CT Scan Generated Material Twins for Composites Manufacturing in Industry 4.0 [1st ed.] 9789811580208, 9789811580215

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Muhammad A. Ali Rehan Umer Kamran A. Khan

CT Scan Generated Material Twins for Composites Manufacturing in Industry 4.0

CT Scan Generated Material Twins for Composites Manufacturing in Industry 4.0

Muhammad A. Ali Rehan Umer Kamran A. Khan •

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CT Scan Generated Material Twins for Composites Manufacturing in Industry 4.0

123

Muhammad A. Ali Department of Aerospace Engineering Khalifa University of Science and Technology Abu Dhabi, United Arab Emirates

Rehan Umer Department of Aerospace Engineering Khalifa University of Science and Technology Abu Dhabi, United Arab Emirates

Kamran A. Khan Department of Aerospace Engineering Khalifa University of Science and Technology Abu Dhabi, United Arab Emirates

ISBN 978-981-15-8020-8 ISBN 978-981-15-8021-5 https://doi.org/10.1007/978-981-15-8021-5

(eBook)

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Acknowledgements

This book is based on research conducted at Khalifa University of Science and Technology (KUST), Abu Dhabi, United Arab Emirates. We are thankful to KUST for providing funding for conducting this research. We are thankful to Prof. Wesley J. Cantwell, Director Aerospace Research and Innovation Center (ARIC), for his valuable guidance throughout this work. We also appreciate the efforts of the research and support staff at KUST, Mr. Pradeep George, Mr. Jimmy Thomas, and Dr. Alia Aziz, for their involvement at various stages of this research. We are eternally indebted to our families for their unconditional support throughout the project. We are also profoundly grateful to fellow researchers, Dr. Yarjan Abdul Samad, Dr. Waqas Waheed, Dr. Adnan Saeed, Dr. Nguyen V. Viet, and Dr. Shaohong Luo. Finally, we would like to thank the publishing team at Springer Nature Scientific who tirelessly worked to put all the bits together. A special thanks to the project coordinator books production, Mr. Sriram Srinivas, for guiding us throughout the publication process. Muhammad A. Ali Rehan Umer Kamran A. Khan

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Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Manufacturing of Fiber Reinforced Polymer Composites . . 1.2.1 Liquid Composite Molding . . . . . . . . . . . . . . . . . 1.3 Reinforcement Characterization . . . . . . . . . . . . . . . . . . . . 1.3.1 Reinforcement Compaction Response . . . . . . . . . . 1.3.2 Reinforcement Permeability . . . . . . . . . . . . . . . . . 1.4 State-of-the-Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Compaction Characterization . . . . . . . . . . . . . . . . 1.4.2 Permeability Characterization . . . . . . . . . . . . . . . . 1.4.3 Material Twin Based Characterization Framework . 1.5 Motivation and Objectives . . . . . . . . . . . . . . . . . . . . . . . 1.6 Book Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Generation of Material Twin Using Micro CT Scanning 2.1 XCT Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Compaction Setup . . . . . . . . . . . . . . . . . . . . . 2.1.2 XCT Machine . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Fabric Samples . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Noise Reducing Filters . . . . . . . . . . . . . . . . . . 2.1.5 Radiographic Projections . . . . . . . . . . . . . . . . 2.2 Data Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 3D Volume Reconstruction . . . . . . . . . . . . . . 2.2.2 Segmentation . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Tow Deformations and Compaction Response . . . . . . 2.4 Model Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Material Twin . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Stochastic Virtual Model . . . . . . . . . . . . . . . . 2.4.3 Simplified Models . . . . . . . . . . . . . . . . . . . . .

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2.5 Virtual Permeability . . . 2.5.1 Numerical Setup 2.6 Summary . . . . . . . . . . . References . . . . . . . . . . . . . .

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3 Experimental Techniques for Reinforcement Characterization . 3.1 Types of Reinforcements . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 3D Reinforcements . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 2D Reinforcements . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 AFP Carbon Tape . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Test Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Compaction Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Permeability Measurement . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Through-Thickness Permeability Measurement . . . . . 3.5.2 In-Plane Permeability Measurement . . . . . . . . . . . . . 3.6 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Compaction Tests . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Through-Thickness Permeability Measurements . . . . . 3.6.3 In-Plane Permeability Measurements . . . . . . . . . . . . . 3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Experimental-Numerical Hybrid Reinforcement Characterization Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The In Situ Computed Tomography System . . . . . . . . . . . . . . . 4.2.1 Micro CT Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 In Situ Compression Fixture . . . . . . . . . . . . . . . . . . . . . 4.2.3 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Load and Displacement Control . . . . . . . . . . . . . . . . . . 4.2.5 3D Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Micro CT Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Scan Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Setting the Scan Resolution . . . . . . . . . . . . . . . . . . . . . 4.4 Data Processing and Digital Experiments . . . . . . . . . . . . . . . . . 4.4.1 Data Pre-processing and Segmentation . . . . . . . . . . . . . 4.4.2 Digital Flow Experiments . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Pore Structure Analysis . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Generation of Digital Material Twins . . . . . . . . . . . . . . . . . . . . 4.5.1 Realistic Voxel Model . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Stochastic Virtual Model of the AFP Carbon Tape . . . . 4.5.3 Ideal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

4.6 Flow Modeling . . . . . . . . . . . . . . . . 4.6.1 Micro-Scale Flow Modeling . 4.6.2 Meso-Scale Flow Modeling . 4.7 Flow and Boundary Conditions . . . . 4.8 Permeability Computation . . . . . . . . 4.9 Summary . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

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5 Reinforcement Compaction Response . . . . . . . . . . . . . . . . . 5.1 Stress Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Multi-stage Compaction . . . . . . . . . . . . . . . . . . . 5.1.2 Multi-cycle Compaction . . . . . . . . . . . . . . . . . . . 5.2 Tow Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 3D Orthogonal Reinforcement . . . . . . . . . . . . . . 5.2.2 3D Angle Inter-lock Reinforcement . . . . . . . . . . 5.3 Gap Analysis of the 3D Reinforcements . . . . . . . . . . . . 5.3.1 Gap Analysis of 3D Orthogonal Reinforcement . . 5.3.2 Gap Analysis of Angle Inter-lock Reinforcement . 5.4 Pore Network Analysis of the Angle Inter-lock Reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Pore Size Distribution . . . . . . . . . . . . . . . . . . . . 5.4.2 Path Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Effect of Multi-cycle Compaction on the Pore Size Distribution . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Application of the Hybrid Characterization Framework to AFP Carbon Tape . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Stress Relaxation . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Microscopic Analysis . . . . . . . . . . . . . . . . . . . . . 5.5.3 Gap Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Flow Fields and Reinforcement Permeability . . . . . . . . . . . . . 6.1 Through-Thickness Permeability of the 3D Reinforcements . 6.1.1 Velocity Fields in Through-Thickness Flow . . . . . . 6.1.2 Through-Thickness Permeability Predictions . . . . . . 6.2 In-Plane Permeability of the 3D Reinforcements . . . . . . . . . 6.2.1 Velocity Fields in In-Plane Flow . . . . . . . . . . . . . . 6.2.2 In-Plane Permeability Predictions . . . . . . . . . . . . . . 6.3 Analysis of Variability in Permeability Predictions . . . . . . . 6.3.1 Variability Among Unit Cells . . . . . . . . . . . . . . . . . 6.3.2 Variability Within Unit Cells . . . . . . . . . . . . . . . . . 6.4 Effect of Unit Cell Size on Permeability Predictions . . . . . . 6.5 Kozeny-Carman Analysis . . . . . . . . . . . . . . . . . . . . . . . . .

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6.6 Effect of Cyclic Compaction on Permeability . . . . . . . . . . . . . . 6.6.1 Velocity Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.2 Virtual Permeability Predictions . . . . . . . . . . . . . . . . . . 6.7 Application of the Hybrid Characterization Framework to Other Reinforcements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.1 Through-Thickness Permeability of AFP Carbon Tape . . 6.7.2 In-Plane Permeability of AFP Carbon Tape . . . . . . . . . . 6.7.3 2D Benchmark Reinforcements . . . . . . . . . . . . . . . . . . 6.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Experimental-Empirical Hybrid Approach . . . . . . . . . . . . . . . 7.1 Basic Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Porous Regions in Parallel . . . . . . . . . . . . . . . . . . . 7.1.2 Porous Regions in Series . . . . . . . . . . . . . . . . . . . . 7.2 Breakdown of 3D Architecture . . . . . . . . . . . . . . . . . . . . . 7.3 In-Plane Permeability of the 3D Architecture . . . . . . . . . . . 7.3.1 Permeability of the Channel and Gap . . . . . . . . . . . 7.3.2 Equivalent Permeability of the Channel-Warp Combination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Equivalent Permeability of the Gap-Binder Combination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.4 Equivalent Permeability of the Region A and C . . . 7.3.5 Equivalent Permeability of the Channel-Gap-Binder Combination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.6 Equivalent Permeability of the Warp-Channel-GapBinder Combination . . . . . . . . . . . . . . . . . . . . . . . . 7.3.7 Equivalent Permeability of Region B . . . . . . . . . . . 7.3.8 Overall Permeability . . . . . . . . . . . . . . . . . . . . . . . 7.4 Through-Thickness Permeability of the 3D Architecture . . . 7.5 Numerical Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Modeling Approach . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Validation of Basic Formulation . . . . . . . . . . . . . . . 7.5.3 Validation of Permeability Models for the 3D Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1

Introduction

1.1 Background The availability of adequate materials has been a significant factor in the advancement of human civilization and technological breakthroughs. Throughout the ages, the constant improvements in the mechanical, electrical and thermal properties of materials have been pushing the limits of technology. In these advancements, composite materials represent a giant leap with promising characteristics for high performance, lightweight and multifunctional applications. The design and manufacturing of composite materials and structures has been pursued in evolutionary as well as revolutionary ways during the past few decades [1]. Constant progress is being made in the development of existing techniques through research and experiments, as well as new ideas that are being presented. These efforts aim at reducing the manufacturing cost and complexity, enhancing the quality of parts produced and minimizing environmental impacts. A composite material is a material made from two or more constituent materials with significantly different physical or chemical properties that, when combined, produce a material with characteristics different from the individual components. The individual components remain separate and distinct within the final part. Fiber Reinforced Polymer Composite (FRPC) is a class of composite materials that consists of a polymer matrix reinforced with high-strength natural or synthetic fibers. The reinforcing fiber adds rigidity and is the main load-bearing component of the composite materials. While high performance composites are dominated by synthetic glass, carbon and aramid fibers, a wide range of natural fibers is also gaining popularity. The matrix binds the fibers together, transmits applied loads to the fibers, prevents propagation of cracks and protects the fibers from damage. Polymer matrices are classified as thermoplastic or thermosetting resins. A thermosetting polymer is irreversiblycured from a soft solid or viscous liquid, whereas a thermoplastic polymer becomes pliable or moldable above a specific temperature and solidifies upon cooling.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 M. A. Ali et al., CT Scan Generated Material Twins for Composites Manufacturing in Industry 4.0, https://doi.org/10.1007/978-981-15-8021-5_1

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1 Introduction

The advantages of FRPC over other materials have attracted many industries such as aerospace, automobile, infrastructure, sports and marine to explore and increase their usage. FRPC materials have been used extensively in a wide variety of aerospace applications, ranging from commercial airliners to deep space vehicles. World-leading organizations in aerospace such as National Aeronautics and Space Administration (NASA), the European Space Agency (ESA), Boeing, Airbus, Bombardier etc. are investing intensively in developing such materials. The commercial airliners, the Boeing 787 and the Airbus A350 are excellent examples, where more than 50% of the structure is comprised of fiber reinforced composites. NASA recently achieved a major milestone in the advancement of space technology by successfully testing a pressurized, large cryogenic propellant tank made entirely out of fiber reinforced composite materials. In automobile manufacturing, FRPC materials offer major advantages over steel and similar metals in producing lighter, safer and more fuel-efficient vehicles. FRPC materials are now being used in automobile body, chassis, interiors and engine components. Recently, several high value vehicles, such as the BMW M, and the i-series have used carbon fiber reinforced composites as primary structures. Apart from these two major industries, fiber reinforced composites have been the focus of many other industries, such as marine, defense, sports and energy sectors. For example, entire wind turbine blades are being manufactured using advanced composite manufacturing techniques. Sport items such as, tennis rackets, snow skis, sail boats, kayaks, helmets, shoe soles, hockey sticks, etc. are also being manufactured from fiber reinforced polymer composites.

1.2 Manufacturing of Fiber Reinforced Polymer Composites Fiber reinforcements are typically in the form of random mats, woven reinforcements, unidirectional or non-crimp stitched reinforcements, knits, braids, or 3D woven reinforcements [2]. Such types of reinforcement are developed into composite parts through various composite manufacturing techniques. The quality of the composite material and the resulting part depends on the manufacturing process, since it is during the manufacturing process that the matrix material and the fiber reinforcement are combined and consolidated to form the composite part. The process depends on the type of resin and reinforcement used. In this book, the focus is mainly on composites manufacturing process characterization which can be used in autoclave and Outof-Autoclave (OoA) manufacturing techniques. In the autoclave process, the fiber reinforcements are used in the form of prepregs. The prepregs are cut and formed into the desired shape and placed on a rigid mold in the desired position, orientation and sequence to form a layup. The layup, sealed with vacuum bag, is then placed into the autoclave and the curing process is performed according to the prescribed temperature-pressure-vacuum-time cycle inside the autoclave. The process is very versatile and gives a very uniform quality, as pressure and heat can be regulated

1.2 Manufacturing of Fiber Reinforced Polymer Composites

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very precisely. On the other hand, it is very costly due to high capital cost and time consuming due to long layup and curing time. The Out-of-Autoclave manufacturing includes processes such as vacuum bag only (VBO) and liquid composite molding (LCM). The main advantage of OoA processing techniques is the low capital cost. LCM techniques are attractive as they provide excellent control over part thickness hence, excellent mechanical properties are possible. However, the process characterization and quality control tools are still under development and require considerable research to achieve repeatable part quality. This book will focus on new LCM reinforcement characterizations (compaction response and permeability) techniques based on micro CT imaging.

1.2.1 Liquid Composite Molding Liquid composite molding (LCM) processes involve impregnation of a reinforcement using a liquid resin with “injectable” viscosity. A generic LCM process utilizes a mold cavity that is in the shape of the part to be manufactured. The fiber reinforcement is placed inside the mold cavity and the mold is closed. A reactive (thermoset) resin is then injected into the mold cavity under pressure, until complete saturation of the reinforcing material is achieved. The resin is then allowed to cure, after curing, the part is de-molded to yield the finished product. These steps are illustrated in Fig. 1.1. Liquid composite molding process has several variants with minor differences. Some of these variants of LCM processes are shown in Fig. 1.2. Resin Transfer (2) Preform Compression

(3) Resin Transfer

(1) Preform Layup

Liquid Composite Molding

(6) Finished Part

(4) Resin Cure

(5) De-Moulding Fig. 1.1 Schematic illustration of various steps involved in manufacturing of composite parts using liquid composite molding process

4 Fig. 1.2 Various variations of liquid composite molding technique

1 Introduction

VARTM

RTM

SCRIMP

CRTM LCM

SRIM

RTM-Light

VIPR

CIRTM

Molding (RTM) and Compression Resin Transfer Molding (CRTM) employ two rigid molds that enable a high compaction force to be applied with minimal mold deflections. RTM-light uses semi-flexible plastic or composite molds, providing a medium level compaction while minimizing mold deflections at a much reduced cost. The Vacuum Assisted Resin Transfer Molding (VARTM) and Seemann’s Composite Resin Infusion Molding Process (SCRIMP) are slight modifications of the process where the top half of the mold is replaced by a vacuum bag. In SCRIMP, a highly permeable layer is introduced at the top or the bottom of the reinforcement to facilitate rapid distribution of the resin throughout the part. Both VARTM and SCRIMP rely on drawing the resin from a container at atmospheric pressure through the fibrous bed by creating a vacuum. These processes have replaced RTM for many applications due to their simplicity, low initial capital investment and the ability to manufacture large structures. The cost is low due to low pressures used in the manufacturing process and the curing reactions being carried out mostly at room temperature. The process requires only a single tool surface, while the top surface is covered with a vacuum bag which also cuts down on tooling costs. The disadvantages of VARTM process is rough surface finish on the bag side, the time required in material preparation, inconsistent dimensional tolerances and the lack of automation.

1.3 Reinforcement Characterization During the LCM process, reinforcement compaction inside the mold cavity and successful injection of the resin plays an important role to ensure part quality. These two steps are primarily dependent on the reinforcement compressibility and permeability. Hence, for better process optimization and modeling, the compaction

1.3 Reinforcement Characterization

5

response and permeability of the reinforcement need to be determined. These two characteristics are considered vital processing parameters in LCM.

1.3.1 Reinforcement Compaction Response In an LCM process, once a dry fiber reinforcement is placed inside the mold cavity, the mold is closed using either a hydraulic press or vacuum bag. The behavior of a reinforcement subjected to a applied force normal to its plane has important consequences on the mold design and equipment specifications for all processes using fiber reinforcements. For example, in VARTM, the clamping force is applied using vacuum pressure under a flexible tooling, while in the RTM process, the reinforcement is compacted between two rigid molds. In either case, the reinforcement compaction response influences the mold clamping force, part thickness and fiber volume fraction, as result, affecting the reinforcement permeability. The compaction step is vital since, prior to manufacture, the fibrous material is not yet at the desired fiber volume fraction. The compaction decreases reinforcement thickness and increases fiber volume fraction. Furthermore, in some cases, the reinforcements are intentionally subjected to transverse compaction in order to “de-bulk” to a high fiber volume fraction [3]. The reinforcements can be subjected to single-cycle, multi-cycle or multi-stage compaction [4]. Moreover, during the resin injection, the reinforcement is not only subjected to the mold clamping force provided by the press, but also to the fluid pressure generated as a result of resin injection. The total stress acting on the reinforcement is governed by Terzaghi’s law which states that the total stress carried by the reinforcement is equal to the sum of the compaction stress taken by the fibers and fluid pressure [5–8]. Due to the compaction stress, the internal architecture of the reinforcement also changes, i.e. the tows flatten, the spaces between the fiber tows decrease, the tows undergo bending while at the micro-scale, the gaps between individual fibers also decrease. The compaction of multiple layered reinforcements (a stack of a number of 2D textiles) possess additional complexity of nesting, interlayering and packing. Hence, during manufacturing of composite parts via LCM process, the transverse compaction is important and it is essential to characterize the compaction response of the reinforcing fabric. In any reinforcement compaction characterization, two major attributes need to be investigated; (1) the relationship between applied load and the resulting thickness or fiber volume fraction (Vf ) and, (2) microscopic geometrical changes of the tows and fibers. The relationship between applied stress and/or resulting thickness (fiber volume fraction) is presented as a stress relaxation curve. The standard stress relaxation curve is a plot of applied stress versus time or fiber volume fraction (Vf ). Typically, the compaction of reinforcement consists of two parts, i.e. dynamic, non-linear compression stage and stress relaxation. The compaction depends on the reinforcement architecture, speed, dry/wet state, and number of reinforcement layers used. The second objective of a compaction study is to document tow deformations and

6

1 Introduction

the inter-tow gap reduction as a result of decreasing thickness, this is achieved by microscopic investigation either in-situ or ex-situ.

1.3.2 Reinforcement Permeability In the LCM process, the impregnation of reinforcement with resin is considered as flow through porous medium phenomena, and the permeability of the reinforcement dictates the “ease” of advancement of the resin flow front through the open channels therefore, influencing mold filling time. As the Reynolds number of the resin flow is considered very low due to low flow velocities, the resin flow through the reinforcement is assumed to be governed by Darcy’s law. The Darcy’s law relates the volume flow rate of a Newtonian fluid through a porous media of given crosssectional area and the pressure gradient along the flow direction. The parameters included in Darcy’s law are illustrated in Fig. 1.3a. Mathematically, Darcy’s law is given as, Q = −K

A P A Pi − Po = −K μ L μ L

(1.1)

where Q is the volume flow rate of the fluid, K is the permeability tensor, μ is the dynamic fluid viscosity, Pi − Po is the pressure gradient or the pressure difference, A is the cross-sectional area, and L is the length of the medium across which the fluid flows. The permeability is a measure of the ability of a porous material to allow fluid to pass through it and is also related to its porous structure and the connectivity of the pores. The Permeability is a directional quantity, described by a tensor in three dimensions. The 3D permeability tensor K can be expressed in Cartesian coordinates as, Pi > Po

Pi

L K

In

(a)

Q

Fabric Roll

Po

2

Out

3

1

Production direction (b)

Fig. 1.3 Illustration of a flow through a porous medium with relevant parameters and b principle permeability directions for a typical reinforcement

1.3 Reinforcement Characterization

7

⎡

⎤ K xx K xy K xz K = ⎣ K yx K yy K yz ⎦ K zy K zy K zz

(1.2)

In case of symmetry, the off-diagonal terms are usually taken equal. If the coordinate system is oriented in the principal directions, the off diagonal elements are taken as zero, and the remaining terms are known as principal permeability values i.e. K 11 , K 22 and K 33 . For fibrous reinforcement, they are divided as the in-plane, K 11 and K 22 , and through-thickness or transverse, K 33 , permeabilities. The principal directions for a typical fibrous reinforcement are illustrated in Fig. 1.3b. The reinforcement permeability is primarily a function of the reinforcement architecture and its fiber volume fraction (V f ) [9]. Incorrect compaction and permeability predictions may lead to an inefficient process design through incorrect fiber volume fractions and mold filling time, dry spots and defects in the manufactured parts [3]. The permeability characterization of fiber reinforcements is important for liquid composite molding process modeling and simulations. The permeability dictates the time taken to fill the mold, the degree of fiber wetting by the resin and flow patterns generated as a result of resin flow.

1.4 State-of-the-Art Existing reinforcement characterization techniques focus on compaction response and permeability characterization separately. A number of analytical, experimental and numerical methods exist for the prediction of both the compaction behavior and permeability of reinforcement as described below.

1.4.1 Compaction Characterization 1.4.1.1

Theoretical Compaction Models

Available theoretical models for reinforcement compaction response relate the compaction stress to fiber volume fraction using an algebraic relationship. There are a number of such models which are derived from micro-, meso- or macro-scale compaction behavior. At the micro-scale, micro-mechanical models based on the elastic beam theory have been developed. Another commonly applied approach is the use of a semi-empirical model based on compaction characterization experiments. In this empirical approach, the compaction behavior is modelled using one or multiple non-linear elastic equations, or few more sophisticated models with parameters that are determined from experiments. The fibrous reinforcements have been found to have visco-elastic, plastic and visco-plastic behavior under compaction.

8

1 Introduction

These efforts are mainly focused on developing models that account for inelastic reinforcement behavior.

1.4.1.2

Experimental Compaction Characterization

The compaction response of fiber reinforcements can be obtained via performing compaction experiments on a universal testing machine equipped with a load cell and displacement measuring sensors. Generic experimental procedures involve; compressing test sample between two flat and rigid platens installed in a universal testing machine. A typical setup of a reniforcement compaction test is shown in Fig. 1.4. The rate of compression, as well as the target load or thickness is controlled via integrated software. The applied load and the displacement of the platens are also recoded. The data is converted into stress versus time or fiber volume fraction curve. The cavity thickness and Vf are related by the following equation, Vf =

Aw N ρs h

(1.3)

where Aw is the areal weight of the reinforcement, N is the number of layers, ρs is the density of fiber and h is the cavity thickness. In order to investigate the internal deformations, composite parts are manufactured using the separate test sample, to which the compaction levels of interest are applied. The part is then cut such that its cross-section may be viewed using an optical microscope or a Scanning Electron Microscope [SEM]. However, this step is independent of the compaction test and requires fresh test samples, as well as a very different set of equipment.

Load Cell

Thickness Control Upper Platent Test Sample

Lower Platent Fig. 1.4 Illustration of experimental setup of reinforcement compaction characterization

1.4 State-of-the-Art

1.4.1.3

9

Numerical Simulations

Numerical simulation using finite element method (FEM), is a powerful tool to predict the compaction behavior of fiber reinforcements. Compaction simulations are carried out by mimicking the loading conditions on a geometrical configuration. The main challenge is to obtain accurate models for the geometry of woven reinforcements while simultaneously being able to describe their mechanical and physical behavior. The prediction quality of an FEM analysis of a fibrous structure strongly depends on the model, i.e. its geometry, the material model, and the associated boundary, as well as contact conditions. The inhomogeneous structure of woven reinforcements and locally varying material properties associated with the fiber architecture make their modeling a complex and challenging task. It is necessary to consider realistic textile geometries in order to accurately predict the performance of 3D woven reinforcements and their composites. Importantly, 3D woven reinforcements feature a more complex architecture than 2D woven fabrics.

1.4.1.4

Micro CT Assisted Compaction

Micro CT has been used by a number of researchers to investigate the internal geometry changes due to compaction and other loading conditions typical of those encountered in LCM processes. Through special experimental procedures, Hemmer et al. [10] quantified the evolution of a given dual-scale fibrous microstructure under controlled infusion. Emerson et al. [11] quantified the fiber re-orientation during axial compression of a composite through time-lapse micro CT imaging and individual fiber tracking. Vanaerschot et al. [12] quantified geometrical variability of laminated composite textiles in terms of geometrical parameters of the tows such as, centroid location, aspect ratio, area, orientation, etc. using the micro CT images of a composite part produced by resin transfer molding. In a similar study by the same authors, a dry 3D reinforcement sample was used for acquiring the micro CT images [13]. Mahadik et al. [14] used five different potted samples at each fiber volume fraction to study the yarn waviness caused by compaction of two 3D angle interlock woven reinforcements. However, all of these studies presented geometrical data for a single V f and the effects of compaction were not included. An in-house designed compression rig was used by Yousaf et al. [15, 16] to obtain micro CT images of an E-glass plain woven reinforcement under different compressive loadings to validate compaction simulation using digital element methods. The micro CT images were used to measure the geometrical features of the meso-structure.

1.4.2 Permeability Characterization Similar to the compaction characterization, there are various approaches to permeability characterization as well. These are described below.

10

1.4.2.1

1 Introduction

Analytical Permeability Models

Current analytical models, such as Gebart’s model [17] and the Kozeny-Carman relation [18], relate permeability to the fiber volume fraction of the reinforcement. Some semi-analytical models link the permeability with geometrical features such as, the fiber tow cross-section, tow gaps in the reinforcement etc. but these models require experimental validation. Analytical models have been developed for simple homogenous structures in order to predict their permeability. These models present permeability as a function of fiber diameter and the fiber volume fraction. One of the models that relates the permeability to the fiber volume fraction is Kozeny-Carman equation [18], given as, 3  1 1 − Vf K = R2 4k V f2

(1.4)

where the permeability K , non-dimensionalized by the square of the fiber radius R, is a function of the fiber volume fraction, Vf . The parameter k is called the Kozeny–Carman constant with values given in Table 1.1. In the various forms of this model, the porous medium is treated as a bundle of parallel capillary tubes that are not necessarily circular in cross-section. This equation was originally developed for granular beds consisting of ellipsoids and it has been assumed that it is also valid for porous media consisting of fibers. The predicted permeability is isotropic, which is obviously not the case in the fiber reinforcements. To address this limitation, researchers have suggested different values for k in different directions and various variations of this equation. Another commonly used set of analytic model, known as Gebart’s model [17], describe the permeability of a unidirectional reinforcement by treating the reinforcement as a set of idealized parallel rods. These models can be used to predict the permeability for flow both parallel and perpendicular to the fibers. These mathematical equations of Gebart’s model are given as, 3  8 1 − Vf K = R2 c V f2 K⊥ = C1 R2



V f max −1 Vf

(1.5)

25 (1.6)

Table 1.1 Numerical values of parameters in Kozeny-Carman and Gebart’s equations Fiber arrangement

C1

V f max

c

k

Quadratic

16 √ 9π 2 16 √ 9π 6

π 4

57

1.78

π √ 2 3

53

1.78

Hexagonal

1.4 State-of-the-Art

11

1.E+01

Gebart - Hexagonal Gebart - Square Kozeny-Carman

K/R

2

1.E-01

1.E-03

1.E-05

1.E-07 0.0

0.2

0.4

Vf

0.6

0.8

1.0

Fig. 1.5 Transverse permeability of a unidirectional bundle of fibers as a function of fiber volume fraction (V f ) predicted by various analytical models

where K  is the permeability parallel to the fibers and K ⊥ permeability transverse to the fibers. For the transverse case, V f max is the fiber volume fraction when the fibers are completely packed together and there are no spaces available for the fluid to pass through. The parameters c and C1 are given in Table 1.1. Note that the curves dive rapidly as the V f max values are approached. The permeability values, predicted by both the Kozeny-Carman and Gebart models are given in Fig. 1.5 as a function of V f . These two models, along with several other similar models, have been found to have varying degrees of accuracy, and are strictly applicable to a unidirectional assembly of fibers [19, 20].

1.4.2.2

Experimental Permeability Measurement

Many experimental methods have been proposed and are in use for the measurement of the permeability of fiber reinforcements. As there is a lack of established norms and the experimental procedures are being standardized, most researchers have developed in-house methods depending on their individual requirements and facilities. In principle, a permeability measurement is rather straightforward and follows Darcy’s law (Eq. 1.1). A fluid of known viscosity is injected under a known pressure differential through a known thickness and area of porous medium. By measuring the volumetric flow rate, permeability is easily calculated by using Darcy’s law (Eq. 1.1). Due to the anisotropic permeability of the reinforcements, the flow of the fluid must be directed along selected orientations to arrive at each of the relevant component(s) of the permeability tensor. The measurement setups for the through-thickness and inplane permeability are different. For through-thickness permeability measurement, techniques used for composite reinforcements are all somewhat similar, being based

12

1 Introduction

on unidirectional saturated flow across a stack of reinforcement layers, as shown schematically in Fig. 1.6a. Various methods are in use for the measurement of inplane permeability. These can be classified into several types: (1) based on the flow geometry, radial versus unidirectional; (2) based on the inlet boundary condition: constant flow rate or constant inlet pressure; and (3) based on the type of flow:

Pressurized Air Line

Thickness Control Flow Front Test Sample Upper Platen

Pressure Regulator Pressure Transducer

Lower Platen Camera

Valve

(a)

Fluid Pot

Pressurized Air Line

Thickness Control Upper Platen

Pressure Regulator Flowmeter

Inlet Pressure Transducer

(b)

To Waste Pot

Flow

Valve

Test Sample

Lower Platent

Outlet Pressure Transducer

Fluid Pot

Fig. 1.6 Illustration of experimental setups for reinforcement permeability measurement, a in-plane permeability using radial flow, b through-thickness permeability

1.4 State-of-the-Art

13

saturated continuous flow, or transient filling of the reinforcement, generally called ‘unsaturated’ measurement. Each of these techniques has its own advantages and disadvantage. As an example, a radial flow experiment is illustrated in Fig. 1.6b. Regardless of the method used, the experimental measurements are timeconsuming and labor-intensive, typically involving the use of a number of pressure transducers, flowmeters, optical sensors, digital cameras, etc. The results can vary significantly, with more than 20% coefficient of variation due to a lack of standardized testing methods, and errors associated in the material handling process.

