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English Pages xvi, 238 pages; 23 cm [258] Year 2018;2019
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites and Hybrid Composites
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Woodhead Publishing Series in Composites Science and Engineering
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites and Hybrid Composites Edited by
Mohammad Jawaid Mohamed Thariq Naheed Saba
Woodhead Publishing is an imprint of Elsevier The Ofﬁcers’ Mess Business Centre, Royston Road, Duxford, CB22 4QH, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, OX5 1GB, United Kingdom Copyright © 2019 Elsevier Ltd. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this ﬁeld are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress CataloginginPublication Data A catalog record for this book is available from the Library of Congress British Library CataloguinginPublication Data A catalogue record for this book is available from the British Library ISBN: 9780081022894 For information on all Woodhead publications visit our website at https://www.elsevier.com/booksandjournals
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Dedicated to Parents of Dr. Mohamed Thariq Sufaidah Binti Mohd MusaMother Haji Hameed Sultan Bin Mohamed SulaimanFather
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Contents
List of contributors About the editors Preface 1
2
3
4
Finite element modeling of natural ﬁberbased hybrid composites A. Karakoti, P. Tripathy, V.R. Kar, K. Jayakrishnan, M. Rajesh and M. Manikandan 1.1 Introduction 1.2 Micromechanical material modeling 1.3 Finite element formulations 1.4 Finite element solutions of natural ﬁberbased hybrid composites 1.5 Conclusions References The effects of cutout on thinwalled plates N. Yidris and M.N. Hassan 2.1 Introduction 2.2 Finite element analysis 2.3 Analysis of results Modeling of crushing mechanisms of hybrid metal/ﬁber composite cylindrical tubes Al Emran Ismail and KamarulAzhar Kamarudin 3.1 Introduction 3.2 Research methodology 3.3 Results and discussion 3.4 Conclusion Acknowledgments References Roles of layers and ﬁber orientations on the mechanical durability of hybrid composites Muhammad Eka Novianta, Al Emran Ismail and KamarulAzhar Kamarudin 4.1 Introduction 4.2 Methodology
xi xv xvii 1 1 5 7 13 16 17 19 19 20 24 27 27 28 33 39 39 39 41 41 43
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Contents
4.3 4.4
5
6
7
Results and discussion Conclusion References Further reading
Numerical modeling of hybrid composite materials Nabil Bouhﬁd, Marya Raji, Radouane Boujmal, Hamid Essabir, MohammedOuadi Bensalah, Rachid Bouhﬁd and Abou el kacem Qaiss 5.1 Introduction 5.2 Classiﬁcation of materials and ﬁller types 5.3 Various modeling techniques of composite mechanical properties 5.4 Numerical modeling of the mechanical behavior of composite material 5.5 Numerical modeling of hybrid composite materials 5.6 Conclusion Acknowledgments References Computationally efﬁcient modeling of woven composites under uniaxial stress KamarulAzhar Kamarudin, Al Emran Ismail, Iskandar Abdul Hamid and Ahmad Suﬁan Abdullah 6.1 Introduction 6.2 Material 6.3 Finite element modeling 6.4 Mesh sensitivity analysis 6.5 Conclusions Acknowledgments References Progressive damage modeling of synthetic ﬁber polymer composites under ballistic impact KamarulAzhar Kamarudin, Mohd Khir Mohd Nor, Al Emran Ismail, Iskandar Abdul hamid and Ahmad Suﬁan Abdullah 7.1 Introduction 7.2 Material properties of carbon ﬁberreinforced plastic 7.3 Finite element modeling using the continuum shell element 7.4 Results and discussion 7.5 Conclusions Acknowledgments References
44 55 56 56 57 57 58 61 83 86 93 93 93 103 103 104 106 109 112 113 113 115 115 118 119 124 131 131 132
Contents
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9
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Investigation of damage processes of a microencapsulated selfhealing mechanism in glass ﬁberreinforced polymers J. Lilly Mercy and S. Prakash 8.1 Introduction 8.2 Chemistry of capsulebased selfhealing materials 8.3 Background to the study 8.4 Fabrication process 8.5 Experimental plan 8.6 Testing 8.7 Study of the effect of a selfhealing agent on mechanical properties 8.8 Study of the effect of selfhealing agent on dynamic mechanical properties 8.9 Microstructural analysis 8.10 Discussion and conclusion References Finite element analysis of natural ﬁberreinforced polymer composites J. Naveen, Mohammad Jawaid, A. Vasanthanathan and M. Chandrasekar 9.1 Introduction 9.2 Basic steps in ﬁnite element analysis 9.3 Finite element analysis of polymer matrix composites 9.4 An overview of ﬁnite element analysis of natural ﬁberreinforced polymer composites 9.5 Finite element analysis of natural ﬁber and natural ﬁberreinforced polymer composites 9.6 Conclusion Notations Acknowledgments References Modeling shock waves and spall failure in composites as an orthotropic materials Mohd Khir Mohd Nor, N. Ma’at, H.C. Sin and M.S.A. Samad 10.1 Introduction 10.2 Constitutive formulation 10.3 Results and analysis 10.4 Conclusion Acknowledgments References Further reading
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133 133 133 134 135 137 138 142 144 147 149 150 153 153 155 157 159 161 167 168 168 168 171 171 173 187 195 195 195 198
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Contents
TOPSIS method for selection of best composite laminate M.R. Sanjay, Mohammad Jawaid, N.V.R. Naidu and B. Yogesha 11.1 Introduction 11.2 TOPSIS method 11.3 Methodology adopted 11.4 Results and discussion 11.5 Conclusion References Deformation characteristics of functionally graded composite panels using ﬁnite element approximation V.R. Kar, S.K. Panda, P. Tripathy, K. Jayakrishnan, M. Rajesh, A. Karakoti and M. Manikandan 12.1 Introduction 12.2 Micromechanical material modeling 12.3 Finite element approximations 12.4 Results and discussions 12.5 Conclusions References
Index
199 199 200 201 202 208 208 211 211 212 214 216 228 228 231
List of contributors
Iskandar Abdul Hamid Crash Reconstruction Unit, Vehicle Safety & Biomechanics Research Centre, Malaysian Institute of Road Safety Research, Kajang, Malaysia Ahmad Suﬁan Abdullah ARTeC, Faculty of Mechanical Engineering, Universiti Teknologi MARA, Permatang Pauh, Malaysia MohammedOuadi Bensalah Rabat, Morocco Nabil Bouhﬁd Morocco
Mohammed VRabat University, Faculty of Science,
Mohammed VRabat University, Faculty of Science, Rabat,
Rachid Bouhﬁd Moroccan Foundation for Advanced Science, Innovation and Research (MAScIR), Institute of Nanomaterials and Nanotechnology (NANOTECH), Laboratory of Polymer Processing, Rabat, Morocco Radouane Boujmal Morocco
Mohammed VRabat University, Faculty of Science, Rabat,
M. Chandrasekar Department of Aerospace Engineering, Universiti Putra Malaysia, Serdang, Malaysia Hamid Essabir Moroccan Foundation for Advanced Science, Innovation and Research (MAScIR), Institute of Nanomaterials and Nanotechnology (NANOTECH), Laboratory of Polymer Processing, Rabat, Morocco M.N. Hassan Malaysia
