti et al. [] and Zhu et al. [40, 4] have hown that bending of cell trut/wall play a major role in foam deformation. Thi reult in a quadratic dependence of foam tiffne on denity, which bear better correlation with actual foam. Gent and Thoma [8] and Lederman [9] conidered model in which trut with random orientation are interconnected via phere; imilarly, Cunningham [5] and Chritenen [4] have alo examined interconnected trut with random orientation. Gent and Thoma [8], Lederman [9], Cunningham [5], and Chritenen [4] analyed thee tructure by determining the average tiffne of the trut that are randomly oriented. They took the tretching of trut to be the main mechanim governing foam deformation, while other mode of trut deformation bending and torion were not conidered. Thi led to a (nearly) linear relationhip 4

39 between tiffne and denity which i found to be unatifactory for actual foam, epecially open cell foam and cloed cell foam with thin cell wall. Gent and Thoma [9] have propoed a imple cubic tructure which Kanakkanatt [7] further developed to include geometric aniotropy. A with Gent and Thoma [8], Lederman [9] Cunningham [5] and Chritenen [4], Gent and Thoma [9] and Kanakkannat [7] conidered the tretching of cell trut a the main deformation mechanim in foam. Thu, they alo found a linear relationhip between tiffne and foam denity. Kanakkannat [7] concluded that hi model differ from experimental reult on actual foam becaue it neglected bending of trut. Warren and Kraynik [] and Wang and Cuitino [8] analyed trut in a precribed interection pattern to repreent foam. Warren and Kraynik [] conidered a tetrahedral arrangement of four trut interecting at one point a the baic repeating unit. They conidered tretching and bending of trut a well a random orientation of the tetrahedral unit to model the variation of cell in actual foam. Their model could be arranged to fill pace in three dimenion but lacked geometrical imilarity with actual foam. Thi model wa extended by Sahraoui [6] to account for aniotropy. Wang and Cuitino [8] derived equation for any number of trut originating from one point in their model and conidered the tretching and bending of trut in their model. Thee model are emi iotropic with equal trut length and hence, cannot be ued for aniotropic foam. They alo lack geometric imilarity with cell in actual foam. The development of cell model alo include employing implified repeating cell unit and reearcher have utilized uch model to identify the main mechanim governing foam deformation and failure, and devie equation to decribe the mechanical propertie of foam. A implified cell model baed on open cubic cell 5

40 have been propoed by Gibon et al. [], Triantafillou et al. [8], Gibon and Ahby [, 0], Maiti et al. [] and Huber and Gibon [6] to decribe the mechanical behaviour of foam (ee Fig..4). From thee cell model, they uggeted that the main mechanim for foam deformation i the bending of cell trut and wall. Other deformation mechanim, uch a cell wall tretching and compreion of fluid inide cell, have alo been incorporated into their model. They alo developed a yield urface/failure model baed on three ditinctive mechanim that can initiate nonlinearity in the tre-train curve for foam, i.e. elatic buckling, platic yielding and brittle cruhing or fracture. They propoed contitutive model for aniotropic foam baed on elongated verion of thee cell model. Mot of the contitutive model propoed for open or cloed cell foam with thin cell wall were able to relate the propertie of actual foam to the propertie of the olid material and relative denity of the foam via power law relationhip of the form P f C where f n f P, P f, P, f and (.) are repectively property of the foam, property of the olid defining the cell, foam denity and denity of the trut material; C f i a contant defined empirically from experiment, while n i a contant derived analytically. Thi model ha been further extended by Andrew et al. [] to analye creep in foam. Although thi cell model manage to decribe the mechanical behaviour of foam, it doe not provide a direct explanation for the value of the contant C f for each mechanical property. The cell model i alo not realitic becaue it cannot be aembled in three dimenion, i.e. thi model cannot be arranged to fill pace. 6

41 Fig..4 Cubic cell model propoed by Gibon et al. [], Triantafillou et al. [8], Gibon and Ahby [, 0], Maiti et al. [], Huber and Gibon [6] Reearcher have alo made ue of polyhedra to model cell in foam. Ko [8] conidered a trapezo rhombic dodecahedron a polyhedron with ix equilateral trapezoid and ix congruent rhombic face arranged in a hexagonal cloet packing arrangement, a well a a rhombic dodecahedron a polyhedron with twelve rhombic face arranged in a face-centred cubic packing arrangement. Ko [8] and Dawon and Shortall [57] uggeted that actual rigid polyurethane foam cell are motly pentagonal dodecahedra, i.e. polyhedra with twelve pentagonal face each. However, pentagonal dodecahedra cannot be aembled in three-dimenion to fill pace. Tetrakaidecahedra a polyhedron with 4 face a hown in Fig..5, have attracted many reearcher uch a Dement ev and Tarakanov [6, 7], Zhu et al. [40], Warren and Kraynik [9], Choi and Lake [], Simone and Gibon [7], Grenetedt [4], Grenetedt and Tanaka [], and Ridha et al. [44-46]. They have utilized thi to model open and cloed cell foam. Thi cell model can be arranged to fill pace in a bodycentred cubic arrangement. Patel and Finnie [] aerted that in actual foam, five ided face are found in abundance, while four and ix ided face are alo found in a coniderable quantitie. Thi ugget that pentagonal dodecahedra and tetrakaidecahedra are imilar to cell in actual foam. 7

42 Menge and Knipchild [0] carried out a theoretical analyi of the tiffne and trength of foam baed on the pentagonal dodecahedron model. They conider bending and axial deformation of trut a the deformation mechanim. Compreive trength wa derived baed on the buckling load, while the tenile trength wa derived from the axial loading of trut. Their aumption of uing only axial loading on trut a the main failure criterion for tenile loading eem to contradict with their aumption of employing both axial loading and bending a the main deformation mechanim. Thi i becaue the bending of beam/trut uually initiate fracture earlier than axial loading. A with Menge and Knipchild [0], Chan and Nakamura [] have alo ued pentagonal dodecahedra for their foam model. They ued it to model the tiffne and yield trength of open and cloed cell foam under compreive loading. The bending of cell trut and wall wa conidered the primary deformation mechanim while buckling wa aociated with yield. Both Menge and Knipchild [0] and Chan and Nakamura [] only tudied iotropic foam uing their model. Zhu et al. [40], Warren and Kraynik [9] and Choi and Lake [] ued a tetrakaidecahedral geometry to model an open cell foam. Zhu et al. [40] aerted that a tetrakaidecahedron i the only polyhedron that can be aembled with identical unit to fill pace, and it nearly atifie the minimum urface energy criterion. Warren and Kraynik [9] alo concurred with thi. They tated that it i the only polyhedral bubble known that fill pace to form dry oap foam (one that contain very little liquid) with perfect order, and alo uggeted that other polyhedra can fill pace but do not atify the energy condition tated in Plateau law []. Both Zhu et al. [40] and Warren and Kraynik [9] obtained analytical expreion for the mechanical propertie, i.e. Young modulu, hear modulu, bulk modulu and Poion ratio, baed on mall deformation of thi cell. Their model how that for 8

43 low denity foam, bending i the main deformation mechanim in the foam. The tiffne of thi model can alo be approximated uing Eq. (.) with value of and n approaching and when the cro ection of the cell trut correpond to a Plateau border [], which i defined by three circular arc that touch each other. Thi i conitent with the emi empirical equation propoed by Gibon et al. [], Triantafillou et al. [8], Gibon and Ahby [, 0], Maiti et al. [], Huber and Gibon [6]. Zhu et al. [40] concluded that the mechanical propertie of thi model are approximately iotropic. The model ha alo been ued to analye large deformation [4] and creep behaviour [58] and appear more realitic than the cubic cell model [, 8, 0,, 6, ] becaue the cell can be arranged to fill pace. However, due to it geometric iotropy, thi model can only be applied to iotropic foam. Mill and Zhu [], Simone and Gibon [7], Grenetedt [4], and Grenetedt and Tanaka [] ued a tetrakaidecahedron cell model for cloed-cell foam. Mill and Zhu [] derived equation that decribe the mechanical propertie of foam for high compreive train. Simone and Gibon [7] performed numerical imulation uing thi model in order to ee the influence of material ditribution in metallic foam. Grenetedt [4] employed FEM imulation to obtain elatic propertie baed on thi model, while the finite element analyi of Grenetedt and Tanaka [] wa directed at examining the effect of cell irregularity on the mechanical propertie of cloed cell foam. A with the model by Zhu et al. [40], thoe by Mill and Zhu [], Simone and Gibon [7], Grenetedt [4] and Grenetedt and Tanaka [] are iotropic. C f 9

44 Fig..5 Tetrakaidecahedral foam cell model Other reearcher uch a Robert and Garboczi [4, 5], Huang and Gibon [5], Zhu et al. [4] and Zhu and Windle [4] preferred to ue a more random cell model becaue actual foam do not comprie identical repeating cell. Robert and Garboczi [4] generated a foam model uing Voronoi teellation of ditributed eed point in pace to form a cloed cell foam with planar face. Fig..6 how the Voronoi model ued by Robert and Garboczi [4] who alo employed other type of random cell model called Gauian random field to model cloed cell foam with curved cell wall; Fig..7 how uch a model. They performed FEM imulation of thee model to obtain their elatic propertie. Both model were for iotropic foam and cannot be applied to aniotropic cell. In contrat to Robert and Garboczi [4] who analyed random cell model for cloed cell foam, Robert and Garboczi [5], Huang and Gibon [5], Zhu et al. [4] and Zhu and Windle [4] have ued random cell to model open cell foam. Robert and Garboczi [5] ued Voronoi teellation, a node-bond model that connect random node to their nearet neighbour and Gauian random field to develop random cell for open cell foam model. They performed FEM imulation baed on thee model to obtain elatic propertie. Zhu et al. [4], Zhu and Windle [4] ued 0

45 the Voronoi teellation technique to build foam model. They applied FEM imulation to thee model to tudy the influence of cell irregularity in foam on their elatic propertie and repone to high train compreive loading. Similarly, Huang and Gibon [5] ued the Voronoi teellation technique to build cloed cell model and employed FEM imulation to tudy creep behaviour. Although random cell model appear to better repreent the microtructure of actual foam, thee model cannot be analyed eaily due to their complex geometrie. Thu, numerical imulation i required. Moreover, the random model propoed have only been applied to iotropic foam and cannot be ued to model aniotropic foam which commonly occur. Fig..6 Voronoi teellation cell model [4] Fig..7 Cloed cell Gauian random field model [4]

46 .4. Contitutive model A knowledge of contitutive material behaviour i eential in tructural deign involving foam. Implementation of appropriate contitutive material model into computer imulation code facilitate the deign and analyi of component. The development of contitutive model for foam ha been of interet to reearcher uch a Dehpande and Fleck [], Schreyer et al. [5], Ehler [47] and Miller [50]. Some of the contitutive model developed have been implemented in finite element package e.g. LS-DYNA and ABAQUS. Hanen et al. [49] have briefly decribed ome contitutive model developed for olid foam, i.e. material model 6, 6, 6 and 75 in LS-DYNA, the cruhable foam model in ABAQUS and the model by Dehpande and Fleck [], Miller [50], Schreyer et al. [5] and Ehler [47]. A comparion among the model i hown in Fig..8 [49]. Fig..8 Comparion of yield urface baed on everal model for foam [49] Material model 6 and 6 in LS-DYNA ue the orthogonal direction of the material (principal direction of aniotropy) a the bai for their hardening function. Yield criterion i defined for each of the ix tre component. The difference between the two i the hardening model: material model 6 ue the engineering

47 volumetric train a the variable, while material model 6 ue the engineering train. The main feature of both i that they can be ued to model aniotropic foam. [49] Material model 6 in LS-DYNA ue the principal tree a the bae of it train hardening function. Thi model eentially employ a maximum principal tre criterion, whereby the compreive principal tre i a function of volumetric tre, while the tenile principal tre follow an elatic-perfectly platic curve. Thi model cannot be ued for aniotropic foam [49]. Material model 75 in LS-DYNA and the cruhable foam model in ABAQUS involve yield function that can be decribed by an ellipe in the von Mie-mean tre plane. The major difference between the two model i that the ellipe in ABAQUS ha a fixed ratio between it major and minor axi, while the ratio i not fixed for material model 75 in LS-DYNA. Hence material model 75 in LS-DYNA require a larger number input parameter than the cruhable foam model in ABAQUS. The ABAQUS cruhable foam model only require the hardening curve in the form of a relationhip between hydrotatic tre and volumetric train, while material model 75 in LS-DYNA require not only a definition of the hardening curve via a relationhip between hydrotatic tre and volumetric train, but alo the uniaxial compreive tre-train hardening curve. [49] Dehpande and Fleck [] modified the von Mie criterion by including the effect of elatic volumetric train energy in the criterion (note that the von Mie criterion only conider elatic hear train energy). Dehpande and Fleck [] criterion reult in an ellipe when it i drawn in the von Mie-mean tre plane. Strain hardening i modelled via a function of equivalent platic train defined a the energy conjugate of Dehpande and Fleck [] equivalent tre. Thi criterion only

48 need data from a uniaxial compreion tet on foam and can only be ued to decribe iotropic foam [, 49]. Miller [50] combined the Drucker-Prager criterion and the yield urface propoed by Gibon et al. [], Triantafillou et al. [8], and Gibon and Ahby [, 0]. The reult wa a criterion in the form of a polynomial of the tre invariant firt order in the von Mie tre and econd order in the mean tre. Miller train hardening model i defined in term of a function of the volumetric train and platic train. [49, 50] Gioux et al. [48] preented a phyical bai for the criterion propoed by Miller [50] and Dehpande and Fleck []. They found that the Miller [50] criterion can be derived from the mechanitic failure urface criterion developed by Gibon et al. [], Triantafillou et al. [8], and Gibon and Ahby [, 0], by taking into account the influence of cell wall curvature. They alo found that Dehpande and Fleck [] criterion can be derived from the Miller [50] criterion by neglecting the effect of the linear term for the mean tre. Gioux et al. [48] uggeted that both criteria, i.e. Dehpande and Fleck [] and Miller [50], are able to decribe the multiaxial failure urface for aluminium foam. Hanen et al. [49] have compared reult from LS-DYNA model 6, 6, 75, and 6 with experimental reult for aluminium foam under pecific loading configuration. They concluded that none of the model can decribe the experimental reult with convincing accuracy. They believe that inaccuracy occur becaue the model do not conider local and global fracture. Hence, an accurate, efficient and robut fracture criterion i needed to enhance the accuracy of the material model. 4

49 Chapter Rigid Polyurethane Foam A highlighted in Chapter, thi tudy focue on the tenile mechanical propertie of rigid polyurethane foam and etablihment of an appropriate cell model. The mechanical and tructural propertie of rigid polyurethane foam are dicued in thi chapter, and the fabrication of rigid polyurethane foam i firt preented.. Fabrication of rigid polyurethane foam Rigid polyurethane foam block were fabricated by mixing polyurethane rein (polyol), a hardener (diiocyanate) and a gaing agent together. The chemical ued were DALTOFOAM MM 6775 for the rein, SUPRASEC 5005 for the hardener and HCFC 4b for the gaing agent. Three foam block were made by firt mixing the rein and the gaing agent together thoroughly, followed by introducing the hardener to the mixture, which wa then poured into a mould and left to cure. Mixing of the component caued a reaction, which reulted in a foaming proce whereby bubble are formed and rie cauing the bubble to elongate in the foam rie direction while the mixture harden. The denity of the three foam block wa varied by uing different amount of gaing agent. The foam made by thi technique compried primarily cloed cell with trut defining the cell edge and thin membrane for cell wall. The cell were elongated in the foam rie direction, generating geometric aniotropy quantified by a geometric aniotropy ratio (i.e. the ratio between the cell length in the rie direction to that in the tranvere direction) that varied with foam denity. The material in each block wa dener and le uniform near the outer urface. Thu, thee portion at the urface were cut away and dicarded. Nearer the middle of each block, the cell dimenion perpendicular to the rie direction were generally imilar, making them roughly 5

50 axiymmetric. The overall denity of the olid foam, average cell ize and cell geometric aniotropy ratio are preented in Table.. The characteritic of the foam are conitent with the concluion by Dawon and Shortall [57] that the cell aniotropy ratio of rigid polyurethane foam decreae with a higher denity. Table. Solid foam data Foam A Foam B Foam C Denity (kg/m ) Average cell ize ( mm mm) Average cell geometric aniotropy ratio.5.7. Quai-tatic Tenile tet Quai tatic tenile tet were performed on foam pecimen to tudy the influence of loading direction, foam denity and cell aniotropy ratio on the mechanical propertie. Dog-bone haped pecimen were cut from foam block according to five direction along the foam rie direction and 0 o, 45 o, 60 o and 90 o to the foam rie direction. Fig.. how the pecimen dimenion. The end of the pecimen were glued to acrylic block to facilitate gripping by an Intron (model 5500) univeral teting machine ued to apply tenion (ee Fig..). Fig.. Dog-bone haped pecimen 6

51 Fig.. Foam pecimen attached to acrylic block Reult from thee tet are preented in the form of engineering tre-train curve and a typical tre-train curve for thee foam i illutrated in Fig... The curve how that the foam ha an initial linear repone followed by a horter nonlinear repone before fracture. σ (MPa) Fig.. Typical tre-train curve The tre-train curve are converted into tiffne and tenile trength data a function of loading direction. Foam tiffne wa determined from the gradient of the initial linear portion, while foam trength wa defined by the maximum tre before failure. Fig..4 how the tiffne a a function of loading direction, while Fig..5 how tenile trength a function of loading direction for the three foam. 7

52 E (MPa) direction (deg) Fig..4 Stiffne Foam C Foam B Foam A σ max (MPa) Foam C Foam B Foam A direction (deg) Fig..5 Tenile trength All the foam exhibited aniotropy in their mechanical propertie the tiffne and tenile trength depend on loading direction. The tiffne i highet in the rie direction (0 o ) and decreae with the angle to the rie direction. Minimum tiffne correpond to loading perpendicular to the foam rie direction, i.e. the tranvere direction. A with tiffne, foam trength i alo highet in the foam rie direction and decreae with angle to the rie direction. Conequently, foam i alo weaket in the tranvere direction. Thee reult are expected becaue the foam cell are aniotropic and elongated in the rie direction. The cell trut that are aligned in the 8

53 foam rie direction are longer; hence, they are relatively eaier to bend when loaded in the tranvere direction, making the foam more compliant in that direction. Thi alo caue the foam to be weaker in the tranvere direction. Moreover, the foam i alo weaker in the tranvere direction becaue fewer trut are required to be broken when failure occur, a there i a maller number of trut per unit area perpendicular to the tranvere direction. Foam C i tiffer and tronger than foam B, while foam B i tiffer than tronger than foam A, i.e. E > fc > E fb E fa and σ fc max σ fb max > σ fa max >. Thi i expected becaue foam C i dener than foam B, while foam B i dener than foam A ( fc fb > fa > ). The amount of olid material to carry the load increae with denity, thu, increaing the tiffne and trength of the foam. Notable catter in the tiffne and trength data wa found from tet on the foam pecimen, a preented via the I-haped bar in Fig..4 and.5. Thi i expected becaue the cell within each type of foam were not uniform. Fig..6 how the average aniotropy ratio for trength and tiffne a a function of cell geometric aniotropy. It indicate that the cell geometric aniotropy ratio govern the degree of aniotropy in the mechanical propertie of the foam aniotropy in the trength and tiffne increae with cell geometric aniotropy. 9

