MULTISCALE MODELING TO DESCRIBE FREE SURFACE RESIN FLOW AROUND FIBERS AND WITHIN FIBER BUNDLES. Michael Yeager

Transcription

1 MULTISCALE MODELING TO DESCRIBE FREE SURFACE RESIN FLOW AROUND FIBERS AND WITHIN FIBER BUNDLES by Michael Yeager A dissertation submitted to the Faculty of the University of Delaware in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mechanical Engineering Spring Michael Yeager All Rights Reserved

2 MULTISCALE MODELING TO DESCRIBE FREE SURFACE RESIN FLOW AROUND FIBERS AND WITHIN FIBER BUNDLES by Michael Yeager Approved: Suresh G. Advani, Ph.D. Chair of the Department of Mechanical Engineering Approved: Babatunde A. Ogunnaike, Ph.D. Dean of the College of Engineering Approved: Ann L. Ardis, Ph.D. Senior Vice Provost for Graduate and Professional Education

3 I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy. Signed: Suresh G. Advani, Ph.D. Professor in charge of dissertation I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy. Signed: R. Valery Roy, Ph.D. Member of dissertation committee I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy. Signed: John W. Gillespie Jr., Ph.D. Member of dissertation committee I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy. Signed: Shridhar Yarlagadda, Ph.D. Member of dissertation committee

4 ACKNOWLEDGMENTS First I would like to acknowledge and thank my advisor, Dr. Suresh Advani. I feel truly blessed to benefit from his guidance, expertise, patience, and enthusiasm. Our weekly interactions have helped me challenge myself, learn how to approach problems in a methodical manner, and develop into a better scientist. I would like to acknowledge Dr. Pavel Simacek for his support and advice on process modeling. I am thankful for fruitful discussions about mechanics with Dr. John Gillespie and Dr. Bazle Haque. I would also like to give thanks to the following professors who have agreed to serve on my dissertation committee, Dr. Shridhar Yarlagadda and Dr. Valery Roy. I am thankful for the mechanical engineering and CCM staff for their support throughout my time as a graduate student. I am also thankful for the friship and support of my fellow graduate students at the CCM, whom I am hesitant to list because I do not want to forget anyone. Particularly, I would like to thank John Gangloff and Jiayin Wang for serving as mentors in my first few years. I thank my parents for always encouraging me to peruse my educational dreams. Their support has meant more than they will ever know. I would also like to thank my sisters, Ashley and Amanda, and brother Charlie for their support and love over the years. I also thank Christine Wilkerson, her love and support has helped me find the iv

5 energy to complete this journey. I feel truly blessed to have her in my life. I would also like to thank all of my family and fris who have supported me during this journey. Finally, I would like to thank the Army Research Laboratory, whose dollars under Cooperative Agreement Number W911NF has made this research possible. v

6 TABLE OF CONTENTS LIST OF TABLES... xi LIST OF FIGURES... xii ABSTRACT... xx Chapter 1 INTRODUCTION Overview of Composites Manufacturing Background and Motivation Objectives and Dissertation Outline MICROSCALE FIBER WETTING BY A FINITE RESIN VOLUME Background Methods Model Setup Axisymmetric Single Fiber Model Three Dimensional Single Fiber Model Unit Cell with Square and Triangular Packing Arrangements Resin Spreading on a Flat Plate Assumptions Governing Equations Fiber and Resin Parameters Static Contact Angle between Fiber and Resin Resin Viscosity Slip Length Fiber Volume Fraction and Packing Experimental Setup and Procedure Results and Discussion vi

7 2.3.1 Validation of the Numerical Model Comparison between Experimental and Numerical Results Mesh Refinement Study Comparison of Final Drop Shape with an Analytical Solution Parametric Study of Axi-symmetric Model Static Contact Angle between Fiber and Resin Fiber Radius Slip Length Three Dimensional Single Fiber Model Square and Hexagonal packing Fiber Unit Cells Fiber Volume Fraction Static Contact Angle Limitations of the Model Summary CHARACTERIZATION OF AVERAGE CAPILLARY PRESSURE FOR RESIN FLOW BETWEEN FIBERS Introduction Model Setup Numerical Capillary Pressure Model Numerical Model Validation Analytical Capillary Pressure Model Averaging the Capillary Pressure Results Influence of Applied Macroscopic Pressure on Capillary Pressure Effect of Fiber Arrangement Effect of Imperfect Fiber Arrangement Capillary Pressure for Unit Cells with Fibers with Different Sizings Summary vii

8 4 CAPILLARY DRIVEN FLOW OF AN INFINITE RESIN VOLUME INTO FIBER TOWS: ROLE OF FIBER PACKING AND AIR EVACUATION Introduction Previous Work Methods Solution Methodology Applied Pressure Boundary Condition Inclusion of Capillary Pressure and Air Compression Validation Compressed Air Pressure Fiber Volume Fraction Distribution Parametric Study and Baseline Properties Results Fiber Volume Fraction Distribution Effect of Air Evacuation Time Influence of Tow location within the Part Influence of Wetting Properties Effect of Fiber Radius Summary PREDICTION OF CAPILLARY DRIVEN FLOW OF A FINITE RESIN VOLUME INTO A FIBER TOW Introduction Motivation Background Information Methods Model Setup Determining Tow-Drop Contact Area Inclusion of capillary pressure Quantification of Interlocking viii

9 5.2.3 Mesh Refinement Study Results Influence of Drop Spacing Effect of Packing Arrangement Charge Type Summary CONCLUSIONS, CONTRIBUTIONS, AND FUTURE WORK Conclusions Contributions Future Work Including Fiber Movement in Microscale Model Manufacturing and Testing of Composites with Optimized Microstructure Determination of Capillary Pressure for Different Microstructures Twisted Tows Hybrid Composites REFERENCES Appix A ANALYTICAL MICROSCALE MODELS A.1 Drop of Resin between Four Fibers A.2 Commingled Hexagonally Packed Unit Cell A.3 Hexagonally Packed Cell with Resin Coating on Center Fiber B MICROVOID DISSOLUTION WITHIN TOWS B.1 Void Dissolution B.2 Results B.2.1 Fiber Volume Fraction B.2.2 Liquid Saturation B.2.3 Processing Temperature and Cure Time B.3 Summary ix

10 C EXPERIMENTAL CHARACTERIZATION OF POROUS AND HYBRID TOWS C.1 Introduction C.2 Material Processing C.3 Mechanical Testing C.3.1 CompressionTesting C.3.2 Tensile Testing C.4 Summary D MATLAB CODE FOR CAPILLARY PRESSUER AND TOW INFUSION MODELS D.1 Capillary Pressure Prediction D.1.1 pcpacked.m: Predicts Capillary Pressure as a Function of Fiber Wetted Length D.1.2 avepc.m: Calculates the Average Capillary Pressure D.2 Impregnation of Two Dimensional Tows by Infinite Resin Volume D.2.1 ParamSimLimGaussVFC.m: Main m-file for ParametricSstudies D.2.2 Subroutine m-files (in the Order in which they are First Called)169 D.3 Impregnation of Three Dimensional Fiber Tows by Multiple Drops D.3.1 simlimsymmetric.m: Main m-file for Parametric Studies D.3.2 Subroutine m-files (in the Order in which they are First Called)202 E PERMISSIONS x

11 LIST OF TABLES Table 2.1: Baseline properties utilized for single fiber and unit cell models Table 3.1: Baseline properties used for capillary pressure prediction Table 3.2: Calculation of capillary pressure and directional body angle for each section...56 Table 4.1: Baseline properties utilized in tow filling model Table 5.1: Baseline properties utilized for drops wicking into tow model Table B.1: Baseline properties utilized for void dissolution model.139 xi

12 LIST OF FIGURES Figure 1.1: Schematic of RTM and VARTM composites manufacturing processes... 4 Figure 1.2: Schematic of composite with thermoplastic resin fibers (red) commingled with glass fibers (purple). A representative unit cell is shown (bottom left), along with variation including symmetry planes (bottom right) to further decrease computational time Figure 1.3: Solution methodology for predicting energy absorbed in a microscale unit cell, given the resin distribution within the unit cell. The initial resin configuration is input into a micrcoscale flow model. Resin flow is predicted and serves as an input into a mechanical model to predict the energy absorbed by the unit cell. [11]... 8 Figure 1.4: Optimization scheme for determining the resin configuration in the unit cell that will result in maximum energy absorption, starting with an initial configuration of resin placement. The initial location of resin placement does not influence the methodology, but can reduce the number of iterations before a satisfactory result is developed [11] Figure 1.5: Schematic of some possible ways to introduce finite amount of resin (red) into a fiber tow (blue), the method will influence the final resin distribution. From left to right: a drop of resin is placed on a fiber tow, resin fibers are commingled with structural fibers, a resin film is placed on a fiber tow, and a tow is surrounded by an infinite volume of resin Figure 1.6: Overview of solution procedure and multi-scale interactions. The transverse capillary pressure (P c ) and permeability (K) are calculated on the microscale. They are then included in mesoscale tow filling models to predict resin flow within a fiber tow using Darcy s law of flow through porous media. Note here the permeability tensor could be anisotropic Figure 2.1: Schematic of the final position of a resin drop on a fiber in the barrel configuration, graphically describing the variables in the analytical model. Recreated from [18] xii

13 Figure 2.2: Phase diagram of drop shape on a single fiber. Drop will adapt barrel or clamshell shape regardless of initial drop configuration for some combinations of fiber radius, drop volume, and contact angle. In the bistable region, the final result deps on the initial resin configuration. Figure used with permission from [19] Figure 2.3: Overview of particle impregnation process and methodology for predicting resin distribution in resulting composite Figure 2.4: Axisymmetric drop spreading on a single-fiber model Figure 2.5: Figure 2.6: Figure 2.7: Figure 2.8: Figure 2.9: Three-dimensional model to predict the wetting of a fiber by a resin drop. The model on the right is the highlighted portion from the figure on the left side, utilizing symmetry planes to reduce computational time. Resin initially begins in a clam shell configuration. Parameters are selected to ensure the resin s in a barrel configuration Schematic of three- and four-fiber unit cell models, with symmetry planes shown. Three and four fiber unit cells represent hexagonal and square fiber packing arrangements respectively Schematic of drop spreading on a flat plate axisymmetric model, which will be utilized to compare numerical and experimental results for a drop spreading on a flat plate Experimental results depicting spreading of the resin drop as a function of time at selected time steps. The results will be used to verify the numerical model Schematic of the experimental setup to record spreading of a resin drop on a flat plate. A thin wire is utilized to deposit the resin onto the glass substrate Figure 2.10: (a) Explanation of slip length and comparison of numerical model and experimental data for a drop of volume (b) mm 3 and (c) mm Figure 2.11: Mesh-refinement study for the axisymmetric (a) and threedimensional (b) drop spreading on single-fiber models Figure 2.12: Comparison of the analytical and numerical solutions for the final shape of the resin-air interface xiii

14 Figure 2.13: Time-depent non-dimensional contact length for selected contact angles Figure 2.14: Evolution of contact length over time for selected fiber radii Figure 2.15 : Evolution of non-dimensional wetting length over time for selected slip lengths, β Figure 2.16: Three-dimensional spreading of a finite volume of resin on a single fiber. Resin is shown in red, air is represented by blue, the interface between the resin and air is shown by the others colors. The fiber is represented by the three horizontal lines, near the center of the model vertically Figure 2.17: Spreading of resin inside triangular-packed (left) and square-packed (right) unit cells at selected time steps. Resin is shown in red, air is represented by blue, the interface between the resin and air is shown by the others colors Figure 2.18: Final fiber-resin contact area as a function of fiber volume fraction Figure 2.19 : Equilibrium fiber-resin contact area for three-fiber triangular and fourfiber square packing arrangements Figure 3.1: Figure 3.2: Schematic of (a) two fiber and (b) five fiber unit cell capillary pressure numerical models. Blue and grey are the resin and air initial positions respectively. In (b), the dotted lines outline the five fiber unit cell Diagram depicting capillary pressure and the application of traction force. The location of the resin is given in red and on the right side blue represents air. The other colors on the right side indicate the different steps of the interface, where the level set function is continuously changing from 1 (resin only) to 0 (air only) Figure 3.3: Wetted length versus time for resin moving between parallel plates for an analytic solution with a contact angle of 30 degrees in Eq.(11). In the numerical model, slip length (β) is varied to find the best fit for that contact angle. (a) broad range (b) fine tuning of (β) to match the analytic solution Figure 3.4: Comparison between numerical and analytical solution for capillary pressure (eq. (2)) as resin moves between two fibers xiv

15 Figure 3.5: Characterizing average capillary pressure solution methodology Figure 3.6: Figure 3.7: Figure 3.8: Figure 3.9: Unit cell model for analytical model, with the expected flow front shape shown in a different color for each section. Resin will start on the left side of the unit cell, and wet the fibers as it moves across the unit cell. The equations utilized to calculate the capillary pressure and directional body angle in each section are given in Table Parameters utilized to describe jump between sections, described in Eqs Comparison between our analytical and numerical solutions for nondimensional capillary pressure as a function of wetted fiber surface area Influence of external pressure gradient on the capillary pressure as resin moves through a unit cell Figure 3.10: Comparing average capillary pressure for presented and previously developed methods in a fiber unit cell Figure 3.11: Average capillary pressure for case where center fiber is moved along the (a) x-axis and (b) y-axis Figure 3.12 : Normalized capillary pressure as a function of the x-offset of the center fiber for selected fiber volume fractions, calculated with the analytical model Figure 3.13: Unit cells with hybrid sized fibers (a) model setup and (b) influence on average capillary pressure Figure 4.1: Schematic of void formation in a dual scale fibrous preform Figure 4.2: Tow filling model schematic, shown at an intermediate time step in which resin is still entering the tow and the macroscale flow front has yet to reach the outlet. Red and grey represent the saturated portion of tow and preform respectively, with blue representing the unsaturated regions of each Figure 4.3: Multiple voids (black) distributed within a fiber tow, with a thickness of ~0.2 mm [Adapted from [75]] Figure 4.4: Methodology for including capillary and void pressure during resin flow into fiber tows xv

16 Figure 4.5: (a) Comparison of analytical and MATLAB + LIMS solution for a cylindrical tow, not considering void pressure and (b) mesh refinement study for elliptic tow including void pressure Figure 4.6: Sample similarity matrix. A 1 indicates that the current void and previous void have elements in common, whereas a 0 indicates that they do not Figure 4.7: Void distribution (blue) within tows with (a) Gaussian (baseline), (b) constant fiber volume fraction distribution, (c) a random fiber distribution, and (d) a linear fiber distribution Figure 4.8: Influence of the type of fiber packing arrangement on tow filling Figure 4.9: Influence of air evacuation time before introducing the resin on the resulting void content in the tow Figure 4.10: Non-dimensional void area as a function of tow location from the inlet, with and without the inclusion of capillary pressure Figure 4.11: Effect of wetting properties on the void volume fraction Figure 4.12: Effect of fiber radius on microvoid content in a fiber tow. Fiber tows with two different fiber radii show that smaller diameter fibers in the center as compared to on the outside result in lower void content Figure 5.1: Diagram of process for depositing drops of resin onto a fiber tow Figure 5.2: Boundary between two different resin types for (a) smooth and (b) interlocking interfaces Figure 5.3: Description of unit cell for resin spreading into fiber tow at initial time step Figure 5.4: Geometric and boundary conditions, shown at the time the second drop is introduced into the system Figure 5.5: Description of parameters necessary for calculating the resin's radius of curvature Figure 5.6: Schematic of final resin distribution within tow. Resin flow front position is given in white and described by y=f(z). The standard deviation of this function, std y, will be utilized in quantifying roughness xvi

17 Figure 5.7: Mesh refinement study for the flow front shape on the yz-plane Figure 5.8: Impregnation of a unit cell which represent half of a tow due to symmetry with two drops of resin (red) at (a) the initial time step, (b) after about half of the first drop was inside the tow, (c) right after introducing of the second drop, (d) while both drops are still spreading into the tow, and (e) at the of the simulation after both drops have completely impregnated into the tow Figure 5.9: Mapping of the LIMS results on the yz-plane into a finite number of points (y,z). Purple represents the resin locations, red represents porous regions, and intermediate colors are utilized to show partially saturated regions. In the bottom picture, the mapping to calculate roughness is shown in white Figure 5.10: Comparison of resin distribution for (a) 5.00 mm between drops and (b) 2.50 mm between drops (purple represents resin, red represents porous areas, other colors indicate partially filled nodes) Figure 5.11: Roughness as a function of resin to tow porosity ratio for selected drop spacings Figure 5.12: Fiber volume fraction distribution and flow front shape for (a) constant fiber volume fraction distribution and (b) linear fiber volume fraction distribution Figure 5.13: Influence of charge type and film aspect ratio on final resin distribution within a tow (shown are the top view, bottom view, and two cross section views of the resin distribution) Figure A. 1: Cross section view of resin wetting square packed fibers Figure A.2: Comparison between the numerical and analytical models for resin wetting a square packed unit cell Figure A.3: Numerical results for commingled resin fiber wetting hexagonally packed glass fibers Figure A. 4: Numerical results for hexagonally packed glass fibers, with the center fiber intially having a thermoplastic coating on it Figure B.1: Saturation of a tow with (a) 0.40 fiber volume fraction and (b) 0.70 fiber volume fraction xvii

18 Figure B.2: Influence of the type of fiber volume fraction on void size within a fiber tow, shown at the of the compression phase and after the part has cured Figure B.3: Influence of relative saturation of the resin on void size Figure B.4: Effect of cure time and processing temperature on void content Figure C.1: Cross-section schematic of fiber tow. Tow will first be partially infused with SC-15 epoxy. It will then either be tested as is, or it will be insfused with polyurethane resin before being tested Figure C.2: Tows with epoxy (dyed orange here) painted on them. They are pulled taught across the frame to minimize variations in fiber volume fraction within and between different tows Figure C.3: Schematic of compression test setup with the tow between two platens. The platens cover the entire top and bottom surface of the tow with room to spare in each direction Figure C.4: Compression testing - examples of measured data (top left), the system compliance curve (top right), and the material load-displacement curve after being corrected for machine compliance (bottom) Figure C.5: Schematic of load-unload cycles, with the peak load (P), the load at which the load in the current cycle passes the peak load from the previous cycle (t), and the fully unloaded state (v) shown for each cycle Figure C.6: Compressive energy dissipated per unit mass by hybrid tows loaded up to kn and back down to 0 kn. Percentage of the tow s porous area filled by epoxy is on the horizontal axis, with the rest of the porous area being impregnated with polyurethane Figure C.7: Average of energy absorbed per unit mass as a function of applied pressure for (a) epoxy/polyurethane hybrid tows and (b) epoxy/porous tows Figure C.8: (a) Tensile test setup with representative failed specimens for (b) hybrid tow and (c) epoxy/porous tow. Hybrid tows have more brittle failure and the dry portion of the porous tows experiences significant fraying xviii

19 Figure C.9: Energy absorbed per unit mass for all samples with an acceptable failure mechanism (failure occurred within the gage length and not in the tabs) xix

20 ABSTRACT Composite materials have been increasingly utilized for ballistic armor, in which they are subjected to high strain rates. The porosity of the composite as well as fiber-resin interface plays a key role in dissipating energy when subjected to high strain rates. This dissertation develops a suite of modelling tools capable of predicting the resin configuration within fiber tows across multiple length scales. Fiber tows consists of thousands of individual aligned fibers bundled together. Fiber tows are of the order of millimeters whereas the fibers are of the order of microns. On the microscale, utilized model is developed to predict the wetting dynamics of resin on an individual fiber and within unit cells containing various arrangements of fibers. The model predicts the spreading of the resin between the fibers within the unit cell and can calculate the contact area between the fibers and the resin. This model is used to characterize the average capillary pressure of resin for various packing arrangements and the influence of imperfections and different surface treatments on the capillary pressure. A novel mesoscale model that accounts for capillary pressure characterized from our microscale model to describe resin impregnation into fiber tows is developed. This model can account for non-uniformly spaced fibers within the tow. The mesoscale model allows for a stochastic fiber distribution which predicts xx

21 formation of many microvoids within the tow as observed in manufacturing as opposed to the single void located in the center of the tow for traditional models which assume spatially uniform capillary pressure. The influence of air evacuation time, hybrid fiber tows, and capillary forces is explored with this model. The mesoscale model is exted to predict the wicking of finite resin volumes into a fiber tow, which allows for the design of composites with controlled resin distributions within individual fiber tows. The model should prove useful to optimize and tailor composite properties on a tow level for desired applications. xxi

22 Chapter 1 INTRODUCTION 1.1 Overview of Composites Manufacturing Composite materials, known for their extremely high strength to weight ratio, are comprised of fibers embedded in a matrix material (resin). The load transfer between the matrix material and the fibers is what causes the synergistic effect on composite properties. One of the advantages of using composites is that one can design parts with location depent properties based on the fiber and matrix materials chosen and the orientation of the fibers. One of the major focusses in composites manufacturing is ensuring that the resin is properly distributed within the part. Any porous areas within a preform not saturated with resin are known as voids. Voids are caused by entrapment of gases due to the resin flow front dynamics during the manufacturing process. The prediction and control of voids is a very active area in composites research [1 4]. Most of the previous resin flow dynamics work has been focused on minimizing voids because porous areas within a composite are seen as initiation of failure sites. Void removal in composites processing is a complicated problem because the preform architecture behaves like a dual scale porous medium. When the applied pressure driving the resin flow is very high, resin moves fast and 1

23 fills the regions between tows of the preform before it saturates the tows, creating voids within the tows. If there was no applied pressure, the resin flow rate would only be driven by capillary forces pulling resin into the tows, which would result in a large number of voids between tows. There has been work done in determining an optimal range of flow rates to minimize void content [1 3]. The capillary number, a ratio of viscous forces to capillary forces including the effect of contact angle, θ, is considered to be crucial [1]: Ca = μv γcosθ (1.1) Here, μ and γ are the viscosity and surface tension of the resin, and v is the average flow velocity. Capillary number is found to play a crucial role in weather the fiber tows will saturate before the regions in between the tows or visa versa which can allow one to address the problem of void formation [2]. Micro-voids dominate the void volume percentage in systems with capillary numbers above a critical value and meso-scale voids dominate systems with low capillary numbers [5]. The critical value is system depent because it does not include important parameters such as fiber packing density. For example, a part whose tows have a high fiber volume fraction would have a lower critical capillary number than one containing tows with a lower fiber volume fraction because the tow permeability would be much lower (the macroscale flow front would need to be moving slower to avoid microscale voids). 2

24 Composites are used in ballistic armor, which subjects the material to high strain rates. Composites for high strain rate applications are usually made using preimpregnated fibers (prepregs) with low resin content concentrated between fabric layers, or by liquid composite molding (LCM) in which resin impregnates the tows in liquid form [6,7]. Prepregs are fiber sheets with resin already infused between the fibers. During processing, the prepreg is placed in the desired shape and orientation and subjected to vacuum while heat is applied. The application of heat causes the resin to become less viscous, flow, and eventually cure, while the vacuum, sometimes with the help of positive pressure being applied to the part, causes the resin to flow and forces voids out of the preform [8]. In LCM, the fibers are placed as a preform that is dry and the resin is then injected or infused into the tow using techniques such as resin transfer molding (RTM) and vacuum assisted resin transfer molding (VARTM) respectively [9]. RTM utilizes a rigid mold on both sides of the preform to maintain its shape while resin is injected into the part. VARTM uses a vacuum to pull the resin through the preform under the pressure gradient of one atmosphere. Overviews of these processes are shown in Figure 1.1. Composites are sometimes manufactured using thermoplastic particle that are inter dispersed between fibers and fiber tows. When the preform is heated the thermoplastic powder become a liquid drop and infuses the space in between the fibers and the fiber tows of the fiber preform. 3

25 Figure 1.1: Schematic of RTM and VARTM composites manufacturing processes 4

26 1.2 Background and Motivation Composite materials utilized in high strain rate applications will typically need to have both structural integrity and energy absorptive capabilities. In continuous fiber composites, energy is dissipated through friction between neighboring fibers, the collapsing of microvoids, fiber breakage, matrix deformation and cracking, delamination, fiber pullout, and fiber-matrix debonding [10]. These mechanisms are depent on the constituent and fiber-resin interfacial properties. There are two possible avenues we explore for increasing energy absorption while maintaining structural integrity: hybrid resin tows and tows designed to have a porous resin structure. Hybrid composites are composite materials comprised of multiple types of fiber or resin materials. Typically, glass fibers are able to absorb more energy than carbon fibers, but at the cost of a decrease in stiffness and strength. It has been shown that commingling glass and carbon fibers within a composite can result in a stiffness and strength greater than glass fibers but energy absorption higher than if only carbon or glass fibers were utilized [11]. The reason for this is because as the brittle carbon fibers break (they contribute to energy absorption), the ductile glass fibers are still able to carry a load, resulting in a higher strain to failure than just the carbon alone but larger stiffness than glass alone. Controlling the resin distribution within tows will allow for the creation of hybrid resin composites (with a high stiffness resin and a high strain to failure resin) which can be optimized for energy absorption in extreme dynamic environments. These hybrid resin composites are traditionally manufactured 5

27 using co-injection resin transfer molding, in which both resins are injected into a fibrous preform separated by a barrier simultaneously [12]. For example, a resulting hybrid composite may contain a specified number of glass fabric plies impregnated with epoxy for structural integrity and polyurethane for energy absorption [13]. It was shown that hybrid resin composites with more polyurethane infused plies dissipated more energy than plies only impregnated with epoxy, but at the cost of structural integrity [13]. If one were to control where the high strain to failure resin was located within tows, it would allow for an additional level of optimization for composites in high strain rate applications. In small percentages, voids, up to a certain concentration, have been shown to increase the energy absorption capabilities of composites under high loading rates [14]. The ability to control the location and size of microscale voids in composites presents a new frontier for one to design parts that will absorb maximum amount of energy under high loading rates. Although Foley was able to show experimentally that there appears to be an optimal void concentration for energy absorption in high strain rate applications [14], understanding the influence of processing parameters on the location and size of the voids within a fiber tow will allow for composites to be designed with pre-determined porosity distribution within the composite. This additional level of design can lead to creation of composites with even more tailored properties. The optimization of the microstructural topology in composites, with the goal of increasing energy absorption, has been studied and is important in understanding the role of porosity and resin distribution in energy absorption [15]. 6

28 Figure 1.2: Schematic of composite with thermoplastic resin fibers (red) commingled with glass fibers (purple). A representative unit cell is shown (bottom left), along with variation including symmetry planes (bottom right) to further decrease computational time. The microstructures are representative of thermoplastic polymer fibers commingled with glass fibers, shown in Figure 1.2. When heated the polymer fibers will melt and impregnate the region in between the fibers. Coupling the resin distribution with mechanical models one can predict the influence of processing and geometric parameters on energy absorption as outlined in Figure 1.3 [15]. A sample of the optimization scheme is shown in Figure 1.4, which found there to be a significant increase in energy absorbed per unit mass for the porous unit cells with designed mircostructures when compared with the fully infused baseline case [15]. 7

29 Both experimental [14] and numerical [15] results show that porosity can be manipulated to increase energy absorption of composites in high strain rate applications. This holds true both on a meso and micro scale. Previous models predicting resin distribution between fibers on the micro scale and within tows on the meso scale are either insufficient or non-existent. Understanding the correlation Figure 1.3: Solution methodology for predicting energy absorbed in a microscale unit cell, given the resin distribution within the unit cell. The initial resin configuration is input into a micrcoscale flow model. Resin flow is predicted and serves as an input into a mechanical model to predict the energy absorbed by the unit cell. [11] between geometric/processing parameters and resin distribution will allow the development of materials with architectures designed for optimal performance under high strain rates, similar to the aforementioned microscale unit cell. This dissertation seeks to develop multi-scale modelling techniques capable of predicting the 8

30 distribution of resin between fibers and within fiber tows. The results of this work can be utilized in conjunction with a mechanical model to open up a novel avenue to design innovative composite materials. It is written such that each chapter can be read as a stand-alone body of work or as part of the overall story, as such some equations and references will appear in multiple chapters. 9

31 Figure 1.4: Optimization scheme for determining the resin configuration in the unit cell that will result in maximum energy absorption, starting with an initial configuration of resin placement. The initial location of resin placement does not influence the methodology, but can reduce the number of iterations before a satisfactory result is developed [11] 10

32 1.3 Objectives and Dissertation Outline The primary objective of this research is to understand the flow of finite volume of resin around fibers and inside fiber tows as a function of the fiber and resin properties as well as the interaction between the fiber surface and resin. Potential ways by which the desired amount of resin can be introduced is schematically shown in Figure 1.5. The final distribution of the resin deps not only on the volume of resin and wetting properties, but also on how the resin is introduced. Predicting the correlation between processing parameters and resin flow will result in the ability to manipulate resin flow behavior inside the fiber tow to create desired void distributions within a fiber tow. This will allow the user to create composites that absorb the Figure 1.5: Schematic of some possible ways to introduce finite amount of resin (red) into a fiber tow (blue), the method will influence the final resin distribution. From left to right: a drop of resin is placed on a fiber tow, resin fibers are commingled with structural fibers, a resin film is placed on a fiber tow, and a tow is surrounded by an infinite volume of resin maximum amount of energy without sacrificing mechanical properties. Strategic 11

33 infusion of a tow with a finite volume of resin would result in a porous structure; this structure could be left porous or infused with a secondary resin, which could be optimized for energy absorption under high loading rates. This dissertation focusses on resin flow on the micro (order of μm) and meso (order of mm) scale. It will start with microscale models to predict resin flow between individual fibers and characterizing average capillary pressure within unit cells containing fibers. It will then bridge scales to include the microscopic average capillary pressure based on fiber arrangement and contact angle of the resin with the fiber surface in mesoscale fiber tow filling simulations. The second chapter will begin with a microscale model, which will explore the flow of resin particles on individual fibers and within unit cells containing fibers packed in common fiber packing arrangements. Predicting three-dimensional resin distribution between fibers is unique to this work. The third chapter will modify the microscale model in an effort to calculate the capillary pressure exerted on resin moving through unit cells containing hexagonally packed fibers. The influence of processing parameters and imperfections in packing arrangements will be explored. The fourth chapter will incorporate the microscale characterized average capillary pressure into a mesoscale model to predict the impregnation of fiber tows surrounded by an infinite volume of resin and is summarized in Figure 1.6. This model will be the first to introduce stochastic fiber distributions into a fiber tow filling 12

34 model. It allows for one to manipulate geometric and processing parameters to obtain a desired size and distribution of microscopic voids within a tow. K Figure 1.6: Overview of solution procedure and multi-scale interactions. The transverse capillary pressure (P c ) and permeability (K) are calculated on the microscale. They are then included in mesoscale tow filling models to predict resin flow within a fiber tow using Darcy s law of flow through porous media. Note here the permeability tensor could be anisotropic The fifth chapter takes the approach developed in chapter four and exts it to three dimensions to study the wicking of resin drops into a fiber tow. This will result in a porous tow that can be infused with a second resin to create a hybrid resin composite, with both resin types throughout the entire part. The final chapter will summarize the important findings and conclusions and list the original contributions of this work. 13

