Mechanical Characterization Using Digital Image Correlation: Advanced Fibrous Composite Laminates 9783030840402, 3030840409

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Matthias Merzkirch

Mechanical Characterization Using Digital Image Correlation Advanced Fibrous Composite Laminates

Mechanical Characterization Using Digital Image Correlation

Matthias Merzkirch

Mechanical Characterization Using Digital Image Correlation Advanced Fibrous Composite Laminates

Matthias Merzkirch Guest Researcher National Institute of Standards and Technology (NIST) Gaithersburg, MD, USA

ISBN 978-3-030-84039-6 ISBN 978-3-030-84040-2 https://doi.org/10.1007/978-3-030-84040-2

(eBook)

© Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are reserved 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 Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Für Opa Hans { 24.02.2017

Learn to fail efficiently. Fail, recognize its part of the process and start again. Devin Townsend

Preface

The book describes theory and application of Digital Image Correlation for selected standardized and common state-of-the-art tests, for the mechanical characterization of advanced fibrous composite laminates under quasi-static loading conditions. The purpose of this book is to provide good practice guidelines for setting up and conducting non-contacting Digital Image Correlation (DIC) measurements in conjunction with mechanical testing of planar test specimens in general-purpose laboratory conditions. In order to obtain accurate and representative material data, it is important being minute and careful when planning, performing, and evaluating experiments. Many inherent difficulties of tensile, shear, flexure, and fracture testing of composite laminates are addressed and efficiently handled where pertinent measurement issues are visualized with DIC. The more you see, the less you can hide. Digital Image Correlation offers a new look at old problems of structural mechanics of composite laminates. The purpose is to present a detailed insight into mechanical testing of complex materials with anisotropic behavior, which cannot be covered by point measurements. The motivation for the book is based on the needs for modeling the material behavior and the accurate experimental determination of material parameters aiming for an interdisciplinary understanding between modelers and experimentalists. The purpose is to contribute to training and educating users of (nonspecific) DIC systems, giving practical advice, and to present a unified approach to analyzing and understanding the results, with a potential future standardization of DIC practice. This is accompanied by the development of programs as appropriate to improve industry measurement techniques and practices for advanced DIC applications. Chapter 1 covers basic background information, tools, terms, and definitions for mechanics of composite laminates and Digital Image Correlation. Chapters 2–5 provide detailed background information and a historical insight into different test methods with focus on basic loading conditions with special cases, such as tensile with off-axis testing and flexure with small span-to-thickness ratios. The in-plane response is presented in Chap. 2 on tensile testing and in Chap. 3 on testing V-notched specimens. The determination of the flexural through-thickness vii

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properties with focus on delamination is covered in Chaps. 4 and 5, the latter presenting fracture testing under different failure modes. Metrological sensitivity analyses, including novel DIC-related uncertainty quantification, are shown in terms of the potential testing bias. Chapter 6 includes a comparative study of interlaminar and intralaminar shear properties, presented in Chaps. 2–4. Besides providing the resulting mechanical properties determined in the previous chapters, e.g., input parameter for analytical and numerical models, Chap. 6 presents a sensitivity study on the input parameters for analytical models in order to develop a sense for weighting up the necessity for conducting more/repetitive experiments and to present a way of data integrity. The results shown are based on many iterations of optimizing setups for mechanical testing of composite laminates using DIC, with the goal to improve the reproducibility and to present the adaptation to composites with different reinforcing architecture. The adaptability, compatibility, and versatility of the methodologies and testing techniques presented should be broadened to different material systems in the future. Key conclusions presented serve as suggestions for good practice and improvements necessary for the materials tested within this treatise and furthermore present an outlook for the use of DIC for testing a variety of structural materials. The end-of-chapter exercises are designed to use the DIC measured data for selfcalculation with the data reduction methodologies presented in the particular chapter. Solutions, together with supplementary informational and instructional resources (e.g., figures, videos, data reduction codes, slides with deeper explanations of selected expressions, and deduction of equations) suitable for lecturing or lab courses, are available to instructors who adopt the book for classroom use. Please visit the book web page at www.springer.com for the password-protected material. Any and all suggestions in this realm are welcome; please email me at [email protected]. Gaithersburg, MD, USA

Matthias Merzkirch

Acknowledgments

This book is based on the research I conducted as a Guest Researcher at the National Institute of Standards and Technology (NIST) in Gaithersburg (MD) at NIST Center for Automotive Lightweighting (NCAL), led by Tim Foecke and later continued by Mark Iadicola, from January 2016 until December 2020. I want to thank Tim Foecke, Jon Guyer, Mark Iadicola, and NIST Foreign Guest Researcher Program for providing me the opportunity and continuous funding over all the years. I am deeply grateful to Tim Foecke for the unique, quick, and stressless hiring process (“You got the job!” after a 45-min online interview); for giving me the opportunity to start my journey; and for his trust and support over all the years. I am especially thankful to Mark Iadicola and Jon Guyer for their support by giving me the opportunity to continue my research during the challenging pandemic COVID19, where most parts of this book have been finalized in the new and lonely working environment aka “home office.” I want to thank Ami Powell for sharing office, opinion, and ears over all the years. Furthermore, I want to thank the permanent “NCALers” for their support and many discussions in and outside the lab: Bill Luecke, Adam Creuziger, Ed Pompa, Evan Rust, Dilip Banerjee, and Steve Mates (in Monday morning meeting appearance). In addition, I am thankful to Aaron Forster for our fruitful collaboration and his thorough review of this manuscript. Furthermore, I want to thank the following people for their input, contribution, discussion, or help during my stay at NIST (in alphabetical and hopefully complete order): R. Ak, C. Amigo, Q. An, K. Atnafu, C. Beauchamp, S. Beauchamp, J. Bonevich, C. Calhoun, A. Coleman, M. Dadfarnia, C. Emeje, S. Freiman, J. Galuardi, T. Gnaupel-Herold, L. Hazel, M. Henn, M. Hoehler, D. Hunston, A. Jennion, U. Kattner, A. Krishnamurthy, J. LaRosa, M. Li, B. Lin, E. Lin, C. Makrides, P. Manescu, W. McDonough, D. Nell, W. Osborn, D. Pitchure, W. Poling, L. Poole, G. Quinn, S. Ridder, R. Singh, J. Steve, M. VanLandingham, and J. Woodcock. My initially planned 1-year excursion started on December 29th, 2015, with nothing else but two suitcases. The journey came to an end 5 years later on ix

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November 25th, 2020, the same way, with in/evitable goodbyes but enriched with new friendships, an uncountable number of experiences, and unforgettable memories of journeys, moments, and people. I dedicate this book to my family and friends: To those who came, To those who left, To those who stayed.

About This Book

In this book, a precise treatment of the experimental characterization of advanced composite materials using Digital Image Correlation (DIC) is presented. The text explains test methods, testing setup with 2D- and stereo-DIC, specimen preparation and patterning, testing analysis, and data reduction schemes to determine and to compare mechanical properties from DIC calculated data, such as modulus, strength, and fracture toughness of advanced composite materials. Sensitivity and uncertainty studies on the DIC calculated data and mechanical properties for a detailed engineering-based understanding are covered instead of idealized theories and sugarcoated results. The book provides students, instructors, researchers, and engineers in industrial or government institutions, and practitioners working in the field of experimental/ applied structural mechanics of materials, a myriad of color pictures from DIC measurements for better explanation, datasets of material properties serving as input parameters for analytical modelling, raw data and computer codes for data reduction, illustrative graphs for teaching purposes, and practice exercises with solutions provided online, and extensive references to the literature at the end of each stand-alone chapter. The Book • Emphasizes practical matters, such as preparation and testing of specimens, design of DIC measurements, and data reduction methodologies • Compares and discusses many tensile, shear, flexure, and fracture test methods that are ASTM, ISO, and DIN standards from direct and real non-idealized measurements • Provides detailed insight into testing conditions including metrological uncertainty and sensitivity studies • Reinforces concepts with photomechanical visualizations, key conclusions/good practice suggestions, and end-of-chapter practice exercises

xi

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About This Book

• Illustrates, visualizes, and explains experimental mechanics with DIC for an interdisciplinary understanding for experimentalists and modelers, providing input parameters for analytical modeling

Contents

1

Introduction and Theoretical Background . . . . . . . . . . . . . . . . . . . . Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fibrous Composite Laminates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Historical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structural Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mechanical Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Test Methods and Materials of Investigation . . . . . . . . . . . . . . . . . . . Digital Image Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Historical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . From Speckles to Displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . Virtual Strain Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resolution and Uncertainty Quantification . . . . . . . . . . . . . . . . . . . . Patterning Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 3 3 4 15 19 23 23 24 26 29 30 35 38 40

2

Tensile Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Specimen Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental Setup and DIC Configuration . . . . . . . . . . . . . . . . . . . . Mechanical Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Data Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fractography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sensitivity Analysis—Angularity . . . . . . . . . . . . . . . . . . . . . . . . . . Sensitivity Analysis—Strain Location and Range . . . . . . . . . . . . . . Visualization of Strains and Displacements . . . . . . . . . . . . . . . . . . . . . Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47 47 52 53 56 56 59 59 61 62 63 69

. . . . . . . . . . . .

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Effect of Specimen Geometry and Fiber Orientation . . . . . . . . . . . . . . . Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . In-Plane Extensional and Intralaminar Shear Properties . . . . . . . . . . . Testing of Assorted Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Woven-CFRP—Uniaxial and
45 Tensile Testing . . . . . . . . . . . . . UD-GFRP—Uniaxial and 10 Off-Axis Tensile Testing . . . . . . . . . . Key Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structural Mechanics and Testing . . . . . . . . . . . . . . . . . . . . . . . . . . Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74 78 78 79 80 84 85 85 86 86 87

3

V-Notched Specimen Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Specimen Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental Setup and DIC Configuration . . . . . . . . . . . . . . . . . . . . . V-Notched Beam Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V-Notched Rail Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mechanical Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Data Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fractography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sensitivity Analysis—Strain Type, Location, and Range . . . . . . . . . . Visualization of Strains and Displacements . . . . . . . . . . . . . . . . . . . . . . Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of Fiber Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of Specimen Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Intralaminar Shear Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Testing of Assorted Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Woven-CFRP—V-Notched Beam Test . . . . . . . . . . . . . . . . . . . . . . UD-GFRP—V-Notched Rail Test . . . . . . . . . . . . . . . . . . . . . . . . . . Key Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structural Mechanics and Testing . . . . . . . . . . . . . . . . . . . . . . . . . . Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91 91 95 96 96 99 103 103 105 106 107 108 118 121 125 128 128 129 129 131 132 132 133 133 134

4

Flexural Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bending Deflection and Bending Strength . . . . . . . . . . . . . . . . . . . Shear Deflection and Shear Strength . . . . . . . . . . . . . . . . . . . . . . . Specimen Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental Setup and DIC Configuration . . . . . . . . . . . . . . . . . . . . Mechanical Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

137 137 137 141 147 149 153

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Contents

5

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Data Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fractography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sensitivity Analysis: Shear Stress . . . . . . . . . . . . . . . . . . . . . . . . . . Sensitivity Analysis: Span . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Visualization of Strains and Displacements . . . . . . . . . . . . . . . . . . . . . . Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of Specimen Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elastic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interlaminar Shear Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Testing of Assorted Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Woven-CFRP: Three-Point, Four-Point, and Five-Point Bending . . . . Resin: Three-Point Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Key Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structural Mechanics and Testing . . . . . . . . . . . . . . . . . . . . . . . . . . Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

153 155 156 157 161 162 182 187 189 189 193 194 194 196 200 200 200 201 202

Delamination Resistance Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear Elastic Fracture Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . Mode I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mode II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Testing Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Specimen Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Testing Setup and DIC Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . Mode I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mode II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mode I—Mechanical Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Data Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fractography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mode I—Visualization of Strains and Displacements . . . . . . . . . . . . . . . Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mode II – Mechanical Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Data Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fractography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mode II—Visualization of Strains and Displacements . . . . . . . . . . . . . . Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interlaminar Fracture Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elastic Flexural Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

205 205 205 207 211 215 216 218 218 219 226 226 230 231 231 237 242 242 245 245 245 250 254 255 257

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Contents

Testing of Assorted Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Woven-CFRP—Mode I DCB Testing . . . . . . . . . . . . . . . . . . . . . . . Key Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structural Mechanics and Testing . . . . . . . . . . . . . . . . . . . . . . . . . . Practice Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

258 258 259 259 260 260 261

Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UD-CFRP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Intralaminar and Interlaminar Shear Properties . . . . . . . . . . . . . . . . Experiment vs. Analytical Model . . . . . . . . . . . . . . . . . . . . . . . . . . Sensitivity Analysis: Input Parameters for Analytical Modeling . . . . Woven-CFRP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Intralaminar and Interlaminar Shear Properties . . . . . . . . . . . . . . . . Experiment vs. Analytical Model . . . . . . . . . . . . . . . . . . . . . . . . . . Sensitivity Analysis: Input Parameters for Analytical Modeling . . . . Comparison: Material Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

265 265 266 269 273 277 277 278 281 283 286

. . . . . . . . . . .

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

Abbreviations1

Coordinate systems 1-2-3

x-y-z ξ-η

Principal orthogonal coordinate system. Material axes (lamina) coordinate system with 1-direction aligned with the fiber orientation and 3-direction normal to the plane of the lamina Reference Cartesian orthogonal coordinate system Transformed orthogonal coordinate system relative to x–y

Greek letters β γ γ 12 γ xy Δ ε ε 1, ε 2 εx , εy ε ξ, ε η η θ κ λ ν σ

Error in modulus Shear strain, engineering (subscripts define coordinate system) Principal (maximum) shear strain Shear strain in x–y coordinate system Correction shift, difference Strain (subscripts define coordinate system) Principal major (1) and minor (2) strains Normal (or nominal) strains in x- and y-direction Strains in transformed ξ-η coordinate system Extension-shear ratio (no subscript) Orientation angle between structural and material axes (x-axis and fiber orientation, resp.) measured positive counterclockwise Curvature Stiffness-related ratio Poisson’s ratio Normal stress (subscripts define plane and direction)

1

This section lists Greek symbols, abbreviations, and acronyms used in the subsequent chapters. Some of the symbols listed might deviate from the standards cited; the aim was to unify and simplify. For correct nomenclature, see the standards cited in each chapter. xvii

xviii

τ φ Φ φ* φ


Abbreviations

Shear stress (subscripts define plane and direction) Angle of rotation for coordinate transformation Notch angle Angle of principal strains Misorientation error in angularity

Roman letters [T] a a0 A A1 Aij ai b B Bij c C Dij d E E* Efl F G h I k l L m M n N q Q Qij R Sij t u, U v, V

Transformation matrix Overall crack length Initial crack length Constant Slope Components of the extension stiffness matrix with i, j ¼ 1–3 Crack propagation length Inner span Constant Components of the extension-bending coupling stiffness matrix with i, j ¼ 1–3 Stiffness of the elastic foundation Compliance (displacement/force) Components of the bending stiffness matrix with i, j ¼ 1–3 Diameter Young’s modulus Apparent measured modulus Flexural modulus Force Shear modulus Thickness Moment of inertia Shear correction factor Overall length Length, gauge length (tensile loading), span (flexural loading) Fracture exponent Moment Slope Number of layers Distributed load Shear force Components of the reduced stiffness matrix with i, j ¼ 1–3 Roughness Components of the compliance matrix with i, j ¼ 1–3 Time In-plane displacement in x-direction In-plane displacement in y-direction

Abbreviations

V* w W

xix

Deflection Width Out-of-plane displacement in z-direction

Subscripts a, v, p B, S comp f fl i k L, R L, T max min s off tot

Arithmetical mean, valley, peak Bending, shear Compression Fracture Flexure Running index Index of summation Left, right Longitudinal (fiber orientation in plane of layer), transverse (normal of fiber orientation in plane of layer) Maximum Minimum specimen (V-notched) Offset Total

Special symbols G J

Energy release rate J-integral

Acronyms 2D 3D 3pt 4pt 5pt ASTM CBTE CC C-ELS CF CFRP CLT COD const CSD CTOD CTSD CV

Two-dimensional Three-dimensional Three-point Four-point Five-point American Society for Testing and Materials Corrected beam theory using effective crack length Compliance calibration Calibrated end-loaded split Correction factor Carbon fiber-reinforced polymer Classical lamination theory Crack opening displacement Constant Crack shear displacement Crack tip opening displacement Crack tip shear displacement Coefficient of variation

xx

DBS DCB DIC DOF ENF FEM FOV FRP GFRP ILS LEFM LiC LVDT MB MCC PC QOI ROI ROM RUC SB SBS SBT SCB SiC SOD UD USS UTS VSG

Abbreviations

Double beam shear Double cantilever beam Digital Image Correlation Depth of field End-notched flexure Finite element method Field-of-view Fiber-reinforced polymer Glass fiber-reinforced polymer Interlaminar shear strength Linear elastic fracture mechanics Loss in correlation Linear variable differential transformer Modified beam Modified compliance calibration Pure compliance Quantity-of-interest Region-of-interest Rule of mixture Representative unit cell Simple beam Short beam shear Short beam test Single cantilever beam Silicon carbide Stand-off distance Unidirectional Ultimate shear strength Ultimate tensile strength Virtual strain gauge

Chapter 1

Introduction and Theoretical Background

Motivation Polymer matrix composites are extensively used in structural lightweighting applications due to their combination of high stiffness and strength and, on the other hand, low weight which makes them attractive for transportation on land, in the air, and on the water (Davies et al., 1998). The use of composite laminates for structural components in automotive, aerospace, naval, and energy transformation (i.e. wind turbine blades) sectors requires a reliable investigation of the mechanical performance aiming for a detailed and robust understanding of the deformation, damage, and fracture behavior under different loading conditions (types of loading and environmental conditions) and loading rates (quasi-static, monotonic, cyclic, and dynamic). There is a need for analytical and numerical models able to predict the structural and crashworthiness capability and performance of fiber reinforced polymer components (Bru et al., 2017). Structural components for aerospace are designed either that outage of the component can be avoided (safe life) or the function of the component can be taken over by another component (fail safe). A third option is damage tolerance, which requires monitoring of the damage within inspection intervals, being a key parameter in determining the safety (Davies et al., 1998). For the analytical description of fiber reinforced composite laminates, the intralaminar (in-plane) and interlaminar (through-thickness) material properties such as modulus and strength are needed. Furthermore, fracture toughness values are needed as input data for numerical models, e.g. based on cohesive zones, within computational materials science. The determination of the interlaminar properties is important to safety modeling where delamination is a major mode of energy release. For the determination of the in-plane and through-thickness tensile/compressive, shear and delamination properties of flat laminate sheet materials, many different mechanical testing procedures have been developed within the last decades to attempt to generate the needed data. The evaluation of an accurate constitutive © Springer Nature Switzerland AG 2022 M. Merzkirch, Mechanical Characterization Using Digital Image Correlation, https://doi.org/10.1007/978-3-030-84040-2_1

1

2

1 Introduction and Theoretical Background

response in shear is a crucial input to numerical material models (Bru et al., 2017) which also includes non-linearity and therewith the need for determining the full stress–strain response. As an example, approximately two dozen measurement techniques exist for the determination of the shear properties, several of which are supported by ASTM and ISO standards. One reason for so many attempts is the difficulty in obtaining a reasonably pure and uniform stress state in the specimen. The brittle nature of carbon fiber reinforced polymer (CFRP) composites and the highly anisotropic character of the unidirectional (UD) architecture poses an additional challenge in testing those materials, even under quasi-static loading conditions. Typically, strain gauges (Ajovalasit, 2011) and/or other contact-based techniques (e.g. extensometer) are recommended by standards to measure deformations and strains for the determination of the elastic modulus and the strain to failure. The advent of optical strain measurement techniques, such as Digital Image Correlation (DIC), provide new opportunities to generate high resolution maps of the displacement and strain field as a function of the globally applied strain. DIC allows for the contact-less easy full-field mapping of in-plane and out-of-plane (for stereo-DIC) displacements and axial, lateral, and shear strains, revealing the pattern of deformation and damage throughout the specimen. Full-field deformation measurements in combination with numerical simulations allow for inverse modeling (Daiyan et al., 2012; Grédiac et al., 1994; Pierron & Grédiac, 2012). In this treatise, in-plane testing of tensile coupon and standardized V-notched specimens will be compared in terms of intralaminar shear moduli and shear strengths. Additionally, the interlaminar shear properties determined via standardized three-point, non-standardized four-point, and standardized five-point shortbeam flexural testing, requiring small span-to-thickness ratios for the shear loading to become dominant, will be compared. Besides a determination and comparison of fracture toughness values, DIC will be used while performing standardized fracture testing under mode I and mode II (including a comparison of two types of standardized test methods) loading conditions, revealing the critical strains and crack tip opening displacements in the vicinity of the crack tip as useful data for numerical models. The goal is a metrology-based lightweighting by implementing DIC for accurate measurements and an improvement of state-of-the-art, quasi-static testing techniques, aiming for an increase in reliability and safety, in advance of future mixed mode, superimposed and rate-dependent testing. With the main material of interest being a unidirectionally carbon fiber reinforced composite, the versatility will be presented on assorted materials such as woven crossply CFRP and unidirectionally glass fiber reinforced composite laminates. The use of the methods presented is not limited to a specific software or material system, but aiming for adaptability and compatibility.

Fibrous Composite Laminates

3

Fibrous Composite Laminates This section presents a short introduction of composite laminates and some fundamentals of the mechanics of composite materials, needed for the subsequent chapters. For a detailed treatise on mechanics of composites, see (Jones, 1999; Herakovich, 1998; Agarwal et al., 2006). For further details on terminology for composite materials, see (ASTM, 2018).

Historical Background More than 6000 years passed from the first man-made fibrous composite laminates, which appears to be the papyrus paper (lay-up of fibrous papyrus plant in 0
/90
orientation) made by the Egyptians, to today’s date. A boost for the introduction of advanced composite structures happened in the second half of the twentieth century (Herakovich, 2012), which is related to the push of new innovations usually for aerospace (military, resp.) purposes (Haka, 2011; Tsai, 2005). The introduction of new types of advanced composite materials with complex architectures has also driven the evolution of new tools and methods for the mathematical description of the composite behavior. A historical review on composite materials is provided in (Herakovich, 2012). A review of the evolution of fiber reinforced composites for aerospace applications, starting in the early twentieth century, can be found in (Palmer, 2012), with a focus on Germany in (Haka, 2011) and the USA in (Scala, 1996; Tsai, 2005).

Configuration A simple classification of composite materials can be done by the type of matrix material, into metal matrix composites (MMC), ceramic matrix composites (CMC), and polymer matrix composites (PMC). A widespread classification can be done by the type and architecture of the reinforcing structure (for structural composites) such as particles, fibers (short and long), and sponge structures for interpenetrating composites (Merzkirch et al., 2015). The latter denoting the architecture of the composite itself, similarly to composite laminates. A composite laminate (lay-up, resp.) is formed by stacking different laminae, with each lamina (layer, ply, resp.) contributing to the mechanical performance. The volume fraction, architecture, stacking sequence (Pagano & Pipes, 1971; Chawla, 2012) and orientation of the single constituents, as well as parameters of the production process lead to process-structure-property relationships. Figure 1.1a delineates the orthogonal principal material axes (1–2–3) for an orthotropic material. For a unidirectional (UD) lamina, the fiber orientation is in

4

1 Introduction and Theoretical Background

Fig. 1.1 (a) Principal material axes 1–2–3, (b) Laminate coordinate system x–y–z and rotation θ of principal material axes

direction 1 (the direction of maximum in-plane Young’s modulus, resp.), transverse direction 2, and lamination direction 3. The relation between principal material axes to (arbitrary) structural x–y–z-axes is depicted in Fig. 1.1b, with positive (counterclockwise) rotation θ of principal material axes.

Structural Mechanics Stress Figure 1.2 delineates the three-dimensional equilibrium stress state for a cubic volume element, including the normal stresses σ and tangential shear stresses τ, for an orthotropic material in principal material axes. Based on the equilibrium of moments (see color code in Fig. 1.2), the stress tensor is symmetric and can be described with six entries

Fibrous Composite Laminates

5

Fig. 1.2 Stress components acting at a threedimensional volume element

0

σ 11 B σ ij ¼ @ τ21

τ12 σ 22

1 0 σ1 τ13 C B τ23 A ¼ @ τ12

τ12 σ2

1 τ13 C τ23 A

τ31

τ32

σ 33

τ23

σ3

τ13

ð1:1Þ

Considering a UD laminate, see Fig. 1.1, σ 1 refers to the longitudinal (“L”) direction with fiber orientation of θ ¼ 0
(e.g. warp for woven laminates), σ 2 refers to the transverse (“T”) direction perpendicular to the fiber orientation (fill for woven laminates), and σ 3 refers to the through-thickness direction. The shear stresses τ31 and τ32 (face side 3) relate to the interlaminar direction, τ21 and τ23 (face side 2) relate to the intralaminar direction, τ13 and τ12 (face side 1) relate to the translaminar direction (Brunner, 2020). For a lamina, the plane stress state simplifies to σ 3 ¼ 0 and τ13 ¼ τ23 ¼ 0. For the principal material axes oriented at an angle θ with respect to the x–y coordinate system, as shown in Fig. 1.1b, the in-plane (1–2) stresses have to be transformed according to 2

3 σx 6 7 6 7 4 σ 2 5 ¼ ½T 4 σ y 5 τxy τ12 with the transformation matrix

σ1

3

2

ð1:2Þ

6

1 Introduction and Theoretical Background

2

m2 6 2 ½T  ¼ 4 n mn

n2 m2 mn

3 2mn 7 2mn 5 m2  n2

ð1:3Þ

in which m ¼ cos θ and n ¼ sin θ (Carlsson and Pipes, 1987).

Strain The deformation of a body is described by the displacement field, with the in-plane deformation for a square surface element (dA ¼ dx  dy) being shown in Fig. 1.3a. For the description of the relative deformation, usually the strain components of the strain tensor (analogous to the stress tensor) are described using the absolute distances of neighboring points in the initial undeformed and deformed state (based on the displacement field, the displacement vector, resp., with the in-plane components u and v) with small deformations, see Fig. 1.3a. For a detailed deduction, see reference (Gross et al., 2018; Gross & Seelig, 2011). The plane strain state is characterized by the two in-plane normal strains, acting perpendicular on the face of an infinitesimal element εx ¼

∂u ∂x

ð1:4Þ

εy ¼

∂v ∂y

ð1:5Þ

∂u ∂v þ ¼ 2 ∙ εxy ∂y ∂x

ð1:6Þ

and

and the shear strain γ xy ¼

When considering the engineering strain (or Cauchy strain, after A. Cauchy (1789–1857) (Timoshenko, 1953; Gross et al., 2018)), the shear strain γ xy represents the angle change depicted in Fig. 1.3a. The tensor shear strain εxy refers to only half the angle change. For shear sign convention and the visualization of the corresponding angle change, it can be started from the engineering shear definition. Assuming an initial square element deforms such that the top and bottom edges move horizontally, i.e. Eq. (1.5) ¼ 0 and Eq. (1.6) focusing on the first term, this turns the square into a parallelogram, see Fig. 1.3b. For the coordinate system shown in Fig. 1.3, the shear strain is positive if the top edge moves to the right (see Fig. 1.3b) and negative if the top edge moves to the left relative to the bottom edge.

Fibrous Composite Laminates

7

Fig. 1.3 (a) Strain components acting at a two-dimensional surface element, (b) pure shear deformation

8

1 Introduction and Theoretical Background

The principal strain axes represent three mutually perpendicular axes in a strained material, which are parallel to the directions of greatest, intermediate, and least elongation, and which describe the state of strain at any particular point. The major (“1”) and minor (“2”) strains in the (in-plane) principal coordinate system 1-2 is expressed by, compare also trigonometric relationships in Mohr’s circle for engineering strain (after O. Mohr (1835–1918) (Timoshenko, 1953)) in Fig. 1.4a, ε1,2

εx þ εy ¼  2

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ε  ε 2 γ 2 x y xy þ 2 2

ð1:7Þ

with the angle of principal strain   γ xy 1 φ ¼ ∙ arctan 2 εx  εy 

ð1:8Þ

The maximum shear strain is expressed by γ 12 ¼ ε1  ε2 ¼ 2 ∙ ε12

ð1:9Þ

Note the engineering shear strain convention at the y-axis in Fig. 1.4a. A transformation from the x–y to ξ–η coordinate system with the angle φ, see Fig. 1.4b, is performed using the following equations (Gross et al., 2018; Gross & Seelig, 2011) γ xy εx þ εy ε x  εy  ∙ cos 2φ  ∙ sin 2φ 2 2 2

ð1:10Þ

  γ ξη ¼  εx  εy ∙ sin 2φ þ γ xy ∙ cos 2φ ¼ 2 ∙ εξη

ð1:11Þ

εξ,η ¼ and

Mohr’s circle for engineering strain for bi-dimensional pure shear strain state is shown in Fig. 1.4c, see also Fig. 1.3b.

Constitutive Relations The law of elasticity describes the linear relationship (for small deformations) between the strain tensor and stress tensor via the compliance matrix Sij εij ¼ Sij ∙ σ ij

ð1:12Þ

The compliance matrix is shown in Fig. 1.5 with denotation of the non-coupled (represented by the diagonal) and coupled entries.

Fibrous Composite Laminates Fig. 1.4 Mohr’s circle for engineering strain. (a) Reference x–y and principal 1-2 coordinate system, (b) arbitrary ξ–η with respect to x–y coordinate system, (c) bi-dimensional pure shear strain state

9

10

1 Introduction and Theoretical Background

Since this 66 matrix (with 36 entries) is symmetric (due to ij ¼ ji), and can be reduced to 21 independent entries. For the description of the in-plane (plane stress, see Eq. (1.2)) behavior, the compliance matrix can be reduced to 0

s11

B sij ¼ @ s21 s61

s12 s22 s62

s16

1

C s26 A s66

ð1:13Þ

The relationship between the stress tensor σ ij and the strain tensor εij for plane stress state is the reduced stiffness matrix Qij (Schürmann, 2007; Jones, 1999) σ ij ¼ Qij ∙ εij

ð1:14Þ

For calculation of the single components of the reduced stiffness matrix, see (Jones, 1999). In accordance with (Herakovich, 2012), the earliest publications (written in the 1930s and 1940s) employing anisotropic constitutive equations for the solution of real problems, considering wood as primary material, appear to be those of S.G. Lekhnitskii (Lekhnitskii, 1968). He showed that an orthotropic material (with three planes of symmetry and 12 entries) has nine independent constants and, for the special case of an orthotropic material, a transversely isotropic, unidirectional material (isotropic properties in one of the planes of symmetry), has five independent material constants for the three-dimensional state. The orientation dependent in-plane coefficients of the compliance matrix, see Eq. (1.13) and Fig. 1.5, based on the transformation relation Eq. (1.3) (Yeow & Brinson, 1978; Chamis & Sinclair, 1978; Herakovich, 1998; Agarwal et al., 2006; Schürmann, 2007) will be presented in the following. Five elastic properties including Young’s moduli in the longitudinal fiber orientation EL, and in the transverse fiber orientation ET, major Poisson’s ratio parallel to the fiber orientation υL, minor Poisson’s ratio perpendicular to the fiber orientation νT, and in-plane shear modulus G

Fig. 1.5 Description of the compliance matrix Sij

Fibrous Composite Laminates

11

are required for the plane stress state. A reduction to four independent elastic in-plane engineering constants is possible due to the dependency of the constants, described by Maxwell–Betti reciprocal theorem (Sideridis & Papadopoulos, 2004) υL υ ¼ T EL ET

ð1:15Þ

The extension compliance in longitudinal orientation is S11 ¼

  cos 4 θ 1 1 υ sin 4 θ  2 ∙ L ∙ sin 2 2θ þ þ ∙ EL 4 G ET EL

ð1:16Þ

and in transverse direction is S22

  sin 4 θ 1 1 υL cos 4 θ ¼ þ ∙  2∙ ∙ sin 2 2θ þ EL 4 G ET EL

ð1:17Þ

The shear compliance is S66 ¼

  cos 2 2θ 1 1 υ þ þ þ 2 ∙ L ∙ sin 2 2θ G EL ET EL

ð1:18Þ

The extension–extension coupling compliance is expressed by S12

   1 1 1 1 υ  ∙ sin 2 2θ  L ∙ sin 4 θ þ cos 4 θ ¼ ∙ þ  4 EL ET G EL

ð1:19Þ

The extension–shear coupling compliances S16 (S26, resp.), which is highest (smallest, resp.) for small (large, resp.) angles are expressed by   2 υ 1 ∙ sin 3 θ ∙ cos θ þ2∙ L  S16 ¼  ET EL G   2 υL 1 þ ∙ cos 3 θ ∙ sin θ þ2∙  EL EL G

ð1:20Þ

and S26

  2 υL 1 ∙ cos 3 θ ∙ sin θ ¼ þ2∙  ET EL G   2 υ 1 þ þ2∙ L  ∙ sin 3 θ ∙ cos θ EL EL G

ð1:21Þ

The coefficient of mutual influence for extension, Poisson’s ratio, is expressed by

12

1 Introduction and Theoretical Background

ν¼

S12 S11

ð1:22Þ

The coefficient of mutual influence between extension and shear is expressed by η¼

S16 S11

ð1:23Þ

With respect to fiber orientation of θ ¼ 0
, Eq. (1.16) leads to S11 ¼ 1/EL, Eq. (1.17) to S22 ¼ 1/ET, Eq. (1.18) to S66 ¼ 1/G, and Eq. (1.19) to S12 ¼ ν/EL. For homogeneous (property independent of the location), isotropic (property independent of the direction) materials (E ¼ EL ¼ ET, ν ¼ νL ¼ νT), with maximum shear at θ ¼ 45
(see Fig. 1.4a), Eq. (1.18) can be simplified to two independent constants G¼

E 2 ∙ ð1 þ νÞ

ð1:24Þ

A sensitivity analysis on the single parameters for the compliances will be presented in Chap. 6.

Classical Lamination Theory Classical lamination theory (CLT) (Herakovich, 2012) considers the linear elastic deformation of a parallel assembly of single and plane laminae of constant thickness that form a laminate (Schürmann, 2007). Typically, the layers are unidirectional fibrous composites with fibers in the Nth layer oriented at a specific angle θ (positive from the global axis), see Fig. 1.6. A laminate is symmetric when the stacking sequence is symmetric with respect to the midplane (the laminae above the midplane are a mirror image of those below the midplane). A laminate is balanced when it has equal numbers of positive (counterclockwise rotation as shown in Fig. 1.6) and negative angled laminae (ASTM, 2018). The fundamental equation includes in-plane forces F and moments M acting on the laminate to the midplane strains ε and curvatures κ (Carlsson & Pipes, 1987; Herakovich, 1998; Jones, 1999; Reddy, 2004)

F M





A ¼ B

B D



ε κ

ð1:25Þ

The extensional stiffness matrix is expressed by Aij ¼

N X k¼1

Qij

 k

ðzk  zk1 Þ

ð1:26Þ

Fibrous Composite Laminates

13

Fig. 1.6 Laminate coordinate system and definition of the lamina coordinates zN

the extension–bending coupling stiffness matrix is expressed by Bij ¼

N  1 X   2 Qij k zk  z2k1 2 k¼1

ð1:27Þ

and the bending stiffness matrix is expressed by Dij ¼

N  1 X   3 Qij k zk  z3k1 3 k¼1

ð1:28Þ

with z0 ¼ h/2 and zk ¼ zk1 + hk (for k ¼ 1 to N). All matrices are functions of the material properties, layers thickness, and stacking sequence of the single layers. Depending on the stacking sequence, the laminate exhibits coupling between in-plane and bending effects. The engineering properties of a symmetric laminate can be deduced from Eq. (1.25) through thought experiments. With the laminate compliance defined as aij ¼ h Aij1, EL ¼ 1/a11, G ¼ 1/a66, ν ¼ a12/a11, and η ¼ a16/a11 (Herakovich, 2012). Symmetric laminates have a zero [B] matrix, with no coupling between in-plane and out-of-plane responses. Asymmetric laminates exhibit in-plane strains when subjected to pure bending. As an example: when an asymmetric laminate is cured at elevated temperature, it will develop curvature (due to the hygrothermal strains introduced (Carlsson & Pipes, 1987)) when cooled down to room temperature and released from all constraints in the curing process. The possible shapes are shown in Fig. 1.7. The plaque shown on the right side has an asymmetric and unbalanced lay-up [+45
, 45
, +45
, 45
] ([452]ns (Merzkirch et al., 2019). Curvature shown in Fig. 1.7b occurs when pushing the center of the plaque in Fig. 1.7a. Depending on

14

1 Introduction and Theoretical Background

z

a

x

y

z

b

x

y

z

c

x

y

Fig. 1.7 Examples for (a) and (b) monoclastic curvatures of an asymmetric [452]ns laminate, (c) anticlastic (hyperbolic paraboloid) curvature

the stacking sequence, more complex lay-up architectures lead to the saddle like curvature shown in Fig. 1.7c. Interlaminar stresses resulting from the stacking sequence (Pagano & Pipes, 1971) at and near free edges are not encompassed by CLT (Schürmann, 2007; Chawla, 2012). Another example for an asymmetric, crossply laminate is Papyrus, with laid-up strips in two layers with a configuration [0/90]ns, whereas a symmetric lay-up would avoid curvature (Herakovich, 2012).

Fibrous Composite Laminates

15

Strength and Failure A quadratic failure criterion for the orientation dependent maximum bearable stress under tensile loading of a laminate, based on maximum work theory acc. to Tsai– Hill, is expressed by (Tsai & Wu, 1971) (Agarwal et al., 2006)   1 cos 4 θ 1 1 1 sin 4 θ ¼ þ ∙  ∙ sin 2 2θ þ 2 2 2 2 UTSθ UTSL 4 USS UTSL UTS2T

ð1:29Þ

For the analytical description of the in-plane properties, usually three strength values are needed: ultimate tensile strengths UTSL and UTST, and ultimate shear strength USS, which can be reduced to two for orthotopic composites. The Tsai–Hill criterion represents the reduction of the Tsai–Wu criterion, assuming tensile and compressive strengths to be equal (Carlsson & Pipes, 1987; Chawla, 2012). Typical failure mechanisms of fibrous composite laminates include matrix and/or reinforcement cracking, visco-elastic/plastic effects, debonding between matrix and reinforcement, and delamination in between layers. For further failure criteria based on transverse and/or shear failure, see (Puck & Schürmann 1998; Schürmann 2007). The description of interlaminar failure, delamination, resp., based on fracture mechanics concepts will be treated in Chap. 5.

Mechanical Characterization This section gives an overview on selected test methods for composite laminates. Note that multiple standardized and state-of-the-art test methods exist. Depending on the type of composite material and the specific properties, a particular test method may be preferred or appropriate relative to others. A detailed treatise on shear test methods is provided and discussed.

Motivation Testing composite materials has proven to be much more challenging than testing homogeneous, isotropic materials. The difficulties are primarily associated with load introduction (see principle of Saint-Venant (Carlsson & Pipes, 1987)), fixture requirements, specimen preparation, obtaining stress–strain results into the non-linear range, and controlling the type and location of failure (Chatterjee et al., 1993c). One key issue is to achieve well-controlled and well-defined conditions in the test region. A good test for the determination of mechanical material properties should have a single, uniform stress component in the gauge section (Zweben et al., 1979). Producing such a state of stress in the laboratory is not a trivial task

16

1 Introduction and Theoretical Background

(Pagano & Halpin, 1968). This is intricate for anisotropic composite materials (Melin & Neumeister, 2006). The fundamental tests of tension, compression, and shear must be extended to include consideration of material anisotropy. In general, in-plane moduli can be determined without difficulty, whereas the measurement of in-plane strengths remains controversial (Davies et al., 1998). Unidirectional, continuous fibers in a polymeric matrix represent the most difficult case in terms of anisotropy (Herakovich, 2012). An overview of different standardized test methods for various types of composites is given in (ASTM, 2016b) with emphasis on tension in (Chatterjee et al., 1993a), compression in (Chatterjee et al., 1993b), and shear in (Chatterjee et al., 1993c). Practical guides for testing composite materials can be found in (Carlsson & Pipes, 1987; Carlsson et al., 2014). For a couple of loading conditions, multiple standards exist, aiming to determine the same properties, as it will be presented on shear testing.

Shear Test Methods The fact that the shear behavior, of anisotropic materials in general and composite laminates in particular, is independent of the tensile properties, raises the need for a specific shear test (Grédiac et al., 1994). Shear testing of anisotropic materials, namely plywood, began at U.S. Forest Products Laboratory in the 1950s (Adams et al., 2003). The determination of the in-plane (intralaminar, intraplanar, intraply) and out-ofplane (interlaminar, interplanar, interply, through-thickness) shear properties (shear modulus, shear strength, and shear strain at failure) of flat laminate sheet materials is of great interest, and many different testing procedures have been developed within the last decades to attempt to generate the needed data. One of the most controversial problems of material strength theories and material testing techniques is the determination of the ultimate strength for pure shear loading, with only shear stresses without normal stresses (compare Fig. 1.4c) in the test section until breaking of the specimen (Iosipescu, 1967). In 1967, a screening of existent shear test methods (e.g. bending of a rectangular beam, torsion of a circular cross-section) including several photoelastic investigations on the shear profiles on cutting, double cutting (punching), single-lap shear (double-notch shear in different configurations), and double-lap shear was presented (Iosipescu, 1967). The conclusion was, that even for metals it is challenging to create a state of pure shear up to fracture of the specimen without additional normal stress components. The photoelastic investigations proved that none of the (at that time) existing procedures for shear testing of metals (and also concrete) fulfills the aim of producing pure shear failure (Iosipescu, 1967). Brittle materials usually fail in tension when subjected to a pure shear stress state (Adams & Walrath, 1987). According to (ASTM, 2012a), no standard test method exists, being capable of producing a perfectly pure shear stress condition to failure for every material, although some test methods can come acceptably close for a specific material.

Fibrous Composite Laminates

17

Table 1.1 Selected standardized shear test methods for the determination of the interlaminar and intralaminar shear modulus G and/or ultimate shear strength USS Year of first standardization 1965

Test method Three-point short-beam bending

1976

45
tension

1979

Double-notch shear Two-rail & Three-rail shear V-notched beam

1983 1993 1993 1999 2005 2018 2018

Torsion of hoop wound cylinder Plate twist V-notched rail Five-point double beam bending Shear frame

Interlaminar X

Intralaminar

G

USS X

X

X

X

X

X

X

X

X

X

X

X X

X

X

X

Reference ASTM (2016a) DIN (1998) ASTM (2013) ISO (1998) ASTM (2015a) ASTM (2015b) ASTM (2012a) ASTM (2016c) ISO (2005) ASTM (2012b) ISO (2018a)

X

X

X

X

X

ISO (2018b)

X

X

X X

X

X X

According to (Purslow, 1977; Lee & Munro, 1986; Chatterjee et al., 1993c; Summerscales, 1987), approximately 26 different test methods (several of which have found their way into standards) exist to determine the interlaminar and intralaminar shear (see Fig. 1.2) properties, including shear modulus G and/or ultimate shear strength USS (Summerscales, 1987) of unidirectionally (UD) and crossply materials. A few selected standardized test methods are listed in Table 1.1. At least one new standard for shear testing has been released every decade since the 1960s. In reference (Lee & Munro, 1986), nine test methods for the determination of in-plane strength and modulus are ranked, using a four-criteria decision analysis (cost of fabrication, cost of testing, data reproducibility, and accuracy of experimental results). The reference was the standardized torsion test of the hoop wound (circumferential, 90
) thin-walled cylinder specimen (Agarwal et al., 2006; ASTM, 2016c), which is considered to be the most reliable test method for determining the shear stress–shear strain response (Lee & Munro, 1986; Chatterjee et al., 1993c; Pierron & Vautrin, 1996), followed by the standardized uniaxial tensile test on crossply (45
) laminates (ASTM, 2013; ISO, 1998). Even though the thinwalled tube subjected to torsion provides pure shear stress and strain in the wall, the specimen is expensive and the method is too time consuming to be efficient (Chamis & Sinclair, 1977). While torsion tests reveal challenges due to the complex loading of the specimen (Summerscales, 1987), the disadvantage of 45
laminates is that

18

1 Introduction and Theoretical Background

they are not free of lamination residual stresses (Pipes & Pagano, 1970; Pagano & Pipes, 1971; Pagano & Pipes, 1978; Chamis & Sinclair, 1977). The disadvantage of performing torsion tests on tubes and tensile tests on crossply coupon specimens for determining the shear properties of unidirectional sheet material is the necessity of manufacturing materials in two geometries (laminate architecture, resp.) (Pierron & Vautrin, 1996). For the determination of the in-plane shear properties of unidirectionally reinforced composite laminate, the standardized V-notched beam test (ASTM, 2012a) (aka “Iosipescu” test) seems to be the best choice, followed by the 10
off-axis tensile test, acc. to (Lee & Munro, 1986). An attractive alternative to the V-notched beam test is the standardized V-notched rail test (Adams et al., 2003; ASTM, 2012b). Reference (Pindera et al., 1987) recommends 45
off-axis tensile testing for the determination of the intralaminar shear modulus and the V-notched beam test for determination of shear strength. According to (Sun & Berreth, 1988), (two- and three-) rail shear tests have the disadvantages, that it is hard to produce a pure shear state over the test section, and that the test fixture introduces large stress concentrations. A comparison of different in-plane test methods via off-axis tensile testing and three-rail shear testing on carbon fiber reinforced epoxy is provided in (Yeow & Brinson, 1978). Determination of the interlaminar properties of composite laminates to date have included a few direct measurement techniques. Many different standardized testing procedures exist besides the flexure-based test methods, such as double notched specimen (ASTM, 2015a) and a modification as inclined version (Melin et al., 2000). The complexity of testing does not lead to satisfying results (“apparent” interlaminar testing). In particular, the determination of the interlaminar shear strain and shear modulus is of great interest, albeit those test methods are limited to a determination of the ultimate shear strength alone. There is a considerable amount of confusion about which shear test method should be used because so many have been developed. An appropriate shear test method for composite materials should ideally produce pure shear, which is difficult to attain due to coupling effects, having a good reproducibility and ease to execute, aiming for a full stress–strain response and therefore provide all shear properties (shear modulus, shear strength, and shear strain at failure) (Sideridis & Papadopoulos, 2004). A particular shear test method may be preferred relative to others, depending on the composite material to be tested and which material properties are desired (Adams et al., 2007). Only a few comparisons between the entire intralaminar and interlaminar shear stress–shear strain curves are available in the literature. Usually, the interlaminar shear modulus is obtained by transversely isotropic assumption (for UD laminates) and dedicated interlaminar shear strengths are extracted without necessarily measuring the shear strain (Bru et al., 2017). It is pretty common to assume the interlaminar shear response to be the same as the in-plane shear response for UD laminates (Cui et al., 1992).

Fibrous Composite Laminates

19

Test Methods and Materials of Investigation Test Methods The necessity of analytically modeling the material behavior asks for a determination of the shear properties, besides the tensile properties, see Constitutive Relations Section. In particular, the strength related values have to be determined experimentally, since a calculation using the properties of the single constituents is considered to be unreliable (Schürmann, 2007). None of the interlaminar shear test methods listed in Table 1.1 provide a determination of the full stress–strain behavior and some test methods are limited to a determination of the elastic properties alone (plate twist testing). Additionally, a few interlaminar shear test methods require the materials of interest (as-received flat blank, resp.) to be “artificially” modified, e.g. by manufacturing thick specimens (V-notched beam and rail testing) and machining notches (double-notch shear testing). The main goal of this treatise is to directly determine the full shear stress–shear shear response of flat sheet materials (laminates) using DIC, with a focus on the interlaminar (through-thickness) properties. Since the scope of most test methods listed in Table 1.1 covers UD laminates, the study is expanded for comparison between selected interlaminar (via flexural test methods) and intralaminar (via tensile and V-notched beam and rail testing) shear test methods on the same UD material, due to the transversely isotropic assumption of UD composite laminates. Furthermore, interlaminar fracture, delamination processes, resp., will be investigated since the related properties serve as input for numerical models.

Materials This section provides some information on the materials used (provided by NIST) in this treatise. UD-CFRP represents the primary material of investigation for the mechanical characterization via all test methods described in the subsequent Chaps. 2–5. Additionally, assorted materials such as CFRP with a woven architecture and UD-GFRP have been used for selected test methods to show the applicability, adaptability, compatibility, and limitations. The investigations rather focus on deformation, damage, and fracture mechanics in contrast to the underlying processstructure-property specific uncertainties. Therefore, a direct materials science-based comparison between the different types of materials and a detailed investigation on the phenomenological micro-mechanisms is not provided.

UD-CFRP Primary material of investigation is a unidirectionally, carbon fiber (with a non-round cross-sectional geometry and a diameter of 6.9 μm) reinforced

20

1 Introduction and Theoretical Background

Fig. 1.8 Representative cross-section of UD-CFRP Table 1.2 Results from quasi-static uniaxial tensile testing of the matrix material (taken from investigations followed from (Powell et al., 2017) and manufacturer’s datasheet (DOWAKSA, 2016) for fibers) (representing standard deviation)

Epoxy (matrix) Carbon Fiber (reinforcing element)

UTS in MPa 75.3  6.5 4200

E in GPa 3.39  0.09 240

ν 0.39  0.03

εUTS in % 3.13  0.59 1.8

G in GPa 1.22  0.05

(non-stitched) composite laminate manufactured via compression molding of epoxy prepregs (Su & Wagner, 2019). Nominally 12 prepreg layers (with a thickness of 200 μm per lamina) were stacked up to a final thickness of approx. h ¼ 2.4 mm of the cured balanced laminate sheet material ([012]s) as a plaque of approximate size 300 mm  300 mm, with a reported reinforcing volume fraction of 50%. Figure 1.8 depicts a representative cross-section, polished in several steps using water and SiC papers to P1500, depicting the wavy nature of the single layers and pores (black spots). The mechanical tensile properties, e.g. Young’s modulus E, ultimate tensile strength UTS, and related strain at failure εUTS, of the single constituents are listed

Fibrous Composite Laminates

21

a

b

Fig. 1.9 Surface profiles of the cutting edges (orientation 0
). (a) Abrasive 80, (b) Abrasive 220 Table 1.3 Quantitative surface quality of selected specimen after waterjet cutting

Orientation 0
0
90


Abrasive 80 220 220

Ra in μm 5.8 2.6 2.0

Rv in μm 22.6 12.0 6.7

Rp in μm 19.4 7.7 7.2

in Table 1.2. The shear modulus G of the epoxy has been determined using Eq. (1.24).

