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Misalignment defects in unidirectional composite materials Stewart, Andrew Lawrence 2018

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Misalignment defects in unidirectional composite materialsbyAndrew Lawrence StewartB.A.Sc. (Mechanical Engineering), University of Ottawa, 2010M.A.Sc. (Mechanical Engineering), University of Ottawa, 2012A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Materials Engineering)The University of British Columbia(Vancouver)December 2018© Andrew Lawrence Stewart, 2018The following individuals certify that they have read, and recommend to the Faculty of Graduateand Postdoctoral Studies for acceptance, the dissertation entitled:Misalignment defects in unidirectional composite materialssubmitted by Andrew Lawrence Stewart in partial fulfillment of the requirements for the degreeof Doctor of Philosophy in Materials Engineering.Examining Committee:Anoush Poursartip, Materials EngineeringSupervisorSteve Cockcroft, Materials EngineeringSupervisory Committee MemberAnasavarapu Srikantha Phani, Mechanical EngineeringUniversity ExaminerDaan Maijer, Materials EngineeringUniversity ExaminerMichael Sutcliffe, Department of Engineering, University of CambridgeExternal ExaminerAdditional Supervisory Committee Members:Reza Vaziri, Civil EngineeringSupervisory Committee MemberiiAbstractThe properties of reinforced composite materials are dominated by the orientation of their rein-forcing fibres. This orientation is typically assumed, often implicitly, to be perfect leading toover-predictions of the final part’s strength. Traditional approaches for measuring misalignmentare laborious which has limited their adoption.Further, as the desire for manufacturers to collect more and more data increases, both the datacollection and reduction techniques must become automated in order to create value.In this thesis, several large data sets are created and ultimately analyzed using purpose-builtautomated scripts. One of these techniques analyzed over 2500 high resolution micrographs andreturned information on over 200,000 fibres. Using the information from this analysis allowed thecreation of an analytical model for the fibre bed. This phenomenological model uses the calcu-lated excess length distribution to individually assign a unique excess length to each fibre in thesystem. It is shown that only with a distribution of excess lengths can the experimental unimodalmisalignment distributions be properly modelled.Homogeneously dispersed variability was associated with each of the measured values whichincluded in-plane and out-of-plane fibre alignment, cured and uncured ply thickness, and fibrevolume fraction. Save the alignment distribution which lacks a standardized quality descriptor, theother metrics bounded the manufacturer’s data sheet values; however, these measurements showediiithat a non-trivial amount of variability should be expected in even high quality, aerospace grade,prepregs.A separate series of tests were developed which were able to impart small compressive strainsinto the compliant uncured prepreg. The localization of the uniformly distributed wrinkles waspartially attributed to the homogeneous variability of the prepreg’s underlying architecture. Theseslow forming wrinkles were shown to have a consistent set of mechanics as fast tool-part debond-ing.A hypoelastic shear-lag relationship was developed which was able to predict the excess lengthintroduced into the prepreg from the tool. This shear-lag approach predicts a zone of influencefor the wrinkles which was experimentally determined using wrinkle initiators. Reducing tool-partinteraction or rapid quenching were proposed as mitigation strategies for wrinkle management.ivLay SummaryFrom the Boeing 787 Dreamliner to the BMW i3, composite materials have fully established them-selves as high performance materials capable of being used in the mass market. These materialsrely on strong fibres which behave much like string in that they are only strong when pulled. Push-ing on string simply causes it to meander and for the material to become strong again, the string’sslack must be removed.This work samples hundreds of thousands of these fibres to fully understand the underlyingstructure. New techniques were developed which halved the amount of work required to performthis characterization. Novel experiments were designed which showed where this meanderingoriginates and how it persists in the manufacturing environment. The majority of this analysis wasdone using automated techniques which not only allowed these data sets to be analyzed but alsoallows them to be quickly implemented in future works.vPrefaceThis thesis entitled “Misalignment defects in unidirectional composite materials” presents the re-search conducted by Andrew Stewart while under the supervision of Professor Anoush Poursartipat the University of British Columbia.All experiments discussed in Chapter 3, Chapter 4, and Chapter 5 were designed and performedby Andrew Stewart. All results were analyzed by Andrew Stewart while under the supervision ofProfessor Anoush Poursartip. The analytical fibre bed model proposed in Chapter 3 Section 3.4was developed by Andrew Stewart while under the supervision of Professor Anoush Poursartip.The hypoelastic tool-part interface formulation was developed by Andrew Stewart while under thesupervision of Professor Anoush Poursartip.A paper based on Chapter 3 Section 3.4 has been published. Stewart, A. L. and Poursar-tip, A. (2018) “Characterization of fibre alignment in as-received aerospace grade unidirectionalprepreg,” Composites Part A: Applied Science and Manufacturing. Andrew Stewart conducted allof the testing and analysis. All parts of the paper were written by Andrew Stewart while under thesupervision of Professor Anoush Poursartip.A version of Section 3.3 and Section 3.4 of Chapter 3 was presented at the 21st InternationalConference on Composite Materials, Xi’an, 2017. A paper was published in the conference pro-ceedings. Stewart, A. L. and Poursartip, A. “An alternative method for advancing glass transitiontemperature in epoxy based composites for softening sensitive measurements,” 21st InternationalviConference on Composite Materials, Xi’an, 2017. Andrew Stewart conducted all of the testing andanalysis. All parts of the paper were written by Andrew Stewart while under the supervision ofProfessor Anoush Poursartip.A version of Chapter 3 Section 3.3 and Chapter 5 was presented at the Canadian Interna-tional Conference on Composite Materials, Edmonton, 2015. A paper was published in the con-ference proceedings. Stewart, A. L., Farnand, K., Fernlund, G., and Poursartip, A. “Experimentalstudy into the parameters affecting the architecture of waviness and wrinkling in prepreg systems,”Canadian International Conference on Composite Materials, Edmonton, 2015. The “Single lam-ina wrinkling evaluation” experiments and analysis were conducted by Andrew Stewart while the“Quasi-isotropic laminate evaluation” experiments were conducted by Kyle Farnand. Most of thispaper was written by Andrew Stewart while under the supervision of Professor Anoush Poursartipwith the “Quasi-isotropic laminate” sections originally drafted by Kyle Farnand under the supervi-sion of Professor Go¨ran Fernlund.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiSymbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiAbbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxxixAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xli1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Literature review and research objectives . . . . . . . . . . . . . . . . . . . . . . . . 42.1 Definition of terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Effect of defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7viii2.3 Sources and controlling factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.1 Fibre and prepreg manufacture . . . . . . . . . . . . . . . . . . . . . . . . 122.3.2 Layup and forming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.3 Debulk and cure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.4 Material characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.4.1 Chemical composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.4.2 Fibre alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.5 Research objectives and scope of thesis work . . . . . . . . . . . . . . . . . . . . 402.5.1 Synthesis of the open literature . . . . . . . . . . . . . . . . . . . . . . . . 402.5.2 Research objectives and scope . . . . . . . . . . . . . . . . . . . . . . . . 412.5.3 Thesis organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 Incoming material characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.2 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.3 Resin characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.3.1 Shear characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.3.2 Cure characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.4 Misalignment characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.5 Fibre volume fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023.6 Tow and ply architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1193.7 Summary of characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1234 Transient wrinkle growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1254.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1254.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133ix4.2.1 Geometric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1334.3 Geometric results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1454.3.1 No initiator (Geometric results) . . . . . . . . . . . . . . . . . . . . . . . 1454.3.2 Wrinkle initiators (Geometric results) . . . . . . . . . . . . . . . . . . . . 1534.4 Stress analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1734.4.1 Modelling parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1734.4.2 Wrinkle initiators (Stress analysis) . . . . . . . . . . . . . . . . . . . . . . 1804.4.3 Critical strain energy release rate . . . . . . . . . . . . . . . . . . . . . . . 1844.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1954.6 Summary of the transient wrinkle growth . . . . . . . . . . . . . . . . . . . . . . 2005 Quasi-static wrinkle growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2025.1 Introduction and methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2025.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2065.3 Discussion and summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2166 Conclusions, contributions, and future work . . . . . . . . . . . . . . . . . . . . . . 2186.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2186.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2226.3 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225A Discovery DSC calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239B Micrograph analysis script . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247B.1 Semi-automated ellipse detection . . . . . . . . . . . . . . . . . . . . . . . . . . . 247B.2 Yurgartis analysis on ellipse data . . . . . . . . . . . . . . . . . . . . . . . . . . . 264xC Wrinkle analysis script . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273D Supplementary transient wrinkle growth data . . . . . . . . . . . . . . . . . . . . . 274D.0.1 No initiator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274D.0.2 Wrinkle initiators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279D.0.3 Peirce cantilever test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291E Supplementary quasi-static wrinkle growth data . . . . . . . . . . . . . . . . . . . . 293F Volumetric contraction of the resin . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298xiList of TablesTable 3.1 Results from sectioning virtual fibres using various through-thickness rotationsgenerated in order to verify that the measured values correctly map to the inputvalues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Table 4.1 Input parameters for hypoelastic modelling. . . . . . . . . . . . . . . . . . . . 179Table 4.2 General procedure for determining the critical strain energy release rate. a)shows a single line running along the lamina’s 1-direction. b) shows the methodfor determining the excess length and crack length from the wrinkle’s pathlength and width. c) shows the general method for estimating the strained en-ergy. d) shows the final step to determine the critical strain energy release rate. . 185Table A.1 Standard values for the melting temperature and heat of fusion of indium, [176],[177] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244xiiList of FiguresFigure 1.1 (a) Schematic representation of an idealized ply featuring all straight fibrescollimated to the 0° direction of the lamina. (b) A more realistic representationof the fibre network with slight perturbations in fibre alignment with respect tothe 0° direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Figure 2.1 3D representation of a unidirectional ply displaying the conventional coordi-nate system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Figure 2.2 (a) Schematic representation of a bundle of wavy fibres/tows/tapes and (b) asection of wrinkled fibres/tows/plies, recreated after Yurgartis [18]. . . . . . . 6Figure 2.3 Schematic representation of wavy fibres/tows/tapes/plies undergoing (a) exten-sion and (b) shear compressive buckling, recreated after Rosen [20]. . . . . . . 7Figure 2.4 Schematic showing the half wavelength of wavy fibres in an otherwise per-fectly straight unidirectional ply, recreated after Garnich and Karami [27] . . . 10Figure 2.5 Flow diagram of the PAN and resin to uncured prepreg process. . . . . . . . . 12Figure 2.6 Flow diagram of the uncured prepreg to final part process. . . . . . . . . . . . 12xiiiFigure 2.7 Schematic of the rolling process of a high shear stiffness beam with inextensi-ble fibres resulting in compression on the inner surface. This compression isrepresented by a buckled section on the inner surface, created after concepts in[9], [22], [39], [40]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Figure 2.8 Predicted maximum angle of misalignment as a function of radial position onthe roll, as calculated by Equation 2.5, tc = 0.25mm. . . . . . . . . . . . . . . 15Figure 2.9 Schematic of the rolling process of the ideal shear bounded case highlightingthe bookend length, created after concepts in [22], [39], [40]. . . . . . . . . . . 16Figure 2.10 (a) Initial weave with 90° warp and weft tows, (b) weave undergoing in-planeshear, and (c) weave at the locking angle, recreated after Prodromou and Chen[42]. Note that the white connecting lines do not change length between stepsin shear. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Figure 2.11 Bending stiffness of uncured IMA-M21, recreated after Belnoue et al. [71]. . . 22Figure 2.12 Bending stiffness of AS4/8552 from two of Dodwell et al.’s publications, recre-ated after [40] and [74]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Figure 2.13 Schematic diagram showing the excess length on the outer surface as a resultof consolidation over a single curvature, recreated after Dodwell et al. [40]. . . 25Figure 2.14 Diagram of the sticking interface condition of a ply separated from the toolthrough an adhesive interface, reproduced from Twigg et al. [60]. . . . . . . . 26Figure 2.15 Diagram of the sliding interface condition showing the tool under load from aconstant shear stress, τsliding, reproduced from Twigg et al. [60]. . . . . . . . . 28Figure 2.16 Critical strain energy release rate from a peel tack test performed on a non-commercial glass fibre reinforced polymer (GFRP) at various temperatures,sample ‘PP2’, recreated from Crossley et al. [84]. . . . . . . . . . . . . . . . . 30xivFigure 2.17 (a) Resin volume change, recreated after Garstka et al. [93], and (b) resin vis-cosity change of a typical epoxy resin during a single hold thermal cure cyclewith highlighted process direction, initial temperature, To, and curing temper-ature, Tc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Figure 2.17 Various components of AS4/8552, epoxy components (a) TGDDM and (b)TGAP, along with the curing agent (c) DDS and toughening agent (d) PES. . . 33Figure 2.18 (a) A perspective view of three straight cylindrical fibres intersected by an or-ange 5° degree cutting plane. The blue, red, and green fibres have an in-planemisalignment angle of −2.5°, 0°, and 2.5°, respectively. (b) Highlights theangles of interest for the green fibre when sectioning. . . . . . . . . . . . . . . 36Figure 2.19 Resulting cross-sections on the cutting surface described in Figure 2.18(a). Themajor and minor diameter, required in Equation 2.14, are highlighted for thegreen fibre. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Figure 2.20 Example sectioning of misaligned fibres cut along the 2-3 plane as would typ-ically be performed for fibre volume fraction or porosity characterization. The0° degree fibre follows the lamina 1-direction and the other fibres are mis-aligned from this fibre by their corresponding values. . . . . . . . . . . . . . . 37Figure 2.21 Example sectioning of misaligned fibres cut at 5° to the 1-3 plane. The 0°degree fibre follows the lamina 1-direction and the other fibres are misalignedrelative to this fibre by their corresponding values. . . . . . . . . . . . . . . . 38Figure 3.1 Diagram tracking some of the intrinsic properties and the extrinsic conditionswhich influence misalignment growth. . . . . . . . . . . . . . . . . . . . . . . 46Figure 3.2 Flow diagram highlighting the interplay of misalignment growth and the in-trinsic material properties of the prepreg. . . . . . . . . . . . . . . . . . . . . 47Figure 3.3 AS4/8552-1 roll information. . . . . . . . . . . . . . . . . . . . . . . . . . . . 48xvFigure 3.4 AS4/8552 roll information. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Figure 3.5 Anton-Paar MCR502 rheometer with disposable parallel plate geometries andPeltier plate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Figure 3.6 Shear storage modulus for both AS4/8552, the average of three trials with stan-dard error bars, and neat 8552 resin. . . . . . . . . . . . . . . . . . . . . . . . 54Figure 3.7 Shear storage modulus for AS4/8552, average of three trials with standard errorbars, and the temperature invariant fibre bed shear property, over a range oftemperatures consistent with forming. . . . . . . . . . . . . . . . . . . . . . . 55Figure 3.8 Photograph of two AS4/8552-1 specimens in preparation for a soak in thetransparent ammonium hydroxide solution. . . . . . . . . . . . . . . . . . . . 57Figure 3.9 Thermogram of an as-received specimen of AS4/8552-1. Exothermic heat flowis plotted as a positive value. . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Figure 3.10 Thermograms of ammonium hydroxide soaked and vacuum dried AS4/8552-1specimen. Exothermic heat flow is plotted as a positive value. (a) 1 hourvacuum dry. (b) 20 hour vacuum dry. . . . . . . . . . . . . . . . . . . . . . . 59Figure 3.11 Residual heat of reaction of the AS4/8552-1 specimens after various soak du-rations in aqueous ammonium hydroxide. . . . . . . . . . . . . . . . . . . . . 60Figure 3.12 Degree of cure evolution due to varying soak times in aqueous ammoniumhydroxide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Figure 3.13 Glass transition temperature increase due to increasing soak times in aqueousammonium hydroxide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62Figure 3.14 Cure rate as a function of degree of cure for (a) 1 h vacuum dry and (b) 20 hvacuum dried specimens after various soak times in aqueous ammonium hy-droxide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63xviFigure 3.15 (a) Physical specimen sectioned into three wedges and (b) a schematic of thespecimen with various geometric features highlighted. . . . . . . . . . . . . . 67Figure 3.16 A series of images depicting the various steps from the automated procedurecreated in order to fit individual ellipses over fibres. Refer to the prior para-graph for a discussion of each image. . . . . . . . . . . . . . . . . . . . . . . 69Figure 3.17 Several steps from the manual procedure for individually fitting ellipses overthe remaining aberrant fibres. Refer to the prior paragraph for a discussion ofeach image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70Figure 3.18 A perspective view of the virtual misaligned fibres in both the in-plane andout-of-plane directions intersected by a 5° cutting plane. The blue, red, andgreen fibres have an in-plane misalignment angle of−2.5°, 0°, and 2.5° and anout-of-plane misalignment angle of −10°, 0°, and 10°, respectively. . . . . . . 73Figure 3.19 (a) Top view of Figure 3.18 highlighting the typical in-plane angles of interestand (b) a front view showing the green fibre’s input 10° out-of-plane misalign-ment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Figure 3.20 Resulting cross-sections on the cutting surface described in Figure 3.18. Themajor diameter, minor diameter, and measured out-of-plane angle are high-lighted for the green fibre. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Figure 3.21 Schematic diagram showing virtual fibres, defined by a single excess length,being sectioned by an orange cutting plane. The number of fibres shown issignificantly fewer than those analyzed for visualization purposes. . . . . . . . 78Figure 3.22 In-plane fibre angle distribution for the virtual set of fibres defined by a singleexcess length shown in Figure 3.21. 14,752 virtual intersections were analyzedto generate this distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . 79xviiFigure 3.23 Flow diagram of the general procedure used to generate the individual fibresof the virtual fibre bed. Refer to the prior paragraph for a discussion of each step. 80Figure 3.24 Schematic diagram of a tow segment featuring fibres with independent excesslengths, amplitudes, wavelengths, phase shifts, and initial positions. The bluefibre features the lowest excess length, the red fibre with intermediate excesslength, and the green fibre with the highest excess length. . . . . . . . . . . . . 81Figure 3.25 Sectioned virtual tow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82Figure 3.26 Individual in-plane fibre misalignment volume fraction distributions for AS4/8552-1roll ‘A’ prepreg specimens at various radial positions, (a) 163 mm, (b) 171 mm,(c) 181 mm, (d) 192 mm, (e) 213 mm, and (f) 224 mm. Each distribution wasgenerated from the W2 wedge. . . . . . . . . . . . . . . . . . . . . . . . . . . 84Figure 3.27 AS4/8552-1 roll ‘A’ in-plane fibre angle probability density functions for thesix radially spaced specimens. . . . . . . . . . . . . . . . . . . . . . . . . . . 85Figure 3.28 Standard deviation of in-plane fibre misalignments, AS4/8552-1 roll ‘A’, withdashed lines indicating the take-up roll core radius and original outer radius. . . 85Figure 3.29 In-plane fibre misalignment volume fraction distributions for wedge W1 at224 mm, (a) typical window with 97.7 % of the volume fraction, (b) the fullvolume fraction distribution, and (c) highlighting the non-zero fraction of fi-bres at large angles to the lamina 1-direction. . . . . . . . . . . . . . . . . . . 87Figure 3.30 Highly misaligned fibres located (a) in the internal structure and (b) along theperiphery of the lamina. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Figure 3.31 AS4/8552-1 roll ‘A’ out-of-plane fibre angle probability density functions forthe six radially spaced specimens. . . . . . . . . . . . . . . . . . . . . . . . . 89Figure 3.32 Standard deviation of out-of-plane fibre misalignments, AS4/8552-1 roll ‘A’,with dashed lines indicating the take-up roll core radius and original outer radius. 90xviiiFigure 3.33 Comparison of the in-plane and out-of-plane standard deviation values for thethree different prepreg rolls studied. . . . . . . . . . . . . . . . . . . . . . . . 90Figure 3.34 Cartoon showing the large number of spatial in-plane angle distributions gen-erated for a single specimen. A select few distributions are shown in greaterdetail in Figure 3.36. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91Figure 3.35 Cartoon showing the corresponding large number of generated spatial excesslength distributions generated from the single specimen used in Figure 3.34. Aselect few distributions are shown in greater detail in Figure 3.36. . . . . . . . 92Figure 3.36 (a), (c), and (e) show various in-plane angle volume fractions with their cor-responding (b), (d), (f) empirical normalized excess length probability densityfunctions for arbitrary windows. Each window had a width of approximately0.9 mm containing nearly 200 fibres, (AS4/8552-1, roll ‘A’, thermally-cured). . 93Figure 3.37 Average normalized excess length for the thermally-cured and soak-cured AS4/8552-1roll ‘A’ specimens, with dashed lines indicating the take-up roll core radius andoriginal outer radius. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94Figure 3.38 Single-sided Student’s t-distribution, dashed red line, overlay with each of theexcess length distributions shown in Figure 3.35. . . . . . . . . . . . . . . . . 95Figure 3.39 Histogram of the virtual fibres normalized excess lengths generated from theabsolute values of a Student’s t-distribution, location parameter of 0, scale pa-rameter of 0.0001, and degrees of freedom of 0.45. . . . . . . . . . . . . . . . 96Figure 3.40 Virtual fibres sectioned by an orange cutting plane. Red markers indicate thelocation at which each fibre intercepts the cutting plane. The number of fibres,and intercepts, shown is drastically fewer than those analyzed for visualizationpurposes with only approximately 1 in 20 fibres shown. . . . . . . . . . . . . . 96xixFigure 3.41 In-plane fibre angle probability density function for the virtual fibre data setsectioned in Figure 3.40. Approximately 2000 fibres were required to ensurethe virtual cutting plane cut fibres along its entire length. . . . . . . . . . . . . 97Figure 3.42 (a) A virtual set of idealized HCP fibres with a translucent grey cutting planeintersecting the fibres parallel to the 2-3 plane. (b) Resultant faces after cuttingwith a 57 % fibre areal fraction. The RVE used to calculate the fibre arealfraction has been highlighted in blue. . . . . . . . . . . . . . . . . . . . . . . 103Figure 3.43 (a) a 48 mm wide mosaic micrograph of an approximately 0.16 mm thick AS4/8552-1lamina sectioned along the 2-3 plane. (b) an arbitrary 0.5 mm wide sectionmagnified from the previous mosaic. (c) the traditional image thresholdingused to determine a fibre volume fraction of 57.1 %. . . . . . . . . . . . . . . 104Figure 3.44 (a), (c), (e), and (g) show a virtual cutting surface along the 1-3 plane as it trans-lates along the 2-direction by 0, rh4 ,rh2 , and3rh4 from an arbitrary fibre centroidof an hexagonal close-packing (HCP) fibre bed. (b), (d), (f), and (h) virtualfibre faces sectioned by their respective cutting surfaces. The correspondingfibre volume fractions are 80 %, 67 %, 0 %, 67 %. . . . . . . . . . . . . . . . . 105Figure 3.45 Fibre volume fraction as determined by percent area of sectioned fibres as the1-3 plane cutting surface translates along the 2-direction. . . . . . . . . . . . . 106Figure 3.46 (a), (c), (e), and (g) show various cutting surfaces inclined at 5° to the 1-3 planeas it translates along the 2-direction. (b), (d), (f), and (h) resultant fibre facesfor a fixed window width. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107xxFigure 3.47 (a) Fibre volume fraction as determined from the areal fraction of fibre facesaccording to the window width and 2-direction shift factor for a 5° cutting sur-face. (b) The fibre volume fraction as a function of 2-direction shift of thecutting surface for a constant window width of 39 µm. (c) The fibre volumefraction as a function of window width for a constant 2-direction shift consis-tent with Figure 3.46(f). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108Figure 3.48 (a) a full 49 mm mosaic micrograph of one wedge specimen. (b) a magnifiedsection of (a) highlighting the resolution by visualizing individual fibres. (c)the digital recreation of the elliptical fibres from the ellipse study performed inSection 3.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109Figure 3.49 (a) a recreated 6 mm wide section. (b) a reduced window for further analysis.(c) the centroid from each fibre. (d) the minimum area bounding box contain-ing all of the centroids. (e) the final resultant fibre volume fraction of 59.4 %for the windowed section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111Figure 3.50 The fibre volume fraction for the approximately 1 mm wide windows as a func-tion of window position for the innermost and outermost specimens cured ei-ther thermally or using the ammonium hydroxide soak. . . . . . . . . . . . . . 112Figure 3.51 The cured and uncured ply thickness for the approximately 1 mm wide win-dows as a function of window position for the innermost and outermost speci-mens cured either thermally or using the ammonium hydroxide soak. . . . . . 113Figure 3.52 (a) Fibre volume fraction and cured ply thickness as a function of position forthe thermally cured outermost specimen. (b) Fibre volume fraction and curedply thickness (CPT) normalized by their respective mean values as a functionof position. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114xxiFigure 3.53 (a) Clustered fibre volume fraction as a function of CPT for the thermally curedspecimens. (b) the fibre volume fraction for both thermally and ammoniumhydroxide cured specimens. (c) Linear relationship between reciprocal of theply thickness and the fibre volume fraction. . . . . . . . . . . . . . . . . . . . 116Figure 3.54 Micrograph showing a section where only half of the ply was imaged due tothe large internal void. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117Figure 3.55 (a), (b), and (c) display the number of total number of fibres, the number offibres with misalignments greater than the standard deviation, and the corre-sponding fraction, as a function of centroid position through thickness fromthree representative 6 mm wide windows. . . . . . . . . . . . . . . . . . . . . 118Figure 3.56 Histogram of the measured fibre diameters for the AS4/8552-1 roll ‘A’ speci-mens. d f ibres = (6.8±0.5)µm with N = 126429 fibres . . . . . . . . . . . . . 121Figure 3.57 Resulting specimen after performing the burn off and applying small transverseloads to separate the tows. The ruler’s major graduations are in cm. . . . . . . 122Figure 4.1 Schematic of the sticking tool-part loading condition with annotations for therelevant geometric properties, recreated after [60]. . . . . . . . . . . . . . . . 127Figure 4.2 (a) Top view, and (b) side view, of a typical specimen during preparation withhighlighted dimensions and materials. The 1-direction of the unidirectionalprepreg is parallel to the 152 mm dimension. . . . . . . . . . . . . . . . . . . 134Figure 4.3 Visual description of the heated debulk process. (a) The expansion due to the65 ◦C ramp and 30 min hold and (b) the contraction as the specimen returns toroom temperature under applied vacuum. . . . . . . . . . . . . . . . . . . . . 136Figure 4.4 Schematic of imaging procedure of the prepreg surface using oblique lightingand an off-axis camera to highlight any surface defects. . . . . . . . . . . . . . 137xxiiFigure 4.5 (a) Nikon Metrology coordinate measuring machine (CMM) and (b) the non-contact Metris XC65D-LS laser scanning head. . . . . . . . . . . . . . . . . . 138Figure 4.6 Example surface profile recreated from the coordinate measuring machine (x,y,z)data with a highlighted blue line running along the prepreg 1-direction. . . . . 139Figure 4.7 (a) The raw data along the blue line described in Figure 4.6 has been overlaidwith a filtered signal to decrease noise. (b) The further refinement after remov-ing the slight curvature to allow for individual maximum and minimum peaksto be identified. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140Figure 4.8 The surface profile from Figure 4.6 has been overlaid with the disparate maxi-mum peaks determined for each step in dy. . . . . . . . . . . . . . . . . . . . 141Figure 4.9 Several steps from the individual wrinkle detection methodology are repro-duced which show the transition from disparate peaks into unique wrinkle re-gions. Refer to the prior paragraph for a discussion of each image. . . . . . . . 142Figure 4.10 (a), (c), and (e) show a reduced surface region as time progresses with thecorresponding (b), (d), and (f) surface highlighted with the maximum peak. (g)Plots the maximum wrinkle height and excess length for the three highlightedpoints against logarithmic time. . . . . . . . . . . . . . . . . . . . . . . . . . 144Figure 4.11 The surface of the prepreg (a) immediately after removal from vacuum bag-ging, (b) after approximately 2 h, and (c) after approximately 8.75 h showingthe transient wrinkle formation centered around the lengthwise midline. . . . . 146Figure 4.12 (a), (b), and (c) show the surface profile, with indicated wrinkles, for a speci-men prepared for the CMM. The wrinkle dispersion measured by the CMM isvery similar to dispersion directly observed in Figure 4.11. . . . . . . . . . . . 148Figure 4.13 Top down view of Figure 4.12(c) with each wrinkle uniquely numerated. . . . 149xxiiiFigure 4.14 (a) and (b) plot the maximum wrinkle height for each wrinkle indicated inFigure 4.13 over the first 9 d and a reduced section over the first 24 h. (c)The legend, starting from tallest final wrinkle to shortest wrinkle, follows thenumbering of Figure 4.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150Figure 4.15 (a), (b), (c), and (d) track the population distribution of wrinkle heights throughfour time steps showing the general reduction in the total number of wrinkles. . 151Figure 4.16 (a), (b), and (c) show the prepreg surface through time with a single 3 mmwide by 0.0375 mm thick film of polymethylpentene (PMP) placed along thelengthwise midline of the prepreg specimen. . . . . . . . . . . . . . . . . . . . 154Figure 4.17 (a), (b), and (c) show the surface profile after analysis of the CMM data fora specimen prepared with a 3 mm wide by 0.0375 mm PMP wrinkle initiatoralong the lengthwise midline. . . . . . . . . . . . . . . . . . . . . . . . . . . 156Figure 4.18 Wrinkle height of the initiated wrinkle as a function of time for various tri-als conducted at slightly different ambient conditions showing a plateau afterapproximately 1 d. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157Figure 4.19 Wrinkle width of the initiated wrinkle as a function of time for various trialsconducted at slightly different ambient conditions showing an overall increasein the wrinkle width. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158Figure 4.20 Excess length consumed in the initiated wrinkle as a function of time for vari-ous trials conducted at slightly different ambient conditions showing a similarprofile as the wrinkle height. . . . . . . . . . . . . . . . . . . . . . . . . . . . 159Figure 4.21 Population distribution of wrinkle heights through four time steps. (d) showsa marked reduction in the number of wrinkles in comparison to the uninitiateddistribution shown in Figure 4.15(d) . . . . . . . . . . . . . . . . . . . . . . . 160xxivFigure 4.22 (a), (b), and (c) show the surface profile for an example specimen preparedwith two 3 mm wide by 0.0375 mm thick PMP wrinkle initiators spaced ap-proximately 40 mm from the edges and 80 mm from center to center. . . . . . . 162Figure 4.23 Population distribution of wrinkle heights through four time steps. . . . . . . . 163Figure 4.24 Wrinkle height of the initiated wrinkles as a function of time for the left andright wrinkles and at slightly different ambient conditions. . . . . . . . . . . . 164Figure 4.25 Wrinkle width of the initiated wrinkles as a function of time for the left andright wrinkles and at slightly different ambient conditions. . . . . . . . . . . . 164Figure 4.26 Wrinkle excess length of the initiated wrinkles as a function of time for the leftand right wrinkles and at slightly different ambient conditions. . . . . . . . . . 165Figure 4.27 Overlay of the single and double initiated trial wrinkle heights conducted atnominally 21 ◦C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165Figure 4.28 (a), (b), and (c) show the surface profile of a specimen prepared with four3 mm wide by 0.0375 mm thick PMP wrinkle initiators spaced approximately30.5 mm from edge-to-center and center-to-center. . . . . . . . . . . . . . . . 167Figure 4.29 Population distribution of wrinkle heights through four time steps. . . . . . . . 168Figure 4.30 Wrinkle height growth for the four initiated wrinkles depicted in Figure 4.28overlaid with the single initiator wrinkle height trial conducted at the sameambient temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169Figure 4.31 Wrinkle width for the four initiated wrinkles depicted in Figure 4.28 overlaidwith the single initiator wrinkle width trial conducted at the same ambient tem-perature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169Figure 4.32 Wrinkle excess length for the four initiated wrinkles depicted in Figure 4.28overlaid with the single initiator wrinkle excess length trial conducted at thesame ambient temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170xxvFigure 4.33 Effect of increasing temperature on the maximum wrinkle height for an initi-ated trial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171Figure 4.34 Stable wrinkle height at various temperatures for the initiated wrinkles withsymmetric left-right surrounding regions. . . . . . . . . . . . . . . . . . . . . 172Figure 4.35 TA Instruments Q400 thermomechanical Analyzer (TMA) machine. . . . . . . 176Figure 4.36 Results for the through thickness linear expansion and contraction of a poly-carbonate specimen between 0 ◦C to 100 ◦C at 1 ◦Cmin−1. . . . . . . . . . . . 177Figure 4.37 (a) Side view of the quadruple initiator trial with the PMP initiators shown inwhite and lines of symmetry in grey. (b) and (c) show the shape of the shear inthe interlayer and in-plane stress in the composite. . . . . . . . . . . . . . . . 180Figure 4.38 Shear stress in the resin interlayer with a reduced interaction lenght due toclose initiators conducted at 24.3 ◦C. . . . . . . . . . . . . . . . . . . . . . . . 181Figure 4.39 In-plane stress in the composite layer with reduced interaction length due toclose initiators conducted at 24.3 ◦C. . . . . . . . . . . . . . . . . . . . . . . . 181Figure 4.40 Excess lengths as a function of temperature; predictions made using Equa-tion 4.18. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182Figure 4.41 Predicted excess lengths consumed in a single wrinkle over a cool down cyclefor various interaction lengths. . . . . . . . . . . . . . . . . . . . . . . . . . . 183Figure 4.42 Wrinkle width and height of a single initiator trial conducted at 21.1 ◦C. . . . . 186Figure 4.43 Wrinkle width and height during the first hour of growth. Monotonically in-creasing wrinkle height with invariant wrinkle width. . . . . . . . . . . . . . . 187Figure 4.44 (a) shows the initial state of the prepreg with a wrinkle initiator. (b) shows ahypothetical situation with highly strained interlayer. . . . . . . . . . . . . . . 188Figure 4.45 (a) shows a snapshot of the wrinkle which has grown during the first hour. (b)shows the interlayer after relieving strain into the wrinkle causing growth. . . . 189xxviFigure 4.46 (a) shows the initial state of the prepreg with a wrinkle initiator. (b) shows ahypothetical situation with compressed prepreg layer. . . . . . . . . . . . . . . 190Figure 4.47 (a) shows the initial state of the prepreg with a wrinkle initiator. (b) shows ahypothetical situation with shear of material into the wrinkle. . . . . . . . . . . 191Figure 4.48 Wrinkle height and width during the transition from mode II dominated tomixed mode failure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192Figure 4.49 Critical strain energy release rate of all of the initiated wrinkles with symmetricleft-right surrounding regions. . . . . . . . . . . . . . . . . . . . . . . . . . . 193Figure 4.50 Critical strain energy release rates compared against Crossley et al.’s GFRP/s-teel results [84]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193Figure 4.51 Critical strain energy release rates compared against Crossley et al.’s GFRP/s-teel results with failure modality overlaid [84]. . . . . . . . . . . . . . . . . . 194Figure 5.1 3D representation of the assembly with clamps at the boundary. (a) in the initialstate and (b) in a strained state. . . . . . . . . . . . . . . . . . . . . . . . . . . 203Figure 5.2 DIC apparatus used in the quasi-static wrinkle growth trials. . . . . . . . . . . 204Figure 5.3 Side profile of a trial through various steps of end-to-end shortening. . . . . . . 206Figure 5.4 Initial state of a representative specimen mounted in the Hounsfield tensometer. 207Figure 5.5 Strained surface of a trial conducted at 22 ◦C. . . . . . . . . . . . . . . . . . . 208Figure 5.6 Strained surface of a trial conducted at 32 ◦C. . . . . . . . . . . . . . . . . . . 209Figure 5.7 Strained surface of a trial conducted at 50 ◦C. . . . . . . . . . . . . . . . . . . 209Figure 5.8 Representative initial surface of a trial as measured on the CMM. . . . . . . . 210Figure 5.9 Three surface profiles digitally flattened from a forming radius of approxi-mately 500 mm. Conducted at approximately (a) 24 ◦C, (b) 40 ◦C, and (c) 62 ◦C. 211Figure 5.10 Three surface profiles digitally flattened from a forming radius of approxi-mately 250 mm. Conducted at approximately (a) 24 ◦C, (b) 40 ◦C, and (c) 62 ◦C. 213xxviiFigure 5.11 Comparison of the total number of wrinkles as a function of temperature. Theroom temperature trial shows significantly more features than the elevated tem-perature trials over a large range of forming steps. . . . . . . . . . . . . . . . . 214Figure 5.12 Comparison of the largest wrinkles as a function of progressive forming stepsshowing the smaller overall wrinkles at room temperature compared to thelarger wrinkles measured for the elevated trials. . . . . . . . . . . . . . . . . . 215Figure A.1 DSC calibration manual: Tzero Calibration (Tzero Verification) desired base-line, [174] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242Figure A.2 Typical Tzero calibrated state of CRN’s Discovery DSC. . . . . . . . . . . . . 243Figure A.3 DSC calibration manual: Cell Constant/Temperature Calibration (Indium Ver-ification), [174] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244Figure A.4 Typical thermogram of indium after Cell Constant/Temperature Calibration ofCRN’s Discovery DSC. Annotated on the figure are the onset melting temper-ature and enthalpy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245Figure A.5 DSC calibration manual: Standard Heat Capacity Calibration (Heat CapacityVerification), [174] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245Figure A.6 Typical thermogram of sapphire after a Standard Heat Capacity Calibration ofCRN’s Discovery DSC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246Figure A.7 Thermograms of empty pan trials performed in a calibrated Discovery DSC,2 ◦Cmin−1 with ±0.318 ◦Cmin−1 modulation. . . . . . . . . . . . . . . . . . 246Figure D.1 Duplicate non-initiated transient wrinkle growth trial conducted at an uncon-trolled ambient temperature of 22 ◦C. (a), (b), and (c) show various time stepsfrom the initial time step to nearly 9 d. . . . . . . . . . . . . . . . . . . . . . . 275xxviiiFigure D.2 Top down view of the final time step of the wrinkled surface with each wrinklegiven a unique identifier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276Figure D.3 (a) and (b) plot the maximum wrinkle height over the first nearly 9 d and thefirst 24 h. (c) indicates the individual wrinkles following the numbering schemeoutlined in Figure D.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277Figure D.4 Population height distribution of wrinkles for the Figure D.1 trial. . . . . . . . 278Figure D.5 Three time steps for a single initiator trial conducted at an isothermal temper-ature of 20.7 ◦C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279Figure D.6 Population distribution of wrinkle heights through four time steps for the Fig-ure D.5 trial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280Figure D.7 Three time steps for a duplicate single initiator trial conducted at an isothermaltemperature of 20.7 ◦C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281Figure D.8 Population distribution of wrinkle heights through four time steps for the Fig-ure D.7 trial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282Figure D.9 Three time steps for a single initiator trial conducted at an isothermal temper-ature of 21.1 ◦C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283Figure D.10 Population distribution of wrinkle heights through four time steps for the Fig-ure D.9 trial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284Figure D.11 Three time steps for a single initiator trial conducted at an isothermal temper-ature of 21.2 ◦C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285Figure D.12 Population distribution of wrinkle heights through four time steps for the Fig-ure D.11 trial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286Figure D.13 Three time steps for a single initiator trial conducted at an isothermal temper-ature of 24.3 ◦C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287xxixFigure D.14 Population distribution of wrinkle heights through four time steps for the Fig-ure D.13 trial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288Figure D.15 Three time steps for a double initiator trial conducted at an isothermal temper-ature of 21.8 ◦C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289Figure D.16 Population distribution of wrinkle heights through four time steps for the Fig-ure D.15 trial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290Figure D.17 (a) 0 h, (b) 24 h, and (c) 48 h of a single ply of AS4/8552-1 cantilevered underself weight at an uncontrolled ambient temperature of (25.0±0.1) ◦C. Theruler’s major graduations are in cm. . . . . . . . . . . . . . . . . . . . . . . . 