54 property aniotropy ratio geometric aniotropy ratio Strength aniotropy ratio Stiffne aniotropy ratio. Dynamic tenile tet Fig..6 Strength and tiffne aniotropy ratio In addition to quai-tatic tenile tet, dynamic tenile tet were alo undertaken. Thee tet were performed on foam B ( 9.5kg m ) to examine if train rate influence it tenile mechanical propertie. The pecimen in thee tet were alo dog-bone haped, imilar to thoe in quai-tatic tenile tet (ee Fig..). Specimen were dynamically tretched uing a tenile plit Hopkinon bar arrangement (SPHB), a hown in Fig..7. The input/output bar ued in thi experiment were made of polycarbonate (with tiffne E. GPa ; diameter 6.5 mm) to reduce mimatch in impedance with the compliant pecimen. A pendulumdriven ytem wa employed, whereby a pendulum trike anvil, generating a tenile pule in the input bar, which propagate and load the pecimen. The load applied to the pecimen wa calculated from the train gauge ignal of the output bar. Deformation of the pecimen wa recorded uing a high peed camera (PHOTRON ultima APX) and the train induced wa derived from thi viual data. 0

55 Fig..7 Split Hopkinon bar arrangement The tet yielded dynamic engineering tre-train data and a typical tretrain curve for foam i hown in Fig..8. The initial part of the curve (up to a train of about 0.008) how a lightly nonlinear increae (note that the tre in the pecimen i not yet uniform during thi early phae and hence the data may not be accurate ee [59]). Neglecting thi initial phae, the curve how that the foam ha an eentially linear initial repone followed by ome nonlinearity before fracture. Thi i imilar to it behaviour under quai-tatic loading. σ (MPa) Fig..8 Typical tre-train curve The tre-train curve were then converted into tenile tiffne and trength data. Fig..9 how tiffne a a function of train rate while Fig..0 how

56 trength a a function of train rate. Data from quai-tatic tenile tet performed uing the Intron (model 5500) univeral teting machine are included for comparion with repone at low train rate. 5 0 E (MPa) o o o train rate (/) Fig..9 Stiffne σ max (MPa) train rate (/) Fig..0 Tenile trength 0 o o o A with the data from quai-tatic tenile tet, the foam alo diplayed aniotropy in mechanical propertie for dynamic loading the tiffne and trength depend on loading direction. Again, the foam are tiffet in their rie direction (0 o ) and the tiffne decreae with angle with the rie direction. Minimum tiffne correpond to loading in the tranvere direction. With regard to trength, thi i alo

57 highet in the foam rie direction and decreae with alignment to the tranvere direction, where it i lowet. The reaon for thee are the ame a thoe for quaitatic tet (Section..) Fig..9 and.0 appear to indicate no obviou dependence of tenile trength and tiffne on train rate. The trength at low train rate (e.g. le than 0./) doe not how any ignificant difference with that at higher train rate (higher than 0/). The tiffne correponding to thee two range do not how any notable difference a well. Although the gla tranition temperature of rigid polyurethane i 00-50K [], the foam exhibit inignificant rate enitivity probably becaue it i loaded in tenion; it i relatively brittle and hence fracture occur before any rate enitivity become apparent..4 Micro CT imaging of rigid polyurethane foam cell A decribed in Chapter, the cell tructure within a foam play an important role in determining it mechanical propertie. Micro CT canning of foam cell wa therefore performed to tudy the cell tructure. Micro CT can wa performed on rigid polyurethane foam ample uing a SkyScan -076 machine with a reolution of 8 μ m ; thi reulted in two-dimenional image of the ample cro-ection, which were then converted to three-dimenional image uing oftware (CTAN). Fig.. how image of cell obtained from CT canning and they reemble an elongated tetrakaidecahedron cell model hown in Fig... Both the actual cell and the model have: horizontal face at the upper and lower end lanted face adjacent to upper and lower face vertical face at the ide Thi probably arie becaue during the foaming proce, the foam bubble generated tend toward the minimum urface energy condition [40] and the energy criteria

58 tated in Plateau law [9]. Tetrakaidecahedra, a uggeted by Zhu et al. [40] and Warren and Kraynik [9], are the only polyhedra that can fill pace (arranged in three dimenion) and that approximately atify both energy criteria. Thi i conitent with Patel and Finnie [] uggetion that cell in actual foam are imilar to pentagonal dodecahedra and tetrakaidecahedra. Fig.. -D image of cell tructure 4

59 Fig.. Elongated tetrakaidecahedron cell model.5 Microcopic obervation of cell trut In addition to microcopic obervation of foam cell uing CT canning, microcopic obervation were made on foam trut that define the cell edge. Thi wa undertaken to examine the hape and ize of their cro-ection. For thi purpoe, pecimen cut from the foam block were examined uing an optical microcope et to a magnification of 000X or 500X and photographic image were taken. 5

60 Fig.. Cro-ection of cell trut in rigid polyurethane foam (foam B; 9.5kg m ) Fig.. how everal typical image of cro-ection of cell trut in rigid polyurethane foam. They indicate that the cro-ection i imilar in hape to that of the Plateau border [] geometry hown in Fig..4, which i defined by three circular arc that touch each other. Thi i conitent with Plateau law [] that propoe that oap bubble urface alway meet in three and do o at an angle of 0 o, forming an edge called a Plateau border. The ame phenomenon i likely to occur in the foaming proce in fabricating rigid polyurethane foam, whereby adjacent bubble in the foam adjut to a table configuration a the polyurethane harden (cure). Thu, a Plateau 6

61 border hape i aumed by the trut a the material et. Zhu et al. [40] alo oberved Plateau border hape in the cro-ection of trut in foam cell. To meaure the ize of thee cro-ection, an incribed circle of radiu r wa drawn for each image (ee Fig..4), and a ample of taking a meaurement i hown in Fig..5. The relationhip between the incribed radiu r and the outer dimenion R, hown in Fig..4, i r R The average reult from thee meaurement are preented in Table. (.) Table. Average dimenion of rigid polyurethane foam trut kg mm R ( μm) Foam A. 49 Foam B Foam C Fig..4 alo ugget that the thickne of membrane that contitute cell wall i much maller than the thickne of the trut. Hence, the preence of membrane ha minimal influence of the mechanical propertie of the foam, which i mainly governed by the repone of the trut. Thi i becaue the very thin cell wall can eaily be broken, folded, or buckled under load [0]. 7

62 Fig..4 Plateau border Fig..5 Size meaurement.6 Microcopic obervation of deformation and failure of polyurethane foam Microcopic obervation of foam were carried out to tudy the nature of it internal tructure, a well a it repone under tenile and compreive loading. Cuboid pecimen cut from foam B ( 9.5kg m ) were placed under a microcope and loaded uing a crew-driven jig (ee Fig..6). During the tet, image were recorded uing a high peed camera, reulting in photographic equence of deformation in the cell. Several mode of loading were carried out, i.e. tenion and 8

63 compreion along the foam rie direction (0 o ), the tranvere direction (90 o ) and 45 o to the rie and tranvere direction. Fig..6 Foam pecimen loaded uing crew driven jig.6. Tenile repone Fig..7 and.8 how foam behaviour under tenion along the rie direction (0 o ), Fig..9 and.0 depict tenion along the tranvere direction (90 o ) and Fig.. and. illutrate tenion along the 45 o direction. The picture indicate that the deformation mechanim in cell comprie bending and tretching of cell wall and trut, with bending being dominant. Bending can be identified from the change in angle between trut at their interconnection. Several report e.g. Ko [8], Gibon et al. [], Triantafillou et al. [8], Gibon and Ahby [, 0], Huber and Gibon [6], Maiti et al. [] and Zhu et al. [40, 4] have alo tated that bending of cell trut i the primary deformation mechanim in foam. It i alo evident from the picture that fracture occur by crack propagation through trut and cell wall membrane, perpendicular to the loading direction. 9

64 rie direction Fig..7 Micrograph of fracture propagation for tenion along the foam rie direction 40

65 rie direction Fig..8 Micrograph of cell deformation for tenion along the foam rie direction rie direction Fig..9 Micrograph of fracture propagation for tenion along the tranvere direction 4

66 rie direction Fig..0 Micrograph of cell deformation for tenion along the tranvere direction Fig.. Micrograph of fracture for tenion along the 45 o to the foam rie direction 4

67 Fig.. Micrograph of cell deformation for tenion along the 45 o to the foam rie direction.6. Compreive repone Fig.. how the deformation in cell for compreion along the rie direction (0 o ), while Fig..4 how the repone for compreion in the tranvere direction (90 o ). Fig..5 depict the repone for compreion along the 45 o direction. Thee picture demontrate that deformation initiate via bending and contraction of trut, followed primarily by buckling of compreed trut or platic bending of trut. The bending and hortening of trut, with bending a the main mechanim, correpond to linear elatic deformation, while the buckling and platic deformation of the trut are aociated with inelatic behaviour in plateau phae of the tre-train repone. Note that during deformation, the egment of the trut at their interconnection do not exhibit much bending. Thi implie that deformation occur away from thee end egment becaue the portion near the interconnection are tiffer and the trut are thicker there. Moreover, the membrane attached to thee interection are alo thicker and thu contribute to the bending reitance (ee Fig..6). The length of thee rigid portion were meaured and Fig..7 how ome of thee meaurement. The average length of the rigid egment i 48.6 μm or about 95 % of the average outer dimenion of the trut cro-ection R (ee Section.5) 4

68 rie direction Fig.. Micrograph of cell deformation for compreion along the foam rie direction 44

69 Fig..4 Micrograph of cell deformation for compreion along the tranvere direction Fig..5 Micrograph of cell deformation for compreion in the 45 o to the foam rie direction 45

70 thicker membrane thicker membrane Fig..6 Thick membrane at trut interconnection Fig..7 Meaurement of rigid trut egment.7 Mechanical propertie of olid polyurethane Other than the cell tructure, the material propertie of the olid material in the trut and wall alo determine the mechanical propertie of the foam. Several mechanical tet were therefore performed on olid polyurethane pecimen to obtain their mechanical propertie i.e. tiffne and tenile trength. The tet included compreion, tenion and three-point bending. Solid polyurethane wa made by mixing the polyurethane rein with the hardener and curing it inide a cloed tube to 46

71 prevent foaming. The reulting ample were eentially olid, but not entirely homogeneou, becaue there wa till a mall amount of ga produced that could not be totally expelled. Neverthele, the ample howed a good degree of homogeneity near the wall of the container. Conequently, the central portion wa cored out to yield tubular ample with an average denity of around 00 kg/m ; thi i conitent with publihed value of the denity of olid polyurethane []. The olid polyurethane ample were then prepared for compreion, tenion, and three point bending tet. For compreion tet Tubular ample were cut to an appropriate length and train gauge were attached to the ide; Fig..8 how one of the pecimen. The pecimen were then loaded in compreion uing a Shimadzu (model AG- 5TB) univeral teting machine. Fig..8 Compreion pecimen For tenile tet Tubular olid polyurethane ample were plit lengthwie and cut into dog-bone haped pecimen. The end of the pecimen were glued onto aluminium plate to facilitate gripping. Strain gauge were then attached to the pecimen; Fig..9 how one of the pecimen. The pecimen were then loaded uing an Intron (model 5500) univeral teting machine. 47

72 Fig..9 Tenion pecimen For three-point bending tet Rectangular pecimen were cut out from the olid tubular polyurethane ample. The pecimen were then loaded tranverely uing the Intron univeral teting machine with a pecial jig to facilitate three-point bending; Fig..0 how one of the pecimen being teted. Fig..0 Three point bending tet Two pecimen were ubjected to elatic compreion, and each pecimen wa loaded three time. Reult from the compreion tet are preented in form of engineering tre-train curve. Fig.. and. how the tre-train repone for loading and unloading of Specimen and. Overlapping of the loading and unloading curve indicate elaticity and the behaviour appear linear. The initial part of the curve ( < for Specimen and < for Specimen ) are not linear probably becaue contact between the loading plate and the pecimen were not perfect. 48

73 σ (MPa) tet tet tet Fig.. Compreion tre-train curve for Specimen σ (MPa) tet tet tet Fig.. Compreion tre-train curve for Specimen The tiffne of olid polyurethane calculated from the tre-train curve i hown in Table.. Specimen and how a difference in their tiffne; on average, Specimen ha a tiffne of.75 MPa, while that of pecimen i.65 MPa. Thi difference i relatively mall and ome variation i expected becaue the olid polyurethane produced for the fabrication of tet pecimen wa not perfectly homogeneou. Even though no gaing agent wa ued in making thee pecimen, the reaction between the polyurethane rein and hardener did generate ome mall bubble which remained trapped inide the pecimen. Moreover, the mechanical 49

74 propertie of polymer alo depend on the molecular polymer-chain alignment, ageing and oxidation which are not eay to control []. The average tiffne of the two pecimen i.65 MPa. Table. Stiffne from compreion tet Specimen (MPa) Specimen (MPa) The reult from the three-point bending tet are in the form of loaddiplacement curve. Fig...5 how the reult obtained from thee tet. In order to obtain the tre-train curve correponding to thi mode of loading, finite element imulation of the tet were performed uing ABAQUS. Structural model having imilar dimenion with the three-point bend pecimen were etablihed uing 00 linear Timohenko beam element available in ABAQUS-Standard (B). The tructure wa loaded tranverely, a hown chematically in Fig..6. Bilinear elatic-platic material propertie were aumed and adjuted to obtain repone imilar to the actual pecimen in term of load-diplacement curve hown in Fig It wa found that the mechanical propertie of olid polyurethane could be approximated by an iotropic elatic-platic material model with a bilinear tretrain repone. Table.4 how the tiffne and the yield trength determined from reult baed on comparing FEM imulation and tet data. 50

75 load (N) diplacement (mm) actual FEM Fig.. Load-diplacement curve for three-point bending tet of Specimen load (N) diplacement (mm) actual FEM Fig..4 Load-diplacement curve for three-point bending tet of Specimen 5

76 load (N) diplacement (mm) actual FEM Fig..5 Load-diplacement curve for three-point bending tet of Specimen (a) Three point bending tet (a) Finite element model Fig..6 Three-point bending tet and it finite element model Table.4 Stiffne and yield trength from three point bending tet Dimenion: length ; width ; thickne (mm) Stiffne (MPa) Yield trength (MPa) Tenile trength (MPa) Specimen 7.9 ;.9 ; Specimen 7.9 ;.89 ; Specimen 7.9 ;.99 ; Average A with compreion tet, engineering tre-train curve were obtained from tenion tet. Fig..7 how the reult, whereby the curve i initially approximately linear, followed by ome non-linearity before fracture. The curve for the three pecimen how little catter in term of their initial linear lope (i.e. the tiffne), but how ignificant catter in term of the fracture train. Thi i due to the lack of 5

77 perfect homogeinity of the pecimen, whereby the ga bubble inide the pecimen introduce tre concentration during the tet and hence affect the fracture propertie. Thee curve were ued to calculate the tiffne and the yield trength of the pecimen. The tiffne wa obtained from the lope of the linear portion of the curve, while the yield trength wa obtained uing a 0.00 offet train line from the linear repone and tenile the trength wa derived from the maximum tre attained σ (MPa) Fig..7 Stre-train curve from tenion tet σ (MPa) Fig..8 Determination of yield trength 5

78 Table.5 Mechanical propertie from tenile tet Stiffne (MPa) Yield Strength Tenile Fracture train (MPa) Strength (MPa) Specimen Specimen Specimen Average Table.5 how the tiffne, yield trength, tenile trength, and failure train value derived from tenile tet on olid polyurethane. Although there i catter in the data, the variation i maller than that from bending tet. The data from tenile tet are more conitent, epecially for the tiffne, the yield trength and the tenile trength. Thu, thi data i ued in ubequent development of a cell model in thi invetigation..8 Summary The mechanical propertie of rigid polyurethane foam were invetigated by performing tenile tet on foam ample. The influence of loading direction, foam denity, and cell aniotropy on thee mechanical propertie wa examined. Reult how that foam tiffne and trength decreae with angle between the loading and the foam rie direction. Thi arie from the orientation of the trut in cell that make them harder to bend for loading in rie direction. Moreover, more trut are required to be broken to initiate failure for loading in the rie direction, making the foam tronger in thi direction. The reult alo how that foam tiffne and trength increae with denity, and that aniotropy in the mechanical propertie increae with aniotropy in cell geometry. Microcopic obervation of the cro-ection of cell trut how that they have a hape imilar to that of a Plateau border []. Thi i probably becaue cell formation during the foaming proce follow Plateau law [] that relate minimum energy with a table tructure. Microcopic obervation of polyurethane foam loaded under 54

79 tenion and compreion how that the main mechanim governing foam deformation i the bending of cell trut and wall; other reearcher [, 8, 0,, 6, 8,, 40, 4] have alo reported imilar finding. The image alo reveal that the portion of the trut near their interconnection with other trut are tiffer and do not flex during deformation, becaue the thickne of thee portion are relatively larger and their deformation i contrained by thicker membrane at the interconnection. They alo how that fracture in the foam occur by crack propagation through trut and cell wall membrane, perpendicular to the loading direction Micro CT can image how that the hape of polyurethane foam cell bear a high degree of reemblance to an elongated tetrakaidecahedron, probably becaue a tetrakaidecahedron i a table polyhedral geometry when the minimum urface energy and the energy criteria defined in Plateau law [] are conidered [9, 40]. Compreion, three-point bending, and tenile tet were performed on olid polyurethane to obtain the tiffne, yield trength and tenile trength propertie. Thee data will be ued in the development of a cell model for foam. The experimental data exhibited ome catter becaue the olid polyurethane ample were not completely homogeneou, a ome tiny ga bubble generated from reaction between the polyurethane rein and the hardener were till trapped inide. Neverthele the data obtained i conidered valid for the formulation of foam model in the following chapter. 55

80 Chapter 4 Analytical Model of Idealized Cell Thi chapter decribe the development of idealized cell model for the rigid polyurethane foam tudied. Simplified cell aemblie compriing identical cell are etablihed to facilitate analyi and derive the eential load-deformation characteritic of foam. Variation to the model will be examined in Chapter 5. Two analytical open cell model are propoed a rhombic dodecahedron and a tetrakaidecahedron. Both are open celled even though the actual rigid polyurethane foam ha cloed cell, becaue the membrane that make up the wall are very thin compared to the thickne of the cell trut and corner; hence, the foam behave a if it i open celled [6, 0] (ee Section.5). Thee two geometrie are elected becaue both of them can be aembled in three dimenion to fill pace. Small deformation analyi baed on linear elatic-brittle fracture material behaviour will be developed. Thi i aimed at obtaining expreion to define the mechanical propertie of foam uch a tiffne, tenile trength, and Poion ratio. 4. Rhombic dodecahedron cell model Fig. 4. how an elongated rhombic dodecahedron open cell (i.e. polyhedron compriing twelve parallelogram). The cell model i elongated in the z-direction, correponding to the foam rie direction, in order to incorporate the geometric aniotropy oberved in actual foam cell. Thi model can be aembled uch that it fill pace in three dimenion, forming a tructure that conit of cell poitioned along an elongated FCC lattice, a hown in Fig. 4.. The foam propertie baed on thi cell model i.e. denity, tiffne, Poion ratio, and tenile trength will be derived and dicued. For thi analyi, the following definition and aumption are made: 56

81 The length Lˆ of the trut at the edge of each rhombic dodecahedron are the ame. The geometric aniotropy ratio, i.e. the ratio between the cell dimenion in the rie direction to that in the tranvere direction, can be defined by tan where i hown in Fig. 4.5 (ee Fig ). The trut have a contant cro-ectional area A and a contant econd moment of area I, uch that I C A (4.) The maximum ditance from the urface of the trut to it centroidal axi i R and the relationhip between R and A i R C A (4.) The trut are aumed to be made of an iotropic linear elatic material with tiffne E and tenile fracture tre σ max. The trut follow the Bernoulli-Euler beam theory when they deform. Failure of the model i aumed to occur when any location in a trut attain the tenile fracture tre σ max. 57