Surface Quality Preliminary studies were conducted to determine the effect of the waterjet cutting parameters (i.e. abrasive grit size, nozzle diameter, pressure, speed) and to optimize the cutting conditions to ensure minimal fabrication damage in the specimen (Shanmugam et al., 2008). For quantitative characterization of the damage due to specimen fabrication via waterjet cutting, profilometry measurements (using a stylus with a 90
conical tip with 5 μm radius) have been carried out, see also (Merzkirch & Foecke, 2020). Figure 1.9 depicts the differences in the surface profile, over a sampling length of 10 mm (sampling every 2.2 μm), between as-waterjet-cut with two abrasive grit sizes (220 and 80), and related waterjet cutting parameters, used within this treatise. The resulting roughness values of the total height (RV, RP) are similar to the fiber diameter of 6.9 μm for abrasive 220, see also Table 1.3.

Assorted Materials Woven-CFRP The second epoxy-based laminate investigated in this treatise is a carbon fiber reinforced composite with woven architecture (22 twill, weave fabric with equal structure in the warp and fill/weft direction), see Fig. 1.10, also manufactured via

22

1 Introduction and Theoretical Background

Fig. 1.10 (a) Midplane twill architecture, (b) schematic 22 twill architecture (one unit cell), (c) representative cross-section of Woven-CFRP

compression molding (Su & Wagner, 2019). The yarn width is approx. 2.4 mm and of elliptical shape with a thickness of approx. 0.5 mm (see Fig. 1.10c). Nominally four prepreg layers (with an average thickness of 0.85 mm per lamina) were stacked up to a final thickness of approx. h ¼ 2.7 mm of the cured balanced laminate sheet material as a plaque of approximate size 300 mm  300 mm, with a reported reinforcing volume fraction of 50%. The prepreg representative unit cell (RUC) has a side length of approx. 9.7 mm (Su and Wagner, 2019).

UD-GFRP The third epoxy-based laminate investigated is a glass fiber reinforced polymer (GFRP) composite with an E-glass unidirectional fabric manufactured via vacuumassisted resin transfer molding (Merzkirch et al., 2019). Approximately 15% of the overall fabric mass are stitching fibers, which are aligned perpendicular to the UD fibers. Nominally four prepreg layers were stacked up to a thickness of approx. h ¼ 0.93 mm of the cured balanced laminate sheet material ([04]s) as a plaque of approximate size 350 mm  250 mm (see Fig. 1.11). The reinforcing fraction of 54.3% by mass was determined through thermal gravimetric measurements (An et al., 2018; Merzkirch et al., 2019).

Digital Image Correlation

23

Fig. 1.11 UD-GFRP plaque

Digital Image Correlation This section presents a short introduction to Digital Image Correlation with practical information, terms and definitions that are needed in the subsequent chapters. For a detailed treatise on Digital Image Correlation, see (Sutton et al., 2009) and a general practical guide for Digital Image Correlation and terminology is provided in (iDICs, 2018; Reu, 2012a, 2012b, 2012c, 2012d, 2012e, 2012f, 2013a, 2013b, 2013c, 2013d, 2013e, 2013f, 2013g, 2014a, 2014b, 2014c, 2014d, 2014e, 2014f, 2015a, 2015b, 2015c, 2015d, 2015e, 2015f; Merzkirch et al., 2020; Relland et al., 2020).

Historical Background Imaging is the most obvious and important way to gain scientific knowledge. Visualization is a prerequisite in applied/experimental mechanics, which has been done with several methods in a lot of different scientific fields over the last centuries. Over the time, single visualization was not enough with quantification getting more important. Photoelastic measurements of specific materials were one way to do mechanical observations in the late nineteenth, early twentieth century. Those were usually done on glass due its property to become doubly refracting when exposed to a mechanical stress (Filon, 1912). Related to photogrammetry, Digital Image Correlation started to evolve for practical use in the 1980s (Reu, 2012d). The first known DIC related work was performed in 1961 (Sutton et al., 2017). The steady increase in the use of DIC is also driven by the continuous increase in resolution with simultaneous decrease in costs for machine vision cameras. That trend also applies for the processing power of the personal computer and further optical devices. Besides hardware-related items, DIC owes the evolution of the underlying mathematical concepts (Reu, 2012d). After 2005, commercialization of DIC systems and codes has become larger which increased the amount of DIC users in comparison to the developers (Reu, 2015d).

24

1 Introduction and Theoretical Background

Configuration The basic principle of Digital Image Correlation (DIC) is to track the motion of a random stochastic pattern applied on a surface of a specimen that is subjected to deformation, motion or load. This pattern is imaged before and during deformation (motion or load) by usually high resolution monochrome machine vision cameras (Reu, 2012e). Thus, the measurement, based on optical pattern recognition, is contact-less and nondestructive. When using a single camera perpendicular to the surface of the specimen, 2D-DIC, the in-plane displacement in x- and y-direction can be calculated, as well as the in-plane (surface) strain components (and thus any derivative such as rotation, rate, etc.). By using two cameras, stereo-DIC, simultaneously observing the surface from different angles (stereo angle), a stereoscopic view is obtained with overlapping views from both cameras. In addition to 2D-DIC, the out-of-plane displacement (in z-direction) is determined. Figure 1.12 delineates the principle of a stereo-DIC setup. Before test acquisition, a calibration of the stable and rigid (Reu, 2014c) DIC setup has to be done using a calibration target, with features of known distance to each other, being moved (tilt, rotation, translation) within the calibration volume (Reu, 2013c, 2013b, 2014e). Calibration allows the system to determine the intrinsic parameters of the single camera with lens, such as sensor center point, lens focal length, camera skew, lens distortions, and image scale (Reu, 2013a). The latter refers the optical resolution of the cameras in pixel into physical units (mm/px or px/mm). Calibration is sometimes skipped when using a 2D-DIC setup with the focus on strains (since strains are unitless) (Reu, 2012c). When using a stereo-DIC setup with two cameras, as depicted in Fig. 1.12, calibration is used to additionally determine the extrinsic camera parameters (relative orientations and positions in space). Those include the stereo angle, therewith the location and orientation of the cameras to each other and the (out-of-plane) stand-off distance (LSOD) between the aperture of the lens to the surface of the specimen. Some of the calibration results provide a general verification in conjunction with the actual DIC setup (Reu, 2014d) (i.e. image center vs. camera resolution, lens focal length, stereo angle, baseline). Configuring the DIC setup is done before calibration and includes, besides the choice of lenses (fixed-focal length or zoom lenses) (Reu, 2013d) and cameras (resolution, type: CCD, CMOS) (Reu, 2012e), simultaneous setting of focus and aperture, lighting and exposure (Reu, 2013c). The smaller the focal length, the smaller the magnification and the larger the FOV (Reu, 2013d). The aperture of the lens, governing the light entering the optical system, controls the depth-of-field (DOF). Reducing the aperture leads to an increase of the DOF (Reu, 2013c, 2013d), which requires appropriate lighting (Reu, 2013e). The position of the light source should not interfere with the cameras in order to avoid heat waves (air turbulences) (Reu, 2013e). Lenses also have a resolution to be considered and adapted to the cameras chosen (Reu, 2013d).

Digital Image Correlation

25

Fig. 1.12 Schematic horizontal stereo-DIC setup with specimen’s coordinate system

Figure 1.13 depicts both field-of-view (FOV, width by height) of images captured from the left (a) and right (b) camera of a horizontally positioned stereo-DIC setup, with a clamped tensile specimen with stochastic speckle pattern before deformation. Based on the stereo angle, the overlapping view will be smaller than the single FOV of each camera. Figure 1.13c highlights the region-of-interest (ROI) within the gauge section of the specimen for the following correlation to happen. In order to keep the uncertainty small (see also Resolution and Uncertainty Quantification Section), the measurement should be kept at the center of the FOV (Reu, 2013f), see also crosshair in Fig. 1.13a and b.

26

1 Introduction and Theoretical Background

Fig. 1.13 FOV (LFOV ¼ 2448 px  2048 px, landscape orientation) with speckle patterned tensile specimen before deformation, reference image of left (a) and right (b) camera with image center, (c) reference image of left camera and ROI broken into subsets (no overlap)

From Speckles to Displacements Figure 1.13c depicts the ROI divided into (usually) square subsets, small sections of the ROI (subset-based method or local DIC in comparison to global DIC (iDICs, 2018)). The subset controls the area within the ROI that is used to track the displacement between images. Each subset, see Fig. 1.14a, consists of a defined amount of pixel, each representing one gray level. Subsets are correlated from the reference image (which is taken before applying any deformation, motion or load) to each subsequent image during deformation, motion or load. The temporal distance of subsequent images is defined by the acquisition rate and has to be adapted to the motion/deformation of the specimen since the previous location is used for starting the search in the subsequent image. Note that for a stereo-DIC setup, synchronization, i.e. the simultaneous acquisition of images for both cameras is important (Reu, 2012e).

Digital Image Correlation

27

Fig. 1.14 Undeformed and cropped upper half of reference images from Fig. 1.13 with a single subset: (a) Left camera image, (b) right camera image, Artificially deformed and cropped images with distorted subset: (c) left image, (d) right image, (e) left image with three subsets and overlap of 50%

The square subset in the reference image (Fig. 1.14a) will have a different shape in the corresponding stereo image (Fig. 1.14b) and will deform as the specimen will be deformed (Fig. 1.14c, d, with an artificial elongation of 30% and narrowing of 10%). A mathematical subset shape function will accommodate for deformation of the subset between cameras and subsequent images. The linear affine shape function

28

1 Introduction and Theoretical Background

(of lower order) can represent translation, stretch, and shear (see Fig. 1.3 for analogy), which is adequate for most experiments on flat surfaces (Reu, 2012c). During the correlation process, first the pattern within a subset is approximated by an interpolation function, which supplies subpixel accuracy (of 0.01 px) for displacement (Reu, 2012a, 2012f). In accordance with (Reu, 2012a), the quality of the interpolation filter is related to its ability to suppress both the interpolation bias and the noise bias. The matching criterion (including subset weights) is used to compare each subset in the reference image with the subset in the deformed image. Beside this temporal matching, the matching criterion is used for the subsets from one camera image to match with the other camera image (cross correlation) in stereo-DIC, see Fig. 1.14a, b). This is realized by minimizing an optical residual based on the change in gray levels between the reference and deformed images. The best match (based on the ratio between image noise and image contrast) results in a displacement and deformation value (Reu, 2012a). Triangulation uses the intrinsic and extrinsic parameters from the calibration process for unambiguously locating the lines projected from the centers of the corresponding subsets to a unique point in space (Reu, 2012b). Since those lines do not intersect, an optimization process, based on the change of the subset location, is used to find the optimum triangulated point. A successful correlation of the pixel location should be close to the epipolar line, which is also an important indicator for the calibration quality (Reu, 2014c, 2014d). In subset-based, local DIC, the measured displacement solution is centered within the subset and defined at regular spacing (steps), see point-based tracking in Fig. 1.14e. The post-analysis ROI is usually smaller than the pre-analysis ROI, covering the edges of the specimen (see Fig. 1.13c), due to the subset size. The larger the subset size, the more missing points at the edges of the specimen (Rossi et al., 2015), see Fig. 1.14e, resulting in an offset between edge of the specimen and post-analysis ROI of the image. Note that smaller subset sizes might lead to a loss in correlation within the ROI. The choice of the subset size is related to the approximate size of the speckle pattern feature, see also Patterning Methods Section. Subsets are typically symmetric with a centered data point (odd integers), which are given by the characteristic length (i.e. one side of the square) (iDICs, 2018). Figure 1.15 depicts the recommended subset sizes (Lsubset) from (Reu, 2014b; iDICs, 2018). Five times the approximate (average) pattern feature size is practical minimum for the subset size, based on fluctuations (see also Patterning Methods Section). Note that a camera with a higher image resolution allows smaller speckles than a camera with a lower resolution. The step size (Lstep) defines the spacing (integer numbers with a minimum of 1 px) of at which the displacements of the subset are calculated. The ratio (Lsubset-Lstep)/Lsubset is defined as overlap (iDICs, 2018). The boundaries (Reu, 2014b; iDICs, 2018) shown in Fig. 1.15 are considered to be estimates, with the non-shaded white area as the recommended DIC analysis parameter settings.

Digital Image Correlation

29



 

 

 



 

            

     





Fig. 1.15 Relation between subset size and step size







 























Virtual Strain Gauge Common quantities-of-interest (QOI) other than the primary measurement, i.e. displacement, include spatial derivatives and temporal derivatives. The in-plane (surface) strain components (and rotations) are determined by deriving the calculated displacement fields, see Strain Section for a formulation of engineering strains and (iDICs, 2021) for examples of strain calculation methods (e.g. shape functions). Those QOI are usually not determined point-based but over a local region of the ROI to make the results less noisy (Reu, 2012b). Besides the analysis parameters for calculation of the displacement, subset, and step, strain calculation requires a finite number of data points over a region, which is specified by the length of the strain window (Lwindow). For square strain windows (see Fig. 1.16), the window size is given by the characteristic length, i.e. one side of the square. Windows are typically symmetric and centered at a data point, thus, are odd integers (iDICs, 2018). The virtual strain gauge (VSG) represents the local region of the ROI that affects the strain value at a specific location. The virtual strain gauge size is expressed by LVSG ¼ ðLwindow  1Þ ∙ Lstep þ Lsubset

ð1:30Þ

with the first term representing the strain window size (Lwindow ). The virtual strain gauge size is divided by the average image scale to determine the size in physical units. The VSG represents the average strain and is comparable but not to confound with the area that a physical strain gauge (including averaging) would cover (Reu, 2015f; iDICs, 2018). Therefore, virtual strain gauge sizes should be smaller than the smallest dimension of the specimen surface to be imaged. An increase in displacement resolution (due to high camera resolution, small FOV, etc.) will not lead to an increase in strain resolution, based on the unitless character of strain (Reu, 2012b).

30

1 Introduction and Theoretical Background

Fig. 1.16 Subset, step, strain window, and virtual strain gauge (iDICs, 2018), including selected parameter variation

At strain concentrations, strain is the higher, the smaller the virtual strain gauge size (the smaller the virtual area of averaging) and vice versa. The disadvantage with a larger VSG is that fine features, large variations, and gradients (e.g. arising from surface undulations and flaws, local material inhomogeneities) in the strain field are smoothed out and thus not distinguishable (Melin et al., 2000).

Resolution and Uncertainty Quantification Resolution Choosing the FOV and the resolution of the cameras defines the image scale, usually determined during a calibration. The spatial resolution is defined as the distance

Digital Image Correlation

31

between independent measurement points (data point spacing) (Reu, 2012f). An alternative definition is the shortest period over which the displacement or strain gradients can be captured without unacceptable bias errors (Reu, 2015f). A small subset size increases the spatial resolution, which can be further increased by reducing the step size (though increasing the solution time) (Reu, 2012f). A small subset size and a small step size lead to an increase of the (displacement) noise-floor. However, non-redundant and independent data can only be achieved with an overlap smaller than approx. 0.5 (the upper limit in Fig. 1.15) (Reu, 2012f). Due to the interpolation function in the correlation algorithm, the corresponding position of each subset in the deformed image is determined with subpixel accuracy. The minimum resolution for the in-plane displacements in x- and y-directions is approx. 0.01 px–0.05 px (Reu, 2012a, 2012f). Regardless of the interpolant, a smoother speckle pattern with good contrast and low image noise will always yield better results (Reu, 2012a). Using stereo-DIC, the accuracy for the out-ofplane displacement in z-direction is worse in comparison to in-plane displacements (Reu, 2012f). Since strain values are derivatives of displacements (so are rotations), the strain results are usually less accurate (i.e. noisier) than the primary measurement of displacements. The same is true for temporal derivatives (Reu, 2012b). Strains in the specimen’s coordinate system are more accurate than strains in the principal coordinate system (since those are dependent on the single strain components in the specimen’s coordinate system, see Strain Section) and shear strain in the specimen’s coordinate system is more accurate than maximum shear strain. Larger stereo angles improve the out-of-plane results but increase in-plane uncertainty. Smaller stereo angles are preferred when focusing on the QOI “strain”. The uncertainty at the image center is smaller than off-center, a long focal length lens leads to a smaller uncertainty than a short focal length lens (Reu, 2013f). One benefit of using stereo-DIC, in comparison to 2D-DIC, is the removal of in-plane assumptions due to the inclusion of out-of-plane motion. This usually affects the normal strains (in all directions), translation, and rotation (Sutton et al., 2008, 2009). Since the SOD is unknown in 2D-DIC, the in-plane assumptions lead to errors especially when large out-of-plane motions occur. An estimate for the strain error when using 2D-DIC (based on 2D, in-plane assumptions) is the ratio between out-of-plane motion and SOD (Reu, 2013a; Sutton et al., 2008). Resolution of a QOI is used interchangeably with noise (variance), representing the random (volatile) error, which gives information on the preciseness (fluctuations) of a measurement, in terms of the standard deviation (or a multiple of). Bias represents a systematic and persistent error, being a metric for the accuracy (or offset) of a measurement, in terms of the mean, and being difficult to measure (Reu, 2015d). Table 1.4 lists some sources of uncertainties that will affect the measurement and therewith the actual resolution. It has to be differentiated between spatial (changes depending on the coordinate) and temporal resolution (changes over time). An advantage of using DIC to measure deformation post-mortem is the ability to acquire and probe the full-field data set for variations in measured strains due to the selection of subset, step, and window sizes, therewith the choice of lengths and locations of the virtual strain gauge (whereas the physical strain gauge has to be

32

1 Introduction and Theoretical Background

Table 1.4 Sources for bias and noise-floor (iDICs, 2018) Bias

Noisefloor

Avoidable Pattern fidelity, system heating, relative camera motion (stereo-DIC), stand-off distance (2D-DIC), signal synchronization, glare, poor lighting, uncorrected lens distortion, air turbulence, image blur Pattern (defects, breakdown, & aliasing), focus variation, contrast & intensity, poor lighting, dirt on optics, vibrations, air turbulence

Fig. 1.17 Qualitative influence of DIC analysis parameters on LVSG

Pattern, image noise, correlation parameters (subset size, step size, filtering)

f(Lsubset )

300

LVSG in px

Unavoidable Pattern, subset shape function, interpolant, correlation parameters (correlation function, subset size, step size, filtering), strain calculation

f(Lstep ) f(Lwindow )

200

100

0 0

5

10

15

20

25

Parameter in px or datapoints chosen before the actual test (Reu, 2015f)). The qualitative sensitivity of the virtual strain gauge is visualized in Fig. 1.17 by the slopes, independent derivatives of Eq. (1.30), with the subset size having the smallest and the step size the largest influence.

DIC Parameter Uncertainty Quantification In the following, a methodology for investigation of variance (noise-floor) and bias errors (iDICs, 2018) with respect to the DIC analysis parameters such as subset, step, and window size (parameters of the VSG) is exemplarily presented in several process steps, focusing on three images of the patterned specimen, to optimize the DIC measurements. The aim of this virtual strain gauge study is to find a compromise between smoothing (large VSG, resp.) and capturing strain gradients (small VSG, resp.) after acquisition of the images (Reu, 2012b, 2015f). 1. At this point, only the static and spatial noise-floor, the measurement resolution, resp., is considered for the ROI of an image (image-pair, resp., for stereo-DIC) at

Digital Image Correlation

33

a

b

c

d

Fig. 1.18 (a and c) Distribution of a strain-based QOI for LVSG ¼ 99 px (VSG(15/6/15)), (b and d) bias and noise-floor of two QOI for different LVSG

zero load (no motion), just prior (and referred to) the reference image. Optionally, the temporal noise-floor can be considered by referring to multiple images with no load. The same image is used for a determination of the bias. The primary QOI has to be defined, usually strain-based for the strain window to be another parameter besides subset and step, although not limited to those only (Reu, 2015f). Figure 1.18a depicts the distribution of a strain-based QOI of an image just prior the reference image, for one selected virtual strain gauge size LVSG. The bias is represented by the mean of the distribution, the offset from zero, resp., where the QOI should be zero for a non-loaded/deformed/moved specimen. Be aware of the influence of clamping (a single) side of the specimen. The noise-floor is represented by (one) standard deviation of the distribution. Note that certain QOI, that include a calculation of other QOI, have a larger bias, i.e. maximum shear strain γ 12 (see also Strain Section) due to the inherent dependence of other QOI such as axial strain εx, lateral strain εy and shear strain γ xy, (see also angle of principal strain), see Fig. 1.18c. 2. An image at large deformation (e.g. just before failure), including a strain gradient (e.g. necking), is chosen. If an image at a specific strain is chosen, note that

34

1 Introduction and Theoretical Background

different DIC analysis parameters also lead to different strains and therewith different time and image when a particular strain is reached. In this case, the investigation is carried out on the same image (same force resp). 3. The subset size is varied with respect to a minimum subset size to be at least three times the speckle size, and the step-to-subset ratio is chosen to be between 1/3 and ½, see Fig. 1.15. Additionally, the window size is varied. The product of each number of variation from each parameter gives the amount of parameter variations. Since the VSG represents the length over which local strains are measured, the resulting virtual strain gauge sizes LVSG should not be larger than the dimensions of the specimen (e.g. width or thickness). Figure 1.18b and d show the bias (mean) and noise-floor (standard deviation) of the QOI for different virtual strain gauge sizes LVSG. The larger virtual strain gauge size, the smaller the noise-floor. Maximum shear strain γ 12 shows comparable values for noise-floor and bias, see Fig. 1.18c, d. Two selected sets of DIC analysis parameters including one with higher precision (small LVSG) and one with less noise (larger LVSG) are compared. Virtual strain gauges with sizes LVSG ¼ 45 px (VSG(Lsubset/Lstep/Lwindow ¼ 15/ 5/7)) and LVSG ¼ 99 px (VSG(15/6/15)) are highlighted in Fig. 1.18b. Figure 1.19 Fig. 1.19 Distribution of strain-based QOI. (a) LVSG ¼ 45 px, VSG(15/5/7), (b) LVSG ¼ 99 px, VSG (15/6/15)

a

b

Digital Image Correlation

35

illustrates the distribution of a strain-based QOI of the two selected sets of DIC analysis parameters (with equal subset size) with the smaller VSG depicting higher fluctuations and the larger VSG leading to a reduction in blurring in strain. 4. Large strain gradients might be located close to the edge of the specimen, where location and magnitude are affected by the subset size and the corresponding reduced post-analysis ROI. Therefore, the focus can be along, e.g. the centerline of the specimen. An extraction of the strain gradients along a line for an image at large deformation will be presented in detail in the subsequent chapters. Smaller VSG sizes typically result in a larger noise but better-resolved gradients, while larger VSG sizes typically result in smaller noise but more smoothed strain gradients. 5. As will be presented in the subsequent chapters, the aim of the virtual strain gauge study is to find a trade-off of the competing objectives, i.e. low strain noise and better-resolved strain gradients, by examining the convergence between noise and strain gradient to find an appropriate signal-to-noise ratio. In accordance with (Reu, 2015f), a lack of convergence can occur at cracks, other failure points and at large strain gradients. A trade-off diagram represents a graphical procedure for a multi-objective decision analysis for optimization, with options (amount of parameter variations) and possible other criteria or constraints (boundary conditions). The boundary conditions can include a specific signal-to-noise ratio, maximum LVSG, etc. Optimization represents a compromise, an optimal balance, resp., with respect to the relative importance of conflicting objectives. Multiple “good” decisions are possible, depending on the amount and distinction of choices. For further information on different methods for uncertainty quantification in DIC, see (iDICs, 2021).

Patterning Methods A prerequisite for the digital correlation process is that the pattern of uniform and small thickness on the surface of the specimen, follows the deformation of the underlying specimen (Reu, 2015e). The pattern is a fundamental turnkey considering the spatial resolution since it is dictating the smallest possible subset size (Reu, 2015f). The speckle pattern has to follow four attributes, such as size, contrast, edge sharpness, and density (spatial distribution, resp.) (Reu, 2014f). As shown in Fig. 1.15, the speckle size should be small, but not too small, whereas a speckle size chosen too big leads to a waste in spatial resolution, due to the correlation between subset size and speckle size. Based on the sampling theorem and aliasing, in order to unambiguously locate the center of a speckle to a single pixel, the minimum speckle pattern feature size should be 3 px (also considering the gap in between the speckles) (Reu, 2014a), the optimum range being 3 px–5 px (iDICs, 2018). Details

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on different types of speckle patterns can be looked up in (Reu, 2014a, 2014b, 2014f, 2015a, 2015b, 2015c, 2015e). Common patterning methods include the use of marker/pencil, stamp roller, toner powder (while the base paint is still wet to ensure proper adhesion), spray can and airbrush for spray-painting, and printed stickers with synthesized, optimized patterns (Bossuyt, 2013) (rather for motion than for deformation), only to name a few, see also (Iadicola, 2013; Reu, 2014f, 2015c). Recommended attributes of the pattern include matte, high gray value contrast, relatively fine speckles (depends on camera used, at least 3 px) (Reu, 2014a), isotropic (no directionality (Reu, 2014b)), non-periodic and non-repetitive (random), stochastic distribution (equally sized dark and light areas) and with a soft edge (Reu, 2015b). Examples for inappropriate patterns include unbalanced density of dark and light speckles, small speckles leading to aliasing, bad contrast and directionality, see also (Reu, 2014a, 2014b, 2015a, 2015c, 2015e). Furthermore, the contrast of the pattern should not change during the measurement, where (no variable) lighting is a key issue, with diffuse lighting and less heating (e.g. LED) being preferred (Reu, 2013e). For paint-based patterns, the ductility of the paint should be aligned with the expected deformation of the specimen. The paint should be ductile for large deformations, so that it stretches with the underlying specimen without cracking or flaking. Therefore, the test should be executed immediately after painting (iDICs, 2018). For small deformations, i.e. brittle specimens, and observation of crack propagation, the paint should be as brittle, while still not flaking or cracking independently of the specimen. Therefore, the paint should be allowed to fully cure to make it more brittle. Alternatively, individual features (e.g. speckles) can be placed directly on the test piece, without a base coat of paint, so that cracking of the specimen can be observed directly (iDICs, 2018). In the following, three patterning methods, based on the commonly used spraypainting and a printed pattern, will be discussed and a methodology will be presented for a quantification of the approximate feature properties. The sticker with printed (synthesized) pattern has been directly placed on the specimen (the test piece of interest, resp.). The specimens for spray-painting were first prepared with commercially available flat/matte white spray-paint as a light base coat that has been applied directly onto the specimen and dried. The first method includes overspray-painting, thereby matte black spray-paint has been applied by intended overspray (drizzle or indirect spraying) to create a random pattern. This has been realized by spraying the area close to the inclined placed specimen and using the drift of the overspray to create a fine speckle pattern. The second method, aiming for a finer speckle pattern, has been done with an airbrush set. Even though overspray and airbrush are used for achieving a fine speckle pattern, it is common that the mist causes an area compromising the contrast, creating small speckles that lead to aliasing (Reu, 2015c). Both spray methods allow for a relatively broad spectrum of speckle sizes, depending on the distance to the specimen and pressure of compressed air, and therefore recommend practicing. The resulting as-applied patterns are shown in Fig. 1.20, taken using a stereo-DIC configuration and 9 Mpx cameras (8-bit) (same setup for spray-based patterns). The

Digital Image Correlation

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Fig. 1.20 Pattern (left) and gray level intensity distribution (right) for (a) printed, (b) overspray, (c) airbrush

gray level distribution of each pattern is shown in Fig. 1.20 (right), for a square section at the center of each pattern (across the height of 2.4 mm for spray-painted speckle patterns). The dynamic range of the image is divided into 256 gray levels (28 ¼ 256 from 0 to 255, based on 8-bit), represented by each pixel. Note that differences in the distribution are results of lighting, aperture, and exposure times. As a rough estimate, the lighting should lead at least 50 gray level counts between light and dark (with the difference defining the contrast (Reu, 2012a, 2015a)), i.e. 20% contrast, and 50% contrast for 130 gray level counts for an 8-bit camera (Reu, 2013c, 2013e, 2015a). As can be seen from Fig. 1.20, the finer the pattern, the

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1 Introduction and Theoretical Background

lower the contrast, with the printed pattern having the highest contrast (with an almost binary pattern). There are three common options for the determination of pattern feature sizes, i.e. by eye, autocorrelation (the most common) and segmentation (which is part of standard image processing codes) (Reu, 2014b). In the following, a thorough investigation of the approximate pattern feature properties is presented via the line intersection method, also used for calculating the fiber volume fraction of composites (Carlsson & Pipes, 1987) and the grain size for metallic and ceramic materials (Macherauch & Zoch, 2011). The approximate pattern feature size is determined using a specific inspection length divided by the number of intersections minus one (the denominator of the ratio representing the number of dark and light pattern features). The methodology is shown in Fig. 1.21 for all patterning methods, with the line profile taken across the height of the spray-painted specimens and the mean intensity shown in Fig. 1.20. Besides the mean intensity, the linear regression of the line profile (“adjusted mean”), shown in Fig. 1.21, accounts for inhomogeneous lighting along the inspection length. Only the intersections between the line profile and the adjusted mean of the inspection length are shown in Fig. 1.21. Furthermore, the slope of the line profile at the intersection is used to determine the ratio dark/light (relating to contrast) as well as the approximate size of dark and light features, see Fig. 1.21 (right). Positive slope at the intersection denotes crossing from dark to light, whereas a negative slope denotes crossing from light to dark, see Fig. 1.21. Table 1.5 lists the results of the determination of the pattern feature characteristics for all patterning methods. The physical feature size is determined by dividing by the image scale, being 78.2 px/mm (0.0128 mm/px) for the spay-painted patterns and 36 px/mm (0.0278 mm/px) for the printed pattern. The inverse of the pattern feature size determines the spatial frequency. For the spray-painted patterns, speckle sizes below 3 px should not be ignored but the determined absolute size should be interpreted with caution, since it is not possible to determine whether a speckle size is 2 px or smaller because of aliasing (Reu, 2014a). If the pattern contains enough appropriately sized speckles, some undersized speckles can be tolerated but will certainly increase noise (Reu, 2014b). Spray-paints in particular cause too much gray in between the speckle causing a loss in contrast (Reu, 2015b), as can be seen in Fig. 1.20 (right). None of the spray-paintbased methods allows for an exact application with a tight distribution of the speckle size (Reu, 2014a), see the high fluctuations, coefficient of variation (CV), resp.

Practice Exercises This section provides several practice exercises including the use of DIC measured data for self-calculation with the data reduction methodologies presented in this chapter. Solutions, together with supplementary instructional resources (e.g. figures, videos, data reduction codes, slides with deeper explanations of selected expressions

Practice Exercises Line Profile Mean

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c Fig. 1.21 Line profile and intersection method (left) and distribution of dark and light speckle pattern (right) for (a) printed, (b) overspray, (c) airbrush

Table 1.5 Approximate feature characteristics from different patterning methods Method Printed Overspray Airbrush

Ratio (dark/light) 46/54 60/40 51/49

Length in px (dark/light) 13.9/14.2 11/5.5 4.5/5

Length in mm (dark/light) 0.385/0.395 0.141/0.071 0.057/0.064

CV in % (dark/light) 23/65 71/76 75/61

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1 Introduction and Theoretical Background

and deduction of equations) suitable for lecturing or lab courses are available to instructors who adopt the book for classroom use. Please visit the book web page at www.springer.com for the password-protected material. Use the material parameters of UD-FRP. Calculate and visualize the fiber orientation (0
–90
) dependent 1. Inverse compliance in tension. 2. Inverse compliance in shear. 3. Compliance ratio describing Poisson’s ratio. 4. Compliance ratio describing extension-shear coupling. 5. Longitudinal strength. Use the material parameters of crossply FRP. Calculate and visualize the fiber orientation (0
–90
) dependent 6–10. Apply questions 1–5 Use the DIC configuration and geometric details of a coupon specimen from tensile testing. 11. Calculate the image scale for a portrait orientation of the cameras (the short and long axes of the detector align with the short and long axes of the specimen). 12. Determine the FOV of each camera. 13. Estimate the expected resolution accuracy. 14. Determine the DIC configuration parameters. 15. Estimate the single parameters of the virtual strain gauge. 16–20. Apply questions 11–15 for landscape orientation of the cameras (the long and short axes of the detector align with the short and long axes of the specimen). Acknowledgments Louise Powell and Qi An for uniaxial tensile results of epoxy specimens. Bill Luecke and Carlos Beauchamp for supporting the profilometry measurements. Steven Mates and Christopher Calhoun for speckle pattern techniques. Dave Pitchure, Ed Pompa and Evan Rust for waterjet cutting the specimens.

References Adams, D. F., & Walrath, D. E. (1987). Current status of the iosipescu shear test method. Journal of Composite Materials, 21(6), 494–507. https://doi.org/10.1177/002199838702100601 Adams, D. O., Moriarty, J. M., Gallegos, A. M., & Adams, D. F. (2003). Development and evaluation of the V-notched rail shear test for composite laminates. U.S. Department of Transportation Federal Aviation Administration. Adams, D. O., Moriarty, J. M., Gallegos, A. M., & Adams, D. F. (2007). The V-notched rail shear test. Journal of Composite Materials, 41(3), 281–297. https://doi.org/10.1177/ 0021998306063369 Agarwal, B. D., Broutman, L. J., & Chandrashekhara, K. (2006). Analysis and performance of fiber composites. Wiley. Ajovalasit, A. (2011). Advances in strain gauge measurement on composite materials. Strain, 47(4), 313–325. https://doi.org/10.1111/j.1475-1305.2009.00691.x

References

41

An, Q., Merzkirch, M., & Forster AM (2018). Characterizing fiber reinforced polymer composites shear behavior with digital image correlation. In American Society for Composites—Thirty-third technical conference, Seattle, WA, September 24–26. DEStech Publications Inc. ASTM. (2012a). ASTM D5379/D5379M—Standard test method for shear properties of composite materials by the V-notched beam method. ASTM International. https://doi.org/10.1520/d5379_ d5379m-12 ASTM. (2012b). ASTM D7078/D7078M—Standard test method for shear properties of composite materials by V-notched rail shear method. ASTM International. https://doi.org/10.1520/d7078_ d7078m-12 ASTM. (2013). ASTM D3518/D3518M—Standard test method for in-plane shear response of polymer matrix composite materials by tensile test of a 45
laminate. ASTM International. https://doi.org/10.1520/d3518_d3518m-13 ASTM. (2015a). ASTM D3846—Standard test method for in-plane shear strength of reinforced plastics. ASTM International. https://doi.org/10.1520/D3846-08R15 ASTM. (2015b). ASTM D4255/D4255M—Standard test method for in-plane shear properties of polymer matrix composite materials by the rail shear method. ASTM International. https://doi. org/10.1520/D4255_D4255M-15 ASTM. (2016a). ASTM D2344/D2344M—Standard test method for short-beam strength of polymer matrix composite materials and their laminates. ASTM International. https://doi.org/10.1520/ D2344_D2344M-16 ASTM. (2016b). ASTM D4762—Standard guide for testing polymer matrix composite materials. ASTM International. https://doi.org/10.1520/D4762-16 ASTM. (2016c). ASTM D5448/D5448M—Standard test method for inplane shear properties of hoop wound polymer matrix composite cylinders. ASTM International. https://doi.org/10.1520/ D5448_D5448M-16 ASTM. (2018). ASTM D3878—Standard terminology for composite materials. ASTM International. https://doi.org/10.1520/D3878-18 Bossuyt, S. (2013). Optimized patterns for digital image correlation. In 2012 Annual Conference on Experimental and Applied Mechanics. Imaging methods for novel materials and challenging application. Bru, T., Olsson, R., Gutkin, R., & Vyas, G. M. (2017). Use of the iosipescu test for the identification of shear damage evolution laws of an orthotropic composite. Composite Structures, 174, 319–328. https://doi.org/10.1016/j.compstruct.2017.04.068 Brunner, A. J (2020). Fracture mechanics characterization of polymer composites in aerospace applications. In Polymer composites in the aerospace industry (pp. 191–230). https://doi.org/10. 1016/B978-0-08-102679-3.00008-3 Carlsson, L. A., Adams, D. F., & Pipes, R. B. (2014). Experimental characterization of advanced composite materials (4th ed.). CRC Press. Carlsson, L. A., & Pipes, R. B. (1987). Experimental characterization of advanced composite materials. Prentice-Hall. Chamis, C. C., & Sinclair, J. H. (1977). Ten-deg off-axis test for shear properties in fiber composites. Experimental Mechanics, 17(9), 339–346. https://doi.org/10.1007/bf02326320 Chamis, C. C., & Sinclair, J. H. (1978). Mechanical behaviour and fracture characteristics of off-axis fiber composites. II—Theory and comparisons. Technical Paper. NASA—National Aeronautics and Space Administration. Chatterjee, S. N., Adams, D. F., & Oplinger, D. W. (1993a). Test methods for composites: A status report. Volume I: Tension test methods. U.S. Department of Transportation Federal Aviation Administration. Chatterjee, S. N., Adams, D. F., & Oplinger, D. W. (1993b). Test methods for composites: A status report. Volume II: Compression test methods. U.S. Department of Transportation Federal Aviation Administration.

42

1 Introduction and Theoretical Background

Chatterjee, S. N., Adams, D. F., & Oplinger, D. W. (1993c). Test methods for composites: A status report. Volume III: Shear test methods. U.S. Department of Transportation Federal Aviation Administration. Chawla, K. K. (2012). Composite materials (3rd ed.). Springer. https://doi.org/10.1007/978-0-38774365-3 Cui, W. C., Wisnom, M. R., & Jones, M. (1992). Failure mechanisms in three and four point short beam bending tests of unidirectional glass/epoxy. Journal of Strain Analysis for Engineering Design, 27(4), 235–243. Daiyan, H., Andreassen, E., Grytten, F., Osnes, H., & Gaarder, R. H. (2012). Shear testing of polypropylene materials analysed by digital image correlation and numerical simulations. Experimental Mechanics, 52(9), 1355–1369. https://doi.org/10.1007/s11340-012-9591-7 Davies, P., Blackman, B. R. K., & Brunner, A. J. (1998). Standard test methods for delamination resistance of composite materials: current status. Applied Composite Materials, 5(6), 345–364. https://doi.org/10.1023/A:1008869811626 DIN. (1998). DIN EN ISO 14130—Fibre-reinforced plastic composites—Determination of apparent interlaminar shear strength by short-beam method. CEN. DOWAKSA. (2016). 24K A-42 technical data sheet. Filon, L. N. G. (1912). The investigation of stresses in a rectangular bar by means of polarized light. Philosophical Magazine, 23(133–8). https://doi.org/10.1080/14786440108637197 Grédiac, M., Pierron, F., & Vautrin, A. (1994). The iosipescu in-plane shear test applied to composites: A new approach based on displacement field processing. Composites Science and Technology, 51(3), 409–417. https://doi.org/10.1016/0266-3538(94)90109-0 Gross, D., Hauger, W., Schröder, J., Wall, W. A., & Bonet, J. (2018). Engineering mechanics 2 (2nd ed.). Springer. https://doi.org/10.1007/978-3-662-56272-7 Gross, D., & Seelig, T. (2011). Fracture Mechanics - With an Introduction to Micromechanics. Mechanical Engineering Series, (2nd ed.). Springer. https://10.1007/978-3-642-19240-1 Haka, A. (2011). Wings of “Black Gold”. History of fibre-reinforced materials. NTM, 19(1), 69–105. https://doi.org/10.1007/s00048-011-0047-4 Herakovich, C. T. (1998). Mechanics of fibrous composites. John Wiley & Sons Inc.. Herakovich, C. T. (2012). Mechanics of composites: A historical review. Mechanics Research Communications, 41, 1–20. https://doi.org/10.1016/j.mechrescom.2012.01.006 Iadicola, M. A. (2013). Augmented use of standard mechanical testing measurements for sheet metal forming: digital image correlation for localized necking. In: Numisheet 2014: The 9th International conference and workshop on numerical simulation of 3d sheet metal forming processes: Part a Benchmark problems and results and Part B General papers (pp. 614–619). AIP Publishing LLC. iDICs. (2018). International digital image correlation society—A good practices guide for digital image correlation. International Digital Image Correlation Society iDICs. https://doi.org/10. 32720/idics/gpg.ed1 iDICs. (2021). Digital image correlation standards, training, and global networking. http://www. idics.org/. Iosipescu, N. (1967). New accurate procedure for single shear testing of metals. Journal of Materials, 2(3), 537–566. ISO. (1998). ISO 14129—Fibre-reinforced plastic composites—Determination of the in-plane shear stress/shear strain response, including the in-plane shear modulus and strength, by the +/-45
tension test method. CEN ISO. (2005). ISO 15310—Fibre-reinforced plastic composites—Determination of the inplane shear modulus by the plate twist method. CEN. ISO. (2018a). ISO 19927—Fibre-reinforced plastic composites—Determination of interlaminar strength and modulus by double beam shear test. ISO. ISO. (2018b). ISO 20337—Fibre-reinforced plastic composites—Shear test method using a shear frame for the determination of the in-plane shear stress/shear strain response and shear modulus. ISO.

References

43

Jones, R. M. (1999). Mechanics of composite materials. Taylor & Francis. Lee, S., & Munro, M. (1986). Evaluation of in-plane shear test methods for advanced composite materials by the decision analysis technique. Composites, 17(1), 13–22. https://doi.org/10.1016/ 0010-4361(86)90729-9 Lekhnitskii, S. G. (1968). Anisotropic plates ([by] S.G. Lekhnitskii. Translated from the Second Russian Edition by S.W. Tsai and T. Cheron). Gordon and Breach. Macherauch, E., & Zoch, H.-W. (2011). Praktikum in Werkstoffkunde. Vieweg + Teubner. Melin, L. G., Neumeister, J. M., Pettersson, K. B., Johansson, H., & Asp, L. E. (2000). Evaluation of four composite shear test methods by digital speckle strain mapping and fractographic analysis. Journal of Composites Technology and Research, 22(3), 161–172. https://doi.org/ 10.1520/CTR10636J Melin, L. N., & Neumeister, J. M. (2006). Measuring constitutive shear behavior of orthotropic composites and evaluation of the modified iosipescu test. Composite Structures, 76(1-2), 106–115. https://doi.org/10.1016/j.compstruct.2006.06.016 Merzkirch, M., An, Q., & Forster, A. M. (2019). In-plane shear response of GFRP laminates by 45
and 10
off-axis tensile testing using digital image correlation. In: American Society for Composites—Thirty-fourth technical conference, Atlanta, GA. DEStech Publications Inc. Merzkirch, M., & Foecke, T. (2020). 10
Off-axis testing of CFRP using DIC: A study on strength, strain and modulus. Composites Part B: Engineering, 196. https://doi.org/10.1016/j. compositesb.2020.108062 Merzkirch, M., Pinter, P., Dietrich, S., & Weidenmann, K. A. (2015). Interpenetrating freeze cast composites: correlation between structural and mechanical characteristics. Materials Science Forum, 825–826, 109–116. https://doi.org/10.4028/www.scientific.net/MSF.825-826.109 Merzkirch, M., Siebert, T., Weikert, T., & Witzel, O. (2020). Praxisleitfaden für die Digitale Bildkorrelation. [A good practices guide for digital image correlation]. https://doi.org/10. 32720/idics/gpg.ed1.de Pagano, N. J., & Halpin, J. C. (1968). Influence of end constraint in the testing of anisotropic bodies. Journal of Composite Materials, 2(1), 18–31. https://doi.org/10.1177/002199836800200102 Pagano, N. J., & Pipes, R. B. (1971). The influence of stacking sequence on laminate strength. Journal of Composite Materials, 5, 50–57. Pagano, N. J., & Pipes, R. B. (1978). Some observations on the interlaminar strength of composite laminates. International Journal of Mechanical Sciences, 15, 679–688. Palmer, R. J. (2012). History of composites in aeronautics. Wiley Encyclopedia of Composites. https://doi.org/10.1002/9781118097298.weoc107 Pierron, F., & Grédiac, M. (2012). The virtual fields method. Springer. https://doi.org/10.1007/9781-4614-1824-5 Pierron, F., & Vautrin, A. (1996). The 10
off-axis tensile test: a critical approach. Composites Science and Technology, 56(4), 483–488. https://doi.org/10.1016/0266-3538(96)00004-8 Pindera, M. J., Choksi, G., Hidde, J. S., & Herakovich, C. T. (1987). A methodology for accurate shear characterization of unidirectional composites. Journal of Composite Materials, 21(12), 1164–1184. https://doi.org/10.1177/002199838702101205 Pipes, R. B., & Pagano, N. J. (1970). Interlaminar stresses in composite laminates under uniform axial extension. Journal of Composite Materials, 4, 538–548. Powell, L. A., Luecke, W. E., Merzkirch, M., Avery, K., & Foecke, T. (2017). High strain rate mechanical characterization of carbon fiber reinforced polymer composites using digital image correlations. SAE International Journal of Materials and Manufacturing, 10(2), 138–146. https://doi.org/10.4271/2017-01-0230 Puck, A., & Schürmann, H. (1998). Failure analysis of FRP laminates by means of physically based phenomenological models. Composites Science and Technology, 58, 1045–1067. https://doi. org/10.1016/S0266-3538(96)00140-6 Purslow, D. (1977). The shear properties of unidirectional reinforced plastics and their experimental carbon fibre plastics and their determination. MoD Procurement Executive—Aeronautical Research Council.

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1 Introduction and Theoretical Background

Reddy, J. N. (2004). Mechanics of laminated composite plates and shells—Theory and analysis (2nd ed.). CRC Press. https://doi.org/10.1201/b12409 Relland, J., Blaysat, B., Amyot, C.-O., Passieux, J.-C., Réthoré, J., & Merzkirch, M. (2020). Guide des Bonnes Pratiques de la Corrélation d’Images Numériques. [A good practices guide for digital image correlation]. https://doi.org/10.32720/idics/gpg.ed1.fr Reu, P. (2012a). Hidden components of 3D-DIC: Interpolation and matching—Part 2. Experimental Techniques, 36(3), 3–4. https://doi.org/10.1111/j.1747-1567.2012.00838.x Reu, P. (2012b). Hidden components of 3D-DIC: Triangulation and post-processing—Part 3. Experimental Techniques, 36(4), 3–5. https://doi.org/10.1111/j.1747-1567.2012.00853.x Reu, P. (2012c). Hidden components of DIC: Calibration and shape function—Part 1. Experimental Techniques, 36, 3–5. Reu, P. (2012d). Introduction to digital image correlation: best practices and applications. Experimental Techniques, 36(1), 3–4. https://doi.org/10.1111/j.1747-1567.2011.00798.x Reu, P. (2012e). Stereo-rig design: Camera selection—Part 2. Experimental Techniques, 36, 3–4. Reu, P. (2012f). Stereo-rig design: Creating the stereo-rig layout—Part 1. Experimental Techniques, 36(5), 3–4. https://doi.org/10.1111/j.1747-1567.2012.00871.x Reu, P. (2013a). Calibration: 2D calibration. Experimental Techniques, 37(5), 1–2. https://doi.org/ 10.1111/ext.12027 Reu, P. (2013b). Calibration: A good calibration image. Experimental Techniques, 37(6), 1–3. https://doi.org/10.1111/ext.12059 Reu, P. (2013c). Calibration: Pre-calibration routines. Experimental Techniques, 37(4), 1–2. https:// doi.org/10.1111/ext.12026 Reu, P. (2013d). Stereo-rig design: Lens selection—Part 3. Experimental Techniques, 37(1), 1–3. https://doi.org/10.1111/ext.12000 Reu, P. (2013e). Stereo-rig design: Lighting—Part 5. Experimental Techniques, 37(3), 1–2. https:// doi.org/10.1111/ext.12020 Reu, P. (2013f). Stereo-rig design: Stereo-angle selection—Part 4. Experimental Techniques, 37(2), 1–2. https://doi.org/10.1111/ext.12006 Reu, P. (2013g). A study of the influence of calibration uncertainty on the global uncertainty for digital image correlation using a Monte Carlo approach. Experimental Mechanics, 53(9), 1661–1680. https://doi.org/10.1007/s11340-013-9746-1 Reu, P. (2014a). All about speckles: Aliasing. Experimental Techniques, 38(5), 1–3. https://doi.org/ 10.1111/ext.12111 Reu, P. (2014b). All about speckles: Speckle size measurement. Experimental Techniques, 38(6), 1–2. https://doi.org/10.1111/ext.12110 Reu, P. (2014c). Calibration: Care and feeding of a stereo-rig. Experimental Techniques, 38(3), 1–2. https://doi.org/10.1111/ext.12083 Reu, P. (2014d). Calibration: Sanity checks. Experimental Techniques, 38(2), 1–2. https://doi.org/ 10.1111/ext.12077 Reu, P. (2014e). Calibration: Stereo calibration. Experimental Techniques, 38(1), 1–2. https://doi. org/10.1111/ext.12048 Reu, P. (2014f). Speckles and their relationship to the digital camera. Experimental Techniques, 38(4), 1–2. https://doi.org/10.1111/ext.12105 Reu, P. (2015a). All about speckles: Contrast. Experimental Techniques, 39(1), 1–2. https://doi.org/ 10.1111/ext.12126 Reu, P. (2015b). All about speckles: Edge sharpness. Experimental Techniques, 39(2), 1–2. https:// doi.org/10.1111/ext.12139 Reu, P. (2015c). All about speckles: Speckle density. Experimental Techniques, 39(3), 1–2. https:// doi.org/10.1111/ext.12161 Reu, P. (2015d). DIC: A revolution in experimental mechanics. Experimental Techniques, 39(6), 1–2. Reu, P. (2015e). Points on paint. Experimental Techniques, 39(4), 1–2. https://doi.org/10.1111/ext. 12147

References

45

Reu, P. (2015f). Virtual strain gage size study. Experimental Techniques, 39(5), 1–3. https://doi. org/10.1111/ext.12172 Rossi, M., Lava, P., Pierron, F., Debruyne, D., & Sasso, M. (2015). Effect of DIC spatial resolution, noise and interpolation error on identification results with the VFM. Strain, 51(3), 206–222. https://doi.org/10.1111/str.12134 Scala, E. P. (1996). A brief history of composites in the U.S.-The dream and the success. JOM:45–48. Schürmann, H. (2007). Konstruieren mit Faser-Kunststoff-Verbunden. Springer. Shanmugam, D. K., Nguyen, T., & Wang, J. (2008). A study of delamination on graphite/epoxy composites in abrasive waterjet machining. Composites Part A: Applied Science and Manufacturing, 39(6), 923–929. https://doi.org/10.1016/j.compositesa.2008.04.001 Sideridis, E., & Papadopoulos, G. A. (2004). Short-beam and three-point-bending tests for the study of shear and flexural properties in unidirectional-fiber-reinforced epoxy composites. Journal of Applied Polymer Science, 93(1), 63–74. https://doi.org/10.1002/app.20382 Su, X., & Wagner, D. (2019). Integrated computational materials engineering development of carbon fiber composites for lightweight vehicles—Ford motor company. https://doi.org/10. 2172/1502875 Summerscales, J. (1987). Shear modulus testing of composites. In I. H. Marshall (Ed.), Composite structures 4, volume 2: Damage assessment and material evaluation (pp. 305–316). Elsevier. https://doi.org/10.1007/978-94-009-3457-3_23 Sun, C. T., & Berreth, S. P. (1988). A new end tab design for off-axis tension test of compositematerials. Journal of Composite Materials, 22(8), 766–779. https://doi.org/10.1177/ 002199838802200805 Sutton, M. A., Matta, F., Rizos, D., Ghorbani, R., Rajan, S., Mollenhauer, D. H., Schreier, H. W., & Lasprilla, A. O. (2017). Recent progress in digital image correlation: Background and developments since the 2013 WM Murray lecture. Experimental Mechanics, 57(1), 1–30. https://doi. org/10.1007/s11340-016-0233-3 Sutton, M. A., Orteu, J.-J., & Schreier, H. W. (2009). Image correlation for shape. Motion and Deformation Measurements. https://doi.org/10.1007/978-0-387-78747-3 Sutton, M. A., Yan, J. H., Tiwari, V., Schreier, H. W., & Orteu, J.-J. (2008). The effect of out-ofplane motion on 2D and 3D digital image correlation measurements. Optics and Lasers in Engineering, 46(10), 746–757. https://doi.org/10.1016/j.optlaseng.2008.05.005 Timoshenko, S. P. (1953). History of strength of materials. Dover Civil and Mechanical Engineering, Tsai, S. W. (2005). Three decades of composites activities at US Air Force materials laboratory. Composites Science and Technology, 65(15-16), 2295–2299. https://doi.org/10.1016/j. compscitech.2005.05.017 Tsai, S. W., & Wu, E. M. (1971). A general theory of strength for anisotropic materials. Journal of Composite Materials, 5(1), 58–80. https://doi.org/10.1177/002199837100500106 Yeow, Y. T., & Brinson, H. F. (1978). A comparison of simple shear characterization methods for composite laminates. Composites, 9(1), 49–55. https://doi.org/10.1016/0010-4361(78)90519-0 Zweben, C., Smith, W. S., & Wardle, M. W. (1979). Test methods for fiber tensile strength, composite flexural modulus, and properties of fabric-reinforced laminates. In S. W. Tsai (Ed.), Composite materials: Testing and design (fifth conference), ASTM STP 674 (pp. 228–262). American Society for Testing and Materials.