292Figure E.1 (a), (b), and (c) show the surface profile of a room temperature trial undergoingprogressive amounts of bending. . . . . . . . . . . . . . . . . . . . . . . . . . 294Figure E.2 (a), (b), and (c) show the surface profile of a 40 ◦C trial undergoing progressiveamounts of bending. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295Figure E.3 (a), (b), and (c) show the surface profile of a 50 ◦C trial undergoing progressiveamounts of bending. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296Figure E.4 (a), (b), and (c) show the surface profile of a 62 ◦C trial undergoing progressiveamounts of bending. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297xxxSymbolsSign Description SI UnitΓ Gamma functionΛ Wavelength of a oscillatory curve mΦi Phase shift of the ith sinusoidal type fibre radΦPC Cutting plane angle radαi Mode number of the ith fibreα1 Linear coefficient of thermal expansion of the part K−1α2 Linear coefficient of thermal expansion of the tool K−1αPC Linear coefficient of thermal expansion of polycarbonate K−1αcs Volumetric cure shrinkageα Linear thermal coefficient of thermal expansion K−1βL Locking angle of a woven fabric radχ Degree of cureεT Thermal strainη Matrix viscosity Pasγa Shear strain amplitude during oscillatory testingγy Yield shear strain of the compositeγ Shear strainxxxiSign Description SI Unitλ Exponential parameterµs Location parameter for the excess length Student’s t-distributionµPP Inter-ply coefficient of frictionµT P Tool-part coefficient of frictionµ Coefficient of frictionν Degrees of freedom constant for the excess length Stu-dent’s t-distributionωi Misalignment angle of a fibre relative to the cutting plane radω Shear strain frequency during oscillatory testing s−1φo Initial misalignment radφ Out-of-plane angle of an individual fibre radρ f Density of the fibre kgm−3σ1 In-plane stress in the part Paσ2 In-plane stress in the tool Paσc Critical compressive stress Paσ Normal stress PaτDebond Shear debond stress of the resin interlayer Paτsliding Sliding shear stress Paτy Yield stress in longitudinal shear Paτ Shear stress PaθF Forming angle radθi In-plane misalignment angle of the ith fibre relative to thenominal 1-directionradxxxiiSign Description SI Unitςs Scale parameter for the excess length Student’s t-distributionς Standard deviation of misalignment radA Amplitude of an oscillatory curve ma Crack length mAc Total area of composite in a sectioned micrograph m2AF Areal weight of the fibre kgm−2A f Total area of fibre in a sectioned micrograph m2Ai Amplitude of the ith sinusoidal type fibre mB Bending stiffness Nm2b Crack width mB∞ Fitting parameter for the bending stiffness Nm2ci Spacing factor of the ith sinusoidal type fibre md Film thickness mdi,ma j Major diameter of the ith fibre mdi,min Minor diameter of the ith fibre mE1,p Young’s modulus of ply in the 1-direction PaE2 Young’s modulus of the tool PaE2,p Young’s modulus of ply in the 2-direction PaxxxiiiSign Description SI UnitE3,p Young’s modulus of ply in the 3-direction PaEB Fitting parameter for the bending stiffness Jmol−1Ec Prepreg core modulus PaEr Young’s modulus of the resin PaEu Unrelaxed Young’s modulus PaFf Friction force NFN Normal force Nftt Number of fibres through thickness of a plyG12,p In-plane shear modulus of ply PaGc Critical strain energy release rate Jm−2Gc Shear modulus of the composite PaG f b Fibre bed stiffness PaGr Shear modulus of the resin PaGu Unrelaxed shear modulus PaH Mass normalized heat of reaction Jkg−1h1 Part thickness mh2 Half of the tool’s thickness mhA Adhesive film thickness mHt Mass normalized total heat of reaction Jkg−1I Second moment of area m4xxxivSign Description SI UnitK Column effective length factorL Interaction length mL1D Total length along the 1-direction mL2, f Post-cure length along the 2-direction mL2,o Initial length along the 2-direction mLc Column length m∆L Excess length in the wrinkle mδL Bookend length m∆̂L Average mormalized excess length for a wedge specimen∆̂Li Normalized excess length of the ith sinusoidal type fibre∆Li, j In-plane excess length contribution of a fibre population, i,in a frame, jmL∞ Distance to the free edge or symmetry condition betweenan adjacent initiatorm∆L j Average excess length the fibres in a window, j mLp Path length of the wrinkle mLu Unwound ply length mLw, j Length of an individual window, j mN f Number of fibres in a town j Number of discrete angle bins in a window, jnp Number of pliesxxxvSign Description SI Unitnw Number of windows, w, measured for a wedge specimenPcr Euler’s critical buckling load NR Specific gas constant JK−1 mol−1Rc Take-up roll core radius mRF Forming radius mr f Fibre radius mrfit Constant radius of the circle fitted to the forming curvature mrh Inradius of a hexagon mRo Initial outer radius of the rolled prepreg mst Intertow spacing mT Temperature KtA Adhesive film thickness used by Twigg et al. mtAS,p Cured ply thickness after the ammonium hydroxide soak mtb Backing paper thickness mtc Combined backing paper and uncured ply thickness m∆T Temperature differential KTf Final resting temperature KTg Glass transition temperature KTo Minimum temperature at which the resin interlay can im-part stressesKxxxviSign Description SI Unitt0 Time between removing vacuum and first CMM scan stp,u Uncured ply thickness mtT,p Cured ply thickness after the thermal cure cycle mU Elastic energy Ju1 Displacement of the part mu2 Displacement of the tool mV Volume of strained material m3v Relative velocity ms−1V c Volume of composite m3V f Volume of fibre m3v f Fibre volume fractionvI Resin interlayer volume fractionV i, j Number of fibres counted within a discrete angle bin, iw Specimen width mwt Tow width mww Wrinkle width mxc Horizontal coordinate of the center of an individual fibre mxo Horizontal coordinate of the center of a circle fitted to thesingle curvature during formingmx′ Horizontal coordinate of the flattened surface mxxxviiSign Description SI Unityc Vertical coordinate of the center of an individual fibre myi In-plane centroid position of the ith sinusoidal type fibre mzo Vertical coordinate of the center of a circle fitted to thesingle curvature during formingmz′ Vertical coordinate of the flattened surface mxxxviiiAbbreviationsµCT X-ray micro–computed tomographyAFP automated fibre placementATL automated tape layingCFRP carbon fibre reinforced polymerCMM coordinate measuring machineCPT cured ply thicknessCTE coefficient of thermal expansionDDS 4,4’-diaminodiphenyl sulfoneDIC digital image correlationDOC degree of cureDSC differential scanning calorimeterDTMA dynamic thermomechanical analyzerE-B Euler-BernoullixxxixGFRP glass fibre reinforced polymerHCP hexagonal close-packingMDSC modulated differential scanning calorimeterMRCC manufacturer’s recommended cure cycleNDI nondestructive inspectionOEM original equipment manufacturerPAN polyacrylonitrilePES polyethersulfonePJN pin-jointed netPMP polymethylpenteneTGAP triglycidyl-p-aminophenolTGDDM tetraglycidyl-4,4’diaminodiphenylmethaneTGMDA tetraglycidyl-4,4’diaminodiphenylmethaneTMA thermomechanical AnalyzerUD unidirectionalUPT uncured ply thicknessxlAcknowledgmentsI would like to thank my supervisor Professor Anoush Poursartip. His insight, intuition, and per-sistence has shaped my character and brought a new level of detail to this work.I would also like to thank Professor Reza Vaziri and Professor Go¨ran Fernlund for their guid-ance and the opportunity to be a part of the Composites Research Network.To my friends and colleagues, Kamyar Gordnian, Renaud Daenzer, Kyle Farnand, Martin Roy,Mohammad Mohseni, and Alex Burns; I couldn’t have asked for better group of people to spend mytime with while at UBC. Alireza Forghani, thanks not only for being a friend but also for the manyfruitful technical discussions. A special thank you to past CRN graduate students, Leyla Farhang,Mina Shahbazi, James Kay, Gabriel Fortin, Sina Amini Niaki, Janna Fabris, Sanjukta Chatterjee,and Amir Garekani. All of whom made the graduate experience something I can cherish. I wishthe current and future graduate students of CRN all the best in their endeavours. I would liketo acknowledge CRN’s professional staff, especially Casey Keulen, Navid Zobeiry, ChristopheMobuchon, Roger Bennett, and Suzana Topic for keeping things running smoothly.I would like to thank my parents, John Stewart and Jo-Anne Trenholm, for inspiring me. With-out their support, none of this would be possible. To my brothers, sister, and Patricia Hirrel-Cyr,thanks for keeping my head screwed on right. Finally, I would like to thank my partner,李超, forher unwavering support and devotion.xliChapter 1IntroductionCarbon fibre reinforced polymer (CFRP) composite materials are well established in domains, suchas aviation, where high specific properties are required. CFRP’s are also seeing increasing use inthe automotive industry as fuel efficiency requirements become more stringent. The promise ofcomposite materials lays in their high performance to weight ratio, environmental stability, andease of manufacturing. This ease of manufacturing, when performed correctly, can drasticallyreduce the number of parts and fasteners allowing for simple, more cost effective, designs [1].Airbus’s A310-300 composite tail fin reduced the number of parts from 2000 to 100 over tradi-tional materials [2]. However, this massive reduction in part count comes with its own set of risks.Specifically, when a part fails to meet the design criteria, the investment into each individual part ismuch higher. Therefore, to unlock all of the benefits associated with fibre reinforced composites,a reduction of all defects is essential. While the line between design defects and manufacturingdefects is not always clearly defined [3], certain defects are the direct result of manufacturing.Having a fundamental understanding of defect initiation and propagation or an automated proce-dure for detecting and tracking these defects is critical as manufacturing shifts to more automatedsolutions [4].1In composites, the manufacturing procedure relies on a number of factors including desiredmaterial, part shape, cost, lead-time etc. Generally, thermosetting composites make up the largestmarket segment of fibre reinforced polymer materials [5] and within this classification, unidirec-tional (UD) prepreg tapes offer benefits in structural performance. Ideally, these UD thermosetprepregs have all of their fibres aligned in a common direction with a uniform distribution of fibrespre-impregnated by the thermosetting matrix, Figure 1.1(a). In reality, perfectly controlling thealignment of millions to billions of 5 µm to 10 µm diameter fibres in a single part is infeasible andslightly misaligned fibres are expected in both the raw material and final parts, Figure 1.1(b).(a) (b)Figure 1.1: (a) Schematic representation of an idealized ply featuring all straight fibres colli-mated to the 0° direction of the lamina. (b) A more realistic representation of the fibrenetwork with slight perturbations in fibre alignment with respect to the 0° direction.All fibre reinforced polymers mechanical properties are orientation dependent. The maximumpart strength and stiffness occur when the fibres are loaded along their axial direction. As thefibres deviate from the loading direction, their contribution to the strength and stiffness of the partdecreases. This effect has been repeatedly shown in the literature with reported reductions of 30 %to compressive strength [6] and smaller knockdowns to tensile strength and stiffness [7], [8]. Thesemisalignments lead to more conservative designs and ultimately increase part cost and weight.The degree to which these fibres are misaligned will contribute to the overall quality of theraw material; however, fibre alignment conformity is not currently a criteria stated on supplier datasheets nor is there a standardized quality metric. As well, current representations of the underlying2fibrous architecture are based on inappropriate assumptions and have not been able to reproduceexperimental fibre alignment data. Poor knowledge and control over other incoming factors suchas porosity, fibre volume fraction, thickness, tack, and other properties can also feed variability intothe final assemblies [9].Another complicating factor which can lead to misalignment defects is the uncured prepreg’shigh compliance at room temperature. This compliance is initially beneficial because it allowsthe prepreg sheets to be formed into complex geometries using moderate loads. Unfortunately,this compliance also makes the prepreg sheets susceptible to defects such as wrinkles. Thesewrinkles are caused by compressive stresses and can be formed under realistic loading conditions.Wrinkling in composites further alters the alignment of the material and reduces many of the partsmechanical properties. This alteration either requires rework or altogether part rejection whichincurs significant extra cost and time.Further, experiments have shown that these defects vary with a number of compounding factorssuch as temperature, forming rate, humidity, ply-ply interaction, tool-ply interaction, forming ge-ometry, etc. Predictive tools which are able to remove or mitigate these defects would be valuableto both design and manufacturing engineers. The mechanics of uncured prepreg along with workdone to capture the effects of each of these variables is reviewed in Chapter 2.The focus of this research is on the initiation and propagation of misalignment defects in uni-directional thermoset prepregs; however, the knowledge gained and principles applied throughoutare foundational and can be extended to other material systems and manufacturing techniques. Thefull scope and research objectives are fully detailed in the closing section of Chapter 2.3Chapter 2Literature review and researchobjectives2.1 Definition of termsIn the manufacturing of composites, a defect can be defined as a tracked outcome which exceedsa predetermined specification, whereby the part is considered to have failed. In composite mate-rials and composite material manufacturing there are a vast array of potential defects. Potter putforth a taxonomy for defects in composite materials which outlines 132 distinct defect types [10].The most common defects include voidage, delamination, human-error introduced defects, inclu-sions and contamination, errors in the cure process, and of most importance to this work, fibremisalignment defects.Working definitions for misalignment defects vary and there exists a lack of consistent nomen-clature. Misalignments can be easily defined as any deviation from the as-designed direction. Com-posite convention dictates a local orientation for unidirectional laminates where the 1-direction ofa lamina follows the nominal axial direction of the aligned fibres, the 2-direction lays in the plane4of the material and is normal to the 1-direction, and the 3-direction is in the through-thicknessdirection, Figure 2.1.123Figure 2.1: 3D representation of a unidirectional ply displaying the conventional coordinatesystem.This definition of misalignments includes user error where laminates are physically laid incor-rectly, for example where the lamina 1-direction is different from the as-designed 1-direction. Thistype of user-error is beyond the scope of this work; hence, misalignments will be defined as anyalignment deviation from the lamina’s nominal 1-direction. Even within this reduced scope, termssuch as waviness [11], wrinkling, buckling, marcelling [12]–[14], puckering [15], [16], blisters,etc. are often used interchangeably to describe similar phenomena.This work will use a nomenclature consistent with Bloom et al. and Lightfoot where wavinessrefers exclusively to in-plane misalignment while wrinkling refers to out-of-plane misalignment[8], [17]. Both of these definitions will hold at the fibre, tow, and lamina level, Figure 2.2. Thesubfigures of Figure 2.2 are recreated after a figure from Yurgartis’s work [18] which had the 1-direction defined but omitted the 2 and 3-direction. The two subfigures created here, one a rotationand crop of the other, have intentionally been created to highlight the fundamental similarity be-tween waviness and wrinkling. While they are fundamentally similar, they develop due to differentdeformation restrictions and a delineation between the two is useful. Both waviness and wrin-5kling are a subset of the more general encompassing misalignment defects and both could occursimultaneously at any length scale.12(a)31(b)Figure 2.2: (a) Schematic representation of a bundle of wavy fibres/tows/tapes and (b) a sec-tion of wrinkled fibres/tows/plies, recreated after Yurgartis [18].Buckling is defined in the mathematical sense as a bifurcation in the solution to the staticequilibrium equations. In other words, two or more deformed shapes precisely satisfy static equi-librium. In the classic example of a fixed-free rigid column under compressive axial loading, thesystem is unstable if an infinitesimal lateral displacement causes the column’s lateral deflection tocontinue to grow rather than return to the column’s initial state. The column is considered buck-led if it enters this growing lateral deflection. This definition is drawn from traditional structuralmechanics [19]. Euler’s critical load, Equation 2.1, states that the maximum compressive loadwhich can be carried by a column before buckling, Pcr [N], is proportional to the flexural rigidity,EI[Pam4], and inversely proportional to the column’s length, Lc [m].Pcr =pi2EI(KLc)2 (2.1)Euler bucklingwhere K is an effective length factor dependent on the mode shape of the buckled column.6Rosen [20] outlined two misalignment sub-classifications which can develop in wavy or wrin-kled features. An “extension” mode, Figure 2.3(a), and a “shear” mode, Figure 2.3(b). For com-posites with a fibre volume fraction greater than 20 % the dominant failure mode is the shear mode.For reference, most high performance composite materials feature volume fractions in excess of50 % [21].12,3(a)12,3(b)Figure 2.3: Schematic representation of wavy fibres/tows/tapes/plies undergoing (a) exten-sion and (b) shear compressive buckling, recreated after Rosen [20].2.2 Effect of defectsTo reduce risk and to deliver on the promised ease of manufacturing, a fundamental understandingof defect generation in composite would be highly beneficial. Misalignments cause a multitude ofissues; the most significant is their alteration to the mechanical properties of the laminate from theas-designed specification. Any alteration from the designed specifications leads to uncertainty andcan have significant impact on the in-service performance of the part and can lead to costly reworkor altogether part rejection [10], [17], [22].The mechanical property which is most impacted from misalignment defects is the compres-sive strength. As part of Rosen’s studies into the modes of compressive failure described above, hereached an expression relating the lamina’s compressive strength to material properties. He explic-7itly assumed a fully cured composite with fibres that are perfectly collimated and aligned with thelamina 1-direction. Any eccentricity will reduce the strength, hence his analysis returns an upperbound on the compressive strength of a unidirectional composite under axial loading. For com-posites with fibre volume fractions greater than 20 %, the maximum compressive stress, σc [Pa], isformulated as Equation 2.2 [20], [23], [24].σc = Gc (2.2)Rosen criteriawhere Gc [Pa] is the shear modulus of the composite.This early type of analysis yielded highly unconservative estimates for real laminates, over-predicting compressive strength values by a factor of 200 % to 400 % [24]. This over-predictionstems from the neglect of misalignment defects [24].Major modifications to this type of analysis were contributed by Argon in 1972 [25] who ac-commodated an initial misalignment value, φo. Argon proposed a rigid-perfectly plastic relation-ship between the compressive stress and fibre alignment, Equation 2.3. This was extended by Budi-ansky in 1983 [23] into an elastic-perfectly plastic composite response described in Equation 2.4.The Budiansky relationship reduces to the Rosen relationship when the initial misalignment isequal to zero and reduces to the Argon relationship for φo much larger than γy.σc =τyφo(2.3)Argon criteriawhere τy is the yield stress in longitudinal shear [26].σc =Gc1+ φoγy(2.4)Argon-Budiansky criteria8where γy is the yield stress of the composite in shear [26].Budiansky and Fleck compiled results from 10 experimental studies conducted on nominallyaligned fibre composites and plotted the critical compressive stress and estimated composite shearmodulus [26]. Fitting the entire set with a constant φo/γy returns a poor fit; however, the compositematerials used were not necessarily similar between studies and likely do not have the same φo/γy.Regardless of correlation, an average strength reduction in excess of 400 % was measured whencompared to Rosen’s perfectly straight fibres prediction which highlights the need for accuratemisalignment information.Garnich and Karami used finite element modelling to analyze the impact of fibre alignmenton a ply’s mechanical properties during tensile loading. These properties were investigated bymodelling wavy fibres in a otherwise unidirectional laminate using several different schemes todefine the fibre waviness [27]. A few of these scenarios include the fully periodic shear structure,depicted in Figure 2.3(b), a single wavy ply centered in a thicker stack of straight plies, and aply featuring a single half wavelength of wavy fibres at its centre, depicted in Figure 2.4. Anamplitude to wavelength ratio of A/Λ = 0.04 was assumed allowing the authors to compute thetensile modulus and tensile failure strengths. All scenarios showed significant knockdowns. Thefully periodic scenario described in Figure 2.3(b) saw the largest reduction with an 82 % reductionin strength and a 33 % decrease in E1 when compared to a perfectly aligned ply.912Figure 2.4: Schematic showing the half wavelength of wavy fibres in an otherwise perfectlystraight unidirectional ply, recreated after Garnich and Karami [27]The effect of wavy fibres on the laminate’s tensile modulus was also studied by Potter et al.They assumed a fibre bed composed of a set of sinusoidal fibres with a single defining amplitudeand wavelength but a random distribution of phase shifts. For their A/Λ ratio of 0.13, the ran-dom phase shift caused a less severe reduction in stiffness, 15 %, compared to the 26 % reductioncorresponding to the fully periodic shear structure. The authors conclude that while 15 % is stilla significant reduction, the actual impact for a quasi-isotropic layup may be insignificant due to abalancing of misaligned regions.Experimentally measuring the impact of fibre misalignment on laminate compressive strengthhas also repeatedly been demonstrated in the literature. Artificially introduced waviness, wavinesssuperimposed on the as-received misaligned values, has been shown to reduce the compressivestrength by 20 % when increasing the standard deviation of wavy misalignment from ±1° to ±6°[7], [28]. Artificially introduced wrinkles, even in the single symmetric central ply of a 22 plylayup, reduced the compressive strength of the laminate by 36 % [6]. Similarly, knockdowns of42 % have been reported for the compressive modulus of thick laminates with embedded wrinkles10with an A/Λ ratio of 0.043 [29], [30]. Similar strength knockdowns, 23 % to 40 %, have beenshown for tensile failure as well [8], [31].More recently, high-speed cameras have been employed to study the damage evolution aroundan artificially wrinkled laminate. The high-speed footage showed inter-ply delamination as thedominant failure mode for high severity wrinkles. This led to a similar, 33 %, reductions in com-pressive strength [32]. These tests showed that wrinkles with amplitudes on the order of single plythickness, 0.2 mm, can have serious ramifications.All of these results show not only the importance in not introducing waviness and wrinkles intoa structure during processing but also the importance of having proper input values for realisticstrength and stiffness predictions.2.3 Sources and controlling factorsThe current section will focus on the mechanisms influencing fibre alignment at each stage of com-posite manufacture. The specific materials of interest are preimpregnated, thermosetting, unidirec-tional, CFRP composites with fibres made from a polyacrylonitrile (PAN) precursor. Material pref-erence and availability of this project’s industrial partner, The Boeing Company, further reducedthe scope of interest to AS4/8552 and AS4/8552-1 prepregs. Topics encompassing all thermosetswill be covered; however, results in the literature for these material systems will be highlighted.A simplified sequence from PAN and resin to final part follows Figure 2.5 and Figure 2.6. Thesediagrams are not exhaustive, nor universal, several other stages can be inserted into the processas necessitated by the final part or assembly requirements. These additional stages often includenondestructive inspection (NDI), co-curing, bonding, drilling, assembly, shimming, and the like.11PAN CFResinPrepregging Storage ShippingFigure 2.5: Flow diagram of the PAN and resin to uncured prepreg process.Receiving Storage Thawing Cutting Layup and formingDebulkBaggingCuringDebaggingMachiningFinishing andshippingFigure 2.6: Flow diagram of the uncured prepreg to final part process.Several of these stages can result in compressive stresses being imparted into the fibres withsufficient magnitude as to satisfy the buckling criteria. The stages with a strong likelihood ofgenerating compressive stresses are discussed in the following subsections. Within each stageseveral mechanisms may occur simultaneously, each mechanism has been further broken out to aidthe discussion.2.3.1 Fibre and prepreg manufactureThe precursor for modern carbon fibres are PAN fibres [33], [34]. Typically, the first stage inprocessing PAN fibres into carbon fibres involves stretching the fibre to improve the orientationof the resulting carbon fibre [33]. This stretching operation already indicates that the carbon fibreprecursor is imperfectly aligned. Subsequent heat and sizing treatments then transform the PAN12into carbon fibres. These fibres can be directly spun onto a cardboard tube or they can be spreadprior to being spun onto the tube, forming a dry fibre creel. This spreading process involves pullingthe fibre bundle through a collimating rake which simultaneously sets the tow pitch and flattensthe circular cross-section into a rectangular cross-section tow (cited in [21]). These creels are thenoverwrapped with a plastic film for protection during transportation and handling [35]. The towstypically contain 3000, 6000, or 12,000 fibres each with a diameter of 5 µm to 10 µm [35]. UsingEuler’s buckling criteria, Equation 2.1, with these fine diameter fibres almost always results in nearzero critical buckling loads. In other words, almost any compressive loading on dry fibres willresult in misalignments.For the materials of interest here, hot-melt unidirectional prepregs, several of these creels areunwound then sandwiched between two resin films in the prepregging process. Comminglingof adjacent tows is ideal as this decreases void space or resin rich areas improving mechanicalperformance. This resin-fibre sandwich is heated and compressed causing the resin to infiltratethe void space between fibres [36], [37]. The prepreg sheet is then quickly cooled and spooledonto a take-up roll. This process can be run semi-continuously, from creel to prepreg, forming theunidirectional sheets provided to original equipment manufacturer (OEM)s.As the fibre bundle, tow, or prepreg ply, is wound onto a spool, stresses will likely developdue to the material’s finite thickness. The literature [9] suggests that following Euler-Bernoulli(E-B) beam theory, the magnitude of these stresses will be dependent on the forming radius and thedistance from the ply’s neutral axis. This will cause the surface closest to the center of curvatureto compress while the opposing surface will be placed under tension. Typical dimensions involvedare: a circular cross-section tow diameter of 1 mm, a rectangular cross-section width of 3 mm to5 mm, uncured ply thickness of 0.1 mm to 0.3 mm, creel core diameter of 75 mm, and a take upcore diameter of 150 mm to 400 mm [9], [35], [38].13Potter and colleagues [9] have proposed that misalignments should be introduced due to thisrolling and that these misalignments should vary along the length of a roll. These authors suggestthat parts cured from plies removed close to the roll core will feature higher excess lengths, andtherefore higher degrees of fibre misalignment. Excess length is defined as the difference in pathlength between the wavy fibre and an ideal straight fibre following the lamina 1-direction. Aschematic for the rolling process on a high shear stiffness material is shown in Figure 2.7 [39].They propose that the expected maximum angular misalignment of an inextensible, sinusoidal,fibre can be determined by equating the wavy fibre path length at the inner ply surface to a straightfibre at the outer ply surface. Although they did not present an equation describing this effect, suchan equation can be easily derived, as in Equation 2.5.∫ θF RF0√1+(dydx)2dx = θF (RF + tc) (2.5)where dydx is the slope of a sinusoidal type fibre, θF is an arbitrary angle over which the fibre followsat a radius RF [m], and tc [m] is the combined backing paper and uncured ply thickness.tρρ+t(a)ρρ+t(b) (c)Figure 2.7: Schematic of the rolling process of a high shear stiffness beam with inextensiblefibres resulting in compression on the inner surface. This compression is represented bya buckled section on the inner surface, created after concepts in [9], [22], [39], [40].14The values Potter et al. used, a 0.25 mm thick unidirectional ply wrapped over a take-up corewith a diameter of 300 mm, results in a derived maximum angle of misalignment of 4.7°; however,a series of solutions can be generated using Equation 2.5 and are presented in Figure 2.8. 3.8 4 4.2 4.4 4.6 4.8 150  160  170  180  190  200  210  220  230Maximum angle  of misalignment [°]Radial position [mm]Figure 2.8: Predicted maximum angle of misalignment as a function of radial position on theroll, as calculated by Equation 2.5, tc = 0.25mm.Farnand outlined several boundary cases for single plies and stacked plies wound over a radius[39]. The high shear stiffness, Figure 2.7, outlines one bounding condition while the ideal shearscenario, where the material has no resistance to shear, forms the opposing bounding condition,Figure 2.9 [39]. The ideal shear scenario results in an angular position difference between theinnermost section of the ply compared to the outermost section of the ply. This effective excesslength is called a bookend, δL [m] [40]. This bookend length is accommodated in Figure 2.7 as theexcess length driving waviness or wrinkling [39].15tρρ+t(a)ρρ+t(b)δL(c)Figure 2.9: Schematic of the rolling process of the ideal shear bounded case highlighting thebookend length, created after concepts in [22], [39], [40].The authors are not aware of any literature which has experimentally characterized the mis-alignment of fibres in the dry tows as received by a prepregger. Further, the prepregging processitself is normally proprietary and the exact processes and conditions are unknown. Characterizationof fibre alignment during the prepreg processing stage is also an area lacking experimental data.2.3.2 Layup and formingThe current manufacturing bottleneck for unidirectional composites, and the focus of the vast ma-jority of published literature relating to the development of misalignment defects, is the layup andforming stages of CFRPs [41]. Much of the literature relates to forming of woven fabrics; however,extensions to unidirectional ply forming have been trialled due to the relative ease in derivation andsimulation. The earliest published piece of analysis concerning the draping of fibrous materialsover relatively complex geometries comes from Mack and Taylor in 1956 [15]. They define a fit-ted, or successfully draped, material as one where the cloth is in contact with the surface over theentire desired region. Their proposed pin-jointed net (PJN) model assumes that the cross-over ofthe warp and weft act as pin joints with straight, rigid, bars connecting adjacent pins. A furtherrestriction is that the smallest radius of curvature to be fit is much larger than the warp-weft spac-ing, allowing the fabrics bending stiffness to be neglected. Using this PJN method, a 2D cloth16can drape a 3D geometry, such as the doubly curved surface of a hemisphere, solely by rotation atthe joints. The resulting in-plane shear values can be quickly computed from the spacing betweentows, st [m], and the tooling geometry. The locking angle of a weave, depicted in Figure 2.10 asβL, defines the angle at which further shearing will produce wrinkling as adjacent tows come intocontact with one another [42].sttwβ(a)β(b)βL(c)Figure 2.10: (a) Initial weave with 90° warp and weft tows, (b) weave undergoing in-planeshear, and (c) weave at the locking angle, recreated after Prodromou and Chen [42].Note that the white connecting lines do not change length between steps in shear.This locking angle can be derived from geometric properties of the weave, Equation 2.6, andagrees well with typical measured values of between 20° to 45° [42]–[45].βL = arcsin(wtst)(2.6)where wt [m] is the width of the tow.The locking angle is derived using the same assumptions as PJN theory, specifically restrictingslip between joints, and can be shown to greatly underestimate the ability of a weave to shear if theboundary conditions are modified via a clamping force [46]. The Boeing Company uses a similarclamping force approach when forming unidirectional prepreg as a wrinkle mitigation strategy17[47]. Regardless of the model’s assumptions, wrinkling will occur when out-of-plane deformationbecomes more energetically favourable than in-plane shear [14].This simple PJN relationship and locking angle have seen such extensive use that efforts havebeen made to extend them to the deformation of uncured unidirectional prepreg layups [48]–[51].While some agreement could be found for early stages of forming, the fundamental lack of strongcontact between plies has led to a large variation in wrinkling onset without a defined locking angleindicator [50], [52].Experimentally, several sources showed that higher forming temperatures and lower formingrates reduce both the load required to form and the likelihood of wrinkling [50], [52]–[54]. Com-paring the relative influence of temperature and loading rate, a 30 ◦C increase in temperature, aboveambient, was found to have a similar impact as a thousand-fold reduction in rate [50].Similarly, layup sequence and orientation have been shown to play a significant role in defectgeneration [41], [55]–[57]. Interaction between plies with different stiffnesses define a set of likelydeformation modes [57], [58]. Johnson et al. [57] outlined a strategy for determining the compati-bility between any two plies by comparing the energy required for the deformation. For example,two plies with the same orientation, e.g., [0°/0°] or [90°/90°], are compatible because their lowenergy deformation modes are identical and a [0°/90°] interface is compatible because they sharea low energy shearing mode. A quasi-isotropic layup, [45°/0°/−45°/90°], has a set of interfaceswhich are incompatible because the relatively low stiffness, E2,p [Pa] and G12,p [Pa], of one ply isaligned with the high stiffness, E1,p [Pa], of another ply in the layup [57], [58]. This incompati-bility leads to an inability to shear and eventual wrinkling. This was corroborated from their ownexperiments, as well as aided in explaining the origin of the wrinkle defects viewed in Hallanderet al.’s work [41], [58]. Hallander et al. formed several layups into a doubly curved shape using ahot drape process at 65 ◦C. They observed wrinkles in areas which were influenced by the layup;layups restricting shear deformation featured wrinkles while those prone to shear were defect free.18This type of analysis was designed to be “simplistic and quick” [57] relying on only the plies in-plane stiffness values and “does not accurately describe the deformation of uncured laminate norany inter-ply slippage” [58].More involved methods for predicting misalignment defects during forming must thereforetake into account more deformation modes including tension, in-plane shear, and bending [59].Further, tool-part [60], ply-ply [61], [62], and ply-bag [17], [56], interaction are also important.Tool-part and part-bag interactions are more aligned with the scope of the debulk and cure section,Section 2.3.3, and will be discussed further therein.Ply-ply interactionPly-ply interaction has seen significant research in the last five years. Ply-ply interaction is mostoften measured by sliding a ply relative to a fixed ply at a constant rate. Early researchers useda ply pull out scheme where a central ply was removed from a stationary stack under an imposedtransverse load, typically a static mass placed on the stack [63], [64]. Either the change in contactarea as the ply was removed needed to be accounted for or the material overhang needed to becharacterized. These proved problematic and led to several modified apparatus designs [34].Larberg et al. designed a symmetric rig where plates, overwrapped in CFRP, could be slid alongone another [48]. Given a larger central plate and smaller moving plates, the contact area could bekept constant. Load-displacement traces were reported for a range of materials, temperatures, andstrain rates. In general, the load-displacement traces showed initially high stiffnesses followed by astable, invariant, response with further displacement. The complex sliding behaviour was regressedusing coefficients of friction calculated from the stable region to varying degrees of success. Above45 ◦C, the coefficient of friction AS4/8552 was found to follow a hydrodynamic friction response,meaning the shear stress is proportional to the strain rate, Equation 2.7. The other material systems,T700/M21 and HTS/977-2, either varied with strain rate or normal force indicating a complex19response deviating from the often assumed Coulomb friction relationship, Equation 2.8 [61]. Thereported ply-ply coefficient of friction, µPP, values for all materials ranged from 0.01 to 0.15.τ =ηdν (2.7)Hydrodynamicwhere τ [Pa] is the shear stress, η [Pas] is the matrix viscosity, d [m] is the film thickness andv[ms−1]is the relative velocity.Ff = µFN (2.8)Coulomb frictionwhere Ff [N] is the friction force, µ is the coefficient of friction, and FN [N] is the normal force.Flanagan reported µPP values for a thermosetting prepreg composed of Fiberite Grade95 fi-bre (PAN precursor) with an Epoxy 934 matrix of approximately 0.01. The author had difficultydetermining an exact value due to the very low forces required to yield [63].Erland et al. used an apparatus modelled after Larberg et al.’s but coupled the analysis with abilinear model to accommodate the initial stiffness followed by a stable response [34]. For theirAS4/8552 material, the transition between initial stiffness and yielded stiffness, named the inter-ply critical shear stress, varied nearly an order of magnitude from 40 ◦C to 80 ◦C. The authors hadthe lowest confidence in these critical shear values and stated they were the least repeatable valuespresented in the work due to the compounding errors in both strain rate and normal force [65]. Thisyield value was also shown to be invariant with fibre interface angle [65]. This body of work isvaluable for capturing the inter-ply interaction occurring during the early stages of forming whichhad typically been neglected.20BendingA 2018 review paper on wrinkling of composite preforms and prepregs highlights the unsolvednature of uncured composite wrinkling and attributes the error to PJN’s negligible bending stiffnessassumption [66]. The review also highlights the disparity of published literature between fabricsand unidirectional composites with only 10 of the 153 citations focused on UD materials.Many research teams have shown that modifying the ply bending stiffness can alter the forma-tion of wrinkles [14], [46], [67], [68]. Decoupling bending stiffness from the in-plane properties isrequired due to the relative inextensibility of the fibres. For dry fibres, the bending stiffness is dras-tically reduced from the nominally high in-plane properties due to the possible slippage betweenfibres [66]. While dry fibres will undergo significantly more slippage, AS4/8552 and AS4/8552-1prepregs have “engineered vacuum channels”, a core of dry fibres to aid in air evacuation, whichcould also undergo slippage [41]. However, even the assumed high values for the in-plane prop-erties is not a simple conclusion. Using standard rule of mixtures to calculate the axial elasticmodulus from fibre modulus, resin modulus, and fibre volume fraction leads to prediction on theorder of 100 GPa, [57], [58], [69]. However, similar to the large reductions in compressive strengthvalues caused from misaligned fibres, the axial tensile modulus will also be greatly reduced [9].During the tensile loading of uncured unidirectional prepregs, the misaligned fibres must alignwith the direction of applied load prior to imparting their relatively high stiffness. Very limitedexperimental work is available in the literature for direct measurement of the uncured axial tensilemodulus; however, lead-in strain values of 0.1 % have been measured [9]. This lead-in region isnon-linear with an upper bound of 5 GPa to 10 GPa for the tensile modulus. Closer to zero strain,the tensile modulus is even further reduced. Alshahrani et al. performed a series of buckling exper-iments on an uncured woven CFRP prepreg and calculated a compressive modulus of 190 MPa forthe warp direction and 140 MPa for the weft direction. The woven prepregs were then run througha cantilever test which showed a viscoelastic response. The applied bending moment to maintain a21constant deflection did not decay to zero, rather the bending moment was asymptotic to 31 % of itsinitial value within 5 min [70]. Compressive modulus data for uncured unidirectional prepreg wasnot found; however, given the drastically reduced tensile modulus and the trends from the wovendata, assuming a high compression modulus as predicted from the rule of mixtures seems unlikely.A large range for the out-of-plane bending property have been reported. Sjo¨lander et al. useda modulus of 100 MPa when modeling a toughened carbon fibre thermoset unidirectional prepreg[14]. Belnoue et al. measured the bending stiffness of IMA-M21, an intermediate modulus fibreand high performance toughened resin system, using a modified Pierce cantilever test [71]. ThePierce cantilever test [72] or the ASTM cantilever test [73] involves horizontally cantilevering astrip of material such that it deflects by 20° to 30° under self-weight. While not a perfect solution,Euler-Bernoulli beam theory is then applied to determine the bending stiffness. Belnoue et al.reported the bending stiffness as a function of temperature, replotted in Figure 2.11. 