1.4.2.3

Numerical Permeability Prediction

The numerical permeability prediction involves digital flow simulation based on a unit cell model. There are a number of approaches that vary according to the solution method chosen to compute fluid flow through the reinforcement, but the strategy is often similar. Unit cells are generated through various modeling techniques which are then converted into a computational mesh. The unit cell contains fiber tows that are assumed to either constitute an impermeable solid or have themselves a porous structure. The relationship between the flow-rate across the unit cell and the pressure drop is calculated through numerical solutions of boundary value problem using the governing equations of fluid dynamics. The flow field data from the numerical solution are used to compute the reinforcement permeability. The general procedure to compute permeability of an array of unidirectional fibers numerically is illustrated in Fig. 1.7. The procedure for numerical permeability predictions shall be discussed in detail in Chap. 2. Nevertheless, these modeling approaches do not capture the real-time geometrical variations of the reinforcing reinforcement architecture during processing, causing variations in computed permeability values.

Unit Cell

R (a)

(b)

(c)

Fig. 1.7 Steps in the numerical permeability predictions, a unit cell model generation, b generation of computational mesh and c velocity fields

14

1.4.2.4

1 Introduction

Micro CT Assisted Virtual Permeability Prediction

Permeability predictions using micro CT based models have been reported in the fields of geology and petroleum engineering [21]. A similar approach has been adopted in composite manufacturing, where the micro CT imaging has been used to create domains for permeability computations of 2D and 3D reinforcements. These approaches either generate voxel meshes based on the reconstruction of the real microstructures or re-modeling using the statistical information of the reinforcement geometrical features. The re-modeling strategy involves taking vital internal geometric variability to create a virtual model in software tools such as WiseTex and TexGen, representing a near to realistic model [3]. The generation of near-realistic stochastic replicas of the reinforcement involves scatter characterization and probabilistic quantification of average trends, standard deviation, and correlation information of tow geometric features [22]. The regenerated model technique is limited by the number of measurements taken. On the other hand, voxel-based models can truly represent the reinforcement’s architecture. Straumit et al. [23] applied the voxelbased strategy to compute the permeability and local variability of a 2D non-crimp reinforcement using test samples cut from a manufactured plate. The use of a cured sample can include tow deformations during pressure driven resin infusion and curing [24]. Soltani et al. [25–27] predicted the permeability of a non-woven layered reinforcement using both micro CT based realistic and virtual models. Most of the data presented in these studies either correspond to a single reinforcement state or use multiple samples. Moreover, these studies have focused on 2D reinforcements and the method has not yet been applied to 3D reinforcements.

1.4.3 Material Twin Based Characterization Framework The characterization methods discussed in the previous sections are the traditional approaches that are being used in both academic research and industry. However, with advanced computing capabilities and interconnectivity of machines and devices, there is a paradigm shift towards smart manufacturing, and we are on the verge of fourth industrial revolution, known as Industry 4.0, which relies on interconnectivity, automation, machine learning and real-time data. The traditional characterization methods are not capable of generating data compatible with Industry 4.0. The characterization approaches based on digital material twins generated from X-ray computed tomography (XCT) images can be well integrated in a smart manufacturing environment of Industry 4.0. The XCT data contains valuable information that can be generated at different stages of the manufacturing process [28]. The information can be extracted in the form of material twins and used to better understand, improve and optimize the overall process. The potential of the XCT for generating numerical models to predict the effect of varying LCM process parameters on the dry fiber compaction behavior, the permeability, and voids may lead to a reduction in the processing time and an

1.4 State-of-the-Art

15

Preprocessing Block of XCT Aided Material Twins

Fabrics/Prepregs

Raw Volume

XCT scans

Reconstructed Volume Characteristics Building Block Interface Pores

Segmentation

Matrix/Fiber/void Fiber Orientation Voxel/Mesh

XCT Aided Material Twin

Computational Model Virtual Design and Processing Block Simulations Tools (BC and Loads)

Compaction, Permeability Flow/Process simulations Simulation Results

Fig. 1.8 Schematic illustration of the characterization framework using material twins generated from XCT images [28]

enhancement in the component quality [15, 29–31]. Three-dimensional (3D) geometrical models are well-suited to perform simulations, characterization, and modeling operations [32]. Figure 1.8 shows key steps required to develop XCT-generated material models for virtual design and processing [28]. In the first step, reconstructed XCT images of fabric/prepregs are converted into a three-dimensional raw volume. As a result, a voxel-based micro-/meso- scale geometrical model is created. In the next step, computational model or material twin is generated from the raw volume through segmentation procedure based on the gray values of each constituent in dry fabric or prepregs. The material twin can then be used to perform simulations (such as compaction, flow, curing, etc.) for virtual manufacturing.

1.5 Motivation and Objectives The use of experimental-numerical hybrid characterization techniques using material twins generated from XCT requires knowledge that spans multiple scientific areas. A good understanding of the physics of X-ray absorption and the mathematical principles of tomographic reconstruction, digital image processing, model selection and numerical treatment of partial differential equations is required to achieve this [33–42]. A researcher working in the area of composites manufacturing may not have a thorough understanding of these areas. It would be convenient for the readers to have all the relevant information in a single publication. The focus of this book is to present an overview of the complete process, illustrating with examples, as

16

1 Introduction

well as highlighting major challenges that are faced in generating and using material twins from CT scans for virtual characterization of fibrous reinforcements. Some insights into the shortcomings of the process are also discussed in relevant chapters and sections.

1.6 Book Outline This book has been divided into chapters as described below. • Chapter 1. Introduction—This chapter introduces the Liquid Composite Molding processes and their common characterization techniques with the motivation and objectives described for the book. • Chapter 2. Micro CT for Material Twin—This chapter presents in-depth discussions on the use of micro CT for virtual model creation through segmentation and permeability prediction with several examples from literature. • Chapter 3. Experimental Materials and Methods—This chapter describes the commonly used experimental setup, procedures and data analysis for the permeability measurements. • Chapter 4. Material Twin based Characterization Framework—This chapter outlines the procedure of generating material twins from XCT data and discusses typical equipment and software used. • Chapter 5. Compaction Analysis and Internal Geometry—This chapter presents the compaction results in the form of relaxation curves and internal geometry changes. • Chapter 6. Flow Fields and Permeability Prediction Results—This chapter presents the flow visualizations, permeability results and relevant discussions. • Chapter 7. Experimental-Empirical Hybrid Approach—This chapter presents an averaging scheme for formulating predictive permeability models of complex geometries.

References 1. Mallick PK (2007) Fiber-reinforced composites: materials, manufacturing, and design. CRC Press 2. Michaud V (2016) A review of non-saturated resin flow in liquid composite moulding processes. Transp Porous Media 115(3):581–601 3. Boisse P (2011) Composite reinforcements for optimum performance. Woodhead, Cambridge 4. Khan KA, Umer R (2017) Modeling the viscoelastic compaction response of 3D woven fabrics for liquid composite molding processes. J Reinf Plast Compos, 1–17 5. Simacek P, Advani SG (2017) Resin flow modeling in compliant porous media: an efficient approach for liquid composite molding. Int J Mater Form, 1–13

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6. Bréard J, Henzel Y, Trochu F, Gauvin R (2003a) Analysis of dynamic flows through porous media. Part I: Comparison between saturated and unsaturated flows in fibrous reinforcements. Polym Compos 24(3):391–408 7. Bréard J, Henzel Y, Trochu F, Gauvin R (2003b) Analysis of dynamic flows through porous media Part II: Deformation of a double-scale fibrous reinforcement. Polym Compos 24(3):409– 421 8. Terzaghi K (1943) Theoretical soil mechanics. Wiley, New York 9. Di Fratta C, Klunker F, Trochu F, Ermanni P (2015) Characterization of textile permeability as a function of fiber volume content with a single unidirectional injection experiment. Compos A Appl Sci Manuf 77:238–247 10. Hemmer J, Burtin C et al (2018) Unloading during the infusion process: direct measurement of the dual-scale fibrous microstructure evolution with X-ray computed tomography. Compos A Appl Sci Manuf 115:147–156 11. Emerson MJ, Wang Y et al (2018) Quantifying fibre reorientation during axial compression of a composite through time-lapse X-ray imaging and individual fibre tracking. Compos Sci Technol 168:47–54 12. Vanaerschot A, Cox BN, Lomov SV, Vandepitte D (2013a) Stochastic framework for quantifying the geometrical variability of laminated textile composites using micro-computed tomography. Compos A Appl Sci Manuf 44:122–131 13. Vanaerschot A, Panerai F et al (2017) Stochastic characterisation methodology for 3-D textiles based on micro-tomography. Compos Struct 173:44–52 14. Mahadik Y, Brown KAR, Hallett SR (2010) Characterisation of 3D woven composite internal architecture and effect of compaction. Compos A Appl Sci Manuf 41(7):872–880 15. Yousaf Z, Potluri P et al (2018) Digital element simulation of aligned tows during compaction validated by computed tomography (CT). Int J Solids Struct 154:78–87 16. Yousaf Z, Potluri P, Withers PJ (2016) Nesting and packing of aligned tows during compaction. Euromech Colloquium 569: Multiscale modelling of textile and fibrous materials, ChatenayMalabry, France 17. Gebart B (1992) Permeability of unidirectional reinforcements for RTM. J Compos Mater 26(8):1100–1133 18. Carman PC (1937) Fluid flow through granular beds. Trans Inst Chem Eng 75:150–166 19. Karaki M, Hallal A et al (2017) A comparative analytical, numerical and experimental analysis of the microscopic permeability of fiber bundles in composite materials. Int J Compos Mater 7(3):82–102 20. Zarandi MAF, Arroyo S, Pillai KM (2019) Longitudinal and transverse flows in fiber tows: evaluation of theoretical permeability models through numerical predictions and experimental measurements. Compos A Appl Sci Manuf 119:73–87 21. Stock SR (2008) Recent advances in X-ray microtomography applied to materials. Int Mater Rev 53(3):129–181 22. Vanaerschot A, Cox BN, Lomov SV, Vandepitte D (2016) Experimentally validated stochastic geometry description for textile composite reinforcements. Compos Sci Technol 122:122–129 23. Straumit I, Hahn C et al (2016) Computation of permeability of a non-crimp carbon textile reinforcement based on X-ray computed tomography images. Compos A Appl Sci Manuf 81:289–295 24. Caglar B, Orgéas L et al (2017) Permeability of textile fabrics with spherical inclusions. Compos A Appl Sci Manuf 99:1–14 25. Soltani P, Zarrebini M, Laghaei R, Hassanpour A (2017) Prediction of permeability of realistic and virtual layered nonwovens using combined application of X-ray µCT and computer simulation. Chem Eng Res Des 124:299–312 26. Soltani P, Johari MS, Zarrebini M (2015) Tomography-based determination of transverse permeability in fibrous porous media. J Ind Text 44(5):738–756 27. Soltani P, Johari MS, Zarrebini M (2014) Effect of 3D fiber orientation on permeability of realistic fibrous porous networks. Powder Technol 254:44–56

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28. Naresh K, Khan KA, Umer R, Cantwell WJ (2020) The use of X-ray computed tomography for design and process modeling of aerospace composites: a review. Mater Des 190:108553 29. Endruweit A, Zeng X, Matveev M, Long AC (2018) Effect of yarn cross-sectional shape on resin flow through inter-yarn gaps in textile reinforcements. Compos A Appl Sci Manuf 104:139–150 30. Larson NM, Zok FW (2018) Insights from in-situ X-ray computed tomography during axial impregnation of unidirectional fiber beds. Compos A Appl Sci Manuf 107:124–134 31. Desplentere F, Lomov SV et al (2005) Micro-CT characterization of variability in 3D textile architecture. Compos Sci Technol 65(13):1920–1930 32. Ali MA, Umer R, Khan KA (2019) A virtual permeability measurement framework for fiber reinforcements using micro CT generated digital twins. Int J Lightweight Mater Manuf 3(3):204–216 33. Flannery BP, Deckman HW, Roberge WG, d’Amico KL (1987) Three-dimensional X-ray microtomography. Science 237(4821):1439–1444 34. Carmignato S, Dewulf W, Leach R (2017) Industrial X-ray computed tomography. Springer 35. Landis EN, Keane DT (2010) X-ray microtomography. Mater Charact 61(12):1305–1316 36. Ketcham RA, Carlson WD (2001) Acquisition, optimization and interpretation of X-ray computed tomographic imagery: applications to the geosciences. Comput Geosci 27(4):381– 400 37. Wildenschild D, Vaz C et al (2002) Using X-ray computed tomography in hydrology: systems, resolutions, and limitations. J Hydrol 267(3–4):285–297 38. Maire E, Withers PJ (2014) Quantitative X-ray tomography. Int Mater Rev 59(1):1–43 39. Moreno-Atanasio R, Williams RA, Jia X (2010) Combining X-ray microtomography with computer simulation for analysis of granular and porous materials. Particuology 8(2):81–99 40. Schlüter S, Sheppard A, Brown K, Wildenschild D (2014) Image processing of multiphase images obtained via X-ray microtomography: a review. Water Resour Res 50(4):3615–3639 41. Bultreys T, De Boever W, Cnudde V (2016) Imaging and image-based fluid transport modeling at the pore scale in geological materials: a practical introduction to the current state-of-the-art. Earth Sci Rev 155:93–128 42. Stock SR (2008) Microcomputed tomography: methodology and applications. CRC Press

Chapter 2

Generation of Material Twin Using Micro CT Scanning

With its roots in the medical sciences, X-ray computed tomography has been used widely in the field of materials characterization. Intensive studies have been carried out focusing on mechanical characterization and damage detection [1–6]. The application of this technique ranges from simple three-dimensional (3D) visualization of various physical attributes to intensive computer simulations and image processing. The basic principle of X-ray tomography is based on the fact that an X-ray beam passing through matter is attenuated and the degree of attenuation is dependent on material’s absorptivity. The degree of attenuation in the intensity of the X-ray beam as it passes through the test sample is measured in the form of radiographic projections which are then reconstructed to obtain 3D volume. One recent example is the increasing use of this technique to compute permeability in porous structures offering an alternative to potentially laborious experimental methods. The 3D volumes obtained from computed tomography are now being used as computational domains in numerical permeability computation. The focus of this chapter is to present an overview of the contextual knowledge necessary for using material twins from CT scans for virtual characterization of fibrous reinforcements. The chapter has been divided into various sections as shown in Fig. 2.1. In the first section, common lab-scale equipment and the associated accessories are discussed. In the subsequent sections, data processing and techniques for generating material twins are described. In the final section, the numerical simulation technique for virtual permeability computation is discussed which includes, establishing governing equations and boundary conditions, followed by identifying a suitable algorithm to solve governing equations on the discretized geometry, and obtaining reinforcement permeability through Darcy’s Law.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 M. A. Ali et al., CT Scan Generated Material Twins for Composites Manufacturing in Industry 4.0, https://doi.org/10.1007/978-981-15-8021-5_2

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2 Generation of Material Twin Using Micro CT Scanning

Load Data

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Numerical Simulation

Stochastic Virtual Model

Boundary Conditions

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Permeability Computation

Tow Deformations and Compaction Response

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Model Generation

Fig. 2.1 Flow chart of the general procedure followed to compute virtual permeability using micro CT generated material twin

2.1 XCT Setup 2.1.1 Compaction Setup While using a micro CT, an in-situ or ex-situ compaction fixture is the most suitable apparatus for obtaining these curves via a load cell. In an in situ experiment the sample is imaged under load using a loading fixture mounted within the X-ray scanner, while for an ex situ experiment the sample is removed from the testing rig before scanning it. In both cases, the capability to track the evolution of structure for single sample means that the architectural features can be studied on a single sample rather than having to extract trends in a statistical manner from a number of samples examined at different stages. There are significant advantages to in-situ compaction, partly because it is easier to correlate and register the different 3D images, and partly because removing a sample from its environment may change the state of the composite [3]. The in-situ testing stage is usually operated through a remote control software with load and displacement control functions [7]. The load and displacement data can be continuously monitored. The pre-defined target displacement or force is achieved by moving platens of the compression stage. However, this requires the deployment of compact loading devices specifically designed to minimize interaction with the X-ray illumination of the sample [3]. In addition, the mounting of loading rigs within laboratory X-ray systems can limit the maximum resolution for cone beam imaging, where the sample needs to be as close as possible to the source.

2.1 XCT Setup

21

2.1.2 XCT Machine A typical setup in a micro CT machine used to obtain X-ray projections consists of four main components; an X-ray source, a detector, a stage to hold the test sample and an operating system with data storage. The hardware comes in various configurations and sizes, ranging from laboratory scale equipment [8] to synchrotron radiation facilities [9]. A schematic of the laboratory scale test set-up is illustrated in Fig. 2.2. More details about these components, various configurations and comparative capabilities can be found in [10]. The main disadvantages of micro CT are; limitations in spatial and temporal resolution and the lack of contrast between phases in the scanned specimen having similar attenuation coefficients e.g. carbon fiber and polymer matrix in a composite. The spatial resolution is determined by the detector resolution and the distance between the sample and the source. Often, the highest resolution is obtained at the expense of a small sample size or a small scanning area, and one needs to adopt a compromise between sample size and resolution. This represents one of the main limitations when using current generation of micro CT systems to study composite materials [3]. For example, if the reinforcement microstructure is to be resolved, the sample must be a few millimeters in size which may not be enough to capture a full representative volume element (RVE) of the composite reinforcement [11]. The sample size limitations can be overcome by stitching multiple scanned images together [12] or selecting a region of interest (ROI) [13]. The data acquisition procedure requires that the internal configuration of the test sample remains unchanged during the scan (typically 30–120 min) which limits the temporal resolution of the scanned images. This restricts the micro CT’s ability to look at fast and dynamic phenomena such as a resin infusion [14–16] and curing processes [17, 18]. Nevertheless, such dynamic phenomena are captured through time series acquisition of micro CT in a timelapse or continuous streaming approach [3, 19]. In time-lapse imaging, in-situ or Fig. 2.2 A schematic illustration of the of the micro CT setup [8]

X-Ray Detector

Rotation Axis Test Section X-Ray Beam

X-Ray Source

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2 Generation of Material Twin Using Micro CT Scanning

ex-situ setups are required, whereas for continuous streaming, a high-flux source is used. The micro CT distinguishes between components of different materials via their differences in attenuation coefficients which are related to material density. If a sample contains components with similar attenuation coefficients or contrast, the micro CT may not provide sufficient information [9, 20]. The level of contrast can be improved by using phase contrast [21–23], or staining the sample using contrast agents [24, 25].

2.1.3 Fabric Samples The sample to be scanned can be in three different forms: (1) a dry reinforcement layup constrained in a container [26] or between plates [8], (2) a cured composite sample [27–29], and (3) a resin potted sample [30]. The resin potted samples are cured samples which are produced by simply dipping dry reinforcement into a resin contained in a mold for polishing. Using a dry reinforcement sample, it was possible to study the effect of loading scenarios, such as transverse compaction [8], or inplane shear [31], using a single sample. When using dry fiber stacks, it is important to keep the stack intact and any un-intended movement among layers should be avoided. The cured and resin potted sample can be useful after mechanical loading and testing, the fiber volume content cannot be altered and these samples are widely used for analyzing voids and cracks in composites. For permeability computations, a separate sample for each fiber volume fraction is required. When dealing with dry reinforcements, the size and handling of test sample is extremely important as distortions induced during cutting, transferring or placing the sample will have a significant impact on the resulting permeability predictions [32, 33].

2.1.4 Noise Reducing Filters When scanning test samples, filters are sometimes employed to reduce the level of noise and enhance the contrast of the scanned images. These filters can be used during acquisition or post-processing. An X-ray filter is a material that is placed in the direct path of an X-ray beam, in order to reduce the intensity of particular wavelengths from its spectrum, and selectively alter the distribution of X-ray wavelengths within a given beam [10]. Filtration is required to absorb the lower-energy X-ray photons emitted by the tube before they reach the target. The use of a filter produces a cleaner image with reduced noise, by absorbing the lower energy X-ray photons that tend to scatter more. For industrial radiography, the X-ray beam filters are generally constructed from high atomic number materials such as, lead, copper, brass, and similar heavy

2.1 XCT Setup

23

metals. The thickness of the filter material is dependent on the atomic number, the X-ray power settings, and the desired filtration factor. On the other hand, there are digital filters that are based on image processing techniques, which can be applied to either the projection or the stack of reconstructed images.

2.1.5 Radiographic Projections The underlying principle of X-ray tomography is based on the physics of interaction of X-rays with matter [1]. The intensity of an X-ray beam passing through matter is attenuated and the degree of attenuation depends on the material’s absorptivity, which depends on the number and types of atoms along the path, and the distance traveled by the X-ray beam within the material [34]. By measuring the degree of attenuation in the intensity of the X-ray beam as it passes through the test sample, a two dimensional map can be generated. Such a map is known as the radiographic projection or a radiograph. A single projection corresponds to one orientation of the sample, and multiple projections are taken by either rotating the test sample or by simultaneous rotation of the X-ray source and detector. In the literature, a set of projections is often referred to as a “sinogram” and is formed by stacking all of the projections acquired at different angles [10]. The creation of X-ray radiographic projections of a 3D object at various orientations is illustrated in Fig. 2.3. In Fig. 2.3a, the gray scale projections are presented in color for illustrative purposes only. On each projection, the attenuation at a given point is the sum of all the attenuations along the path, since each point in the test specimen contributes to the total attenuation [10]. As seen in Fig. 2.3a, the X-ray beam has to pass through different paths depending on the orientation of the sample. This has a direct effect on the quality of scanned images of fiber reinforcements having dense and oriented packing structure. Less contrast was likely to be obtained for fiber boundaries oriented in the direction orthogonal to the rotation axis [35]. To further illustrate this phenomenon, formation of X-ray projections of three 2D objects at 0° and 90° orientations is illustrated in Fig. 2.3b, c. The two dimensional objects with differing densities, depicted by their grey scale color, are placed among two pairs of mutually perpendicular sources and detectors, as shown in Fig. 2.3b. The resulting projections at 0° and 90° orientations, which are simply curves in this case, are depicted in Fig. 2.3c. It can be clearly seen in Fig. 2.3c that the peak of the curve corresponds to the path along which the objects density and absorption are highest, and vice versa. Mathematically, each point on the X-ray projection describes the line integral of the distribution of attenuation coefficients along a particular X-ray path. This idea forms the basis of reconstruction of radiographs through back-projection techniques, which is discussed in the next section in detail.

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2 Generation of Material Twin Using Micro CT Scanning

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Fig. 2.3 a 3D Illustration of formation of X-ray projection maps, b a 2D specimen with a source and a detector, c projection maps with 2D specimen at 0° and 90°

2.2 Data Processing The raw images obtained from the scan consist of multiple projections that need to undergo a number of processes which include 3D volume reconstructions, segmentation and filtering [36]. Filtering can be incorporated within 3D volume reconstructions and segmentation, or can be applied as a separate procedure. Filtering at this stage is digital in nature and is based on image processing procedures.

2.2 Data Processing

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2.2.1 3D Volume Reconstruction The radiographs acquired from the CT equipment are merely projections, and an image reconstruction procedure is required to convert the set of radiographs into a 3D volume of the object under investigation. The reconstructed 3D volume consists of voxels, the volumetric representation of pixels, and can be sliced in all three principle directions, or along any arbitrary cutting plane, to obtain 2D cross-sectional images. The general idea of reconstruction is depicted in Fig. 2.4a. The projections may undergo a series of image processing steps in order to remove artifacts, to smooth them and apply filters. In fibrous reinforcements, the artefacts may appear due to the orientation of the sample with respect to the rotation axis, which can be potentially removed by using a Gaussian filter [35]. It should be noted that in literature, the term

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Fig. 2.4 a 3D Illustration of back projection, b 2D specimen, back-projection of 0° and 90° radiographs and their superposition c back-projections from multiple projections

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2 Generation of Material Twin Using Micro CT Scanning

reconstruction may also refer to other procedures that generate 3D volumes from 2D cross-sectional images obtained through microscopy of physically cross-sectioned objects [1]. The principle of reconstruction from projections is based on the fact that mathematically, each X-ray projection can be described as a line integral of the distribution of attenuation coefficients, along a particular X-ray path, and it is possible to reconstruct a function from such a line integral [37]. For a rigorous derivation of the mathematical formulation, and various types of reconstruction algorithms, readers are referred to Hsieh [38] and Buzug [39]. For a quick overview, we look at the founding work established in 1986 by Radon [40]. The reconstruction of this function is performed by applying an inverse transform, referred to as inverse Radon transform, to recover the function from its set of line integrals. The reconstruction using inverse Radon transformation of three simple 2D objects is shown in Fig. 2.4b, c. The back-projection of the test object using single projections at 0° and 90°, as well as their combination, are shown in Fig. 2.4b. The reconstruction is improved as the number of projections at different angles is increased, as shown in Fig. 2.4c [10]. The reconstructed volume presents the test sample as a single volume and gives a 3D digital visualization of the test sample. The 3D images, however, cannot be used directly as virtual representation of the internal reinforcement structure, because the image pixels lack directionality, which is paramount characteristic of fiber reinforced composites with oriented fibrous assemblies [10]. For more elaborate analyses and generation of computer models, the reconstructed volume needs to be partitioned into various sub-domains representing each constituent of the test sample and the directionality of each image pixel needs to be determined. This is achieved through segmentation of the reconstructed volume.

2.2.2 Segmentation The reconstruction of tomographic projected images yields a 3D map of X-ray absorption in the form of gray scale digital image [41–43]. The next step is to identify, and separate different phases and components within the scan. The digital nature of the data facilitates the use of image processing techniques to extract a wide array of quantitative measurements on internal structures [34]. The image stack can be regarded as a three dimensional function, and mathematical principles of linear algebra, statistics and data analysis can be applied to perform this segregation. This step is known as “segmentation” and involves the manipulation of the digital images. Segmentation is the procedure by which a continuous grey scale values are assigned to certain discrete groups, usually based solely on their grey scale levels. In the characterization of woven composites, this step means separating matrix and fibers so as to isolate or identify fibers with specific orientations (weft, warp, etc.). The segmentation process may be performed manually [8, 32], by analyzing directional gradients [44–49], or using learning algorithms [50–52].

2.2 Data Processing

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From an image processing point of view, the image segmentation can be divided into seven categories [53]. The classifications vary based on the field of application, and one technique may fall into multiple categories. Here, for composites application, we divide them in three categories, 1st order, 2nd order and techniques based on artificial intelligence and learning algorithms. The 1st order category involves using the gray scale data, and includes techniques such as manual, k-means [54], and Otsu’s method [55]. In 2nd order techniques, in addition to the gray values, the phase division can be made using texture analysis for images. Through this technique, additional variables, such as structural anisotropy, average gray values, components of fiber orientation vectors, contrast, correlation, energy and homogeneity can be generated and analyzed. The techniques based on artificial intelligence and learning algorithms utilize data driven learning algorithms for woven fibrous reinforcements [50–52]. However, such algorithms are in the developmental stages and practical applications are yet to be achieved. The manual method in the 1st order category involves simply selecting a gray value as the dividing line, defined as a user input [8, 32], as shown in Fig. 2.5a–c . Both kmeans (where k is the number of clusters) and Otsu’s methods are based on clustering of data, and minimizing the variance. The concept behind the Otsu’s method is to

Voxel Count (×103)

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Fig. 2.5 Histogram based image segmentation of a 3D orthogonal reinforcement, a grayscale intensity histogram with a global thresholding value, b a raw grayscale volume, c a binary volume with applied thresholding (green = pores, red = fiber tows)

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2 Generation of Material Twin Using Micro CT Scanning

find a threshold value which minimizes the within-class variances of background and foreground voxel classes; this is equivalent to maximizing the variance between the means of the two clustered classes. K-mean uses an iterative approach to find the threshold, as the midpoint between means of background and foreground. In 2nd order techniques, variables are extracted from the gray scale data, which contain directional information. These techniques use either the Grey Level Cooccurrence Matrix (GLCM) [45], or the structure tensor of the 3D gray scale data [46– 48, 56]. Both GLCM and structure tensor have to be evaluated in a certain window of the integration limit, and the accuracy of the method is highly dependent on the parameters of this window. Statistical parameters (contrast, correlation, energy and homogeneity) are extracted from a normalized GLCM. Examples of these parameters are shown in Fig. 2.6a–d. The raw and segmented volumes are shown in Fig. 2.6e–f respectively.

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Fig. 2.6 3D volume segmentation using GLCM, a contrast, b, correlation, c energy, d homogeneity, e 3D raw volume and f 3D segmented volume [45] [reproduced with permission]

2.2 Data Processing

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The structure tensor is used to calculate the degree of structural anisotropy and orientation vectors by solving for its Eigen values and vectors [47]. The structure tensor can be either 2 × 2 [46] or 3 × 3 [47, 48]. The procedure involves the calculation of these additional parameters and then, thresholding by using any of the 1st order techniques discussed above. The degree of anisotropy or average grey values can only separate fiber tows and the matrix. However, by using the anisotropy directions, which are obtained from largest eigenvector of the structure tensor, tows having different orientations can also be distinguished. The anisotropy directions of a 2D reinforcement are shown in Fig. 2.7a, b, where the warp and weft tows are clearly distinguised and color-coded. The local predominant orientation in the region under investigation corresponds to the direction of the largest eigenvector of the structure tensor [46]. Figure 2.7c, d respectively show the final segmented 2D section and the 3D volume obtained after cleaning using a median filter and morphological closing operations. As stated earlier, in virtual permeability computation with dual scale flow, it is fundamentally important that the fiber tows with specific orientations may also be

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(b)

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Fig. 2.7 3D volume segmentation using structure tensor a The raw image, b anisotropy directions, c final segmented 2D section and d 3D model [46] [reproduced with permission]

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distinguished. A robust segmentation algorithm should be able to achieve this as well. In a given reinforcement, the fibers are made from a single material, i.e. glass, carbon, aramid etc., having the same X-ray attenuation at different orientations. The permeability predictions may be based on the assumption that the tows are solid and impermeable. It is justified where the micro-scale permeability is assumed to be lower than macro-scale permeability by approximately two order of magnitude [8, 57]. The 1st order techniques discussed above may not be able to separate tows into warp and weft directions. These techniques are only useful if the reinforcement is a hybrid, meaning that the warp and weft tows are made of a different material. The second order techniques and learning algorithms are considered superior.