Chapter 2

Tensile Testing

Background Starting in the late 1940s, extensive tensile (off-axis) testing (Erickson & Norris, 1955; Werren & Norris, 1956) on glass fiber reinforced polymer (GFRP) sheet panels in different directions with respect to the fiber orientation were performed on non-tabbed dogbone specimens to determine the directional properties and were compared to analytical solutions for the orientation dependent moduli and strengths. A classification of the orientation (θ) dependent failure modes of GFRP (Sinclair & Chamis, 1977; Chamis & Sinclair, 1978) confirmed that intralaminar shear (matrix shear fracture) appears in the θ ¼ 5 –20 (Chamis & Sinclair, 1978) (θ ¼ 5 –30 (Sinclair & Chamis, 1977)) load-fiber range, whereas mixed mode (intralaminar shear and transverse tensile) failure appears up to a load-fiber angle of 45 (Sinclair & Chamis, 1977). Small orientation angles of θ ¼ 10 –20 (Pierron & Vautrin, 1996) (θ ¼ 10 –15 (Chatterjee et al., 1993)) lead to shear-dominated failures. Investigations on off-axis loading mainly focused on the determination of elastic properties, until in the earlier 1970s, shear strength related values were investigated (Pipes & Cole, 1973). Further investigations on the orientation dependent intralaminar shear strain lead to the result that it approaches maximum under θ ¼ 10 (Chamis & Sinclair, 1976, 1977). Based on the amount of induced shear stress, researchers (Daniel & Liber, 1975; Chamis & Sinclair, 1977; Pindera & Herakovich, 1986) have converged to an off-axis angle of θ ¼ 10 and θ ¼ 45 (Ganesh & Naik, 1997), even though the latter only provides a value for stiffness (Pindera et al., 1987). Since then, the 10 off-axis tensile test represents a common method for determining the intralaminar shear properties such as modulus and strength due to the actual shear stress at failure (see also Chap. 6). According to (Chamis & Sinclair, 1977), two of eleven listed advantages for using 10 off-axis testing are the applicability of the test for fatigue and high-rate loading. Furthermore, a familiar test procedure for the determination of the © Springer Nature Switzerland AG 2022 M. Merzkirch, Mechanical Characterization Using Digital Image Correlation, https://doi.org/10.1007/978-3-030-84040-2_2

47

48

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Tensile Testing

Fig. 2.1 (a) Schematic specimen with dimensional parameters, (b) scheme of a loaded off-axis specimen with indication of bending moments and shear forces due to axial forces (for the fiber orientation shown)

quasi-static tensile properties on specimens with fiber orientations θ ¼ 0 and 90 can be used, (ASTM, 2017) and (DIN, 2010), first standardized in 1971. In references (Chamis & Sinclair, 1976, 1977) a comparison between 10 off-axis tensile testing, 45 laminate tensile testing and thin tube torsion testing revealed that off-axis tensile testing was found to have the highest strength and a comparable strain at failure as the thin tube torsion test. The tensile test method of a 45 crossply laminate (see also (ISO, 1998)) has been first standardized in 1976 (ASTM, 2013). For tensile testing, the nominal stress in the loading axis refers the applied force F to the cross-sectional area with width w and thickness h (see Fig. 2.1a) σx ¼

F w∙h

ð2:1Þ

The minimal aspect ratio (length-to-width) for fracture within the gauge length L is L/w (min) ¼ 5.7 for a fiber orientation with respect to the specimen’s loading (x-) axis of θ ¼ 10 acc. to (see Fig. 2.1a)

Background

49

Fig. 2.2 Mohr’s circle for stress for off-axis tensile testing

  L ¼ cot θ w min

ð2:2Þ

For the determination of the shear stress, Fig. 2.2 depicts the coordinate transformation on basis of Mohr’s circle for stress. According to (Pindera & Herakovich, 1986), the shear stress in the ξ–η coordinate system at the midpoint of the specimen is
1 τξη ¼  ∙ σ x  σ y ∙ sin 2θ þ τxy ∙ cos 2θ 2

ð2:3Þ

with τxy being the shear stress in the laminate coordinate system induced by the end constraints in the presence of extension–shear coupling and the transverse stress σ y ¼ 0. Compare also to strain transformation relations in Chap. 1. In practice, the shear stress contribution (second term in Eq. (2.3)) and the non-uniformity in the tensile stress across the width of the specimen is usually ignored. This leads to an apparent shear stress in the 1-2 plane, neglecting the “parasitic” (Pierron & Vautrin, 1996) shear stress in the second term of Eq. (2.3) τ12 ð¼ τ21 Þ ¼

1 ∙ σ ∙ sin 2θ 2 x

ð2:4Þ

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Tensile Testing

which is larger in magnitude than the shear stress expressed by Eq. (2.3) (Pindera & Herakovich, 1986). The apparent shear modulus therefore represents an upper bound. Finite element investigations for the calculation of the stress distribution have been done in (Pierron & Vautrin, 1996; Nemeth et al., 1983; Sun & Chung, 1993). A state of pure shear relative to the principal material axes cannot be produced using off-axis testing (Sun & Berreth, 1988) due to a large extension–shear coupling effect that anisotropic materials inhibit. The most prominent extension–shear coupling occurs between θ ¼ 10 –15 for graphite/epoxy systems (Sun & Berreth, 1988). Since the early 1940s, off-axis loading of anisotropic materials (e.g., wood (Hearmon, 1943)) has been investigated analytically (Hearmon, 1943; Pagano & Halpin, 1968) and numerically (Rizzo, 1969). The aforementioned extension–shear coupling could be experimentally demonstrated using nylon reinforced rubber specimens (Pagano & Halpin, 1968), due to the relatively soft material that is capable of sustaining deformations that can be detected visually. When the direction of applied traction does not coincide with the principal material axes (Wu & Thomas, 1968), i.e. the fibers are neither perpendicular nor parallel to the loading axis, the extension–shear coupling causes the upper and lower edges of the coupon specimen to rotate under an applied normal stress and causes the specimen to distort into a parallelogram for an unconstrained condition (Halpin & Pagano, 1968). Constant displacements of the ends, by constraining the upper and lower edges of the coupon specimen to remain horizontal and restraining in-plane rotation via rigid clamping, induce shear forces and superimposed bending couples (see Fig. 2.1b). This has been referred to “end-constraint effect” (Pindera & Herakovich, 1986), which results in a non-uniform deformation, leading to an “Sshape” (Rizzo, 1969) due to extension–shear coupling based on anisotropic material behavior (Pagano & Halpin, 1968; Rizzo, 1969). Furthermore, this results in stress concentrations (Pagano & Halpin, 1968; Chang et al., 1984) that do not develop a pure shear condition within the test section (Ganesh & Naik, 1997) but a state of bi-axial tension and shear (Richards et al., 1969). An approximate analytical solution based on idealized boundary conditions, namely on the centerline of a rigidly clamped specimen with no end rotation, was first proposed by (Pagano & Halpin, 1968) for large specimens for the axial (U ) U ðx, yÞ ¼ εx ∙ x þ

  2  6 ∙ S16 ∙ εx ∙ y x x  ∙ S11 L L

ð2:5Þ

and lateral (V ) displacements V ðx, yÞ ¼

  2  S12 S ∙ε ∙x x x ∙ εx ∙ y þ 16 x ∙ 1  3 ∙ þ 2 ∙ S11 L L S11

ð2:6Þ

as a function of the elastic material properties (extension–extension coupling S12/S11, extension–shear coupling S16/S11), axial strain εx, and specimen geometry.

Background

51

Displacement gradients disappear for the gauge length L ! 1. For the calculation of the single compliances, see Chap. 1. The extent of the bending is affected by the specimen’s aspect ratio (Pagano & Halpin, 1968), off-axis orientation and the material parameters (degree of anisotropy) (Pindera & Herakovich, 1986; Nemeth et al., 1983). Due to the non-uniform stress field, the modulus E* is erroneous E ¼ E ∙

1 ð1  β Þ

ð2:7Þ

with β being a measure of the error between the apparent measured modulus E* and actual modulus E which is expressed by (Pagano & Halpin, 1968) β¼



3 ∙ S216

S11 ∙ 3 ∙ S66 þ 2 ∙ S11 ∙


L 2 w



ð2:8Þ

as a function of the elastic material properties (incl. shear compliance S66), aspect ratio of the specimen and assumptions of the approximate solution (Nemeth et al., 1983). This leads to a large aspect ratio L/w (Pagano & Halpin, 1968; Pindera & Herakovich, 1986; Rizzo, 1969; Nemeth et al., 1983) for the error to approach zero, resulting in the need for a large area of uniform material to generate the specimens. For the 10 off-axis test, generally slender coupon specimens with large aspect ratios L/w ¼ 12–15 (Sun & Berreth, 1988) are recommended. In (Pierron & Vautrin, 1996), an aspect ratio of L/w ¼ 9 was used, while in (Chamis & Sinclair, 1976, 1977), an aspect ratio of L/w ¼ 14 was recommended, and in (Pindera et al., 1987), an aspect ratio L/w ¼ 16 with rotating end-grip fixture was used. Gripping conditions of the specimen using complete rigid clamping with and without end (in-plane) rotation were investigated experimentally in (Wu & Thomas, 1968; Richards et al., 1969; Chang et al., 1984), analytically and numerically in (Rizzo, 1969). Non-rotating grips should be avoided when testing short specimens unless accurate strain gauge measurements can be made, acc. to (Rizzo, 1969). Reference (Pagano & Halpin, 1968) states that for off-axis testing of structural composites to provide reliable strength data, a suitable modification of the ends of the specimen is needed. Investigations concerning an improved loading condition, aiming for a uniform shear stress in the gauge section, include different materials and geometries for the tabs (see Fig. 2.1a). According to (Wu & Thomas, 1968; Richards et al., 1969), a strong, stiff in the thickness direction and compliant material between specimen and clamp can reduce the inhomogeneous state of stress. A gradual transition of the load from the clamps to the specimen is provided by long tapered tabs (Pipes & Cole, 1973; Chang et al., 1984). The use of fiberglass knit in silicon rubber was investigated in (Sun & Berreth, 1988), aiming for a shear deformation between the specimen and the grip, for the determination of shear modulus and strength while using hydraulic gripping. Woven fiberglass cloth was found to exhibit the desired characteristics (Sun & Berreth, 1988). Aluminum (Pierron & Vautrin,

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Tensile Testing

1996; Sun & Chung, 1993) and GFRP (Pierron & Vautrin, 1996) tabs with straight and oblique (by imposing displacements on the theoretical line of iso-displacements) geometry were used for the determination of shear strength (Pierron & Vautrin, 1996) and modulus (Sun & Chung, 1993) of CFRP. Oblique GFRP tabs lead to the highest values, while straight aluminum tabs show the lowest strength values (Pierron & Vautrin, 1996). Further details can be found in the tabbing guide for composite test specimens (Adams & Adams, 2002). In the past, strain gauge rosettes have been used to measure the shear strain, while between five (Chamis & Sinclair, 1977) up to nine (Wu & Thomas, 1968) strain gauges have been used to measure Young’s modulus and the strain non-uniformity across the specimen’s gauge section as well on the back of the specimen in order to eliminate bending errors (Sun & Berreth, 1988). See also Hiel (1987) and Ajovalasit (2011) for strain gauge measurements on composite materials. Beside a pure optical investigation of the occurring deformation and rotational effects during off-axis testing on nylon reinforced rubber (Pagano & Halpin, 1968) also Moiré interferometry was used to visualize and quantify the displacement field (Nemeth et al., 1983) for off-axis tensile testing.

Specimen Geometry Figure 2.3a shows rectangular coupon specimens that have been waterjet cut from a plaque of approximate size 300 mm  300 mm and an average thickness h ¼ 2.4 mm. Note that the fiber orientation is in vertical direction. For the initial preparation of the 10 off-axis direction from fiber orientation, the edge of the plaque served as reference for the determination of the angle (using a digital protractor). The adherent faying surfaces of the future specimen head areas and the tabs (made of woven fiberglass in an epoxy resin, “G11”) with a thickness of approx. 1.6 mm and a length of approx. 40 mm, see Fig. 2.1a) were lightly abraded using P400 silicon carbide abrasive paper. The chamfer of the tabs, see Fig. 2.3b, of approx. 16 (Adams & Adams, 2002) was formed by grinding. Metallic wires, with a diameter of approx. 220 μm, were used as spacers between plaque and the tab surfaces (see Fig. 2.3b) before applying a commercially available acrylic adhesive and cured at room temperature. Finally, the tabbed plaque has been waterjet cut (using abrasive 220). The geometric details of the coupon specimens are given in Table 2.1 in comparison to standard geometries for longitudinal (in θ ¼ 0 ) and transverse in (θ ¼ 90 ) fiber orientations. The specimens with an aspect ratio L/w ¼ 14 have been tested in clockwise fiber orientation (θ ¼ 10 ). Another aspect ratio of L/w ¼ 9 has been investigated, with coupon specimens of width w ¼ 19 mm (see Fig. 2.3a) and w ¼ 12.7 mm, with gauge length L ¼ 120 mm. The influence of the specimen geometry and fiber orientation (clockwise θ ¼ 10 and counterclockwise θ ¼ 10 ) will be presented in Effect of Specimen Geometry and Fiber Orientation Section.

Experimental Setup and DIC Configuration

53

Fig. 2.3 (a) UD plaque with fibers orientated in vertical direction and the waterjet cut tabbed specimens, (b) Tabbed specimen head (with 4 spacer wires) Table 2.1 Dimensions of the tensile specimens Orientation θ 10

Width w in mm 12.7 (Chamis & Sinclair, 1976, 1977)

Length l in mm 260

Gauge length L in mm 180

0

15 (ASTM, 2017; DIN, 2010) 25 (ASTM, 2017; DIN, 2010)

250 (ASTM, 2017) 175 (ASTM, 2017)

138 (ASTM, 2017) 125 (ASTM, 2017)

90

Aspect ratio L/w 14 (Chamis & Sinclair, 1976, 1977) 9 (ASTM, 2017) 5 (ASTM, 2017)

The geometry for 45 (crossply) coupon specimens is comparable to θ ¼ 90 specimens, with a longer gauge length of L ¼ 200–300 mm, see (ASTM, 2013).

Experimental Setup and DIC Configuration The quasi-static tests were executed on an MTS servo hydraulic testing machine with a maximum load capacity of 100 kN. The displacement-controlled tests were conducted at a nominal rate of 0.00014 1/s (1.5 mm/min for L ¼ 180 mm, based on 1 mm/min for L ¼ 120 mm acc. to (Pierron & Vautrin, 1996)).

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Tensile Testing

Fig. 2.4 (a) FOV of left camera focusing on speckle pattern on specimen, (b) cropped FOV and ROI with inspection rectangles (colored area in equidistant partitions of 20 mm from 0 to 170 mm)

The loading has been induced by the actuator located at the bottom of the test frame. Before each test, a precision steel block has been used for rotational alignment of the actuator to reduce out-of-plane misorientation. The specimens were rigidly hydraulically gripped using diamond jaw surfaces and a pressure of 4 MPa for specimens of width w ¼ 12.7 mm. The gripping length on each side ranged between 30 and 40 mm (lower limit for uniaxial testing of θ ¼ 0 coupon specimens (Adams & Adams, 2002)). The specimens have been aligned with specimen stops on one side of the grip. The use of anti-rotation (out-of-plane) collars should take into account plaque curvature on the mechanical results. A horizontal stereo-DIC camera arrangement in portrait orientation has been chosen, for the depth-of-field (DOF) being almost constant along the gauge length for both cameras. Figure 2.4a shows the field-of-view (FOV) of the left camera, details about the setup can be taken from Table 2.2. The reference image has been chosen at the end of a 3 s time interval of acquisition for verification of proper camera synchronization, while the specimen has only been clamped by the top grip. For further analysis, the coordinate system has been adapted, so that the abscissa correlated with the specimen axis, loading direction, resp., with the origin being at the bottom centerline and the ordinate shows along the width of the specimen, see Fig. 2.4b. For the following analysis, the average of two inspection rectangles (see Fig. 2.4b) were investigated and compared to the average of the region-of-interest (“ROI”) within the specimen gauge section. Even though post-analysis ROI does not cover the whole gauge section, it might still include end effects due to gripping, acc.

Experimental Setup and DIC Configuration

55

Table 2.2 DIC hardware parameters (for stereo-DIC) Cameras/Image resolution Lenses LFOV Image scale Stereo-angle LSOD Image acquisition rate Patterning technique Approximate pattern feature size

Table 2.3 DIC analysis parameters

Value Point Gray 9.1 Mpx (2704 px  3376 px) CCD, portrait Schneider 35 mm fixed focal length 225 mm  180 mm 15 px/mm, 66 μm/px 28 0.7 m 2 Hz (0.5 s interval) matte white primer and matte black protective enamel by overspray method 200 μm (4 px)

Lsubset Lstep Lwindow LVSG Strain formulation Subset shape function Software

Value 15 px, 1 mm 6 px, 0.4 mm 15 datapoints 99 px, 6.6 mm Engineering Gaussian weights Vic-3D Correlated Solutions

to principle of Saint-Venant (Timoshenko & Goodier, 1951), which also depends on the anisotropy of the material to be tested (Carlsson & Pipes, 1987; Horgan & Simmonds, 1994). Therefore, an inspection rectangle “Aspect” (“A”) was applied with respect to the aspect ratio of the specimen’s gauge section. The breadth was chosen to be approx. 4 mm (in analogy to the center) and the length to be approx. (4 mm  L/w ¼) 56 mm. The inspection rectangle “Center” (“C”) represents the strain measurement with a strain gauge rosette within an area of approx. 4 mm  4 mm. The line inspection for measuring the pattern feature size has been done along the width of the specimen. Note this being close to the critical size of 3 px, where inspection of smaller speckles is recommended in order to avoid aliasing. Table 2.3 lists the specific DIC analysis parameters. Overall, three specimens with an aspect ratio L/w ¼ 14, with acceptable failures in the gauge section, have been loaded up to fracture. Additional tests on selected specimens with an aspect ratio L/w ¼ 9 have been conducted. Note that the virtual strain gauge size LVSG, based on the default DIC analysis parameters listed in Table 2.3, is approx. half the width of the specimen and larger than the smallest size of the inspection rectangles “Aspect” and “Center”. A detailed sensitivity analysis on the DIC analysis parameters will be presented in Sensitivity Analysis—DIC Parameter Uncertainty Quantification Section, including an indication of the resolution of different relative and absolute quantities-of-interest (QOI).

56

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Mechanical Response This section focusses on the deduction of the stress–strain response including data reduction schemes and interpretation of the deformation and damage behavior of specimens with fiber orientation of θ ¼ 10 (fiber orientation clockwise to loading direction). An advantage of using DIC is the ability to acquire and probe the full-field data set post-mortem for variations in determined strains due to the choice of size and location of inspection rectangles, which will also be compared to different types of strain (axial strain εx and maximum shear strain γ12).

Data Reduction Figure 2.5a shows the temporal evolution of the different DIC calculated nominal and shear strains in the specimen’s coordinate system (x–y) and in the principal coordinate system (1-2), obtained from the average strains from inspection rectangle “ROI”. A detailed comparison of the different strain locations will be discussed in Sensitivity Analysis—Strain Location and Range Section. The axial strain εx and maximum shear strain γ 12 are primary QOI. The temporal evolution of the maximum shear strain depicts a non-linearity, leading to a variable shear rate. The presence of the shear strain γ xy results from off-axis testing of an anisotropic material. In Fig. 2.5b, the angle of principal strain φ* varies over the entire test duration with the angle φ* 45 almost reaching 10 (clockwise, compare to Fig. 2.1) at fracture of the specimen. When using a bi-axial strain gauge rosette for determining the maximum shear strain, it is mounted 45 to the fiber orientation, equal to þ35 (direction of major strain) and 55 (direction of minor strain) to loading direction for fiber orientation of θ ¼ 10 (clockwise). The orientation of the fibers being

a

b

Fig. 2.5 Evolution of (a) Nominal ε and shear γ strains vs. time, (b) Angle of principal strain φ* vs. axial strain εx (all from “ROI”)

Mechanical Response

57

Fig 2.6 Mohr’s circle for engineering strain with qualitative values from Fig. 2.5a

clockwise with respect to the loading axis (compare to Fig. 2.1), the shear strain γ xy. is positive due to shear sign convention. The single strain components are shown in Mohr’s circle for engineering strain in Fig. 2.6 with qualitative values from Fig. 2.5a. Note the engineering shear strain convention at the y-axis. Figure 2.7 depicts the maximum shear strain γ 12 against the axial strain εx up to fracture of the specimen. Since the 10 off-axis test is not standardized, guidance for the strain ranges to be used for the determination of moduli has been taken from standards for tensile testing of composites specimens (ASTM, 2013, 2017) and shear testing of V-notched composite specimens (ASTM, 2012a, 2012b) (see also Chap. 3). According to (ASTM, 2012a, 2012b, 2013), the shear modulus should be extracted between a range of γ 12 ¼ 0.2 %–0.6 %, which is depicted in Fig. 2.7 in comparison to the range of axial strain in loading direction, for determination of Young’s modulus, between εx ¼ 0.1%–0.3 % (ASTM, 2017). The same strain ranges have been used to extract the axial ε_ x and shear γ_ 12 strain rates from the temporal evolution in Fig. 2.5a. Furthermore, the same axial strain range has been used to extract the coefficients of mutual influence Poisson’s ratio ν, see Fig. 2.8a, and extension–shear coupling ratio η, see Fig. 2.8b. Note that for the given fiber orientation θ ¼ 10 (clockwise),

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Tensile Testing

Fig. 2.7 Evolution of maximum shear strain γ 12 vs. axial strain εx

a

b

Fig. 2.8 Evolution of (a) lateral strain εy, (b) shear strain γ xy vs. axial strain εx

shear strain in the x–y coordinate system is positive, see Fig. 2.5a (and negative for θ ¼ 10 counterclockwise as shown in Fig. 2.1). Figure 2.9a depicts the nominal stress–axial strain curve (referring to the initial cross-sectional area using Eq. (2.1) and elongation in loading direction x, resp.) up to fracture of the specimen. The resulting shear stress–shear strain curve, using Eq. (2.4), is depicted in Fig. 2.9b. Note that the shear stress is 17.1% of the nominal stress, acc. to Eq. (2.4). The strain ranges and slopes in the elastic region are shown for the deduction of the Young’s modulus and the shear modulus, which will be presented in Sensitivity Analysis—Strain Location and Range Section. Both curves show a non-linear behavior before fracture, the latter defining the ultimate tensile strength UTS and ultimate shear strength USS.

Mechanical Response

a

59

b

Fig. 2.9 Representative (a) nominal stress–axial strain curve, (b) shear stress–shear strain curve

Fractography Figure 2.10 shows a representative specimen that failed neatly in the gauge section (specimens that failed in the tab area have been discarded). Measurements of the angle between fracture surface and edge of the specimen showed values in the range 9.9  0.1 (coefficient of variation CV ¼ 1%). Furthermore, the fractured surfaces randomly show a non-planar (not parallel to out-of-plane vector) fracture. This might result from a deviated crack growth due to the fiber alignment across the thickness or any slight non-flat character of the specimens and tabs that resulted from production of the source plaques. The clean indents in the tabs resulting from the diamond wedge gripping show that no slipping occurred between tabs and grips. A closer post-mortem investigation of the interface between the tabs and specimen heads does not show any irreversible relative movement (i.e. no “pull out”). The distance of the ROI within the specimen gauge section is at approx. 8.8 mm from the first gripping indent in the tabs (for both top and bottom). A detailed fractographic comparison to the DIC measured data will be presented in Failure Investigation Section.

Sensitivity Analysis—Angularity Since the purpose of the off-axis tensile test is to measure failure stresses and strains, an analysis of the effect of errors and variations in the fiber orientation with respect to the loading axis is illustrative of the quality of data that can be obtained using this technique. Based on the previously presented quantitative fractographic investigations, where the angle of the fracture surface with respect to the loading direction revealed that there were slight deviations (0.1 ) from the nominal 10 , a sensitivity analysis of the effect of misorientation on the expected values of the shear stress– shear strain curve has been performed by choosing an artificial misorientation error in angularity (offset angle) of φ in Eq. (2.4) with θ ¼ 10 + φ for shear stress and

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Fig. 2.10 (a) Representative specimen after failure, (b) gripping area, (c) tabbed area top and bottom

with φ ¼ φ* + φ (see also corresponding strain transformation equations εξ,η and γ 12 ¼ f(εξ,η) in Chap. 1) for shear strain (e.g. by simulating a misorientation of a bi-axial strain gauge rosette mounted nominally 45 to the fiber orientation). Figure 2.11a shows the calculated influence of the deviation for shear stress and shear strain. A misorientation error of φ ¼ 1 leads to a substantial variation in stress of approx. 10%. Figure 2.11b depicts a misorientation error of φ ¼ 10 only in strain to visualize the sensitivity of strain gauge measurements (see also Chamis & Sinclair,

Mechanical Response

a

61

b

Fig. 2.11 Representative shear stress–shear strain curves depicting the difference in angle for a misorientation of (a) φ ¼ 1 in stress and strain, (b) φ ¼ 10 only in strain

1976, 1977; Hiel, 1987; Ajovalasit, 2011). In comparison to stress, the calculated expected error in strain due to specimen misorientation is much smaller, proving that strain is less prone to an offset angle.

Sensitivity Analysis—Strain Location and Range Figure 2.12a compares nominal stress–axial strain curves obtained from average strains each from inspection rectangles “ROI”, “Aspect”, and “Center”. The strain from the inspection rectangle “Aspect” is comparable to the strain measured within “ROI”. The more localized measurement location, inspection rectangle “Center” shows a pronounced zigzag pattern in strain in comparison to the global strain measurement locations (“Aspect” and “ROI”). Note that small local perturbations in strain are averaged by the global strain measurement locations. Regarding maximum shear strain, the strain at failure deviates less between the different locations, only a slightly higher strain at failure can be seen for “ROI”. The advantage of using the global strain measurement location “ROI” is capturing strain perturbations/ inhomogeneities within the whole gauge section, indeed including Saint-Venant effects due to gripping and load introduction. The ranges for the determination of the elastic properties have been shown in Fig. 2.7. For the current investigation, the interval has been kept constant but the extrema have been varied by 0.05% (ASTM, 2013). The sensitivity in determining both shear modulus and Young’s modulus, due to the strain location (inspection rectangle, resp.) and strain range, is shown in Fig. 2.13. The location of the inspection rectangle and the extrema seem to have a higher sensitivity on Young’s modulus, whereas the shear modulus shows less variation. Note that the highest sensitivity on the shear modulus is related to the angularity (see Fig. 2.13b), presented in the previous section.

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b

Fig. 2.12 (a) Nominal stress–axial strain curves, (b) shear stress–shear strain curves for different inspection rectangles

a

b

Fig. 2.13 Influence of different strain locations and ranges on (a) Young’s moduli, (b) Shear moduli

Visualization of Strains and Displacements This section provides a full-field investigation on the occurring nominal and shear strain distribution to obtain insight into the deformation and damage behavior. Due to the previously presented deduction of the strain related properties, strains (derivatives of displacements) are shown first. Furthermore, absolute displacements (in-plane axial U and lateral V, out-of-plane W ) in three different (x, y, z) provide insight into the clamping and loading conditions during 10 off-axis tensile testing. The in-plane displacements will be compared to analytical solutions. The postanalysis ROI of the image is smaller than the ROI of the specimen resulting from the chosen subset size, where only a smaller area within the gauge section L  w can be used for DIC data analysis. This results in an offset between edge of the specimen (and tab line) and post-analysis ROI of the image which is shown as white areas in the contour plots. The enclosing border represents the gauge section of the coupon

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63

specimen. If nothing else is stated, the contour plots in this section refer to the undeformed reference coordinates.

Strains Elastic Behavior Figure 2.14 depicts the non-uniformity of the axial strain εx (a), maximum shear strain γ 12 (b) and shear strain in the specimen’s coordinate system γ xy (c) within the elastic region at γ 12 ¼ 0.6%. The “parasitic” (Pierron & Vautrin, 1996) shear strain γ xy is induced due to off-axis testing of an anisotropic material. The whole test section shows an inhomogeneous strain distribution with high strain concentration in the corner of the specimen, close to the grips with severe gradients near the grips across the width. At an average axial strain of εx ¼ 0.23 %, the overall standard deviation is approx. 0.03%. At an average shear strain of γ 12 ¼ 0.6 %, the standard deviation in maximum shear strain is approx. 0.07%. Both types of strain show largest gradients in the top section. There is variation in shear strain along the centerline, whereas only in a small part of the central portion of the specimen, the shear strain is independent of the axial coordinate. This is consistent with the

a

b

c

Fig. 2.14 Strain distribution at γ 12 ¼ 0.6% (“ROI”) (a) axial strain εx, (b) maximum shear strain γ 12, (c) Shear strain γ xy

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observations based on FEM calculated stress made in (Nemeth et al., 1983; Pierron & Vautrin, 1996). According to (Pagano & Halpin, 1968), a region of uniform strain is only attained in the central test section of the specimen, if the aspect ratio is large enough, see also Effect of Specimen Geometry and Fiber Orientation Section on effect of specimen geometry. For both shear strains, the highest values can be found in the upper right part of the specimen.

Failure Investigation The full-field investigations hitherto focused on elastic strains; this section hereinafter provides a detailed fractographic comparison using different DIC calculated data just (one image) before fracture of the specimen. Figure 2.15 depicts the non-uniformity of the axial strain (a) and shear strains (b, d) just before fracture of the specimen. The whole test section shows an inhomogeneous strain distribution with high strain concentrations in the corner of the specimen, whereas fracture occurred in the upper specimen part (see Fig. 2.15c). Perturbations in the strain fields represented by a standard deviation of 0.1% in axial strain εx and 0.5% in maximum shear strain γ 12 can be seen with having the largest values, largest strain gradients, resp., at the top, close to the grips. Shear strain concentrations in the corner of the coupon specimen are more obvious close to fracture of the specimen, compared to the elastic investigation. For a detailed investigation of the strain in fiber transverse direction, a coordinate transformation has been performed (see also strain transformation equations for εξ,η in Chap. 1). The angle of interest is chosen to be φ ¼ 10 with respect to the x–y coordinate system, see Fig. 2.1b. The strain εξ(φ ¼ 10 ) represents the strain in fiber orientation and εη(φ ¼ 10 ) is the strain perpendicular to fiber orientation (fiber transverse strain, resp.), which is shown in Fig. 2.15e. The highest transverse strains appear close to the edge of the specimen, near the gripping area, having the maximum in a small portion on the upper right side. In summary, Fig. 2.15 depicts high strain concentrations close to the edge of the specimen (near the tabs, gripping, resp.) just before fracture which overlap with the actual location of fracture.

Sensitivity Analysis—DIC Parameter Uncertainty Quantification In this section, a detailed virtual strain gauge study of the DIC analysis parameters, such as subset, step, and window size, is performed in several process steps to optimize the DIC measurements. 1. At this point, only the static and spatial noise-floor is considered for the ROI of an image just prior to the reference image, for a single side clamped specimen at zero load (during verification of camera synchronization). The same image is used for

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65

a

b

d

e

c

Fig. 2.15 Strain distribution just before fracture. (a) Axial strain εx, (b) maximum shear strain γ 12, (c) fractography (focusing on post-mortem gauge section), (d) shear strain γ xy, (e) fiber transverse strain εη

66

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a determination of the bias, since the QOI should be close to zero for a single side clamped specimen. Primary QOI are axial strain εx and maximum shear strain γ 12. In the following, the focus is on an image just before fracture of the specimen (at εx, USS(“ROI”) ¼ 0.78%, with large strain gradients), see Fig. 2.15. Alternatively, the investigation can be done at εx ¼ 0.3% (see Merzkirch and Foecke (2020b)). Note that different DIC analysis parameters also lead to different strains and therewith different time and image when a particular strain is reached. Therefore, the investigation is carried out on the same image, at the same force, displacement, resp. The subset size is varied between 15:4:31 px based on a minimum subset size to be at least three times the speckle size. A small subset size is desirable to capture information close to the edges of the coupon specimen, also for other QOI such as fiber transverse strain. Note that a small subset size might lead to a loss in correlation within the ROI. The step-to-subset ratio is chosen to be 1/3, 0.4, and ½ with the window size varying between 7:4:15 datapoints. This leads to overall 45 parameter variations. Fig. 2.16a and b depict the bias (mean) and noise-floor (standard deviation) of both QOI for different virtual strain gauge sizes LVSG with a minimum of approx. 3 mm and the maximum being larger than the width of the specimen (w ¼ 12.7 mm). With increasing virtual strain gauge size, a decrease of the noise-floor for both QOI can be attested. Maximum shear strain γ 12 (b) shows comparable values for noise-floor and bias, whereas axial strain εx (a) shows a lower and almost non-variant bias. Note that maximum shear strain has an inherent dependence of other QOI such as axial strain εx, lateral strain εy and shear strain γ xy. The resulting virtual strain gauge size, with the default DIC analysis parameters listed in Table 2.3, is LVSG ¼ 6.6 mm (99 px, VSG(Lsubset/Lstep/Lwindow ¼ 15/ 6/15)) and is compared to two additional sets of parameters including one with higher precision (small LVSG) and one with less noise (larger LVSG). Virtual strain gauges with sizes LVSG ¼ 3.0 mm (45 px, VSG(15/5/7)) and LVSG ¼ 10.7 mm (161 px, VSG(31/13/11)), the latter almost covering the whole width of the specimen, are highlighted in Fig. 2.16. Large strain gradients (for most QOI) are located close to the edge of the specimen, see Fig. 2.15, where location and magnitude are affected by the subset size and the corresponding reduced post-analysis ROI. Therefore, the maxima of both QOI have been extracted from the centerline (at y ¼ 0 mm) along the gauge length, see Figs. 2.17 and 2.18. Figure 2.16c and d show that the smaller the virtual strain gauge size, the larger the strain maxima of both types of QOI. Centerline plots for axial strain (see Fig. 2.18) for different DIC analysis parameters are shown in Fig. 2.17. Beside the inhomogeneous strain distribution with highest strains at the top of the specimen, increasing fluctuations with smaller virtual strain gauge size are evident. Figure 2.16e and f depict the maxima of the QOI against the noise-floor, representing the signal-to-noise ratio, where no convergence is obtained with increasing noise-floor (decreasing LVSG, resp.). Due to the degression of

Visualization of Strains and Displacements

67

a

b

c

d

e

f

Fig. 2.16 Bias and noise-floor vs. LVSG for (a) axial strain εx, (b) maximum shear strain γ 12, Centerline peak values vs. LVSG for (c) εx (d) γ 12 and centerline peak values vs. noise-floor for (e) εx, (f) γ 12

the signal-to-noise ratio, the optimal balance, best compromise, resp., is within the area of the curvature, with the signal-to-noise ratio for axial strain being on the order of 200 and even higher for maximum shear strain.

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Fig. 2.17 Centerline plots along gauge length just before fracture. (a) All parameter variations, (b) selected parameter variations (Lsubset/Lstep/ Lwindow)

a

b

6. Figure 2.18 illustrates the axial strain distribution of two selected sets of DIC analysis parameters with the smaller VSG depicting higher fluctuations (see Fig. 2.17b and compare to Fig. 2.15a with default DIC analysis parameters). Note that due to the larger subset size, the post-analysis ROI is smaller for the larger LVSG, the offset between edge of the gauge section and ROI of the image is larger, resp. Table 2.4 lists the bias and noise-floor for the default DIC analysis parameters. As a rough estimate, LVSG ¼ 6.6 mm corresponds to approx. 54 physical strain gauges covering the ROI (two across the width and 27 along the gauge length of the coupon specimen).

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69

Fig. 2.18 Distribution of axial strain εx just before fracture. (a) LVSG ¼ 45 px, VSG(15/5/7), (b) LVSG ¼ 161 px, VSG(31/13/ 11)

a Table 2.4 Bias and noisefloor (spatial and static) of QOI from ROI with default DIC analysis parameters

QOI U in mm V in mm W in mm εx in % εy in % γxy in % ε1 in % ε2 in % γ12 in %

b Bias 0.0000 0.0005 0.0015 0.0011 0.0010 0.0020 0.0061 0.0041 0.0102

Noise-floor 0.0006 0.0005 0.0017 0.0058 0.0080 0.0105 0.0074 0.0068 0.0095

Displacements This section focusses on the investigation of the load introduction and absolute deformation response for tensile testing. Furthermore, clamping conditions of the specimen are investigated since those play a huge role in tensile testing of brittle epoxy composite systems (Chatterjee et al., 1993; Adams & Adams, 2002).

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The calculation of the compliances (see Chap. 1) needed for the approximate analytical solutions of axial and lateral displacements has been done with quasistatic values listed in Table 2.5.

Axial Displacement U Figure 2.19 depicts iso-displacement contour plots in the testing (x-) direction at an average strain (“ROI”) in the elastic region of γ 12 ¼ 0.6 %. The DIC calculated axial displacements U show the non-uniformity with a large deformation gradient along the width of the specimen. The initially rectangular specimen deforms into a parallelogram (the longer diagonal with direction of major strain, see Fig. 2.5) with the highest (absolute) displacement on the lower right side. Due to shear sign convention, the shear strain in the specimen’s coordinate system is positive for fiber orientation of θ ¼ 10 (clockwise), see Fig. 2.14c, and compare to orientation of θ ¼ 10 (counterclockwise) in Effect of Specimen Geometry and Fiber Orientation Section. The slopes of the iso-displacement lines (corresponding to strain) at the top and bottom are slightly different (less steep) in comparison to the center part of the specimens. The largest absolute values are located at the bottom due to the actuator movement. The axial displacement close to the top grip leads to the assumption that some amount of pull out from the grips (due to Poisson’s contraction in the thickness direction and compliance of tab material and bonding agent) must have occurred, whereas a post-mortem investigation of the fractured specimen did not show any irreversible pull out (see Fig. 2.10). The corresponding analytically determined axial displacements for the elastic region are shown in Fig. 2.19b, where the boundary conditions for Eq. (2.5) have been modified to take into account the load application from the bottom, so that U(0,y) ¼ εx  Lgrip and U(Lgrip,y) ¼ 0 at the top, where Lgrip (with L < Lgrip < l, see Fig. 2.1a) represents the distance between the first indents in the upper and lower tabs (see Fig. 2.10). Close to the gripping area, the calculated axial displacements do not show a gradient across the width, as the DIC determined displacements, due to the assumption of constant displacement. The assumption U(Lgrip,y) ¼ 0 for the approximate solution cannot be verified with DIC due to the lack in visual accessibility near the grips. The stepwise evolution of the axial displacement along the centerline of the specimen is shown in Fig. 2.19c and compared to the approximate solution (for the elastic region) (Pagano & Halpin, 1968) between the grips. Note that the second term of Eq. (2.5) can be neglected at the centerline (y ¼ 0 mm). There is an offset of approx. 8.8 mm (each side) between gripping (indents in the tabs) to the tab line, see Fig. 2.10, and an additional offset of approx. 2.2 mm (each side) to the ROI, which is larger than the subset size (Lsubset ¼ 1 mm). A unifying condition was imposed, the centerline axial displacement was made equal to that of the real specimen (Nemeth et al., 1983), by an artificial shift of 11.9 mm of the calculated values from the top

Visualization of Strains and Displacements

71

Anal. Soln.

x in m m

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Fig. 2.19 Comparison between DIC determined and calculated axial displacement U. (a) Distribution at γ 12 ¼ 0.6% (“ROI”), (b) corresponding analytical solution, (c) centerline plots for an interval of Δγ ¼ 0.15% up to γ 12 ¼ 0.6% (“ROI”)

Grip Tab ROI

DIC Anal. Soln.

100 50 0 -0.5

-0.4

-0.3

-0.2

-0.1

0

U in mm c grip, to match the DIC calculated data (see Fig. 2.19c), with x ¼ 0 mm being within the ROI. Furthermore, Fig. 2.19c delineates the different boundaries and distances between grip, tab, and ROI. A good qualitative and quantitative agreement between analytical model and DIC calculated data can be attested (Pagano & Halpin, 1968; Rizzo, 1969; Merzkirch & Foecke, 2020a).

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Lateral Displacement V Figure 2.20a depicts iso-displacement contour plots in lateral (y-) direction at an average strain (“ROI”) in the elastic region of γ 12 ¼ 0.6 %. The DIC calculated lateral displacement contour plots show a high non-uniformity over the specimen length and width, displaying an inverse “S-shape” deformation (due to clockwise fiber orientation, opposite to Fig. 2.1b) depicting an in-plane bending of the specimen. Overall, the lateral displacements are smaller than the axial displacements. DIC

Anal. Soln.

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Fig. 2.20 Comparison between DIC determined and calculated lateral displacement V. (a) Distribution at γ 12 ¼ 0.6% (“ROI”), (b) corresponding analytical solution, (c) centerline plots for an interval of Δγ ¼ 0.15% up to γ 12 ¼ 0.6% (“ROI”)

Grip Tab ROI

100 50

DIC Anal. Soln.

0 0.2

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V in mm c

-0.1

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Visualization of Strains and Displacements

73

A comparison to the approximate analytical solution is given in Fig. 2.20b, where the same modifications (adaptation of boundary conditions) have been done as described earlier for the axial displacements. Only a qualitative agreement can be attested, whereas the overestimation of the lateral displacements by the theory can also be seen in differences in the slopes (corresponding to strain (Nemeth et al. 1983)). It is not expected that the approximate analytical solution would provide good agreement off the centerline since the boundary conditions were specified only there, at one point at each end (Richards et al., 1969). Full and fixed grip clamping of the ends requires the boundary condition to be applied across the entire width of the specimen which provides a considerably greater restraint, and therewith predicts larger lateral displacements than the less constraint solution based on idealized boundary conditions from (Pagano & Halpin, 1968; Richards et al., 1969; Nemeth et al., 1983). Note that the latter is affected by the use of tabs, the choice of the tabbing adhesive, gripping pressure and Poisson’s ratio (in thickness direction) which may affect contraction at the grips (Rizzo, 1969). The use of tabs and bonding agent could cause deformation at the specimen ends which vary from the straightline assumptions made in the analytical model (Rizzo, 1969). In comparison to the theory, the DIC calculated data represents the less constrained solution. Results for the centerline lateral displacement obtained from DIC and the approximate analytical solution from (Pagano & Halpin, 1968) are compared in Fig. 2.20c, showing the overestimation by the theory. Note that the boundary conditions for Eq. (2.6) imply no lateral displacement at the centerline (y ¼ 0 mm) at the first indents of the grips with V(0,0) ¼ V(Lgrip,0) ¼ 0 and the first term of Eq. (2.6) can be neglected. The center is not affected by the bending moment and shear force induced by the gripping constraint. As a conclusion, the displacement results show both axial deformation of the specimen into a parallelogram and lateral deformation into an inverse “S-shape.” Both are in good qualitative agreement to the investigations made in (Pagano & Halpin, 1968; Rizzo, 1969; Merzkirch & Foecke, 2020a).

Out-of-Plane Displacement W Figure 2.21 gives a detailed insight into the clamping conditions of a coupon specimen. W represents the out-of-plane displacement of the specimen surface in the z-direction, with the z-origin defined as the surface with only top clamping the coupon specimen. Fig. 2.21 shows a quantification of the out-of-plane displacement of the specimen while being clamped only at the top (a) and from both sides forcefree (b). The contour plots depicting both ends clamped show a twisting (out-ofplane movement) that might be a result of the curvature of plaque due to residual stresses, stacking sequence (here symmetrical, see (Merzkirch et al., 2019) for asymmetrical laminate), and tabbing inaccuracies (e.g., thickness of the adhesive layer). An overall displacement gradient of approx. 0.5 mm can be seen.

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Fig. 2.21 Out-of-plane displacement W, (a) only top clamped, (b) fully clamped and force-free, (c) just before fracture (non-uniform coordinates for better out-of-plane visualization, z-axis exaggerated)

Based on the investigations on out-of-plane displacements W within this section and on fiber transverse strains in Failure Investigation Section, it can be concluded that a mixed mode failure is present that includes the intended mode II (in-plane shear) failure, as well as mode I (normal) due to fiber transverse strain and mode III (out-of-plane shear) due to twisting of the specimen (as a results of clamping and specimen compliance) affecting the determination of the shear modulus and shear strength, with a presumably premature failure of the specimen.

Effect of Specimen Geometry and Fiber Orientation In this section, the influence of different specimen geometries and fiber orientations (θ ¼ 10 and θ ¼ 10 ) on the mechanical response and especially strain response is investigated. Selected contour plots are shown on different specimen geometries, aspect ratios, resp., than discussed before, namely different widths (w ¼ 13 mm and w ¼ 19 mm), with an aspect ratio of L/w ¼ 9 and therewith closer to the standard (ASTM, 2017) for 0 specimen. A thorough investigation on the influence of

Effect of Specimen Geometry and Fiber Orientation

a

75

b

Fig. 2.22 Representative (a) nominal stress–axial strain curve, (b) shear stress–shear strain curve for specimens with L/w ¼ 9 in comparison to L/w ¼ 14

different specimen geometries on strength, strain and moduli can be looked up in (Merzkirch & Foecke, 2020a). The mechanical response of different coupon specimen geometries in comparison to the results from Data Reduction Section are shown as nominal stress–axial strain curves in Fig. 2.22a and shear stress–shear strain curves in Fig. 2.22b. Even though all curves show comparable stress levels, the strain at failure of the configuration with smallest aspect ratio is obvious. Figures 2.23 and 2.24 depict the non-uniformity of the axial strain εx (a) and maximum shear strain γ 12 (b) just before fracture of the specimen for different specimen geometries. The whole test section shows an inhomogeneous strain distribution representing a complex bi-axial (Carlsson & Pipes, 1987) strain state with high strain concentrations in the corner of the specimen. By comparing both specimen geometries with Failure Investigation Section, more pronounced strain concentration within the test section can be seen, with highest strain close to the edge of the specimen. Smaller aspect ratios and larger widths provide a smaller region of uniform strains than larger aspect ratios. According to (Pagano & Halpin, 1968), the strain field approaches uniformity with increasing specimen length (note that those investigations did not include the use of tabs). The shear strain γ xy in the shorter specimen is high over a large region away from the tab, see Fig. 2.24c. Note also the influence of the fiber orientation θ ¼ 10 counterclockwise, shown in Fig. 2.24, resulting in negative shear strain γ xy. This can be explained by the shear sign convention and the initial rectangular deforming into a parallelogram with the largest (absolute) displacement on the lower left side. Another specimen with aspect ratio L/w ¼ 9 has been tested without any tabs (at gripping pressure of 2.5 MPa) in order to investigate if the additional tabbing leads to changes in the boundary conditions for the analytical solution. Since the investigation before fracture of the specimen in the gripping area showed comparable results, those are not presented here.