82 Fig. 4. Elongated rhombic dodecahedron cell Fig. 4. Elongated FCC tructure made from rhombic dodecahedron cell 4.. Relative denity The length of the trut in the cell model i aumed to be much longer than the ize of the vertice; hence, the volume of the trut can be approximated by taking the total length of the trut multiplied by their cro-ection area. By uing the repeating unit hown in Fig , the foam relative denity, i.e. ratio of the overall foam denity ( ) to the denity of the olid material in the cell trut and wall ( ) defined., can be 58

83 4LA ( 4L co ) co co Lin A 4in co L (4.) (4.4) 4.. Mechanical propertie in the z-direction The z-direction i the direction of cell elongation in thi model, which correpond to the foam rie direction in actual foam. To derive the elatic tiffne and the Poion ratio for thi cell model in the z-direction, an analyi of tructural deformation for loading in thi direction i performed. From ymmetry and imilarity, the analyi can be implified uch that only the fundamental repeating unit (hown in bold line in Fig. 4.) i conidered. Symmetry of thi repeating unit with it neighbour will be conidered in impoing boundary condition. More detailed view are hown in Fig. 4.4 and 4.5, which depict three-dimenional and two-dimenional view repectively. Fig. 4.5 alo define the geometrical quantitie ued in the analyi. Fig. 4. Repeating unit for the analyi of an elongated rhombic dodecahedron cell loaded in the z-direction 59

84 Fig. 4.4 Three-dimenional view of repeating unit in the analyi of an elongated rhombic dodecahedron cell loaded in the z-direction Fig. 4.5 Two-dimenional view of repeating unit in the analyi of an elongated rhombic dodecahedron cell loaded in the z-direction Taking into conideration imilarity and ymmetry, the analyi can be further implified to the analyi of jut one of the trut e.g. OC and then applying it to other imilar trut (Fig. 4.4). Fig. 4.6(a) and (b) how repectively, a -dimenional view of OC and it projection onto the plane ODCB. For the purpoe of analyi, the angle α i defined a a parameter. It can be een from Fig. 4.6(a) that the geometric quantitie L, Lˆ,, and α are interrelated via the following: 60

85 tan α tan inα in co coα co co ˆ L L co (4.5) (4.6) (4.7) (4.8) Fig. 4.6 Strut OC Fig. 4.7 Deformation of trut OC in plane OBCD For convenience, analyi of the deformation of trut OC can be done with repect to the plane ODCB (ee Fig. 4.6 and 4.7) and then expreed in term of the 6

86 Carteian pace defined by the coordinate xyz. When the rhombic dodecahedron tructure i loaded in the z-direction, point C move upward and left to point C (Fig. 4.7). Note that there i no force in the horizontal plane (i.e. parallel to the xy-plane) becaue the trut are free to move in the x and y direction. Becaue of ymmetry about the line CD (Fig. 4.6) with another trut in the adjacent repeating unit, the lope of the trut at point C with repect to the plane ODCB doe not change during deformation, i.e. the lope remain equal to α with repect to the xy-plane. Hence, the bending moment M in the trut at point C i related to the vertical force F z at that cro-ection by F ˆ z L coα ML 0 E I E I (4.9) F ˆ z L coα M (4.0) Thu, the repective diplacement of point C in the axial (n) and tranvere (t) direction are u n and Lˆ AE F z inα (4.) u t F Lˆ z coα ML E I E I F Lˆ z coα E I F Lˆ z coα Fz L coα L E I E I Lˆ u t E F I z coα (4.) (4.) 6

87 Thee diplacement can be related to the deformation of the repeating unit in the x, y and z-direction by uing the geometrical relationhip inferred from Fig. 4.6 and 4.7, and conidering ymmetry and imilarity of the trut in Fig The reult are: δ δ x z δ y Lˆ AE ( u coα u inα ) n ˆ L E I ( u inα u coα ) n Lˆ AE in t t F Lˆ α co E I z inα coα α F z (4.4) (4.5) Subtituting Eq. (4.5)-(4.8) into Eq. (4.4) and (4.5) replacing Lˆ and α by L and : L δ x δ y L δ z L AE co AE L E co in AE I ( L co ) E I F in co ( co ) Fz ( L co ) E I z co co in co F co ( co ) co Fz co co z (4.6) L L in AE 6E I co (4.7) Conequently, the train are obtained by dividing the deformation by the initial length. xx yy δ x L co L AE E I in ( co ) Fz co (4.8) 6

88 zz δ z Lin in AE L 6E I ( co ) co Fz in co (4.9) The tre in the z-direction i given by the ratio of the applied force A z over which it act 4 Fz to the area 4Fz Fz σ zz Az L co (4.0) The tiffne in the z-direction i obtained by dividing the tre by the train in the z- direction, while the Poion ratio are obtained from the ratio between the appropriate train. σ zz E E zz 4 4 zz in co L co co L co A in I (4.) ( co ) L in yy A I xx υ υ zx zy zz zz ( ) L co co in A 6 I (4.) L 4in co For foam with a low overall denity, the term i much maller A co than L 4 I becaue L i much larger than the dimenion of the cro-ection of the cell trut; hence, the elatic tiffne and the Poion ratio can be approximated by E υ zz zx co 4 in E I 4 co L υ tan zy (4.) (4.4) 64

89 The tiffne and the Poion ratio can alo be defined in term of the relative denity (ratio between the overall denity of the foam to that of the olid material in the cell) by ubtituting Eq. (4.) and (4.4) into Eq. (4.) and (4.). The reult are: E υ zz xz υ yz ( co ) xx zz 48in E C ( co ) in in 48co ( co ) in 4in C C (4.5) (4.6).5 bending moment (x M max) poition (x L ) Fig. 4.8 Bending moment ditribution along trut OC The tenile trength of thi cell model in the z-direction can alo be derived uing the repeating unit hown in Fig Again, the analyi i implified by conidering the failure of one trut, a hown in Fig It can be een that the maximum normal tre in a trut i attained at the outer urface at both end becaue the bending moment i highet there (note that the bending moment along a beam loaded a hown in Fig. 4.7 varie linearly with ditance from it end, a hown in 65

90 Fig. 4.8; the maximum bending moment thu occur at both end). The tre at thee point can be derived by conidering a combination of the axial force and the bending moment. F z inα M R σ A I (4.7) Subtitution of Eq. (4.0) into (4.7) and equating the tre σ to the failure tre σ max yield σ inα Lˆ R coα A I max F z Lˆ and α are replaced by L and by uing Eq. (4.5)-(4.8): (4.8) in co L R F z A I σ max co (4.9) Finally, by ubtituting Eq. (4.0), which define the overall tre, into (4.9), the tenile trength in the z-direction i defined by σ max σ z max in co L L R co co A I (4.0) Eq. (4.0) can alo be expreed in term of the relative denity by ubtituting Eq. (4.)-(4.4) into it. σ z max 4 max 4 ( co ) σ ( in ) C C (4.) 4.. Mechanical propertie in the y-direction The y-direction i the tranvere direction for thi cell model, which correpond to the direction perpendicular to the rie direction in actual foam. In order to derive the tiffne and the Poion ratio for the y-direction, loading of the cell in thi 66

91 direction i analyed. A with loading in the z-direction, the analyi of loading in the y-direction can alo be reduced to the analyi of a ingle repeating unit, a hown in bold line in Fig Fig. 4.0 and 4. how repectively, more detailed threedimenional and two-dimenional view of the unit. Although the repeating unit look the ame a that ued for the analyi of loading in the z-direction, thi unit cell i extracted in a different way to implify the analyi (ee Fig. 4.0). Fig. 4.9 Repeating unit for the analyi of an elongated rhombic dodecahedron cell loaded in the y-direction Fig. 4.0 Three-dimenional view of repeating unit for analyi of an elongated rhombic dodecahedron cell loaded in y-direction 67

92 Fig. 4. Two-dimenional view of repeating unit for analyi of an elongated rhombic dodecahedron cell loaded in the y-direction Thi analyi can alo be further implified to the analyi of one trut i.e. OC and then applying it to the other trut. Fig. 4.(a) and (b) how repectively, the trut OC in a -dimenional view and it projection onto the plane OGCH. For the purpoe of analyi, an angle β i defined, a illutrated in Fig. 4., which alo how that the geometric parameter L, Lˆ,, and β are related by tan β co in β co β ˆ co co co L L co (4.) (4.) (4.4) (4.5) 68

93 Fig. 4. Strut OC Fig. 4. Deformation of trut OC in plane OGCH Analyi of the deformation of trut OC can be done with repect to the plane OGCH (ee Fig. 4. and 4.) and then relating it to the x, y and z coordinate axe. When the rhombic dodecahedron i loaded in the y-direction, point C move to point C (ee Fig. 4.). Note that there i no force in the xz-plane becaue the trut are free to move in the x and z direction. Becaue of ymmetry about the line CG (Fig. 4.) with another trut in the adjacent repeating unit, the lope of the trut at point C with repect to the plane OGCH doe not change during the deformation, i.e. the lope remain equal to β with repect to the xz-plane. Thu, the relationhip between the bending moment M and the horizontal force F y at point C i given by 69

94 F ˆ y L co β ML 0 E I E I (4.6) F ˆ y L co β M (4.7) Hence, the diplacement of point C in the axial (n) and tranvere (t) direction are repectively, u n and Lˆ AE F y in β (4.8) u t F Lˆ y co β ML E I E I F Lˆ y co β Fz L co β L E I E I F Lˆ y co β E I (4.9) Lˆ u t Fy co β E I (4.40) Thee diplacement can be related to the deformation of the repeating unit in the x, y and z direction by determining the change in length of the repeating unit projected onto thoe axe. The reult are δ x y ( u coα u inα ) n Lˆ AE δ ˆ L E t F I ( u in β u co β ) n Lˆ AE in t y co Lˆ β co E I in β co β co β F y (4.4) (4.4) 70

95 δ z ( u co β u in β ) n Lˆ AE ˆ L E t F I y in in β co β in Replacing all Lˆ and β with L and by uing Eq. (4.)-(4.5): L δ x L AE L δ y L AE L δ z co AE L E I ( L co ) E I co F co co ( co ) Fy L E I co co AE co co co AE co ( L co ) E ( co ) Fy ( L co ) E I I co in co F co in co ( co ) Fy y F co y y (4.4) (4.44) (4.45) L L AE E I co (4.46) Hence, the train can be obtained by dividing the deformation by the appropriate initial length. xx yy zz δ L co ( co ) Fy x L co AE E I co δ L ( co ) Fy y co L co AE E I co co δ L co ( co ) Fy z Lin AE E I co The tre in the y-direction i (4.47) (4.48) (4.49) σ 4F A y yy y L F y in co (4.50) 7

96 The tiffne and the Poion ratio can then be obtained from conidering the tre and train in the repective direction uing Eq. (4.47)-(4.50) E υ υ yy σ yy yy E in co L in co L co A 6 I ( co ) L co A I ( co ) L co A I xx yx yy ( co ) L co A I ( co ) L co A I zz yz yy A highlighted, for foam with a low denity, the term 4 L A (4.5) (4.5) (4.5) 4in co i co much maller than I L 4 ; hence, the tiffne and Poion ratio can be approximated by E yy in υ co yx υ co yz 6 E I 4 co L (4.54) (4.55) (4.56) Eq. (4.5)-(4.54) can be expreed in term of the relative denity by ubtituting Eq. (4.)-(4.4) into them, reulting in: E yy E 96in co ( co ) 4 C (4.57) 7

97 υ υ yx yz xx yy ( co ) co 48in co 48in co ( co ) ( co ) C C (4.58) co 48in C zz yy ( ) co co 48in co C (4.59) The repeating unit hown in Fig can alo be ued to analye the tenile trength for loading in the y-direction. A with the analyi of the tiffne and the Poion ratio, determination of the tenile trength can be implified by conidering one trut, a hown in Fig. 4.. Following the ame approach for the tenile trength in the z-direction, the maximum tre in the trut alo occur at the outer urface at both end; it value being: F y in β M R σ A I (4.60) Subtituting Eq. (4.7) into (4.60) and equating the tre σ to the tenile failure tre σ max : σ L R A I in β ˆ co β max F y Eliminating Lˆ and β uing Eq. (4.)-(4.5) reult in (4.6) co L R F y A I σ max co (4.6) Finally by ubtituting Eq. (4.50), which define the overall tre, into Eq. (4.6), the tenile trength in the z-direction i given by 7

98 σ max σ y max in co L L R in co co A I (4.6) Eq. (4.6) can alo be written in term of the relative denity by uing Eq. (4.)-(4.4). σ max σ y max 4 ( co ) C 8 in co C (4.64) 4..4 Correction for rigid trut egment It wa highlighted in Section.6 that the trut egment adjoining the interconnection (cell corner) with adjacent trut do not exhibit noticeable bending during deformation. Thi hortening of the effective flexural length of the trut influence the mechanical propertie of the foam. Thu, the equation derived in Section 4.. and 4.. that define the deformation of the rhombic dodecahedron cell hould be modified to incorporate the influence of thee rigid trut egment. Thi modified model will be referred to a the emi-flexible trut model. A dicued in Section.6, hortening of the effective flexural length of the trut occur becaue the trut are larger near their interconnection. Thu, in thi tudy, the length of the rigid portion of each trut i aumed to be determined by the cro-ectional dimenion R, which i the maximum ditance from the urface of the trut to it centroidal axi. Baed on thi, the trut egment of length d immediately adjoining the interconnection with neighbouring trut i aumed to be rigid (they do not tretch nor flex) during cell deformation, where d i related to R by d CR (4.65) The foam tiffne, Poion ratio and tenile trength baed on thi cell model then take on the following expreion: 74

99 E E υ υ υ σ σ zz yy σ σ υ zz zz yy yy in co co 4 ( L d ) L co co ( L d ) A E in I E in co co A A in ( L d ) L in co ( L d ) ( L d ) ( co ) ( L d ) A I xx zx zy zz ( L d ) ( co ) co ( L d ) ( L d ) ( co )( L d ) A co ( L d ) ( co )( L d ) A 6 I co I I xx yx yy ( L d ) ( co )( L d ) A co zz yz yy z max y max σ max in co L co co A ( L d ) ( co )( L d ) σ max in co L in co co A A co I I ( L d ) I ( L d ) I L R L R I in L L (4.66) (4.67) (4.68) (4.69) (4.70) (4.7) (4.7) Denoting X a the fraction of the trut that deform; i.e. X d 4 L in co C C ( ) 4 co (4.7) 75

100 76 Eq. (4.66)-(4.7) can alo be expreed in term of the relative denity by ubtituting Eq. (4.)-(4.4) and (4.65) into them; thu: ( ) 48in co zz C X X E E (4.74) ( ) 4 co 96in co yy C X X E E (4.75) ( ) ( ) υ υ zy zx C X X C X X 4in co in 48co co in in (4.76) ( ) ( ) υ yy xx yx C X X C X X co 48in co co 48in co co (4.77) ( ) ( ) υ yy zz yz C X X C X X co 48in co co 48in co co (4.78) ( ) ( ) 4 max max in 4 co σ σ z C C X (4.79) ( ) 4 max max co in 8 co σ σ y C C X (4.80)

101 4. Tetrakaidecahedron cell model The preceding ection decribed an elongated rhombic dodecahedron cell model. Another polyhedron that can be aembled in three-dimenion i the tetrakaidecahedron, which ha hexagonal and quare face, and can be arranged in a BCC lattice to fill pace. Zhu et al. [40] and Warren and Kraynik [] have analyed an iotropic tetrakaidecahedron open cell model in term of it elatic propertie and aerted that a tetrakaidecahedron i the only polyhedron that can be aembled in three dimenion to fill pace and atify approximately the urface energy criterion and Plateau law []. Tetrakaidecahedra are alo imilar to cell in actual foam []. Zhu et al. [40] and Warren and Kraynik [] howed that the mechanical propertie of their model are approximately iotropic; hence, it cannot be ued to decribe the foam in the preent tudy. A with the rhombic dodecahedron cell model, the tetrakaidecahedron model i elongated in the z-direction (ee Fig. 4.4) to facilitate invetigation of the actual foam cell which are larger in the foam rie direction. The elongated tetrakaidecahedron cell can be packed in an elongated BCC lattice, a hown in Fig

102 Fig. 4.4 Elongated tetrakaidecahedral cell The following definition and aumption are made in deriving foam denity, tiffne, Poion ratio and tenile trength baed on thi model: The length of trut other than thoe that lie on horizontal plane (parallel to the xy-plane) i L. The geometric aniotropy ratio, i.e. the ratio between the cell length in the rie direction to that in the tranvere direction, can be decribed by defined in Fig. 4.8 (ee Fig ). tan, where i The trut lying on horizontal plane have the ame length l related to L by l L co (4.8) The trut have a contant cro-ectional area A and a contant cro-ectional econd moment of area I; relationhip between I and A being I C A (4.8) The maximum ditance from the urface of a trut to it centroidal axi i R and the relationhip between R and A i 78

103 R C A (4.8) The trut are aumed to be made from an iotropic material with elatic tiffne E and tenile fracture tre σ max. The trut follow Bernoulli-Euler beam theory when they deform. Failure occur when any location in a trut attain the tenile fracture tre σ max. Fig. 4.5 Elongated BCC tructure made from tetrakaidecahedron cell 4.. Relative denity A with the rhombic dodecahedron cell, the length of the trut in thi model are aumed to be much larger than the ize of the cell vertice. Thu, the relative denity of the foam baed on the repeating unit hown in Fig can be approximated by conidering the trut length. 4LA ( 4L co ) LAco Lin (4.84) 79

104 co A 8in co L (4.85) 4.. Mechanical propertie in the z-direction A with the rhombic dodecahedron model, the z-direction i alo the direction of cell elongation for thi model, which correpond to the foam rie direction in actual foam. An analyi of tructural deformation for loading in the z-direction i performed to derive the elatic tiffne and the Poion ratio of thi cell model for thi direction. The analyi i implified uch that it only conider the fundamental repeating unit (hown in bold line in Fig. 4.6) due to ymmetry and imilarity. Fig. 4.7 and 4.8 how repectively, the three-dimenional and two-dimenional view of the unit. The geometrical quantitie ued in the analyi are defined in Fig Fig. 4.6 Repeating unit for the analyi of an elongated tetrakaidecahedron cell loaded in the z-direction 80

105 Fig. 4.7 Three-dimenional view of repeating unit in the analyi of an elongated tetrakaidecahedron cell loaded in the z-direction Fig. 4.8 Two-dimenional view of repeating unit for the analyi of an elongated tetrakaidecahedron cell loaded in the z-direction Becaue there i no loading in the horizontal plane (i.e. plane parallel to the xy-plane), all the trut lying on the horizontal plane do not deform. Hence, the analyi can be implified to jut analyzing trut OB, OS, OT and OU (Fig. 4.7). Taking into conideration imilarity and ymmetry of thee trut, the analyi can be further reduced to an analyi of jut one of the trut e.g. OB and then applying it to the other (ee Fig. 4.9). 8

106 Fig. 4.9 Deformation of trut OB Fig. 4.9 how trut OB in the yz-plane. When the tetrakaidecahedron tructure i loaded in the z-direction, point B move upward and left to point B, but due to ymmetry along the z-direction, the lope of the trut at point B doe not change. Conequently, the bending moment M and vertical force F z at point B are related by Fz L co ML 0 E I E I (4.86) Fz L co M (4.87) Hence, the diplacement of point B in the axial (n) and tranvere (t) direction are: u u n t L AE F z in Fz L co ML E I E I Fz L co Fz L co L E I E I L u t E F I z co (4.88) (4.89) (4.90) 8

107 Thee diplacement can be related to the deformation of the repeating unit in the x, y and z-direction by projecting them onto the correponding axe and conidering the ymmetry and imilaritie of the unit a hown in Fig The reult are: δ δ x z y δ ( u co u in ) n L AE L E ( u in u co ) n L AE in t t Fz in co I L co E I F The train are obtained by dividing the deformation by the initial length. z (4.9) (4.9) xx zz δ x yy 4L co L F AE 4E I z in δ z Lin L co in Fz AE E I in The tre in the z-direction i obtained by dividing the applied force (4.9) (4.94) by the area Fz A z over which it act. Fz Fz σ zz Az 4L co (4.95) The tiffne and the Poion ratio of thi model can then be found by dividing the tre by the train in the z-direction, and taking the ratio between the appropriate train: E σ 4in co zz zz 4 zz E 4 L co L A in I (4.96) 8