0 5 10 15 20 25 20  40  60  80  100  120 293  313  333  353  373  393Bending stiffness [Nmm]Temperature [°C]Temperature [K]Figure 2.11: Bending stiffness of uncured IMA-M21, recreated after Belnoue et al. [71].Dodwell et al. measured the bending stiffness of AS4/8552 in a dynamic thermomechanicalanalyzer (DTMA) over a range of temperatures consistent with forming. Datasets from two dif-22ferent papers have been condensed into Figure 2.12 [40], [74]. The inconsistent use of bendingstiffness units, Nmm2 and Nmm, is a reflection of the original data sets. 0 20 40 60 80 20  40  60  80  100 0 4 8 12 16 293  313  333  353  373Bending stiffness [Nmm2 ]Bending stiffness [Nmm]Temperature [°C]Temperature [K]Dodwell et al. 2015Dodwell et al. 2015Dodwell et al. 2014Dodwell et al. 2014Figure 2.12: Bending stiffness of AS4/8552 from two of Dodwell et al.’s publications, recre-ated after [40] and [74].Both of Dodwell et al.’s datasets measured the bending stiffness of a single uncured ply ofAS4/8552 with a measured nominal thickness of 0.2 mm and width of approximately (5.0±0.2)mm[75]. Both ends of the samples were clamped and a center point displacement was applied. Fig-ure 2.12 highlights the difficulty in measuring this bending property. At 20 ◦C, the two datasets aredifferent by a factor of two while at 80 ◦C the 2014 dataset is larger than the 2015 dataset by a factorof greater than five. Regardless of the precise value, both of the datasets show a strong temperaturedependence which is asymptotic close to zero. The reported best fit for the bending stiffness, with aB∞ of 4.46×10−9 Nmm2, collected from the 2015 dataset were fit according to Equation 2.9 [74].The equivalent bending stiffness varies from 24 GPa at 20 ◦C to 150 MPa at 100 ◦C.B = B∞ · eEBRT (2.9)23where B[Nm2]is the bending stiffness, EB is 5.75×104 Jmol−1 and R is the ideal gas constant,approximately 8.314 JK−1 mol−1.Scenario specific complicationsSeveral other features can serve as initiation sites for misalignment defects. One scenario which ismore common due to the rise in automated tape laying (ATL) and automated fibre placement (AFP)is laps and gaps [4]. Both layup strategies involve laying down material using a robot control head.ATL uses wide rolls, 75 mm to 300 mm, while AFP uses slit tapes from 3.2 mm to 12.7 mm wide.Gaps, unfilled space between two adjacent tapes, and laps, the stacking of two adjacent tapes, are anatural consequence of imperfectly precise tools [76]. These features are typically 0.5 mm to 1 mmand occur more frequently in AFP due to the increased number of tapes per part width. Gaps orlaps both cause subsequent layer(s) in a stack to deviate through thickness, causing a wrinkle toform, as the subsequent layer(s) conform to the altered surface profile [77], [78]. Gaps will alterthe ply’s interface condition with sufficiently large gaps completely removing any local interaction.2.3.3 Debulk and cureDebulkConsolidation, used to remove air trapped between plies, creates a similar scenario to forming overa radius. With several plies already laid up onto a curved tool, the reduction in thickness causedfrom consolidation causes another bookend problem shown in Figure 2.13, which is similar to casesdescribed in Figure 2.7 and Figure 2.9 [40], [74]. This reduction, typically 7 % to 20 %, is knownas the bulk factor [41]. For plies with non-zero inter-ply and intra-ply shear stiffness, constrainedconsolidation will take place causing some of the bookend distance to be accommodated into thematerial by fibre misalignment.24δLUnconsolidatedConsolidatedFigure 2.13: Schematic diagram showing the excess length on the outer surface as a result ofconsolidation over a single curvature, recreated after Dodwell et al. [40].Dodwell et al. developed a model to capture the buckling of plies due to constrained consoli-dation. The model incorporates consolidation behaviour of the stack, bending stiffness of each ply,and inter-ply shear properties [40]. This process is not significantly different from the forming pro-cess previously discussed and more emphasis will be placed on the other compressive mechanismsobserved during debulk and cure.Tool-part interactionCompressive stresses can also arise from tool-part interaction. Twigg et al. developed an analyticalmodel to describe the internal stresses of a composite adhered to a thin tool undergoing a thermalcure cycle [60], [79], [80]. The stresses arise from the difference in coefficients of linear expan-sion of the tool and part, α2 and α1, respectively. This relationship was derived for a linear elasticresponse with no spatial temperature gradients. This model further assumes that the adherents un-dergo no shear while the adhesive undergoes no extensional strains. Two scenarios can occur; asticking condition and a sliding condition. The sticking condition arises where in-plane stressesdevelop over a shear interface coupling, τ . The sticking condition is similar to the stress develop-25ment in the shear-lag approach wherein in-plane stresses build logarithmically from zero, at a freeedge, to a far-field stress, if the interaction length is sufficient [81], [82]. The derivation followsthe free body diagram shown in Figure 2.14 with the shear and in-plane stresses described usingEquation 2.10 and Equation 2.11, respectively. The correct form of the λ parameter is also shownin Equation 2.12.Figure 2.14: Diagram of the sticking interface condition of a ply separated from the toolthrough an adhesive interface, reproduced from Twigg et al. [60].τ =(eλx− e−λx)· Gr · (α2−α1) ·∆TtA ·λ ·(eλL+ e−λL) (2.10)σ1 =1h1∫ xL−τdx (2.11)where:26λ =√GrtA·(1h1E1,p+1h2E2)(2.12)where L [m] is the length over which the interaction develops, ∆T [K] is the temperature difference,Gr [Pa] is the shear modulus of the interlayer, tA [m] is the thickness of the interlayer; α1[K−1]and α2[K−1]are the linear coefficients of thermal expansion, σ1 [Pa] and σ2 [Pa] are the in-planestress, h1 [m] and h2 [m] are the thickness, and E1,p [Pa] and E2 [Pa] are the elastic modulus of thecomposite ply along the 1-direction, ‘1’, or the tool, ‘2’.Kugler and Moon [83] corroborated the influence of mismatched tool-part coefficient of ther-mal expansion (CTE) on fibre waviness by performing an experimental study comparing wavinessin thermoplastic parts cured on a variety of tooling materials. Tool-part interaction was found tobe a significant driver in developing waviness and was proportional to the magnitude of the CTE.Plates cured on low CTE graphite or invar tools did not suffer any gross waviness defects whilesteel tooling, with an intermediate CTE, suffered moderate waviness. High CTE tooling, brass oraluminium, suffered the largest density of wavy regions. For their thermoplastic material, wavinessdid not occur above the glass transition temperature, rather waviness developed between the glasstransition temperature and deflection temperature.If the shear stress reaches the critical shear stress, τDebond [Pa], the interface debonds leadingto the sliding condition. Twigg et al. showed a similar derivation for the sliding condition wherethe shear stress is constant along the tool-part interface, τsliding [Pa]. This shear stress resultsin a linearly increasing in-plane stress from the stress free state at the edge of the specimen to amaximum at the center. Twigg et al.’s solution derived the stresses in the tool, shown in Figure 2.15;however, the part’s in-plane stress can be determined from equilibrium, Equation 2.13.27Figure 2.15: Diagram of the sliding interface condition showing the tool under load from aconstant shear stress, τsliding, reproduced from Twigg et al. [60].σ1 =τsliding · (L− x)h1(2.13)The sliding shear stress is often taken to follow a Coulomb friction response, described inEquation 2.8, where the interfacial shear stress is equal to a coefficient of friction multiplied bythe transverse pressure. The tool part coefficient of friction, µT P, is not as well characterized asthe ply-ply coefficient of friction, µPP; however, Flanagan has reported values for a large set ofexperimental conditions using the same Fiberite Grade95 fibre and an Epoxy 934 matrix [63].Flanagan reported a nearly 5 fold increase in µT P when switching from a release film to a Frekotecoated tool at room temperature. Further, a nearly 6 fold increase in µT P when decreasing thetemperature from 60 ◦C to 20 ◦C. Comparing the relative magnitude of the µT P to µPP, the µT P isat least 20 fold larger than µPP [63]. This indicates that the tool has a much larger ability to transferstresses into the part than the plies can impart on one another and that the tool surface conditioncan also drastically alter the magnitude of imparted stresses.28The Coulomb friction approach requires a normal load to be applied to resist translation. Inother words, without a normal force, there is no resistance to slipping. This is not physically repre-sentative of the tacky resin interface. Tack is defined as the pressure sensitive adhesive propertiesof the uncured prepreg matrix [84]. This property is a function of both the resin properties and thesurface properties. Temperature and strain rate have been shown to play a critical role on tackiness,with increasing temperature resulting in lower debonding forces [84], [85]. Characterization oftack is commonly performed using a probe tack test where a relatively small probe, several mmin diameter, is forced into a sample, resin or prepreg, left for a dwell period, and removed fromthe specimen at a constant rate [85], [86]. The output of this test is typically a stress-displacementcurve which defines the strain energy release rate required to debond the specimen from the probe.Wohl et al. used a design of experiments to study the effect of dwell time, normal force, crossheadspeed, temperature, and relative humidity to determine each parameters influence on a series ofoutcomes, including the scaled adhesion force, for IM7/8552-1 [86]. A scaled adhesion force, theforce measured by the instrument normalized by the probe’s cross sectional area, was reported asthe contact area in a probe tack test is difficult to determine leaving the true stress state unknown.The model with the least complexity and highest accuracy followed a quadratic model, hence manyof the responses had local maximum or minimum values. For example, the maximum scaled ad-hesion force increased with increasing relative humidity up to an RH of 50 % to 60 %, explainedas an increase viscoelasticity and “tack”, while continued increase in relative humidity decreasedthe scaled adhesion force, explained as a decrease in viscosity. Collectively, the parameters whichhad a strongest influence on tack was temperature and humidity. The tests spanned 25 ◦C to 80 ◦Cand 20 % to 80 % with maximum tack occurring at the lower temperature and moderate humidityvalues [86].Another test for determining prepreg tack which is more representative of both hand and auto-mated layup strategies is the peel test. Rather than using a probe, the peel test uses a floating roller29[87], [88] or a fixed roller [89] to separate a bonded material from a substrate. Crossley et al. useda fixed roller peel test to determine the force required to debond a non-commercial GFRP prepregfrom a cleaned but untreated steel substrate. The results from one of the tested GFRP samples isshown in Figure 2.16 [84]. This figure shows the strong relationship between the force required todebond and the temperature and feed rate. Peaking behavior, similar to Wohl et al. [86] can alsobe seen, where slow and fast feed rate correspond with relatively low tack while intermediate feedrates correspond with much larger forces. 0 5 10 15 20 1  10  100  1000 0 50 100 150 200 250Tack [N 75mm⁻¹]Critical strain energy release rate [J/m2 ]Feed rate [mm min⁻¹]19.9 °C25.4 °C28.2 °C30.9 °C34.4 °C38.6 °C40.9 °CFigure 2.16: Critical strain energy release rate from a peel tack test performed on a non-commercial GFRP at various temperatures, sample ‘PP2’, recreated from Crossley etal. [84].Bag-ply interactionOne other recently developed interaction which has been attributed to wrinkle and waviness growthduring processing of composites is bag-ply interaction. The example given is of a bridged ply laidonto a conforming stack which undergoes consolidation during debulk and cure. This consolidationcauses the bridged ply to drag the adjacent conforming ply into the radius creating an excess lengthproblem resulting in a misalignment defect. A study into this type of defect using unidirectional30IM7/8552 formed on an aluminium female tool showed that consolidation of ply bridging canlead to waviness, in excess of 40°, and wrinkling, ranging from 2 to 12 CPT [56]. Reducingthe amount of bridging was found to decrease, or entirely remove, the process induced wrinkles[17]. Lightfoot recommends reduced ply bridging by using rollers with radii smaller than the tool’sminimum radius or using compliant rollers to allow the roller to conform to the tool radius. Using aroller also reduced the level of waviness relative to the samples laid up by hand. This was attributedto the even distribution of pressure provided by the roller and the decreased variability in operatordexterity.CureFibre alignment is also affected by the cure cycle itself. The cure cycle inherently modifies theresin system which acts as a viscoelastic foundation upon which any stressed fibre acts against. Arepresentation of the resin volume and viscosity during each stage of a single-hold cure cycle isshown in Figure 2.17. The heating stage, 1 , of a cure cycle decreases the resin’s viscosity whilesimultaneously causing the resin to expand relative to the carbon fibres, which have a very low axialcoefficient of thermal expansion [90]–[93]. During the isothermal hold, 2 , the resin’s ability toimpart loads into the fibres develops concurrently with resin cure shrinkage. In 1964, Rosen brieflydiscussed viewing an ‘elastic instability of glass fiber’ associated with resin shrinkage [20]. Muchmore recently, Jochum et al. was able to directly show that cure shrinkage can cause single fibresto buckle [94]. This effect was only observed prior to glass transition indicating the gel offeredsufficient stiffness to halt continued buckling. Lightfoot tried to replicate the compressive effect ofcure shrinkage on a multi-fibre system of unidirectional carbon fibre with woven glass fibre yarns;however, instrument resolution did not allow signal to be separated from noise [17].31(a) (b)Figure 2.17: (a) Resin volume change, recreated after Garstka et al. [93], and (b) resin vis-cosity change of a typical epoxy resin during a single hold thermal cure cycle withhighlighted process direction, initial temperature, To, and curing temperature, Tc.2.4 Material characterizationGiven all of the mechanisms occurring throughout the entire manufacturing process, any investiga-tion into the prepreg’s initial values must be carefully interpreted from post-cure characterization.Vice versa, any uncured prepreg property which is a function of the fibre and resin state mustjudiciously use post-cure values.For thermoset prepregs, accurately measuring incoming quality is difficult due to the rela-tively soft properties of the matrix with many characterization techniques relying on sectioned mi-croscopy of stiff materials [18], [95]. Modifying these existing techniques for application towardssoft composite materials remains a challenge [96]. Traditionally, in preparation for microscopy,thermosets are run through a cure cycle but this inherently modifies the state of the fibres, matrix,void location and content, and so forth. For example, the shear modulus of a resin decreases byseveral orders of magnitude during the heating ramp of a thermal cure cycle [97]. Therefore, anyinvestigation into pre-cure values which are dependent on resin modulus and cure shrinkage mustbe carefully interpreted from post-cure characterization.322.4.1 Chemical compositionSimilar to other resin systems used in the aerospace industry, the exact chemical composition ofHexcel’s 8552 and 8552-1 is proprietary and is unknown to the end users. However, Hexcel’s8552, has been studied extensively in the literature and its chemical composition, Figure 2.17, hasbeen documented as a stoichiometrically epoxy-rich tetraglycidyl-4,4’diaminodiphenylmethane(TGDDM), also sometimes abbreviated as TGMDA, premixed with an amine curing agent 4,4’-diaminodiphenyl sulfone (DDS) [98], [99]. An additional epoxy component, triglycidyl-p-aminophenol(TGAP), has more recently been discussed, along with polyethersulfone (PES) acting as the tough-ener which is described in the product data sheet [100], [101].(a) (b)(c) (d)Figure 2.17: Various components of AS4/8552, epoxy components (a) TGDDM and (b)TGAP, along with the curing agent (c) DDS and toughening agent (d) PES.A full description of the reaction pathways can be found in the literature [99], [102] and isbeyond the scope of this work, however, a brief description of the main pathway and reactantsis warranted. During a traditional thermal cure cycle, primary amines, DDS, react with epoxides,TGDDM, to form secondary amines. These secondary amines can go on to react with other primaryamines to complete the reaction and form tertiary amines. With three, or more, functional groups,33the compound can form expansive 3-dimensional networks from this primary-secondary-tertiaryreaction path.At room temperature, the DDS-epoxy reaction is extremely slow [103]; therefore an alternativestrategy is required to advance cure without increasing the resin temperature. Substituting a morereactive trifunctional amine, specifically gaseous ammonia, to advance cure has been suggestedas a potential method. This technique has been used to advance cure of epoxy powders [104]and thin epoxy coatings on animal fibres [105]. Rather than using gaseous ammonia at elevatedtemperatures, room temperature aqueous ammonium hydroxide will be explored in this work as apotential means of stiffening epoxy prepreg in preparation for characterization.2.4.2 Fibre alignmentCharacterization of fibre angle in formed and processed fibre reinforced composite parts, proposedby Fakirov et al. [106] but formally established by Yurgartis for continuous composites [18], isone popular methodology seen in the literature, e.g. [56], [107]–[109]. This method uses opticalmicroscopy images and compares the major and minor diameters of sectioned fibres to determinetheir misalignment, Equation 2.14. According to Google Scholar, over 200 papers have cited Yur-gartis’s original work at the time of this writing. Of these approximately 200 papers, roughly onefifth of them had applied the method, generating standard deviation values, ς , of the in-plane fibremisalignment for their respective material systems. Nearly three-quarters of the 200 papers haveinterpreted this standard deviation value, ς , as a characteristic ‘misalignment value’, e.g. [110]–[112], and implemented it as a mean misalignment, initial fibre misalignment, or maximum mis-alignment for subsequent strength, failure, or simulation analysis. However, this reinterpretationof the standard deviation as an average fibre descriptor is inconsistent with the underlying idea thatthe standard deviation is a description of a varying population. In a subsequent paper, Yurgartisclearly states that “by definition, the average angle is zero” [113]. This redefinition of the standarddeviation value will also be further addressed in Chapter 3.34Equation 2.15, and Figure 2.18 outline this procedure. The figure has three straight fibresrepresented by blue, red, and green cylinders. These fibres have a misalignment of −2.5°, 0°, and2.5° relative to the nominal 1-direction of the virtual system, respectively.sin(ωi) =di,mindi,ma j(2.14)where ωi is the individual fibre angle relative to the cutting plane, di,min is the minor diameter, anddi,ma j is the major diameter of the sectioned ith fibre.θi = ωi−ΦPC (2.15)where θi is the individual fibre angle relative to the nominal 1-direction and ΦPC is angle of thecutting plane relative to the nominal 1-direction. The recommended cutting angle is 5°, however,the precise value cannot be known a priori and is taken to be the mean value of the misalignmentdistribution.35132(a)ΦPCωiθi12(b)Figure 2.18: (a) A perspective view of three straight cylindrical fibres intersected by an or-ange 5° degree cutting plane. The blue, red, and green fibres have an in-plane misalign-ment angle of −2.5°, 0°, and 2.5°, respectively. (b) Highlights the angles of interestfor the green fibre when sectioning.di,maj di,minFigure 2.19: Resulting cross-sections on the cutting surface described in Figure 2.18(a). Themajor and minor diameter, required in Equation 2.14, are highlighted for the greenfibre.The justification for the 5° section relative to the 1-direction rather than the typical 90° section,often used for porosity determination, lies in the relative change in diameters. According to Yur-36gartis, 83 % of the fibres studied in his prepreg system fell within ±1° of the nominal 1-direction[18]. A 1° misalignment relative to a 90° section would show only a 0.02 % increase in the el-lipse’s major diameter in relation to its minor diameter. Even a very large misalignment of 25°would only result in a major-to-minor diameter aspect ratio of 1.1. The resultant section faces forseveral misalignments relative to a 90° section are shown in Figure 2.20. This figure highlights thevery small differences in major to minor diameter aspect ratios for misaligned fibres sectioned 90°to the nominal 1-direction. This near unity aspect ratio provides little room for error in diameterdetermination to return proper alignment values.Figure 2.20: Example sectioning of misaligned fibres cut along the 2-3 plane as would typi-cally be performed for fibre volume fraction or porosity characterization. The 0° degreefibre follows the lamina 1-direction and the other fibres are misaligned from this fibreby their corresponding values.By comparison, if the section is made at a shallow angle, such as the proposed 5° angle, thevast majority of fibres will have aspect ratios between 9.6 to 14.3 for the same ±1° misalignmentboundary. However, this raises an issue for a fibre intersecting the cutting plane at the same angleas the cutting plane. For this example,−5°, the major diameter would tend towards infinity makingmeasurement impossible. Also, this will eventually violate the assumption that “short sections offibers are approximately straight” over the distance they are sectioned [18]. Several virtual mis-aligned fibres sectioned at 5° to the nominal 1-direction are shown in Figure 2.21. In contrast to37Figure 2.20, Figure 2.21 highlights the large aspect ratio of fibres within a few degrees of mis-alignment to the nominal 1-direction. Another issue using this technique is that any misalignmentangles less than the negative cutting plane angle, ω <−φPC, fold back on the distribution [18]. Forexample, a misaligned fibre with a true misalignment of −6° returns a measured misalignment of1° using Equation 2.14. For well aligned fibres, this error is small as number of highly misalignedfibres is typically on the order of 0.01 % [18].Figure 2.21: Example sectioning of misaligned fibres cut at 5° to the 1-3 plane. The 0° degreefibre follows the lamina 1-direction and the other fibres are misaligned relative to thisfibre by their corresponding values.This Yurgartis technique has been modified to generate more information by several authors.Hine et al. [114], [115] captured both in-plane and out-of-plane information from a single crosssection and also studied the effect of sectioning angle, 0°, 30°, 45°, and 60°, on the returneddistributions. Out-of-plane angle information was measured directly from the fibres orientation,however, very high cutting plane angles were required due to the highly scattered fibres. Low con-fidence data was generated for the 0° cutting plane due to the large number of fibres with extremelyhigh aspect ratios. Higher angle cutting planes returned more reasonable results with most fibreslaying close to the lamina’s 1-direction. Another modified technique which has been be used onmaterials with a non-opaque resin and fibre systems, such as glass fibre or low volume fractionCFRP, is confocal laser scanning microscopy [116]–[118]. This specialized piece of equipmentallows information to be gathered through the depth of a sample. For GFRP’s, the useable depthvaries from 150 µm to 30 µm when increasing the fibre volume fraction from 30 % to 50 %. Byadding some depth information, this technique removes the ambiguity for fibres with misalign-38ments less than the negative of the cutting angle as the positive or negative value can be inferredfrom fibre volume below the surface.While the Yurgartis technique describes reducing all spatial information into a single distribu-tion, a windowing technique has been used by some authors to retain some spatial information [17],[56], [119]. This windowing technique has been applied to capture the ‘overall’ wavy response offibres in proximity to a wrinkle. These wavy regions showed very large misalignments, in excessof 40° [17], [56]. These authors did not provide distribution data making it difficult to determinethe degree of local coordination, and the lack of utility of an averaged fibre descriptor has alreadybeen discussed. Several groups have used a labour intensive serial grinding technique [120]–[123].The serial grinding technique follows the same grinding, polishing, and imaging sequence, exceptthis process is repeated several times to extract depth information. Joyce et al. used this techniquewith a similar ellipse aspect ratio study to measure the volume of a large, localized, high degree oflocal coordination, wavy set of fibres. Several of these localized regions were found from surfaceinspection and cut from the laminate for further inspection. Serial grinding and imaging, performedapproximately every 0.5 mm returned localized wavy regions on the order of 4mm× 6mm in thein-plane direction with depths in the 3-direction near 1 mm. Paluch combined serial grinding andYurgartis analysis over 40 sections ground at 20 µm intervals. They found wavelengths associatedwith 50 wavy fibres fell within 500 µm to 840 µm for a sample of thermoplastic AS4/PEEK. Forcomparison, Potter measured a wavelength of 3 mm in a thermoset prepreg [9].All of the techniques described so far have required the physical destruction of the samples.NDI has clear benefits for monitoring production parts through the manufacturing cycle. The mostprominent technique, X-ray micro–computed tomography (µCT), has been trialled for wrinklingand waviness detection.µCT, offers the resolution required to differentiate individual fibres; however, equipment costand sample size limitations drastically reduce their applicability. To differentiate individual fibres,39sample volumes less than 2mm× 2mm× 2mm for industrially relevant fibre volume fractionsare typical [124]. Preliminary investigations using µCT on larger volumes, 100mm× 115mm×100mm, showed the technique’s effectiveness at tracking tow wrinkling and waviness [17]. Un-fortunately, their resolution was incapable of differentiating individual fibres by several orders ofmagnitude. Other studies show similar trade-off between sample size and resolution [41], [125].For example, Denos and Pipes studied a volume 65mm×65mm×65mm with a voxel edge lengthof 53 µm [41]. For reference, a 7.1 µm fibre in a square close-packed arrangement has an inter-fibregap of 1.8 µm at 50 % fibre volume fraction. Accurately differentiating fibres would require severalvoxels for each gap necessitating a sub 1 µm voxel edge length.Another technique, ultrasonic NDI, used to measure void content or delamination [126], [127],has also been trialled for misalignment detection. The published work does not have the resolutionrequired to measure individual tows, nor fibres. However, the technique was succesful in measuringa large, 50 mm wavelength and 3 mm amplitude, resin rich zone surrounding an internal wrinkle ina representative GFRP wind turbine blade [128]–[131].2.5 Research objectives and scope of thesis work2.5.1 Synthesis of the open literatureBased on the preceding sections, several main conclusions can be drawn from the literature. Firstis that almost every stage of processing involving uncured material can introduce some degreeof compressive stresses. These compressive stresses can ultimately lead to misalignment defectsdue to the compliant nature of the plies. The processing stages which influence misalignmentsthe most are layup, forming, debulk, bagging, and curing. The uncured laminates mechanicalproperties, which are a function of the underlying architecture, will define the laminates responseto compressive loading. The material’s underlying architecture will not necessarily reset at the end40of a processing stage which implies that in order to correctly predict the outcome of one processingstage, the output of the preceding stage must be known.The existing literature lacks a convincing definition of fibre alignment which is compatible withthe experimental evidence. There currently exists conflicting reports on the quality of as-receivedprepregs along with inconsistent and inappropriate use of experimentally measured fibre angledescriptors. There is a significant gap in knowledge as the material response is highly dependenton the reinforcing fibres orientation.The literature review has shown that generating reproducible misalignment defects in unidirec-tional prepregs is a highly non-trivial task. Nomenclature and classification has also been poorlyadopted, further increasing the difficulty for characterization. While many variables play a role,those which can be more easily controlled and have a large impact on outcomes are temperature,strain rate, relative humidity, and forming geometry.One area which has not previously been discussed but remains problematic for our industrialpartner is the slow formation of wrinkles observed after forming. One main goal is to develop anapparatus to impose small strains into uncured prepreg and monitor the prepregs response.2.5.2 Research objectives and scopeThe end goal of this work is to study the initiation and propagation of misalignment defects duringthe early uncured stages of composites processing. One overarching theme is to use as much infor-mation from each individual test in order to gain statistical confidence in their results. These largedata sets will necessitate the creation of automated data collection and reduction techniques. Basedon the literature review presented, several intermediate objectives have been defined to achievethese goals. These objectives are as follows:• Characterize the fibre architecture of a high quality prepreg within a single roll of material,between rolls of the same material, and between rolls of different materials.41• Develop a cold cure method to allow traditional characterization techniques to be used onthe as-received material form. Characterize the fibre architecture in its as-received state anddetermine the effect of the manufacturer’s recommended cure cycle on fibre alignment.• Create a robust and automated analysis tools to allow all of the fibre alignment characteriza-tion to be performed on large datasets.• Create a phenomenological model for the fibre bed which is both physically representativeand can properly regenerate the results of the fibre architecture characterization.• Characterize the uncured materials, AS4/8552, AS4/8552-1, and their neat resin forms, todetermine their correlation to any observed phenomena.• Develop an experimental methodology capable of reproducibly imparting small strains intouncured prepregs. Investigate the time scales involved for the wrinkle growth observed post-forming. Develop a technique for characterizing all of the individual features.• Develop a mitigation strategy or manufacturing aid to reduce these types of wrinkles.• Create a tool-part interaction model using the temperature dependent properties of the un-cured prepreg which is capable of predicting the stress and strain state of the prepreg andtool.422.5.3 Thesis organizationThe thesis is organized into the following sections:• Chapter 3: Incoming material characterization — This chapter details the characterizationof both AS4/8552 and AS4/8552-1 prepregs and 8552 neat resin. Results for fibre alignment,fibre volume fraction, ply thickness, and shear modulus are presented. A novel strategyfor curing the resin system at room temperature is presented and differences between theroom temperature cure and traditional manufacturer’s recommended cure cycle (MRCC) areshown. A phenomenological fibre alignment model is proposed which is able to correctlyreproduce the experimentally measured misalignment distributions.• Chapter 4: Transient wrinkle growth — An experimental apparatus and procedure whichis able to impart small strains into uncured prepregs is detailed. The transient prepreg re-sponse is captured and analyzed using automated methods. A stress-strain description of thesandwich structure is derived and a strain rate dependent wrinkle growth criteria is proposed.Several mitigation strategies are proposed in order to reduce the transient wrinkle height.• Chapter 5: Quasi-static wrinkle growth — An apparatus designed to mimic individual stepsin a forming process is developed. The previously developed automated wrinkle detectioncode was successfully used to regress the forming data. The effect of temperature and form-ing radius on wrinkle morphology is described.• Chapter 6: Conclusions, contributions, and future work — This chapter summarizes thethesis and formally states the contributions of this work. Several future projects are outlinedbased on the presented results and data reduction techniques.• Appendix A: Discovery DSC calibration — The method for calibrating the differential scan-ning calorimeter (DSC) used to determine the effect of the cold curing strategy is formallyoutlined.43• Section B.1: Semi-automated ellipse detection — The semi-automated code used in analyz-ing the misalignment micrographs is documented.• Appendix C: Wrinkle analysis script — This appendix provides information on the onlinelocation of the wrinkle analysis scripts developed for Chapter 4.• Appendix D: Supplementary transient wrinkle growth data — Supplementary results fromduplicate transient wrinkle trials are compiled. These supplementary results pertain to infor-mation presented in Chapter 4.• Appendix E: Supplementary quasi-static wrinkle growth data — Supplementary results fromadditional forming steps discussed in Chapter 5.• Appendix F: Volumetric contraction of the resin — Derivation of the volumetric cure shrink-age equation used in Chapter 3.44Chapter 3Incoming material characterizationPray, Mr. Babbage, if you put into the machine wrong figures, will the right answerscome out?— Charles Babbage, Passages from the Life of a Philosopher, 18643.1 IntroductionFrom the information garnered in the literature review, the two classifications of misalignmentdefects are in-plane, waviness, and out-of-plane, wrinkling. A map of the important elementsinfluencing fibre misalignment has been created in Figure 3.1 which delineates the intrinsic materialproperties from the extrinsic imposed properties.45Misalignment growthIn-plane Out-of-planeIntrinsic material propertiesMisalignmentsModuliFibre/Ply geometryFibre volume fractionPly-ply interactionTack. . .Extrinsic conditionsToolingPart geometryStrain/Temperature ratePressureTemperature. . .Figure 3.1: Diagram tracking some of the intrinsic properties and the extrinsic conditionswhich influence misalignment growth.Both the in-plane and out-of-plane misalignment defects will be a function of the materialsintrinsic properties and the extrinsic conditions imposed on the material. Given a processing con-dition, the intrinsic material properties cannot be easily modified, without substituting the materialsystem, and serve as the input conditions for any representative model. Intrinsic misalignments willbe imparted into the material prior to the part manufacturer’s receipt and cannot be feasibly altered,rather it can only be characterized. Therefore, an in-depth characterization of these misalignmentswill be useful in developing test methodologies for misalignment growth in forming scenarios.46However, prior to diving into a discussion on the alignment of fibres in as-received prepreg,a discussion on their own dependents is warranted. As previously mentioned, uncured prepregare often simplified as a three-phase system, fibre, resin, and void/gas. Strains imposed onto theprepreg system are imparted from the resin system into the fibre system; conversely any strainin the fibre system will have to ultimately be accommodated by the surrounding resin and fibrebed system. Gaining an appreciation for the mechanical properties of the fibre system and resinsystem will aid the discussion of fibre alignment during prepreg production and part manufacture.Figure 3.2 outlines the dependent nature of the misalignment growth on underlying properties andthe bottom up approach taken to study the defect.MisalignmentgrowthIntrinsicmisalignmentsSurroundingpropertiesFigure 3.2: Flow diagram highlighting the interplay of misalignment growth and the intrinsicmaterial properties of the prepreg.473.2 MaterialsThroughout this study, two prepreg products have been used, the unidirectional prepreg formsof AS4/8552 and AS4/8552-1, and the neat resin form of 8552. Access to neat resin forms ismore difficult and neat 8552-1 was unavailable. Both of the prepreg forms feature Hextow® AS4fibres. While some fibre and prepreg specifications have been discussed in the preceding chapter,the fibre specification sheet states that AS4 fibres are cylindrical carbon fibres with a filamentdiameter of 7.1 µm [35] and a density of 1.79 gcm−3. For reference, the fibres are “a continuous,high strength, high strain, PAN based fiber [. . . ] sized to improve its interlaminar shear properties,handling characteristics, and structural properties” [35]. The resin specification sheet states that the8552 resin is an amine cured, toughened system [101] with a density of 1.30 gcm−3. The 8552-1resin is a catalyst modified version of 8552 offering lower tack for ATL applications. The rollinformation for the AS4/8552-1 used in this study is reproduced in Figure 3.3 and states an arealweight of 190 gm−2 and a resin content of 35 % by weight. The resultant fibre volume fraction,v f , is 57.42 %. The exact same areal weight, resin content, and resulting fibre volume fraction wasreported for the AS4/8552 prepreg, Figure 3.4.Figure 3.3: AS4/8552-1 roll information.48Figure 3.4: AS4/8552 roll information.493.3 Resin characterizationThis section documents the characterization of the complex shear modulus, glass transition temper-ature, and initial degree of cure of the previously discussed materials. Additionally, a novel roomtemperature curing strategy is documented showing the ability to cure a prepreg sample withoutaltering the temperature and therefore monotonically increasing the resin’s modulus.As described in the literature review Section 2.4, the most widely used method for character-izing fibre alignment, the Yurgartis method, involves sectioning, grinding, polishing, and imagingof cured laminates. The main difficulty in sectioning and grinding uncured composites lies in therelatively soft properties of the matrix. These properties do not provide sufficient restraint for thefibres to be cut cleanly and instead allow the fibres to deform into the matrix prior to cutting. Sec-tioning uncured laminates without severely influencing the fibre architecture, and alignment, is stillnovel and suffers significant challenges [96].The traditional approach for imaging involves running the composite through an MRCC inorder to increase the resin’s stiffness and provide the required restraint. This MRCC inherentlymodifies the state of the matrix, void location, and void content, and therefore the state of thefibres.The temperature associated with an amorphous material’s shift from a glassy solid to a morecompliant rubbery material is the glass transition temperature, Tg [K]. To mitigate fibre deformationinto the matrix during sectioning, grinding, and polishing, the material’s glass transition temper-ature should be well above the working temperature. Thomas [132] documented standard testingprocedures for measuring glass transition temperature in a DSC or modulated differential scanningcalorimeter (MDSC). This standard outlines plotting the specific heat capacity as a function oftemperature and sets Tg as the midpoint of the quasi-step change in specific heat capacity. Thismethod is normally effective; however, it suffers from interference if the Tg lays close or inside the50temperature of any other underlying reaction which would modify the specific heat capacity of thematerial.The degree of cure (DOC), χ , of a thermosetting resin system is an important state variable forcomposites indicating the extent of reacted resin and the cross-linking density. Formally, the DOCis a measure of the fraction of reacted material and can be determined using Equation 3.1 [91],[133].χ = 1− HHt·100% (3.1)where χ is the degree of cure, H[Jkg−1]is the mass normalized heat of reaction, and Ht[Jkg−1]is the mass normalized total heat of reaction. The ‘mass normalized’ terminology will be impliedin all further discussion.The shrinkage associated with a curing resin is also an important metric within the contextof fibre alignment. The volumetric cure shrinkage for a full cure cycle of AS4/8552 has beenreported as 4.94 % [93]. This contraction of the resin has been shown to cause compressive stresses,and buckle, individual fibres [94]. However, the change in 1-direction length associated with thebuckled fibres is often neglected and the total contraction is assumed to be equally borne in the 2and 3-direction of the lamina [93]. The volumetric cure shrinkage, αcs, can therefore be determinedusing Equation 3.2, outlined in Appendix F. While the change in length in the fibre direction isnegligible relative to the several percent change in the transverse direction, this should not beinterpreted as having a negligible effect on fibre misalignment.αcs = 2 · L2, f −L2,o(1− v f ) ·L2,o (3.2)where v f is the fibre volume fraction, L2,o [m] is the initial length, and L2, f [m] is the post-curelength along the 2-direction.513.3.1 Shear characterizationMethodCharacterization of the complex shear response of both a neat resin and a prepreg were performedusing an Anton-Paar MCR502 rheometer, shown in Figure 3.5. All experiments were performedusing disposable 25 mm parallel plate geometries using a Peltier plate and shroud for temperaturecontrol.Figure 3.5: Anton-Paar MCR502 rheometer with disposable parallel plate geometries andPeltier plate.A series of tests were performed to characterize the shear response as a function of temperatureof unidirectional AS4/8552 as well as the neat resin form of 8552. All tests were performed undera dynamic oscillation procedure with an applied frequency, ω , of 6.28 rads−1. The neat resinsample procedure used a constant normal force, FN , of 0.05 N and a shear strain amplitude, γa,of 0.1 %. The prepreg sample procedure used a constant normal force of 3 N and a shear strainamplitude of 0.003 %. The modification to the normal force between the neat resin and prepreg52was to minimize any squeeze out of the very low viscosity neat resin. The difference in the shearstrain amplitude was required to accommodate the significantly reduced linear viscoelastic regionof the prepreg [134]. The specimens were approximately 2 mm thick and were slightly oversized,diameter of 25.4 mm, to ensure complete coverage of the disposable geometry. The temperatureregion of interest was from below initial glass transition temperature, −5 ◦C, to an upper limit ontypical forming temperatures, 90 ◦C. To ensure proper adhesion between the specimens and theparallel plates, an initial heating ramp was applied, up to 60 ◦C, at a rate of 2 ◦Cmin−1. This initialheating ramp had a negligible effect on the degree of cure, 0.1 % [135]. This was followed by amachine maximum cooling rate to −35 ◦C which was held for 5 minutes to ensure equilibrium.Finally, a standard heating rate of 2 ◦Cmin−1 was used to heat the specimen from −35 ◦C to 90 ◦C;several samples were run past 90 ◦C to determine the minimum resin shear storage modulus and toview the effect of curing on the ply modulus.Results and discussionResults of the rheometry trials are compiled in Figure 3.6 and Figure 3.7. Figure 3.6 shows a muchlarger temperature range which captures some of the curing behaviour occurring between 170 ◦C to180 ◦C. Three trials were averaged for the AS4/8552 prepreg overlaid with the standard error bars.The standard error was very low indicating a very reproducible result. The neat resin has a verylarge decrease in shear storage modulus, greater than 5 orders of magnitude. A much lower, yetstill significant, decrease in shear storage modulus was observed for the prepreg samples, a changeof 1 order of magnitude, followed by a plateau independent of further increasing temperature.This plateau is where the contribution of the resin to the shear response is very small relative tothe fibre bed shear property, G f b [Pa]. This change in behaviour has been previously describedin the literature [136] and calls for G f b to be set to match the value of this plateau. This valuewill be dependent on the normal force applied due to increased fibre interaction, a consequence ofthe increased fibre volume fraction, under compacting pressure [137]. Hence, the fibre bed shear53property value reported here, 2.60 MPa, is only valid for a compaction pressure of 6.1 kPa; a low,albeit non-zero, applied pressure.100101102103104105106107108-50  0  50  100  150  200  250Shear storage modulus [Pa]Temperature [°C]AS4/8552 prepreg8552 neat resinFigure 3.6: Shear storage modulus for both AS4/8552, the average of three trials with stan-dard error bars, and neat 8552 resin.A magnified region of Figure 3.6 is shown in Figure 3.7 which clearly shows the importanceof temperature control during forming. Exponential functions have been fitted to two regions priorto the plateau, 15 ◦C ≤ T (R1) ≤ 20 ◦C and 35 ◦C ≤ T (R2) ≤ 45 ◦C. R1 could readily correspondwith an open factory warehouse while R2 enters the automated forming temperature region as thisrange is at the limit of human pain tolerance [138], [139]. These regions are linear in log-space andhighlight the rate of change of the shear storage modulus. The maximum difference between therate of change with respect to temperature is a factor of 30 greater between R1 and R2. Thereforeany parameter exclusively sensitive to shear storage modulus will change much greater over R1than R2 and should not vary in the plateau region, T (R3)> 50 ◦C.54106107108 10  20  30  40  50  60  70  80  90A1 e-0.14 TA2 e-0.030 TShear storage modulus [Pa]Temperature [°C]AS4/8552 prepregFibre bed stiffnessFigure 3.7: Shear storage modulus for AS4/8552, average of three trials with standard errorbars, and the temperature invariant fibre bed shear property, over a range of temperaturesconsistent with forming.These tests confirm the resin response as described in Chapter 2 as well as the combined resinand fibre response described in the literature [136]. This section provides a foundation to discussthe forthcoming fibre alignment pre- and post-cure analysis.3.3.2 Cure characterizationMethodThomas [132] and ASTM-E2160-04(2012) [133] both use a DSC to measure a materials Tg andDOC, respectively. A TA Instruments Discovery DSC was used to measure the heat flow intoand out of an as-received specimen of the various prepregs over a specified temperature range.Integrating the mass normalized heat flow with respect to time, using an appropriate baseline,returns the heat of reaction. If this process is performed on an as-received prepreg, the resultingheat of reaction is assumed to be the resins total heat of reaction, or the zero value of the DOC.55Subsequent modifications can be made to the prepreg, then run through a DSC cycle, in orderto determine the modifications effect on the DOC. The DSC was calibrated prior to each seriesof experiments, a full discussion of the DSC calibration, limitations, and baselines, can found inAppendix A.The recommended amount of reactive material for the Discovery DSC was approximately 2 mg[133]; however, larger reactive masses, 5 mg, have been used in the CRN laboratory to increase thesignal to noise ratio [140]. Given a 57 % volume fraction of inert fibres, approximately 14 mg ofprepreg was required, and used, in all trials. Samples constructed from flat, rectangular, plies werestacked, placed, and crimped inside Tzero® hermetic aluminium DSC pans.The test procedure followed a dynamic modulated ramp from −80 ◦C to 300 ◦C at a rate of2 ◦Cmin−1 with a modulation of ±0.318 ◦Cmin−1. This fell within the test procedure parametersgiven by [141] and was consistent with the MRCC heating rate of 2 ◦Cmin−1 [101]. By modulatingthe temperature of the cell, the reversing heat flow can be separated from the average heat flowresponse [142].Chapter 2 Section 2.4.1 outlined potential strategies for advancing cure by introducing a morereactive amine group which should allow cure to occur at room temperature. Therefore, in additionto the as-received AS4/8552-1 specimens, several samples of AS4/8552-1 unidirectional prepregwere left to soak at room temperature in an 14.8 molL−1 ammonium hydroxide solution [143].Samples and solution are depicted in Figure 3.8. These samples were a single ply thick in order tomaximize the surface area to volume ratio in order to minimize concentration and/or cure gradientsin the material. A room temperature vacuum drying stage was used to drive off any remnantaqueous solution. Residual aqueous solution would undergo a phase change during the temperaturecycle of the DSC which would distort the heat flow response. Two sets of samples were left to soakfor 1 h, 4 h, 8 h, 12 h, 21 h, or 24 h in the ammonium hydroxide solution. The first set was vacuumdried for 1 h while the other was left under vacuum for 20 h.56Figure 3.8: Photograph of two AS4/8552-1 specimens in preparation for a soak in the trans-parent ammonium hydroxide solution.Cure shrinkage is typically measured in a TMA which measures dimensional changes as asample is heated or cooled. Using a TMA was not feasible due to the risk of contamination fromthe corrosive aqueous solution. Therefore, measurement of volumetric resin chemical shrinkage,αcs, was determined by measuring the width of a full roll width sample. The sample was initially30.5 cm wide and 15 cm in length. The geometry of the sample was measured pre- and post- a 33 hsoak in the ammonium hydroxide.Results and discussionThe thermogram, a plot of heat flow as a function of temperature, for the as-received prepreg isshown in Figure 3.9. Thermograms for the series of ammonium hydroxide soaked solutions areshown in Figure 3.10.57Figure 3.9: Thermogram of an as-received specimen of AS4/8552-1. Exothermic heat flowis plotted as a positive value.The as-received prepreg shows a typical thermogram for an uncured epoxy. The step changein heat flow at approximately 2.6 ◦C corresponds to the initial glass transition followed by a largeexotherm, corresponding with the formation of bonds as the epoxy forms a 3D network. A sig-moidal baseline with tangent-tangent conditions was applied to determine the baseline after whichthe heat of reaction could be calculated. The total heat of reaction for AS4/8552-1 was approxi-mately 588 Jg−1 which is is well within one standard deviation of the documented value for thesimilar 8552, (597.89±21.65) Jg−1 [135].58(a)(b)Figure 3.10: Thermograms of ammonium hydroxide soaked and vacuum dried AS4/8552-1specimen. Exothermic heat flow is plotted as a positive value. (a) 1 hour vacuum dry.