35 Chapter 2 MICROSCALE FIBER WETTING BY A FINITE RESIN VOLUME 2.1 Background The partial wetting of cylindrical surfaces by a finite volume of resin is an important phenomenon in many industrial applications such as composites manufacturing, MEMS, and textile engineering. A constitutive equation governing the partial wetting of a finite volume of liquid on a flat plate has been formulated and reported [16]. The equilibrium shape of resin on single fibers has also been studied in depth [17 20]. Carroll was the first to develop an analytical solution for the equilibrium shape of a resin drop on a single fiber [17]. A drop at rest on a solid fiber surface will either conform to a barrel geometry, where the drop wraps around the fiber, or a clamshell geometry in which the fiber rests on the fiber s surface without wrapping around it. A phase diagram predicting which of these configurations a particular drop will adopt has been constructed, and is presented in Figure 2.2 [20]. Wu and Dzenis later developed an analytical solution to this problem using an energy approach [19]. Both of these solutions assume an axisymmetric shape. The equations necessary to determine the maximum height of the drop on the fiber and the length of contact between the fiber and resin are given in [17,19] and reproduced below: y 2 = y 2 0 (1 k 2 sin 2 ϕ) (2.1) x = ±[λrf(k, ϕ) + y 0 E(k, ϕ)] (2.2) 14

36 Figure 2.2: Phase diagram of drop shape on a single fiber. Drop will adapt barrel or clamshell shape regardless of initial drop configuration for some combinations of fiber radius, drop volume, and contact angle. In the bistable region, the final result deps on the initial resin configuration. Figure used with permission from [20]. Figure 2.1: Schematic of the final position of a resin drop on a fiber in the barrel configuration, graphically describing the variables in the analytical model. Recreated from [19]. Here x is the location on the axis of the fiber measured from the center of the 15

37 drop, and y is the height of the drop, measured from the axis of the fiber as seen in Figure 2.1. y 0 is the maximum height of the drop, measured from the axis of the fiber and r is the fiber radius. E(k,φ) and F(k,φ) are the Legre s elliptical functions of the second and first kind respectively. Here and k are defined as follows: λ = y 0cosθ r y 0 rcosθ (2.3) k 2 = 1 λ 2 ( r 2 ) y 0 (2.4) Here, θ is the static contact angle between the fiber and resin. The final wetted length, L, which is also defined in 2.4, can be calculated using the known volume V once y 0 is solved for with Eqs. (2.3) and (2.4) evaluated at y = y 0 along with: L = 2[λrF(k, ϕ 0 ) + y 0 E(k, ϕ 0 )] (2.5) y 0 y 2 (y 2 + λry 0 ) V = 2π (y 2 0 y 2 )(y 2 λ 2 r 2 ) dy πr 2 L r (2.6) φ 0 is found through setting y = 0 in Eq. (2.1), There have also been other investigations with resin spreading within multiple fibers, for example final resin configuration between two parallel fibers has been studied with relation to static contact angle, filament spacing, resin volume, and fiber diameter [21]. The axial wetting of a single fiber from a reservoir of resin has been experimentally examined and constitutive equations have been developed to describe this phenomenon [22]. The dynamics of a finite volume of resin spreading on a single fiber has yet to be explored and is a novel element of this work. 16

38 Figure 2.3: Overview of particle impregnation process and methodology for predicting resin distribution in resulting composite A numerical model is formulated to study the movement and spreading of a finite volume of resin on any planar or curvilinear surface. The method and accuracy is verified by comparing the model results with experiments conducted with a drop of resin spreading on a flat plate. The method is then used to describe the wetting dynamics of a finite drop of resin on a single fiber. The model is further exted to investigate the flow of finite volume resin within multi-fiber unit cells representing 17

39 square and hexagonal fiber packing arrangements. These models can be utilized to predict the resin distribution in fiber tows impregnated through particle impregnation as outlined in Figure 2.3. This investigation should prove useful in tailoring the interface properties between fibers and resin as a function of the resin and fiber surface properties and the fiber arrangement. 2.2 Methods Model Setup The numerical models were developed using COMSOL Multiphysics and the Microfluidics Module to investigate the dynamics of wetting over a single fiber and within unit cells of multiple fibers with a finite volume of resin and are presented below [23]. A model was also constructed of a drop spreading on a flat plate with the goal of experimentally validating the solution method Axisymmetric Single Fiber Model Figure 2.4 shows the axisymmetric single fiber model. In this model, a spherical drop of resin is initially enveloping the fiber. This simplifies the resin movement to be along the fiber surface in the axial direction. An axis of symmetry is utilized to increase computational efficiency. The axis of symmetry is the center of the fiber with a full slip condition (symmetry condition), which sets the derivative of the tangential velocity equal to zero along the axis. The no slip boundary condition is applied along the walls shown in Figure 2.4 where the velocity is set to zero. The results would not change if the no slip walls were given a slip condition because the walls only come into contact with air. The air density is orders of magnitude lower than the resin density, meaning small changes in the air velocity profile will have no 18

40 effect on the resin flow. The pressure is set equal to zero, the reference pressure, at a single point to ensure that the pressure solution is unique [24]. This is needed because Figure 2.4: Axisymmetric drop spreading on a single-fiber model the Navier-Stokes equations only solve for the gradient of pressure. The wetting wall is the surface of the fiber along which the resin moves and employs a slip length and the final static contact angle to drive the wetting and spreading of the resin, both of which will be discussed in a later section. The fiber diameter was chosen to be 9 micrometers. These baseline values, shown in Table 2.1, were selected based on resin and fiber systems used in composites processing Three Dimensional Single Fiber Model The three dimensional representation of a resin drop spreading on a single fiber is depicted in Figure 2.5. The resin drop diameter to fiber radius ratio in this model is intentionally large. This is because we desire the equilibrium position of the resin to be in the barrel shape. A fiber radius of 3 micrometers, resin volume of 2280 μm 3, and contact angle of 15 degrees were selected to ensure the final shape is a barrel 19

41 guided by the studies performed by Eral et al [20]. Using the baseline radius would Figure 2.5: Three-dimensional model to predict the wetting of a fiber by a resin drop. The model on the right is the highlighted portion from the figure on the left side, utilizing symmetry planes to reduce computational time. Resin initially begins in a clam shell configuration. Parameters are selected to ensure the resin s in a barrel configuration. cause the drop to become much to large to be realistic for a composite system. The other important properties are the same as the baseline values described in Table 2.1. Table 2.1: Baseline properties utilized for single fiber and unit cell models Property Value Resin Density (g/cm 3 ) 1.17 Resin Viscosity (Pa s) 0.95 Static Contact Angle (degrees) Resin Surface Tension (0.07 J/m 2 ) 0.07 Slip Length [β] (μm) 0.10 Fiber radius (μm) Unit Cell with Square and Triangular Packing Arrangements The square and triangular fiber packing arrangement unit cells are shown in Figure 2.6. Flow in each unit cell is simplified through use of symmetry planes to 20

42 increase computational efficiency. The four fiber unit cell has three planes of symmetry due to the assumption that gravity is negligible, which will be discussed later. The fiber surfaces are wetting walls and walls other than fiber surfaces and symmetry planes are no slip walls. As mentioned previously, the choice of slip or no slip for these walls has no significant influence on resin flow, so either would yield similar results. Both models also include a point where the pressure is set equal to zero. The pressure is set equal to zero at a reference location, to ensure that the pressure solution is unique. The fiber radius was 4.5 micrometers. The baseline values for the resin and interaction parameters can be found in Table 2.1. Figure 2.6: Schematic of three- and four-fiber unit cell models, with symmetry planes shown. Three and four fiber unit cells represent hexagonal and square fiber packing arrangements respectively Resin Spreading on a Flat Plate Figure 2.5 shows the schematic of the initial drop along with the boundary conditions modeled using COMSOL to predict the spreading of a drop of glycerin. The spreading dynamics were then compared with an experiment we conducted and recorded. The glycerin properties were found using traditional characterization 21

43 techniques, described and reported in the experimental methods section. The properties for the air phase were taken from the COMSOL material library. The two dimensional model takes advantage of the axisymmetric property of the process being modeled. Figure 2.7: Schematic of drop spreading on a flat plate axisymmetric model, which will be utilized to compare numerical and experimental results for a drop spreading on a flat plate Assumptions The Reynolds number in this problem is on the order of 10-5, thus it can be assumed that the inertial forces are negligible relative to the viscous forces and the Navier-Stokes equations can be reduced to Stokes flow. The ratio of gravitational to capillary forces, represented by the bond number, is on the order of 10-6, making it acceptable to neglect gravity. Determination of Reynolds and bond numbers are found 22

44 in Eqs. 2.7 and 2.8. For the axisymmetric model, due to the geometry being symmetric Re = ρvl μ = (1170 Bo = ρgl2 γ = (1170 kg m 3) ( m (0.95 s ) ( m) = kg m s 2) kg m 3) (9.81 m s 2) ( m) 2 (2.7) 0.07 kg s 2 = (2.8) about the fiber axis and our assumption of no gravity, the flow is considered axisymmetric about the axis of the fiber. It is assumed that the resin used does not cure during the wetting process, allowing us to maintain a constant viscosity value during the flow. It is also assumed that the fibers are rigid and do not move as the resin flows Governing Equations The governing equations (Eqs. 2.7 and 2.8) in the model are the Stokes and continuity equations. The interface between the two fluids is tracked using the level set method [25]. The level set method creates an interface with a finite thickness, described by the level set variable (φ) which continuously changes from 0 to 1 across the interface [25]. These equations (Eq. (2.9)), modified to account for the stated assumptions, are given by [26]: ρ u t = P + μ 2 u + F st (2.9) u = 0 (2.10) ϕ t + u ϕ = γ (ε ϕ ϕ(1 ϕ) ϕ ϕ ) (2.11) 23

45 Where u is the velocity vector, the subscript denotes the partial derivative with respect to that variable, μ is the viscosity, P is the pressure, F st is the force due to surface tension, γ is the reinitialization parameter for the interface, ε is the interface thickness, and ϕ is the level set variable. To minimize computational cost, the interface thickness is set to one half of the largest element length [26]. The density and viscosity within the interface between the resin and air are found using rule of mixtures [26]: ρ = ρ Resin + (ρ Air ρ Resin )ϕ (2.12) μ = μ Resin + (μ Air μ Resin )ϕ (2.13) Fiber and Resin Parameters The properties of the resin and the fiber-resin interactions play an important role in the wetting of the fibers by the resin. The viscosity of the resin has a large impact on the rate of wetting, but not a significant effect on the final shape of the drop. The bond number is the ratio of surface forces to body forces, providing a good indication if the resin flow is driven by surface forces or gravity. This study focusses on flows with low bond numbers. The contact angle between the fiber and resin, largely impacted by the surface tension of the resin, represents the principle force driving wetting at the micro scale. The fiber diameter will be an important geometrical parameter when investigating drops spreading on the fiber surfaces. When the model is exted to include multiple fibers, the fiber spacing and packing arrangement will influence the wetting dynamics. 24

46 Static Contact Angle between Fiber and Resin Wetting describes the spreading of a liquid on a solid substrate [27]. The wettability of a substrate by a liquid is quantified by the static contact angle, a force balance at the line of contact between the fiber surface (solid (s)), resin (liquid (l)), and air (vapor (v)) and is given by Young s equation [27]: cosθ = γ sv γ sl γ lv (2.14) In Young s equation, γ ij represents the surface energy at the i-j interface. As shown in equation (2.12), the final static contact angle takes into account both the resin surface tension and the difference in interfacial energies of the solid-vapor and solid-liquid interfaces. The solid-vapor and solid-liquid surface energies can be manipulated by modifying the fiber sizing, which is a coating that is applied to the fiber surface. The final static contact angle of the resin on the fiber surface has been shown to have a direct relationship with the interfacial shear strength of the resulting composite [28] Resin Viscosity The viscosity of the resin does not affect the final position of the resin on the fibers since it is assumed to be constant. As Stokes solution is linear, the time it takes to wet the fiber surface will be directly proportional to the viscosity of the resin Slip Length There are stress and velocity singularities at the three-phase contact line when solving the Stokes equations with a no-slip condition at the solid surface [29]. A way to handle this boundary condition is to move the no-slip condition to a plane located a distance β (slip length) below the solid surface and assume simple shear flow in the 25

47 region between the wall and the no slip plane [30]. The frictional force at the wall is scaled with the slip length [26]. Not unlike viscosity, changing the slip length will influence the wetting rate, but not the final distribution and configuration of the resin on the fiber surface. The slip length is a parameter that models the interactions at the fiber-resin interface Fiber Volume Fraction and Packing The fiber volume fraction is an important property of composites when considering their strength and stiffness. The fiber volume fraction will be controlled in this study through manipulating the distance between fibers, measured from axis to axis. Two common packing arrangements for fibers are square and hexagonal packing. The hexagonally packed fibers are modeled with a unit cell in which lines connecting the center of each fiber would form an equilateral triangle. The relationship between the fiber volume fraction, (v f ), fiber radius (r), and distance between fiber axes, (d), will be: v f = π ( r d ) 2 (2.15) for fibers in a square packing arrangement. arrangement, the relationship will be: v f = π (r d ) 2 For fibers in a triangular packing (2.16) Experimental Setup and Procedure The experimental setup, shown in Figure 2.9, includes a substrate on which a drop of liquid can be deposited by using a thin wire and a camera to capture time stamped images of the process. 26

48 This experiment was performed by depositing glycerin drop on a flat glass substrate. Glycerin was used as the test liquid because it has similar properties to the epoxy ultimately being used in the drop spreading experiment. The surface tension Figure 2.9: Schematic of the experimental setup to record spreading of a resin drop on a flat plate. A thin wire is utilized to deposit the resin onto the glass substrate. measured with a dynamic contact analyzer to be 0.07 N/m. A Brookfield DV-E viscometer measured the viscosity of the glycerin to be Pa s. The density of the Figure 2.8: Experimental results depicting spreading of the resin drop as a function of time at selected time steps. The results will be used to verify the numerical model. 27

49 glycerin, measured using a precision scale and flask, was g/cm 3. Sample images of the drop spreading are shown in Figure 2.8. The static contact angle between the glass and glycerin, measured using image analysis software on the drop in equilibrium, is 28.5 degrees. 2.3 Results and Discussion First experimental and analytical validation of the model used will be provided in the next section before parametric studies are conducted to investigate the dynamics of resin spreading [23]. The capillary number, described in Eq. 1.1, is calculated to be 6.9x10-4 for the baseline case with resin properties given in Table 2.1 and a characteristic velocity of 44 μm/s, which clearly indicates that the capillary forces dominate in the process Validation of the Numerical Model The physics involved in the preceding models is multiphase fluid flow with a high surface to gravitational force ratio. A model of a drop spreading on a flat plate was developed to experimentally verify that the governing equations could predict an acceptable numerical solution to a multiphase wetting dynamics dominated by surface forces Comparison between Experimental and Numerical Results Two experimental trials were conducted of a drop spreading on a flat plate. The first experiment was used to determine the value of which defines the resin fiber surface characteristics. A value of zero would correspond to the case where the liquid will not wet the substrate and an infinite value would describe the scenario where the liquid would reach its final configuration instantaneously. The β value in 28

50 Figure 2.10: (a) Explanation of slip length and comparison of numerical model and experimental data for a drop of volume (b) mm 3 and (c) mm 3 29

51 real systems will fall between the preceding cases and can be determined experimentally by comparing the numerical and experimental solutions using a range of β values. As β is increased or decreased, the wetting rate in the numerical solution will become higher or lower. The value that describes the liquid-substrate system is found by adjusting the value until the dimensionless length, defined as the drop length at time t divided by the final length of the drop, matches the experimental results. This experiment used the resin volume of mm 3. The value for slip length from this case was determined to be 0.25 μm. The experimental and COMSOL results for this value are shown in Figure 2.10 along with an inset that describes distance β (slip length) below the solid surface and assumes simple shear flow in the region between the wall and the no slip plane. Having determined the value, the next experiment was conducted with a drop volume of mm 3. The dimensionless length of the COMSOL simulation is compared to the experimental dimensionless length in Figure 2.10 using the characterized value. Comparing the results verify the numerical model used to describe the dynamics of resin spreading on a surface for a large surface force to body force ratio Mesh Refinement Study A mesh refinement study was performed to ensure that the numerical results converged as the number of elements in the mesh was increased. Wetting length, as shown in Figure 2.4, will be used as the characteristic output parameter studied for the axisymmetric model of resin wetting a single fiber. The area of the fiber-resin interface will be used as the characteristic output parameter for the three dimensional model of resin spreading on a single fiber. Comparing the four solutions for each, 30

52 depicted in Figure 2.11, confirms that the numerical output converges and the lowest Figure 2.11: Mesh-refinement study for the axisymmetric (a) and threedimensional (b) drop spreading on single-fiber models mesh density used provides an acceptable result Comparison of Final Drop Shape with an Analytical Solution The equilibrium solution for the axisymmetric model of resin spreading on a fiber was compared to the resin configuration predicted by Carroll [17]. In the numerical solution, resin volume, fiber diameter, and final contact angle are all 31

53 known. These values were substituted into Eqs. (2.3-5) and then substituted into Eq.(2.2-6) to create an equation with one unknown, allowing one to solve for y 0. Once y 0 is known, it can be substituted back into Eqs. (2.1-4) to develop a parametric equation for x and y. φ was varied for values corresponding to y > r. The resin-air interface shape at the yx-plane is solved by using this method and is compared to the numerical solution in Figure The closeness of the two solutions provides further validation of the numerical model. Figure 2.12: Comparison of the analytical and numerical solutions for the final shape of the resin-air interface Parametric Study of Axi-symmetric Model The wetting physics in the axisymmetric model was influenced by the static contact angle, slip length, fiber and resin geometry, and viscosity. 32

54 Static Contact Angle between Fiber and Resin The evolution of non-dimensional wetting length over time is shown in Figure Figure 2.13: Time-depent non-dimensional contact length for selected contact angles 10 for a range of static contact angles. The baseline values, given in Table 2.1, are used for all other properties. Here, the non-dimensional wetting length is normalized by the initial wetting length. Fiber-resin combinations with high contact angles reach their equilibrium position faster because the resin does not travel very far. As the wetting properties are increased, evidenced by a lower contact angle, the amount of the fiber surface covered by the resin increases. The trs found in the contact angle study can be translated to changes in the surface energy of the solid-resin, solid-air, or resin-air through Equation Fiber Radius The non-dimensional wetting length, as a function of time for various fiber radii is depicted in Figure The baseline values are used for the fiber and resin 33

55 properties listed in Table 2.1. All of the initial wetting lengths were slightly different due to the radius of the resin drop changing slightly to keep the resin volume constant Figure 2.14: Evolution of contact length over time for selected fiber radii for a varying fiber radius. At low times, the resin moves at a similar rate for all trials. With increasing time, the resin reaches equilibrium on the smaller fibers first because it has to travel less and the capillary forces are stronger. The final wetting length increases as the fiber radius was increased due to the resin trying to minimize its surface area Slip Length With the exception of the slip length, all properties were equal to their baseline values for this study. The slip length, which characterizes the resin fiber interface property, did not impact the final wetting length of the resin on the fiber. It did impact the wetting rate as shown in Figure As one would expect, the system reached 34

56 equilibrium at a faster rate when the slip length was increased due to the increase in slip velocity at the fiber surface. The slip length found in the experimental validation of the physics would fall close to the middle of the values studied. A methodology for determining the slip length for a fiber-resin combination is detailed in the following chapter. Figure 2.15 : Evolution of non-dimensional wetting length over time for selected slip lengths, β Three Dimensional Single Fiber Model In the three dimensional model, resin wets the top of the fiber faster than it wets around the circumference of the fiber, depicted in Figure The curvature of the surface slows the rate of wetting on the outside of a concave surface because more resin surface area is created per unit length traveled. The wetted length on the top of the fiber decreases slightly after the resin begins to spread along the bottom of the 35

57 fiber, the time of which is indicated by the plateau of the circumferential spreading curve. 36

58 Figure 2.16: Three-dimensional spreading of a finite volume of resin on a single fiber. Resin is shown in red, air is represented by blue, the interface between the resin and air is shown by the others colors. The fiber is represented by the three horizontal lines, near the center of the model vertically. 37

59 Square and Hexagonal packing Fiber Unit Cells Figure 2.17: Spreading of resin inside triangular-packed (left) and squarepacked (right) unit cells at selected time steps. Resin is shown in red, air is represented by blue, the interface between the resin and air is shown by the others colors. 38

60 The spreading of a finite volume of resin within a three (hexagonal packing) and four fiber unit cells (square packing) with a fiber volume fraction of 30%, static contact angle of 30 degrees, and fiber radius of 4 μm are shown in Figure The interface is described by the level set function, described by the scale bar where a value of one represents purely resin. The resin spreads axially and circumferentially along the fiber. The four fiber unit cell spreads and reaches equilibrium at a much faster rate (0.38 seconds) than the three fiber unit cell (1.47 seconds). This is a result of there being an increased fiber-resin contact area in the four fiber unit cell Fiber Volume Fraction For this study, the base parameters, found in Table 2.1, were used and the spacing between fibers was varied. The normalized fiber-resin contact area is the calculated fiber-resin contact area multiplied by the fiber radius divided by the resin volume. The effect of changing fiber volume on the fiber-resin contact area increased as the fiber volume fraction was decreased, as shown in Figure As the volume fraction is changed for the three and four fiber unit cells, the spacing between fibers changes at different rates, described in equations (2.15) and (2.16). The change in fiber spacing, governed by equations (2.15) and (2.16), for the two types of unit cells causes the capillary pressure to change, which can be modeled using the Young- Laplace pressure equation [27]. At larger fiber volume fractions, the capillary pressure increases at a much faster rate, leading to an increased wetted area. At lower fiber volume fractions, the rate of change of capillary pressure is not as high, resulting in similar increase in wetted area for both types of unit cells. The triangular packing arrangement had a larger fiber-resin contact area per fiber, thus it would be the 39

61 preferred packing arrangement if one were to create a network of resin micro drops within a fiber tow with the goal of maximizing fiber-resin contact area. Figure 2.18: Final fiber-resin contact area as a function of fiber volume fraction Static Contact Angle The static contact angle had a large effect on the final fiber-resin contact area. The contact area was linear with the cosine of the static contact angle, shown in Figure A linear increase with cos(θ) makes sense because the final results shown are with a fiber volume fraction of 30%. It is clear that for a given fiber volume fraction, the square packing arrangement is preferred for increasing fiber-resin contact area. For this particular combination of resin volume, fiber volume fraction, and fiber size the ratio of fiber-resin contact area for the triangular and square packing arrangements ranged between 1.07 and 1.11 for the given static contact angles. The static contact 40

62 angle did not have as significant of an impact on the resin spreading as the packing arrangement did. Resin went from its initial to final position inside square packed fibers about five times faster than with the triangular packing arrangement. This indicates that when the same volume of resin wets fibers in a square packing arrangement, the resulting composite will have a higher fiber-resin contact area and faster processing time when compared to a triangular packing arrangement. Figure 2.19 : Equilibrium fiber-resin contact area for three-fiber triangular and four-fiber square packing arrangements 2.4 Limitations of the Model A limitation on this model is imposed by the assumption of a microscopic length scale. This is because when the diameter of the fiber or the volume of the resin is increased by a large amount, the inertial and gravitational forces are no longer considered negligible. This would invalidate the axisymmetric assumption in the axisymmetric fiber model. In the four fiber model, one would no longer be able to use 41

63 the symmetry plane orthogonal to the direction of gravity. The trs seen in these models may not hold for models with extremely large contact angles because they only examine the case where the liquid will wet the fiber s surface. 2.5 Summary Numerical models describing the partial wetting of a finite volume of resin on a single fiber and in triangular and square packed unit cells were presented and validated. The static contact angle affected both the rate of axial spreading as well as the final fiber-resin contact area. Both the wetting length and final fiber-resin contact area increased with increasing fiber diameter. This claim is only for the case when the resin is in a barrel shape around the fiber as the clamshell shape was not investigated. The slip length had a defined effect on the rate of wetting, but did not impact the final fiber-resin contact area. This indicates that the slip length will not impact the composite properties. Fiber volume fraction had a significant impact on fiber-resin contact area, being more influential at higher fiber volume fractions. The final fiberresin contact area was larger for square packed unit cells than triangular packed unit cells. In unit cells with triangular or square packing arrangements, the static contact angle had a large impact on the final fiber-resin contact area. The effect of static contact angle on wetting rate was small compared to the impact of packing arrangement on wetting rate. These models can be used to predict the impact of manipulating fiber and resin surface properties, interaction, and geometry on the wetting of fibers by a finite volume of resin. This chapter emphasized the final configuration of resin within unit cells containing fibers. The following chapter will utilize a numerical model of resin flowing between fibers to characterize the microscale capillary pressure. The model 42

64 will be employed to explore unique scenarios which have not previously been studied, including the effect of packing imperfections and tows containing fibers subject to different surface treatments. An analytical capillary pressure model will also be developed with the goal of rapidly being able to compute the average capillary pressure of resin moving between fibers. 43

65 Chapter 3 CHARACTERIZATION OF AVERAGE CAPILLARY PRESSURE FOR RESIN FLOW BETWEEN FIBERS 3.1 Introduction Composite materials are comprised of fibers embedded in a resin matrix. The resin is usually introduced into the fibrous preform in the liquid form. During liquid composite molding (LCM), a pressure gradient drives the flow of resin into the preform. These preforms are a dual scale porous medium in which the resin will usually fill the macroscopic pores between fiber tows much faster than it saturates the microscopic pores inside the fiber tows for most composite applications (although it is possible to fill the tows first deping on the capillary number and flow front velocity) [31,32]. In addition to the applied pressure gradient, the microscopic flow of resin into fiber tows, especially in regions far from the inlet, is driven by the capillary pressure. The influence of capillary pressure on composites processing is an active area of research [33 36]. Increasing capillary pressure will result in less microscopic voids inside of fiber tows [1,37]. This will increase the fiber-matrix interfacial area and improve the mechanical properties of the resulting composite because it will eliminate many stress concentration regions. On the other hand, presence of microscopic voids can increase energy absorption of the composites by dissipating the impact energy through friction between fibers within the tows devoid of resin [38]. Hence by understanding the role of capillary action within fiber tows at the microscopic level, one can tailor the composite properties for the desired application. The capillary pressure is the pressure differential across the interface of immiscible fluids. It is depent on the shape of the interface and can be found by 44

66 examination of the radii of curvature of the surface and the surface tension, using the Young-Laplace equation [27]: ΔP = γ ( 1 R R 2 ) (3.1) The radii of curvature of the interface between the immiscible fluids are given by R 1 and R 2 and γ is the surface tension. The radii are measured in orthogonal planes and considered positive if the circle s center is inside the liquid [27]. Bayram and Powell used Eq. 3.1 along with geometric quantities and the contact angle between the fibers and resin to derive an equation to describe the capillary pressure [39]: P c = γcos (θ + α) r(1 cosα) + d (3.2) Here, γ is the surface tension, θ is the contact angle between the fiber surface and resin, r is the fiber radius, d is half of the gap between fibers, and α is the directional body angle (which is described later in Figure 3.6). An average value of the capillary pressure is used for a practical way to simulate mesoscopic flow within a fiber tow because using the exact capillary pressure values between fibers would require the use of numerical methods and would not be solvable in a reasonable time due to the extremely large number of elements that would be necessary to determine the microscopic motion of the fluid between three to twelve thousand fibers in a single tow [14,40]. With the assumption that the fibers are either packed in a square or hexagonal packing arrangement, the only parameter describing capillary pressure that changes during resin flow between fibers is the directional body angle. Foley integrated this capillary pressure as a function of directional body angle to obtain an average capillary pressure term [14]. The approximation for the maximum and minimum directional body angles were found by setting the capillary pressure equal to zero and taking the first positive and negative value respectively. An alternative approach was formulated by Neacsu et al., which uses a more complicated weighting function [40,41]. Ahn et al used an equivalent pore diameter as the input radius for Eq. 3.1 and calculated the 45

67 average capillary pressure for transverse flow between hexagonally packed fibers using [42]: v f ΔP = γ cosθ (3.3) r 1 v f This approximation does not take the microflow details into account. The approaches by Neacsu et al [40] and Foley [14] only examine flow between two fibers, without taking into account the influence of other neighboring fibers. The capillary pressure contributions have also been determined utilizing experimental methods, but these only find the average capillary pressure, and do not allow for the capillary pressure to be a function of location within a tow [43]. It is desirable to develop a methodology to calculate the average capillary pressure of resin moving through a unit cell representing the common hexagonal packing within the fiber tow. This calculated average capillary pressure could then serve as input at the resin flow front for a mesoscale model of tow filling. This section introduces a method to numerically calculate the average capillary pressure for a resin as it fills a unit cell. In addition, a faster method, using the analytical expression for capillary pressure in Eq. 3.2, is also developed and compared with the numerical method. The average capillary pressure for both of these is found using the same concepts that Foley et al has shown to be acceptable [14]. The influence of fiber packing as well as utilizing different sizings on the fibers within the unit cell will also be investigated in this chapter. In both the numerical and analytical models, fiber surface roughness will be neglected. This assumption is valid as long as the surface roughness is much smaller than both the fiber diameter and the gaps between fibers. Utilizing atomic force microscopy, the surface roughness for S-2 glass fibers was found to be on the order of 10 nm (1000 orders of magnitude smaller than the fiber diameter). 46

68 3.2 Model Setup Numerical Capillary Pressure Model A multiphase flow model was developed to describe the flow of resin between two fibers and through a unit cell containing five fibers which is a more common Figure 3.1: Schematic of (a) two fiber and (b) five fiber unit cell capillary pressure numerical models. Blue and grey are the resin and air initial positions respectively. In (b), the dotted lines outline the five fiber unit cell. arrangement of fibers within a fiber tow, as shown in Figure 3.1. The geometry is simplified to two dimensions and resin flow is only considered across the fibers because the resin flow between two fibers is assumed to be uniform in the axial direction. The slight curvature in the axial direction will be orders of magnitude less than the curvature 47

69 of the flow front across the fibers. Thus ΔP in Eq. 3.1 is decided by the very small inplane radius of curvature across the fibers compared to the axial radius of curvature R 2 which will be very large. The flow is assumed to be Stokes flow due to its small scale and the low Reynolds number, allowing the effect of inertia to be neglected. The walls are periodic because the unit cell is geometrically repetitive and hence the flow pattern will be repeating. The wetting walls describe the partial wetting of the fiber surfaces by the resin. To avoid singularities at the contact line, where there is a triple point, the model uses a slip boundary condition at the fiber surface [29]. This is done through use of the slip length, β. The slip velocity on the fiber surface is expressed as the product of the slip-length and the local velocity gradient normal to the surface (the Navier-slip). The no-slip condition, instead of being applied at the fiber surface, is assumed at a distance β below the fiber surface and simple shear flow is assumed over the depth β: u = β(n w u), with n w being the normal vector on the fiber surface [30]. The interface between the resin and air is described using the level set function [44]. The level set function creates an interface with a finite thickness, defined by the signed distance function, φ. The smeared-out delta function is then defined by the levelset function [25]: δ(ϕ(x)) = 6 ϕ ϕ(1 ϕ) (3.4) The delta function is later integrated to introduce a smeared Heaviside function to change from 0 to 1 across the interface, as is done similarly with the volume-of-fluid method to differentiate the resin and air [25]. The equations governing the resin flow, implemented using COMSOL Multiphysics, are [25,26]: ρu t + ρu u = P + μ 2 u + ρ g + F st (3.5) u = 0 (3.6) ϕ t + u ϕ = λ (ε ϕ ϕ(1 ϕ) ϕ ϕ ) (3.7) Here, u is the velocity vector, μ* is the viscosity, ρ is the density, P is the fluid pressure, g is the gravity vector, and F st is the distributed body force over the interface, which is 48