2.3 Tow Deformations and Compaction Response Reinforcement permeability can be predicted using various modeling approaches, as discussed in the next section. It is widely recognized that permeability is a function of fiber volume fraction and reinforcement architecture [58], both of which are strongly related to the compressibility of the reinforcement [59]. Hence, any study focusing on the permeability of a composite reinforcement is incomplete without understanding the compaction behavior of the reinforcement [60, 61]. Dry reinforcements experience compaction and shear during the molding and forming processes. They are also intentionally compacted to achieve a target thickness or desired fiber volume fraction in some cases. Such reinforcements can be subjected to single-cycle, multi-cycle or multi-stage compaction, as illustrated in Fig. 2.8 [60, 62]. The stress relaxation curve obtained via experiments or using compaction models that range from linear to visco-elastic to visco-plastic approaches. During the compaction of reinforcements, the evolution of reinforcement internal structure is also documented in addition to the load deflection curve, meso- and micro-structural evolution of tows for different loading conditions in both the dry and wet states is also observed [63]. Here, we discuss studies that use micro CT to analyze internal geometrical features of reinforcements under compaction. A stochastic framework known as the reference period collation method [64, 65] for the statistical representation of a woven textile has been proposed by Vanaerschot et al. [28]. They decomposed the spatial variations into non-stochastic and periodic (or systematic) trends, and non-periodic stochastic fluctuations. The stochastic character of every tow was analyzed in terms of centroid position, aspect ratio, area, and orientation of its cross-section, as shown in Fig. 2.9. The collation method was tested using X-ray micro-computed tomography data for a seven-ply 2/2 twill woven carbon-epoxy composite produced by the resin transfer molding technique. All of the tow characteristics, with exception of the in-plane centroid position, exhibited systematic trends that show only mild differences between plies. The statistical characteristics of a sample were used to calibrate the recently proposed “Markov Chain” algorithm for textile reinforcements, and a virtual unit cell was generated in WiseTex [67]. The Markov Chain formulation is very efficient and suited well

2.3 Tow Deformations and Compaction Response

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Fig. 2.8 Different types of compaction that effects reinforcement internal structure, a single-step compaction, b multi-stage compaction and c multi-cycle or cyclic compaction [60] [reproduced with permission]

for generation of virtual stochastic unit cells, provided the dominant correlations are those along a tow, with correlations between tows relatively weak [68]. The Markov Chain formulation is based on the assumptions that the equilibrium probability density matrix for the chain variable has the datum root mean square deviation (RMSD) of the stochastic coordinate deviation being regenerated and the correlation length between the values taken by the stochastic quantity at different points along a tow matches the datum correlation length. For the in-plane centroids, realizations of these fields were generated using a framework based on the “Karhunen–Loève” series expansion that had been calibrated with experimental information from prior work [69]. The method was experimentally validated [70] and extended to 3D reinforcements [66]. The authors validated the use of the normal, log-logistic, log-normal and gamma distributions for the tow thickness, tow width, intra-tow fiber volume fraction and intra-tow permeability, respectively. A negative correlation was observed

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(a)

(b) Fig. 2.9 a Identification and segmentation of tow cross-sections, and b Variation of cross-sectional parameters along a tow length (note the periodic nature of all the parameters except the in-plane centroid coordinate “y”) [66] [reproduced with permission]

between the thickness and width of the tows. The distribution of intra-tow fiber volume fractions was found to be skewed to the right, with a lognormal distribution representing a better fit to this data. Karahan et al. [29] presented measurements of the internal geometry of a carbon fiber non-crimp 3D orthogonal woven composite. The measurements included yarn waviness, yarn cross-section and local fiber volume fraction. The obtained detailed parameters of the internal fiber architecture of one representative non-crimp 3D orthogonal composite demonstrate high level of straightness of the in-plane yarns,

2.3 Tow Deformations and Compaction Response

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with high uniformity of the geometric features of the reinforcement, as shown in Fig. 2.10. The variations are much lower than respective variability parameters observed for typical 2D woven carbon reinforcement composites. The measured geometrical parameters of the 3D orthogonal reinforcement provide very useful input data for permeability computation. Mahadik et al. [30] studied two angle interlock 3D woven carbon reinforcements to characterize yarn waviness and resin-rich regions, and how they change under increasing levels of compaction. The study focused on the out-of-plane crimp of the yarns, and the size and shape of resin rich-regions in the consolidated panels. The appearance of resin channels was found to be heavily dependent on weave style, with large resin pockets appearing in weaver yarn planes (plane that passes through the weaver or binder or interlacing yarn), which decreased significantly in size under compaction. Local and out-of-plane yarn crimp was recorded for each of the yarn directions in both reinforcements over the volume fraction range, as shown in Fig. 2.11. Barburski et al. [31] presented an micro CT analysis of the internal structure of 2D woven reinforcements in a sheared state. Both manual and automated procedures were employed to extract data from the micro-CT images. Changes in the fiber orientations within the yarns, and overall yarn geometrical parameters both before and after the shear deformation were quantified. It was noted that the structure of the reinforcement becomes more condensed following shear deformation, the reinforcement density increases and the yarn spacing decreases, causing the permeability to drop considerably. The geometrical data presented are well suited for verification of meso-scale models of textile deformation, and for calculations of permeability of the reinforcement and mechanical behavior of composites [44]. In a similar investigation of a 3D orthogonal reinforcement, Pazmino et al. [71] concluded that binder yarns remained unchanged under in-plane shear loading. Consequently, the width of the yarn cross-section also maintained low variability, particularly in the weft orientation. Hemmer et al. [14] used a downsized in-situ infusion setup, shown in Fig. 2.12, to quantify the evolution of a quasi-unidirectional non-crimp reinforcement under several controlled flow-induced compaction states. Using digital image processing

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Fig. 2.10 Tow cross-sections of a 3D orthogonal reinforcement, a weft, b warp and c binder [29] [reproduced with permission]

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FABRIC A length

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Fig. 2.11 Schematic of the dimension definition and CT scan views of the weaver resin channel in two types of 3D reinforcements [30] [reproduced with permission]

Vacuum Vent Breathing Material

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Fig. 2.12 Illustration of the in-situ infusion setup [14] [reproduced with permission]

2.3 Tow Deformations and Compaction Response

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tools, the corresponding 3D images were analyzed to quantify the unloading phenomenon occurring during the infusion process at mesoscopic scale. They found it challenging to separate the warp tows using simple edge detection algorithms, such as Canny or Sobel, which was overcome by concatenating 100 slices. Applying edge detection on the concatenated images, dramatically improved the tow visibility and detection, as shown in Fig. 2.13a–f. It was also observed that, while the stack thickness increased between the dry and saturated states, the tows also swell and consequently displace in the saturated state. Somashekar et al. [72]. suggested that permanent (and elastic) deformations in a biaxial stitched glass reinforcement occur mainly by means of change in the fiber bundle cross-sectional shape and size. The results demonstrated that the fiber bundle cross-sections became thinner and elongated, particularly under repeated compaction. Yousaf et al. [73] conducted a meso-scale compaction study of the tow geometrical changes by using micro CT under compression loading.

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Fig. 2.13 a, b Dry tows near the inlet and outlet, c, d saturated tows near the inlet and outlet, e, f percentage swelling of tows near the inlet and outlet [14] [reproduced with permission]

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In this section, we have discussed several studies that uses micro CT to investigate the changes in internal geometrical features of reinforcements under compaction. The literature suggested that majority of these studies focused on post-cured composite panels and not dry reinforcements. The statistics of tow geometrical changes reported in these studies are useful in generating stochastic models for virtual permeability predictions and potentially, numerically model key mechanical properties of the cured composite part. There is an opportunity to explore more about the in-situ compaction and flow visualization in dry reinforcements, however more sophisticated arrangements are required to establish link between the micro-structural changes and their effects on reinforcement permeability.

2.4 Model Generation Models for computing virtual permeability are generated from the stack of micro CT images in three different ways, (1) material twin, (2) stochastic virtual model, and (3) simplified model. The first two methods involve digital flow simulation using finite element or finite volume methods. The last method does not require any numerical simulation, rather it requires some geometry simplification and assumptions that will lead to the use of empirical or closed-form solutions of permeability models. Here, we discuss one by one each type of model.

2.4.1 Material Twin A material twin is a 3D realistic voxel model, which is a regular rectangular grid of 3D elements with certain characteristics, such as different material properties, associated with each voxel (an element of a voxel model). A generic three phase material twin is shown in Fig. 2.14a. Each phase is separated by neighboring phases by an interface, as shown in Fig. 2.14b. The generation of such a model is possible from any geometric representation of a reinforcement including a 3D stack of images,

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Fig. 2.14 Material twin, a Illustrative representation of a material twin with three phases, b the material twin with one phase hidden to show the interface, and c an example of the material twin (with matrix hidden) taken from CT scan [74]

2.4 Model Generation

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an example material twin of a 3D reinforcement is also shown in Fig. 2.14c. The defining parameter of a material twin is the voxel size, which is chosen based on the degree of required geometrical details. If the microstructure of a reinforcement is to be resolved, the voxel size should be approximately 1/5th of the fiber diameter. In case, where meso-structure is to be resolved, the resolution could be taken as 5–10 times that of a single fiber diameter. The first step in converting the 3D stack of images into material twin is segmenting the images using the procedures discussed in previous section. For the permeability computation, the 1st order segmentation methods are used when the intra-tow flow is ignored, and the tows are assumed to be solid. On the other hand, in case of dual scale flow modeling, second order methods are used in cases where it is possible to isolate tows of different orientations, as well as extract local fiber volume fraction and orientation vectors. The local fiber volume fraction and orientation vectors are used to compute local permeability tensor to be assigned to the voxels representing the tows [46, 75]. The segmentation process allows one to assign phases appropriately to the segmented volume. Two of the main phases to be identified in the permeability computation are the inter-tow and intra-tow regions. The fiber tows are generally considered as a solid phase and can be further divided into weft or warp tows with assigned orientations. The ratio of the volume of solid phase to the total volume is known as “solid volume fraction” and is an important parameter of a material twin. Flow simulations are performed on unit cells which are extracted from the segmented volume by recognizing the periodic pattern of the reinforcement structure. The unit cell extraction is achieved by cropping images. The generation of material twin from micro CT images is highly dependent on the segmentation process which significantly affects the features of the material twin, such as phase allocation, solid volume fraction and quality of the mesh. The segmentation algorithm needs to be robust and validated. Since noise and artifacts are inherent to the micro CT images, the resulting material twin may have isolated voxels which are totally separated from their actual phase. To avoid this, a cleanup operation is required, which may be based on image filtering [45] or the use of sophisticated cellular automata [47]. In a material twin, the elements are regular with a fixed size, poorly resolved curvatures and rough interfaces. Unlike traditional finite element/volume models, the mesh size stays uniform throughout the model, and was primarily determined by the resolution of the micro CT images. Extraction of unit cells from the segmented volume is also challenging, since maintaining the size, boundaries, and periodicity of the unit cells is difficult [76, 77]. The inconsistencies encountered during unit cell extraction are highlighted in Fig. 2.15. Figure 2.15a, b shows the differences in the boundaries of the unit cells at the top and the bottom surfaces respectively. Size differences in units cells extracted from the same test sample are shown in Fig. 2.15b. Figure 2.15c, d shows the dissimilarities between the opposing faces of Unit Cell 1, shown in Fig. 2.15a. The size and boundary differences are not of major concern; however, the periodicity of the unit cell face needs to be addressed in subsequent computations, and the unit cells must be selected carefully, keeping the opposite faces as identical as possible.

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D

Fig. 2.15 Unit cell assignment procedure, a extraction of unit cells from the image stack showing the cell boundaries on the top face, b cell boundaries and in-plane sizes on the bottom face, c left and right boundary faces of the unit cell showing the differences and d left and right boundary faces of the unit cell

2.4.2 Stochastic Virtual Model Stochastic virtual modelling procedures do not require any segmentation, can generate meshes with desired element sizes and shapes, as well as the periodicity of the unit cells can be well maintained [70]. However, the model may not be a true representation of the actual structure, and requires additional tools for geometric modeling, and generating a computational mesh is required. The geometric modeling tools generally generate ideal models, which have constant tow cross-section and uniform geometric features as can be seen in Fig. 2.16a. In reality, composite reinforcements have complex hierarchical structures with high local tow variations [78, 79], as can be observed from the micro CT model shown in Fig. 2.16b. It is known that permeability can strongly vary due to the inherent structural variations. The micro CT image stacks are used to quantify the geometrical variations and generate stochastic virtual models with a high level of geometrical detail, including systematic local variations in tow paths and tow cross-sections [78]. In a stochastic model, one does not only want to know the average number of randomize models, but also the standard deviation. Such a stochastic model is shown in Fig. 2.16c and the computational

2.4 Model Generation

39

(a)

( b)

(c)

( d)

Fig. 2.16 Three different types of representation of a 3D reinforcement a ideal model, b micro CT reconstructed volume, c stochastic virtual model and d mesh generated from the stochastic virtual model [81] [reproduced with permission]

mesh generated from this model is shown in Fig. 2.16d. The geometrical parameters are quantified by taking measurements from the stack of reconstructed micro CT images prior to, or after segmentation of the 3D volume. Microscopic sections are also used but in such cases, a limited number of cross-sections are available, hence limiting the range of generated data. Quantitative evaluation of the images include measurements of the area and height of gaps in a cross-section, yarn spacing, and the distance between the centroids of two neighboring gaps, these measurements are taken from the stack of reconstructed micro CT images either manually [80] or through an automatic procedure [78]. Manual measurements are obtained by either using a graphic user interface or hand-picked tools that are available in any image processing software package. Measurements are taken from multiple slices carved in all three directions, as shown in Fig. 2.17a. The measurements are taken automatically by using image processing tools which involve noise reduction, binarization and feature detection, as shown in Fig. 2.17b. The automatic feature detection algorithms are capable of extracting geometric features such as area, perimeter, centroid, etc. A number of software packages, such as MATLAB, Fiji, ImageJ, etc. are used to obtain the measurements, both manually and automatically. The measured data is statistically analyzed and

40

2 Generation of Material Twin Using Micro CT Scanning In-Plane Parameters Wp=0.41 3.31 3.16

Wt

hint

Original XCT Image

weft

3.33 warp 0.9 mm

Transverse Parameters

Binary Image with Isolated Gaps

thickness

0.07

(a)

2.29

0.22

weft

hr

hy

(b)

Fig. 2.17 Two different measurement approaches a manual method [80] and b automatic method [78] [reproduced with permission]

stochastically equivalent unit cell models are generated using modelling tools such as, TexGen or WiseTex for numerical flow simulations and permeability computation.

2.4.3 Simplified Models In the simplified approach, the numerical flow simulations can be completely avoided by making certain geometrical simplifications, which can lead to use of empirical or closed form solutions of flow models. The micro CT images are first analyzed to identify flow channels/paths from which their shapes and sizes are extracted, similar to the example shown in Fig. 2.18a. Once the channels are identified, the procedure follows a hierarchical order, with a divide and conquer strategy. The unit cell is divided into meso- and micro-channels. The meso-channels are then divided into sub-regions depending on their shapes; an equivalent region is constructed by combining the individual regions, as shown in Fig. 2.18b. The permeability values of individual regions are estimated using analytical or empirical relations which require information about the shape and size of the channels and tows. This information is also obtained from the micro CT images via manual or automatic measurements. The micro-scale permeability is estimated by analytical models such as, KozenyCarman or Gebart’s model, or through micro-scale numerical simulations. These models require the diameter of the fibers and the fiber volume fraction of the tows. The fiber diameter is either obtained from the material supplier or can be measured from micrographs. The fiber volume fraction of the tows is calculated from the fiber count in a single tow, the fiber diameter and the tow cross-sectional area, which was generally estimated from the micro CT images. The permeability of the meso

2.4 Model Generation

41

Channel Identification

Binarization and Channel Isolation

(a) Rectangular Channel

Triangular Channel

Rectangular Channels

Triangular Channel

(b)

Fig. 2.18 Geometry simplification based on micro CT images, a channel identification and isolation, and b simplified geometries and their combination [82] [reproduced with permission]

channels are calculated by relating the pressure- drop across the length of channel to its hydraulic resistance [82–84] or friction factor derived from Hagen–Poiseuille flow [14]. Both the hydraulic resistance and friction factor can be related to the cross-sectional shape and size of the channels. The equivalent permeability of the homogenized unit cell is then determined by making the fluid flow through the porous unit cell in a manner that was analogous to the flow of electric current through electric circuits. The sub-regions are arrangements in parallel or series combinations, and the permeability is calculated by a weighted-averaging scheme, depending on this arrangement. If two sub-regions are parallel, the equivalent permeability is the arithmetic mean of the permeability of the individual regions, whereas if they are in series, the equivalent permeability is the harmonic mean [85–88]. The homogenization procedure for the through-thickness permeability of a 3D reinforcement is illustrated in Fig. 2.19. The 3D reinforcement, Fig. 2.19a, was divided into N sub-layers, Fig. 2.19b. Each sub-layer was further divided into inter-tow and intra-tow sub-regions, as shown in Fig. 2.19c. These subregions were assumed to be parallel to each other and the permeability of each

(a)

( b)

(c)

Fig. 2.19 Homogenization of a 3D reinforcement for estimating through-thickness permeability, a reinforcement geometrical description, b partitioning into sub-layers and c sub-regions of the sub-layer [57] [reproduced with permission]

42

2 Generation of Material Twin Using Micro CT Scanning

sub-layer was estimated as the area-weighted arithmetic mean of the permeability of each sub-region. For through-thickness flow, the fluid was assumed to penetrate the sub-layers successively. Thus, the equivalent through-thickness permeability of the reinforcement was estimated as thickness-weighted harmonic mean of the permeability of each sub-layer [57]. The method is a very crude approximation and is suitable for early design explorations and parametric studies.

2.5 Virtual Permeability 2.5.1 Numerical Setup The numerical simulation is the solution of boundary value problems using governing equations of fluid dynamics. An important parameter is to define the working fluid, which was assumed to be a Newtonian fluid with a constant density and viscosity [89– 94]. In the case of non-Newtonian resins, special flow models, such as the Ericksen model, can be employed [95]. The relationship between the flow-rate across the unit cell and the pressure drop was calculated by a numerical method (computational fluid dynamics or the finite element method) and a permeability value is computed. The models vary according to the solution method chosen to compute fluid flow through the reinforcement, but the strategy was often similar [96]. Apart from conventional CFD methods, other techniques, such as the Lattice Boltzman method (LBM) [97, 98], meshless LBM [99] and smooth particle hydrodynamics [100], have also been reported. Usually, textile modeling techniques are used to generate such unit cell models, however meshing of such models can be a challenge [75], hence micro CTbased techniques provide a good alternative to the geometrical modeling and meshing steps. There are very few commercial software packages available to compute permeability of porous media. However, these packages are for general porous media, where the distinction of multi-scales, anisotropy, and heterogeneity are not important as compared to fiber networks.

2.5.1.1

Tow Modeling Approaches

For permeability computations, the flow can be treated as dual scale, i.e. inter-tow and intra-tow flows. As mentioned previously, in some cases, intra-tow flow is ignored and the simulation is only performed through spaces available around the meso-scale tows assuming that the tows are impermeable [101]. In these cases, there is absolutely no flow through a tow, as shown in Fig. 2.20a. This approach is valid when the intratow pores contribute negligibly to the overall flow. The Stokes equations can be used in this case, with an impermeable no-slip boundary condition at the tow surface to ensure that there is no flow through it [102–104], as shown in Fig. 2.20a.

2.5 Virtual Permeability

43

No Flow

(a)

(b)

K⊥

K⊥

                 

        

K0

     

K

    

      

   





(c) Fig. 2.20 Intra-tow flow modeling strategies, a solid tows with no intra-tow flow, b intra-tow microstructure resolved and c homogenized porous media model with transversely isotropic permeability tensor

In the dual scale approach, the reinforcement tows are considered as a porous structure and are included in the flow domain. The intra-tow flow can then be incorporated in the analysis by either modelling the individual fibers in the tows, as shown in Fig. 2.20b, or using a sink term in the governing equations. Modelling the fiber structure in the tows and solving the governing equations require a much finer mesh, which substantially increases the computational effort and expense. A simpler approach would be to use the Brinkman formulation, for which it is necessary to estimate the local permeability of the individual tows, referred to as K 0 in Fig. 2.20c. The tow permeability can be estimated using analytical formulations, such as the Gebart’s model. The local tow permeability is treated as transversely-isotropic, as illustrated in Fig. 2.20c [105, 106].

2.5.1.2

Governing Equations

The governing equations to be solved are in the form of partial differential equations, derived from the conservation laws of mass and momentum [107–109]. The choice of the governing equations relies on the Reynolds number, and tow modeling approaches. For Re  1, the simplified Stokes equation can be used, whereas for faster flows, the Navier-Stokes equation can be employed [110]. Some researcher

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2 Generation of Material Twin Using Micro CT Scanning

Table 2.1 Governing equations for various flow regimes and intra-tow flow models Equations

Flow regimes Linear

Tow models Non-linear Solid

Resolved Porous media

Stokes

✓

✗

✓

✓

✗

Navier-Stokes

✗

✓

✓

✓

✗

Stokes-Brinkman

✓

✗

✗

✗

✓

Navier-Stokes-Brinkman ✗

✓

✗

✗

✓

prefer to combine Darcy’s law with the governing equations and solve for DarcyStokes coupled equations [111]. The Stokes and Navier-Stokes equations are applicable when the tows are either assumed to be solid or the microstructure was fully resolved. The tows can be modeled in three ways, (1) by assuming them to be solid (2) by taking them to be porous media and (3) by resolving the tow microstructure. When the tows are assumed to be homogenized porous media, the Brinkman term needs to be added to the two fundamental equations [112]. While an effective fluid viscosity is defined, which is used to match the shear stress values at the interface between the free flowing fluid and the porous medium [113]. A summary of the usual combination of the equations under specific flow regimes and tow modelling approach are given in Table 2.1 [114–117].

2.5.1.3

Boundary Conditions

The next step in the simulation process is to assign boundary conditions to the micro CT-based unit cells. The simulation is carried out on a unit cell which is a representative volume element with three sets of faces. One of these sets was normal to the flow direction whereas the other two are tangential. Normally, periodic boundary conditions are applied on all the faces. A pressure drop is also applied across the faces normal to the flow direction. The pressure drop may be defined as a pressure gradient, or as nominal pressures at the inlet or outlet. These boundary conditions are valid for through-thickness flow but may be slightly varied for inplane flow cases where the top and bottom faces may be assumed to be non-slip impermeable walls. As an alternative to the pressure drop, one may also define the net flow rate and solve for the pressure drop [113, 118–120]. Another important boundary encountered is the tow surface, which separates the tows with the pore spaces [121]. The boundary condition to be defined here depends on the tow model that is adopted. When the tows are assumed to be solid, the non-slip impermeable wall is applied to the tow surfaces. When the microstructure was fully resolved, this boundary condition is transferred to the surface of individual fibers [122]. In case of porous tow model, the shear stress values at the interface between the free flowing fluid and the porous medium need to be matched.

2.5 Virtual Permeability

2.5.1.4

45

Solvers

The final step in setting-up the numerical flow simulation is to choose the discretization scheme for the governing equations. The discretization schemes may be the general finite element method, finite difference method, control volume schemes, boundary element method [123] as well as dedicated schemes, such as explicit jump (EJ) [124], Fast Fourier Transform (SimpleFFT) [125], left identity right (LIR) [126] solvers. Usually, the EJ solver is fast and runs on low memory, but can only be applied to flows when the pressure drop/flow velocity dependence is linear (laminar). Thus, although the SimpleFFT solver can be used, a slow fluid flow that follows the Stokes equation is best computed with the EJ Solver. The LIR solver uses a non-uniform adaptive grid which results in very low memory requirements. While the LIR solver’s speed is comparable to the EJ solver for low porous materials, it is very fast (and should be chosen) for highly porous materials. It can be used to solve the Stokes, Stokes-Brinkman, Navier-Stokes, and Navier-Stokes-Brinkman equations. The SimpleFFT (Simple Fast Fourier Transform) solver takes longer to converge and requires more memory, but it can deal with non-linear (non-laminar) fluid flow, as well as with linear flows. Therefore, faster flows which are not necessarily laminar, and are modeled by the Navier-Stokes equation, can be computed with the SimpleFFT solver. The solver can also deal with porous voxels, and thus solves the Stokes-Brinkman and Navier-Stokes-Brinkman equations [125]. A finite difference Navier–Stokes solver, NaSt3DGP, employs a Chorin projection method on a staggered grid for the solution of the Navier–Stokes equations [22, 23]. In the staggered grid approach, the pressure is discretized at the center of the cells, while the velocities are discretized on the side surfaces. This discretization leads to a strong coupling between pressure and velocities, and therefore avoids the occurrence of non-physical oscillations in the pressure [127]. There are no comparative studies available where the performance of these solvers was evaluated.

2.6 Summary This chapter presents a detailed discussion on the use of micro CT based material twins for the computation of reinforcement permeability for liquid composite molding processes. The use micro CT based material twins offers a great potential to be used as a characterization tool, however there are several challenges associated with this methodology. The potential issues with the micro CT include; sample size restrictions, temporal resolution, and noise. Improvements in both hardware and software can make the methodology more robust and more widely applicable. The most challenging step in the permeability computation process is segmentation. In the virtual permeability computation, it is important that the fiber tows with specific orientations are clearly distinguished. The segmentation algorithm based on artificial intelligence and machine learning needs to be further developed and their applicability must be demonstrated.

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2 Generation of Material Twin Using Micro CT Scanning

Three modeling approaches were discussed for the permeability computations which vary significantly in terms of implementation and accuracy. The realistic voxel model is considered to be true representation of the complex reinforcement architecture. However, the voxel model may need some clean-up operations. Unlike traditional finite element/volume models, the mesh size stays uniform throughout the model, and is primarily determined by the resolution of the micro CT images. Extraction of unit cells from the segmented volume is also challenging. The periodicity of the unit cell face needs to be addressed in subsequent computations, and the unit cells must be carefully selected. The stochastic virtual modelling procedure does not require any segmentation, provided that measurements are taken manually, and can generate mesh with desired element sizes and shapes, as well as the periodicity of the unit cells can be well maintained. However, the model may not be a true representation of the actual structure, and requires additional tools for geometric modeling, and generating a computational mesh. The simplified geometry models are a very crude approximation and are suitable for early design evaluations and parametric studies. In the numerical simulation for permeability computation of fibrous reinforcements, the validity of Darcy’s laws needs to be achieved at both micro and meso scales. The relationship between the flow-rate across the unit cell and the pressure drop is calculated using numerical methods (computational fluid dynamics, or finite element method) and a permeability value is thus computed for the cells through postprocessing. Example permeability studies show the utility of the micro CT technique for virtual permeability computation through comparisons with experimental data.

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Chapter 3

Experimental Techniques for Reinforcement Characterization

There are many experimental methods that have been proposed and are in use for the measurement of permeability of fiber reinforcements. As there is a lack of established norms, many researchers have developed in-house methods depending on their individual requirements and facilities. The results obtained from these methods can vary significantly, with a 20% coefficient of variation, due to a lack of standardized testing methods, and errors associated in the material handling process [1, 2]. Another study suggests the relative variations in the principal permeability values to lie in a range between 9.1 and 29.3% [3]. In an effort to standardize the procedures, several smaller regional benchmark studies [4] were conducted which were then followed by two international benchmark studies [1, 2]. In these benchmark exercises, in-plane permeability values were studied by a number of research groups across the world. In the first exercise [1], the same reinforcement was used by all participants, but no specifications were made regarding the measurement method and the test parameters. This resulted in a level of scatter in the measured permeability values that was greater than one order of magnitude. In the second international benchmark exercise [2], the participants were required to apply an unsaturated linear injection method. This benchmark exercise showed—for this specific test method—that by defining minimum requirements for the equipment, measurement procedure and analysis, satisfactory reproducibility of data obtained using different systems can be achieved. Motivated by the outcome of these two benchmark exercises, a new benchmark exercise was proposed not only for the in-plane permeability measurements but also for the through-thickness permeability measurements and the compaction test. The experimental procedures for the compaction test and permeability measurements are described in detail in this chapter. The same setup and procedures were used for the validation of virtual permeability values obtained from the proposed hybrid framework. Before describing the experimental setups and procedures, the test materials used for various activities are also discussed.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 M. A. Ali et al., CT Scan Generated Material Twins for Composites Manufacturing in Industry 4.0, https://doi.org/10.1007/978-981-15-8021-5_3

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3 Experimental Techniques for Reinforcement Characterization

3.1 Types of Reinforcements Several 2D and 3D reinforcements have been described here. The details are given in Table 3.1 below. The focus here is mainly on 3D reinforcements and hence the 3D reinforcements are discussed in detail. The 2D reinforcements along with the carbon AFP tape will be discussed in brief where deemed necessary.

3.1.1 3D Reinforcements Three different types of 3D reinforcement, supplied by Sigmatex, UK, are investigated in this study. These three reinforcements differ in their reinforcement architectures, and presumably exhibit different behaviors under various compression loading histories. The z-binder yarns in all architectures have different paths or orientations to hold the structure together. The architecture of these test reinforcements are orthogonal, angle interlock and layer-to-layer. The 3D test reinforcements are shown in Fig. 3.1a–c. The 3D orthogonal carbon reinforcement has an areal density of 3353 g/m2 . The reinforcement has eight layers of 12 K carbon fiber warp tows, nine layers of weft tows, and 6 K through-thickness z-yarns acting as binder yarns. The warp and weft tows are alternatively placed in the form of cross-layers (0°/90°) and are interlaced by the z-binder, which travels vertically through the layers from top to bottom, as shown in Fig. 3.1a. The interlacing z-binder is basically the ninth warp layer. As per the specifications provided by the supplier, Sigmatex UK, the reinforcement in its pristine state has a Vf of 37% which was also verified by calculating the Vf from the reinforcement thickness and areal weight [50]. The 3D angle interlock carbon reinforcement has an areal density of 3045 g/m2 . Similar to the orthogonal reinforcement, the angle interlock reinforcement has eight layers of 12 K carbon fiber warp tows, nine layers of weft tows, and 6 K throughthickness z-yarns acting as binder yarns. One in three warp yarns is an interlacing Table 3.1 Specifications of different reinforcements Reinforcement

Reinforcement type

Material

Areal weight (g/m2 )

Density (kg/m3 )

Code

Orthogonal

3D woven

Carbon

3353

1770

3D_ORTHO

Angle inter-lock

3D woven

Carbon

3260

1770

3D_ANGIL

Layer-to-layer

3D woven

Carbon

3045

1770

3D_LR2LR

Hexcel twill weave

2D woven

Glass

295

2550

2D_HEXTW

Saertex NCF

±45 NCF

Glass

444

2550

2D_SXNCF

Carbon AFP

±45 AFP

Carbon

–

1770

2D_CBAFP

3.1 Types of Reinforcements

55

We

Z-Binder

Warp

We

(a)

Warp

We

(b)

(c)

Warp

Fig. 3.1 The 3D reinforcements. (a) orthogonal, (b) angle inter-lock, and (c) 3D layer-to-layer

weaver with the rest being warp stuffers. The layer-to-layer reinforcement has six weft layers and five warp layers acting as interlacing yarns. It has an areal density of 3260 g/m2 .

3.1.2 2D Reinforcements The 2D test reinforcements are: (1) a biaxial (±45°) glass fiber non-crimp reinforcement (NCF) from Saertex and (2) a 2/2 twill weave glass fiber woven reinforcement (WF) from Hexcel. Both the 2D test reinforcements are shown in Fig. 3.2. The Saertex NCF has a nominal areal weight of 444 g/m2 (217 g/m2 in +45° and in −45° direction and additionally 1 g/m2 and 2 g/m2 in the 0° and 90°, respectively, for stabilization) as well as 6 g/m2 polyester stitching yarn (76 dtex) with a warp pattern at a stitch length of 2.6 mm and a gauge length of 5 mm. The Hexcel woven reinforcement has a nominal areal weight of 295 g/m2 equally distributed in weft and warp direction. Both reinforcements are nominally balanced. The actual construction of the WF is somewhat different in warp and weft direction: ends (warp) count is 7.13 yarns/cm, picks (weft) count is 7.00 yarns/cm. For the NCF, the bundle

56

3 Experimental Techniques for Reinforcement Characterization

Weft

Stitching

(a)

(b) Production Direction

Warp

Fig. 3.2 The test materials, (a) 2D 2 × 2 twill woven reinforcement, and (b) 2D multi-axial non-crimped reinforcement

count is 4.2 bundles/cm in the production direction and 4.1 bundles/cm perpendicular to it.