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a

b

d

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c

Fig. 2.23 Strain distribution for L/w ¼ 9, w ¼ 19 mm, θ ¼ 10 , just before fracture. (a) Axial strain εx, (b) maximum shear strain γ 12, (c) fractography (focusing on post-mortem gauge section), (d) shear strain γ xy (e) fiber transverse strain εη

Effect of Specimen Geometry and Fiber Orientation

a

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77

c

Fig. 2.24 Strain distribution for L/w ¼ 9, w ¼ 13 mm, θ ¼ 10 , just before fracture. (a) Axial strain εx, (b) maximum shear strain γ 12, (c) fractography (focusing on post-mortem gauge section), (d) shear strain γ xy, (e) fiber transverse strain εη

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Fig. 2.25 Orientation dependent error of modulus measurement for different aspect ratios

By using the components of the compliance matrix (Sij, see Chap. 1), Eq. (2.8) is used to visualize the error of modulus measurement for different aspect ratios in Fig. 2.25. Depending on the aspect ratio, the approximate solution predicts that extension–shear coupling can be significant over a range of fiber orientation (Nemeth et al., 1983). It can be stated that the largest error occurs for the smallest aspect ratio (L/w ¼ 6) with an error of approx. 0.17 for 10 off-axis testing. Larger aspect ratios show smaller error in the determination of the modulus. By comparing the moduli for the different aspect ratios from (Merzkirch & Foecke, 2020b), barely a difference can be seen for the moduli of L/w ¼ 9 and 14 with the fluctuations being within a 5% range, whereas the smallest aspect ratio L/ w ¼ 6 shows higher moduli.

Summary and Discussion This section presents the repeatability of the DIC measurements by a summary of results of feature extraction from stress–strain responses from 10 off-axis tensile testing.

In-Plane Extensional and Intralaminar Shear Properties Extensional and intralaminar shear properties deduced from 10 off-axis tensile testing of three specimens are summarized in Table 2.5. The strain related values were deduced from the inspection rectangle “ROI” in order to account for the inhomogeneous strain distribution shown in Fig. 2.14 and Fig. 2.15, depicting a

Testing of Assorted Materials

79

Table 2.5 Results from three tests each on UD-CFRP (representing standard deviation) Orientation θ 10

E in GPa 64.31.4

0

1250.3

90

8.12 0.26

ν 0.40 0.00 0.33 0.05 0.021 (calc.)

η 2.54 0.02 N/A N/A

UTS in MPa 381.6 8.3 1875 146 66 1.4

εx, UTS

in % 0.91 0.13 1.4 0.1 0.85 0.07

USS in G in GPa MPa 4.030.1165.3 1.42 N/A N/A

γ 12, USS in % 3.49 0.67 N/A

N/A

N/A

N/A

complex bi-axial (Carlsson & Pipes, 1987) strain state with high strain concentration in the corner of the specimen. Beside axial and shear strains, also fiber transverse strains are present. Consequently, the ultimate shear strength captured does not represent failure stress, but represents a conservative value (lower bound). It rather refers to the applied stress at which a relieving of these local shear stress concentrations by local cracking occurred. For detailed investigations of the influence of specimen geometry and aspect ratio, surface preparation and clamping conditions on the strain and strength related properties, see (Merzkirch & Foecke, 2020b). Furthermore, results from uniaxial tensile testing in longitudinal and transverse directions are taken from investigations followed from (Powell et al., 2017) and listed in Table 2.5. The stiffness related degree of orthotropicity EL/ET is approx. 15 (6%), the strength related degree of orthotropicity UTSL/UTST is approx. 28 (35 %), representing a high anisotropy. Young’s modulus in the transverse direction is more than two times the modulus of the matrix material (see Chap. 1). Poisson’s ratio in transverse direction has been calculated using Maxwell–Betti reciprocal theorem (see Chap. 1).

Testing of Assorted Materials This section focusses on the adaptability and applicability of DIC measurements for different types of composite laminates and the related conspicuous photomechanical results. A slight deviation from the DIC hardware parameters listed in Table 2.2 was chosen (4.2 Mpx cameras), see also (Merzkirch et al., 2019). The image scale is 13 px/mm (77 μm/px) and the virtual strain gauge size is LVSG ¼ 12.4 mm (161 px, Lsubset/Lstep/Lwindow ¼ 21/10/15).

80

2

Tensile Testing

Woven-CFRP—Uniaxial and 45 Tensile Testing Hereinafter, the use of DIC for describing the strain response while performing standardized tensile tests on 45 woven coupon specimens with geometry acc. to (ASTM, 2013) is presented. The specimens were waterjet cut (using abrasive 220) to a nominal width w ¼ 25 mm and total length l ¼ 268 mm, including tabs with a length of approx. 60 mm. This results in a gauge length of approx. L ¼ 140 mm and an aspect ratio L/w ¼ 5.6 (acc. to Eq. (2.2) the minimal aspect ratio is 1). Note that at least two unit cells are present across the gauge width and the virtual strain gauge size is larger than one unit cell (approx. 9.7 mm) of the woven architecture. For a detailed investigation on virtual strain gauge size and related local strain across the unit cell of woven composites, see (Koohbor et al., 2017). For a DIC-based study of in-plane mechanical response of woven-CFRP, see (Koohbor et al., 2014). Figure 2.26a shows the temporal evolution of nominal and shear strains. Due to the testing angle of 45 with respect to the warp and fill directions, the shear strain in the specimen’s coordinate system γ xy is close to zero, due to non-existing extension– shear coupling. This is confirmed by the evolution of the angle of principal strain φ*, shown in Fig. 2.26b, being approx. 0 . When using a bi-axial strain gauge rosette, it is mounted 0 and 90 to the loading direction, equal to 45 to the fiber (bundle) orientation. In comparison to 10 off-axis tensile testing (Fig. 2.5a), the initial evolution of the strains shows a linear trend. Mohr's circle for engineering strain is shown in Fig. 2.27 with qualitative values from Fig. 2.26a. Note that the major principal strain ε1 is equal to the axial strain εx and the minor principal strain ε2 is equal to lateral strain εy, which means that the x–y coordinate system coincides with principal material axes. The state of strain (stress, resp.) is not pure shear due to normal strains present in addition the desired shear strain γ 12.

a

b

Fig. 2.26 Evolution of (a) nominal ε and shear γ strains vs. time, (b) angle of principal strain φ* vs. axial strain εx (all from “ROI”)

Testing of Assorted Materials

81

Fig. 2.27 Mohr’s circle for engineering strain and qualitative values from Fig. 2.26a

a

b

Fig. 2.28 Representative (a) nominal stress–axial strain curve, (b) shear stress–shear strain curve

For the shear stress–shear strain curve shown in Fig. 2.28b, Eq. (2.4) has been used with an angle of 45 resulting in a shear stress half the nominal stress, see Fig. 2.28a. The ultimate tensile strength (UTS), ultimate shear strength (USS), resp., is taken from the first maximum within the plateau region. In accordance with (ASTM, 2013), USS can be taken at or below 5% of maximum shear strain. Note that the ultimate strength values rather relate to a stress where relieving mechanisms are active, than to an ultimate strength since the specimen did not fully fail.

82

2

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Table 2.6 Results from two tensile tests each on woven-CFRP ( representing standard deviation) Orientation θ 45

0 (warp)

DIC

E in GPa 15.7 0.2

ν 0.80 0.01

η 0.02 0.00

58.5 1.2

0.05 0.01

N/A

Anal. Soln.

140

140

120

120

100

100

80

80

UTS in MPa 117.4 1.8

εx, UTS in % 1.60 0.14

669.3 38.5

1.40 0.00 DIC

0 -0.05 -0.1

G in GPa 4.27 0.06 4.37 (calc.) 27.99 (calc.)

USS in MPa 58.7 0.9

γ 12, USS in % 2.96 0.30

N/A

N/A

Anal. Soln.

140

140

120

120

100

100

80

80

60

60

40

40

20

20

0

0

0.05 0.04 0.03

-0.15

0.02

60

-0.3

40

40

-0.35

0.01 0 -0.01

-0.02 -0.03

-0.4

20

20

-0.04

-0.45

0

a

0

V in mm

60

x in m m

-0.25

U in mm

x in m m

-0.2

-0.5

-0.05

10 0 -10

10 0 -10

10 0 -10

10 0 -10

y in mm

y in mm

y in mm

y in mm

b

Fig. 2.29 Comparison between DIC determined (left) and calculated (right) distribution at γ 12 ¼ 0.6 % (“ROI”) of (a) axial displacement U, (b) lateral displacement V

Table 2.6 summarizes the extensional and shear properties deduced from tensile testing of two specimens per orientation. The results in warp direction are taken from (Powell et al., 2017). In accordance with (Rosen, 1972; Carlsson & Pipes, 1987), the 45 laminate is a special laminate from which the in-plane shear modulus can be calculated from the measured axial strain and lateral strain, Young’s modulus and Poisson’s ratio, resp. (see also Chap. 1, shear modulus for isotropic materials). In comparison to 10 off-axis tensile testing, a homogenous axial deformation across the width can be seen in Fig. 2.29a, which is in good agreement to the analytical solution. The same can be attested for the lateral deformation in Fig. 2.29b. For both analytical solutions, the second terms of Eqs. (2.5) and (2.6) can be neglected (extension–shear coupling is zero). Figure 2.30 shows the strain distribution with localization just before reaching UTS. Strain concentration is visible at approx. φ ¼ +45 (counterclockwise) to the loading axis. Note that fractography shown depicts the specimens tested beyond

Testing of Assorted Materials

83

a

b

c

d

Fig. 2.30 Strain distribution at UTS. (a) Axial strain εx, (b) maximum shear strain γ 12, (c) fractography (focusing on post-mortem gauge section), (d) shear strain γ xy, (e) fiber bundle transverse strains εη

84

2

Tensile Testing

Table 2.7 Results from two tensile tests each on UD-GFRP ( representing standard deviation) Orientation θ 10 0 90

E in GPa 28.4 0.0 31.2 1.3 11.5 0.00

ν 0.32 0.02 0.28 0.01 0.1 0.00 0.12 (calc.)

η 0.84 0.01 N/A N/A

UTS in MPa 285.8 2.5 696.5 31 47.9 0.7

εx, UTS in % 1.61 0.07 2.4 0.00 0.44 0.01

G in GPa 2.91 0.11 N/A

USS in MPa 48.8 0.43 N/A

γ 12, USS in % 4.79 0.35 N/A

N/A

N/A

N/A

UTS. In accordance with (Rosen, 1972) the shear stress is considered to be constant through the thickness and uniform throughout the specimen, except for laminate edge effects. A state of rather combined strains than pure shear exists and, acc. to (Sun & Berreth, 1988), it is difficult to detect the first ply failure due to an additional interlaminar shear stress at free edges (Pipes & Pagano, 1970).

UD-GFRP—Uniaxial and 10 Off-Axis Tensile Testing Since tensile testing of UD coupon specimens has been discussed in detail before, this section only gives a short summary of conspicuousness and challenges in testing thin (h < 1 mm) GFRP specimens (with high compliance), see also (Merzkirch et al., 2019) for details on tensile testing of an asymmetric [452]ns laminate. Table 2.7 summarizes the extensional and shear properties deduced from tensile testing of two specimens each per orientation. The extensional properties in 0 and 90 orientation are taken from investigations followed from (An et al., 2018a, 2018b). Poisson’s ratio in transverse direction is compared to the calculated value using Maxwell–Betti reciprocal theorem (see Chap. 1). Figure 2.31 depicts iso-displacement contour plots and centerline plots for the axial and lateral direction. The calculated axial displacements U show a non-uniformity with a gradient in the tensile direction due to the extension–shear coupling. A comparison to the approximate analytical solution of the lateral displacements shows a non-uniformity along the length and across the width of the specimen, displaying an inverse “S-shape” deformation (Merzkirch et al., 2019). In comparison to CFRP (where higher lateral displacements at the same strain are expected by the analytical model), shown in Figs. 2.19 and 2.20, the approximate analytical solution seems to be in better agreement for the lateral displacement, even though still being overestimated.

Key Conclusions

85 Anal. Soln. 120

120

100

100

80

80

60

60

40

40

20

20

Anal. Soln.

DIC

0

0.05

120

120

-0.05

0.04

-0.1

100

100

80

80

60

60

40

40

20

20

0

0

0.03 0.02

-0.25

x in m m

-0.2

U in mm

x in m m

-0.15

0.01 0 -0.01

-0.3

V in mm

DIC

-0.02

-0.35 -0.03

-0.4

0

0

-0.45

-0.04 -0.05

5 0 -5

5 0 -5

y in mm

y in mm

a

5 0 -5

y in mm

y in mm

b Grip Tab ROI

DIC Anal. Soln.

100

x in mm

100

x in mm

5 0 -5

50

0 -0.6

Grip Tab ROI

50

DIC Anal. Soln.

0 -0.4

-0.2

0

0.2

U in mm

c

0.1

0

-0.1

-0.2

V in mm

d

Fig. 2.31 Comparison between DIC determined and calculated (a) axial displacement U, (b) lateral displacement V at γ 12 ¼ 0.6% (“ROI”), (c and d) corresponding centerline plots for an interval of Δγ ¼ 0.15% up to γ 12 ¼ 0.6% (“ROI”)

Key Conclusions DIC • The reference image should be captured with the specimen being clamped from one side only (and increasing gripping pressure afterwards). • For capturing the strain inhomogeneity during off-axis tensile testing, including end effects due to gripping, the use of “ROI” is recommended. • The uncertainty quantification on DIC analysis parameters focused on an inhomogeneous strain field with highest strains appearing close to the edge of the specimen. A similar situation occurs to open hole tensile testing (ASTM, 2018) where highest strains are expected near the hole.

86

2

Tensile Testing

• Investigation of the temporal noise-floor should be done when using hydraulic gripping and actuation for testing, due to vibrations transferred to the cameras. • Fiducial marks can be applied for the centerline axis to adapt the reference coordinate system.

Structural Mechanics and Testing • Fracture surface angles should be quantified post-mortem for determining the shear stress with regard to uncertainties during specimen manufacturing. • A sensitivity analysis on strength, strain, and modulus is presented in (Merzkirch & Foecke, 2020b), based on variation of specimen geometry, aspect ratio, surface finishing, and clamping conditions. Thereby the shear strengths and moduli range within the boundaries of a misorientation error of 1 (CV ¼ 10%). Preliminary investigations on specimens with oblique tabs, only gripped at the straight part of the tab, barely show a difference to specimens with straight tabs. This is controversial to the observations made in (Pierron & Vautrin 1996), where oblique tabs lead to a higher shear strength. One possible explanation is that the complete oblique tab needs to be gripped. • Gripping conditions include the specimen and its curvature, thickness and type (compliance, resp.) of the adhesive, and tabbing material (Sun & Berreth, 1988; Adams & Adams, 2002), gripping morphology and pressure. Tabbing length (Richards et al., 1969) affects the gradual redistribution of the load which might reduce stress concentrations (principle of Saint-Venant) (Sun & Chung, 1993). • The tabbing procedure and waterjet cutting with the given parameters presented seem to be well chosen for manufacturing of 10 off-axis coupon specimens. • Anticipatory to Chap. 6, 10 off-axis tensile testing is an adequate method for determination of the intralaminar properties of UD-CFRP and is a competitor to standardized V-notched specimen testing. • The consequences of end constraint are more serious in compression testing for small aspect ratios of the coupon specimen acc. to (Pagano & Halpin, 1968).

Practice Exercises This section provides several practice exercises including the use of DIC measured data for self-calculation with the data reduction methodologies presented in this chapter. Solutions, together with supplementary instructional resources (e.g., figures, videos, data reduction codes, slides with deeper explanations of selected expressions, and deduction of equations) suitable for lecturing or lab courses, are available to instructors who adopt the book for classroom use. Please visit the book web page at www.springer.com for the password-protected material.

References

87

Use the measured data and geometric details of a coupon specimen from 10 offaxis tensile testing. 1. Calculate the aspect ratio. 2. Calculate nominal stress and shear stress. 3. Calculate maximum shear strain using major strain and minor strain. 4. Calculate major strain and minor strain. 5. Calculate the angle of principal strain. 6. Draw Mohr’s circle for engineering strain. 7. Explain the sign of the shear strain γ xy. 8. Illustrate nominal stress–axial strain and shear stress–shear strain curves. 9. Determine E, UTS, G, and USS. 10. Determine Poisson’s ratio. 11. Determine the extension–shear coupling ratio. 12. Determine the axial strain at failure and shear strain at failure. 13. Determine the axial strain rate and shear strain rate. 14. Determine axial strain, transverse strain, and maximum shear strain determined from a tri-axial strain gauge rosette. Use the measured data and the geometric details of a coupon specimen from tensile testing of a 45 woven laminate. 15–28. Apply questions 1–13. 29. Calculate G. Acknowledgments Edward Pompa and Dave Pitchure for waterjet cutting, Alex Jennion for fractography, Evan Rust for providing code for UQ. Louise Powell for uniaxial tensile testing results of UD-CFRP in longitudinal and transverse direction, and woven-CFRP in warp direction. Qi An for uniaxial tensile testing results of UD-GFRP in longitudinal and transverse direction.

References Adams, D. O., & Adams, D. F. (2002). Tabbing guide for composite test specimens. U.S. Department of Transportation Federal Aviation Administration. Ajovalasit, A. (2011). Advances in strain gauge measurement on composite materials. Strain, 47(4), 313–325. https://doi.org/10.1111/j.1475-1305.2009.00691.x An, Q., Merzkirch, M., & Forster, A. M. (2018a). Characterizing fiber reinforced polymer composites shear behavior with digital image correlation. In American Society for Composites— Thirty-third technical conference, Seattle, WA, September 24–26. DEStech Publications Inc. An, Q., Tamrakar, S., Gillespie, J. W., Rider, A. N., & Thostenson, E. T. (2018b). Tailored glass fiber interphases via electrophoretic deposition of carbon nanotubes: Fiber and interphase characterization. Composites Science and Technology. https://doi.org/10.1016/j.compscitech. 2018.01.003 ASTM. (2012a). ASTM D5379/D5379M—Standard test method for shear properties of composite materials by the V-notched beam method. ASTM International. https://doi.org/10.1520/d5379_ d5379m-12 ASTM. (2012b). ASTM D7078/D7078M—Standard test method for shear properties of composite materials by V-notched rail shear method. ASTM International. https://doi.org/10.1520/d7078_ d7078m-12

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ASTM. (2013). ASTM D3518/D3518M—Standard test method for in-plane shear response of polymer matrix composite materials by tensile test of a 45 laminate. ASTM International. https://doi.org/10.1520/d3518_d3518m-13 ASTM. (2017). ASTM D3039/D3039M—Standard test method for tensile properties of polymer matrix composite materials. ASTM International. https://doi.org/10.1520/d3039_d3039m-14 ASTM. (2018). ASTM D5766/D5766M—Standard test method for open-hole tensile strength of polymer matrix composite laminates. ASTM International. https://doi.org/10.1520/D5766_ D5766M-11R18 Carlsson, L. A., & Pipes, R. B. (1987). Experimental characterization of advanced composite materials. Prentice-Hall. Chamis, C. C., & Sinclair, J. H. (1976). 10 off-axis tensile test for intralaminar shear characterization of fiber composites. Technical note. NASA—National Aeronautics and Space Administration. Chamis, C. C., & Sinclair, J. H. (1977). Ten-deg off-axis test for shear properties in fiber composites. Experimental Mechanics, 17(9), 339–346. https://doi.org/10.1007/bf02326320 Chamis, C. C., & Sinclair, J. H. (1978). Mechanical behaviour and fracture characteristics of off-axis fiber composites. II—Theory and comparisons. Technical paper. NASA—National Aeronautics and Space Administration. Chang, B.-W., Huang, P.-H., & Smith, D. G. (1984). A pinned-end fixture for off-axis testing. Experimental Techniques, 8(6), 28–30. https://doi.org/10.1111/j.1747-1567.1984.tb02400.x Chatterjee, S. N., Adams, D. F., & Oplinger, D. W. (1993). Test methods for composites: A status report. Volume III: Shear test methods. U.S. Department of Transportation Federal Aviation Administration. Daniel, I. M., & Liber, T. (1975). Lamination residual stresses in fiber composites. Interim report. NASA—National Aeronautics and Space Administration. DIN. (2010). DIN EN ISO 527-5—Plastics—determination of tensile properties—Part 5: Test conditions for unidirectional fibre-reinforced plastic composites. CEN. Erickson, E. C. O., & Norris, C. B. (1955). Tensile properties of glass-fabric laminates with laminations oriented in any way. Forest Products Laboratory, Forest Service— U. S. Department of Agriculture, Ganesh, V. K., & Naik, N. K. (1997). (45) Degree off-axis tension test for shear characterization of plain weave fabric composites. Journal of Composites Technology and Research, 19(2), 77–85. https://doi.org/10.1520/CTR10018J Halpin, J. C., & Pagano, N. J. (1968). Observations on linear anisotropic viscoelasticity. Journal of Composite Materials, 2(1), 68–80. Hearmon, R. F. S. (1943). The significance of coupling between shear and extension in the elastic behaviour of wood and plywood. Proceedings of the Physical Society, 55, 67–80. Hiel, C. C. (1987). Errors associated with the use of strain gages on composite materials. In I. H. Marshall (Ed.), Composite structures 4, Volume 2: Damage assessment and material evaluation. Elsevier. Horgan, C. O., & Simmonds, J. G. (1994). Saint-venant end effects in composite structures. Composites Engineering, 4(3), 279–286. https://doi.org/10.1016/0961-9526(94)90078-7 ISO. (1998). ISO 14129—Fibre-reinforced plastic composites—Determination of the in-plane shear stress/shear strain response, including the in-plane shear modulus and strength, by the +/-45 tension test method. CEN Koohbor, B., Mallon, S., Kidane, A., & Sutton, M. A. (2014). A DIC-based study of in-plane mechanical response and fracture of orthotropic carbon fiber reinforced composite. Composites Part B: Engineering, 66, 388–399. https://doi.org/10.1016/j.compositesb.2014.05.022 Koohbor, B., Ravindran, S., & Kidane, A. (2017). Experimental determination of representative volume element (RVE) size in woven composites. Optics and Lasers in Engineering, 90, 59–71. https://doi.org/10.1016/j.optlaseng.2016.10.001

References

89

Merzkirch, M., An, Q., & Forster, A. M. (2019) In-plane shear response of GFRP laminates by 45 and 10 off-axis tensile testing using digital image correlation. In American Society for Composites—Thirty-fourth technical conference, Atlanta, GA. DEStech Publications Inc. Merzkirch, M., & Foecke, T. (2020a). 10 off-axis tensile testing of carbon fiber reinforced polymers using digital image correlation. In R. Singh (Ed.), Mechanics of composite and multifunctional materials, conference proceedings of the society for experimental mechanics series (Vol. 5). Springer Nature. https://doi.org/10.1007/978-3-030-30028-9_8 Merzkirch, M., & Foecke, T. (2020b). 10 off-axis testing of CFRP using DIC: A study on strength, strain and modulus. Composites Part B: Engineering, 196. https://doi.org/10.1016/j. compositesb.2020.108062 Nemeth, M. P., Herakovich, C. T., & Post, D. (1983). On the off-axis tensile test for unidirectional composites. Journal of Composites Technology Review, 5(2). https://doi.org/10.1520/ CTR10794J Pagano, N. J., & Halpin, J. C. (1968). Influence of end constraint in the testing of anisotropic bodies. Journal of Composite Materials, 2(1), 18–31. https://doi.org/10.1177/002199836800200102 Pierron, F., & Vautrin, A. (1996). The 10 off-axis tensile test: a critical approach. Composites Science and Technology, 56(4), 483–488. https://doi.org/10.1016/0266-3538(96)00004-8 Pindera, M. J., Choksi, G., Hidde, J. S., & Herakovich, C. T. (1987). A methodology for accurate shear characterization of unidirectional composites. Journal of Composite Materials, 21(12), 1164–1184. https://doi.org/10.1177/002199838702101205 Pindera, M. J., & Herakovich, C. T. (1986). Shear characterization of unidirectional composites with the off-axis tension test. Experimental Mechanics, 26(1), 103–112. https://doi.org/10.1007/ Bf02319962 Pipes, R. B., & Cole, B. W. (1973). On the off-axis strength test for anisotropic materials. Journal of Composite Materials, 7(2), 246–256. https://doi.org/10.1177/002199837300700208 Pipes, R. B., & Pagano, N. J. (1970). Interlaminar stresses in composite laminates under uniform axial extension. Journal of Composite Materials, 4, 538–548. Powell, L. A., Luecke, W. E., Merzkirch, M., Avery, K., & Foecke, T. (2017). High strain rate mechanical characterization of carbon fiber reinforced polymer composites using digital image correlations. SAE International Journal of Materials and Manufacturing, 10(2), 138–146. https://doi.org/10.4271/2017-01-0230 Richards, G. L., Airhart, T. P., & Ashton, J. E. (1969). Off-axis tensile coupon testing. Journal of Composite Materials, 3, 586–589. https://doi.org/10.1177/002199836900300322 Rizzo, R. R. (1969). More on the influence of end constraints on off-axis tensile tests. Journal of Composite Materials, 3, 202–219. Rosen, B. W. (1972). A simple procedure for experimental determination of the longitudinal shear modulus of unidirectional composites. Journal of Composite Materials, 6, 552–554. Sinclair, J. H., & Chamis, C. C. (1977). Mechanical behaviour and fracture characteristics of off-axis fiber composites. I—Experimental investigations. Technical Paper. NASA—National Aeronautics and Space Administration. Sun, C. T., & Berreth, S. P. (1988). A new end tab design for off-axis tension test of compositematerials. Journal of Composite Materials, 22(8), 766–779. https://doi.org/10.1177/ 002199838802200805 Sun, C. T., & Chung, I. (1993). An oblique end-tab design for testing off-axis composite specimens. Composites, 24(8), 619–623. https://doi.org/10.1016/0010-4361(93)90124-Q Timoshenko, S., & Goodier, J. N. (1951). Theory of elasticity. McGraw-Hill. Werren, F., & Norris, C. B. (1956). Directional properties of glass-fabric-base plastic laminate panels of sizes that do not buckle. Forest Products Laboratory, Forest Service— U. S. Department of Agriculture. Wu, E. M., & Thomas, R. L. (1968). Note on the off-axis test of a composite. Journal of Composite Materials, 2(4), 523–526.

Chapter 3

V-Notched Specimen Testing

Background One of the most prominent shear test methods for composite laminates is the V-notched beam test, first standardized in 1993 (ASTM 2012a). The testing principle is based on the setup for shear testing of (isotropic) metallic materials and welds presented by N. Iosipescu in 1967 (Iosipescu 1967). About the same time, a similar test method was developed, also in Rumania, by M. Arcan (Adams and Walrath 1987a). The development of the original shear test method was accompanied by a screening of existent test methods and a photoelastic investigation on the shear profiles. Starting condition was the fact that one can obtain pure shear loading by generating a shear force in the zero-moment section of a beam under flexural loading (see also Chap. 4). Aiming for the most uniform possible shear stress distribution, it is necessary to weaken the specimen in the desired failure section without generating stress concentrations. Since shear stresses must have maximum values in the plane of that section, isostatic lines must cross that section at an angle of 45
, leading to a notch angle of 90
. This was experimentally confirmed with qualitative photoelastic studies on Plexiglass. Back at that time, the testing procedure has already been applied to wooden specimens to measure the ultimate shear strength along the fibers, as well as to concrete (Iosipescu 1967). After applying the test method to ceramic matrix composites in the early 1970s, the test method has been expanded and modified for testing different polymers and polymer-based fibrous composite materials by D. Walrath and D. Adams in 1983 (Walrath and Adams 1983a, 1983b). The authors honored the original inventor by retaining the name “Iosipescu shear test”. Due to the full shear stress-shear strain response, the relatively small specimen size, and the general ease of performing the test, the V-notched beam test became so popular. The testing principle can be idealized as asymmetric four-point flexural loading of a rectangular (ws  l) double notched specimen, see Fig. 3.1a, where a set of counteracting displacements (with inner span b) is applied in order to accentuate a © Springer Nature Switzerland AG 2022 M. Merzkirch, Mechanical Characterization Using Digital Image Correlation, https://doi.org/10.1007/978-3-030-84040-2_3

91

92

3 V-Notched Specimen Testing

M

a

Inverse Config.

b

Q

Inverse Config.

c

Fig. 3.1 (a) Schematic double notched rectangular specimen with dimensional parameters, (b) idealized bending moment diagram, (c) Idealized shear force diagram (solid line for compressive loading)

state of predominant shear stress (Grédiac et al. 1994). Due to the bending moment, see different slopes along the length in Fig. 3.1b, a high shear region (first derivative of the bending moment) appears in the middle of the specimen, see Fig. 3.1c (for further details on flexure and shear, see Chap. 4). The notches (with angle Φ) affect the shear strain along the loading direction, making the distribution more uniform than would evolve without the notches, while the degree of uniformity is a function of material orthotropy (ASTM 2012a). The shear stress relates the applied force F to the cross-sectional area between two notches with width w and thickness h acc. to (Iosipescu 1967; Walrath and Adams 1983b)

Background

93

τ¼

F w∙h

ð3:1Þ

While the idealization indicates constant pure shear loading and zero bending moment at the notched section of the specimen, see Fig. 3.1b, c, the actual load application is distributed and imperfect which contributes to an asymmetric shear strain distribution and to an additional component of normal strain that is particularly deleterious to specimens with fiber orientation along the loading direction (ASTM 2012a). Investigations of optimization have been performed on the test fixture design (including load introduction, esp. inner loading points to be farther from center to avoid specimen crushing), on specimen geometry (Walrath and Adams 1983a, 1983b; Adams and Walrath 1987a, 1987b) including notch depth, radius, and angle and on specimen preparation (Lee and Munro 1990), including edge preparation (Pierron et al. 1995). Changes in notch dimensions, such as angle Φ, depth w, and radius are acceptable acc. to (ASTM 2012a). Numeric calculations were done on notch angles of Φ ¼ 90
and Φ ¼ 110
in (Adams and Walrath 1987a), stating that an increased notch angle is advantageous when testing orthotropic materials. The influence of the clamping conditions have been investigated in detail in (Pierron et al. 1995; Pierron 1998; Melin and Neumeister 2007). In terms of specimen clamping, different parasitic moments act on the V-notched beam specimen, such as in-plane and out-of-plane bending and twisting around the horizontal axis (Pierron 1998). In the past, correction factors based on strain, when using strain gauges at the center of the specimen (Grédiac et al. 1994; Pierron et al. 1995; He et al. 2002b), and based on stress (Chatterjee et al. 1993; Grédiac et al. 1994; Pierron et al. 1995; He et al. 2002a) have been recommended to account for the shear concentrations at the center of the specimen. An expression for the correction factor relating to the measured shear strain γ at the center referred to the average shear strain between the notches (Grédiac et al. 1994; Pierron et al. 1995; He et al. 2002b) is CFγ ¼

γ < γN >

ð3:2Þ

The standard specimen is for high modulus fibers parallel (fiber orientation θ ¼ 90
) and perpendicular (fiber orientation θ ¼ 0
) to the loading axis, and for woven with the warp direction parallel or perpendicular to the loading axis. In (He et al. 2002a), different correction factors for θ ¼ 0
and θ ¼ 90
specimens are reported for the determination of the shear modulus. The best overall results when testing in the 1-2 plane have been obtained on 0
\90
laminates (ASTM 2012a). One of the advantages of this test method is the use for measuring both the in-plane and interlaminar shear properties (see also Chap. 4) of unidirectional composites. The latter can be realized by either gluing or manufacturing thick laminates.

94

3 V-Notched Specimen Testing

A limitation of the V-notched beam test is the magnitude of the load that may be applied through the top and bottom edges of the specimen without producing localized failures at the loading points. Tabs are recommended when specimen thickness is less than 2.5 mm (ASTM 2012a) (details about tabbing can be found in (Adams and Adams 2002)). For unidirectional composites, both the in-plane and the interlaminar shear strengths are relatively low and edge loading of the specimen is usually not a problem. However, for multidirectional composite laminates and some textile composites, much higher shear strengths are possible, and thus, require a greater loading capability than is possible with the edge loading by the beam test fixture (Adams et al. 2007). The relatively small gauge section of the V-notched beam specimen is not well suited for some textile composites with coarse fiber architectures, large unit cell sizes or fabrics using large filament count tows. For such materials, a scale up of the specimen (larger gauge section) and test fixture is recommended (ASTM 2012a). Like the V-notched beam test, the V-notched rail test uses a “butterfly” shaped specimen that has been shown to generate a uniform shear stress across the gauge section. This too is dependent on specimen configuration (fiber orientation) and the shape of the V-notch used to concentrate shear stresses in the gauge section (Adams et al. 2003). The V-notched rail test has first been standardized in 2005 (ASTM 2012b) and incorporates attractive features of both the V-notched beam and the two-rail shear test methods (Adams et al. 2007). The specimen is clamped mainly along the face sides with additional c-clamping along the edges. For tensile loading, the inverse configuration is shown in Fig. 3.1b, c. In comparison to the V-notched beam specimen, the V-notched rail specimen has an increased gauge width (by a factor of three) which is better suited for some composites. Regardless, both the V-notched beam and V-notched rail test methods are applicable to obtaining in-plane shear properties including shear modulus and shear strength measurements for certain fiber orientations. Note that force eccentricity results in a twisting of the specimen during loading due to an out-of-tolerance test fixture or from specimens that are too thin. The amount of twist can be investigated by back-to-back strain gauge rosettes (ASTM 2012a, 2012b) in order to account for Saint-Venant effects (Pierron 1998). Usually, a length of the strain gauge of 1.5 mm is recommended whereas larger sizes might be more suitable for woven fabrics. In that case, the active gauge length should be at least as great as the characteristic repeating unit of the weave (ASTM 2012a, 2012b). In (He et al. 2002a), two different strain gauge sizes with square lengths of 4 and 6 mm have been compared on V-notched beam specimens. The use of specialized shear strain gauges, which span the notch width and allow the average shear strain to be measured, are recommended (ASTM 2012b). Photoelastic investigations on composite specimens using the V-notched beam test have been performed (Walrath and Adams 1983b; Broughton et al. 1990). Nowadays, the use of Digital Image Correlation for strain determination with the V-notched beam test is widespread (Daiyan et al. 2012; Bru et al. 2017). As early as 2000, the use of DIC on the V-notched beam test for visualization of shear zones in a 4 mm thick CFRP specimen, using a single 0.5 Mpx camera, was already successfully shown (Melin et al. 2000). Both, 2D-DIC (Murakami and Matsuo 2015) and

Specimen Geometry

95

stereo-DIC (An et al. 2018) has been used for determination of the properties using the V-notched rail test. A comparison of V-notched beam and rail test methods on unreinforced polymers using DIC is presented in (Daiyan et al. 2012), where distributions of shear strain and strain states are analyzed. Reference (Melin and Neumeister 2006) used 2D-DIC on front and back of the V-notched beam test fixture for investigating the out-of-plane deformation of the specimen and test fixture, aiming to improve and to prevent undesired specimen twisting. Additionally, different notch angles were investigated using DIC with the purpose to modify the notch geometry of the specimen in order to achieve more uniform stress field, depending on material anisotropy and fiber orientation with respect to the loading direction (Melin and Neumeister 2006). The rescaling operation for the notch angle Φ, based on a commonly used optimal notch angle of Φopt ¼ 110
(102.6
in theory (Melin and Neumeister 2006)) for isotropic materials, can be determined via the following equation
 tan Φopt=  Φ ffiffiffiffiffiffiffiffiffi2 tan ¼ p 4 Ex 2 =Ey

ð3:3Þ

with the fraction in the denominator representing the degree of anisotropy, with the moduli E in loading x-direction and perpendicular, along the specimen y-axis, see coordinate system in Fig. 3.1. In (Melin and Neumeister 2006), it was stated that the standard notch angle of Φ ¼ 90
only works well for anisotropic material testing in the more compliant direction.

Specimen Geometry Figure 3.2 shows the waterjet cut (using abrasive 220) V-notched beam (top) and rail (bottom) specimens with an average thickness h ¼ 2.4 mm, see also geometric dimensions in Table 3.1. In accordance with (ASTM 2012b), fiber orientation of θ ¼ 0
corresponds to fiber alignment (horizontal, along the specimen y-axis in Fig. 3.1) perpendicular to the (vertical) loading direction, whereas fiber orientation of θ ¼ 90
represents longitudinal fiber alignment parallel to the loading direction. Nominal fiber orientation of θ ¼ +45
is considered as fiber alignment 45
counterclockwise from the horizontal direction and θ ¼ 45
as fiber alignment 45
clockwise from the horizontal direction. The notch width w is 60% of the specimen width ws for the beam specimen and approx. 55% for the rail specimen. Specimens with different notch angles than listed in Table 3.1 have been investigated acc. to Eq. (3.3), see Effect of Specimen Geometry Section.

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3 V-Notched Specimen Testing

Fig. 3.2 Waterjet cut V-notched beam (top) and V-notched rail specimen (bottom) with schematic fiber orientations θ Table 3.1 Dimensions of the V-notched specimens Test method Beam (ASTM 2012a) Rail (ASTM 2012b)

Notch width w in mm 11.4

Specimen width ws in mm 18.9 ( 5  h (DIN, 1998) 13¼3pt 9.6¼4 < 5  h (ISO, 2018)

Length l in mm 15 6  h (ASTM, 2016b) 24¼10  h (DIN, 1998) 36¼L + 20% 35.915  h (ISO, 2018)

Span L in mm 12.3 14.6 30 28.3

Span-to-thickness L/h 5>4 (ASTM, 2016b) 6>5 (DIN, 1998) 12.54  L(3pt)/2 11.8>10  h (ISO, 2018)

In addition to the investigation on interlaminar shear (those specimens have been waterjet cut using abrasive 220), elastic measurements solely for the determination of flexural moduli for different span-to-thickness ratios (L/h) have been conducted in three-point and four-point bending, see Table 4.2. Note that those specimens with width w ¼ 13.1 mm (acc. to ASTM (2015)) have been waterjet cut using abrasive 80. Reference (ASTM, 2015) lists span-to-thickness ratios of L/h ¼ 16/20/40/60 with suggestion L/h ¼ 32, whereas DIN (2011) suggests L/h ¼ 40. The overall specimen length l with respect to the span L is l ¼ L + 20% in accordance with ASTM (2015).

Experimental Setup and DIC Configuration

149

Experimental Setup and DIC Configuration The quasi-static tests were executed on an Instron electromechanical universal testing system with a maximum load capacity of 5 kN. The displacement-controlled tests were conducted in compressive loading at a nominal rate of 1 mm/min acc. to ASTM (2015, 2016b), independent of the span. Span ranges between L ¼ 10 mm and 150 mm for the setup shown in Fig. 4.6. The diameter of the (non-fixed) rollers was d ¼ 6.35 mm (d/h ¼ 2.7) where usually all standards, either for interlaminar shear testing or for elastic flexural testing, suggest different diameters, see DIN (1998, 2011), ASTM (2015, 2016b), and ISO (2018), which are in the range 5–6 mm for the loading roller (2 mm for four-point). Specimens are loaded in symmetric three-point, four-point (“quarter-point loading,” inner/loading span is one-half of the support span L acc. to ASTM (2015)) and five-point (with three support rollers at the bottom) bending, see geometric details about the testing setup in Tables 4.1, 4.2, and Fig. 4.6. The alignment of the support Table 4.2 Dimensions of the specimens for elastic flexural investigation and geometric details about testing setup Test method 3pt 3pt 3pt and 4pt 3pt 3pt

Length l in mm 40 50 75 120 180

Span L in mm 23 40 62 101 151

Span-to-thickness L/h 9 17 26 and 25 41 62

Fig. 4.6 Horizontal stereo-DIC setup for 3pt/4pt/5pt bending (3pt bending setup shown)

150

4 Flexural Testing

Fig. 4.7 FOV (left) of left camera and additional ROI (right) (colored area in equidistant partitions of 4 mm from 0 mm to l ) with inspection rectangles for testing under (a) 3pt, (b) 4pt, (c) 5pt loading

rollers with respect to the loading roller(s) has been done with a caliper (center to center of each roller). Figure 4.7 shows the field-of-view (FOV) of the left camera with specimen and setups for different test methods with focus on interlaminar shear. It is recommended

Experimental Setup and DIC Configuration

151

to position the center of each camera to match the center of the specimen below the loading roller(s). The different brightness for the three test methods is a result of different exposure times. Since the focus of short-beam bending is the determination of the interlaminar shear behavior, the FOV has been chosen to capture the whole specimen length, the span resp., with the focus being in the center of the specimen for the deflection and the shear dominant regions left and right from the loading roller(s). Technically, for flexural testing, a vertical stereo-DIC camera arrangement is preferred in order to have the region-of-interest (ROI) within the same depth-of-field (DOF) of both cameras. However, the loading and support rollers might cover the ROI, depending on the thickness of the specimen, the geometry of the rollers, and the occurring deflection. For this purpose, a horizontal stereo-DIC camera arrangement in landscape orientation has been chosen. Disadvantage is the limitation of capturing the specimen length due to the chosen DOF. Loss in correlation might occur especially at the right and left ends of longer specimens due to the limited DOF (and therewith being out of focus). Therefore it is recommended to choose a small aperture for the DOF to be large (iDICs, 2018) taking into account the perspective difference between the two camera views (Reu, 2013). A small stereo-angle, within the limits of iDICs (2018), for the large focus length of the lenses chosen, has been used for testing with small span (see Table 4.3). Note that a smaller stereo-angle leads to better in-plane displacement accuracy, at the cost of increased out-of-plane uncertainty. The coordinate system has been chosen so that the abscissa (x-axis) is in direction of the specimen length, which corresponds to the fiber orientation and perpendicular to the loading direction (y-axis).

Table 4.3 DIC hardware parameters (for stereo-DIC) Cameras/Image resolution Lenses

Value (Shear test) Point Grey/Flir 9.1 Mpx (3376 px  2704 px) CCD, landscape Sigma 105 mm macro lens

LFOV Image scale Stereo-angle LSOD Image acquisition rate Patterning technique

44 mm  35 mm 79 px/mm, 13 μm/px 16 0.4 m 2 Hz

Approximate pattern feature size

52 μm (4 px)

Matte white primer and matte black protective enamel by airbrush method

Value (Elastic flexural test) Point Grey 5 Mpx (2448 px  2048 px) CCD, landscape Schneider 50 mm/16 mm fixed focal length

14 /27 0.9 m/0.6 m 2 Hz (1 Hz for L > 100 mm) Matte white primer and matte black protective enamel by overspray method 141 μm (see Chap. 1)

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4 Flexural Testing

Fig. 4.8 FOV of left camera with 3pt flexural setup with larger L/h. (a) L ¼ 62 mm (L/h ¼ 26, 50 mm lenses, SOD ¼ 0.85 m), (b) L ¼ 101 mm (L/h ¼ 42, 16 mm lenses, SOD ¼ 0.6 m)

The ROI is shown on the right of Fig. 4.7, depicting inspection rectangles/points located at the center between the support rollers, “Center” (“C”), for the determination of the maximum deflection for three-point and four-point bending. For compensation of system compliance (indentation of the loading pins, compression of the specimen, and machine compliance), additional inspection rectangles/points at the supports (challenges in locating due to stereo vision) are used for determining the relative displacement (to maximum deflection). See Accessory Loading Conditions Section for using the contact strain to allocate the inspection rectangles at the rollers. For five-point bending, the deflection at the center, “Left” (“L”) and “Right” (“R”), between loading roller and middle support roller (where fracture occurred) has been used. The use of a single linear variable differential transformer (LVDT) for measuring the deflection at the center (3/8 L or 5/8 L ) of one of the high-shear zones is suggested by ISO (2018). Deflection of both sides can be easily measured using DIC (or using a second LVDT), leading to an additional result for the interlaminar shear modulus. Overall, three specimens for each test method, with acceptable failures, have been loaded beyond delamination. Sensitivity Analysis: Span Section presents additional tests, which have been rejected for the determination of the interlaminar shear properties due to unintended failure. Since for the elastic flexural investigation with varying span-to-thickness ratio, only the maximum deflection below the loading roller is of interest, the DIC setup shown in Fig. 4.6 (and Fig. 4.7) is sufficient to capture the quantity-of-interest (QOI) absolute displacement measurement (no strain calculation, therefore, no details for elastic flexural testing have been listed in Table 4.4). If the deflection, or other QOI,

Mechanical Response Table 4.4 DIC analysis parameters

153

Lsubset Lstep Lwindow LVSG Strain formulation Subset shape function Software

Value (Shear test) 21 px, 0.3 mm 7 px, 0.1 mm 11 datapoints 91 px, 1.15 mm Engineering Gaussian weights Vic-3D Correlated Solutions

along the whole specimen length is of interest, e.g., determining the relative displacement at the supports, different lenses and stand-off-distances (SOD) have to be chosen, see Fig. 4.8 and Table 4.3. The specimens for the elastic flexural investigations have been tested twice for each span-to-thickness ratio listed in Table 4.2. For calibration, especially for testing with small spans, the support at the bottom of the test fixture has been removed to allow positioning of the calibration grid. Due to the chosen small FOV and high magnification, calibration may require more DOF than the actual experiment (Reu, 2013). Tables 4.3 and 4.4 list the specific DIC parameters for all test methods. The line inspection for measuring the pattern feature size has been done across the thickness of the specimen. Reference image has been chosen at the end of a 3 s time interval of acquisition for verification of proper camera synchronization. Note that the virtual strain gauge size LVSG, based on the default DIC analysis parameters listed in Table 4.4, is smaller than the thickness of the specimen. A larger sensitivity due to vibrations exists based on the choice of the small FOV and macro lenses. A detailed sensitivity analysis on the DIC analysis parameters will be presented in Sensitivity Analysis: DIC Parameter Uncertainty Quantification Section on three-point short-beam bending, including an indication of the resolution of different relative and absolute QOI for all shear test methods.

Mechanical Response This section focuses on the deduction of the force- and shear stress-deflection response including data reduction schemes as well as interpretation of the deformation and damage behavior.

Data Reduction Figure 4.9 depicts the force–deflection curves from inspection points within the ROI, see Fig. 4.7. For three-point and four-point bending, the maximum absolute

154

4 Flexural Testing

3000

L/h = 9

5000

F in N

F in N

4000

2000 4pt, L/h = 25

2000

L/h = 17

1000

L/h = 26

3000

1000

L/h = 41 L/h = 62

0 0

1

2

Left Right Lin. Regr. Limits

0 0

3

0.1

V* in mm

a

0.2

0.3

0.4

V* in mm

b

Fig. 4.9 Representative (a) force-maximum deflection curves for different L/h for 3pt and 4pt bending, (b) force-deflection curves for both high-shear zones for 5pt bending

Table 4.5 Parameters from one double beam shear test for calculation of shear modulus G31 Side Left Right

3 -1 D1 11 in (GPa mm ) 0.007306

c1 in mm/N 0.001404

c2 in 1/mm 0.011710

C in mm/N 25237 23147

G12 in GPa (input) 4.0 or 4.7

G31 in GPa 4.39 4.40

deflection V* at the center of the specimens, between the support rollers, has been used. For five-point bending, absolute deflection V* at both centers between loading rollers and middle support roller has been used. The determination of the compliance (reciprocal of the force–deflection curve) has been done from 10% to 100% of the maximum force for elastic flexural investigation and 10% to 50% of the maximum force subjected to the specimen for interlaminar shear investigation. For L/h ¼ 9, a deviation from linear behavior at higher force is obvious, which corresponds to a flexural stress of approx. 1000 MPa (see Eq. (4.6)). Therefore, a maximum force with lower corresponding flexural stress was chosen for the elastic flexural investigation. The smaller the span-to-thickness ratio, the lower the compliance, being higher for three-point bending than for four-point bending at similar L/h (25, 26 resp.). Flexural moduli deduced from force-deflection curves of three-point and four-point bending for different spans will be shown in Sensitivity Analysis: Span Section. Figure 4.9b depicts the force-deflection curves of five-point bending for both high-shear zones (at 3/8 L and 5/8 L ). A slight asymmetric behavior is indicated by the early increase of the force-deflection curve on the right side. Since the five-point double beam shear test allows for a calculation of the interlaminar shear modulus, Table 4.5 lists the single parameters and results. Note that the calculation based on CLT (see Chap. 1) has been done with input parameters determined from the balanced and symmetric [012]s laminate (therefore the entire [B] matrix is zero). The interlaminar shear modulus is insensitive to small variations of the intralaminar shear modulus as input parameter (G12 ¼ 4 GPa from Chap. 2 and G12 ¼ 4.7 GPa

Mechanical Response

155

Fig. 4.10 Representative maximum shear stressdeflection curves for 3pt, 4pt, and 5pt bending

from Chap. 3). Since the interlaminar shear modulus is only marginally affected by differences in compliance C on the left (at 3/8 L ) and right (at 5/8 L ) side, the influence of the system compliance is not further investigated. Figure 4.10 shows the maximum shear stress-deflection curves from three-point and four-point short-beam tests as well as from five-point double beam shear test (including an initial setting effect) up to the onset of delamination. Except five-point bending, all curves show a non-linear plateau region. The indicated ultimate shear strength (USS) represents shear failure/delamination which occurred after reaching the maximum of the maximum shear stress for all test methods, after the force drop. Bias in determining the ultimate shear strength can be found in deviations in specimen geometry and Hertzian contact pressure that may affect the onset of delamination. See also Sensitivity Analysis: Span Section on investigations on small variations in span-to-thickness ratio for three-point bending aiming for intended delamination failure. For three-point bending, the maximum force is considered as the corresponding short-beam strength acc. to ASTM (2016b), whereas either the maximum force or the force at failure is considered to be the apparent interlaminar shear strength acc. to DIN (1998). For the five-point double beam shear test, the critical load corresponding to the occurrence of delamination is considered as the corresponding interlaminar shear strength acc. to ISO (2018).