70 represented by the divergence of the interfacial stress tensor T due to interfacial tension (i.e., F st = T). The re-initialization parameter for the interface,λ, is set to the approximate maximum interface speed. The interface thickness is given by ε. A value of μm, one half of the largest element s height, was selected for ε. The capillary pressure solution was unchanged upon further refinement of the mesh, thus this is an acceptable value of ε. Eqs. 3.5 and 3.6 are the Stokes and mass conservation equations respectively. Eq. 3.7 describes the level set function, which is utilized to define the resinair interfacial movement. There are multiple ways to formulate the level set method, this form was available in COMSOL and is effective in solving this problem as we are dealing with creeping flow at low Reynolds number also evidenced by the validation of our method. The viscosity and density are interpolated by the Heaviside function according to the regime where the fluid material is present: i.e., resin or air. The reinitialization has been introduced to normalize the distance function property of the level-set function. We remark that the force from capillary pressure, acting on the resin interface, is assumed to be equal in magnitude as the traction force on the surface of the fibers, shown in Figure 3.2, since the interfacial tensions that always acts in the tangential Figure 3.2: Diagram depicting capillary pressure and the application of traction force. The location of the resin is given in red and on the right side blue represents air. The other colors on the right side indicate the different steps of the interface, where the level set function is continuously changing from 1 (resin only) to 0 (air only). 49

71 direction will be cancelled along the interface, except for the contact point. The continuous surface stress tensor of the resin acting on the fiber surface is [45]: T = γ(i nn)δ(x) (3.8) The traction force within 0.5 ε from the solid surface, denoted by t w, is given, using the normal vector on the fiber surface equilibrium with the normal vector n on the interface: n w and assuming that n n w = cosθ in t w = T n w = γδ(ϕ(x))(i nn) n w = γδ(ϕ(x))(n w ncosθ) (3.9) The force due to the partial wetting is the integral of the traction force, t w, over the fiber surfaces in the flow direction (x-direction). The average capillary pressure is then computed by integrating the t w component in the flow direction along the fiber surface and dividing it by the interface length. The t w component in the flow direction is given by: t w,x = γ( 1 + n x n x n x n y )δ dγ (3.10) Γ The interface length between the resin and air can be obtained easily by the integrating the smeared delta function over the domain Numerical Model Validation The contact angle between a fiber and resin can be found experimentally using a contact angle analyzer and geometric parameters may be measured, but the slip length is a parameter based on interactions between the fiber surface and resin that is difficult to determine. We propose a methodology to determine the slip parameter as follows; the slip length introduces a slip velocity at the contact surface in the numerical model. The slip length parameter is determined by comparing the resin flow front movement with a simple channel flow between partially wetted parallel plates, whose analytical solution is known. An analytic solution that can predict the wetted length as a function of time for flow between parallel plates is given by [46]: 50

72 L(t) = hγcos(θ) t (3.11) 3μ Here L is the distance traveled by the flow front along the length of the plates (wetted length) and h is the distance between the two plates. The surface tension of the fluid,, is Figure 3.3: Wetted length versus time for resin moving between parallel plates for an analytic solution with a contact angle of 30 degrees in Eq.(11). In the numerical model, slip length (β) is varied to find the best fit for that contact angle. (a) broad range (b) fine tuning of (β) to match the analytic solution. 51

73 known and to relate the contact angle to the slip parameter, the slip length (β) can be found through fitting the numerical and analytical solutions for the flow between parallel plates, an example of which is shown in Figure 3.3 for a contact angle of 30 degrees. Figure 3.3-a shows a broad range of slip lengths to approximate the slip length. In Figure 3.3(b), this approximate solution is fine tuned to find a more precise value (β=0.125 μm), as determined by comparing the numerical and analytical solutions. The slip length is depent on the contact angle and fiber curvature but is not a function of fiber spacing. The slip length found using parallel plates can be converted to the slip length for a fiber with radius r using [47]: 1 β = 1 β 0 1 r (3.12) β 0 is the slip length found using parallel plates since the radius of curvature for a flat plate is infinite. To confirm this method, the slip length found using Figure 3.3 was input into Eq. 12 to determine the slip length for a fiber with a radius of 4 micrometers (other parameters listed in Table 3.1). The results for capillary pressure, found analytically using Eq. 3.2 and numerically using the model described in Figure 3.1-a and Eq. 3.10, as a function of directional body angle are shown in Figure 3.4. The curves in Figure 3.4: Comparison between numerical and analytical solution for capillary pressure (eq. (2)) as resin moves between two fibers 52

74 Figure 3.4 are non-symmetric because the capillary pressure, defined by Eq. 3.2, is not symmetric about a directional body angle of zero. The analytical and numerical results for capillary pressure match very closely, indicating that this method can accurately predict the capillary pressure of resin moving between fibers. The capillary number for this flow is 2.3x10-7, indicating it is dominated by surface forces. Table 3.1: Baseline properties used for capillary pressure prediction Property Fiber Radius (μm) 4 Fiber Volume Fraction 0.6 Value Surface Tension (N/m) 0.07 Contact Angle ( ⁰ ) Analytical Capillary Pressure Model It is desirable to develop a methodology for calculating average capillary pressure without the computational effort required to solve the Stokes equations for micro flow within a unit cell. An alternate method will analytically predict capillary pressure of resin moving within a unit cell based on Eq Foley et al averaged the capillary pressure as a function of the directional body angle for resin flow between two fibers [14]. Averaging the capillary pressure as a function of the total wetted fiber surface area yields the same result as averaging it as a function of directional body angle. With this in mind, when including more than two fibers, the capillary pressure will be averaged as a function of total wetted fiber surface area. The analytical method will yield a matrix of capillary pressure values and the corresponding total fiber surface area wetted by the resin. The fiber surface area wetted, denoted by s, will be equal in value to the wetted length along the circumference of the fiber since the length of the fiber can be assumed to be unity for simplicity. The analytical method is outlined in Figure

75 Figure 3.5: Characterizing average capillary pressure solution methodology The total fiber surface area wetted by the resin will be increased by specified increments from zero until all of the fiber surfaces in the unit cell are wetted by the resin, with the capillary pressure being recorded at each increment. The capillary pressure will be calculated using Eq. 3.2, which will be multiplied by sin(π/3) when the flow is not in the x-direction because we are only interested in how the capillary pressure is driving the resin flow in the x-direction (resin flow will be at an angle of π/3 with respect to the x-axis). When this value is then included as an input into the homogenized unit cell, it will accurately represent the flow dynamics. The unit cell will be broken into sections; a new section is created when a new surface is contacted 54

76 Figure 3.6: Unit cell model for analytical model, with the expected flow front shape shown in a different color for each section. Resin will start on the left side of the unit cell, and wet the fibers as it moves across the unit cell. The equations utilized to calculate the capillary pressure and directional body angle in each section are given in Table 3.2. or when a surface is fully wetted, as shown in Figure 3.6. The directional body angle, α, which is a function of flow front location, is shown at time t i. The equation for capillary pressure, which is a piecewise continuous function over the entire domain, and directional body angle in each section is given in Table 3.2. The directional body angle calculations will be depent on which section the flow is in. In section B, the directional body angle is the angle of the flow front with respect to the line connecting the centers of the top left and center fibers. The equation for directional body angle must include a π/6 term to account for the fact that the line connecting the centers of the top left and center fibers is at an angle of π/6. 55

77 Table 3.2: Calculation of capillary pressure and directional body angle for each section Section Capillary Pressure Directional Body Angle A P c (α) = γ r cos(α + θ) 1 cos(α) + d r B P c (α) = γ r cos(α + θ) 1 cos(α) + d sin ( π 3 ) r C P c (α) = γ r cos(α + θ) 1 cos(α) + d r D P c (α) = γ r cos(α + θ) 1 cos(α) + d sin ( π 3 ) r E P c (α) = γ r cos(α + θ) 1 cos(α) + d r α = s Top Left r α = s Center 2r α = s Center 2r α = s Center 2r α = s Top Right r π 6 π 2 5π 6 π 2 The condition for each section, which signifies that the flow front is in the following section, will be different for each of the five sections. Section A and C when the resin contacts the center and far right fibers respectively. The resin is considered to be in Section B when the flow front contacts the center fiber. The condition for section A, derived utilizing the geometry of the hexagonally packed unit cell and curvature of the resin flow front (which can be determined utilizing Eqs. 3.1 and 3.2), is when the directional body angle satisfies: 56

78 sin(α) 1 cos(α) + d r cos(α + θ) [1 cos ( = sin (π 3 ) 3v f 2 3π sin 1 [d + r(1 cos(α))]cos(α + θ) ( r (1 cos(α) + d ) r ) 1 ) ] (3.13) The smallest positive directional body angle from Eq. 3.2 can be converted to total fiber surface area wetted, s, using the definition in Table 3.2 corresponding to Section A. The condition for Section B will be when the fibers on the left side of the unit cell are fully wetted by resin. The resin will transition from Section C to Section D based on Eq (same as A to B). The of Section D will be signaled when the resin has fully wetted the center fiber. The algorithm will when all of the fiber surfaces in the unit cell are fully covered by resin. The transition between sections is important to define. When the resin fully wets or first contacts a new surface, the flow front configuration will be effected. The two transition types seen in this unit cell are (i) when the resin contacts a new surface, seen when the resin transitions from Section A to B as well as C to D, and (ii) when two resin flow fronts collide, such as the transition from Section B to C and D to E. The transition from Section A to B will be utilized as the example for how to handle resin contacting a new surface. The first assumption made is that the two fibers on the far left side have identical wetted surface areas. The additional condition that should be met is that the total surface area wetted within the unit cell is conserved during this transition. The final assumption made is that the directional body angle, measured between the resin flow front, center fiber, and top left fiber, is the same for both the center and the top left fiber. These assumptions lead to the following three 57

79 equations respectively, which will be employed to solve for the three unknowns, s Top Left (s Center, s Bottom Left ): s Top Left = s Bottom Left and s Top Left s Total = s Top Left + s Bottom Left = s Center + s Top Left = s Bottom Left + s Bottom Left + s Increment (3.14) (3.15) s Top Left = ( r fiber 2 ) (s Top Left s Increment π ) (3.16) 6 Here, s Increment is the incremental increase of surface area wetted during each iteration. s Top Left and s Bottom Left are the surface areas wetted of the two fibers on the left side before the transition. s Center, s Top Left, and s Bottom Left are the surface areas wetted of the center and two left fibers after the transition. The position of the resin Figure 3.7: Parameters utilized to describe jump between sections, described in Eqs

80 flow front before and after the transition along with necessary geometric terms is shown in Figure 3.7. It may be noted that s Center is not included in Eq because there is no resin wetting the center fiber before the jump. This jump will also occur between sections C and D. The transition occurring when two resin flow fronts collide, using the jump from Section B to C for example, is more straight forward. The equations governing this transition are: s Top Left = s Bottom Left and s Top Left = s Bottom Left = 1 2 π r fiber (3.17) s Total = s Top Left + s Bottom Left + s Center = π r fiber + s Center + s Increment (3.18) This transition will also occur between Sections D and E. The algorithm will once all of the fibers in the unit cell are fully wetted. Figure 3.8: Comparison between our analytical and numerical solutions for non-dimensional capillary pressure as a function of wetted fiber surface area 59

81 The output will be a matrix of total wetted surface area values, increasing in user defined increments, and the corresponding capillary pressure values. The results for capillary pressure as a function of wetted area are in reasonably good agreement between the numerical and analytical models, shown in Figure 3.8, especially when considering that the average capillary pressure is what will be used in process models when addressing flow at the macro scale Averaging the Capillary Pressure It has been shown that taking the average value of the capillary pressure as a function of directional body angle curve is an acceptable input for macroscale flow simulations [14]. The capillary pressure, averaged as a function of fiber surface area wetted, yields the same result. In this dissertation, the capillary pressure will be averaged as a function of fiber surface area wetted. This allows the calculation of an average capillary pressure as resin moves through a unit cell containing any number of fibers; in this chapter we present results for five fibers. The trapezoidal rule will be used to integrate the capillary pressure function and an average capillary pressure will be obtained using: P c = n S max S min 2 (P c,i + P c,i+1 )Δs 1 (3.19) Here, s is the fiber surface area wetted. The maximum and minimum permissible surface area wetted in the unit cell is given by S max and S min respectively. The number of data points in P c versus s dataset is represented by n. The output from the analytical (Table 3.2) and numerical capillary pressure models previously presented will be used to calculate the corresponding P c for a finite number of s values. These values will be the inputs into Eq

82 3.3 Results Influence of Applied Macroscopic Pressure on Capillary Pressure The indepence of capillary pressure from the applied pressure gradient across a multi-fiber unit cell has been assumed but not verified in literature [14,40]. This leads to the assumption that the average capillary pressure also is indepent of applied pressure in macroscale flow. It is not intuitively obvious that the capillary pressure should be indepent of applied pressure because the shape of the interface will deform based on the applied pressure, which may affect the capillary pressure. The influence of applied pressure gradient across a hexagonally packed unit cell, with properties given in Table 3.1, for selected pressure gradients was investigated by applying a positive pressure at the inlet and keeping the outlet pressure Figure 3.9: Influence of external pressure gradient on the capillary pressure as resin moves through a unit cell. 61

83 equal to zero for the unit cell utilized in the numerical model, shown in Figure 3.1-b. Comparing the capillary pressure as a function of fiber surface area covered for selected applied pressures is shown in Figure 3.9. The capillary pressure is indeed indepent of the applied pressure. There are some deviations when there is an applied pressure of 101 kpa, but they have an insignificant effect on the average capillary pressure and that large of a pressure gradient over a few micron unit cell would not be expected in a practical situation. Capillary pressure is a pressure across an interface between two fluids that is driven by the surface tension between the two fluids. The pressure across the resin-air interface is driven by the surface energies of the fibers and resin as well as the fiber radius and packing arrangement Effect of Fiber Arrangement Capillary pressure, being depent on the curvature of the resin-air interface, is greatly influenced by fiber packing. The fibers are packed much closer with higher fiber volume fractions, which lead to a lower radius of curvature for the resin-air Figure 3.10: Comparing average capillary pressure for presented and previously developed methods in a fiber unit cell 62

84 interface. The volume fraction is varied by changing the distance between fibers, while still maintaining the hexagonal packing arrangement. The influence of decreasing the radius of curvature is seen in Eq. 3.1 and can be quantified utilizing Eq The effect this has on the average capillary pressure can be seen in Figure Figure 3.10 also compares the numerical and analytical solutions for average capillary pressure to those found in literature [14,40,42]. There is not a significant difference in average capillary pressure for lower fiber volume fractions. For higher fiber volume fractions, the average capillary pressures from our analytical and numerical approaches are much higher than the ones found in literature. This is because Foley et al [14] and Neacsu et al [40] did not include the effect of neighboring fibers on capillary pressure and Ahn et al [42] did not consider the shape of the resin-air interface or the direction of the resin flow between fibers. The influence of neighboring fibers on average capillary pressure is very significant, particularly when the fibers are closely packed. Without including neighboring fibers, the capillary pressure would approach zero as the flow front moved far enough (seen in Figure 3.4). When the fibers are packed closely and neighboring fibers are included, the resin flow front will contact and wet the next row of fibers before the capillary pressure is driven down towards zero. The capillary pressure is driven up when the resin contacts the next row of fibers, causing there to not be a large portion of the capillary pressure versus surface area covered curve that is very low, shown in Figure 3.9. With the fibers are packed further away from each other, the capillary pressure will approach zero before the next row of fibers is contacted, shown in Figure 3.8 where a new row of fibers is contacted between Sections A and B. The average capillary pressure increased at a higher rate as the fiber volume fraction approaches its maximum limit for the packing arrangement because the effect of neighboring fibers cuts off more of the lower portion of the capillary pressure versus surface area covered curve (corresponding to the higher α values in Figure 3.4). 63

85 3.3.3 Effect of Imperfect Fiber Arrangement Previous work has been aimed at predicting capillary pressure for resin moving within a hexagonally packed unit cell. In real systems, the fibers will not be packed perfectly in a hexagonal arrangement, making it important to understand the effect of imperfections on the average capillary pressure. To predict the influence of fibers shifting away from perfect fiber packing arrangement, the center fiber in Figure 3.1-b was moved to different locations along the x and the y axis. When the location of the center fiber is changed along the x-axis, the average capillary pressure is significantly affected, depicted in Figure 3.11-a. As the fiber is moved further to the left, the capillary pressure is increased due to two reasons. The fibers are packed closer on the left of the unit cell which causes a large increase in capillary pressure due to decreasing the resin-air interface radius of curvature. The fibers being packed closer on the left side of the unit cell increases the capillary pressure by a more substantial amount than the capillary pressure is decreased on the right side because the capillary pressure does not linearly change with fiber spacing. This drives up the average capillary pressure. The second reason is that the neighboring fiber effect, discussed previously, which become stronger as the neighboring fiber moves closer. The increase in average capillary pressure is much higher as the fiber reaches its maximum allowed displacement because the equation for capillary pressure (Eq. 3.2) is not linearly related to fiber spacing. As seen in Figure 3.11-b, changing the center fiber location in the direction perpicular to the flow direction does not significantly influence the average capillary pressure. As the center fiber is moved up and down, it moves closer to the top and bottom two fibers respectively. This results in an increase in capillary pressure, but due to the geometry of the unit cell, this increase is mostly seen in the y-direction because that is the direction of the flow. In the calculation of average capillary pressure, only the capillary pressure driving flow in the x-direction is considered since that is the flow 64

86 path and the goal is to determine the average capillary pressure driving the resin through a unit cell. Figure 3.11: Average capillary pressure for case where center fiber is moved along the (a) x-axis and (b) y-axis 65

87 The average capillary pressure will be used in macro-scale flow predictions, making it useful to calculate the effect of fibers not perfectly placed in fiber packing for various fiber volume fractions. Only moving the center fiber in the x-direction will be considered because it was shown that moving the center fiber in the y-direction had Figure 3.12 : Normalized capillary pressure as a function of the x-offset of the center fiber for selected fiber volume fractions, calculated with the analytical model. a negligible influence on the average capillary pressure. To study the influence of these variations, the average capillary pressure will be normalized by the capillary pressure for a perfectly packed unit cell and the offset of the center fiber will be normalized by the maximum offset allowed by the geometry of the unit cell. The normalized capillary pressure as a function of the normalized center fiber offset was shown to be indepent of fiber volume fraction (Figure 3.12). This is a key result because a fiber tow will consist of a statistical distribution of fiber volume fractions as 66

88 well as defects in packing arrangements. A single curve can be utilized in conjunction with the average capillary pressure as a function of fiber volume fraction curve to predict the capillary pressure of resin moving through a unit cell with variations in packing arrangements for any fiber volume fraction Capillary Pressure for Unit Cells with Fibers with Different Sizings The average capillary pressure of resin moving through a unit cell containing fibers can be manipulated by having different sizings on the fibers. Utilizing different sizings on fibers within a tow will result in fibers with different surface energies; therefore they will have different contact angles when wetted by resin. The unit cell shown in Figure 3.13-a contains fibers with different sizings, half of the fibers are sized such that they will have a contact angle of 30 degrees and the other half will have a contact angle of 60 degrees. The numerical model was utilized to determine the influence of varying the surface treatments on the average capillary pressure, shown by the hybrid sizing values in Figure 3.13-b. The average capillary pressure for the hybrid sizing tows fall between those of the uniformly sized tows with 30 and 60 degree contact angles. This is because the wettability of the hybrid unit cells falls between that of the two uniform cells, causing the capillary pressure to fall as well since the geometry of the cells are identical. Fiber tows with differently sized fibers would be useful if one sizing was optimized for bonding with the resin and the other increased the saturation level of the tow. They could also be utilized to manipulate the capillary pressure in a manner that results in the tow being filled with a desired volume of resin during LCM processes. 67

89 Figure 3.13: Unit cells with hybrid sized fibers (a) model setup and (b) influence on average capillary pressure 3.4 Summary Novel computational methods to predict capillary pressure of resin moving through a fiber unit cell was presented and validated with analytic methods. The analytical and numerical predictions agreed well with those found in literature and 68

90 were able to account for presence of neighboring fibers which was neglected in previous efforts. The capillary pressure of resin moving between fibers was shown to be indepent of the pressure gradient applied to the unit cell and increase with fiber volume fraction. The influence of arrangement imperfections within a hexagonally packed system of fibers was explored. The effect of these imperfections on average capillary pressure of resin moving through unit cells containing fibers was found to be depent on the direction the fibers were offset, which was previously unreported. These results can be utilized to develop models of resin flowing into fiber tows with a statistical distribution of fibers. The influence of including fibers with different surface treatments within a unit cell was studied. It was found that the capillary pressure of these unit cells will fall between that of unit cells uniformly sized with each individual sizing. Developing a methodology to quickly and accurately predict the average capillary pressure of resin moving between fibers allows one to include more accurate average capillary pressures in macroscopic tow filling simulations. The next step is to include this capillary pressure in a mesoscale tow filling model, which will be the focus of the next chapter. A methodology will be developed to include capillary pressure in modelling the infusion of tows with stochastically spaced fibers. This will allow prediction of microvoid formation and distribution within the tow. The microvoid distribution will dep on a number of factors including fiber spacing, wetting properties, and how long the preform was under vacuum before infusion begins. 69

91 Chapter 4 CAPILLARY DRIVEN FLOW OF AN INFINITE RESIN VOLUME INTO FIBER TOWS: ROLE OF FIBER PACKING AND AIR EVACUATION 4.1 Introduction Most composite manufacturing processes impregnate the fiber preform by driving the resin to flow and saturate the open spaces between fibers. The resin flow is induced by a pressure gradient comprised of an applied pressure, capillary forces, or a combination of the two. The empty spaces between the fibers that are not occupied by the resin after impregnation are called dry spots or voids. Voids are caused by the entrapment of air or volatiles due to the resin flow front dynamics and filling patterns during the resin impregnation process as shown in Figure 4.1. Voids are usually Figure 4.1: Schematic of void formation in a dual scale fibrous preform. 70

92 viewed as negative because they cause stress concentration regions upon being loaded, which reduces the strength and stiffness of the resulting composite. The prediction and control of voids is a very active area in composites research [1 4,48]. A part with more than 2% void content is often discarded in the aerospace industry, which has significant financial consequences [49]. Preforms are fiber tows either woven or stitched together. Each fiber tow will have 1K to 24K fiber filaments bundled together. Preforms are considered dual scale porous media because the spacing between the fiber tows is usually one to two orders of magnitude larger than the spacing between the fibers. Due to the dual scale pore sizes, resin will flow around fiber tows much faster than into fiber tows [50]. The resin will continue to impregnate the tow and compress the void until the gas pressure inside the void is equal to the sum of the applied and capillary pressures. The compressed gas within the tow, if unable to escape the tow, will become a microvoid in the final composite part. The void size and morphology will be a function of the fiber tow geometry and the distribution of the fibers within tows. Hence there is a need to develop a methodology capable of modeling resin impregnation into fiber tows that may contain non-uniformly spaced fibers, which will more accurately predict the formation of these microvoids. It is also desirable to determine the correlation between the duration of time a part is subjected to a vacuum to evacuate the air before impregnation and the microvoid concentration within the tows. The novelty of this work is that it will account for the role of capillary forces and vacuum 71

93 hold times on the void formation and distribution within a fiber tow which has a nonuniform fiber distribution Previous Work The impregnation of fiber tows has been the subject of many research efforts, a review of which compares some of the prominent ones [51]. The outer surfaces of tows are usually surrounded by a volume of resin significantly larger than the empty space between fibers, making it justifiable to assume that an infinite resin source is available to impregnate the tow. This problem can be simplified by ignoring gravitational forces since they are negligible when compared to the applied and capillary pressures. The infusion of resin into fiber tows has been modeled using Darcy s Law [40,52]. Darcy s Law and the conservation of mass are given below [9]: v = K P (4.1) μ v = 0 (4.2) The resin pressure is given by P, μ is the resin viscosity, K is the anisotropic permeability of the porous media expressed in a tensorial form, and v is the volume averaged resin velocity. The impregnation of cylindrical tows with resin has been assumed to be one dimensional when the fiber volume fraction is constant throughout the tow. Resin is introduced from the circumference of the fiber tow under a constant pressure and the tow permeability and capillary pressure are assumed to be uniform throughout the tow [40,52]. Darcy s Law is invoked to describe the resin flow into the porous tow. Foley validated her model, incorporating an average capillary pressure 72

94 term, experimentally by measuring the weight change over time of a tow submerged in resin [14]. A closed form solution correlating a non-dimensional time and flow front radius was developed by Neacsu et al. for radial flow into a cylindrical fiber tow with uniformly spaced fibers [40]: τ = 1 ε 2 f (1 2 ln ε f ) (4.3) Here the non-dimensional flow front radius (ε f ) is normalized by the tow radius and the non-dimensional time (τ) is written in terms of the pressure difference between the tow s surface and the flow front (ΔP), resin viscosity (μ), fiber volume fraction (v f ), tow radius (R 0 ), and tow transverse permeability (K T ) [40]: τ = 4tK TΔP R 2 0 μ(1 v f ) (4.4) Neacsu et al. provided experimental support for their model by constructing a very large fiber tow with a pipe in the center and flow monitoring sensors along the radius to determine when the flow front reached predetermined locations [53]. After the tow was fully saturated, the mass flow rate through the pipe was measured to determine the radial permeability of the tow [53]. This analytical model will be utilized to validate our novel method of including capillary pressure at the flow front. Necessary inputs for Eq. (4.4) include the tow permeability and the average capillary pressure of resin moving between fibers, which are included in the pressure difference. It would be expected for this equation to underestimate tow fill times because it does not take into account the effect of air pressure within the tow. A constitutive model for the transverse permeability of the tow, K T, can be expressed as 73

95 a function of fiber radius, r fib, and volume fraction, v f, using the Gebart equation for hexagonally packed tows [54]: K T = 16 9π 6 r fib 2 ( π 1) 2 3v f 5/2 (4.5) The methodology presented in Chapter 3 (Table 3.2 and Eq. 3.19) will be employed to calculate the transverse capillary pressure associated with the fiber properties and arrangement in each element. Another important parameter in tow filling is the amount of air evacuated from the tow during the application of vacuum on the mold cavity prior to injecting the resin. Cer et al. developed a procedure to calculate the air pressure remaining within a fiber tow by satisfying continuity equation that accounts for compressibility in porous media with the volume averaged Darcy s velocity [55]. As the gas becomes rarified a correction can be added to the intrinsic permeability (K i ) in order to ext the validity of Darcy s Law by introducing a Klinkenberg parameter, b, which has units of pressure. Power law models have been successful at predicting b from K i as b only deps on the porous structure [55]: b = ( K i ϕ ) 0.5 (4.6) 74

96 Here, ϕ is its porosity. The Klinkenberg parameter corrects for the intrinsic permeability through [55]: K gas = K i (1 + b P ) (4.7) For our purposes, we calculate b << P and assume perfect vacuum pressure at the of the part, which allows the pressure at any location to be predicted with [56]: P = P P 0 = t + 1 (1 L L part ) (4.8) Here, P 0 is the initial pressure at the inlet at t=0 which usually is the atmospheric pressure, L part is the part length, and L is the distance from the inlet, as shown in Figure 4.2. The non-dimensional time, expressed as a function of part length, is given by [56]: t = K gasδp ϕμl part 2 t (4.9) Here, ΔP is the pressure difference between the inlet and the vacuum. The influence of air compression and capillary pressure has been examined in one-dimensional flow into cylindrically shaped tows [40,52,57]. Capillary driven flow into tows without air compression has also been studied numerically and experimentally [58]. Kang et al. showed that voids were larger in tows farther from the inlet location [58]. None of the papers surveyed in literature included capillary pressure and air compression on the impregnation of elliptical tows. Another large 75

97 gap in literature is that no models were found capable of including location depent Figure 4.2: Tow filling model schematic, shown at an intermediate time step in which resin is still entering the tow and the macroscale flow front has yet to reach the outlet. Red and grey represent the saturated portion of tow and preform respectively, with blue representing the unsaturated regions of each. fiber volume fractions. This is of interest because neither the capillary pressure nor permeability of unit cells is linear with fiber volume fraction, thus using their average values will lead to less accurate results. Realistic fiber tows have been shown to exhibit a stochastic distribution of spacing between fibers [59]. This dissertation utilizes a novel methodology to address the filling of elliptical tows with nonuniformly spaced fiber volume fractions, which includes the effect of both capillary and air pressure. The inclusion of a statistical distribution of fiber spacing allows for the prediction of multiple microvoids within the tow as opposed to predicting one large void in the center, which is consistent with experimental results [60]. A micrograph of void distribution in fiber tows is shown in Figure 4.3. Correlating void 76

98 content within tows with air evacuation time has yet to be explored but is important for manufacturing applications, particularly in the aerospace industry. Figure 4.3: Multiple voids (black) distributed within a fiber tow, with a thickness of ~0.2 mm [Adapted from [77]]. 4.2 Methods The resin flow modeling software used to describe resin infusion into fiber tows was Liquid Injection Molding Simulation (LIMS). LIMS is a finite element software that uses a control volume approach to simulate resin flow in a porous domain [61]. The pressure distribution within the porous domain is calculated by combining Darcy s Law with the conservation of mass equation [61]: ( K P) = 0 (4.10) μ The pressure gradient and Darcy s Law are utilized to calculate the flow rate at each node and advance the flow front. This is repeated at each time step until the condition is satisfied. LIMS allows the modification of both material properties and boundary conditions in each node and element after each time increment. This model, like the aforementioned models, simplifies the tow filling to a two dimensional geometry, represented by the cross section of the tow. One of the advantages of this 77

99 model is that it can be employed to predict filling on any realistic tow shape and is not limited to a cylindrical or elliptic geometry Solution Methodology As the tow permeability is orders of magnitude lower than the permeability between the tows, it is justifiably assumed that the fiber tow is surrounded by the resin before it impregnates the tow. Hence in the simulation it is assumed that the resin will enter the tow from the entire perimeter, as shown with necessary parameters at an intermediate time step in Figure 4.2. Here P app is the applied pressure on the tow where x = 0 (center of the tow), ΔP is the pressure gradient across the tow, and P v is the void pressure. Deping on the preform structure, it is possible that tows may be in contact with each other. This model may be employed to predict filling in such scenerios by only classifying nodes that are not part of the tow-tow contact as inlets. The tow location and resin flow front location in the part, measured from the inlet are given by L tow and L ff (t) respectively. The air pressure in the unsaturated region of the tow is denoted by P void. The shape of the flow front in Figure 4.2 is not elliptical like the fiber tow because the fiber volume fraction changes as a function of location in the tow. Elements with higher fiber volume fractions will exhibit lower permeability and higher capillary pressures, causing some elements to fill at a much faster rate than others. The tow initially does not contain any resin as this model is utilized to predict microvoid formation due to resin flow front dynamics. The perimeter of the tow is modeled as inlets with a specified pressure boundary condition, which is linear with 78

100 respect to x as the resin flows from the inlet towards the vent under a constant pressure gradient. The initial void pressure will be calculated from Eq. (4.8), which deps on how long the preform was under vacuum before introducing the resin [55]. The condition for the algorithm is when the pressure in every void is higher than the surrounding resin pressure plus the corresponding average capillary pressure and the resin flow front has reached the of the preform. The latter condition is necessary because if the resin flow front has not reached the of the preform, the applied pressure experienced by the tow will continue to increase, further compressing the void Applied Pressure Boundary Condition The pressure in the resin surrounding the fiber tows will change with time as the flow front moves from the inlet towards the vent. This will be incorporated into our model by defining a time depent pressure boundary condition around the circumference of the tow. By assuming a linear pressure gradient across the part from inlet to the flow front, one can calculate the time depent resin pressure applied along the surface of the tow at x=0 as a function of the tow location within the part (L Tow ): P app (t, x = 0) = P 0 (1 L Tow L ff (t) ) (4.11) The flow front location at a given time, L ff (t), is given by: 79