3.1.3 AFP Carbon Tape The dry fiber tapes (PRISM TX1100) used in this study were supplied by Solvay, Wrexham, UK and is shown in Fig. 3.3. The dry tape uses a proprietary veil and binder technology which is incorporated to improve resin flow and tack. The reinforcements were manufactured using the AFP manufacturing process. A robot with a 6 kW laser mounted on the Coriolis end-effector, was used to place the fibers on a flat mold. The heat generated by the laser was sufficient to activate the binder that ensured bonding of adjacent tows. The panels were manufactured after adjusting the tow gap to facilitate resin flow. Two types of such reinforcements were used each with a tape width of ¼ inch (6.35 mm) and ½ inch (12.7 mm). Each reinforcement consists of 24 layers with a stacking sequence of [+45/0/−45/90]3s , giving an overall thickness 2 mm Gap

+45°

-45°

Tape Width

2.5 mm

90°

Fig. 3.3 The 2D carbon tape produced by automated fiber placement

0°

3.1 Types of Reinforcements

57

Fig. 3.4 Viscosity of the test fluid as a function of temperature

200

Viscosity (mPa.s)

TUM

Shell

150 100 50 0 10

20

30 40 Temperature (°C)

50

60

of 5.15 mm. The tapes consist of unidirectional Tenax IMS65 carbon fibers with a filament diameter of 5 µm were used.

3.2 Test Fluid Industrial hydraulic oil, Shell Tellus S2 M46 was used as a test fluid. To have confidence in the viscosity of the test fluid supplied by Shell, a small quantity of the test fluid was sent to the Technical University of Munich (TUM) for rheological measurements. At TUM, the viscosity was measured over a temperature range from 15 to 40 °C using an Anton Paar MCR 302 rheometer. The viscosity measurements made at TUM, as well as data supplied by Shell are shown in Fig. 3.4. The comparison shows that both sets of data agree quite well.

3.3 Sample Preparation The reinforcement was rolled out and cut in the transverse direction of required size. The cut pieces were stacked with the same face up and then cut in the longitudinal direction to the desired size. The reinforcement roll out, cutting and layup procedure is shown in Fig. 3.5a. It is preferable to use circular samples which can be cut from the layup using circular cutters shown in Fig. 3.5b. The circular cutters have diameters ranging from 40 to 250 mm. The cutter with the appropriate diameter was used as per the specific experiment. For cutting the samples, the reinforcement layup was placed on the top of the bench-press, as shown in Fig. 3.5c. The cutter is then placed on the top of the reinforcement layup as shown in Fig. 3.5d. The cutter was pressed down on the sample. The extra material was removed to yield the final test sample, shown in Fig. 3.5e. Finally, each reinforcement stack was weighed using an electronic balance,

58

3 Experimental Techniques for Reinforcement Characterization

Fig. 3.5 Sample preparation steps. (a) reinforcement roll out, cutting and layup, (b) circular cutter, (c) bench-press cutting machine, (d) reinforcement layup with the cutter place on the bench-press, (e) final sample, (f) sample weighing, and (g) inspecting the alignment of reinforcement layers

3.3 Sample Preparation

59

as shown Fig. 3.5f, and recorded carefully for calculating the fiber volume fraction of the stack. The reinforcement stack needed to be examined for proper alignment and the layers were adjusted carefully as necessary as shown Fig. 3.5g. The alignment check was particularly necessary for the in-plane permeability measurements. For fluid injection in the in-plane radial permeability measurements, a hole was punched in the middle of the test sample.

3.4 Compaction Tests The compaction tests were carried out on an INSTRON 5980 dual column Universal Testing System with a maximum testing load of 50 kN. The test setup with its various components and software interface are shown in Fig. 3.6. Tests can be set up and run directly from an integral control panel. Load and extension data were acquired through the Bluehill software. The current tests were carried out using a 5 kN load cell on 150 mm round stainless steel platens. Circular test samples with a diameter of 100 mm were cut according to the procedures described in the previous section. Pre-tests were conducted in order to evaluate the compliance of both the load cell and the fixture, with and without samples. The load profile was defined in the Bluehill software as per the instructions. The platens were brought together with no gap to set zero gauge length and then separated sufficiently for sample placement. The sample was carefully placed on the lower platen ensuring that it was at the center of the platen. The upper platen was brought down so that the gap was 10 mm. The test was started and controlled by the predefined load profile in the Bluehill software.

Crosshead

Load Cell Upper Platen Test Sample Lower Platen

(a)

(b)

Fig. 3.6 Compaction characterization setup. (a) compaction platens installed on the UTM, and (b) software interface

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3 Experimental Techniques for Reinforcement Characterization

3.5 Permeability Measurement 3.5.1 Through-Thickness Permeability Measurement The through-thickness permeability measurement setup is based on unidirectional saturated flow across a stack of reinforcement layers under a constant injection pressure. The test fixture consists of two aluminum cylindrical platens that fit together in a piston-cylinder fashion vertically shown in Fig. 3.7. The top platen was inserted in the lower platen forming an adjustable air tight cavity in between sealed with rubber O-rings. The platens have internal flow paths in a conical shape for injection and removal of the test fluid. The platens also contain a circular pattern of 1 mm diameter holes creating a through-thickness flow area with diameter of 100 mm. Two pressure transducers were attached at the periphery of the top and bottom cylindrical platens. These transducers give the inlet and vent pressures. A flow meter was also attached before the inlet pressure transducer. Analog electrical signals were captured from the transducers and flow meter through the NI Elvis II module and fed into LabVIEW software. The electrical signals were then converted into pressure and volume flow according to the calibrations of the measuring instruments. For the final computations, only those measurements were considered for which the inlet pressure had achieved a steady level. The whole setup was installed on a 50 kN MTS universal testing machine as shown in Fig. 3.8. This allows for accurate control of sample thickness, and the potential for simultaneous compressibility and permeability measurements. The upper platen was attached to the MTS crosshead making possible for accurate thickness adjustments. A digital LVDT, with a precision of 10 µm, was also installed on the upper platen for the thickness measurement. The accuracy of LVDT measurements was confirmed by measuring the thickness of slip gauges of known thickness. This

Fig. 3.7 The cylindrical platens used in through-thickness permeability measurement. (a) upper cylinder (shown inverted here) and (b) lower cylinder

3.5 Permeability Measurement

61

Pressure Transducer II LVDT

Thickness Control Flow DirecƟon To Waste Pot

(a)

Flowmeter

(b) Pressure Transducer I From Resin Pot

Fig. 3.8 The through-thickness permeability measurement setup. (a) thickness control and measurement and (b) pressure and flow measurements

measurement was taken after installing the LVDT on the test fixture and the verticality was adjusted accordingly. Samples were cut using a stamping press to the desired diameter. Once the sample was compressed to the required thickness for flow measurements, hydraulic shell oil was injected from the bottom platen. Using the pressure drop and volume flow rate, the through-thickness permeability K 33 is given by, K 33 = μ

Q L A P

(3.1)

where Q is the volumetric flow rate through the sample, L is the imposed cavity thickness, μ is the test fluid (hydraulic oil) viscosity, A is the cross-sectional area, and ΔP is the applied pressure difference. The test fluid was contained in a pressurized pot with a sealed lid. The pot has an inlet for the compressed air and two outlets, one for releasing pressure and the other for letting out the test fluid. The pot was pressurized by compressed air. The pressure inside the pot was measured using a dial gauge attached with the pot and was maintained as 100 kPa (~1 atm). In order to reduce the influence of the reinforcement deformation because of the fluid pressure, a low injection pressure was used [5]. The test fluid was injected through the test specimen by opening the outlet valve and collected in a waste pot. To achieve a saturated flow, the test fluid was allowed to flow for some time and the injection pressure and flow rate was continuously monitored during this time. Meanwhile, the fluid coming out of the test fixture was examined for air bubbles. The flow rate and pressure measurements were recorded once the pressure readings were stable and there were no bubbles in the out-flowing fluid. Measurements were recoded for one minute and the valve was closed. The used test sample was removed from the cavity and remaining fluid within the pipes was flushed out.

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3 Experimental Techniques for Reinforcement Characterization

3.5.2 In-Plane Permeability Measurement The in-plane permeability measurements can be classified as the “rectilinear flow” and the “radial flow” methods. The rectilinear flow method is based on onedimensional fluid flow through the test sample placed in a specifically designed test fixture. This method is prone to race tracking [6, 7], which is very likely to occur in gaps along the side edges of the material, deteriorating the rectilinear flow front advancement. The race tracking issue can be mitigated typically by sealing the preform material at the side walls of the test fixture [7–9]. Secondly, at least three experiments are required to fully characterize the in-plane permeability tensor which causes increased experimental effort. The experimental efforts can be reduced by using “multi-cavity parallel flow cells”, which enable the simultaneous characterization of samples cut along different material directions [10, 11]. In contrast to the linear flow technique, the radial flow method allows for a full characterization of the in-plane permeability tensor from a single experiment and avoids race tracking effects. In this method, the test fluid is injected from an inlet port in the middle of the test sample, typically with circular in shape. The size and shape of the inlet port play significant role in the correct estimation of the permeability. The smaller size of the inlet port lead to more inaccurate measurements [7]. The radially advancing flow front is then typically tracked by means of a video camera [12, 13] or electric/dielectric sensors [14–16] or capacitive sensors [17]. When the flow front is captured using a video camera, one side of the mold needs to be optically transparent which also needs to be stiffened in order to avoid extensive mold deflection during the experiments. The flow front data is then fitted with an elliptic geometric model to reconstruct the overall shape of the flow front as a function of time. The choice of elliptic geometry model fitted to the data can easily have an impact on the resulting in-plane permeability values as reported by Fauster et al. [18]. The final step involves computing the permeability using specifically developed mathematical algorithms [18–21]. Fauster et al. [22] introduced a novel approach for evaluating flow front data based on the fitting of an elliptic paraboloid model, which directly and robustly delivers an estimate of the orientation angle as well as principal in-plane permeability values characteristic for the reinforcing material under test. The experimental setup and procedure for in-plane radial permeability measurements using the constant injection pressure are explained below.

3.5.2.1

Experimental Setup

Similar to the through-thickness permeability measurement setup, the in-plane permeability measurements setup can also be installed on a 50 kN MTS universal testing machine. The test setup is shown in Fig. 3.9. The fixture consists of a 300 × 300 mm2 toughened glass plate fixed in an aluminum frame. A circular steel plate of 250 mm diameter was used as the top platen to compact the reinforcement. The top plate incorporates a 10 mm central injection hole drilled from the side through

3.5 Permeability Measurement

63

Crosshead

Pressure Transducer

Upper Platen Lower Platen

Injection Port Compressed Air

Support Frame

Test Sample

(a)

Camera

Fluid Pot

(b)

Fig. 3.9 The in-plane permeability measurement setup. (a) fluid injection and pressure measurement mechanism along with the test sample, and (b) upper and lower platens with support structure and video camera

which test fluid was injected. In order to ensure the parallelism between the top and bottom surfaces, the cavity thickness was measured at four different locations using precision gauge blocks. The measurements were taken with the empty cavity as well as filled with a sample. Necessary adjustments were made to keep the difference in the thickness at these locations to within few microns. A pressure transducer was connected to the top plate which measures the injection pressure. The pressure given by this transducer was taken as the inlet pressure. The test fluid was injected using a pressure pot connected to compressed air. The samples were placed on the top of a transparent glass plate and the flow progression was recorded via a high definition camera placed under the glass plate movie camera. Appropriate lighting needed to be ensured to clearly distinguish the flow front without any reflections. The thickness of the test cavity was controlled by the crosshead of the MTS and was measured using a digital LVDT. The injection pressure was recorded by using LabVIEW data acquisition software. A 12 mm hole was punched in the center of the test sample to ensure radial in-plane flow. A compressive pre-load was applied and the reinforcement was compacted at 2 mm/min to the target fiber volume fraction. The extension was held constant to allow for stress relaxation for about 5 min. After this, the test fluid was injected at constant pressure.

3.5.2.2

Flow Front Detection

The data from the pressure transducer, MTS machine and camera were extracted for post-processing. The captured video of flow front progression is digitized through image analysis using MATLAB®. The various steps in this procedure are shown in Fig. 3.10. Video frames were extracted at a time interval of 5 s. The extracted video frames are the raw images, Fig. 3.10a, that were processed further. The alignment of

64

3 Experimental Techniques for Reinforcement Characterization

Fig. 3.10 Sequence of steps in image processing for flow front tracking. (a) raw image extracted from the video, (b) enhanced image, (c) binarized image, (d) isolated flow region, (e) flow front detection, and (f) detected flow front superimposed on the original image

the raw image was adjusted whenever necessary and the image was cropped to an appropriate size. The image was then enhanced by contrast adjustment and sharpening. This was achieved via the imadjust, imsharpen and stretchlim functions in MATLAB®. An example of an enhanced image is shown in Fig. 4.8b. The enhanced image was binarized into a black and white image, Fig. 3.10c. The flow region was then isolated by removing the other regions through morphological filling, opening and closing operations. The boundary of the isolated flow region, Fig. 3.10d, was then traced using the bwboundaries function. The coordinates of traced boundary, Fig. 3.10e, was stored in the memory and used to fit an ellipse. The accuracy of flow front tracing can be seen in Fig. 3.10f where the traced flow front is superimposed on the original raw image.

3.5.2.3

Computing In-Plane Permeability

The principal in-plane permeability values were determined based on the measurement of the principal axes of the elliptically shaped flow front as a function of time. This technique is well documented by Weitzenbock ¨ et al. [23, 24] and calculates the permeability without prior knowledge of the orientation of the principal axes. The approach assumes that the inlet pressure is constant and the pressure at the flow front was ambient pressure. The basic formulation is derived from the governing equation for two dimensional flows in porous media in conjunction with Darcy’s law. The in-plane permeability are related to the flow front position and time by,

3.5 Permeability Measurement

      a 1 μφ − 1 + x02 a 2 2 ln t 4P x0       b 1 μφ 2 2 = − 1 + y0 b 2 ln t 4P y0

65

Kxx =

(3.2)

K yy

(3.3)

where a and b are the major and minor axes of the ellipse fitted on the flow front and x 0 and y0 are the coordinates of the injection hole in the respective directions. Φ = 1-V f is the porosity of the reinforcement, t is the time, ΔP is the pressure difference and μ is the viscosity of the test fluid. The viscosity of the test fluid was assumed to remain constant during the injection. The major and minor axes of the ellipse, as well as its orientation, are determined by fitting an ellipse on the flow front coordinate obtained during previous step using least square fitting method. The method also gives the orientation of the ellipse which was used to define the principle axes of the in-plane permeabilities. A number of ellipses are fitted to the flow front at multiple time steps. The major and minor axes of each ellipse are used to compute the permeability at each time step. The overall permeability is then obtain by taking the arithmetic mean of the permeability calculated at each time step. Since the flow is not well established in the beginning of the experiment, the data for the initial moments are ignored. The time required to achieve a stable flow can be estimated by visually inspecting the shape of the flow front.

3.5.2.4

Evaluation of Cavity Deformation

In order to estimate the deflection behavior, a Finite Element simulation was performed. Geometrical models representing both the top and bottom platens were generated as shown in the Fig. 3.11a. Note that the upper plate was made of stainless steel with six blind holes. The lower plate was made of glass which was supported by aluminum blocks. The region of mold surfaces covered by the test sample was separated from the remaining area. As per the guidelines, a pressure of 1.0 MPa was applied in the region covered by the test sample whereas only 0.1 MPa (~1 atm) was applied on the remaining area of the mold surfaces. The structure was constrained on the appropriate surfaces as to mimic the actual supports. The loading conditions and constraints are shown in Fig. 3.11b. The deflections of both the top and bottom mold surfaces are shown in Fig. 3.11c. The results confirm that the rigidity of the structure was sufficient with negligible deflections. While the obtained deformations do not fully reflect the actual deformation during the experiments, they give a good impression of the tendency of a system to deviate from the target cavity thickness.

66

3 Experimental Techniques for Reinforcement Characterization

Upper Platen

Lower Platen Glass Plate

Area Outside Test Sample

Area Covered by Test Sample

Aluminum Support

Area Covered by Test Sample

(a)

0

50

0

100 mm

100

1.0 MPa

200 mm

0.1 MPa

Fixed Support

1.0 MPa

(b) 0

50

0.1 MPa

100 mm

Fixed Support 0

100

200 mm

50

100 mm

Transverse Deflecon (μm)

Transverse Deflecon (μm)

27.3

2.0

19.3

1.0

(c)

9.20

0.5 0

50

100 mm

0

Fig. 3.11 Fixture analysis. (a) geometric model of the fixture, (b) load conditions, and (c) transverse deflection in the area covered by the test sample

3.6 Discussions 3.6.1 Compaction Tests The stress relaxation curves during dry compaction are shown in Fig. 3.12. The peak stress required to achieve the target thickness is almost twice for the Saertex-NCF showing relatively more stiffness as compare the Hexcel-WF. The stress relaxation curves during wet compaction are shown in Fig. 3.13. Both the dry and wet relaxation

3.6 Discussions

0.35 0.30

Stress (MPa)

Fig. 3.12 Stress relaxation curves during the dry compaction tests. (a) Saertex-NCF, and (b) Hexcel-WF

67

0.25 0.20 0.15 0.10 0.05 0.00

(a)

0

500

1000 Time (s)

1500

2000

0

500

1000 Time (s)

1500

2000

Stress (MPa)

0.15

0.10

0.05

0.00

(b)

curve show typical behavior. The curves were obtained for five test samples. There was a 4–8% variation among the test samples in the peak stress required to achieve the target thickness. Similar variations was also noted for the steady-state stress level. Comparing the dry and wet compaction, it can be noted that the Hexcel-WF require 68% less peak stress in the wet compaction. However, for the Seartex-NCF, the reduction in peak stress during wet compaction is only 17.5%. This means that the Seartex-NCF retain its stiffness even after soaking worth the test fluid.

3.6.2 Through-Thickness Permeability Measurements The through-thickness permeability measurements as a function of fiber volume fraction are shown in Fig. 3.14. The data shown in the Fig. 3.14 shows a typical exponential relationship between the permeability and V f . The through-thickness permeability of the Saertex-NCF is relatively higher than the Hexcel-WF. This is primarily due to the stitching which produces gaps in between the fiber bundles

68

3 Experimental Techniques for Reinforcement Characterization

Fig. 3.13 Stress relaxation curves during the wet compaction tests. (a) Saertex-NCF, and (b) Hexcel-WF

0.3

Stress (MPa)

0.2

0.1

0.0 0

500

1000 Time (s)

1500

2000

0

500

1000

1500

2000

(a) 0.05

Stress (MPa)

0.04 0.03 0.02 0.01 0

(b) Fig. 3.14 Throughthickness permeability measurements versus fiber volume fraction (V f )

Time (s)

5.0E-12

K33 (m²)

Hexcel-WF Saertex-NCF

5.0E-13 0.45

0.50 Vf

0.55

3.6 Discussions

69

and creates flow path transverse to the plane of fabric. In Fig. 3.14, each data point represents the arithmetic mean of the five repeat measurements and the error bars represent the standard deviation for permeability and V f . The coefficient of variation among these five measurements were 5.98% and 3.16% for the Saertex-NCF and the Hexcel-WF, respectively. The coefficient of variation for the Saertex-NCF is slightly higher the Hexcel-WF which is contrast with the overall global results [25]. The measurements for the Saertex-NCF, being stiffer than the Hexcel-WF, should have been less prone to defamations induced by fluid pressure. Nevertheless, the overall scatter in the through-thickness permeability measurements is very low.

3.6.3 In-Plane Permeability Measurements Figure 3.15a,b summarize the measured permeability values for the Saertex-NCF

Permeability (m2)

1.E-10

1.E-11 K11 K22 1.E-12

(a)

0.40

0.45

0.50 Vf

0.55

0.60

0.50 Vf

0.55

0.60

1.E-10

Permeability (m2)

Fig. 3.15 In-plane permeability measurements versus fiber volume fraction (Vf ). (a) Saertex-NCF and (b) Hexcel-WF

1.E-11 K22 K11 1.E-12

(b)

0.40

0.45

70

3 Experimental Techniques for Reinforcement Characterization

and the Hexcel-WF, respectively, showing the highest (K 11 ) and lowest (K 22 ) inplane permeability value (logarithmic scale on y-axis) as a function of the V f . The data shown in the Fig. 3.15 shows a typical exponential relationship between the permeability and V f . In Fig. 3.15, each data point represents the arithmetic mean of the five repeat measurements and the error bars represent the standard deviation for permeability and V f . For each data point, the coefficient of variation of the measured permeability values was calculated as the ratio of the standard deviation and the arithmetic mean. The overall coefficients of variation were 6.13 and 10.37% for the Saertex-NCF and the Hexcel-WF, respectively, which were significantly lower than the overall global coefficients of variation (32.3 and 43.9%) [25, 26]. In addition to the permeability values K 11 and K 22 , the orientation angle of K 11 relative to the fiber directions values were also determined in the tests. On average, the orientation angle were 15.27° and 5.2° for the Saertex-NCF and the HexcelWF, respectively. The orientation angle for the Seartex-NCF had relatively higher coefficient of variance. This could be attributed to the degree of anisotropy of the permeability which is defined as the ratio K 22 /K 11 . This ratio determines the circularity of the flow front. For a perfectly circular flow front, the ratio is equal to one and the flow front becomes elliptical as the ratio deviates from a value of one. For the Seartex-NCF, this ratio is close to one as can been seen from Fig. 3.15a in which both the lines for K 11 and K 22 are close to each other. As the orientation angle is derived from the ellipse fitted to the flow front, a near-circular shape increases the influence of irregularities in the flow front shape. The degree of anisotropy were calculated to be 0.69 and 0.13 for the Saertex-NCF and the Hexcel-WF, respectively. Since there are no significant differences in fiber bundle geometry for the two main fiber directions in the Saertex-NCF, degree of anisotropy for the Saertex-NCF fabric was expected to be close to one with near-circular flow fronts. However, the slight anisotropy of the Saertex-NCF is likely to be caused by the presence of stitching, oriented in the production direction. On the other side, the Hexcel-WF shows relatively high anisotropy presumably resulting from the deviations from the balanced fabric construction. In summary, the results show that for radial flow measurements the error in orientation angle determination increases with decreasing anisotropy. The variability is small when the anisotropy is high. This seems acceptable, since the relevance of the orientation angle decreases as the flow front becomes more circular. Both orientation angle and anisotropy did not show a clear dependence on V f .

3.7 Summary Experimental compaction study and permeability measurements require three different test setups each requiring a fresh test samples. This chapter gives the description of a typical in-house test facility for the compaction study and permeability measurements. The compaction tests are carried out on an INSTRON 5980 dual column Universal Testing System with a maximum testing load of 50 kN. Load and extension data are acquired through the Bluehill software. The throughthickness permeability measurement setup is based on unidirectional saturated flow

3.7 Summary

71

across a stack of reinforcement layers. The experimental in-plane permeability values are obtained using constant injection pressure radial flow method. The in-plane permeability measurements setup is installed within a 50 kN MTS universal testing machine. The permeability values are computed using the Weitzenbock approach which uses the flow front progression data recorded via video camera.

References 1. Arbter R, Beraud J, Binetruy C et al (2011) Experimental determination of the permeability of textiles: a benchmark exercise. Compos Part A Appl Sci Manuf 42(9):1157–1168 2. Vernet N, Ruiz E, Advani S et al (2014) Experimental determination of the permeability of engineering textiles: benchmark II. Compos Part A Appl Sci Manuf 61:172–184 3. Endruweit A, McGregor P, Long A, Johnson M (2006) Influence of the fabric architecture on the variations in experimentally determined in-plane permeability values. Compos Sci Technol 66(11):1778–1792 4. Lundström T, Stenberg R, Bergström R et al (2000) In-plane permeability measurements: a nordic round-robin study. Compos Part A Appl Sci Manuf 31(1):29–43 5. Danzi M, Klunker F, Ermanni P (2017) Experimental validation of through-thickness resin flow model in the consolidation of saturated porous media. J Compos Mater 51(17):2467–2475 6. Bickerton S, Advani SG (1999) Characterization and modeling of race-tracking in liquid composite molding processes. Compos Sci Technol 59(15):2215–2229 7. Gauvin R, Trochu F, Lemenn Y, Diallo L (1996) Permeability measurement and flow simulation through fiber reinforcement. Polym Compos 17(1):34–42 8. Bickerton S, Sozer E, Graham P, Advani SG (2000) Fabric structure and mold curvature effects on preform permeability and mold filling in the RTM process. Part I Exp Compos Part A Appl Sci Manuf 31(5):423–438 9. Bickerton S, Sozer E, Šimácek P, Advani S (2000) Fabric structure and mold curvature effects on preform permeability and mold filling in the RTM process. Part II Predictions Comparisons Exp Compos Part A Appl Sci Manuf 31(5):439–458 10. Gebart BR, Lidström P (1996) Measurement of in-plane permeability of anisotropic fiber reinforcements. Polym Compos 17(1):43–51 11. Lundström TS, Gebart BR, Sandlund E (1999) In-plane permeability measurements on fiber reinforcements by the multi-cavity parallel flow technique. Polym Compos 20(1):146–154 12. Adams KL, Rebenfeld L (1987) In-plane flow of fluids in fabrics: structure/flow characterization. Text Res J 57(11):647–654 13. Adams KL, Miller B, Rebenfeld L (1986) Forced in-plane flow of an epoxy resin in fibrous networks. Polym Eng Sci 26(20):1434–1441 14. Liu Q, Parnas RS, Giffard HS (2007) New set-up for in-plane permeability measurement. Compos Part A Appl Sci Manuf 38(3):954–962 15. Morren G, Bossuyt S, Sol H (2008) 2D permeability tensor identification of fibrous reinforcements for RTM using an inverse method. Compos Part A Appl Sci Manuf 39(9):1530–1536 16. Hoes K, Dinescu D, Sol H et al (2002) New set-up for measurement of permeability properties of fibrous reinforcements for RTM. Compos Part a Appl Sci Manuf 33(7):959–969 17. Kissinger C, Mitschang P, Neitzel M et al (2000) Continuous on-line permeability measurement of textile structures, 45th International SAMPE Symposium. CA, USA, Long Beach, pp 2089– 2096 18. Fauster E, Berg DC, Abliz D et al (2019) Image processing and data evaluation algorithms for reproducible optical in-plane permeability characterization by radial flow experiments. J Compos Mater 53(1):45–63

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19. Adams K, Russel W, Rebenfeld L (1988) Radial penetration of a viscous liquid into a planar anisotropic porous medium. Int J Multiph Flow 14(2):203–215 20. Chan AW, Hwang ST (1991) Anisotropic in-plane permeability of fabric media. Polym Eng Sci 31(16):1233–1239 21. Bear J, (2013) Dynamics of fluids in porous media, courier corporation. ISBN 978-0-48665675-5 22. Fauster E, Berg DC, May D et al (2018) Robust evaluation of flow front data for in-plane permeability characterization by radial flow experiments. Adv Manuf Polym Compos Sci 4(1):24–40 23. Weitzenböck J, Shenoi R, Wilson P (1999a) Radial flow permeability measurement. Part A: Theory Compos Part A Appl Sci Manuf 30(6):781–796 24. Weitzenböck J, Shenoi R, Wilson P (1999b) Radial flow permeability measurement. Part B Appl Compos Part A Appl Sci Manuf 30(6):797–813 25. May D, Aktas A, Yong A (2018) Workshop: international benchmark exercises on textile permeability and compressibility characterization, 14th international conference on flow processes in composite materials, Lulea, Sweden 26. May D, Aktas A, Advani SG et al (2019) In-plane permeability characterization of engineering textiles based on radial flow experiments: A benchmark exercise. Compos Part A Appl Sci Manuf 121:100–114

Chapter 4

Experimental-Numerical Hybrid Reinforcement Characterization Framework

4.1 Overview In the experimental permeability measurements, a number of test samples are prepared for each fiber volume fraction and tested in test tools having a number of flow and pressure measuring sensors. The flow related data such as flow rate, pressure difference, and flow front progression etc. obtained from the sensors installed on the test rig are then used to compute the permeability. On the other hand, in a numerical permeability computation approach, a computational model is setup from reinforcement geometrical models. Computational fluid dynamics (CFD) codes are used to generate flow field data that is used to compute the reinforcement permeability. Here, in the proposed hybrid approach, we have combined the first step of the experimental procedure with the second step of the numerical method, as shown in Fig. 4.1. The hybrid approach consists of three major steps, (i) the non-destructive acquisition of X-ray micro computed tomography data of the test reinforcement at different levels of compaction utilizing a single sample, (ii) reconstruction of the micro CT data and extraction of a unit cell, (iii) Numerical solutions of boundary value problems on a unit cell using governing equations of fluid dynamics. The methodology is described in detail in the following sections. The characterization framework consists of a laboratory-sized micro CT device and a miniaturized in situ compression fixture. The compression fixture is specifically designed for in situ micro CT applications capable of tensile, compression and torsion testing. The compression fixture is remotely controlled via an integrated software through which displacement and load data can be acquired. The compression fixture consists of 40 mm diameter steel plates attached to a 5 kN load cell at the bottom. This compression module is mounted on the testing platform of the micro CT device for obtaining X-ray images as seen in the flow chart in Fig. 4.2.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 M. A. Ali et al., CT Scan Generated Material Twins for Composites Manufacturing in Industry 4.0, https://doi.org/10.1007/978-981-15-8021-5_4

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Fig. 4.1 Illustration of the hybrid characterization framework

Geometry Model

Mesh

Voxel Model from XCT

Hybrid Method

Test Sample

Flow Simulation

Numerical Method

Fluid Injection

Test Fixture

Permeability

Permeability

Experimental Method

X-ray Source

Compaction Fixture

X-ray Detector 3D Model

Slicing

Segmentation

1 2

Voxel Model

Test Sample

Stress

Warp Peak Stresses

1 2

Flow Simulation Weft

Time

CompactionCurve

Geometry Variability

Permeability

Fig. 4.2 Overview of the hybrid characterization framework illustrating its versatility

4.2 The In Situ Computed Tomography System

75

0.2 m

Detector

0.5 m

Source

Stage

Fig. 4.3 The experimental setup for the in situ micro CT image acquisition

4.2 The In Situ Computed Tomography System The in situ computed tomography package should be capable of not only acquiring micro CT images but should also be able to create and control different loading conditions. Thus, it requires several hardware and software integrated in an efficient manner, each component is described in below sections.

4.2.1 Micro CT Equipment The fundamental equipment in the hybrid reinforcement characterization framework used is the micro CT equipment in which the in situ compression testing stage is housed. A typical setup in a micro CT equipment consists of four main components; an X-ray source, a detector, a stage to hold the test sample and an operating system with data storage. An example of a laboratory scale equipment and its components is shown in Fig. 4.3. Such type of equipment is primarily used for scientific and industrial computed tomography and 3D metrology with a minimum resolution of up to 300 nm which is very close to synchrotron CT. Since the test chamber is enclosed for safety purposes, there is a constraint on the maximum size (e.g. 240 mm Ø × 250 mm in height) of the test samples.