(b) 20 hour vacuum dry.For the ammonium hydroxide soaked specimens shown in both Figure 3.10 (a) and (b), eachthermogram features at least one distinct exothermic peak with a discontinuity in the first derivativeof the heat flow to the right of the maximum peaks. The discontinuities at elevated temperatures are59likely the result of a degradation reaction rather than a curing reaction and were therefore chosenas an upper limit for the baseline determination. A sigmoidal baseline, with tangent and horizontalconstraints on the lower and upper limits, respectively, was used to determine the heat of reactionfor each curve, shown in Figure 3.11.Figure 3.11: Residual heat of reaction of the AS4/8552-1 specimens after various soak dura-tions in aqueous ammonium hydroxide.Viewing Figure 3.11, a clear correlation can be seen between the soak time and a decrease inthe heat of reaction. After a soak of 21 h to 24 h the residual heat of reaction decreases to less than10 % of its initial value.The DOC can then be calculated by using the information from Figure 3.10 and applyingthem into Equation 3.1. The results from this are shown in Figure 3.12. After 21 h to 24 h thereaction plateaus to a maximum value of (91.6±1.9)%. For reference, the final degree of cure forAS4/8552 using the MRCC is approximately 83 % [144].60Figure 3.12: Degree of cure evolution due to varying soak times in aqueous ammonium hy-droxide.The Tg values, determined from the reversing heat capacity and plotted in Figure 3.13, increasefrom the 8552-1’s as-received state of 2.6 ◦C to maximum of 76.6 ◦C. The samples prepared witha 1 h vacuum dry showed only a minor increase for the first 4 h soaked in solution followed by anincrease which plateaued near the 24 h mark. Samples prepared using a 20 h vacuum dry showedan increase in Tg without any lag.61Figure 3.13: Glass transition temperature increase due to increasing soak times in aqueousammonium hydroxide.Plotting the cure rate as a function of degree of cure displays the different reaction kinetics.The 1 h and 20 h vacuum dry results are plotted in (a) and (b) of Figure 3.14, respectively.620 E01 E-42 E-43 E-44 E-45 E-46 E-47 E-4 0  0.2  0.4  0.6  0.8  1Cure rate [s-1 ]Degree of cureSoak timeAs-received0 hour1 hour4 hour8 hour12 hour21 hour24 hour(a)0 E01 E-42 E-43 E-44 E-45 E-46 E-47 E-4 0  0.2  0.4  0.6  0.8  1Cure rate [s-1 ]Degree of cureSoak timeAs-received0 hour1 hour4 hour12 hour21 hour24 hour(b)Figure 3.14: Cure rate as a function of degree of cure for (a) 1 h vacuum dry and (b) 20 hvacuum dried specimens after various soak times in aqueous ammonium hydroxide.Advancing degree of cure by exposing the specimens of AS4/8552-1 prepreg to aqueous am-monium hydroxide is a simple and effective technique for advancing DOC and Tg. After 24 hsuspended in solution, the degree of cure reaches (91.6±1.9)%, approximately 10 % higher thanthe final resin state after a traditional MRCC, 83 % [144]. This implies most of the epoxides in the63system have been consumed. As this reaction is occurring at room temperature, the added ammo-nium hydroxide is most likely reacting with the epoxides while the latent amines, DDS, have littlecontribution. This can be viewed in Figure 3.10, but is very visible in Figure 3.14, as the speci-ation occurring due to the different soak times. This speciation is visible as double peaks in thetemperature-heat flow diagrams and the cure-cure rate diagrams of the 1 h soak trial with a 1 h vac-uum dry and both 4 h soak trials, 1 h and 20 h vacuum dry. Any remnant, lower activation energy,ammonium hydroxide will react first creating its distinct exothermic peak. Before all of the firstamine reactant has been consumed, the primary DDS/TGDDM reaction initiates. This can be seenby the first peak not decaying to the baseline before the beginning of the secondary peak. However,preparing samples using only the soak technique, i.e. without the added heat from the DSC, theDDS compounds will remain in the system without contributing to the growing macromolecule.As such, the macromolecule will be much less efficiently packed in its final state. Further, this am-monium hydroxide reaction is occurring at room temperature, hence the molecular spatial freedomis reduced, leading to a less efficient arrangement when compared to the traditional MRCC.This decrease in density and arrangement efficiency is evident from the change in glass tran-sition temperature. From the prepreg’s initial 2.6 ◦C, the maximum glass transition temperaturerecorded was 76.6 ◦C which is significantly lower than an MRCC glass transition temperature of212 ◦C [144].Insufficient vacuum dry times also show a clear reduction in glass transition temperature. Thiscan likely be attributed to the failure to remove water absorbed during the aqueous solution soak.Water will diffuse into the epoxy structure and further decrease the molecular packing efficiency.With the macromolecule otherwise unchanged, this inefficiency creates a softer material as demon-strated by a decrease in glass transition temperature. This process is referred to as plasticization.While for all cases the glass transition temperature is lower than MRCC values, the ammoniumsoak technique is still a very valuable for characterization as it is still much higher than room tem-64perature. The key issue that this approach has solved is that using this cold curing allows traditionalmisalignment characterization to be used without modification.As for the cure shrinkage, the 2-direction path length reduction, as read from the meter stick,was 1 mm, resulting in an ammonium hydroxide soak volumetric cure shrinkage of 1.5 %. This iswell below the recorded thermal cure volumetric cure shrinkage of 4.94 % [93]. Minor undulationsof the prepreg surface after the soak did not allow the true 2-direction path length to be determined,as a result the 1.5 % cure shrinkage is likely an upper estimate.This ammonium hydroxide soak methodology has increased the glass transition temperature ofthe prepreg well above room temperature. These gains come with a fraction of the cure shrinkageassociated with an MRCC and without the associated decrease in resin properties due to the thermalcycling. This technique is simple, inexpensive, and only required equipment typically associatedwith a standard chemical laboratory.653.4 Misalignment characterizationThis section documents the largest study to date on the characterization of fibre alignment boththermally cured and soak cured composite materials. This characterization followed and extendedthe Yurgartis method described in Section 2.4.2 [18].MethodMisalignment defects in flat, oven-cured, but otherwise off-the-roll prepregs were investigated forthree different rolls of prepreg using two different prepreg systems, AS4/8552-1 and AS4/8552.Roll ‘A’ and ‘B’ were AS4/8552-1 and roll ‘C’ was AS4/8552. The first set of specimens werecreated from AS4/8552-1 roll ‘A’ to determine the roll radial dependence of fibre misalignment.Another set of specimens were created from AS4/8552-1 roll ‘B’ to determine if there was sig-nificant variation between rolls. Other than initial outer radius, the two rolls ‘A’ and ‘B’ werefunctionally identical. A second material system, AS4/8552 roll ‘C’, was also studied to determineif, for the same fibre type, there was a difference in the fibre alignment caused by the modified resinsystem.Additionally, another two specimens of AS4/8552-1 roll ‘A’ specimens were soaked in aqueousammonium hydroxide for 18 to 21 hours to determine the effect of the decrease in shear storagemodulus and the cure shrinkage associated with the traditional MRCC.In-plane fibre misalignments in the as-received AS4/8552-1 roll ‘A’ specimens of unidirec-tional prepreg were determined by preparing the specimens according to the Yurgartis method[18]. Six single ply specimens, each approximately 170 mm long in the 1-direction and 30 mmin the 2-direction, were cut from roll ‘A’ at various positions as the roll was unwound, shown inFigure 3.15(a). The mid-width of these specimens were coincident with the mid-width of the roll.These specimens were cured flat on an aluminium tool with a release film at the interface so asto reduce any tool part interaction which could impact the alignment of the fibres. The cure cycle66was consistent with manufacturer’s recommendations, save an oven was substituted in place of anautoclave. As such, vacuum was continuously drawn to yield an applied pressure of approximately101 kPa rather than the recommended 586 kPa to 690 kPa.(a)(b)Figure 3.15: (a) Physical specimen sectioned into three wedges and (b) a schematic of thespecimen with various geometric features highlighted.Post cure, the specimens were cut at approximately 5° to the nominal 1-3 plane, cut into two60 mm and one 50 mm wedges, schematically described in Figure 3.15(b), and potted in a coldcure mounting epoxy suitable for the subsequent grinding and polishing stages. Digital optical mi-croscopy images were taken at 200× magnification using a Nikon Epiphot 300 and then manuallystitched into a full width mosaic for each wedge using Adobe Photoshop. The pixel to micrometreconversion ratio for the digital images was determined using a Nikon 1 mm objective micrometerwith 100 graduations.A mosaic micrograph image was manually created from the individual micrographs for each ofthe six 60 mm wide wedge ‘2’ specimens. This procedure involved importing each of the sample’smicrographs as layers in Adobe Photoshop CS6, decreasing the opacity to 50 %, and manually67repositioning the micrographs to form a single, continuous, high resolution mosaic. These mosaicmicrographs were then imported and analyzed using a purpose-built script written in WolframMathematica. The full script is documented in Appendix B. Briefly, this script performed a seriesof automated and semi-automated image manipulation and ellipse fitting techniques, shown inFigure 3.16.First, the 60 mm wide mosaic micrograph was sectioned into approximately 6 mm wide im-ages. This was done solely to increase computational speed and decrease the required memory,a small subsection of the working image is shown in Figure 3.16(a). (b) shows the output after agreyscale and thresholding operation, the final output of which is a binary image. This image isthen stretch by a factor of 9 in the vertical direction. This transforms the high aspect ratio ellipsesinto more circular objects which image recognition tools are better suited. This step has been omit-ted in the images of Figure 3.16 as it obfuscates the operations. A distance transform is applied onthe stretched and binarized image, shown in (c), which shows progressively lighter pixels based ontheir distance to the nearest boundary pixel. (d) applies a local maximum transform on the distancetransform returning the likely centroids of the ellipses. A popular image segmentation, watershedsegmentation, is then applied, using the likely centroid positions, to segment the greyscale imageinto individual ellipses, the colors in (e) are solely to differentiate the individual ellipses. For errordetermination, the largest and smallest 10 % of the fibres were highlighted in blue and red, respec-tively, (f). These ellipses were manually checked and removed from the watershed componentsarray as required. The final image from the mainly automated technique, (g), shows well fittedellipses with few missing ellipses.68(a) (b)(c) (d)(e) (f)(g)Figure 3.16: A series of images depicting the various steps from the automated procedurecreated in order to fit individual ellipses over fibres. Refer to the prior paragraph for adiscussion of each image.69These missing ellipses from Figure 3.17(a) were then manually fitted by defining 8 to 10 pointsalong the periphery, (b), and performing a best fit to the generalized ellipse equation, (c). (d) showsthe final components of the array with each fibre uniquely fitted. Ellipses could only be fitted ontofibres which were mostly elliptical, certain fibres, such as those cut along their own 0° or fibreswhich were grossly damaged were removed and left unfitted. These fibres make up a small 1 % to2 % of the overall micrographs.(a) (b)(c) (d)Figure 3.17: Several steps from the manual procedure for individually fitting ellipses over theremaining aberrant fibres. Refer to the prior paragraph for a discussion of each image.The final array output had the centroid position, fi (xc,yc), the major and minor diameters ofthe fitted ellipse, fi (dma j,dmin), and the orientation of the ellipse relative to constant horizontal ref-erence, fi (φ), for each fibre, fi. The remaining two wedges from only the innermost and outermostradial positions were analyzed to estimate variance for wedges taken at nominally the same radialposition. This strategy, coupled with a very large sample size, leads to a very high confidence in70the misalignment data. In all, 2653 micrograph images from 20 wedges of the 10 specimens wereanalyzed.With the determined dma j and dmin for each ellipse in the micrograph, the in-plane alignmentof each fibre was determined using Equation 2.14. After this, a frequency table was created bypopulating 0.25° wide bins with the appropriate number of fibres. Accounting for bias in countinghighly misaligned fibres and removing the mean cutting angle,ΦPC the volume fraction distributionof in-plane fibre misalignments can be created [18]. Probability density functions were created bydividing these volume fractions by the bin width which could then be linearly interpolated to forma continuous distribution.The same methodology was used to analyze a single AS4/8552 roll ‘C’ specimen, an AS4/8552-1roll ‘B’ specimen, as well as two ammonium hydroxide soaked AS4/8552-1 roll ‘A’ specimens.To determine the through-thickness fibre alignment distribution, the traditional Yurgartis methoduses another specimen cut orthogonally to the first specimen, making a desired 5° angle with the1-2 plane. This new specimen is analyzed following the exact same procedure used to determinethe in-plane fibre alignment distribution described previously. However, there is no reason why theout-of-plane misalignment cannot be extracted from the traditional in-plane micrographs. Modify-ing Figure 2.18 to include an additional out-of-plane misalignment component, as has been donein Figure 3.18, allows the through-thickness fibre angle information to be extracted by measuringthe orientation of the sectioned ellipses.Figure 3.18 has removed Figure 2.18’s vertical shift between the green, red, and blue fibres,keeping their respective 2.5°, 0°, and −2.5° 1-3 plane misalignments, while adding a 10°, 0°, and−10° 1-2 misalignment. Figure 3.19(b) shows the out-of-plane orientation for each fibre whileFigure 3.20 shows the sectioned faces highlighting how the green fibres out-of-plane orientation,φi, can be directly determined from the cross-sections. To verify this approach, a virtual sectioninganalysis was conducted on fibres sectioned at 5° to the 1-3 plane and all integer angle values71between−10° and 10° to the 1-2 plane. This virtual analysis showed that even for fibres with largeout-of-plane misalignments, error only manifests itself in the determination of the major ellipsediameter. This slight corruption in the major ellipse diameter modifies the determination of thein-plane angle. A corrected equation for the in-plane angle implementing the out-of-plane angleis Equation 3.3. However, this resulting error, even for a highly misaligned out-of-plane fibre at10°, only results in an in-plane error of less than 0.1°; with decreasing errors as the out-of-planemisalignment approaches 0°. The tabulated results from this virtual study are in Table 3.1. Thisvirtual analysis was repeated for a −5° misalignment to the 1-3 plane and the exact same resultswere obtained. This ellipse orientation method is a modification of the method proposed by Hine etal. [114] and was proposed as a possibility but not investigated by Krishnamurthy [145]. The Hine’smethod has been used for short-fibre samples, however, the literature review has not uncovered anypublished literature which captures through-thickness information from samples generated usingthe Yurgartis method without using multiple samples cut orthogonally.sin(ωi) =di,mindi,ma j cos(φi)(3.3)72132Figure 3.18: A perspective view of the virtual misaligned fibres in both the in-plane and out-of-plane directions intersected by a 5° cutting plane. The blue, red, and green fibreshave an in-plane misalignment angle of −2.5°, 0°, and 2.5° and an out-of-plane mis-alignment angle of −10°, 0°, and 10°, respectively.7312ΦPCωiθi(a)13ϕi(b)Figure 3.19: (a) Top view of Figure 3.18 highlighting the typical in-plane angles of interestand (b) a front view showing the green fibre’s input 10° out-of-plane misalignment.ϕidi,majdi,minFigure 3.20: Resulting cross-sections on the cutting surface described in Figure 3.18. Themajor diameter, minor diameter, and measured out-of-plane angle are highlighted forthe green fibre.74Table 3.1: Results from sectioning virtual fibres using various through-thickness rotationsgenerated in order to verify that the measured values correctly map to the input values.Input in-planeangle, ωiInputout-of-planeangle, φiCalculated ωifrom YurgartisEquation 2.14Calculated ωifromEquation 3.3Measured φi5.◦ −10.◦ 4.92° 5.00° −10.0°5.◦ −9.◦ 4.94° 5.00° −9.0°5.◦ −8.◦ 4.95° 5.00° −8.0°5.◦ −7.◦ 4.96° 5.00° −7.0°5.◦ −6.◦ 4.97° 5.00° −6.0°5.◦ −5.◦ 4.98° 5.00° −5.0°5.◦ −4.◦ 4.99° 5.00° −4.0°5.◦ −3.◦ 4.99° 5.00° −3.0°5.◦ −2.◦ 5.00° 5.00° −2.0°5.◦ −1.◦ 5.00° 5.00° −1.0°5.◦ 0.◦ 5.00° 5.00° 0.0°5.◦ 1.◦ 5.00° 5.00° 1.0°5.◦ 2.◦ 5.00° 5.00° 2.0°5.◦ 3.◦ 4.99° 5.00° 3.0°5.◦ 4.◦ 4.99° 5.00° 4.0°5.◦ 5.◦ 4.98° 5.00° 5.0°5.◦ 6.◦ 4.97° 5.00° 6.0°5.◦ 7.◦ 4.96° 5.00° 7.0°5.◦ 8.◦ 4.95° 5.00° 8.0°5.◦ 9.◦ 4.94° 5.00° 9.0°5.◦ 10.◦ 4.92° 5.00° 10.0°75To extract more information from the 170 mm wide micrographs, a novel windowing tech-nique was developed and applied to the innermost and outermost thermally-cured and soak-curedspecimens from AS4/8552-1 roll ‘A’. Rather than collapsing the spatial information from the mi-crographs into a single distribution, a series of windows, approximately 1 mm wide, were indi-vidually analyzed using the in-plane misalignment technique. By integrating the resultant angledistributions, excess length distributions for the fibres in each window can then be determinedusing Equation 3.4.∆Li, j =∫ Lw, j0√1+(dydx)2i, jdx−Lw, j (3.4)where ∆Li, j [m] is the in-plane excess length contribution of a fibre population, i, at a discrete angleover the length of an individual frame, Lw, j [m]. An assumption made to solve this integral is thatany wavelength associated with the fibres is much longer than the window width. This will bediscussed in a later section.The excess length distributions were then averaged by weighting the excess lengths by thenumber of contributing fibres using Equation 3.5.∆L j =∑n ji=1∆Li, jV i, j∑n ji=1V i, j(3.5)where ∆L j [m] is the average excess length in a window, j, n j is the number of discrete angle binsin a window, and V i, j is the number of fibres counted in the discrete angle bin, i. Vi, jFinally, the average normalized excess length for a specimen, ∆̂L, can be determined by sum-ming the average excess lengths of all windows and normalizing with respect to the width of allwindows, Equation 3.6.∆̂L =∑nwj=1∆L j∑nwj=1 Lw, j(3.6)76where nw is the number of windows, w, measured.Care must be taken when interpreting the raw normalized excess length after initial transfor-mation. The angular misalignment bin widths will have a constant value; however, after the trans-formation to excess length, the bin widths will be significantly smaller near zero than the longerexcess length values. Applying an area-preserving method using the smallest excess length bin sizereturns an easily interpretable histogram [146].Further care must be taken when interpreting the fibre alignment distributions and mean val-ues. Tracking the mean value of sequential windows along the 1-direction and encapsulating theresultant information as an average fibre is fraught with undesirable consequences when analyzingmean values within the bulk of expected angular values. Take for example a simplified scenariowith two straight fibres. Let one fibre be perfectly aligned with the 1-direction, and the other 45° tothe 1-3 plane. The total excess length, over a unit distance, is 0+0.41 while averaging the anglesinto two average fibres at 22.5° returns a total excess length per unit distance of only 0.16. Estimat-ing the excess length in this manner vastly underestimates the average excess length in the prepreg,often by an order of magnitude. For fibres where there is a high degree of local coordination, adja-cent fibres following similar trajectories, this average fibre technique may provide information onwavelength and amplitude; however, it will always result in an underestimate on excess length inthe fibre system.All misalignment distributions for unidirectional composites which have been reported in theliterature, including this study, have a unimodal distribution shape. Several plausible fibre pathdescriptions can satisfy this condition. The most simple solution is a set of perfectly straightfibres aligned predominantly along the 1-direction with a decaying number of perfectly straight,but misaligned to the 1-direction, fibres. However, fibres within a tow must remain within a towwidth for the solution to be physically representative. This gives an upper bound on the length ofa fibre for this perfectly straight fibre solution. A fibre at 1° to the 1-direction would be limited to77approximately 23 cm in length for a tow width of 4 mm. This solution is therefore not appropriateas fibres in these prepregs are continuous over much longer distances.Another solution is to represent the fibres by a sinusoidal fibre model, similar to Equation 3.7.However, treating the fibre system as an average fibre defined by a single ‘maximum angle’ raisesa significant issue. Characterizing in this way results in a bimodal distribution regardless of am-plitude, wavelength, and phase-shift; save the trivial case where the amplitude is zero. Several ofthese solutions were trialled and the in-plane misalignment distribution for the fibre system, Fig-ure 3.21, is presented in Figure 3.22. This system had a single small imposed excess length and aset of fibres with a uniform distribution of wavelengths and phase shifts.yi = Ai sin(2piαixL1D+Φi)+ ci (3.7)where yi [m] is the centroid, Ai [m] is the amplitude of a sinusoidal curve, x is the distance alongthe nominal 1-direction, αi is the mode number, L1D [m] is the 1-direction total length, Φi is thephase shift, and ci [m] is a spacing factor, of a given fibre, i. The spacing factor is the distancebetween two adjacent fibre centroids and was derived to be 8.3 µm from the AS4/8552-1 averagefibre diameter, 7.1 µm, and volume fraction, 57.42 %.0 50 100 150-20-15-10-501-direction [mm]2-direction [mm]Figure 3.21: Schematic diagram showing virtual fibres, defined by a single excess length,being sectioned by an orange cutting plane. The number of fibres shown is significantlyfewer than those analyzed for visualization purposes.78 0 0.05 0.1 0.15 0.2-2 -1.5 -1 -0.5  0  0.5  1  1.5  2Volume fractionFibre angle, θ, [°]Figure 3.22: In-plane fibre angle distribution for the virtual set of fibres defined by a singleexcess length shown in Figure 3.21. 14,752 virtual intersections were analyzed togenerate this distribution.Only by treating the fibre as a system defined by a distribution of path lengths, Equation 3.7,can we generate a virtual set of fibres which satisfy the experimentally determined unimodal mis-alignment distributions.Figure 3.23 shows the general procedure for generating each individual virtual fibre for the fi-bre bed model. First, the distributions described by Equation 3.4 are normalized by their respectivewindow width’s, Lw, j. A probability density function is fitted to the shape of these normalized ex-cess length distributions. This normalized excess length probability density function is sampled toobtain a set of unique normalized excess lengths, ∆̂Li, for the desired number of virtual fibres. Theproposed fibre bed model, Equation 3.7, requires that each fibre has an individual amplitude, Ai,mode number, αi, phase shift, Φi, and spacing factor ci. The mode number and phase shifts valueswere sampled from separate uniform distributions while the spacing factor followed the 8.3 µmspacing as previously discussed. The amplitude of an individual fibre can then be determined byequating the path length of the individual fibre with its associate normalized excess length. This79process fully defines the centroid of each fibre along the virtual part length, L1D. Finally, to extend-ing the mathematical description into a solid body requires sampling the fibre diameter distribution,described in Section 3.6, to add a volume to the individual fibres.Normalized excess lengthProbability density∆̂L1 ∆̂L2 . . .y1 = A1 sin(2piα1xL1D+Φ1)+ c1 y2 = A2 sin(2piα2xL1D+Φ2)+ c2 . . .∫ L1D0√1+(dy1dx)2dx = ∆̂L1 ·L1D∫ L1D0√1+(dy2dx)2dx = ∆̂L2 ·L1D . . .Figure 3.23: Flow diagram of the general procedure used to generate the individual fibres ofthe virtual fibre bed. Refer to the prior paragraph for a discussion of each step.A schematic representation of a tow segment featuring a fibre system described in Figure 3.23is shown in Figure 3.24. This figure shows the importance of the interdependent features, such asamplitude and wavelength, on the fibres excess length. The red fibre, with a smaller wavelength,80still has a smaller excess length than the green fibre due to the red fibre’s smaller amplitude. Theblue fibre, with a large, yet finite, wavelength, has the lowest excess length.Figure 3.24: Schematic diagram of a tow segment featuring fibres with independent excesslengths, amplitudes, wavelengths, phase shifts, and initial positions. The blue fibrefeatures the lowest excess length, the red fibre with intermediate excess length, and thegreen fibre with the highest excess length.Performing the Yurgartis technique on a virtual set of fibres, shown schematically in Fig-ure 3.25, will show the robustness of this excess length technique.81Figure 3.25: Sectioned virtual tow.82ResultsIn total, 227,038 fibres were characterized in this study. 126,429 fibres from the thermally-curedAS4/8552-1 roll ‘A’ specimens were analyzed with an average of 34,844 fibres from the innermostand outermost radial position and an average of 14,185 fibres from the intermediate specimens. Theresultant in-plane fibre angle misalignment distributions are individually shown in Figure 3.26 withtheir corresponding in-plane probability density functions overlaid in Figure 3.27. All of the figuresshow symmetric distributions with little error caused by mirroring of the distribution as ωi <ΦPC.Further, all of the distributions display with very similar profiles. This is best evidenced by thestandard deviations which are plotted with respect to their original roll location in Figure 3.28.83(a) (b)(c) (d)(e) (f)Figure 3.26: Individual in-plane fibre misalignment volume fraction distributions forAS4/8552-1 roll ‘A’ prepreg specimens at various radial positions, (a) 163 mm, (b)171 mm, (c) 181 mm, (d) 192 mm, (e) 213 mm, and (f) 224 mm. Each distribution wasgenerated from the W2 wedge.840.00.10.20.30.4-4 -3 -2 -1  0  1  2  3  4f (θ), [°]-1Fibre angle, θ, [°]r = 163 mmr = 171 mmr = 181 mmr = 192 mmr = 213 mmr = 224 mmFigure 3.27: AS4/8552-1 roll ‘A’ in-plane fibre angle probability density functions for the sixradially spaced specimens.0.00.51.01.52.0 150  160  170  180  190  200  210  220  230Standard deviationof misalignment [°]Radial position [mm]r = 163 mmr = 171 mmr = 181 mmr = 192 mmr = 213 mmr = 224 mmFigure 3.28: Standard deviation of in-plane fibre misalignments, AS4/8552-1 roll ‘A’, withdashed lines indicating the take-up roll core radius and original outer radius.Analyzing the upper bounds of one of the in-plane misalignment distributions shows a non-zeronumber of highly misaligned fibres, Figure 3.29. Examples in the micrographs of these infrequent,85but countable, highly misaligned fibres are shown in Figure 3.30. Figure 3.30(a) shows a fibre inthe through-thickness center of a tow with an in-plane misalignment in excess of 35° while (b)displays highly misaligned fibres on the periphery, both on the side closest to the tool as well as theside closest to the bagging material. Highly misaligned fibres were uniformly dispersed throughthe ply and no clear trend could be ascertained as to their location.86(a)(b)(c)Figure 3.29: In-plane fibre misalignment volume fraction distributions for wedge W1 at224 mm, (a) typical window with 97.7 % of the volume fraction, (b) the full volumefraction distribution, and (c) highlighting the non-zero fraction of fibres at large anglesto the lamina 1-direction.87(a) (b)Figure 3.30: Highly misaligned fibres located (a) in the internal structure and (b) along theperiphery of the lamina.8814,515 fibres from the AS4/8552-1 roll ‘B’ specimen were analyzed resulting in an in-planemisalignment standard deviation value of ±1.65° while the analysis of 27,467 fibres from theAS4/8552 roll ‘C’ specimen resulted in a in-plane misalignment standard deviation value of±1.59°.In-plane misalignment standard deviation values for the soak-cured AS4/8552-1 roll ‘A’ speci-mens were ±1.43° and ±1.08° for the inner and outer radial specimens, respectively.Out-of-plane misalignments in the thermally-cured AS4/8552-1 roll ‘A’ specimens were sig-nificantly lower than their in-plane counterparts with a pooled standard deviation of ±0.53°. Theprobability density function for these specimens are displayed in Figure 3.31 and their standarddeviations as a function of roll position are displayed in Figure 3.32.0.01.02.03.04.05.0-4 -3 -2 -1  0  1  2  3  4f (Φ), [°]-1Fibre angle, Φ, [°]r = 163 mmr = 171 mmr = 181 mmr = 192 mmr = 213 mmr = 224 mmFigure 3.31: AS4/8552-1 roll ‘A’ out-of-plane fibre angle probability density functions forthe six radially spaced specimens.890.00.51.01.52.0 150  160  170  180  190  200  210  220  230Standard deviationof misalignment [°]Radial position [mm]r = 163 mmr = 171 mmr = 181 mmr = 192 mmr = 213 mmr = 224 mmFigure 3.32: Standard deviation of out-of-plane fibre misalignments, AS4/8552-1 roll ‘A’,with dashed lines indicating the take-up roll core radius and original outer radius.A comparison of the in-plane and out-of plane pooled standard deviations for 3 different rollsis shown in Figure 3.33.Figure 3.33: Comparison of the in-plane and out-of-plane standard deviation values for thethree different prepreg rolls studied.90The normalized excess length was determined for thermally-cured and soak-cured specimens atboth the inner and outer radial positions. The series of 63 windowed misalignment distributions fora single wedge specimen is shown in Figure 3.34. The corresponding excess length distributions,Figure 3.35, conveys the large volume of data compiled for a single wedge specimen.Figure 3.34: Cartoon showing the large number of spatial in-plane angle distributions gen-erated for a single specimen. A select few distributions are shown in greater detail inFigure 3.36.91Figure 3.35: Cartoon showing the corresponding large number of generated spatial excesslength distributions generated from the single specimen used in Figure 3.34. A selectfew distributions are shown in greater detail in Figure 3.36.A sample volume fraction of the in-plane misalignments and the corresponding normalizedexcess length probability density function for three arbitrary windows, of the approximately 200windows per sample, are shown in Figure 3.36. The computed average normalized excess lengthsfor the thermally-cured specimens were 0.041 % and 0.039 % for the inner and outer radial speci-mens. In comparison, the soak-cured specimens had average normalized excess lengths of 0.029 %92and 0.030 % for the inner and outer radial specimens. These values are plotted as a function oftheir initial radial position in Figure 3.37.(a) (b)(c) (d)(e) (f)Figure 3.36: (a), (c), and (e) show various in-plane angle volume fractions with their corre-sponding (b), (d), (f) empirical normalized excess length probability density functionsfor arbitrary windows. Each window had a width of approximately 0.9 mm containingnearly 200 fibres, (AS4/8552-1, roll ‘A’, thermally-cured).93Figure 3.37: Average normalized excess length for the thermally-cured and soak-curedAS4/8552-1 roll ‘A’ specimens, with dashed lines indicating the take-up roll core ra-dius and original outer radius.The distributions in Figure 3.35, (and Figure 3.36(b), (d), and (f)), imply a much larger volumefraction of small excess length fibres, shown schematically as the blue fibres in Figure 3.24, withfewer large excess length fibres, the green fibres.Joining each of the normalized excess length distributions allowed the set to be fitted with arepresentative distribution. Several symmetric and asymmetric distributions were trialed. Symmet-ric distributions were accommodated by halving the frequencies and mirroring the data set aboutzero. Fitting the distributions with a location-scale family Student’s t-distribution, Equation 3.8,yielded the lowest discrepancy between experimental data and fitted distributions trialed. The ex-tracted parameters from one data set had a location parameter, µs, of 0; a scale parameter, ςs, of0.0001; and a degrees of freedom constant, ν , of 0.45. A plot showing an overlay of the fitted,single sided, Students-T distribution and the excess length distributions of the outermost thermallycured specimen is shown in Figure 3.38.p(x|ν ,µs,ςs) =Γ(ν+12)Γ(ν2)√piνςs(1+1ν((x−µs)ςs)2)−( ν+12 )(3.8)94where p(x|ν ,µs,ςs) is the probability density and Γ is the gamma function.Figure 3.38: Single-sided Student’s t-distribution, dashed red line, overlay with each of theexcess length distributions shown in Figure 3.35.Generating a sample of 37,000 normalized excess length values from this Student’s t-distribution,then multiplying by a virtual specimen length of 191 mm yields a distribution of fibre excesslengths, Figure 3.39. Solving Equation 3.7 for each fibre’s unique amplitude, using the speci-men length and fibre’s individual excess length, generates a virtual fibre bed shown in Figure 3.40.For reference, the blue curves represent the fibres, the straight orange line represents a 5° cuttingplane, and the red points represent locations of intersection between the fibres and cutting plane.To improve clarity, only 1 in 20 fibres are shown, as such, many of the red intersecting points arealso hidden.95Figure 3.39: Histogram of the virtual fibres normalized excess lengths generated from theabsolute values of a Student’s t-distribution, location parameter of 0, scale parameterof 0.0001, and degrees of freedom of 0.45.Figure 3.40: Virtual fibres sectioned by an orange cutting plane. Red markers indicate thelocation at which each fibre intercepts the cutting plane. The number of fibres, andintercepts, shown is drastically fewer than those analyzed for visualization purposeswith only approximately 1 in 20 fibres shown.Coming full circle by applying the Yurgartis methodology on the virtual fibre bed results in theprobability distribution shown in Figure 3.41. The first significant result is that the distribution isunimodal and relatively symmetric, which is not possible by representing the fibre bed as beingdefined by a single excess length value. The second important result is that the standard deviationsfall between ±1.5° and ±1.7° which very closely bound the experimentally measured standarddeviation of ±1.6°.96Figure 3.41: In-plane fibre angle probability density function for the virtual fibre data setsectioned in Figure 3.40. Approximately 2000 fibres were required to ensure the virtualcutting plane cut fibres along its entire length.97DiscussionContrary to the best working hypotheses in the literature, both in-plane and out-of-plane misalign-ments were not significantly impacted by their original location on the roll for the prepreg systemsstudied herein after a thermal cure cycle. This is specifically evidenced in Figure 3.28 and Fig-ure 3.32 where the standard deviation of fibre alignment is invariant with respect to radial position.The in-plane pooled standard deviation of the thermally-cured specimens was ±1.60°. Directlyanalyzing the population-weighted distributions show that 20 % of fibres fall outside of ±1.99°with 5 % of fibres falling outside of ±3.28°. A large fibre sample size, in excess of 200,000 fibres,gives extremely high confidence in these results. For a normally distributed set with a standard de-viation of ±1.60°, 20 % fall outside ±2.05° and 5 % fall outside ±3.14°. The difference betweenthe experimental set and a truly normally distributed set is real; however, for practical applications,the misalignment data can be approximated by a normally distributed set.Since the prepregs follow a probability distribution similar to a normally distributed set, thiseffectively nullifies the ability to characterize a composite by a single ‘misalignment angle’ or‘maximum misalignment angle’. This is further evidenced in micrographs of the highly misalignedfibres, Figure 3.30. An aerospace grade unidirectional prepreg with a sufficiently large populationof fibres will include a finite number of fibres that have more than 45° of misalignment. Given thefact that even a relatively small production part will have far in excess of a million fibres, theseparts will, with almost certainty, have a finite number of fibres at very large misalignments.Although the result of the rolling process on an inextensible fibre in a finite thickness plyshould result in highly misaligned fibres at the inner surface with nominally straight fibres at theouter surface of a single ply, this trend was not observed in any consistent manner for any of thethermally-cured specimens or the soak-cured specimens. This suggests that the effect of the rollingprocess is negated during the unrolling process for the range of radial positions studied in this work.98The out-of-plane misalignments were significantly more aligned than their in-plane counter-parts, with a pooled standard deviation of ±0.53°. The fibres themselves are axisymmetric andtherefore have no preferred misaligning direction, however, during a typical pre-gregging pro-cesses, a normal force is applied to the resin as it is forced through-thickness to fill the void spaceof the fibre architecture. Also, during the debulk and curing stages of processing, atmospheric orautoclave pressures are applied through-thickness which results in a constraining force on the fibresability to deviate through-thickness.As the ammonium hydroxide soak-cured specimens did not require the use of a vacuum bag tocure, they clearly show that the prepregging process itself is more significantly involved in reducingout-of-plane misalignments, as these specimens had only a marginally higher pooled out-of-planestandard deviation of ±0.57°. However, the impact of the thermal-cure stage on the alignment ofthe fibres is important. As Jochum et al. [94] have shown using a single fibre, the cure shrinkagewill impose compressive stresses on fibres as modulus develops during the hold stage of a typicalcure cycle. Using both the ammonium hydroxide soak-cure method along with the windowingtechnique described above, not only can this effect be observed but the excess length imposed bythe thermal-cure cycle can be quantitatively determined on a realistic multi-fibre unidirectional car-bon fibre component. As the ammonium hydroxide soak-cure has a much smaller cure-shrinkagethan thermal-curing and does not involve the thermal excursion, implying the viscosity only in-creases, the as-received uncured prepreg should closely mimic the soak-cured specimens fibrealignment values. The difference in excess length between these specimens and the thermally-cured specimens can therefore be attributed to the resin volumetric expansion during heating andthe volumetric contraction during cure of a typical cure cycle. The percent difference from thesoak-cured average normalized excess length, 0.029 %, to the thermally-cured average normalizedexcess length, 0.040 %, is 31 %. This indicates an average normalized excess length of 0.011 % hasbeen imparted by the resin cure shrinkage into further misaligning the fibres. Although measurable99and significant compared to the overall misalignment, it is small compared to the typical linearcure shrinkage of 8552 resin which, converted from the reported volumetric value of 4.94 % [93],is 1.6 %.This methodology assumes fibres are straight as they pass through the cutting plane and thattheir wavelength is larger than the window width. Neither of these assumptions were grosslyviolated as only a small fraction of the total fibres counted in the micrographs had distortionsconsistent with being cut at multiple angles along the cutting plane. Also, no short undulations inthe fibres were observed in SEM images taken of the burn-off specimens.Relatively uniform processes, such as cure shrinkage or thermal loading, should result in aconstant excess length imposed on the fibre. Therefore, while the cure cycle has impacted the fibrein-plane alignment, the shape of the excess length distribution, or the misalignment distribution,cannot originate from the compressive stresses generated from the curing process. The origin ofthe distribution of excess lengths likely lies in early prepregging stages or in the manufacture of thecarbon fibre itself.The roll-to-roll variation between AS4/8552-1 roll ‘A’ and ‘B’ was marginal with a differencein in-plane misalignment standard deviations of±1.60° and±1.65°, respectively. Further, the vari-ation between resin systems, AS4/8552-1 and AS4/8552, was even smaller, ±1.60° and ±1.59°.While these three values are different, the implication that 5 % of fibres lay outside of approxi-mately ±3.1° range remains the same for all of the material systems studied herein. This impliesthat the response of this family of high quality aerospace grade prepregs should be consistent re-gardless of roll.Extracting distribution parameters from the experimental normalized excess length distribu-tions allows a virtual set of fibres to be generated using a reasonable sinusoidal solution for thefibre centroid. As seen in the micrographs, there was no local coordination between adjacent fibreswhich would have been expected from Rosen’s predictions for the buckling solution of composites100with volume fractions in excess of 20 % [20]. Conveniently, this allowed for a simpler randomphase shift sinusoidal solution to be applied as there was no requirement for local fibres to featuresimilar trajectories.The in-plane misalignment values were interrogated from the virtual fibre bed system and var-ied depending on the input distribution type for the normalized excess lengths. Normal distri-butions fitted to the mirrored data produced very aligned virtual fibres, ς = ±1.00°, because thefitted normal distributions quickly decayed to near-zero frequencies for more misaligned fibres.Exponential distributions suffered a similar issue. Student’s t-distribution represented the symmet-ric data most accurately and generated in-plane misalignment standard deviations between 1.5° to1.7°. This variance is an artefact of the wavelength and phase shifts randomly chosen for eachfibre. While the amplitude is determined from the wavelength and excess length, no correlation forthe misalignment probability density functions was observed for all longer wavelengths, all shorterwavelengths, randomly chosen, or generated from a non-linear distribution functions. Note that noeffort was made to mitigate interference or contact between fibres as this would greatly increasethe complexity of this type of model without significantly increasing the information garnered fromthese tests.There was very good agreement on the misalignment distributions and standard deviationsbetween the virtual analysis, using only the distribution of excess lengths and known constantsfrom the material datasheet, and the experimentally observed data, ±1.5° to ±1.7° and ±1.6°,respectively. This corroborates the use of a simple sinusoidal fibre descriptor within a fibre bedfeaturing a distribution of excess lengths.1013.5 Fibre volume fractionIn a similar vein to fibre alignment, fibre volume fraction plays a critical role in determining com-posite stiffness and also suffers from variability [147]. This stiffness variability will alter theprepreg’s response during compressive loading and ultimately influence the location and severityof misalignments.Fibre volume fraction is defined using Equation 3.9 and has traditionally been determined usingacid digestion, ignition, or carbonization, according to ASTM [148]. However, determining fibrevolume fraction using optical microscopy images, sectioned at 90° to the 1-direction, is a robust,accurate [149] and commonly used technique [150].v f =V fV c(3.9)where V f[m3]is the volume of fibre contained in a volume of composite, V c[m3].In contrast to a direct measurement of volume, measurement of fibre volume fraction usingoptical microscopy typically involves comparing light and dark regions of a micrograph, Equa-tion 3.10.v f =A fAc(3.10)where A f[m2]is the area of fibres and Ac[m2]is the area of the composite.VfVc102MethodFigure 3.42 shows an idealized set of fibres in a hexagonal close-packing (HCP) formation. Theinradius of the hexagon, rh = 7.73µm, can be determined from the AS4/8552-1 fibre diameter,r f = 7.1µm, and the fibre volume fraction, v f = 57%. By sectioning the composite normal tothe 1-direction, along the 2-3 plane, any repeating unit, an example being the blue portions ofFigure 3.42(b), will return the correct fibre volume fraction.