108 υ L in A 4I co L in A in I xx yy zx υ zy zz zz A highlighted in the rhombic dodecahedron model, the term A L (4.97) i much maller than L 4 I for low denity foam becaue L i much larger than the lateral dimenion of the trut cro-ection; hence, the tiffne and the Poion ratio can be approximated by E υ zz in E I 4 4 co L (4.98) υ tan (4.99) Eq. (4.96) and (4.97) can alo be expreed in term of the relative denity by zx zy ubtituting Eq. (4.8) and (4.85) into them. E E zz ( ) co co 9in C (4.00) in co 9 co C xx υ zx υ zy zz co in 96in C (4.0) The tenile trength of thi cell model in the z-direction can be derived from the repeating unit hown in Fig The analyi i implified by conidering the failure of one trut, a hown in Fig A with the rhombic dodecahedron model, the maximum normal tre in a trut i attained at it outer urface at both end of the 84

109 trut and it value i obtained by conidering a combination of the axial force and bending moment. F z in M R σ A I (4.0) Subtitution of Eq. (4.87) into Eq. (4.0) and equating the tre σ to the failure tre σ max yield: in L R co σ max F z A I (4.0) Finally, by ubtituting Eq. (4.95), which define the overall tre, into Eq. (4.0), the tenile trength in the z-direction i defined. σ max σ z max L L R 4in co co A I (4.04) Eq. (4.04) can alo be expreed in term of the relative denity by incorporating Eq. (4.8), (4.8), and (4.85) into it. σ z max max co co C σ in 8 C (4.05) 4.. Mechanical propertie in the y-direction A with the analyi of the rhombic dodecahedron cell model, the y-direction for the tetrakaidecahedron cell model correpond to the tranvere direction for actual foam. In order to derive the tiffne and the Poion ratio in the y-direction, loading of the cell in thi direction i analyed. A with loading in the z-direction, the analyi of loading in the y-direction can alo be reduced to the analyi of the ingle repeating unit hown in bold line in Fig The deformation of trut lying in plane parallel to the vertical xz-plane need not be conidered becaue there i no 85

110 force in thoe plane. Thi analyi i further implified to an analyi of two trut one lying in the horizontal xy-plane and one in the yz-plane (ee Fig. 4. and 4.). Thu, the analyi of the mechanical propertie of thi cell model in the y-direction i performed in two part an analyi of a trut in the vertical yz-plane and one in the xy-plane, followed by a combination of the two. In thi analyi, it i aumed that a force F i pulling the repeating unit in the y-direction; hence, each trut ha to utain a load of F. The overall tre in the y-direction can then be expreed a σ F A y yy x 4L F y in co (4.06) Fig. 4.0 Repeating unit for the analyi of an elongated tetrakaidecahedron cell loaded in the y-direction 86

111 Fig. 4. Three-dimenional view of repeating unit for the analyi of an elongated tetrakaidecahedron cell loaded in the y-direction Fig. 4. Two-dimenional view of repeating unit ued for the analyi of elongated tetrakaidecahedron cell loaded in the y-direction 87

112 Analyi of trut in the vertical yz-plane Fig. 4. Deformation of trut OS Ariing from ymmetry, the analyi of trut lying in the yz-plane can be further implified to an analyi of jut one trut e.g. OS (Fig. 4. and 4.) and then applying it to other imilar trut. When the tetrakaidecahedron tructure i loaded in the y-direction, Fig. 4. how that point S move in the axial (n ) and tranvere (t ) direction. However, the lope of the trut at point F doe not change due to ymmetry with another repeating unit in the y-direction. Thu, the bending moment M and the horizontal force F y at point S are related by Fy L in ML 0 E I E I (4.07) Fy Lin M (4.08) The diplacement of point S in the axial (n ) and tranvere (t ) direction are: u L F n y AE co (4.09) 88

113 u t Fy L in ML E I E I Fy L in Fy Lin L E I E I (4.0) L u t Fy in E I (4.) Therefore, the diplacement of point S in the y and z-direction can be obtained by projecting them onto thee axe. δ δ y z u u n co u L AE n co in u t in L in E I t co F y (4.) L L Fy in co AE E I (4.) Strut OF in Fig. 4. can alo be ued to determine the tenile trength of thi cell model, governed by failure of trut in the vertical yz-plane. The maximum tre occur at the outer urface of trut OS at both end, where the bending moment i highet, and it value being: F y co M R σ A I (4.4) Subtitution of Eq. (4.08) into Eq. (4.4) and equating the tre σ to the tenile failure tre σ max yield co L Rin σ max F y A I (4.5) Finally, by incorporating Eq.(4.06) for tre in the y-direction into Eq. (4.5), the tenile trength in the y-direction i given by. 89

114 σ max σ y max L L R 4in co in co A I (4.6) Eq. (4.6) can alo be expreed in term of the relative denity by uing Eq. (4.8), (4.8) and (4.85). σ y max σ max co in 8 co C Analyi of trut in the xy-plane ( co ) C (4.7) Fig. 4.4 Deformation of trut OH A with the analyi of trut in the yz-plane, the analyi for trut lying in the xy-plane can alo be implified to the analyi of one trut e.g. OH a hown in Fig When the tetrakaidecahedron tructure i loaded in the y-direction, point H move in the axial (n ) and tranvere (t ) direction. However, the lope of the trut at point H doe not change due to repeating ymmetry in the y-direction. Conequently the bending moment M and the horizontal force F y at point A are related by F y ( L co ) E I M F y L co M ( L co ) 0 E I (4.8) (4.9) 90

115 The diplacement of point H in the axial (n ) and tranvere (t ) direction are: u u u L co Fy L co F AE n y AE t t F y F y ( L co ) L E E I co I L co Fy 6E I M Fy L co ( L co ) E I ( L co ) E Therefore, the diplacement of point H in the x and y-direction are I (4.0) (4.) (4.) δ δ x y ( u u ) L n AE t L co co E I ( u u ) n t F y (4.) L L co co F y AE E I (4.4) Strut OH can alo be ued to acertain the tenile trength in the y-direction, which correpond to the failure of trut in the xy-plane. Following the previou analyi, the maximum tre in thi trut i F y σ A M R I (4.5) Subtitution of Eq. (4.8) into Eq. (4.5) and equating the tre σ to the failure tre σ max yield: σ L R co A I max F y (4.6) 9

116 Finally by uing Eq. (4.06), which decribe the overall tre in the y-direction, in the preceeding expreion, the tenile trength in the y-direction i defined by. σ max σ y max L L r in co in co A I (4.7) Eq. (4.7) can alo be expreed in term of the relative denity by ubtituting Eq. (4.8), (4.8) and (4.85) into it. σ y max Summary co σ max ( co ) C 8 in co C (4.8) The total deformation of the tetrakaidecahedron cell model in the x, y and z- direction i obtained by combining the deformation of the trut in the vertical yz and horizontal xy-plane. L L δ x δ x co co Fx AE E I δ δ δ y y L AE δ y co y co L in E I co L AE F y L AE L co co E I L ( in co ) Fy E I F y (4.9) (4.0) (4.) L L δ z δ z Fx in co AE E I (4.) The train are then calculated by dividing the deformation by their correponding initial length. yy δ y co L co AE in co co L Fy 4E I (4.) 9

117 xx zz δ x 4L co 4AE co δ z L co 4 in AE L F E I 4 L Fy E I co 4 Thu, the tiffne i given by the ratio of the tenile tre y (4.4) (4.5) σ (Eq. (4.06)) to the yy correponding train yy (Eq. (4.5)) E yy σ yy yy E 4 L L ( in co in co ) ( in in co ) 6 I The Poion ratio are derived by taking the ratio of the appropriate train. A (4.6) υ υ yx yz xx yy co co L A 6 I in co A co L I (4.7) L co A I zz yy in L co co A co I (4.8) Eq. (4.6)-(4.8) can alo be expreed in term of the relative denity by ubtituting Eq. (4.8), (4.8) and (4.85) into them. E υ yy yx in ( co ) ( co ) 4 co 8 co in co 84C xx yy co E (4.9) co 48in C in co co co 96in co C (4.40) 9

118 υ yz co co 96in co C zz yy in co co co co 96in co C (4.4) The overall tenile trength i determined by comparing Eq. (4.6) and (4.9), which decribe the tenile trength correponding to failure of trut in the yz and xy-plane, repectively. The trut in the yz-plane are more vulnerable than the trut in the xy-plane, i.e. σ y max σ y max, if the geometric aniotropy ratio tan. For foam with a low denity, the ratio between the two failure tree can be approximated by: σ σ y max y max tan (4.4) Hence, the overall tenile trength of thi model i governed by the failure of trut in the vertical yz-plane. σ σ σ σ max L L R 4in co in co A I y max y max y max σ max co in 8 co C ( co ) C (4.4) (4.44) 4..4 Correction for rigid trut egment A with the rhombic dodecahedron cell model, the equation defining the mechanical propertie of the tetrakaidecahedron model derived in Section 4.. and 4.. hould be modified to include the influence of rigid egment at the end of each trut. Following the rhombic dodecahedron model, thi modified model will alo be referred to a the emi-flexible trut model. Similar aumption are ued, i.e. trut 94

119 egment of length d adjacent to the interconnection with neighbouring trut do not deform during cell deformation, where d i related to R by d CR (4.45) Conequently, the foam tiffne, Poion ratio and tenile trength baed on thi cell model are modified to: E E υ σ 4in co zz zz 4 z ( L d ) L co ( L d ) yy σ yy yy A E in L in ( in co ( L d ) in ( L co d ) in ( L d ) ( L co d ) υ in E A ( L d ) ( L d ) A ( L d ) co ( L d ) A 4I in I in xx zx zy zz xx υ yx υ σ σ yz yy zz yy co co 4in co I L ( L co d ) ( L co d ) Aco ( ) ( L co d ) in ( ) ( L co d ) L d L d co A co ( L d ) ( L d ) A 4co I co ( ) ( L co d ) in ( ) ( L co d ) L d L d L A co I A max z max ( L d ) 4in co L A σ co L r I I co max y max ( L d ) σ in co L r co co (4.46) L 6I (4.47) (4.48) I (4.49) I (4.50) (4.5) (4.5) 95

120 96 Thee expreion can alo be written in term of the relative denity by ubtituting Eq. (4.8)-(4.86) and (4.45) into them, thu: ( ) ( ) ( ) 9in co co zz C X X E E (4.5) ( ) ( ) ( ) ( ) 4 84 co co in co co in 8 co co co yy C X X X X E E (4.54) ( ) ( ) ( ) ( ) υ υ zz xx zy zx C X X C X X 96in co in 9 co co in (4.55) ( ) ( ) υ yy xx yx C X X X X C X X co 96 in co co co co in co co 48 in co co co (4.56) ( ) ( ) ( ) ( ) υ yy zz yz C X X X X C X X co 96in co co co co in co co co 96in co co (4.57) ( ) max max 8 in co co σ σ z C C X (4.58) ( ) ( ) max max co 8 co in co σ σ y C C X (4.59) where C C L d X co co 8in (4.60) denote the fraction of trut that that i rigid.

121 4. Contant C, C and C A uggeted in Section.5, the cro-ection of the trut in rigid polyurethane foam i imilar in hape to the Plateau border [] hown in Fig Thu, the model developed will adopt the Plateau border a the hape of the trut cro-ection. Baed on thi, the contant C and C defined in Section 4. and 4. can be determined. 0 π C 0.8 6( π ) (4.6) C.478.5π (4.6) The value of C i calculated baed on the maximum ditance R from the centroid of the Plateau border (Fig. 4.5). Failure however might not occur at that point; it depend on the direction and orientation of the trut cro-ection relative to the applied bending moment. To account for thi variation, C hould be determined by uing the average maximum ditance R from the centroid of the Plateau border, thu: C C R.5π R.89.5π π.5π π 0 co d π (4.6) 97

122 Fig. 4.5 Plateau border C, which i the ratio of the rigid trut egment length to the maximum ditance of the trut urface from the centroidal axi of the cro-ection, i determined from experimental obervation of the ize of the trut and the length of the rigid egment at the vertice of the cell inide actual foam (ee Section.5 and.6). Thi howed that on average C However, becaue C i determined by uing the average maximum ditance ( R ) from the cro-ection centroidal axe, C alo ha to be baed on thi average ditance o that the length of the rigid egment and the area of the trut cro-ection are till related by d CC A Hence, C become: (4.64) C π The contant C, C and (4.65) C and the data derived from the experiment dicued in Chapter facilitate calculation of the mechanical propertie of foam baed on the model. Parametric tudie and comparion of thee model with actual foam can then be performed; thi will be dicued in Section

123 4.4 Reult and dicuion 4.4. Cell geometry and parametric tudie Fig. 4.6 how an elongated rhombic dodecahedron and an elongated tetrakaidecahedron cell, while Fig. 4.7 how cell in actual foam. The figure indicate that the geometry of the rhombic dodecahedron cell model ha ome ditinct difference with the cell in rigid polyurethane foam. Actual cell have lanted face adjacent to the upper and lower end, vertical face on the ide and horizontal face at the top and bottom; the rhombic dodecahedron cell model only ha lanted face and pointed apice at the top and bottom. On the other hand, a highlighted in Section.4, an elongated tetrakaidecahedron cell diplay great imilarity with actual foam cell, much better than that of a rhombic dodecahedron cell. a. Rhombic dodecahedron b. Tetrakaidecahedron Fig. 4.6 Elongated rhombic dodecahedron and tetrakaidecahedron cell 99

124 Fig. 4.7 Actual foam cell The mechanical propertie of an idealized foam model baed on rhombic dodecahedron and tetrakaidecahedron cell in term of it tiffne, Poion ratio and tenile trength can be determined by uing the equation derived in Section The parameter required are the elatic tiffne ( E ) and tenile trength ( σ max ) of the olid material that contitute the trut, the relative denity of the foam and the geometric aniotropy ratio of the cell ( tan ). A parametric tudy wa undertaken by varying thee parameter to analye their influence on the overall mechanical propertie of foam baed on thee model. 00

125 0 Stiffne The tiffne of foam predicted by the rhombic dodecahedron cell model can be calculated uing the following equation defined in Section 4.: Neglecting the correction for rigid egment at the trut end (fully flexible trut) o The tiffne in the rie direction i given by ( ) 48in co zz C E E (4.66) o The tiffne in the tranvere direction i ( ) 4 co 96in co yy C E E (4.67) Conidering the correction for rigid egment at the trut end (emi-flexible trut) o The tiffne in the rie direction i ( ) 48in co zz C X X E E (4.68) o The tiffne in the tranvere direction i ( ) 4 co 96in co yy C X X E E (4.69) where ( ) C C X 4 co co in 4 (4.70)

126 0 The tiffne of foam predicted by the tetrakaidecahedron cell model derived in Section 4. are given by: For fully flexible trut o The tiffne in the rie direction i ( ) 9in co co zz C E E (4.7) o The tiffne in the tranvere direction i ( ) ( ) 4 84 co co in co in 8 co co yy C E E (4.7) For emi-flexible trut o The tiffne in the rie direction i ( ) ( ) ( ) 9in co co zz C X X E E (4.7) o The tiffne in the tranvere direction i ( ) ( ) ( ) ( ) 4 84 co co in co co in 8 co co co yy C X X X X E E (4.74) where C C X co co 8in (4.75) (Difference ariing from conideration of rigid trut egment will be highlighted later.)

127 Note that all the expreion have the term and in their denominator, which repreent repectively the contribution from the axial force and the bending moment in the trut. For low denity foam, i much larger than howing that the dominant deformation mechanim in foam i bending of the trut. Thi i conitent with the obervation decribed in Section.6 a well a with reult by ome previou reearcher [, 8, 0,, 8,, 40, 4]. It i alo worth noting that cell ize doe not affect foam tiffne, which depend only on the tiffne E of the olid trut material, the relative denity, the cell geometric aniotropy ratio ( tan ), the trut cro-ection hape repreented by C and C, and the length of the rigid trut egment repreented by C. 0.0 E /E rie (emi-flexible trut) tranvere (emiflexible trut) rie (fully flexible trut) tranvere (fully flexible trut) / Fig. 4.8 Variation of foam tiffne with relative denity baed on an iotropic rhombic dodecahedron cell model 0

128 E /E rie (emi-flexible trut) tranvere (emiflexible trut) rie (fully flexible trut) tranvere (fully flexible trut) / Fig. 4.9 Variation of foam tiffne with relative denity baed on an iotropic tetrakaidecahedron model Fig. 4.8 and 4.9 how repectively the variation of foam tiffne with relative denity baed on an iotropic ( tan ) rhombic dodecahedron and an iotropic tetrakaidecahedron cell model. A expected, the tiffne in the foam rie direction i the ame a that in the tranvere direction becaue the model are iotropic. Another expected reult i that the foam tiffne baed on both cell with rigid trut egment (emi-flexible trut) i higher than that without thi correction and thi difference increae with denity becaue the length of the rigid trut egment alo increae with denity. Foam tiffne predicted by the model increae with relative denity, a hown in Fig. 4.8 and 4.9. Thi i expected, a a higher denity mean that there i more olid material to bear the applied load, making the tructure tiffer. The tiffne E of foam baed on a cell model without rigid egment (fully flexible trut) can be approximated by a power law relationhip in term of the relative denity. E E C f n (4.76) 04

129 where E i the tiffne of the olid trut material, and and are repectively the denitie of the foam and the olid trut material, and C f and n are contant that are governed by the hape of the cell and the trut cro-ection, a well a the deformation mechanim in the foam (bending and tretching of the trut). It turn out that C. 75 and n. 99 for the fully flexible rhombic dodecahedron cell model, f while C f and n for the fully flexible tetrakaidecahedron cell model are 0.89 and.98, repectively. Thi i imilar to the value for the emi-empirical model by Gibon et al. [], Triantafillou et al. [8] and Gibon and Ahby [, 0], which had C f and n, but were not derived from any particular cell tructure and trut cro-ection. Hence, the propoed model ha yielded the bai for the value of and n, which the earlier reearcher determined empirically from experimental reult. In contrat with the tiffne of cell without the correction for rigid trut egment, foam tiffne baed on cell with rigid trut egment (Fig. 4.8 and 4.9) cannot be approximated uing a power law equation of the form of Eq. (4.76) becaue introduction of the correction for rigid trut egment to the expreion for C f tiffne reult in an additional term m with different pecific value of m. The value of the tiffne are alo not imilar to thoe of the emi-empirical model by Gibon et al. [], Triantafillou et al. [8], Gibon and Ahby [, 0] if the length of the rigid trut egment i ignificant compared to the length of cell trut (greater than 0. of the trut length). Thi i probably becaue the foam they tudied in deriving their empirical equation have relatively hort rigid trut egment compared to the trut length. 05

130 E /E rie (emi-flexible trut) rie (fully flexible trut) tranvere (emiflexible trut) tranvere (fully flexible trut) / Fig. 4.0 Variation of foam tiffne with relative denity baed on an aniotropic rhombic dodecahedron cell model 0.06 E /E / rie (emi-flexible trut) rie (fully flexible trut) tranvere (emiflexible trut) tranvere (fully flexible trut) Fig. 4. Variation of foam tiffne with relative denity baed on an aniotropic tetrakaidecahedron cell model Fig. 4.0 and 4. how repectively the variation of foam tiffne with relative denity baed on the aniotropic rhombic dodecahedron and the aniotropic tetrakaidecahedron cell model. The aniotropy ratio ( tan ) in thi intance i taken a and a with the iotropic model, the aniotropic model alo yield a higher value of tiffne when the relative denity i higher. The tiffne i greater in the foam rie direction becaue trut in the cell are more aligned in thi direction, making them harder to bend when loaded in thi direction but eaier to bend when loaded 06