4.2.2 In Situ Compression Fixture The second equipment in the hybrid reinforcement characterization framework is the in situ compression testing stage, which should be small enough to be housed inside a micro CT device, as shown in Fig. 4.4. Using such a miniaturized testing stage makes the hybrid characterization framework versatile, adaptable and integrated test solution. The testing system is to be controlled remotely using a control software, that can also display values of load and extension in real-time. The in situ testing

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(a)

(b)

Fig. 4.4 The in situ compression fixture. a sample placement and assembly of the compression fixture, and b compression fixture placed inside a micro CT device

stage used in this study has a 3 mm thick glass tube with low X-Ray attenuation that encloses the test sample as well as supports the top jaw. The test sample of up to 40 mm diameter can be placed on the lower compression platen of the in situ testing stage, Fig. 4.4. From this test sample, approximately 8 and 4 unit cells can be extracted for the 3D reinforcements. The testing stage is placed inside the micro CT machine for image acquisition.

4.2.3 Data Acquisition The micro CT data acquisition is usually fully automated using integrated software which minimizes operator time and influence, while highly increasing the repeatability and reproducibility of micro CT results. Once the appropriate setup is programmed, the whole scan and reconstruction process including volume optimization features (e.g. automatic beam hardening correction) or surface extraction can be fully automated. The user interface of a typical micro CT data acquisition software, Phoenix Datos|x [1], is shown in Fig. 4.5.

4.2.4 Load and Displacement Control The in situ compression fixture is remotely controlled using a control software, which gives a real-time display of load and extension [2]. The load and displacement data is continuously displayed on the screen as shown in Fig. 4.6. The user can setup various types of loading conditions. A target displacement or force is defined which is then achieved by moving the lower platen of the compression fixture.

4.2 The In Situ Computed Tomography System

77

Fig. 4.5 The user interface of a typical micro CT data acquisition software

Fig. 4.6 The user interface of control software displaying the applied load

4.2.5 3D Visualization Once the scanning is complete, the micro CT data is saved on a dedicated computer drive for preliminary data processing, visualization and subsequently exporting for further processing. The scanned data is used to create three-dimensional model of the volume that can be viewed in 3D or in 2D sliced images as shown in Fig. 4.7.

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Fig. 4.7 a Graphical user interface of a 3D visualization software

This step can be achieved by using commercially available software, such as the VGStudio MAX from Volume Graphics [3]. These type of software packages also allow volume data to be imported or exported in various commonly used formats. In the current application, the micro CT volume data was exported as a stack of images to be used in further processing.

4.3 Micro CT Data Acquisition 4.3.1 Sample Preparation Circular test samples of 40 mm diameter were used for all the test materials. From this test sample, approximately 4–8 unit cells can be extracted for the all the 3D reinforcements respectively. The teste samples were cut by a punch. The punch used to cut the sample, along with a test sample, is shown in Fig. 4.8. The test samples were then placed on the lower compression platen of the in situ testing stage.

4.3 Micro CT Data Acquisition

79

Fig. 4.8 a Punch used to cut the test sample, and b a test sample of the 3D orthogonal reinforcement

40 mm (a)

(b)

4.3.2 Scan Settings A series of 2D radial slices of X-ray absorption maps, each with a resolution of 2400 × 1600 pixels, were captured by rotating the sample to full 360° in 1500 steps. This compilation of 2D radial slices was then used to reconstruct the threedimensional tomographic geometry. The applied voltage on the X-ray tube is 120 kV with a current of 200 μA.

4.3.3 Setting the Scan Resolution The resolution of the scanned images determines the final voxel size of the computational model. For the current study, the resolution of the micro CT was kept at 25 μm for fast data acquisition. The voxel size of 25 μm was chosen following a preliminary study using voxel sizes of 25, 12.5 and 5.667 μm. Samples of scanned images at each resolution are shown in Fig. 4.9. High resolution scans resulted in complexities in computation, such as voxel mesh discontinuities, with many isolated elements and noise, as a result the simulations failed to converge. Here, the objective was to scan the full test sample (40 mm) from which multiple unit cells capturing key features of the reinforcement could be extracted, hence a trade-off between sample size and resolution is achieved. Moreover, faster acquisition means lower man hours

(a)

(b)

(c)

Fig. 4.9 Tow details as resolved at voxel sizes. a 25 μm, b 12.5, and c 5.667 μm

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and machine time resulting in a greater life of the X-ray tube. This voxel size of 25 μm was found to be suitable for sufficiently resolving the meso-scale structure of 3D reinforcements [4].

4.4 Data Processing and Digital Experiments 4.4.1 Data Pre-processing and Segmentation The next step in the hybrid reinforcement characterization framework is to create three-dimensional geometric models from the stack of images. This step involves advanced 3D pre-processing and phase segmentation. The segmented volume in which the distinct phases are clearly identified and labeled or indexed is known as the “digital twin”. The basic segmentation algorithms are based on the Otsu, Kmeans and manual methods which uses a global thresholding grey value for the segmentation [6], as shown in Fig. 4.10. These algorithms are very simple in nature and cannot distinguish the tows with different orientations. Prior to segmentation, the images may be subjected to advanced 3D pre-processing which include cropping, resizing, rotating and filtering.

4.4.2 Digital Flow Experiments Permeability can be computed through digital flow experiments and invoking Darcy’s law. Two categories of experiments can be performed; prediction of mean flow velocity for a given pressure drop or prediction of pressure drop for a given mean flow velocity. The relationship between the predicted mean flow velocity (or pressure drop), and the fluid viscosity and media thickness expressed in Darcy’s law is then used to compute the permeability of the porous medium. The typical user input options to set up a digital flow experiment for permeability computation and sample computed values are shown in Fig. 4.11. To solve the partial differential equations (PDEs) describing the fluid flow, three solution methods, called solvers, have been specifically developed [7, 8]. These dedicated solvers are the explicit jump (EJ) [9], Fast Fourier Transform (SimpleFFT) [10] and left identity right (LIR) [11] solvers. The EJ (Explicit Jump) solver is fast and has low memory requirements, but can only be applied to flows when the pressure drop/flow velocity dependence is linear (laminar). The SimpleFFT (Simple Fast Fourier Transform) solver takes longer to converge and requires more memory, but it can deal with non-linear (non-laminar) fluid flow as well as with linear flow. The LIR (Left Identity Right) solver uses a non-uniform adaptive grid which results in very low memory requirements. While the LIR solver’s speed is comparable to the EJ solver for low porous materials, it is very fast (and should be chosen) for highly porous materials.

4.4 Data Processing and Digital Experiments

81

Fig. 4.10 Applying thresholding Importing micro CT images

(a)

(b)

Fig. 4.11 Setting up a digital flow experiment and computing permeability. a defining flow and boundary conditions, and b viewing final results

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4.4.3 Pore Structure Analysis The process of fluid flow and transport through any porous medium at the meso-scale are strongly dependent on its internal morphology, which includes the size, shape, distribution and connectivity of pores. Two of the main characteristics of the internal morphology of a porous medium are; the pore size distribution, and the percolation paths. The size distribution gives an idea of the free space available, whereas, the percolation path gives the connectivity of these pores. In order to compute the size distribution, binary images are subjected to a series of morphological openings via a procedure known as “granulometry”. An example of the pore size distribution calculate via granulometry is shown in Fig. 4.12. The percolation theory can be applied to establish the connectivity of voxels in the binary micro CT model. Here, the PoroDict module of GeoDict [5] was used to estimate the pore structure characteristics on three-dimensional models of porous media obtained from micro CT [12, 13].

Fig. 4.12 An example of the pore size distribution obtained using granulometry

4.5 Generation of Digital Material Twins

83

4.5 Generation of Digital Material Twins 4.5.1 Realistic Voxel Model The micro CT gives a series of 2D slices containing the X-ray absorption maps. These 2D slices are converted into a volume and mesh data for further analysis. The pixels in each slice, combined together with adjacent slices, can be treated as volume elements, called voxels. These voxels can be utilized as mesh grid. Here, it is important to differentiate between the various phases present in the test sample. In the current case, the voxels representing the reinforcements are differentiated from the free spaces between them through a bimodal segmentation and the results of this procedure can be seen in Fig. 4.13. The outcome of this segmentation is the “digital twin” in which the distinct phases are clearly identified and labeled or indexed.

4.5.1.1

Phase Segmentation

The segmentation step is very crucial in correctly identifying the phases and this will have a direct impact on the permeability computations. The segmentation methods, such as K-means or Otsu, usually underestimate the solid volume fraction and hence larger errors are created in the computed permeability values. Here, the threshold gray values are selected manually for the segmentation. In the manual segmentation method, a global grey values is selected for as the threshold as shown in Fig. 4.14. The value is selected based on visual inspection of the tow cross-sections. The value that optimally extract tow cross-sections is selected as the thresholding values. The resulting segmented volumes of the 3D orthogonal and angle interlock reinforcements are shown in Fig. 4.15. Fiber Tows Pores

(a)

(b)

Interface Pores

(c)

Fig. 4.13 Generation of computational models. a 3D grey scale data, b digital twin, and c a unit cell extracted from the digital twin

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Voxel Count (×103)

35 Solid (Fibers)

Pores (Air)

30 25 20

15 10

5 0

0

1

2

3

4

5

6

Gray Values (×103) Fig. 4.14 Selection a global thresholding grey value for phase differentiation

→

+

→

+

(a)

(b) Fig. 4.15 Phase differentiation in the micro CT images. a orthogonal reinforcement (Vf = 47.36%), and b angle interlock reinforcement (Vf = 32.77%)

4.5.1.2

Unit Cell Extraction

Flow simulations are performed on unit cells which are extracted from the digital twin by recognizing the periodic pattern of the reinforcement structure. The unit cell extraction is achieved by cropping images. For computing the virtual permeability of the orthogonal and angle interlock reinforcements, four unit cells were used, as shown in Fig. 4.16. The typical sizes of unit cells of the orthogonal and angle interlock reinforcements are also shown in Fig. 4.16. Due to the smaller size, it was possible to extract exclusive unit cells from different locations for the orthogonal reinforcement. However, it was not possible for the angle inter-lock reinforcement and the unit cells were extracted in this case had significant overlap.

4.5 Generation of Digital Material Twins RVE 3

RVE 1

RVE 2

85

8 mm

RVE 4

(a)

(b) Fig. 4.16 Extraction of unit cells extracted from the digital twin. a angle inter-lock reinforcement, and b orthogonal reinforcement

4.5.1.3

Realistic Voxel Model of the AFP Carbon Tape

Using the similar appraoch described above, unit cells of the AFP carbon tape were also generated. The unit cells constitute 3D spatial domians coressponding to both carbon fiber tapes and the gaps, both being separated through image segmentation and different gray scale assignment. Since the gaps were mostly quadrilateral shape, two types of unit cells were extracted. The edges of both types of unit cells are aligned either with 0°/90° or +45°/−45° orientations and shown with dimensions in Fig. 4.17a–d. These models were used to compute the in-plane permeability values in 0°/90° and +45°/−45° orientations.

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¼ inch Tape

(a)

(c)

½ inch Tape

90°

00°

-45°

+45°

90° 90°

(b)

(d)

°

-45°

00° 00°

+45° +45°

Fig. 4.17 Micro CT based unit cell models. (a, b) 0°/90° orineted micro CT models, and (c, d) +45°/−45° orinetated micro CT models

4.5.1.4

Mesh Export

To show the compatibility of the methodology with other commercial software packages, CFX and FLUENT modules of ANSYS were also used for the flow simulations. For this purpose, the unit cell were exported as a mesh file readable in the ANSYS modules. The exported mesh contains distinguishable domains representing the fiber tows and the inter-tows spaces. The exported mesh of the 3D orthogonal and angle inter-lock reinforcements are shown in Fig. 4.18. Since the mesh is voxel based, the grid cells were all hexahedral, having an orthogonality of 1 and 0 skewness. This approach is used only for computing the through thickness permeability of the 3D orthogonal and angle inter-lock reinforcements and the AFP carbon tape.

(a)

(b)

Fig. 4.18 ANSYS FLUENT models of unit cells for a orthogonal reinforcement at (Vf = 47.36%), b angle interlock reinforcement at (Vf = 32.77%)

4.5 Generation of Digital Material Twins

87

Fig. 4.19 Measurement of the gap. a extracted 2D slice, b identification of the gap cross-section, and c measured parameters

4.5.2 Stochastic Virtual Model of the AFP Carbon Tape From the geometry analysis obtained from the micro CT images of the AFP carbon tape, it was noted that the gap width had significant variation. A stochastic modeling approach is also adopted to complement the realistic voxel model. Here, the procedure of this method was explained.

4.5.2.1

Measurement of Gap Width and Height

In order to generate statistical data of the gaps, the gap sizes were carefully measured and recorded. A close examination of the cross-sections revealed that most of the gaps are of trapezoidal shapes. Image slices normal to four orientations were taken and gaps sizes were measured using MATLAB image processing tool, as described in Fig. 4.19a, b and the dimensions are shown in Fig. 4.19c. The parameters a1 , a2 , a3 and h were measured from the acquired images, and the angle β was calculated using a3 and h. For a complete statistical analysis, mean of a1 and a2 were taken as gap width, and h is taken as gap height.

4.5.2.2

Statistical Analysis and Stochastic Modeling

The measured gap width and height were statistically analyzed. The variation in the height of the gaps are statistically insignificant and hence have been treated as uniform throughout the sample with a value of 0.2 mm. The gap width, however, shows significant variation. The variation is more significant in the ½ inch tape. The probability density of the gap width for both the reinforcements are shown in Fig. 4.20. Based on this data, a bimodal distribution seems to be appropriate which is given a, f (x) = p0 g0 (x) + p1 g1 (x) + p2 g2 (x)

(4.1)

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Fig. 4.20 Probability distribution of gap width. a ¼ inch tape, and b ½ inch tape

(a)

20

Density

15

10

5

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Gap Width (mm)

(b)

Density

6

3

0

0.2

0.4

0.6

0.8

1.0

1.2

Gap Width (mm)

The above model represent a combination of one uniform distribution and two normal distribution. The values for each coefficient is given in Table 4.1. The normal Table 4.1 Parameters of the multimodal stochastic model of gap width Parameter

Description

¼ inch tape

½ inch tape

p0

Mixing parameter for gap width 0 mm

0.33

0.20

g0 (x)

Uniform distribution

1

1

p1

Mixing parameter for gap width 0–0.5 mm

0.67

0.76

g1 (x)

Normal distribution

mean = 0.15 σ = 0.05

mean = 0.17 σ = 0.08

p2

Mixing parameter for gap width > 1.0 mm

0

0.04

g2 (x)

Normal distribution

–

μ = 1.11; σ = 0.06

4.5 Generation of Digital Material Twins

89

Define gap position

Define gap periodicity

Define unit cell domain

Define gap orientation

Interpolate varied gap width

Generate AFP model

Define gap height

Define gap width (stochastic)

Export Mesh

Fig. 4.21 Modeling strategy in TexGen

Fig. 4.22 0°/90° oriented TexGen stochastic models

distributions are defined by their mean and standard deviation (σ). The normal distribution for the gap width of the ¼ inch tape shows only one peak with mean around 0.15 mm and hence the coefficient of second normal distribution was 0. The normal distribution for the gap width of the ½ inch tape shows two distinct peaks with mean around 0.15 mm and 1.11 mm. In order to compare the results with the scanned images, models were reconstructed using TexGen software. The TexGen unit cell models were generated based on statistical measurements of tape widths and gap dimensions obtained from micro CT image slices. The model generation steps are shown in Fig. 4.21. The unit cells generated in TexGen are oriented in 0°/90° orinetation, Fig. 4.22. These unit cells were then used in the FlowDict [7] module of GeoDict [5] software and digital flow simulations using Stoke’s equations were used for the in-plane permeability computation.

4.5.3 Ideal Model For modelling the ideal tape placement, the dry tape geometries are modelled as per the manufacturer’s data, the veil and the binder is ignored in this model for simplicity. The ideal tape placement with uniform gaps of 0.2 mm. Figure 4.23a, b show top surface views of the ANSYS models for ¼ inch and ½ inch tapes, respectively.

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(a)

(b)

Fig. 4.23 Ideal models of the carbon AFP tape for through thickness permeability predictions. a Top view of the ¼ inch tape, and b Top view of the ½ inch tape

4.6 Flow Modeling Typically, numerical permeability computations involve modeling the flow of a viscous fluid through a reinforcement, which can be described as Newtonian fluid flow; hence the flow is governed by the conservation laws of fluid dynamics. These governing equations can be solved numerically to compute the permeability of the reinforcement, and to simulate mold filling during an LCM process. The permeability computations are based on steady-state saturated flow.

4.6.1 Micro-Scale Flow Modeling The micro and meso-scale computations are performed separately, as it was computationally expensive to address both meso and micro-scales simultaneously. The micro-scale permeability, defined at the intra-tow level, is computed by assuming the individual fibers in the tows to be parallel rods with different packing arrangements (square, hexagonal, random etc.), and solving for flow both parallel and transverse to the fibers. Alternatively, analytical relations can be used to model the intra-tow flow for calculating the micro-scale permeability. A number of models are available [15–17], of which the most widely used is Gebart’s equation [14] . Gebart’s modelwas used to estimate the micro-scale intra-tow permeability for which the tow fiber volume fraction V f tow is required. The tow fiber volume fraction can be computed from, Vf

tow

=

N f π R2 At

(4.2)

4.6 Flow Modeling

91

(a)

(b)

Fig. 4.24 Image analysis for the permeability computations. a fiber radius calculations, and b horizontal slice of the orthogonal reinforcements showing the cross-section area of the z-binder tows (in red)

where N f is the total number of fibers in a tow, R is the radius of the fiber, and At is the cross-sectional area of the tows. The radius of the fiber was measured using optical microscopy of a polished sample, and the cross-sectional area of the tows was measured through MATLAB image processing as shown in Fig. 4.24. The micro-scale intra-tow permeabilities were found to be in the order of 10−12 to 10−13 m2 , which were approximately two orders of magnitude lower than the experimentally-determined reinforcement permeability values obtained in a previous study [20]. Similar observations were made following investigations on a carbon tape reinforcement [21], 2D woven reinforcements [22] and 3D woven reinforcements [19]. Xiao et el. observed that the permeability of inter-tow gaps accounts for approximately 90% of the through-thickness permeability of orthogonal and angle interlock reinforcements [18]. Hence, in all subsequent computations, the tows were assumed to be solid in nature.

4.6.2 Meso-Scale Flow Modeling For the meso-scale computations, saturated flow of a Newtonian fluid was simulated using the Navier-Stokes equations, which in their steady-state incompressible formulation are presented as partial differential equations, as given below , ∇ ·v =0

(4.3)

ρv(∇.v) = μ∇ 2 v − ∇ p

(4.4)

where v is the velocity vector, ρ is the fluid density, μ is the fluid viscosity and P is the fluid pressure. Equation (4.3) is the mass conservation law and Eq. (4.4) is the conservation law for linear momentum without body forces. These equations can be modified by including dominant terms or ignoring negligible terms. One of

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the important criteria for modifying these equations was a quantitative comparison between the viscous and inertial terms, involving a dimensionless parameter known as the Reynold’s number. At low Reynolds number (less than unity), the viscous forces dominate the inertial forces, resulting in a laminar flow. Such kind of scenarios occurs when the fluid velocities are either very low, the viscosities are very large, or the characteristic lengths of the flow are very small. After incorporating these effects, the governing equation can be rewritten for a steady-state incompressible case given as, ∇ ·v =0

(4.5)

μ∇ 2 v − ∇ p = 0

(4.6)

Equations (4.5) and (4.6) are known as the Stokes equations and are valid for flows with Reynold’s number less than unity. Flow field variables, such as velocity and pressure, can be obtained by numerical solutions of either Eqs. (4.3) and (4.4) or Eqs. (4.5) and (4.6). Within a porous medium, the flow field variables and the permeability are related by Darcy’s law.

4.7 Flow and Boundary Conditions As the unit cells are a representative structure with in-plane periodicity, periodic boundary conditions were applied on the faces normal to the reinforcement plane. In addition to periodic boundary condition, a pressure differential of 100 kPa was also defined across the faces normal to the flow. The upper and lower faces of the unit cells were treated as non-slip impermeable walls for the in-plane flow. A no-slip boundary condition was applied at the tow surfaces, which imposes zero-velocity condition within the tows. Details of the boundary conditions are given in Fig. 4.25.

4.8 Permeability Computation To compute the numerical permeability, Darcy’s law needs to be invoked. In principle, Darcy’s law relates the net volume flow rate of a given fluid through a porous medium of specified cross-sectional area. The flow is driven by an applied pressure difference across the length of the medium in a particular direction. By defining a superficial velocity, the Darcy’s law can be written as q=−

K ∇P μ

(4.7)

4.8 Permeability Computation

93

Inlet-Outlet Faces: PBC with ΔP

Flow

Transverse Tangentail Faces: No-Slip Wall

Z X Z

Interface: No-Slip Wall

X Y

(a)

Tangentail Faces: PBC

(b)

Y

Interface: No-Slip Wall

Inlet-Outlet Faces: PBC with ΔP

In-plane Tangentail Faces: PBC

Fig. 4.25 Boundary conditions for the flow in a through-thickness, and b in-plane directions

where q is the superficial velocity, ∇ P is the pressure gradient, and K is the permeability. The superficial velocity is not the velocity at which the fluid travels through the pores rather it’s related to the volume averaged fluid velocity v¯ factored by the porosity, ϕ = 1 − V f given as, v¯ =

q ϕ

(4.8)

The flux is divided by the porosity (inter-yarn gaps in the current case) to account for the fact that only a fraction of the total volume was available for flow. In the computation of permeability using Darcy’s law, the volume averaged fluid velocity was obtained from numerical simulations based on conservation laws of fluid dynamics under a known pressure gradient and fluid viscosity.

4.9 Summary This chapter gives detailed description of the hybrid characterization framework. Specific details about the hardware accessories and software packages used in the virtual permeability predictions are presented. All the three modeling approaches i.e. realistic, stochastic and ideal, are used in different applications. The micro-scale flow is ignored and the meso-scale flow is modeled as Stoke’s flow.

References 1. GE Measurement and Control Phoenix Datos|x, (2019). https://www.industrial.ai/inspectiontechnologies/radiography-ct/x-ray-computed-tomography/phoenix-datosx. (Accessed 27 Nov 2019)

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2. DEBEN, MICROTEST (2019) https://deben.co.uk/. (Accessed 27 Nov 2019) 3. Volume Graphics, VGStudio Max (2019) https://www.volumegraphics.com/en/products/vgs tudio-max.html. (Accessed 27 Nov 2019) 4. Ali M, Umer R, Khan K et al (2018) Non-destructive evaluation of through-thickness permeability in 3D woven fabrics for composite fan blade applications. Aerosp Sci Technol 82–83:520–533 5. Math2Market GmbH, GeoDict (2019) https://www.math2market.com/. (Accessed 27 Nov 2019) 6. Math2Market GmbH, ImportGeo (2019) https://www.math2market.com/Modules/Interfaces/ ImportGeo-Vol.php. (Accessed 27 Nov 2019) 7. Math2Market GmbH, FlowDict (2019) https://www.math2market.com/Modules/Dicts/FlowDi ct.php. (Accessed 27 Nov 2019) 8. Wiegmann A, Glatt E (2018) FlowDict user guide, release 2018, Math2Market GmbH 9. Wiegmann A, Bube KP (2000) The explicit-jump immersed interface method: finite difference methods for PDEs with piecewise smooth solutions. SIAM J Num Anal 37(3):827–862 10. Wiegmann A (2007) Computation of the permeability of porous materials from their microstructure by FFF-Stokes. Kaiserslautern, Fraunhofer ITWM, pp 1–24 11. Linden S, Wiegmann A, Hagen H (2015) The LIR space partitioning system applied to the stokes equations. Graph Models 82:58–66 12. Math2Market GmbH, PoroDict (2019) https://www.math2market.com/Modules/Dicts/PoroDi ct.php. (Accessed 27 Nov 2019) 13. Wiegmann A, Glatt E (2018) PoroDict user guide, release 2018, Math2Market GmbH 14. Gebart B (1992) Permeability of unidirectional reinforcements for RTM. J Compos Mater 26(8):1100–1133 15. Karaki M,Hallal A et al (2017) A comparative analytical, numerical and experimental analysis of the microscopic permeability of fiber bundles in composite materials. Int J Compos Mater 7(3):82–102 16. Zarandi MAF, Arroyo S, Pillai KM (2019) Longitudinal and transverse flows in fiber tows: evaluation of theoretical permeabilitymodels through numerical predictions and experimental measurements. Compos A Appl Sci Manuf 119:73–87 17. Carman PC (1937) Fluid flow through granular beds. Trans Inst Chem Eng 75:150–166 18. Xiao X, Endruweit A et al (2015) Through-thickness permeability study of orthogonal and angle-interlock woven fabrics. J Mater Sci 50(3):1257–1266 19. Zeng X, Brown LP et al (2014) Geometrical modelling of 3D woven reinforcements for polymer composites: prediction of fabric permeability and composite mechanical properties. Compos A Appl Sci Manuf 56:150–160 20. Alhussein H, Umer R, Rao S et al (2016) Characterization of 3D woven reinforcements for liquid composite molding processes. J Mater Sci 51(6):3277–3288 21. Aziz AR, Ali MA, Zeng X et al (2017) Transverse permeability of dry fiber preforms manufactured by automated fiber placement. Compos Sci Technol 152:57-67 22. Swery EE, Meier R, Lomov SV et al (2016) Predicting permeability based on flow simulations and textile modelling techniques: comparison with experimental values and verification of FlowTex solver using ansys CFX. J Compos Mater 50(5):601-615

Chapter 5

Reinforcement Compaction Response

This chapter discusses the compaction response of the test fabrics in terms of the relaxation curve, tow deformations and internal geometrical changes. The data for 3D orthogonal and angle inter-lock reinforcements are presented under multistage compaction. The data for 3D orthogonal and layer-to-layer reinforcements are presented under multi-cycle compaction. Tow deformations, inter-tow gaps and geometric variability within the fabric meso-structure extracted from the scanned images are also presented.

5.1 Stress Relaxation The data for 3D orthogonal and angle inter-lock reinforcements are presented under multi-stage compaction. The data for 3D orthogonal and layer-to-layer reinforcements are presented under multi-cycle compaction.

5.1.1 Multi-stage Compaction In the multi-stage compaction tests, the reinforcements were initially scanned in their pristine states. This state is referred to as h0 . Subsequently, they were subjected to three stages of compaction (h1 –h3 ). During the compaction, once the target thickness was obtained, the crosshead was held stationary to maintain a constant thickness. Stress relaxation was allowed for 30 min before the compression cycle for next thickness was initiated.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 M. A. Ali et al., CT Scan Generated Material Twins for Composites Manufacturing in Industry 4.0, https://doi.org/10.1007/978-981-15-8021-5_5

95

96

5 Reinforcement Compaction Response

4.0

3.5

Vf = 58.3% (h1) Vf = 67.1% (h3)

3.0 Vf = 61.6% (h2)

2.5

(a)

1.5

1800

3600

5400

7200

Time (s)

h2

1.0

(b)

h1

h0

0.5 0.0

0

h3

2.0

Stress (MPa)

Thickness (mm)

2.5 Vf = 47.4% (h0)

0

1800

3600

5400

7200

Time (s)

Fig. 5.1 Compaction data of 3D angle interlock reinforcement, a thickness vs. time and b stress vs. time

5.1.1.1

Compaction Response of 3D Orthogonal Reinforcement

The multistage stress relaxation curve for the 3D orthogonal reinforcement is shown in Fig. 5.1a, b. After each loading increment, the stress drops instantaneously, and then relaxes until a nominally-constant value of stress is observed. The higher stiffness of the orthogonal reinforcement is a direct result of the z-binder yarn extending vertically through the thickness of the reinforcement. The compaction mechanism in the fibrous reinforcement begins with a process of gap reduction between the weft and warp tows. Once the inter-tow gap has reduced, the tows begin to flatten. This flattening continues until individual fibers are in contact with neighboring tows.

5.1.1.2

Compaction Response of 3D Angle Inter-lock Reinforcement

During the multi-thickness in situ micro CT compaction test on the 3D angle interlock reinforcement, the time history of the cavity thickness and the load (converted to stress) on the reinforcement were recorded, and is shown in Fig. 5.2a, b. The thicknesses at each compaction level were also measured through micro CT images. 1.5 Vf = 32.77% (h0)

4.5 Vf = 50.60% (h1)

3.5

2.5

(a)

Vf = 56.40% (h3)

Vf = 53.34% (h2)

0

1800

3600

Time (s)

Stress (MPa)

Thickness (mm)

5.5

7200

h2

h1

0.5 h0

0.0 5400

h3

1.0

(b)

0

1800

3600

5400

7200

Time (s)

Fig. 5.2 Compaction data of 3D angle interlock reinforcement, a thickness vs. time and b stress vs. time

5.1 Stress Relaxation

97

Comparing both measurements, it was noticed that the thickness obtained at each fiber volume fraction were different from the target thickness, with a compliance factor of 5–10%. The highest error was observed at the highest fiber volume fraction. The thickness measurements obtained from the fixture were corrected as per the micro CT images at all the thicknesses and the Vf were calculated using the corrected thickness values. However, the load data obtained were accepted as is. It can be observed that the applied stress increased until the required thickness is achieved. Once the target thickness was obtained, the crosshead was held stationary to maintain a constant thickness. As with a typical relaxation curve, the applied stress reaches a peak value during each step and then drops rapidly. Stress relaxation was allowed for 30 min before the compression cycle for next thickness was initiated.

5.1.2 Multi-cycle Compaction During the multi-cycle compaction, the reinforcements were subjected to 15 cycles of compaction to a target fiber volume fraction of 60% without any relaxation. After the 15th cycle, the test samples were held at the target thickness and scanned. The rate of loading and unloading was maintained at 0.5 mm/min in all scenarios. The load versus fiber volume fraction data for both the orthogonal and layer-to-layer reinforcements for the first and last three cycles is shown in Fig. 5.3a, b. In Fig. 5.3, the improvement in the volume fraction has been highlighted. It is clear that permanent deformation after cyclic compaction is much greater than following single-cycle compaction, the peak stress during the final cycle being much less than that during the first cycle. As the loading and unloading cycle’s progress, the reinforcements reach a point at which there is no significant difference in peak stresses and the cycles become repeatable. In the pristine state both reinforcements have similar fiber volume fractions. Once the reinforcements are allowed to fully recover after the multi-cycle compaction, the orthogonal reinforcement almost recovered to the initial fiber volume fraction with less than 1% difference. The layer-to-layer reinforcement, on the other hand, did not recover fully and the reinforcement achieved 4% increment in the fiber volume fraction. The difference in the behavior is clearly because of the z-binder in the orthogonal reinforcement.

5.2 Tow Deformations One of the key advantages of images obtained using a micro MICRO CT is that they provide detailed 3D geometrical features of the reinforcement. Using these images, tow deformations, inter-tow gaps and geometric variability can be assessed and quantified.