Aerospace Engineering Department, Universiti Putra Malaysia,
Al Emran Ismail Crashworthiness and Collisions Research Group (COLORED), Mechanical Failure Prevention and Reliability Research Centre (MPROVE), Faculty of Mechanical and Manufacturing Engineering, Universiti Tun Hussein Onn Malaysia, Batu Pahat, Johor, Malaysia Mohammad Jawaid Laboratory of Biocomposite Technology, Institute of Tropical Forestry and Forest Products (INTROP), Universiti Putra Malaysia, Serdang, Malaysia K. Jayakrishnan
School of Mechanical Engineering, VIT, Vellore, India
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List of contributors
KamarulAzhar Kamarudin Crashworthiness and Collisions Research Group (COLORED), Mechanical Failure Prevention and Reliability Research Centre (MPROVE), Faculty of Mechanical and Manufacturing Engineering, Universiti Tun Hussein Onn Malaysia, Batu Pahat, Johor, Malaysia V.R. Kar
Department of Mechanical Engineering, NIT, Jamshedpur, India
A. Karakoti
School of Mechanical Engineering, VIT, Vellore, India
N. Ma’at Crashworthiness and Collisions Research Group (COLORED), Mechanical Failure Prevention and Reliability Research Centre (MPROVE), Faculty of Mechanical and Manufacturing Engineering, Universiti Tun Hussein Onn Malaysia, Batu Pahat, Johor, Malaysia M. Manikandan Department of Mechanical Engineering, Amrita College of Engineering and Technology, Nagercoil, Tamil Nadu, India J. Lilly Mercy School of Mechanical Engineering, Sathyabama Institute of Science and Technology, Chennai, India N.V.R. Naidu Visvesvaraya Technological University, Belagavi, India; Department of Industrial Engineering & Management, Ramaiah Institute of Technology, Bengaluru, India J. Naveen Department of Mechanical and Manufacturing Engineering, Universiti Putra Malaysia, Serdang, Malaysia Mohd Khir Mohd Nor Crashworthiness and Collisions Research Group (COLORED), Mechanical Failure Prevention and Reliability Research Centre (MPROVE), Faculty of Mechanical and Manufacturing Engineering, Universiti Tun Hussein Onn Malaysia, Batu Pahat, Johor, Malaysia; Centre for General Studies and CoCurricular, Universiti Tun Hussein Onn Malaysia, Batu Pahat, Johor, Malaysia Muhammad Eka Novianta Crashworthiness and Collision Research Group (COLORED), Mechanical Failure Prevention and Reliability Research Centre (MPROVE), Faculty of Mechanical and Manufacturing Engineering, Universiti Tun Hussein Onn Malaysia, Batu Pahat, Malaysia S.K. Panda
Department of Mechanical Engineering, NIT, Rourkela, India
S. Prakash School of Mechanical Engineering, Sathyabama Institute of Science and Technology, Chennai, India Abou el kacem Qaiss Moroccan Foundation for Advanced Science, Innovation and Research (MAScIR), Institute of Nanomaterials and Nanotechnology (NANOTECH), Laboratory of Polymer Processing, Rabat, Morocco M. Rajesh School of Mechanical Engineering, VIT, Vellore, India
List of contributors
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Marya Raji Mohammed VRabat University, Faculty of Science, Rabat, Morocco; Moroccan Foundation for Advanced Science, Innovation and Research (MAScIR), Institute of Nanomaterials and Nanotechnology (NANOTECH), Laboratory of Polymer Processing, Rabat, Morocco M.S.A. Samad Department of Computer Aided Engineering, Vehicle Development and Engineering, Perusahaan Otomobil Nasional Sdn Bhd, Shah Alam, Selangor, Malaysia M.R. Sanjay Department of Mechanical Engineering, Ramaiah Institute of Technology, Bengaluru, India; Visvesvaraya Technological University, Belagavi, India H.C. Sin Crashworthiness and Collisions Research Group (COLORED), Mechanical Failure Prevention and Reliability Research Centre (MPROVE), Faculty of Mechanical and Manufacturing Engineering, Universiti Tun Hussein Onn Malaysia, Batu Pahat, Johor, Malaysia P. Tripathy Department of Mechanical Engineering, NIT, Rourkela, India A. Vasanthanathan Department of Mechanical Engineering, Mepco Schlenk Engineering College, Sivakasi, India N. Yidris
Aerospace Engineering Department, Universiti Putra Malaysia, Malaysia
B. Yogesha Visvesvaraya Technological University, Belagavi, India; Department of Mechanical Engineering, Malnad College of Engineering, Hassan, India
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About the editors
Dr. Mohammad Jawaid is currently working as a fellow researcher (associate professor) at the Biocomposite Technology Laboratory, Institute of Tropical Forestry and Forest Products (INTROP), Universiti Putra Malaysia (UPM), Serdang, Selangor, Malaysia, and has also been a visiting professor at the Department of Chemical Engineering, College of Engineering, King Saud University, Riyadh, Saudi Arabia, since June 2013. He is also a visiting scientist at the TEMAG Laboratory, Faculty of Textile Technologies and Design at Istanbul Technical University, Turkey. He has more than 14 years of experience in teaching, research, and industries. His area of research interests includes hybrid reinforced/filled polymer composites, advance materials, (graphene/nanoclay/fireretardant, lignocellulosic reinforced/filled polymer composites), modification and treatment of lignocellulosic fibers and solid wood, biopolymers and biopolymers for packaging applications, nanocomposites and nanocellulose fibers, and polymer blends. So far, he has published 20 books, 45 book chapters, more than 250 peerreviewed international journal papers, and 5 review papers under the top 25 hot articles in Science Direct during 2013e18. Dr. Jawaid worked as a guest editor for special issues for Current Organic Synthesis and Current Analytical Chemistry, Bentham Publishers, UK; International Journal of Polymer Science, Hindawi Publishing; Inderscience Enterprises Ltd.; and IOP Conference Proceedings. He is an editorial board member of the Journal of Asian Science Technology and Innovation and Recent Innovations in Chemical Engineering Journal. In addition, he is also a reviewer of several highimpact international peerreviewed journals for Elsevier, Springer, Wiley, Saga, etc. Presently, he is supervising 18 PhD students (6 PhD as main supervisor and 12 as a member of the supervisory committee) and 8 Master’s students (3 Master’s as main supervisor and 5 as a member of the supervisory committee) in the field of hybrid composites, green composites, nanocomposites, natural fiberreinforced composites, nanocellulose, etc. Additionally, 13 PhD (3 PhD as main supervisor and 10 as a member of the supervisory committee) and 6 Master’s students (2 Master’s as main supervisor and 4 as a member of the supervisory committee) have graduated under his supervision from 2014 to 2018. He has several research grants at university, national, and international levels on polymer composites of around RM 3 million (USD 700,000). He has also delivered plenary and invited talks at international conferences related to composites in India, Turkey, Malaysia, Thailand, UK, France, Saudi Arabia, and China. Also, he is a member of technical committees of several national and international conferences on composites and material science. His Hindex is 40 (Google Scholar) and 35 (Scopus).