131 tranverely. Thee trend are in agreement with the experimental reult decribed in Section E / E cell geometric aniotropy ratio rie (emi-flexible trut) tranvere (emiflexible trut) rie (fully flexible trut) tranvere (fully flexible trut) Fig. 4. Variation of foam tiffne with cell aniotropy baed on a rhombic dodecahedron cell model E / E cell geometric aniotropy ratio rie (emi-flexible trut) tranvere (emiflexible trut) rie (fully flexible trut) tranvere (fully flexible trut) Fig. 4. Variation of foam tiffne with cell aniotropy baed on a tetrakaidecahedron cell model Fig. 4. and 4. how repectively, the variation of tiffne with geometric cell aniotropy ratio for foam with a relative denity , baed on a rhombic dodecahedron and a tetrakaidecahedron cell model. A the geometric aniotropy increae, the tiffnee in the rie and tranvere direction diverge, 07

132 reulting in an increae in the aniotropic tiffne ratio, a hown in Fig. 4.4 and 4.5. Fig. 4. and 4. alo how that the tiffne in the rie direction increae with the geometric aniotropy ratio, but the rate of increae diminihe. In the cae of rhombic dodecahedron cell with emi flexible trut, the tiffne actually decreae after a certain point. The elevation in tiffne i due mainly to the orientation of cell trut, whereby a higher aniotropy ratio implie that trut are more aligned toward the rie direction, making them harder to bend for loading in thi direction. Thi enhancement of tiffne due to trut orientation i however reduced by the reduction in cro-ectional area of the trut due to a higher aniotropy ratio, a implied by Eq. (4.4) and (4.85). The effect of a maller cro-ection ultimately dominate, reulting in the convex curve hown in Fig. 4. and 4.. On the other hand, the tiffne in the tranvere direction decreae with cell geometric aniotropy and thi arie from the orientation of the trut a well a the decreae in the cro-ectional area a the cell geometric aniotropy increae. With a larger geometric aniotropy ratio, trut are more aligned toward the rie direction, making them eaier to bend when the foam i loaded in the tranvere direction, reulting in a lower tiffne. E zz/e yy cell geometric aniotropy ratio fully flexible trut emi-flexible trut GAZT Fig. 4.4 Variation of aniotropy in foam tiffne with cell aniotropy baed on a rhombic dodecahedron cell model 08

133 E zz/e yy cell geometric aniotropy ratio fully flexible trut emi-flexible trut GAZT Fig. 4.5 Variation of aniotropy in foam tiffne with cell aniotropy baed on a tetrakaidecahedron cell model 0 E zz/e yy fully flexible trut emi-flexible trut GAZT / Fig. 4.6 Variation of aniotropy in foam tiffne with relative denity baed on a rhombic dodecahedron cell model 09

134 E zz/e yy fully flexible trut emi-flexible trut GAZT / Fig. 4.7 Variation of aniotropy in foam tiffne with relative denity baed on a tetrakaidecahedron cell model Fig. 4.4 and 4.5 depict repectively the variation of foam tiffne aniotropy ratio with geometric aniotropy for foam with a relative denity baed on a rhombic dodecahedron and a tetrakaidecahedron cell model, while Fig. 4.6 and 4.7 illutrate the variation of aniotropy in tiffne with relative denity for foam with a geometric aniotropy ratio of tan baed on the ame two cell model. The graph indicate that the influence of the relative denity on the tiffne aniotropy ratio i relatively mall compared to that of the cell geometric aniotropy (Fig. 4.5). Thi i imilar to the prediction baed on the cell model by Gibon et al. [], Triantafillou et al. [8], Gibon and Ahby [, 0], marked a GAZT, in the graph. However, the preent tudy give higher value of the tiffne aniotropy ratio due to the difference in cell geometry with the GAZT model. Fig. 4.4 and 4.5 alo how that cell model with and without correction for rigid trut egment predict imilar trend for the tiffne aniotropy ratio. Tenile trength 0

135 The tenile trength of foam baed on the model i governed by failure of the mot vulnerable trut. The foam tenile trength baed on the rhombic dodecahedron cell model, a derived in Section 4., are: For fully flexible trut o Tenile trength in the rie direction σ z max 4 max 4 ( co ) σ ( in ) C C o Tenile trength in the tranvere direction σ y max 8 For emi-flexible trut σ max 4 ( co ) C in co C (4.77) (4.78) o Tenile trength in the rie direction σ z max X max ( co ) 4 σ ( in ) 4 C C o Tenile trength in the tranvere direction (4.79) where σ y max X max 4 ( co ) C 8 σ in co C (4.80) ( ) 4 in co X CC 4 co (4.8) Correpondingly, the expreion for foam tenile trength baed on the tetrakaidecahedron cell model, a derived in Section 4., are: For fully flexible trut

136 o Tenile trength in the rie direction σ z max max co co C σ in o Tenile trength in the tranvere direction σ y max For emi-flexible trut 8 C σ max co in 8 co C ( co ) C (4.8) (4.8) o Tenile trength in the rie direction σ z max max co co C ( ) X o Tenile trength in the tranvere direction σ in 8 C (4.84) where σ y max σ max co in ( ) ( co ) C X 8 co C (4.85) X CC 8in co co Thee expreion have the term and (4.86) in their denominator, and they repreent repectively the effect of axial tretching and bending of trut. For low denity foam, i much larger than indicating that the dominant factor for tenile fracture in the foam i the bending of trut. A with the elatic tiffne, the tenile trength i not influenced by the ize of the cell; it i determined

137 by the tenile trength σ max of the cell trut material, the relative denity geometric aniotropy ratio ( tan ), cell, the hape of the trut cro-ection repreented by C and C, and the length of the rigid trut egment repreented by C σ max /σ max rie (fully flexible trut) tranvere (fully flexible trut) rie (emi-flexible trut) tranvere (emiflexible trut) / Fig. 4.8 Variation of foam tenile trength with relative denity baed on a rhombic dodecahedron cell model σ max /σ max rie (emi-flexible trut) tranvere (emiflexible trut) rie (fully flexible trut) tranvere (fully flexible trut) / Fig. 4.9 Variation of foam tenile trength with relative denity baed on a tetrakaidecahedron cell model Fig. 4.8 and 4.9 how repectively the variation of foam tenile trength with relative denity predicted by an iotropic ( tan ) rhombic dodecahedron and an

138 iotropic tetrakaidecahedron cell model. Coincidence of the trength value in the foam rie and tranvere direction confirm iotropy. The tenile trength of the model incorporating the correction for rigid trut egment i higher than that of the model without thi correction. Thi i becaue the deformable portion of the trut i reduced when their end are conidered rigid, thu changing the poition of the maximum bending moment from the end of the trut to the boundary between the rigid and the flexible portion, hence reulting in a higher trength. Another imilarity between foam trength and tiffne predicted by thee model i that the tenile trength increae with relative denity becaue more material i required to be broken at failure when the denity i higher. The tenile trength σ max of foam baed on the fully flexible trut model can be approximated by σ σ max max C f n (4.87) where σ max i the tenile trength the olid material in the cell trut, and are repectively the denity of the foam and the olid trut material, and C f and n are contant that are governed by the hape of the cell and the trut cro-ection, a well a the mechanim governing the failure (bending and tretching of trut). The value of the contant baed on the fully flexible rhombic dodecahedron cell are C 0.65 and n. 49, while thoe baed on the fully flexible tetrakaidecahedron f cell are C 0. 5 and n Gibon et al. [], Triantafillou et al. [8] and f Gibon and Ahby [, 0] have expreed the tenile platic collape of foam, which i eentially imilar to tenile fracture in thi tudy, uing the ame expreion, but with C f 0. and n. 5 ; however, the value of C f in their model i determined uing empirical data intead of an analytical olution. The value of n baed on the 4

139 three model are cloe to each other becaue the main caue of failure in all the model i common the bending of the trut. A with foam tiffne, the foam trength baed on the cell model with the correction for rigid trut egment cannot be approximated by uch a power law equation becaue the introduction of thi correction make the relationhip between the trength and the denity more complicated becaue of the additional term m which ha different pecific value of m. σ max /σ max rie (emi-flexible trut) rie (fully flexible trut) tranvere (emiflexible trut) tranvere (fully flexible trut) / Fig Variation of foam tenile trength with relative denity baed on a rhombic dodecahedron cell model 5

140 σ max /σ max rie (emi-flexible trut) rie (fully flexible trut) tranvere (emiflexible trut) tranvere (fully flexible trut) / Fig. 4.4 Variation of foam tenile trength with relative denity baed on a rhombic dodecahedron cell model Fig and 4.4 how repectively the variation of foam tenile trength with relative denity baed on a rhombic dodecahedron and a tetrakaidecahedron cell model. The aniotropy ratio ( tan ) i, in thi intance. The tenile trength exhibit a trend imilar to that for foam tiffne (Fig. 4.4 and 4.5), i.e. it increae with foam denity and i higher in the foam rie direction. Thi correlate with the experimental reult decribed in Section.. The tenile trength i lower in the tranvere direction becaue trut are more aligned toward the rie direction, making them eaier to bend for loading in the tranvere direction and hence, eaier to fail. Moreover, more trut need to be broken for loading in the rie direction, reulting in a higher trength in thi direction. 6

141 σ max / σ max rie (fully flexible trut) tranvere (fully flexible trut) rie (emi-flexible trut) tranvere (emiflexible trut) cell geometric aniotropy ratio Fig. 4.4 Variation of foam tenile trength with cell aniotropy baed on a rhombic dodecahedron cell model σ max / σ max cell geometric aniotropy ratio rie (emi-flexible trut) tranvere (emiflexible trut) rie (fully flexible trut) tranvere (fully flexible trut) Fig. 4.4 Variation of foam tenile trength with cell aniotropy baed on a tetrakaidecahedron cell model Fig. 4.4 and 4.4 how repectively the variation of foam tiffne with cell geometric aniotropy ratio for foam with a relative denity , baed on a rhombic dodecahedron and a tetrakaidecahedron cell model. A with tiffne, thee graph illutrate that the tenile trength in the rie and tranvere direction diverge, cauing an increae in the tenile trength aniotropy ratio with geometric aniotropy, a hown in Fig and Another expected imilarity with tiffne i that 7

142 although the trength in the rie direction increae with the geometric aniotropy ratio, the rate of increae diminihe. In the ame way, the trength in the tranvere direction decreae with geometric aniotropy ratio. The reaon for thi are alo the orientation of the trut and the reduction in their cro-ectional area with greater cell aniotropy. The increae in cell geometric aniotropy ratio mean that the trut are more aligned in the rie direction, thu making them harder to bend and tronger when loaded in that direction; however they bend more eaily and are weaker when loaded in the tranvere direction. The reduction in trut cro-ectional area when the cell aniotropy increae decribed by Eq. (4.4) and (4.85), caue the rate of increae in trength in the rie direction to diminih. 7 6 σ zz/σ yy 5 4 fully flexible trut emi-flexible trut GAZT cell geometric aniotropy ratio Fig Variation of foam aniotropy in tenile trength with cell aniotropy baed on a rhombic dodecahedron cell model 8

143 σ zz/σ yy cell geometric aniotropy ratio fully flexible trut emi-flexible trut GAZT Fig Variation of foam aniotropy in tenile trength with cell aniotropy baed on a tetrakaidecahedron cell model 4.5 σ zz/σ yy / fully flexible trut emi-flexible trut GAZT Fig Variation of foam tenile trength aniotropy with relative denity baed on a rhombic dodecahedron cell model 9

144 σ zz/σ yy / fully flexible trut emi-flexible trut GAZT Fig Variation of foam tenile trength aniotropy with relative denity baed on a tetrakaidecahedron cell model Fig and 4.45 how repectively the variation of foam tenile trength aniotropy with geometric aniotropy for a foam with a relative denity 0.046, baed on a rhombic dodecahedron and a tetrakaidecahedron cell model, while Fig and 4.47 how the variation of foam tenile trength aniotropy with relative denity for foam with a cell geometric aniotropy ratio of tan for the ame two cell model. A with tiffne, the graph indicate that the influence of relative denity on the aniotropic tenile trength ratio i relatively low and that aniotropy in foam trength i primarily dependent on cell geometric aniotropy. Gibon et al. [], Triantafillou et al. [8], Gibon and Ahby [, 0] found imilar reult, which are denoted by GAZT in the graph. Fig and 4.45 alo how that cell model with and without the correction for rigid trut egment exhibit imilar behaviour in term of their tenile trength aniotropy ratio. However, the value of the aniotropic tenile trength ratio baed on the rhombic dodecahedron and tetrakaidecahedron cell model are higher than thoe baed on the cubic cell model by Gibon et al. [], Triantafillou et al. [8] and Gibon and Ahby [], becaue different 0

145 aumption were incorporated in determining the failure in the model tenile failure of the model in the preent tudy i aumed to occur when any trut fail, while failure in the GAZT model i aumed to occur at the average tre needed to break all loaded trut. The aumption made in the GAZT model mainly influence the trength in the tranvere direction, whereby two type of trut with different length are deformed (Fig. 4.48). Baed on their aumption, the overall tenile trength in the tranvere direction i the average tre needed to break both type of trut; thi i higher than the tre needed to break the longer trut which are more vulnerable. On the other hand, baed on the aumption in the preent tudy, the tenile trength of the foam i governed by the tre to break the mot vulnerable trut. Thi i more reaonable becaue once the more vulnerable trut break, the overall trength of the foam i decreaed. Fig Open celled cubic model (GAZT) loaded in the tranvere direction Poion ratio The Poion ratio for foam baed on a rhombic dodecahedron cell model can be calculated uing the following expreion: For fully flexible trut υ zx υ zy xx zz ( co ) in in 48co ( co ) in 4in C C (4.88)

146 ( ) ( ) υ yy xx yx C C co 48in co co 48in co co (4.89) ( ) ( ) υ yy zz yz C C co 48in co co 48in co co (4.90) Note that z denote the foam rie (cell elongation) direction For emi-flexible trut ( ) ( ) υ υ zy zx C X X C X X 4in co in 48co co in in (4.9) ( ) ( ) υ yy xx yx C X X C X X co 48in co co 48in co co (4.9) ( ) ( ) υ yy zz yz C X X C X X co 48in co co 48in co co (4.9) where ( ) C C X 4 co co in 4 (4.94) While the Poion ratio for foam baed on a tetrakaidecahedron cell are:

147 For fully flexible trut υ υ zz xx zy zx C C 96in co in 9 co co in (4.95) υ yy xx yx C C co 96in co co co in co 48in co (4.96) υ yy zz yz C C co 96in co co co in co co 96in co co (4.97) For emi-flexible trut ( ) ( ) ( ) ( ) υ υ zz xx zy zx C X X C X X 96in co in 9 co co in (4.98) ( ) ( ) υ yy xx yx C X X X X C X X co 96 in co co co co in co co 48 in co co co (4.99) ( ) ( ) ( ) ( ) υ yy zz yz C X X X X C X X co 96in co co co co in co co co 96in co co (4.00) where C C X co co 8in (4.0)

148 A with foam tiffne and trength, the Poion ratio baed on thee model are not influenced by cell ize. All the expreion for the Poion ratio have two term in their numerator and denominator; the firt term doe not contain the relative denity while the econd term doe. The firt term repreent the influence of the axial force in the trut, while the econd term capture the effect of bending in the trut. For low denity foam, the term i large and hence, the econd term in the numerator and denominator become dominant, howing that bending of the trut i the main mechanim governing the Poion ratio. If the axial force (firt term) i neglected, the expreion implify to: For the rhombic dodecahedron cell model υ zx υ tan zy υ co yx υ co yz For the tetrakaidecahedron cell model (4.0) (4.0) (4.04) υ zx υ zy tan (4.05) υ yx in co (4.06) υ yz tan co (4.07) Eq. (4.0)-(4.07) how that for low denity foam, the main factor determining the Poion ratio i the geometric aniotropy ratio defined by tan ; the denity of the foam, the hape of the trut cro ection, the length of the rigid edge, 4

149 and the propertie of the olid material do not have much influence. Gibon and Ahby [] reported that the Poion ratio of a foam depend on the hape of it contituent cell rather than the propertie of the olid material, but they did not explain how the Poion ratio vary with cell aniotropy. In fact, an analyi baed on their open celled cubic model ugget that the Poion ratio for foam are zero and do not depend on cell aniotropy υ cell geometric aniotropy ratio zy (fully flexible trut) zy (emi-flexible trut) yx (fully flexible trut) yz (fully flexible trut) yx (emi-flexible trut) yz (emi-flexible trut) Fig Variation of Poion' ratio with cell geometric aniotropy ratio for a rhombic dodecahedron cell model υ cell geometric aniotropy ratio zy (fully flexible trut) zy (emi-flexible trut) yz (fully flexible trut) yz (emi-flexible trut) yx (fully flexible trut) yx (emi-flexible trut) Fig Variation of Poion' ratio with cell geometric aniotropy ratio for a tetrakaidecahedron cell model 5

150 Fig and 4.50 how the variation of the Poion ratio with cell geometric aniotropy ratio for a foam with a relative denity of , baed repectively on a rhombic dodecahedron and a tetrakaidecahedron cell model. υ zy ( υ zx ) increae while υ yx and υ yz decreae with aniotropy becaue of the difference in trut orientation length, which can be accounted for a follow: The increae in υ zy ( υ zx ) with aniotropy can be explained by Fig. 4.5, which how the deformation of repeating unit in rhombic dodecahedron and tetrakaidecahedron cell, together with the projection of a deformed cell trut on the yz-plane when the cell i loaded in the z-direction (note that the z-direction correpond to the foam rie/cell elongation direction). Fig. 4.5 how that the ratio between foam deformation in the y-direction to that in the z-direction increae with cell aniotropy, reulting in a larger value of υ zy ( υ zx ). 6

151 D cell deformation rhombic dodecahedron Strut deformation in the yz-plane iotropic aniotropic tetrakaidecahedron Fig. 4.5 Influence of cell aniotropy on υ zy ( υ zx ) The influence of cell aniotropy on υ yx for tetrakaidecahedron and rhombic dodecahedron cell can be explained uing Fig. 4.5 and 4.5, repectively. Fig. 4.5 how the deformation of the trut in a repeating unit of a tetrakaidecahedron cell, with repect to the yz and xy-plane when the cell i loaded in the y-direction. Note that the total deformation of the unit can be conidered a the um of four component deformation of trut OS to OS and OT to OT in the yz-plane, a well a deformation of trut OU to OU and OV to OV in the xy-plane. For an elongated rhombic dodecahedron, the repeating cell unit ha eight trut which are eentially identical, a hown Fig Hence, the deformation of a cell comprie the um of the contribution from thee trut. Fig. 4.5 how that the deformation of each trut can be conidered a um of two 7

152 component one in the yz-plane (for trut OC, deformation in the yz-plane caue point C to move to point C ) and the other in the xy-plane (deformation in the xyplane move point C to C ). Fig. 4.5 and 4.5 how that the total deformation of the cell in the x-direction doe not change with cell aniotropy, while the deformation in the y-direction increae with aniotropy becaue the trut are more aligned toward the z-direction and hence are more compliant when loaded in the y-direction; thi reult in a lower value of υ yx. D cell deformation Strut deformation in the yz and xy-plane iotropic aniotropic Fig. 4.5 Influence of cell aniotropy on υ yx and υ yz for tetrakaidecahedron cell 8