5 Reinforcement Compaction Response

1.5 1.0

46.01% Vf 54.68% Vf

0.5

Stress (MPa)

1.5

CYC 1 CYC 2 CYC 3 CYC 13-15

2.0

~ 50 %

Stress (MPa)

2.5

1.0

45.77 % Vf 55.22 % Vf

0.5

100 kPa

100 kPa

0.0 0.40

CYC 1 CYC 2 CYC 3 CYC 13-15

~ 50 %

98

0.45

0.50

0.55

0.0 0.40

0.60

Vf

0.45

0.50

0.55

0.60

Vf

(a)

(b)

2.5

Orthogonal

Stress (MPa)

2.0

Layer-to-Layer

1.5 1.0 0.5 0.0 0 1 2

(c)

3

4

5 6

7

8

9 10 11 12 13 14 15

No of Cylces (N)

Fig. 5.3 a–b Stress vs. fiber volume fraction of the orthogonal and layer-to-layer reinforcements respectively and c peak stress vs. number of cycles of both reinforcements

5.2.1 3D Orthogonal Reinforcement The local tow deformations, as well as gaps in between the tows, are strongly influenced by the presence of the z-binder. Figure 5.4 shows cross-sections of locations where the z-binder is absent and present. Initially, at h0 i.e. without any compaction, the weft and warp tows are mostly in contact with each other at places away from the z-binder in both the reinforcements. In the region where the z-binder passes, the warp tow is absent. The evolution of the z-binder yarn as the reinforcement compacts and its impact on nearby weft tows is shown in Fig. 5.5. The weft tows flatten to the point where they are restricted by the z-binder yarn, which deforms into an “S” shape at increasing compaction levels. The weft tows present in the open-end region of the z-binder yarn are squeezed and thickened by the buckling effect of z-binder yarn whereas, the weft tows lying in the closed-end region (just below the loop) are allowed to expand freely as the compaction forces increase. The tows in the center somewhat retain their shapes with minor elongation. The flattening of warp and weft tows are

5.2 Tow Deformations

99

Fig. 5.4 Tow cross-sections taken at different locations at h0 for the orthogonal reinforcement

quantified in terms of changes in the width and height (aspect ratio) of the tow crosssection and presented in Fig. 5.6. The width and height were measured manually from 10 slices. The aspect ratio of the tow cross-section increased steadily for the orthogonal reinforcement. The variability in the aspect ratio of the tow cross-sections quantified in terms of percentage standard deviation was 15% for the 3D orthogonal reinforcement.

5.2.2 3D Angle Inter-lock Reinforcement Similar to the orthogonal reinforcement, the z-binder yarn in angle interlock reinforcement also influences the deformation behavior of neighboring tows and causes nesting where the angle of the yarn aligns with the horizontal tows. The cross-sections of various neighborhoods, taken at different locations at h0 for the angle inter-lock reinforcement, are shown in Fig. 5.7. The weft tows situated away from the z-binder yarn expand freely and nest with neighboring tows. The shape of z-binder yarn in the angle interlock reinforcements becomes like “steps” navigating through the thickness.

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5 Reinforcement Compaction Response

Fig. 5.5 Evolution of z-binder yarn in 3D reinforcements at different compaction stages

12

Fig. 5.6 Changes in the aspect ratio of tow cross-sections at each compaction level Aspect Ratio

10 8 6 4 2 0 h0

h1

h2

h3

Compaction Levels

It was noted that the tows in the angle interlock reinforcement were elliptical and thick just before compaction. The deformation of the tows, the evolution of the z-binder yarn as the reinforcement compacts, and its impact on nearby weft tows are shown in Fig. 5.8a, b. The z-binder yarns deform into a “stair” type structure navigating through the thickness. The warp tows are flattened because of the compaction

5.2 Tow Deformations

101

Fig. 5.7 Tow cross-section at different locations at h0 for the angle inter-lock reinforcement

force, becoming more elongated and rectangular. The weft tows also show very similar behavior. In addition to flattening, the weft tows also adjust themselves under the influence of the z-binder, causing a nesting effect, where the angle of yarns aligns with horizontal tows. The flattening of the warp and weft tows are quantified in terms of changes in the width and height (aspect ratio) and is presented in Fig. 5.9. The width and height were measured manually from 10 slices. There is an exponential increase in the tow aspect ratio in the angle interlock reinforcements. The variability in the aspect ratio of the tows quantified in terms of percentage standard deviation 30% for the angle interlock reinforcement at almost all compaction levels.

5.3 Gap Analysis of the 3D Reinforcements 5.3.1 Gap Analysis of 3D Orthogonal Reinforcement The overall volumetric gap reduction in the reinforcement is presented in Fig. 5.10, which shows a linear drop in the inter tow gap sizes as the V f increases. In the

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5 Reinforcement Compaction Response

Fig. 5.8 Cross-sectional images of tow deformations and gap reductions in a warp and b weft directions respectively at all compaction levels

25

Tow Aspect Ratio

Fig. 5.9 Aspect ratio of tow cross-sections at all compaction levels

20 15 10 5 0

h0

h1

h2

h3

Compaction Levels

initial compaction stage, the percentage of inter tow gaps reduced by 37% as the V f is reduced from 47.4% to 58.3%. The subsequent compaction levels show a gap reduction of 17% and 19% from the previous level. These gap reductions are a result of transverse compaction which forces the warp, weft, and z-binder yarns to move closer together as well as elongate the yarns in the reinforcement plane, filling the

5.3 Gap Analysis of the 3D Reinforcements

103

15

% Inter Yarn Gaps

Weft Gaps Warp Gaps 10

5

0

h0 T0

(a)

h1 T1

h2 T2

h3 T3

Compaction Levels 15

% Inter Yarn Gaps

Weft Gaps Warp Gaps 10

5

0 0.4 (b)

0.5

0.6

0.7

Vf

Fig. 5.10 Percentages areal inter yarn gaps vs a compaction level, and b Vf

available gaps. This behavior is in contrast to the compaction of the 2D reinforcements where the tow spacing remains unchanged and tow nesting is dominant. Since the channels run along weft and warp directions, it is expected that the principle in-plane directions will also be aligned to these channels. The permeability measurement experiments have also indicated the exact orientation of the principle axes. In order to predict which principle direction will have a higher permeability, we can compare the cross-sectional gaps present in each direction. Figure 5.10 shows that the gap area in the weft direction is consistently greater and in some cases, almost twice the size of the warp direction gaps. Hence, K x x is expected to be well aligned with the weft yarns, this finding was also confirmed by Zeng et al. [1]. The larger gaps are a direct result of z-binder yarns pushing the weft yarns to create larger gaps.

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5 Reinforcement Compaction Response

Following this, the inter yarn gaps along the weft and warp orientations, and their distribution in the thickness direction, were analyzed. Resin flow predominantly occurs through the gaps present between each fiber tow. A further analysis of these gaps yielded insights into the resin flow paths. The images indicate the presence of available channels between the fiber yarns and their varying cross-sections in both the in-plane and through-thickness directions. At all compaction levels, the channels are larger at the top and bottom as compared to the channels in the middle of the reinforcement. This variation in the channel size for the initial thickness (h0 ) is shown in Fig. 5.11a, b. At the initial thickness (h0 ), the larger gaps at the top and bottom are due to bending of the z-binder yarn during the weaving process. At each compaction level, the analysis of the reduction in the gaps is presented in Fig. 5.11. By approximating the gaps as rectangular cross-sections, the percentage of gaps within each layer have been computed by manual measurements. Figure 5.11c, d shows variations in gaps along the through-thickness direction in different layers, and at each compaction level. The figure shows that the gaps in the middle layers are almost half the size of the gaps at those in the top and bottom layers. There is up to 30% variability in the gap size between these layers. The layer-wise variation in gaps

(a)

(b)

L-1

L-1

L-2

L-2

L-3

T0 h0 T1 h1

L-4

L-6

h3 T3

L-5

L-7

L-6

L-8 0.00

T1 h1 h2 T2

L-4

T2 h2 T3 h3

L-5

h0 T0

L-3

L-7 0.05

0.10

0.15

Gap Area (mm2)

(c)

0.20

0.25

0.00

0.05

0.10

0.15

0.20

Gap Area (mm2)

(d)

Fig. 5.11 Inter yarn gaps in the a warp, b weft directions, c, d layer-wise distribution of the area gaps in the respective directions

5.3 Gap Analysis of the 3D Reinforcements

105

is a concern for the quality of the product produced. It can be concluded from this distribution that the resin will flow faster through the top and bottom layers which may result in dry spots in the middle layers, appropriate measures need to be taken during the weaving process to ensure uniformity of the gap sizes in each layer.

5.3.2 Gap Analysis of Angle Inter-lock Reinforcement The percentage reduction in the inter tow areal gaps in each direction is shown in Fig. 5.12. In Fig. 5.12, the areal gaps mean the free spaces in a cross-section taken in that particular direction, i.e. weft areal gaps mean gaps on a slice cut normal to the weft tows. The gaps in the weft direction are due to the un-deformed binder yarn, which keeps the warp and weft tows pushed apart. Once deformed, the z-binder is no more capable of holding the weft tows, and they move relatively freely under transverse loading, blocking the available open channels.

5.4 Pore Network Analysis of the Angle Inter-lock Reinforcement Moreover, to acquire a greater understanding of the internal pore structure of a complex 3D reinforcement for full flow path analysis, granulotmetry and percolation path analyses are also presented. The process of fluid flow and transport through any porous medium at the meso-scale are strongly dependent on its internal morphology,

Areal Gaps (%)

40

Warp Gaps (Kxx) Weft Gaps (Kyy) Tranverse Gaps (Kzz)

30

20

10

0

h0

h1

h2

Compaction Levels Fig. 5.12 Areal gaps at different thicknesses in all directions

h3

106

5 Reinforcement Compaction Response

which includes the size, shape, distribution, and connectivity of pores [2]. Two of the main characteristics of the internal morphology of a porous medium are; the pore size distribution, and the percolation paths. The size distribution gives an idea of the free space available, whereas, the percolation path gives the connectivity of these pores. The pore size distribution can be calculated experimentally [3] as well as by analyzing images of the porous medium using image processing tools. In order to compute the size distribution, binary images are subjected to a series of morphological openings via a procedure known as “granulometry”. In the PoroDict module of the GeoDict® software package, an algorithm calculates the distribution by fitting spheres into the pore volume and determining the diameter of the fitted sphere [4]. The percolation theory can be applied to establish the connectivity of voxels in the binary micro CT model [5, 6]. The technique is also implemented in the PoroDict module and can be used to determine the maximal diameters of spherical particles that can move through the medium and extract the shortest path followed [7].

5.4.1 Pore Size Distribution An in-depth analysis of interaction of the resin flow with porous reinforcement can be achieved by studying the pore-scale properties via statistical means [8]. A pore size distribution can be obtained from the 3D model, reconstructed from the micro CT data and by using granulometry analysis [9–11]. In this method, pore diameters were calculated by fitting spheres into the pore spaces using morphological image processing operations on the stack of images. In order to extract the effect of compaction on the pore size distribution, the geometries of unit cells were analyzed at h1 and h3 . The size distribution and cumulative pore distributions are shown in Fig. 5.13a, b. The plots show the percentage of the pore volume of a given size to the total volume of the pores. Due to compaction, the pore sizes become smaller and the cumulative curve shifts to the left towards smaller pore sizes, Fig. 5.13b. It is very likely that some of the small pores are disappearing in the acquired images, as the tows come into contact with each other. The reduction in number of available large pores (diameter greater than the median diameter of 200 µm) at h1 and h3 was due to compaction as shown in Fig. 5.13c, d.

5.4.2 Path Analysis For resin flow through these pore networks, connectivity of pores was very important which can be investigated by analyzing the percolation paths [12]. For the calculation of percolation paths, the method used in the software determines the maximum diameters of spherical particles that can move through the medium [3]. In addition, the shortest paths of the largest particle were also calculated, and are shown in Fig. 5.14. Figure 5.14 shows 15 percolation paths with maximum pore diameters of

100

% Volume Fraction

Cum. Vol. Fraction (%)

5.4 Pore Network Analysis of the Angle Inter-lock Reinforcement 90th Percentile

h1

80

h3

60 40 20

(a)

0

0

200

400

600

800

Pore Diameter (μm)

1000

40

Small Pores (200 μm)

h1 h3

20

0

(b)

50 150 250 350 450 550 650 750 850 950

Pore Diameter (μm)

Fig. 5.13 Pore size distribution, a cumulative percentage volume fraction vs. pore diameter, b percentage volume fraction vs. pore diameter, c, d pore structures with diameters larger than the median diameter (>200 µm) at h1 and h3 respectively

Fig. 5.14 Percolation paths in two in-plane directions a, b at h1 thickness and c, d at h3 thickness

150 µm and 125 µm at thickness h1 . The mean diameters of these 15 paths were found to be 112 µm at h1 and 80 µm at h3 . The average length of the path that a fluid particle has to travel from one face to other face was around 20–30% more than a unit cell length. This means that there are no direct straight paths available for the

108

5 Reinforcement Compaction Response

given pore size and fluid particle has to travel a distance that was 20–30% more than the length of the unit cell in the flow direction.

5.4.3 Effect of Multi-cycle Compaction on the Pore Size Distribution The gap reduction in the 3D orthogonal and angle inter-lock reinforcements as a result of multi-cycle compaction was quantitatively investigated by analyzing the meso-scale pore structure. The investigation was carried out for both the singlecycle and multi-cycle cases. The results for both the 3D reinforcements are shown as pore size distributions in Fig. 5.15. The pore size distributions are presented in terms of the maximum diameter of a sphere that can be fitted in the pore spaces. The results clearly suggest that larger gaps are present in the single-cycle compaction case as compared to the multi-cycles compaction. Both 3D reinforcements have a median pore diameter of 250 µm under single-cycle compaction. The orthogonal reinforcement under single-cycle compaction has pore diameters as large as 1 mm mostly in the outer most layers. The median diameter is reduced to 150 µm under multi-cycles compaction. The reduction in the pore sizes will have a significant effect on the flow field and permeability of the reinforcements.

5.5 Application of the Hybrid Characterization Framework to AFP Carbon Tape

Fig. 5.15 Pore size distributions after single-cycle and multi-cycle compaction of a the orthogonal and b the layer-to-layer reinforcements

5.5 Application of the Hybrid Characterization Framework to AFP Carbon Tape

109

0.15

Stress (MPa)

¼ inch tape ½ inch tape 0.1

0.05

0 0

1000

2000

3000

4000

5000

Time (s) Fig. 5.16 Stress relaxation curves of the ¼ inch (6.35 mm) and ½ inch (12.7 mm) tapes during in situ compression test

The hybrid characterization framework was also employed to study the microstructure of the carbon tape, specifically the gaps between the tapes. A detailed micro CT analysis of the dry AFP tape compacted inside the micro CT machine was undertaken, highlighting the variability in the gaps produced by the AFP machine. The following section discusses the findings in this study.

5.5.1 Stress Relaxation The specimen had a thickness of 4.8 mm under full vacuum, and the experimental data shown in Fig. 5.16 were obtained by decreasing the thickness from the uncompressed value to 4.8 mm, and then maintaining the thickness at this value during the relaxation process. The first phase starts with a rapid increase in the elastic region with time. The stresses reached a peak of approximately 0.12 MPa for both ¼ inch and ½ inch tape. An abrupt drop in stress occurred and the reinforcements continue to relax at a constant value with increasing time, until there was no further decrease.

5.5.2 Microscopic Analysis The reinforcement microstructure was investigated by examining tape cross-sections under optical microscopy. The cured reinforcement samples were examined under an optical microscope to physically study their microstructure. Figure 5.17a illustrates a typical cross-section micrograph image for the ¼ inch tape reinforcement following resin infusion. It can be observed that the unidirectional fibers form an almost circular

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5 Reinforcement Compaction Response

Fig. 5.17 Cross-sections of the ¼ inch reinforcements after resin infusion, a close-up of two plies, and b close-up of individual fibers

shape for the 90° (out of plane) tape and close to an oval shape as it was placed at 45° angle. A gap due to the veil and binder can be clearly identified as a darker region of approximately 20 µm size in between the 90° and the 45° tapes. Figure 5.17b shows a cross-section image consisting of the 90° fibers analyzed using the ImageJ software. The fibers can be seen to be scattered, with the more compact fibers being packed in a hexagonal arrangement. Measurements yielded an average fiber diameter of 5 µm. By calculating the area of a focus part of the tape, the area of each fiber and the number of fibers, the average fiber volume fraction of the entire individual tape was estimated to be 0.74 in a 0.585 fiber volume fraction reinforcement.

5.5.3 Gap Analysis To further investigate the irregularities in the gap sizes, a pore size distribution study was conducted on the reinforcements via micro CT scan. The re-constructed images of the reinforcements taken from the micro CT machine were visually analyzed in GeoDict. Figure 5.18a, c shows typical 3D micro CT scan images with the total volume of tapes and gaps captured. These images were transferred to GeoDict software, where the software interprets the tapes as a solid section and pores as gaps. Figure 5.18b, d shows the gap analysis images for the ¼ and ½ inch tapes from GeoDict. A closer inspection of Fig. 5.17d image reveals that some gaps are noticeably wider than others. From these images, very fine isolated gaps can also be observed, highlighting the binder material region. From the gap distribution analysis, it was found that the measured % gap of ¼ and ½ inch tapes were approximately 1.5% and 3.5% respectively. Figure 5.19 shows the micro CT image slices of the reinforcements at selected depths, corresponding to plies oriented at +45°, 0°, 90° and −45° to the reference direction. An examination of the figure indicates that the gaps in ¼ inch tape are typically smaller than the anticipated value of 0.2 mm. The average gap widths in the

5.5 Application of the Hybrid Characterization Framework to AFP Carbon Tape

111

Fig. 5.18 GeoDict analysis showing the distribution of pores in a, b ¼ inch tape, and c, d ½ inch tape

+45°, 0°, 90° and −45° layers were 0.14, 0.14, 0.15 and 0.15 mm, respectively. A number of large gaps are apparent in the ½ inch tape, with values as high as 1.0 mm in some cases. In this case, the average values for the gap size in the 45°, 0°, 90° and 45° layers were 0.59, 0.58, 0.25 and 0.26 mm, respectively. However, note that for both reinforcements, some gaps are clearly visible while others are difficult to distinguish.

5.6 Summary The hybrid framework was employed to obtain the compaction response, as well as to investigate the effect of compaction on the internal geometry of the reinforcements. During the multi-stage compaction, the stress drops instantaneously from its peak value, and then relaxes until a nominally-constant value of stress was observed. As a result of the multi-cycle compaction, 5–10% improvement in the fiber volume fraction of 3D reinforcements was observed. During the multi-cycle compaction, peak stress levels decreased significantly during the initial cycles. However, after few cycles, the change in the peak stress became negligible.

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5 Reinforcement Compaction Response

Fig. 5.19 Micro CT images taken under compression of the ¼ and ½ inch tapes at a +45°, b 0°, c 90° and d −45° orientations

5.6 Summary

113

Under compaction, the internal structure of the fibrous reinforcements evolves with a process of gap reduction between the weft and warp tows, and deformation of these tows, including the z-binder. In the 3D reinforcements, the local tow deformations, as well as gaps in between the tows, are strongly influenced by the presence of the binder yarn. Under transverse compaction, the binder yarns in the orthogonal and angle inter-lock deform into an “S” shape and a “stair” type structure respectively.

References 1. Zeng X, Endruweit A, Brown LP, Long AC (2015) Numerical prediction of in-plane permeability for multilayer woven fabrics with manufacture-induced deformation. Compos A Appl Sci Manuf 77:266–274 2. White EV, Fullwood D, Golden KM, Zharov I (2019) Percolation analysis for estimating the maximum size of particles passing through nanosphere membranes. Phys Rev E 99(2):1–9 3. Bonnard B, Causse P, Trochu F (2017) Experimental characterization of the pore size distribution in fibrous reinforcements of composite materials. J Compos Mater 51(27):3807–3818 4. Silin D, Patzek T (2006) Pore space morphology analysis using maximal inscribed spheres. Phys A Stat Mech Appl 371(2):336–360 5. Jarvis N, Larsbo M, Koestel J (2017) Connectivity and percolation of structural pore networks in a cultivated silt loam soil quantified by X-ray tomography. Geoderma 287:71–79 6. Hunt A, Ewing R, Ghanbarian B (2014) Percolation theory for flow in porous media, Springer. ISBN 978-3-319-03771-4 7. Wiegmann A, Glatt E (2018) PoroDict user guide, Release 2018, Math2Market GmbH 8. Baychev TG, Jivkov AP, Rabbani A et al (2019) Reliability of algorithms interpreting topological and geometric properties of porous media for pore network modelling, Transp Porous Media p 271–301 9. Vernet N, Trochu F (2016) Analysis and modeling of 3D Interlock fabric compaction behavior. Compos A Appl Sci Manuf 80:182–193 10. Vernet N, Trochu F (2016) Analysis of mesoscopic pore size in 3D-interlock fabrics and validation of a predictive permeability model. J Reinf Plast Compos 35(6):471–486 11. Vernet N, Trochu F (2016) In-plane and through-thickness permeability models for threedimensional Interlock fabrics. J Compos Mater 50(14):1951–1969 12. Trochu F, Ruiz E, Achim V, Soukane S (2006) Advanced numerical simulation of liquid composite molding for process analysis and optimization. Compos A Appl Sci Manuf 37(6):890–902

Chapter 6

Flow Fields and Reinforcement Permeability

One of the primary aims of the hybrid framework is to provide a single platform for faster compaction and permeability characterization of complex reinforcements, such as 3D reinforcements, with minimal material consumption and labor costs, where conventional methods require tedious and labor -intensive experiments and material consumption. The X-ray images obtained at various compaction levels are converted into computational models, followed by a numerical flow simulation using different unit cells for permeability predictions. The numerical simulation computes the three -dimensional velocity fields within the computational domain due to the applied pressure gradient in a given direction. The velocity field can be visualized to assess the flow paths and is used to compute the permeability. This chapter presents the flow visualizations and the computed virtual permeability values of the 3D reinforcements as well as 2D reinforcements and AFP carbon tape. The flow visualizations are presented in the form of three -dimensional velocity fields. The predicted permeability values are compared against existing experimental results. Variability of the permeability among various unit cells and within the unit cells is also discussed. The permeability data is presented as through-thickness permeability and in-plane permeability. The discussions are primarily focused on the results of the 3D reinforcements and some prominent results from 2D reinforcements and AFP carbon tape are also presented.

6.1 Through-Thickness Permeability of the 3D Reinforcements The through-thickness permeability is crucial in VARTM of large structures, where a resin distribution medium is used on top of the reinforcement to facilitate throughthickness flow. With increasing reinforcement thickness, the influence of throughthickness permeability on LCM processing of composites becomes increasingly © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 M. A. Ali et al., CT Scan Generated Material Twins for Composites Manufacturing in Industry 4.0, https://doi.org/10.1007/978-981-15-8021-5_6

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significant. When resin injection is through the thickness, or in cases when a distribution medium is placed on top of the reinforcement for flow enhancement, such as in SCRIMP, Kzz becomes the dominant material parameter. In this section, the through-thickness permeability values of the orthogonal and angle inter-lock 3D reinforcements, predicted through virtual flow simulations using the FlowDict module of GeoDict software and ANSYS FLUENT, are presented.

6.1.1 Velocity Fields in Through-Thickness Flow The velocity profiles predicted by GeoDict and ANSYS FLUENT software packages, for both types of 3D reinforcements at compaction level (h0 ) are shown in Fig. 6.1. For the sake of clarity, flow fields with very low velocities (almost equal to zero) were “clipped”, and shown in Fig. 6.1c, d. The profiles highlight the presence of a dominant flow around the z-binder yarn as well as in the channels between the in-plane tows. It is very clear from the clipped flow fields that, the resin is forced to follow a tortuous path through the reinforcement because of intricate reinforcement architecture, and there are well -connected flow paths in three dimensions, which may lead to skew terms in the overall permeability tensor [1]. Comparing velocities in Fig. 6.1a, b, the highest velocities in the orthogonal and angle interlock reinforcements were 5.0 × 10−7 m/s and 3.29 × 10−7 m/s respectively. Although the orthogonal reinforcement has a higher peak velocity, the angle interlock reinforcement exhibits a higher volume flow rate, as shown in the clipped flow fields of Fig. 6.1d. As a result, the angle interlock reinforcement has a higher overall flow rate and hence high permeability at this low fiber volume fraction. Figures 6.1e, f show the predicted velocity profiles generated by ANSYS FLUENT at three different cross-section locations in both 3D reinforcements. Similar to GeoDict predictions, the flow field data suggests a dominant flow regime around the binder fibers that extend upwards from the lower surface of the reinforcement. This suggests that the binder fiber is acting as a resin guide, similar to the discrete channels or pinholes in a 2D reinforcement [2]. The flow velocity becomes much higher at locations where the gaps are very small. In comparison with the flow field predicted by GeoDict, the scatter in the velocity distribution predicted by ANSYS FLUENT is much higher. The mean and peak velocities obtained from GeoDict have approximately the same order of magnitude. To comprehend the influence of compaction on the flow path, the flow fields at the compaction level h3 for both reinforcements are shown in Fig. 6.2. By comparing the flow field within the inter-tow gaps of the two reinforcements at compaction levels h0 and h3 , the wider flow channels available at h0 disappear as the thickness was decreased. Although the gaps available were reduced for both reinforcements, the orthogonal reinforcement ensured that the gaps remained connected, thereby providing a continuous flow path. On the other hand, the gaps in the angle interlock

6.1 Through-Thickness Permeability of the 3D Reinforcements

117

Fig. 6.1 Velocity distribution within the unit cells of the orthogonal and angle inter-lock 3D reinforcements at h1 , a, b full 3D velocity field, c, d clipped flow field, and e, f velocity field at different three different cross-sections

reinforcement were separated at some points where the nesting effect was prominent. These nesting regions are clearly seen as empty white spaces in Fig. 6.2f, which are formed because of the fusion of tows to form a larger irregular shaped nested tow that was taken as an impermeable solid region in the flow software.

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Fig. 6.2 Velocity distribution within the unit cells of the orthogonal and angle inter-lock 3D reinforcements at h3 , a, b full 3D velocity field, c, d clipped flow field, and e, f velocity field at different three different cross-sections

6.1.2 Through-Thickness Permeability Predictions The through-thickness permeability (Kzz ) was calculated after a series of steady state flow simulations. The computed values of Kzz for all the reinforcements are plotted as a function of fiber volume fraction in Fig. 6.3, along with the experimentallydetermined values taken from previous experimental studies. The figure indicates that the reduction in Kzz with increasing Vf is significant for both type of reinforcements. This reduction is due to a decrease in the number of connected channels through which the resin can flow within the reinforcement; hence, lower permeability was experienced in the highly compacted reinforcements. The permeability of the angle

6.1 Through-Thickness Permeability of the 3D Reinforcements

119

1.E-10 GeoDict FLUENT Experimental Equation

Kzz (m2)

1.E-11

1.E-12

1.E-13 0.4

0.5

0.6

0.7

Vf

(a) 1.E-09

GeoDict FLUENT Experimental Equation

Kzz (m2)

1.E-10

1.E-11

1.E-12

1.E-13 0.3

(b)

0.4

Vf

0.5

0.6

Fig. 6.3 Comparison between hybrid experimental-numerical (GeoDict) and experimentally measured through-thickness permeability values, a orthogonal and b angle inter-lock reinforcements

interlock reinforcement is lower than orthogonal reinforcement at similar Vf , on average by approximately 40% at 0.6 Vf , due to more nesting of tows. The two different prediction methods give similar results with approximately 25% variations using ANSYS FLUENT and 35% using GeoDict with the average experimental permeability values. From Fig. 6.3, it should be noted that GeoDict slightly over-predicts while ANSYS FLUENT slightly under-predicts the permeability. Although the two numerical approaches yielded very close predictions, the differences between the two numerical computations methods may arise from the fundamental laws used in each case. GeoDict, based on the Stoke’s equation, may have not fully captured the high local velocities at congested locations due to the

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exclusion of inertia terms in the Stoke’s equation. Nevertheless, both simulation results are well within the experimental deviations.

6.2 In-Plane Permeability of the 3D Reinforcements Flow simulations in which the pressure difference is applied across in-plane faces of the unit cell lead to the identification of the two in-plane values of the permeability tensor, Kxx and Kyy , which are necessary for almost all LCM processes where inplane flow predominantly occurs. With similar approach as for the through-thickness, in-plane flow simulations were performed in GeoDict only. The following sections discusses the flow fields in the in-plane flow case.

6.2.1 Velocity Fields in In-Plane Flow The velocity fields obtained as a result of pressure difference across in-plane faces are presented below.

6.2.1.1

3D Orthogonal Reinforcement

The flow field visualizations show that in case of 3D orthogonal reinforcement with binder yarns, once the flow reaches the z-binder yarn, it changes the path to flow around the z-binder. Unlike flow through the thickness where the z-binder yarns act like pathways for the resin [2], the z-binder yarns cause a major obstruction for inplane flow. For flow in weft direction, the gaps in the warp direction also participate in the fluid transport process which can be seen in Fig. 6.4c, d. The obstruction caused by the z-binder yarns can be clearly seen in Fig. 6.4a, b, where the streamlines change the path, navigating around the z-binder yarn. As embedded in Darcy’s law, the flow rate is directly linked to the pressure gradient. Local pressure gradient is linked to the areal porosity. Areal porosity means the percentage of inter-tow spaces at a given cross-section. The obstruction caused by z-binder yarns is also evident from the pressure drop in the flow direction along weft or warp directions. To illustrate this phenomenon, the pressure drop in the weft direction at all compaction levels is shown in Fig. 6.5. The pressure drop is approximately linear in the regime corresponding to the weft channels as shown in Fig. 6.5. The pressure drops abruptly near z-binder yarns and interestingly, the pressure drop is dependent on the shape of the z-binder. In region A, the shape of the z-binder is an “S” shape that produces gaps at the top and bottom layers, allowing the flow to pass through with relatively low resistance. On the other hand, in region B, the z-binder yarn deforms in such a way that the gaps are squeezed, and hence a large pressure drop is observed at that point. Figure 6.5 shows the average pressure drop across

6.2 In-Plane Permeability of the 3D Reinforcements

121

Fig. 6.4 Flow fields showing flow paths through the reinforcement in, a x-direction at h1 , b ydirection at h1 , c x-direction at h3 and d y-direction at h3

the length of the unit cell. The pressure gradient is steep where the areal porosity is low. The pressure drop in other directions does not show any pattern but are still correlated to the areal porosity.

6.2.1.2

3D Angle Inter-lock Reinforcement

Figure 6.6 shows that the resin mainly flows through the channels with some in-plane and transverse interactions. This connectivity is more prominent at initial stages of

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6 Flow Fields and Reinforcement Permeability

Fig. 6.5 Pressure drop across the flow direction along warp orientation of the orthogonal reinforcement at the compaction levels h0 , h1 , h2 and h3

Fig. 6.6 Flow fields showing flow paths through the reinforcement in, a x-direction at h1 , b ydirection at h1 , c x-direction at h3 and d y-direction at h3

the compaction, but the pore connectivity is lost at higher compaction loads. Similar trends can be seen for the streamlines shown in Fig. 6.7a–d. The loss of connectivity between the pores may result in the formation of macroscopic voids in these regions as it would be difficult to fill these regions with resin within a short amount of time. The resin reaches to these regions only in cases where the micro-flow is reasonably high. Increasing the mold filling time will decrease the productivity.