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About the editors
Assoc. Prof. Ir. Ts. Dr. Mohamed Thariq Bin Haji Hameed Sultan is a professional engineer (PEng) registered under the Board of Engineers Malaysia (BEM), a professional technologist (PTech) registered under the Malaysian Board of Technologists, a charted engineer (CEng) registered with the Institution of Mechanical Engineers, UK, and is currently attached to the Universiti Putra Malaysia as the Head of the Biocomposite Technology Laboratory, Institute of Tropical Forestry and Forest Products (INTROP), UPM Serdang, Selangor, Malaysia. Being Head of the Biocomposite Technology Laboratory, he is also appointed as an independent scientiﬁc advisor to the Aerospace Malaysia Innovation Centre (AMIC) based in Cyberjaya, Selangor, Malaysia. He received his PhD. from the University of Shefﬁeld, UK. He has about 10 years of experience in teaching as well as in research. His area of research interests includes hybrid composites, advance materials, structural health monitoring, and impact studies. So far he has published more than 100 international journal papers and received many awards locally and internationally. In December 2017, he was awarded a Leaders in Innovation Fellowship (LIF) by the Royal Academy of Engineering (RAEng), UK. He is also the Honourable Secretary of the Malaysian Society of Structural Health Monitoring (MSSHM) based in UPM Serdang, Selangor, Malaysia. Currently, he is also attached to the Institution of Engineers Malaysia (IEM) as Deputy Chairman in the Engineering Education Technical Division (E2TD). Dr. Naheed Saba completed her PhD. in Biocomposites Technology from the Institute of Tropical Forestry and Forest Products (INTROP), Universiti Putra Malaysia, Serdang, Selangor, Malaysia, in 2017. She completed her Master’s in Chemistry and also her postgraduate diploma in Environment and Sustainable Development from India. She has published over 40 scientiﬁc and engineering articles in advanced composites. She edited one book from Elsevier and also published more than 15 book chapters for Springer, Elsevier, and Wiley publications. She has also attended a few international conferences and presented research papers. Her research interest areas are nanocellulosic materials, ﬁreretardant materials, natural ﬁberreinforced polymer composites, biocomposites, hybrid composites, and nanocomposites. She is a recipient of the International Graduate Research Fellowship, UPM. She is a reviewer of several international journals such as Cellulose, Constructions and Building Materials, Journal of Materials Research and Technology, BioResources, Carbohydrate Polymers, etc. Her H Index is 14.
Preface
Modeling of damage processes of biocomposites, ﬁberreinforced composites, and hybrid composites is part of a subseriesdTesting, Modeling and Analysis Under Composite Science and Technology Series. The modeling of damage to composite materials is necessary when considering automotive, aerospace, construction, and building components. This book ﬁlls the gap in the published literature on the modeling of damage to biocomposites, ﬁberreinforced composites, and hybrid composites and provides a reference material for future research in natural and hybrid composite materials, which are currently in great demand due to their sustainable, recyclable, and ecofriendly nature as required in different applications. The book is focused on ﬁnite element modeling and damage modeling of natural ﬁbers, synthetic ﬁbers, and hybrid material composites and covers upto date techniques for aerospace, automotive, and construction applications. This book covers topics such as ﬁnite element modeling of natural ﬁberbased hybrid composites, the effect of cutout on thin wall plates, directional damage gradient modeling of polymeric composites, micromechanical modeling of damage to and partially debonded interfaces of polymer composite materials, numerical modeling of hybrid composite materials, computationally efﬁcient modeling of woven composites under uniaxial stress, progressive damage modeling of natural/synthetic ﬁber polymer composites under ballistic impact, investigation of damage processes of microencapsulated selfhealing mechanisms in glass ﬁberreinforced polymers, element analysis of natural ﬁberreinforced polymer composites, modeling of shock waves and spall failure in composites as orthotropic materials, and the TOPSIS method for selection of best composite laminate. We are highly thankful to all authors from different parts of world who have contributed book chapters to this edited book and supported it by providing valuable ideas and knowledge. We would also like to appreciate their proﬁciency in scattered information from diverse ﬁelds in modeling of damage to biocomposites, ﬁbrereinforced composites, and hybrid composites and in accepting editorial suggestions to produce this venture that we will hope will be a success. We are also grateful to the Elsevier, UK, support team, especially Gwen Jones, Thomas Poulouse, and Sandhya for helping us to ﬁnalize this book. Mohammad Jawaid Universiti Putra Malaysia, Serdang, Malaysia Mohamed Thariq Universiti Putra Malaysia, Serdang, Malaysia Naheed Saba Universiti Putra Malaysia, Serdang, Malaysia
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Finite element modeling of natural ﬁberbased hybrid composites
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A. Karakoti 1 , P. Tripathy 2 , V.R. Kar 3 , K. Jayakrishnan 1 , M. Rajesh 1 , M. Manikandan 4 1 School of Mechanical Engineering, VIT, Vellore, India; 2Department of Mechanical Engineering, NIT, Rourkela, India; 3Department of Mechanical Engineering, NIT, Jamshedpur, India; 4Department of Mechanical Engineering, Amrita College of Engineering and Technology, Nagercoil, Tamil Nadu, India
1.1
Introduction
Humans have known the existence of composite materials for several thousand years and applied the innovation for improving the quality of life. Natural ﬁbers were used for the ﬁrst time 3000 years ago in composite systems in ancient Egypt. However, it is not clear how humankind discovered that mud bricks can be used for construction purposes if they are lined with straw in a regular pattern. This method was used to construct buildings that have lasted for a fairly long time. Approximately in the early 1900s, many resources for producing goods and technical products were obtained from natural ﬁbers. Textiles, ropes, canvas, paper, etc., were made from locally available natural ﬁbers. In 1908, for the ﬁrst time, composite materials were fabricated on a large scale for sheets, tubes, and pipes. In 1896, fuel tanks and seats were made of natural ﬁbers with polymeric binders [1]. The automobile industry has always faced numerous challenges to improve fuel economy, ergonomics, and performance, while keeping in mind vehicle emissions. Reducing weight plays a vital role in improving fuel economy. The possibility of substituting conventional materials to improve performance and ergonomics has encouraged the development of polymer composites speciﬁcally for applications in the automobile industry. The growing automobile industry demands cost reductions that can only be possible by selecting suitable materials [1]. However, when it comes to automobile engineering, reducing weight alone will not solve the problems, even though it is required from the point of view of inertia. The use of natural ﬁberreinforced composites in automobile applications is expected to improve mechanical properties, stability, machining, building safety design, reliability, etc. There must be coordination between those who design composite materials and those who design and manufacture engineering components to obtain the desired output [2].
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites and Hybrid Composites https://doi.org/10.1016/B9780081022894.000011 Copyright © 2019 Elsevier Ltd. All rights reserved.
2
1.1.1
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
Natural ﬁbers (or cellulosebased ﬁbers)
“Natural ﬁbers” is a distinctive term that indicates various ﬁbers that are naturally obtained by plants, animals, and minerals (Fig. 1.1). India has abundant availability of natural ﬁbers such as jute, nettle, sisal, banana, etc., and always focuses on the development of natural ﬁber composites due to their vast application in different industries. These types of natural ﬁber composites are well suited for the replacement of various materials such as wood, plastic, glass ﬁber, etc. Natural ﬁber composites are known to be very costeffective materials for various industries such as building, packaging, automobiles, railway coaches, etc. With the increase in global energy crises and environmental risk, the unique advantages of plant ﬁbers such as abundance, nontoxicity, nonirritation of the skin, eyes, or respiratory system, and noncorrosive properties, plantbased ﬁberreinforced polymer composites have attracted much interest because of their potential to serve as alternative reinforcements for synthetic materials [3]. Plant ﬁbers are the most popular natural ﬁbers used for reinforcement in natural ﬁber composites. Plant ﬁber includes stem ﬁber, leaf ﬁber, seed, fruit, wood straw, and other grass ﬁbers. The chemical composition and its constituents are fairly complicated. Plant ﬁbers are a kind of composite material made by nature. Fibers are basically comprised of cellulose, hemicellulose, lignin, pectin, waxes, and several watersoluble compounds; here cellulose, hemicellulose, and lignin are the major constituents. Most plant ﬁbers contain 65%e70% of cellulose, which is the main constituent of plant ﬁbers and has relatively high modulus and ﬁbril component. Nanoﬁllers are highly potential materials that can improve the mechanical or physical properties of polymer composites. Nanoﬁllers have various advantages over other
Natural fiber
Vegetable (cellulose or lignocellulose)
Seed
Bast (flax, jute)
(cotton)
Fruit (coir)
Animal
Stalk (wheat)
Leaf (pineapple)
Figure 1.1 Classiﬁcation of natural ﬁbers [4].
Wool/hair?