101 L ff (t) = 2K L,PreformP 0 t (4.12) μϕ Here, K L,Preform is the fabric permeability in the flow direction. When the flow front reaches the of the part, the applied pressure will become constant with respect to time. However, while the flow front is moving towards the vent, the applied pressure will change as a function of the x-location of the tow and the flow front location because there will be a pressure gradient across the tow. The change in pressure across the tow is calculated by: P Tow = P 0 2a Tow L ff (t) (4.13) assuming that the pressure at the flow front is perfect vacuum. Here, a Tow is the half length of the ellipse axis parallel to the flow direction. Combining Eqs. ( ) yields the equation governing the applied pressure boundary condition along the perimeter of the tow in our model: P app (t, x) = P app (t, x = 0) + P Tow 2a Tow x (4.14) This equation will determine the applied pressure at each inlet node around the surface of the tow until the flow front reaches the of the preform, thus introducing the macro resin flow coupling with the mesoscale level of fiber tow filling. Note that it is a one way coupling where the external fluid pressure influences the tow filling behavior but the tow filling does not influence the change in macroscopic pressure. 80

102 As our goal is to study the tow filling and not the macro filling of the mold, we feel our assumption is justified Inclusion of Capillary Pressure and Air Compression The capillary pressure, which LIMS is unable to accommodate, is usually direction and location depent. Hence, a novel multiscale method has been developed to include the microscopic capillary pressure in a mesoscopic tow filling model. This method modifies the permeability of elements on the flow front of the fiber tow to account for the capillary and air pressures acting on the flow front. The flow front velocity should be conserved, which will insure that our methodology does not incorrectly influence the flow front movement. The resin velocity within a tow at the flow front can be expressed as: v r = K T P μϕ r = K T(P c + P app P void ) μϕ r (4.15) Here, r is the radial direction as defined in Figure 4.2. To account for the capillary and void pressure at the flow front, we modify the permeability of the flow front elements at each time step as follows: K modified = K T(P app + P c P void ) P app (4.16) 81

103 Elements that were part of the flow front in the previous time step but become completely filled will have their permeability values reset to the original value. As the Figure 4.4: Methodology for including capillary and void pressure during resin flow into fiber tows. flow front progresses, the void within the fiber tow may split into multiple parts due to the non-uniformity of element permeability and capillary pressure values. The pressure in each void will be tracked and updated individually. A void is defined as any region containing one or more elements that is not fully saturated by the resin. The algorithm will identify each void and assign the void pressure for the next time 82

104 step accordingly; more details about this process are described later. An outline of the iterative solution methodology is shown in Figure Validation To validate our numerical approach, we confirmed that the solution was mesh indepent as well as it agreed with the analytical solution for the simplified case. The simplified case had full vacuum inside of a cylindrically shaped tow (pressure at the flow front is zero). The results of this, compared with the analytical solution in Figure 4.5: (a) Comparison of analytical and MATLAB + LIMS solution for a cylindrical tow, not considering void pressure and (b) mesh refinement study for elliptic tow including void pressure. Eq. (4.3), are shown in Figure 4.5-a. This verifies that the manipulation of the permeability of nodes on the flow front is an acceptable method to include pressures acting at the flow front. In addition to addressing the capillary pressure applied at the flow front, we use the same method to account for the change in the void pressure, 83

105 resulting from the compression of air within the tow. It is necessary to ensure the solution is mesh indepent when considering an ellipsoid tow shape and accounting for change in the void pressure due to compression of the tow, which is shown in Figure 4.5-b. Each element in all meshes employed is comprised of three nodes. The 1313 node mesh was selected to be utilized in the parametric studies Compressed Air Pressure The algorithm employed in this work can keep track of which elements represent void regions, record the change in size of each void at every time step, and update the air pressure in the void for the next time step. The initial void is comprised of the porous region in all of the tow, with pressure calculated from Eq. (4.8). At subsequent time steps, the void region is tracked using the fill factors associated with each node and corresponding elements. If the fill factor is less than one, that element is associated with the void region. With time, the void region may change size, split into multiple void regions or multiple voids may coalesce deping on the movement of the resin flow fronts. For example, void splitting results by two flow fronts meetings and cleaving a larger void into multiple smaller ones. Resin surface tension is not included in predicting these events because the fibers make the resin-void interface have many discontinuities. To allow us to determine the state of each void, a comparison of the state of the elements and their connectivity is queried at each time step. The algorithm will check to see if any of the constituent elements from each current void were constituent 84

106 elements in any voids from the previous time step. A similarity matrix is constructed to track which current voids have elements in common with previous voids. If current and previous voids share the same element, the similarity matrix for the corresponding voids will have a 1. An example of this matrix is shown in Figure 4.6. Examination of this matrix can tell us which of the three classifications the void will be: 1. A void in the current time step having elements in common with multiple voids from the previous time step indicates that the two voids have coalesced 2. Multiple voids in the current time step having elements in common with the same void from the previous step signals that the void from the previous time step has split 3. If a void in the current time step and previous time step only have elements Figure 4.6: Sample similarity matrix. A 1 indicates that the current void and previous void have elements in common, whereas a 0 indicates that they do not. common with each other, then it is known that the void has simply changed size between the two time steps Once the voids have been grouped and any major events (coalescence, splitting, only size change) have been identified, it is necessary to calculate the 85

107 pressure in each of the voids at the current time step to serve as inputs for the next time step. The ideal gas law is utilized to govern pressure changes within voids. The resin-void boundary is a series of fiber surfaces connected by small menisci of resinvoid interfaces. This breaks up the resin-void boundary into many small boundaries, thus the surface tension of the resin is only relevant in deriving the capillary pressure term. Without the fibers breaking up the resin-air boundary, the pressure compressing the void due to resin surface tension would need to be included as a function of void size. The gaseous or void area in each void is calculated by summing the void area in each element: A Void,i = # Elements in i A element,j j=1 (1 v f,j )f j (4.17) Here, f j is the average fill factor of all of the nodes that comprise element j. The pressure change is predicted utilizing the ideal gas law, which for the case when the void only changes size can be simplified to: P new = P olda old A new (4.18) For voids splitting, it is assumed that each of the new voids has the same pressure at the time step immediately after the split and therefore the pressure in each of the new voids can be calculated: P new = P olda old n i=1 A new,i (4.19) When voids coalesce, the pressure in the resulting void is given by: 86

108 P new = n i=1 P old,ia old,i A new (4.20) The resulting P new for each void will be used for the void pressure, P void, in Eq. (4.16) Fiber Volume Fraction Distribution The baseline case of fiber volume fraction distribution follows a Gaussian curve, with an average fiber volume fraction of 0.6 with a standard deviation of 0.1. The linear distribution of fiber volume fractions defines the fiber volume fraction as a function of y location within the tow. The fiber volume fraction at y = 0 is equal to the maximum fiber volume fraction for a hexagonal packing arrangement (v f,max ) and at other locations can be described by a linear relationship: v f (y) = v f,max 3π 4b (v f,max v f)y (4.21) Here, v f is the average volume fraction. The derivative of fiber volume fraction with respect to y location is set to ensure that the average value of the volume fraction is equal to the input v f, which for the basline case is 0.6. The random distribution assigns each element a random fiber volume fraction, but the overall average is the input average fiber volume fraction. For the baseline case, the upper bound of the random number generator was set to the maximum fiber volume fraction for hexagonal packing (0.907) and the minimum bound was set such that the mean of the bounds is equal to the average fiber volume fraction. 87

109 Parametric Study and Baseline Properties In addition to including both void and capillary pressure in elliptic tows, this work also explores the role of non-uniform fiber distributions within a tow on tow filling and void formation. Neither permeability nor capillary pressure are linear with fiber volume fraction so using average values for both in tow filling leaves room for error, in particular with regard to the formation of multiple microvoids. A correlation is developed between the duration of time a part is under vacuum before infusion and the microvoid content within fiber tows. We examine the relationship between tow location within a part and the tow impregnation. The role of fiber wettability and diameter in microvoid formation is also explored. A parametric study is conducted to examine the sensitivity of various parameters on void formation and size. The baseline properties for these studies are provided in Table 4.1 and are all reasonable values for composites processing. Table 4.1: Baseline properties utilized in tow filling model Fiber diameter (μm) 9 Contact angle (degrees) 30 Fiber volume fraction (tow) 0.6 Tow aspect ratio 8 a Tow (mm) 2 P 0 (kpa) 100 Resin viscosity (Pa s) 0.95 Resin surface tension (N/m) 0.07 Air viscosity (Pa s) L Part (cm) 30 L Tow (cm) 15 88

110 Time under vacuum before infusion (seconds) 59.9 K L,Preform (m 2 ) Results One of the novel elements of this methodology is the ability to predict the formation of multiple voids within the fiber tow. Figure 4.7 compares the void distribution between a tow with a Gaussian fiber volume fraction distribution and one with a constant distribution. The blue regions represent microvoids and red regions are fully saturated with resin. With a constant distribuiton (Figure 4.7-b), there is one large void in the center of the tow. In the Gaussian case (Figure 4.7-a) the smaller voids are due to one or more tow elements having high fiber volume fractions. The larger voids have a jagged shape because the elements will fill at different rates due to the non-uniform and varying permeability and capillary pressure values. Prediction of formation of multiple voids and non-symmetric large voids are both novel outcomes of this methodology. The voids size and distribution within a tow is attributed to the non-uniform fiber distribution. For Gaussian and random fiber distribution within a tow, the quantity of small voids is much greater than that of large voids as seen in Figure 4.7-a, which is consistent with experimental results [60]. The random fiber distribution (Figure 4.7-c) has a similar void distribution to the Gaussian distribution. Increasing the staandard deviation of the fiber volume fraction distribution in the Gaussian case would not have a large influence on the void distribution as the random 89

111 case represents the upper limit for standard deviation and yields similar results. The smaller voids found in tows with statistical fiber distributions are likely to diffuse into the resin before curing. Also seen in Figure 4.7, many of the voids for the tow with a Figure 4.7: Void distribution (blue) within tows with (a) Gaussian (baseline), (b) constant fiber volume fraction distribution, (c) a random fiber distribution, and (d) a linear fiber distribution. Gaussian fiber distribution are very close to the surface of the tow as opposed to the constant fiber distribution where the void is located in the center of the tow. This indicates that the voids in the tow with the Gaussian fiber distribution have a higher potential to escape the tow than the constant fiber volume fiber distribution. Thus, the 90

112 final composite part would be expected to have less micovoids in a tow with a statistical fiber distribution. The capillary number for this flow, with a characteristic velocity of 0.56 cm/s and resin properties given in Table 4.1, is 8.7x10-2. The velocity was calculated utilizing the preform properties in Table 4.1 and Darcy s Law Fiber Volume Fraction Distribution When the macroscopic flow front is very close to the tow, the applied pressure experienced by the tow is very small as the pressure at the macroscopic flow front is close to vacuum pressure. This results in low initial resin flow rates inside the tow which translates into slow void reduction rates as seen to varying extents with all of the fiber distribution types in Figure 4.8, with one simulation shown for each distribution type. At later times when the tow experiences higher pressures due to the macroscopic flow front moving further downstream, the rates of filling and hence the void reduction rate increase dramatically. The tow with a linear fiber volume fraction distribution fills at a faster rate initially due to the higher permeability of the tow in the outer region compared to the inner region. Different packing arrangement also caused the voids to form in different locations. The constant and linear packing arrangements featured a single void in the center of the tow, whereas the random and Gaussian distributions had larger voids near the center of the tow and other smaller voids, distributed throughout the tow. Although the final distribution of void sizes may be a function of how nonuniform is the fiber distribution within the fiber tow, the final void content is not 91

113 Figure 4.8: Influence of the type of fiber packing arrangement on tow filling. significantly influenced by the fiber distribution. The void will stop compressing when the air pressure is equal to the capillary pressure acting on the void plus the applied pressure acting on the tow. At the, each void is assumed to have approximately the same final pressure. In accordance with the ideal gas law, the summation of the void volume times pressure for each void is set to a constant, resulting in the total volume being similar for each distribution type Effect of Air Evacuation Time The duration of time for which the fiber preform is under vacuum has a significant influence on the amount of air removed from the fiber tows, calculated using Eq. (4.8). The amount of air removed from a tow is linearly proportional to the 92

114 Figure 4.9: Influence of air evacuation time before introducing the resin on the resulting void content in the tow. initial void pressure in the tow. Lower initial void pressures correspond to higher tow saturation levels, which is shown in Figure 4.9. Under perfect vacuum, the tow would saturate completely. Leaving the part under vacuum for a longer period of time before introducing the resin corresponds to a higher cost of production. There will be a tradeoff between the cost of part production and production rate (influenced by vacuum time) and quality of the part (tow saturation). Charts such as Figure 4.9 can be utilized to determine the optimal time under vacuum which will yield the best combination of tow saturation and production time Influence of Tow location within the Part We define the non-dimensional location within a part as the distance from the inlet to the tow location divided by the length of the part. The applied pressure, as 93

115 seen by the tow, is much higher in areas near the inlet than near the of the part, as is the pressure gradient across the tow at early time steps. This leads to higher rates of void compression and convection. Figure 4.10 shows the void volume fraction as a function of tow location with and without the inclusion of capillary pressure. The inclusion of capillary pressure has a large influence on the resulting void volume Figure 4.10: Non-dimensional void area as a function of tow location from the inlet, with and without the inclusion of capillary pressure. fraction, particularly for areas close to the vent. This is because the ratio of the capillary pressure to the applied pressure acting on the tow s surface increases when the tow location is closer to the vent. The void content near the inlet is smaller compared to that at the vent as the applied pressure experienced by the tow near the 94

116 inlet is higher enabling it to compress the void to a larger extent, which is supported by experimental results [62]. The void compression rate follows a similar tr Influence of Wetting Properties The wettability of a substrate by a fluid is quantified using the contact angle between the two. Low contact angles indicate better wetting properties. For fibers, the contact angle can be manipulated by changing the sizing on the fiber. Lower contact angles correspond to higher capillary pressures thus increased tow saturation. The influence of fiber wettability on the resulting void volume fraction within the fiber tows is exhibited in Figure The void volume fraction is driven up at a faster rate as the contact angle approaches 90 degrees because the depence of capillary pressure on contact angle is non-linear and contact angles above 90 are considered de-wetting. Figure 4.11: Effect of wetting properties on the void volume fraction. 95

117 The influence of wetting properties is much more critical at areas far from the inlet as the capillary pressure is the driving force Effect of Fiber Radius The fiber radius varies based on the type of fiber utilized and how the fiber is manufactured. Higher fiber diameters yield increased microvoid content because the capillary pressure is lower for larger fiber diameters, which can be seen in Figure Figure 4.12: Effect of fiber radius on microvoid content in a fiber tow. Fiber tows with two different fiber radii show that smaller diameter fibers in the center as compared to on the outside result in lower void content. There are cases, such as hybrid composites, where there will be two different fiber 96

118 types packed within the same tow. We explore two cases of tows packed with different types of fibers. In the first case, the inner half of the tow contains fibers of 2 μm radius (common for carbon fibers) and the outer half contains fibers with a 4.5 μm radius (typical for glass fibers). In the second case, they are switched. There is less void content with the smaller fibers in the center because the center contains the largest voids and a larger capillary pressure associated in regions with the 2 μm radius fibers will compress the void further. 4.4 Summary A novel methodology was presented to include the influence of fiber spacing, air evacuation time, and capillary pressure on tow filling. It also allows for a realistic time depent applied pressure function on the surface of the tow. This model was validated using an analytical solution for a simplified geometry. The elliptic model represents a realistic fiber tow geometry. The time a part is left under vacuum before introducing the resin has a substantial effect on tow saturation. This is because the more air removed from a tow before initiating resin flow, the easier it is to saturate. The location of a tow within the part was shown to have a large influence on the final saturation of the tow. The capillary pressure was also found to significantly influence the tow saturation, particularly at locations farther from the inlet. More importantly, being able to apply different capillary pressures at different locations within the tow, resulting from the non-uniform fiber distribution allowed us to model the formation of multiple smaller voids within the fiber tow, which has been observed experimentally. 97

119 This understanding can be useful to create distributions of fibers in a tow that will result in voids more along the periphery of the tow making it easier to flush them out of the system during processing. The prediction of microvoid formation during the infusion a fiber tow by an infinite volume of resin is the first half of the mesoscale modelling objectives. In the next chapter, a three-dimensional model of tow impregnation by a finite volume of resin will be developed. The directionally depent capillary pressure will be included in this model through modification of the permeability tensor for elements on the flow front. This model would be necessary to develop a process model to predict resin distribution in tows with multiple types of resin. 98

120 Chapter 5 PREDICTION OF CAPILLARY DRIVEN FLOW OF A FINITE RESIN VOLUME INTO A FIBER TOW 5.1 Introduction Motivation Additive manufacturing is utilized to create complex structures while minimizing wasted material. Fuse deposition modelling (FDM) is an additive manufacturing process in which a material in the liquid form is deposited onto a substrate by a 3D printer [63]. This allows for the precise placement of material, layer by layer, creating a complex three-dimensional structure. FDM has begun to be utilized to manufacture composite materials, whose strength and stiffness are much higher than parts comprising of the polymer without reinforcement [64]. There are many applications for composite materials where it is desirable to have multiple types of resin systems, such as ballistic protection as well barriers for smoke and toxins, which are of particular importance in military environments [65]. These hybrid resin composites can be manufactured using co-injection resin transfer molding, in which both resins are injected into a fibrous preform simultaneously, separated by a barrier [12]. The resulting composite contains a specified number of glass fabric plies impregnated with epoxy for structural integrity and polyurethane for 99

121 energy absorption [13]. It was shown that hybrid resin co-injected composites dissipated more energy than plies only impregnated with epoxy [13]. A novel avenue for designing composites can be developed by additively manufacturing hybrid resin composites. Impregnating tows within a fibrous preform with a resin utilizing FDM will result in a partially impregnated preform. This preform can then be impregnated with a second type of resin to create a hybrid resin composite with controlled resin topography. This process, described in Figure 5.1 for the case of a single fiber bundle, utilizes a 3D printing head to dispense drops with a specified drop size and spacing. The size and spacing can be controlled by manipulating the size of the nozzle opening, flow rate of resin through the nozzle, and velocity of the preform/printing head (deping on which has a fixed location). The first advantage of this method is that it does not require a layer dividing the part found in traditional co-injection resin transfer molding processes [66]. Elimination of the separation layer allows for a continuous boundary between all plies, thus decreasing interply issues. This manufacturing technology would also allow for tailoring of resin Figure 5.1: Diagram of process for depositing drops of resin onto a fiber tow 100

122 content within tows to improve shear performance by allowing interlocking between resins, shown in Figure 5.2. Interlocking has been shown to increase the strength of composites through the hook and loop interlocking found in Han-3D-Fabrics [67]. This work develops a methodology for predicting the influence of processing parameters on how resin drops will be distributed within a tow. We will examine the roughness of the resin distribution as a means of quantifying interlock between resin types. It will also explore how utilizing resin films instead of drops will affect the resin topography. Figure 5.2: Boundary between two different resin types for (a) smooth and (b) interlocking interfaces Background Information Flow through porous media, including flow into fiber tows, is traditionally predicted utilizing Darcy s Law [40,52], which is given with the conservation of mass equation below [9]: v = K P (5.1) μ v = 0 (5.2) 101

123 The velocity of the resin flow front is given by v and the pressure and viscosity of the resin are given by P and μ respectively. K is the permeability tensor, which represents the porous medium s ability to allow passage of liquid through it. Under ambient conditions, the resin will flow due to a pressure gradient created due to capillary forces between the fibers and the resin. Gravitational forces are neglected because they are much smaller than the capillary forces, shown in Eq. 5.3, with the characteristic length being the initial resin drop radius for the baseline case. Bo = ρgl2 γ = (1170 kg m 3) (9.81 m s 2) ( m) kg s 2 = (5.3) Necessary inputs for Darcy s Law can be found through analytical methodologies. The permeability of a hexagonally packed fiber bundle can be described for axial (along the fiber direction) and transverse flow (perpicular to the fiber direction) [54]: K Axial = 8r fib 2 (1 v f ) 3 (5.4) c v 2 f K Transverse = 16 9π 6 r fib 2 ( π 1) 2 3v f 5/2 (5.5) Here, v f is the fiber volume fraction within the bundle r fib is the radius of the individual fibers, and c is a geometrical constant. For flow through an equilateral triangle pipe the Darcy friction factor, c, is 53. These equations are valid for fiber volume fractions greater than The axial and transverse components of capillary 102

124 pressure, which is the primary driving force of the resin into the tow, can also be determined analytically. The axial capillary pressure as a function of resin surface tension (γ) and the contact angle between the fibers and resin (θ) is given by [42]: P c,axial = 8γv fcosθ r fib (1 v f ) (5.6) Ahn et al [42] utilize an equivalent pore diameter in their description of axial capillary pressure in Eq. (5.6), which is applicable because the cross section is constant in the axial direction. In the transverse direction, flow is more complicated due to a location depent pore size and the flow front contacting new surfaces as it moves, thus an equivalent pore diameter approach is not suitable. Yeager et al employed this equation as part of a methodology to predict the average capillary pressure of resin moving through a unit cell containing fibers, which is presented in Chapter 3 [68]. This method will be utilized to predict the transverse capillary pressure in this chapter similar to the methodology used in Chapter Methods The flow of resin into a three dimensional tow with directionally depent properties cannot be solved analytically. Liquid Injection molding Simulation (LIMS), a control volume based finite element software, was selected to predict resin flow in our porous domain [61]. LIMS solves for the pressure distribution in the domain through combination of Darcy s Law and the conservation of mass [61]: ( K P) = 0 (5.7) μ 103

125 Once the pressure gradient is calculated, Darcy s Law is employed to calculate the flow rate at each node and advance the flow front. LIMS is an ideal choice because it allows the user to modify material properties and boundary conditions at each node and element at each time increment. The model presented in this chapter will investigate the flow of finite resin volumes into fiber tows, taking into account the directionally depent permeability and capillary forces Model Setup A model was developed to predict the spreading of drops with selected volumes into a unit cell, as depicted in Figure 5.4. The simulation is initiated when the first Figure 5.3: Description of unit cell for resin spreading into fiber tow at initial time step 104

126 drop contacts the tow within the unit cell. Each node where the resin and tow contact each other will be set as an inlet, with the pressure driving the flow equal to the average capillary pressure of all the inlet nodes. After a specified amount of time, which is influenced by the flow rate of resin through the nozzle and the surface tension of the resin, a second drop will be introduced into the unit cell. This allows us to include the effect of neighboring drops in our model. The spacing between drops will dep on the time between drops and the speed at which the nozzle is moving with respect to the tow. A setup of the geometric and boundary conditions in the unit cell, at the instant the second drop is released, is shown in Figure 5.3. The first drop will continue to permeate the tow until its entire volume is inside of the tow. The size of the drops and corresponding inlet nodes will change over time as the drop becomes smaller as described in section The volume of each drop is tracked separately by computing how much resin impregnated the tow at the current time step by multiplying the flow rates with the time step at the inlet nodes. The condition for the program is when there is no resin remaining outside of the tow. There is a no flow condition across the symmetry planes. The baseline properties are given in Table

127 Figure 5.4: Geometric and boundary conditions, shown at the time the second drop is introduced into the system Table 5.1: Baseline properties utilized for drops wicking into tow model Fiber diameter (μm) 9 Contact angle (degrees) 30 Fiber volume fraction (tow) 0.4 Tow aspect ratio 4 a Tow (mm) 2 Resin viscosity (Pa s) 0.95 Resin surface tension (N/m) 0.07 Resin to porous volume ratio 0.20 Gap between drops (mm) Determining Tow-Drop Contact Area The inlet nodes are defined as nodes on the surface of the tow, within the tow-resin contact area. The tow-resin contact area is influenced by the wetting properties 106

128 between the tow and the resin, the remaining volume of the resin outside the tow, and the tow dimensions. It is assumed that the radius of curvature of the resin drop is equal in the xy and yz planes. This assumption is common for wetting scenarios on small length scales because the surface tension of the resin is stronger than gravitational forces. The resin can be treated as a spherical cap resting on top of the fiber tow, with the sphere having a radius r and center at x = 0 and y = y c as depicted in Figure 5.5. The center of the sphere in the axial direction will be depent on the unit cell dimensions, as already explained in Figure 5.3. The y-location of the center of the sphere, which the resin spherical cap is a part of, can be calculated with the following equations: m 2 = tanβ 2 = tan(β 1 + θ) = tan( θ) + m 1 1 m 1 tan( θ) = x 1 y 1 y c (5.8) m 1 = x 1 y 1 ( btow atow ) 2 (5.9) Here the slopes and unknown angles are described in Figure 5.5. The y-location as a function of x 1 and y 1 can then be substituted into the following two equations to solve for the two unknowns (x 1 and y 1 ): (y 1 y c ) 2 + x 2 1 = r 2 (5.10) V = 4 [ r 2 x 2 y 2 0 x 1 0 r 2 x 2 + y c b tow 1 ( x 2 ) a tow ] dzdx (5.11) V is the volume of the resin drop that has yet to impregnate the tow. The dimensions of the tow-resin contact area are utilized in conjunction with the known center in the z- 107

129 direction to determine which nodes on the tow s surface can be denoted as inlets. The tow-resin contact dimensions are recalculated at each time step as the resin impregnates the tow. Figure 5.5: Description of parameters necessary for calculating the resin's radius of curvature Inclusion of capillary pressure This model is designed such that the directionally depent capillary pressure may also be location depent. The previous chapter described and verified a 108

130 methodology for including capillary pressure in two-directional flow through manipulating the permeability of nodes on the flow front. Once the flow front fills the elements with modified permeability value, their permeability value was reset to the original value. We modify the governing equation of this method for the case where capillary pressure is the only driving force. The new governing equations, with permeability modified in the transverse and axial directions indepently are: K Transverse,modified = K TransverseP c,transverse P c,ave (5.12) K Axial,modified = K AxialP c,axial P c,ave (5.13) Here Pc,ave is the average of the axial and transverse capillary pressure over all of the elements Quantification of Interlocking One of the novel elements of this work is that it allows for the prediction of the distribution of a finite volume of resin within a tow. As described in Figure 5.2, the interlocking will be quantified by the roughness of the resin flow front shape. The Figure 5.6: Schematic of final resin distribution within tow. Resin flow front position is given in white and described by y=f(z). The standard deviation of this function, std y, will be utilized in quantifying roughness. 109

131 results for the flow front location from the LIMS results will be mapped into a finite number of points to reconstruct the flow front shape at the yz-plane, shown connected by line segments in Figure 5.6. The average and standard deviation (std y ) from the set of y data points are calculated, representing the depth of infusion and roughness respectively. To ascertain a more meaningful quantity to describe roughness, we introduce the non-dimensional flow front roughness as: Roughness = std y d tow (5.14) Where d tow is the depth of the tow (maximum distance the resin can infuse) Mesh Refinement Study The final distribution of resin within the fiber tow should be mesh indepent to be considered a valid solution. Figure 5.7 shows the results for the flow front shape Figure 5.7: Mesh refinement study for the flow front shape on the yz-plane 110

132 over a number of meshes with different element sizes, confirming that the solution is mesh indepent as the number of elements is increased. The yz-plane was selected for this because it shows both axial (z-direction) and transverse (y-direction) flow. The baseline properties were utilized for this study. 5.3 Results One of the novel elements of this work is the ability to predict the distribution of multiple resin drops, placed at different times, within a fiber tow. Figure 5.8 shows Figure 5.8: Impregnation of a unit cell which represent half of a tow due to symmetry with two drops of resin (red) at (a) the initial time step, (b) after about half of the first drop was inside the tow, (c) right after introducing of the second drop, (d) while both drops are still spreading into the tow, and (e) at the of the simulation after both drops have completely impregnated into the tow 111

133 this infusion process at selected time steps. It is noted that the second drop spreads in faster because there is a larger flow front area. The second drop also has a larger infusion depth, which is because its flow front reaches the first drops flow front in the axial direction and therefore has less space to axially spread. For an increasing fiber volume fraction, the ratio of axial to transverse capillary pressure decreases and permeability increases. These effects negate each other since flow is related to the ratios multiplied by each other, as seen in Darcy s Law. Thus, the fiber volume fraction, for a given resin to pore volume ratio, does not have a large effect on the resin distribution. The capillary number for this flow is 1.6x10-3, with a characteristic velocity of 0.1 mm/s and resin properties given in Table 5.1. Verification that the mapping function is properly implemented with the LIMS results is shown in Figure 5.9. Figure 5.9: Mapping of the LIMS results on the yz-plane into a finite number of points (y,z). Purple represents the resin locations, red represents porous regions, and intermediate colors are utilized to show partially saturated regions. In the bottom picture, the mapping to calculate roughness is shown in white. 112

134 5.3.1 Influence of Drop Spacing The spacing between the drops, which can be influenced by the resin flow rate and the rate at which the nozzle and tow are moving with respect to each other, plays a large role in resin distribution. As shown in Figure 5.10, for lower resin to porosity volume ratios, less spacing between drops will lead to decreased roughness values because the flow fronts from each drop coalesce and smooth out faster. The results an optimal resin to porosity volume ratio for each drop spacing studied when the goal is to increase roughness, shown in Figure For low ratios, the drops do not penetrate deep in the y-direction, leading to low roughness values. As more resin is added, each drop penetrates deeper and deeper in the y-direction, leading to a greater difference in depth of infusion on the sides and right below where the drop is placed Figure 5.10: Comparison of resin distribution for (a) 5.00 mm between drops and (b) 2.50 mm between drops (purple represents resin, red represents porous areas, other colors indicate partially filled nodes) 113

135 and higher roughness values. After sufficient resin has been added, the resin reaches the bottom of the tow, and the resin flow front begins to flatten out, leading to roughness values decreasing with increasing volume. The optimal resin to initial porosity ratio did not dep on the gap between drops, although more resin is required to reach the optimal roughness when the drops are spaced farther apart. Figure 5.11: Roughness as a function of resin to tow porosity ratio for selected drop spacings Effect of Packing Arrangement If one were to desire a uniform spacing without the flow front reaching the bottom of the tow, they would have to deposit many drops per millimeter which is unrealistic. With permeability and capillary pressure both being depent on fiber volume fraction, it would stand to reason that one would be able to manipulate fiber volume 114

136 fraction as a function of location within the tow to achieve a smooth flow front of a desired depth. Tension or pressure on the fibers can be utilized to vary fiber spacing as a function of location within the tow. If the fiber tow were to be packed more tightly in the center than near the edges, the flow front shape would become more uniform as shown in Figure The resin fills the areas with lower fiber volume fractions first because they are much more permeable. The roughness, as one would expect, decreases drastically when the fiber spacing is changed from constant to linear in the y-direction (roughness reduces 0.22 to 0.03 for drops that are spaced 2.50 mm). Figure 5.12: Fiber volume fraction distribution and flow front shape for (a) constant fiber volume fraction distribution and (b) linear fiber volume fraction distribution 115