Fractography Figure 4.11 depicts selected and representative fractographic images across the width of three-point and four-point specimens focusing only on the intended interlaminar shear failures, delamination resp. Since five-point specimens did not fully delaminate along the width (as will be discussed in Failure Investigation Section), no fractography is shown here. Both three-point and four-point specimens show an eccentric delamination, with the crack deviating from the midplane where

156

4 Flexural Testing

Fig. 4.11 Representative specimens after delamination (with speckle patterned side on the left). (a) 3pt, (b) 4pt

the interlaminar planes below and above the midplane have delaminated. Note that the specimens have been loaded beyond the onset of delamination.

Sensitivity Analysis: Shear Stress Since the fractographic analyses revealed deviations from a midplane delamination, this section presents a sensitivity analysis in terms of the shear stress in different layers of three-point and four-point specimens. Figure 4.12 visualizes a potential bias in determining the interlaminar shear stress using the normalized version of Eq. (4.12) for the symmetrical through-thickness shear stress distribution for three-point and four-point bending, with 12 idealized (straight) layers (considering the actual material of interest). The deviation from the

Mechanical Response

157

Fig. 4.12 Idealized and normalized throughthickness shear stress distribution for 3pt and 4pt bending with indication of layers (for a 12-layer laminate)

maximum shear stress at the midplane is given in percent for each layer. The maximum shear stress at the midplane (100%) has been investigated hitherto for determination of the (apparent) interlaminar shear strength. Due to the parabolic character of the shear stress evolution, the interlaminar shear stress of the first layer (s) beside the midplane is only 2.8% smaller than the maximum shear stress. Delamination between layer two and three off the midplane leads to strength that is 11.1% smaller. At a distance of a quarter thickness of the specimen from the midplane, a decrease of 25% can be observed.

Sensitivity Analysis: Span In this section, the influence of varying span-to-thickness ratio L/h on the elastic flexural properties is investigated for three-point and four-point bending. Additionally, the influence on interlaminar shear strength is investigated (compare with accuracy in spans in Table 4.1) for three-point short-beam bending, which represents a special case for flexural loading with small span-to-thickness ratio being the dominating factor. Those extensive preliminary studies included the thorough search for the appropriate critical span, in order to avoid tensile failure at the bottom and/or compressive failure at the top of the specimen, including high contact stresses due to the loading roller, for achieving the results shown in the previous and subsequent sections. The apparent flexural modulus determined from the maximum deflection (which includes bending and shear deflection (Zweben et al., 1979)) is depicted in Fig. 4.13a for varying span-to-thickness ratio. The moduli from three-point bending, determined using Eq. (4.3), asymptotically approach a convergence state, starting at a span-to-thickness ratio of approx. L/h ¼ 25, being 50% higher than the lower limit of L/h ¼ 16 for flexural elastic testing acc. to DIN (2011) and ASTM (2015). Four-

158

4 Flexural Testing

Efl in GPa

Fig. 4.13 (a) Apparent flexural moduli for different spans from 3pt and 4pt bending in comparison to recommended L/h acc. to ASTM (2015), (b) opposing processes of deflection in relation to span-to-thickness ratio for 3pt bending, (c) opposing processes of stress in relation to span-tothickness ratio for 3pt bending

130 120 110 100 90 80 70 60 50 40 30

3pt 4pt L/h = 16/20/32/40/60

0

10

20

30

40

50

60

L/h in mm/mm

a

VB,S /V in %

100

VB/V (E/G = 3)

50

VB/V (E/G = 30) VS/V (E/G = 3) VS/V (E/G = 30)

0 0 5 10 15 20 25 30 35 40 45 50 55 60

b

c

L/h in mm/mm

Mechanical Response

159

point bending leads to higher flexural moduli than three-point bending. Note that for four-point bending at L/h ¼ 12.5, the span of the outer shear areas (see Fig. 4.2b) was chosen to be similar to three-point bending at L/h ¼ 6. At L/h ¼ 25, the flexural modulus from four-point bending even exceeds the modulus from three-point bending. A detailed comparison between tensile and flexural moduli and the effect of system compliance will be presented in Summary and Discussions Section. Note that for the determination of the flexural moduli shown in Fig. 4.13a, the shear term in Eq. (4.10) (consisting of bending displacement VB and shear displacement VS) has been ignored, which is pretty common under the proposition that the span-to-thickness ratio should be large, for the influence of shear to be negligible. For smaller spans, shear becomes dominant while opposing processes of deflection (bending and shear) are acting in relation to the span-to-thickness ratio. The amount of bending deflection with respect to overall deflection with varying span-to-thickness ratio, using Eq. (4.10), is expressed by (Mullin and Knoell (1970)) VB 1 ¼
2 E V 1 þ G ∙ Lh ∙

1 k

ð4:19Þ

which is depicted in Fig. 4.13b for two ratios E/G ¼ 3 (close to homogeneous isotropic materials) and E/G ¼ 30 (representing a highly anisotropic material). Note that this equation is independent of the width of the specimen. With increasing span-to-thickness ratio, a decrease in shear deflection, increase in bending deflection resp., can be attested, resulting in a convergence of the flexural modulus. The contribution of shear increases with increasing E/G ratio, leading to a smaller curvature. A good qualitative agreement can be attested by comparing the opposing processes of deflection (Fig. 4.13b) to the apparent flexural moduli (Fig. 4.13a). An equal amount of bending and shear (50%) appears at L/h ¼ 6, with shear being the dominant deflection mechanism below this value (for E/ G ¼ 30). Figure 4.13c depicts measured ultimate shear strengths for three-point short-beam bending in relation to different span-to-thickness ratios. As been mentioned in Specimen Geometry Section, the initial spans have been taken from the recommendations of the corresponding standards, but have been modified in order to aim for an appropriate interlaminar failure, delamination resp. The overall specimen length has not been changed, a detailed investigation on the effect of the overhang is omitted at this point. Interlaminar shear is difficult to achieve at larger span-to-thickness ratios. Additional bending failure upon interlaminar failure was registered for larger spans, leading to smaller ultimate shear strength values. Warping of the ends of the specimen was registered before the occurrence of delamination for smaller spans, leading to higher ultimate shear strength values. Based on the opposing processes of bending and shear, using Eq. (4.6) and Eq. (4.13) (Mullin & Knoell, 1970)

160

4 Flexural Testing

τmax ¼

σmax 2 ∙ L=h

ð4:20Þ

assures shear dominant failure for small span-to-thickness ratios. Note that this equation is independent of the width of the specimen. The hyperbolic relation is shown in Fig. 4.13c, using Eq. (4.20) for shear dominant behavior and assuming flexural failure occurring in tension (σ max ¼ UTS ¼ 1875 MPa, see Chap. 2) as a first estimate. The interlaminar shear strength is approx. 185 MPa for a span-to-thickness ratio L/h ¼ 5, suggested in DIN (1998) (230 MPa, for three-point ASTM, L/h ¼ 4 (ASTM, 2016b)). Note that the recommendations from standards of L/h ¼ 4–5 are considerably smaller than the estimation of the critical span-to-thickness ratio of L/h ¼ 11, determined by Eq. (4.20) and using the ultimate shear strength presented in Data Reduction Section. The compression strength of carbon epoxy systems is often reported as the limiting condition of flexure, being on the order of 2/3 of the ultimate tensile strength (UTS) (Mullin & Knoell, 1970; Carlsson & Pipes, 1987). Matching the experimentally determined span-to-thickness ratio would require a value of 1/2 UTS though. Furthermore, the trials depicted in Fig. 4.13c show a rather smooth transition zone from bending failure (superimposed upon shear) to shear dominant failure, than a single abrupt critical value for L/h, which was already stated by Mullin and Knoell (1970). From Fig. 4.13 it can be concluded that shear dominant failure occurs for small span-to-thickness ratios of approx. L/h ¼ 5, with a transition zone consisting of a decrease in measured shear strength due to opposing failure processes (Fig. 4.13c). The transition zone for the determination of the elastic properties (Fig. 4.13a) lasts up to L/h ¼ 25, where at larger span-to-thickness ratios, flexural deflection predominates (Fig. 4.13b). Comparison of the estimated transition span-to-thickness ratio, L/h ¼ 6 from deflection-based calculation (using Eq. 4.19), being equal to the actual value for shear strength testing (see Table 4.1), and L/h ¼ 11 from strengthbased calculation (using Eq. (4.20)), gives a difference with a factor of almost two, which was already stated in Kim and Dharan (1995). Hereinafter a small extract of the extensive preliminary investigation on the search for the appropriate critical span in order to avoid unintended loading conditions such as warping or bending failure, is presented. Examples for span-to-thickness ratios chosen too small and too large are shown in Fig. 4.14. Even though all specimens show delamination failure, they suffered from superimposed (preceded, simultaneous or subsequent) bending failure (tensile on bottom or compressive on top, as well as contact pressure due to loading roller) upon interlaminar failure (delamination). Those types of failure have been discarded for the determination of the ultimate shear strength. Figure 4.14a shows a specimen tested at a recommended ratio of L/h ¼ 5 (DIN, 1998) with indications of delamination off the midplane right from the loading roller, suffering from severe squeezing by the loading roller and warping. Delamination was observed far beyond maximum stress. Figure 4.14b depicts a specimen, tested at a slightly larger ratio than for the actual

Visualization of Strains and Displacements

161

Fig. 4.14 Cropped FOV of 3pt short-beam specimens. (a) L/h ¼ 5 (right camera), (b) L/h ¼ 7.3 (left camera)

determination of USS, with initial bending failure at the bottom and subsequent delamination at the midplane. The following, mostly simultaneously occurring, mechanisms have to be considered for short-beam bending. Roller induced Hertzian contact pressure is omnipresent, dependent on the force introduction (higher on top loading roller for three-point than for four-point) and roller diameter. For large L/h, deflection is dominant with tension on bottom and compression on top. For small L/h, roller induced shear becomes predominant, with bending deflection being less dominant. For very small L/h, further inelastic deformation such as severe warping, squeezing (dependent on the roller diameter) with any notable delamination failure, and roller induced through-thickness shear occur simultaneously.

Visualization of Strains and Displacements This section provides a full-field investigation on the occurring through-thickness strain distribution of specimens tested in three-point, four-point, and five-point bending to obtain insight into the deformation and damage behavior. Besides the intended delamination under shear loading, also unavoidable loading conditions will be visualized that are accessory to flexural testing. For a deduction of the full stressstrain response and strain related properties, strains (derivatives of displacements) are shown first. Furthermore, absolute displacements (in-plane U and V ) in two directions (x, y) provide insight into the loading conditions and are compared to the analytical solution of the deflection. The post-analysis ROI of the image is smaller than the ROI of the specimen resulting from the chosen subset size, where only a smaller area of the specimen side l  h can be used for DIC data analysis. This results in an offset between edge of the specimen and post-analysis ROI which is shown as white areas in the contour plots. The enclosing border, the axes limits resp.,

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4 Flexural Testing

were chosen to be identical with the specimen dimensions. For better comprehension, the contact points from loading and support rollers have been added. If nothing else is stated, the contour plots in this section refer to the undeformed reference coordinates.

Strains Shear Strain: Intended Loading Condition This section focuses on the uncertainty in symmetric alignment of the support with respect to the loading roller(s) for short-beam bending. Alignment is aggravated by increasing the number of rollers and by deviations of the thickness of the specimen along its length. Different types of shear strain, namely shear strain in the specimen’s coordinate system γ xy and maximum shear strain γ 12 are used as QOI to obtain insight into the loading symmetry for three-point, four-point, and five-point bending. For the current investigation, contour plots for all test methods are shown within the elastic region at τmax ¼ 50 MPa, see Fig. 4.10, and with identical strain scales for better comparison. Figure 4.15a depicts the distribution of the shear strain γ xy for three-point bending, showing antisymmetric extrema left and right from the loading roller with the highest values being at the right and the smallest (negative) values being at the left half-span. Figure 4.15b compares idealized and real extrema of the shear strain γ xy along the span, which serves as control for symmetric alignment, attesting a neat symmetric loading condition for three-point bending. Figure 4.15c depicts highest values for the maximum shear strain with extrema in the vicinity of the midline of the specimen. The loading roller induces the highest shear strain concentrations in comparison to the shear strain perturbations from the support rollers. Four-point bending with quarter-point loading shown in Fig. 4.16, depicts a slight asymmetry due to aggravated alignment of the support and additional loading rollers. An almost shear free zone appears in the constant moment middle section (see Fig. 4.2) of the specimen, see Fig. 4.16b. The distribution of maximum shear strain in Fig. 4.16c shows perturbations with concentrations close to the support rollers. Figure 4.17a depicts the shear strain γ xy distribution for five-point bending, which shows alternating shear strain along the span with the highest (absolute) shear strain between both loading rollers in the middle section of the specimen, see Fig. 4.17c. The loss in correlation only at the right end of the specimen can be explained by the limited DOF (loss in focus resp.). Figure 4.17b reveals challenges in alignment of five rollers, where a slight asymmetry can be stated. The idealization of the high-shear zones is twice the value of the low-shear zones (compare to Fig. 4.4c). The left high-shear zone and the right low-shear zone are close to the idealization, whereas for the given alignment, the right high-shear zone is overestimated and the low-shear zone on the left

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a

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Fig. 4.15 Shear strain distribution for 3pt at τmax ¼ 50 MPa (a) γ xy, (b) Midline plot of γ xy, (c) γ 12

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Fig. 4.16 Shear strain distribution for 4pt at τmax ¼ 50 MPa. (a) γ xy, (b) Midline plot of γ xy, (c) γ 12

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a

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Fig. 4.17 Shear strain distribution for 5pt at τmax ¼ 50 MPa. (a) γ xy, (b) Midline plot of γ xy, (c) γ 12

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underestimated. Note also the overall smaller shear strain compared to three-point and four-point bending for similar load levels.

Shear Strain: Stress–Strain Response Standards for three-point short-beam bending are limited to the determination of the interlaminar shear strength and do not include a quantification of the shear modulus and the strain at failure, delamination resp. In this section, a precise quantitative investigation of the DIC determined shear strains for three-point short-beam bending will be presented. The aim is to use the shear strains for a deduction of the full stressstrain response including data reduction schemes for a determination of the interlaminar shear modulus and strain at delamination. Figure 4.18 shows 3D surface plots of maximum shear strain γ 12 (a) and shear strain in the specimen’s coordinate system γ xy (b), see also Fig. 4.15. Besides the strain perturbation induced by the loading roller, the shear extrema in the vicinity of the midline along the length of the specimen can be seen. For emphasis, throughthickness line plots for each half-span (taken at left half-span at ¼ L, x ¼ 8.4 mm, and at right half-span at ¾ L, x ¼ 15.7 mm, see Fig. 4.15) are shown in Fig. 4.18c, visualizing the parabolic distribution of different types of shear strain. Due to the symmetric alignment (see also Fig. 4.15b), the distribution of maximum shear strain from both sides (left and right) almost overlaps. Since the post-analysis ROI does not cover the whole thickness of the specimen, the shear strain does not extend all the way to the edge. Extrapolation of the shear strain to the free edges is assumed to lead to values of zero. The parabolic distribution depicts the extrema of shear to be close to the midline of the specimen. Note that y ¼ 0 mm does not perfectly coincide with the specimen’s midline and does not represent the neutral axis. For a detailed investigation of the position of the neutral axis, see Neutral Axis Section. The spatial resolution with the datapoints shown across the thickness is approx. 88 μm for the default DIC analysis parameters, see also Sensitivity Analysis: DIC Parameter Uncertainty Quantification Section. Line plots of shear strain for selected load levels with indication of the extrema are depicted in Fig. 4.19, representatively using shear strain γ xy from the left side at ¼ L and from the right side at ¾ L (compare with Fig. 4.18a and b). Using the shear strain extrema over the whole test duration up to delamination failure, the temporal evolution of different types of shear strain for both half-spans (¼ L and ¾ L) is shown in Fig. 4.20a. Combining the shear strain extrema with maximum shear stress, the full stress-strain response as shear stress-shear strain curves can be deduced. Figure 4.20b depicts different types of shear strain and locations with indication of onset of delamination (even though the specimen has been tested beyond delamination), which occurred within the right half-span. The full shear stress-shear strain curves show a comparable behavior within the elastic region, whereas differences in strain at highest shear stresses are obvious due to the location of delamination. The shear modulus is extracted from a shear strain range of γ ¼ 0.2%–0.6%, based on ASTM (2012a, 2012b, 2013).

Visualization of Strains and Displacements Fig. 4.18 Shear strain distribution for 3pt at τmax ¼ 50 MPa. (a) γ 12, (b) γ xy, (c) through-thickness line plots of shear strains at left half-span ¼ L and right half-span ¾ L

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Fig. 4.19 Through-thickness line plots of shear strain γ xy for selected load levels from τmax ¼ 10 MPa up to delamination. (a) At left half-span at ¼ L, (b) at right half-span at ¾ L

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Fig. 4.20 (a) Evolution of shear strains vs. time, (b) representative shear stress-shear strain curves for different types of shear strain from left half-span ¼ L and right half-span ¾ L

The procedure described is implemented in a self-written code and expanded along selected regions of each half-span for both types of shear strain, see Fig. 4.21. Localized perturbations in the strain field resulting from the support rollers and the loading roller in the middle of the three-point specimen, leading to non-parabolic shear strain distribution across the thickness, have been discarded. It can be concluded that the DIC calculated shear strains can be used to get a full stress-strain response for three-point short-beam bending. Note that average shear moduli from both half-spans should be considered in order to account for misalignment. A shorter half-span with less pronounced shear will lead to higher moduli and vice versa. The adaptability of the methodology to four-point short-beam bending showed that strain concentrations on the right side resulting from a slight asymmetric alignment, see also Fig. 4.16, lead to higher moduli for the left quarterspan compared to the right quarter-span. A summary of the extracted properties will be presented in Interlaminar Shear Properties Section.

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Fig. 4.21 Distribution of calculated shear moduli from two types of shear strain for 3pt short-beam bending

Failure Investigation The full-field investigations hitherto focused on shear strains for the determination of the full stress-strain response; this section hereinafter provides detailed fractographic investigations and comparison of DIC calculated data just (one image) before delamination. A comparison between the distribution of maximum shear strain γ12 just before delamination and fractographic investigations is shown for three-point specimen in Fig. 4.22, four-point specimen in Fig. 4.23, and five-point specimen in Fig. 4.24. Furthermore, the distribution of fiber transverse strain εy (perpendicular to horizontal fiber orientation) is shown at identical strain scales for better comparison of all test methods. Note that the specimens have been loaded beyond the force drop and therefore represent a post-delamination state, whereas the contour plots represent a pre-delamination state. The post-analysis ROI is reduced in comparison to ROI within the elastic region shown in previous section. This is based on loss in correlation due to mainly the moving loading roller(s) covering area of the ROI (of each camera). This effect is even more crucial for large deflections. The contour plots in this section are shown in the deformed state. Therefore, the enclosing border, the axes limits resp., are not identical with the specimen dimensions. For three-point bending, shown in Fig. 4.22a, the shear strain concentration on the left side predicts failure, notably resulting from the loading roller. The fractographic investigation in Fig. 4.22c depicts multi-layer delamination of the layers close to the midplane at the upper and lower part. As shown in Fig. 4.11a, no straight delamination across the width occurred, even though it is not clear to this point what is considered to be primary or secondary failure and how the first delamination affected the neighboring layers. The distribution of fiber transverse strain in Fig. 4.22b depicts an area of concentrations in the vicinity of the midline at the left half-span. The fractographic investigation of a four-point specimen, shown in Fig. 4.23c, depicts an eccentric apparent single-layer delamination. Roller induced shear from

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Fig. 4.22 Strain distribution of 3pt specimen just before delamination. (a) Maximum shear strain γ 12, (b) fiber transverse strain εy, (c) fractography from DIC opposite side (specimen has been flipped horizontally for better comparison to contours)

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Fig. 4.23 Strain distribution of 4pt specimen just before delamination. (a) Maximum shear strain γ 12, (b) fiber transverse strain εy, (c) fractography from DIC opposite side (specimen has been flipped horizontally for better comparison to contours)

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Fig. 4.24 Strain distribution of 5pt specimen just before delamination. (a) Maximum shear strain γ 12, (b) fiber transverse strain εy, (c) fractography from DIC opposite side (specimen has been flipped horizontally for better comparison to contours)

both loading rollers is shown in Fig. 4.23a, which seems to be more concentrated at the left loading roller but more pronounced over a larger area at the right quarterspan. Contrary to the highest shear strain concentration at the left quarter-span, delamination occurred on the right side. A comparable observation can be attested for fiber transverse strain, shown in Fig. 4.23b, where concentrations are visible over a larger area, compared to the left quarter-span where strains are more localized. Figure 4.24c shows the irreversible overall deformation of the five-point specimen. Delamination occurred at the right side of the middle roller, depicting the highest shear strains, shown in Fig. 4.24a. Since the five-point specimen did not fully delaminate along the half-length and across the whole width of the specimen, a detailed fractographic investigation giving indication on single- or multidelamination processes is missing. Shear strain concentrations can be seen coming from both loading rollers (Fig. 4.24a). Fiber transverse strain shown in Fig. 4.24b is less pronounced in comparison to three-point and four-point specimens. All test methods show roller induced shear failure coming from the loading rollers with superposition of fiber transverse strain, both initiating the delamination process. The ultimate shear strength can be considered as the applied stress at which a relieving of these local strain concentrations by local cracking through delamination occurs.

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Stress concentrations induced by the rollers never fully dissipate, as the principle of Saint-Venant is not satisfied in highly orthotropic specimens with small span-tothickness ratios (Sideridis & Papadopoulos, 2004; Tolf, 1985).

Accessory Loading Conditions This section gives an overview on accessory loading resulting from the rollers, using (vertical) contact strain εy, in order to visualize Hertzian contact pressure. Furthermore, (horizontal) bending strain εx in fiber orientation is shown with symmetrical strain scale for better comparison of tensile and compressive loading. Depending on the focus of investigation, bending strain can be the intended loading condition for large span-to-thickness ratios (instead of shear strain). The relating post-mortem specimen pictures give indication on the proper and symmetrical loading across the width of the specimen. For comparison of three-point (Fig. 4.25), four-point (Fig. 4.26), and five-point (Fig. 4.27) flexural testing, the same line load (F/w) has been chosen (the reference at τmax ¼ 50 MPa for three-point bending). For better comparison, the strain scales have been chosen to be identical for all test methods. Note that post-mortem specimens have been loaded beyond force drop and delamination and therefore represent a progressed stage of deformation and damage compared to the DIC calculated data shown. Figure 4.25b shows high contact strain concentrations coming from the loading roller on top of the three-point specimen due to the higher acting force compared to the supports. The post-mortem investigation of the top of the specimen shows indentations that might result in the plateau region for shear stress-deflection curves (Fig. 4.10). Furthermore, high tensile strain causing deformations at the bottom of the specimen can be seen in Fig. 4.25c, with the compressive strain being perturbated by the loading roller. The bottom of the specimen does not show any tensile or contact failure. Figure 4.26 depicts the top and bottom surfaces of the four-point specimen indicating Hertzian contact pressure with emphasized indentations on top from the loading rollers that might result in the plateau region for maximum shear stressdeflection curves (Fig. 4.10). Due to the larger deflection of four-point specimen, it is likely that slippage occurred at the support rollers (higher contact strains are obvious in Fig. 4.26b), whereas the loading rollers dominate the deflection of the specimen and therefore barely any slippage might occur there. This could explain the more pronounced indentations on top. Furthermore, high tensile strain, being almost constant at the bottom surface over the half-span between the loading rollers can be seen in Fig. 4.26c. Note the bottom of the specimen indicating maximum bending moment (see Fig. 4.2a) along the half-span coinciding with the position of the top rollers. As expected, the loading roller in three-point bending has a higher influence on contact pressure than the support rollers, which is less pronounced in four-point bending.

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Fig. 4.25 Strain distribution of 3pt specimen at F/w ¼ 165 N/mm (τmax ¼ 50 MPa). (a) Top view (post-mortem), (b) roller contact strain εy, (c) bending strain εx, (d) bottom view (post-mortem)

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Fig. 4.26 Strain distribution of 4pt specimen at F/w ¼ 165 N/mm (τmax ¼ 51.4 MPa). (a) Top view (post-mortem), (b) roller contact strain εy, (c) bending strain εx, (d) bottom view (post-mortem)

The pictorial results of five-point bending are shown in Fig. 4.27, where indentations can be seen on top which are more pronounced than the contact of the support rollers at the bottom. Overall, the contact strain is less pronounced in comparison to three-point and four-point specimens. Similar observation can be attested for the bending strain, shown in Fig. 4.27c, being low in both high-shear zones. Note that

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the results taken are at a similar line load for all test methods, although being at smaller maximum shear stress for five-point specimen. Five-point bending shows pronounced contact pressure at the middle support roller. Note that this behavior can be affected by variations in thickness and loading (asymmetry) for multiple rollers, for all test methods. Asymmetric effects can be seen using bending strain for five-point bending. The contact strain can be used of locating additional inspection rectangles/points at the supports for determining the relative displacement (to maximum deflection).

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Fig. 4.27 Strain distribution of 5pt specimen at F/w ¼ 165 N/mm (τmax ¼ 35.6 MPa). (a) Top view (post-mortem), (b) roller contact strain εy, (c) bending strain εx, (d) bottom view (post-mortem)

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Neutral Axis This section focuses on a methodology for the investigation of the position of the neutral axis of three-point short-beam bending, using the maximum shear strain and the bending strain (see also Fig. 4.3). Figure 4.28a depicts the through-thickness distribution of the bending strain εx at the right half-span ¾ L. It can be seen that the compressive side (top) allows more strain than the bottom side (indicating a lower compressive modulus in comparison to the tensile modulus, see also Flexural Modulus vs. Tensile Modulus Section). Note that y ¼ 0 mm does not perfectly coincide with the midline of the specimen and does not represent the neutral axis. The black circle in Fig. 4.28b denotes the smallest absolute bending strain determined with DIC, whereas the red circle denotes the intersection via interpolation with zero bending strain. Figure 4.29 compares the localization of the neutral axis across the thickness of the specimen using the locations of the extrema of the maximum shear strain γ 12 (see

a

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Fig. 4.28 Through-thickness line plots of bending strain εx for 3pt at right half-span ¾ L. (a) Line plots for selected load levels between τmax ¼ 10 MPa–50 MPa, (b) methodology for determination of neutral axis at τmax ¼ 50 MPa Fig. 4.29 Comparison of different methods for identification of neutral axis during 3pt short-beam bending at left half-span ¼ L between τmax ¼ 10 MPa up to delamination

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Shear Strain: Stress–Strain Response Section) and the smallest absolute bending strain εx determined with DIC, as well as the intersection via interpolation with zero bending strain for a range between 10 MPa and the onset of delamination (which occurred at the left half-span). Note the DIC data locations shown as layers with a distance of approx. 88 μm, which is dependent on the DIC analysis parameters (see subsequent Sensitivity Analysis: DIC Parameter Uncertainty Quantification Section), that should not be confused with the position of the layers (with approximate thickness of 200 μm) of the composite laminate, see Fig. 4.12. The intersection method shows fluctuations in between the DIC data locations. The method of using maximum shear strain shows less fluctuation in comparison to the other methods. The extrema of the maximum shear strain and the smallest absolute bending strain differ between load levels of 30 MPa and 80 MPa (the latter being in the non-linear region) between DIC data locations with a spatial resolution of approx. 88 μm. Using the same method for determination of bias for the strength-based values (see Fig. 4.12 and Eq. (4.12)), this leads to an offset of approx. γ/γ max ¼ 0.5% between the middle and the first neighboring data layer, being less than the strength-based values with an offset of 2.8% between the middle and the first neighboring lamina, see Fig. 4.12. No direct conclusion can be drawn if the neutral axis is in the geometrical middle (midline) of the three-point short-beam specimen, since reference points at the edge of the specimen, due to reduced post-analysis ROI, are missing. Applying a fiducial mark at the midline with a painter is considered as too uncertain (in comparison to spatial resolution of 88 μm). Another option includes the use of the pixel-based data of the DIC reference image.

Sensitivity Analysis: DIC Parameter Uncertainty Quantification In this section, a detailed virtual strain gauge study of the DIC analysis parameters, such as subset, step, and window size, is performed for three-point short-beam bending in several process steps to optimize the DIC measurements. 1. At this point, only the static and spatial noise-floor is considered for the ROI of an image just prior the reference image at zero load (during verification of camera synchronization). The same image is used for a determination of the bias, since the QOI should be zero for a non-clamped specimen. Primary QOI are shear strain in the specimen’s coordinate system γ xy and maximum shear strain γ 12. Since bending strain εx has its local maxima (extrema resp.) at the top and bottom edge of the specimen (see previous section), it is not considered to be an appropriate QOI for DIC sensitivity analysis because it is highly affected by the subset size and the corresponding reduced post-analysis ROI. 2. In the following, the focus is on an image at τmax ¼ 50 MPa, see Fig. 4.15, (related to force), independent of any DIC related parameters. Alternatively, the

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Fig. 4.30 Bias and noise-floor vs. LVSG for shear strains (a) γ xy, (b) γ 12, Through-thickness peak values (c) γ xy and γ 12 vs. LVSG, (d) through-thickness peak values vs. noise-floor for γ xy (center of right half-span)

investigation could also be done just before failure or delamination of the specimen. 3. The subset size is varied between 15:2:23 px based on a minimum subset size to be at least three times the speckle size. A small subset size is desirable to capture information close to the edges of the bending specimen, also for other QOI such as bending strain. The step-to-subset ratio is chosen to be 1/3, 0.4, and ½ with the window size varying between 7:4:15 datapoints. This leads to overall 45 parameter variations. Figure 4.30a and b depict the bias (mean) and noise-floor (standard deviation) of both QOI for different virtual strain gauge sizes LVSG with a minimum of approx. 0.6 mm, a quarter thickness of the specimen, and a maximum being comparable to the thickness of the specimen. Note that the x-axis limit in Fig. 4.30a–c is chosen to match the specimen thickness (h ¼ 2.4 mm). With increasing virtual strain gauge size, a decrease of the noise-floor for both QOI can be attested. Maximum shear strain γ 12 (b) shows comparable values for noise-floor and bias, whereas shear strain γ xy (a) shows a lower and almost

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Fig. 4.31 Throughthickness line plots of shear strain γ xy at right half-span ¾ L and τmax ¼ 50 MPa. (a) All parameter variations, (b) selected parameter variations (Lsubset/Lstep/ Lwindow)

a

b non-variant bias. Note that maximum shear strain has an inherent dependence of other QOI such as axial strain εx, lateral strain εy and shear strain γ xy. The resulting virtual strain gauge size, with the default DIC analysis parameters listed in Table 4.4, is LVSG ¼ 1.2 mm (91 px, VSG(Lsubset/Lstep/Lwindow ¼ 21/ 7/11)), being half the thickness of the specimen, and is compared to two additional sets of parameters including one with higher precision (small LVSG) and one with less noise (larger LVSG). Virtual strain gauges with sizes LVSG ¼ 0.7 mm (53 px, VSG(17/6/7)) and LVSG ¼ 1.5 mm (119 px, VSG(19/10/11)) are highlighted in Fig. 4.30. 4. Large shear strain gradients are located across the thickness of each half-span, see Fig. 4.15. The extrema of both types of shear strain have been extracted across the thickness at the centerline of each half-span (left at ¼ L and right at ¾ L), see Figs. 4.31 and 4.32. Figure 4.30c shows that the smaller the virtual strain gauge size, the larger the strain extrema of both types of shear strain, with no significant

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a

b Fig. 4.32 Shear strain distribution γ xy at τmax ¼ 50 MPa. (a) LVSG ¼ 53 px, VSG(17/6/7), (b) LVSG ¼ 119 px, VSG(19/10/11)

increase for smallest virtual strain gauge sizes. Through-thickness line plots across the center of the right half-span ¾ L for shear strain γxy (see Fig. 4.32) for different DIC analysis parameters are shown in Fig. 4.31. Increasing fluctuations with smaller virtual strain gauge size are evident. The larger the virtual strain gauge size, the smaller the strain extrema (compare to Fig. 4.30c) and the smaller the amount of data points. The smaller the spatial resolution (acc. to the step size Lstep); e.g., VSG(17/6/7) has a spatial resolution of approx. 75 μm and VSG(19/10/11) has a spatial resolution of approx. 125 μm in comparison to the default DIC analysis parameters with VSG(21/7/11) having a spatial resolution of approx. 88 μm. This has to be taken into account for determining the shear strain for shear modulus and strain at failure, see Shear Strain: Stress–Strain Response Section and the neutral axis, see Neutral Axis Section. Smaller strain extrema will lead to a higher shear modulus and a smaller strain at failure (due to a larger averaged area). 5. Figure 4.30d depicts selected extrema of the QOI against the noise-floor, representing the signal-to-noise ratio, where barely a convergence is obtained with decreasing LVSG (increasing noise-floor resp.). Due to the degression of the signal-to-noise ratio, the optimal balance, best compromise resp., is within the area of the curvature, with the signal-to-noise ratio being on the order of 100. Figure 4.32 illustrates the shear strain γxy distribution of two selected sets of DIC analysis parameters with the smaller VSG depicting higher fluctuations (see

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Table 4.6 Bias and noisefloor (spatial and static) of QOI from ROI for default DIC analysis parameters for 3 pt short-beam bending

QOI U in mm V in mm W in mm εx in % εy in % γxy in % ε1 in % ε2 in % γ12 in %

Bias 0.0001 0.0003 0.0006 0.0015 0.0003 0.0027 0.0120 0.0107 0.0227

Noise-floor 0.0002 0.0002 0.0017 0.0095 0.0215 0.0205 0.0178 0.0140 0.0219

Table 4.7 Bias and noisefloor (spatial and static) of QOI from ROI for default DIC analysis parameters for 4 pt short-beam bending

QOI U in mm V in mm W in mm εx in % εy in % γxy in % ε1 in % ε2 in % γ12 in %

Bias 0.0002 0.0005 0.0044 0.0005 0.0023 0.0039 0.0114 0.0087 0.0201

Noise-floor 0.0003 0.0003 0.0048 0.0113 0.0161 0.0179 0.0131 0.0135 0.0157

Fig. 4.31b and compare to Fig. 4.15a with default DIC analysis parameters). Note that due the larger subset size, the post-analysis ROI is smaller for the larger VSG, the offset between edge of the specimen and ROI of the image is larger resp., compare extrema to y-axis limits in Fig. 4.31. Tables 4.6, 4.7, and 4.8 list the bias and noise-floor for three-point, four-point, and five-point short-beam bending for the default DIC analysis parameters. Results for three-point and four-point bending at larger L/h, solely focusing on absolute displacements, are not shown here. Note the higher values in comparison to Chaps. 2–3, possibly due to the applied fine speckle pattern (via airbrush) and related issues (e.g., aliasing due to subcritical pattern features, contrast, and intensity resp.). Further explanations include the use of macro lenses (high magnification can be sensitive to small motions) and image blur.

182 Table 4.8 Bias and noisefloor (spatial and static) of QOI from ROI for default DIC analysis parameters for 5 pt short-beam bending

4 Flexural Testing QOI U in mm V in mm W in mm εx in % εy in % γxy in % ε1 in % ε2 in % γ12 in %

Bias 0.0006 0.0002 0.0011 0.0024 0.0030 0.0047 0.0165 0.0111 0.0276

Noise-floor 0.0004 0.0003 0.0031 0.0131 0.0234 0.0243 0.0186 0.0173 0.0235

Displacements This section focuses on the investigation of the absolute deformation response during three-point, four-point, and five-point short-beam bending. Detailed horizontal and vertical displacements for three-point bending are investigated and compared to the analytical solution of the deflection for small and larger span-to-thickness ratios.

Horizontal Displacement U Figures 4.33a and 4.34 show the horizontal (antisymmetric) displacement fields U for three-point, four-point, and five-point short-beam bending in the undeformed reference coordinates. Detailed through-thickness line plots in the vicinity of the support rollers are shown in Fig. 4.33b and c for three-point short-beam bending at selected load levels, depicting plane (linear) deformation outside and (non-linear) warping within the shear zone, within the span resp. Note the horizontal displacement present in the shear zones for four-point (Fig. 4.34a) and only in the small shear zones for five-point bending (Fig. 4.34b).

Vertical Displacement V Figure 4.35 shows the vertical deformation V, deflection resp., for three-point, fourpoint, and five-point short-beam bending in the undeformed reference coordinates. For three-point and four-point bending, the maximum absolute deflection V* is observable at the center between the support rollers, which has been used for the deduction of force-deflection curves (see Data Reduction Section) and the flexural moduli (see Sensitivity Analysis: Span Section). For five-point bending, the absolute deflection V* at the center between left and right loading roller and middle support

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Fig. 4.33 (a) Distribution of horizontal displacement U at τmax ¼ 50 MPa, Through-thickness line plots for selected load levels up to τmax ¼ 50 MPa close to (b) left and (c) right support for 3pt, L/h ¼ 6

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35

x in mm Fig. 4.35 Distribution of vertical displacement V at τmax ¼ 50 MPa for (a) 3pt, (b) 4pt, (c) 5pt

roller has been used for the deduction of force-deflection curves and the calculation of the shear moduli (see Data Reduction Section). Figure 4.35c depicts a slight asymmetric alignment, visible by the larger deflection located on the left compared to the right side (where delamination occurred, see Fig. 4.24). For a quantitative description of bending and shear deflection for different spanto-thickness ratios, the DIC calculated vertical displacement provided in Figs. 4.36 and 4.37 aim for a thorough investigation of the midline deflection. A three-point loaded specimen at L/h ¼ 26 is shown in Fig. 4.36. The vertical dashed lines depict the points of no deflection for the midline deflection shown in Fig. 4.36b, representing the supports with a span of 62.5 mm. The difference to the initial (pre-test) measurement of 62 mm using a caliper is 0.5 mm, being larger than the required accuracy of 0.1 mm from ASTM (2015) for spans smaller than 63 mm (0.3 mm for L > 63 mm). Additionally, Fig. 4.36b depicts the analytical solution for the deflection between the supports at selected load levels. Using Eq. 4.1 (neglecting shear), the second derivative for both regions of integration, left (“L”) and right (“R”) half-span with variable xi, due to a bending moment is

Visualization of Strains and Displacements V in mm y in m m

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

1 0 -1 0

10

20

30

40

a

50

60

70

x in mm

DIC Anal. Soln.

0

V in mm

Fig. 4.36 (a) Distribution of vertical displacement V in the deformed state at Fmax, (b) midline deflection in comparison to analytical solution for selected load levels up to Fmax, (c) analytical solution for bending and shear at Fmax for 3pt bending at L/h ¼ 26

185

-0.5 -1 -1.5 Support

-2 0

20

40

60

x in mm

b DIC

f(Efl)

f(G)

f(Efl, G)

V in mm

0 -0.5 -1 -1.5 Support

-2 0

20

V00L ð0 < xi < L=2Þ ¼  V00R ðL=2 < xi < LÞ ¼ 

40

60

x in mm

c

F=2 ∙ xi

E∙I

F=2 ∙ ðL

 xi Þ E∙I

ð4:21Þ ð4:22Þ

The deflection V for both regions is expressed by integrating twice   1 F 1 3 L2 VL ¼  ∙ ∙ ∙ xi  ∙ xi 4 E∙I 3 4

ð4:23Þ

186

4 Flexural Testing

Fig. 4.37 (a) Midline deflection for selected load levels up to delamination, (b) Midline deflection at τmax ¼ 50 MPa in comparison to analytical calculation for bending and shear deflection for 3pt short-beam bending at L/ h¼6

V in mm

0

-0.2

-0.4 Support

-0.6 0

5

10

15

20

x in mm

a DIC

f(G)

f(Efl)

f(Efl, G) + Voff

V in mm

0

-0.1

-0.2 Support

0

b

5

10

15

20

x in mm

  1 F 1 L2 ∙ ∙ ð L  xi Þ 3  ∙ ð L  xi Þ VR ¼  ∙ 4 E∙I 3 4

ð4:24Þ

The integration constants are determined using the boundary conditions at the supports with VL(0) ¼ VR(L ) ¼ 0 and at mid-span (for continuity) VL(L/2) ¼ VR(L/ 2) ¼ V* (maximum absolute deflection), where both equations simplify to Eq. (4.3) (Gross et al., 2018). For the analytical solution, a flexural modulus of Efl ¼ 116 GPa has been chosen (anticipatory to the results shown in Fig. 4.40 from convergence state) and compared to the DIC calculated data, see Fig. 4.36b. The offset of approx. 5% between DIC calculated data and analytical solution results from the lower flexural modulus determined at L/h ¼ 26, being approx. Efl ¼ 110 GPa, compare with Fig. 4.13a. This can be verified by using the analytical solution for a regression analysis of the DIC calculated data, resulting in slightly smaller and different moduli for each load

Effect of Specimen Geometry

187

level. Note that the flexural modulus deduced from the force-maximum deflection curve relates to the compliance (slope of the force-deflection curve), multiple load levels resp. In order to investigate the offset between DIC calculated data and analytical solution, Fig. 4.36c depicts the single contribution of each bending and shear (index S) deflection, using Eq. (4.8) for both regions of integration expressed by 1 F ∙ ∙x 2 k∙G∙w∙h i 1 F ∙ ð L  xi Þ VR,S ðxi Þ ¼ ∙ 2 k∙G∙w∙h VL,S ðxi Þ ¼

ð4:25Þ ð4:26Þ

At the loading point, with VL,S(L/2) ¼ VR,S(L/2), both equations simplify to the second term in Eq. (4.10). By using an interlaminar shear modulus of G ¼ 3.9 GPa (from the investigation presented in Shear Strain: Stress–Strain Response Section, see Table 4.10) and a shear correction factor of k ¼ 5/6, the resulting sum of both bending and shear contribution (with approx. 5% shear deflection) matches the DIC calculated data, see Fig. 4.36c. Figure 4.37a shows midline deflection from three-point short-beam bending for a span-to-thickness ratio of L/h ¼ 6 at selected load levels. Note that the whole specimen moves downward with an offset of approx. Voff ¼ 70 μm, resulting from a setting effect (compare with Fig. 4.10). In general, the midline deflection shows a “V” shape, resulting from shear deflection. A thorough investigation on the contribution of shear deflection within the elastic region (at τmax ¼ 50 MPa) is provided in Fig. 4.37b. The corresponding contour plot is shown in Fig. 4.35a. Applying the previously presented analytical solution including each contribution of both bending and shear deflection, the resulting curve including the offset Voff, is shown in Fig. 4.37b and matches the DIC calculated data. Note the contribution of shear of approx. 50%, which is in good agreement with the equal amount of bending and shear presented in Fig. 4.13b with Efl/G ¼ 116 GPa/3.9 GPa ¼ 30 at L/h ¼ 6.

Effect of Specimen Geometry Figure 4.38 compares the maximum shear stress-deflection curves from three-point short-beam bending for two specimen geometries, see details in Fig. 4.5a and Table 4.1. Note the smaller setting effect for the standardized three-point ASTM specimen (w/h ¼ 2, L/h ¼ 5) (ASTM, 2016b). Both curves show a plateau region with delamination occurring after reaching the maximum shear stress. For the threepoint ASTM specimen, an almost straight delamination at the midplane across the width can be attested (not shown here, see (Merzkirch & Foecke, 2020)). Figure 4.39a depicts the distribution of the shear strain γ xy for a three-point ASTM specimen attesting a symmetric alignment, see Fig. 4.39b.

Fig. 4.38 Representative maximum shear stress– maximum deflection curves for 3pt short-beam bending with different width-tothickness ratios of the specimens

a

b Fig. 4.39 (a) Shear strain distribution γ xy at τmax ¼ 50 MPa, (b) corresponding midline plot

Summary and Discussion

189

Summary and Discussion This section focuses on a detailed investigation and interpretation of the elastic flexural properties, i.e., flexural modulus. Furthermore, the results of feature extraction from shear stress-shear strain responses for all test methods are summarized.

Elastic Properties

130 120 110 100 90 80 70 60 50 40 30

1 0.9 EL

0.8

Efl (A)

0.7

Efl (3pt)

0.6

Efl (4pt)

0.5

Efl, cor (3pt) 0.4 Efl, cor (4pt) 0.3

0

10

20

30

40

L/h in mm/mm

50

60

E/EL in GPa/GPa

Fig. 4.40 Comparison of apparent flexural moduli for different span-to-thickness ratios from 3pt and 4pt bending to tensile modulus (with standard deviation)

E in GPa

Figure 4.40 depicts similar information as Fig. 4.13a, including the evolution of the apparent flexural moduli and an additional comparison to the tensile modulus. Furthermore, the corrected apparent flexural moduli (up to L/h ¼ 26 for 3pt bending) are shown, determined by the relative displacement between maximum absolute deflection V* and displacement at the supports, to account for system compliance (see setting effect in Fig. 4.10 and Voff in Fig. 4.37, and indentation in Failure Investigation Section). The larger the span-to-thickness ratio, the smaller the influence of the system compliance on the flexural moduli. The true flexural modulus Efl (A), i.e., when shear deflection is significantly lower than bending deflection (Zweben et al., 1979) (see also Fig. 4.13b), reaches a maximum of approx. 93% of the tensile modulus in the convergence state. Four-point bending leads to higher flexural moduli at smaller span-to-thickness ratios than three-point bending with the flexural modulus at a span-to-thickness ratio of approx. L/h ¼ 25 being close to the tensile modulus. Even though the bending strains are distributed in the constant moment and shear free section, in comparison to three-point bending, (compare with Figs. 4.2, 4.16, and 4.26), the contribution of the outer shear zones cannot be neglected.

190

4 Flexural Testing

Flexural and Shear Properties The following methodology (Carlsson et al., 2014; Tolf & Clarin, 1984; Wagner & Marom, 1982) is used for determining flexural (ASTM, 2016a) and shear properties of sandwich core structures. The compliance for three-point bending including bending and shear deflection, using Eq. (4.10), leads to C¼

V 1 L3 1 L ¼ ∙ þ ∙ F 4 E ∙ w ∙ h3 4 k ∙ G ∙ w ∙ h

ð4:27Þ

which is relocated to a linear equation  2 4∙w∙h 1 L 1 ∙ þ ∙C ¼ L Efl h k∙G

ð4:28Þ

and used as linear regression analysis, see Fig. 4.41a, with Efl being the inverse slope and with k G being the reciprocal of the intercept (which has per se a high sensitivity), representing method A. A similar method B with a slight change of Eq. (4.27) (Carlsson et al., 2014) to  2 4 ∙ w ∙ h3 1 h 1 ∙C ¼ þ ∙ 3 k ∙ G L E L fl

ð4:29Þ

is used as linear regression analysis, see Fig. 4.41b, with k G being the inverse slope and Efl being the reciprocal of the intercept (which has per se a high sensitivity). The resulting elastic properties of flexural and shear moduli from both regression analysis methods are listed in Table 4.9, including the effect of the correction to account for system compliance. The uncertainty of the elastic properties determined from the intercept from both regression analysis methods (method A for G and x 10

0.03 0.02 0.01 Data (3pt) Lin. Regr.

C*4*w*h3/L3 in mm2/N

C*4*w*h/L in mm2/N

0.04

0 0

1000

2000

3000

2 1.5 1 0.5

Data (3pt) Lin. Regr.

0

4000

0

2

0.01

0.02

0.03

(h/L)2 in mm/mm

(L/h) in mm/mm

a

-5

b

Fig. 4.41 Compliance-based methods for the determination of the flexural and shear modulus. (a) Method A, (b) method B

Summary and Discussion Table 4.9 Results from linear regression from two methods (minimum and maximum in brackets)

191 Method A (uncorrected) A (corrected) B (uncorrected) B (corrected)

Efl in GPa 116.4 (115.2, 117.7) 116.0 (114.7, 117.2) 116.4 (113.3, 119.6) 113.2 (109.5, 117.2)

G in GPa 2.6 (1.9, 3.7) 3.3 (2.3, 5.6) 2.5 (2.5, 2.6) 4.0 (3.7, 4.3)

method B for Efl) is higher. The true flexural modulus Efl shows comparable values for all methods. A range of the resulting true flexural modulus (Efl (A) with correction) is shown in Fig. 4.40, which is comparable to the flexural modulus determined at a span-to-thickness ratio of L/h ¼ 62. Note the huge difference of the determined shear moduli from different regression analysis methods. Both regression analysis methods include the use of the shear correction factor k ¼ 5/6, usually used for isotropic beams. A comparison of the interlaminar shear modulus will be discussed in subsequent Interlaminar Shear Properties Section.

Flexural Modulus vs. Tensile Modulus The flexural moduli from three-point bending, even for large span-to-thickness ratios, is approx. 7% lower than the tensile modulus, due to the heterogeneous cross-section acc. to Tolf and Clarin (1984). This section presents possible explanations for this discrepancy, which has already been discussed in Zweben et al. (1979) and He et al. (2002). Those include resin rich layers, the effect of the compressive modulus (Carlsson & Pipes, 1987), the number of laminae in the laminate, non-uniform layer thickness, and alignment issues leading to an off-axis angle (He et al., 2002). Four explanations will be presented hereinafter.

Resin Rich Layers Since the stress distribution in flexural loading is inhomogeneous across the thickness, see Fig. 4.3a, compared to tensile loading, the outermost planes, where bending stress is highest, contribute to the flexural modulus, especially if considering the inhomogeneous structure of composite laminates. For quantification of the effect of resin rich layers of thickness Δh at the outermost surface (e.g., due to manufacturing), an effective flexural modulus (Zweben et al., 1979) can be expressed by  3 Δh Efl,eff ¼ EL ∙ 1  h

ð4:30Þ

Using the specimen thickness h ¼ 2.4 mm and an effective (true resp.) flexural modulus listed in Table 4.9, the thickness of the resin layer is approx. Δh ¼ 56 μm.

192

4 Flexural Testing

Note that the thickness of all specimens tested, shown in Fig. 4.40, is h ¼ 2.44 mm  0.024 mm.