Mineral
Silk
Asbestos wollastonite
Finite element modeling of natural ﬁberbased hybrid composites
3
reinforcing materials such as high interfacial area, homogeneous dispersion in a matrix, gradual load transfer capabilities, as well as enhancing thermal stability and mechanical properties [5,6]. Generally, very few macro or microparticles are present in the plastic zone but nanoﬁllers are present in the minor zone; this improves the fracture and mechanical properties of the matrix, which is brittle in nature [7]. Nanoﬁllers could be organic or inorganic in nature. Nanoparticles such as silica (SiO2), titanium dioxide (TiO2), carbon nanotubes, etc., are inorganic; however, organic ﬁllers such as cellulose nanoﬁbres/whiskers can be extracted from the raw ﬁbers using chemical or mechanical techniques. There are several advantages of natural ﬁbers over other reinforcements: • • • •
They are renewable raw materials and their availability is more or less unlimited. When natural reinforced plastics are subjected at the end of their life cycle to a combustion process or landﬁll, the amount of CO2 released from the ﬁbers is neutral with respect to the assimilated amount during their growth. The abrasive nature of natural ﬁbers is much lower compared to that of glass ﬁbers, which leads to advantages regarding machining, material recycling, or the processing of composite materials in general. Natural ﬁberreinforced plastics using biodegradable polymers as matrix materials are the most environmentally friendly materials and can be composted at the end of their life cycle.
1.1.1.1
Mechanical properties
Natural ﬁbers are suitable for reinforcement because of their relative high strength, stiffness, and low density. Natural ﬁbers can be processed in different ways to yield reinforcing elements having different mechanical properties. Their hydrophilic nature is a major problem for all cellulose ﬁbers if used as reinforcement in plastics. The moisture content of the ﬁbers, dependent on the content of noncrystalline parts and void content of the ﬁber, amounts to 10 wt% under standard conditions [8]. The hydrophilic nature of natural ﬁbers inﬂuences the overall mechanical properties as well as other physical properties of the ﬁber itself [9].
1.1.1.2
Chemical composition
Climatic conditions, age, etc., inﬂuence not only the structure of ﬁbers but also their chemical composition. The components of natural ﬁbers are cellulose, hemicellulose, lignin, pectin, waxes, and watersoluble substances, with cellulose, hemicellulose, and lignin as the basic components with regard to the physical properties of the ﬁbers. On average, different kinds of natural ﬁbers contain 60%e80% of cellulose, 5%e20% of lignin, and up to 20% moisture [10]. The cell wall consists of a hollow tube with four different layers: one primary cell wall, three secondary cell walls,
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
and a lumen, which is an open channel in the center of the microﬁbril. Each layer is composed of cellulose embedded in a matrix of hemicellulose and lignin, a structure that is analogous to artiﬁcial ﬁberreinforced composites. Hemicellulose is made up of highly branched polysaccharides, including glucose, mannose, galactose, xylose, and others [11]. Lignin is made up of aliphatic and aromatic hydrocarbon polymers positioned around ﬁbers. The cell wall differs widely between different species and between different parts of the plants. The strength and stiffness of the ﬁbers are provided by cellulose components via hydrogen bonds and other linkages. Hemicellulose is responsible for biodegradation, moisture absorption, and thermal degradation of the ﬁbers. On the other hand, lignin (pectin) is thermally stable, but responsible for UV degradation of the ﬁbers [4].
1.1.2
Composite materials
Composites consist of two or more chemically distinct constituents, with different chemical and physical properties, having a distinct interface separating them. One or more discontinuous phases therefore are embedded in a continuous phase to form a composite. The discontinuous phase is usually harder and stronger than the continuous phase and is called the reinforcement, whereas the continuous phase is termed the matrix. The matrix material can be metallic, polymeric or ceramic. The signiﬁcance of composite materials development is that scientists and engineers are able to place strong ﬁbers in exact orientations, in exact places, and with the exact amount of volume fraction to achieve the desired properties [2]. A major driving force behind the development of composites has been to produce materials with improved speciﬁc mechanical properties over existing materials. Also, composites are useful in applications where the environment would be detrimental to other materials. Cost is ever present in the engineering equation and it is the balance of cost and performance that determines whether or not a composite will be preferred over an alternative structural material [12]. The main advantages of composites over conventional materials are as follows: • • • • • • •
High strength. Lower weight. Great freedom of shape or ﬂexibility in design. High dielectric strength. Dimensional stability and fatigue endurance. Corrosion and environmental resistance. Low tooling costs.
1.1.3
Hybrid composites
Natural ﬁber has poor ﬁber/matrix adhesion, low durability, and lower water resistance capabilities. The mechanical properties of natural ﬁberreinforced composites are
Finite element modeling of natural ﬁberbased hybrid composites
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affected due to the hydrophilic nature of natural ﬁber. However, several factors such as climatic conditions, maturity, harvesting time and conditions, the retting process, decortications, disintegration, ﬁber modiﬁcation, textiles, etc., are responsible for the characteristic properties of ﬁbers [13]. Nevertheless, various coupling agents as well as surface modiﬁcation techniques have been utilized to improve ﬁber/matrix compatibility [14]. Surface modiﬁcation can be achieved mechanically or chemically but homogeneous distribution, orientation, adhesion, and aspect ratio also affect the mechanical properties. However, in composite fabrication, it is difﬁcult to achieve the homogeneous distribution of the ﬁller materials. A hybrid composite is a good substitute and excludes these disadvantages. A hybrid composite can be developed by reinforcing any one ﬁller in a mixture of two different matrices, or reinforcing two or more ﬁlling materials in a single matrix, or both approaches can be clubbed together [10,15,16]. Hybridization leads to a reduction in water absorption and enhances the mechanical properties due to the combined effect of nanoﬁller and natural ﬁber when reinforced in a matrix [17].
1.2
Micromechanical material modeling
Generally, the mechanical or physical properties of a natural ﬁberreinforced composite can almost emulate the properties of glassﬁberreinforced composites but properties can be improved by using hybrid composites [8]. Parameters such as volume fraction of ﬁber, ﬁber length, orientation, ﬁber/matrix adhesion, lamination scheme, failure strain of individual ﬁbers, etc., affect the properties of hybrid composites. Highperformance composites can be fabricated using natural ﬁbers with easy availability [18,19]. The selection of components for hybridization depends on the required properties of the material to be obtained. It is a major task to blend one natural ﬁber with another natural ﬁber to obtain a material that can reduce the cost of the end product [20]. By the simple rule of mixture, the properties of a hybrid system can be deﬁned as follows [10]: Ph ¼ P1 V1 þ P2 V2
(1.1)
V1 þ V2 ¼ 1
(1.2)
where Ph is the property of the hybrid component, P1 and P2 are the corresponding properties of the different components such as longitudinal modulus E1 and Poisson’s ratio n12, and V1 and V2 are the volume fractions of the ﬁrst and second components, respectively [21].