153 D cell deformation Strut deformation in the yz and xy-plane iotropic aniotropic Fig. 4.5 Influence of cell aniotropy on υ yx and υ yz for rhombic dodecahedron cell Fig. 4.5 and 4.5 alo illutrate the influence of cell aniotropy on υ yz. The figure how that cell deformation in the y-direction increae with cell aniotropy becaue the trut become more compliant in that direction. Thu, the ratio between the deformation in the z-direction to that in the y-direction decreae with greater cell aniotropy, cauing a decreae in υ yz. The Poion ratio for model with emi-flexible trut tend to be lower than thoe with fully-flexible trut becaue the axial elongation of trut in the emiflexible trut model contribute proportionally more to the deformation of cell trut. The axial elongation of trut enlarge the foam cell in all direction, thu, decreaing the Poion ratio (Fig. 4.54). On the other hand, bending of the trut increae the cell dimenion in one direction while decreaing it in another, cauing an increae in 9

154 the Poion ratio (Fig. 4.54). Thi i becaue cell deformation aociated with the bending of trut decreae more markedly with a longer rigid trut egment, compared to the reduction in tretching deformation via bending decreae by the cube of the length of the rigid trut egment, while deformation via tretching decreae linearly ee Eq. (4.68)-(4.70) and (4.48)-(4.50); the term that contain the econd moment of area I are aociated with bending while thoe with the croectional area A are linked to tretching. The difference between the emi-flexible trut model and the fully flexible trut model increae with denity, becaue the length of the rigid trut egment in the emi-flexible trut model alo increae with a higher denity, a implied by Eq. (4.7) and (4.60) (ee Fig and 4.56). Thi trend applie to all the Poion ratio except for υ yz for the tetrakaidecahedron cell model, becaue calculation of υ yz for thi model involve two type of trut (Fig. 4.5) and hence the mechanim cannot be imply decribed by Fig In thi cae, there are two competing effect that influence υ yz : Mechanim that decreae υ yz A decribed previouly, the greater contribution of trut elongation in the yz-plane to cell deformation becaue of longer rigid trut egment decreae υ yz. Mechanim that increae υ yz For tetrakaidecahedron cell loaded in the y- direction, deformation in the z-direction arie only from the deformation of trut lying in the yz-plane e.g. trut OS and OT in Fig. 4.5 while deformation in the y-direction i due not only to the deformation of trut lying in the yz-plane, but alo to that of trut in the xy-plane e.g. trut OU and OV in Fig Becaue the trut in the xy-plane for aniotropic tetrakaidecahedron cell are horter than thoe in the yz-plane, the compliance of thee trut in the xy-plane 0

155 decreae more ignificantly when the rigid trut egment are longer, compared to the compliance of the trut in the yz-plane. Thi caue a greater decreae in cell deformation in the y-direction than in the z-direction when the length of the rigid trut egment increae, thu, reulting in an increae in υ yz. Thi effect i more ignificant when the trut in the yz-plane are longer than the one in the xyplane, i.e. when the cell ha a higher geometric aniotropy Since the econd effect i more prominent when the cell geometric aniotropy i higher, υ yz for aniotropic tetrakaidecahedron cell with emi-flexible trut i larger than that for fully flexible trut when the cell geometric aniotropy ratio of the cell exceed a threhold value (e.g.. in Fig. 4.50). a. axial elongation b flexure Fig Influence of axial elongation and flexure of trut on Poion' ratio

156 .5.5 υ / zy (fully flexible trut) zy (emi-flexible trut) yx (fully flexible trut) yz (fully flexible trut) yx (emi-flexible trut) yz (emi-flexible trut) Fig Variation of Poion' ratio with relative denity for a rhombic dodecahedron cell model ( tan ).5.5 υ / zy (fully flexible trut) zy (emi-flexible trut) yz (fully flexible trut) yz (emi-flexible trut) yx (fully flexible trut) yx (emi-flexible trut) Fig Variation of Poion' ratio with relative denity for a tetrakaidecahedron cell model ( tan ) 4.4. Comparion between model and actual foam Fig and 4.58 how repectively comparion between the tiffne of actual foam and that predicted by the rhombic dodecahedron and tetrakaidecahedron cell model. Fig and 4.60 how the normalized verion of Fig and 4.58, whereby the predicted tiffnee are divided by the tiffne of the actual foam. Experimental catter in the tiffne of the actual foam i preented via the I-haped bar. The cell model without the correction for rigid trut egment (fully flexible

157 trut model) predict lower tiffnee than the actual foam. The tetrakaidecahedron cell model with the rigid trut egment (emi-flexible trut model) how fairly good agreement with the actual foam, epecially for foam B. Thi i becaue the hape of the actual cell i more imilar to a tetrakaidecahedron than to a rhombic dodecahedron. Thi alo confirm the ignificance of the rigid trut egment oberved in the deformation of actual foam (ee Chapter ). The difference in tiffne predicted by the tetrakaidecahedron cell model with that of actual foam can be attributed to variation in the tiffne of the olid polyurethane material contituting the trut and membrane in actual foam. Moreover, Gibon and Ahby [] aerted that the tiffne of the olid material in polymeric foam are rarely known with preciion becaue it depend on the degree of polymer-chain alignment, on chemical change brought about by the foaming agent and on gradual ageing and oxidation of the polymer. E (MPa) fully flexible trut emi-flexible trut actual rie tranvere rie tranvere rie tranvere Foam A Foam B Foam C Fig Stiffne of actual foam and that baed on a rhombic dodecahedron cell model

158 rie tranvere rie tranvere rie tranvere E (MPa) fully flexible trut emi-flexible trut actual Foam A Foam B Foam C Fig Stiffne of actual foam and that baed on a tetrakaidecahedron cell model.5 E/E act fully flexible trut emi-flexible trut actual rie tranvere rie tranvere rie tranvere Foam A Foam B Foam C Fig Normalized tiffne of actual foam and that baed on a rhombic dodecahedron cell model 4

159 rie tranvere rie tranvere rie tranvere E/E act fully flexible trut emi-flexible trut actual Foam A Foam B Foam C Fig Normalized tiffne of actual foam and that baed on a tetrakaidecahedron cell model The trength of actual foam and that predicted by the analytical model cannot be compared directly becaue the model aume that the olid material ha linear elatic mechanical and brittle fracture propertie while actual foam exhibit ome nonlinearity before fracture (ee Section.7). The Poion ratio determined from the cell model are not compared becaue uch data i difficult to obtain for actual foam Summary Thi chapter decribed the development of an analytical cell model to define the mechanical behaviour of rigid polyurethane foam under tenile loading. Two open cell model were etablihed and invetigated a rhombic dodecahedron cell and a tetrakaidecahedron cell. Both cell model can be arranged in three-dimenion to fill pace and form a large tructure. An open cell tructure i aumed becaue even though actual rigid polyurethane foam ha primarily cloed cell, the membrane defining the wall are very thin compared to the ize of the trut and vertice (ee Section.5). Hence, the foam behave a if it i open celled; uch a perpective and approach have alo been accepted and adopted by other [, 0]. The mechanical 5

160 propertie of the propoed model tenile tiffne, Poion ratio and tenile trength were derived analytically, reulting in expreion that relate the cell geometric aniotropy ratio, foam relative denity, mechanical propertie of the olid trut material and mechanical propertie of foam to one another. The exitence of rigid trut egment (ee Chapter ) wa incorporated into the model. A comparion of the geometry of the cell model with cell in actual foam how that the tetrakaidecahedron cell ha greater imilarity than the rhombic dodecahedron cell. Thi i probably becaue a tetrakaidecahedron i a more table geometry, baed on atifying minimum urface energy condition a well a energy condition tated in Plateau law [,, 40]. A tudy of the expreion decribing the mechanical propertie of foam how that the bending of trut i the primary mechanim governing deformation and failure in low denity foam. Thi i in line with obervation on the deformation of actual foam dicued in Chapter, and with the finding of other [, 8, 0,, 6, 8,, 40, 4]. The expreion for Poion ratio ugget that for low denity foam, Poion ratio depend primarily on cell geometric aniotropy, indicating that the Poion effect arie from a tructural geometry repone rather than from material behaviour. A expected, the tiffne and trength of foam increae with relative denity, becaue there i more olid material to utain the load. Alo, aniotropy in tiffne and trength increae with cell geometric aniotropy becaue of the change in the orientation of cell trut a the cell become more elongated in the rie direction. For iotropic foam, a comparion between the tiffne predicted by both cell model without correction for rigid trut egment (fully flexible trut) with a emiempirical model baed on a open cubic cell propoed by Gibon et al. [], Triantafillou et al. [8], Gibon and Ahby [, 0] how a trong agreement. Thi 6

161 demontrate that the expreion developed in thi tudy are able to provide an explanation for the numerical contant in the earlier emi-empirical model. The tiffne of foam predicted by the tetrakaidecahedron cell model with correction for rigid trut egment exhibit good agreement with actual foam. Thi i probably becaue the hape of actual foam cell are imilar to a tetrakaidecahedron cell. Moreover, the reult how that incluion of the correction for rigid trut egment i important in arriving at good correlation with experimental reult. The foam trength predicted by the two model developed in thi tudy were not compared with that of actual foam becaue the aumption of linear elatic-brittle failure material for the cell trut i not totally applicable, a the polyurethane in actual foam exhibit ome nonlinearity in it tre-train repone. The cell model in thi invetigation are able to decribe the mechanical behaviour of actual foam, epecially in term of the tiffne; moreover the idealized cell geometrie are reaonably realitic becaue they are imilar to thoe of cell in actual foam and the cell can be aembled together in three dimenion. However, thee analytical model have only been utilized to decribe the mechanical propertie for two direction, i.e. the foam rie and tranvere direction. The analytical equation defining the mechanical propertie of thee model derived are not amenable to decribing the mechanical repone of foam when loaded in other direction or loaded multi-axially a thee ituation are too complex. The tenile trength criterion etablihed i alo pecific to linear elatic-brittle failure material and not fully applicable to the rigid polyurethane foam teted. Neverthele, the analyi i till ueful in demontrating how the mechanical propertie of foam relate to parameter uch a foam denity, cell aniotropy and the trength and tiffne of the trut material. Moreover, the expreion can be ued to provide an etimate of the 7

162 mechanical propertie of foam. Thee geometric cell model can be utilized to examine loading in direction other than the foam rie and tranvere direction, to tudy multi-axial loading and cell made of non-linear material. Thee can be effected via finite element modelling and thi i dicued in the following chapter. 8

163 Chapter 5 Finite Element Model To augment the theoretical analyi decribed in Chapter 4, finite element modelling of a cellular tructure baed on tetrakaidecahedral cell wa undertaken. The oftware ABAQUS-Explicit wa ued to facilitate a tudy of the effect of loading direction, the incorporation of non-linear mechanical behaviour of the olid cell material, a well a random variation in cell parameter. The FEM reult are compared to thoe from experiment on actual foam, preented in Chapter. 5. Modelling of cell Cellular tructure compriing elongated tetrakaidecahedron open cell were modelled uing linear Timohenko beam element (type B) available in the explicit verion of ABAQUS. Thi verion of the code for tranient dynamic analyi wa employed although the imulation wa for quai-tatic loading, becaue ABAQUS- Standard, which i normally ued to imulate quai-tatic ituation, doe not accommodate material failure. The FE model conit of elongated tetrakaidecahedron cell packed together in an elongated BCC lattice (Fig. 5.) and ubjected to tenion in five direction rie (0 o ), tranvere (90 o ), a well a 0 o, 45 o and 60 o to the foam rie direction for comparion with the experimental reult in Section.. Thee cell model were formulated to correpond to the three foam denitie of the pecimen decribed in Section.. Value for the parameter in thee model are preented in Table 5.. Note that the trut cro-ectional dimenion (R) baed on thi model i not exactly the ame a that of the actual foam dicued in Section.5 becaue the hape of the cell in the actual foam are not exactly identical to tetrakaidecahedra and the cell ize vary. 9

164 Fig. 5. Elongated tetrakaidecahedron cell packed together in an elongated BCC lattice Table 5. Value of parameter in finite element cell model Foam A Foam B Foam C Relative denity Cell ize mm mm mm Geometric cell aniotropy ratio.5.7 ( tan ) Strut cro ectional area ( A ) mm mm mm nd area moment of inertia of I mm 4 mm 4 mm 4 trut cro-ection ( ) Strut thickne ( R )(ee Fig..4) 5 μ m 46 μ m 5 μ m The cro-ection of the beam element defining the flexible egment of the trut i a twelve pointed tar compriing two component, a hown in Fig. 5.. It ha a total area A, a econd moment of area I and the average ditance R from it centroid to the point farthet away in it cro-ection (ee Section 4.) which are equal to that for the Plateau border cro-ection calculated (Table 5.). A Plateau border wa not employed a the cro-ection becaue it only ha three axe of ymmetry; the beam element would have been enitive to orientation if they were defined uing thi 40

165 hape, epecially with regard to fracture, thu making it neceary to define the orientation of each element. On the other hand, the twelve pointed tar ha axe of ymmetry and hence i much le enitive to orientation. Conequently, the croectional orientation of every trut doe not have to be defined. Simpler cro-ection uch a a circle and an annulu cannot yield the ame cro-ectional area A and econd moment of area I imultaneouly a the Plateau border. The rigid egment at the trut end have an annular circular cro-ection with a much higher area and econd moment of area than the flexible egment; thi yield a higher rigidity. Five B beam element were ued to model the flexible egment of each trut. A larger number of element wa avoided to limit the cot of the imulation in term of running time and computing reource. On the other hand, only one element wa ued for each rigid trut egment becaue it deformation i conidered inignificant. Fig. 5. how the element in a tetrakaidecahedral cell model Fig. 5. Element a tetrakaidecahedral cell model 4

166 Fig. 5. Star hape for beam cro ection An iotropic material with a bilinear tre-train relationhip wa elected for the trut. Fracture wa defined uing a hear failure option in ABAQUS-Explicit, whereby fracture i aumed to occur when the platic train at an element integration point attain a critical value (note that a fracture criterion baed on tre and train yield identical reult becaue the tre tate i uniaxial). The value of the parameter in the material model were derived from tet on olid polyurethane pecimen (Section.7), i.e.: Stiffne E GPa R Denity 00 kg m Yield trength σ 5 MPa y Tenile fracture trength σ max 64 MPa Platic train at fracture pl max 0. 0 Loading of the cell aembly wa precribed through boundary condition defining a imple upport at one ide and impoing a contant velocity condition at the other (Fig. 5.5). Periodic boundary condition were impoed on the model for loading in the rie and tranvere direction by precribing rotational contraint at the 4

167 boundarie. Thi make the analyi eentially the ame a modelling an aembly of infinite number of cell. Strain rate were limited to a maximum of 60/ to minimize non-uniformity of tre due to tre wave propagation within the tructure. Thi facilitate approximation of quai-tatic loading condition (train rate which are too low were alo avoided to limit computational time). A localied area of weakne wa introduced into the model to initiate failure. Thi wa done by lowering the tenile trength of trut in a cell at the edge of the model (Fig. 5.4). localied weakne Fig. 5.4 Localied area of weakne in a finite element model 0 o 0 o 45 o 60 o 90 o Fig. 5.5 Loading condition in the finite element model 4

168 5. Reult and dicuion 5.. Repone to tenile loading σ (MPa) o 0 90 o Fig. 5.6 Stre-train curve for foam B ( 9.5kg m ; geometric aniotropy ratio ) Fig. 5.6 how the tre-train curve for the model correponding to foam B ( 9.5kg m ) loaded in the rie (0 o ) and tranvere (90 o ) direction. The graph how that a with the actual foam (Section.), the tre-train curve exhibit a linear repone followed by a horter phae of non-linearity before fracture. The figure alo how that the tiffne (i.e. gradient of the initial linear portion) and the tenile trength (maximum tre before fracture) are higher in the rie direction. Unloaded, D view Unloaded, iometric view Failure initiation, D view Failure initiation, iometric view Fig. 5.7 Crack pattern for tenion in the cell elongation/rie direction 44

169 Unloaded, D view Unloaded, iometric view Failure initiation, D view Failure initiation, iometric view Fig. 5.8 Crack pattern for tenion in the tranvere direction Fig. 5.7 and 5.8 how repectively, the repone of the model correponding to foam B for loading in the rie/elongation and tranvere direction. The figure illutrate that the crack for loading in the tranvere direction follow a lanted/zigzag pattern. Thi contrat with the crack propagation in actual foam, whereby a crack propagate perpendicular to the direction of loading (ee Fig..7,.9 and. in Section.6.). Thi phenomenon and it implication, particularly with regard to the fracture characteritic of the model, will be dicued in the next ection. 5.. Influence of cell wall membrane on crack propagation Initial imulation reult baed on the propoed model (Fig. 5.6) howed an overetimation of the ratio of the tenile trength in the rie direction to that in the tranvere direction for foam B ( 9.5kg m ; geometric aniotropy ratio ); the predicted aniotropy ratio wa 4.05, while that of the actual foam wa.95. It i alo noted that the platic collape model for foam by Gibon et al. [], Triantafillou et al. [8], and Gibon and Ahby [, 0], which i eentially imilar to the fracture model in thi tudy, alo overetimate thi aniotropy ratio with a value of.6. An examination of how fracture develop in the model indicate that the behaviour differ omewhat from that in actual foam. In actual foam, the crack propagate equentially through cell wall membrane and trut, reulting in crack propagation that i perpendicular to the loading direction (ee Fig..7,.9 and. in Section 45

170 .6.), while fracture predicted by the model occur by failure of the mot highly loaded trut without any accompanying crack propagation through membrane, reulting in a zigzag or lanted crack pattern, particularly for loading in the tranvere direction (Fig. 5.7 and 5.8). In actuality, a crack which i initiated in a membrane facilitate continued crack propagation perpendicular to the direction of loading, thu driving the crack tip to the centre of the trut the membrane i connected to. On the other hand, cracking in the FEM model follow any direction that lead to vulnerable trut, i.e. trut that attain the highet tre. For loading in the tranvere direction, two type of trut are loaded trut lying in horizontal plane (plane that are parallel to the xy-plane) and trut lying in vertical plane (plane that are parallel to the yz-plane). Thee two type of trut have different length and orientation and thu they attain different tre level when the foam i loaded in the tranvere direction. According to the analyi in Section 4.., the trut lying in the vertical plane attain a higher maximum tre than thoe lying in the horizontal plane and thu the trut in the vertical plane fail earlier i.e. they are more vulnerable (ee Fig. 5.9). Thi reult in crack propagation that follow a zigzag or lanted pattern, cutting the trut lying in the vertical plane (Fig. 5.8). On the other hand, for loading in the rie direction, the trut that deform are of the ame type and hence have the ame vulnerability (Fig. 5.0). FEM imulation were carried out uing ABAQUS-Explicit to examine the influence of the preence of membrane on the direction of crack propagation. A ingle tetrakaidecahedron cell with beam element (type B) defining it edge and membrane element (type S4R) for it wall wa modelled to invetigate loading in the rie and tranvere direction. The ize of the cell wa mm x 0.5 mm. Each trut wa defined uing 0 beam element and the total number of membrane element wa 46

171 8800. The cro-ection of the beam element i aumed to be circular with a radiu of 0. mm and the cell wall membrane were 0.00 mm thick. A linear elatic-brittle failure material model wa aumed for the trut and cell wall membrane, and the failure of the element wa defined uing a tre failure criterion whereby an element i aumed to fail whenever the hydrotatic tre value at it integration point attain a critical value. The following parameter value were ued to define the material in the trut and wall. Stiffne E 000 GPa Denity 000 kg m Tenile fracture trength σ max 0. MPa Loading of the cell aembly wa precribed through boundary condition by defining a imple upport at one ide and impoing a contant velocity condition at the other. Fig. 5. and 5. illutrate repectively, imulation reult for loading in the rie and tranvere direction. They how that crack propagation through the membrane tend to contrain the crack to cut trut that are perpendicular to loading direction, although thee trut are le vulnerable according to the analyi in Section 4.. and 4... Fig. 5. how that for loading in the tranvere direction, crack initiate from the lanted trut but continue to cut the trut lying in the horizontal plane (i.e. plane parallel to the xy-plane) reulting in a crack perpendicular to the loading direction. (Note that imulation for loading in the tranvere direction were performed in two tep the firt tep reulted in initiating failure of the membrane parallel to the rie direction, thi wa then followed by the econd tep whereby thi membrane wa deleted and failure in a trut wa triggered by deleting an element at it centre, where the tip of the crack from the membrane met 47