137 5.3.3 Charge Type An alternative methodology to introduce resin into a fiber tow with a finite volume of resin would be to strategically place resin films on the surface instead of drops. This is desirable when dealing with viscous thermoplastic resins which can be formed into films and then melted to impregnate the tow. The resin distribution within the tow can be manipulated by utilizing films with different dimensions (ie long thin films, short fat films, etc.). The inlet dimensions were assumed to be constant for predicting the impregnation of films into fiber tows. Figure 5.13 compares the final resin distribution for when a resin charge is placed on a tow in the form of films of Figure 5.13: Influence of charge type and film aspect ratio on final resin distribution within a tow (shown are the top view, bottom view, and two cross section views of the resin distribution) 116

138 varying dimensions; all scenarios contained the same volume of resin. Film dimensions can be manipulated to create more even resin distributions in the axial or through thickness direction of the tow, based on whichever yield more desirable properties for the particular application. 5.4 Summary A novel model was developed to simulate the wicking of a finite volume of resin into a fiber tow. This process model can predict the final distribution of resin deposited on the surface of a tow via 3D printing. The influence of tow fill percentage on resin surface roughness was explored. It was found that there was an optimal amount of resin that could be added to the tow to increase resin roughness for a given tow geometry. This work showed that the roughness could be influenced through introducing a location depent fiber volume fraction. The form (film or drop) in which resin was introduced was also shown to influence the configuration of resin within the tow. This work will be useful to anyone interested in developing hybrid composites with a controlled resin distribution. 117

139 Chapter 6 CONCLUSIONS, CONTRIBUTIONS, AND FUTURE WORK 6.1 Conclusions This work focused on developing a suite of models capable of predicting resin flow between fibers and within fiber tows spanning from the micro to meso scale. Emphasis was placed on the influence of processing parameters on the final resin configuration. First, the dynamics of wetting individual fibers was examined with an experimentally verified numerical model. This provided insight into the rate at which a drop would spread on an individual fiber as well as how far along the fiber the resin would spread before reaching equilibrium. The model was then exted to unit cells containing multiple fibers to study the influence of packing arrangement and neighboring fibers on the wetting process. Three and four fiber models represented the common square and hexagonally packed fiber arrangements. Parametric studies were conducted to gain insight into the relationship between processing parameters and resin configuration for the selected packing arrangements. This knowledge will allow for the design of composites with novel microstructures to achieve a desired set of mechanical properties. In addition to predicting microstructure, microscale unit cell models are also critical in predicting properties such as the transverse capillary pressure resin will experience during infusion of fiber tows. The microscale unit cell numerical model was modified to predict capillary pressure of resin moving between fibers. This model 118

140 showed that capillary pressure was indepent of applied pressure. It was also utilized to validate an analytical model for predicting the average capillary pressure of resin moving through a unit cell containing fibers. These models were utilized to examine the effect of packing imperfections on capillary pressure. The effect of including fibers with different surface treatments was also studied. A method to rapidly predict the average capillary pressure of resin moving through a unit cell will lead to more accurate mesoscale tow filling models. Then, a novel methodology was developed for including the microscale capillary pressure in a two-dimensional mesoscale tow filling simulations. The model developed was capable of predicting void formation in fiber tows with stochastic fiber volume fraction distributions. The formation of multiple smaller voids was predicted as observed in practice. The ability to predict microvoid formation within a tow is important because smaller voids will dissolve into the resin and voids located near the edges of tows are more likely to escape during processing. The influence of air evacuation time, capillary pressure, and location of different fiber types for hybrid tows on microvoid distribution was studied. This understanding is important for anyone desiring to control microvoid size through manipulating fiber surface treatment, arrangement, or the time a preform is under vacuum before infusion. Finally a model was developed capable of predicting the three-dimensional wicking of a finite resin volume into a fiber tow. The process model can be utilized to predict the distribution of resin within a tow for resin drops or films deposited on the surface of the tow. Parametric studies were conducted to examine the influences of resin volume and spacing between drops on the configuration of resin within the tow. 119

141 This work is useful to anyone designing hybrid resin composites with controlled resin distribution within tows. The suite of models developed in this work can predict the distribution of resin within tows over multiple length scales. The result would be a porous composite, with porosity depent on the selected process and processing parameters. In some cases, the porous part will have the desired set of properties; in others the porous structure will be infused with a secondary resin to meet the design specifications. These models can be employed in conjunction with mechanical models to study the interplay between processing parameters and mechanical performance and optimize composite materials for selected applications. Experiments such as the quasi static punch shear test and impacting the part with a projectile could be utilized to quantify the energy absorbed by the composites. 6.2 Contributions This work yielded unique contributions over multiple length scales, which are summarized below. On the microscale, this is the first work to study the dynamics of a finite volume of resin wetting an individual fiber. The three-dimensional model was the first to predict the final rein configuration within three and four fiber unit cells, packed in common fiber packing arrangements. Second, this work was the first to predict the average capillary pressure of resin moving through unit cells containing hexagonally packed fibers. The model is capable of determining the influence of neighboring fibers on capillary pressure because it includes the case where the resin contacts new fiber surfaces. It was determined that capillary pressure is not influenced by an applied pressure gradient, 120

142 which has been assumed but not confirmed before. This model allowed for examining the influence of imperfect packing arrangements on capillary pressure. It also was utilized to analyze the capillary pressure between fibers with different surface treatments. Developing an analytical model which can rapidly (much less than a second) calculate the average capillary pressure of resin moving through a unit cell with a given arrangement and surface treatment is extremely useful when modelling tows with stochastic fiber volume fraction distributions. A novel methodology was developed to include the microscopic capillary pressure in mesoscale tow impregnation models. The manner in which this is done permits location and direction depent capillary pressures to be incorporated into the models. This allows for studying the formation of multiple microvoids in tows with stochastic fiber volume fraction distributions, which is representative of what occurs experimentally. This work studied the influence of air evacuation time, hybrid fiber tows, and capillary pressure on microvoid formation. Finally, a novel three-dimensional model was developed to predict the wicking of a finite resin volume into a fiber tow. This was the first of its kind and is able to predict how drops and films of resin spread into fiber tows, with the user being able to control material, geometric, and process parameters. This opens up a novel ability to control the distribution of resin within a tow in hybrid resin composites. 6.3 Future Work The following are opportunities for expanding upon this work in the future. 121

143 6.3.1 Including Fiber Movement in Microscale Model The microscale model of resin moving between fibers presented in Chapters 1 and 2 assume that the fibers are fixed in space. As the resin moves between fibers, the capillary pressure applies a force on the fibers, which may cause fiber rearrangement that in turn will change the capillary pressure. This coupling between capillary pressure and fiber movement has yet to be explored and quantified. Developing a multiphysics model (mechanical and fluid flow) would lead to increased accuracy in predicting resin distribution between fibers on a microscale. In such a model, resin movement, the capillary pressure, and fiber position could to be calculated at each time step. It also must be considered that the time scale for each event (resin reconfiguration, fiber movement, etc.) may rer it unnecessary to recalculate both fiber position and resin configuration at each time step (For example, if the resin reaches equilibrium in 0.1 second and it takes the fiber 10 seconds to move to its new location, it could be possible to do the model in sequence as opposed to coupling the physics) Manufacturing and Testing of Composites with Optimized Microstructure This work has developed a process model capable of predicting resin distribution in unit cells containing a small number of fibers. The numerical model can be utilized in conjunction with a mechanical model to determine which processing parameters are required to manufacture microcomposites tailored for specific performance applications. These composites would then need to be manufactured. Once the process parameters are determined, the first challenge would be to determine a methodology for placing fibers in desired location with respect to each other, which is a complicated problem due to the static forces between individual filaments. One 122

144 would also need to either find a resin pump capable of dispensing extremely small drops or develop a method for susping individual solid particles of resin between fibers. This would allow for the creation of microcomposites with pre-determined microstructures. Another opportunity to advance this work would be to use the process model to drive particle sizes for particle impregnation methods. This problem has the added complexity of determining how to get the optimal concentration of particles inside of the tows Determination of Capillary Pressure for Different Microstructures With framework for capillary pressure, the numerical model could be exted to study a variety of different cases relevant to composites processing. In experimental results, the fibers are not uniform in the axial direction; they intertwine as a function of location within the tow. The numerical capillary pressure model could be modified to calculate the capillary pressure of resin moving between fibers as a function of tortuosity. This would allow for three dimensional models of resin spreading into tows, and a correlation between microvoid distribution and fiber misalignment Twisted Tows In addition to a three dimensional void formation model, it would also be useful to include void dissolution in the model. An approximation for void concentration is given in Appix B, but this model does not include resin curing or the effect of concentration changes in the resin due to neighboring voids dissolving. It 123

145 would also be worthwhile to examine how twisting the tows in the direction perpicular to the axis would influence microvoid formation Hybrid Composites A mechanical model could be developed with the output from the model presented in Chapter 5 as an input. This could be utilized to determine which processing parameters yield the best composite tow. Once identified, tows could be manufactured with these attributes and compared to the fully infused baseline through the quasi-static punch shear test. A 3D printer could then be utilized to place the desired volume of resin, with the selected spacing between drops, on a single glass fiber ply. The ply can then be impregnated with a secondary resin, the order of which would be determined utilizing a mechanical model. These plies could then be impacted with a projectile to compare performance in high strain rate applications. Some of this preliminary work has been done and presented in Appix C. 124

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153 ANALYTICAL MICROSCALE MODELS A.1 Drop of Resin between Four Fibers The numerical model works well for resin spreading between four fibers, but is computationally expensive. Similar to the capillary pressure model, it is desirable to have a rapid methodology for predicting the final resin configuration of a drop wetting fibers packed in a square packing arrangement. This will allow for the development of stochastic models in the future. The particle of resin wets the fibers in both the axial and transverse direction. The main assumption in this model is that the drop reaches an equilibrium state when the capillary pressures in the axial and transverse direction are equal. In the transverse direction, the capillary pressure can be calculated between the fibers as [39]: P c,transverse = γcos (θ + α) r(1 cosα) + d (A.1) The pressure across any air-liquid interface can be calculated with the Young-Laplace equation [27]: ΔP = γ ( 1 R R 2 ) (A.2) Utilizing geometry to derive an equation for the radius of curvature in the two orthogonal planes selected, the capillary pressure in the axial direction can be found as: P c,axial = γcosθ 2r (1 + π 4v f 1) (A.3) 132

154 This assumes that the angle of contact between the resin and the fibers is the static contact angle. The orthogonal planes selected to derive Eq. (A.3) were the two diagonals of the square, exted in the axial direction. The maximum fiber volume fraction for square packed fibers is π/4, so there is not a plausible scenario where there could be a negative number inside the radical of Eq. (A.1). Figure A. 1: Cross section view of resin wetting square packed fibers Now, the task is to see how far the resin will spread around the fibers. To do this, the capillary pressures in both directions are set equal to each other, leaving one equation 133

155 and one unknown (the directional body angle, described in depth in Chapter 3). After algebraic simplification, the directional body angle can be found from: cos (θ + α) 1 2 π v f cosα = 2v fcosθ π (A.4) Being periodic, there will be multiple solutions for this equation. Since the resin is only assumed to begin in contact with the fibers at one point per fiber, it is assumed that the lowest positive solution for the directional body angle (α). It is assumed that the resin is uniform in the axial direction (this will introduce a small amount of error since the s of the resin have some curvature, but the error is negligible except for the case of very low wetting lengths). With this assumption, the wetted length (L) in the axial direction can be computed: L = V Resin 4(r + d + rsinα) 2 (π + 4α)r 2 4r 2 sinα (A.5) Here, 2d is the gap between fibers. This does take into account the curvature of the resinair interface in the cross section perpicular to the axial direction. Finally, the contact area between the fibers and resin can be calculated: A c = (2π + 8α)V Resin r[4(r + d + rsinα) 2 (π + 4α)r 2 4r 2 sinα] (A.6) The non-dimensional contact area, as presented in Chapter 3, is given by: A c,nd = A cr V (A.7) To validate this analytical model, the non-dimensional contact area between the fibers and resin is compared over different fiber volume fractions, shown in Figure A.2. This allows the rapid prediction of resin distribution between fibers for a range of fiber volume fractions and resin particle sizes, which is critical in the development of stochastic models including thousands of unit cells with unique properties. 134

156 Figure A.2: Comparison between the numerical and analytical models for resin wetting a square packed unit cell A.2 Commingled Hexagonally Packed Unit Cell Commingled fiber tows have thermoplastic resin fibers dispersed throughout the fiber tow. Upon heating, the resin fibers will melt, wet the structural fibers, and cure to create a composite. This problem is assumed to be axially symmetric since the fibers are uniform in the axial direction. In this case, the resin will reconfigure itself as it wets the surface of the fibers, stopping when the directional body angle satisfies: A Resin,0 = 6 [(4r 3π 7vf (d + r)) cos π 3 + rsinα(2r + 2d + rcosα)] 3r 2 ( π 3 + 2α) (A.8) 135

157 The initial area of the resin fiber can be calculated by multiplying its radius squared by pi. The directional body angle (α), d, and r are described in Figure A. 1. The final fiber-resin contact area can then be calculated: A c = 6r ( π 3 + 2α) (A.9) Sample results from the numerical model are shown in Figure A.3, with the percent error between the numerical and analytical model being 2.98%. Figure A.3: Numerical results for commingled resin fiber wetting hexagonally packed glass fibers A.3 Hexagonally Packed Cell with Resin Coating on Center Fiber A surface treatment, known as a sizing, is commonly applied to fibers during processing. In addition to a surface treatment, if some fibers had a thermoplastic resin applied to their surface during processing, that resin would melt and wet other fibers upon heating. Hexagonal packing is assumed, with the center fiber having a 136

158 thermoplastic coating on it. The resin will wet the surface of the other fibers until the following condition is satisfied: A Resin,0 = 6 [(4r 3π 6vf (d + r)) cos π 3 + rsinα(2r + 2d + rcosα)] 3r 2 ( π 3 + 2α) πr2 (A.10) Here, the initial resin area can be computed through comparing the resin radius and the radius of the fiber it coats. Once α is known, the final resin configuration is known, and the fiber-resin contact area can be described by: A c = 6r ( π + 2α) + 2πr 3 (A.11) A typical result from this method is shown in Figure A. 4. Here the percent difference between numerical and analytical solution was 4.8%. Figure A. 4: Numerical results for hexagonally packed glass fibers, with the center fiber intially having a thermoplastic coating on it 137

159 Appix A MICROVOID DISSOLUTION WITHIN TOWS B.1 Void Dissolution We begin with the results from the model discussed in Chapter 4, which would be a microvoid distribution within a fiber tow. The air comprising voids for a given system will either partially or fully dissolve into the resin, deping on the size of the void and the length of time before the resin cures. Wood and Bader predict this dissolution, neglecting the effect of surface tension, with [69]: R 2 = R 0 2 2DC S ρ (1 C C S ) t (B.1) Here, R is the bubble radius at time t, R 0 is the initial bubble radius, D is the diffusion coefficient, C S is the saturation concentration of the gas dissolved in the liquid, and C is the concentration of the gas dissolved in the bulk liquid, both having units of density. Eq. (B.1) justifiably neglects the transient term in Epstein and Plesset s differential equation describing the bubble radius because of the long processing time for composites [69,70]. The quantity C /C S is defined as the relative saturation of a gas in a liquid. For our purposes, the relative saturation can range from 0 (no air dissolved in the resin) to 1 (resin is fully saturated by air). It has been shown that the relative saturation 138

160 of the resin may be manipulated by passing bubbles through it, which creates nucleation sites [71]. The saturation of the liquid at the bubble/liquid interface is equal to C s, which is the equilibrium concentration of the gas dissolved in resin for a given partial gas pressure in the bubble. The concentration of gas dissolved at the resin/air interface can be found with Henry s Law, which relates the partial pressure of a gas and the amount of gas dissolved in a liquid [72]: C S = HP G (B-2) Where H is Henry s constant, and P G is the partial pressure of the gas in the bubble. As C approaches C S, the air takes much longer to dissolve (or infinitely long if they are equal). Henry s constant for air in epoxy, which will be utilized in this paper, was 7 kg found experimentally to be m 3 Pa 1 [73]. The temperature depent diffusion coefficient can be described by (derived by performing a logarithmic fit on the Wood and Bader experimental data [69]): D = 2.93e ( 7761 T ) (B-3) Here T is the processing temperature of the composite (in Kelvin) and the units of D are m 2 /s. The baseline properties for our model are provided in Table B.1. Table B.1: Baseline properties utilized for void dissolution model Fiber radius (μm)

161 Contact angle (degrees) 30 Fiber volume fraction 0.6 Tow aspect ratio 8 a Tow (mm) 2 P Inlet (kpa) 100 P Outlet (kpa) 0 Resin viscosity (Pa s) 0.95 Resin surface tension (N/m) 0.07 L part (cm) 30 L tow (cm) 15 Initial air pressure in tow (kpa) 10 Relative saturation 0.5 Temperature (K) 295 Resin cure time (Hours) 8 K L,Preform (m 2 ) B.2 Results B.2.1 Fiber Volume Fraction The fiber volume fraction was shown to have a significant influence on the resulting void size within a tow. The void volume fraction at the of the void Figure B.1: Saturation of a tow with (a) 0.40 fiber volume fraction and (b) 0.70 fiber volume fraction. 140

162 compression phase is shown in B.1 for volume fractions of 0.4 and 0.7, with the red representing fully saturated areas and blue signifying voids. The size of the voids is decreased for higher fiber volume fractions because the capillary pressure driving void compression is much higher, shown graphically in Figure B.2. Smaller voids will also dissolve into the resin in a much shorter amount of time as Eq. (B.1) shows a linear relationship between the time it takes a bubble to dissolve and R 2 0. The bubbles are assumed to stop dissolving when the resin is cured, making the final void content Figure B.2: Influence of the type of fiber volume fraction on void size within a fiber tow, shown at the of the compression phase and after the part has cured. influenced by the rate at which the bubble dissolves. 141

163 B.2.2 Liquid Saturation The dissolution of air into the resin is driven by the concentration gradient of compressed air as you move from the bubble to the unsaturated resin. The dissolution process is slowed as the relative saturation of the air in the resin increases, resulting in a higher void content as seen in Figure B.3. This effect is seen to a greater extent for tows with lower fiber volume fractions because the starting bubble sizes are larger for tows with a decreased capillary pressure. Figure B.3: Influence of relative saturation of the resin on void size. 142

164 B.2.3 Processing Temperature and Cure Time The diffusion constant in Eq. (B.1) is sensitive to processing temperature changes. Higher temperatures yield increased diffusion constants, speeding up the dissolution of bubbles into the resin, shown in Figure B.4. Conversely, increasing temperatures will also increase the rate at which the thermoset resin will crosslink, decreasing the cure time. Understanding the interplay between cure time and bubble dissolution time can allow for processing temperatures to be optimized for minimum void content. Figure B.4: Effect of cure time and processing temperature on void content. 143

165 B.3 Summary The flow of resin into fiber tows will result in a microvoid distribution within the tow. As the flow patterns lead to the formation of an increased number of microvoids, the average size of the microvoid decreases. Smaller microvoids dissolve into the resin at a much faster rate, leading to decreased void content when compared with a method which yields just a single centralized void. The extent at which this occurs was shown to be largely influenced by the fiber volume fraction. The void content was also significantly affected by the initial air concentration within the resin. Increased processing temperatures were shown to have a strong influence on the rate at which the bubbles dissolved into the resin. This understanding is of particular interest to those desiring a more accurate methodology of predicting microvoid content within fiber tows. 144

166 Appix B EXPERIMENTAL CHARACTERIZATION OF POROUS AND HYBRID TOWS C.1 Introduction Individual tows within a composite will be subject to loading during impact from a projectile. At short time scales the impacted tow will be transversely compressed by the projectile and on longer time scales it will be under axial tension [72,73]. To characterize the response of the tows to loading, compression and tensile tests will be performed. Samples that will be tested include S2-glass fiber tows: 1. Fully saturated with SC-15 epoxy 2. Partially impregnated with SC-15 epoxy and partially porous 3. Infused with SC-15 epoxy and polyurethane. A cross-section view of the resulting composite tow is shown in Figure C.1. Figure C.1: Cross-section schematic of fiber tow. Tow will first be partially infused with SC-15 epoxy. It will then either be tested as is, or it will be insfused with polyurethane resin before being tested. 145

167 C.2 Material Processing The glass fibers chosen were S-2 glass fibers, which are known for their energy absorbing capabilities. SC-15 epoxy and Adiprene L-100 (polyurethane) were selected as the resin systems for their strength and strain to failure respectively. The SC-15 epoxy was mixed with a ratio of 100 g Part A to 30 g Part B. It was cured for 24 hours at 25 ºC. The polyurethane utilized was Adiprene L-100, cured with the 4,4-methylene-bis (2-chloroaniline) (MBCA) curing agent. Mixing and curing was in accordance with the manufacturers recommations. The resin was mixed with the Adiprene at 70 ºC and the MBCA at 125 ºC (121 ºC is minimum temperature for the MBCA to transition from solid to liquid form). The mixing ratio of the Adiprene to MBCA was 100:12.4 parts by mass. The polyurethane was cured for 1 hour at 100 ºC and post-cured for 16 hours at 70 ºC. The tows were pulled taught across a frame before being infused to minimize variation in how tight the tow was packed. The tows were first infused with SC-15 epoxy. The resin was applied along the length of the tow and evened out along the length with a cotton swab. The amount of time it took for the resin to wick into the tow was long enough that there was sufficient time spread the resin along the tow before the resin wicked into the tow. This was verified by cutting the tow into equal length segments and comparing the weight of each segment. The tows were left undisturbed for a minimum of 24 hours after the resin was applied. Samples of tows infused with epoxy resin (dyed orange since it is difficult to see it bls in with the fibers in its natural color) are shown in Figure C.2. The mass of the tow before and after infusion was recorded along with the length of the tow that was infused with resin. If the tows were meant to be part of the epoxy/porous sample set, they were set aside until it was time to prepare them for testing. Otherwise, the polyurethane resin 146

168 was applied in the same manner and cured as previously described and the tow s mass was recorded again. The tow width and height were also recorded for all samples. The epoxy had a long enough pot life that over 20 samples could be prepared with each pot. The polyurethane resin was found to have a very short service life, thus many pots were heated in the oven simultaneously and were mixed and used one at a Figure C.2: Tows with epoxy (dyed orange here) painted on them. They are pulled taught across the frame to minimize variations in fiber volume fraction within and between different tows. time. It was noted that the viscosity of the polyurethane increased significantly as the temperature decreased. C.3 Mechanical Testing C.3.1 CompressionTesting The tows were tested in transverse compression testing where they were subject to load-unload cycles up to predetermined loads. The loads, when converted 147

169 to stresses for the baseline tow dimensions of 50 mm in the axial direction and 0.4 mm in the width direction, are characteristic of impact velocities ranging from m/s for composites comprised of S2 glass impregnated with SC-15 epoxy resin. The stress predicted for a transverse impact velocity (u) can be approximated with [76]: σ = (ρ E 33 ρ ) u (C.1) Here ρ is the density of the tow (1.85 g/cm 3 ) and E 33 is the modulus of elasticity in the through thickness direction (11.8 GPa). The test setup, depicted schematically in Figure C.3, includes platens on the top and bottom of the tow. The platens are circular, with a radius much larger than the length of the fiber tow. The top platen will compress the tow at a rate of 0.25 mm/minute and the bottom platen will have a fixed position. During this process, Figure C.3: Schematic of compression test setup with the tow between two platens. The platens cover the entire top and bottom surface of the tow with room to spare in each direction. there will be some compliance from the machine, which can be measured by running the test without the tow present (platens will press against each other). The load vs. 148

170 displacement output from the Instron testing machine will be corrected for the system compliance using Eq. C.2. An example of correcting the load-displacement cycles is shown in Figure C.4, the result of which represents the actual load-displacement curve for the sample. δ measured = δ sample + δ system (C.2) Figure C.4: Compression testing - examples of measured data (top left), the system compliance curve (top right), and the material loaddisplacement curve after being corrected for machine compliance (bottom) 149

171 Once the corrected load displacement curve is obtained, it is converted to the engineering stress-strain curve utilizing the measured dimensions of the tow from before it is compressed. The total energy dissipated is calculated form this curve utilizing Eq. C.2. P i E Total = σ(ε)dε t i P i 1 = E Previous peak + (ε n+1 ε n )( σ n+1 + σ n 2 n=t i ) (C.3) Here the stress and strain are given by σ and ε, P i represents the peak stresses for each cycle, t i represents the point on the reload cycle when the stress passes the previous peak. These values are graphically explained in Figure C.5. The stored energy within Figure C.5: Schematic of load-unload cycles, with the peak load (P), the load at which the load in the current cycle passes the peak load from the previous cycle (t), and the fully unloaded state (v) shown for each cycle 150

172 the tow is then calculated with: v i 1 P i E Stored = σ(ε)dε = (ε n+1 ε n )( σ n+1 + σ n ) (C.3) v i 2 n=p i Here v i is valley i, as depicted in Figure C.5. Sample results for a load of kn (roughly 100 m/s impact) are shown for epoxy/polyurethane samples in Figure C.6. There is not a very strong tr for this Figure C.6: Compressive energy dissipated per unit mass by hybrid tows loaded up to kn and back down to 0 kn. Percentage of the tow s porous area filled by epoxy is on the horizontal axis, with the rest of the porous area being impregnated with polyurethane data set, but when looking at the averaged energy absorbed per unit mass for select epoxy infusion percentages (the rest of the porosity is filled with polyurethane) as a function of the applied pressure in Figure C.7-a, it can be seen that the fully infused 151

173 (a) (b) Figure C.7: Average of energy absorbed per unit mass as a function of applied pressure for (a) epoxy/polyurethane hybrid tows and (b) epoxy/porous tows with epoxy data set performs best. This holds true for the epoxy/porous sample set 152

174 too, shown in Figure C.7-b. C.3.2 Tensile Testing During the tension testing, the tows will undergo axial tension until they break. The gage length for the tows is 150 mm, in accordance with ASTM standards for fiber bundle testing. The tabs, measuring 40 mm by 40 mm are attached using Hysol, an epoxy based adhesive. The tab adhesive was cured for a minimum of three days before testing. The displacement rate for the testing was set to 25 mm per minute. The testing setup along with representative failure mechanisms are shown in Figure C.8. The energy absorbed by the tow was calculated by integrating the Figure C.8: (a) Tensile test setup with representative failed specimens for (b) hybrid tow and (c) epoxy/porous tow. Hybrid tows have more brittle failure and the dry portion of the porous tows experiences significant fraying. 153

175 engineering stress strain curve (initial tow dimensions were utilized to calculate the stress and strain from the load-displacement curve). The energy absorbed per unit mass, as a function of the percentage of the tow infused with epoxy, is shown in Figure C.9. Even if one only examined the tows with more than 50% epoxy the conclusion would still that the tows fully saturated with epoxy performed the best. Figure C.9: Energy absorbed per unit mass for all samples with an acceptable failure mechanism (failure occurred within the gage length and not in the tabs) C.4 Summary The overall result for the testing is summarized below: Summary Tension Compression ND Total Epoxy Porous Hybrid All Epoxy

176 The energy absorbed per unit mass is normalized by the fully infused tows energy absorbed per unit mass. The fully infused epoxy tows performed the best. The results from this test do not agree with the results discussed in the motivation section. The testing discussed in the motivation included all of the energy dissipation mechanisms, as opposed to this work only examining two. There are a lot of mechanics not included when the problem is simplified as much as it has been here. This oversimplification of a ballistic impact rers these results not useful. It would be prudent for work in the future to focus on testing mechanisms which combine the transverse and axial loading, such as the quasi static punch shear test. 155

177 Appix C MATLAB CODE FOR CAPILLARY PRESSUER AND TOW INFUSION MODELS D.1 Capillary Pressure Prediction This set of programs calculates the average capillary pressure of resin moving through a unit cell. It implements the methodology discussed in Chapter 3. C.1.1 pcpacked.m: Predicts Capillary Pressure as a Function of Fiber Wetted Length function [resmat] = pcpacked(rfib, vf, theta, surften, sstep) format long; %Calculates the capillary pressure as a function of fiber wetted length for resin moving through a unit cell containing fibers %Inputs include: Fiber radius, fiber volume fraction, contact angle, surface tension of the resin, and step size (very small step sizes yield more accurate results) %Here we calculate some parameters d = rfib/(2*sqrt(3*vf/(2*sqrt(3)*pi)))-rfib; smax = 4*pi*rFib; s = zeros(5,1); cmat = zeros(4,1); stepct = 1; stepstaken = floor(smax/sstep); resmat = zeros((stepstaken),2); for scovered = sstep:sstep:smax %Check to see which of the 4 conditions are met so we know which section to move the resin in and which formula to use to calculate the capillary pressure if cmat(1,1) == 0 alpha1 = s(1,1)/rfib; beta1 = asin((d+rfib*(1-cos(alpha1)))*cos(alpha1+theta)/(rfib*(1- cos(alpha1)+d/rfib))); 156

178 xff1 = (sin(alpha1)-(1-cos(alpha1)+d/rfib)/cos(alpha1+theta)*(1- cos(beta1)))*rfib; xfib3min = (sin(pi/3)/sqrt(3*vf/(2*sqrt(3)*pi))-1)*rfib; if (xff1 >= xfib3min) cmat(1,1) = 1; adjneeded = 1; elseif (cmat(2,1) == 0) if s(1,1) >= pi*rfib/2 cmat(2,1) = 1; elseif (cmat(3,1) == 0) alpha3 = s(3,1)/(2*rfib)-pi/2; beta3 = asin((d+rfib*(1-cos(alpha3)))*cos(alpha3+theta)/(rfib*(1- cos(alpha3)+d/rfib))); xff3 = (sin(alpha3)-(1-cos(alpha3)+d/rfib)/cos(alpha3+theta)*(1- cos(beta3)))*rfib; xfib3min = (sin(pi/3)/sqrt(3*vf/(2*sqrt(3)*pi))-1)*rfib; if (xff3 >= xfib3min) cmat(3,1) = 1; adjneeded = 1; elseif (cmat(4,1) == 0) if s(3,1) >= 2*pi*rFib cmat(4,1) = 1; %Now that we know which of conditions were met, we know which section the flow is in so we can calculate the new s matrix and corresponding Pc for this time step conditionsmet = sum(cmat); sect = conditionsmet + 1; %Moving it in section 1, touching the third fiber if the first two are completely covered stot = sum(s); smove = scovered - stot; if (sect == 1) s(1,1) = s(1,1)+smove/2; s(2,1) = s(2,1)+smove/2; alpha1 = s(1,1)/rfib; pcap = surften/rfib*cos(alpha1+theta)/(1-cos(alpha1)+d/rfib); resmat(stepct,1) = scovered; resmat(stepct,2) = pcap; elseif (sect == 2) 157