Tensile Modulus vs. Compressive Modulus Another source of uncertainty is the inequality of the tensile EL and compressive EL,comp moduli, which results in an apparent flexural modulus, acc. to Carlsson and Pipes (1987) Efl ¼

1þ

2 ∙ EL pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EL=E

ð4:31Þ

L,comp

This results in a compressive modulus of EL,comp ¼ 95.1 GPa (EL,comp ¼ 0.76 EL), which qualitatively agrees with the observations shown in Neutral Axis Section. As a comparison, reference (Makeev et al., 2012) reports a number of EL,comp ¼ 0.91 EL, whereas reference (Carlsson & Pipes, 1987) reports a number of EL,comp ¼ 1.09 EL for carbon/epoxy composites.

Number of Layers Since the stress distribution over the thickness of composite laminates is only continuous and linear within each layer, the flexural modulus is determined from the tensile modulus of each layer EL,k (Tolf & Clarin, 1984), with zk being the distance to the midline of the kth layer (Jones, 1999), using lamination theory (see Chap. 1 and D-matrix entry in Table 4.5) Efl ¼

N X k¼1

EL,k ∙

  h3k 12 2 ∙ h ∙ z þ k k 12 h3

ð4:32Þ

For a single-layer composite, lamina resp., the flexural modulus will be equal to the tensile modulus (h1 ¼ h, z1 ¼ 0, Efl ¼ EL).

Off-Axis Angle A further explanation for the discrepancy in offset between flexural and tensile moduli is the fiber tow offset effect (He et al., 2002) resulting in an offset modulus. Note that an offset-angle of approx. 2.5 leads to a decrease in modulus of approx. 7%, see also Chap. 6, orientation dependent inverse compliance S11.

Summary and Discussion

193

Table 4.10 Results from three short-beam tests each ( representing standard deviation) Test method 3pta 3pt ASTMa 4pta 5pt

G (γ xy) in GPa 3.93  0.09 3.73  0.05 3.58  0.51 4.5  0.09 (calc.)

G (γ 12) in GPa 3.94  0.10 3.69  0.12 3.60  0.53 N/A

USS in MPa 82.4  2.8 88.8  0.4 84.8  2.7 114.1  1.7

γ xy, USS in % 8.52  0.39 9.54  2.15 7.87  0.77 N/A

γ 12, USS in % 8.54  0.39 9.56  2.16 7.89  0.77 N/A

a

Strain related values from one test, moduli from both shear zones, strain at failure from delamination side

Interlaminar Shear Properties Three-point, four-point, and five-point short-beam tests have been investigated using DIC calculated shear strains and absolute displacements to examine the elastic and delamination shear properties. Table 4.10 summarizes the extracted shear properties for all test methods. Note that four-point bending showed a slight asymmetric alignment leading to higher shear moduli on the side with lower shear strains and vice versa. In order to account for this asymmetry, the average value from both sides have been taken for all test methods. Despite the challenging smaller size of the ASTM specimen, it can be concluded that the procedure described for the determination of the full stress-strain response (see Shear Strain: Stress–Strain Response Section) is applicable to different specimen geometries and test methods (three-point and four-point short-beam bending). It is noteworthy that the determined shear moduli from different tests are in good agreement with the shear modulus determined via the linear regression method B for three-point bending and accounting for a correction of the system compliance, being G ¼ 4 GPa, see Table 4.9. All values for three-point and four-point bending show comparable ultimate shear strengths, whereas the value for five-point bending is considerably higher, approx. 30% (in agreement to (ISO, 2018) for epoxy-based UD composites). Three-point ASTM shows slightly higher ILSS values than three-point and four-point shortbeam bending. One possible explanation is the smaller volume tested. The line load of the loading roller is similar for both types of three-point specimens, despite the smaller width of the three-point ASTM specimen. In accordance with Cui et al. (1992), the failure mode in four-point bending (with 5% higher values for USS) is purely interlaminar shear failure with roller induced damage being observed in threepoint bending. Since all force-deflection curves in three-point and four-point shortbeam bending showed a pronounced plateau region, it is assumed that the interaction between roller and specimen cannot be neglected. For all test methods, delamination mostly occurred after reaching the load maximum. In summary, the shear properties from five-point bending show the highest values. Future investigation should clarify if the alternating shear strain along the length of the specimen (see Fig. 4.17c) or more contact (due to five rollers, see also

194

4 Flexural Testing

investigations presented in Kim and Dharan 1995) leads to a higher constraint and therefore to a higher ultimate shear strength and shear modulus. Albeit the determination of the “apparent” (DIN, 1998) shear strength is based on the fact that the result obtained is not considered to be an absolute value and can be affected by, e.g., different specimen sizes/geometries, two different specimen geometries for three-point and four-point bending show comparable interlaminar shear strength values.

Testing of Assorted Materials This section focuses on the applicability of DIC measurements for a different type of composite laminate and the related conspicuous photomechanical results. Furthermore, flexural properties in terms of flexural and shear moduli of a woven composite laminate and the epoxy-based matrix material are investigated.

Woven-CFRP: Three-Point, Four-Point, and Five-Point Bending Three-point, four-point, and five-point short-beam bending has been carried on specimens with an average thickness h ¼ 2.7 mm and average width w ¼ 13 mm, resulting in a width-to-thickness ratio of w/h ¼ 5. Note that the ASTM specimen is not representative for the unit cell (approx. 9.7 mm) due to the small width (w ¼ 2 mm). Additionally, elastic flexural investigations were carried out under three-point and four-point loading, see Table 4.11 for further details on geometry and setup.

Elastic Flexural Properties Figure 4.42 compares the resulting flexural moduli (including compensation of system compliance) from three-point and four-point bending to the tensile modulus. Table 4.11 Dimensions of the specimens and geometric details about testing setup Test method 3pt (shear) 5pt (shear) 3pt 3pt and 4pt 3pt and 4pt 3pt

Length l in mm 26.9 39.8 40.0 50.0 75.0 120.0

Span L in mm 16.0 28.3 23.7 40.1 62.0 100.8

Span-to-thickness L/h 5.9 10.3 9 15 23 37

Testing of Assorted Materials

70 1

0.9

50

0.8

40 30

EL

0.7

Efl (A)

0.6

Efl (3pt)

0.5

Efl (4pt)

0.4

20

E/EL in GPa/GPa

1.1 60

E in GPa

Fig. 4.42 Comparison of apparent flexural moduli for different span-to-thickness ratios from 3pt and 4pt bending to tensile modulus (with standard deviation)

195

0 5 10 15 20 25 30 35 40 45 50 55

L/h in mm/mm

Table 4.12 Results from linear regression for flexural and shear moduli for woven (minimum and maximum in brackets)

Method A B

Efl in GPa 54.5 (54.3, 54.8) 54.0 (53.0, 55.0)

G in GPa 2.7 (2.4, 3.2) 3.0 (2.8, 3.3)

Both flexural moduli from three-point and four-point bending show comparable values at L/h ¼ 23, starting to asymptotically converge. The resulting true flexural modulus is approx. 6% below the tensile modulus. Table 4.12 lists the calculated (corrected) flexural and shear moduli deduced using the methodology presented in Flexural and Shear Properties Section. An explanation for the discrepancy between flexural and tensile moduli is presented in Flexural Modulus vs. Tensile Modulus Section, including resin rich layers (50 μm, acc. to Eq. (4.30)) and differences in tensile and compressive moduli (EL,comp ¼ 46 GPa ¼ 0.78 ET, acc. to Eq. (4.31)).

Interlaminar Shear Properties Figure 4.43 shows maximum shear stress-deflection curves from three-point and five-point short-beam bending, with indication of onset of delamination. DIC calculated strain distribution for three-point short-beam bending is shown in Fig. 4.44a. A symmetric alignment can be attested via the midline plot in Fig. 4.44b. Both shear zones depict a non-parabolic distribution across the thickness, see Fig. 4.44c. This can be explained by the inhomogeneous architecture of the woven composite, resulting in a rather complicated failure mode in the left half-span, see also Fig. 4.44d. Figure 4.45a visualizes the shear distribution for five-point short-beam bending. The midline plot in Fig. 4.45b depicts some irregularity in shear strain along the

196

4 Flexural Testing

Fig. 4.43 Maximum shear stress-deflection curves for 3pt and 5pt short-beam bending

span. Failure of the specimen occurred within the right high-shear zone, indicated by the shear strain distribution shown in Fig. 4.45c. Preliminary results are listed in Table 4.13, which have to be handled with caution. The maximum stress for the determination of the ultimate shear strength relates to the extrema at the neutral axis of the specimen. Doubtful that Eq. (4.13) can be used for a proper characterization of the shear properties, such as USS and shear modulus due to highly inhomogeneous distributions of shear strain along the length for three-point and five-point short-beam bending. The determined shear modulus from five-point bending is comparable to the shear modulus determined via the linear regression method for three-point bending, see Table 4.12.

Resin: Three-Point Bending The aim for testing the epoxy-based resin, representing the matrix material of both UD and woven-CFRP laminates, is the investigation of the contribution of shear on the flexural properties of isotropic materials. Therefore, three-point flexural testing has been carried out on specimens with an average thickness h ¼ 2.5 mm and an average width w ¼ 12 mm, with a nominal displacement rate of 1 mm/min, independent of the span (be aware of the time-dependent material behavior). A comparison of the moduli from three-point flexural testing and tensile testing is shown in Fig. 4.46. In comparison to UD composites, the effect of shear can be neglected for L/h > 10, compare with Fig. 4.13b, since the flexural modulus asymptotically approaches the tensile modulus. Table 4.14 lists the calculated flexural and shear moduli deduced using the methodology presented in Flexural and Shear Properties Section. The high uncertainty in determining the shear modulus is based on the intercept method as well as the sparse data. The shear modulus of the epoxy (G ¼ 1.22 GPa) falls within the ballpark of the calculated shear moduli, method B, using a shear correction factor k ¼ 5/6.

Testing of Assorted Materials

197

a

b

c

d Fig. 4.44 Shear strain distribution at τmax ¼ 30 MPa. (a) γ 12, (b) midline plot of γ xy, (c) throughthickness line plots of shear strains at left half-span ¼ L and right half-span ¾ L, (d) shear strain distribution γ 12 just before failure (in the deformed state)

198

4 Flexural Testing

a

b

c Fig. 4.45 Shear strain distribution for 5pt at τmax ¼ 30 MPa. (a) γ 12, (b) midline plot of γ xy, (c) shear strain distribution γ 12 just before failure (in the deformed state)

Key Conclusions

199

Table 4.13 Results from short-beam bending on woven-CFRP ( representing standard deviation)

Test method 3pta 5ptb a

USS in MPa 41.5 52.3  7.6

G in GPa N/A 2.77  0

Only one test Two tests

b

Fig. 4.46 Comparison of apparent flexural moduli for different spans from 3pt bending to tensile modulus

E in GPa

1.1 3.5 1 0.9

EL

3

Efl (3pt) Efl (A)

E/EL in GPa/GPa

4

0.8

2.5 0

5

10

15

20

L/h in mm/mm

Table 4.14 Results from linear regression for flexural and shear moduli for epoxy resin (minimum and maximum in brackets)

Method A B

Efl in GPa 3.50 (3.44, 3.56) 3.51 (3.43, 3.59)

G in GPa 1.44 (0.77, 10.4) 1.43 (1.15, 1.89)

Key Conclusions DIC • For large deflections expected, a severe loss in correlation might be accompanied as a result from rollers covering the ROI. Remedy could be the use of 2D-DIC. • High resolution cameras must be chosen for recognizing the fine speckle pattern applied, e.g., by airbrush, which will lead to subcritical speckle sizes ( 10), the compliance is simplified to C ffi 64 ∙

a3 E ∙ w ∙ h3

ð5:7Þ

When experimentally determining the compliance, the crack length is calculated via rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 3 C∙E∙w∙h a¼ 64

ð5:8Þ

but requires the value of the flexural (Hashemi et al., 1989) modulus E in 0 fiber orientation. A non-standardized procedure, with the advantage of requiring fewer measured inputs (i.e., no need to measure the crack length by using Eq. (5.8)) is the pure compliance (PC) method (Hashemi et al., 1989), using Eqs. (5.1) and (5.7), for the determination of the energy release rate G PC IC ¼

96 ∙ F 2 ∙ a2 E ∙ w 2 ∙ h3

ð5:9Þ

The simple beam (SB) theory method (Hashemi et al., 1989; ASTM, 2013a), which will overestimate G IC due to the simplification of assuming built-in conditions at the delamination front (ASTM, 2013a), uses the pure compliance method and substitutes E with the compliance expressed in Eq. (5.7) to G SB IC ¼

3∙F∙V 2∙w∙a

ð5:10Þ

In order to compensate the former assumption that the deflection and rotation at the cantilever beam root (at the crack tip) is zero (Hashemi et al., 1989), the modified beam (MB) theory (ASTM, 2013a) or corrected beam theory (CBT) (ISO, 2001) method (Hashemi et al., 1989), recommended by (ISO, 2001; ASTM, 2013a) G MB IC ¼

3∙F∙V 2 ∙ w ∙ ða þ jΔjÞ

ð5:11Þ

includes a shift Δ of the crack length to zero the third root of the compliance C1/3 at a ¼ 0 mm (Hashemi et al., 1989). This correction to the crack length, to account for

210

5

Delamination Resistance Testing

the imperfectly clamped beam boundary condition (Davies et al., 1998), allows for a determination of the flexural modulus (Hashemi et al., 1989) (compare to Eq. (5.7)) ða þ jΔjÞ3 w ∙ h3 ∙ C

E I,fl ¼ 64 ∙

ð5:12Þ

The compliance calibration (CC) method, recommended by (ASTM, 2013a) G CC IC ¼

n∙F∙V 2∙w∙a

ð5:13Þ

includes the slope n of the relationship between the logarithm of the compliance C and the logarithm of the crack length a. Note that for n ¼ 3, the CC method becomes identical to the SB method, see Eq. (5.10). The modified compliance calibration (MCC) method, recommended by (JIS, 1997; ISO, 2001; ASTM, 2013a) G MCC ¼ IC

3 ∙ F 2 ∙ C2=3 2 ∙ A1 ∙ w ∙ h

ð5:14Þ

includes the slope A1 between the third root of the compliance C1/3 and the crack length-to-thickness ratio a/h. The Area method (Hashemi et al., 1989), which does not provide an initiation value (ASTM, 2013a), is recommended by (DIN, 2013a) R G Area IC

¼

irrev F ∙ dV

w ∙ ai

ð5:15Þ

includes the (irreversible) work determined from the force–displacement curve (see also Fig. 5.1), needed to create a crack surface (of area w  ai). For verification of the theoretical stability of crack propagation under displacement-controlled loading conditions, using Eqs. (5.2) and (5.10), the inequality dG IC 3∙F∙V 0 ¼ da 2 ∙ w ∙ a2

ð5:16Þ

is always given and therefore leads to stable crack growth (whereas force-controlled loading conditions lead to unstable crack growth) (Carlsson & Pipes, 1987; Hashemi et al., 1990; Gross & Seelig, 2011).

Background

211

Mode II Determination of the energy release rate under mode II loading conditions includes a few methods, such as three-point, four-point and single cantilever beam testing, see Davies et al. (1998) and Blackman et al. (2006) for an overview. Two test methods, the end-notched flexural (ENF) test under three-point loading and the calibrated end-loaded split (C-ELS) test (single cantilever beam loading) with a variety of data reduction schemes will be presented hereinafter.

End-Notched Flexural Test The ENF test, initially developed for wood (Davies et al., 1998; Brunner et al., 2008), has been standardized as composite test method for the first time in 1993. Three commonly used data reduction methods, including two standardized methods, exist which will be presented here. The testing principle is based on three-point loading, see Fig. 5.2b, with the deflection (Carlsson et al., 1986; Wang & Qiao, 2004) (with contradictory shear terms in the references) consists of a term from bending and a term from shear deflection (see also Chap. 4)  3  1 F ∙ L þ 12 ∙ a3 1 F∙L V¼ ∙ þ ∙ 3 4 4 k ∙ G ∙w∙h E∙w∙h

ð5:17Þ

Relocation leads to the compliance C¼

V 1=4 ∙ L3 þ 3 ∙ a3 1 L ¼ þ ∙ 3 F 4 k ∙ G ∙w∙h E∙w∙h

ð5:18Þ

and by neglecting the shear deflection term, the compliance is simplified to Cffi

  1 ∙ 1=4 ∙ L3 þ 3 ∙ a3 3 E∙w∙h

ð5:19Þ

When experimentally measuring the compliance, the crack length is determined via rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 3 C ∙ E ∙ w ∙ h  1= ∙ L3 4 a¼ 3

ð5:20Þ

but requires the value of the flexural modulus E in 0 fiber orientation. A nonstandardized procedure, with the advantage of requiring fewer measured inputs (i.e., no need to measure the crack length by using Eq. (5.20)) is the pure compliance (PC) method using Eqs. (5.1) and (5.19) for the determination of the energy release rate

212

G PCENF ¼ IIC

5

Delamination Resistance Testing

9 ∙ F 2 a2 2 ∙ E ∙ w2 ∙ h3

ð5:21Þ

By substituting E using Eq. (5.19), the simple beam (SB) method, recommended by (DIN, 2013b; JIS, 1997), is determined according to G SBENF ¼ IIC

9 ∙ F ∙ a2 ∙ V   2 ∙ w ∙ 1=4 ∙ L3 þ 3 ∙ a3

ð5:22Þ

Note that the force is referred to initiation of crack growth, reference (DIN, 2013b) relates a to the initial crack length a0, whereas (JIS, 1997) relates to the calculated, compliance-based, crack length. The compliance calibration method (CC), recommended by (ASTM, 2014), is determined according to G CCENF ¼ IIC

3 ∙ m ∙ F 2 a2 2∙w

ð5:23Þ

and includes the slope m between the cube of the crack length a3 and the compliance C. Usually compliance calibration tests, by varying a0 through a shift of the specimen, are carried out beforehand. Note that the force relates to the maximum force (ASTM, 2014) and a to the initial crack length a0. For verification of the theoretical stability of crack propagation during displacement-controlled loading conditions, using Eqs. (5.2) and (5.22), the inequality (Carlsson & Pipes, 1987) " # dG IIC 9 ∙ a ∙ V2 9 ∙ a3  0 ¼ ∙ 1 3 da 1=4 ∙ L þ 3 ∙ a3 8 ∙ w 2 ∙ h3 ∙ C 2 ∙ E

ð5:24Þ

9 ∙ a3 1 3 1=4 ∙ L þ 3 ∙ a3

ð5:25Þ

leads to 

a

p ffiffiffiffiffiffiffi 3

1=24 ∙ L

ð5:26Þ

stating that stable crack growth is expected for crack lengths a > 0.347 L (whereas force-controlled testing always leads to unstable crack growth (Carlsson & Pipes, 1987; Hashemi et al., 1990).

Background

213

Calibrated End-Loaded Split Test The testing principle of the end-loaded split (ELS) test has already been described in the late 1980s (Williams, 1989), standardization as calibrated ELS (C-ELS) test has been done for the first time in 2014. Four commonly used data reduction methods exist, which will be presented here. The testing principle is based on single cantilever beam loading, see Fig. 5.2c. The deflection of a pristine single cantilever beam (SCB) (Timoshenko, 1940a; Wagner & Marom, 1982; Tolf, 1985) is expressed by V ¼ 4∙

F ∙ L3 F∙L þ E ∙ w ∙ h3 k ∙ G ∙ w ∙ h

ð5:27Þ

Substitution using crack length a gives the compliance of a pre-cracked specimen (de Moura & de Morais, 2008) C¼

V L3 þ 3 ∙ a3 L ¼ 4∙ þ F k∙G∙w∙h E ∙ w ∙ h3

ð5:28Þ

and neglecting the shear term, the compliance is simplified to (Wang & VuKhanh, 1996; ISO, 2014) C ffi 4∙

L3 þ 3 ∙ a3 E ∙ w ∙ h3

ð5:29Þ

When experimentally measuring the compliance, the crack length is determined via sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 3 C ∙ E ∙ w ∙ h  L3 4 a¼ 3

ð5:30Þ

but requires the value of the flexural modulus E in 0 fiber orientation. The determination of the energy release rate using the experimental compliance method (EC) (ECM in (ISO, 2014)), is comparable to the previously shown compliance calibration (CC) method for ENF testing (ASTM, 2014) using Eq. (5.23). The procedure differs to that extent, that the slope m between the cube of the crack length a3 and the compliance C, refers to the actually test (no calibration in advance needed). This data analysis is very sensitive to errors in crack length measurements and may return erroneous results (Blackman et al., 2006). A procedure, with the advantage of requiring fewer measured inputs (i.e., no need to measure the crack length by using Eq. (5.30)) is the pure compliance (PC) method (“simple beam theory” acc. to (ISO, 2014)), using Eqs. (5.1) and (5.29)

214

5

G PCELS ¼ IIC

Delamination Resistance Testing

18 ∙ F 2 a2 E ∙ w 2 ∙ h3

ð5:31Þ

and by substituting E using Eq. (5.29), the simple beam (SB) method (not mentioned in (ISO, 2014)) can be deduced G SBELS ¼ IIC

9 ∙ F ∙ a2 ∙ V   2 ∙ w ∙ L3 þ 3 ∙ a3

ð5:32Þ

The recommended procedure by (Blackman et al., 2006) is the corrected beam theory using effective crack length (CBTE) (ISO, 2014) including a calculation of the crack length based on clamp calibration data. Beforehand, a specimen is tested at a variation of span-to-thickness ratio L/h via “inverse ELS testing”, with the specimen being a single cantilever beam (SCB) with the crack fully held within the clamp (Blackman et al., 2006). The clamp correction is interpreted as the shift Δ of the span to zero the third root of the compliance C1/3 at L ¼ 0 mm. The flexural modulus is determined by E ELS II,fl ¼ 4 ∙

ðL þ jΔjÞ3 w ∙ h3 ∙ C

ð5:33Þ

(compare to first term in Eq. (5.27)). Relocation of Eq. (5.33) refers to the slope of the clamp calibration data, i.e., the slope of the linear regression to C1/3 vs. L, representing the corrected flexural modulus ECBTE (ISO, 2014). The same correction Δ is included to the span L for the calculation of the effective crack length, see Eq. (5.30). For the determination of the energy release rate G CBTE IIC , Eq. (5.31) is used, substituting by the corrected effective crack length and the corrected flexural modulus. For verification of the theoretical stability of crack propagation during displacement-controlled C-ELS testing, using Eqs. (5.2) and (5.31), the inequality " # dG IIC 9 ∙ a ∙ V2 9 ∙ a3  ¼ ∙ 1 3 da L þ 3 ∙ a3 8 ∙ w 2 ∙ h3 ∙ C 2 ∙ E

ð5:34Þ

9 ∙ a3 1 L þ 3 ∙ a3

ð5:35Þ

leads to 

3

a

p ffiffiffiffiffi 3

1=6 ∙ L

ð5:36Þ

Background

215

stating that stable crack growth is expected for crack lengths a > 0.55 L (whereas force-controlled testing always leads to unstable crack growth).

Testing Uncertainties By inserting the release film, with a recommended thickness less than 13 μm (Blackman et al., 2006) during manufacturing, matrix material can cluster in front of the crack tip (Beehag & Ye, 1996). Therefore, pre-cracking (less than 5 mm (Blackman et al., 2006)) is recommended before the actual test (Davies et al., 1998). Further influences on the energy release rate include fiber bridging (based on nesting), multiple cracking in mode I loading (Davies et al., 1998) and the span-tothickness ratio L/h (Davies, 1997). The effect of friction in mode II loading (Carlsson et al., 1986) due to the contact of the upper and lower crack faces, has been investigated for ENF testing in (Davies, 1997), leading to an overestimation of the energy release rate of approx. 20% when not using an additional PTFE film between the crack faces. Further practical difficulties include the loading roller diameter (for ENF testing), specimen geometry and fixture compliance (Brunner et al., 2008). The use of correction factors, based on large deflections, related to material and geometry specific compliance, and stiffening effect by the use of load-blocks, can be looked up in the referenced standards for mode I DCB (ASTM, 2013a) and mode II C-ELS (ISO, 2014). For materials with a low flexural modulus or high interlaminar fracture toughness, it may be necessary to respect a minimum specimen thickness or maximum delamination length to avoid large deflections. Besides the possibility of calculating crack propagation based on the increase in compliance and known flexural modulus, delamination lengths are usually determined visually during the test. For this purpose the specimen is painted white, fiducial marks as equidistant vertical lines are applied and a traveling microscope for more accurate readings is used (Brunner et al., 2008). For mode II delamination, shearing of these lines is the only observable surface feature (Blackman et al., 2006). According to (Blackman et al., 2006), defining and accurately measure crack propagation during mode II delamination is challenging and very difficult, thus leading to a large scatter. Especially at high magnification, it is considered to be a non-trivial task to distinguish between damage and crack growth. A mechanical approach for measuring the crack shear displacement in mode II loading is presented in (JIS, 1997) using gauges in three-point end-notched flexural (ENF) testing for a local determination of an energy release rate just after fracture initiation, which can also serve as real-time test control (stabilized ENF test). In order to circumvent the necessity of monitoring crack propagation under mode I loading, the width-tapered DCB specimen is alternative, e.g., at high loading rates (Daniel & Ishai, 1994). One possibility of crack tip tracking using Digital Image Correlation is based on the contrast between a white painted specimen and black background. A more advanced procedure is based on subset splitting (Poissant & Barthelat 2010). Other methodologies are based on a critical value for crack opening displacement

216

5

Delamination Resistance Testing

(Sutton et al., 1999, 2009; Rajan et al., 2018) and strain in front of the crack tip (Khudiakova et al., 2018, 2020). In order to reduce the noise and improve the DIC measurement, strain around a crack tip can be integrated to find stress intensity factors (Reu, 2012). Reference (Rajan et al., 2018) used stereo-DIC to capture the strain and displacement fields in the vicinity of the crack tip during mode I loading of a DCB specimen aiming for determination of the crack opening displacement and visualization of the process zone of an adhesively bonded laminate.

Specimen Geometry Figure 5.3 depicts the rectangular specimens with an average thickness h ¼ 2.4 mm, for mode I DCB (a), mode II ENF (b) and mode II C-ELS (c) testing, with identical geometric dimensions, see Table 5.1. Note that the fiber orientation is in vertical direction. The specimens have been waterjet cut (using abrasive 80) from a plaque of approximate size 300 mm  300 mm, including a PTFE release film with a length of 50 mm and a thickness of 55 μm (whereas (Davies et al., 1998; Blackman et al., 2006) suggest a thickness of approx. 13 μm). The release film acting as starter crack has been symmetrically inserted (in the middle of 12 layers) along the longitudinal edges of the plaque during production. For mode I DCB and mode II C-ELS specimens, the adherent faying surfaces of the piano hinges and load-block (both same width as the specimen) surfaces were lightly abraded using P400 SiC abrasive paper before applying a commercially available acrylic adhesive and cured at room temperature. Note that hinge-specimen interfaces are located as shown in Figs. 5.2 and 5.3 in order to avoid a stiffening of the single cantilever beams of the DCB specimen. The size of the hinges, with a pin diameter of 2.2 mm, was approx. 15 mm, covering the specimen. The edge size of Fig. 5.3 (a) Mode I DCB specimen with glued on piano hinges, (b) blank specimen for mode II ENF, (c) mode II C-ELS specimen with glued on load-block

a

120 (>160)

120 (190)

20 (15–30)

120 (>160 (ASTM, 2014), >110 (DIN, 2013b))

Length l in mm 120 (>125 (ASTM, 2013a; ISO, 2001), 250 (DIN, 2013b))

Width w in mm 20 (20–25 (ASTM, 2013a), 25 (DIN, 2013b), 15–30 (ISO, 2001)) 20 (19–26 (ASTM, 2014), 25 (DIN, 2013b)) 20 (19–26)

Varies with crack growth since ai ¼ L

Mode II ENF Calibration (ASTM, 2014) Mode II C-ELS (ISO, 2014) Mode II C-ELS Calibration (ISO, 2014)

Mode II ENF

Test method Mode I DCB

75 (100) 40–65 (50–100)

>50 N/A

80 (100)

37.5 (33) 17–27 (17–33)

33 ((DIN, 2013b), 21–29 (ASTM, 2014)) 33 (21–29)

a

80 (100 (ASTM, 2014; DIN, 2013b))

L/h a

Span L in mm

14–30 (20–40)

28 (30 (ASTM, 2014), 40 (DIN, 2013b))

Pre-crack length a0 in mm 40.08 (50 (ASTM, 2013a), 0.55) N/A

0.35 (0.3 (ASTM, 2014), 0.4 (DIN, 2013b)) 0.18–0.38 (0.2–0.4)

a

N/A

21

6–12.5 (4–12)

(13 (DIN, 2013b))

a0/h 17 (10–17 (ASTM, 2013a), 0.35, the maximum theoretical crack propagation length is ai,max ¼ L/2  0.35 L ¼ 0.15 L (which is smaller due to the loading roller). For obtaining representing crack growth data, span L should be large, with a possible adaptation of the DIC setup. Based on suggestions from (ASTM, 2014; DIN, 2013b) on specimen thickness (h ¼ 3 mm (DIN, 2013b) or h ¼ 3.4 mm–4.7 mm (ASTM, 2014) resp.), and related span-to-thickness ratios L/h ¼ 21–33, a span L ¼ 80 mm (L/h ¼ 33) has been chosen, compare to flexural modulus results in Chap. 4. The resulting maximum theoretical crack propagation length is ai,max ¼ 12 mm. See also geometric details listed in Table 5.1. Two types of experiments, calibration and crack propagation experiments,

Testing Setup and DIC Configuration

221

Table 5.2 DIC Hardware parameters (for 2D- and stereo-DIC) Cameras/Image resolution

Lenses LFOV Image scale Stereo-angle LSOD Image acquisition rate Pattern technique Approximate pattern feature size

Value (Mode I DCB) Point Grey 5 Mpx (2448 px  2048 px) CCD, landscape (left) Point Grey/Flir 9.1 Mpx (3376 px  2704 px) CCD, landscape (right) Sigma 105 mm macro lens 50 mm  42 mm (left) 25 mm  20 mm (right) 49 px/mm, 20 μm/px (left) 137 px/mm, 7 μm/px (right) 0 (no stereo vision) 0.7 m (left) 0.6 m (right) 1 Hz

Value (Mode II ENF and C-ELS) Point Grey/Flir 9.1 Mpx (3376 px  2704 px) CCD, landscape

Sigma 105 mm macro lens 63 mm  50 mm 54 px/mm, 19 μm/px 12 0.7 m

matte white primer and matte black protective enamel by airbrush method 53 μm (7 px), crack region 74 μm (4 px), ENF 80 μm (5 px), C-ELS

have been conducted. For the calibration experiments, the location of the tip of the pre-crack has been moved with respect to the left support roller. Figure 5.6 depicts the flexural fixture with horizontal stereo-DIC camera arrangement in landscape orientation, see also Table 5.2. The light sources shown might be too close to the FOV of the cameras, see also Mode II - Sensitivity Analysis—DIC Parameter Uncertainty Quantification Section. Figure 5.7 shows the FOV (left camera) with specimen for mode II ENF testing. Since QOI is the deflection under the loading roller and crack propagation within the left half-span, the DIC setup has been chosen to capture the left half-span (support roller to loading roller). In comparison to flexural testing shown in Chap. 4, both cameras have been laterally shifted toward the left half-span, with the center of each camera being out of center of the span. Technically, for flexural testing (see also Chap. 4), a vertical stereo-DIC camera arrangement is preferred in order to have the ROI, focusing on the crack growth area, within the same DOF of both cameras. However, the loading and support rollers might cover the ROI, depending on the thickness of the specimen, the geometry of the rollers and the occurring deflection. For this purpose, a horizontal stereo-DIC camera arrangement in landscape orientation has been chosen. Disadvantage is the limitation of capturing the specimen length due to the chosen DOF. Loss in correlation might occur especially at the right and left ends of longer specimens due to the limited DOF (and therewith being out of focus). Therefore it is recommended to choose a small aperture for the DOF to be large (iDICs, 2018) taking into account the perspective difference between the two camera views (Reu, 2013b). A small stereoangle, within the limits of (iDICs, 2018), for the large focus length of the lenses

222

5

Delamination Resistance Testing

Fig. 5.6 Horizontal stereo-DIC setup for investigations under mode II loading (3pt setup for ENF testing shown)

chosen, has been used (see Table 5.2). Note that a smaller stereo-angle leads to better in-plane displacement accuracy, at the cost of increased out-of-plane uncertainty. The coordinate system has been chosen so that the abscissa (x-axis) is in direction of the specimen length, which corresponds to the fiber orientation and perpendicular to the loading direction (y-axis), with the origin being at the tip of the pre-crack (x (a0) ¼ 0 mm). The ROI is shown in Fig. 5.7b where the deflection has been taken from the inspection rectangle/point below the loading roller, located at the center (“C”) between the support rollers.

Calibrated End-Loaded Split Test Figure 5.8 shows the self-designed C-ELS test fixture with a possible span ranging between L ¼ 37 mm and 73 mm. The modular character of the setup allows for adaptation to other specimen geometries and spans. It is possible to realize the load introduction as tensile force gripping the specimen from below and additionally as

Testing Setup and DIC Configuration

223

Fig. 5.7 (a) FOV of left camera (b) ROI on specimen (colored area in equidistant partitions of 3 mm from 0 mm to 18 mm) with inspection rectangle for ENF testing

compressive force gripping the specimen from above (the load-block with pin diameter of 6 mm has to be adjusted to the loading condition, the specimen has to be flipped resp.). The latter is limited to small deformations due the design shown. It also allows for self-adjustment with a rotational degree of freedom around the x-axis. The DIC setups and DIC parameters for both types of mode II testing configurations were comparable for the sake of consistency. In comparison to ENF testing, the DIC setup for C-ELS testing has been laterally shifted to the right of the load introduction. The light sources shown might be too close to the FOV of the cameras, see also Mode II - Sensitivity Analysis—DIC Parameter Uncertainty Quantification Section. Two types of experiments, calibration and crack propagation experiments, have been conducted. For the calibration experiments, the specimen has been clamped in inverse arrangement, along the length of the starter crack. Taking into account the block size, the maximum span for calibration of cracked specimen in inverse arrangement is L ¼ 65 mm (L/h ¼ 27). The span has been varied in four intervals from L ¼ 40 mm–65 mm, see Table 5.1. Even though the smallest span-to-thickness ratio is L/h ¼ 15 due to the setup chosen, single cantilever beam testing requires larger deflections which causes a reduction of the span. For crack propagation experiments, based on the requirement for stable crack growth a0/L > 0.55, with a pre-crack length of approx. a0 ¼ 50 mm (measured from the center of the loading pin to the tip of the pre-crack), the length should be approx. L < 91 mm (L/h < 38).

224 Fig. 5.8 (a) C-ELS test fixture with indication of degrees of freedom, (b) horizontal stereo-DIC setup

5

Delamination Resistance Testing

Testing Setup and DIC Configuration

225

Fig. 5.9 (a) FOV of left camera. (b) ROI on specimen (colored area in equidistant partitions of 2 mm from 0 mm at pre-crack to 6 mm) with inspection rectangle for C-ELS testing

The chosen span-to-thickness ratio of L/h ¼ 31 mm (approx. L ¼ 74 mm) is comparable to ENF experiments. This results in a clamping length of approx. 40 mm. After clamping the specimen (with a recommended torque of 8 Nm (ISO, 2014)), the span has been marked on the specimen along the edge of the sliding clamp. A maximum force of 250 N is recommended for calibration (ISO, 2014). Figure 5.9 shows the FOV with specimen for C-ELS testing, which needs to be larger when aiming for larger crack propagation lengths. For mode II C-ELS testing, the optimal stereo-DIC camera arrangement would be vertical with a comparable DOF for both cameras, focusing on the crack growth area. Displacement measurement can be done with either a third camera or an LVDT. For the calibration experiments, QOI is deflection (maximum and along the span) of the cantilever beam (with the starter crack being clamped). The coordinate system has been chosen so that the abscissa (x-axis) is in direction of the specimen length, which corresponds the fiber orientation and perpendicular to the loading direction (y-axis), with the origin at the tip of the pre-crack (x (a0) ¼ 0 mm). The ROI is shown in Fig. 5.9b where the deflection has been taken from the inspection rectangle/point above the center of the loading pin. For DIC calibration, the C-ELS fixture has to be removed for placing the calibration target. Reference image has been chosen at the end of a 3 s time interval of acquisition for verification of proper camera synchronization for both types of mode II tests. Thereby the C-ELS specimen has been clamped force free.

226

5

Delamination Resistance Testing

Table 5.3 DIC analysis parameters Lsubset Lstep Lwindow LVSG Strain formulation Subset shape function Software

Value (Mode I DCB) 17 px, 0.12 mm 6 px, 0.04 mm 11 datapoints 138 px, 0.55 mm Engineering Gaussian weights Vic-2D Correlated Solutions

Value (Mode II ENF and C-ELS)) 15 px, 0.28 mm 6 px, 0.11 mm 11 datapoints 75 px, 1.39 mm

Vic-3D Correlated Solutions

Tables 5.2 and 5.3 summarize the DIC related parameters for mode I (using 2D-DIC) and different types of mode II test methods (using stereo-DIC). Since the specimen is brittle and observation of crack propagation is important, the paint should be as brittle as possible, while still not flaking/debonding or cracking independent of the specimen. In this case, the paint should be allowed to fully cure with adequate dry timing (iDICs, 2018). The line inspection for measuring the pattern feature size has been done at the midline of the specimen along the crack propagation direction. Due to comparable DIC hardware parameters, the default DIC analysis parameters for ENF testing have been applied to C-ELS testing for the sake of consistency and comparison purposes. A detailed sensitivity analysis on the DIC analysis parameters will be presented in Mode I - Sensitivity Analysis—DIC Parameter Uncertainty Quantification Section on mode I DCB testing and in Mode II - Sensitivity Analysis—DIC Parameter Uncertainty Quantification Section on mode II ENF testing, including an indication of the resolution of different relative and absolute QOI. Overall, two acceptable crack propagation experiments have been done for each loading condition.

Mode I—Mechanical Response This section focusses on the deduction of the force-deflection from flexural loading of a DCB specimen with relating crack propagation response including a variety of data reduction schemes for obtaining multiple crack resistance (R-) curves.

Data Reduction Figure 5.10a depicts the force-deflection curve with indications for the determination of the compliance. According to (Davies et al., 1998), the onset of non-linearity in the force-deflection curve is the most common definition of crack initiation. A

Mode I—Mechanical Response

227

Fig. 5.10 (a) Representative forcedeflection curve, (b) evolution of crack growth by various compliancebased methods, (c) comparison of different methods for determination of crack propagation

40

F in N

30 20 Data Lin. Regr. 95 % Lin. Regr. Limits

10 V*(F

0 0

2

max

)

4

6

8

10

8

10

8

10

V* in mm

a 20

V*(Fmax )

ai in mm

15

f(EL) f(EL) - Offset

10

f(Efl) f(Efl) - Offset

5 0 0

2

4

6

V* in mm

b 20

V*(Fmax )

ai in mm

15 f(Efl) - Offset DIC (LiC) By Eye

10 5 0 0

c

2

4

6

V* in mm

228

5

Delamination Resistance Testing

quantitative estimate is given by an increase in compliance of 5% acc. to (ISO, 2001), which intersects the force-deflection curve before reaching the maximum force. Crack growth is accompanied by a steady decrease in force. Crack propagation is shown in Fig. 5.10b, determined by the compliance-based method (Eq. (5.7)) using the tensile modulus EL ¼ 125 GPa (see Chap. 2) and the true flexural modulus Efl ¼ 116 GPa (see Chap. 4). Usually, a correction of the calculated crack length is done using the offset to the known pre-crack length a0. In both cases, the initial crack length is overestimated and the resulting corrected crack propagation data overlaps. A comparison of different methods for the determination of crack propagation lengths is given in Fig. 5.10c including selected visual measurements by eye (in approx. 2 mm spacings) on the DIC speckle side. This rather subjective method (differentiating between opening and deformation) is additionally hindered in comparison to a white base specimen for crack inspection (having the inherent problem of deformation of the paint vs. specimen). This has been chosen for the sake of comparison to the DIC-based method of automated crack tip tracking, referring to the loss in correlation (LiC, including a smoothing filter), see also Mode I— Visualization of Strains and Displacements Section. Both visual methods focus on the horizontal crack lengths, limited by the ROI. A comparison between LiC to selected crack lengths measured from the image by eye are in good agreement. The compliance-based method shows larger crack lengths compared to by eye measurement as well as by the loss in correlation method. This results from the compliance-based method being an integral method which takes into account the crack length across the width of the specimen and deviations across the thickness (but also includes the deflection of the single cantilever beams), whereas both optical methods only focus on the crack tip on one side of the specimen. The crack propagation rate is represented by the slope, based on displacement-controlled testing. In accordance with (ISO, 2001), a correction factor is needed for large deflection (V/a > 0.4), which is not needed at this point (V*/amax ¼ 0.2 leading to a correction of approx. 2%). Figure 5.11 depicts various methods for the deduction of parameters needed for the determination of the energy release rate. From Fig. 5.11a, it can be concluded that approx. 50% of the energy is dissipated as fracture energy by the end of the test. The compliance-based methods are shown in Fig. 5.11b–d. Note that the intercept and therewith the correction Δ for MB is sensitive. The crack length-to-thickness ratio (analogous to L/h) shown in Fig. 5.11d respects the limit a/h > 10 acc. to (ASTM, 2013a). Typical delamination resistance curves are shown in Fig. 5.12 as evolution of the energy release rate G IC over crack propagation ai, determined with deflection and crack propagation measurements using DIC (except for PC method). The crack resistance (R-) curves are determined from various methods, depicting the sensitivity. As already stated, the Area method does not allow for determination of an initiation value. Since the initiation value can be affected by placing a release film during manufacturing (Beehag & Ye, 1996) (e.g., leading to atypical behavior with

Area in Nmm / W ork in mJ

Mode I—Mechanical Response

229

250 All Area Irreversible Work

200

V*(Fmax )

150 100

50 0 0

2

4

6

8

10

V* in mm

a

b 0

-2

Data Lin. Regr. Limits

24

a/h in mm/mm

log C

-1

Data Lin. Regr. Limits

n = 2.7

-3

22

A1 = 33.6

20 18 16

-4 3.7

3.8

3.9

4

0

4.1

0.2

0.4

0.6

0.8

1

C1/3 in (mm/N)1/3

log a

c

d

Fig. 5.11 Data reduction methods. (a) Area method, Compliance-based methods (b) MB, (c) CC, (d) MCC Fig. 5.12 R-curves using different methods from one mode I DCB test

500

IC

in J/m2

400 300 200 Area SB MB

100

PC(EL)

CC MCC PC(Efl)

0 0

5

10

ai in mm

15

20

230

5

Delamination Resistance Testing

initial decreasing delamination resistance (ISO, 2001), starting from an initiation value higher than propagation value), no conclusions from other methods will be drawn at this point. The standardized compliance-based methods MB, CC, and MCC include a correction of the crack length, so the curves barely show any difference to each other. The MB method shows smaller values in comparison to the SB method due to a smaller factor n (see Fig. 5.11c). All methods presented show crack propagation bounded by the Area method (maximum) and the pure compliance method (minimum, using the flexural modulus). Since the standardized compliance-based methods MB, CC, and MCC include a correction of the crack length, the values of the energy release rate show barely any sensitivity to an artificial change of the crack length, see (Merzkirch et al., 2017).

Fractography Figure 5.13 shows representative fractography of a DCB specimen with indications of the different crack lengths with the origin of the pre-crack a0 starting from the loading pin of the hinge. Note the different contrast at transition from pre-crack release film to crack propagation.

Fig. 5.13 Representative fractography of a DCB specimen

Mode I—Visualization of Strains and Displacements

231

Mode I—Visualization of Strains and Displacements This section provides a full-field investigation on the occurring through-thickness strain distribution of a DCB specimen under mode I loading conditions to obtain insight into the deformation and delamination behavior. Besides the intended delamination, also unavoidable loading conditions will be visualized that are accessory to flexural testing. Furthermore, absolute displacements are used for crack opening displacement (COD) measurements and compared to the analytical solution of the deflection. The post-analysis ROI of the image is smaller than the ROI of specimen resulting from the chosen subset size, where only a smaller area of the specimen side l  h could be used for DIC data analysis. The contour plots in this section refer to the deformed state and focus on delamination during crack propagation (at a selected calculated crack propagation length ai(Efl)), rather than crack initiation.

Strains Intended Loading Condition Figure 5.14 shows the through-thickness distribution of normal (laminae transverse resp.) strain εy with strain concentrations in the vicinity of the crack tip. Along the crack faces and the crack tip some subsets lose correlation and these sections of the ROI no longer show data. The high strains along the crack faces are partially artifacts of subset decorrelation. Crack growth leads to a loss in correlation (LiC) in the DIC data, which is used as DIC-based automated crack tip tracking methodology using the quality of correlation (“sigma” in the software used). By following a defined threshold value for the match (which corresponds to sigma ¼ 1, no correlation), the crack path is determined in absolute reference coordinates, for a further deduction of the evolution of crack propagation, shown in Fig. 5.10c. The theoretical accuracy is related to the

Fig. 5.14 Strain distribution of normal strain εy at ai(Efl) ¼ 13.42 mm

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Fig. 5.15 Midline plot of normal strain εy in the vicinity of the crack tip at ai(Efl) ¼ 13.42 mm

subset size, half a subset resp., being Lsubset/2 ¼ 60 μm for the default DIC analysis parameters (Table 5.3) and therewith only approx. a tenth of the recommended resolution of 0.5 mm (ASTM, 2013a). Note that also the horizontal shift of the specimen due to deflection can easily be included in the DIC determined crack propagation, see Mode I – Horizontal Displacement U Section. The distribution of the midline normal strain in the vicinity of the crack tip is shown in Fig. 5.15, depicting the strain concentration (due to the singularity) in front of the crack tip (and compressive strains beyond). Additionally, the calculated crack length a(Efl) and the crack length determined using the loss in correlation, a(LiC), methodology are shown. The difference between both methods is obvious with the calculated crack length being more advanced. The decay in normal strain in front of the crack tip up to intersection with zero strain (εy ¼ 0%) depicts the process zone under mode I loading, with a length of approx. 1.14 mm. As an alternative to the LiC methodology, a critical strain can also be used as a threshold value for an approximate automated crack tip tracking. Note that the magnitude of the maximum strain as well as the decay of the strain is dependent on the DIC analysis parameters, see Mode I - Sensitivity Analysis—DIC Parameter Uncertainty Quantification Section.

Accessory Loading Conditions Figure 5.16 shows the distribution of shear strain in the specimen’s coordinate system γ xy (a) and maximum shear strain γ 12 (b) as a metric for impurity in mode I loading. Especially maximum shear strain γ 12 in the vicinity of the crack tip proves an additional shear loading superimposed upon the intended normal strain (Fig. 5.14). The shear strain γ xy depicts shear zones due to bending of both cantilever beams (see also Chap. 4).

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233

a

b

Fig. 5.16 Shear strain distribution at ai(Efl) ¼ 13.42 mm (a) γ xy, (b) γ 12

Sensitivity Analysis: DIC Parameter Uncertainty Quantification In this section, a detailed virtual strain gauge study of the DIC analysis parameters, such as subset, step, and window size, is performed in several process steps to optimize the DIC measurements. 1. At this point, only the static and spatial noise-floor is considered for the ROI of an image just prior the reference image. The same image is used for a determination of the bias, since the QOI should be close to zero for a clamped and non-loaded specimen. Primary QOI is normal strain εy in the vicinity of the crack tip. 2. In the following, the focus is on an image at a calculated crack length of approx. a (Efl) ¼ 13.42 mm (related to force and deflection), independent of any DIC related parameters. 3. The subset size is varied between 15:2:23 px based on a minimum subset size to be at least three times the speckle size. The step-to-subset ratio is chosen to be 1/3, 0.4 and ½ with the window size varying between 7:4:15 datapoints. This leads to overall 45 parameter variations. Figure 5.17a and b depict the bias (mean) and noise-floor (standard deviation) of the QOI for different virtual strain gauge sizes LVSG with a minimum of approx. 0.3 mm. Note that the x-axis limit in Fig. 5.17a, b is chosen to match the thickness of the specimen h ¼ 2.4 mm. With increasing virtual strain gauge size, a decrease of the noise-floor of the QOI can be attested.

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a

b

c

Fig. 5.17 (a) Bias and noise-floor vs. LVSG for normal strain εy, (b) peak values vs. LVSG, (c) peak values vs. noise-floor

The resulting virtual strain gauge size of the default DIC analysis parameters in this chapter is LVSG ¼ 0.6 mm (77 px, VSG(Lsubset/Lstep/Lwindow ¼ 17/6/11)), being a quarter the thickness of the specimen. Two additional sets of parameters, including one with higher precision (small LVSG) and one with less noise (larger LVSG), have been compared to parameters used within this chapter, see Figs. 5.17, 5.18 and 5.19. Virtual strain gauges with sizes of LVSG ¼ 0.5 mm (¼ 65 px, VSG (15/5/11)), and LVSG ¼ 0.7 mm (91 px, VSG(21/7/11)) have been highlighted in Fig. 5.17. 4. Large normal strain gradients are located in the vicinity of the crack tip, see Figs. 5.14 and 5.15. The extrema of the normal strain have been extracted along the midline in the vicinity of the crack tip, see Figs. 5.18 and 5.19. Figure 5.17b shows that the smaller the virtual strain gauge size, the larger the strain extrema. Midline plots for different DIC analysis parameters are shown in Fig. 5.18. The larger the virtual strain gauge, the smaller the strain extrema, gradient (slope resp.) and fluctuations. 5. Figure 5.17c shows the maxima of the QOI against the noise-floor, representing the signal-to-noise ratio, where no convergence is obtained with decreasing LVSG

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235

Fig. 5.18 Midline plots of normal strain εy at ai(Efl) ¼ 13.42 mm. (a) All parameter variations, (b) Selected parameter variations (Lsubset/Lstep/ Lwindow)

a

b (increasing noise-floor resp.). In accordance with (Reu, 2015), a lack of convergence can occur at cracks, other failure points and at large strain gradients. Note that multiple objectives have to be considered such as maximum strain, strain gradient (slope in front of the crack tip resp.) and decorrelation for the loss in correlation methodology (see subsequent section). Figure 5.19 illustrates a comparison of two selected sets of DIC analysis parameters with the smaller VSG depicting the more concentrated normal strain εy in the vicinity of the crack tip (see Fig. 5.18b and compare to default DIC analysis parameters in Fig. 5.14). Note that due the relatively high strain (and corresponding scale), fluctuations can vaguely be seen, except in the vicinity of the crack tip and the crack faces.