172 the trut.). Thi indicate that the influence of crack propagation in the membrane ha to be incorporated in the FEM model for foam. In thi tudy, thi i achieved by auming that the trength of the trut that are broken during crack propagation i reduced by the crack that propagate through the membrane. Thu, the following aumption are made. Loading in the tranvere direction the crack that propagate through the membrane tend to cut the trut lying in horizontal plane (Fig. 5.); thu the trength of thee trut i reduced o that they fail before the inclined trut do. From the analytical olution dicued in Section 4., the trut trength hould be reduced by at leat about tan (ee Eq. (4.4)) i.e. the trength hould be made maller than the original trength divided by the cell geometric aniotropy ratio, to match the vulnerability of the inclined trut. Conequently, the tenile trength and yield trength of the material in trut lying in the horizontal plane are reduced by. (Note that failure of a trut occur whenever an tan integration point in any element attain a critical platic train value correponding to the tenile trength). Loading in the rie (cell elongation) direction reult from the ingle cell FEM analyi how that the crack initiated in a membrane i likely to propagate to and cut the inclined trut. Thu, to be conitent with the modification for loading in the tranvere direction, the trength of thee trut i reduced by lowering both the tenile and yield trength of the material by. tan Loading 0 o, 45 o, and 60 o to the rie direction the crack i aumed to cut the inclined trut whenever the crack propagation perpendicular to the loading 48

173 direction, doe not require it to cut trut which are parallel to the xy-plane (ee Fig. 5.4). Thi reult in crack propagating through the inclined trut for all the loading direction conidered in thi tudy except for loading in the tranvere direction. Hence, a with loading in the rie direction, the trength of the inclined trut for loading 0 o, 45 o, and 60 o to the rie direction i alo reduced by. FEM imulation were alo performed to examine thi aumption. A tan ingle tetrakaidecahedron cell, imilar to the one ued to examine loading in the rie and tranvere direction, wa tretched at 0 o, 45 o, 60 o, and 8.5 o to the rie direction. Loading wa impoed through a contant velocity boundary condition at the cell edge, o that the deformation reemble uniform extenion in the loading direction. Fig how the reult of the imulation and confirm the aumption made. Thee figure indicate that, for cell correponding to foam B (cell aniotropy ratio ) loaded 0 o, 45 o, and 60 o to the rie direction, the crack propagate through the inclined trut. On the other hand, for loading 8.5 o to the rie direction, the crack firt interect the inclined trut and then propagate through the membrane to meet the trut in the horizontal plane. Fig. 5.9 Cell model loaded in the tranvere (y) direction 49

174 Fig. 5.0 Cell model loaded in the rie (z) direction z y Fig. 5. Single cell loaded in the cell elongation (foam rie) direction 50

175 z y Fig. 5. Single cell loaded in the tranvere direction Fig. 5. Strut in a tetrakaidecahedron cell 5

176 Fig. 5.4 Crack propagation for loading in the 0 o, 45 o, 60 o, and 8.5 o direction z y Fig. 5.5 Single cell loaded 0 o to the cell elongation (foam rie) direction z y Fig. 5.6 Single cell loaded 45 o to the cell elongation (foam rie) direction 5

177 z y Fig. 5.7 Single cell loaded 60 o to the cell elongation (foam rie) direction z y Fig. 5.8 Single cell loaded 8.5 o to the cell elongation (foam rie) direction 5

178 5.. Repone to tenile loading after modification Fig how FE imulation reult for cell aemblie correponding to foam A, B, and C; they depict the foam tructure initially load-free, then undergoing loading. The figure how that for loading in the tranvere direction, the crack propagate in a direction approximately perpendicular to the line of load application, confirming that the correction incorporated to account for the influence of crack propagation in membrane reult in correlation with experimental reult (Fig..7,.9 and.). For loading in the rie direction a well a 0 o, 45 o, and 60 o to thi direction, the crack propagate generally perpendicular to the load direction with minor deviation in the middle The FEM imulation are employed to generate tre-train relationhip for loading in the variou direction; Fig how predicted tre-train curve correponding to foam A, B and C. They exhibit an initial linear repone followed by a horter non-linear phae before fracture. Thi i conitent with reult from tet on actual foam preented in Chapter. A with the experimental data, the tiffne and trength predicted by FEM modelling are highet in the rie direction and decreae a the angle between the loading and rie direction increae. There are mall ocillation in the tre-train curve, epecially for loading in the 0 o, 45 o, and 60 o direction, becaue of premature failure in ome trut at the loading boundarie due to variation in the load ditribution among the trut. Thee ocillation appear noticeable becaue the number of cell in the model i relatively mall and hence, the failure of individual trut caue ocillation in the tre-train curve. The tretrain curve for loading in the rie and tranvere direction are much moother becaue the load are uniformly ditributed among the trut at the loading boundarie and hence, early failure of trut at thee edge doe not occur. Note that the 54

179 boundarie where the load are applied for loading in the rie and tranvere direction conit of repeating cell urface which are regular, and thu the velocity boundary condition applied yield an evenly ditributed load among the trut in that area. Stiffne and tenile trength value were derived from the tre-train curve the tiffne wa obtained from the gradient of the initial linear portion of the curve, while the trength correpond to the maximum tre attained before failure. Fig how comparion between tiffne value from the FEM model with experimental data for foam A, B, and C. There i reaonably good agreement for foam B and C, but noticeable dicrepancy for foam A, which exhibit a higher tiffne than the FEM model. Thi i probably becaue the tiffne of the cell trut material of foam A i higher than that in the model (the tiffne value in the model wa baed on tet on olid polyurethane ample, a dicued in Section.7). Thi might be becaue of a greater degree of polymer molecular chain alignment caued by cell tretching during the foaming proce, a well a chemical change due to the effect of a high gaing agent content []. Fig. 5.8 how a comparion of the foam tiffne predicted by FEM and by the analytical olution; reaonable agreement i oberved. The tiffne predicted by FEM imulation tend to be lower becaue the rigid trut egment in the FEM model are not fully rigid; they were imulated by making the cro-ectional area and the econd moment of area much larger than the Plateau border. Moreover, a localied area of weakne wa introduced in the FEM model by reducing the trength of ome trut in a cell at the edge of the foam model. Thee trut thu yielded earlier than the other, decreaing the overall tiffne of the model. 55

180 Unloaded, D view Unloaded, iometric view Failure initiation, D view Failure initiation, iometric view 0 o 0 o 45 o 60 o 90 o Fig. 5.9 FEM imulation reult for foam A (.kg ratio.5) m ; geometric aniotropy 56

181 Unloaded, D view Unloaded, iometric view Failure initiation, D view Failure initiation, iometric view 0 o 0 o 45 o 60 o 90 o Fig. 5.0 FEM imulation reult for foam B ( 9.5kg ratio ) m ; geometric aniotropy 57

182 Unloaded, D view Unloaded, iometric view Failure initiation, D view Failure initiation, iometric view 0 o 0 o 45 o 60 o 90 o Fig. 5. FEM imulation reult for foam C ( 5.kg ratio.7) m ; geometric aniotropy 58

183 σ (MPa) o 0 o 0 o 45 o 60 o Fig. 5. Stre-train curve for foam A (.kg ratio.5) m ; geometric aniotropy σ (MPa) o 0 o 0 o 45 o 60 o 90 Fig. 5. Stre-train curve for foam B ( 9.5kg ) m ; geometric aniotropy ratio 59

184 σ (MPa) o 0 o 0 o 45 o 60 o Fig. 5.4 Stre-train curve for foam C ( 5.kg.7) m ; geometric aniotropy ratio 4 0 E (MPa) direction (deg) emi flexible trut model actual Fig. 5.5 Stiffne of foam A (.kg m ; geometric aniotropy ratio.5) 60

185 E (MPa) direction (deg) emi flexible trut model actual Fig. 5.6 Stiffne of foam B ( 9.5kg m ; geometric aniotropy ratio ) E (MPa) direction (deg) emi flexible trut model actual Fig. 5.7 Stiffne of foam C ( 5.kg m ; geometric aniotropy ratio.7) 6

186 rie tranvere rie tranvere rie tranvere E (MPa) Analytical FEM Foam A Foam B Foam C Fig. 5.8 Comparrion between tiffne predicted by FEM and analytical model Fig how comparion between the tenile trength predicted by the FEM model with actual value for foam A, B, and C. The model underetimate the actual tenile trength, probably becaue the yield and tenile trength data obtained from tet on olid polyurethane ample (Section.7) do not reflect fully the actual material in the cell trut becaue: The olid polyurethane pecimen teted (Chapter ) contain tiny bubble that reduce the actual cro-ectional area and generate tre concentration; hence, the meaured trength i lower. The tretching of polyurethane during the foaming proce might align the molecular polymer chain preferentially, reulting in a higher trength compared to the olid polyurethane pecimen, in which foaming wa prevented. The gaing agent ued to enhance foaming might have affected the property of the foam []. Neverthele, when comparion are made in term of the normalized tenile trength, i.e. the tenile trength divided by the trength in the rie direction, agreement between the model and actual foam i quite good, a hown in Fig

187 5.4. Thi demontrate that the adjutment of the trength of cell trut to take into account the influence of crack propagation in cell wall i valid and yield a cloer correlation in term of the aniotropic tenile trength ratio. Thi alo indicate that although the thin cell wall membrane do not have much effect on foam tiffne [6, 0], they play an important part in determining it fracture characteritic by influencing the direction of fracture propagation. In contrat with actual foam, whereby the tenile trength decreae with angle between the loading and the rie direction, the tenile trength predicted by the FEM model for loading 60 o to the rie direction i lightly lower than that in the tranvere direction (Fig ). Thi i becaue different trut fail for loading in the two direction due to the adjutment of trut trength to incorporate the effect of crack propagation in cell wall for loading 60 o to the rie direction, the inclined trut in the FEM model fail, wherea for loading in the tranvere direction, trut lying parallel to the horizontal xy-plane fail. Actual foam do not have cell with geometrie exactly identical to the idealized FEM model and hence there i no abrupt change in the trut that fail; thu, the crack propagation direction and the tenile trength change gradually with loading direction. σ max (MPa) direction (deg) emi flexible trut model actual Fig. 5.9 Tenile trength for foam A (.kg m ; geometric aniotropy ratio.5) 6

188 σ max (MPa) direction (deg) emi flexible trut model actual Fig. 5.0 Tenile trength for foam B ( 9.5kg m ; geometric aniotropy ratio ) σ max (MPa) direction (deg) emi flexible trut model actual Fig. 5. Tenile trength for foam C ( 5.kg m ; geometric aniotropy ratio.7) 64

189 . σ max / σ z max direction (deg) emi flexible trut model actual Fig. 5. Normalized tenile trength for foam A (.kg aniotropy ratio.5) m ; geometric.4. σ max / σ z max direction (deg) emi flexible trut model actual Fig. 5. Normalized tenile trength for foam B ( 9.5kg aniotropy ratio ) m ; geometric 65

190 . σ max / σ z max direction (deg) emi flexible trut model actual Fig. 5.4 Normalized tenile trength for foam C ( 5.kg aniotropy ratio.7) m ; geometric 5..4 Influence of randomne in cell geometric aniotropy and hape The influence of randomne in cell parameter on the mechanical behaviour of a tetrakaidecahedron cell aembly wa examined uing finite element imulation. Two mode of randomization were employed, i.e. Variation in cell geometric aniotropy the four trut forming the edge of quare at the top and bottom of each tetrakaidecahedron cell were randomly moved in the rie (z-) direction, reulting in variation in cell geometric aniotropy of individual cell. Thee adjutment were limited o that the modified trut poition do not reult in interpenetration between trut and the general hape of the tetrakaidecahedron i maintained. The following procedure wa employed: o The coordinate of the node at the corner of the quare were modified uing the following expreion: ( h, 0. h) znew zold rand (5.) rand a, b i a random number that i evenly ditributed between a where ( ) and b, while h i the height of the original cell. 66

191 o The coordinate of the other node were adjuted uch that each trut remain traight and the length of the rigid egment i the ame a that before the randomization proce. Fig. 5.5 how a cell aembly incorporating thi variation in cell geometric aniotropy. To facilitate the application of loading, the top and bottom urface of the model were not adjuted. Variation in the interconnection between trut the poition of each vertex or corner of a cell, where the trut are interconnected, wa randomly adjuted in three dimenion. Again, thee adjutment were limited o that the modified trut poition do not reult in interpenetration between trut and the general hape of the tetrakaidecahedron i maintained. The following procedure wa employed: o The coordinate of the node at the vertice were modified via the following expreion: ( x, 0. x ) xnew xold rand 0.5 flex 75 flex (5.) ynew yold rand( 0.5 y flex, y flex ) (5.) znew zold rand( 0.5 z flex, z flex ) (5.4) rand a, b i a random number evenly ditributed between a where again, ( ) and b, while x flex, y flex, and z flex are the length of the flexible egment of the inclined trut in the original model projected onto the x, y, and z axe, repectively. o The coordinate of the other node were adjuted uch that each trut i traight and the length of the rigid egment remain the ame a that before the randomization proce. 67

192 A with the variation in cell geometric aniotropy ratio, no adjutment were made at the outer urface of the cell aembly. Fig. 5.6 how a cellular tructure correponding to thi mode of cell randomization. Thee two mode of randomization were applied to the cell model correponding to foam B ( 9.5kg m ). Fig. 5.7 how the repone of the model with random cell variation, for loading in the rie and tranvere direction. The direction of crack propagation i approximately perpendicular to the loading direction and doe not exhibit any obviou zigzag path in the middle. The modified model exhibit better reemblance with the behaviour of actual foam compared to the original model which how ome unevenne in the fracture line. Thi cloer correlation i due to randomne in the trut length which caue variation in their trength. Thu the crack i not contrained to run at pecific angle in order to cut particular trut, but can caue failure in any trut that i oriented cloe to the loading direction, reulting in a fracture line perpendicular to the loading direction. Fig. 5.5 Model with random variation in cell geometric aniotropy ratio 68

193 Fig. 5.6 Model with random variation in cell vertex location 69

194 Unloaded, D view Unloaded, iometric view Failure initiation, D view a. Model with variation in geometric aniotropy (loading in the rie direction) Failure initiation, iometric view b. Model with variation in cell vertex location (loading in the rie direction) c. Model with variation in geometric aniotropy (loading in the tranvere direction) d. Model with variation in cell vertex location (loading in the tranvere direction) Fig. 5.7 Random cell model for loading in the rie and tranvere direction Fig. 5.8 and 5.9 how repectively comparion between the original model and the one with cell variation, in term of the tre-train repone for loading in the rie and tranvere direction. The figure how that the tiffne defined by the initial lope for loading in each direction remain approximately the ame, but the tenile trength of the model with random cell variation tend to be lower than that 70

195 of the original model; Fig and 5.4 how repectively comparion between the model in term of their tiffne and tenile trength. The reult indicate that the overall foam tiffne i independent of randomne in cell; however, foam trength i affected by uch randomne. Thi i becaue the length and orientation of the trut in cell with random variation poe different vulnerabilitie and hence, failure i determined by the weaker trut. Thi caue the overall tenile trength to be lower than that of the model with uniform cell. 0. σ (MPa) uniform cell variation in cell aniotropy variation in cell vertex poition Fig. 5.8 Stre-train curve for uniform and random cell model for loading in the rie direction σ (MPa) uniform cell variation in cell aniotropy variation in cell vertex poition Fig. 5.9 Stre-train curve for uniform and random cell model for loading in the tranvere direction 7

196 4 E (MPa) uniform cell variation in cell aniotropy random variation in cell vertex poition 0 rie tranvere Fig Elatic tiffne of uniform and random cell model σ max (MPa) uniform cell variation in cell aniotropy random variation in cell vertex poition 0 rie tranvere Fig. 5.4 Tenile trength of uniform and random cell model 5. Summary Finite element imulation of foam behaviour baed on tetrakaidecahedron cell were undertaken to examine foam tiffne and tenile trength for comparion with value from experimental tet. The modelling of cell wa baed on beam element available in ABAQUS-Explicit and material propertie derived from actual foam (Chapter ). The model wa loaded in the rie (0 o ) and tranvere (90 o ) direction, a well a 0 o, 45 o and 60 o to the rie direction. Model that do not conider the effect of crack propagation in cell membrane overetimate the trength aniotropy ratio of 7

197 actual foam. Subequently, thi influence wa incorporated by reducing the trength of trut at location where the crack in the membrane meet them. The tre-train characteritic predicted by the model are imilar to thoe of actual foam: There i an initial linear elatic repone followed by ome nonlinearity before fracture. The tiffne and trength are highet in the foam rie (cell elongation) direction and decreae with angle between the loading and the foam rie direction. Foam tiffne predicted by the model diplay generally good agreement with actual foam depite ome dicrepancie which arie becaue the tiffne of the trut material in the model may not be identical to that of the polyurethane in the trut and wall of actual foam cell. Although the model underetimate the tenile trength, there i good agreement in term of value normalized with repect to the trength in the foam rie direction. A with the tiffne, the underetimation of foam tenile trength might be becaue the yield and tenile trength defined in the model are not exactly the ame a thoe of the polyurethane in actual foam cell. Agreement between the FEM model and the actual foam in term of their normalized tenile trength how that although cell membrane do not have much influence on the tiffne of cloed cell foam with very thin cell wall uch a rigid polyurethane foam [6, 0], they are ignificant in determining the tenile trength by influencing the direction of crack propagation. An examination of the influence of randomne in cell parameter on tiffne and tenile trength indicate that crack propagation in FEM model with ome degree of randomne in cell reemble that in actual foam more cloely, whereby a crack propagate along a line perpendicular to the loading direction without any ignificant 7

198 deviation in path. Thi i becaue model with random cell have trut with a range of vulnerabilitie to failure, thu the crack i not contrained to cut particular trut, but can caue failure in any trut that i oriented cloe to the loading direction, reulting in a fracture line perpendicular to the loading direction. Comparion between the tiffne and tenile trength predicted by the uniform and random cell model how that the effect of variation in cell geometry on tiffne i negligible, but caue a noticeable decreae in foam trength. Thi i becaue tenile trength i governed by the weaker trut and the random cell model have a range of trut vulnerabilitie compared to the uniform propertie of the model with identical cell. 74

199 Chapter 6 for future work Concluion and Recommendation Thi tudy ha focued on identifying and explaining the mechanical propertie of rigid polyurethane foam under tenile loading, through an experimental invetigation and the development of geometric cell model. Quai-tatic and dynamic tenile loading in variou direction on foam ample were undertaken to examine the influence of train rate and loading direction on the mechanical propertie of the foam tudied. Obervation uing optical microcopy and micro-ct canning were undertaken to examine the tructure of foam cell, a well a their behaviour under tenile and compreive load. The experimental invetigation wa accompanied by development of geometric model for the cell within the foam and thee were tudied both analytically and via finite element imulation. 6. Concluion Experimental tet and obervation of the repone of polyurethane foam pecimen to tenile loading applied at variou angle relative to the foam rie (cell elongation) direction were conducted to identify the overall mechanical propertie a well a the behaviour of the cell within them. The reult have yielded a fuller undertanding of the tructure of polyurethane foam and and it behaviour. The primary finding are a follow: The tenile trength and tiffne of the rigid polyurethane foam tudied are highet in the foam rie (cell elongation) direction and decreae with angle between the loading and foam rie direction, indicating that the mechanical propertie are aniotropic becaue of aniotropy in the cell geometry. 75