179 fib1max = pi*rfib/2; if adjneeded == 1 s1orig = s(1,1)+smove/2; s1 = rfib/2*(s1orig/rfib+pi/6); s3 = 2*(s1orig - s1); s(1,1) = s1; s(2,1) = s1; s(3,1) = s3; adjneeded = 0; elseif ((s(1,1)+smove/4) < fib1max) s(1,1) = s(1,1) + smove/4; s(2,1) = s(2,1) + smove/4; s(3,1) = s(3,1) + smove/2; else fib1and2sneed = 2*(fib1Max - s(1,1)); s(1,1) = fib1max; s(2,1) = fib1max; s(3,1) = smove - fib1and2sneed; alpha3 = 0.5*s(3,1)/rFib-pi/6; pcap = surften/rfib*cos(alpha3+theta)/(1-cos(alpha3)+d/rfib)*sin(pi/3); resmat(stepct,1) = scovered; resmat(stepct,2) = pcap; elseif (sect == 3) s(3,1) = s(3,1)+smove; alpha3 = s(3,1)/(2*rfib)-pi/2; pcap = surften/rfib*cos(alpha3+theta)/(1-cos(alpha3)+d/rfib); resmat(stepct,1) = scovered; resmat(stepct,2) = pcap; elseif (sect == 4) fib3max = pi*rfib*2; if adjneeded == 1 s3orig = s(3,1)+smove; s3 = s3orig/2+2*rfib*pi/3; s4 = 1/2*(s3orig - s3); s(3,1) = s3; s(4,1) = s4; s(5,1) = s4; adjneeded = 0; elseif ((s(3,1)-smove/2) < fib3max) s(3,1) = s(3,1) + smove/2; s(4,1) = s(4,1) + smove/4; s(5,1) = s(5,1) + smove/4; 158

180 else fib3sneed = (fib3max - s(3,1)); s(3,1) = fib3max; s(4,1) = 1/2*(sMove-fib3SNeed); s(5,1) = 1/2*(sMove-fib3SNeed); alpha3 = 0.5*s(3,1)/rFib-pi/2-pi/3; pcap = surften/rfib*cos(alpha3+theta)/(1-cos(alpha3)+d/rfib)*sin(pi/3); resmat(stepct,1) = scovered; resmat(stepct,2) = pcap; else s(4,1) = s(4,1)+smove/2; s(5,1) = s(5,1)+smove/2; alpha4 = s(4,1)/rfib-pi/2; pcap = surften/rfib*cos(alpha4+theta)/(1-cos(alpha4)+d/rfib); resmat(stepct,1) = scovered; resmat(stepct,2) = pcap; stepct = stepct + 1; C.1.2 avepc.m: Calculates the Average Capillary Pressure function [PcAve] = avepc(resmat) %This function finds the average value of capillary pressure as a function of fiber surface covered using trapezoidal integration %Input is the results from pcpacked.m format long; %Find the average value of the capillary pressure using trapezoids matsize = size(resmat); numpoints = matsize(1,1); totarea = 0; for trapezoidn = 1:(numPoints-1) thisarea = 1/2*( resmat (trapezoidn,2)+ resmat((trapezoidn+1),2))*(resmat((trapezoidn+1),1)-resmat(trapezoidn,1)); totarea = totarea + thisarea; wettedlength = vals(numpoints,1) - vals(1,1); PcAve = totarea/wettedlength; 159

181 C.2 Impregnation of Two Dimensional Tows by Infinite Resin Volume This set of programs models the flow of an infinite volume of resin into a fiber tow. Each void is tracked individually as the tow is infused. It implements the methodology discussed in Chapter 4. C.2.1 ParamSimLimGaussVFC.m: Main m-file for ParametricSstudies function resmatrix = ParamSimLimGaussVFC(p,e,t) %Output matrix (resmatrix) will be the parameter value in column 1, the %time of part 1 in column 2, and the time of part 2 in column 3. %Secondary variables will be in columns 4, etc. tic; %Start timer for simulation glbct = 1; %Iteration number %Enter number of nodes per element eletype = 3; format long %Enter values for parametric study, note: only one grouping of terms should be uncommented at a time. Other values will be input later airleftmat = [.1;.25;.5;.75; 0.9]; %fibervfmat = [.4;.5;.6;.7]; %contangmat = [15; 45; 60] %xtowmat = [0.05;.1; 0.15;.2; 0.25]; for thisnumber = 1:1:1 airleft= airleftmat(thisnumber,1); changingterm = airleft; stringchterm = 'airleft'; %xtow= xtowmat(thisnumber,1); %changingterm = xtow; %stringchterm = 'xtow'; %contang= contangmat(thisnumber,1); %contactang = contang*pi/180; %changingterm = contang; %stringchterm = 'contactang'; 160

182 aspectratiooftow = 8; %Enter tow aspect ratio %Enter the part length (in m) LenPart =.3; %Tow location in part (in m): xtow = 0.15; %Enter fiber radius, resin surface tension, and static contact angle (will %be utilized to calculate the capillary pressure) rfib = 4*10^(-6); surften = 0.07; contactang = 30*pi/180; %Enter resin viscosity viscosity = 0.95; %Enter the percent of the air left (by mass) in the tow after vacuum was applied %airleft = 0.1; %Temperature Temper = 275; %Kelvins %Enter pressure gradient across the part (Pgrad = Pinlet - Pvent) Pgrad = ; %Input the thickness and fiber volume fraction of the specimin thickness = 1; volumefraction = 0.6; %Setting our average fiber volume fraction vftolerance = 0.01; %Setting how close we need our fiber volume fraction to be to our desired fiber volume fraction stdevvf = 0.1; %Standard deviation in our gaussian distribution %How many nodes can be maximum left empty (or select 0 to fill all) emptyallow = 1; %We set it at one so there will be a non-zero void matrix %Input number of nodes you would like to fill each time step in LIMS nodesperstep = 1; pauset1 = 1; % seconds %Input center of ellipse centerx = 0; centery = 0; 161

183 %Fabric information kxxply = 8.3*10^(-10); %~~~~~~~~~~~~~~~End of Inputs~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ %INPUTS COMPLETE - running simulation (and applying permeability) triggertrip = 0; currnodefilltime = 0.01; %This sets it such that analysis can be done on resulting data, with 0 fill time indicating never filled globaltime = 0.01; %Setting comstants for calculating the capillary pressure r = rfib; st = surften; theta = contactang; sstep = ; %Use the following mesh definition for working with the ellipse part of %matlab pde toolbox - three nodes per element nodes = points2nodes(p); edgematrix = generateedgematrix(e); elements = generateelements(t, thickness, volumefraction); %Lets get some basic information from the mesh n = length(nodes); % number of nodes szeelemat = size(elements); m2d = szeelemat(1,1); % number of 2D elements VoidSize = 0; %Default value for void size - if it is zero, we know either fully filled or something went wrong, would need to inspect result to see which ellipsea = max(nodes(:,2)) - centerx; %Retrieving major ellipse axis ellipseb = max(nodes(:,3)) - centery; %Retrieving minor ellipse axis [avelength, edgeelements] = getaveedgelenth(nodes,elements, edgematrix); gapbetweentows = 0.001; volunitcell = (2*(ellipseA+ellipseB))^2; voltow = pi*ellipsea*ellipseb*0.5; vfply = volumefraction*ellipsea/(ellipsea+gapbetweentows); Ladv = sqrt(4*vfply/pi); %Calculating the pressure gradient across the tow when the flow front %reaches the vent PGrad = Pgrad*(2*ellipseA/LenPart); 162

184 %We assign our fiber volume fractions to be within our set parameters elements = assigngaussvfdist(m2d,volumefraction,stdevvf,vftolerance,elements); %elements = assignconstantvfdist(m2d,volumefraction,stdevvf,vftolerance,elements); %elements =assignrandomvfdist(m2d,volumefraction,stdevvf,vftolerance,elements); %elements =assignlinearvfdist(m2d,volumefraction,stdevvf,vftolerance,elements,nodes,ellip seb,edgeelements); %Doing some calculations using Darcy's Law: tfillmold = viscosity * (1-vfPly) * LenPart^2 / (2 * kxxply * Pgrad); %Time it will take to fill the mold t0= viscosity * (1-vfPly) * xtow^2 / (2 * kxxply * Pgrad); %Time it will take the flow front to reach the tow globaltime = t0+currnodefilltime; %Setting the global time (starting when part is first infused) PappMax = Pgrad*(LenPart-xTow)/LenPart; %Maximum applied presure seen by the tow %Getting flow front location and applied pressure at this time xff = getfflocation(kxxply,pgrad,globaltime,viscosity,vfply,lenpart); %Location of the flow front at this time Papp = Pgrad*( 1 - xtow/sqrt(2*kxxply*pgrad*(globaltime)/(viscosity*(1-vfply)))); %Applied pressure at this time %Calculating void area in the ellipse vvoid1 = getvoidvolume(m2d, elements, nodes); vvoidnow = vvoid1 %Calculating initial void pressure nigl = *vVoid1/(.082*Temper); %n of PV = nrt before air removal naftervac = nigl*airleft; %n after removal PvoidInitial = naftervac*.082*temper/vvoid1; %Initial void pressure %Here we set the initial void matrix prevvoidmat = zeros(m2d,6); for num = 1: m2d prevvoidmat(num,1) = 1; prevvoidmat(num,2) = num; prevvoidmat(num,3)= elements(num,3); prevvoidmat(num,4) = elements(num,4); prevvoidmat(num,5) = elements(num,5); 163

185 n1 = elements(num,3); n2 = elements(num,4); n3 = elements(num,5); x1 = nodes(n1,2); x2 = nodes(n2,2); x3 = nodes(n3,2); y1 = nodes(n1,3); y2 = nodes(n2,3); y3 = nodes(n3,3); TEvolumeFraction = elements(num,7); aele = abs((x1*(y2-y3)+x2*(y3-y1)+x3*(y1-y2))/2); vthisele = aele*(1-tevolumefraction); prevvoidmat(num,6) = vthisele; prevvoidmat(num,7) = PvoidInitial; voidmat = prevvoidmat; prevsum = sum(prevvoidmat(:,6)); %Setting the nodes that will serve as gates (all of them) szeedge = size(edgematrix); numedgepts = szeedge(1,1); gate1ct = 1; for edpt = 1:numEdgePts nodenum = edgematrix(edpt,1); gate1matrix(gate1ct,1) = nodenum; gate1ct = gate1ct + 1 ; %Assigning initial permeability values based on Gebart Equations ktowave = 16/(9*pi*sqrt(6))*(sqrt((pi/4)/volumeFraction)-1)^2.5*(rFib)^2; elezeros = zeros(m2d,1); elements = assign2dperms(m2d, rfib, elements); %Creating copies of the matricies for storing information rawnodes = nodes; rawelements = elements; dynkele = elements; ghostnodes = nodes; ghostelements = elements; fillfactor(1,1)=n; fillfactor(2,1)=0; %Setting the capillary pressure for each element 164

186 for num = 1: m2d vf = elements(num,7); pcvss = pcpacked(rfib, vf, theta, surften,sstep); averagepc = avepc(pcvss); Pcap = averagepc; elepc(num,1) = Pcap; %Calculating the average capillary pressure for the tow Pcap = mean(elepc); avepctow = Pcap; pcapa = Pcap; %Setting fill factors for nodev = 1:n ghostnodes(nodev,5)=0; fillfactor((nodev+2),1)=0; ghostnodes(nodev,6)=0; %Setting the fill factor of the gate nodes to 1 gate1mat = gate1matrix; gt1size = size(gate1matrix); numgates = gt1size(1,1); for g1ind = 1:numGates gatenode = gate1mat(g1ind,1); ghostnodes(gatenode,5) = 0; ghostnodes(gatenode,6) = 1; fillfactor((gatenode+2),1) = 1; %Running the first time step dynkele = assignffperms(m2d, rawelements, ghostnodes, dynkele, Papp, voidmat, elepc); writedmp(nodes,dynkele,m2d,eletype,viscosity); writelb(ghostnodes, gate1matrix, PGrad, nodesperstep, pcapa, Papp,ellipseA); runlims(pauset1); load fillfactors.dmp load pressures.dmp load flowrates.dmp emptynodes = fillfactors(1,1); steptime = fillfactors(2,1); numsteps = 1; 165

187 currnodefilltime = currnodefilltime + steptime; globaltime = globaltime + steptime; %Updating the fill factors and pressures at each node for indct = 1:n if ((fillfactors((indct+2),1) == 1) && (fillfactors((indct+2),1) > ghostnodes(indct,6))) ghostnodes(indct,5) = currnodefilltime; ghostnodes(indct,6) = fillfactors((indct+2),1); ghostnodes(indct,7) = pressures((indct+2),1); %Getting the void matrix at this time step and archiving it voidmat = getvoidmatrix(ghostnodes, elements, prevvoidmat); ctddime = round(currnodefilltime*100); fvoidmat = [stringchterm,num2str(changingterm),'voidmatat',num2str(ctddime),'.txt']; dlmwrite(fvoidmat,voidmat); %Getting the element fill percentages to determine which elements are on %the flow front elefillpct = getelefillpcts(elements,ghostnodes); %Getting the applied pressure and global time for the next time step [Papp, globaltime, flagger] = getappliedp(globaltime,pappmax, Pgrad,xTow,kxxPly,viscosity,vfPly,voidMat,elePc, elefillpct); currnodefilltime = globaltime - t0; avevoidp = mean(voidmat(:,6)); PdriveNow = Papp + Pcap - avevoidp %Setting our data output file frct = 1; datacol = zeros(1,1); datacol(frct, 1) = currnodefilltime; datacol(frct, 2) = vvoidnow; datacol(frct, 3) = vvoidnow/vvoid1; datacol(frct, 4) = Papp; frct = frct + 1; %Looping for future time steps while ((numsteps < n) && (flagger == 0) && emptynodes > 1) 166

188 %Updating the flow front permeabilties and retuning the filled elements %back to their original permeability dynkele = assignffperms(m2d, rawelements, ghostnodes, dynkele, Papp, voidmat, elepc); %We run the next time step since the pressure driving the void into the %tow is larger than the resisting pressure writedmp(nodes,dynkele,m2d,eletype,viscosity); writelb(ghostnodes, gate1matrix, PGrad, nodesperstep, pcapa, Papp,ellipseA); runlims(pauset1); pause(0.5) load fillfactors.dmp load pressures.dmp load filltimes.dmp %Updating the filled nodes and time steps emptynodes = fillfactors(1,1); steptime = fillfactors(2,1); numsteps = numsteps + 1; currnodefilltime = currnodefilltime + steptime; globaltime = globaltime + steptime; %Setting the number of empty nodes and setting the fill factors as well %as times and pressures for indct = 1:n if filltimes((indct+2),1) > 0 ghostnodes(indct,5) = currnodefilltime + filltimes((indct+2),1); ghostnodes(indct,6) = fillfactors((indct+2),1); ghostnodes(indct,7) = pressures((indct+2),1); %Getting the element fill percentages to determine which elements are on %the flow front elefillpct = getelefillpcts(elements,ghostnodes); %Updating the void matrix and archiving the previous time step prevvoidmat = voidmat; voidmat = getvoidmatrix(ghostnodes, elements, prevvoidmat); avevoidp = mean(voidmat(:,7)); ctddime = round(currnodefilltime*100); fvoidmat = [stringchterm,num2str(changingterm),'voidmatat',num2str(ctddime),'.txt']; 167

189 dlmwrite(fvoidmat,voidmat) %Getting the applied pressure and global time for the next time step [Papp, globaltime, flagger] = getappliedp(globaltime,pappmax, Pgrad,xTow,kxxPly,viscosity,vfPly,voidMat,elePc, elefillpct); currnodefilltime = globaltime - t0; %Our important quantities we want to keep track of vvoidnow = sum(voidmat(:,6)); PdriveNow = Papp + Pcap - avevoidp %Advancing the time step for next run numsteps = numsteps + 1; emptynodes numsteps changingterm datacol(frct, 1) = currnodefilltime; datacol(frct, 2) = vvoidnow; datacol(frct, 3) = vvoidnow/vvoid1; datacol(frct, 4) = Papp; frct = 1 + frct; %This will output our ghost node matrix at the selected time currtime = currnodefilltime; pressureequalizationtime = currtime VFof100 = round(volumefraction*100); writesol(rawnodes, rawelements, ghostnodes, gate1matrix,currtime,stringchterm,changingterm) fname2 = [stringchterm,num2str(changingterm),'resar',num2str(aspectratiooftow),'randv F',num2str(VFof100),'.txt']; dlmwrite(fname2,ghostnodes) writesol(rawnodes, rawelements, ghostnodes, gate1matrix,currnodefilltime,stringchterm,changingterm) vvoidpart1 = vvoidnow; voidtransportmat = voidtransportfromtow(voidmat, nodes, ghostnodes, edgematrix, elements,papp, ellipsea, PGrad, Pcap,currNodeFillTime,pauseT1,stringChTerm,changingTerm); snumvoids = size(voidtransportmat); 168

190 numvoids = snumvoids(1,1); %Now we output our void size versus time VFof100 = round(volumefraction*100); fnamevoidsize = [stringchterm,num2str(changingterm),'voidsizear',num2str(aspectratiooftow),'ra ndvf',num2str(vfof100),'.txt']; dlmwrite(fnamevoidsize,datacol) vvoidfin = vvoidnow; avoidell = sqrt(vvoidfin/(pi*aspectratiooftow)); if VoidSize == 0 VoidSize = vvoidnow; resmatrix(glbct, 1) = volumefraction; %Fiber volume fraction of tow resmatrix(glbct, 2) = vfply ; %Ply volume Fraction resmatrix(glbct, 3) = ktowave; %Permeability resmatrix(glbct, 4) = Pcap; %Capillary Pressure resmatrix(glbct, 5) = ellipsea; %Ellipse A resmatrix(glbct, 6) = changingterm; %changingterm resmatrix(glbct, 7) = t0; %Time at which tow is contacted resmatrix(glbct, 8) = tfillmold; %Analytical part fill time resmatrix(glbct, 9) = pressureequalizationtime; %End of part 1 time resmatrix(glbct, 10) = vvoidpart1; %Void size when equilibrium is reached resmatrix(glbct, 11) = vvoidpart1/vvoid1; %Void percent remaining glbct = glbct + 1; simtime = toc C.2.2 Subroutine m-files (in the Order in which they are First Called) Note: in first 3 m-files, the inputs are the P,E,T matrix when outputting the 2D ellipse mesh from the MATLAB PDE toolbox function [nodes] = points2nodes(p) %Converts the points output from MATLAB pde toolbox into nodal descriptions pts = transpose(p); szepts = size(pts); numnodes = szepts(1,1); nodes = zeros(numnodes, 3); 169

191 for thisnode = 1:numNodes nodes(thisnode, 1) = thisnode; nodes(thisnode, 2) = pts(thisnode,1); nodes(thisnode, 3) = pts(thisnode,2); nodes(thisnode, 4) = 0; function [edgemat] = generateedgematrix(e) %Converts the edge output from MATLAB pde toolbox into a matrix with the %edge points edgmat = transpose(e); szeed = size(edgmat); numedge = szeed(1,1); edgepoints = zeros(numedge,2); for num = 1:numEdge edgepoints(num,1) = edgmat(num,1); edgepoints(num,2) = edgmat(num,2); edgemat = unique(edgepoints); function [elements] = generateelements(t, thickness, volumefraction) %Generates a rectangular mesh with a height of yht and width of xht. The %mesh will have npux nodes per unit in the x direction and a nodal density %of npuy in the y direction %Setting a dummie value to the permeabilties since they are manipulated in %the SIMLIM.m function Kxx = 1*10^(-10); Kyy = 1*10^(-10); Kxy = 0; %Converts the points output from MATLAB pde toolbox into nodal descriptions els = transpose(t); szeels = size(els); numels = szeels(1,1); elements = zeros(numels, 3); for thisel = 1:numEls elements(thisel, 1) = thisel; elements(thisel, 2) = 3; elements(thisel, 3) = els(thisel,3); elements(thisel, 4) = els(thisel,2); elements(thisel, 5) = els(thisel,1); elements(thisel, 6) = thickness; 170

192 elements(thisel, 7) = volumefraction; elements(thisel, 8) = Kxx; elements(thisel, 9) = Kxy; elements(thisel, 10) = Kyy; function [avelength, edgeelements] = getaveedgelenth(nodes,elements, edgematrix) %Calculates the average length of the sides of the elements edcount = 1; szel = size(elements); numele = szel(1,1); for el = 1:numEle n1 = elements(el,3); n2 = elements(el,4); n3 = elements(el,5); check = sum(sum(ismember(edgematrix,n1)))+sum(sum(ismember(edgematrix,n2)))+sum(su m(ismember(edgematrix,n3))); if check == 2 einfomat = zeros(3,2); elepn = 1; if sum(sum(ismember(edgematrix,n1))) > 0 einfomat(elepn,1) = nodes(n1,2); einfomat(elepn,2) = nodes(n1,3); elepn = elepn + 1; else einfomat(3,1) = nodes(n1,2); einfomat(3,2) = nodes(n1,3); if sum(sum(ismember(edgematrix,n2))) > 0 einfomat(elepn,1) = nodes(n2,2); einfomat(elepn,2) = nodes(n2,3); elepn = elepn + 1; else einfomat(3,1) = nodes(n2,2); einfomat(3,2) = nodes(n2,3); if sum(sum(ismember(edgematrix,n3))) > 0 einfomat(elepn,1) = nodes(n3,2); einfomat(elepn,2) = nodes(n3,3); else 171

193 einfomat(3,1) = nodes(n3,2); einfomat(3,2) = nodes(n3,3); if check == 1 einfomat = zeros(3,2); elepn = 1; if sum(sum(ismember(edgematrix,n1))) == 0 einfomat(elepn,1) = nodes(n1,2); einfomat(elepn,2) = nodes(n1,3); elepn = elepn + 1; else einfomat(3,1) = nodes(n1,2); einfomat(3,2) = nodes(n1,3); if sum(sum(ismember(edgematrix,n2))) == 0 einfomat(elepn,1) = nodes(n2,2); einfomat(elepn,2) = nodes(n2,3); elepn = elepn + 1; else einfomat(3,1) = nodes(n2,2); einfomat(3,2) = nodes(n2,3); if sum(sum(ismember(edgematrix,n3))) == 0 einfomat(elepn,1) = nodes(n3,2); einfomat(elepn,2) = nodes(n3,3); else einfomat(3,1) = nodes(n3,2); einfomat(3,2) = nodes(n3,3); if check > 0 L1 = sqrt((einfomat(1,1) - einfomat(2,1))^2+(einfomat(1,2) - einfomat(2,2))^2); L2 = sqrt((einfomat(3,1) - einfomat(2,1))^2+(einfomat(3,2) - einfomat(2,2))^2); L3 = sqrt((einfomat(3,1) - einfomat(1,1))^2+(einfomat(3,2) - einfomat(1,2))^2); sel = 0.5*(L1+L2+L3); AEl = sqrt(sel*(sel - L1)*(sEl-L2)*(sEl-L3)); del = 2*AEl/L1; edgeelements(edcount,1) = el; dmat(edcount,1) = del; edcount = edcount + 1; 172

194 avelength = mean(dmat); function [elements] = assigngaussvfdist(numelements,volumefraction,stdevvf,vftolerance,elements,edg eelements) %This function assigns a Gaussian distribution of fiber volume fraction to all of the elements. %Inputs: Number of elements, average desired fiber volume fraction, desired standard deviation of fiber volume fractions, tolerance for fiber volume fraction (how much can the actual average deviate from the desired one before it is considered unacceptable) %Output: Fiber volume fraction matrix (each row in matrix corresponds to an element in the mesh %We start by setting fiber volume fraction as unacceptable and have gaussdist function assign fiber volume fractions in a while loop until it is acceptable notacceptablevf = 1; maxvf = 0.9; minvf = volumefraction - (0.9-volumeFraction); if minvf < 0 minvf =0.01; ; szel = size(elements); numele = szel(1,1); while notacceptablevf == 1 vfmatrix = normrnd(volumefraction,stdevvf,numele,1); for num = 1:numEle happy = 0; while happy < 1 vf = vfmatrix(num,1); if (vf > minvf && vf < maxvf) happy = 1; else vfmatrix(num,1) = normrnd(volumefraction,stdevvf,1,1); vfave = mean(vfmatrix); if ((vfave + vftolerance) > volumefraction && (vfave - vftolerance) < volumefraction) notacceptablevf = 0; 173

195 vfave szed = size(edgeelements); numedel = szed(1,1); for edct = 1:numEdEl thisel = edgeelements(edct,1); vfmatrix(thisel,1) = 0.01; for num = 1:numElements elements(num,7) = vfmatrix(num,1); function [elements] = assignconstantvfdist(numelements,volumefraction,stdevvf,vftolerance,elements,e dgeelements) %This function assigns a constant distribution of fiber volume fraction to all of the elements. %Inputs: Number of elements, average desired fiber volume fraction, desired standard deviation of fiber volume fractions, tolerance for fiber volume fraction (how much can the actual average deviate from the desired one before it is considered unacceptable) %Output: Fiber volume fraction matrix (each row in matrix corresponds to an element in the mesh for num = 1:numElements vfmatrix(num,1) = volumefraction; szed = size(edgeelements); numedel = szed(1,1); for edct = 1:numEdEl thisel = edgeelements(edct,1); vfmatrix(thisel,1) = 0.01; for num = 1:numElements elements(num,7) = vfmatrix(num,1); function [elements] = assignrandomvfdist(numelements,volumefraction,stdevvf,vftolerance,elements,e dgeelements) %This function assigns a Random distribution of fiber volume fraction to %all of the elements. %Inputs: Number of elements, average desired fiber volume fraction, desired %standard deviation of fiber volume fractions, tolerance for fiber volume 174

196 %fraction (how much can the actual average deviate from the desired one %before it is considered unacceptable) %Output: Fiber volume fraction matrix (each row in matrix corresponds to an %element in the mesh maxbound = 0.9; minbound = volumefraction - (maxbound-volumefraction); notacceptablevf = 1; while notacceptablevf == 1 for num = 1:numElements vfmatrix(num,1) = rand()*(maxbound-minbound)+minbound; vfave = mean(vfmatrix); if ((vfave + vftolerance) > volumefraction && (vfave - vftolerance) < volumefraction) notacceptablevf = 0; vfave szed = size(edgeelements); numedel = szed(1,1); for edct = 1:numEdEl thisel = edgeelements(edct,1); vfmatrix(thisel,1) = 0.01; for num = 1:numElements elements(num,7) = vfmatrix(num,1); function [elements] = assignlinearvfdist(numelements,volumefraction,stdevvf,vftolerance,elements,nod es,ellipseb,edgeelements) %This function assigns a linear distribution of fiber volume fraction to all of the elements. %Inputs: Number of elements, average desired fiber volume fraction, desired standard deviation of fiber volume fractions, tolerance for fiber volume fraction (how much can the actual average deviate from the desired one before it is considered unacceptable) %Output: Fiber volume fraction matrix (each row in matrix corresponds to an element in the mesh maxbound = 0.9; vfmax = maxbound; vfave = volumefraction; m = 3*pi/(4*ellipseB)*(vfMax-vfAve); 175

197 vfmatrix = zeros(numelements,1); for num = 1: numelements n1 = elements(num,3); n2 = elements(num,4); n3 = elements(num,5); y1 = nodes(n1,3); y2 = nodes(n2,3); y3 = nodes(n3,3); eley = (y1+y2+y3)/3; vfmatrix(num,1) = vfmax - m*abs(eley); szed = size(edgeelements); numedel = szed(1,1); for edct = 1:numEdEl thisel = edgeelements(edct,1); vfmatrix(thisel,1) = 0.01; for num = 1:numElements elements(num,7) = vfmatrix(num,1); function [xff] = getfflocation(kxxply,pgrad,globaltime,viscosity,vfply,lenpart) %Calculates the part level flow front location at this time step xff = sqrt(2*kxxply*pgrad*(globaltime)/(viscosity*(1-vfply))); %Location of the flow front at this time if xff > LenPart xff = LenPart; function [vvoid] = getvoidvolume(m2d, elements, nodes) %Calculates the total void volume elefill = zeros(m2d,1); for num = 1: m2d n1 = elements(num,3); n2 = elements(num,4); n3 = elements(num,5); x1 = nodes(n1,2); x2 = nodes(n2,2); x3 = nodes(n3,2); y1 = nodes(n1,3); y2 = nodes(n2,3); 176

198 y3 = nodes(n3,3); TEvolumeFraction = elements(num,7); aele = abs((x1*(y2-y3)+x2*(y3-y1)+x3*(y1-y2))/2); elefill(num,1) = aele*(1-tevolumefraction); vvoid = sum(elefill); function [elements] = assign2dperms(m2d, rfibmat,elements) %Assigns permeability values for num = 1: m2d %Transverse K For square packing %kgabt = 16/(9*pi*sqrt(2))*(sqrt((pi/4)/vf)-1)^2.5*(rFib)^2; %Transverse K For Hexagonal packing volumefraction = elements(num,7); kgabt = 16/(9*pi*sqrt(6))*(sqrt((pi/4)/volumeFraction)- 1)^2.5*(rFibMat(num,1))^2; if kgabt < 1*10^(-25) kgabt = 1*10^(-25); %Axial K (Not used here since 2D radial flow) %kgaba = 8/57*(1-volumeFraction^3)/(volumeFraction^2)*rFib^2; elements(num,8) = kgabt; %Both are transverse since 2D radial flow elements(num,9) = 0; elements(num,10) = kgabt; function [dynkele] = assignffperms(m2d, rawelements, ghostnodes, dynkele, Papp, voidmat, elepc) %This function manipulates the capillary pressure of elements on the flow %front pressdif = 1000; %Difference in pressure required for flow front to not be considered impermeable maxfill = 2.97; %Sum of the three nodal fill factors required to be considered filled %Getting required information elevoidvals = zeros(m2d,1); szvoidmat = size(voidmat); numvoidele = szvoidmat(1,1); %Setting the void pressure for each element for vel = 1:numVoidEle voidelen = voidmat(vel,2); elevoidvals(voidelen,1) = voidmat(vel,7); 177

199 %Assigning the permeability values to nodes on the flow front and resetting %those for nodes no longer on flow front for num = 1: m2d n1 = rawelements(num,3); n2 = rawelements(num,4); n3 = rawelements(num,5); ff1 = ghostnodes(n1,6); ff2 = ghostnodes(n2,6); ff3 = ghostnodes(n3,6); fftot = (ff1 + ff2 + ff3); if ((fftot > 0) && (fftot < maxfill)) krealt = rawelements(num,10); pcapthisele = elepc(num,1); pvoidthisele = elevoidvals(num,1); if (pcapthisele + Papp - pvoidthisele) > pressdif kmodt = krealt*(papp+pcapthisele-pvoidthisele)/papp; if kmodt < 1*10^(-20) kmodt = 1*10^(-25); dynkele(num,8) = kmodt; dynkele(num,10) = kmodt; else kmodt = 1*10^(-25); dynkele(num,8) = kmodt; dynkele(num,10) = kmodt; else dynkele(num,8) = rawelements(num,8); dynkele(num,10) = rawelements(num,10); function writedmp(nodes, elements, m2d, eletype,viscosity) %Writes LIMS dump input file n = length(nodes); % number of nodes %Writing the.dmp file for the updated nodes and elements fname = sprintf('run'); filename = sprintf('%s.dmp',fname); o = fopen(filename, 'w'); fprintf(o,'# \r\n'); fprintf(o,'number of nodes : %5.0f \r\n',n); 178