(a) (b)Figure 3.42: (a) A virtual set of idealized HCP fibres with a translucent grey cutting planeintersecting the fibres parallel to the 2-3 plane. (b) Resultant faces after cutting with a57 % fibre areal fraction. The RVE used to calculate the fibre areal fraction has beenhighlighted in blue.This procedure has been followed in for a sample of AS4/8552-1 sectioned at nominally 90° tothe 1-direction, Figure 3.43.103(a)(b)(c)Figure 3.43: (a) a 48 mm wide mosaic micrograph of an approximately 0.16 mm thickAS4/8552-1 lamina sectioned along the 2-3 plane. (b) an arbitrary 0.5 mm wide sec-tion magnified from the previous mosaic. (c) the traditional image thresholding usedto determine a fibre volume fraction of 57.1 %.However, this procedure is only correct if the section is made along the 2-3 plane. This rela-tionship becomes sensitive to position if the sectioning is performed in either the 1-2 or 1-3 plane.Using the same virtual fibre system described in, Figure 3.42(a), a simple example of where Equa-tion 3.10 returns a grossly incorrect value is the section shown in Figure 3.44(f). Translating thecutting plane along the 2, or 3, plane alters the determined fibre volume fraction from 0 % to 80 %,shown in Figure 3.45.104(a) (b)(c) (d)(e) (f)(g) (h)Figure 3.44: (a), (c), (e), and (g) show a virtual cutting surface along the 1-3 plane as ittranslates along the 2-direction by 0, rh4 ,rh2 , and3rh4 from an arbitrary fibre centroid ofan HCP fibre bed. (b), (d), (f), and (h) virtual fibre faces sectioned by their respectivecutting surfaces. The corresponding fibre volume fractions are 80 %, 67 %, 0 %, 67 %.105Figure 3.45: Fibre volume fraction as determined by percent area of sectioned fibres as the1-3 plane cutting surface translates along the 2-direction.Returning to a system similar to the micrographs used in Section 3.4, the same virtual fibreshave been sectioned at 5° to the 1-3 plane, Figure 3.46. Again, the volume fraction determined byarea comparisons is not constant. Extending the width of the analysis window to extremely largevalues will effectively return the true volume fraction. However, small window sizes will vary dueto the location of the cutting plane, similar to Figure 3.45. The results from a virtual analysis,Figure 3.47, show that the reported volume fraction changes significantly, greater than 30 %, forsmall window sizes, 100 µm, and stabilize for sufficiently large windows, greater than 500 µm. TheComposite Materials Handbook-17 recommends a minimum of 20 images with a minimum of 30fibres in the field of view which results in a minimum bound of approximately 750 µm [150]. Thistype of analysis will scale with fibre diameter, for example a larger fibre diameter will require largerwindows, and vice versa. For this work, a 1 mm wide window provided sufficient spatial resolutionwhile minimizing errors introduced from window size.Radford commented on the fibre shape but not any impact from off-axis sectioning when deter-mining fibre volume fraction from sections cut at 45° degrees and commented “this results in fibercross-sections that appear slightly elliptical; however, this does not affect the measured volumefraction which is determined as an area fraction.” [151].106(a) (b)(c) (d)(e) (f)(g) (h)Figure 3.46: (a), (c), (e), and (g) show various cutting surfaces inclined at 5° to the 1-3 planeas it translates along the 2-direction. (b), (d), (f), and (h) resultant fibre faces for a fixedwindow width.107(a) 0 0.2 0.4 0.6 0.8 1 0  2  4  6  8  10  12  14  16Fibre volume fractionShift along 2-Direction [μm]Window width39 μm(b) 0 0.2 0.4 0.6 0.8 1 0  100  200  300  400Fibre volume fractionWindow width [μm]2-direction shift3.86 μm(c)Figure 3.47: (a) Fibre volume fraction as determined from the areal fraction of fibre facesaccording to the window width and 2-direction shift factor for a 5° cutting surface.(b) The fibre volume fraction as a function of 2-direction shift of the cutting surfacefor a constant window width of 39 µm. (c) The fibre volume fraction as a function ofwindow width for a constant 2-direction shift consistent with Figure 3.46(f).108The fibre position, orientation, and major and minor diameters were used to regenerate themicrographs in digital form for the innermost and outermost specimens which had been thermallycured and ammonium hydroxide soak cured. A sample section has been reproduced here for visu-alization, Figure 3.48.(a)(b)(c)Figure 3.48: (a) a full 49 mm mosaic micrograph of one wedge specimen. (b) a magnifiedsection of (a) highlighting the resolution by visualizing individual fibres. (c) the digitalrecreation of the elliptical fibres from the ellipse study performed in Section 3.4.109To determine fibre volume fraction, Equation 3.10 requires not only the area corresponding tothe fibre but also the area corresponding to the composite. This composite area is less well defined.This value was determined by initially setting 1 mm wide bounding windows over the dataset,Figure 3.49(a). Applying a minimum area bounding box function over the fibre centroids, (c) and(d), returns a reproducible composite area, as well as the ply thickness. The area for the fibresis determined by rasterizing the mathematical ellipses into pixel image data, then masked by thecorresponding bounding box, and counting the total number of coloured pixels. This rasterizationis required due to the partial clipping of any periphery fibres by the minimum area bounding box.As a comparison to the experimentally measured values, the cured ply thickness can be esti-mated from roll information using the areal weight, AF[kgm−2], the number of plies, np, the fibrevolume fraction, v f , and the density of the fibre, ρ f[kgm−3], using Equation 3.11.tp =npAFρ f v f(3.11)110(a)(b)(c)(d)(e)Figure 3.49: (a) a recreated 6 mm wide section. (b) a reduced window for further analysis.(c) the centroid from each fibre. (d) the minimum area bounding box containing all ofthe centroids. (e) the final resultant fibre volume fraction of 59.4 % for the windowedsection.111Results and discussionFibre volume fraction for the 12 wedge specimens, as a function of position along the wedge length,is shown in Figure 3.50. The thermally cured specimens had a volume fraction of v f = (55.9±5.1)%.The ammonium hydroxide soaked specimens, exhibiting uncured prepreg features, had a fibre vol-ume fraction of v f = (30.0±9.7)%. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0  20  40  60  80  100  120  140  160  180Fibre volume fractionPosition along wedge length [mm]r = 224 mmr = 163 mmr = 224 mmr = 163 mmWedge 1 Wedge 2 Wedge 3Soak cureThermal cureFigure 3.50: The fibre volume fraction for the approximately 1 mm wide windows as a func-tion of window position for the innermost and outermost specimens cured either ther-mally or using the ammonium hydroxide soak.Ply thickness for the same samples are plotted in Figure 3.51. Ply thickness for the thermallycured specimens was tT,p = (172.0±12.9)µm while the ammonium hydroxide cured specimenswas tAS,p = (263.0±55.6)µm.Applying Equation 3.11 to determine the manufacturer’s estimate of cured ply thickness returnsa value of 184.9 µm. Therefore, the measured fibre volume fraction and thermally cured ply thick-112ness results are in good agreement with their expected values; however, the experimental valuesshow that variability should be associated with any of the reported values. 50 100 150 200 250 300 350 400 450 500 0  20  40  60  80  100  120  140  160  180Thickness [µm]Position along wedge length [mm]r = 224 mmr = 163 mmr = 224 mmr = 163 mmWedge 1 Wedge 2 Wedge 3Soak cureThermal cureFigure 3.51: The cured and uncured ply thickness for the approximately 1 mm wide windowsas a function of window position for the innermost and outermost specimens curedeither thermally or using the ammonium hydroxide soak.For a constant fibre area cross section and window width, the fibre volume fraction will beproportional to the inverse of the ply thickness. However, this relationship was not respected inthe thermally cured samples, and was only observed very loosely for the ammonium hydroxidesoaked specimens. For example, the outermost, thermally cured, specimen thickness and fibrevolume fraction are plotted in Figure 3.52(a) and normalized values, with respect to their respectivemean values, are plotted in (b). No consistent trend was observed, implying that the variability isnot the result of applied loads and temperature modifications during processing. Rather, the lackof a relationship between local ply thickness and local fibre volume fraction indicates that thevariability must stem from the prepreg manufacture itself. This variability is also observed upon113further inspection of Figure 3.48(b) and Figure 3.43(b). The fibre arrangement in each figure varysignificantly from their ideally packed representations in Figure 3.48 and Figure 3.42, respectively. 0 0.2 0.4 0.6 0.8 0  20  40  60  80  100  120  140  160  180 100 150 200 250 300Fibre volume fractionThickness [µm]Position along wedge length [mm]r = 224 mmThermal curevftWedge 1 Wedge 2 Wedge 3(a)0.7511.25 0  20  40  60  80  100  120  140  160  180 0.75 1 1.25Normalized fibrevolume fractionNormalized thicknessPosition along wedge length [mm]r = 224 mmThermal curevftWedge 1 Wedge 2 Wedge 3(b)Figure 3.52: (a) Fibre volume fraction and cured ply thickness as a function of position for thethermally cured outermost specimen. (b) Fibre volume fraction and CPT normalizedby their respective mean values as a function of position.Directly comparing fibre volume fraction to cured ply thickness, Figure 3.53(a), no significantcorrelation can be drawn. The 6 wedge specimens were very consistent about their ply thickness,114all samples fell within a maximum bound of±0.025 mm. Figure 3.53(b) introduces the ammoniumhydroxide cured samples; data points in the annotated ‘A’ region are the result of imaging artefactscaused by only half of the ply being properly imaged due to the presence of a large void. Anexample of this imaging artefact is shown in Figure 3.54 where only the top half is imaged and thebottom half is absent.115 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0  50  100  150  200  250  300  350  400  450  500  550Fibre volume fractionPly thickness [µm]r = 224 mm, TC/W1r = 224 mm, TC/W2r = 224 mm, TC/W3r = 163 mm, TC/W1r = 163 mm, TC/W2r = 163 mm, TC/W3(a) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0  50  100  150  200  250  300  350  400  450  500  550AFibre volume fractionPly thickness [µm]r = 224 mm, TC/W1r = 224 mm, TC/W2r = 224 mm, TC/W3r = 163 mm, TC/W1r = 163 mm, TC/W2r = 163 mm, TC/W3r = 224 mm, AC/W1r = 224 mm, AC/W2r = 224 mm, AC/W3r = 163 mm, AC/W1r = 163 mm, AC/W2r = 163 mm, AC/W3(b) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0  0.001  0.002  0.003  0.004  0.005  0.006  0.007  0.008Fibre volume fractionReciprocal of ply thickness [1/µm]r = 224 mm, TC/W1r = 224 mm, TC/W2r = 224 mm, TC/W3r = 163 mm, TC/W1r = 163 mm, TC/W2r = 163 mm, TC/W3r = 224 mm, AC/W1r = 224 mm, AC/W2r = 224 mm, AC/W3r = 163 mm, AC/W1r = 163 mm, AC/W2r = 163 mm, AC/W3(c)Figure 3.53: (a) Clustered fibre volume fraction as a function of CPT for the thermally curedspecimens. (b) the fibre volume fraction for both thermally and ammonium hydroxidecured specimens. (c) Linear relationship between reciprocal of the ply thickness andthe fibre volume fraction.116Figure 3.54: Micrograph showing a section where only half of the ply was imaged due to thelarge internal void.Combining the misalignment data collected in Section 3.4 with the positional and thicknessinformation, through-thickness misalignment data can be investigated. Each nominally 1 mm widewindow was subdivided into 20 µm regions through thickness. Fibres whose centroids were boundby these subregions were collected, represented by dark blue ‘Total’ points in Figure 3.55. Fibreswhose misalignment values were highly misaligned, greater than one standard deviation from themean, are represented by purple ‘> |σ |’ points. The corresponding fraction of highly misalignedfibres to total fibres in each subregion and window are represented by teal ‘Fraction > |σ |’ points.Three of these windows are shown in Figure 3.55. No correlation between highly misaligned fibresand their through thickness position was found for either the thermally cured or ammonium hy-droxide soak cure specimens. This further displays the lack of effect between fibre misalignmentsand initial roll radial position as, if the rolling induced strains were retained, larger misalignmentswould be expected on the inner surface of the ply, with unaffected, straight, fibres on the outersurface of the ply. This experimental misalignment data has shown that for the radial positionstested, neither the absolute magnitude nor the positional misalignment has been affected by theinitial position.117 0 75 150 225 300 0  50  100  150  200 0 0.25 0.5 0.75 1Number of fibresFraction of fibresPosition through thickness [µm]Total> |σ|Fraction > |σ|(a) 0 75 150 225 300 0  50  100  150  200 0 0.25 0.5 0.75 1Number of fibresFraction of fibresPosition through thickness [µm]Total> |σ|Fraction > |σ|(b) 0 75 150 225 300 0  50  100  150  200 0 0.25 0.5 0.75 1Number of fibresFraction of fibresPosition through thickness [µm]Total> |σ|Fraction > |σ|(c)Figure 3.55: (a), (b), and (c) display the number of total number of fibres, the number offibres with misalignments greater than the standard deviation, and the correspondingfraction, as a function of centroid position through thickness from three representative6 mm wide windows.1183.6 Tow and ply architectureMethodAnother source of variability which is not often discussed, yet is easily returned from the misalign-ment study, is the fibre diameter. All of the fibre diameters from the thermally cured AS4/8552-1specimens have been compiled into a histogram shown in Figure 3.56.Further, some supplementary information, such as average uncured ply thickness, UPT, can bedetermined from information gathered during the roll unwinding by assuming the original windingprocess results in a ply which follows an Archimedean spiral, Equation 3.12. Measuring the take-up roll core radius, Rc [m], the outer radius, Ro [m], along with the unwound ply length, Lu [m],the number of windings, T , can be determined from Equation 3.13. Assuming a negligible gapspacing between plies in the 3-direction allows the combined thickness, tc [m], to be calculatedfrom Equation 3.14 . The combined thickness being equal to the sum of the UPT, tp,u [m], and thebacking paper thickness, tb [m]. The backing paper thickness was measured at several locationsusing calipers returning an average thickness of 0.112 mm.r (θF) = Rc+Ro−Rc2piTwθF (3.12)Lu =∫ 2piTw0√r2+(drdθF)2dθF (3.13)tc =Ro−RcTw(3.14)Finally, burn off tests were performed according to a modified [9] ASTM D2584-11 [152] todetermine the tow width of the underlying prepreg structure. The tow width can be estimated fromthe datasheet values assuming a packing factor; for square close packing, the number of fibres119through thickness, ftt , and the width of the tow, wt [m], can be calculated using Equation 3.15 andEquation 3.16.ftt =CPT −d f√pid2f4v f+1 (3.15)tw =N fftt·√pid2f4v f+d f (3.16)where N f is the number of fibres per tow. For these prepregs, the roll information states 12,000fibres per tow.Results and discussionThe fibre diameter distribution is reasonably symmetric with a mean value of 6.8 µm and a standarddeviation of ±0.5 µm. The distribution is very similar to the distribution shown by Yurgartis inhis paper [18]. An average AS4 fibre diameter of 6.8 µm compares well with the manufacturer’snominal filament diameter value of 7.1 µm [35]. Yurgartis measured a similar manufacturer overestimate for the XAS fibre diameter, Yurgartis’ value of (6.9±0.3) µm relative to the XAS supplierreported value of 7 µm. Others have reported similar values, (6.890±0.273) µm, for AS4 fibreswith a sample size of 91 fibres [121].120 0 2000 4000 6000 8000 10000 12000 14000 16000 5  5.5  6  6.5  7  7.5  8  8.5  9N = 126'429Number of fibresFibre diameter [µm]Figure 3.56: Histogram of the measured fibre diameters for the AS4/8552-1 roll ‘A’ speci-mens. d f ibres = (6.8±0.5)µm with N = 126429 fibresThe initial radial position, Ro, was measured at 224 mm while the core radius was 163 mm.Approximately 230 m of material was unwound during this process which when substituted intoEquation 3.13 returns a value of 189.4 revolutions around the core. The combined backing paperand ply thickness, determined using Equation 3.14, is 0.319 mm. Subtracting the backing paperthickness, 0.112 mm, returns a value of 0.206 mm for the average uncured ply thickness.Results from the burn-off test, Figure 3.57, show how the tow width was measured. The towwidth was approximately (4.0±0.2)mm with approximately 23 fibres through thickness. Usingthe datasheet values, Equation 3.15 reduces to 22 fibres through thickness and Equation 3.16 returnsa value of 4.5 mm. These values are in good agreement with the experimentally measured valuesand are further improved by substituting the experimentally measured fibre diameter. Using a fibrediameter of (6.8±0.5) µm the tow width becomes (4.1±0.6)mm.121Figure 3.57: Resulting specimen after performing the burn off and applying small transverseloads to separate the tows. The ruler’s major graduations are in cm.1223.7 Summary of characterizationThe ultimate takeaway from this characterization is that all of the measured composite propertiesfeature some variability. This chapter has documented several novel strategies for capturing andmeasuring this variability. A new curing method has been used which allowed traditional tech-niques to measure the laminates properties without the traditional cure’s associated temperaturemodification or cure shrinkage. This is significant as both the temperature modification and cureshrinkage will alter the stress state, and therefore alignment, of the fibres.Using this strategy, the excess length introduced from the cure cycle has been measured to beapproximately 31 % of the total excess length. This additional excess length is net effect of thecoefficient of thermal expansion (CTE) and the cure shrinkage. This shows the dominant feature isthe cure shrinkage as the CTE would decrease the overall excess length. With 31 % of the excesslength stemming from the cure cycle, the majority of the excess length likely originates from theprepreg manufacture itself.Writing and implementing semi-automated data reduction techniques allowed the largest dataset of its kind to be created and analyzed. This large dataset, >200,000 elements, allowed novelspatial techniques to be used. The fibre alignment, while consistent among different different rollsand resin types, was associated with variability. The in-plane misalignments had a measured±1.6°standard deviation while the out-of-plane misalignments were significantly more aligned at±0.53°.These values did not vary significantly between rolls or resin system as long as the fibre systemremained constant. The fibre alignment was not correlated with the position at which the prepregwas removed from the roll, nor was misalignment correlated to the through thickness position.The proposed fibre bed model is able to reproduce the misalignment distributions measuredexperimentally. Only by allowing each fibre in the system to have a unique excess length is itpossible to properly describe the underlying fibre architecture.123Variability was also associated with fibre volume fraction, (55.9±5.1)%. This value boundedthe nominal 57.42 % but casts slight doubt on the significant digits provided by the manufacturer.Similarly, the cured ply thickness bounded the data sheet value of 185 µm but had regions slightlythicker than 200 µm and other regions thinner than 160 µm.While variability is a fundamental attribute of CFRPs available today, the variability is homo-geneously dispersed throughout the entire composite. This means that defects, such as wrinkling,seen during further processing should not be readily attributable to a single defect originating inthe raw material which would be expected if variability manifested itself only in localized regions.124Chapter 4Transient wrinkle growth4.1 IntroductionGenerating wrinkles in uncured composite materials is not difficult; doing so in a reproducible andcontrolled manner is where the difficulty arises. Much of the literature has discussed wrinklingdeveloping during forming and consolidation where the lamina or laminate is unable to undergoappreciable in-plane [42] or through-thickness shear [39], [153] resulting in compressive stressesand wrinkling. For woven fabrics this is more understood whereas the fundamental dynamics ofunidirectional prepregs are still areas under active research. The key parameters governing wrinkledevelopment in this process are the mechanical properties of the ply and the interactions of theply on it’s surroundings, e.g., ply-ply interaction, ply-bag interaction, and tool-part interaction. Ofthese, the tool-part interaction is the least studied with regard to wrinkle generation; this type ofloading is also an important area for this project’s industrial partner.One method for reproducibly generating small strains is through temperature alteration of non-zero CTE materials. Without mechanical constraint, linear thermal strains can be easily deter-mined from the imposed thermal cycle and the materials linear CTE, Equation 4.1. Measuringthe 1-direction linear CTE of uncured prepregs is difficult; however, the 1-direction CTE of cured125AS4/8552 has been previously studied and is reported as “practically zero” along the fibre direc-tion, with a measured value of 0.21×10−6 K−1 [92]. During cure, the 1-direction CTE will be lowdue to the contribution of the reinforcing carbon fibres with high axial stiffness and near zero axialCTE, −0.63×10−6 ◦C [35], [154].εT = α∆T (4.1)where εT is the thermal strain, α[K−1]is the linear coefficient of thermal expansion, and ∆T [K]is the imposed temperature differential.Residual stresses due to tool-part interaction caused by difference in CTE are well documented.One seemingly straightforward solution is to match the tool’s CTE to the part’s CTE, unfortunately,low CTE tooling materials such as Invar are prohibitively expensive for many manufacturers [155],[156]. Further, the anisotropic nature of composites also implies that without conscious considera-tion, the part’s CTE can change over the part geometry making closely matched tooling very diffi-cult. More common, cost-effective, aluminium tooling has a CTE in the range of 23.5×10−6 ◦C−1[83]. Other strategies for diminishing tool-part interaction is release coat or release films. Theformer decreases interaction while the latter effectively removes the interaction entirely [56], [60].One stress-strain relationship between the tool and part was devised by Twigg et al. who de-veloped an analytical relationship between the tool and part’s geometric and mechanical propertiesand the stress and strain in the respective components [60], [79], [80], [157]. As discussed inSection 2.3.3, either a sliding or sticking regime develop according to the stage of the cure cycleand the material properties. During the heat-up portion of the cure cycle, the sliding condition isdominant [79]. If the debond stress, τDebond , is sufficiently high, the sliding condition will changeto a sticking condition [79]. A schematic has been recreated in Figure 4.1 which shows the loadingcondition during the sticking regime. This model was developed assuming invariant part moduluswith respect to temperature and degree of cure.126Tool (E2,α2)CFRP (E1,α1)Adhesive layerx=0Tool centrelinesymm.xzLh1h2hAdxσ2 + dσ2σ1 + dσ1σ1σ2ττFigure 4.1: Schematic of the sticking tool-part loading condition with annotations for therelevant geometric properties, recreated after [60].Given the very large temperature dependence of resin properties, shown in Section 3.3.1, andthe significant reduction in prepreg modulus, discussed in Figure 2.12, the later assumption needsto be amended. The material will be above its glass transition temperature during processing andcannot be assumed to have a linear elastic response. A viscoelastic formulation can potentiallyprovide more accurate results; however, viscoelastic characterization is significantly more complexand time consuming [158]. One strategy with intermediate complexity which has been successfulin capturing residual stresses during cure is the so called cure hardening instantaneously linearelastic (CHILE) model [91], [158], developed around Equation 4.2. The CHILE model assumesthat the material is invariant with time and only changes with temperature and degree of cure [158].This approach has been shown to model stress development during the curing process where the127resin’s modulus increases monotonically. During the cool-down of a heated debulk or forming step,the uncured resin modulus will also increase monotonically; hence, this approach has been takenin deriving the tool and part stresses during the uncured part cool-down in Equation 4.2 throughEquation 4.18.σ˙ = Eu · ε˙ (4.2)where σ [Pa] is the normal stress, Eu [Pa] is the unrelaxed [140], or instantaneous [158], elasticmodulus and the dot notation refers to a time-derivative, e.g., σ˙ = dσ/dt.The relationship between the rate of shear stress, τ˙ , and the rate of shear strain, γ˙ , becomesEquation 4.3τ˙ = Gu · γ˙ (4.3)where Gu [Pa] is the unrelaxed shear modulus and γ is the shear strain.This derivation uses several of the same assumptions that Twigg et al. used, specifically thatthere is no spatial gradient for temperature, the tool and part undergo no shear deformation, and theadhesive interlayer undergoes no extensional strains. Implicitly it was also assumed that there is nochange in through thickness geometry of all of the bodies and the coefficients of thermal expansionwere invariant. By using a symmetric layup, where one ply is bonded to either side of the tool, itis assumed that there is no bending. The derivation proposed here requires these assumptions andan additional assumption that the temperature range is well below the working temperature of thetool; the tool’s modulus is invariant, E2 6= f (t,T,χ).Analyzing the three free body diagrams shown in the callout of Figure 4.1, the following tworelationships hold during equilibrium.128τ =−dσ1dx·h1 (4.4)τ =dσ2dx·h2 (4.5)where h1 [m] is thicknesses of the part and h2 [m] is half the thickness of the tool.Adding the thermal strain rate to the composites mechanical strain rate relationship of Equa-tion 4.2 returns the strain rate function:du˙1dx= ε˙1 =σ˙1Eu1+α1 · T˙ (4.6)where T˙ is the temperature rate, du˙1dx , anddu˙2dx are the strain rates of the part and tool.For the linear elastic tool, the stress-strain relationship follows Hooke’s law, Equation 4.7,which if differentiated with respect to time becomes Equation 4.8.σ2 = E2 · εM,2 (4.7)σ˙2 = E2 · ε˙M,2+0E˙2 · εM,2 (4.8)Adding the thermal strain into Equation 4.8 returns:du˙2dx= ε˙2 =σ˙2E2+α2T˙ (4.9)Differentiating Equation 4.4 and Equation 4.5 with respect to time, Equation 4.6 and Equa-tion 4.9 with respect to position, and performing substitutions results in the following two equa-tions:129d2u˙1dx2=1Eu1· dσ˙1dx=− 1Eu1 h1· τ˙ (4.10)d2u˙2dx2=1E2· dσ˙2dx=1E2h2· τ˙ (4.11)Introducing the relative displacement rates into the shear rate relationship of Equation 4.3 andassuming the adhesive layer does not change thickness over time, the shear rate relationship relativeto the displacement rates of the tool and part becomes:τ˙ = Gu · u˙2− u˙1hA(4.12)where hA [m] is the interlayer thickness.Differentiating Equation 4.12 twice with respect to position and substituting Equation 4.10 andEquation 4.11 returns a differential equation for the shear stress rate:d2τ˙dx2=GuhA(1Eu1 h1+1E2h2)τ˙ (4.13)Recognizing that a first order time differential appears in all of the τ terms, Equation 4.13 canbe treated as an ordinary differential equation with respect to position. This differential equationcan be solved using the symmetry boundary condition, τ (x = 0) = τ˙ (x = 0) = 0, and the free edgeboundary condition, σ (x = L(T )) = σ˙ (x = L(T )) = 0.τ˙ =dτdTdTdt=Gu (α2−α1) dTdtλhA· eλx− e−λxeλL(T )+ e−λL(T )(4.14)Finally, the shear stress at temperature and position can be determined by integrating over thethermal cool-down:130τ =∫ TfToGu (T ) · (α2−α1)λ (T )hA· eλ (T )x− e−λ (T )xeλ (T )L(T )+ e−λ (T )L(T )dT (4.15)where To [K] is the minimum temperature at which the resin can appreciably impart stresses, Tf [K]is the final resting temperature after the cool down, and λ (T ) takes the form:λ (T ) =√Gu (T )hA·(1Eu1 (T )h1+1E2h2)(4.16)Substituting the shear stress solution into Equation 4.4 and integrating with respect to positionreturns the solution for in-plane stress in the composite part:σ1 =∫ xL(T )− 1h1· τ (T )dx (4.17)Finally, the strain in the composite ply during the cooling process can be determined fromEquation 4.18.ε1 =∆LL(Tf )=∫ TfTo∫ 0L(T )− 1h1· 1Eu1 (T )· Gu (T ) · (α2−α1)hA ·λ (T ) ·eλ (T )x− e−λ (T )xeλ (T )L(T )+ e−λ (T )L(T )dx dT (4.18)Therefore, to estimate strain induced into the uncured composite during a cool-down causedby tool-part interaction, four geometric properties (h1,h2,hA,L(T )), five mechanical properties(α1,α2,Eu1 ,Gu1,E2), and the two temperatures (To,Tf ) must be known.Wrinkle generation will ultimately be related to the ability of the part to locally debond fromthe tool. This tool-part debond has been shown to be strain rate dependent [84], [86]. At the debondfront, continued failure at the interface can occur from the interface edge crack propagating alongthe interface [89], [159]. For a crack to propagate, the energy released during crack propagationmust provide the energy required for the crack to grow [160]. For wrinkle growth, the change in131internal energy must therefore provide the energy to propagate the wrinkle. Knowing the internalenergy of the prepreg, Equation 4.19, and the wrinkle geometry, specifically the width, the crit-ical strain energy release rate, Gc[Jm−2], can be calculated using Equation 4.20. The internalenergy state of the wrinkle itself is unknown; however, the post-buckled load required for furtherdisplacement decreases drastically [161] which implies the internal energy of the wrinkle is alsovery low.U =12V Eu1ε2 (4.19)where V[m3]is the volume of the strained material.Gc =− dUb ·da (4.20)where U [J] is the elastic energy, a [m] the crack length, and b [m] is the width of the crack.V u1 u2 λ I1324.2 Method4.2.1 GeometricOne of the material systems characterized previously, AS4/8552-1, was used to study transientwrinkle growth.To generate these compressive stresses, a 152 mm by 76 mm single ply was centered on a200 mm by 104 mm by 1.58 mm thick piece of polycarbonate. The polycarbonate has a relativelylarge CTE, 70×10−6 ◦C−1 [162], which should exacerbate the tool-part interaction allowing mea-surements to be made over relatively small, lab-scale, distances.The 1-direction of the lamina follows the long direction of the polycarbonate; the full layup isshown in Figure 4.2. To inhibit bending caused by asymmetry, a second ply was centered on theopposing face to form a symmetric sandwich. The polycarbonate was received with a protectivefilm which was removed just prior to the lamina application and care was taken to not touch orcontaminate the surface so that the surface condition would be consistent between trials. The layupwas then placed in a breather and peel ply envelope to ensure that there was no tool part interactionbetween the layup and the aluminium tool.To test the hypothesis that the prepreg is under a shear lag condition, a second series of testswere performed using a PMP release film to modify the surface condition in a localized region.By modifying the location of these PMP initiators, the debond strength and effective length ofinteraction can be determined using Equation 4.15 and Equation 4.18. PMP is sold as a releasefilm and prevents any degree of tool-part interaction [60]. The 0.0375 mm thick PMP [163] filmwas manually cut to 3 mm wide strips for each trial and placed between the polycarbonate substrateand the lamina such that the film spanned the full length along the 2-direction. One, two, or four,initiators were used in these trials. The center-to-center spacing between initiators for the two andfour initiator trials was approximately 68 mm and 31 mm, respectively.13376 mm152 mm200 mm104 mm(a)BreatherPrepregPeel plyPolycarbonateAluminiumTacky tapeVacuum portVacuum bag(b)Figure 4.2: (a) Top view, and (b) side view, of a typical specimen during preparation withhighlighted dimensions and materials. The 1-direction of the unidirectional prepreg isparallel to the 152 mm dimension.A vacuum was drawn on the system and the setup as placed into the same laboratory ovenused in the characterization trials. A 1.5 ◦Cmin−1 ramp was used to bring the temperature of thespecimen to 65 ◦C and was followed by a hold at temperature for 30 min while still under dynamicvacuum. This hold temperature is consistent with hot drape forming [53] while also having aninsignificant effect on the degree of cure [135], [164]. After the 30 min hold, the specimen was left134to cool to room temperature, again under dynamic vacuum. Once equilibrated with room tempera-ture, the specimen was removed from the dynamic vacuum, remaining under static vacuum in thesealed bagging arrangement, and quickly moved from the oven area to the analytical laboratory.The polycarbonate and prepreg specimens were quickly removed from vacuum and their envelopewhereby the surfaces were measured using either a digital camera or a coordinate measuring ma-chine (CMM). The time between removing the static vacuum and beginning of the first image orscan is represented in several figures as t0 and is on the order of 1 min to 5 min. The nomenclaturefor the timing is Ad BB:CC:DD where A is the number of days, BB is the number of hours, CC isthe number of minutes, and DD is the number of seconds elapsed since the first image or scan. Thereported times have been rounded to the nearest 5 min increment for easier comparison.135(a)(b)Figure 4.3: Visual description of the heated debulk process. (a) The expansion due to the65 ◦C ramp and 30 min hold and (b) the contraction as the specimen returns to roomtemperature under applied vacuum.Several specimens were imaged using a Nikon D7000 digital camera to directly view theprepreg response over an 8 h period. The surface of the single layer of CFRP prepreg does notoffer significant contrast, hence oblique lighting was used to accentuate any wrinkling by elongat-ing their cast shadows, Figure 4.4.136Figure 4.4: Schematic of imaging procedure of the prepreg surface using oblique lighting andan off-axis camera to highlight any surface defects.A Nikon Metrology CMM with a non-contact XC65D-LS laser scanning head was used toquantitatively measure the prepreg surface profile. This non-contact method was required as anycontact could easily alter the stress state of the compliant specimen. The CMM relies on a gantrysystem where the specimen to be measured must reside inside the instrument and the laser scannermust have an unobstructed view of the specimen. These restrictions limited the ability to controlthe environment’s temperature and humidity as an enclosed controlled environment could not becreated. Each test was run in an analytical laboratory without fine control over the ambient temper-ature and humidity. Temperature was recorded throughout the duration of each test and remainedwithin ±0.5 ◦C for any given test. Between tests, the average temperature ranged from 20.7 ◦C to24.3 ◦C. The relative humidity fell within (40±7)%. The CMM offers high fidelity reconstruc-tions for matte white surfaces with an accuracy of 12 µm. To measure the semi-gloss black prepregsurfaces using the CMM, standard practice dictates coating the surface with a fine layer of talcpowder. The quantitative output from the CMM was an (x,y,z) point cloud with a typical (δx,δy)grid size of approximately 0.05 mm by 0.05 mm. The measured (x,y,z) points did not necessar-ily correspond with an evenly spaced grid, hence, a linear interpolation scheme, using a grid size137smaller than the measured grid size, was implemented for further data regression. Any extrapo-lated points were removed from the data set to remove possible errors. A rigid body rotation wasperformed to orient the point cloud x-direction with the 1-direction of the lamina. The 2-directionand 3-directions correspond with the reported y-direction and z-direction, respectively. This set ofdefinitions holds for all figures reported herein. An example surface from one of the trials is shownin Figure 4.6. A line running along a constant value of y has been highlighted in blue which isanalyzed in Figure 4.7 to display the methodology used for capturing peaks of interest.(a) (b)Figure 4.5: (a) Nikon Metrology CMM and (b) the non-contact Metris XC65D-LS laser scan-ning head.138Figure 4.6: Example surface profile recreated from the coordinate measuring machine (x,y,z)data with a highlighted blue line running along the prepreg 1-direction.Figure 4.7(a) shows the raw data for the highlighted blue line along with a filtered line toimprove the signal to noise ratio. A circle was fitted to each line and a detrended line was createdto aid the peak detection by having the minimum peaks lay on a flat reference, Figure 4.7(b).Maximum and minimum peaks were identified using an inbuilt MATLAB ‘findpeaks’ function. Aminimum peak prominence of 0.0625 mm was used to remove small peaks associated with noise orgeneral surface roughness. This minimum detectable defect size was determined when additionalthin talc coatings did not significantly alter the total wrinkle count. The (x,y) associated with theminimum and maximum peaks were used to probe the original grid data to determine the unfilteredminimum and maximum z values. This was performed for each of the lines of constant y andthe maximum positions for each wrinkle have been overlaid in red in Figure 4.8. For the wrinkleinitiator trials, the maximum wrinkle height for each line is tabulated and presented as an averagemaximum wrinkle height with an associated standard deviation. The technique for coalescingdisassociated adjacent maximum peaks into a single wrinkle descriptor for each individual wrinkleis non-trivial and steps from this process have been outlined in Figure 4.9.139-789.05-789-788.95-788.9-788.85-788.8 0  20  40  60  80  100  120  140  160z [mm]x [mm]Grid dataFiltered(a)-0.05 0 0.05 0.1 0.15 0.2 0  20  40  60  80  100  120  140  160z [mm]x [mm]Detrended gridDetrended filteredMax peaksMin peaksMax peak(b)Figure 4.7: (a) The raw data along the blue line described in Figure 4.6 has been overlaidwith a filtered signal to decrease noise. (b) The further refinement after removing theslight curvature to allow for individual maximum and minimum peaks to be identified.The left column of figures in Figure 4.9 display the process on the whole specimen, from atop-down view, while the right column display the process on an individual wrinkle, from an off-axis projected view. Figure 4.9(a) and (b) display the peak regions on the surface described earlier.These peak regions were converted into a logical mask by replacing the peak z-values with binaryhigh values leaving all other z-values in the (x,y) grid as binary low values. The logical high valueswere expanded over the (x,y) grid to include neighboring points within 1.3 mm of the x-directionand 2 mm in the y-direction. These values were determined through trial and error in an effort tomerge any very close wrinkle peaks which would otherwise not be connected; an example of thisis Figure 4.9(b) where there the wrinkle peak does not perfectly fall along a constant line in the140y-direction. The logical mask is shown in (c) and (d), (the continuous colour markings in (d) are fordisplay purposes only with the real mask only containing dimensionless 0 or 1 values). An imageanalysis technique was applied to the mask to convert connected regions into labeled components.These individually defined components are represented using an arbitrary colour scheme in (e)and (f). For each component, the maximum peak contained inside the individual mask was thenextracted allowing the corresponding minimum peaks to be determined. The position and identifierof each individually defined wrinkle is overlaid on top of the original surface in (g) and (h).Figure 4.8: The surface profile from Figure 4.6 has been overlaid with the disparate maximumpeaks determined for each step in dy.141(a) (b)(c) (d)(e) (f)(g) (h)Figure 4.9: Several steps from the individual wrinkle detection methodology are reproducedwhich show the transition from disparate peaks into unique wrinkle regions. Refer tothe prior paragraph for a discussion of each image.142The transient nature of the problem was characterized by performing this same scanning andwrinkle identification procedure on the specimen over a period of several days until the systemreached a steady state response. Three frames from the scanning process are depicted, with andwithout the maximum winkle position, in Figure 4.10. Figure 4.10(g) displays the maximum wrin-kle height, and corresponding excess length, as a function of time for the three selected wrinkles.This excess length was determined by subtracting the distance between the two minor peaks asso-ciated with the wrinkle from the surface path length between those two points.143(a) (b)(c) (d)(e) (f) 0 0.1 0.2 0.3 0.4 0.5 0.001  0.01  0.1  1  10 0 0.02 0.04 0.06 0.08 0.1Wrinkle height [mm]Excess length [mm]Time [days]Wrinkle heightExcess length(g)Figure 4.10: (a), (c), and (e) show a reduced surface region as time progresses with the cor-responding (b), (d), and (f) surface highlighted with the maximum peak. (g) Plots themaximum wrinkle height and excess length for the three highlighted points againstlogarithmic time.1444.3 Geometric results4.3.1 No initiator (Geometric results)Figure 4.11 shows three frames taken from a nine hour time-lapse recording of a specimen pre-pared using the previously described heated debulk methodology, Figure 4.3 and Figure 4.4. Thespecimen was removed from active vacuum once it had reached the ambient laboratory tempera-ture of approximately 22 ◦C. The single ply of prepreg was 152 mm in the 1-direction by 76 mm inthe 2-direction; however, due to the angle and lighting scheme used to highlight the out-of-planefeatures, the surface appears as a trapezoid caused from perspective distortion. Qualitatively, Fig-ure 4.11 shows that wrinkle growth is time dependent. All of the wrinkles are centered aroundthe midline of the specimen with an overall wrinkle effected zone width of approximately 70 mm.Another important observation is that each of the large features in the final time step can be tracedto smaller features in the original time step.Specifically, the final time step features on the order of 60 distinct wrinkles with the largestwrinkles having a 1-direction length on the order of 3 mm to 6 mm. All of these features can betraced to the initial time step implying that the number of features in the original time step mustbe equal or greater than 60. This is corroborated upon close inspection of Figure 4.11(a) and (c)as several features in the first time step do not appear in the final time step. The height of thewrinkles cannot be interpreted from the images because the oblique lighting is not collimated.The polycarbonate substrate laid at approximately 45° away from the light source, illustrated inFigure 4.4, will result in muted shadows cast by features along the lower half and accentuatedshadows cast along the upper half of the surface when interpreted from Figure 4.11.145(a)(b)(c)Figure 4.11: The surface of the prepreg (a) immediately after removal from vacuum bagging,(b) after approximately 2 h, and (c) after approximately 8.75 h showing the transientwrinkle formation centered around the lengthwise midline.146Figure 4.12 shows the CMM surface profiles of a single ply sample prepared in the same man-ner as Figure 4.11 with the required addition of a thin coating of talc powder. As seen in Fig-ure 4.12, the surfaces measured using the CMM are qualitatively similar to the surfaces measureddirectly using a digital camera. Therefore, the CMM procedure and surface preparation did notgreatly influenced the wrinkle growth patterns and is an effective tool for measuring these features.The data presented in Figure 4.12 spans a much longer time frame than the direct imaging, nearly9 d, in comparison to 9 h, which allowed sufficient time for the surface to reach a steady state re-sponse. The individual wrinkle methodology also appears to work very well; tracking 127 wrinklefeatures from the initial time step down to 65 wrinkle features in the final time step. A slightcurvature can be seen in Figure 4.12(a) and (b); however, the associated strains are on the order of10−6 and are negligible when compared to the strains associated with the wrinkles themselves. Themagnitude of the wrinkle’s associated strains will be discussed in a following section. The resultsfrom a single trial are presented below; a duplicate trial run under similar conditions showing verysimilar results can be found in Appendix D Figure D.1 though Figure D.4.147(a)(b)(c)Figure 4.12: (a), (b), and (c) show the surface profile, with indicated wrinkles, for a specimenprepared for the CMM. The wrinkle dispersion measured by the CMM is very similarto dispersion directly observed in Figure 4.11.A top-down view of the final surface is presented in Figure 4.13. Similar to the direct imaging,the wrinkles are bounded within the center of the surface, with all of the wrinkles appearing within51mm≤ x≤ 106mm. The tallest three wrinkles, identified with marker’s 45, 41, and 36, lay closeto the boundary indicating that a free boundary promotes wrinkle growth.148Figure 4.13: Top down view of Figure 4.12(c) with each wrinkle uniquely numerated.Each of the wrinkles appearing in the final time step can be traced backwards to the initialtime step. These wrinkle’s respective heights are plotted as a function of time in Figure 4.14.The identifiers used in Figure 4.14(c) correspond with the identifiers shown in Figure 4.13. Themaximum wrinkle height is 0.55 mm with 10 wrinkles exceeding 0.2 mm. While at first theseheights seem quite small, in comparison to the ply’s uncured ply thickness (UPT) of 0.2 mm thesedefect sizes are important and could significantly alter the strength and stiffness of a thin laminate.The average wrinkle 1-direction length was (5.1±1.7)mm and (5.9±2.4)mm for the repeattrial. No wrinkle had a length below 3 mm and taller wrinkles, greater than 1 UPT, consistentlycorresponded with longer lengths, 8 mm to 12 mm; however, shorter wrinkles, less than 1 UPT, hada random length between 3.5 mm to 10 mm.149 0 0.1 0.2 0.3 0.4 0.5 0.6 0  1  2  3  4  5  6  7  8  9Wrinkle height [mm]Elapsed time [days](a) 0 0.1 0.2 0.3 0.4 0.5 0.6 0  6  12  18  24Wrinkle height [mm]Elapsed time [hours](b)4541365348273847712262250615259253423186421529573346594224603583062145837652154132819663314443495135617553911016402032114Wrinkle identifier(c)Figure 4.14: (a) and (b) plot the maximum wrinkle height for each wrinkle indicated in Fig-ure 4.13 over the first 9 d and a reduced section over the first 24 h. (c) The legend,starting from tallest final wrinkle to shortest wrinkle, follows the numbering of Fig-ure 4.13Plotting the population distribution for four selected time steps shows the dynamic growthof the wrinkles. Figure 4.15(a) shows the average wrinkle height as 0.09 mm and a maximum150wrinkle height of 0.16 mm. As time elapses, the maximum wrinkle heights increase, (b) 0.25 mm,(c) 0.43 mm, and (d) 0.55 mm, respectively. Simultaneously the number of wrinkles decreasessignificantly, from 127 to 65, with the mode wrinkle height remaining constant at the minimumdetectable defect size of 0.0625 mm.Interestingly, the 10 largest wrinkles in the steady state response did not grow from the 10largest wrinkles in the initial time step. Only four of the largest initial wrinkles remained in the 10largest steady state wrinkles. The number of common wrinkles increases to 7 out of 10 after the2 h mark. By the next recorded time step, t0+0d17 : 20 : 00, all of the top 10 largest wrinkles sharea common (x,y) position with the steady state wrinkles; however, the exact ranking order changesslightly between these two time steps. 