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a

b Fig. 5.19 Distribution of normal strain εy at ai(Efl) ¼ 13.42 mm. (a) LVSG ¼ 65 px, VSG(15/5/11), (b) LVSG ¼ 91 px, VSG(21/7/11) Table 5.4 Bias and noisefloor (spatial and static) of QOI from ROI for default DIC analysis parameters for mode I DCB testing

QOI U in mm V in mm W in mm εx in % εy in % γxy in % ε1 in % ε2 in % γ12 in %

Bias 0.0046 0.0052 N/A 0.0035 0.0010 0.0024 0.0120 0.0166 0.0286

Noise-floor 0.0003 0.0009 N/A 0.0147 0.0296 0.0406 0.0283 0.0268 0.0445

The bias and noise-floor for mode I DCB testing for the default DIC analysis parameters in this chapter are listed in Table 5.4. Note that due to the 2D-DIC setup, no information on out-of-plane displacement W is given. Note that an uncertainty quantification can also focus on the QOI “loss in correlation (LiC)” for automated crack tip tracking. Since this QOI is solely dependent on the absolute values in the reference coordinates x and y, only subset and step

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237

size are relevant parameters. Optionally, the threshold for correlation/matching represents an additional DIC analysis parameter. Based on preliminary investigations focusing on automated crack tip tracking using the loss in correlation as primary QOI, it could be shown that besides a small subset, due to a theoretical accuracy of half a subset, a small step size of Lstep ¼ 1 px is desirable. Besides leading to a high calculation time, the strain related values will be highly affected. Note that if both QOI, strain and loss in correlation, are determined via two separate correlation calculations using different step sizes, the spatial resolution of the absolute values (x, y, U, and V ) will differ.

Displacements This section focusses on the investigation of the absolute deformation response of a DCB specimen under mode I loading at a selected (calculated) crack length. Detailed horizontal and vertical displacements are investigated and compared to the analytical solution of the deflection.

Horizontal Displacement U The horizontal shift of the specimen toward the load application line, due to crack growth, and relating deflection of both cantilever beams is shown in Fig. 5.20. The calculated crack length is a(Efl) ¼ 53 mm with a measured horizontal shift of approx. 0.5 mm toward the load application line, leading to a reduction in the measured overall (horizontal) crack length. The lateral shift has to be considered if outside the limits of (ISO, 2001) (see large deflection V/a calculation in Mode I – Data Reduction Section.



















   







 Fig. 5.20 Distribution of horizontal displacement U at ai(Efl) ¼ 13.42 mm



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

















   









 Fig. 5.21 Distribution of vertical displacement V at ai(Efl) ¼ 13.42 mm

Vertical Displacement V The vertical shift and deflection of a DCB specimen are shown in Fig. 5.21. Reference (Hashemi et al., 1989) and (ASTM, 2013a) stated that rotations are possible at the delamination front, being equal to the derivative of U with respect to V. The methodology of the crack opening displacement (COD) is presented in Fig. 5.22a. First, reference lines at a distance hCOD/2 from the crack line (in the following the focus is on the midline of each cantilever beam, hCOD/2 ¼ h/4 from the crack line) are chosen. The vertical displacement along those reference lines for each cantilever beam is determined. Figure 5.22b depicts the relative deformation of both cantilever beams (for sign convention: subtracting the bottom deflection from the top deflection) via half crack opening displacement measurements (for a single cantilever beam (COD/2) resp.) for different crack lengths, shown in Fig. 5.22a. Furthermore, deflection at the crack tip COD(ai), i.e., the crack tip opening displacement (CTOD) is shown in Fig. 5.22b. A thorough investigation of the flexural modulus by the analytical solution for selected COD lines at different crack-length-to-half-thickness ratios a/(h/2), crack lengths ai resp., is shown in Fig. 5.22b. The second derivative of the deflection of a single cantilever beam (index SCB) with span L and thickness h/2 (and variable xi) due to a bending moment is V 00SCB ðxi Þ ¼ 

F ∙ ðL  xi Þ E∙I

ð5:37Þ

The deflection V is expressed by integrating the single region twice (Gross et al., 2018) V SCB ðxi Þ ¼

1 F 1 F ∙ ∙ x3  ∙ ∙ x2 ∙ L þ const1 ∙ xi þ const2 6 E∙I i 2 E∙I i

ð5:38Þ

Applying Eq. (5.38) using the boundary condition for COD(xi ¼ a0, x ¼ 0), a flexural modulus from regression analysis of the COD lines shown in Fig. 5.22b, is

Mode I—Visualization of Strains and Displacements Fig. 5.22 (a) COD methodology for selected crack lengths at hCOD ¼ h/2, (b) corresponding half crack opening displacement in comparison to analytical solution

239

h h

COD

COD

Crack Line V(DIC)

a DIC Anal. Soln. (Regression) a i(LiC)

COD/2 in mm

0.6

0.4

0.2

0 0

b

5

10

15

20

x in mm

extracted to be Efl ¼ 110 GPa  2 GPa. A comparison of flexural moduli from different methods will be presented in Summary and Discussion Section. Usually, the integration constants for single cantilever beam bending are determined using the boundary conditions at the root of the cantilever beam with VSCB(0) ¼ 0 and VSCB(L) ¼ V* (maximum deflection at load introduction). Note that the span differs from crack length (with deformation CTOD) and the “clamping” length changes with crack propagation, see Fig. 5.22b. Subsequently, a thorough investigation of the crack opening displacement with focus on the vicinity of the crack tip is presented in Fig. 5.23a. The calculated crack length and the crack length determined using the loss in correlation (LiC) methodology are shown. Note that the latter is determined at the crack plane, where decorrelation due to crack propagation occurs. The current investigation focusses on a distance hCOD/2 from the crack plane; hence no decorrelation is observed at the midplane of the single cantilever beam. The distance between the crack tip opening

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5



 



 

  

a









Delamination Resistance Testing













b Fig. 5.23 (a) COD for both cantilever beams at ai(Efl) ¼ 13.42 mm in the vicinity of the crack tip, (b) schematic process zone with idealized linear traction-separation relationship

displacement (CTOD) and the point of no deformation (COD ¼ 0) represents the process zone with an approximate length of 0.97 mm and therewith slightly smaller than determined via the strain distribution in front of the crack tip (Fig. 5.15). If considering the deflection and slope (first derivative) to be zero at the root of the cantilever beam, Eq. (5.38) can be simplified to the first two terms and representing simple beam theory for xi ¼ L (compare to single cantilever beam deflection in Eq. (5.27)). Figure 5.23a visualizes the slope of the deflection being zero in front of the zone tip, at the calculated crack length a(Efl). The correction Δ used for the MB method (see Fig. 5.11b) is larger than the difference between calculated crack length (a(Efl)) and DIC determined crack length (a(LiC)). The correction from the MB method represents an integral method along the whole width of the specimen, whereas the length of the process zone focusses only on one side of the specimen.

Mode I—Visualization of Strains and Displacements

241

Since numerical models for delamination use cohesive zone models, Fig. 5.23b delineates a linear traction-separation relationship for a description of the material behavior in front of the crack tip. Using the J -integral as fracture criterium and previously determined parameters, namely CTOD at hCOD ¼ h/2 (including the uncertainties presented in Table 5.4) and the energy release rate, Eq. (5.3) is used for the calculation (within LEFM) of the critical/fracture stress in the vicinity of the crack tip. The maximum traction at the tip of the process zone resp. σf ¼

G CC 233 J=m2 IC ¼ ¼ 13:3 MPa ð5 MPaÞ CTOD 18:8 μm ð5 μmÞ

ð5:39Þ

results in a stress concentration factor of 3.6–8 (9 from (Carlsson & Pipes, 1987) for open hole tensile testing (ASTM, 2018)), by comparison to the in-plane transverse UTST ¼ 66 MPa (assuming transversely isotropic behavior). Note that this is a rough estimate, since fluctuations adhere the determination of the crack tip as well as the non-straight crack line across the width of the specimen, and that COD measurements (at a chosen inital distance from the crack line) include (steady) crack propagation (non-stationary crack). Furthermore, analytical models applied to the experimentally determined displacement between the crack tip opening and the point of no deformation (the process zone representing the material behavior) can be exploited to determine the interlaminar normal strength based on the line load, depicted in Fig. 5.23b. The model represents an Euler–Bernoulli beam (neglecting shear effects) on Winkler foundation (Kanninen, 1973; Olsson 1992). The flexural line can be described as distributed load acting on a beam by a differential equation of fourth order (based on investigations from L. Euler) (Williams & Hadavinia, 2002; Timoshenko, 1940b) 4 ∂ V qð V Þ ¼ E∙I ∂x4

ð5:40Þ

qð V Þ ¼ c ∙ V

ð5:41Þ

with the load distribution

for ideal elastic behavior with c being the stiffness of the elastic foundation (Flügge, 1962; Williams, 1989). The solution for a semi-infinite beam on an elastic foundation (Timoshenko, 1940b; Ungsuwarungsri & Knauss, 1987) is V ðxÞ ¼ eλ ∙ x ∙ ðA ∙ cos ðλ ∙ xÞ þ B ∙ sin ðλ ∙ xÞÞ

ð5:42Þ

with λ¼ and appropriate boundary conditions.

rffiffiffiffiffiffiffiffiffiffiffiffiffi c 4 4∙E∙I

ð5:43Þ

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Note that even in a brittle material, there is a zone immediately in front of the crack tip in which the material is not linearly elastic (Freiman & Mecholsky, 2012). Further experimental-analytical procedures for the approximate determination of non-linear material characteristics (including adhesives) are provided in (Ungsuwarungsri & Knauss, 1987).

Mode II – Mechanical Response This section focusses on the deduction of the force-deflection with relating crack growth response including a variety of data reduction schemes for obtaining multiple crack resistance (R-) curves for two types of mode II test methods.

Data Reduction Figure 5.24 depicts the force-deflection curves for ENF (a) and C-ELS (b) testing. The linear regression analysis is applied to data between 90 N (for ENF, 25% Fmax 600

150 Data Lin. Regr. 95 % Lin. Regr. Limits

100

F in N

F in N

400

Data Lin. Regr. 95 % Lin. Regr. Limits

200

50

V*(F

0 0

1

2

max

)

V*(F

0

3

4

0

5

V* in mm

)

b 45

80 amax = L/2

40

amax = L

V*(Fmax )

V*(F

max

)

70

35

a in mm

a in mm

max

15

V* in mm

a

30 a

25 20

0

f(EL)

f(Efl)

f(EL) - Offset

f(Efl) - Offset

0

1

2

3

60 50 40

15 4

a

0

f(EL)

f(Efl)

CBTE

f(EL) - Offset

f(Efl) - Offset

CBTE - Offset

0

V* in mm

c

10

5

10

15

V* in mm

d

Fig. 5.24 Representative force-deflection curve. (a) ENF, (b) C-ELS, Evolution of crack growth by various methods (c) ENF, (d) C-ELS

Mode II – Mechanical Response

243

for C-ELS) up to Fmax for the calibration test and up to 50% Fmax for fracture testing (ASTM, 2014). C-ELS testing leads to smaller forces and higher displacements compared to ENF testing. Preliminary investigations on ENF specimens tested without pre-cracking showed a sudden force drop with the onset of delamination at higher forces than shown in Fig. 5.24a. This goes along with (Davies et al., 1998), stating that higher initiation values are expected from release films than from pre-cracks, the latter yielding the more conservative values (Blackman et al., 2006). A quantitative estimate for crack initiation is determined in the same way as specified for mode I, by an increase in compliance of 5%. In order to circumvent the inherent challenge in visual determination of the delamination length under mode II loading (Blackman et al., 2006), including the DIC determined loss in correlation (LiC) methodology (see Mode I – Visualization of Strains and Displacements Section), the focus is on the calculated compliance-based crack propagation for both test methods. Therefore, the tensile modulus EL ¼ 125 GPa and the flexural moduli Efl ¼ 111 GPa (for ENF testing, see Fig. 5.25a), Efl ¼ 116 GPa (from Chap. 4) and ECBTE method (for C-ELS testing, see hereinafter) have been used. The initial compliance-based crack lengths have been adapted to match the pre-crack length. The evolution of the crack propagation depicts the higher displacements needed for crack initiation, going along with initiation time, for C-ELS testing, Fig. 5.24d, compared to ENF testing, Fig. 5.24c. The increase in force after the first force drop during ENF testing results from the interaction between crack and the center section in the vicinity of the loading roller (see also location of maximum deflection in Mode II – Vertical Displacement V Section). Note the steeper slope after the force maximum during ENF testing (a possible indication for unstable crack growth). Figure 5.25 depicts various compliance-based methods for the deduction of parameters needed for the determination of the energy release rate from ENF testing (a) and C-ELS testing (b). The results from the calibration (Fig. 5.25b) during x 10-3

C in mm/N

6

27.8

5

29.2 30.4 28.3

22.8 18.1

4

22.1

13.7

Data Lin. Regr. C0(a0 = 0)

3 2

0

0.5

1

1.5

a30 in mm3

a

2

2.5 x 104

b

Fig. 5.25 Compliances and data reduction methods. (a) ENF calibration tests on cracked specimens with indications of pre-crack lengths a0 (in mm), (b) C-ELS calibration tests on an uncracked specimen

244

5

in J/m2

1500

1000

500

1000

IIC

PC(EL)

IIC

in J/m2

1500

Delamination Resistance Testing

PC(EL)

500

PC(Efl)

PC(Efl)

SB CC

CBTE

0

0 0

2

4

6

8

10

0

ai in mm

a

5

10

15

20

ai in mm

b

Fig. 5.26 R-curves using different methods from one mode II test each (a) ENF, (b) C-ELS

inverse ELS testing with the crack fully held within the clamp have been determined using the whole possible variation in length limited by the setup (span Lmin) and the remaining non-pre-cracked length of the specimen used. For ENF testing (Fig. 5.25a), with m ¼ 1.01  107, extrapolation of the linear regression of the compliance to a0 ¼ 0 mm leads to a flexural modulus of Efl(C0) ¼ 111 GPa (compare Eq. (5.19) with a ¼ 0 mm), representing a pristine specimen under three-point loading at L/h ¼ 33. This is close to the true flexural modulus from three-point bending, Efl ¼ 116 GPa (from Chap. 4), even though (Davies et al., 1998) states that very small changes in compliance make it difficult to obtain reliable experimental compliance calibration of the ENF specimen (which is impeded by the sensitivity of the intercept). By using the correction from the calibration of an uncracked single cantilever beam specimen, see Fig. 5.25b, the flexural modulus is ECBTE ¼ 80 GPa (compare to Eq. (5.29) with a ¼ 0 mm). Note that the crack length determined with ECBTE has to be adjusted upwards, see Fig. 5.24d. A comparison of the flexural moduli determined will be presented in Summary and Discussion Section. Note that the experimental compliance method (EC) using the measured crack length demands the determination of the slope of the compliance against the cube of the crack length, comparable to Fig. 5.11b for mode I testing. The EC method returns erroneous results due to the high sensitivity in crack length measurements (Blackman et al., 2006) and is therefore not shown here. The delamination resistance is shown in Fig. 5.26 as evolution of the energy release rate G IIC over crack growth ai depicting typical crack resistance curves (R-) curves for various methods for ENF (a) and C-ELS (b) testing. ENF testing shows the two methods focusing on the pre-crack length a0 instead the calculated crack propagation ai. Note the limited crack growth, based on the chosen span, for ENF testing compared to C-ELS testing. The increase of the crack resistance curve for ENF testing with (calculated) crack propagation results from the interaction of the crack at the center section in the vicinity of the loading roller (resulting in an increase in force, see Fig. 5.24a).

Mode II—Visualization of Strains and Displacements

245

The CBTE method leads to values for the energy release rate, which are approx. 50% higher than the pure compliance (PC) methods. The origin of this is the low modulus determined during calibration of inverse ELS testing. If not relating to the pre-crack length a0 (SB and CC method) but measured crack propagation, unstable behavior occasionally occurs for ENF testing, affected by e.g., the material and rate, even though following the geometry-based stability criterion (Eq. (5.26)).

Fractography Figure 5.27 shows representative fractography of ENF and C-ELS specimens (manual mode I delamination after testing) with indications of the different crack lengths with the origin of the pre-crack a0 starting from the left support roller (ENF), the loading pin of the load-block (C-ELS) resp. Note the different contrast at transition from pre-crack release film (which has been removed post-mortem for C-ELS) to crack propagation.

Mode II—Visualization of Strains and Displacements This section provides a full-field investigation on the occurring through-thickness strain distribution of ENF and C-ELS specimens under mode II loading conditions to obtain insight into the deformation and delamination behavior. Besides the intended delamination, also unavoidable loading conditions will be visualized that are accessory to flexural testing. Furthermore, absolute displacements are used for crack shear displacement (CSD) measurements and compared to the analytical solution of the deflection. The contour plots in this section refer to the deformed state and are focusing on the area between the pre-crack a0 and the loading roller (ENF testing), up to the clamp (C-ELS testing) resp. Furthermore, the focus of this investigation is on delamination during crack propagation (at a selected calculated crack propagation length ai(Efl)), rather than crack initiation. In case unstable crack growth or momentary crack arrest occurs, it is worth looking into subsequent images for observation of variation in critical strains.

Strains Intended Loading Condition Since the intended QOI is shear, Figs. 5.28 and 5.29 show different types of shear strains, namely shear strain in the specimen’s coordinate system γ xy and maximum

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Fig. 5.27 Representative fractography of (a) ENF specimen, (b) C-ELS specimen

shear strain γ 12, that are used to obtain insight into the loading condition in the vicinity of the crack tip. For better comparison, the strain scales have been chosen to be identical for both test methods. The calculated crack propagation length is approx. ai(Efl) ¼ 6.8 mm for ENF testing. The loss in correlation on the left side is based on the fiducial mark for the position of the pre-crack a0. Note that the strain in the specimen’s coordinate system γ xy is negative, due to shear sign convention in the chosen coordinate system (see also three-point bending in Chap. 4). The high strains along the crack faces are partially artifacts of subset decorrelation. Even though mode II loading also leads to a loss in correlation during

Mode II—Visualization of Strains and Displacements

247

Fig. 5.28 Shear strain distribution for ENF at ai(Efl) ¼ 6.8 mm. (a) γ xy, (b) γ 12

crack propagation, it was found to be more challenging and less reliable than for mode I testing. Figure 5.29 shows the through-thickness distribution of different types of shear strain in the vicinity of the crack tip for C-ELS testing. Similar as for ENF testing, shear strain in the specimen’s coordinate system γ xy is negative, due to shear sign convention in the chosen coordinate system. A comparison of the midline strain in the vicinity of the crack tip is shown in Fig. 5.30 for different types of shear strain, depicting the concentration (due to the singularity) in front of the crack tip. The decay in shear strain in front of the crack tip up to intersection with zero strain (γ xy ¼ 0, only possible for shear strain in the specimen’s coordinate system, since maximum shear strain is always positive considering the bias) depicts the process zone under mode II ENF loading with a length of approx. 5.6 mm. Note that, as already shown in Fig. 5.29, the magnitude in shear strain is higher for C-ELS testing than for ENF testing. This could also be a result of momentary crack arrest. Figure 5.30b shows a process zone with a length larger than 5 mm for C-ELS testing, although the exact size cannot be determined due to limited visual

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Fig. 5.29 Shear strain distribution for C-ELS at ai(Efl) ¼ 9.7 mm, (a) γ xy, (b) γ 12

access. Note that the calculated crack length overestimates the real crack length for C-ELS testing, whereas only a small offset can be attested for ENF testing. The size of the process zone is larger in mode II than in mode I, being even larger than the transverse dimension (thickness) of the specimen.

Mode II—Visualization of Strains and Displacements

a

249

b

Fig. 5.30 Midline plots of different types of shear strain in the vicinity of the crack tip (a) ENF, (b) C-ELS

Accessory Loading Conditions This section gives an overview on accessory loading resulting from flexural testing of pre-cracked specimens. For better comparison, the strain scales have been chosen to be identical for both test methods. The existence of the normal strain εy shown in Fig. 5.31 in front of the crack tip, proves the mode impurity for flexural testing for both test methods under mode II loading conditions. Shear loading induces a superimposed mode I component which might be a result from touching of the crack faces behind the crack tip.

Sensitivity Analysis—DIC Parameter Uncertainty Quantification This section provides a short virtual strain gauge study on the results from mode II ENF testing at a calculated crack length of at ai(Efl) ¼ 6.8 mm. The same variations in subset, step and window size have been performed as described in detail in Mode I - Sensitivity Analysis—DIC Parameter Uncertainty Quantification Section for mode I DCB testing. The primary QOI are shear strain γ xy and maximum shear strain γ 12 in the vicinity of the crack tip. Figure 5.32a depicts the bias and noise-floor of the shear strain γ xy for different virtual strain gauge sizes LVSG. The midline plots in the vicinity of the crack tip are shown in Fig. 5.32b. VSG with sizes of LVSG ¼ 0.8 mm (45 px, VSG(15/5/7)) and LVSG ¼ 2.2 mm (119 px, VSG(19/10/11)) have been highlighted for a comparison to the default DIC analysis parameters with LVSG ¼ 1.4 mm (75 px, VSG(15/6/11)). Tables 5.5 and 5.6 list the bias and noise-floor for ENF and C-ELS testing. Note the relatively high values (compare to flexural testing in Chap. 4), likely due to larger SOD and lights positioned too close to the FOV of the cameras. The signalto-noise ratio is considered to still be large enough for the current investigation.

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Fig. 5.31 Normal strain εy distribution (focus on crack tip). (a) ENF at ai(Efl) ¼ 6.8 mm, (b) C-ELS at ai(Efl) ¼ 9.7 mm

Displacements This section focusses on the investigation of the absolute deformation response of ENF and C-ELS specimens under mode II loading. Furthermore, loading conditions are investigated and the analytical solution is compared to the actual deflection for C-ELS testing of an uncracked single cantilever beam specimen.

Mode II—Visualization of Strains and Displacements

a

251

b

Fig. 5.32 (a) Bias and noise-floor vs. LVSG for γ xy, (b) midline plots of γ xy for all parameter variations

Table 5.5 Bias and noisefloor (spatial and static) of QOI from ROI for default DIC analysis parameters for mode II ENF testing

QOI U in mm V in mm W in mm εx in % εy in % γxy in % ε1 in % ε2 in % γ12 in %

Bias 0.0006 0.0036 0.0029 0.0052 0.0382 0.0038 0.0616 0.0287 0.0903

Noise-floor 0.0017 0.0037 0.0217 0.0484 0.0593 0.0788 0.0565 0.0514 0.0475

Table 5.6 Bias and noisefloor (spatial and static) of QOI from ROI for default DIC analysis parameters for mode II C-ELS testing

QOI U in mm V in mm W in mm εx in % εy in % γxy in % ε1 in % ε2 in % γ12 in %

Bias 0.0004 0.0001 0.0039 0.0021 0.0005 0.0048 0.0188 0.0204 0.0391

Noise-floor 0.0004 0.0002 0.0065 0.0182 0.0357 0.0346 0.0258 0.0280 0.0350

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U in mm y in m m

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

-1 -2 -3 -10

-5

0

a

5

10

15

20

x in mm

U in mm 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

y in m m

15 10

5

-50

-40

b

-30

-20

-10

0

x in mm

Fig. 5.33 Distribution of horizontal displacement U. (a) ENF at ai(Efl) ¼ 6.8 mm, (b) C-ELS at ai(Efl) ¼ 9.7 mm

Horizontal Displacement U Figure 5.33 depicts the horizontal displacement for both ENF and C-ELS testing. Note that the C-ELS specimen includes a shift toward the load application line with increasing deflection (see clamping on the right). The lateral shift of the C-ELS specimen might affect crack growth measurement due to a reduction in the measured overall crack length, which has to be taken into account via correction factors (ISO, 2014). The interlaminar shear stress for three-point bending, in accordance with (DIN, 1998; ASTM, 2016), refers to the maximum shear stress at the midline of the specimen (at the neutral axis, see Chap. 4) τ max ,3pt ¼

3∙F 4∙w∙h

ð5:44Þ

which is used in (JIS, 1997) for the determination of the energy release rate using a crack shear displacement gauge.

Mode II—Visualization of Strains and Displacements

a

253

b

Fig. 5.34 Shear stress-CSD curves. (a) ENF and (b) C-ELS

The maximum shear stress for a single cantilever beam is τ max ,SC ¼

3∙F 2∙w∙h

ð5:45Þ

By focusing on equidistant points in the upper and lower portion of the beam from the crack line (e.g., midline of each half-beam hCSD/2 ¼ h/4), the relative motion as crack shear displacement (CSD) and crack tip shear displacement (CTSD) is extracted, analogous to COD and CTOD measurements shown in Mode I Vertical Displacement V Section. For sign convention (analogous to COD), the relative motion relates to subtracting the bottom displacement from the top displacement. The resulting maximum shear stress-crack shear displacement curves are shown in Fig. 5.34. The initial increase of absolute CSD for C-ELS is a result of the deflection, leading to a rotation. The increase in shear stress after the first force drop during ENF testing results from the interaction between crack and the center section in the vicinity of the loading roller. For both test methods, the determination of the exact CTSD point represents the inherent challenge in visual delamination length under mode II loading.

Vertical Displacement V Figure 5.35a shows the vertical deformation of an ENF specimen, depicting maximum deflection located left from the loading roller. Figure 5.35b depicts the vertical deformation of a C-ELS specimen. Note that rotations, which are equal to derivatives of U with respect to V, are possible at the delamination front. The analytical solution for the deflection of a pristine (inverse ELS resp. for calibration at L/h ¼ 23, L ¼ 54.4 mm) single cantilever beam specimen, shown in Fig. 5.36a, is presented in Fig. 5.36b using Eq. (5.38). Applying the boundary

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Delamination Resistance Testing

V in mm y in m m

-2.6

-2.4

-2.2

-2

-1.8

10

15

-1.6

-1 -2 -3 -10

-5

0

a

5

20

x in mm

V in mm 0

5

10

15

y in m m

15 10

5

-50

-40

b

-30

-20

-10

0

x in mm

Fig. 5.35 Distribution of vertical displacement V. (a) ENF at ai(Efl) ¼ 6.8 mm, (b) C-ELS at ai(Efl) ¼ 9.7 mm

condition at xi ¼ 0 being the measured maximum deflection V*, an average flexural modulus (from the regression for selected load levels shown in Fig. 5.36b) Efl ¼ 116 GPa  1 GPa is extracted. The determination of the flexural modulus is comparable to that of the DCB test (with half-beam thickness), whereas in this case, 12 layers are considered. Even though single cantilever beam testing provides a fixed boundary condition at the clamp, deflection at the clamp is not zero, see Fig. 5.36b, indicating non-rigid clamping.

Summary and Discussion This section summarizes and compares the delamination properties obtained from flexural testing under mode I and mode II loading conditions. The flexural moduli from different types of flexural tests will be compared and used for interpretation of the uncertainty (“reality check”) of the energy release rates obtained.

Summary and Discussion

255

V in mm y in m m

0

0.5

1

1.5

2

2.5

24 23 22 0

5

10

15

20

a

25

30

35

40

45

50

x in mm

3

L

2 ROI

V in mm

DIC Anal. Soln. (Regression)

1

0 0

b

10

20

30

40

50

x in mm

Fig. 5.36 (a) Contour plot with vertical deformation on non-cracked calibration specimen for C-ELS. (b) Corresponding midline deflection in comparison to analytical solution for selected load levels

Interlaminar Fracture Properties Table 5.7 lists the average values of the energy release rate from two tests each per test method. Comparison of standardized and non-standardized methods under mode I loading shows that the Area method (followed by SB method) provides the highest values. The standardized compliance-based methods barely show a sensitivity either to a change in crack growth or displacement measurement, as already confirmed by (Starke et al., 1996; Merzkirch et al., 2017). The crack length correction could be visualized with DIC measured COD, noting that the Area method is the only method relating to the real crack length.

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Table 5.7 Comparison of different methods for determination of energy release rate G C for mode I and mode II, in J/m2 ( representing standard deviation) Method PC SB MB CC MCC Area CBTE

Mode I DCB 235  12 252  3 234  0 233  0 235  1 289  27 N/A

Fig. 5.37 Semi-empirical relationship between mode I and mode II testing results

Mode II ENF 1044  117 913  9 N/A 926  96 N/A N/A N/A

Mode II C-ELS 1102  64 1217  73 N/A N/A N/A N/A 1601  81

1200 Mode I DCB (CC) Mode II ENF (CC) Mode II C-ELS (PC)

tot,C

in J/m2

1000 800 600

m=2

400 m=3

200

0 0

20

40

60

/ II

in %

tot

80

100

The values for mode II loading are higher than for mode I loading, with the mode G ratio being approx. G CI ¼ 1=5 , stating mode I is the more critical fracture mode, CII requiring less energy for crack propagation. C-ELS testing leads to slightly higher values, in comparison to ENF testing, with the CBTE representing the highest value. As already stated in (Davies et al., 1998), mode II testing is always dominated by mode I (on the micro-scale). Mode impurity could be visualized using the DIC calculated shear strain in mode I and normal strain in mode II loading. Figure 5.37 compares the results from mode I and mode II testing listed in Table 5.7. Additionally the semi-empirical relationship between mode I and mode II is expressed by the total critical energy release rate (Benzeggagh & Kenane, 1996) 

G tot,C

G II ¼ G IC þ ðG IIC  G IC Þ ∙ G tot

m ð5:46Þ

with G tot being the sum of mode I and mode II, and the exponent m to be determined via mixed mode testing conditions. Since there is scatter in the high mode II region, errors in extrapolation may be important (Davies et al., 1998). Examples for two

Summary and Discussion

257

different exponents are shown in Fig. 5.37. Studies on woven composites have shown that the exponent depends on whether the resin is brittle (m ¼ 2) or ductile (m ¼ 3) (Benzeggagh & Kenane, 1996). Usually, mode-mixity must contain a large proportion of mode II before significant change is observed. The standardized mixed mode bending (MMB) test (ASTM, 2013b), for mixed mode I–mode II, is based on a combination of double cantilever beam with threepoint flexural testing of pre-cracked specimens (Reeder & Crews, 1988). Another non-standardized possibility is based on ELS testing, the asymmetric DCB test (ADCB (Davies et al., 1998)) or mixed mode ELS (MMELS (Banks-Sills, 2018)) with pulling the top cantilever beam of the specimen (Banks-Sills, 2018) (or pushing the bottom cantilever beam (Williams, 1988, 1989; Hashemi et al., 1990)), see Figs. 5.2c and 5.8a. Both types of mixed mode tests allow for use of the same specimen geometry as for mode I and mode II testing. The MMB test is a variable ratio mixed mode test over the range from pure mode I to pure mode II (Davies et al., 1998). The MMELS test is considered as fixed ratio mixed mode test (Williams, 1988), with G I =G II ¼ 4=3 (Hashemi et al., 1990) due to geometric conditions (Williams, 1988), resulting in G II =G tot ¼ 42:9% . Another option is Arcan mixed mode testing (Carlsson & Pipes, 1987).

Elastic Flexural Properties Figure 5.38 compares flexural moduli determined from mode I and II testing to the true flexural modulus from three-point bending (Efl ¼ 116 GPa, see Chap. 4), all with fiber orientation parallel to loading span, and to the tensile modulus (EL ¼ 125 GPa, see Chap. 2). The flexural moduli are determined from mode I loading using Eqs. (5.7) and (5.12) including the correction Δ (from MB method). By using Eq. (5.12), the calculated flexural modulus is larger than the modulus determined excluding the

a

b

Fig. 5.38 Flexural moduli in comparison to tensile modulus (a) Mode I DCB testing with increasing crack growth, (b) Mode II ENF and C-ELS testing at various span-to-thickness ratios

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Delamination Resistance Testing

correction Δ, and is in good agreement to the tensile modulus. Fiber bridging might affect the flexural modulus, which might explain the slight increase of EI,fl with increasing crack propagation. The resulting average flexural modulus of the regression of the DIC calculated deflection, shown in Fig. 5.22b, is Efl ¼ 110 GPa  2 GPa comparable to the flexural modulus determined via three-point bending (Efl). Figure 5.38b depicts a comparison of the different methods of calculating the flexural moduli from mode II ENF and C-ELS testing in comparison to the flexural and tensile moduli. The flexural modulus from ENF testing, by extrapolation of the compliance to a ¼ 0 mm, see Fig. 5.25a, representing a pristine specimen at L/h ¼ 33, leads to a comparable flexural modulus determined via three-point bending (see Chap. 4). The different flexural moduli for C-ELS testing include simple beam theory (first term of Eq. (5.27)) and Eq. (5.33) including the correction Δ. The flexural modulus (without clamp correction Δ) from inverse ELS testing of a single cantilever beam is known to be always lower compared to values obtained via three-point flexural testing, acc. to (Blackman et al., 2006). For details on the effect of shear deformation at small span-to-thickness ratios, see Chap. 4. The resulting average flexural modulus of the regression of the DIC calculated deflection, shown in Fig. 5.36, for L/h ¼ 23, is Efl ¼ 116 GPa  1 GPa and therewith larger than moduli determined via maximum deflection method, leading to energy release rates (CBTE in Table 5.7) that are considered to be too large. As could be shown on the DIC data (Fig. 5.36), deflection at the clamp was not zero, which might be affected by the clamping force or the fixture compliance, due to rotational effects.

Testing of Assorted Materials This section focusses on the applicability of DIC measurements for a different type of composite and the related conspicuous photomechanical investigations.

Woven-CFRP—Mode I DCB Testing The force-deflection curve in Fig. 5.39a depicts an irregular, stick-slip behavior leading to an R-curve (Fig. 5.39b) with oscillating pattern. This stick-slip behavior is characterized by crack growth (local force maximum) and crack arrest (local force minimum) (Starke et al., 1996). Fractography shown in Fig. 5.39c delineates the interaction of the crack front with the warp and fill direction. The underlying crack propagation from Fig. 5.39b reflects a two-dimensional crack propagation, ignoring the three-dimensional crack branching across the woven architecture. The woven laminate shows higher values for the energy release rate in comparison to the UD laminate composite, which are considered to represent a lower bound with more conservative values, acc. to (Davies et al., 1998; Brunner et al., 2008).

Key Conclusions

259

40 Lin. Regr. 95 % Lin. Regr. Limits

in J/m2

600

20

400

IC

F in N

30

200

10

MB V*(F

0 0

5

max

)

10

CC

PC(Efl)

0 15

0

20

10

20

30

ai in mm

V* in mm

a

b

c Fig. 5.39 (a) Force-deflection curve, (b) R-curves from one mode I test, (c) fractography

Key Conclusions DIC • Speckle pattern should be applied with inserted blade when wedge pre-cracking to avoid paint sticking and covering the pre-crack. • A small subset size and Lstep ¼ 1 for automated crack tip tracking is preferred due to high spatial resolution but also requires high solution time and the related strain calculation are prone to high noise-floor. • DIC (LiC) method is in good agreement with pure visual and compliance-based crack length determination, but statements about the accuracy (see requirements of 0.5 mm in (JIS, 1997)) need further investigation. • Automated crack tip tracking for mode II delamination remains challenging using DIC. • When testing non-brittle materials, e.g., adhesives under mode I loading (ISO, 2009), stereovision is essential for accurate measurements of the COD in the vicinity of the crack tip, due to large out-of-plane response of the adhesive layer

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Delamination Resistance Testing

and the relatively small FOV. Errors larger than 50% of the measured value are observed when using a single camera system (Rajan et al., 2018). • For further information on the challenges due to the specimen geometry (thickness) and the use of high-resolution cameras and fine speckle patterning, see also key conclusions in Chap. 4.

Structural Mechanics and Testing • It is recommended to perform more calibration tests at a wider span range (e.g., by minimizing Lmin), with the pre-crack being fully clamped, in order to reduce the uncertainty of the correction Δ (note the high sensitivity due to the intercept). • (Wedge) pre-cracking for mode II is essential in order to avoid unstable crack growth and high initiation energy release rates. • Depending on the material and related specimen compliance, correction factors might be needed to account for stiffening of the specimen by load-blocks (or piano hinges) and large deflection for mode I DCB and mode II C-ELS testing.

Practice Exercises This section provides several practice exercises including the use of DIC measured data for self-calculation with the data reduction methodologies presented in this chapter. Solutions, together with supplementary instructional resources (e.g., figures, videos, data reduction codes, slides with deeper explanations of selected expressions and deduction of equations) suitable for lecturing or lab courses are available to instructors who adopt the book for classroom use. Please visit the book web page at www.springer.com for the password-protected material. Use the measured data and the geometric details for mode I DCB testing. 1. Calculate the crack propagation using the true flexural modulus Efl. 2. Illustrate crack resistance curve using the PC method. 3. Illustrate crack resistance curve using the SB method. 4. Illustrate crack resistance curve using the MB method. 5. Illustrate crack resistance curve using the CC method. 6. Illustrate crack resistance curve using the MMC method. 7. Illustrate crack resistance curve using the Area method. Use the measured data and the geometric details for mode II ENF testing. 8. Calculate the crack propagation using the true flexural modulus Efl. 9. Illustrate crack resistance curve using the PC method. 10. Illustrate crack resistance curve using the SB method. 11. Illustrate crack resistance curve using the CC method.

References

261

Use the measured data and the geometric details for mode II C-ELS testing. 12. Calculate the crack propagation using the true flexural modulus Efl. 13. Illustrate crack resistance curve using the PC method. 14. Illustrate crack resistance curve using the SB method. 15. Illustrate crack resistance curve using the CBTE method. Acknowledgments Thanks to Dave Pitchure for waterjet cutting the specimens. Alex Jennion for fractography. Christopher Amigo for support in manufacturing of the testing setups. Evan Rust for providing code for UQ. Mark Iadicola for input on the loss in correlation methodology and Chris Calhoun for input on the airbrush spray-painting method.

References ASTM. (2013a). ASTM D5528—Standard test method for mode I interlaminar fracture toughness of unidirectional fiber-reinforced polymer matrix composites. ASTM International. https://doi. org/10.1520/D5528-13 ASTM. (2013b). ASTM D6671/D6671M—Standard test method for mixed mode I-mode II interlaminar fracture toughness of unidirectional fiber reinforced polymer matrix composites. ASTM International. https://doi.org/10.1520/D6671_D6671M-13E01 ASTM. (2014). ASTM D7905/D7905M  Standard test method for determination of the mode II interlaminar fracture toughness of unidirectional fiber-reinforced polymer matrix composites. ASTM International. https://doi.org/10.1520/D7905_D7905M-14 ASTM. (2016). ASTM D2344/D2344M—Standard Test Method for Short-Beam Strength of Polymer Matrix Composite Materials and Their Laminates. ASTM International. https://doi.org/10. 1520/D2344_D2344M-16 ASTM. (2018). ASTM D5766/D5766M—Standard test method for open-hole tensile strength of polymer matrix composite laminates. ASTM International. https://doi.org/10.1520/D5766_ D5766M-11R18 Banks-Sills, L. (2018). Interface fracture and delaminations in composite materials. Springer. https://doi.org/10.1007/978-3-319-60327-8 Barenblatt, G. I. (1962). The mathematical theory of equilibrium cracks in brittle fracture. Advances in Applied Mechanics, 7, 55–129. https://doi.org/10.1016/S0065-2156(08)70121-2 Beehag, A., & Ye, L. (1996). Consolidation and interlaminar fracture properties of unidirectional commingled CF/PEEK composites. Journal of Thermoplastic Composite Materials, 9(2), 129–150. https://doi.org/10.1177/089270579600900203 Benzeggagh, M. L., & Kenane, M. (1996). Measurement of mixed-mode delamination fracture toughness of unidirectional glass/epoxy composites with mixed-mode bending apparatus. Composites Science and Technology, 56, 439–449. Blackman, B. R. K., Brunner, A. J., & Williams, J. G. (2006). Mode II fracture testing of composites: A new look at an old problem. Engineering Fracture Mechanics, 73(16), 2443–2455. https://doi.org/10.1016/j.engfracmech.2006.05.022 Blackman, B. R. K., Hadavinia, H., Kinloch, A. J., & Williams, J. G. (2003). The use of a cohesive zone model to study the fracture of fibre composites and adhesively-bonded joints. International Journal of Fracture, 119(1), 25–46. https://doi.org/10.1023/A:1023998013255 Brunner, A. J., Blackman, B. R. K., & Davies, P. (2008). A status report on delamination resistance testing of polymer–matrix composites. Engineering Fracture Mechanics. Carlsson, L. A., Adams, D. F., & Pipes, R. B. (2014). Experimental characterization of advanced composite materials (4th ed.). CRC Press.

262

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Carlsson, L. A., Gillespie, J. W., & Pipes, R. B. (1986). On the analysis and design of the end notched flexure (ENF) specimen for mode II testing. Journal of Composite Materials, 20. https://doi.org/10.1177/002199838602000606 Carlsson, L. A., & Pipes, R. B. (1987). Experimental characterization of advanced composite materials. Prentice-Hall. Daniel, I. M., & Ishai, O. (1994). Engineering mechanics of composite materials. Oxford. Davies, P. (1997). Influence of ENF specimen geometry and friction on the mode II delamination resistance of carbon/PEEK. Journal of Thermoplastic Composite Materials, 10(4), 353–361. https://doi.org/10.1177/089270579701000404 Davies, P., Blackman, B. R. K., & Brunner, A. J. (1998). Standard test methods for delamination resistance of composite materials: Current status. Applied Composite Materials, 5(6), 345–364. https://doi.org/10.1023/A:1008869811626 de Moura, M. F. S. F., & de Morais, A. B. (2008). Equivalent crack based analyses of ENF and ELS tests. Engineering Fracture Mechanics, 75(9), 2584–2596. https://doi.org/10.1016/j. engfracmech.2007.03.005 DIN. (1998). DIN EN ISO 14130—Fibre-reinforced plastic composites—Determination of apparent interlaminar shear strength by short-beam method. CEN. DIN. (2013a). DIN EN 6033—Luft- und Raumfahrt—Kohlenstofffaserverstärkte Kunststoffe— Prüfverfahren—Bestimmung der interlaminaren Energiefreisetzungsrate, Mode I—GIC. , DIN. (2013b). DIN EN 6034—Luft- und Raumfahrt—Kohlenstofffaserverstärkte Kunststoffe— Prüfverfahren: Bestimmung der interlaminaren Energiefreisetzungsrate, Mode II – GII. , Dugdale, D. S. (1960). Yielding of steel sheets containing slits. Journal of the Mechanics and Physics of Solids, 8(2), 100–104. https://doi.org/10.1016/0022-5096(60)90013-2 Elices, M., Guinea, G. V., Gomez, J., & Planas, J. (2002). The cohesive zone model: Advantages, limitations and challenges. Engineering Fracture Mechanics, 69(2), 137–163. https://doi.org/ 10.1016/S0013-7944(01)00083-2 Flügge, W. (1962). Handbook of engineering mechanics. McGraw Hill. Freiman, S., & Mecholsky, J. (2012). The fracture of brittle materials: Testing and analysis. Wiley. Griffith, A. A. (1921). The phenomena of rupture and flow in solids. Philosophical Transactions of the Royal Society of London. Series A, 221, 163–198. https://doi.org/10.1098/rsta.1921.0006 Gross, D., Hauger, W., Schröder, J., Wall, W. A., & Bonet, J. (2018). Engineering mechanics 2 (2nd ed.). Springer. https://doi.org/10.1007/978-3-662-56272-7 Gross D, Seelig T (2011) Fracture Mechanics - With an Introduction to Micromechanics. Mechanical Engineering Series, (2nd ed.). Springer. https://10.1007/978-3-642-19240-1 Hashemi, S., Kinloch, A. J., & Williams, J. G. (1989). Corrections needed in double-cantilever beam tests for assessing the interlaminar failure of fibre-composites. Journal of Materials Science Letters, 8(2), 125–129. https://doi.org/10.1007/Bf00730701 Hashemi, S., Kinloch, A. J., & Williams, J. G. (1990). The analysis of interlaminar fracture in uniaxial fiber-polymer composites. Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, 427(1872), 173–199. https://doi.org/10.1098/rspa.1990. 0007 iDICs. (2018). International digital image correlation society—A good practices guide for digital image correlation. International Digital Image Correlation Society iDICs. https://doi.org/10. 32720/idics/gpg.ed1 Irwin, G. R., & Kies, J. A. (1954). Critical energy rate analysis of fracture strength. Welding Journal Research Supplement, 33, 193–198. ISO. (2001). ISO 15024—Fibre-reinforced plastic composites—Determination of mode I interlaminar fracture toughness, GIC, for unidirectionally reinforced materials. ISO. ISO. (2009). ISO 25217—Adhesives—Determination of the mode 1 adhesive fracture energy of structural adhesive joints using double cantilever beam and tapered double cantilever beam specimens. ISO.

References

263

ISO. (2014). ISO 15114—Fibre-reinforced plastic composites—Determination of the mode II fracture resistance for unidirectionally reinforced materials using the calibrated end-loaded split (C-ELS) test and an effective crack length approach. ISO. JIS. (1997). JIS K 7086—Japanese industrial standard—Testing methods for interlaminar fracture toughness of carbon fibre reinforced plastics. Japanese Standards Association. Kanninen, M. F. (1973). An augmented double cantilever beam model for studying crack propagation and arrest. International Journal of Fracture, 9, 83–92. Khudiakova, A., Grasser, V., Blumenthal, C., Wolfahrt, M., & Pinter, G. (2020). Automated monitoring of the crack propagation in mode I testing of thermoplastic composites by means of digital image correlation. Polymer Testing, 82. https://doi.org/10.1016/j.polymertesting. 2019.106304 Khudiakova, A., Wolfahrt, M., Godec, D., & Pinter, G. (2018). Determination of the mode I strain energy release rate in carbon fibre reinforced composites by means of digital image correlation technique. In 18th European Conference on Composite Materials ECCM18, Athens (Greece). Merzkirch, M., Powell, L. A., & Foecke, T. (2017). Measurements of mode I interlaminar properties of carbon fiber reinforced polymers using digital image correlation. Key Engineering Materials, 742, 652–659. https://doi.org/10.4028/www.scientific.net/KEM.742.652 Moore, D. R., Pavan, A., & Williams, J. G. (2001). Fracture mechanics testing methods for polymers, adhesives and composites (Vol. 28). ESIS. Elsevier. Olsson, R. (1992). A simplified improved beam analysis of the DCB specimen. Composites Science and Technology 43, 329–338. https://doi.org/10.1016/0266-3538(92)90056-9 Poissant, J. & Barthelat, F. (2010). A Novel “Subset Splitting” Procedure for Digital Image Correlation on Discontinuous Displacement Fields. Experimental Mechanics, 50(3), 353–364. https://doi.org/10.1007/s11340-009-9220-2 Rajan, S., Sutton, M. A., Fuerte, R., & Kidane, A. (2018). Traction-separation relationship for polymer-modified bitumen under mode I loading: Double cantilever beam experiment with stereo digital image correlation. Engineering Fracture Mechanics, 187, 404–421. https://doi. org/10.1016/j.engfracmech.2017.12.031 Reeder, J. R., & Crews, J. H. (1988). A mixed mode bending apparatus for delamination testing. NASA—National Aeronautics and Space Administration. Reu, P. (2012). Hidden components of 3D-DIC: Triangulation and post-processing—Part 3. Experimental Techniques, 36(4), 3–5. https://doi.org/10.1111/j.1747-1567.2012.00853.x Reu, P. (2013a). Calibration: 2D calibration. Experimental Techniques, 37(5), 1–2. https://doi.org/ 10.1111/ext.12027 Reu, P. (2013b). Calibration: Pre-calibration routines. Experimental Techniques, 37(4), 1–2. https:// doi.org/10.1111/ext.12026 Reu, P. (2013c). Stereo-rig design: Lens selection—Part 3. Experimental Techniques, 37(1), 1–3. https://doi.org/10.1111/ext.12000 Reu, P. (2015). Virtual strain gage size study. Experimental Techniques, 39(5), 1–3. https://doi.org/ 10.1111/ext.12172 Rice, J. R. (1968). A path independent Integral and the approximate analysis of strain concentration by notches and cacks. Journal of Applied Mechanics, 35, 379–386. Starke, C., Beckert, W., & Lauke, B. (1996). Characterization of the delamination behaviour of composites under mode I- and mode II-loading. Materialwissenschaft und Werkstofftechnik, 27 (2), 80–89. https://doi.org/10.1002/mawe.19960270209 Sutton, M. A., Orteu, J.-J., & Schreier, H. W. (2009). Image correlation for shape. Motion and Deformation Measurements. https://doi.org/10.1007/978-0-387-78747-3 Sutton, M. A., Zhao, W., McNeill, S. R., Helm, J. D., Piascik, R. S., & Riddell, W. T. (1999). Local crack closure measurements: Development of a measurement system using computer vision and a far-field microscope. American Society for Testing and Materials, 1343, 145–156. https://doi. org/10.1520/Stp15755s Timoshenko, S. (1940a). Strength of materials—Part I: Elementary theory and problems (2nd ed.). D. Van Nostrand Company Inc.. Timoshenko, S. (1940b). Strength of materials—Part II: Advanced theory and problems (2nd ed.). D. Van Nostrand Company Inc..