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
Longitudinal modulus E1 can be calculated for different volume fractions for any combination of composite. Banerjee and Sankar [22] observed that with the increase in volume fraction, E1 also increases linearly when the volume fraction of carbon was varied from 0 to 0.6. Then E1 for different ratios was obtained by the rule of hybrid mixture: E1 ¼ EfA VfA þ EfB VfB þ .. þ EM VM
(1.3)
where EfA and EfB are the modulus of different natural ﬁbers A and B, respectively, which are used for reinforcing, EM is the modulus for the matrix material, VfA and VfB are volume fractions of natural ﬁbers A and B, respectively, and VM is the volume fraction of the matrix. Reinforcing ﬁbers could be two or more than two and the relation would change accordingly. A and B could be any natural ﬁbre such as jute, banana, hemp, ﬂax, etc. However, it is difﬁcult to get the transverse modulus from Eq. (1.3). The transverse strength varies with the variation in the location of ﬁbers in the case of hybrid composites. HalpineTsai equations are very helpful in determining the transverse and shear moduli when the properties of the individual components in a hybrid composite are known. So, the transverse modulus E2 can be determined using the HalpineTsai equation [22,23] as: 1 þ laVf E2 ¼ EM 1 aVf
(1.4)
ððE =E Þþ1Þ
where a ¼ ððEff =EMM ÞþlÞ, Vf is the volume fraction of the reinforcing ﬁber and l depends on the packing arrangement of the ﬁber. However, Eq. (1.4) is only applicable for composites having only one type of reinforcement. The foregoing equations can be modiﬁed, which would contain the volume fraction of different ﬁllers and it will be suitable for hybrid composites. The modiﬁed relation is as follows: 1 þ lðaA VfA þ aB VfB Þ E2 ¼ 1 ðaA VfA þ aB VfB Þ EM
(1.5)
where: aA ¼
ððEfA =EM Þ þ 1Þ ððEfB =EM Þ þ 1Þ and aB ¼ . ððEfA =EM Þ þ lÞ ððEfB =EM Þ þ lÞ
Here, subscripts A and B represent the two different natural ﬁbers, l is the curveﬁtting parameter, and l ¼ 1.165 is the optimum value obtained by the least squares method [22].
Finite element modeling of natural ﬁberbased hybrid composites
7
Banerjee and Sankar [22] also found that Poisson’s ratio varies linearly as the volume fraction increases gradually. Therefore from the simple rule of mixture, the following relation can be written for Poisson’s ratio: n12 ¼ n12fA VfA þ n12fB VfA þ nM VM
(1.6)
where n12fA and n12fB are the Poisson’s ratio of any natural ﬁber A and B and nM is the Poisson’s ratio of the matrix. Similarly, shear moduli G12, G13 and G23 can be predicted using modiﬁed Halpine Tsai equations as follows: 1 þ lðaA VfA þ aB VfB Þ G ¼ GM 1 ðaA VfA þ aB VfB Þ
(1.7)
where: aA ¼
ððGfA =GM Þ þ 1Þ ððGfB =GM Þ þ 1Þ and aB ¼ . ððGfA =GM Þ þ lÞ ððGfB =GM Þ þ lÞ
For every shear moduli, aA and aB have to be calculated separately. l can be taken as 1.01 for G12 and G13 and 0.9 for G23.
1.3
Finite element formulations
Fiberreinforced hybrid composites of various geometries are designed and used as structural components in many weightsensitive and highperformance engineering applications. Not only during manufacturing but also throughout the service period, these structural components are subjected to uniform loading and exhibit large amplitude bending and vibration. Altogether, the material constitutive relation of laminated structure and their vibration, bending and stability behavior are affected adversely. To solve the complex problems, various approximate techniques such as the ﬁnite difference method, ﬁnite element method (FEM), meshfree method, etc. have been utilized in the past to evaluate the desired responses by incorporating them into reallife situations. Out of all approximated analyses, FEM has dominated engineering computations since its invention and has also expanded into a variety of engineering ﬁelds. In this regard, a general mathematical model can be developed based on the classical laminated plate theory, ﬁrstorder shear deformation theory, or higherorder shear deformation theory (HSDT) midplane kinematics. Some of the past studies also indicate that moisture and temperature effects should be taken into consideration while analyzing composite structures, especially ﬁberreinforced composite panels. Panda and his collaborators [24e28] developed a nonlinear ﬁnite element model using HSDT midplane kinematics and Greene Lagrange strain terms. They analyzed the deformation and frequency characteristics
8
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
of a laminated composite singly/doubly curved shell panel under individual and/or combined (hygrothermomechanical) loading conditions. Katariya et al. [29] observed that the higherorder kinematic model is not only cost effective but also easy to implement without affecting accuracy when implemented on a sandwich composite panel to obtain frequency responses. Sharma et al. [30,31] investigated the vibroacoustic behavior of a laminated composite ﬂat panel in an inﬁnite rigid bafﬂe using HSDT.
1.3.1
Displacement ﬁeld
In this study, a hybrid composite plate consisting of N numbers of uniformly thick orthotropic layers with length a, width b and thickness h is considered (Fig. 1.2). Zk is the top and Zke1 is the bottom surface in z coordinates of kth lamina. The midsurface of the shell is assumed to be the reference surface (z ¼ 0). The kinematic model in the HSDT framework is assumed to deﬁne the midplane deformation behavior of ﬁberreinforced laminated composite panels under uniform pressure. So, the displacements at any point for an arbitrary laminated panel with respect to the midplane and along the x, y and z directions is given by: 9 uðx; y; z; tÞ ¼ u þ zf1 þ z2 j1 þ z3 q1 > > =
h h z vðx; y; z; tÞ ¼ v þ zf2 þ z j2 þ z q2 > 2 2 > ; wðx; y; tÞ ¼ w 2
3
(1.8)
a
b
y, 2 z, 3
h/2 h/2 Midplane
Z x, 1 N
K
2 1
X Zk1 Zk
Figure 1.2 Geometry and stacking sequence of a hybrid composite plate.
Finite element modeling of natural ﬁberbased hybrid composites
9
where ðu; v; wÞ are the displacements at any point of the panel along the (x, y, z) coordinates, (u, v, w) are the displacements associated with a point on the midplane of the panel, f1 and f2 are the rotations about the y and x axes, respectively, and t is the time. The functions of j1, j2, q1, and q2 are the higherorder terms of Taylor series expansion deﬁned at the midplane to account for the parabolic distribution of shear stress across the thickness.
1.3.2
Strainedisplacement relation
The strainedisplacement relations of the shear deformable hybrid composite panel can be expressed as: 8 9 8 9 ε1 > > > > u; x > > > > > > > > > > > > > > > > > > > > ε 2> > > > v; y > > > > < = < = fεg ¼ ε6 ¼ u; y þ v; x > > > > > > > > > > u; z þ w; x > > > >ε > > > > > > 5 > > > > > > > > > > > > : ; : v; z þ wy ; ε4
(1.9)
where {ε} is the global strain tensor. Substituting Eq. (1.8) into Eq. (1.9) the strainedisplacement relations of the hybrid composite panel in terms of individual strain terms are expressed as: 8 9 0 1 2 2 3 3> > > ε þ zk þ z k þ z k > > 1 1 1 1> > > > > > > > 0 1 2 2 3 3 > ε2 þ zk2 þ z k2 þ z k2 > > > > > > < = 0 1 2 2 3 3 fεg ¼ ε6 þ zk6 þ z k6 þ z k6 . > > > > > > ε0 þ zk 1 þ z2 k2 þ z3 k 3 > > > > > > 5 5 5 5 > > > > > > > 0 1 2 2 3 3 : ε þ zk þ z k þ z k > ; 4
4
4
(1.10)
4
By separating the thickness coordinates and midplane displacements the ﬁnal equation becomes: fεg ¼ ½Hfεg where fεg ¼
n
ε01 ε02 ε06 ε05 ε04 k11 k21 k61 k51 k41 k12 k22 k62 k52 k42 k13 k23 k63 k53 k43
(1.11) oT is the linear
midplane strain term that is the function of the x and y coordinates and the linear thickness coordinate matrices. The superscripts 0e3 in the individual midplane strain terms stand for the extension, bending, curvature, and higherorder terms, respectively.