200 fprintf(o,' Index x y z\r\n'); fprintf(o,'===================================================\r\n'); for i = 1: n % loop to write node data fprintf(o,'%6.0f %12.6f %12.6f %12.6f \r\n',nodes(i,:)); fprintf(o,'number of elements : %5.0f \r\n',m2d); fprintf(o,' Index NNOD N1 N2 (N3) (N4) (N5) (N6) (N7) (N8) h(a) Vf Kxx Kxy Kyy Kzz Kzx Kyz\r\n'); fprintf(o,'====================================================== ============================================================= ===========================================================\r\n '); for i = 1: m2d % loop[ to write 2D elements data fprintf(o,'%6.0f %5.0f %6.0f %6.0f %6.0f %42.6f %15.6f %15.4e %15.4e %15.4e \r\n',elements(i,:)); fprintf(o,'resin Viscosity model NEWTON\r\n'); fprintf(o,'viscosity : %d\r\n',viscosity); fclose(o); function writelb(ghostnodes, gate1matrix, PGrad, nodesperstep, PcapA, Papp,ellipseA) % Generate lb file Pin = PGrad; fname = sprintf('run_simulate.lb'); fid2 = fopen(fname,'w+'); fprintf(fid2,'proc simu\r\n'); fprintf(fid2,'solve\r\n'); fprintf(fid2,'endproc\r\n'); fprintf(fid2,'\r\n'); %Creating file for fill factor output fprintf(fid2,'proc PFF\r\n'); fprintf(fid2,'defint i\r\n'); fprintf(fid2,'defint count\r\n'); fprintf(fid2,'let i=fifreefile()\r\n'); fprintf(fid2,'fiopen i,"fillfactors.dmp","w"\r\n'); fprintf(fid2,'fiprint i,sonumberempty()\r\n'); fprintf(fid2,'fiprintnl 1\r\n'); fprintf(fid2,'fiprint i,socurrenttime\r\n'); fprintf(fid2,'for count=1 TO SONUMBERNODES\r\n'); fprintf(fid2,'fiprintnl 1\r\n'); fprintf(fid2,'fiprint i,sofillfactor(count)\r\n'); 179

201 fprintf(fid2,'next count\r\n'); fprintf(fid2,'ficlose i\r\n'); fprintf(fid2,'endproc\r\n'); fprintf(fid2,'\r\n'); %Creating file for pressure output fprintf(fid2,'proc PP\r\n'); fprintf(fid2,'defint i\r\n'); fprintf(fid2,'defint count\r\n'); fprintf(fid2,'let i=fifreefile()\r\n'); fprintf(fid2,'fiopen i,"pressures.dmp","w"\r\n'); fprintf(fid2,'fiprint i,sonumberempty()\r\n'); fprintf(fid2,'fiprintnl 1\r\n'); fprintf(fid2,'fiprint i,socurrenttime\r\n'); fprintf(fid2,'for count=1 TO SONUMBERNODES\r\n'); fprintf(fid2,'fiprintnl 1\r\n'); fprintf(fid2,'fiprint i,sopressure(count)\r\n'); fprintf(fid2,'next count\r\n'); fprintf(fid2,'ficlose i\r\n'); fprintf(fid2,'endproc\r\n'); fprintf(fid2,'\r\n'); %Creating file for fill time output fprintf(fid2,'proc PFillT\r\n'); fprintf(fid2,'defint i\r\n'); fprintf(fid2,'defint count\r\n'); fprintf(fid2,'let i=fifreefile()\r\n'); fprintf(fid2,'fiopen i,"filltimes.dmp","w"\r\n'); fprintf(fid2,'fiprint i,sonumberempty()\r\n'); fprintf(fid2,'fiprintnl 1\r\n'); fprintf(fid2,'fiprint i,socurrenttime\r\n'); fprintf(fid2,'for count=1 TO SONUMBERNODES\r\n'); fprintf(fid2,'fiprintnl 1\r\n'); fprintf(fid2,'fiprint i,sotimetofill(count)\r\n'); fprintf(fid2,'next count\r\n'); fprintf(fid2,'ficlose i\r\n'); fprintf(fid2,'endproc\r\n'); fprintf(fid2,'\r\n'); %Creating file for flowrate output fprintf(fid2,'proc PFR\r\n'); fprintf(fid2,'defint i\r\n'); fprintf(fid2,'defint count\r\n'); fprintf(fid2,'let i=fifreefile()\r\n'); fprintf(fid2,'fiopen i,"flowrates.dmp","w"\r\n'); fprintf(fid2,'fiprint i,sonumberempty()\r\n'); 180

202 fprintf(fid2,'fiprintnl 1\r\n'); fprintf(fid2,'fiprint i,socurrenttime\r\n'); fprintf(fid2,'for count=1 TO SONUMBERNODES\r\n'); fprintf(fid2,'fiprintnl 1\r\n'); fprintf(fid2,'fiprint i,soflowrate(count)\r\n'); fprintf(fid2,'next count\r\n'); fprintf(fid2,'ficlose i\r\n'); fprintf(fid2,'endproc\r\n'); fprintf(fid2,'\r\n'); fprintf(fid2,'read "run.dmp"\r\n'); %Setting fill factors and pressures nodematsize = size(ghostnodes); numnodes = nodematsize(1,1); for nodenum = 1:numNodes fprintf(fid2,'setfillfactor %d %d \r\n',nodenum, ghostnodes(nodenum,6)); %Setting Gates gatsize = size(gate1matrix); numgates = gatsize(1,1); for i = 1:numGates nodenum = gate1matrix(i,1); fprintf(fid2,'setgate %d, 1, %d \r\n',gate1matrix(i,1), (Papp+0.5*PGrad/ellipseA*(ghostNodes(nodeNum,2))));%Here PGrad is pressure gradient across the tow fprintf(fid2,'\r\n'); fprintf(fid2,'settime 0\r\n'); fprintf(fid2,'\r\n'); fprintf(fid2,'call simu\r\n'); fprintf(fid2,'\r\n'); fprintf(fid2,'call PFF\r\n'); fprintf(fid2,'\r\n'); fprintf(fid2,'call PFR\r\n'); fprintf(fid2,'\r\n'); fprintf(fid2,'call PP\r\n'); fprintf(fid2,'\r\n'); fprintf(fid2,'call PFillT\r\n'); fprintf(fid2,'\r\n'); fprintf(fid2,'print "# empty nodes =", sonumberempty\r\n'); fprintf(fid2,'\r\n'); fprintf(fid2,'setouttype "dump"\r\n'); fprintf(fid2,'write "run_res.dmp"\r\n'); fclose all; 181

203 function runlims(pauset) % Run the lb file, uses limsslv application and lcmaster file, written by Ali Gokce fname = sprintf('run_simulate.lb'); filename = sprintf('load run_simulate.lb'); lims(3,6,2000); % set time-out to 2000 ms lims(3,3,50); %Set option to value (2, opt, value) lims(1,1); %Starts the slave, second 1 is slave ID (start, ID) lims(5,1,'setoutputlevel 2'); lims(5,1,'setmessagelevel 0'); lims(5,1,filename); %Put string to slave ID (lims(5,id,string)) lims(5,1,fname); output = 'ini'; pause(2); while ~isempty(output) output = lims(4,1,350); %Get line from slave with ID (lims(4,id, maxw)) lims(5,1,'print sonumberempty'); %Records # of empty nodes pause(pauset) %Manipulate this pause based on single simulation time lims(2,1) %Ends LIMS (Kills the slave) fclose all; function [voidmatrix] = getvoidmatrix(ghostnodes, elements, prevvoidmat) %Tracks each void individually including which elements make up the void, %and what the new void pressure should be. Function identifies and accounts for key events %such as voids splitting, coalescing, or simply changing sizes. sizeele = size(elements); numele = sizeele(1,1); numberelements0 = numele; dumelements = elements; %Making a matrix of all of the elements that are part of voids for thiselement = numele:(-1):1; thisvf = elements(thiselement,7); n1 = elements(thiselement,3); n2 = elements(thiselement,4); n3 = elements(thiselement,5); ff1 = ghostnodes(n1,6); ff2 = ghostnodes(n2,6); ff3 = ghostnodes(n3,6); ffpercent = (ff1 + ff2 + ff3)/3; 182

204 if ffpercent > 0.99 dumelements = removerows(dumelements,thiselement); else x1 = ghostnodes(n1,2); x2 = ghostnodes(n2,2); x3 = ghostnodes(n3,2); y1 = ghostnodes(n1,3); y2 = ghostnodes(n2,3); y3 = ghostnodes(n3,3); TEvolumeFraction = elements(thiselement,7); aele = abs((x1*(y2-y3)+x2*(y3-y1)+x3*(y1-y2))/2); porousareafilled = ffpercent*aele*(1-tevolumefraction); dumelements(thiselement,6) = aele*(1-tevolumefraction)-porousareafilled; %Next step is to group the elements by void, with the index of the void %being the first column of the matrix voidcount = 1; dumsize = size(dumelements); numdum = dumsize(1,1); groupedvoids = zeros(numdum,6); %We start by setting the first void element as part of the first void groupedvoids(1,1) = voidcount; %Void index groupedvoids(1,2) = dumelements(numdum,1); %Element index groupedvoids(1,3) = dumelements(numdum,3); %Element node 1 groupedvoids(1,4) = dumelements(numdum,4); %Element node 2 groupedvoids(1,5) = dumelements(numdum,5); %Element node 3 groupedvoids(1,6) = dumelements(numdum,6); %Void area in element dumelements = removerows(dumelements,numdum); % We remove this element from the "to assign" list dumsize = size(dumelements); numdum = dumsize(1,1); numdum0 = numdum; rcount = 1; %Row in grouped element matrix (which row you look at to see if there are any nodes in common) gvoiden = 2; %Grouped voids element number (which row to go to in the grouped voids matrix) flagloop = 0; while (numdum > 0)&& (gvoiden <= (numdum0+1)) %We also want to check to see if we are in the last row of the grouped %matrix (if not, we will move to the next node if this element has no %more nodes in common with other elements... if so, then we will start %a new void if this elemens has no more nodes in common with other 183

205 %elements dumsize = size(dumelements); numdum = dumsize(1,1); %Finding the similiar number of nodes in common between the element we %are examining and any of the elements that have not been assigned a %group coltots = sum(ismember(groupedvoids(rcount,3:5),dumelements(1:numdum,2:4))); nodescommon = coltots;%coltots(1,1)+coltots(1,2)+coltots(1,3); if nodescommon > 0 %Since we know there are nodes in common with other elements, we %will loop through the elements, adding the ones in common to our %list for thise = (numdum):(-1):1; coltots = sum(ismember(groupedvoids(rcount,3:5),dumelements(thise,2:4))); istouching = coltots; %coltots(1,1)+coltots(1,2)+coltots(1,3); if istouching > 0 groupedvoids(gvoiden,1) = voidcount; %Void index groupedvoids(gvoiden,2) = dumelements(thise,1); %Element index groupedvoids(gvoiden,3) = dumelements(thise,3); %Element node 1 groupedvoids(gvoiden,4) = dumelements(thise,4); %Element node 2 groupedvoids(gvoiden,5) = dumelements(thise,5); %Element node 3 groupedvoids(gvoiden,6) = dumelements(thise,6); %Void area in element dumelements = removerows(dumelements,thise); % We remove this element from the "to assign" list gvoiden =gvoiden + 1; elseif (nodescommon == 0) && (rcount < gvoiden) %Here we tell it to move to the next element in our list but still %stay on the same void number rcount = rcount + 1; else %We move to the next element that has not been assigned to a void, %and since we know it will be part of a different void, we also %change the void number voidcount = voidcount + 1; groupedvoids(gvoiden,1) = voidcount; %Void index groupedvoids(gvoiden,2) = dumelements(1,1); %Element index groupedvoids(gvoiden,3) = dumelements(1,3); %Element node 1 groupedvoids(gvoiden,4) = dumelements(1,4); %Element node 2 groupedvoids(gvoiden,5) = dumelements(1,5); %Element node 3 184

206 groupedvoids(gvoiden,6) = dumelements(1,6); %Void area in element dumelements = removerows(dumelements,1); gvoiden = gvoiden + 1; %We also want to check to see if we are in the last row of the grouped %matrix (if not, we will move to the next node if this element has no %more nodes in common with other elements... if so, then we will start %a new void if this elemens has no more nodes in common with other %elements dumsize = size(dumelements); numdum = dumsize(1,1); %Here we create a summary chart of the previous voids - including their void number, void pressure, and void size pvnums = unique(prevvoidmat(:,1)); pvsz = size(pvnums); numprevoids = pvsz(1,1); szpe = size(prevvoidmat); preelements = szpe(1,1); prevvoidsum = zeros(numprevoids,3); %prevvoidsum matrix: [prevvoidnumber prevvoidpressure prevvoidarea] for vn = 1:numPreVoids prevvoidsum(vn,1) = vn; preveleinvoid(vn,1) = sum(ismember(prevvoidmat(:,1), vn)); for pe1 = 1:preElements if prevvoidmat(pe1,1) == vn prevvoidsum(vn,2) = prevvoidmat(pe1, 7); prevvoidsum(vn,3) = prevvoidsum(vn,3) + prevvoidmat(pe1,6); prevvoidsum(vn,3) %Here we create a summary chart of the current voids - including their void %number, void size, but not void pressure yet cvnums = unique(groupedvoids(:,1)); cvsz = size(cvnums); numcurrvoids = cvsz(1,1); szce = size(groupedvoids); currelements = szce(1,1); currvoidsum = zeros(numcurrvoids,3); for vn = 1:numCurrVoids currvoidsum(vn,1) = vn; curreleinvoid(vn,1) = sum(ismember(groupedvoids(:,1), vn)); 185

207 for pe1 = 1:currElements if groupedvoids(pe1,1) == vn currvoidsum(vn,2) = 0; currvoidsum(vn,3) = currvoidsum(vn,3) + groupedvoids(pe1,6); %Here we make our similiarity matrix cvnpre = 1; similiarmatrix = zeros(numcurrvoids, numprevoids); %Set zeros for the number of rows of current void and columns of previous void for cvn = 1:numCurrVoids prevstel = 1; %Resetting the ing of the "previous void state" element counting matrix elethisvd = curreleinvoid(cvn,1); %Number of elements in this void for pvn = 1:numPreVoids eleprevvoid = preveleinvoid(pvn,1); similiarmatrix(cvn,pvn) = sum(ismember(groupedvoids((cvnpre):(cvnpre+elethisvd-1),2), prevvoidmat((prevstel):(prevstel+eleprevvoid-1),2))); prevstel = prevstel + eleprevvoid; cvnpre = cvnpre + elethisvd; %Now if there are more than one element in similair, we want to set the %value to 1 in the similiar matrix (this way we can have it binary) for cvn = 1:numCurrVoids for pvn = 1:numPreVoids if similiarmatrix(cvn,pvn) > 0 similiarmatrix(cvn,pvn) = 1; %At this point we have the similiarity matrix - any column with multiple %1's in the same row indicate that voids have joined together. Similairly, %any with multiple 1's in the same column indiciate that voids have split. %If a 1 is the only one in its row and column, then the void has simply %changed size. %prevvoidsum matrix: [prevvoidnumber prevvoidpressure prevvoidarea] %Here we will handle the cases where a void has split~~~~~~void SPLITTING splitlist = 0; %Resetting split list from previous iteration szsim = size(similiarmatrix); numrowssim = szsim(1,1); 186

208 %We need to do this next if loop to prevent issues if there is only one %current void but multiple previous voids if numrowssim == 1 for tallyme = 1:numPreVoids splitcheck(1,tallyme) = 1; else splitcheck = sum(similiarmatrix); %Summing the columns of the similiar matrix - anything columns above1 indicate that the void has split for spv = 1:numPreVoids if splitcheck(1,spv) > 1 newsplitarea = 0; splitprevarea = prevvoidsum(spv, 3); splitprevpres = prevvoidsum(spv, 2); for newspv = 1:numCurrVoids if (similiarmatrix(newspv, spv) == 1) newsplitarea = newsplitarea + currvoidsum(newspv,3); newsplitpressure = splitprevarea*splitprevpres/newsplitarea; for newspv = 1:numCurrVoids if (similiarmatrix(newspv, spv) == 1) currvoidsum(newspv,2) = newsplitpressure; %Here we handle the cases wehre voids come together~~~~~void COALESCENCE for cov = 1:numCurrVoids coalcheck = sum(similiarmatrix(cov,:)); if coalcheck > 1 oldcopv = 0; for oldv = 1:numPreVoids if similiarmatrix(cov, oldv) == 1 oldcopv = oldcopv + prevvoidsum(oldv, 3)*prevVoidSum(oldV, 2); newvoidp = oldcopv / currvoidsum(cov,3); currvoidsum(cov,2) = newvoidp; %Now that we have covered void splitting and coming together, we handle the 187

209 %case where the void only changes size/pressure. Since we have been %manipulating the pressure values in the previous steps, we know if %pressure is equal to the default, then the void size simply changes for csv = 1:numCurrVoids sizeonlycheck = currvoidsum(csv, 2); if sizeonlycheck == 0 for oldv = 1:numPreVoids if similiarmatrix(csv, oldv) == 1 oldpv = prevvoidsum(oldv, 3)*prevVoidSum(oldV, 2); newvoidp = oldpv/currvoidsum(csv,3); currvoidsum(csv,2) = newvoidp; szege = size(groupedvoids); numvoidelements = szege(1,1); for elepass = 1:numVoidElements voidnumberassign = groupedvoids(elepass,1); groupedvoids(elepass,7) = currvoidsum(voidnumberassign, 2); voidmatrix = groupedvoids; function [elefillpct] = getelefillpcts(elements,ghostnodes) %Gets the fill percentage for each element numeles = size(elements); m2d = numeles(1,1); elefillpct = zeros(m2d,1); for num = 1: m2d n1 = elements(num,3); n2 = elements(num,4); n3 = elements(num,5); ff1 = ghostnodes(n1,6); ff2 = ghostnodes(n2,6); ff3 = ghostnodes(n3,6); fftot = (ff1 + ff2 + ff3); x1 = ghostnodes(n1,2); x2 = ghostnodes(n2,2); x3 = ghostnodes(n3,2); y1 = ghostnodes(n1,3); y2 = ghostnodes(n2,3); y3 = ghostnodes(n3,3); 188

210 elefillpct(num,1) = fftot/3; function [Papp, globaltime, flagger] = getappliedp(globaltime, PappMax, Pgrad,xTow,kxxPly,viscosity,vfPly,voidMat,elePc,eleFillPct,PappPrev) %Calculates the applied pressure at a given global time pressdiff = 2500; %How much larger the Papp + Pcap has to be than void P for it to be acceptable timeinc = 1; %How many seconds are added each time the loop has to redo Papp flagger = 0; Papp = Pgrad*( 1 - xtow/sqrt(2*kxxply*pgrad*(globaltime)/(viscosity*(1-vfply)))); %Applied pressure at this time if Papp > PappMax Papp = PappMax; %Counting the number of voids and number of elements that are part of voids voiddesignat = unique(voidmat(:,1)); sizevodes = size(voiddesignat); numvoids = sizevodes(1,1); %Number of unique voids voidsze = size(voidmat); numvoidele = voidsze(1,1); %Number of elements that are constituents of voids voidvspcap = ones(numvoids,3); %our blank matrix for our comparing matrix %Here we make a list of the void pressures (col 1) and capillary pressures %(col 2) for each void for voidn = 1:numVoids pcapv = 0; pcapvct = 1; thisvoidnum = voiddesignat(voidn,1); voidpthisv = 0; for velnum = 1:numVoidEle voidelementnumber = voidmat(velnum,2); if ((voidmat(velnum,1) == thisvoidnum) && (elefillpct(voidelementnumber,1)>0) && (elefillpct(voidelementnumber,1)<0.99)) pcapv(pcapvct,1) = elepc(voidelementnumber,1); pcapvct = pcapvct + 1; voidpthisv = voidmat(velnum,7); pcapave = mean(pcapv); voidvspcap(voidn,1) = voidpthisv; voidvspcap(voidn,2) = pcapave; 189

211 %Now we check to see if we need to advance the global time for voidn = 1:numVoids if ((voidvspcap(voidn,2) + Papp) > (voidvspcap(voidn,1) + pressdiff)) voidvspcap(voidn,3) = 0; numunsatisfied = sum(voidvspcap(:,3)); %Number of voids this is satisfied for while (numunsatisfied == numvoids && Papp < PappMax) globaltime = globaltime + timeinc; Papp = Pgrad*( 1 - xtow/sqrt(2*kxxply*pgrad*(globaltime)/(viscosity*(1- vfply)))); if Papp > PappMax Papp = PappMax; for voidn = 1:numVoids if ((voidvspcap(voidn,2) + Papp) > (voidvspcap(voidn,1) + pressdiff)) voidvspcap(voidn,3) = 0; numunsatisfied = sum(voidvspcap(:,3)); %Number of voids this is satisfied for if (numunsatisfied == numvoids && PappPrev == PappMax) flagger = 1; function writesol(nodes, elements, ghostnodes, gatemat,currtime,stringchterm,changingterm) %Writes dump output file, readable by LIMS UI n = length(nodes); % number of nodes m2d = length(elements); % number of 2D elements load pressures.dmp load flowrates.dmp %Writing the matrix for the results for num = 1: n resmat(num,1) = num; resmat(num,2) = pressures((num+2),1); resmat(num,3) = flowrates((num+2),1); resmat(num,4) = ghostnodes(num,6); resmat(num,5) = ghostnodes(num,5); gtsize = size(gatemat); 190

212 numgates = gtsize(1,1); currentt = currtime; %Writing the.dmp file for the updated nodes and elements fname = [stringchterm,num2str(changingterm),'mlimsres']; %fname = sprintf(stringchterm,num2str(changingterm),'mlimsres'); filename = sprintf('%s.dmp',fname); fina = fopen(filename, 'w'); fprintf(fina,'# \r\n'); fprintf(fina,'number of nodes : %5.0f \r\n',n); fprintf(fina,' Index x y z\r\n'); fprintf(fina,'===================================================\r\ n'); for i = 1: n % loop to write node data fprintf(fina,'%6.0f %12.6f %12.6f %12.6f \r\n',nodes(i,:)); fprintf(fina,'number of elements : %5.0f \r\n',m2d); fprintf(fina,' Index NNOD N1 N2 N3 (N4) (N5) (N6) (N7) (N8) h Vf Kxx Kxy Kyy Kzz Kzx Kyz\r\n'); fprintf(fina,'===================================================== ============================================================= ============================================================\r\ n'); for i = 1: m2d % loop[ to write 2D elements data fprintf(fina,'%6.0f %4.0f %6.0f %6.0f %6.0f %42.6f %15.6f %15.4e %15.4e %15.4e \r\n',elements(i,:)); fprintf(fina,'resin Viscosity model NEWTON\r\n'); fprintf(fina,'viscosity : 0.005\r\n'); fprintf(fina,'results at %d \r\n', currtime); fprintf(fina,'number of Current Gates : %d \r\n', numgates); fprintf(fina,' Type Node Value \r\n'); fprintf(fina,'============================== \r\n'); for i = 1:numGates fprintf(fina,'pressure at %5.0f, p= \r\n', gatemat(i,1)); fprintf(fina,'nodal results \r\n'); fprintf(fina,' Index Pressure Flow Rate Fill Factor Fill Time \r\n'); fprintf(fina,'===================================================== =================== \r\n'); for i = 1: n % loop to write results fprintf(fina,'%6.0f %15.0f %15.6f %15.6f %15.6f \r\n',resmat(i,:)); fclose(fina); 191

213 C.3 Impregnation of Three Dimensional Fiber Tows by Multiple Drops This set of programs models the impregnation of a fiber tow unit cell by two drops with a finite resin volume. Each drop is tracked individually as the tow is infused. It implements the methodology discussed in Chapter 5. C.3.1 simlimsymmetric.m: Main m-file for Parametric Studies % Required input: Nodal matrix with column format: (1: node number, 2-4: node x,y,z coordinates). Element matrix with column format: (1: element number, 2-5: nodes in element). function resultsmat = simlimsymmetric(nodes,elements) tic; % Starting the timer to see how long simulation runs format long %Enter the tow and resin information rfib = 4*10^(-6); % meters surften = 0.07; % N/m contactang = 30*pi/180; % Radians viscosity = 0.95; % Pa*s volumefraction = 0.5;%0.417; % Average fiber volume fraction desired %Enter the volume of resin in each drop (as percentage of porous volume) vres1pct = 0.1; %ratio of drop 1 volume to porous volume in the tow vres2pct = 0.1; %ratio of drop 2 volume to porous volume in the tow gappct = 0.5; %Which percentage of the tow length do you want the gap between drops (0.5 is used as default since it is a repeating unit cell) %Enter the amount of time between when the first drop is dropped and second tdelay2 = 7; % seconds %How many nodes to solve for each time step nodesthisstep = 1; % use 1 as default, increasing this may lead to issues with permeability modification %Input the fiber volume fraction tolerances vftolerance = 0.01; %Setting how close we need our fiber volume fraction to be to our desired fiber volume fraction stdevvf = 0.1; %Standard deviation if gaussian distribution is employed 192

214 %Here we input some matrices for parametric studies rfibmat = [4.5; 2; 7; 9; 11]; vfmat = [.41;.5;.6;.7;.8]; contangmat = [15; 45; 60; 75]; vres1mat = [0.05;.1;.15;.20;.25]; gappercentmat = [.5]; %~~~~~~~~~~~~~~~~~~~~~INPUTS COMPLETE~~~~~~~~~~~~~~~~~~~ for thisnumber = 1:1:4 %Be sure to only make this number go up to the desired number of parameters to be studied in input matrix %We only want to have one grouping uncommented out for parametric study, other values will be the default ones previously defined. %volumefraction= vfmat(thisnumber,1); %porousvolume = volumetow*(1-volumefraction); %changingterm = volumefraction; %stringchterm = 'volfra'; %gappct= gappercentmat(thisnumber,1); %changingterm = gappct; %stringchterm = 'gappct'; %rfib= rfibmat(thisnumber,1)*10^(-6); %changingterm = rfib; %stringchterm = 'rfib'; vres1pct= vres1mat(thisnumber,1);%*10^(-9); changingterm = vres1pct; stringchterm = 'len15vres'; vres2pct = vres1pct; %contang= contangmat(thisnumber,1); %contactang = contang*pi/180; %changingterm = contang; %stringchterm = 'contactang'; vres1 = vres1pct * porousvolume; vres2 = vres2pct * porousvolume; %Setting the center angles for resin contact (initial) 193

215 res1alphaave = pi/2; % Radians %Getting some stats from the mesh n = length(nodes); % number of nodes szeelemat = size(elements); m2d = szeelemat(1,1); % number of elements centerx = 0; %Center of the tow horizontally centery = 0.5*(max(nodes(:,3)) + min(nodes(:,3))); %Center of the tow vertically centerz = 0.5*(max(nodes(:,4)) + min(nodes(:,4))); %Center of the tow vertically ellipsea = max(nodes(:,2)) - centerx; %Retrieving major ellipse axis ellipseb = max(nodes(:,3)) - centery; %Retrieving minor ellipse axis lentow = (max(nodes(:,4)) - min(nodes(:,4))); %Calculating the length of the tow volumetow = ellipsea*ellipseb*pi*lentow/2; %Divide by 2 since we are dealing with only half of the tow due to symmetry plane porousvolume = volumetow*(1-volumefraction); eletype = 4; % 4 Nodes per element flagger = 0; % This flag can be utilized to signal of program currnodefilltime = 0.01; %This sets it such that analysis can be done on resulting data, with 0 fill time indicating never filled nodes vresorig1 = vres1; % Total volume of drop 1 vresorig2 = vres2; % Total volume of drop 2 resouttow = vres1 + vres2; % Initial volume of resin outside of tow sstep = ; %This is utilized for the step size in our capillary pressure function resvolmat = 1; % Initializing our resin volume matrix so we can track changes over time % The nodes residing on the surface of the tow are identified edgematrix = getsurfacenodes(nodes); %We assign our fiber volume fractions to be within our set parameters Each of the following sets the fiber volume fractions for all elements, with the desired volume fraction distribution elements = assignconstantvfdist(m2d,volumefraction,elements); %elements = assigngaussvfdist(m2d,volumefraction,stdevvf,vftolerance,elements); %elements =assignrandomvfdist(m2d,volumefraction,stdevvf,vftolerance,elements,edgeele ments); 194

216 %elements =assignlinearvfdist(m2d,volumefraction,stdevvf,vftolerance,elements,nodes,ellip seb); %Assigning initial permeability values based on Gebart Equations and volume fractions elements = assign3dperms(m2d, rfib, elements); %Setting the capillary pressure for each element and calculating average, using the same m-files as previously for capillary pressure prediction for num = 1: m2d vf = elements(num,7); % Fiber volume fraction of element pcvss = pcpacked(rfib, vf, contactang, surften,sstep); averagepc = avepc(pcvss); % Average Transverse capillary pressure Pcap = averagepc; elepc(num,1) = Pcap; axialpc = 8/rFib*(vf)/(1-vf)*surfTen*cos(contactAng); %Average axial Pc elepc(num,2) = axialpc; meansofpcs = mean(elepc); pcapave = mean(meansofpcs); % Results in matrix with row number being the element number, column 1 being the % Transverse capillary pressure, and column 2 being the axial capillary pressure %Getting the coordinate values for drop gap centerz1 = centerz - (gappct/2)*lentow; centerz2 = centerz + (gappct/2)*lentow; %Creating copies of the matrices for storing information rawnodes = nodes; rawelements = elements; dynkele = elements; ghostnodes = nodes; fillfactor(1,1)=n; %Number of empty nodes fillfactor(2,1)=0.01; %Start time %Setting fill factors for all nodes (will go back to change gate ones) and our initial fill factor matrix for nodev = 1:n ghostnodes(nodev,5)=0; fillfactor((nodev+2),1)=0; ghostnodes(nodev,6)=0; ghostnodes(nodev,7) = 0; %Pressures 195

217 ghostnodes(nodev,8) = 0; %flow rates %Getting the amount of resin that will flow into the tow as a result of %each drop's inlets having their fill factor set to one [volts1, volts2] = getinitialtimestepvolsforeachdrop(ellipsea, ellipseb, lentow, contactang,edgematrix, ghostnodes, elements, centerx, centery,vres1,vres2, centerz1, centerz2); %Determining the case %Case 1: Drop 1 is spreading into the tow only %Case 2: Drop 2 is spreading into the tow only %Case 3: Both drops are spreading into the tow %Case 4: Drop 1 is over, but drop 2 has yet to be dispensed %Case 5: Simulation should be over, both drops are fully in the tow dropcase = getdropcase(vres1, vres2, currnodefilltime, tdelay2); %determining the starting location for gate matrix if dropcase == 1 gate1matrix = 1; [xresw1,zresw1] = getresininletdim(ellipsea, ellipseb, lentow, contactang, (2*vRes1)); [gate1matrix, numgates1] = getgatematrixdropsym(edgematrix, nodes, centerx, centery, centerz1, xresw1, zresw1,ellipsea,ellipseb); gatematrix = gate1matrix; numgates = numgates1; numgates2 = 0; elseif dropcase == 2 gate2matrix = 1; [xresw2,zresw2] = getresininletdim(ellipsea, ellipseb, lentow, contactang, (2*vRes2)); [gate2matrix, numgates2] = getgatematrixdropsym(edgematrix, nodes, centerx, centery, centerz2, xresw2, zresw2,ellipsea,ellipseb); gatematrix = gate2matrix; numgates = numgates2; numgates1 = 0; elseif dropcase == 3 gate1matrix = 1; [xresw1,zresw1] = getresininletdim(ellipsea, ellipseb, lentow, contactang, (2*vRes1)); [gate1matrix, numgates1] = getgatematrixdropsym(edgematrix, nodes, centerx, centery, centerz1, xresw1, zresw1,ellipsea,ellipseb); gate2matrix = 1; 196