6.2 In-Plane Permeability of the 3D Reinforcements

123

Fig. 6.7 Streamlines showing flow paths through the reinforcement, a x-direction at h1 , b ydirection at h1 , c x-direction at h3 and d y-direction at h3

6.2.2 In-Plane Permeability Predictions 6.2.2.1

3D Orthogonal Reinforcement

Figure 6.8 shows numerically-computed permeability values compared with experimentally measured in-plane permeability, the experimental data are fitted with an exponential curve. The reinforcement permeability values agree well with the experimental data except for values obtained at initial thickness, h0 . The maximum difference between extrapolated permeability values at initial thickness, h0 is around 372% for Kyy . The difference at h0 for Kxx is approximately 98%. For these comparisons, the mean permeability values among all the four unit cells were used. This means that at low fiber volume fractions, the predictions do not agree well with experimental results mainly due to difference in sample size and preparations in accuracies. The micro CT test sample has a diameter of 40 mm, as compared to experimental permeability sample, which has a diameter of 200 mm. The smaller sample was more fragile as compared to the bigger sample. Once compressed to a higher fiber volume fraction, the fiber tows were forced to settle into position, hence better permeability predictions (less than 50% difference) were obtained at subsequent thicknesses. Similar observations were reported by Endruweit et al. [3] in a study on the effect of specimen history on the in-plane permeability of multi-layer 2D woven reinforcements, where higher in-plane permeability was predicted due to material handling and involuntary deformation which were partially reversed by through-thickness reinforcement compression.

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6 Flow Fields and Reinforcement Permeability

Permeability (m2)

Experiment

GeoDict

1.E-10

1.E-11

1.E-12 0.4

0.5

0.6

0.7

Vf

(a)

Permeability

(m2)

Experiment

GeoDict

1.E-10

1.E-11

1.E-12 0.4

(b)

0.5

0.6

0.7

Vf

Fig. 6.8 Comparison between hybrid experimental-numerical (GeoDict) and experimentally measured in-plane permeability values, a Kxx and b Kyy

6.2.2.2

3D Angle Inter-lock Reinforcement

The permeability values are compared with experimental data and are shown in Fig. 6.9. For this comparison, the experimental data are fitted with an exponential curve. For these comparisons, a mean permeability value between four unit cells was used. Overall, the predicted permeability values were in good agreement with the values obtained via experimental measurements. The permeability at h1 was found to be one order magnitude higher than experimental value for in-plane flow. This means that at low fiber volume fractions, the predicted results did not match well

6.2 In-Plane Permeability of the 3D Reinforcements

125

1.E-10 Hybrid

Kxx (m2)

Exper.

1.E-11 0.4

0.5

0.6

0.7

Vf

(a) 1.E-10

Hybrid

Kyy (m2)

Exper.

1.E-11 0.4

(b)

0.5

0.6

0.7

Vf

Fig. 6.9 Comparison between hybrid experimental-numerical (GeoDict) and experimentally measured in-plane permeability values of the angle inter-lock reinforcement, a Kxx and b Kyy

with the experimental measurements, this being due to different sample sizes used and cutting inaccuracies. The micro CT test sample used had a diameter of 40 mm, whereas the experimental permeability sample diameter was 200 mm. The smaller sample was more fragile compared to larger sample. Once the small sample was compressed at a higher fiber volume fraction, the fiber tows were forced to settle into positions, hence improved in-plane permeability. Moreover, at higher compaction levels, the fiber tows become denser with a higher tow volume fraction. At such tow volume fractions, the tows are practically impermeable and the assumption of negligible micro-flow becomes more reasonable. So, as the tow becomes denser at high Vf , the predicted permeability values match the experimental values. On the

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6 Flow Fields and Reinforcement Permeability

other hand, at low Vf , the tows are not sufficiently dense and the role of micro-flow is not negligible.

6.2.2.3

Deviation of In-Plane Permeability Predictions at Low Vf

For both the orthogonal and angle inter-lock 3D reinforcements, at low fiber volume fractions, the predicted results did not match well with the experimental measurements. This deviation of the predicted permeability values from the measured ones is primarily due to different sample sizes used and preparation inaccuracies. The micro CT test sample used had a diameter of 40 mm, whereas the experimental permeability sample diameter was 200 mm [4]. The smaller sample was more fragile compared to larger sample. Once the small sample was compressed at a higher fiber volume fraction, the fiber tows were forced to settle into positions, hence improved in-plane permeability [3]. Moreover, at higher compaction levels, the fiber tows become denser with a higher tow volume fraction. At such tow volume fractions, the tows are practically impermeable and the assumption of negligible micro-flow becomes more reasonable. So, as the tow becomes denser at high Vf , the predicted permeability values match the experimental values. On the other hand, at low Vf , the tows are not sufficiently dense and the role of micro-flow is not negligible. These effects were extensively studied by Syerko et al. [5] and Salvatori et al. [6]. Syerko et al. [5] proposed a scale separation criterion for in-plane flow in terms of the ratio between the micro-scale permeability and the characteristic meso-pore size to decide when the microscopic intra-yarn flow can be neglected compared to the mesoscopic inter-yarn flow. According to this criterion, when the fiber tows become very dense, especially at higher global fiber volume fractions, it becomes more reasonable to ignore the micro-scale flow. Salvatori et al. [6] established that the permeability for fibrous reinforcements is dominated by viscous flow along the meso-channels for a broad range of capillary numbers. As a result, provided that the capillary number exceeds a threshold value, the permeability can be estimated by neglecting microflow and capillary effects. However, the through thickness permeability values were in excellent agreement with experimental results.

6.3 Analysis of Variability in Permeability Predictions The permeability of composite reinforcements is not deterministic rather a stochastic value with significant scatter in both experimental measurements and numerical predictions. This scatter is primarily a result of inherent geometrical variability in the reinforcements at all the scales (micro, meso, and macro). Through the hybrid framework, the macro scale variability can be studied by studying multiple unit cells taken from different locations in the test sample. The meso-scale variability can be investigated by looking at the flow patterns and internal structure of a given unit cell.

6.3 Analysis of Variability in Permeability Predictions

127

6.3.1 Variability Among Unit Cells The purpose of using multiple unit cells was to study the spatial variability of permeability. The unit cells were extracted from different locations in the reinforcement sample; hence each unit cell has a slightly different solid volume fraction (Vs ). The value of Vs for each unit cell was obtained from the segmentation procedure of the micro CT images. To study the variability in reinforcement permeability throughout the reinforcement, four different unit cells were analyzed for both the orthogonal and angle inter-lock 3D reinforcements. Figure 6.10 presents the predicted values of Kxx and Kyy for each unit cell at four compaction levels (h0 - h3 ), along with their standard deviations for the orthogonal 3D reinforcement. Each unit cell has a different solid volume fraction (Vs ), which is the ratio of the volume of the solid tows to the total volume. The numerical value of Vs for each unit cell was obtained from the segmentation of the micro CT images. The overall variation in permeability among the unit cells was around 25%, with high variations at low fiber volume fractions. Straumit et al. [7] correlated the variation of the permeability in different unit cells to their Vs . Although Vs varies less than 3% in the current case, the 25% variation was in addition to the influence of Vs and can be attributed to geometrical variability causing flow path discontinuities within the unit cells. A similar trend was also found for the angle inter-lock reinforcement. Four unit cells were also used in this case. The Vs values for each unit cell at all thicknesses are given in Table 6.1. As expected, the Vs increased as the fiber volume fraction increases. The lower Vs for unit cell 1 at h3 (Vs = 0.864) as compared to h2 (Vs = 0.863) was possibly due to local in-plane shifting of tows in the location from where unit cell 1 was extracted. The movement of tows may have increased the gaps in h0 1.E-10

Permeability (m2)

Kxx K11 Kyy K22

h1 h2

h3 1.E-11

0.91 0.92 0.92 0.92

0.89 0.89 0.90 0.91

0.87 0.87 0.87 0.90

0.77 0.79 0.83 0.83

1.E-12

Solid Volume Fraction (Vs) Fig. 6.10 Variation of predicted permeability values among various unit cells of the orthogonal 3D reinforcement

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6 Flow Fields and Reinforcement Permeability

Table 6.1 Solid volume fraction of the unit cells of the angle inter-lock reinforcement

90

Unit cell

Vs h1

h2

h3

1

0.853

0.864

0.863

2

0.829

0.868

0.892

3

0.841

0.868

0.887

4

0.838

0.874

0.878

5 (Mean)

0.840

0.869

0.880

Standard deviation (%)

0.87

0.34

1.09

1

75

25

2 5

30

4

Kxx (×10-12 m2)

3

(b)

4

Kyy (×10-12 m2)

h1 h2 h3

1

10

2 5

2

0

0

0

(a)

20

50

60 5

1

3

(c)

4

Kzz (×10-12 m2)

3

Fig. 6.11 Permeability variation among different unit cells of the angle inter-lock reinforcement a Kxx , b Kyy and c Kzz

the neighborhood and hence lower Vs was obtained. At a given thickness, the solid volume fraction among unit cells varied by less than 2%. The variability can be seen in Fig. 6.11 which presents the predicted values of Kxx , Kyy , and Kzz for each unit cell at three compaction levels (h1 -h3 ) in the form of a radar chart. The chart is used to visualize the statistical variability, identify clusters and outliers in the data. A data set with no variability will assume the exact shape of the polygon. In the radar charts in Fig. 6.11, the first four radii (spokes or corners) represent unit cells and the fifth radius being their mean value. The permeability values are plotted along each radius. The coefficients of variance were 8%, 15% and 23% for Kxx , Kyy and Kzz respectively. The other cause of variation was due to the effect of geometrical variability causing flow path discontinuities within the unit cells [8]. Since the unit cells were extracted from different locations of the test sample, the solid volume fraction of each unit cell was different even at the same Vf . The variation of permeability among the unit cells, as a function of Vs , has been investigated via a modified Kozeny-Carman correlation [9], which is discussed in Sect. 6.5.

6.3 Analysis of Variability in Permeability Predictions

129

6.3.2 Variability Within Unit Cells The permeability values presented in the previous sections are based on the average volume flow rate through the unit cell. A close examination of the geometry of the unit cell reveals that within a unit cell, there are regions of dominant or high flow as well as regions with a low volume flow rate as depicted in Fig. 6.12. This was also supported from the flow fields and pressure drop presented in the previous flow analysis section. Based on these observations, it can be inferred that within a unit cell, permeability and flow field has some local variations. In the vicinity of the z-binder yarn, the flow rate seems to be low and hence, a low permeability region around its neighborhood was observed. Having more gaps in the top and bottom layers, we expect higher permeability in those layers, this presents a very similar case of race tracking effect in RTM or having distribution medium in a SCRIMP process. With the help of textile manufacturers, this variation in gaps needs to be addressed and further investigated as this may lead to the formation of macro voids in the middle layers of the reinforcement.

6.4 Effect of Unit Cell Size on Permeability Predictions The virtual permeability predictions are also likely to depend on the dimensions of the unit cell. In practice, a single unit cell is sufficient for numerical computation using periodic boundary conditions. However, computational models consist of unit cell repetition of 2, 3, and 4 were extracted from the model, as shown in Fig. 6.13a. For each case, simulations were completed using the same properties of the fluid, boundary conditions, and pressure gradient. From Fig. 6.13b, we can observe that

Fig. 6.12 Variation of flow field with in a unit cell of the orthogonal reinforcement, a In-plane variations and b through thickness variations

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6 Flow Fields and Reinforcement Permeability

(a) 1.E-11

Permeability (m2)

Kzz Experimental Value

1.E-12

(b)

1

2

3

4

No of Unit Cells

Fig. 6.13 Through-thickness permeability computation using repeated unit cells, a repetition of unit cells in different order, b permeability values based on different adjacent unit cells

the use of multiple unit cells did not affect the accuracy of the computed results. The predicted permeability values varied among themselves but were within the experimental variations (gray-shaded area). The scatter in the prediction was approximately 19% for the single unit cell, and 17%, 16%, and 13% for 2, 3, and 4 adjacent unit cells respectively. The number of iterations required for convergence increased linearly as the number of unit cells was increased. This shows that a single representative unit cell can be used effectively for through -thickness permeability computations.

6.5 Kozeny-Carman Analysis

131

6.5 Kozeny-Carman Analysis Predictive models for permeability, such as Gebart’s [5], Kozeny-Carman [6], Bruschke and Advani [7] etc. relate permeability to the fiber volume fraction (Vf ). Here, the Kozeny-Carman analytical model is being adopted to relate the permeability to the Vf and expressed as, ⎡

⎤ ⎡ ⎤ Kxx C x x 1 − V 3 f ⎣ K yy ⎦ = ⎣ C yy ⎦ V f2 K zz C zz

(6.1)

where, C x x , C yy and C zz are coefficients determined by curve-fitting of available data through regression analysis. The values of the coefficients are given in Table 6.2 for each case, along with the degree of fitness in terms of the correlation coefficient, R 2 . The resulting estimation using the Kozeny-Carman relation is shown in Figure 6.14. The experimental data do not fit well with this type of approximation, giving R 2 value of 0.850 and 0.765 for K x x and K yy respectively. However, the hybrid experimental-numerical data for both K x x and K yy fit very well in the Kozeny-Carman approximation, with R 2 value of 0.995 and 0.902 for K x x and K yy respectively. In the numerical permeability prediction presented here, the fiber tows were assumed to be solid. Since the unit cells were extracted from different locations in the test sample, the solid volume fraction of each unit cell can be different even at the same Vf . In this case, we can modify the above relationship and replace Vf by Vs , which is given as, ⎡

⎤ ⎡ ⎤ Kxx Cx x 3 ⎣ K yy ⎦ = ⎣ C yy ⎦ (1 − Vs ) Vs2 K zz C zz

(6.2)

For the numerical data of the orthogonal reinforcement, a simple least square regression yields the values of C x x and C yy to be 1.32 × 10−8 and 8.50 × 10−9 respectively. Using these coefficients, the predictions are shown in Fig. 6.15 which represents an excellent fit to the permeability values obtained from different unit cells. Table 6.2 Values of parameters in Kozeny-Carman model for the orthogonal reinforcement Orientation

K x x (Weft)

Parameters

Cx x

R2

C yy

K yy (Warp) R2

Experimental Data

1.29E-10

0.850

2.96E-11

0.765

GeoDict Data with Vf

2.55E-10

0.995

1.28E-10

0.902

GeoDict Data with Vs

1.34E-08

0.935

7.19E-09

0.989

6 Flow Fields and Reinforcement Permeability

Permeability (m2)

132

Experiment Numerical K-C - Experi. K-C - Numeri.

1.E-10

1.E-11

1.E-12 0.4

0.5

Permeability (m2)

(a)

Vf

0.6

0.7

Experiment Numerical K-C - Experi. K-C - Numeri.

1.E-10

1.E-11

1.E-12

(b)

0.4

0.5

0.6

0.7

Vf

Fig. 6.14 Kozeny-Carman approximations for numerical permeability values for different unit cells a Kxx and b Kyy

Similarly, for the numerical data of the angle inter-lock reinforcement, a simple least square regression yields the values of the coefficients C x x , C yy and C zz to be 1.32 × 10−8 , 8.50 × 10−9 and 2.09 × 10−9 respectively. Using these coefficients, K x x and K yy values given by the Kozeny-Carman approximation fit very well with R 2 values of 0.939, and 0.969 respectively. However, the predictions fit poorly for K zz , with the R 2 value of being 0.695. Using these coefficients, the predictions are shown in Fig. 6.16 which represents an excellent fit to the permeability values obtained from different unit cells. The Kozeny-Carman model predicts the permeability very well for a short range of fiber volume fractions. This is useful when the model is used to predict the permeability of various unit cells with the same fiber volume fraction. However, this require

6.5 Kozeny-Carman Analysis

133

1.E-09

Permeability (m2)

Numerical K-C 1.E-10

1.E-11

1.E-12

(a)

0.7

0.8

0.9

Vs

Permeability (m2)

1.E-09

K-C 1.E-10

1.E-11

1.E-12

(b)

Numerical

0.7

0.8

0.9

Vs

Fig. 6.15 Kozeny-Carman approximations for numerical permeability values for different unit cells of the orthogonal reinforcement a Kxx and b Kyy

existing data, from either experimental measurements or numerical simulations, for determining the coefficients through curve fitting.

6.6 Effect of Cyclic Compaction on Permeability Vacuum -assisted resin transfer infusion is a type of liquid composite molding, during which the maximum load that can be applied on the reinforcement is only 1 atm. Due to the high stiffness of the 3D reinforcements, achieving high fiber volume fractions,

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6 Flow Fields and Reinforcement Permeability

Kxx (m2)

Predicted Values (m2)

1.E-10

1.E-11 0.80

Hybrid Kozeny-Carman

0.85

0.90

(a)

0.95

8.E-11 6.E-11 4.E-11 2.E-11 2.E-11

1.E-11 0.80

(b)

0.85

0.90

Predicted Values (m2)

Kyy (m2)

Hybrid Kozeny-Carman

0.95

1.E-11 1.E-12 1.E-13 0.80

0.85

0.90

Vs

0.95

8.E-11

1.E-10

5.E-11 3.E-11 1.E-11 1.E-11

3.E-11

5.E-11

Original Values Predicted Values (m2)

Kzz (m2)

Hybrid Kozeny-Carman

6.E-11

7.E-11

Vs 1.E-10

4.E-11

Original Values (m2)

Vs 1.E-10

(c)

1.E-10

7.E-11

(m2)

1.9E-11 1.3E-11 7.0E-12 1.0E-12 1.0E-12

7.0E-12

1.3E-11

Original Values

1.9E-11

(m2)

Fig. 6.16 Kozeny-Carman approximations for numerical permeability values for different unit cells of the angle inter-lock reinforcement a K x x b K yy and c K zz

as desired in the aerospace industry, is challenging. The fiber volume fractions for 3D reinforcements can be improved by multi-cycle compaction of reinforcement prior to infusion. This improvement has been established by investigating the effect of cyclic loading on the compaction behavior, internal structure, and permeability of two 3D reinforcements using the hybrid framework. The four loading conditions are illustrated in Fig. 6.17. Prior to any loading, the reinforcements were scanned initially in its as-received pristine state (hi ). The reinforcements were then subjected to 15 cycles of compaction to a target fiber volume fraction of 60% without any relaxation. After the 15th cycle, the test samples were held at the target thickness (hf ) and scanned. The load was then removed and the reinforcement allowed to fully spring back (h0a ). Finally, the same test samples were compacted under 1 atm states to simulate vacuum bag conditions. The test samples were held at the thickness (h1a )

6.6 Effect of Cyclic Compaction on Permeability

Spring-back State (h0a)

Thickness

Pristine State (hi)

135

Relaxed @ 1 atm (h1a) Target Thickness

N = 15

Relaxed State @ Target Thickness (hf)

Time Fig. 6.17 Illustration of the cyclic loading conditions

achieved under a pressure of 1 atm and was allowed to fully relax. The rate of loading and unloading was kept to 0.5 mm/min in all scenarios.

6.6.1 Velocity Fields The flow paths do not appear to have been significantly affected by cyclic compaction. The flow patterns in the orthogonal reinforcement are very similar to the multistage compaction. The in-plane flow is dominantly in the inter-tow channels and is obstructed by the z-binder, Fig. 6.18a, b. The through-thickness flow is however guided by the z-binder, Fig. 6.18c. A similar pattern was also seen in the flow through the layer-to-layer reinforcement. As embedded in Darcy’s law, flow rate is directly linked to the pressure gradient. Local pressure gradient is linked to the areal porosity. By areal porosity, we mean the percentage of inter-tow spaces at a given cross-section. Figure 6.19 shows the average pressure drop across the length of the unit cell. The pressure gradient is steep where the areal porosity is low. The pressure drops in other directions do not show any pattern but are still correlated to the areal porosity.

6.6.2 Virtual Permeability Predictions Through the procedure define previously, the permeability values in all principle directions were computed. In order to validate the results, the Kzz values obtained

136

6 Flow Fields and Reinforcement Permeability

Fig. 6.18 Velocity fields during flow in the a weft, b warp and c through-thickness directions for layer-to-layer reinforcement

6.6 Effect of Cyclic Compaction on Permeability

137

Fig. 6.19 Pressure drop across the flow direction a along weft orientation of the orthogonal reinforcement and b along warp orientation of the layer-to-layer reinforcement

from the predictions are compared against experimental values obtained at 60% Vf , shown in Fig. 6.20. The values are for the cyclically compacted case. The values for the orthogonal reinforcement deviate significantly and this is attributed to a minor difference in the Vf . The values for the layer-to-layer reinforcement agree very well, Fig. 6.20b. However, for the current comparison, they are reliable enough. The computed principle permeability values are given in Table 6.3. Although increased internal deformations were observed under the cyclic compaction, the penalty in the permeability is not significant, Table 6.4. Hence, subjecting the fiber reinforcement to cyclic compaction can yield higher Vf during composite manufacturing via VARTM.

138

6 Flow Fields and Reinforcement Permeability

Fig. 6.20 Comparison of predicted through-thickness permeability values with the results from experimental measurements at Vf = 0.60 of the a orthogonal reinforcement and b the layer-to-layer reinforcement

GeoDict

Experimental

Kzz (m2)

1.E-11

1.E-12

1.E-13 1

2

3

4

GeoDict

Kzz (m2)

5

Unit Cell

(a)

Experimental

1.E-12

1.E-13 1

(b)

2

3

4

5

Unit Cell

Table 6.3 Permeability predictions at various loading conditions Reinforcement

Orthogonal

Compaction State

Vf (%)

Layer-to-Layer

Permeability (× 10−12 m2 ) Kxx

Kyy

Kzz 39.3

hi

42.1

899

258

hf

59.2

150

31.3

6.19

Vf (%)

Permeability (× 10−12 m2 ) Kxx

Kyy

Kzz 50.8

41.6

476

344

60.4

34.6

3.27

1.79

h0a

43.3

919

247

4.52

45.5

288

175

24.6

h1a

51.2

173

60.4

5.38

50.5

132

42.8

11.9

6.7 Application of the Hybrid Characterization …

139

Table 6.4 Permeability comparisons between single cycle and multi-cycle compactions Reinforcement

Orthogonal

Compaction Type

Vf (%)

Permeability (× 10−12 m2 )

Layer-to-Layer

Kxx

Kyy

Kzz

Single Cycle

60.0

54.2

13.8

12.9

Multi Cycle

59.2

150

31.3

Single Cycle

50.0

48.2

13.5

Multi Cycle

51.2

173

60.4

6.19 13.4 5.38

Vf (%)

Permeability (× 10−12 m2 )

60.0

14.6

2.11

60.4

34.6

3.27

1.79

50.0

32.3

6.82

8.58

50.5

132

Kxx

Kyy

42.8

Kzz 1.45

11.9

6.7 Application of the Hybrid Characterization Framework to Other Reinforcements The hybrid characterization framework was employed for the virtual permeability predictions of the AFP carbon tape and the 2D reinforcements used in the benchmark exercise. The following section discusses the findings in this study.

6.7.1 Through-Thickness Permeability of AFP Carbon Tape The hybrid framework was used to study two different width dry AFP carbon tapes with the focus on the through thickness permeability determination. A parametric study on the effect of different gap sizes on the permeability was also presented. In addition, a detailed X-ray computerized tomographic (MICRO CT) analysis of the dry tape reinforcements was presented, highlighting the AFP process variability on the through thickness permeability. Geometric models containing flow channels of two different width dry tape carbon reinforcements have been created in the TexGen modeler. Computational fluid dynamics (CFD) simulation has been undertaken to obtain the predicted through thickness permeability of the dry tape reinforcement. A parametric study on the effect of different dry tape gap sizes on the permeability of the reinforcement was presented. An in situ compaction study carried out in the micro CT machine revealed that the gap sizes are irregular throughout the manufactured reinforcements. The permeability prediction based on the micro CT reconstructed geometric model has been validated by the experimental data.

6.7.1.1

CFD Prediction on the Effect of Gaps on the Permeability

The experimental through-thickness permeability characteristics of the AFP reinforcements based on the two different tape widths are summarized in Table 6.5. The average permeability of ½ inch tape was approximately double that of ¼ inch

140

6 Flow Fields and Reinforcement Permeability

Table 6.5 Gap and permeability values for the dry tapes, simulations versus experiments Tape Tape % Gaps Permeability (m2 ) width width Design MICRO Experiment Design (in) (mm) CT ¼ ½

6.35 3.1 12.7

1.5

% Error MICRO CT

Design MICRO CT

1.5

8.31 × 10−14 3.87 × 10−13 7.47 × 10−14 366

10

3.5

1.85 × 10−14 3.65 × 10−14 2.04 × 10−13

10

80

tape, with the former being 1.85 × 10−13 m2 and the latter 8.31 × 10−14 m2 . It was noted that the permeability variation in ½ inch tape was more evident than in the ¼ inch tape, due to limited availability of the raw materials and reinforcements, only an average of 3 experiments was presented. The unpaired two-tail Student’s t-test was performed to measure the statistical significance between the two groups of samples. The two-tailed p-value (= 0.07) shows that the difference was statistically insignificant. The higher degree of variation in ½ inch tape causes the p-value to be slightly higher than the conventional criteria of 0.05. The permeabilities derived from CFD (geometrical design based) simulations are compared with experimental data in Table 6.5. An examination of the table indicates that the CFD model for ¼ inch tape overestimates the average experimental data by over 4.7 times. This relatively large discrepancy was related to the idealized geometry that was employed in the CFD models, where the actual dimensions of the test sample were not considered. An examination of Reinforcement A in the micro CT scan revealed that the percentage gap in this sample was approximately 2.1 times lower than that in the CFD models. In contrast, Table 6.5 shows that the CFD prediction for ½ inch tape underestimates the experimental value of permeability by approximately 5.1 times. As before, this can be explained by the difference in the measured and calculated percentage gaps, where the percentage gap in the micro CT scan being approximately 2.3 times greater than that in the CFD model. Table 6.5 also lists the CFD predictions based on the reconstructed micro CT geometric models. For both reinforcements, the CFD models accurately predicted the average experimental values with the greatest error being approximately 10%. Permeability inherently exhibits a large standard deviation, due to reinforcement deformation and measurement procedures. The 10% discrepancy associated with the numerical prediction was considered as the extremely good result. It has always been a challenging task to accurately predict permeability of reinforcements due to reinforcement architectural variability. With advanced geometric and stochastic modelling approaches, the numerical predictions have significantly progressed, bringing predictions much closer to experimental values. In the present study, the micro CT scans provide exact fiber architecture and variations in gap sizes in the actual reinforcement sample. Also, the AFP dry reinforcements have simple geometric features compared to a woven or non-crimp reinforcement. The micro CT scans and relatively simple fiber architecture allow accurate geometric representations in the numerical modelling to achieve permeability predictions as close to experimental values.

6.7 Application of the Hybrid Characterization …

141

For the ½ inch tape, in comparison to the idealized model based on the material geometrical design, the re-constructed geometric model differed in the gap width by 20% narrower in average. The number of gaps was 27% less in the re-constructed geometric model due to the large number of tape overlaps. Tape overlap occurred in all the ply orientations. Here, we define the Tape Overlap Rate as the number of missing gaps over the total number of gaps based on the material design, times 100%. The ½ inch tape was more prone to in-plane misalignment, than ¼ inch tape due to its narrower width. Hence, the tape placement process caused greater levels of tape overlap in ½ inch tape. The narrower gap and the tape overlap decreased the permeability to just 20% of the idealized material design. For ¼ inch tape, the re-constructed geometric model had the much wider gaps, on average 300% that of the idealized model. The robotic arm failed to achieve the 0.2 mm tape gap in the ¼ inch tape. As a result, the realistic permeability was four times greater than the target permeability value obtained from the idealized design. The large gaps, measuring up to 1.5 mm in ¼ inch tape, would likely to become resin-rich regions in the finished composite part. The resin-rich regions would represent potential paths for crack propagation, leading to a compromised mechanical performance.

6.7.1.2

Effect of Tape Placement Variability on the Permeability

As discussed in previous chapter, the micro CT analysis showed the significant geometric difference between the idealized reinforcement model and the actual reinforcement. This geometric difference resulted in the fivefold deviation between the permeability prediction and the experimental data. This finding strongly suggests that more consistent control of tape placement process would be desirable, in order to achieve the target permeability. The validated re-constructed TexGen model was adapted to perform parametric studies on the tape placement variability. The Tape Overlap Rate was 27% in the ½ inch tape model. By adding the missing gaps in sequence to the ply orientations the Tape Overlap Rate gradually decreased to 18%, 9% and 0%. Also, by removing the existing gaps from the validated TexGen model, the Tape Overlap Rate increased to 41%. The adapted models were then analyzed using the CFD analysis. Figure 6.21a presents the sensitivity of predicted permeability to the Tape Overlap Rate. The permeability decreases linearly with the Tape Overlap Rate. Permeability reached zero when the Tape Overlap Rate exceeds 50% and the gap connectivity was eliminated. Through-thickness permeability was highly dependent on the connectivity of gaps. Figure 6.21b highlights the quantified effect of tape placement variability. Taking the idealized reinforcement design as the baseline, the re-constructed model with varied gap sizes had the most significant impact, with a 60% decrease on the through-thickness permeability. Further, to combine the varied gap size and the Tape Overlap Rate (modelling the actual reinforcement), the Tape Overlap Rate led to a further 18% decrease in permeability. The parametric studies suggest a consistent control of gap width was critical in the tape placement process to achieve the design target for permeability.

142

6 Flow Fields and Reinforcement Permeability

Fig. 6.21 The effect of tape placement variability on through-thickness permeability based on the CFD prediction, a parametric study of tape overlap rate on permeability, b combined effect of gap size and tape overlap

6.7.2 In-Plane Permeability of AFP Carbon Tape The virtual in-plane permeability values of the carbon AFP tape were computed using both realistic and stochastic approach. For the stochastic model, measurements were taken which were then used in TexGen to model the geometry. The stochastic model was used in both GeoDict and ANSYS CFX.

6.7 Application of the Hybrid Characterization …

143

Fig. 6.22 Flow field visualization a ¼ inch tape and b ½ inch Tape

6.7.2.1

Flow Fields

The numerical simulation solves for the velocity field within the unit cell. The velocity field for the flow in the direction parallel to 0° orientation was shown in Fig. 6.22, where the gaps act as the required flow channels. All the gaps participate in the transport of fluid regardless of their orientation. It was interesting to note that, at the overlapping areas of gaps, the fluid was transferred from one gap to another as shown in Fig. 6.22.

6.7.2.2

In-Plane Permeability Predictions

The virtual in-plane permeability values of the carbon AFP tape were computed using both realistic and stochastic approach. For the stochastic model, measurements were taken which were then used in TexGen to model the geometry. The stochastic model was used in both GeoDict and ANSYS CFX. The computed values are compared with experimental values and given in Table 6.6. The values differ significantly depending on the modeling approach. This shows the level of complexity that the AFP presents.

6.7.2.3

Variability in the Permeability Predictions

As discussed in the previous section, there was always variability in the results, this was more prominent in the case of the AFP carbon tape. The variability in the gaps was very high; hence the permeability was also high. Figure 6.23 shows this variability from the stochastic models.

144

6 Flow Fields and Reinforcement Permeability

Table 6.6 Gap and permeability values Reinforcement

½ in tape

Orientation

+45°

Experiment

µMICRO CT model

TexGen model GeoDict

TexGen model CFX

1.42 × 10−12

2.44 × 10−12

–

2.28 × 10−12

10−12

0°

–

3.51 ×

−45°

1.83 × 10−13

6.58 × 10−12

90° ¼ in tape

Permeability (m2 )

–

+45°

5.41 ×

0°

–

−45° 90°

2.84 × –

10−13

4.21 × 10−12 1.46 × 10−12

–

4.29 ×

10−12

9.65 ×

1.52 ×

10−12

–

6.12 × 10−13

5.57 × 10−13

6.66 × 10−13

–

7.70 × 10−13

2.20 × 10−12 10−13

9.58 ×

10−12

1.07 ×

10−12

2.04 ×

10−12

9.80 ×

10−12

10−13

2.57 × 10−12

6.28 × 10−13

Fig. 6.23 Permeability variation among different unit cells a ¼ inch tape and b ½ inch tape

6.7.3 2D Benchmark Reinforcements In order to demonstrate the universality of the method, we applied it to two 2D reinforcements.