0 10 20 30 40 50 0  0.1  0.2  0.3  0.4  0.5  0.6N = 127Number of wrinklesWrinkle height [mm]t0 + 0d 00:00:00(a) 0 10 20 30 40 50 0  0.1  0.2  0.3  0.4  0.5  0.6N = 123Number of wrinklesWrinkle height [mm]t0 + 0d 02:15:00(b) 0 10 20 30 40 50 0  0.1  0.2  0.3  0.4  0.5  0.6N = 80Number of wrinklesWrinkle height [mm]t0 + 0d 17:20:00(c) 0 10 20 30 40 50 0  0.1  0.2  0.3  0.4  0.5  0.6N = 65Number of wrinklesWrinkle height [mm]t0 + 8d 20:30:00(d)Figure 4.15: (a), (b), (c), and (d) track the population distribution of wrinkle heights throughfour time steps showing the general reduction in the total number of wrinkles.151The exact values for the repeated trial vary slightly; however, the general trend is identical.Initially a large population of wrinkles, 189, decays to 57 wrinkles after a period of nearly 9 d.The largest wrinkles lay along the edge with the vast majority of wrinkles laying within the cen-tral ±35 mm. 13 wrinkles grow above 0.2 mm and the largest 10 wrinkles remain the largest 10wrinkles after 2.75 h.1524.3.2 Wrinkle initiators (Geometric results)Single initiatorSimilar to the non-initiated trials, Figure 4.16 clearly shows the transient nature of the initiatedwrinkles. Other than of the center wrinkle, the surface after 7 h appears relatively feature free.153(a)(b)(c)Figure 4.16: (a), (b), and (c) show the prepreg surface through time with a single 3 mm wideby 0.0375 mm thick film of PMP placed along the lengthwise midline of the prepregspecimen.154Several repeat single initiator wrinkle trials were conducted using the CMM. For each trialthe temperature was recorded with the temperature remaining stable throughout the test, ±0.5 ◦C.There was a maximum average temperature difference of 4 ◦C between all of the different tests. Thesurface profile for a 23 ◦C ambient trial is shown in Figure 4.17 with the surface profiles for thesimilar trials shown in Appendix D Figure D.5 through Figure D.11. The qualitative results from allof these trials are very similar showing an initial fairly uniform distribution of wrinkles, not unlikethe non-initiated surface profiles. The initiated surface profiles then show a response dominated bythe initiated wrinkle which becomes the only significant defect for each trial. Figure 4.17(a) showsan initially disconnected wrinkle at the center implying the initiator is sufficiently thin to allowthe wrinkle to develop according to the modified interface condition rather than a thick initiatorwhich could grossly deform the ply due to its geometry. Then by the next time step, to+5min, thedisconnected center wrinkle connected forming a continuous center wrinkle. Several trials had aconnected wrinkle in the first scan; this is likely due to the variation in to and the time of the scanitself. This very low time to debond also corroborates the negligible interaction between the releasefilm and ply observed by Twigg et al. [60].155(a)(b)(c)Figure 4.17: (a), (b), and (c) show the surface profile after analysis of the CMM data for aspecimen prepared with a 3 mm wide by 0.0375 mm PMP wrinkle initiator along thelengthwise midline.In the non-initiated trials, the wrinkles can be compared to domed features. These domes havea single maximum height, and a single corresponding length in the 1-direction. In contrast, the ini-tiated wrinkle can be visualized as a half cylinder where the height and length can be determined156for each grid step along the cylinders axial direction, the 2-direction of the laminate. The averagewrinkle height of the initiated wrinkle, along with the error bars indicating the heights standarddeviation, are shown in Figure 4.18. The trends are very similar for each of the trials showing anasymptotic growth profile converging to a plateau wrinkle height. The time to reach 95 % of thesteady state wrinkle height was used to determine the plateau. Comparing the steady state wrinkleheight, an initial relationship shows that increasing temperature corresponds with increasing steadystate wrinkle response. This is incompatible with a simple thermal strain mismatch problem, Equa-tion 4.1, because an increased thermal mismatch would cause an increase in excess length whichindicates a more involved solution is required. The trials conducted below 22 ◦C reached a plateauafter (0.5±0.1) d with the trials conducted above 22 ◦C reaching a plateau after (1.5±0.1) d. 0 0.1 0.2 0.3 0.4 0.5 0  0.5  1  1.5  2  2.5  3  3.5  4  4.5Wrinkle height [mm]Elapsed time [days]20.7°C20.7°C21.1°C21.2°C23°C24.3°CFigure 4.18: Wrinkle height of the initiated wrinkle as a function of time for various trialsconducted at slightly different ambient conditions showing a plateau after approxi-mately 1 d.The transient width of the initiated wrinkle is shown in Figure 4.19. Accurate measurementand control of this value was difficult. This difficulty is reflected in the larger standard deviations.157The most important information in regard to the wrinkle width is that in every trial the width waslarger in the steady state response than the initial state. 4 5 6 7 8 9 0  0.5  1  1.5  2  2.5  3  3.5  4  4.5Wrinkle width [mm]Elapsed time [days]20.7°C20.7°C21.1°C21.2°C23°C24.3°CFigure 4.19: Wrinkle width of the initiated wrinkle as a function of time for various trialsconducted at slightly different ambient conditions showing an overall increase in thewrinkle width.The excess length, shown in Figure 4.20, was calculated by measuring the path length of thewrinkle and subtracting the width of the wrinkle. This value inherently incorporates informationfrom the wrinkle width and height.158 0 0.025 0.05 0.075 0.1 0  0.5  1  1.5  2  2.5  3  3.5  4  4.5Excess length in the wrinkle [mm]Elapsed time [days]20.7°C20.7°C21.1°C21.2°C23°C24.3°CFigure 4.20: Excess length consumed in the initiated wrinkle as a function of time for varioustrials conducted at slightly different ambient conditions showing a similar profile as thewrinkle height.The population distribution, shown in Figure 4.21, mirror the non-initiated trials’ trends whilesimultaneously showing the large effect from the modified interface condition. Similar to the non-initiated trials, the early time steps show a large number of small wrinkles distributed over thesurface. There is a large degree of scatter with respect to the initial number of wrinkles betweentrials; however, the steady state response always features significantly fewer wrinkles than theinitial time step. Unlike the non-initiated trials, the steady state response is dominated by thesingle wrinkle described above. The single initiated trials featured only 1.2±0.4 wrinkles largerthan 0.2 mm in contrast to the 11±2 wrinkles observed in the non-initiated trials. In the steadystate response, the total number of wrinkles was also significantly reduced, from 61.0±5.7 in thenon-initiated trials to 8.8±4.4 in the initiated trials.159 0 10 20 30 40 50 60 70 0  0.1  0.2  0.3  0.4  0.5  0.6N = 95Number of wrinklesWrinkle height [mm]t0 + 0d 00:00:00(a) 0 10 20 30 40 50 60 70 0  0.1  0.2  0.3  0.4  0.5  0.6N = 59Number of wrinklesWrinkle height [mm]t0 + 0d 01:50:00(b) 0 10 20 30 40 50 60 70 0  0.1  0.2  0.3  0.4  0.5  0.6N = 46Number of wrinklesWrinkle height [mm]t0 + 0d 20:15:00(c) 0 10 20 30 40 50 60 70 0  0.1  0.2  0.3  0.4  0.5  0.6N = 11Number of wrinklesWrinkle height [mm]t0 + 6d 01:45:00(d)Figure 4.21: Population distribution of wrinkle heights through four time steps. (d) shows amarked reduction in the number of wrinkles in comparison to the uninitiated distribu-tion shown in Figure 4.15(d)160Double initiatorThe surface response with two initiators, spaced 70 mm apart and 40 mm from the free edge, can beseen in Figure 4.22. Results from a duplicate trial, run at the same time and conditions, were verysimilar with a response shown in Appendix D Figure D.15. These profiles are consistent with all ofthe previous tests in that an initially large number of small wrinkles reduces to a small number oflarge wrinkles. Several wrinkles remained in the steady state response; however, the distributionsshown in Figure 4.23 show the steady state response is dominated by the intiated wrinkles. Similarto the previous initiated trials, for both two-initiated trials, only the initiated wrinkles had heightsabove 0.2 mm in the steady state response.161(a)(b)(c)Figure 4.22: (a), (b), and (c) show the surface profile for an example specimen preparedwith two 3 mm wide by 0.0375 mm thick PMP wrinkle initiators spaced approximately40 mm from the edges and 80 mm from center to center.162 0 10 20 30 40 50 60 70 0  0.1  0.2  0.3  0.4  0.5  0.6N = 70N = 132Number of wrinklesWrinkle height [mm]t0 + 0d 00:00:00(a) 0 10 20 30 40 50 60 70 0  0.1  0.2  0.3  0.4  0.5  0.6N = 104Number of wrinklesWrinkle height [mm]t0 + 0d 02:00:00(b) 0 10 20 30 40 50 60 70 0  0.1  0.2  0.3  0.4  0.5  0.6N = 49Number of wrinklesWrinkle height [mm]t0 + 0d 19:20:00(c) 0 10 20 30 40 50 60 70 0  0.1  0.2  0.3  0.4  0.5  0.6N = 36Number of wrinklesWrinkle height [mm]t0 + 4d 10:20:00(d)Figure 4.23: Population distribution of wrinkle heights through four time steps.The transient wrinkle heights are shown in Figure 4.24. The position denoted in the legendcorrespond with the wrinkle peak’s x-position, consistent with Figure 4.22. The green points arefrom the trial measured in Figure 4.22 while the purple points were generated from the duplicatetrial. All of these wrinkles show a similar initially growth pattern followed by a plateau which wasreached after (1.50±0.19) d.163 0 0.1 0.2 0.3 0.4 0.5 0  0.5  1  1.5  2  2.5  3  3.5  4  4.5Wrinkle height [mm]Elapsed time [days]21.8°C @ 43 mm21.8°C @ 110 mm21.8°C @ 42 mm21.8°C @ 110 mmFigure 4.24: Wrinkle height of the initiated wrinkles as a function of time for the left andright wrinkles and at slightly different ambient conditions. 4 5 6 7 8 9 0  0.5  1  1.5  2  2.5  3  3.5  4  4.5Wrinkle width [mm]Elapsed time [days]21.8°C @ 43 mm21.8°C @ 110 mm21.8°C @ 42 mm21.8°C @ 110 mmFigure 4.25: Wrinkle width of the initiated wrinkles as a function of time for the left and rightwrinkles and at slightly different ambient conditions.164 0 0.025 0.05 0.075 0.1 0  0.5  1  1.5  2  2.5  3  3.5  4  4.5Excess length in the wrinkle [mm]Elapsed time [days]21.8°C @ 43 mm21.8°C @ 110 mm21.8°C @ 42 mm21.8°C @ 110 mmFigure 4.26: Wrinkle excess length of the initiated wrinkles as a function of time for the leftand right wrinkles and at slightly different ambient conditions.Figure 4.27 overlays all of the initiated wrinkles heights for the sub 22 ◦C trials. While there isscatter, the initiated wrinkles in these trials grow to an average value of (0.32±0.02)mm. 0 0.1 0.2 0.3 0.4 0.5 0  0.5  1  1.5  2  2.5  3  3.5  4  4.5Wrinkle height [mm]Elapsed time [days]20.7°C20.7°C21.1°C21.2°C21.8°C @ 43 mm21.8°C @ 110 mm21.8°C @ 42 mm21.8°C @ 110 mmFigure 4.27: Overlay of the single and double initiated trial wrinkle heights conducted atnominally 21 ◦C.165Quadruple initiatorThe quadruple initiator trial was conducted at the same time as the 24.3 ◦C single initiator trialpreviously discussed, Appendix D Figure D.13. The quadruple initiator trial had PMP initiatorsspaced at 30.5 mm increments from the free edge. Figure 4.28 shows the surface profile for thistrial.166(a)(b)y(c)Figure 4.28: (a), (b), and (c) show the surface profile of a specimen prepared with four 3 mmwide by 0.0375 mm thick PMP wrinkle initiators spaced approximately 30.5 mm fromedge-to-center and center-to-center.167 0 10 20 30 40 50 60 70 0  0.1  0.2  0.3  0.4  0.5  0.6N = 28Nu mb er  o f wr i nk le sWrinkle height [mm]t0 + 0d 00:00:00(a) 0 10 20 30 40 50 60 70 0  0.1  0.2  0.3  0.4  0.5  0.6N = 18Number of wrinklesWrinkle height [mm]t0 + 0d 01:55:00(b) 0 10 20 30 40 50 60 70 0  0.1  0.2  0.3  0.4  0.5  0.6N = 11Number of wrinklesWrinkle height [mm]t0 + 0d 19:25:00(c) 0 10 20 30 40 50 60 70 0  0.1  0.2  0.3  0.4  0.5  0.6N = 12Number of wrinklesWrinkle height [mm]t0 + 4d 02:05:00(d)Figure 4.29: Population distribution of wrinkle heights through four time steps.Comparing the two trials conducted at 24.3 ◦C, Figure 4.30 shows a clear difference betweenthe steady state wrinkle heights.168 0 0.1 0.2 0.3 0.4 0.5 0  0.5  1  1.5  2  2.5  3  3.5  4  4.5Wrinkle height [mm]Elapsed time [days]24.3°C24.3°C @ 31 mm24.3°C @ 62 mm24.3°C @ 91 mm24.3°C @ 122 mmFigure 4.30: Wrinkle height growth for the four initiated wrinkles depicted in Figure 4.28overlaid with the single initiator wrinkle height trial conducted at the same ambienttemperature. 4 5 6 7 8 9 0  0.5  1  1.5  2  2.5  3  3.5  4  4.5Wrinkle width [mm]Elapsed time [days]24.3°C24.3°C @ 31 mm24.3°C @ 62 mm24.3°C @ 91 mm24.3°C @ 122 mmFigure 4.31: Wrinkle width for the four initiated wrinkles depicted in Figure 4.28 overlaidwith the single initiator wrinkle width trial conducted at the same ambient temperature.169 0 0.025 0.05 0.075 0.1 0  0.5  1  1.5  2  2.5  3  3.5  4  4.5Excess length in the wrinkle [mm]Elapsed time [days]24.3°C24.3°C @ 31 mm24.3°C @ 62 mm24.3°C @ 91 mm24.3°C @ 122 mmFigure 4.32: Wrinkle excess length for the four initiated wrinkles depicted in Figure 4.28overlaid with the single initiator wrinkle excess length trial conducted at the sameambient temperature.170Temperature modificationAfter reaching a steady state wrinkle height, the temperature was increased in each of the trials.This temperature modification was done in an effort to estimate the minimum temperature at whichstresses can be imparted into the laminate through the resin interlayer. The increase in temperatureleads to a violation of the time-invariant property assumption, used in the CHILE approach, anda complex stress state due to the resin’s ability to relax stresses. There was significant variationin the wrinkle height due to the increasing temperature between trials; one characteristic profile isshown in Figure 4.33 with each trial’s profile shown in Appendix D. 0 0.1 0.2 0.3 0.4 0.5 0  1  2  3  4  5  6  7  8  9  10Wrinkle height [mm]Elapsed time [days](21.8 ± 0.49) °C(29.2 ± 0.36) °C(34.9 ± 0.35) °C(40.7 ± 0.59) °CFigure 4.33: Effect of increasing temperature on the maximum wrinkle height for an initiatedtrial.The steady state wrinkle height for each of the isothermal holds is shown in Figure 4.34. The1σ confidence interval bounds the linear best fit using the standard deviation from each trial. Therelationship between temperature and wrinkle height is non-linear and affected by the resin’s relax-ation; however, this type of analysis provides a first estimate to the minimum temperature wheretool-part interaction can induce part stresses. The zero wrinkle height temperature intersection be-171tween 40.7 ◦C to 46.1 ◦C is consistent with the 45 ◦C temperature Larberg et al. described for 8552resin as featuring a strain rate dependent response [48]. 0 0.1 0.2 0.3 0.4 0.5 0  10  20  30  40  50Wrinkle height [mm]Temperature [°C]20.7°C20.7°C21.1°C21.2°C21.8°C @ 43 mm21.8°C @ 110 mm21.8°C @ 42 mm21.8°C @ 110 mm23°C24.3°C1σ confidenceFigure 4.34: Stable wrinkle height at various temperatures for the initiated wrinkles withsymmetric left-right surrounding regions.1724.4 Stress analysis4.4.1 Modelling parametersTo estimate the tool-part stresses, 11 parameters must be accounted for to use Equation 4.18. Thevalues used for the geometric properties, thermal properties, mechanical properties, and two bound-ing temperatures are each discussed in the following subsections and tabulated in Table 4.1.Geometric propertiesThe required geometric properties are the ply thickness, h1, half the tooling thickness, h2, adhesivelayer thickness, hA, and interaction length, L. The average uncured ply thickness was previously de-termined to be 0.21 mm, Section 3.6. The full polycarbonate thickness was nominally 1.5875 mmand several measurements corroborated a thickness of (1.59±0.01)mm. Hence, the average halftool thickness, h2, used was 0.80 mm.The adhesive thickness is one of the lower confidence parameters. This value could not be di-rectly measured with ease, rather this value will be drawn from the literature. Erland et al. estimateda non-debulked interlayer thickness of AS4/8552 at approximately 1/16 of the ply thickness [22].Assuming an even interply thickness on both surfaces of the ply, the interlayer thickness would beapproximately 6.6 µm. SEM micrographs [22] showed a clear reduction in the interlayer thicknessafter a 80 ◦C debulk; hence, the 6.6 µm thickness value is a large upper estimate. Alternatively,Wolh et al. [86] showed a stress-displacement plot for a probe tack test of uncured IM7/8552-1prepreg conducted at 54 ◦C. While the prepreg sample did not undergo a traditional vacuum de-bulk, the test used a short duration, 5 s, large applied normal force, 800 N. From Wohl et al.’s data,the through-thickness tensile modulus, E3,p [Pa], was calculated at 4.9 MPa. Equation 4.21 showsthe standard relationship between the transverse modulus of a composite and its two constituents.This derivation will assume that rather than the deformation occurring in the fibres and resin thatthe deformation will occur in the resin interlayer and a prepreg core. This is justifiable because of173the relative tensile stiffnesses of the resin interlayer, Er (54 ◦C) = 16kPa, and the transverse modu-lus, E2 (54 ◦C) = 10MPa [165]. Composites are often assumed to be transversely isotropic, hence,E2,p is a reasonable value to use for the prepreg core modulus, Ec [Pa]. The discrepancy betweenthe measured E3,p and E2,p also shows the knockdown effect of the resin interlayer. The inter-layer’s tensile stiffness of 16 kPa was determined assuming isotropic properties for the resin, witha Poisson’s ratio of 0.4 [68], and using the resin shear storage modulus at 54 ◦C, 5.8 kPa, which wasreported in Section 3.3 Figure 3.6. Assuming that the resin interlayer is evenly divided along bothsides of the prepreg, the interlayer thickness can be determined from Equation 4.22 using Wohl etal.’s sample thickness of 0.15 mm. This analysis results in an interlayer thickness of 0.13 µm.E3,p =(ErE2,pvIE2,p+(1− vI)Er)(4.21)where vI is the resin interlayer volume fraction and Er [Pa] is the Young’s modulus of the resin.hA =12h1vI (4.22)The interaction length will vary according to the placement of the initiators in the experiment.For the quadruple initiator trial, a clear decrease in the measured excess length, compared to thesingle initiator trial, was observed which indicates interference between the adjacent initiators.Assuming similar stress fields surrounding the initiators, the interaction length is half the distancebetween the initiators, 15.3 mm. For the other trials, the interaction length will be the length atwhich the interlayer shear stress reaches the critical debond shear stress. At first, this analysisassumed a temperature independent value which was determined from Equation 4.18 using eachtrial’s measured excess length. This returned a set of interaction lengths which, for the range ofisothermal temperatures studied, was linearly related to temperature. The slope and intersectionvalues were then iterated in Equation 4.18 to reduce error between the measured and predicted174excess lengths. At elevated temperatures, > 40 ◦C, the linear relationship returned an interactionlength which exceeded the length to the free edge for the single initiator trials, or the symmetrycondition for the double initiator trials. A piecewise function was used to constrain the upper boundof the interaction length to the physical length scales involved for each test. The temperature variantinteraction length relationship which produced the lowest error is given in Equation 4.23.L(T ) =5.5mm ◦C−1T −94.6mm L(T )< L∞L∞ L(T )≥ L∞(4.23)where L∞ [m] is the distance to the free edge or the symmetry condition between an adjacentinitiator.Thermal propertiesThe coefficient of linear thermal expansion along the 1-direction of both cured and uncured carbonfibre prepreg is often assumed to be near zero [60] and invariant over the typical working tempera-tures. Ersoy et al. measure the CTE of fully cured AS4/8552 at 0.21×10−6 ◦C−1 [92]. The valueused in this modelling is 0.21×10−6 ◦C−1; however, the exact 1-direction CTE is non-critical dueto the tool’s large CTE.The polycarbonate tool’s CTE was measured following ASTM E831-14 [166]. The procedureinvolved placing a small sample, 5 mm by 5 mm by 1.58 mm thick, into a TA Instruments Q400TMA, Figure 4.35, and running a temperature cycle consisting of two heat and cool cycles between0 ◦C to 100 ◦C at a rate of 1 ◦Cmin−1. This temperature span and rate ensured accurate informationfor the range of temperatures associated with forming composites [53], [167], [168]. The TMAaccurately measures temperature and displacement, from which the coefficient of linear expansioncan be determined. Following E831-14, a small load, 20 mN, was applied to the specimen from themacro-expansion probe to ensure proper contact.175Figure 4.35: TA Instruments Q400 TMA machine.The expected thermal strain in polycarbonate for a temperature difference of 100 ◦C is 7.00×10−3;this is in very good agreement with the experimentally measured maximum strain of 7.01×10−3,shown in Figure 4.36. Following ASTM-831 - 14 [166], separating data collected from 20 ◦C abovethe initial ramp temperature during heating, and below for cooling, the linear CTE is calculated asαPC = (68±1)×10−6 K−1. This value is within 5 % error of manufacturers documented value of70.0×10−6 K−1 [162]. Some noise can be seen in the initial ramp which was likely caused byvibrations from people walking by the instrument. The remaining ramps occurred while the lab-oratory was free of personnel and do not suffer from this noise. This error does not significantlycontribute to the overall response nor the calculated CTE.176 0 20 40 60 80 100 0  1  2  3  4  5  6  7 0 0.2 0.4 0.6 0.8 1Temperature [°C]Strain [%]Time [hrs]TemperatureStrainFigure 4.36: Results for the through thickness linear expansion and contraction of a polycar-bonate specimen between 0 ◦C to 100 ◦C at 1 ◦Cmin−1.Mechanical propertiesThe polycarbonate tool has a Young’s modulus of 2.3 GPa and a glass transition temperature of150 ◦C [162], [169]. The maximum temperature observed during the trials was 65 ◦C, well belowthe Tg, and in a region where the tensile modulus is relatively flat [170].The temperature variant shear storage modulus of the 8552 resin measured in Section 3.3 wasused for the interlayer unrelaxed shear modulus, Gu (T ). The resin’s shear storage modulus variedbetween 620 kPa at 20 ◦C to 1.4 kPa at 65 ◦C.The in-plane composite properties are determined from data presented by Dodwell et al. [40]and replotted in Figure 2.12 of Chapter 2. The bending stiffness was fitted according to the author’smethodology and converted to in-plane properties using beam theory; Equation 4.24. The in-planemodulus varied between 7.2 GPa to 2.3 GPa over the tests temperature range’s.E1 (T ) =B∞eEBRT112 wh31(4.24)177where the modelling parameters B∞ and EB are 2.2×10−3 Nmm2 and 310 Jmol−1; w is the speci-men width, 5.0 mm [75]; and h1 is the specimen thickness, 0.2 mm [40].A series of Pierce cantilever bending tests were performed to determine if the time invariantprepreg modulus assumption was reasonable at room temperature Appendix D Section D.0.3. Thetests show no significant change in modulus over the testing period of 48 h at an uncontrolledambient temperature of (25.0±0.1) ◦C.Bounding temperaturesEach of the trials were performed under recorded ambient conditions allowing the test temperature,Tf , to be accurately known and directly used.The temperature where the sticking condition begins to dominate and the resin can apprecia-bly transfer stresses into the laminate was taken as 45 ◦C. This is within the zero wrinkle heighttemperature bounds shown in Figure 4.34.178Table 4.1: Input parameters for hypoelastic modelling.Parameter Value Referenceh1 0.21 mm Section 3.6h2 0.80 mmhA 0.13 µm [86]L(T ) 19 mm to 76 mm Equation 4.23α1 0.21×10−6 ◦C−1 [92]α2 68×10−6 ◦C−1 Figure 4.36Eu1 (T ) Equation 4.24 [40]E2 2.3 GPa [162]Gu (T ) Linear interpolation table Section 3.3To 45 ◦C Figure 4.34Tf 20.7 ◦C to 24.3 ◦C Varies by test1794.4.2 Wrinkle initiators (Stress analysis)The quadruple initiator trial, where the stress fields of adjacent initiators interfered, has all ofthe parameters clearly defined. A schematic of a side profile is shown in Figure 4.37(a). Therepresentative interlayer shear stress and composite in-plane stress are shown in (b) and (c).(a)τ(b)σ1(c)Figure 4.37: (a) Side view of the quadruple initiator trial with the PMP initiators shown inwhite and lines of symmetry in grey. (b) and (c) show the shape of the shear in theinterlayer and in-plane stress in the composite.Substituting the values from Table 4.1 into Equation 4.18 using the quadruple initiator’s finaltemperature of 24.3 ◦C returns an excess length of −0.029 mm. The interlayer shear stress, plottedin Figure 4.38, shows a stress transfer length, using a 99 % far field stress criteria, of 7.5 mm. Thezero position of Figure 4.38 and Figure 4.39 corresponds with the midpoint between two wrinkleinitiators.180-1-0.8-0.6-0.4-0.2 0 0  2  4  6  8  10  12  14  16Shear stress in resin interlayer [MPa]Position along x [mm]Figure 4.38: Shear stress in the resin interlayer with a reduced interaction lenght due to closeinitiators conducted at 24.3 ◦C.-5-4-3-2-1 0 0  2  4  6  8  10  12  14  16In-plane stress in the uncured ply [MPa]Position along x [mm]Figure 4.39: In-plane stress in the composite layer with reduced interaction length due toclose initiators conducted at 24.3 ◦C.The predicted excess lengths, for the single and double initiated trials, are shown in Figure 4.40.The L∞ = ∞ mm interaction length provides an upper bound on the magnitude of excess lengthwhich can be imparted. The two predicted data points which do not lay on the infinite interac-181tion length are reduced in magnitude due to the boundaries reducing the interaction length andconsequently reducing the imparted strain.-0.075-0.065-0.055-0.045-0.035 20  21  22  23  24  25Excess length [mm]Temperature [°C]MeasuredPredictedPredicted L∞ = ∞ mmFigure 4.40: Excess lengths as a function of temperature; predictions made using Equa-tion 4.18.The predicted excess length for the full processing window is shown in Figure 4.41. Stressesstart to be imparted during the cool down at approximately 45 ◦C. Further cooling the specimencauses excess length to be introduced into the wrinkle at a nearly linear rate for an unboundedinteraction length. The bounded interaction lengths reduce the length over which strain can beimparted and therefore have a lower magnitude of imparted strain. For the unbounded case, atapproximately 32 ◦C the interface failure outpaces the strain from the temperature differential andthe net magnitude of excess length begins to decrease. The intersection of the unbounded andbounded curves is the point at which the length to debond is equal to the bounded length.The measured excess length value with two associated interaction lengths, 19 mm and 38 mm,is the double initiator trial where the region between the two initiators was twice the distance asbetween the initiator an the free edge. Therefore the predicted excess length for this condition isthe midpoint of the excess lengths associated with the 19 mm and 38 mm interaction length.182-0.1-0.08-0.06-0.04-0.02 0 15  20  25  30  35  40  45Excess length [mm]Temperature [°C]Predicted L∞ = ∞ mmPredicted L∞ = 38 mmPredicted L∞ = 19 mmPredicted L∞ = 15 mmMeasured L∞ = 38 mmMeasured L∞ = (19,38) mmMeasured L∞ = 15 mmFigure 4.41: Predicted excess lengths consumed in a single wrinkle over a cool down cyclefor various interaction lengths.1834.4.3 Critical strain energy release rateTable 4.2 outlines the general procedure for estimating the energy in the system driving wrinklegrowth. Table 4.2 a) shows a single line running along the 1-direction of the steady state ply.Table 4.2 b) defines the wrinkle width, w, and the path length of the wrinkle, Lp. Both the excesslength consumed by the wrinkle, ∆L, and the width of the crack, a, can be determined from thesetwo parameters. This excess length has been released by the bulk of the material into the wrinkledregions. Therefore, the excess length driving the wrinkle growth will be highest in the first timestep and decay to zero once the surface has reached its steady state. This driving excess length canbe determined by subtracting each time step’s released excess length from the steady state excesslength. Both the strain field to the left and right of the wrinkle will contribute to this excess length.Therefore, the driving strain field will be equal to the excess length divided by twice the interactionlength, i.e., the contribution from either side. Table 4.2 c) shows how the energy driving wrinklegrowth can be determined using the known strained volume, Young’s modulus, and driving strain.Finally, Table 4.2 d) restates Equation 4.20.184a)-0.05 0 0.05 0.1 0.15 0.2 0  20  40  60  80  100  120  140  160z [mm]x [mm]Detrended filteredMin peaksMax peakb)∆L = Lp−ww2a = ww-0.05 0 0.05 0.1 0.15 0.2 76  78  80  82  84z [mm]x [mm]Lpwwc) U =12V E1ε2σ1d) Gc =− dUb·da Equation 4.20Table 4.2: General procedure for determining the critical strain energy release rate. a) showsa single line running along the lamina’s 1-direction. b) shows the method for determiningthe excess length and crack length from the wrinkle’s path length and width. c) shows thegeneral method for estimating the strained energy. d) shows the final step to determinethe critical strain energy release rate.Using the change in wrinkle width as the normalizing factor is a simplification of the com-plicated debonding process. To illustrate how this can affect the results, Figure 4.42, shows the185wrinkle height and wrinkle width as a function of time. This is shown for a single trial; however,the following analysis holds for many of the trials. This plot shows the height of the wrinkle asmonotonically increasing while the width increases over the full elapsed time scale. 0 0.2 0.4 0.6 0.8 0  0.5  1  1.5  2  2.5  3  3.5  4  4.5 3 4 5 6 7 8Wrinkle height [mm]Wrinkle width [mm]Elapsed time [days]21.1°CHeightWidthFigure 4.42: Wrinkle width and height of a single initiator trial conducted at 21.1 ◦C.Figure 4.43 shows the same trial results as Figure 4.42 except the time scale has been reducedto only the first hour. This plot shows the monotonically increasing wrinkle height which impliesa release of energy of the system into the wrinkled region, dU > 0. However, over this 1 h timescale, the wrinkle width is invariant, da= 0. Therefore, using Equation 4.20 to calculate the energyrelease rate will return an indeterminate value.186 0 0.2 0.4 0.6 0.8 0  0.2  0.4  0.6  0.8  1 3 4 5 6 7 8Wrinkle height [mm]Wrinkle width [mm]Elapsed time [hours]21.1°CHeightWidthFigure 4.43: Wrinkle width and height during the first hour of growth. Monotonically in-creasing wrinkle height with invariant wrinkle width.One scenario which could cause wrinkle growth without significant alteration to wrinkle widthis a highly strained interlayer. Figure 4.44 shows an idealized ply, interlayer, and substrate duringthe initial time step.187ww(a) t1 ≈ 5min(b) t1 ≈ 5minFigure 4.44: (a) shows the initial state of the prepreg with a wrinkle initiator. (b) shows ahypothetical situation with highly strained interlayer.Figure 4.45 shows the idealized wrinkle growth resulting from a highly strained interlayer. Theinterlayer releases excess length into the wrinkle leading to a larger wrinkle without appreciablyaltering the width of the wrinkle. The purple points of Figure 4.44(b) and Figure 4.45(b) arecoincident. Each of the other highlighted points move relative to this purple point. The whiteline connecting the orange and white dot in both time steps indicate that the prepreg does notappreciably strain due to its relatively high modulus.188ww(a) t1 < t2 < 1h(b) t1 < t2 < 1hFigure 4.45: (a) shows a snapshot of the wrinkle which has grown during the first hour. (b)shows the interlayer after relieving strain into the wrinkle causing growth.However, even over the first hour, approximately 10 µm of excess length must be borne by the0.13 µm thick interlayer. This results in very large strain values which would cause the interlayerto immediately fail.Another scenario which can cause wrinkle growth without increasing the wrinkle width ismode II failure leading to material movement into the wrinkle. This is shown in Figure 4.46 andFigure 4.47. Figure 4.46 shows a similar first time step except the interlayer is not under significantshear. In this scenario, the driving force is the compressed prepreg.189ww(a) t1 ≈ 5min(b) t1 ≈ 5minFigure 4.46: (a) shows the initial state of the prepreg with a wrinkle initiator. (b) shows ahypothetical situation with compressed prepreg layer.Figure 4.47 shows the growth whereby the interlayer has locally failed at the tool surface andmaterial from the bulk relieves some compressive strain forcing material into the wrinkle. This isindicative of mode II dominated failure.190ww(a) t1 ≈ 5min(b) t1 ≈ 5minFigure 4.47: (a) shows the initial state of the prepreg with a wrinkle initiator. (b) shows ahypothetical situation with shear of material into the wrinkle.Figure 4.48 shows the full time scale with a transition region highlighted. In this transitionregion, excess length is introduced into the wrinkle through an increase in both wrinkle height andwidth.191 0 0.2 0.4 0.6 0.8 0  0.5  1  1.5  2  2.5  3  3.5  4  4.5 3 4 5 6 7 8Wrinkle height [mm]Wrinkle width [mm]Elapsed time [days]21.1°CHeightWidthFigure 4.48: Wrinkle height and width during the transition from mode II dominated to mixedmode failure.Comparing the change in energy with change in wrinkle width in this transition region, usingEquation 4.20, allows this mixed mode strain energy release rate to be calculated. These points areshown in Figure 4.49 as the solid points. The corresponding feed rate is calculated from the wrinklewidth and the elapsed time spent in transition zone. The muted points in Figure 4.49 correspondto a running average of 23 time steps. This averaging was done in an effort to show the transitionbetween high energy failure, indicative of shear, into the lower energy failure, indicative of a mixedmode of failure.192 0 20 40 60 80 100 12010-6 10-5 10-4 10-3 10-2Critical strain energy release rate [J/m2 ]Feed rate [mm min⁻¹]20.7°C20.7°C21.1°C21.2°C23°C24.3°C24.3°C @ 31 mm24.3°C @ 62 mm24.3°C @ 91 mm24.3°C @ 122 mmFigure 4.49: Critical strain energy release rate of all of the initiated wrinkles with symmetricleft-right surrounding regions.For comparison, the critical strain energy release rates from Figure 4.49 have been replottedwith the critical strain energy release rates presented by Crossley et al. [84]. The experimentalapparatus, composite material, tooling material, and tooling surface finish were different betweenthe two datasets; however, general trends will be discussed in the next section. 0 50 100 150 200 25010-5 10-4 10-3 10-2 10-1 100 101 102 103Critical strain energy release rate [J/m2 ]Feed rate [mm min⁻¹]Crossley et al.This workFigure 4.50: Critical strain energy release rates compared against Crossley et al.’s GFRP/steelresults [84].193The exact failure mode which the material undergoes in the peel tack test used by Crossley etal. [84] is unknown. The failure is not exclusively mode I or mode II but will be a mixed modefailure. The energy required to undergo mode II failure is higher than mode I failure which impliesdepending on the weighting of the mixed mode failure, an increase or decrease in the critical strainenergy release rate would be expected. This hypothesis would imply a family of curves whichcould resemble Figure 4.51. 