10
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
1.3.3
Constitutive relations
The desired elastic equation for any general kth orthotropic composite lamina with ﬁber orientation angle is given by: h ik fsij gk ¼ Qij fεij g
(1.12) T
ε 6 ε 5 ε 4 gT h ik are the stress and strain tensors, respectively, for any kth layer and Qij is the where fsij gk ¼
s1
s2
s6
s5
s4
and fεij gk ¼ f ε1
ε2
transformed reduced elastic constant. So, Eq. (1.12) can also be rewritten as: 2 6 Q11 6 6 Q21 6 6 k fsij g ¼ 6 Q16 6 6 0 6 4 0
Q12
Q16
0
Q22
Q26
0
Q26
Q66
0
0
0
Q55
0
0
Q45
3 k 8 9k ε1 > > > 0 7 > > > > > > 7 > > ε > 2> > 0 7 < > = 7 > 7 . 0 7 ε6 > 7 > > > > > 7 Q54 7 > > ε5 > > > > 5 > > > > Q44 : ; ε4
(1.13)
The total strain energy of the hybrid composite panel can be expressed as: 1 U¼ 2
8 ZZ > N1 ¼ x x h h N2 ¼ x þ x h h > > > 4 4 > > > > > > 1 1 > 2 2 2 2 N4 ¼ x x h þ h > N3 ¼ x þ x h þ h > > > 4 4 > > = 2 . 1 1 (1.18) 2 2 2 N5 ¼ 1 x h h N6 ¼ x þ x 1 h > > 2 2 > > > > > > 1 1 2 2 2 2 > > N7 ¼ 1 x h þ h N8 ¼ x x 1 h > > > 2 2 > > > > > 2 2 > ; N9 ¼ 1 x 1 h
12
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
Through FEM, the domain is discretized into a set of ﬁnite elements and the displacement vector over each of the elements may be expressed as: u¼
9 X
N i ui ; v ¼
i¼1
f1 ¼ j1 ¼
9 X i¼1
9 X
i¼1 9 X
Ni f1i ; f2 ¼
i¼1
i¼1
9 X
9 X
Ni j1i ; j2 ¼
i¼1
q1 ¼
Ni vi ; w ¼
9 X
9 X
Ni f2i ;
> > > > > > > > > > > > > > > > > > ;
Ni j2i ;
i¼1
Ni q1i ; q2 ¼
i¼1
9 X
9 > > Ni wi ; > > > > > > > > > > > > > > > > =
Ni q2i
.
(1.19)
i¼1
This equation can also be rewritten in general form as: fd g ¼ ½Nifdig
(1.20)
where [Ni] and {di} are the nodal shape functions and displacement vector for the ithnode, respectively. Similarly, fd g ¼ f u v w f1 f2 j1 j2 q1 q2 gT is the displacement vector for any node. Now, the midplane strain vector in terms of nodal displacement vector can be written as: fεgi ¼ ½Bi fdi g
(1.21)
where [Bi] is the strain displacement relation matrix. The elemental equation for strain energy may be expressed by substituting Eq. (1.21) into Eq. (1.16), and the strain energy expression can be rewritten as: 1 U ¼ 2
Z
e
fd gTi ½BTi ½D½Bi fd gi dA
(1.22)
1 (1.23) fd0i gT ½Ke fd0i g 2 R1 R1 where ½Ke ¼ 1 1 ½BT ½D½BjJjdxdh is the elemental stiffness matrix. Similarly, the elemental equation of work done due to uniform pressure may be written in the following form as: Ue ¼
W e ¼ fd0i gT fFge where fFge ¼
R1 R1
T 1 1 ½N fpgjJjdxdh
(1.24) is the mechanical load vectors.
Finite element modeling of natural ﬁberbased hybrid composites
1.3.4
13
Governing equation
The governing equation of the hybrid composite panel is obtained by minimizing the total energy expression: dP¼0
(1.25)
where P ¼ (U W). The equilibrium equation for any element within the panel can be obtained by substituting Eqs. (1.23) and (1.24) in Eq. (1.25): ½Ke fdge ¼ fFge
(1.26)
where [K]e and {F}e are the elemental stiffness matrix and elemental force vector, respectively. The foregoing elemental equation can be rewritten in global form as: ½Kfdg ¼ fFg
(1.27)
where [K] and {F} represent the system stiffness matrix and system force vector of the hybrid composite panel under uniform pressure.
1.4
Finite element solutions of natural ﬁberbased hybrid composites
In this section, the ﬁnite element solutions of natural ﬁberbased hybrid composite plates are examined for different materials (volume fractions), number of layers, and geometrical (sidetothickness ratio, a/h and sidetolength ratio, a/b) and support conditions. A homemade customized code in an FEM framework via a ninenoded isoparametric element based on the HSDT midplane kinematics is developed in a MATLAB environment. A (5 5) mesh is utilized to discretize the proposed model. For computational purposes, jute and ﬂax are considered as ﬁber materials, whereas epoxy is considered as the matrix material. The hybrid composite panel is subjected to uniform pressure (P). The following sets of parameters are considered throughout the analysis: a ¼ 0.1 m, b ¼ 0.1 m, h ¼ 0.01 m, p¼ 100 MPa, singlelayered hybrid composite/doublelayered composite (0 /90 )/threelayered composite (0 /90 /0 ), if not stated otherwise. Young’s modulus (E), shear modulus (G), Poisson’s ratio (y), and density (r) of the following materials are as follows [8,33,34]: Epoxy: Em ¼ 0.0474 GPa, Gm ¼ 1.8 GPa, ym ¼ 0.2, rm ¼ 1150 kg/m3 Jute: EfA ¼ 0.773 GPa, G12fA ¼ 26.5 GPa, y12fA ¼ 0.17, rfA ¼ 1300 kg/m3 Flax: EfB ¼ 1.05 GPa, G12fB ¼ 27.6 GPa, y12fB ¼ 0.17, rfB ¼ 1400 kg/m3
14
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
Here, four different material models are considered by varying volume fractions of the ﬁber and matrix materials, as mentioned here: M1: M2: M3: M4:
VfA ¼ 0.15, VfB ¼ 0.15, Vm ¼ 0.7 VfA ¼ 0.2, VfB ¼ 0.2, Vm ¼ 0.6 VfA ¼ 0.1, VfB ¼ 0.3, Vm ¼ 0.6 VfA ¼ 0.3, VfB ¼ 0.1, Vm ¼ 0.6
where subscripts fA, fB and m represent jute, ﬂax, and epoxy materials, respectively. The purpose of the boundary condition is to avoid rigid body motion as well as reduce the number of unknowns for the system, which eases the calculation, and also the singularity in the matrix equation can be avoided. The following sets of boundary conditions are used here to avoid the rigid body motion of the hybrid composite panel: (a) All edges are simply supported condition (SSSS): u ¼ w ¼ f2 ¼ j2 ¼ q2 ¼ 0 at x ¼ 0; a v ¼ w ¼ f1 ¼ j1 ¼ q1 ¼ 0 at y ¼ 0; b (b) All edges are clamped condition (CCCC): u ¼ v ¼ w ¼ f1 ¼ f2 ¼ j1 ¼ j2 ¼ q1 ¼ q2 ¼ 0 at x ¼ 0; a and y ¼ 0; b (c) Opposite edges are clamped and simply supported (SCSC): u ¼ w ¼ f2 ¼ j2 ¼ q2 ¼ 0 at x ¼ 0; a u ¼ v ¼ w ¼ f1 ¼ f2 ¼ j1 ¼ j2 ¼ q1 ¼ q2 ¼ 0 at y ¼ 0; b
Table 1.1 exhibits the deﬂection behavior of a fully clamped (CCCC) hybrid composite plate under uniform pressure (p¼ 100 MPa) for a moderately thick (a/h ¼ 10) to thin (a/h ¼ 100) plate structure. Also, responses are computed for different material models (M1, M2, M3, and M4) for singlelayered, doublelayered (0 /90 ), and threelayered (0 /90 /0 ) hybrid composites. It is observed that the central deﬂection is enhanced with the increase in sidetothickness ratio because a thin composite plate has less stiffness value. Also, an increment in the number of layers stiffens the hybrid composite panel under uniform pressure. However, out of four different material models (M1 to M4), maximum deﬂection is observed in the M1 material model, whereas minimum deﬂection is observed in the M3 material model. Table 1.2 represents the deﬂection behavior of a fully clamped (CCCC) hybrid composite (a/h ¼ 10) plate under uniform pressure (p ¼ 100 MPa) for different sidetolength ratios (a/b ¼ 1, 1.5, 2, 2.5), material models (M1, M2, M3, and M4), and for single and multilayered structures. It can be seen that the central deﬂection reduces with the increase in sidetolength ratio. Table 1.3 represents the deﬂection behavior of a moderately thick (a/h ¼ 10) hybrid composite square plate under uniform pressure (p ¼ 100 MPa) at different support cases (CCCC, SSSS, SCSC), material models (M1, M2, M3, and M4), and for singleand multilayered structures. The fully clamped (CCCC) hybrid composite plates have minimum central deﬂection, whereas maximum central deﬂection is observed in fully simply supported (SSSS) hybrid composite plates.