6.7.3.1

Sample Preparation

The cut pieces were layered up with the same face up and then cut in the longitudinal direction of the desired size. The circular cutters have diameter ranging from 40 mm. The reinforcement stack was examined for proper alignment and the layers were adjusted carefully as shown Fig. 6.24. The alignment of the reinforcement layers

6.7 Application of the Hybrid Characterization …

145

Fig. 6.24 Inspecting the alignment of the layers a Hexcel WF and b Saertex NCF

Fig. 6.25 The layered test sample

were checked again and adjusted carefully if necessary. The alignment check was necessary for the in-plane permeability measurements. Unlike the 3D reinforcements, which are relatively stiff even in the dry state, it’s very challenging to handle the 2D reinforcements. Particularly, the alignment of each layer and shear deformations. It becomes also very difficult as the sample here was merely 40 mm diameter. The final stack was shown in Fig. 6.25.

6.7.3.2

Alignment Check

The alignment of the reinforcement layers was confirmed by looking at the images layer by layer. Some layers are shown in Fig. 6.26. It can be seen that the misalignment was less than 5°. There was only one layer which looked misaligned by around 20°. However, these are also expected in the samples used in the experimental permeability measurements. Hence, the misalignment can be ignored. The misalignment in the Hexcel WF was negligible as shown in Fig. 6.27.

146

6 Flow Fields and Reinforcement Permeability

Fig. 6.26 Inspecting the alignment of the layers Saertex NCF a top layer, b middle layer, c bottom layer and d the misaligned layer

6.7 Application of the Hybrid Characterization …

147

Fig. 6.27 Inspecting the alignment of the layers Hexcel WF a top layer, b middle layer, and c bottom layer

6.7.3.3

Flow Fields

Flow field visualization was a powerful tool for achieving a better understanding complex three dimensional flow patterns within the reinforcement architecture. Here we use this technique on the reinforcements to find out the flow pattern. The flows in various directions are shown in Fig. 6.28. It can be seen from these figures that the in-plane flow was behaving as expected. The flow was guided by the inter tow channels. The flow through the thickness, on the other hand, shows an anomalous behavior. The through-thickness flow was expected to move through the pinholes. However, in the cases here, they flow in-plane through inter laminar gaps as well. That means that the fluid stays in the in-plane channel until it finds a way downwards. The inter-laminar gaps are negligible as because of extensive nesting. The role of the inter tow gaps in the in-plane flow was further strengthened by Fig. 6.29. This figure shows flow in +45° and -45° directions. The flow simply goes through the channels in straight paths with some flow going via inter layer gaps.

6.7.3.4

Comparison of Segmentation Methods

One of the crucial steps in generating a voxel model from a stack of images was the segmentation. This was the step where the domain was divided into constituent regions. Three different global segmentation methods were compared. All these methods use a global thresholding grey value to partition the domain. In the first two methods, Otsu and K-means, this thresholding computed automatically based on the statistics of the data. The third technique this value was determined by the user based on experience or visual inspection. Here, the value selected was the one that correctly identified the tow cross-section. The choice of the method determines the solid volume fraction. The predicted permeability values are shown in Table 6.7. The comparison shows that the automatic methods, i.e. Otsu and K-means, under predict the Vs and hence a higher error. Based on this comparison, the manual method was used for the subsequent models. The segmentation process of the 2D fabrics were

148

6 Flow Fields and Reinforcement Permeability

Fig. 6.28 Three dimensional velocity fields in the 2D benchmark reinforcements, a Kxx , b Kyy and c Kzz

relatively more difficult than the 3D fabrics. This issue arises because of nesting among the layers of the 2D fabric stack. Also, the fiber bundles in the 2D fabrics are smaller in size as compared to the 3D fabrics and the number of layers in the 2D fabric stack are relatively high.

6.7 Application of the Hybrid Characterization …

149

Fig. 6.29 Three dimensional velocity fields in the Saertex NCF in a +45° and b −45° orientations

Table 6.7 Comparison of three global segmentation methods Reinforcement Vf Type

Segmentation Vs method

Permeability (× 10−12 m2 ) Virtual Kxx

% Error

Experimental Kyy

Kxx Kyy

Saertex NCF

0.55 Otsu

0.73 41.72 39.91

26.63

17.93

57

123

Saertex NCF

0.55 K-mean

0.72 44.71 43.24

26.63

17.93

68

141

Saertex NCF

0.55 Manual

0.78 33.04 29.67

26.63

17.93

24

66

Hexcel WF

0.56 Otsu

0.81 42.69

3.507 24.72

2.068

73

70

Hexcel WF

0.56 Manual

0.86 30.28

1.623 24.72

2.068

22

21

6.7.3.5

Kyy

Kxx

Comparison with Experimental Results

Comparing the virtual permeability values predicted through the hybrid framework will give us more confidence in the proposed method. The experimental values were obtained as part of the international benchmark exercise. The permeability was computed for all the Vf as in the benchmark exercise. The computed values are compared with the experimental values in Fig. 6.30. The comparison shows a really good agreement with experimental for the in-plane permeability. But the predicted through-thickness permeability deviates significantly from the experimental values. The deviation was more significant at the lowest and highest Vf . The anomaly can be explained by looking at the flow field. The throughthickness flow was helped by the inter-laminar gaps rather than by just only the pinholes. This could have resulted in a different layering sequence for the throughthickness flow. In the in-plane experimental case, the layering was done carefully for aligning each layer. However, this was not done with the experimental throughthickness case.

150

6 Flow Fields and Reinforcement Permeability

Permeability (m2)

1.E-10

1.E-11

1.E-12 Kxx - Exper. Kyy - Exper. Kzz - Exper.

1.E-13

(a)

0.40

0.45

Kxx - GeoDict Kyy - GeoDict Kzz - GeoDict

0.50

0.55

0.60

0.55

0.60

Vf

Permeability (m2)

1.E-10

1.E-11

1.E-12

1.E-13

(b)

Kxx - Exper. Kyy - Exper. Kzz - Exper.

0.40

0.45

Kxx - GeoDict Kyy - GeoDict Kzz - GeoDict

0.50

Vf

Fig. 6.30 Numerical permeability of 2D benchmark reinforcements a SAERTEX NCF and b HEXCEL twill weave

6.8 Summary The hybrid framework was employed to predict the permeability values, as well as to investigate the flow pattern of various fibrous reinforcements. Visualizing the flow fields gave insights in the ways in which fluid was transported within the inter-tow pore spaces. The fluid was forced to follow a tortuous path through the reinforcement because of intricate reinforcement architecture in all the cases investigated. The binder yarns in the 3D orthogonal and angle inter-lock reinforcements act like guide ways for the flow through the thickness but cause a major obstruction for in-plane flow. In the in-plane flow through the AFP tape, all the gaps participate in the transport of fluid regardless of their orientation. The in-plane flow through the 2D reinforcements dominantly through the inter tow channels. The flow through

6.8 Summary

151

the thickness of the 2D reinforcements, on the other hand, showed an anomalous behavior by flowing in-plane through inter laminar gaps as well. The predicted permeability values agree well with the experimental data in almost all the cases. The exception was for the orthogonal and angle inter-lock 3D reinforcements at low fiber volume fractions, primarily due to different sample sizes used and preparation inaccuracies. The through-thickness permeability of the 2D reinforcements also deviate from the experimental value at both low and high fiber volume fractions. The permeability predictions did not appear to be dependent on the size of the unit cells. However, there was a significant scatter of the predictions among the unit cells showing a very strong dependence on the solid volume fraction of the unit cells.

References 1. Yun M, Sas H, Simacek P, Advani SG (2017) Characterization of 3D fabric permeability with skew terms. Compos A Appl Sci Manuf 97:51–59 2. Yun M, Simacek P, Binetruy C, Advani S (2018) Random field generation of stochastically varying through the thickness permeability of a plain woven fabric. Compos Sci Technol 159:199–207 3. Endruweit A, Zeng X, Long AC (2014) Effect of specimen history on structure and in-plane permeability of woven fabrics. J Compos Mater 49(13):1563–1578 4. Alhussein H, Umer R, Rao S et al (2016) Characterization of 3D woven reinforcements for liquid composite molding processes. J Mater Sci 51(6):3277–3288 5. Syerko E, Binetruy C, Comas-Cardona S, Leygue A (2017) A numerical approach to design dual-scale porosity composite reinforcements with enhanced permeability. Mater Des 131:307– 322 6. Salvatori D, Caglar B, Teixidó H, Michaud V (2018) Permeability and capillary effects in a channel-wise non-crimp fabric. Compos A Appl Sci Manuf 108:41–52 7. Straumit I, Hahn C, Winterstein E et al (2016) Computation of permeability of a non-crimp carbon textile reinforcement based on X-ray computed tomography images. Compos A Appl Sci Manuf 81:289–295 8. Bodaghi M, Lomov SV, Simacek P et al (2019) On the variability of permeability induced by reinforcement distortions and dual scale flow in liquid composite moulding: A review. Compos A Appl Sci Manuf 120:188–210 9. Carman PC (1937) Fluid flow through granular beds. Trans Inst Chem Eng 75:150–166

Chapter 7

Experimental-Empirical Hybrid Approach

The stack of micro CT images can be used in three different ways to compute virtual permeability. Voxel and stochastic models can be generated from the stack of micro CT images which are then used in digital flow simulations using finite element or finite volume methods. However, the numerical flow simulations can be completely avoided by making certain geometrical simplifications, which can lead to the use of empirical or closed -form solutions of flow models. The micro CT images are first analyzed to identify flow channels/paths from which their shapes and sizes are extracted. Geometrical features, such as shape and size of the channels and tows, are also obtained from the micro CT images via manual or automatic measurements. Once the channels are identified, the procedure follows a hierarchical order, with a “divide and conquer” strategy. The unit cell is divided into meso- and microchannels. The meso-channels are then divided into sub-regions depending on their shapes, equivalent regions are then constructed by combining the individual regions in a bottom-up sequence. The permeability values of individual regions are estimated using analytical or empirical relations. The micro-scale permeability can be estimated by analytical models such as Kozeny-Carman or Gebart’s model, or through micro-scale numerical simulations. These models require the diameter of the fibers and the fiber volume fraction of the tows. The fiber diameter can be either obtained from the material supplier or can be measured from micrographs taken via optical microscopes. The fiber volume fraction of the tows can be calculated from the fiber count in a single tow, the fiber diameter, and the tow cross-sectional area, which is generally estimated from the micro CT images. The permeability of the meso-channels are calculated by relating the pressure drop across the length of the channel to its hydraulic resistance [2–4] or friction factor derived from Hagen–Poiseuille flow [5]. Both the hydraulic resistance and friction factor can be related to the cross-sectional shape and size of the channels. The equivalent permeability of the homogenized unit cell is then determined by making the fluid flow through the porous unit cell in a manner that is analogous to the flow of electric current through electric circuits. The sub-regions are © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020 M. A. Ali et al., CT Scan Generated Material Twins for Composites Manufacturing in Industry 4.0, https://doi.org/10.1007/978-981-15-8021-5_7

153

154

7 Experimental-Empirical Hybrid Approach

arrangements in parallel or series combinations, and the permeability is calculated by a weighted-averaging scheme, depending on this arrangement [6–9]. In this chapter, the basic formulation for the homogenization of adjacent porous regions is explained using the analogy of resistance in electrical circuits. The formulation is then extended to a 3D complex architecture which has been validated by comparing with results from numerical simulations.

7.1 Basic Formulation In order to explain the basic formulation, the analogy of electrical resistance in series and parallel has been applied to formulate models for equivalent permeability of three adjacent porous regions [3, 10]. The electrical quantities of voltage and current are taken as analogous to the pressure drop and volume flow rate of the fluid respectively. As in the case of electrical resistance in parallel, the voltage drop across all the resistance is same, the pressure drop across parallel porous regions is equal. Moreover, the total flow rate is the sum of flow rates in individual regions, just like the total current. For the porous regions in series, the flow rate is constant through each region similar to the current passing through each resistor in series and the total pressure drop is the sum of pressure drops across each porous region. By using the above resemblances, the equivalent permeability of regions in parallel and series arrangements can be formulated that is based on the Darcy’s law.

7.1.1 Porous Regions in Parallel For the derivation of equivalent permeability of porous regions in parallel, let’s take three regions with isotropic permeabilities K 1 , K 2 and K 3 , as shown in Fig. 7.1a. For the sake of simplicity, the cross-sectional area A of each region is taken to be equal. Now, Darcy’s law can be written for each region separately as, Q 1 = −K 1

Fig. 7.1 a Porous regions in parallel and b an equivalent region

A µ

P0

(a)

Q1 Q2 Q3



P1 − P0 L

L K1 K2 K3

 (7.1)

P0

P1 A A A

Qeq

(b)

L Keq

P1 3A

7.1 Basic Formulation

155

Q 2 = −K 2

A µ

A Q 3 = −K 3 µ

 



P1 − P0 L P1 − P0 L

(7.2)  (7.3)

where, P0 and P1 are the pressure at the inlet and outlet respectively. The three regions can be lumped into a single porous region with an equivalent permeability K eq as shown in Fig. 7.1b. For this equivalent region with flow rate Qeq , Darcy’s law takes the form,   3A P1 − P0 (7.4) Q eq = −K eq µ L Now, the flow rate in the equivalent regions should be same as the sum of flow rates through each region, i.e. Q eq = Q 1 + Q 2 + Q 3

(7.5)

Using the above equations and simple algebra, the final relation for the equivalent permeability is, K eq =

K1 + K2 + K3 3

(7.6)

This equation can be extended to any number of porous regions as, K eq =

N 1  K1 + K2 + · · · + K N = Ki N N i=1

(7.7)

The above equation is derived for regions with equal cross-section area. In cases where the cross-sectional area is not same for all the regions, the equivalent permeability is the harmonic mean weighted by the cross-sectional area Ai given as, K eq =

N  A 1 K 1 + A 2 K 2 + · · · + Ai K i + · · · + A N K N 1 = N Ai K i (7.8) A 1 + A 2 + · · · + Ai + · · · + A N i=1 Ai i=1

7.1.2 Porous Regions in Series Similar to the parallel case, for the derivation of equivalent permeability for porous regions in series, let’s take three regions with isotropic permeabilities K 1 , K 2 and K 3 ,

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7 Experimental-Empirical Hybrid Approach

Fig. 7.2 a Porous regions in series and b an equivalent region

L/3

L/3

K1

K2

Q0

(a)

P1

P0

L

L/3

K3 A P2

Keq

Qeq

P3

(b)

P0

A P3

as shown in Fig. 7.2a. For the sake of simplicity, the length of each region is taken to be equal to L/3. Again, Darcy’s law can be written for each region separately as, 3A Q 0 = −K 1 µ Q 0 = −K 2

3A µ

Q 0 = −K 3

3A µ

  

P1 − P0 L P2 − P1 L P3 − P2 L

 (7.9)  (7.10)  (7.11)

For the equivalent permeability, shown in Fig. 7.2b, Q eq = −K eq

A µ



P3 − P0 L

 (7.12)

The equivalent pressure drop P3 –P0 should be sum of the pressure drops across each region, i.e. P3 − P0 = (P3 − P2 ) + (P2 − P1 ) + (P1 − P0 )

(7.13)

Using the above equations and simple algebra, the final relation for the equivalent permeability is, 3 1 1 1 = + + K eq K1 K2 K3

(7.14)

This equation can be extended to any number of porous regions as,  1 N 1 1 1 = + + ··· + = K eq K1 K2 KN Ki i=1 N

(7.15)

The above equation is derived for regions with equal length. In case the crosssectional area is not same for all the regions, the equivalent permeability is the arithmetic mean weighted by the length (li ) given as,

7.1 Basic Formulation

157

 li l1 + l2 + · · · + li + · · · + l N l1 l2 li lN = + + ··· + + ··· + = K eq K1 K2 Ki KN Ki i=1 N

(7.16)

7.2 Breakdown of 3D Architecture The basic formulation presented above can be extended to any combination of porous regions. To apply this formulation, the geometrical features and the permeability of individual regions must be known. To illustrate this application, an expression for the in-plane and through-thickness permeability of a 3D complex architecture, shown in Fig. 7.3, is derived here. The 3D architecture has been chosen to mimic a 3D orthogonal reinforcement. The architecture has two porous regions oriented in the xy-plane which replicate the weft and warp tows of the orthogonal reinforcement. Similarly, the binder tow is represented by a porous region parallel to the z-axis and is placed in the middle. The empty spaces can be regarded as channels and gaps. The channels are the spaces in between the weft and warp tows whereas the gap refers to the free space surrounding the binder tow. All the fiber tows are assumed to have constant cross-sections and transversely isotropic permeability (K  and K ⊥ ). In Fig. 7.3, the regions which have been assigned two permeability values (K  or K ⊥ ), the former is the case when the flow is in-plane and the latter is the case for through-thickness flow. Both the weft and warp tows have rectangular cross-sections with the same width (wt ) and height (ht ). The weft and warp tows are placed on the top of each other in such a way that there are no gaps present in between them. The binder yarn has a square cross-section with a side length of wtz . The channel is also assumed to have a square cross-section with a side length of c. The 3D architecture is divided into three regions A, B, and C which are continuously divided into sub-regions until a single homogenous sub-regions is left behind. The breakdown of the 3D architecture is shown in Fig. 7.3 with its analogous circuit diagrams in Figs. 7.4 and 7.5 for in-plane and through-thickness flows respectively. Each single homogenous sub-region is assigned a permeability value and then combined with adjacent regions according to their sequence of arrangement to obtain a homogenized region with an equivalent permeability.

7.3 In-Plane Permeability of the 3D Architecture The homogenization steps for deriving the expression for in-plane permeability is explained below.

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7 Experimental-Empirical Hybrid Approach

Fig. 7.3 Breakdown of the 3D architecture into sub-regions

7.3 In-Plane Permeability of the 3D Architecture

Fig. 7.4 Equivalent circuit diagram for the 3D architecture for in-plane flow

159

160

7 Experimental-Empirical Hybrid Approach

Fig. 7.5 Equivalent circuit diagram for the 3D architecture for through-thickness flow

7.3.1 Permeability of the Channel and Gap The permeability of the gap and channel is formulated based on the flow resistance model and defining a hydraulic radius. One of the channels (K c1 ) is rectangular in shape while the other two channels (K c2 and K c3 ) are assumed to be square with a side length of c. The expressions for permeability of all the channels are given as,

7.3 In-Plane Permeability of the 3D Architecture

161

   2c c2 1 − 0.63 K c1 = 12 wt K c2 = K c3 =

c2 32

(7.17)

(7.18)

The gaps here are assumed to be rectangular and the permeability is given as, K g1 =

 g  g2  1 − 0.63 12 c

(7.19)

K g2 =

 g  g2  1 − 0.63 12 2c

(7.20)

7.3.2 Equivalent Permeability of the Channel-Warp Combination The channel and warp tow are in series, so the combined effective permeability is the harmonic mean of the transverse permeability of the tow (K ⊥ ) and the channel permeability (K c1 ), weighted by their respective lengths (wt and c). The expression for the combined effective permeability (K*) can be given as, K∗ =

(c + wt )K c1 K ⊥ cK ⊥ + wt K c1

(7.21)

7.3.3 Equivalent Permeability of the Gap-Binder Combination There are two instances where the gap and binder form one region. In both instances, the flow path is perpendicular to the binder. The gap and binder are arranged in a parallel sequence. Hence, the combined effective permeability (K ** ) is the arithmetic mean of the transverse permeability of the binder (K ⊥ ) and the gap permeability (K g1 ), weighted by their respective cross-sectional area (cwtz and cg). After algebraic simplifications, the combined effective permeability for both these instances can be given as, K ∗∗ =

wt z K ⊥ + g K g1 wt z + g

(7.22)

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7 Experimental-Empirical Hybrid Approach

K ∗∗∗ =

wt z K ⊥ + g K g2 wt z + g

(7.23)

7.3.4 Equivalent Permeability of the Region A and C Both the region A and C consist of the channel-warp combination and the weft tows which are in parallel arrangement with an equivalent permeability K* and K || respectively. Also, both the region A and C have the same dimensions, they will have the same equivalent permeability and is given by, 

1 1 K1 + K ∗ K = 2 2 

(7.24)

7.3.5 Equivalent Permeability of the Channel-Gap-Binder Combination The gap-binder combination with an equivalent permeability K** is in series arrangement with the channel with permeability K c2 . Note that the channel has a length of wt + g whereas the gap-binder combination length is wtz . The equivalent permeability of this parallel arrangement is, 

K =

(wt + c)K c2 K ∗∗ (wt + g)K ∗∗ + wt z K c2

(7.25)

7.3.6 Equivalent Permeability of the Warp-Channel-Gap-Binder Combination The gap-binder combination with an equivalent permeability K*** is in series arrangement with the channel with permeability K c3 and the warp tow with permeability K ⊥ . Note that the channel has a length of g, the gap-binder combination has a length of wtz and the warp two has a length of wt . The equivalent permeability of this parallel arrangement is, K  =

wt K c3

(wt + c)K ⊥ K c3 K ∗∗∗ + g K ⊥ K ∗∗∗ + wt z K ⊥ K c3

K ∗∗∗

(7.26)

7.3 In-Plane Permeability of the 3D Architecture

163

7.3.7 Equivalent Permeability of Region B Region B consists of the channel-gap-binder combination and warp-channel-gapbinder combination arranged in parallel. The channel-gap-binder combination and warp-channel-gap-binder combination have a cross-sectional area of c2 and ht respectively. The equivalent permeability of region B is thus given as, K 4 ==

 1   K +K 2

(7.27)

7.3.8 Overall Permeability The overall in-plane permeability is given by the arithmetic mean of the permeability of region A, B and C weighted by their respective cross-sectional area. The permeability of region A and C is K whereas the permeability of region B was K 4 . The total cross-sectional area of region A and C is 2wt ht whereas the cross-sectional area of region B is cht . The overall in-plane permeability of the 3D complex architecture is thus given as, Kxx =

 1

 wt K + cK 4 wt + c

(7.28)

7.4 Through-Thickness Permeability of the 3D Architecture With similar procedures, the through-thickness permeability can also be estimated. The same breakdown, as shown in Fig. 7.3, has been used for the derivation. Care must be taken in identifying the correct cross-sectional area and arrangements of the regions. The equivalent circuit diagram for the through-thickness flow is shown in Fig. 7.5. The final expression for the through-thickness permeability is given as, K zz =

2 (wt K a + cK b ) c + wt

(7.29)

where, c is the width of the channel, wt is the width of the tow and c + wt is the total width of the unit cell. Also,

K ⊥ wt K ⊥ + cK c1

(7.30) Ka = c K ⊥ + K c1 + 2wt K ⊥

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7 Experimental-Empirical Hybrid Approach



 g K g +wt z K  g K g +wt z K  w K + g K + w (wt + g)K c2 + wt z t ⊥ c t z 3 wt z +g wt z +g 

  Kb = g K g +wt z K  {wt + g + wt z } (wt + g)K c2 + 2wt z + w K + g K t ⊥ c 3 wt z +g 

(7.31)

7.5 Numerical Validation The above two formulations are predictive permeability models for highly simplified cases. Their genesis is an analogy rather than actual physical principles and hence they need to be validated. Due to the over-simplification of the problem, it may be very challenging to experimentally validate the models. However, the model predictions can be assessed against results obtained from numerical simulations.

7.5.1 Modeling Approach Numerical simulation was carried out based on the Stokes-Brinkman equations [1]. The Stokes-Brinkman equations enable one to model porous media flow accounting for the pressure loss in terms of flow resistivity. The Brinkman equations are derivatives of the Navier–Stokes equations with additional terms to compensate for the flow resistance offered by the porous media. The flow resistance is accounted in terms of the dissipation of kinetic energy by viscous shear. Mathematically, ∇.v = 0 µe ∇ 2 v − ∇ p −

(7.32) µ v = 0 K

(7.33)

where v is the velocity vector, µ is the fluid viscosity, and p is the fluid pressure. The symbol ∇ is the vector gradient operator. The brackets denote properties averaged over the entire volume of the unit cell, while µe is an effective viscosity, which is used to match the shear stress values at the interface between the porous regions given by,   dv  dv  = µe µ dn n=0+ dn n=0−

(7.34)

where n is the normal to the interface, n = 0 + represents the boundary on the external surface of the porous region and n = 0− represents the boundary on the external surface [11]. In Eq. 7.33, the term µ/K is a penalty term to account for the

7.5 Numerical Validation

165

resistance offered by the porous medium. Equation 7.32 is the continuity equation and Eq. 7.33 is the modified momentum equation.

7.5.2 Validation of Basic Formulation Using the above formulation, digital flow simulations were carried out in FlowDict module of commercial software package GeoDict. Two different arrangements were analyzed, as shown in Fig. 7.6. For the numerical simulation, a working fluid was defined with a density 1 kg/m3 and a dynamic viscosity of 1 Pa.s. Periodic boundary conditions were applied in all directions with a pressure drop on 100 kPa in the flow direction. The porous regions were assumed to have isotropic permeability given in Table 7.1 below in five different cases. Case I-III uses the first arrangement whereas Case IV-V uses the second arrangement. The permeability values predicted by the theoretical models are compared with those obtained from numerical simulations and are discussed here. The permeability predictions for parallel regions agree well in all cases, except for Case III, as shown in Table 7.2. In Case III, K2 was define to be equal to 1, which means that the corresponding region is a free channel with no porous resistance. A parametric study to show the effect of K2 is shown in Fig. 7.7, which shows that when permeabilities differ by more than three orders of magnitude, the model does not perform very well. As the order of magnitude of K2 is increased, the error also

Fig. 7.6 Unit cells for the numerical validation a three in-plane regions and b two sets of out-ofplane regions

Table 7.1 Simulation parameters Cases K1

(m2 )

K2

(m2 )

K3 (m2 ) K4

(m2 )

Case I

Case II

Case III

Case IV

Case V

2×

10–10

1×

10–09

1×

10–03

2×

10–10

5 × 10–10

4×

10−10

1×

10–10

1×

10–00

4×

10–10

1 × 10–10

1 × 10–04

6 × 10–10

5 × 10–11

–

8×

1 × 10–11

6 × 10−10 –

1 × 10−11 –

10–10

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7 Experimental-Empirical Hybrid Approach

Table 7.2 Permeability comparison in parallel Cases

Case I

Case II

Case III

Case IV

Case V

Theoretical—Keq (×10–10 m2 )

4.00

3.70

3.3 × 10+9

5.00

1.65

Numrical—Keq (×10–10 m2 )

3.99

3.63

34.5

4.96

1.61

% Error

0.24

2.00

–

0.77

2.52

Fig. 7.7 The equivalent permeability (Keq ) as a function of K2

1.E+00

Keq (m2)

1.E-03

1.E-06

1.E-09

1.E-12 1.E-12

1.E-09

1.E-06

K2

1.E-03

1.E+00

(m2)

increases with the same order. On the other hand, the permeability predictions agree well in all cases for regions in series as shown in Table 7.3. The pressure drop is an important parameter in the formulation of the models presented here. Hence it is important to check if the simulation results match with the assumption made earlier. For the parallel case, the pressure drop is uniform, as shown in Fig. 7.8a, b. In the series case, the pressure drop is different in each region. However, the total pressure drop is sum of all the drops, shown in Fig. 7.8b. The pressure drop is largest in the region with permeability K2 , which is the lowest permeability. The slope of the pressure vs length curve is proportional to the permeability of that region. Table 7.3 Permeability comparison in series Cases

Case I

Case II

Case III

Case IV

Case V

Theoretical—Keq (×10–10 m2 )

3.2727

0.27027

2.7273

4.7619

0.92167

3.2731

0.27029

2.7299

4.7557

0.90755

0.01

0.01

0.10

0.13

0.99

Numrical—Keq % Error

(×10–10

m2 )

7.5 Numerical Validation

167

Fig. 7.8 a Pressure contours in the parallel arrangement and b pressure drop along the flow direction in both parallel and series

Fig. 7.9 Unit cell of the 3D architecture for numerical validation

7.5.3 Validation of Permeability Models for the 3D Architecture With the similar approach, the models for 3D architecture has also been validated. A unit cell of the model with dimensions 2.5 × 2.5 × 1 mm3 was created in GeoDict, shown in Fig. 7.9. The width and height of the weft and warp tows were kept as 2 mm and 0.5 mm respectively. The side length of the channel was 0.5 mm. As mentioned in the previous section, the model was sensitive to the permeability of the gap. To investigate this effect, the gap size sizes were varied by changing the width and height of the binder tow. The tows were assumed to have a fiber volume fraction of 0.60. The corresponding axial and transverse permeability values were computed using Gebart’s model and are given in Table 7.4. For the numerical simulation, a working fluid was defined with a dynamic viscosity of 1 Pa.s and a density 1 kg/m3 . Periodic boundary conditions were applied in all directions with a pressure drop on 100 kPa in the flow direction. The values of both in-plane and through-thickness permeability predicted by the empirical models is compared with values obtained from the numerical simulation and is shown in Fig. 7.10. The permeability values are presented as a function of the global fiber volume fraction Vf . The global fiber volume fraction was obtained

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7 Experimental-Empirical Hybrid Approach

Table 7.4 Intra-tow permeability of each tow Permeability

Fiber tows Weft

Kxx

(m2 )

Kyy

(m2 )

Kzz

(m2 )

Warp

1.667 ×

10–13

3.640 ×

10–14

3.640 ×

10–14

Binder

3.640 ×

10–14

3.640 × 10–14

1.667 ×

10–13

3.640 × 10–14

3.640 ×

10–14

1.667 × 10–13

Fig. 7.10 Comparison of permeability predictions between the empirical models with numerical simulation as a function of global fiber volume fraction (Vf ), a in-plane permeability and b throughthickness permeability

7.5 Numerical Validation

169

from the solid volume fraction (Vs ) of the GeoDict model and the assumed tow fiber volume fraction. The results show very good agreement in the permeability values in both in-plane and through-thickness cases with correlation coefficients of 0.997 and 0.922 respectively. The predictions for through-thickness permeability begin to deviate from each other at lower values of Vf . Lower values of Vf were achieved by increasing the gap size. Therefore, this discrepancy at lower values of Vf was a direct result of the inherent limitation of the model.

7.6 Summary A formulation for homogenization of multi-scale porous regions using the analogy of electrical resistance in series and parallel has been presented. Using this analogy, a relationship has been developed for computing the equivalent permeability of three adjacent porous regions. The equivalent permeability was the arithmetic mean of the permeabilities of the individual regions in the case of parallel arrangement whereas the harmonic mean in the series case. Using numerical simulations, the model has been validated. In case of adjacent parallel regions, where permeabilities differ by more than three orders of magnitude, the model does not perform very well. Nevertheless, the models presented here are applicable to the homogenization of multiscale porous regions encountered in composite manufacturing. The formulation was extended to a complex 3D architecture as well. Although the experimental-empirical approach present a simple approach for estimating the permeability. However, there are inherent limitations associated with this method. The fundamental assumption in this approach is the unidirectional flow which is rarely encountered in real life. Moreover, the shapes of the channels, gaps and tows are assumed to be regular polygons with constant cross-sectional area. In fact, the geometry have been significantly idealized which may lead to substantial error in the permeability predictions. Also, the permeability models for the isolated homogenous regions have their own assumptions and limitations which will supplement the error.

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