0 50 100 150 200 25010-5 10-4 10-3 10-2 10-1 100 101 102 103Critical strain energy release rate [J/m2 ]Feed rate [mm min⁻¹]Mode II weightedMixed modeMode I weightedFigure 4.51: Critical strain energy release rates compared against Crossley et al.’s GFRP/steelresults with failure modality overlaid [84].1944.5 DiscussionAll of the results indicate a strong ability of the tool to impose compressive stresses into the compli-ant composite during the cool down stage of forming or debulking. The steady state surface profilesof the non-initiated trials show a large number of wrinkles, approximately 1 wrinkle per square cen-timeter, dispersed over the lengthwise midline of the specimen. The observed uniformly dispersedwrinkles are in complete agreement with the homogeneously dispersed variability measured in thepervious material characterization chapter. The coalescing of wrinkles along the midline also cor-roborates the shear-lag approach where the predicted highest in-plane stress, prior to debond, is atthe midline. This compressive in-plane stress eventually relieves itself as the material buckles andforms a wrinkle. This procedure was repeatable and consistently generated transient wrinkles.All of the trials showed an initially rapid wrinkle growth, i.e., occurring faster than the CMM’s5 min scan time, which resulted in a large initial wrinkle distribution. After the early time steps,0 min to 30 min, no new wrinkles formed; rather the majority of wrinkles decayed below the min-imum detectable defect size of 0.0625 mm in nearly all trials. The largest steady state wrinklesdid not correspond with the largest wrinkles in the initial time steps and several hours needed topass before the largest wrinkles remained the largest wrinkles. This is likely a consequence of thematerial’s local variability. Any variation in the tooling surface condition or the ply’s thickness,stiffness, impregnation level, degree of cure, etc., will result in a localized stress response alteringthe resulting wrinkle size and growth rates. For example, a locally compliant region surrounded byhigher than average stiffness material may grow at a faster rate than an average stiffness material.However, the surrounding higher stiffness material may halt further growth allowing the averagestiffness material to reach a larger steady state height.At first, the non-initiated trials had on the order of 2 to 4 wrinkles per square centimeter whichdecayed to a steady state value of 1 wrinkle per square centimeter. This decay could be due tothe post-buckling stiffness of the wrinkled regions. The largest wrinkles will require the lowest195loads to further deform. As a consequence, any non-local influence will preferentially cause thelarger wrinkles to grow which will ultimately dominate the material response. A similar trendwas observed for each of the initiated trials. At the start of the initiated trials, the total number ofwrinkles was large and showed significant scatter, 50±46 wrinkles. However, after the surfaceshad reached a steady state response, the most significant wrinkles, wrinkles in excess of 1 UPT,became approximately 1 for the single initiator, 2 for the double initiator, and 4 for the quadrupleinitiator trials. This is a significant reduction in large wrinkles relative to the non-initiated trial’slarge wrinkle count of 12±2. Further, adding the thin initiator did not increase the overall heightof the maximum wrinkles implying that at worst the initiators should not reduce the performanceof the laminate. The reduction in the number of large wrinkles using the initiators, coupled with theability to tailor their locations, allows design engineers more tools to improve part conformance.The > 1 UPT wrinkles all showed similar growth profiles. The higher isothermal temperatures,23 ◦C to 24 ◦C, corresponded with only a slightly faster growth rate. Each surface reached a steadystate response after approximately 1 d which implies that any inspection done immediately afterforming will not provide accurate information unless subsequent bagging, or potentially subsequentplies in an AFP process, and curing occurs rapidly. For a manufacturing engineer, this testing hasshown that, at a minimum, time between forming and bagging should be kept to a minimum anduncured parts left to dwell at room temperature will be more prone to wrinkling defects.The initiation of the wrinkle site will coincide with the failure of the resin-tool interface. Theshear debond stress values calculated ranged between 10 kPa to 840 kPa over the 20 ◦C to 24 ◦Ctemperature range. However; even a large variation in the debond stress value did not greatlyalter the observed wrinkle sizes for two reasons. First, the exponential shear relationship impliesthat the large shear values occur only over a very short distance. Second, the in-plane stressesleading to the wrinkle’s excess length are proportional to the lengthwise integral of this shearprofile. This means that the overall in-plane shear stress is not largely affected by phenomena196occurring over short distances. This large change in shear debond stresses is consistent with theliterature. For comparison, Wohl et al. used a probe tack test to measure the maximum stress valueswhich ranged between 3 kPa to 470 kPa at an isothermal 25 ◦C, with longer contact times, slowercrosshead speeds, and lower temperatures corresponding with higher maximum stress values [86].Wohl et al.’s temperatures were all at or above 25 ◦C. Ahn et al. [171] performed probe tack testsfrom 0 ◦C to 80 ◦C and showed the maximum debond stress of two different epoxy prepregs. Bothof these prepregs displayed peaking behaviour with a strength of effectively 0 kPa at 0 ◦C, peakingto 200 kPa between 20 ◦C to 30 ◦C, and falling to below 50 kPa around 80 ◦C. Wohl et al. and Ahnet al.’s tests measured the tensile debond strength or mode I failure; however, they show similarlarge variation and temperature dependence as the shear or mixed mode observed in the initiatedtrials.In the case of a shear-lag problem, the shear debond stress will define the length of interaction.With the wrinkles clustered around the midline and the measured excess lengths falling well belowthe perfectly bonded hypoelastic case’s 0.14 mm, these experiments fit the expected behavior of theshear-lag problem. The hypo-elastic shear-lag relationship requires several properties which canbe determined with high confidence, e.g., the tool and part’s thickness and CTE; however, severalof the properties are far more difficult to measure, e.g., prepreg modulus, interlayer thickness, andshear debond stress. Using reasonable values from the literature, Figure 4.41 shows a processingwindow where the steady state excess length in one wrinkle can be determined from the ambientconditions. The shear-lag approach correctly captures the initially counter intuitive decrease insteady state wrinkle height with increasing temperature differential observed as the material coolsbelow approximately 30 ◦C. In either the linear elastic or hypoelastic perfectly bonded case, theexcess length will increase monotonically with an increasing driving mechanism, namely the in-creasing temperature differential. However, these models fail to capture the interface failure whichcauses the length over which strain can be imparted to decrease. As the interlayer begins to fail,197further increasing the temperature differential causes the interaction length to reduce faster thanthe driving CT E ·∆T differential resulting in a net decrease in excess length. The processing win-dow shows that in order to minimize tool-part interaction, either the tool temperature should notbe allowed to cool to room temperature after forming or the tool temperature should be quenchedbelow room temperature so that the entire interface fails without imparting significant strains intothe thermoelastically stiffer material. These two scenarios are ideal as the predicted strains arezero. In practice, quenching may be impractical as large tools would likely suffer thermal gradientsintroducing sites for defects to arise. Maintaining tool temperature closer to forming temperaturecould be a useful mitigation strategy especially as more automated post-forming strategies loosenthe temperature limit required for human safety [139].When neither quenching nor high temperature tooling is an acceptable solution, introducinginteraction length interfering initiators can reduce the impact of a wrinkle. The fitted interactionlength varied from 153 mm to 15 mm from 45 ◦C to 20 ◦C. Spacing initiators outside of thesebounds will allow the wrinkles to grow to their maximum size while also increasing the potentialfor non-initiated wrinkles to occur. This is a worst case scenario as the maximum height wrinklescause the largest strength reductions while random wrinkles increase uncertainty and further re-duce strength. Using multiple initiators, with interfering stress fields, can significantly reduce theexcess length in an individual wrinkle, leading to lower wrinkle heights. A laminate with manysmall wrinkles will have a smaller misalignment angles and the laminate’s predicted strength willbe higher than a laminate with fewer large amplitude and angle wrinkles [27]. Therefore, theseinitiated wrinkles could ultimately be beneficial to part performance.The results also show a clear transient nature with the wrinkles growing, or decaying, overhours and days rather than the behaviour typically associated with buckling; whereby failure issudden. The critical strain energy release rate required to debond uncured prepreg from a tool haspreviously been shown to be rate dependent [84]. Using the internal stress state and geometry of the198ply at known instances in time, the energy required to debond the ply can be estimated. Assumingthe ply does not change thickness, the debonded area is proportional to the width of the wrinkle;unfortunately the wrinkle width was the lowest confidence value with typical standard deviationson the order of 25 % of the mean width. As a result, the precise critical strain energy release ratevalues should be analyzed for general trends. During the initial wrinkle growth, approximately thefirst 1 h, the wrinkle grew without increase in width. This is indicative of mode II dominated failurewhere material is shearing into the wrinkle. This leads to very high critical strain energy releaserates. During the transition period, approximately 6 h to 18 h, excess length is introduced into thewrinkle through a growth in both the wrinkle height and width. This is indicative of a mixed modeof failure. During this transition, the mixed mode critical strain energy release rates measured wereall below 10 Jm−2 and given the growth over many hours, corresponded to very slow feed rates,10×10−4 mmmin−1 to 10×10−5 mmmin−1. Crossley et al. show that fast feed rates typical informing or layup, 10 mmmin−1 to 100 mmmin−1, correspond to critical strain energy release ratesranging between 50 Jm−2 to 250 Jm−2 for their GFRP material on steel tooling. This shows that acompatible relationship could exists where a very slow decay in critical strain energy release rateslink the two feed rates during the mixed mode regime.1994.6 Summary of the transient wrinkle growthWrinkles are often discussed as a defect which arises immediately after forming [39], [41], [55].This work has clearly shown that a secondary, transient, type of wrinkling can occur due to thetool-part interaction and follows a rate dependent growth profile.A simple and effective apparatus was designed which was capable of repeatably impartingsmall strains into the compliant prepreg capable of generating these small wrinkles. Using a non-contact coordinate measuring machine proved to be an effective tool to capture the entire 3D surfaceprofile at sufficiently high resolution to capture all of the significant wrinkles. A semi-automateddata reduction technique was developed which returns easily interpretable information from the10’s of gigabyte point clouds.Several key insights arise from this type of testing and analysis. The most effective methodto minimize compressive strains created from the mismatch in coefficients of thermal expansionis to not allow the tool and part to return to room temperature after forming. If the temperaturedifferential is zero then driving pressure is nullified. A good predictor for the minimum isothermaltemperature which can cause these transient defects is near 45 ◦C. The other predicted strategy isto quickly cool the tool and part causing the interlayer to fail and minimizing imparted strains.Another strategy is to use initiators to drive excess lengths into many smaller wrinkles; ulti-mately benefiting the laminate strength. Using these initiators, the wrinkle’s finite zone of influencewas measured. For the high CTE tooling used, wrinkles more than 40 mm to 80 mm apart will haveno effect on each other. This is because of the proposed shear-lag interaction where the in-planestress develops over non-zero length. The number and spacing of the initiators is highly temper-ature dependent; however, the shear-lag approach is able to properly predict the required spacingfor a desired wrinkle excess length. This adds another tool to mitigate these types of defects whichcan be used by design and manufacturing engineers.200This shear-lag approach predicts a maximum in-plane stress along the lengthwise midline ofthe specimen with near zero stresses at the 1-direction free edges. This is precisely the observerbehaviour as no wrinkles were observed along the 1-direction boundary and a large agglomerationof wrinkles were seen along the lengthwise midline.Finally, this set of experiments has shown potential continuity to peel test experiments per-formed at high feed rates. A failure mode and strain rate dependent critical strain energy releaserate is a solution which could be used to predict the calculated values from the transient trials. Morework is required to fully explore this proposal but a consistent set of mechanics is occurring duringthe fast debond as the slow debond and a properly formulated model should be able to capture bothsets of these transient wrinkles.201Chapter 5Quasi-static wrinkle growth5.1 Introduction and methodologyA second series of experiments were run in order to test the automated data reduction techniqueon a more traditional forming example. Forming experiments are typically done on a fixed toolwhere material is laid up and a single set of measurements are made [17], [39], [55]. Severalof these authors have used tools with multiple radii or doubly curved geometries to extract moreinformation from each test [17], [55].In order to obtain more information as the material progressively deforms, prepreg was adheredto the tool in a flat state and the tool progressively bent into an arc, both states are depicted in Fig-ure 5.1. Measurements were made after each step in bending resulting in a quasi-static response.This bending will impart compression into the prepreg while the tool-part adhesion will inhibit theprepreg from failing in a global manner. Without tool-part interaction, the prepreg will debond en-tirely from the tool and compression will resolve as the characteristic mode 1 Euler buckling shapedefined by a single large deflection of the layer. The prepreg was clamped along the lengthwiseedge to mimic the shear restriction of a long flange. A thin slice of rubber was placed between therigid clamps and compliant prepreg to provide a uniform normal force.202(a)(b)Figure 5.1: 3D representation of the assembly with clamps at the boundary. (a) in the initialstate and (b) in a strained state.The same polycarbonate tooling and preparation methodology as described in Chapter 4 wasused. This methodology included the second prepreg ply on the opposing polycarbonate surfacerequired to inhibit premature bending from the heated debulk. The polycarbonate tool was usefulbecause of its moderate stiffness allowing it to remain rigid during layup while also bending ap-preciably under moderate loads. These loads were imparted into the polycarbonate and prepregstack using a Hounsfield Tensometer with hinged attachments. The full apparatus with a loadedspecimen is shown in Figure 5.2203DIC camerasLightingHounseldtensometerSpecimenFigure 5.2: DIC apparatus used in the quasi-static wrinkle growth trials.The exact stress and strain state of the specimen is unknown and digital image correlation (DIC)was trialled in an attempt to directly measure the surface strain profile. The DIC procedure involvescomparing the displacement of two points on the surface through time. In order to differentiateunique points, the DIC methodology typically involves coating the surface in a random black andwhite speckle pattern. This involved coating the prepreg surface with a thin base layer of whitespray paint and subsequently applying a speckle pattern in black spray paint [172]. The DIC usestwo offset cameras, shown in Figure 5.2, to create a three dimensional representation of the surface.Each of the DIC cameras had an individual resolution of 16 megapixels.The CMM with the non-contact laser scanning head was again used to measure the surfaceprofiles at each stage of forming. The interesting surface details will lay in the relative differencebetween the prepreg and the tool. Therefore, in order to measure the surface details, the globalcurvature needs to be removed. This was performed using the change of space described in Equa-tion 5.1.204{x′,z′}={rfit arctan(x− xoz− zo),rfit−√(x− xo)2+(z− zo)2}(5.1)where {x′ [m] ,z′ [m]} are the coordinates of the flattened surface and {xo [m] ,zo [m]} are thecoordinates of the center of the fitted circle with a constant radius, rfit [m].Ambient temperature was uncontrolled but measured. Several trials were conducted at elevatedtemperatures using flexible silicone heaters. These heaters were mounted and clamped directly ontothe rear side of the sandwich specimen. Trials were conducted between 24 ◦C to 62 ◦C.This series of tests used AS4/8552 in contrast to the previous tests due to the inability to receiveAS4/8552-1.2055.2 ResultsThe side profile from an example sequence is shown in Figure 5.3. The initially flat surface is thetopmost white line and surfaces with progressively larger curvatures have been shifted downwardsin the image. As can be seen in the side profiles, each of the surfaces lay on a thin arc indicatingthe specimen has not undergone significant twist or saddling.Figure 5.3: Side profile of a trial through various steps of end-to-end shortening.The initial surface of a DIC trial is shown in Figure 5.4. While the DIC speckle pattern coatsthe entire surface, a faint line at the bottom and top of the prepreg delineates the prepreg from thebacking polycarbonate.206Figure 5.4: Initial state of a representative specimen mounted in the Hounsfield tensometer.Unfortunately the DIC was unable to measure the strains involved during this forming processand the computed results did not reflect the observed response. The DIC results returned the generalshape of the curved surface but the visible wrinkles were not identifiable in the computed output.However, the images created during the DIC process can be directly analyzed. The followingfigures, Figure 5.5, Figure 5.6, and Figure 5.7, show the surface response for trials conducted at22 ◦C, 32 ◦C, and 50 ◦C. Each of these figures corresponds to a similar amount of end shorteningallowing for a direct comparison on the effect of temperature.Figure 5.5 shows the surface at 22 ◦C. At first glance, the surface appears to only have afew defects. Upon closer inspection, a large distribution of small wrinkles can be observed. Incontrast to the transient growth wrinkles, the wrinkles in this trial span the entire surface betweenthe clamped edges which implies that the compression state is relatively uniform.207Figure 5.5: Strained surface of a trial conducted at 22 ◦C.Figure 5.6 shows the surface at 32 ◦C. The wrinkles on this surface are much more visible,implying their heights are larger. The lengthwise midline of the specimen appear to have fewerdefects; however, this is a consequence of the lighting. Similar to the direct imaging of the transienttrials, the oblique, non-collimated, lighting will accentuate and mute the shadows cast by wrinklesbased on their positions on the surface. This limits the ability to measure the geometry of thewrinkle features.208Figure 5.6: Strained surface of a trial conducted at 32 ◦C.Figure 5.7 shows the surface of a test performed at 60 ◦C. This surface shows wrinkles withboth the largest height and width.Figure 5.7: Strained surface of a trial conducted at 50 ◦C.209Using the CMM, the specimen’s surface can be recreated in digital form. The initial surfacefor a CMM trial is shown in Figure 5.8. A slight amount of tension was imparted into the specimenwhile in the tensometer in order to ensure the first scan was on a flat surface.Figure 5.8: Representative initial surface of a trial as measured on the CMM.The following two blocks of figures, Figure 5.9 and Figure 5.10, show the surface profiles ofdifferent trials conducted between 24 ◦C to 62 ◦C. The profiles in Figure 5.9 were fitted by a circlewith a radius of approximately 500 mm while the profiles in Figure 5.10 were fitted by 250 mm.Qualitatively, both figures shows that the number of wrinkles is inversely proportional to thetemperature at fitted radii between 250 mm to 500 mm. The surface at 24 ◦C has a homogeneouslydispersed number of wrinkles with maximum wrinkle heights smaller than the surface at 40 ◦Cor 62 ◦C. Increasing the temperature to 40 ◦C causes a reduction in the total number of wrinkleswith the surface showing a less evenly distributed location of wrinkles. This trend is continued tothe 62 ◦C trial which shows the fewest number of wrinkles as well as a localized region along thelengthwise midline of the surface. The large feature on the lower right hand corner of the (c) is amagnet which was used in an attempt to create a constant origin for post processing and is not awrinkle.210(a)(b)y(c)Figure 5.9: Three surface profiles digitally flattened from a forming radius of approximately500 mm. Conducted at approximately (a) 24 ◦C, (b) 40 ◦C, and (c) 62 ◦C.211The surfaces fitted by a circle with a radius of 250 mm are shown in Figure 5.10. At 24 ◦C, thehomogeneously dispersed wrinkles continue without significant localization. Intuitively, decreas-ing the radius has caused the individual wrinkles to increase in height. Similar to the transient trial,each of the wrinkles has grown from a pre-existing wrinkle. At 40 ◦C and above, the wrinkles growsignificantly larger and several wrinkles coalesce into a single wrinkle range.212(a)(b)y(c)Figure 5.10: Three surface profiles digitally flattened from a forming radius of approximately250 mm. Conducted at approximately (a) 24 ◦C, (b) 40 ◦C, and (c) 62 ◦C.213Quantitatively, Figure 5.11, tallies the total number of wrinkles as a function of the fitted circleradius for various temperatures. As a reference, a perfectly bonded Euler-Bernoulli (E-B) beamwith a 2 mm thickness formed to a radius of 500 mm will have a maximum strain of 0.2 % and amaximum strain of 0.4 % at 250 mm. These trials do not have a perfect bond and will undergosome internal shear; hence, the E-B only provides a general estimate of the overall strain. 0 50 100 150 200 250 300 350 400 0  200  400  600  800  1000Number of wrinklesFitted radius [mm]24 °C40 °C50 °C62 °CFigure 5.11: Comparison of the total number of wrinkles as a function of temperature. Theroom temperature trial shows significantly more features than the elevated temperaturetrials over a large range of forming steps.The tallest five wrinkles from each of the trials have been tracked through the bending process.Figure 5.12 shows the wrinkle height for these five wrinkles.214 0 0.5 1 1.5 2 2.5 3 3.5 4 0  200  400  600  800  1000Wrinkle height [mm]Fitted radius [mm]24 °C40 °C50 °C62 °CFigure 5.12: Comparison of the largest wrinkles as a function of progressive forming stepsshowing the smaller overall wrinkles at room temperature compared to the larger wrin-kles measured for the elevated trials.2155.3 Discussion and summaryThese trials were run in order to test if the wrinkle analysis script could be used to track wrinklesrelative to the tooling geometry. The automated wrinkle detection procedure developed in Chap-ter 4 was able to analyze the forming trials without modification proving the utility of this approach.The forming in these trials was done over a single, constant radius, curvature allowing the prepregsurface relative to the tool to be easily calculated. Calculating the surface geometry relative to thetool is not significantly more difficult for complex tooling geometries and can be automated [173].Wrinkles were successfully identified and tracked even as they began to coalesce.In tandem with the CMM approach, a DIC was trialled as a method for directly measuringstrains. Unfortunately, this approach did not prove to be as effective as only the general curvaturecould be measured. This is a consequence of the camera resolution which was unable to resolvethe very small strains, less than 0.1 % to 0.2 %, likely observed by the prepreg surface.This work has shown that even forming related wrinkles are not instantaneous events. Consis-tent with the tool-part wrinkles, forming induced wrinkles are a progressive phenomena. Formingspecimens over a radius while limiting the ability of the specimen to shear always resulted in wrin-kles. Increasing temperature was shown to increase the height of the wrinkles when compared toroom temperature isothermal trials. This increase in height was also associated with a decrease inthe total number of wrinkles. Above 40 ◦C, the wrinkle morphology of the largest five wrinklesshowed very similar heights for the same amount of end shortening. This response was starklydifferent from the growth response during the room temperature trial. At room temperature, thetotal number of wrinkles was consistently higher over the majority of the range of forming radiiwhile the individual heights were smaller and grew more steadily. The steady, but slightly erratic,growth observed by the higher temperature wrinkles was associated with the wrinkles coalescingcausing the recorded heights for two wrinkle to merge and resolve to the highest value.216Interestingly, these experiments have shown that long parts which cannot undergo significantshear may be less prone to large wrinkle geometries if the forming is done at lower temperatures.This would result in many more wrinkles; however, their individual heights would be significantlysmaller resulting in an overall stronger part.217Chapter 6Conclusions, contributions, and futurework6.1 ConclusionsMisalignment defects in composite materials are a cascading problem where imperfectly aligneddry carbon fibres in the earliest stages of manufacture track through into the final stages of partassembly. Both types of misalignment defects, wrinkling and waviness, can cause severe reduc-tions in the final part’s mechanical properties and decades of effort have gone into understandinghow to manage each stage of the manufacturing chain to reduce their impact. These works wereoften predicated on the material having properties without variability and perfect fibre alignmentwhile discussions on the true state of the underlying architecture had several conflicting hypothe-ses. Wrinkling and waviness occurring during processing is ultimately a response of the materialunder compressive loading and is defined by the uncured material’s mechanical properties. Thesemechanical properties are themselves defined by many variables, some of which are the fibre align-ment, fibre volume fraction, and resin stiffness.218In this work, a novel phenomenological model was developed for the fibre bed which properlymatches our own experimental data as well as data found in the literature. This was done on thelargest dataset of its kind with measurements on over 200,000 fibres spanning three rolls and twomaterial systems. Data collection and reduction at this scale was only possible by automatingthe analysis. These scripts were written to scrape several thousand micrographs, returning allof the relevant geometric information. The data showed that even high quality aerospace gradeunidirectional prepregs feature some intrinsic variability.In terms of fibre alignment, the variation was consistent between different rolls and resin typesbut the measured variation showed the fibres were not the perfectly collimated set of fibres as isusually assumed. The misalignment distributions, for both the in-plane and out-of-plane compo-nent, were effectively normally distributed and showed that the out-of-plane misalignments weresignificantly more aligned than their in-plane counterparts. Through a novel approach, the amountof work required to capture the through-thickness angle information was halved from the traditionalapproach and was also included in this automated procedure.These methods collapsed all of the spatial information into an overall population descriptor.Given the large sample sizes, spatial information was retained using a windowing technique wherethe automated fibre analysis was applied over 1 mm sections in series. Reintroducing the spatialinformation significantly increased the value of these micrographs. This windowing techniqueallowed the spatial misalignment variation to be directly viewed; showing a relatively homogeneousdistribution. This technique also allowed the excess length consumed by the fibres to be calculated.This distribution of excess lengths proved valuable as it defined the individual fibre lengths requiredby the proposed fibre bed model. These individually defined fibres is the only reasonable solutionwhich was able to regenerate the observed normally distributed misalignment distributions.Both the fibre alignment and excess length techniques relied on optical microscopy which lim-ited their applicability to post cured specimens. Uncured prepreg at room temperature is very com-219pliant which does not allow the samples to be sectioned without influencing the underlying fibrearchitecture. A solution of ammonium hydroxide was shown to stiffen the material by advancingthe sample degree of cure without the associated cure shrinkage or temperature modification; thecombination of which have been shown to alter fibre orientation. The as-received fibre alignmentand excess length was inferred from a set of these ammonium hydroxide soaked samples whichshowed a clear reduction in the system’s excess length. This extra excess length, caused from thetraditional thermal cure, is a result of the cure shrinkage out-pacing the thermal expansion resultingin a net compression on the fibres. Overall, while fibre misalignment is a undesirable outcome, thefibre alignment in these prepregs was consistent and corresponded to very low excess lengths whenanalyzed in aggregate.The proper method for determining fibre volume fraction from off-axis micrographs was pre-sented and results for both the pre- and post-cure materials were presented. The fibre volumefraction was shown to bound the 57.37 % value stated by the manufacturer; however, variabilityshowed regions both significantly higher and lower with an overall standard deviation of ±5 %.Pre-cure the thickness was highly variable, a consequence of internal voids and vacuum channels,which settled to the manufacturers reported value after the thermal cure. The post-cure thicknessalso suffered from some variability which likely stems from the prepreg manufacture itself. Thesemeasurements were pulled from automated, purpose built, scripts allowing this procedure to beapplied to other materials or existing data sets with ease.The main takeaway from all of the material characterization is that even high quality aerospacegrade unidirectional prepregs feature some intrinsic variability. While variability is a fundamentalproperty of CFRPs available today, the variability is homogeneously dispersed throughout the en-tire composite. This means that defects, such as wrinkling, will not likely be readily attributable toa single defect originating in the raw material.220This homogeneously dispersed set of defects was observed in the transient wrinkling trials. Anon-contact and high resolution apparatus was devised and implemented which generated a highfidelity map of the entire prepreg surface. These large datasets were reduced, again using mostlyautomated scripting, which returned geometric information for each of the surface defects. Withoutspecial surface conditions, a small compressive strain resulted in many wrinkles evenly distributedover the midline of the specimens. Hundreds of these features were measured with an averagesteady state density of 1 wrinkle per square centimeter. The centering of the defects and absenceof defects near the free edges indicated a shear-lag phenomena.Using a surface modifier, strips of PMP, wrinkles could be initiated and the shear-lag approachwas investigated. The initiated trials showed the potential effectiveness of localizing and control-ling defect formation. These initiated wrinkles dominated the overall response of the prepreg andwere the only wrinkles of significance on the steady state surfaces. Using multiple initiators, theeffective length over which stress could be transferred was determined and found to be a strongfunction of temperature. Therefore, using multiple initiators to lower the steady state wrinkleheight could be a useful tool; however, knowledge of the tool working environment is critical.Two other potential strategies for minimizing tool-part induced wrinkles were found. Thedifferential between the tool and part’s coefficient of thermal expansion coupled with the cooldown from forming and debulk was the driver of stresses. By maintaining the tool-part assembly atelevated temperatures, above 45 ◦C, no excess lengths are expected. Counter-intuitively, the otherpotential approach is to quench the tool and part. This rapid increase in the part stiffness and theultimately decreasing interface strength is expected to cause the entire interlayer to fail; effectivelyreducing the length over which stresses can be imparted to zero.Using classic linear elastic shear-lag was not appropriate and a hypoelastic shear-lag relation-ship was developed. This relationship was able to predict the excess lengths observed in the initi-221ated trials. The relationship correctly predicted the increasing excess length during small tempera-ture differentials with a peaking behaviour caused by the interlayer failure.All of these results and techniques should allow prepreg manufacturers to provide better in-sight into their material systems and for part manufacturers to better track their incoming material.This research has also expanded the knowledge base in early stages of composite manufacture andshould improve manufacturing techniques aiding in the promised ease of manufacturing.6.2 ContributionsThis work’s contributions can be highlighted in three major areas.• Better and faster data analysis techniques are required as larger and larger data sets becomeavailable. This research has documented the procedure and opened up several automatedtechniques for quickly returning actionable information from these large datasets.• A phenomenological model for the prepreg fibre bed misalignment has been proposed. Thismodel correctly predicts the unimodal misalignment distributions seen in this work and thebroader literature. This model provides a better descriptor which can be used for betterpredictions of CFRP mechanical properties.• A set of experiments has clearly shown transient wrinkles which arise over long time scalesfrom tool-part interaction. This work has shown that all of the wrinkles, including the mostsignificant large wrinkles, were visible in the earliest measured time steps and grow asymp-totically rather than a sudden initiation event. The ranking order of wrinkle’s height didnot remain constant, highlighting the need for fine measurements of the surface at regularintervals.2226.3 Future workSeveral future experiments and projects can be established using the process and data reductiontechniques documented herein. The following tasks would strengthen the proposed models as wellas expand their utility for composites manufacturing.• The procedure for characterizing the composites internal structure has been documented andautomated. Characterizing other composites would provide insight into the generality ofobservations presented here.• Ideally, working directly with a prepreg manufacturer or developing a lab scale prepregline would allow the fibre alignment to be characterized at various stages of manufactureto study their effects on fibre architecture. This could directly influence future developmentof prepregs.• Developing a new serial grinding technique to measure the amplitude and wavelength ofeach fibre would validate the individual excess length fibre bed model.• Predicting wrinkles generated from tool-part interaction requires knowledge of many un-cured material properties. The lower confidence prepreg properties were the interlayer thick-ness and the unrelaxed, temperature variant, compressive modulus. Independently measuringthese values would further improve the proposed modelling technique.• Using automated data reduction, the choice of potential experimental features to study islarge. Several of these features include:– Prepreg material– Tooling material– Tooling surface condition223– Stack layup– Ply thickness– Ply length and width• While µCT is currently limited to very small sample volumes, advancements in this non-destructive method should be closely watched as it has the potential to allow direct obser-vation of the fibre bed architecture. Digital volume correlation could also prove to be aneffective technique to directly measure strains associated with transient misalignment de-fects.• This study resulted in several large datasets, both in prepreg characterization and in prepregsurface profiles. 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The four steps: Tzero® calibration, enthalpy (Cell) constantcalibration, temperature calibration, and heat capacity calibration, were run in sequence with veri-fication stages inserted after each calibration. The document provides criteria for each verificationstage except the heat capacity calibration which only offers a reference curve to ensure the machineoutput is similar. The cell constant, temperature, and heat capacity calibrations used materials en-closed in Tzero® pans which had been crimped in hermetic lids.The Tzero® technology uses an additional thermocouple placed between the reference andsample sensors. Using this additional thermocouple allows the heat flow between the furnace andthe sample and reference to be calculated independently. This allows the relative heat flow betweenthe sample and reference pans to be calculated while properly accounting for the resistance andcapacitance of the system [175]. The Tzero® calibration is performed in two steps. The first stepuses an empty cell while the second step uses large, 95 mg sapphire disks on both the sample andreference positions. Neither the empty cell or sapphire disks use pans.239The cell constant and temperature calibration is performed by running a reference materialthrough its melting transition and comparing the instruments measured values from the referencevalues, Figure A.3. Indium is a commonly used reference material which has a melting transitionin the region of interest, Tm = 156.5985 ◦C, and a heat of fusion of (28.57± 0.17)Jg−1 [176], [177].Once the proper calibration values have been determined from the calibration test; a verificationtest is run on the same sample to ensure the correct heat of fusion and melting temperature aremeasured, Figure A.4.The heat capacity calibration is similar to the cell constant and temperature calibration in thatthe heat flow into a reference material is measured over the instruments working temperature.For the −80 ◦C to 400 ◦C temperature range, a sapphire disk is used as the reference material. Thecalibration curve is determined by dividing the theoretical value of the sapphire’s heat capacity fromthe measured value, Figure A.5. A subsequent verification trial is performed to ensure repeatability,Figure A.6.This procedure, documented by TA Instruments, was followed and the instrument passed eachverification stage. Interestingly, at no point does the procedure call for an empty pan to be com-pared to another empty pan. This is intuitively the effective baseline as all of the materials ofinterest must be enclosed in the pans prior to testing. As a sanity check, the empty reference panused in the calibration previously was compared against an equivalent empty Tzero® pan crimpedwith a hermetic lid. The output heat flows from two tests, each with two heating cycles, is shownin Figure A.7. The data shows an approximately 300 µW heat flow differential from −80 ◦C to250 ◦C between the reference pan and each of the empty sample pans. For reference, the DSC testsdescribed in Chapter 3 had a maximum heat flow differential of approximately 1700 µW between−80 ◦C to 250 ◦C. This 300 µW is much smaller than the tests 1700 µW; however, the unaccountedfor heat flow is still significant. Within one trial, the response was nearly identical between sub-sequent heating cycles. While the exact response between the two trials was different, the general240trends were similar. Therefore, the analysis described in Chapter 3 should be consistent betweentrails and the discussion remains constant.In a response from a TA Instruments representative [178], the response indicated that this typeof analysis at the forefront of the state-of-the-art with respect to MDSC experiments. The repre-sentative indicated that an empty cell baseline is best used as a representation of the performanceof the instrument; useful to TA Instruments for servicing. The dual empty pan baselines show thecontribution of the pans which are more relevant to the analyst. The direct response was that noanalyst is concerned about optimizing the total, reversing, and non-reversing, heat flows and thatthis type of performance optimization is beyond the state-of-the-art.The testing and analysis performed in Chapter 3 was run using the outlined, and recommended,procedure described above. This was done in an effort to ensure consistency within this work aswell as with the greater literature.241Figure A.1: DSC calibration manual: Tzero Calibration (Tzero Verification) desired baseline,[174]242Figure A.2: Typical Tzero calibrated state of CRN’s Discovery DSC.243Figure A.3: DSC calibration manual: Cell Constant/Temperature Calibration (Indium Verifi-cation), [174]Table A.1: Standard values for the melting temperature and heat of fusion of indium, [176],[177]Material system Melting temperature Heat of fusionIndium 156.5985 ◦C 28.57 Jg−1 ±0.17244Figure A.4: Typical thermogram of indium after Cell Constant/Temperature Calibration ofCRN’s Discovery DSC. Annotated on the figure are the onset melting temperature andenthalpy.Figure A.5: DSC calibration manual: Standard Heat Capacity Calibration (Heat CapacityVerification), [174]245Figure A.6: Typical thermogram of sapphire after a Standard Heat Capacity Calibration ofCRN’s Discovery DSC.Figure A.7: Thermograms of empty pan trials performed in a calibrated Discovery DSC,2 ◦Cmin−1 with ±0.318 ◦Cmin−1 modulation.246Appendix BMicrograph analysis scriptB.1 Semi-automated ellipse detectionThe following several pages are the scripts used to analyze the mosaic micrograph images usedfor characterization of alignment in Chapter 3. All of the digital images were taken with using thesame Nikon Epiphot 300 optical microscope at the same 200× magnification resulting in a resolu-tion of 192 pixels per 49.67 µm as measured using a Nikon 1 mm objective micrometer with 100graduations. The approximately 60 mm mosaic micrographs were sectioned into approximately 11to 13 images with widths of 22,500 pixels or approximately 5.7 mm. This was done solely in aneffort to optimize memory usage and efficiency. Therefore, the scripts were reused, with minormodifications, for each of the 5.7 mm wide images. The only changes made to the script betweenimages were the sections relating to the manual removal and manual addition of ellipses which hadbeen fitted automatically in error. Only one of these scripts is reproduced below due to the majorredundancy between scripts.247PreambleIn[121]:= Remove["Global`*"]In[122]:= StartTime = AbsoluteTime[];Image processingIn[123]:= WorkingDirectory = "\\\\ubccmps-sfpp01.ead.ubc.ca\\Comp\\Users\\Stewart,Andrew\\Output\\Experiments\\F12\\";In[124]:= OutputFolder = "2015-12-20_F12_Microscopy\\2016-02-05_F12_c_02_Microscopy_analysis\\";In[125]:= ImageFolder = "2015-12-20_F12_Microscopy\\2016-02-05_F12_c_02_Microscopy\\";In[126]:= WorkingFile = "2016-02-05_F12_c_02_Stiched_Left_22116px_44616px";Input imageTwo common file types used have been .tif and .jpg, comment out the non-used one for continued use.In[127]:= image = WorkingDirectory <> ImageFolder <> WorkingFile <> ".tif";In[128]:= (*image=WorkingFolder<>WorkingFile<>".jpg";*)Initial image for which all subsequent calculations are performed.In[129]:= img = Import[image];Pixel to micron conversion taken from the stage micrometer data.In[130]:= μm = 49.67192 ;MathsWidth and horizontal scaling factors to stretch the ellipses into more circular shapes, respectivelyIn[131]:= nw = 13 ;nh = 3;Dimensions to determine output sizing of the images.  This step also stretches the original image by the scaling factors.In[133]:= {w, h} = ImageDimensions[img];stretched = ImageResize[img, {w * nw, h * nh}];{ws, hs} = ImageDimensions[stretched];In[136]:= bwStretched = ColorNegate[Binarize[stretched]];Printed by Wolfram Mathematica Student Edition248In[137]:= ellipseSeparation =WatershedComponents[ColorNegate[ImageAdjust[DistanceTransform[Binarize[stretched]]]],Method -> {"MinimumSaliency", 0.4}] * ImageData[Binarize[stretched]];Tuning parameter if there are a lot of edge effects, ie. ellipses created at the boundary of an image which are incorrect.In[138]:= noborders = DeleteBorderComponents[ellipseSeparation];Computes the values of the centroid, semiaxes, and orientation of the stretched ellipses for both the case where allellipses are present and the other case where ellipses clipped by the border are removed.  Removed.In[139]:= componentsNB = ComponentMeasurements[noborders, {"Centroid", "SemiAxes", "Orientation"}];MathsGeneral form of the ellipse equation.In[140]:= Ellipse[x_, y_, θ_, rmajor_, rminor_, xoffset_, yoffset_] =(x - xoffset) * Cos[θ] + (y - yoffset) * Sin[θ]rmajor2 +(y - yoffset) * Cos[θ] - (x - xoffset) * Sin[θ]rminor2 - 1;No border ellipsesMathsRecreates the ellipses from the stretched values.In[141]:= For[i = 1, i ≤ Length[componentsNB], i++,StretchedEllipseNB[x_, y_][i] = Ellipse[x, y, componentsNB[[i]][[2]][[3]],componentsNB[[i]][[2]][[2]][[1]], componentsNB[[i]][[2]][[2]][[2]],componentsNB[[i]][[2]][[1]][[1]], componentsNB[[i]][[2]][[1]][[2]]]]Creates the original, (unstretched), ellipses from the stretched ellipses.In[142]:= For[i = 1, i ≤ Length[componentsNB], i++,OriginalEllipseNB[x_, y_][i] = StretchedEllipseNB[x * nw, y * nh][i]]Displays the recreated, stretched, ellipses from their base equations.In[143]:= (*Show[Table[ContourPlot[StretchedEllipseNB[x,y][i]⩵1,{x,0,ws},{y,0,hs},AspectRatio→Automatic,PlotPoints→50],{i,1,Length[componentsNB]}],ImageSize→{ws,hs}]*)Displays the created, unstretched ellipses from their base equationsIn[144]:= (*Show[Table[ContourPlot[OriginalEllipseNB[x,y][i]⩵1,{x,0,w},{y,0,h},AspectRatio→Automatic,PlotPoints→50],{i,1,Length[componentsNB]}],ImageSize→{w,h}]*)Overlays the created ellipses onto the original ellipse image2     2016-02-09_F12_c_02_Ellipse_Counting_and_Determination_Left_22116px_44616px_Thesis.nbPrinted by Wolfram Mathematica Student Edition249In[145]:= (*Showimg,GraphicsRed,componentsNB/.(n_→{centroid_,semiAxes_,orientation_})⧴Textn,centroid* 1nw , 1nh ,Table[ContourPlot[OriginalEllipseNB[x,y][i]⩵1,{x,0,w},{y,0,h},AspectRatio→Automatic,PlotPoints→50,ContourStyle→Red],{i,1,Length[componentsNB]}],ImageSize→{2*w,2*h}*)Generalized form of the ellipse equation http://mathworld.wolfram.com/Ellipse.htmla * x2 + 2 * b * x * y + c * y2 + 2 * d * x + 2 * f * y + g = 0In[146]:= Fori = 1, i ≤ Length[componentsNB], i++,eaNB[i] = Coefficient[OriginalEllipseNB[x, y][i], x, 2];ebNB[i] = 12 * Coefficient[OriginalEllipseNB[x, y][i], x * y, 1];ecNB[i] = Coefficient[OriginalEllipseNB[x, y][i], y, 2];edNB[i] = 12 * Coefficient[Coefficient[OriginalEllipseNB[x, y][i], x, 1], y, 0];efNB[i] = 12 * Coefficient[Coefficient[OriginalEllipseNB[x, y][i], y, 1], x, 0];egNB[i] = Coefficient[Coefficient[OriginalEllipseNB[x, y][i], x, 0], y, 0]In[147]:= Fori = 1, i ≤ Length[componentsNB], i++,apNB[i] =  2 * eaNB[i] * efNB[i]2 + ecNB[i] * edNB[i]2 + egNB[i] * ebNB[i]2 - 2 * ebNB[i] *edNB[i] * efNB[i] - eaNB[i] * ecNB[i] * egNB[i] ebNB[i]2 - eaNB[i] * ecNB[i] *(eaNB[i] - ecNB[i])2 + 4 * ebNB[i]2 - (eaNB[i] + ecNB[i]) ;bpNB[i] =  2 * eaNB[i] * efNB[i]2 + ecNB[i] * edNB[i]2 + egNB[i] * ebNB[i]2 - 2 * ebNB[i] *edNB[i] * efNB[i] - eaNB[i] * ecNB[i] * egNB[i] ebNB[i]2 - eaNB[i] * ecNB[i] *- (eaNB[i] - ecNB[i])2 + 4 * ebNB[i]2 - (eaNB[i] + ecNB[i])In[148]:= Fori = 1, i ≤ Length[componentsNB], i++,IfebNB[i] ⩵ 0 && eaNB[i] < ecNB[i], ϕNB[i] = 0,IfebNB[i] ⩵ 0 && eaNB[i] > ecNB[i], ϕNB[i] = π2 ,IfebNB[i] ≠ 0 && eaNB[i] < ecNB[i], ϕNB[i] = 12 * ArcCot eaNB[i] - ecNB[i]2 * ebNB[i] ,IfebNB[i] ≠ 0 && eaNB[i] > ecNB[i],ϕNB[i] = π2 + 12 * ArcCot eaNB[i] - ecNB[i]2 * ebNB[i] , ϕNB[i] = "ERROR"Outputs, to a table, the ellipse number, its centroid, it’s orientation off of the x-axis (counterclockwise), and its major andminor diameters.2016-02-09_F12_c_02_Ellipse_Counting_and_Determination_Left_22116px_44616px_Thesis.nb     3Printed by Wolfram Mathematica Student Edition250In[149]:= CompleteOutputNBwOutlier = Array[OutputNB, Length[componentsNB] + 1];OutputNB[1] = {"Ellipse", "Centroid (x)","Centroid (y)", "Orientation, [°]", "Major Diameter", "Minor Diameter"};Fori = 1, i ≤ Length[componentsNB], i++, OutputNB[i + 1] =i, componentsNB[[i]][[2]][[1]][[1]] * μm * 1nw ,componentsNB[[i]][[2]][[1]][[2]] * μm * 1nh , ϕNB[i] * 180π , μm * 2 * apNB[i], μm * 2 * bpNB[i]In[152]:= CompleteOutputNB = Prepend[Select[CompleteOutputNBwOutlier, #[[5]] > 14.5 * μm && #[[6]] > 14.5 * μm &], {"Ellipse","Centroid (x)", "Centroid (y)", "Orientation, [°]", "Major Diameter", "Minor Diameter"}];Graphically shows the ellipse, the number of the ellipse, as well as the points for which the major and minor diametersare determined from.PixelVersion simply removes any factor length scaling factor (from pixels to microns) to have 1pixel = 1pixel for plottingthings onto an image.In[153]:= (*PixelVersionNB=Drop[CompleteOutputNB,1]/.{n_,centroidx_,centroidy_,orientation_,major_,minor_}→n,centroidx* 1μm ,centroidy* 1μm ,orientation,major* 1μm ,minor* 1μm ;*)In[154]:= PixelVersionNB =Drop[CompleteOutputNB, 1] /. {n_, centroidx_, centroidy_, orientation_, major_, minor_} →n, centroidx * 1μm , centroidy * 1μm , orientation, major * 1μm , minor * 1μm ;In[155]:= (*Visualization3=Showimg,GraphicsRed,PixelVersionNB/.({n_,centroidx_,centroidy_,orientation_,majordiameter_,minordiameter_})⧴RotateCircle{centroidx,centroidy}, majordiameter2 , minordiameter2 ,orientation* π180 ,{centroidx,centroidy},Text[n,{centroidx,centroidy}],TableGraphicsPointSize[Large],Blue,PointapNB[i]*Cos[ϕNB[i]]+componentsNB[[i]][[2]][[1]][[1]]* 1nw ,apNB[i]*Sin[ϕNB[i]]+componentsNB[[i]][[2]][[1]][[2]]* 1nh ,{i,1,Length[componentsNB]},TableGraphicsPointSize[Large],Blue,PointbpNB[i]*CosϕNB[i]+ π2 +componentsNB[[i]][[2]][[1]][[1]]* 1nw ,bpNB[i]*SinϕNB[i]+ π2 +componentsNB[[i]][[2]][[1]][[2]]* 1nh ,{i,1,Length[componentsNB]},ImageSize→{w,h};*)Graphically shows the ellipse, the number of the ellipse, as well as the points for which the major and minor diametersare determined from. (More exact than previous Visualization3 but much slower).4     2016-02-09_F12_c_02_Ellipse_Counting_and_Determination_Left_22116px_44616px_Thesis.nbPrinted by Wolfram Mathematica Student Edition251In[156]:= (*Visualization3=Showimg,GraphicsRed,componentsNB/.