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5

Delamination Resistance Testing

Tolf, G. (1985). Saint-Venant bending of an orthotropic beam. Composite Structures, 4(1), 1–14. https://doi.org/10.1016/0263-8223(85)90017-0 Ungsuwarungsri, T., & Knauss, W. G. (1987). The role of damage-softened material behavior in the fracture of composites and adhesives. International Journal of Fracture, 35(3), 221–241. https://doi.org/10.1007/Bf00015590 Wagner, H. D., & Marom, G. (1982). Analysis of several loading methods for simultaneous determination of Young’s and shear moduli in composites. Fibre Science and Technology, 16, 61–65. Wang, H., & VuKhanh, T. (1996). Use of end-loaded-split (ELS) test to study stable fracture behaviour of composites under mode II loading. Composite Structures, 36(1-2), 71–79. https:// doi.org/10.1016/S0263-8223(96)00066-9 Wang, J. L., & Qiao, P. Z. (2004). Novel beam analysis of end notched flexure specimen for modeII fracture. Engineering Fracture Mechanics, 71(2), 219–231. https://doi.org/10.1016/S00137944(03)00096-1 Williams, J. G. (1984). Fracture mechanics of polymers. E. Horwood. Williams, J. G. (1988). On the calculation of energy-release rates for cracked laminates. International Journal of Fracture, 36(2), 101–119. https://doi.org/10.1007/Bf00017790 Williams, J. G. (1989). The fracture mechanics of delamination tests. Journal of Strain Analysis for Engineering Design, 24(4), 207–214. https://doi.org/10.1243/03093247v244207 Williams, J. G., & Hadavinia, H. (2002). Analytical solutions for cohesive zone models. Journal of the Mechanics and Physics of Solids, 50(4), 809–825. https://doi.org/10.1016/S0022-5096(01) 00095-3

Chapter 6

Summary and Discussion

UD-CFRP Only a few comparisons between the entire intralaminar and interlaminar shear stress-shear strain curves of unidirectionally reinforced laminates are available in the literature. For UD laminates, the interlaminar shear modulus is usually obtained by transversely isotropic assumption and dedicated interlaminar shear strengths are extracted without necessarily measuring the shear strain (Bru et al., 2017). For the same reason, it is pretty common to assume the interlaminar shear response to be the same as the in-plane shear response acc. to Cui et al. (1992). Intralaminar shear properties determined via three in-plane test methods on a balanced and symmetric unidirectionally carbon fiber reinforced epoxy matrix laminate are compared to the interlaminar shear properties determined from three through-thickness test methods under flexural loading. Therewith the comparison covers two of the three preferred in-plane shear test methods for unidirectionally reinforced sheet materials (Lee & Munro, 1986; Chatterjee et al., 1993; Summerscales, 1987). A phenomenological investigation on micro-mechanisms during deformation and damage has not been pursued in detail. A direct comparison of the interlaminar and intralaminar properties of UD laminates should be done with caution due to process–structure–property relationships. Manufacturing related influences such as processing temperature and pressure affect fiber distribution homogeneity (matrix rich regions resp., related fiber volume fraction and deviations in thickness of the laminate), fiber-matrix bonding and porosity (Schürmann, 2007).

© Springer Nature Switzerland AG 2022 M. Merzkirch, Mechanical Characterization Using Digital Image Correlation, https://doi.org/10.1007/978-3-030-84040-2_6

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6 Summary and Discussion

Fig. 6.1 Comparison of representative shear stressshear strain curves up to failure/delamination from five different test methods on UD-CFRP

Intralaminar and Interlaminar Shear Properties Figure 6.1 compares shear stress-shear strain curves up to failure, delamination resp., deduced from intralaminar and interlaminar shear testing, presented in the previous chapters: state-of-the-art 10
off-axis tensile testing, standardized V-notched beam and rail testing for the in-plane behavior and short-beam testing under standardized three-point and four-point bending for the interlaminar behavior. The novelty includes the use of DIC for a deduction of the full stress-strain response along selected regions of the span for three-point and four-point short-beam testing. To date, those methods were limited to the determination of the (apparent) ultimate shear strength. Since standardized five-point short-beam testing does not allow for a determination of the full stress-strain response, it is not shown. Maximum shear strain γ 12 is taken for the intralaminar and shear strain in the specimen’s coordinate system γ xy for the interlaminar values. For simplification, only shear stress τ is depicted. For 10
off-axis tensile testing, the results shown in this chapter refer to an aspect ratio of L/w ¼ 14 and strain from the ROI, to capture the inhomogeneous strain distribution within the gauge section of the specimen. For V-notched beam and rail testing, the results were deduced from strain measurements from the notched section, since stress refers to the same cross-sectional area, fiber orientation being parallel to loading direction (θ ¼ 90
), and an orthogonal notch angle. For detailed information on stress calculation, determination of strain, and effect of specimen geometry, see corresponding chapters. All shear stress-shear strain curves show a considerable amount of non-linearity before failure. In reference (Merzkirch & Foecke, 2020), interrupted 10
off-axis tensile testing has been conducted up to the non-linear region with a dwell time and an unloading cycle. Both, nominal stress and shear stress showed a relaxation at constant axial strain but with an additional increase in shear strain. The unloading resulted in a non-reversible strain and a macroscopic investigation of the specimen

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267

did not show any damage. The same can be stated for V-notched beam specimens followed from the investigations shown in Chap. 3. Though detailed investigation on the deformation and damage mechanisms on the microscopic scale is not the focus of this treatise, a few possible explanations will be presented hereinafter. The shear deformation of the composite is usually considered to be matrix dominated (He et al., 2002). However, the non-linearity in deformation response was speculated to origin from micro-cracks, micro-damages, or local fiber instability before the major failure events, rather than due to plastic deformation in the resin. The reason for this was believed that the epoxy resin does not exhibit plasticity at the shear strain range of the measurements, since shear yield strain for epoxies is generally around 6 % (He et al., 2002). Reference (Melin & Neumeister, 2006) strengthened the argument, stating that non-linear mechanisms possibly appear together with damage of various kinds, such as micro-cracks, debonding, voiding, and rearrangements on the molecular level. The extracted shear properties are shown in Fig. 6.2 for the shear modulus G (a), the ultimate shear strength USS (b), and the shear strain at failure γ USS (c). The value on top represents the average, and the value on bottom the coefficient of variation (CV), relative standard deviation resp., in percent. In summary, interlaminar and intralaminar shear moduli are comparable (despite the high fluctuations for four-point short-beam testing due to challenges in alignment). Flexural testing with variable spans also provides a calculation of the interlaminar shear modulus (Gfl ¼ 4 GPa, see Chap. 4) which is in good agreement with the strain-based values from three-point and four-point short-beam testing, and the shear moduli for each half-span calculated from five-point double beam shear testing. The comparison of the ultimate shear strengths shows a clear trend, interlaminar to be higher than intralaminar (more conservative resp.), with the latter being comparable. The same can be attested for strain at failure with a possible explanation given before on the origin of the non-linearity. Note also that specimen volumes tested differ. V-notched beam testing shows less fluctuations in ultimate shear strength than rail testing, with comparable strength. The interlaminar ultimate shear strength from short-beam testing barely shows differences between threepoint and four-point testing, whereas five-point testing leads to higher ultimate shear strength. According to Melin et al. (2000), measured interlaminar ultimate shear strengths (ILSS resp.) are on the conservative side, implying that values of the material property are higher than measured values. Consequently, test methods which measure higher USS are deemed to be more accurate. Relating the longitudinal modulus EL ¼ 125 GPa to the intralaminar shear modulus, leads to a ratio of approx. EL/G ¼ 31. In comparison, the ratio of the epoxy resin is E/G ¼ 2.8 with the shear modulus G(matrix) ¼ 1.22 GPa being approx. a third of the reinforced composite. Reference (Pierron & Vautrin, 1997) states that in-plane shear strengths often reported may well be grossly underestimated as a consequence of inappropriate testing procedures. In accordance with Cui et al. (1992), the interlaminar ultimate shear strength is a critical parameter in fibrous composite laminates because of its

268

6 Summary and Discussion

Fig. 6.2 Comparison of (a) Shear moduli G, (b) Ultimate shear strengths USS, (c) Shear strain at failure γ USS for different intralaminar and interlaminar test methods, Top: average value, Bottom: coefficient of variation

5

Intralaminar

Interlaminar

4.7 4.5

G in GPa

4.5

4.5

1.7 % 1.9 %

0.93 %

4

3.9

4 2.7 %

2.3 %

3.5

3.6 14 %

3

a 120

Intralaminar

Interlaminar 114

USS in MPa

110

1.5 %

100 90

82.4

80

3.4 %

70

65.3

67

60

2.2 %

2.9 %

62.9

8.2 %

50

b

c

84.8 3.2 %

UD-CFRP

269

relatively low value compared to the longitudinal strength UTSL ¼ 1875 MPa, being UTSL/USS(interlaminar) ¼ 16–23, compare to UTSL/USS(intralaminar) ¼ 29. Compare also the ultimate shear strengths with the transverse tensile strength UTST ¼ 66 MPa, being UTST/USS(interlaminar) ¼ 0.6–0.8, and UTST/USS (intralaminar) ¼ 1. Zhou et al. (1995) compared interlaminar and intralaminar shear moduli of carbon fiber epoxy UD composites using V-notched beam testing. The interlaminar shear modulus was approx. 14 % higher than the intralaminar shear modulus, whereas the interlaminar ultimate shear strength was lower than the intralaminar ultimate shear strength. In summary, ultimate shear strength is controversial and should rather be interpreted as the maximum stress the material can bear under the loading conditions, the shear stress at failure resp. All test methods include non-pure shear loading conditions revealing superimposed and undesired mixed mode failure of the specimens. DIC was successfully used to analyze a complex state of different strains even though the difficulty in isolating a single failure mode in the specimens. Strain perpendicular to fiber orientation is considered as a possible failure mechanism in all test methods, as well as strains resulting from loading conditions, leading to premature failure.

Experiment vs. Analytical Model The experimentally determined results are compared to analytical predictions derived from tensile properties of the UD composite laminate with an in-plane angle θ of the principal fiber axis to the loading axis. Note that the calculation is based on input parameters determined from the [012]s laminate.

Elastic Properties Based on four needed in-plane parameters, the longitudinal tensile modulus (EL) and transverse tensile modulus (ET), the intralaminar shear modulus (G) and longitudinal Poisson’s ratio (νL), the orientation dependent elastic properties (see compliance calculations in Chap. 1) are compared to the average results from tensile testing in longitudinal (0
), transverse (90
), and off-axis (10
) directions, see Fig. 6.3. For the analytical investigation, the shear modulus from 10
off-axis tensile testing has been used. A decrease of 50% in comparison to the longitudinal modulus occurs within 11
, see Fig. 6.3a.

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6 Summary and Discussion

Fig. 6.3 Analytical orientation dependent elastic parameters in comparison to experimentally determined parameters (a) Young’s modulus, (b) Poisson’s ratio, (c) Extension-shear coupling

a

b

c

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271

Fig. 6.4 Analytical orientation dependent shear modulus in comparison to experimentally determined parameters

Figure 6.3b depicts the extension-extension ratio (Poisson’s ratio ν), with its maximum at approx. θ ¼ 20
(almost reaching 0.5, being the theoretical maximum for homogenous isotropic materials (Gross et al., 2018)). Figure 6.3c depicts the extension-shear coupling ratio (η), with its maximum at θ ¼ 10
, which is in agreement to Sun and Berreth (1988) stating that the most prominent extension-shear coupling occurs between θ ¼ 10
–15
for graphite/epoxy systems. The inverse of the compliances S22 and S26 are symmetric to 1/S11 and 1/S16 and therefore not shown here. The orientation dependent shear modulus is compared to the intralaminar shear test results, see Fig. 6.4. For the analytical investigations, the shear modulus from 10
off-axis tensile testing has been used and is compared to experimental results from V-notched beam and rail testing for different fiber orientations θ (θ ¼ 90
representing loading direction parallel to fiber orientation). Due to the equilibrium of moments, the shear moduli for fiber orientations θ ¼ 90
and θ ¼ 0
are similar, independent of the type of V-notched specimen testing. Note the high modulus for fiber orientation of θ ¼ 45
, which is in good agreement to the experimental value from V-notched rail testing. The sensitivity analysis in the subsequent section will present further explanations for possible fluctuations (e.g., Fig. 6.9b shows that variations in the transverse modulus affect the shear modulus for fiber orientations θ ¼ 30
–60
).

Strength The dependency of fiber orientation on strength is shown in Fig. 6.5a, using three parameters: ultimate shear strength (USS) from 10
off-axis tensile testing,

272

6 Summary and Discussion

Fig. 6.5 (a) Analytical orientation dependent ultimate tensile strength in comparison to experimentally determined parameters, (b) Normalized curve of analytical model

a

b longitudinal ultimate tensile strength (UTSL) and transverse ultimate tensile strength (UTST). Figure 6.5a depicts the transition from longitudinal failure, shear dominant failure, to mixed mode with combined intralaminar shear and transverse failure by a decrease of about 50 % in strength (compared to the longitudinal direction θ ¼ 0
) within approx. 4
. Combining the quadratic failure criterion (Chap. 1) with the shear stress equation for off-axis testing (Chap. 2), the ratio (see Fig. 6.5b) between the induced shear stress via off-axis tensile testing and ultimate shear strength, is determined τ ¼ UTSθ ∙ sin θ ∙ cos θ USS

ð6:1Þ

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273

The calculated ultimate shear strength has its maximum at an off-axis angle of θ ¼ 10.7
. Shear strength values of 90 % of the maximum and higher can be expected between off-axis angles of θ ¼ 3.7
–29.5
, which shows that off-axis testing is relatively insensitive for small variations of the angle in that range which diminishes the sensitivity on the needed fracture angle measurements discussed in Chap. 2. Figure 6.5b underlines that, based on the contribution of the actual shear stress to failure and the amount of induced shear stress, researchers (Pindera & Herakovich, 1986; Chamis & Sinclair, 1977) have converged to an off-axis angle of 10
(Ganesh & Naik, 1997). It is evident that an off-axis angle of θ ¼ 45
leads to smaller error in the determination of the modulus (due to a small extension-shear coupling, see Fig. 6.3c) (Pindera & Herakovich, 1986), but does not allow for a determination of the ultimate shear strength, see Fig. 6.5b. In summary, Figs. 6.3, 6.4, and 6.5 depict the highly anisotropic character of UD-CFRP composites.

Sensitivity Analysis: Input Parameters for Analytical Modeling The current section provides a sensitivity analysis in order to avoid redundant (repetitive) testing and explanations in case experimental data does not match the expected values from the analytical solutions. An artificial deviation, coefficient of variation (CV) resp., of 10% for each input parameter has been chosen for investigation of the sensitivity on the elastic (with four parameters) and strength (with three parameters) properties. For better comparison, plots have been normalized to the value at fiber orientation of θ ¼ 0
.

Elastic Properties (Figs. 6.6–6.9) Note that the coefficient of variation (CV) leads to inverse trend in extensionextension coupling in Fig. 6.7d, in extension-shear coupling in Fig. 6.8b and in shear in Fig. 6.9c. Additionally, crossing of upper and lower limit in extension-shear coupling is obvious from Fig. 6.8d.

a

b

c

d

Fig. 6.6 Analytical solution for the evolution of the inverse longitudinal compliance 1/S11 vs. fiber orientation θ, effect of coefficient of variation (CV) of 10 % in (a) longitudinal modulus EL, (b) transverse modulus ET, (c) Poisson’s ratio νL, (d) Shear modulus G

a

b

c

d

Fig. 6.7 Analytical solution for the evolution of the extension-extension ratio S12/S11 vs. fiber orientation θ, effect of coefficient of variation (CV) of 10% in (a) longitudinal modulus EL, (b) transverse modulus ET, (c) Poisson’s ratio νL, (d) Shear modulus G

a

b

c

d

Fig. 6.8 Analytical solution for the evolution of the inverse extension-shear ratio S16/S11 vs. fiber orientation θ, effect of coefficient of variation (CV) of 10% in (a) longitudinal modulus EL, (b) transverse modulus ET, (c) Poisson’s ratio νL, (d) Shear modulus G

a

b

c

d

Fig. 6.9 Analytical solution for the evolution of the inverse shear compliance 1/S66 vs. fiber orientation θ, effect of coefficient of variation (CV) of 10% in (a) longitudinal modulus EL, (b) transverse modulus ET, (c) Poisson’s ratio νL, (d) Shear modulus G

276

6 Summary and Discussion

Strength (Figs. 6.10 and 6.11)

a

b

c

Fig. 6.10 Analytical solution for the evolution of the ultimate tensile strength UTS vs. fiber orientation θ, effect of coefficient of variation (CV) of 10 % in (a) longitudinal ultimate tensile strength UTSL, (b) transverse ultimate tensile strength UTST, (c) ultimate shear strength USS

a

b

c

Fig. 6.11 Analytical solution for the evolution of the ratio τ/USS vs. fiber orientation θ, effect of coefficient of variation (CV) of 10 % in (a) longitudinal ultimate tensile strength UTSL, (b) transverse ultimate tensile strength UTST, (c) ultimate shear strength USS

Woven-CFRP

277

Woven-CFRP Intralaminar shear properties determined from two in-plane test methods and interlaminar shear properties determined from two through-thickness test methods on a balanced and symmetric woven carbon fiber bundle reinforced epoxy matrix laminate are summarized. A direct comparison between intralaminar and interlaminar shear properties will not be presented and is inappropriate due to the orthotropic character of the composite laminate.

Intralaminar and Interlaminar Shear Properties Figure 6.12 depicts shear stress-shear strain curves of selected test methods for the determination of the intralaminar properties, investigated in the previous chapters: standardized tensile testing of a 45
laminate and V-notched beam testing. The results from through-thickness testing via standardized three-point short-beam bending of this complex architecture are highly doubtful, therefore only the in-plane properties are discussed. Since standardized five-point short-beam bending does not allow for a determination of the full stress-strain response, it is not shown here. Note that the V-notched specimen did not fully fracture, therefore the shear stressshear strain curve up to γ ¼ 5 % is shown. For simplification, only shear stress τ and maximum shear strain γ 12 are depicted. For detailed information on stress calculation and determination of strain, see corresponding chapters. Table 6.1 summarizes the intralaminar and interlaminar shear properties determined via four test methods. Note that the ratio between ultimate tensile strength (UTS ¼ 669 MPa) and ultimate shear strength UTS/USS is approx. UTS/USS (intralaminar) ¼ 10 and UTS/USS(interlaminar) ¼ 14–17, being smaller than for UD laminates. Fig. 6.12 Comparison of representative shear stressshear strain curves up to failure

278

6 Summary and Discussion

Table 6.1 Results from two tests each on woven-CFRP (coefficient of variation in brackets) Test method 45
Tension V-Notched Beama 3pt SBTb 5pt DBS a

G in GPa 4.27 (1.5%) 4.39 (4.2%) N/A 2.77 (0%)

USS in MPa 58.7 (1.5%) 73.8 (0.1%) 41.5 52.3 (15%)

γ USS in % 2.96 (10%) 5.04 (0.6%) N/A N/A

USS related value from γ ¼ 5% Only one test

b

Only a slight difference between the in-plane shear moduli from two test methods exist, with the difference in ultimate shear strength being large. Interlaminar shear properties are smaller than the intralaminar properties.

Experiment vs. Analytical Model The experimentally determined results are compared to analytical predictions derived from tensile properties of the woven composite with an in-plane angle θ of the principal fiber axis to the loading axis. Note that the calculation is based on input parameters determined from the woven composite laminate.

Elastic Properties Based on three needed parameters, the longitudinal modulus (with the transverse modulus being equal EL ¼ ET), the intralaminar shear modulus G as well as longitudinal Poisson’s ratio νL, the orientation dependent elastic properties are compared to the tensile tests in warp (0
), fill/weft (90
) and 45
directions, see Figs. 6.13 and 6.14. For the analytical investigation, the shear modulus from 45
tensile testing has been used. Figure 6.13c depicts the anti-symmetric evolution of the extension-shear coupling being close to zero for θ ¼ 45
. A slight underestimation by the model can be stated in comparison to the average values for θ ¼ 45
. The inverse of the compliances S22 and S26 are symmetric to 1/S11 and 1/S16 and therefore not shown here. Note the high shear modulus for fiber orientation of 45
in comparison to the warp and fill direction with a factor of approx. 7, in Fig. 6.14.

Strength The orientation dependent strength is compared to the tensile tests in warp (0
), fill/ weft (90
) and 45
directions, see Fig. 6.15a. For the analytical investigation, the shear strength from 45
tensile testing has been used. Figure 6.15b shows the

Woven-CFRP

279

Fig. 6.13 Analytical orientation dependent elastic parameters in comparison to experimentally determined parameters (a) Young’s modulus, (b) Poisson’s ratio, (c) extension-shear coupling

a

b

c

280

6 Summary and Discussion

Fig. 6.14 Analytical orientation dependent shear modulus in comparison to experimentally determined parameters

Fig. 6.15 (a) Analytical orientation dependent ultimate tensile strength in comparison to experimentally determined parameters, (b) normalized curve of analytical model

a

b

Woven-CFRP

281

normalized strength, see Eq. (6.1), depicting the small sensitivity to offset in orientation.

Sensitivity Analysis: Input Parameters for Analytical Modeling The current section provides a sensitivity analysis in order to avoid redundant (repetitive) testing and explanations in case experimental data does not match the expected values from the analytical solutions. An artificial deviation, coefficient of variation (CV) resp., of 10% for each input parameter has been chosen for investigation of the sensitivity on the elastic (with three parameters) and strength (with two parameters) properties. For better comparison, plots have been normalized to the value at fiber orientation of θ ¼ 0
.

Elastic Properties (Fig. 6.16–6.19) Note that the coefficient of variation (CV) leads to an inverse trend in extensionextension coupling, Fig. 6.17d and crossing of upper and lower limit in extensionshear coupling, Fig. 6.18d. a

b

c

d

Fig. 6.16 Analytical solution for the evolution of the inverse longitudinal compliance 1/S11 vs. fiber orientation θ, effect of coefficient of variation (CV) of 10 % in (a) longitudinal modulus EL, (b) transverse modulus ET, (c) Poisson’s ratio νL, (d) Shear modulus G

a

b

c

d

Fig. 6.17 Analytical solution for the evolution of the extension-extension ratio S12/S11 vs. fiber orientation θ, effect of coefficient of variation (CV) of 10% in (a) longitudinal modulus EL, (b) transverse modulus ET, (c) Poisson’s ratio νL, (d) Shear modulus G

a

b

c

d

Fig. 6.18 Analytical solution for the evolution of the inverse extension-shear ratio S16/ S11 vs. fiber orientation θ, effect of coefficient of variation (CV) of 10% in (a) longitudinal modulus EL, (b) transverse modulus ET, (c) Poisson’s ratio νL, (d) Shear modulus G

Comparison: Material Anisotropy

283

a

b

c

d

Fig. 6.19 Analytical solution for the evolution of the inverse shear compliance 1/S66 vs. fiber orientation θ, effect of coefficient of variation (CV) of 10% in (a) longitudinal modulus EL, (b) transverse modulus ET, (c) Poisson’s ratio νL, (d) Shear modulus G

Strength (Figs. 6.20 and 6.21)

Comparison: Material Anisotropy This section presents a comparison of the orientation dependent behavior of three fibrous composite laminates investigated in this treatise. The plots have been normalized to the value at a fiber orientation of θ ¼ 0
. Note the effect of stitching fibers (in transverse direction) for UD-GFRP, affecting the transverse modulus (Figs. 6.22a, 6.23).

a

b

c

Fig. 6.20 Analytical solution for the evolution of the ultimate tensile strength UTS vs. fiber orientation θ, effect of coefficient of variation (CV) of 10% in (a) longitudinal ultimate tensile strength UTSL, (b) transverse ultimate tensile strength UTST, (c) ultimate shear strength USS

a

b

c

Fig. 6.21 Analytical solution for the evolution of the ratio τ/UTS vs. fiber orientation θ, effect of coefficient of variation (CV) of 10% in (a) longitudinal ultimate tensile strength UTSL, (b) transverse ultimate tensile strength UTST, (c) ultimate shear strength USS

Comparison: Material Anisotropy

285

a

b

c

d

Fig. 6.22 Comparison of the orientation dependent normalized elastic properties for three different types of fibrous composite laminates (a) inverse longitudinal compliance (b) extension-extension ratio (c) inverse extension-shear ratio (d) inverse shear compliance

Fig. 6.23 Comparison of the orientation dependent normalized tensile strength for three different types of fibrous composite laminates

286

6 Summary and Discussion

References Bru, T., Olsson, R., Gutkin, R., & Vyas, G. M. (2017). Use of the iosipescu test for the identification of shear damage evolution laws of an orthotropic composite. Composite Structures, 174, 319–328. https://doi.org/10.1016/j.compstruct.2017.04.068 Chamis, C. C., & Sinclair, J. H. (1977). Ten-deg off-axis test for shear properties in fiber composites. Experimental Mechanics, 17(9), 339–346. https://doi.org/10.1007/bf02326320 Chatterjee, S. N., Adams, D. F., & Oplinger, D. W. (1993). Test methods for composites: A status report. Volume III: Shear test methods. U.S. Department of Transportation Federal Aviation Administration. Cui, W. C., Wisnom, M. R., & Jones, M. (1992). Failure mechanisms in three and four point short beam bending tests of unidirectional glass/epoxy. Journal of Strain Analysis for Engineering Design, 27(4), 235–243. Ganesh, V. K., & Naik, N. K. (1997). (45) Degree off-axis tension test for shear characterization of plain weave fabric composites. Journal of Composites Technology and Research, 19(2), 77–85. https://doi.org/10.1520/CTR10018J Gross, D., Hauger, W., Schröder, J., Wall, W. A., & Bonet, J. (2018). Engineering mechanics 2 (2nd ed.). Springer. https://doi.org/10.1007/978-3-662-56272-7 He, J. M., Chiang, M. Y. M., Hunston, D. L., & Han, C. C. (2002). Application of the V-notch shear test for unidirectional hybrid composites. Journal of Composite Materials, 36(23), 2653–2666. https://doi.org/10.1177/002199802761675566 Lee, S., & Munro, M. (1986). Evaluation of in-plane shear test methods for advanced composite materials by the decision analysis technique. Composites, 17(1), 13–22. https://doi.org/10.1016/ 0010-4361(86)90729-9 Melin, L. G., Neumeister, J. M., Pettersson, K. B., Johansson, H., & Asp, L. E. (2000). Evaluation of four composite shear test methods by digital speckle strain mapping and fractographic analysis. Journal of Composites Technology and Research, 22(3), 161–172. https://doi.org/ 10.1520/CTR10636J Melin, L. N., & Neumeister, J. M. (2006). Measuring constitutive shear behavior of orthotropic composites and evaluation of the modified iosipescu test. Composite Structures, 76(1-2), 106–115. https://doi.org/10.1016/j.compstruct.2006.06.016 Merzkirch, M., & Foecke, T. (2020). 10
Off-axis testing of CFRP using DIC: A study on strength, strain and modulus. Composites Part B: Engineering, 196. https://doi.org/10.1016/j. compositesb.2020.108062 Pierron, F., & Vautrin, A. (1997). Measurement of the in-plane shear strengths of unidirectional composites with the iosipescu test. Composites Science and Technology, 57(12), 1653–1660. Pindera, M. J., & Herakovich, C. T. (1986). Shear characterization of unidirectional composites with the off-axis tension test. Experimental Mechanics, 26(1), 103–112. https://doi.org/10.1007/ Bf02319962 Schürmann, H. (2007). Konstruieren mit Faser-Kunststoff-Verbunden. Konstruieren mit FaserKunststoff-Verbunden. Springer. https://doi.org/10.1007/978-3-540-72190-1 Summerscales, J. (1987). Shear modulus testing of composites. In I. H. Marshall (Ed.), Composite structures 4, Volume 2: Damage assessment and material evaluation (pp. 305–316). Elsevier. https://doi.org/10.1007/978-94-009-3457-3_23 Sun, C. T., & Berreth, S. P. (1988). A new end tab design for off-axis tension test of compositematerials. Journal of Composite Materials, 22(8), 766–779. https://doi.org/10.1177/ 002199838802200805 Zhou, G., Green, E. R., & Morrison, C. (1995). In-plane and interlaminar shear properties of carbon/ epoxy laminates. Composites Science and Technology, 55(2), 187–193. https://doi.org/10.1016/ 0266-3538(95)00100-x

Index

A Accessory loading conditions, 232, 233 C-ELS, 249, 250 ENF, 249, 250 Adhesives, 216, 242, 259 Anisotropic composite materials, 16 Anisotropy, 95 Area method, 210 Asymmetric shear strain distribution, 93

B Back-lit glass calibration target, 218 Boundary condition, 73

C Calibrated end-loaded split (C-ELS) test calibration and crack propagation experiments, 223 CBTE, 214 CC, 213 coordinate system, 225 data reduction, 242–245 degrees of freedom, 224 DIC analysis parameters, 226 DIC hardware parameters, 221, 226 DIC setups and parameters, 223 displacement-controlled C-ELS testing, 214 displacements horizontal displacement U, 252, 253 vertical displacement V, 253–255 DOF, 225 ECM, 213 flexural modulus, 214

FOV, 225 fractography, 245, 246 PC, 213 principle, 213 QOI, 225 ROI, 225 SB method, 214 SCB, 213, 214 self-designed, 222 shear strain distribution, 248 standardization, 213 strains accessory loading conditions, 249, 250 intended loading condition, 245–249 sensitivity analysis, 249 through-thickness distribution, 247 Cantilever beam specimens, 207 Cantilever beam test, 137 Carbon fiber reinforced polymer (CFRP), 2, 52, 80–83 Castigliano’s second theorem, 146 Ceramic matrix composites (CMC), 3, 91 Clamping conditions, 93, 98, 118 Classical lamination theory (CLT), 12, 13, 146 Coefficient of variation (CV), 273–275 Cohesive zone models (CZM), 205 Compliance-based methods, 228, 230 Compliance calibration (CC) method, 210, 212, 213 Constitutive relations, 8, 10, 11 Core materials, 200 Corrected beam theory (CBT), 209 Corrected beam theory using effective crack length (CBTE), 214 Correction factors, 93, 111, 112, 130, 132

© Springer Nature Switzerland AG 2022 M. Merzkirch, Mechanical Characterization Using Digital Image Correlation, https://doi.org/10.1007/978-3-030-84040-2

287

288 Crack growth, 207 Crack initiation, 226 Crack length, 209 Crack opening displacement (COD), 206, 215, 216, 231, 238–241, 253, 255 Crack propagation, 228 Crack shear displacement (CSD), 215, 245, 253 Crack tip opening angle (CTOA), 207 Crack tip opening displacement (CTOD), 207, 238–241, 253 Crack tip shear displacement (CTSD), 253 Crack tip tracking, 215, 231, 232, 236, 237, 259

D Data reduction, 103–105, 133, 153–155, 211 C-ELS test, 242–245 DCB, 226–230 ENF test, 242–245 Degrees of freedom, 96, 97, 100, 121, 133 Delamination resistance testing correction factors, 215 crack opening displacement, 215 elastic flexural properties, 257, 258 energy release rate, 215 flexural modulus, 215 interlaminar fracture properties, 255–257 LEFM, 205–208 mode I (see Mode I) mode II (see Mode II) quasi-static tests, 218 real-time test control, 215 Depth-of-field (DOF), 24, 54, 99, 102, 120, 133, 151, 219 DIC configuration analysis parameters, 153 calibration, 153 coordinate system, 151 DOF, 151 hardware parameters, 151 horizontal stereo-DIC setup, 149 interlaminar shear behavior, 151 lenses, 153 LVDT, 152 quasi-static tests, 149 ROI, 150 sensitivity analysis, 153 virtual strain gauge, 153 DIC parameter uncertainty quantification analysis parameters, 32 boundary conditions, 35 deformation, 33 distribution, 33, 34

Index large strain gradients, 35 measurements, 32 multi-objective decision analysis, 35 optimization, 35 parameter variations, 34 static and spatial noise-floor, 32 strain-based QOI, 33 variance and bias errors, 32 virtual strain gauge, 32, 35 Digital Image Correlation (DIC), 2 adaptability and applicability, 129 advantage, 103 analysis parameters, 99, 102, 112, 125, 130 boundaries, 28 calibration, 98 configuration basic principle, 24 calibration, 24 cameras, 24 deformation, 24 extrinsic camera parameters, 24 FOV, 25 intrinsic parameters, 24 light source, 24 measurement, 24 optical resolution, 24 stereo-DIC setup, 25 data analysis, 108 hardware parameters, 98, 99, 102, 121, 125, 129 interpolation filter, 28 linear affine shape function, 27 matching criterion, 28 measured data, 105 measurements, 114 mechanical stress, 23 notch geometry, 95 parameters, 102 quasi-static tests, 96 reference and deformed images, 28 reference images, 27 ROI, 26, 28 stereo image, 27 strain determination, 94 subset size and step size, 29 subsets, 26 triangulation, 28 uncertainty quantification, 114–118 visualization, 23 V-notched beam test, 94, 96–99 V-notched rail test, 99–102 Displacement-controlled C-ELS testing, 214 Displacement-controlled loading, 206

Index Displacement-controlled tests, 96 Displacements C-ELS horizontal displacement U, 252, 253 vertical displacement V, 253–255 clamps, 99 counteracting, 91 DCB horizontal displacement U, 237 vertical displacement V, 238–242 ENF horizontal displacement U, 252, 253 vertical displacement V, 253–255 horizontal displacement U, 182, 183 out-of-plane displacement W, 120, 121 test fixture, 119, 120 vertical deformation V, 182, 184–187 Double beam shear test, 144–147 Double cantilever beam (DCB) area method, 210 CBT, 209 CC method, 210 composite materials, 207 crack length, 209 data reduction, 226–230 deflection, 207, 208 deformation and delamination behavior, 231 DIC analysis parameters, 226 DIC configuration, 218 DIC hardware parameters, 221, 226 displacements horizontal displacement U, 237 vertical displacement V, 238–242 flexural modulus, 210 FOV, 218, 220 fractography, 230 hinges, 207 MB theory, 209 MCC method, 210 PC method, 209 QOI, 218 ROI, 218 SB theory, 209 shear deflection, 208 stable crack growth, 210 stereo-DIC configuration, 219 strains accessory loading condition, 232, 233 intended loading condition, 231, 232 sensitivity analysis, 233–237 through-thickness strain distribution, 231 2D-DIC setups, 218, 219 woven-CFRP, 258, 259

289 E Edge preparation, 93 Elastic behavior, 63 Elastic deformations, 138 Elastic flexural properties, 257, 258 Elastic properties, 10, 191, 269, 271, 273–275, 278 compressive modulus, 192 flexural and shear properties, 190, 191 flexural moduli, 189, 191 number of layers, 192 off-axis angle, 193 resin rich layers, 192 shear deflection, 189 span-to-thickness ratio, 189 tensile modulus, 189, 191, 192 Elastic region, 138, 162 End-loaded split (ELS) test principle, 213 End-notched flexural (ENF) test calibration and crack propagation experiments, 220 CC, 212 data reduction, 211, 242–245 diameter, 219 DIC analysis parameters, 226 DIC hardware parameters, 221, 226 disadvantage, 221 displacements horizontal displacement U, 252, 253 vertical displacement V, 253–255 DOF, 221 FOV, 221, 223 fractography, 245, 246 horizontal stereo-DIC setup, 221, 222 PC, 211 QOI, 221 ROI, 221–223 SB method, 212 shear deflection, 211 stable crack growth, 220 strains accessory loading conditions, 249, 250 intended loading condition, 245–249 sensitivity analysis, 249 three-point loading, 211 Energy density, 207 Energy release rate, 205, 215 Euler’s differential equation, 138 Euler–Bernoulli beam, 138, 241 Experimental Compliance Method (ECM), 213, 244

290 Extension-shear coupling, 11, 49, 50, 78, 82, 87, 270, 271, 273, 278, 279, 281

F Failure criterion, 272 Failure investigation, 112–114 Fiber orientation, 5, 93–96, 98, 102, 103, 107, 121–129, 131, 133, 266, 269, 271, 273–276, 278, 281–284 Fiber transverse strains, 112–114, 121, 122, 127, 132, 171 Fibrous composite laminates assorted materials midplane twill architecture, 22 UD-GFRP, 22, 23 composite materials, 3 configuration, 3 laminate coordinate system, 4 materials, 3 mechanical characterization motivation, 15, 16 multiple standardized method, 15 shear test methods, 16–18 state-of-the-art test method, 15 mechanical performance, 3 orthogonal principal material axes, 3 papyrus paper, 3 principal material axes, 4 reinforcing structure, 3 structural mechanics CLT, 12, 13 constitutive relations, 8, 10–12 strain, 6, 8, 9 strength and failure, 15 stress, 4–6 test methods, 19 UD-CFRP epoxy, 20 mechanical tensile properties, 20 quasi-static uniaxial tensile testing, 20 representative cross-section, 20 shear modulus G, 21 surface quality, 21 Young’s modulus E, 20 unidirectional (UD) lamina, 3 Field-of-view (FOV), 25, 54, 98, 101, 150, 218, 220, 221, 223, 225 Flexural moduli, 182, 210, 214, 215 delamination resistance testing, 257, 258 Flexural testing accessory loading conditions, 172, 175 advantages, 137

Index bending deflection, 137–141 bending strength, 137–141 DIC, 200 failure investigation, 169–172 interlaminar shear properties, 193, 194 neutral axis, 176, 177 resin, 196, 199 specimen geometry, 147, 148 structural mechanics, 200 testing, 200 Force-or displacement-controlled loading conditions, 206 Force-deflection curve, 153, 154, 182, 226 Force–displacement curve, 206 Force-maximum deflection curve, 187 Fractography, 59, 105, 112, 113, 122–124, 127, 128, 130, 155, 156 C-ELS test, 245, 246 DCB, 230 ENF test, 245, 246 Fracture behavior, 112 Fracture energy, 205 Fracture mechanics development, 205 LEFM, 205–208 Fracture resistance, 206 Full-field deformation measurements, 2 Full-field investigation, 107, 112

G Geometric constant, 146 Geometries, 18 Glass fiber reinforced polymer (GFRP), 47

H Horizontal displacement U, 237 C-ELS, 252, 253 ENF, 252, 253 Hydrodynamics, 138

I In-plane, 2, 15, 93, 94, 97, 99, 107, 121, 265, 269, 277, 278 Instron electromechanical universal testing system, 96 Integration constants, 186 Intended loading condition, 231, 232 C-ELS, 245–249 ENF, 245–249 Interlaminar fracture testing, 205

Index Interlaminar properties, 1, 18 Interlaminar shear modulus, 146, 154, 155 Interlaminar shear properties, 2 Interlaminar shear strength (ILSS), 144 Interlaminar shear stress, 156, 157 Interlaminar shear test methods, 19 Intralaminar shear properties elastic properties, 269, 271 fiber orientation, 269 fibrous composite, 267 flexural testing, 267 inhomogeneous strain distribution, 266 in-plane behavior, 266 maximum shear strain, 266 microscopic scale, 267 nominal stress, 266 non-linear mechanisms, 267 non-linearity, 266, 267 shear stress, 266 short-beam testing, 266, 267 strain measurements, 266 strength, 271, 273 stress-shear strain curves, 266 V-notched beam testing, 269 Inverse ELS testing, 214 Iosipescu shear test, 91 Irwin–Kies equation, 206

L Lamination residual stresses, 18 Law of elasticity, 8 Linear behavior, 154 Linear elastic fracture mechanics (LEFM), 205–208, 241 Linear equation, 190 Linear regression analysis, 190, 191 Linear relation, 138 Linear variable differential transformer (LVDT), 152, 219 Longitudinal ultimate tensile strength (UTSL), 272 Loss in correlation (LiC), 221, 228, 231, 232, 235–237, 239, 243, 246

M Maximum shear stress, 157 Mechanical response data reduction, 56–58, 153–155 fractography, 59, 155, 156 sensitivity analysis shear stress, 156 stress–strain response, 56

291 Metal matrix composites (MMC), 3 Metrology-based lightweighting, 2 Mixed mode, 207, 256, 257 Mixed mode bending (MMB) test, 257 Mixed mode ELS (MMELS), 257 Mode I DCB (see Double cantilever beam (DCB)) multiple cracking, 215 specimen geometry, 216–218 Mode II C-ELS test (see Calibrated end-loaded split (C-ELS) test) crack propagation, 215 crack shear displacement, 215 ENF test (see End-notched flexural (ENF) test) friction, 215 specimen geometry, 216–218 Modified beam (MB) theory, 209 Modified compliance calibration (MCC) method, 210 Mohr’s circle, 9, 104, 134

N Neutral axis, 176, 177 Notch dimensions, 92, 93 Numerical material models, 2

O Out-of-plane displacement W, 120, 121

P Patterning methods characteristics, 38, 39 contrast, 36, 38 digital correlation process, 35 fiber volume fraction, 38 gray level distribution, 36 inhomogeneous lighting, 38 large deformations, 36 line profile and intersection method, 39 overspray-painting, 36 paint-based patterns, 36 printed pattern, 36 spatial resolution, 35 speckle pattern, 35, 36 spray-painted patterns, 38 spray-painted specimens, 38 spray-painting, 36 Photoelastic investigations, 16, 91, 94 Plexiglass, 91

292 Poisson’s ratio, 10, 11, 57, 73, 79, 82, 84, 87, 146, 269, 270, 274, 275, 278, 279, 281–283 Polymer matrix composites (PCM), 1, 3 Post-analysis ROI, 107, 108, 117, 133 Principal coordinate system, 103, 107 Principal strain, 104, 134 Pure compliance (PC) method, 209, 211, 213, 245

Q Quantities-of-interest (QOI), 29, 55, 102, 114, 115, 117, 118, 152, 218, 221, 225, 233, 236, 237 Quarter-point loading, 162 Quasi-static crack propagation, 207 Quasi-static tests, 53, 96, 218

R R-curve, 206, 229, 244, 258, 259 Real-time test control, 215 Region-of-interest (ROI), 25, 54, 98, 101, 102, 107–109, 114, 151, 218, 221–223, 225, 228 Regression analysis methods, 187, 191 Representative unit cell (RUC), 22 Resolution bias and noise-floor, 31, 32 calibration, 30 displacement/strain gradients, 31 in-plane displacements, 31 interpolation function, 31 principal coordinate system, 31 QOI, 31 qualitative sensitivity, 32 spatial, 30 stereo-DIC, 31 strain values, 31 2D-DIC, 31

S Saint-Venant, 55, 61, 86, 94, 125 Self-designed C-ELS, 222 Semi-infinite beam, 241 Sensitivity analysis, 12 bending strain, 178 bias and noise-floor, 178, 181, 182 C-ELS, 249, 250 DCB, 233–237 DIC, 114–118, 121, 125

Index elastic properties, 273–275 ENF, 249, 250 maximum shear strain, 179 shear strain, 178, 180 signal-to-noise ratio, 180 span bending deflection, 159 carbon epoxy systems, 160 deflection-based calculation, 160 flexural moduli, 158, 159 hyperbolic relation, 160 inelastic deformation, 161 interlaminar shear, 157, 159 maximum deflection, 157 maximum stress, 160 mechanisms, 161 shear dominant failure, 160 shear strength, 160 tensile failure, 157 warping/bending failure, 160 static and spatial noise-floor, 177 strain, 106, 107 strength, 276 virtual strain gauge, 177–179 Shear concentrations, 93 Shear deflection, 208, 211 bending moment, 141, 142 correction factor, 142 cross-sections, 142 double beam shear test, 144–147 elasticity, 143 force, 142 forces, 141 influence, 142 maximum shear stress, 143 SBT, 143, 144 stress distribution, 141 three-point bending, 142 Shear loading, 91, 93, 99 Shear moduli, 61, 93, 105, 112, 121, 129, 133, 184, 265, 267, 274, 275, 278, 281–283 Shear properties, 19 Shear strain distribution, 197 Shear strains, 103, 108–112, 129 loading condition, 162 response, 166–168 distributions, 95 types, 104, 105, 130 Shear strength, see Shear deflection Shear stress, 49, 92, 112, 129, 156 distribution, 91 Shear stress-shear strain

Index data reduction, 103–105 deduction, 103 fiber orientations, 121–126 fractography, 105 sensitivity analysis, 106, 107 Shear testing development, 91 setup, 91 Shear test methods, 16–18 Short-beam shear (SBS), 144 Short-beam test (SBT), 143, 144, 209, 212, 214 Simple beam theory, 213 Single cantilever beam (SCB), 207, 208, 211, 213, 214, 216, 223, 228, 238–240, 244, 250, 253, 254, 258 Spatial resolution, 30 Specimen clamping, 93, 101 Specimen geometry, 52, 53, 147, 148 effect, 187, 188 V-notched beam test, 95, 96, 125–128 Stability of crack propagation/growth, 210, 212, 214 Stabilized ENF test, 215 Stable crack growth, 210 Standardized compliance-based methods, 255 Stand-off-distances (SOD), 153 Stereo-DIC, 94, 97, 99–101, 216, 219 Sticker pattern, 98, 102, 119 Strain distribution, 170, 175 Strain gauges, 93, 94, 98, 111 Strains, 6, 8, 9, 206 C-ELS accessory loading conditions, 249, 250 intended loading condition, 245–249 sensitivity analysis, 249 DCB accessory loading condition, 232, 233 intended loading condition, 231, 232 sensitivity analysis, 233–237 distribution, 107 ENF accessory loading conditions, 249, 250 intended loading condition, 245–249 sensitivity analysis, 249 failure investigation, 112–114 location, 106, 107 range, 106, 107 sensitivity analysis, 102, 106–107, 114–118 shear strains, 108–112 type, 106, 107 Stress, 4–6, 206 Stress distribution, 139

293 Stress tensor, 6, 10 Structural components, 1 Surface quality, 21

T Tensile modulus, 176 Tensile properties, 19 Tensile testing advantages, 47 aluminum, 51 anisotropic materials, 50 axial forces, 48 bi-axial tension, 50 boundary conditions, 50 classification, 47 DIC configuration, 53–55 displacements axial displacements U, 70, 71 fiber orientation, 74, 75, 78 lateral displacement V, 72, 73 out-of-plane displacement W, 73, 74 specimen geometry, 74, 75, 78 elastic behavior, 63 elastic material properties, 50, 51 end-constraint effect, 50 experimental setup, 53–55 extensional and shear properties, 82 failure investigation, 64 fiber orientation, 47 fibers, 50 fractography, 82 gauge, 80 GFRP, 52 gradual transition, 51 gripping conditions, 51 hydraulic gripping, 51 in-plane displacements, 62 in-plane extensional and intralaminar shear properties, 78, 79 Mohr's circle, 80 nominal and shear strain distribution, 62 nominal and shear strains, 80 non-rotating grips, 51 non-uniform stress field, 51 nylon reinforced rubber, 52 off-axis loading, 47 off-axis tensile test, 47, 49, 82, 87 off-axis testing, 51 principal material axes, 50 quasi-static tensile properties, 47 sensitivity analysis angularity, 59, 61

294 Tensile testing (cont.) DIC parameter uncertainty quantification, 64, 66, 68, 69 strain location and range, 61, 62 shear strain, 47 shear stress, 47, 49 specimen geometry, 52, 53 strain distribution, 82, 83 stress distribution, 50 structural mechanics, 86 testing, 86 thin tube torsion test, 48 UD-GFRP, 84, 85 uniform shear stress, 51 woven-CFRP, 82 Young’s modulus, 52 Test fixture motion, 118–120, 133 Testing composite materials, 15 Three-point loading, 211 Through-thickness, 1, 2, 5, 16, 19, 205, 231, 245, 247 Through-thickness test methods, 265, 277 Torthotropy, 92 Traction-separation, 205, 240, 241 Transformation matrix, 5 Transverse modulus, 271 Transverse ultimate tensile strength (UTST), 272 Tsai–Hill criterion, 15

U UD-CFRP, 129 compliance, 269, 275, 283 fiber distribution homogeneity, 265 interlaminar shear properties, 265 intralaminar shear properties, 265 material anisotropy, 283 process–structure–property relationships, 265 shear strain, 265 UD-GFRP V-notched rail test, 131, 132 Ultimate shear strength (USS), 58, 79, 81, 91, 105, 112, 121–131, 133, 134, 155, 266, 267, 269, 271, 273, 276, 284 Ultimate tensile strength (UTS), 58, 81, 272, 276, 277, 284 Uncertainty quantification, 64, 66, 69, 177, 179–181 delamination resistance testing, 233–237 DIC, 114–118, 121, 124 Unidirectional (UD) architecture, 2, 53, 84

Index V Vertical displacement V C-ELS, 253–255 DCB, 238–242 ENF, 253–255 Virtual strain gauge (VSG), 29, 55, 66, 80, 102, 112, 114–116, 118, 130, 233, 249 Visualization displacements (see Displacements) strains (see Strains) V-notched beam test advantages, 93 asymmetric shear strain distribution, 93 bending moment, 92 butterfly shaped specimen, 94 ceramic matrix composites, 91 correction factors, 93 DIC (see Digital Image Correlation (DIC)) double notched specimen, 91, 92 fiber orientations, 121–125 intralaminar shear properties, 128, 129 laminates, 91 limitation, 94 multidirectional composite, 94 notches, 92 optimization, 93 photoelastic investigations, 94 in-plane and out-of-plane bending, 93 in-plane shear properties, 94 principle, 91 Saint-Venant effects, 94 shear force, 92 shear strain, 108–112 shear stress, 92 specimen geometry, 95, 96, 125–128 strain gauge, 94 structural mechanics, 133 tensile loading, 94 test fixture, 94 unidirectional composites, 94 uniformity, 92 woven-CFRP, 129–131 V-notched rail test butterfly shaped specimen, 94 DIC, 99–102 in-plane shear properties, 94 intralaminar shear properties, 128, 129 properties, 95 shear strain, 108–112 tensile loading, 94 UD-GFRP, 131, 132

Index W Warp orientation, 130 Waterjet cut V-notched beam, 95, 96 Winkler foundation, 241 Woven-CFRP DCB, 258, 259 DIC measurements, 194 elastic flexural properties, 195 experiment vs. analytical model elastic properties, 278 strength, 278, 280 interlaminar shear properties, 195–198, 277 intralaminar shear properties, 277

295 sensitivity analysis coefficient of variation (CV), 281 elastic properties, 281–283 strength, 283, 284 test method, 194 V-notched beam test, 129–131 Wyoming Test Fixtures Inc., 96, 97, 99

Y Young’s modulus E, 82, 138