Finite element modeling of natural ﬁberbased hybrid composites
15
Table 1.1 Central deﬂection (in mm) of a hybrid composite plate under uniform pressure for different sidetothickness ratios Sidetothickness ratio (a/h) Material models
No. of layers
10
20
50
100
M1
1
1.6794
8.5949
97.1677
698.148
2
1.0470
6.4329
87.9501
685.437
3
0.9168
5.9906
84.9236
667.611
1
1.3646
7.0202
79.5882
572.44
2
0.8596
5.2857
72.3039
563.577
3
0.7515
4.9133
69.6688
547.717
1
1.3437
6.9503
78.9906
568.560
2
0.8561
5.2668
72.0628
561.736
3
0.7464
4.8812
69.2207
544.201
1
1.3860
7.0903
80.1888
576.346
2
0.8631
5.3050
72.5498
565.457
3
0.7567
4.9455
70.1206
551.263
M2
M3
M4
Table 1.2 Central deﬂection (in mm) of a hybrid composite plate under uniform pressure for different sidetolength ratios Sidetolength ratio (a/b) Material models
No. of layers
1
1.5
2
2.5
M1
1
1.6794
0.8178
0.4701
0.3028
2
1.0470
0.5165
0.3057
0.1956
3
0.9168
0.4406
0.2598
0.1709
1
1.3646
0.6682
0.3849
0.2482
2
0.8596
0.4234
0.2498
0.1593
3
0.7515
0.362
0.2134
0.1403
1
1.3437
0.6629
0.3831
0.2473
2
0.8561
0.4209
0.2476
0.1573
3
0.7464
0.3606
0.2127
0.1397
1
1.386
0.6733
0.3867
0.2490
2
0.8631
0.4258
0.2521
0.1614
3
0.7567
0.3634
0.2142
0.1409
M2
M3
M4
16
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
Table 1.3 Central deﬂection (in mm) of a hybrid composite plate under uniform pressure for different support cases Support conditions Material models
No. of layers
SSSS
CCCC
SCSC
M1
1
2.3317
1.6794
1.9976
2
1.5592
1.0470
1.2298
3
1.4394
0.9168
1.1119
1
1.9088
1.3646
1.6360
2
1.2816
0.8596
1.0103
3
1.1834
0.7515
0.9146
1
1.8985
1.3437
1.6282
2
1.2782
0.8561
1.0069
3
1.1798
0.7464
0.9122
1
1.9187
1.3860
1.6436
2
1.2852
0.8631
1.0138
3
1.1870
0.7567
0.9169
M2
M3
M4
1.5
Conclusions
This study revealed that the numerical solutions (especially FEM) accomplished in the past were mostly on synthetic ﬁberbased composite structures. However, the modeling and analysis of natural ﬁberbased hybrid composites are still unexplored. To bridge this gap, natural ﬁberbased hybrid composites are modeled micromechanically and solved computationally using the higherorder FEM approach. Two different reinforce materials, i.e., jute and ﬂax, and one matrix material, i.e., epoxy, are considered as hybrid composite constituents. To evaluate the overall material properties, the simple rule of hybrid mixture for Poisson’s ratio and longitudinal elastic modulus, and the modiﬁed HelpineTsai scheme for transverse and shear moduli, are utilized. For discretization purposes, a ninenoded isoparametric Lagrangian element with eightyone degrees of freedom is employed. The ﬁnal form of equilibrium equation of the hybrid composite panel under uniform pressure is governed through the minimum total potential energy principle. Finally, the ﬂexural responses of hybrid composite panels are computed for different sets of parameters/conditions, and it is observed that all the parameters affect the ﬂexural response of hybrid composites signiﬁcantly as follows: • •
The increase in numbers of layers in the laminate enhances the structural stiffness. The ﬂexural strength of the hybrid composite panel improves with the increase in sidetolength ratio, and degrades with the increase in sidetothickness ratio.
Finite element modeling of natural ﬁberbased hybrid composites
• •
17
The fully clamped hybrid composite panel exhibits minimum center deﬂection, whereas maximum deﬂection is observed in the case of a simply supported panel. The desired structural stiffness and strength can be attained by varying the volume fractions of composite constituents.
References [1] Gassan J, Bledzki AK. 6th Internationales techtexil symposium. July 15e17, 1994. Frankfurt. [2] Hull D, Clyne TW. An introduction to composite materials. Cambridge University Press; August 13, 1996. [3] Khoathane MC, Vorster C, Sadiku ER. Hemp ﬁbrereinforced 1pentene/polypropylene copolymer: the effect of ﬁbre loading on the mechanical and thermal characteristics of the composites. J Reinf Plast Compos 2008;27(14):1533e44. [4] Akil HM, Omar MF, Mazuki AAM, Saﬁee S, Ishak ZAM, Bakar AA. Kenaf ﬁbre reinforced composites: a review. Mater Des 2011;32(8e9):4107e21. [5] De Azeredo HMC. Nanocomposites for food packaging applications. Food Res Int 2009; 42:1240e53. [6] Schadler LS, Brinson LC, Sawyer WG. Polymer nanocomposites: a small part of the story. JOM 2007;59(3):53e60. [7] Saba N, Tahir PM, Jawaid M. A review on potentiality of nano ﬁller/natural ﬁbre ﬁlled polymer hybrid composites. Polymers 2014;6(8):2247e73. [8] Bledzki AK, Gassan J. Composites reinforced with cellulose based ﬁbres. Prog Polym Sci 1999;24(2):221e74. [9] Dr.SpohrVerlag, Wuppertal EW. Die textile Rohstoffe. 1981. Frankfurt. [10] Thwe MM, Liao K. Durability of bambooeglass ﬁbre reinforced polymer matrix hybrid composites. Compos Sci Technol 2003a;63(3e4):375e87. [11] Mohamed A, Bhardwaj H, Hamama A, Webber C. Chemical composition of kenaf (Hibiscus cannabinus L.) seed oil. Ind Crop Prod 1995;4:157e65. [12] Karande SV. Thesis: polymer composites based on cellulosic nanomaterials. 2013. [13] Van De Velde K, Kiekens P. Thermoplastic pultrusion of natural ﬁbre reinforced composites. Compos Struct 2001;54:355e60. [14] Kalia S, Kaith B, Kaur I. Pretreatments of natural ﬁbres and their application as reinforcing material in polymer compositesda review. Polym Eng Sci 2009;49:1253e72. [15] Fu SY, Xu G, Mai YW. On the elastic modulus of hybrid particle/shortﬁbre/polymer composites. Compos B Eng 2002;33(4):291e9. [16] KargerKocsis J. Reinforced polymer blends. In: Polymer blends. New York: John Wiley & Sons; 2000. [17] Borba PM, Tedesco A, Lenz DM. Effect of reinforcement nanoparticles addition on mechanical properties of SBS/Curauaﬁbre composites. Mater Res Bull 2014;17:412e9. [18] Jacob M, Thomas S, Varughese KT. Mechanical properties of sisal/oil palm hybrid ﬁbre reinforced natural rubber composites. Compos Sci Technol 2004;64(7e8):955e65. [19] Bakare IO, Okieimen FE, Pavithran C, Abdul Khalil HPS, Brahmakumar M. Mechanical and thermal properties of sisal ﬁberreinforced rubber seed oilbased polyurethane composites. Mater Des 2010;31(9):4274e80. [20] Basu G, Roy AN. Blending of jute with different natural ﬁbres. J Nat Fibers 2007;4(4): 13e29.