(n_→{centroid_,semiAxes_,orientation_})⧴Textn,centroid* 1nw , 1nh ,Table[ContourPlot[OriginalEllipseNB[x,y][i]⩵0,{x,0,w},{y,0,h},AspectRatio→Automatic,PlotPoints→50,ContourStyle→Red],{i,1,Length[componentsNB]}],TableGraphicsPointSize[Large],Blue,PointapNB[i]*Cos[ϕNB[i]]+componentsNB[[i]][[2]][[1]][[1]]* 1nw ,apNB[i]*Sin[ϕNB[i]]+componentsNB[[i]][[2]][[1]][[2]]* 1nh ,{i,1,Length[componentsNB]},TableGraphicsPointSize[Large],Blue,PointbpNB[i]*CosϕNB[i]+ π2 +componentsNB[[i]][[2]][[1]][[1]]* 1nw ,bpNB[i]*SinϕNB[i]+ π2 +componentsNB[[i]][[2]][[1]][[2]]* 1nh ,{i,1,Length[componentsNB]},ImageSize→{2*w,2*h};*)Output table with the ellipse number, centroid, and major and maximum diameters.In[157]:= (*CompleteOutputNB//TableForm*)Aberration detectionIn[158]:= smalltodrop = Ceiling0.1 * Length[CompleteOutputNB] - 1;largetodrop = Ceiling0.1 * Length[CompleteOutputNB] - 1;Small drop valuesIn[160]:= Remove[SmallEllipsesToDropSelection, CentroidXYSmallRAW, CentroidXYSmallIndividual,CentroidXYSmall, SmallToDropValue, SmallToDropValueIndividual]After removing the obviously erronious ellipses.  Manually go through the image using the coordinate selection tool andselect the centroid of each ellipse you wish to remove.  Edit->Paste those values into the following array.In[161]:= SmallManuallySelectedCoordinates = {{1.438*^4, 255.1}};In[162]:= SmallCoordinateRemoval = Table[PixelVersionNB[[Position[PixelVersionNB[[All, 2 ;; 3]], Nearest[PixelVersionNB[[All, 2 ;; 3]],SmallManuallySelectedCoordinates[[i]]][[1]]][[1, 1]]]][[1]], {i, 1, Length[SmallManuallySelectedCoordinates], 1}];Ellipses to drop based on ellipse number, then returns the x and y position with units of pixels.  This value should notchange regardless of the kernel state of mathematica, ie. should not change if mathematica crashes.In[163]:= SmallEllipsesToDropSelection = Sort[DeleteDuplicates[SmallCoordinateRemoval]];If you get errors when manually adding values to the to drop array, it’s most likely because an element value you’readding is not in the list.  Use the below function with the values from the SmallDrop array to make sure that every newelement is in the SmallDrop set.In[164]:= (*CheckerFunction[i_]:={i,MemberQ[SmallDrop[[All,1]],i]}*)2016-02-09_F12_c_02_Ellipse_Counting_and_Determination_Left_22116px_44616px_Thesis.nb     5Printed by Wolfram Mathematica Student Edition252In[165]:= (*Map[CheckerFunction,{2284,2283,410,418,237,164,1418,447,2064,229,192,1308,1059,1876,462,465,345,770,1420,1145,1274,259,277,2010,1733,2156,24,27,25,45,64,71,36,34,2221,2226,235,1995,1968,1914,2285,2286,392,387,2198,2219,2229,2214,1396,118,95,54,1576,2294,2295,2189,2195,250,266,261,258,238,214,92,545,678,693,701,1903,416,1903,2101,1891,2074,1335,654,2110,1202,470,502,504,680,2013,1999,1974,38,70,22,17,36,112,107,26,85,150,46,57,115,68,55,23,3,702,695,556,497,4,9,101,510,488,2215,2212,2004,488,1065,1547,2146,2103,2114,2183,1285,1937,2052,2043,1936,2161,2126,1753,93,62,161,132,162,145,151,2152,2132,158,124,117,2118,2121}]//TableForm*)In[166]:= CentroidXYSmallRAW = Array[CentroidXYSmallIndividual, Length[SmallEllipsesToDropSelection]];For[i = 1, i ≤ Length[SmallEllipsesToDropSelection],i++, CentroidXYSmallIndividual[i] = PixelVersionNB[[Position[PixelVersionNB, SmallEllipsesToDropSelection[[i]]][[1, 1]]]][[{2, 3}]]]In[168]:= CentroidXYSmallRAWOut[168]= {{14383., 254.993}}This is the value to keep permanently.  It is independent of ellipse number and mm to pixel conversion ratio.In[169]:= CentroidXYSmall = {{14382.956193353475`, 254.99325276938566`}};This finds the new ellipse number corresponding to the pixel placed centroid values.In[170]:= SmallToDropValue = Array[SmallToDropValueIndividual, Length[CentroidXYSmall]];Fori = 1, i ≤ Length[CentroidXYSmall], i++,SmallToDropValueIndividual[i] = SelectPixelVersionNB,#[[2]] == CentroidXYSmall[[i, 1]] && #[[3]] == CentroidXYSmall[[i, 2]] &[[1, 1]]In[172]:= Sort[SmallToDropValue] == Sort[SmallEllipsesToDropSelection]Out[172]= TrueLarge drop valuesEllipses to drop based on ellipse number, then returns the x and y position with units of pixels.  This value should notchange regardless of the kernel state of mathematica, ie. should not change if mathematica crashes.In[173]:= Remove[LargeEllipsesToDropSelection, CentroidXYLargeRAW, CentroidXYLargeIndividual,CentroidXYLarge, LargeToDropValue, LargeToDropValueIndividual]In[174]:= LargeManuallySelectedCoordinates = {};In[175]:= LargeCoordinateRemoval = Table[PixelVersionNB[[Position[PixelVersionNB[[All, 2 ;; 3]], Nearest[PixelVersionNB[[All, 2 ;; 3]],LargeManuallySelectedCoordinates[[i]]][[1]]][[1, 1]]]][[1]], {i, 1, Length[LargeManuallySelectedCoordinates], 1}];In[176]:= LargeEllipsesToDropSelection = DeleteDuplicates[Flatten[Join[{LargeCoordinateRemoval}]]];6     2016-02-09_F12_c_02_Ellipse_Counting_and_Determination_Left_22116px_44616px_Thesis.nbPrinted by Wolfram Mathematica Student Edition253If you get errors when manually adding values to the to drop array, it’s most likely because an element value you’readding is not in the list.  Use the below function with the values from the SmallDrop array to make sure that every newelement is in the SmallDrop set.In[177]:= CentroidXYLargeRAW = Array[CentroidXYLargeIndividual, Length[LargeEllipsesToDropSelection]];For[i = 1, i ≤ Length[LargeEllipsesToDropSelection],i++, CentroidXYLargeIndividual[i] = PixelVersionNB[[Position[PixelVersionNB, LargeEllipsesToDropSelection[[i]]][[1, 1]]]][[{2, 3}]]]In[179]:= CentroidXYLargeRAWOut[179]= {}This is the value to keep permanently.  It is independent of ellipse number and mm to pixel conversion ratio.In[180]:= CentroidXYLarge = {};This finds the new ellipse number corresponding to the pixel placed centroid values.In[181]:= LargeToDropValue = Array[LargeToDropValueIndividual, Length[CentroidXYLarge]];Fori = 1, i ≤ Length[CentroidXYLarge], i++,LargeToDropValueIndividual[i] = SelectPixelVersionNB,#[[2]] == CentroidXYLarge[[i, 1]] && #[[3]] == CentroidXYLarge[[i, 2]] &[[1, 1]]In[183]:= Sort[LargeEllipsesToDropSelection] ⩵ Sort[LargeToDropValue]Out[183]= True2016-02-09_F12_c_02_Ellipse_Counting_and_Determination_Left_22116px_44616px_Thesis.nb     7Printed by Wolfram Mathematica Student Edition254Math to drop valuesIn[184]:= SmallDrop = Sort[PixelVersionNB, #1[[6]] < #2[[6]] &][[1 ;; smalltodrop, {1, 6}]];SmallDrop[[All, 2]] *= μm;SmallDrop // MatrixForm;SmallDrop = Select[SmallDrop, ! MemberQ[SmallToDropValue, #[[1]]] &];SmallDrop[[1 ;; 20]];LargeDrop = Sort[PixelVersionNB, #1[[6]] > #2[[6]] &][[1 ;; largetodrop, {1, 6}]];LargeDrop[[All, 2]] *= μm;LargeDrop // MatrixForm;LargeDrop = Select[LargeDrop, ! MemberQ[LargeToDropValue, #[[1]]] &];LargeDrop[[1 ;; 20]];Grid[{{Style["Small", 24], Style["Large", 24]}, {SmallDrop[[1 ;; 20]] // MatrixForm,LargeDrop[[1 ;; 20]] // MatrixForm}}]Out[188]=Small Large1499 4.678841381 5.068421180 5.070941525 5.09914272 5.14322127 5.205371533 5.28111349 5.36556795 5.413111517 5.4157836 5.45608835 5.46812236 5.468715 5.475911526 5.48026323 5.485311507 5.522741459 5.5265257 5.550681424 5.5668422 7.69726585 7.62072720 7.5500513 7.50624425 7.5024133 7.4351684 7.42596287 7.42022108 7.41425490 7.41009982 7.38238359 7.38235504 7.3679372 7.3554540 7.35496564 7.35417183 7.34573712 7.34119178 7.30292247 7.29378In[189]:= ETDSmall = SmallDrop[[All, 1]];(*Transpose[SmallDrop[[All,1]]][[1]];*)In[190]:= ETDLarge = LargeDrop[[All, 1]];(*Transpose[LargeDrop[[All,1]]][[1]];*)In[191]:= PixelVersionNBSmallError = {};PixelVersionNBLargeError = {};For[i = 1, i ≤ Length[ETDSmall], i++, AppendTo[PixelVersionNBSmallError,PixelVersionNB[[Position[PixelVersionNB, ETDSmall[[i]]][[1]][[1]]]]]]For[i = 1, i ≤ Length[ETDLarge], i++, AppendTo[PixelVersionNBLargeError,PixelVersionNB[[Position[PixelVersionNB, ETDLarge[[i]]][[1]][[1]]]]]]8     2016-02-09_F12_c_02_Ellipse_Counting_and_Determination_Left_22116px_44616px_Thesis.nbPrinted by Wolfram Mathematica Student Edition255In[195]:= (*Visualization4=Showimg,GraphicsRed,PixelVersionNBSmallError/.({n_,centroidx_,centroidy_,orientation_,majordiameter_,minordiameter_})⧴RotateCircle{centroidx,centroidy}, majordiameter2 , minordiameter2 ,orientation* π180 ,{centroidx,centroidy},GraphicsOpacity[0.5],Red,PixelVersionNBSmallError/.({n_,centroidx_,centroidy_,orientation_,majordiameter_,minordiameter_})⧴RotateDisk{centroidx,centroidy}, majordiameter2 , minordiameter2 ,orientation* π180 ,{centroidx,centroidy},Graphics[{Red,PixelVersionNBSmallError/.{({n_,centroidx_,centroidy_,orientation_,majordiameter_,minordiameter_})⧴{Text[n,{centroidx,centroidy}]}}}],GraphicsBlue,PixelVersionNBLargeError/.({n_,centroidx_,centroidy_,orientation_,majordiameter_,minordiameter_})⧴RotateCircle{centroidx,centroidy}, majordiameter2 , minordiameter2 ,orientation* π180 ,{centroidx,centroidy},GraphicsOpacity[0.5],Blue,PixelVersionNBLargeError/.({n_,centroidx_,centroidy_,orientation_,majordiameter_,minordiameter_})⧴RotateDisk{centroidx,centroidy}, majordiameter2 , minordiameter2 ,orientation* π180 ,{centroidx,centroidy},Graphics[{Blue,PixelVersionNBLargeError/.{({n_,centroidx_,centroidy_,orientation_,majordiameter_,minordiameter_})⧴{Text[n,{centroidx,centroidy}]}}}],ImageSize→w1,h1*)Arbitrary drop valuesEllipses to drop based on ellipse number, then returns the x and y position with units of pixels.  This value should notchange regardless of the kernel state of mathematica, ie. should not change if mathematica crashes.In[196]:= Remove[ArbitraryEllipsesToDropSelection, CentroidXYArbitraryRAW,CentroidXYArbitraryIndividual, CentroidXYArbitrary,ArbitraryToDropValue, ArbitraryToDropValueIndividual]In[197]:= ArbitraryManuallySelectedCoordinates = {{2807, 351.8}, {2698, 675.1}, {812.2, 602.9},{1.454*^4, 251.8}, {1.513*^4, 769.6}, {1.461*^4, 740.7}, {1.515*^4, 746.2}};In[198]:= ArbitraryCoordinateRemoval = Table[PixelVersionNB[[Position[PixelVersionNB[[All, 2 ;; 3]], Nearest[PixelVersionNB[[All, 2 ;; 3]],ArbitraryManuallySelectedCoordinates[[i]]][[1]]][[1, 1]]]][[1]], {i, 1, Length[ArbitraryManuallySelectedCoordinates], 1}];In[199]:= ArbitraryEllipsesToDropSelection =DeleteDuplicates[Flatten[Join[{ArbitraryCoordinateRemoval}]]];In[200]:= (*CheckerFunction3[i_]:={i,MemberQ[PixelVersionNB[[All,1]],i]}*)In[201]:= (*Map[CheckerFunction3,{597,585,393,358,1244,1869,415,122,2175,206,1136,1359,1361,1718,1724,1949,1858,1428,1483,1836,762,818,808,1745,2040,1202,2010,863,1218,1320,1309,1811,846,401,1610}]//TableForm*)2016-02-09_F12_c_02_Ellipse_Counting_and_Determination_Left_22116px_44616px_Thesis.nb     9Printed by Wolfram Mathematica Student Edition256In[202]:= CentroidXYArbitraryRAW =Array[CentroidXYArbitraryIndividual, Length[ArbitraryEllipsesToDropSelection]];For[i = 1, i ≤ Length[ArbitraryEllipsesToDropSelection], i++,CentroidXYArbitraryIndividual[i] = PixelVersionNB[[Position[PixelVersionNB, ArbitraryEllipsesToDropSelection[[i]]][[1, 1]]]][[{2, 3}]]]In[204]:= CentroidXYArbitraryRAWOut[204]= {{2811.45, 350.098}, {2702.74, 674.759}, {811.134, 602.149},{14539.3, 252.185}, {15137., 765.064}, {14607., 739.22}, {15 149.7, 737.948}}This is the value to keep permanently.  It is independent of ellipse number and mm to pixel conversion ratio.In[205]:= CentroidXYArbitrary = {{2811.4525773195874`, 350.09805269186705`},{2702.742304656669`, 674.7588792423047`}, {811.1340349987319`, 602.1494209147011`},{14539.327355623103`, 252.18497129348188`}, {15137.001285347043`, 765.063553270494`},{14607.02657935285`, 739.2202105803799`}, {15149.712716464202`, 737.9478332041009`}};This finds the new ellipse number corresponding to the pixel placed centroid values.In[206]:= ArbitraryToDropValue = Array[ArbitraryToDropValueIndividual, Length[CentroidXYArbitrary]];Fori = 1, i ≤ Length[CentroidXYArbitrary], i++,ArbitraryToDropValueIndividual[i] = SelectPixelVersionNB,#[[2]] == CentroidXYArbitrary[[i, 1]] && #[[3]] == CentroidXYArbitrary[[i, 2]] &[[1, 1]]In[208]:= ArbitraryToDropValueOut[208]= {1495, 1325, 101, 697, 1540, 1517, 1307}In[209]:= Sort[CentroidXYArbitrary] ⩵ Sort[CentroidXYArbitraryRAW]Out[209]= TrueMath to dropThese fibres are dropped from the analysis because they are aberrations.  When looking at the output visualization thisbecomes quick to detect. In[210]:= EllipsesToDrop = DeleteDuplicates[Sort[Flatten[Join[SmallToDropValue, LargeToDropValue, ArbitraryToDropValue]]]]Out[210]= {101, 697, 1307, 1325, 1495, 1517, 1535, 1540}10     2016-02-09_F12_c_02_Ellipse_Counting_and_Determination_Left_22116px_44616px_Thesis.nbPrinted by Wolfram Mathematica Student Edition257MathsIn[211]:= CompleteOutputNB1 = DeleteCasesCompleteOutputNB, _?MemberQ[EllipsesToDrop, #[[1]]] &;CompleteOutputNB1[[1 ;; 10, All]] // MatrixFormOut[211]//MatrixForm=Ellipse Centroid (x) Centroid (y) Orientation, [°] Major Diameter Minor Diameter1 4562.58 159.903 -1.20271 92.1453 7.159562 3250.58 120.29 -1.76914 87.3287 6.490313 4717.07 157.461 -0.032412 71.7746 6.954984 4907.44 156.75 0.230051 97.6734 6.803785 4882.11 140.271 -0.751835 69.9061 6.459766 5172.15 119.949 -1.46147 86.3279 7.000467 4480.86 163.059 -0.303711 74.5038 6.210988 3015.35 92.9663 -0.995307 73.734 7.274869 5378.67 61.1556 -1.28133 228.008 6.61195In[212]:= PixelVersionNB1 =Drop[CompleteOutputNB1, 1] /. {n_, centroidx_, centroidy_, orientation_, major_, minor_} →n, centroidx * 1μm , centroidy * 1μm , orientation, major * 1μm , minor * 1μm ;VisualizationIn[213]:= (*Visualization5=Showimg,GraphicsOpacity[0.5],Red,PixelVersionNB1/.({n_,centroidx_,centroidy_,orientation_,majordiameter_,minordiameter_})⧴RotateDisk{centroidx,centroidy}, majordiameter2 , minordiameter2 ,orientation* π180 ,{centroidx,centroidy},GraphicsRed,PixelVersionNB1/.({n_,centroidx_,centroidy_,orientation_,majordiameter_,minordiameter_})⧴RotateCircle{centroidx,centroidy}, majordiameter2 , minordiameter2 ,orientation* π180 ,{centroidx,centroidy},Text[n,{centroidx,centroidy}],ImageSize→{w,h}*)2016-02-09_F12_c_02_Ellipse_Counting_and_Determination_Left_22116px_44616px_Thesis.nb     11Printed by Wolfram Mathematica Student Edition258Manual ellipse additionIn[214]:= (*Visualization6=Showimg,GraphicsBlack,PixelVersionNB1/.({n_,centroidx_,centroidy_,orientation_,majordiameter_,minordiameter_})⧴RotateDisk{centroidx,centroidy}, majordiameter2 , minordiameter2 ,orientation* π180 ,{centroidx,centroidy}(*,Text[n,{centroidx,centroidy}]*),ImageSize→{w,h}*)Inputting of added ellipsesIn[215]:= Remove[AE, A, AN, NewMatrix]In[216]:= AE = {{1.102*^4, 401.8}, {1.12*^4, 392.9}, {1.111*^4, 410.7}, {1.11*^4, 386.2},{1.106*^4, 410.7}, {1.105*^4, 390.7}, {1.116*^4, 404}, {1.116*^4, 387.3}, {1.118*^4, 19.56},{1.432*^4, 256.2}, {1.468*^4, 250.7}, {1.463*^4, 259.6}, {1.462*^4, 244}, {1.451*^4, 242.9},{1.437*^4, 248.4}, {1.438*^4, 262.9}, {1.45*^4, 265.1}, {1.48*^4, 15.11}};In[217]:= A = DeleteCases[AE, {_, y_} /; y < 0];In[218]:= PixelCutOff = 50;In[219]:= ManuallyCountedEllipses = Length[Select[A, #[[2]] < PixelCutOff &]];In[220]:= AN = A;For[i = 1, i <= ManuallyCountedEllipses, i++,NewMatrix[i] =Drop[AN[[1 ;; Position[AN, Select[AN, #[[2]] < PixelCutOff &][[1]]][[1, 1]]]], -1];AN = Drop[AN, Position[AN, Select[AN, #[[2]] < PixelCutOff &][[1]]][[1, 1]]];]In[222]:= {"Ellipse", "Centroid (x)", "Centroid (y)","Orientation, [°]", "Major Diameter", "Minor Diameter"}Out[222]= {Ellipse, Centroid (x), Centroid (y), Orientation, [°], Major Diameter, Minor Diameter}In[223]:= Remove[lin, pa, ManualCompleteOutputNB, ManualeaNB, ManualebNB, ManualecNB, ManualedNB,ManualefNB, ManualegNB, Manualx0NB, Manualy0NB, ManualapNB, ManualbpNB, ManualOutputNB]In[224]:= Fori = 1, i <= ManuallyCountedEllipses, i++,lin[i] = NewMatrix[i] /. {x_, y_} → x2, x, y, 2 * x * y, y2;pa[i] = LinearModelFit[lin[i],{1, afit, bfit, cfit, dfit}, {afit, bfit, cfit, dfit}]["BestFitParameters"];12     2016-02-09_F12_c_02_Ellipse_Counting_and_Determination_Left_22116px_44616px_Thesis.nbPrinted by Wolfram Mathematica Student Edition259In[225]:= Fori = 1, i ≤ ManuallyCountedEllipses, i++,ManualeaNB[i] = Coefficient[pa[i].{1, x^2, x, y, 2 x y} - y^2, x, 2];ManualebNB[i] = 12 * Coefficient[pa[i].{1, x^2, x, y, 2 x y} - y^2, x * y, 1];ManualecNB[i] = Coefficient[pa[i].{1, x^2, x, y, 2 x y} - y^2, y, 2];ManualedNB[i] = 12 * Coefficient[Coefficient[pa[i].{1, x^2, x, y, 2 x y} - y^2, x, 1], y, 0];ManualefNB[i] = 12 * Coefficient[Coefficient[pa[i].{1, x^2, x, y, 2 x y} - y^2, y, 1], x, 0];ManualegNB[i] = Coefficient[Coefficient[pa[i].{1, x^2, x, y, 2 x y} - y^2, x, 0], y, 0]In[226]:= Fori = 1, i ≤ ManuallyCountedEllipses, i++,Manualx0NB[i] = (ManualecNB[i] * ManualedNB[i] - ManualebNB[i] * ManualefNB[i]) (ManualebNB[i])2 - ManualeaNB[i] * ManualecNB[i];Manualy0NB[i] = (ManualeaNB[i] * ManualefNB[i] - ManualebNB[i] * ManualedNB[i]) (ManualebNB[i])2 - ManualeaNB[i] * ManualecNB[i];ManualapNB[i] =  2 * ManualeaNB[i] * ManualefNB[i]2 + ManualecNB[i] * ManualedNB[i]2 +ManualegNB[i] * ManualebNB[i]2 - 2 * ManualebNB[i] * ManualedNB[i] * ManualefNB[i] -ManualeaNB[i] * ManualecNB[i] * ManualegNB[i]ManualebNB[i]2 - ManualeaNB[i] * ManualecNB[i] *(ManualeaNB[i] - ManualecNB[i])2 + 4 * ManualebNB[i]2 -(ManualeaNB[i] + ManualecNB[i]) ;ManualbpNB[i] =  2 * ManualeaNB[i] * ManualefNB[i]2 + ManualecNB[i] * ManualedNB[i]2 +ManualegNB[i] * ManualebNB[i]2 - 2 * ManualebNB[i] * ManualedNB[i] * ManualefNB[i] -ManualeaNB[i] * ManualecNB[i] * ManualegNB[i]ManualebNB[i]2 - ManualeaNB[i] * ManualecNB[i] *- (ManualeaNB[i] - ManualecNB[i])2 + 4 * ManualebNB[i]2 -(ManualeaNB[i] + ManualecNB[i])2016-02-09_F12_c_02_Ellipse_Counting_and_Determination_Left_22116px_44616px_Thesis.nb     13Printed by Wolfram Mathematica Student Edition260In[227]:= Fori = 1, i ≤ ManuallyCountedEllipses, i++,IfManualebNB[i] ⩵ 0 && ManualeaNB[i] < ManualecNB[i], ManualϕNB[i] = 0,IfManualebNB[i] ⩵ 0 && ManualeaNB[i] > ManualecNB[i], ManualϕNB[i] = π2 ,IfManualebNB[i] ≠ 0 && ManualeaNB[i] < ManualecNB[i],ManualϕNB[i] = 12 * ArcCot ManualeaNB[i] - ManualecNB[i]2 * ManualebNB[i] ,IfManualebNB[i] ≠ 0 && ManualeaNB[i] > ManualecNB[i],ManualϕNB[i] = π2 + 12 * ArcCot ManualeaNB[i] - ManualecNB[i]2 * ManualebNB[i] , ManualϕNB[i] = "ERROR"In[228]:= ManualCompleteOutputNB = Array[ManualOutputNB, ManuallyCountedEllipses + 1];ManualOutputNB[1] = {"Ellipse", "Centroid (x)","Centroid (y)", "Orientation, [°]", "Major Diameter", "Minor Diameter"};Fori = 1, i ≤ ManuallyCountedEllipses, i++, ManualOutputNB[i + 1] =CompleteOutputNB1[[-1, 1]] + i, Manualx0NB[i] * μm, Manualy0NB[i] * μm,ManualϕNB[i] * 180π - 90, μm * 2 * ManualbpNB[i], μm * 2 * ManualapNB[i]In[231]:= ManualCompleteOutputNB // TableFormOut[231]//TableForm=Ellipse Centroid (x) Centroid (y) Orientation, [°] Major Diameter Minor Diameter1540 2872.94 103.019 -2.52856 46.1782 6.225981541 3751.72 65.673 -0.749502 92.2836 5.6466814     2016-02-09_F12_c_02_Ellipse_Counting_and_Determination_Left_22116px_44616px_Thesis.nbPrinted by Wolfram Mathematica Student Edition261In[232]:= (*Visualization6=Showimg,GraphicsBlack,PixelVersionNB1/.({n_,centroidx_,centroidy_,orientation_,majordiameter_,minordiameter_})⧴RotateDisk{centroidx,centroidy}, majordiameter2 , minordiameter2 ,orientation* π180 ,{centroidx,centroidy}(*,Text[n,{centroidx,centroidy}]*),GraphicsOpacity[0.5],Red,Drop[ManualCompleteOutputNB,1]/.({n_,centroidx_,centroidy_,orientation_,majordiameter_,minordiameter_})⧴RotateDisk centroidxμm , centroidyμm , majordiameter2*μm , minordiameter2*μm ,orientation* π180 , centroidxμm , centroidyμm ,GraphicsRed,Drop[ManualCompleteOutputNB,1]/.({n_,centroidx_,centroidy_,orientation_,majordiameter_,minordiameter_})⧴RotateCircle centroidxμm , centroidyμm , majordiameter2*μm , minordiameter2*μm ,orientation* π180 , centroidxμm , centroidyμm ,Textn, centroidxμm , centroidyμm ,ImageSize→{w,h}*)OutputIn[233]:= CompleteOutputNB1[[1 ;; 30, All]] // MatrixFormOut[233]//MatrixForm=Ellipse Centroid (x) Centroid (y) Orientation, [°] Major Diameter Minor Diameter1 4562.58 159.903 -1.20271 92.1453 7.159562 3250.58 120.29 -1.76914 87.3287 6.490313 4717.07 157.461 -0.032412 71.7746 6.954984 4907.44 156.75 0.230051 97.6734 6.803785 4882.11 140.271 -0.751835 69.9061 6.459766 5172.15 119.949 -1.46147 86.3279 7.000467 4480.86 163.059 -0.303711 74.5038 6.210988 3015.35 92.9663 -0.995307 73.734 7.274869 5378.67 61.1556 -1.28133 228.008 6.6119510 2448.06 67.8326 0.0708301 236.082 5.9585711 5534.07 165.202 -0.849249 80.4202 6.2053812 4834.84 127.092 0.0780185 68.5677 6.3799113 3641.03 99.3633 -1.14983 74.6576 7.5062414 4223.52 108.131 1.96542 175.951 6.8649215 4649.64 182.978 -0.16636 223.747 5.4759116 4398.39 164.074 -0.583388 83.424 6.8130117 5032.13 184.162 0.0704894 198.705 5.9906118 4338.22 177.814 -0.00854461 179.81 6.2476119 3368.83 194.802 -0.971855 69.1414 5.6883920 3414.55 89.2082 -0.0389782 87.7067 6.6064421 2701.69 53.6172 -0.294879 174.559 5.9782822 3582.63 132.089 1.36481 161.355 6.4660623 4753.07 164.137 0.0275109 81.0021 7.0621324 4006.11 67.5762 0.133645 90.5402 7.1043325 1379.45 125.269 0.562152 147.778 6.8163826 4558.66 177.064 -0.700439 151.417 6.5863227 2890.46 186.174 -0.802556 63.8441 6.4259328 4832.07 82.3564 0.263903 154.183 6.388729 3226.84 182.261 0.0387449 65.9689 6.423322016-02-09_F12_c_02_Ellipse_Counting_and_Determination_Left_22116px_44616px_Thesis.nb     15Printed by Wolfram Mathematica Student Edition262In[234]:= CompleteOutputNBwManual = Join[CompleteOutputNB1, Drop[ManualCompleteOutputNB, 1]];CompleteOutputNBwManual[[-30 ;; -1, All]] // MatrixFormOut[234]//MatrixForm=1510 5612.83 129.18 -0.555337 35.1619 6.693991511 1768.4 200.951 0.810358 40.7901 5.816291512 4151.37 135.24 0.282465 38.5113 6.073891513 5576.84 128.674 0.472321 35.3691 6.522271514 2241.39 139.706 -2.02566 32.4445 7.093541515 3158.58 210.35 1.88152 34.0487 6.661531516 3351.38 132.696 -1.8978 38.3467 5.808361518 1222.61 208.847 0.373487 35.6676 6.226731519 4544.05 38.8457 -1.32995 36.9096 6.038281520 2449.27 198.171 -2.26343 38.617 5.711051521 2356.8 145.833 -4.2581 34.1838 6.437251522 5541.64 128.768 0.503893 32.8576 6.664131523 5109.76 36.1986 -0.624335 37.3632 5.80071524 3076.13 144.782 -3.70047 35.2975 5.945151525 3404.31 199.739 1.62836 40.9983 5.099141526 1140.66 210.467 -0.9502 37.6132 5.480261527 5529.2 33.0134 -0.639846 33.0661 6.058511528 1377.95 182.734 0.368525 32.1777 6.197461529 5766.02 66.902 -0.342595 30.5445 6.365411530 2658.22 216.483 -1.80669 30.9539 6.12751531 3125.62 211.256 0.634298 29.7131 6.256841532 2591.13 133.369 -2.02544 27.1192 6.823181533 2823.15 96.9152 -1.69508 32.3226 5.28111534 2700.12 217.188 -1.41354 28.9653 5.837281536 1906.95 133.6 -0.940854 22.2129 6.240251537 3129.33 149.368 -3.83093 20.3935 6.790651538 1925.51 213.287 -5.80872 19.4909 6.413331539 3304.94 125.993 -4.04674 18.1629 5.853221540 2872.94 103.019 -2.52856 46.1782 6.225981541 3751.72 65.673 -0.749502 92.2836 5.64668ExportCSVRAW = WorkingFolder <> WorkingFile <> "_RAW.csv";Export[CSVRAW, CompleteOutputRAW]Run time calculationsIn[235]:= FinishTime = AbsoluteTime[];In[236]:= IfFinishTime - StartTime ≤ 60, RunTime = Quantity[N[FinishTime - StartTime, 2], "seconds"],RunTime = QuantityN FinishTime - StartTime60 , 4, "minutes"Out[236]= 32. s16     2016-02-09_F12_c_02_Ellipse_Counting_and_Determination_Left_22116px_44616px_Thesis.nbPrinted by Wolfram Mathematica Student Edition263B.2 Yurgartis analysis on ellipse dataThe following script imports the fibre information scraped from the preceding scripts for each ofthe 5.7 mm reduced mosaic images and compiles them into a single radial position table. Thisradial position is then analyzed using the Yurgartis method. The script for a single radial positionis shown below.264PreambleIn[675]:= Remove["Global`*"]Yurgartis analysisInput fileIn[676]:= WorkingDirectory ="\\\\ubccmps-sfpp01.ead.ubc.ca\\Comp\\Users\\Stewart, Andrew\\Output\\Experiments\\F5\\";In[677]:= OutputFolder = "2015-12-17_F5_Micrography\\2016-01-24_F5_c_03_Analysis\\";In[678]:= ImageFolder = "2015-12-17_F5_Micrography\\2016-01-22_F5_c_03_Microscopy\\";In[679]:= WorkingName = "2016-01-24_F5_c_03";In[680]:= RadialPosition = "22.35cm";In[681]:= WorkingFile1 = "2016-01-24_F5_c_03_0px_22500px_NB";WorkingFile2 = "2016-01-24_F5_c_03_22063px_44563px_NB";WorkingFile3 = "2016-01-24_F5_c_03_44078px_66578px_NB";WorkingFile4 = "2016-01-24_F5_c_03_66115px_88615px_NB";WorkingFile5 = "2016-01-24_F5_c_03_88248px_110748px_NB";WorkingFile6 = "2016-01-24_F5_c_03_110112px_132612px_NB";WorkingFile7 = "2016-01-24_F5_c_03_131807px_154307px_NB";WorkingFile8 = "2016-01-24_F5_c_03_153916px_176416px_NB";WorkingFile9 = "2016-01-24_F5_c_03_175537px_193602px_NB";No boundariesInput fileIn[690]:= CSVNB1 = WorkingDirectory <> OutputFolder <> WorkingFile1 <> ".csv";CSVNB2 = WorkingDirectory <> OutputFolder <> WorkingFile2 <> ".csv";CSVNB3 = WorkingDirectory <> OutputFolder <> WorkingFile3 <> ".csv";CSVNB4 = WorkingDirectory <> OutputFolder <> WorkingFile4 <> ".csv";CSVNB5 = WorkingDirectory <> OutputFolder <> WorkingFile5 <> ".csv";CSVNB6 = WorkingDirectory <> OutputFolder <> WorkingFile6 <> ".csv";CSVNB7 = WorkingDirectory <> OutputFolder <> WorkingFile7 <> ".csv";CSVNB8 = WorkingDirectory <> OutputFolder <> WorkingFile8 <> ".csv";CSVNB9 = WorkingDirectory <> OutputFolder <> WorkingFile9 <> ".csv";Shows a small section of the full imported table (from rows 1 to 16).Printed by Wolfram Mathematica Student Edition265In[699]:= FibreValues1 = Import[CSVNB1];FibreValues2 = Import[CSVNB2];FibreValues3 = Import[CSVNB3];FibreValues4 = Import[CSVNB4];FibreValues5 = Import[CSVNB5];FibreValues6 = Import[CSVNB6];FibreValues7 = Import[CSVNB7];FibreValues8 = Import[CSVNB8];FibreValues9 = Import[CSVNB9];In[708]:= FibreValues = Join[FibreValues1, Drop[FibreValues2, 1], Drop[FibreValues3, 1],Drop[FibreValues4, 1], Drop[FibreValues5, 1], Drop[FibreValues6, 1],Drop[FibreValues7, 1], Drop[FibreValues8, 1], Drop[FibreValues9, 1]];FibreValues[[1 ;; 16, All]] // TableFormOut[709]//TableForm=Ellipse Centroid (x) Centroid (y) Orientation, [°] Major Diameter Minor Diameter15 177.031 568.878 0.580409 41.0422 7.1067318 222.325 566.957 0.300875 44.3322 7.7477820 341.099 563.439 0.636489 60.6778 7.3414322 271.793 559.821 0.264748 60.4904 7.238723 195.23 559.277 -0.361435 56.2316 7.0519825 382.723 558.245 -1.08228 52.829 6.9701427 437.277 556.142 0.742206 57.1736 7.0427228 332.93 555.628 1.13792 52.5717 7.68431 233.575 553.106 0.146942 62.3352 7.5734833 532.672 551.037 1.67063 71.8982 7.2034734 749.671 548.796 -3.57802 174.551 6.9403636 290.833 550.195 -0.205467 58.9087 7.2373737 372.265 549.923 0.0410675 46.5568 7.2054438 462.717 549.829 0.705254 59.7295 6.9891240 634. 546.555 0.0482982 46.5951 7.09171MathsManiputlation of input data to workable dataDrops the header line and converts the angles from degrees into radians.In[710]:= WorkingValues =Drop[FibreValues, 1] /. {n_, centroidx_, centroidy_, orientation_, major_, minor_} →n, centroidx, centroidy, orientation * π180 , major, minor;WorkingValues[[1 ;; 15, All]] // TableFormOut[710]//TableForm=15 177.031 568.878 0.0101301 41.0422 7.1067318 222.325 566.957 0.00525126 44.3322 7.7477820 341.099 563.439 0.0111088 60.6778 7.3414322 271.793 559.821 0.00462072 60.4904 7.238723 195.23 559.277 -0.00630824 56.2316 7.0519825 382.723 558.245 -0.0188894 52.829 6.9701427 437.277 556.142 0.0129539 57.1736 7.0427228 332.93 555.628 0.0198604 52.5717 7.68431 233.575 553.106 0.00256462 62.3352 7.5734833 532.672 551.037 0.0291581 71.8982 7.2034734 749.671 548.796 -0.0624482 174.551 6.9403636 290.833 550.195 -0.00358608 58.9087 7.2373737 372.265 549.923 0.000716764 46.5568 7.2054438 462.717 549.829 0.012309 59.7295 6.9891240 634. 546.555 0.000842963 46.5951 7.09171Useful term used throughout the analysis so a variable name is used to make the writing more clear.2     2016-01-24_F5_c_03_Yurgatis_Method_All_changing_diameter_Thesis.nbPrinted by Wolfram Mathematica Student Edition266In[711]:= n = Length[WorkingValues];Ellipse identifier: WorkingValues[[i]][[1]]Major diameter: WorkingValues[[i]][[5]]Removes any rows which have a complex angle.In[712]:= ωT = Array[ω, {n, 3}];Fori = 1, i ≤ n, i++, ω[i, 1] = WorkingValues[[i]][[1]];ω[i, 2] = ArcSin WorkingValues[[i]][[6]]WorkingValues[[i]][[5]] ;ω[i, 3] = 180π * ArcSin WorkingValues[[i]][[6]]WorkingValues[[i]][[5]] ωT = DeleteCases[ωT, a_ /; ! FreeQ[a[[2]], _Complex]];Prepend[ωT[[1 ;; 15, All]], {"Ellipse", "Angle [rad]", "Angle [°]"}] // TableFormOut[715]//TableForm=Ellipse Angle [rad] Angle [°]15 0.174034 9.971418 0.175669 10.065120 0.121287 6.9492622 0.119954 6.8728823 0.125741 7.2044125 0.132324 7.5815827 0.123495 7.0757528 0.146688 8.4045831 0.121797 6.9784533 0.100358 5.750134 0.0397718 2.2787636 0.123169 7.0570537 0.155391 8.9032738 0.117281 6.7197340 0.152793 8.75437Resets the length to accomodate any ellipses which were removed for having a complex angle.In[716]:= n = Length[ωT];Histogram generationWidth of the bins to collect the angles into, binsize [°]In[717]:= binsize = 0.25;Calculates the number of boxes required to display the histogram using the binsize.In[718]:= ceiling = NCeiling 1binsize * Max[ωT[[All, 3]]] * binsize, 3;floor = NFloor 1binsize * Min[ωT[[All, 3]]] * binsize, 3;NSteps = Rationalize ceiling - floorbinsize ;Counts the number of occurences of an angle being contained in the binsize from the minimum value to the maximumvalue.In[721]:= Bins = BinCounts[ωT[[All, 3]], {floor, ceiling + binsize, binsize}];Creates an array to store the data for the histogram.2016-01-24_F5_c_03_Yurgatis_Method_All_changing_diameter_Thesis.nb     3Printed by Wolfram Mathematica Student Edition267In[722]:= HistogramData = Arrayvalue, Rationalize ceiling - floorbinsize , 2;Stores the given number of counts, ie Bins, next to the appropriate containing angle.In[723]:= Fori = 1, i ≤ NSteps, i++, value[i, 1] = floor + 2 * i - 12 * binsize ;value[i, 2] = Bins[[i]]Creates an array to store the values after accounting for counting bias.In[724]:= Fv,Yurgartis = Array[volumefraction, {NSteps, 2}];Equation 8 from Yurgartis, (1987), Measurement of small angle fibre misalignments in continuous fibre composites.In[725]:= denominatorValue = j=1Length[HistogramData] HistogramData[[j]][[2]]TanHistogramData[[j]][[1]] * π180  ;In[726]:= Fori = 1, i ≤ NSteps, i++, volumefraction[i, 1] = HistogramData[[i]][[1]];volumefraction[i, 2] =HistogramData[[i]][[2]]TanHistogramData[[i]][[1]]* π180 denominatorValue In[727]:= Fv,mean,Yurgartis = Fv,Yurgartis[[All, 1]].Fv,Yurgartis[[All, 2]]Total[Fv,Yurgartis[[All, 2]]] ; N[Fv,mean,Yurgartis, 10]Out[727]= 4.28017In[728]:= (*Export["S:\\dissertation\\xx-gnuplotdata\\incoming_characterization\\misalignments\\individual_volumefraction_plots\\F5_c_03_vf.dat",{#[[1]]-Fv,mean,Yurgartis,#[[2]]}&/@Fv,Yurgartis,"TSV"]*)In[729]:= Fv = Fv,Yurgartis /. {x_, y_} → {x, y / binsize};In[730]:= Total[Fv[[All, 2]] * binsize]Out[730]= 1.Determines the mean of the “volume fraction of the total volume of fiber that is at an angle ωi to the plane-cut surface”.In[731]:= Fv,mean = Fv[[All, 1]].Fv[[All, 2]]Total[Fv[[All, 2]]] ; N[Fv,mean, 10]Out[731]= 4.28017Determines the mode of the volume fraction.In[732]:= Fv,mode = First[Sort[Fv, #1[[2]] > #2[[2]] &]][[1]];Determines the standard deviation of the group off of the mean.In[733]:= Fv,standarddeviation = Fv[[All, 2]].Fv[[All, 1]] - Fv,mean2Total[Fv[[All, 2]]] ; N[Fv,standarddeviation, 10]Out[733]= 1.633714     2016-01-24_F5_c_03_Yurgatis_Method_All_changing_diameter_Thesis.nbPrinted by Wolfram Mathematica Student Edition268Displays the histogram not corrected for the cutting angle, MeanFvIn[790]:= OriginalFigure = ListPlotFv, PlotRange → All(*{{0,10},All}*), Filling → Axis, FillingStyle →Directive[{Opacity[0.45], Darker[Darker[Orange]], Thickness[0.005 * 0.8], CapForm[None]}], PlotMarkers → {""}, ImageSize → {540, 400}, AspectRatio → 400540, BaseStyle → FontSize → 20, Frame → True, FrameLabel → {"Fibre Angle, ω (degrees)", "(ω)"},Epilog → Inset[Framed[Grid[{{Style["Mean = " <> ToString[NumberForm[Fv,mean, 3]] <> "°", 20]},{Style["s = " <> ToString[NumberForm[Fv,standarddeviation, 3]] <> "°", 20]},{Style["N = " <> ToString[n], 20]}}],Background → Lighter[LightGray]], ImageScaled[{0.855, 0.865}]];Manual Calculation to double check mean and standard deviation outputhttp://publib.boulder.ibm.com/infocenter/db2luw/v9r5/index.jsp?topic=%2Fcom.ibm.db2.luw.admin.wlm.doc%2Fdoc%2Fc0052459.htmlIn[735]:= Fv // MatrixForm;In[736]:= Fvc = MapThread[Insert,{Fv, Accumulate[Fv[[All, 2]] * binsize], Table[3, {Length[Accumulate[Fv[[All, 2]]]]}]}];https://wikimedia.org/api/rest_v1/media/math/render/svg/0242a61c35625b029f890b9114fac6be4c60293bIn[737]:= MEAN = Fv[[All, 1]].Fv[[All, 2]]Total[Fv[[All, 2]]]Out[737]= 4.28017In[738]:= S1 = {};Fori = 1, i ≤ Length[Fv], i++, AppendToS1, Fv[[i, 2]] * Fv[[i, 1]] - MEAN2In[740]:= STANDARDDEVIATION = Total[S1]Total[Fv[[All, 2]]]12Out[740]= 1.63371In[741]:= CummulativePlot = ListPlot[Fvc[[All, {1, 3}]],PlotRange → {{0, 90}, {0, 1}}, ImageSize → {1000, 600}, PlotStyle → Blue];Transformation about zero degrees according to the meanIn[742]:= Fv,AboutZero = Transpose[{Fv[[All, 1]] - Fv,mean, Fv[[All, 2]]}];2016-01-24_F5_c_03_Yurgatis_Method_All_changing_diameter_Thesis.nb     5Printed by Wolfram Mathematica Student Edition269Chop simply sets any value less than 10-10 as strictly equal to zero.In[743]:= Fv,AboutZero,mean = Chop Fv,AboutZero[[All, 1]].Fv,AboutZero[[All, 2]]Total[Fv,AboutZero[[All, 2]]] ;In[744]:= Fv,AboutZero,mode = Last[Sort[Fv,AboutZero, #1[[2]] < #2[[2]] &]][[1]]; Standard deviationhttp://ime.math.arizona.edu/g-teams/Profiles/VP/MeanVarSDRandomVarsSlides.pdfRemember that the expected value is sum(x*p/sum(p)) != sum(x*p)In[745]:= s =  Fv,AboutZero[[All, 1]] - Fv,AboutZero[[All, 1]].Fv,AboutZero[[All, 2]]Total[Fv,AboutZero[[All, 2]]] 2.Fv,AboutZero[[All, 2]]Total[Fv,AboutZero[[All, 2]]]Out[745]= 1.63371In[774]:= AboutZeroFigure = ListPlotFv,AboutZero, PlotRange → {{-4, 4}, {0, 0.4}}, Filling → Axis, FillingStyle →Directive[{Opacity[0.45], Darker[Darker[Orange]], Thickness[0.03], CapForm[None]}], PlotMarkers → {""}, ImageSize → {600, 400}, AspectRatio → 400600, BaseStyle → FontSize → 20, Frame → True, FrameLabel → {"Fibre Angle, θ (degrees)", "(θ)",WorkingName <> "\n" <> "Approximate radial position: " <> RadialPosition}, Epilog → Inset[Framed[Grid[{{Style["Mean = " <> ToString[NumberForm[Fv,AboutZero,mean, 3]] <>"°", 20]}, {Style["Mode = " <> ToString[NumberForm[Fv,AboutZero,mode, 3]] <> "°", 20]},{Style["s = " <> ToString[NumberForm[s, 3]] <> "°", 20]},{Style["N = " <> ToString[n], 20]}}],Background → Lighter[LightGray]], ImageScaled[{0.8125, 0.69}]];6     2016-01-24_F5_c_03_Yurgatis_Method_All_changing_diameter_Thesis.nbPrinted by Wolfram Mathematica Student Edition270Output VisualizationsIn[788]:= Show[OriginalFigure]Out[788]=0 10 20 30 400.000.050.100.150.200.250.30Fibre Angle, ω (degrees)(ω)Mean = 4.28°s = 1.63°N = 9938In[789]:= Show[AboutZeroFigure]Out[789]=-4 -2 0 2 40.00.10.20.30.4Fibre Angle, θ (degrees)(θ)2016-01-24_F5_c_03Approximate radial position: 22.35cmMean = 0°Mode = -0.405°s = 1.63°N = 9938Assuming normally distributed data set:68% fall within2016-01-24_F5_c_03_Yurgatis_Method_All_changing_diameter_Thesis.nb     7Printed by Wolfram Mathematica Student Edition271In[749]:= 0.994458 * STANDARDDEVIATIONOut[749]= 1.6246695% fall withinIn[750]:= 1.959964 * STANDARDDEVIATIONOut[750]= 3.2020199% fall withinIn[751]:= 2.575829 * STANDARDDEVIATIONOut[751]= 4.20816Export[WorkingDirectory <> OutputFolder <> WorkingName <> "_Fv_Distibution.csv",Prepend[Fv,AboutZero, {"Sample size", n}], "CSV"]\\ubccmps-sfpp01.ead.ubc.ca\Comp\Users\Stewart,Andrew\Output\Experiments\F5\2015-12-17_F5_Micrography\2016-01-24_F5_c_03_Analysis\2016-01-24_F5_c_03_Fv_Distibution.csv8     2016-01-24_F5_c_03_Yurgatis_Method_All_changing_diameter_Thesis.nbPrinted by Wolfram Mathematica Student Edition272Appendix CWrinkle analysis scriptTwo sample scripts written using Mathworks MATLAB and were uploaded to the University ofBritish Columbia’s supplementary materials online repository when the document was uploadedto the Faculty of Graduate Studies. The sample scripts were written and used to measure theindividual wrinkle features from the transient and quasi-static wrinkle growth trials. The output ofthese scripts were presented in Chapter 4 and Chapter 5.The example code has not been appended here due to their length.273Appendix DSupplementary transient wrinklegrowth dataD.0.1 No initiatorResults for a duplicate no initiator trial are presented in Figure D.1 through Figure D.4. The samematerial system and sample size were used as presented in Chapter 4.The surface profile for three time steps are shown in Figure D.1 with the individual wrinklesoverlaid on the surface.274(a)(b)(c)Figure D.1: Duplicate non-initiated transient wrinkle growth trial conducted at an uncon-trolled ambient temperature of 22 ◦C. (a), (b), and (c) show various time steps fromthe initial time step to nearly 9 d.The individual wrinkle identifiers are shown in Figure D.2 corresponding to the final recordedtime step with an elapsed time of nearly 9 d.275Figure D.2: Top down view of the final time step of the wrinkled surface with each wrinklegiven a unique identifier.The individual wrinkle heights for the tracked features from Figure D.2 are shown as a functionof time in Figure D.3.276 0 0.1 0.2 0.3 0.4 0.5 0.6 0  1  2  3  4  5  6  7  8  9Wrinkle height [mm]Elapsed time [days](a) 0 0.1 0.2 0.3 0.4 0.5 0.6 0  6  12  18  24Wrinkle height [mm]Ellaspsed time [hours](b)172914924219550152227113140102573642325212216133781261843453285130485533443954564316464520414757384935233Wrinkle identifier(c)Figure D.3: (a) and (b) plot the maximum wrinkle height over the first nearly 9 d and the first24 h. (c) indicates the individual wrinkles following the numbering scheme outlined inFigure D.2.Four time steps of the population plots for this non initiated trial are shown Figure D.4.277 0 10 20 30 40 50 60 70 80 90 0  0.1  0.2  0.3  0.4  0.5  0.6N = 189Number of wrinklesWrinkle height [mm]t0 + 0d 00:00:00(a) 0 10 20 30 40 50 60 70 80 90 0  0.1  0.2  0.3  0.4  0.5  0.6N = 199Number of wrinklesWrinkle height [mm]t0 + 0d 02:00:00(b) 0 10 20 30 40 50 60 70 80 90 0  0.1  0.2  0.3  0.4  0.5  0.6N = 71Number of wrinklesWrinkle height [mm]t0 + 2d 00:20:00(c) 0 10 20 30 40 50 60 70 80 90 0  0.1  0.2  0.3  0.4  0.5  0.6N = 57Number of wrinklesWrinkle height [mm]t0 + 8d 22:00:00(d)Figure D.4: Population height distribution of wrinkles for the Figure D.1 trial.278D.0.2 Wrinkle initiatorsSingle initiatorSurface profiles and population distribution results from additional single initiator trials are shownin Figure D.5 through Figure D.14.(a)(b)(c)Figure D.5: Three time steps for a single initiator trial conducted at an isothermal temperatureof 20.7 ◦C279 0 10 20 30 40 50 60 70 0  0.1  0.2  0.3  0.4  0.5  0.6N = 23Number of wrinklesWrinkle height [mm]t0 + 0d 00:00:00(a) 0 10 20 30 40 50 60 70 0  0.1  0.2  0.3  0.4  0.5  0.6N = 27Number of wrinklesWrinkle height [mm]t0 + 0d 02:00:00(b) 0 10 20 30 40 50 60 70 0  0.1  0.2  0.3  0.4  0.5  0.6N = 31Number of wrinklesWrinkle height [mm]t0 + 0d 19:20:00(c) 0 10 20 30 40 50 60 70 0  0.1  0.2  0.3  0.4  0.5  0.6N = 13Number of wrinklesWrinkle height [mm]t0 + 5d 00:10:00(d)Figure D.6: Population distribution of wrinkle heights through four time steps for the Fig-ure D.5 trial.280(a)(b)(c)Figure D.7: Three time steps for a duplicate single initiator trial conducted at an isothermaltemperature of 20.7 ◦C281 0 10 20 30 40 50 60 70 0  0.1  0.2  0.3  0.4  0.5  0.6N = 6Number of wrinklesWrinkle height [mm]t0 + 0d 00:00:00(a) 0 10 20 30 40 50 60 70 0  0.1  0.2  0.3  0.4  0.5  0.6N = 3Number of wrinklesWrinkle height [mm]t0 + 0d 02:00:00(b) 0 10 20 30 40 50 60 70 0  0.1  0.2  0.3  0.4  0.5  0.6N = 1Number of wrinklesWrinkle height [mm]t0 + 0d 19:20:00(c) 0 10 20 30 40 50 60 70 0  0.1  0.2  0.3  0.4  0.5  0.6N = 1Number of wrinklesWrinkle height [mm]t0 + 5d 00:10:00(d)Figure D.8: Population distribution of wrinkle heights through four time steps for the Fig-ure D.7 trial.282(a)(b)(c)Figure D.9: Three time steps for a single initiator trial conducted at an isothermal temperatureof 21.1 ◦C283 0 10 20 30 40 50 60 70 0  0.1  0.2  0.3  0.4  0.5  0.6N = 29Number of wrinklesWrinkle height [mm]t0 + 0d 00:00:00(a) 0 10 20 30 40 50 60 70 0  0.1  0.2  0.3  0.4  0.5  0.6N = 23Number of wrinklesWrinkle height [mm]t0 + 0d 02:00:00(b) 0 10 20 30 40 50 60 70 0  0.1  0.2  0.3  0.4  0.5  0.6N = 10Number of wrinklesWrinkle height [mm]t0 + 0d 20:20:00(c) 0 10 20 30 40 50 60 70 0  0.1  0.2  0.3  0.4  0.5  0.6N = 7Number of wrinklesWrinkle height [mm]t0 + 4d 00:45:00(d)Figure D.10: Population distribution of wrinkle heights through four time steps for the Fig-ure D.9 trial.284(a)(b)(c)Figure D.11: Three time steps for a single initiator trial conducted at an isothermal tempera-ture of 21.2 ◦C285 0 10 20 30 40 50 60 70 0  0.1  0.2  0.3  0.4  0.5  0.6N = 8Number of wrinklesWrinkle height [mm]t0 + 0d 00:00:00(a) 0 10 20 30 40 50 60 70 0  0.1  0.2  0.3  0.4  0.5  0.6N = 12Number of wrinklesWrinkle height [mm]t0 + 0d 02:00:00(b) 0 10 20 30 40 50 60 70 0  0.1  0.2  0.3  0.4  0.5  0.6N = 59Number of wrinklesWrinkle height [mm]t0 + 0d 20:20:00(c) 0 10 20 30 40 50 60 70 0  0.1  0.2  0.3  0.4  0.5  0.6N = 12Number of wrinklesWrinkle height [mm]t0 + 4d 00:45:00(d)Figure D.12: Population distribution of wrinkle heights through four time steps for the Fig-ure D.11 trial.286(a)(b)(c)Figure D.13: Three time steps for a single initiator trial conducted at an isothermal tempera-ture of 24.3 ◦C287 0 10 20 30 40 50 60 70 0  0.1  0.2  0.3  0.4  0.5  0.6N = 61Number of wrinklesWrinkle height [mm]t0 + 0d 00:00:00(a) 0 10 20 30 40 50 60 70 0  0.1  0.2  0.3  0.4  0.5  0.6N = 35Number of wrinklesWrinkle height [mm]t0 + 0d 01:55:00(b) 0 10 20 30 40 50 60 70 0  0.1  0.2  0.3  0.4  0.5  0.6N = 28Number of wrinklesWrinkle height [mm]t0 + 0d 19:25:00(c) 0 10 20 30 40 50 60 70 0  0.1  0.2  0.3  0.4  0.5  0.6N = 9Number of wrinklesWrinkle height [mm]t0 + 4d 02:05:00(d)Figure D.14: Population distribution of wrinkle heights through four time steps for the Fig-ure D.13 trial.288Double initiatorSurface profiles and population distribution results from an additional double initiator trial areshown in Figure D.15 and Figure D.16, respectively.(a)(b)(c)Figure D.15: Three time steps for a double initiator trial conducted at an isothermal temper-ature of 21.8 ◦C289 0 10 20 30 40 50 60 70 0  0.1  0.2  0.3  0.4  0.5  0.6N = 125Number of wrinklesWrinkle height [mm]t0 + 0d 00:00:00(a) 0 10 20 30 40 50 60 70 0  0.1  0.2  0.3  0.4  0.5  0.6N = 75Number of wrinklesWrinkle height [mm]t0 + 0d 02:00:00(b) 0 10 20 30 40 50 60 70 0  0.1  0.2  0.3  0.4  0.5  0.6N = 33Number of wrinklesWrinkle height [mm]t0 + 0d 19:20:00(c) 0 10 20 30 40 50 60 70 0  0.1  0.2  0.3  0.4  0.5  0.6N = 11Number of wrinklesWrinkle height [mm]t0 + 4d 10:20:00(d)Figure D.16: Population distribution of wrinkle heights through four time steps for the Fig-ure D.15 trial.290D.0.3 Peirce cantilever testThe Peirce cantilever test is a test used to determine the stiffness of fabrics [72] and is beginningto see use in determining the bending stiffness of prepregs [71]. The test involves horizontallycantilevering a known length of material under self-weight. This method is sensitive to sampledimensions and the values used here are consistent with testing performed at Convergent Manufac-turing Technologies [179]. Specifically, the free hanging length is 75 mm and a sample width of50 mm.While this testing method may prove to be effective at determining the temperature dependentbending stiffness of prepregs, the current goal is only to determine if a time independent stiffnessis an appropriate assumption. Three samples were prepared using AS4/8552-1 with the testingconducted at an uncontrolled ambient temperature of (25.0±0.1) ◦C. The deflection of one rep-resentative trial at progressive time steps are shown in Figure D.17. Some slight adjustment tothe sample can be seen between the initial time step and the 1 d time step; however, no differencebetween the 1 d and 2 d side profiles can be seen.Gross deflection would imply a significant contribution from viscoelastic effects. This was notobserved; hence, treating the AS4/8552-1 as time independent is a reasonable assumption.291(a)(b)(c)Figure D.17: (a) 0 h, (b) 24 h, and (c) 48 h of a single ply of AS4/8552-1 cantilevered underself weight at an uncontrolled ambient temperature of (25.0±0.1) ◦C. The ruler’smajor graduations are in cm.292Appendix ESupplementary quasi-static wrinklegrowth dataAdditional steps in the forming trials are shown in Figure E.1 through Figure E.4.Figure E.1 shows three snapshots of the digitally flattened surface at 24 ◦C.293(a)(b)y(c)Figure E.1: (a), (b), and (c) show the surface profile of a room temperature trial undergoingprogressive amounts of bending.294Figure E.2 shows three snapshots of the digitally flattened surface at 40 ◦C.(a)(b)y(c)Figure E.2: (a), (b), and (c) show the surface profile of a 40 ◦C trial undergoing progressiveamounts of bending.295Figure E.3 shows three snapshots of the digitally flattened surface at 50 ◦C.(a)(b)y(c)Figure E.3: (a), (b), and (c) show the surface profile of a 50 ◦C trial undergoing progressiveamounts of bending.296Figure E.4 shows three snapshots of the digitally flattened surface at 62 ◦C.(a)(b)y(c)Figure E.4: (a), (b), and (c) show the surface profile of a 62 ◦C trial undergoing progressiveamounts of bending.297Appendix FVolumetric contraction of the resinThe volumetric cure shrinkage of the resin is defined as the change in volume after cure normalizedby the initial resin volume.αcs =∆V RV RThe volume of the composite, neglecting the volume of voids, is:VC =V R+V FIntroducing the fibre volume fraction definition returns:VC =V R+V FVC·VC =V R+ v fVCor,V R = (1− v f )VC298Also, recognizing that the volume of fibres does not undergo any change as the resin undergoescure, the change in volume of the composite can be determined from the change of volume of theresin.∆VC = ∆V R+*0∆V FSubstituting these into the cure shrinkage definition returns:αcs =∆VC(1− v f )VCExpanding returns:αcs =(xC,1+δxC,1)(xC,2+δxC,2)(xC,3+δxC,3)− xC,1xC,2xC,3(1− v f )xC,1xC,2xC,3Expanding and neglecting small terms, i.e., terms with multiple δxC,i elements, returns:αcs =1(1− v f )(δxC,1xC,1+δxC,2xC,2+δxC,3xC,3)Assuming there is negligible change in the 1-direction of the laminate and that the laminate istransversely isotropic, i.e., the strains in the 2- and 3- direction are equivalent, results in:αcs =2δxC,2(1− v f )xC,2Finally, in terms of laminate width before cure, L2,o, and after cure, L2, f , returns the equationpresented in Chapter 3 Equation 3.2:αcs = 2 · L2, f −L2,o(1− v f ) ·L2,o299

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