218 [xresw2,zresw2] = getresininletdim(ellipsea, ellipseb, lentow, contactang, (2*vRes2)); [gate2matrix, numgates2] = getgatematrixdropsym(edgematrix, nodes, centerx, centery, centerz2, xresw2, zresw2,ellipsea,ellipseb); numgates = numgates1+numgates2; for num = 1:numGates1 gatematrix(num,1) = gate1matrix(num,1); for num = 1:numGates2 gatematrix((num+numgates1),1) = gate2matrix(num,1); elseif dropcase == 4 currnodefilltime = tdelay2; gate2matrix = 1; [xresw2,zresw2] = getresininletdim(ellipsea, ellipseb, lentow, contactang, (2*vRes2)); [gate2matrix, numgates2] = getgatematrixdropsym(edgematrix, nodes, centerx, centery, centerz2, xresw2, zresw2,ellipsea,ellipseb); gatematrix = gate2matrix; numgates = numgates2; numgates1 = 0; else flagger = 1; % If flow is done, flag the simulation for ing %Computing the initial time step values resct = 1; % Time step number resvolmat(resct,1) = 0; %Time resvolmat(resct,2) = 0; %Resin inside of tow resvolmat(resct,3) = resouttow; %Resin outside of tow resvolmat(resct,4) = vres1; % Volume of drop 1 outside of the tow resvolmat(resct,5) = vres2; % Volume of drop 2 outside of the tow resvolmat(resct,6) = numgates1; %Number of gates for inlet of drop 1 resvolmat(resct,7) = numgates2; %Number of gates for inlet of drop 2 resvolmat(resct,8) = vresorig1 - vres1; %Volume drop 1 in the tow resvolmat(resct,9) = vresorig2 - vres2; %Volume drop 2 in the tow resct = resct + 1; %Setting the fill factor of the gate 1 nodes to 1 since they are instantly filled gatemat = gatematrix; for g1ind = 1:numGates gatenode = gatemat(g1ind,1); ghostnodes(gatenode,5) = 0.1; 197

219 ghostnodes(gatenode,6) = 1; fillfactor((gatenode+2),1) = 1; numsteps = 2; % Time step emptynodes = n; stepmax = emptynodes; % Max steps to be run in simulation %Looping for future time steps (assume drop 2 flows after drop 1 is done) while ((numsteps < emptynodes) && (vres2 > 0) && flagger == 0) emptynodesprev = emptynodes; %Updating the flow front permeability and retuning the filled elements %back to their original permeability dynkele = assignffperms(m2d, rawelements, ghostnodes, dynkele, pcapave, elepc); %Running LIMS writedmp(nodes,dynkele,m2d,eletype,viscosity); %Matlab generates dmp file writelb(ghostnodes, gatematrix, pcapave, nodesthisstep); %LB file generated runlims();% LIMS is ran pause(0.2) % Pause for 0.2 seconds %Load the necessary results from LIMS load fillfactors.dmp load pressures.dmp load filltimes.dmp load flowrates.dmp %Updating our flow information emptynodes = fillfactors(1,1); %Fill factor matrix 1 st line is empty nodes steptime = fillfactors(2,1); % Fill factor matrix 2 nd line is the step time currnodefilltime = currnodefilltime + steptime; % Updating global time numsteps = numsteps + 1; %Updating the number of steps %Updating nodal fill factors, pressures, flow rates, and fill times for indct = 1:n if ((fillfactors((indct+2),1) == 1) && (fillfactors((indct+2),1) > ghostnodes(indct,6))) ghostnodes(indct,5) = currnodefilltime; if fillfactors((indct+2),1) > 0.5 && fillfactors((indct+2),1) < 0.99 ghostnodes(indct,5) = currnodefilltime + filltimes((indct+2),1); 198

220 ghostnodes(indct,6) = fillfactors((indct+2),1); ghostnodes(indct,7) = pressures((indct+2),1); ghostnodes(indct,8) = flowrates((indct+2),1); %Updating the volume of each drop by looking at flow rate through inlet nodes if dropcase == 1 volflow1 = getflowthisstep(ghostnodes, steptime, gate1matrix, numgates1); vres1 = vres1 - volts1 - volflow1; volts1 = 0; elseif dropcase == 2 volflow2 = getflowthisstep(ghostnodes, steptime, gate2matrix, numgates2); vres2 = vres2 - volts2 - volflow2; volts2 = 0; elseif dropcase == 3 volflow1 = getflowthisstep(ghostnodes, steptime, gate1matrix, numgates1); vres1 = vres1 - volts1 - volflow1; volflow2 = getflowthisstep(ghostnodes, steptime, gate2matrix, numgates2); vres2 = vres2 - volts2 - volflow2; volts1 = 0; volts2 = 0; elseif dropcase == 4 volflow2 = getflowthisstep(ghostnodes, steptime, gate2matrix, numgates2); vres2 = vres2 - volts2 - volflow2; volts2 = 0; %Updating global volumes (in and out of tow) resintow = (vresorig1+vresorig2) - (vres1 + vres2); resouttow = (vres1 + vres2); %Updating our results matrix resvolmat(resct,1) = currnodefilltime; %Time resvolmat(resct,2) = resintow; %Amount of resin in the tow resvolmat(resct,3) = resouttow; %Amount of resin outside the tow resvolmat(resct,4) = vres1; %Volume of drop 1 outside of the tow resvolmat(resct,5) = vres2; %Volume of drop 2 outside of the tow resvolmat(resct,6) = numgates1; %Number of gates for drop 1 resvolmat(resct,7) = numgates2; %Number of gates for drop 2 resvolmat(resct,8) = vresorig1 - vres1; %Volume of drop 1 inside of the tow resvolmat(resct,9) = vresorig2 - vres2; %Volume of drop 2 inside of the tow resct = resct + 1; 199

221 %Providing some information as to how far simulation is to the user through the MATLAB interface PercentComplete = (1-(vRes1+vRes2)/(vResOrig1 + vresorig2))*100 numsteps %Determining the case and corresponding starting location for the gates dropcase = getdropcase(vres1, vres2, currnodefilltime, tdelay2); if dropcase == 1 gate1matrix = 1; [xresw1,zresw1] = getresininletdim(ellipsea, ellipseb, lentow, contactang, (2*vRes1)); [gate1matrix, numgates1] = getgatematrixdropsym(edgematrix, nodes, centerx, centery, centerz1, xresw1, zresw1,ellipsea,ellipseb); gatematrix = gate1matrix; numgates = numgates1; numgates2 = 0; elseif dropcase == 2 gate2matrix = 1; [xresw2,zresw2] = getresininletdim(ellipsea, ellipseb, lentow, contactang, (2*vRes2)); [gate2matrix, numgates2] = getgatematrixdropsym(edgematrix, nodes, centerx, centery, centerz2, xresw2, zresw2,ellipsea,ellipseb); gatematrix = gate2matrix; numgates = numgates2; numgates1 = 0; elseif dropcase == 3 gate1matrix = 1; [xresw1,zresw1] = getresininletdim(ellipsea, ellipseb, lentow, contactang, (2*vRes1)); [gate1matrix, numgates1] = getgatematrixdropsym(edgematrix, nodes, centerx, centery, centerz1, xresw1, zresw1,ellipsea,ellipseb); gate2matrix = 1; [xresw2,zresw2] = getresininletdim(ellipsea, ellipseb, lentow, contactang, (2*vRes2)); [gate2matrix, numgates2] = getgatematrixdropsym(edgematrix, nodes, centerx, centery, centerz2, xresw2, zresw2,ellipsea,ellipseb); numgates = numgates1+numgates2; for num = 1:numGates1 gatematrix(num,1) = gate1matrix(num,1); for num = 1:numGates2 gatematrix((num+numgates1),1) = gate2matrix(num,1); 200

222 elseif dropcase == 4 currnodefilltime = tdelay2; gate2matrix = 1; [xresw2,zresw2] = getresininletdim(ellipsea, ellipseb, lentow, contactang, (2*vRes2)); [gate2matrix, numgates2] = getgatematrixdropsym(edgematrix, nodes, centerx, centery, centerz2, xresw2, zresw2,ellipsea,ellipseb); gatematrix = gate2matrix; numgates = numgates2; numgates1 = 0; else flagger = 1; if numgates < 1 flagger = 1; %Here we change the fill time of the unfilled nodes to a later time to make results easily viewable in LIMS UI maxt = currnodefilltime; for num = 1:n ft = ghostnodes(num,5); if ft == 0 ghostnodes(num,5) = maxt*1.1; changingtermval = changingterm; %Write output file for our nodal results and our volume tracking matrix as well as our LIMS output DMP file All file names including identifying information. A TECPlot version of the solution is also written so results may be viewed in TECPlot fnamenodes = ['ghostnodesfor',stringchterm,num2str(changingterm),'.txt']; fnameresout = ['resouttowvstimefor',stringchterm,num2str(changingterm),'.txt']; currtime = currnodefilltime; writesol(rawnodes, rawelements, ghostnodes, gate1matrix,currtime,stringchterm,changingterm) writetecplotsol(ghostnodes,rawelements,n,m2d,stringchterm,changingterm) dlmwrite(fnamenodes,ghostnodes) 201

223 dlmwrite(fnameresout,resvolmat); % We make a results matrix of the average and standard deviation for the flow front location (y direction), evaluated for each case ran filledvolpct = findfilledvf(ghostnodes, elements); resultsmat(thisnumber,1) = changingtermval; [avet stdt] = getroughnesse(ghostnodes,elements); resultsmat(thisnumber,2) = avet; resultsmat(thisnumber,3) = stdt; simtime = toc %Stopping the clock to see how long simulation took C.3.2 Subroutine m-files (in the Order in which they are First Called) function [surfacenodes] = getsurfacenodes(nodes) %Input is the nodal matrix from main m-file %For elliptic tows, returns the nodes on the tow's outer surface %Axial direction = z for this program %Note: does not return the nodes on the symmetry plane surfacenodes = 0; format long; tolerateme = 1*10^(-2); %Tolerance to be included (error < 1% here) ellipsea = 0.5*(max(nodes(:,2))-min(nodes(:,2))); ellipseb = 0.5*(max(nodes(:,3))-min(nodes(:,3))); sznodes = size(nodes); numnodes = sznodes(1,1); surfnodect = 1; for num = 1:numNodes nodeindex = nodes(num,1); nodex = nodes(num,2); nodey = nodes(num,3); thistot = (nodex/ellipsea)^2 + (nodey/ellipseb)^2; if (thistot > (1-tolerateMe)) && (thistot < (1+tolerateMe)) surfacenodes(surfnodect,1) = nodeindex; surfnodect = surfnodect + 1; function [elements] = assignconstantvfdist(numelements,volumefraction,elements) %This function assigns a constant distribution of fiber volume fraction to all of the elements. 202

224 function [elements] = assignlinearvfdist(numelements,volumefraction,stdevvf,vftolerance,elements,nod es,ellipseb,edgeelements) %This function assigns a linear distribution of fiber volume fraction to all of the elements. %Inputs: Number of elements, average desired fiber volume fraction, desired, standard deviation of fiber volume fractions, tolerance for fiber volume, fraction (how much can the actual average deviate from the desired one, before it is considered unacceptable) %Output: Fiber volume fraction matrix (each row in matrix corresponds to an element in the mesh %Note: Average fiber volume fraction required to be > 0.45 maxbound = 0.9; vfmax = maxbound; vfave = volumefraction; m = 3*pi/(4*ellipseB)*(vfMax-vfAve); %Calculating the slope at which it will change with respect to y vfmatrix = zeros(numelements,1); for num = 1: numelements n1 = elements(num,3); n2 = elements(num,4); n3 = elements(num,5); y1 = nodes(n1,3); y2 = nodes(n2,3); y3 = nodes(n3,3); eley = (y1+y2+y3)/3; vfmatrix(num,1) = vfmax - m*abs(eley); for num = 1:numElements elements(num,7) = vfmatrix(num,1); function [elements] = assigngaussvfdist(numelements,volumefraction,stdevvf,vftolerance,elements) %This function assigns a Gaussian distribution of fiber volume fraction to all of the elements. %Inputs: Number of elements, average desired fiber volume fraction, desired standard deviation of fiber volume fractions, tolerance for fiber volume fraction (how much can the actual average deviate from the desired one before it is considered unacceptable) %Output: Fiber volume fraction matrix (each row in matrix corresponds to an element in the mesh 203

225 %We start by setting fiber volume fraction as unacceptable and have gaussdist function assign fiber volume fractions in a while loop until it is acceptable notacceptablevf = 1; maxvf = 0.9; minvf = volumefraction - (0.9-volumeFraction); if minvf < 0 minvf =0.01; ; szel = size(elements); numele = szel(1,1); while notacceptablevf == 1 vfmatrix = normrnd(volumefraction,stdevvf,numele,1); for num = 1:numEle happy = 0; while happy < 1 vf = vfmatrix(num,1); if (vf > minvf && vf < maxvf) happy = 1; else vfmatrix(num,1) = normrnd(volumefraction,stdevvf,1,1); vfave = mean(vfmatrix); if ((vfave + vftolerance) > volumefraction && (vfave - vftolerance) < volumefraction) notacceptablevf = 0; for num = 1:numElements elements(num,7) = vfmatrix(num,1); function [volts1, volts2] = getinitialtimestepvolsforeachdrop( ellipsea, ellipseb, lentow, contactang,edgematrix, ghostnodes, elements, centerx, centery,vres1,vres2, centerz1, centerz2) %Here we get the volume of resin filled into the tow for each drop resulting from setting inlet fill factors equal to one %For drop one: [xresw1,zresw1] = getresininletdim(ellipsea, ellipseb, lentow, contactang, vres1); 204

226 [gate1matrix, numgates1] = getgatematrixdrop(edgematrix, ghostnodes, centerx, centery, centerz1, xresw1, zresw1); gatemat = gate1matrix; for g1ind = 1:numGates1 gatenode = gatemat(g1ind,1); ghostnodes(gatenode,5) = 0.1; ghostnodes(gatenode,6) = 1; volts1 = getresvolume(ghostnodes, elements); %Getting the values for drop 2: [xresw2,zresw2] = getresininletdim(ellipsea, ellipseb, lentow, contactang, vres2); [gate2matrix, numgates2] = getgatematrixdrop(edgematrix, ghostnodes, centerx, centery, centerz2, xresw2, zresw2); gatemat = gate2matrix; for g1ind = 1:numGates2 gatenode = gatemat(g1ind,1); ghostnodes(gatenode,5) = 0.1; ghostnodes(gatenode,6) = 1; voltot = getresvolume(ghostnodes, elements); volts2 = voltot - volts1; function [dropcase] = getdropcase(vres1, vres2, currnodefilltime, tdelay2) %Determines which of the 5 possible cases it is: %Case 1: Drop 1 is spreading into the tow only %Case 2: Drop 2 is spreading into the tow only %Case 3: Both drops are spreading into the tow %Case 4: Drop 1 is over, but drop 2 has yet to be dispensed %Case 5: Simulation should be over, both drops are fully in the tow if vres1 > 0 sig1 = 1; else sig1 = 0; if vres2 > 0 sig2 = 1; else sig2 = 0; if currnodefilltime > tdelay2 205

227 sig3 = 1; else sig3 = 0; %Case 1: Drop one is the only one spreading into the tow if sig1 == 1 && sig2 == 0 dropcase = 1; elseif sig1 == 1 && sig3 == 0 dropcase = 1; %Case 2: Drop 2 is spreading into the tow only elseif sig1 == 0 && sig2 == 1 && sig3 == 1 dropcase = 2; %Case 3: Both drops are spreading into the tow at teh same time elseif sig1 == 1 && sig2 == 1 && sig3 == 1 dropcase = 3; %Case 4: Drop 1 is over, but drop 2 has not started yet elseif sig1 == 0 && sig3 == 0 dropcase = 4; %Case 5: No drops spreading - simulation should be over else dropcase = 5; function [xmax,zmax] = getresininletdim(ellipsea, ellipseb, lentow, contactang, vres) %Note: permeter of ellipse is given by P=2*pi*sqrt(a^2+b^2)/sqrt(2), we are looking only at x1 values along top half as otherwise it would all flow to bototm of tow format long x1maxval = ellipsea*0.97; z2maxval = lentow; %Length of the tow (in m). Resin cannot be longer than tow %Finds the x1 bestx1 = 0; x1tincrement = ellipsea/1000; z2increment = lentow/100; xct = 1; for x1t =x1tincrement:x1tincrement:x1maxval y1t = ellipseb*sqrt(1-(x1t/ellipsea)^2); m1 = 0-(x1T/y1T)*(ellipseB/ellipseA)^2; m2 = (m1-tan(contactang))/(1+m1*tan(contactang)); yc = x1t*(m1*tan(contactang)+1)/(m1-tan(contactang))+y1t; rres = sqrt(x1t^2+(y1t-yc)^2); fun (sqrt(rres^2-x.^2-z.^2))+yc; 206

228 zmax (sqrt(rres^2-x.^2)); volquart = integral2(fun,0,x1t,0,zmax); volthis = 4*abs(volQuart); xzchart(xct,1) = x1t; xzchart(xct,2) = y1t; xzchart(xct,3) = rres; xzchart(xct,4) = volthis; xzchart(xct,5) = (abs(vres-volthis)/vres)*1/ ; xzchart(xct,6) = yc; xct = xct + 1; szch = size(xzchart); numptst = szch(1,1); xzchart; bestvoldif = 1000; for n = 1:numPtsT if xzchart(n,5) < bestvoldif xbest = xzchart(n,1); ycbest= xzchart(n,6); rbest = xzchart(n,3); bestvoldif = xzchart(n,5); xmax = xbest; rbest; ellipseb; zmax = sqrt(rbest^2-(ellipseb-ycbest)^2); function [gatematrix, numgates] = getgatematrixdropsym(edgematrix, nodes, centerx, centery, centerz, xresw, zresw,ellipsea,ellipseb) %This function determines which nodes are part of the gate matrix, given the dimensions and location of the drop szed = size(edgematrix); numedgepts = szed(1,1); gatematrix = -1; gatect = 1; for edpt = 1:numEdgePts nodenum = edgematrix(edpt,1); nodex = nodes(nodenum,2); nodey = nodes(nodenum,3); nodez = nodes(nodenum,4); if nodey >= 0 207

229 zmax = centerz + zresw; zmin = centerz - zresw; if nodez > zmin && nodez < zmax xmax = centerx + xresw*sqrt(1-(nodez-centerz)^2/zresw^2); xmin = centerx - xresw*sqrt(1-(nodez-centerz)^2/zresw^2); if nodex == 0 difren = abs(ellipseb - nodey); %Need to make sure it is not one of the nodes on %the side face of symmetry plane if difren < 0.01*ellipseB gatematrix(gatect,1) = nodenum; gatect = gatect + 1; elseif nodex > xmin && nodex < xmax gatematrix(gatect,1) = nodenum; gatect = gatect + 1; numgates = gatect - 1; function writedmp(nodes, elements, m2d, eletype,viscosity) % Write dmp will write the dmp input file for LIMS to run, given the current state of the tow n = length(nodes); % number of nodes fname = sprintf('run'); filename = sprintf('%s.dmp',fname); o = fopen(filename, 'w'); fprintf(o,'# \r\n'); fprintf(o,'number of nodes : %5.0f \r\n',n); fprintf(o,' Index x y z\r\n'); fprintf(o,'===================================================\r\n'); for i = 1: n % loop to write node data fprintf(o,'%6.0f %12.6f %12.6f %12.6f \r\n',nodes(i,:)); fprintf(o,'number of elements : %5.0f \r\n',m2d); if eletype == 4 fprintf(o,' Index NNOD N1 N2 (N3) (N4) (N5) (N6) (N7) (N8) h(a) Vf Kxx Kxy Kyy Kzz Kzx Kyz\r\n'); fprintf(o,'====================================================== 208

230 ============================================================= ===========================================================\r\n '); for i = 1: m2d % loop[ to write 2D elements data fprintf(o,'%6.0f T',elements(i,1)); fprintf(o,'%6.0f %4.0f %6.0f %6.0f %29.0f %15.4f %15.4e %15.4e %15.4e %15.4e %15.4e %15.4e \r\n',elements(i,2:13)); fprintf(o,'resin Viscosity model NEWTON\r\n'); fprintf(o,'viscosity : %d\r\n',viscosity); fclose(o); function writelb(ghostnodes, gate1matrix, PcapA, nodesthisstep) % Generate lb file for LIMS to run fname = sprintf('run_simulate.lb'); fid2 = fopen(fname,'w+'); % Writing process for LIMS to run fprintf(fid2,'proc simu\r\n'); for thisnum = 1:nodesThisStep fprintf(fid2,'solve\r\n'); ; fprintf(fid2,'endproc\r\n'); fprintf(fid2,'\r\n'); %Creating file for fill factor output fprintf(fid2,'proc PFF\r\n'); fprintf(fid2,'defint i\r\n'); fprintf(fid2,'defint count\r\n'); fprintf(fid2,'let i=fifreefile()\r\n'); fprintf(fid2,'fiopen i,"fillfactors.dmp","w"\r\n'); fprintf(fid2,'fiprint i,sonumberempty()\r\n'); fprintf(fid2,'fiprintnl 1\r\n'); fprintf(fid2,'fiprint i,socurrenttime\r\n'); fprintf(fid2,'for count=1 TO SONUMBERNODES\r\n'); fprintf(fid2,'fiprintnl 1\r\n'); fprintf(fid2,'fiprint i,sofillfactor(count)\r\n'); fprintf(fid2,'next count\r\n'); fprintf(fid2,'ficlose i\r\n'); fprintf(fid2,'endproc\r\n'); fprintf(fid2,'\r\n'); %Creating file for pressure output fprintf(fid2,'proc PP\r\n'); 209

231 fprintf(fid2,'defint i\r\n'); fprintf(fid2,'defint count\r\n'); fprintf(fid2,'let i=fifreefile()\r\n'); fprintf(fid2,'fiopen i,"pressures.dmp","w"\r\n'); fprintf(fid2,'fiprint i,sonumberempty()\r\n'); fprintf(fid2,'fiprintnl 1\r\n'); fprintf(fid2,'fiprint i,socurrenttime\r\n'); fprintf(fid2,'for count=1 TO SONUMBERNODES\r\n'); fprintf(fid2,'fiprintnl 1\r\n'); fprintf(fid2,'fiprint i,sopressure(count)\r\n'); fprintf(fid2,'next count\r\n'); fprintf(fid2,'ficlose i\r\n'); fprintf(fid2,'endproc\r\n'); fprintf(fid2,'\r\n'); %Creating file for fill time output fprintf(fid2,'proc PFillT\r\n'); fprintf(fid2,'defint i\r\n'); fprintf(fid2,'defint count\r\n'); fprintf(fid2,'let i=fifreefile()\r\n'); fprintf(fid2,'fiopen i,"filltimes.dmp","w"\r\n'); fprintf(fid2,'fiprint i,sonumberempty()\r\n'); fprintf(fid2,'fiprintnl 1\r\n'); fprintf(fid2,'fiprint i,socurrenttime\r\n'); fprintf(fid2,'for count=1 TO SONUMBERNODES\r\n'); fprintf(fid2,'fiprintnl 1\r\n'); fprintf(fid2,'fiprint i,sotimetofill(count)\r\n'); fprintf(fid2,'next count\r\n'); fprintf(fid2,'ficlose i\r\n'); fprintf(fid2,'endproc\r\n'); fprintf(fid2,'\r\n'); %Creating file for flowrate output fprintf(fid2,'proc PFR\r\n'); fprintf(fid2,'defint i\r\n'); fprintf(fid2,'defint count\r\n'); fprintf(fid2,'let i=fifreefile()\r\n'); fprintf(fid2,'fiopen i,"flowrates.dmp","w"\r\n'); fprintf(fid2,'fiprint i,sonumberempty()\r\n'); fprintf(fid2,'fiprintnl 1\r\n'); fprintf(fid2,'fiprint i,socurrenttime\r\n'); fprintf(fid2,'for count=1 TO SONUMBERNODES\r\n'); fprintf(fid2,'fiprintnl 1\r\n'); fprintf(fid2,'fiprint i,soflowrate(count)\r\n'); fprintf(fid2,'next count\r\n'); 210

232 fprintf(fid2,'ficlose i\r\n'); fprintf(fid2,'endproc\r\n'); fprintf(fid2,'\r\n'); % Read the DMP file fprintf(fid2,'read "run.dmp"\r\n'); %Setting fill factors and pressures nodematsize = size(ghostnodes); numnodes = nodematsize(1,1); for nodenum = 1:numNodes fprintf(fid2,'setfillfactor %d %d \r\n',nodenum, ghostnodes(nodenum,6)); %Setting Gates gatsize = size(gate1matrix); numgates = gatsize(1,1); for i = 1:numGates nodenum = gate1matrix(i,1); fprintf(fid2,'setgate %d, 1, %d \r\n',gate1matrix(i,1), (PcapA)); % Setting the time to 0 and calling our processes fprintf(fid2,'\r\n'); fprintf(fid2,'settime 0\r\n'); fprintf(fid2,'\r\n'); fprintf(fid2,'call simu\r\n'); fprintf(fid2,'\r\n'); fprintf(fid2,'call PFF\r\n'); fprintf(fid2,'\r\n'); fprintf(fid2,'call PFR\r\n'); fprintf(fid2,'\r\n'); fprintf(fid2,'call PP\r\n'); fprintf(fid2,'\r\n'); fprintf(fid2,'call PFillT\r\n'); fprintf(fid2,'\r\n'); fprintf(fid2,'print "# empty nodes =", sonumberempty\r\n'); fprintf(fid2,'\r\n'); % Setting our output to DMP file and writing the results fprintf(fid2,'setouttype "dump"\r\n'); fprintf(fid2,'write "run_res.dmp"\r\n'); fclose all; function runlims() % Run the lb files, uses limsslv application and lcmaster file, written by Ali Gokce fname = sprintf('run_simulate.lb'); 211

233 filename = sprintf('load run_simulate.lb'); lims(3,6,2000); % set time-out to 2000 ms lims(3,3,50); %Set option to value (2, opt, value) lims(1,1); %Starts the slave, second 1 is slave ID (start, ID) lims(5,1,'setoutputlevel 2'); lims(5,1,'setmessagelevel 0'); lims(5,1,filename); %Put string to slave ID (lims(5,id,string)) lims(5,1,fname); output = 'ini'; pause(2); while ~isempty(output) output = lims(4,1,350); %Get line from slave with ID (lims(4,id, maxw)) lims(5,1,'print sonumberempty'); %Records # of empty nodes pause(2) %Manipulate this pause based on single simulation time lims(2,1) %Ends LIMS (Kills the slave) fclose all; function [vresthisstep] = getflowthisstep(ghostnodes, steptime, gatematrix, numgates) %Calculates how much resin flowed for specific drop during this time step totalflowrate = 0; for n = 1:numGates nodenum = gatematrix(n,1); flowratethisnode = abs(ghostnodes(nodenum,8)); totalflowrate = totalflowrate + flowratethisnode; vresthisstep = totalflowrate*steptime; function writesol(nodes, elements, ghostnodes, gatemat,currtime, stringchterm,changingterm) %This function writes a DMP files with the solution, readable by LIMS UI n = length(nodes); % number of nodes m2d = length(elements); % number of 2D elements load pressures.dmp load flowrates.dmp %Writing the matrix for the results for num = 1: n resmat(num,1) = num; resmat(num,2) = pressures((num+2),1); resmat(num,3) = flowrates((num+2),1); resmat(num,4) = ghostnodes(num,6); 212

234 resmat(num,5) = ghostnodes(num,5); gtsize = size(gatemat); numgates = gtsize(1,1); currentt = currtime; %Writing the.dmp file for the updated nodes and elements fname = [stringchterm,num2str(changingterm),'mlimsres']; filename = sprintf('%s.dmp',fname); fina = fopen(filename, 'w'); fprintf(fina,'# \r\n'); fprintf(fina,'number of nodes : %5.0f \r\n',n); fprintf(fina,' Index x y z\r\n'); fprintf(fina,'===================================================\r\ n'); for i = 1: n % loop to write node data fprintf(fina,'%6.0f %12.6f %12.6f %12.6f \r\n',nodes(i,:)); fprintf(fina,'number of elements : %5.0f \r\n',m2d); fprintf(fina,' Index NNOD N1 N2 (N3) (N4) (N5) (N6) (N7) (N8) h(a) Vf Kxx Kxy Kyy Kzz Kzx Kyz\r\n'); fprintf(fina,'===================================================== ============================================================= ============================================================\r\ n'); for i = 1: m2d % loop[ to write elements data fprintf(fina,'%6.0f T',elements(i,1)); fprintf(fina,'%6.0f %4.0f %6.0f %6.0f %29.0f %15.4f %15.4e %15.4e %15.4e %15.4e %15.4e %15.4e \r\n',elements(i,2:13)); fprintf(fina,'resin Viscosity model NEWTON\r\n'); fprintf(fina,'viscosity : 0.005\r\n'); fprintf(fina,'results at %d \r\n', currtime); fprintf(fina,'number of Current Gates : %d \r\n', numgates); fprintf(fina,' Type Node Value \r\n'); fprintf(fina,'============================== \r\n'); for i = 1:numGates fprintf(fina,'pressure at %5.0f, p= \r\n', gatemat(i,1)); fprintf(fina,'nodal results \r\n'); fprintf(fina,' Index Pressure Flow Rate Fill Factor Fill Time \r\n'); fprintf(fina,'===================================================== =================== \r\n'); 213

235 for i = 1: n % loop to write results fprintf(fina,'%6.0f %15.0f %15.6f %15.6f %15.6f \r\n',resmat(i,:)); fclose(fina); function writetecplotsol(ghostnodes,rawelements,n,m2d,stringchterm,changingterm) %Writes solution in TecPlot format ndele = [n m2d]; for num = 1:n TPnodeMat(num,1) = ghostnodes(num,2); TPnodeMat(num,2) = ghostnodes(num,3); TPnodeMat(num,3) = ghostnodes(num,4); TPnodeMat(num,4) = ghostnodes(num,7); TPnodeMat(num,5) = ghostnodes(num,5); TPnodeMat(num,6) = ghostnodes(num,6); for num = 1:m2d TPeleMat(num,1) = rawelements(num,2); TPeleMat(num,2) = rawelements(num,3); TPeleMat(num,3) = rawelements(num,4); TPeleMat(num,4) = rawelements(num,4); TPeleMat(num,5) = rawelements(num,5); TPeleMat(num,6) = rawelements(num,5); TPeleMat(num,7) = rawelements(num,5); TPeleMat(num,8) = rawelements(num,5); fname = [stringchterm,num2str(changingterm),'restplt']; filename = sprintf('%s.tec',fname); fina = fopen(filename, 'w'); fprintf(fina,'title = "%s" \r\n',filename); fprintf(fina,'variables = "x", "y", "z", "p", "Time", "Fill"\r\n'); fprintf(fina,'zone N = %d, ',n); fprintf(fina,'e = %d, F = FEPOINT, ET = BRICK\r\n',m2d); for num = 1:n fprintf(fina, '%f %f %f %f %f %f\r\n', TPnodeMat(num,:)); for num = 1:m2d fprintf(fina, ' %d %d %d %d %d %d %d %d\r\n', TPeleMat(num,:)); fclose(fina); 214

236 Appix D PERMISSIONS Prediction of capillary pressure for resin flow between fibers (Full article-yeager et al) 215

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240 Role of fiber distribution and air evacuation time on capillary driven flow into fiber tows (Full article Yeager et al) 219

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246 Image from A non-local void filling model to describe its dynamics during thermoplastic composites 225

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250 Image from: Drops on functional fibers: from barrels to clamshells and back 229

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254 Image from: Droplet on a fiber: geometrical shape and contact angle 233

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