200 The primary mechanim governing foam deformation i the bending of cell trut and wall. A ignificant obervation i that a hort egment at each end of a trut, where it i connected to neighbouring trut, eentially doe not deform. Thi i becaue the trut cro-ection there i larger and the interconnection with adjacent trut and cell membrane at thee cell vertice retrict movement. The tudy ha hown that thi feature mut be conidered in developing geometric cell model for foam. Micro CT-can image reveal that the cell in rigid polyurethane foam bear a reaonable reemblance with an elongated tetrakaidecahedron, while the croection of cell trut ha a geometry imilar to that of a Plateau border []. Two analytical model baed on elongated rhombic dodecahedral and elongated tetrakaidecahedral cell were formulated to facilitate analyi of the mechanical propertie of foam. The tiffne, tenile trength, and Poion ratio in the rie (cell elongation) and tranvere direction were calculated and compared to actual foam. The preence of rigid trut egment near their end, a oberved in actual foam pecimen, wa taken into account in the analyi and hown to influence the reult predicted by the model. A parametric tudy of the mechanical propertie of foam baed on the analytical model how that: The elatic tiffne and trength of foam are not influenced by cell ize; they are governed by denity, geometric aniotropy of the cell, cell hape, trut croection geometry, a well a the length of the rigid trut egment. The bending of cell trut i the primary deformation mechanim in foam. Tenile trength and tiffne increae with denity. Aniotropy of mechanical propertie tenile trength and tiffne increae with aniotropy in the geometry of the contituent cell. 76

201 The Poion ratio of a cellular tructure i governed by the geometry of the internal tructure and doe not depend on the material it i made of. A comparion between the cell model and actual foam how that an elongated tetrakaidecahedron reemble the cell in actual foam more cloely than a rhombic dodecahedron. Moreover, the prediction of foam tiffne baed on elongated tetrakaidecahedron cell incorporating the correction for rigid trut egment agree well with experimental reult from tet on the foam tudied. Thi how that imilarity between the idealized cell model geometry and the geometry of actual foam cell determine the effectivene of a model in predicting actual foam behaviour. The analytical model enable identification of the influence of everal parameter overall foam denity, cell aniotropy, tiffne and trength of the olid material in cell trut and wall on the overall mechanical behaviour of a foam. Moreover, the model erve a a mean to etimate mechanical property value of foam Finite element imulation of foam behaviour baed on elongated tetrakaidecahedron cell, were alo undertaken to examine the mechanical propertie for loading direction that were not amenable to analytical olution, and to facilitate incorporation of non-linear material propertie and variation in cell geometry. The imulation yielded the following finding: Although cell membrane have little influence on the tiffne of cloed cell foam with very thin cell wall, they play an important part in determining the tenile trength by influencing the direction of crack propagation. Thi mut be taken into account in utilizing idealized cell geometrie to model foam. The tiffne predicted by the tetrakaidecahedron cell model agree quite well with value from tet on actual foam. Difference occur becaue tiffne value 77

202 obtained from tenile tet on olid polyurethane may not be identical to that of the polyurethane in the cell trut and wall of the foam pecimen. The model underetimate the actual tenile trength for the different direction of loading; however, agreement i good when normalized tenile trength i conidered (i.e. tenile trength normalized with repect to the tenile trength in the foam rie direction). A with foam tiffne, thi underetimation i probably becaue the yield and tenile trength data obtained from tet on olid polyurethane pecimen may not be exactly the ame a that of the polyurethane in the actual foam. Cloe correlation between the model and actual foam behaviour in term of normalized tenile trength how that incorporation of the influence of crack propagation in cell wall membrane i jutified and neceary. FEM modelling demontrate that random variation in the geometry of individual cell do not exert a ignificant influence on the overall tiffne of foam, but decreae the predicted tenile trength. Finite element imulation baed on tetrakaidecahedron cell are able to model pecific feature of rigid polyurethane foam under tenile loading i.e. tiffne and fracture characteritic. Thi method can be ued for analyi of more complex problem in foam behaviour that are not amenable to analytical olution, uch a incluion of non-linear material propertie, large deformation, variation of cell geometry, influence of cell membrane, etc; however, thee may require extenive computational reource and time. In eence, thi tudy demontrate that ue of detailed experimental obervation of the repone of cellular material to formulate an idealized cell model, followed by ubequent finite element imulation, i an effective approach to undertanding the 78

203 mechanical behaviour of foam. The cell model etablihed in thi tudy contitute a bai for further development of contitutive model for foam 6. Recommendation for future work Thi tudy ha focued primarily on the mechanical behaviour of rigid polyurethane foam under tenion and the development of idealized geometric cell model. Thi can be extended to examine other apect of the mechanical behaviour of foam: The bulk and hear moduli can be analytically derived from the cell model. For thi purpoe, the method employed by Zhu et al. [40] for analyzing the bulk modulu and hear modulu for iotropic tetrakaidecahedron cell can be ued a a bai, but ha to be modified to incorporate cell aniotropy. The Poion ratio of foam can be meaured and compared with value predicted by the model. A non-contact technique to meaure train could be ued for thi purpoe, becaue the urface of polyurethane foam pecimen are not flat and are eaily deformed/cruhed, making attachment of meaurement device difficult. Failure criteria baed on the buckling of cell trut for foam under compreion can be analyed uing the cell model; Euler buckling analyi could be applied to cell trut. Compreion involving large deformation and multi-axial loading can be examined to identify foam behaviour and to determine if the cell model propoed i able to repreent the compreive repone of actual foam. However, thi require non-linear deformation analyi; the method ued by Zhu et al. [4] in analyzing high train compreion of an iotropic tetrakaidecahedron cell model could facilitate uch a tudy. Numerical method uch a finite element modelling could alo be employed; however, FEM imulation might be cotly in term of 79

204 computational time and reource a much longer imulation time would be needed to model large deformation and more element are required for accurate reult. The cell model can be examined for applicability to other type of foam with imilar characteritic, i.e. open celled or cloed celled with thin wall, uch a flexible polyurethane foam, open celled aluminium foam, etc. 80

205 Lit of Reference. Plateau, J., Statique expérimentale et théorique de liquide oumi aux eule force moléculaire. 87, Pari: Gauthier-Villar.. Gibon, L.J. and M.F. Ahby, Cellular Solid: Structure and Propertie. nd ed. 997, Cambridge: Cambridge Univerity Pre.. Dehpande, V.S. and N.A. Fleck, Iotropic contitutive model for metallic foam. Journal of the Mechanic and Phyic of Solid, (6-7): p Dehpande, V.S. and N.A. Fleck, Multi-axial yield behaviour of polymer foam. Acta Materialia, (0): p Doyoyo, M. and T. Wierzbicki, Experimental tudie on the yield behavior of ductile and brittle aluminum foam. International Journal of Platicity, 00. 9(8): p Gdouto, E.E., I.M. Daniel, and K.-A. Wang, Failure of cellular foam under multiaxial loading. Compoite Part A: Applied Science and Manufacturing, 00. (): p McIntyre, A. and G.E. Anderton, Fracture propertie of a rigid polyurethane foam over a range of denitie. Polymer, (): p Triantafillou, T.C., et al., Failure Surface for Cellular Material Under Multiaxial Load. II. Comparion of Model With Experiment. International Journal of Mechanical Science, 989. (9): p Zalawky, M., Multiaxial-tre tudie on rigid polyurethane foam. Experimental Mechanic, 97. (): p

206 0. Zhang, J., et al., Contitutive modeling of polymeric foam material ubjected to dynamic crah loading. International Journal of Impact Engineering, 998. (5): p Andrew, E.W., L.J. Gibon, and M.F. Ahby, The creep of cellular olid. Acta Materialia, (0): p Chan, R. and M. Nakamura, Mechanical propertie of platic foam. Journal of Cellular Platic, : p Choi, J.B. and R.S. Lake, Analyi of elatic modulu of conventional foam and of re-entrant foam material with a negative Poion' ratio. International Journal of Mechanical Science, (): p Chritenen, R.M., Mechanic of low denity material. Journal of the Mechanic and Phyic of Solid, (6): p Cunningham, A., Modulu aniotropy of low-denity cellular platic: an aggregate model. Polymer, 98. (7): p Dement'ev, A.G. and O.G. Tarakanov, Effect of cellular tructure on the mechanical propertie of platic foam. Mechanic of Compoite Material, (4): p Dement'ev, A.G. and O.G. Tarakanov, Model analyi of the cellular tructure of platic foam of the polyurethane type. Mechanic of Compoite Material, (5): p Gent, A.N. and A.G. Thoma, The deformation of foamed elatic material. Journal of Applied Polymer Science, 959. (): p Gent, A.N. and A.G. Thoma, Mechanic of foamed elatic material. Rubber Chemitry and Technology, 96. 6(): p

207 0. Gibon, L.J. and M.F. Ahby. The Mechanic of Three-Dimenional Cellular Material. in Proceeding of the Royal Society of London. Serie A, Mathematical and Phyical Science. 98: The Royal Society.. Gibon, L.J., et al., Failure Surface for Cellular Material Under Multiaxial Load. I. Modelling. International Journal of Mechanical Science, 989. (9): p Warren, W.E. and A.M. Kraynik, The linear elatic propertie of open-cell foam. Journal of Applied Mechanic, (): p Grenetedt, J.L. and K. Tanaka, Influence of cell hape variation on elatic tiffne of cloed cell cellular olid. Scripta Materialia, (): p Grenetedt, J.L., Effective elatic behavior of ome model for perfect cellular olid. International Journal of Solid and Structure, (0): p Huang, J.-S. and L.J. Gibon, Creep of open-cell Voronoi foam. Material Science and Engineering A, 00. 9(-): p Huber, A.T. and L.J. Gibon, Aniotropy of foam. Journal of Material Science, 988. (8): p Kanakkanatt, S.V., Mechanical aniotropy of open-cell foam. Journal of Cellular Platic, 97. 9: p Ko, W.L., Deformation of foamed elatomer. Journal of Cellular Platic, 965. : p Lederman, J.M., The prediction of the tenile propertie of flexible foam. Journal of Applied Polymer Science, 97. 5(): p

208 0. Menge, G. and F. Knipchild, Etimation of mechanical propertie for rigid polyurethane foam. Polymer Engineering & Science, (8): p Maiti, S.K., L.J. Gibon, and M.F. Ahby, Deformation and energy aborption diagram for cellular olid. Acta Metallurgica, 984. (): p Mill, N.J. and H.X. Zhu, The high train compreion of cloed-cell polymer foam. Journal of the Mechanic and Phyic of Solid, (): p Patel, M.R. and I. Finnie, Structural feature and mechanical propertie of rigid cellular platic. Journal of Material, (4): p Robert, A.P. and E.J. Garboczi, Elatic moduli of model random threedimenional cloed-cell cellular olid. Acta Materialia, (): p Robert, A.P. and E.J. Garboczi, Elatic propertie of model random threedimenional open-cell olid. Journal of the Mechanic and Phyic of Solid, (): p Sahraoui, S., E. Mariez, and M. Etcheahar, Linear elatic propertie of aniotropic open-cell foam. Journal of the Acoutical Society of America, 00. 0(): p Simone, A.E. and L.J. Gibon, Effect of olid ditribution on the tiffne and trength of metallic foam. Acta Materialia, (6): p Wang, Y. and A.M. Cuitino, Three-dimenional nonlinear open-cell foam with large deformation. Journal of the Mechanic and Phyic of Solid, (5): p

209 9. Warren, W.E. and A.M. Kraynik, Linear elatic behavior of a low-denity Kelvin foam with open cell. Journal of Applied Mechanic, (4): p Zhu, H.X., J.F. Knott, and N.J. Mill, Analyi of the elatic propertie of open-cell foam with tetrakaidecahedral cell. Journal of the Mechanic and Phyic of Solid, (): p Zhu, H.X., N.J. Mill, and J.F. Knott, Analyi of the high train compreion of open-cell foam. Journal of the Mechanic and Phyic of Solid, (-): p Zhu, H.X., J.R. Hobdell, and A.H. Windle, Effect of cell irregularity on the elatic propertie of open-cell foam. Acta Materialia, (0): p Zhu, H.X. and A.H. Windle, Effect of cell irregularity on the high train compreion of open-cell foam. Acta Materialia, (5): p Ridha, M., V.P.W. Shim, and L.M. Yang. An Elongated Tetrakaidecahedral Cell Model for Fracture in Rigid Polyurethane Foam. in Proceeding of The 6th International Conference on Fracture and Strength of Solid Ridha, M., V.P.W. Shim, and L.M. Yang. An Elongated Tetrakaidecahedron Cell Idealization of the Microtructure of Rigid Polyurethane Foam. in Proceeding of JSME/ASME International Conference on Material and Proceing 005 (The th JSME Material and Proceing Conference) Seattle, Wahington, USA. 46. Ridha, M., V.P.W. Shim, and L.M. Yang, An Elongated Tetrakaidecahedral Cell Model for Fracture in Rigid Polyurethane Foam. Key Engineering Material, /08(): p

210 47. Ehler, W., H. Mullerchon, and O. Klar. On the behaviour of aluminium foam under uniaxial and multiaxial loading. in the Conference on Metal Foam and Porou Metal Structure, MetForm 99, 4-6 June Bremen, Germany: MIT Verlag. 48. Gioux, G., T.M. McCormack, and L.J. Gibon, Failure of aluminum foam under multiaxial load. International Journal of Mechanical Science, (6): p Hanen, A.G., et al., Validation of contitutive model applicable to aluminium foam. International Journal of Mechanical Science, (): p Miller, R.E., A continuum platicity model for the contitutive and indentation behaviour of foamed metal. International Journal of Mechanical Science, (4): p Schreyer, H.L., Q.H. Zuo, and A.K. Maji, Aniotropic Platicity Model for Foam and Honeycomb. Journal of engineering mechanic, (9): p Banhart, J. and J. Baumeiter, Deformation characteritic of metal foam. Journal of Material Science, 998. (6): p Motz, C. and R. Pippan, Deformation behaviour of cloed-cell aluminium foam in tenion. Acta Materialia, (): p Zhang, J., et al., Contitutive Modeling and Material Characterization of Polymeric Foam. Journal of engineering material and technology, (): p Tai, S.W. and E.M. Wu, A General Theory of Strength for Aniotropic Material. Journal of Compoite Material, 97. 5(): p

211 56. McCullough, K.Y.G., N.A. Fleck, and M.F. Ahby, Toughne of aluminium alloy foam. Acta Materialia, (8): p Dawon, J.R. and J.B. Shortall, The microtructure of rigid polyurethane foam. Journal of Material Science, 98. 7(): p Zhu, H.X. and N.J. Mill, Modelling the creep of open-cell polymer foam. Journal of the Mechanic and Phyic of Solid, (7): p Yang, L.M. and V.P.W. Shim, An analyi of tre uniformity in plit Hopkinon bar tet pecimen. International Journal of Impact Engineering, 005. (): p

212 Appendix A: SPHB experiment data proceing procedure Fig. A. Split Hopkinon bar arrangement Calculation for tre in SPHB pecimen were performed uing the following procedure. Strain-time data from the train gauge on the output bar wa obtained (ee Fig. A.) The data wa then converted into tre impoed on the pecimen uing the following expreion: A σ A b E b b where σ i the tre in the pecimen, A and b (A.) A are repectively the croectional area of the pecimen and the input/output bar, bar, and b i the train in the output bar. The train in the pecimen wa calculated a follow. E b i the tiffne of the Two reference point were marked along the centre-line of the pecimen (ee Fig. A.) 88

213 The initial ditance and ubequent relative diplacement between the two point were obtained from high-peed photograph, uing a PHOTRON ultima APX high-peed camera, operating at a framing rate of 0,000 frame per econd. The pecimen train wa then calculated by dividing the relative diplacement by the initial ditance between the two reference point. It wa found that the variation of train with time wa eentially linear; hence, linear regreion wa employed to calculate the train rate (ee Fig. A.). Thi train rate wa integrated to calculate the pecimen train correponding to the tre data. Fig. A. SPHB pecimen with two reference point along the centre-line 0.05 y 99.00x R time () Fig. A. Example of train-time data and application of linear regreion 89

214 Appendix B: Figure and Table Fig. B.-B.5 how the reult of quai-tatic tenile tet on foam A, B and C. σ (MPa) Fig. B. Stre-train curve for loading in the rie direction (foam A.kg m ; geometric aniotropy ratio.5) σ (MPa) Fig. B. Stre-train curve for loading 0 o to the rie direction (foam A.kg m ; geometric aniotropy ratio.5) 90

215 σ (MPa) Fig. B. Stre-train curve for loading 45 o to the rie direction (foam A.kg m ; geometric aniotropy ratio.5) σ (MPa) Fig. B.4 Stre-train curve for loading 60 o to the rie direction (foam A.kg m ; geometric aniotropy ratio.5) 9

216 σ (MPa) Fig. B.5 Stre-train curve for loading in the tranvere direction (foam A.kg m ; geometric aniotropy ratio.5) 0.6 σ (MPa) train rate ( - ) Fig. B.6 Stre-train curve for loading in the rie direction (foam B geometric aniotropy ratio ) 9.5kg m ; 9

217 σ (MPa) train rate ( - ) Fig. B.7 Stre-train curve for loading 0 o to the rie direction (foam B 9.5kg m ; geometric aniotropy ratio ) 0. σ (MPa) train rate ( - ) Fig. B.8 Stre-train curve for loading 45 o to the rie direction (foam B 9.5kg m ; geometric aniotropy ratio ) 9

218 σ (MPa) train rate ( - ) Fig. B.9 Stre-train curve for loading 60 o to the rie direction (foam B 9.5kg m ; geometric aniotropy ratio ) 0. σ (MPa) train rate ( - ) Fig. B.0 Stre-train curve for loading in tranvere direction (foam B 9.5kg m ; geometric aniotropy ratio ) 94

219 σ (MPa) Fig. B. Stre-train curve for loading in the rie direction (foam C 5.kg m ; geometric aniotropy ratio.7) σ (MPa) Fig. B. Stre-train curve for loading 0 o to the rie direction (foam C 5.kg m ; geometric aniotropy ratio.7) 95

220 σ (MPa) Fig. B. Stre-train curve for loading 45 o to the rie direction (foam C 5.kg m ; geometric aniotropy ratio.7) σ (MPa) Fig. B.4 Stre-train curve for loading 60 o to the rie direction (foam C 5.kg m ; geometric aniotropy ratio.7) 96

221 σ (MPa) Fig. B.5 Stre-train curve for loading in the tranvere direction (foam C 5.kg m ; geometric aniotropy ratio.7) Fig. B.6-B.8 how the tre-train curve obtained from Split Hopkinon bar tet on foam B. σ (MPa) 8.00E-0 train rate ( - ) 7.00E E E E E E E E Fig. B.6 Stre-train curve for loading in the rie direction (foam B 9.5kg m ; geometric aniotropy ratio ) 97

222 σ (MPa) train rate ( - ) Fig. B.7 Stre-train curve for loading in the 45 o direction (foam B 9.5kg m ; geometric aniotropy ratio ) 0. σ (MPa) train rate ( - ) Fig. B.8 Stre-train curve for loading in tranvere direction (foam B 9.5kg m ; geometric aniotropy ratio ) Fig. B.9 and B. how microcopic image of the cro-ection of trut in foam A, B and C. 98

223 Fig. B.9 Cro-ection of trut in rigid polyurethane foam A (.kg m ; geometric aniotropy ratio.5) 99

224 Fig. B.0 Cro-ection of trut in rigid polyurethane foam B ( 9.5kg m ; geometric aniotropy ratio ) 00

225 Fig. B. Cro-ection of trut in rigid polyurethane foam C ( 5.kg m ; geometric aniotropy ratio.7) Table B. how meaurement of trut cro-ection dimenion obtained from microcopic obervation dicued in Section.5. Table B. how meaurement of the length of rigid egment in trut in foam B ( 9.5kg m ; geometric aniotropy ratio ), obtained from the obervation dicued in Section.6.. 0