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
[21] Sreekala MS, George J, Kumaran MG, Thomas S. The mechanical performance of hybrid phenolformaldehydebased composites reinforced with glass and oil palm ﬁbres. Compos Sci Technol 2002;62(3):339e53. [22] Banerjee S, Sankar BV. Mechanical properties of hybrid composites using ﬁnite element method based micromechanics. Compos B Eng 2014;58:318e27. [23] Gibson RF. Principles of composite material mechanics. 3rd ed. CRC Press; 2011. [24] Mahapatra TR, Kar VR, Panda SK. Large amplitude bending behaviour of laminated composite curved panels. Eng Comput 2016;33(1):116e38. [25] Mahapatra TR, Kar VR, Panda SK. Nonlinear free vibration analysis of laminated composite doubly curved shell panel in hygrothermal environment. J Sandw Struct Mater 2015; 0(00):1e35. [26] Kar VR, Mahapatra TR, Panda SK. Nonlinear ﬂexural analysis of laminated composite ﬂat panel under hygrothermomechanical loading. Steel Compos Struct 2015;19(4):1011e33. [27] Mahapatra TR, Kar VR, Panda SK. Nonlinear ﬂexural analysis of laminated composite panel under hygrothermomechanical loading d a micromechanical approach. Int J Comput Meth 2016;13(3):1650015. [28] Mahapatra TR, Panda SK, Kar VR. Geometrically nonlinear ﬂexural analysis of hygrothermoelastic laminated composite doubly curved shell panel. Int J Mech Mater Des 2016;12:153. [29] Katariya PV, Panda SK, Mahapatra TR. Prediction of nonlinear eigen frequency of laminated curved sandwich structure using higherorder equivalent singlelayer theory. J Sandw Struct Mater 2017:1e24. [30] Sharma N, Mahapatra TR, Panda SK. Vibroacoustic behaviour of shear deformable laminated composite ﬂat panel using BEM and the higher order shear deformation theory. Compos Struct 2017;180:116e29. [31] Sharma N, Mahapatra TR, Panda SK. Numerical analysis of acoustic radiation properties of laminated composite ﬂat panel in thermal environment: a higherorder ﬁniteboundary element approach. Proc Inst Mech Eng C 2017;0(0):1e15. [32] Cook RD, Malkus DS, Plesha ME, Witt RJ. Concepts and applications of ﬁnite element analysis. 4th ed. Singapore: John Wiley & Sons Pvt. Ltd.; 2009. [33] Agarwal M, Arif M, Bisht A, Singh VK, Biswas S. Investigation of toughening behavior of epoxy resin by reinforcement of depolymerized latex rubber. Sci Eng Compos Mater 2015; 22(4):399. 4. [34] Gope PC, Singh VK, Rao DK. Mode I fracture toughness of bioﬁbre and bioshell particle reinforced epoxy biocomposites. J Reinf Plast Comp 2015;34(13):1075e89.
The effects of cutout on thinwalled plates
2
N. Yidris, M.N. Hassan Aerospace Engineering Department, Universiti Putra Malaysia, Malaysia Generally, the web of a structural beam is susceptible to shear failure where in the event of shear buckling the loadcarrying capacity of the web reduces. The presence of shear buckle, which involves outofplane deﬂections on the web, is accompanied by changes in the stress distribution within the crosssection. In beam design, openings are frequently used to reduce the structural component weight and also to provide entrance for inspection and maintenance services. The cutout will cause the shear buckling load to reduce even further and consequently affect the ultimate load of the web. Although there has been much research in the past that studied the effects of cutout in a web, this present study, which uses more advanced numerical simulation tools compared to the past, presents improved data and understanding of the shear buckling capacity of thinwalled web panels. A series of numerical analyses has been carried out to investigate the effects of openings on the buckling capacity of web panels. The ﬁnite element method has been used to compute the buckling coefﬁcient for square and rectangular plates containing cutouts and subjected to inplane shear. The cases considered in this study are: 1. Square plates with varying dimensions of central circular cutouts. 2. Rectangular plates with varying aspect ratios and multiple central circular cutouts.
2.1
Introduction
Basically, web panels of a beam are designed to resist shearing forces. This is the primary function of the web and thus it is prone to failure by shear buckling. Web buckling is considered a local buckling phenomenon. For designers to choose the dimensions and thicknesses of the web panel they are required to accurately estimate and evaluate the elastic shear buckling strength of the web plate. The sharing edges between the web and ﬂange plates can be considered as simply supported, clamped, or in between these two basic support conditions. Assumptions on the boundary conditions of the web panel depend on many geometric parameters of the web and ﬂanges. Simple design equations have been proposed by researchers in the past to take into account the inﬂuence of the geometry on the boundary support conditions at the juncture
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites and Hybrid Composites https://doi.org/10.1016/B9780081022894.000023 Copyright © 2019 Elsevier Ltd. All rights reserved.
20
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
between the ﬂange and web plate. Essentially, the buckling strength of a simply supported plate is lower than the ﬁxed support plate. Generally, to increase the buckling strength of the web, transverse stiffeners are introduced on the web, which is in between the top and bottom ﬂanges. Thus depending on the length of the spacing between the stiffeners, the web panel can be treated as a square plate or rectangular plate in which the square plate has the highest buckling strength compared to the rectangular plate. The edges that are shared between the web and stiffeners can be considered as simply supported. Hence the web panel could have two edges adjacent to the ﬂanges with ﬁxed support boundary conditions and two other edges with simply supported conditions. Nevertheless, designers for so time have been using a conservative approach by assuming that all four edges of the web panel are simply supported. This is practiced by engineers as it gives them increased security in design although there might be considerable differences in the behavior of the component.
2.2
Finite element analysis
The ﬁnite element procedures involving the discretization of web plates into ﬁnite element models, the application of boundary conditions, and material deﬁnitions will be explained in this section. Finite element analysis was carried out by employing the generalpurpose ABAQUS package developed by Dassault Systemes Simulia Corp.
2.2.1
Analytical model
Fig. 2.1 shows a typical square and rectangular perforated web panel of a beam looking from the side position. The ﬂanges are at the top and bottom and the stiffeners are placed on the right and left side of the web panel with spacing, a. The depth of the web is denoted as b and the diameter of the opening is d. Fig. 2.2 illustrates a rectangular web panel with multiple openings. The openings are placed so that the center
(a)
a – 2
a – 2
b – 2
(b)
a – 2
b – 2 d
b – 2
a – 2
d b – 2
Figure 2.1 Typical square (a) and rectangular (b) perforated web panel.
The effects of cutout on thinwalled plates a – 4
21
a – 4
a – 2
a – 2
a – 4
a – 4
b – 2
b – 2 d
d
d
d
d
b – 2
b – 2 a Rectangular plate, – = 2 b
a Rectangular plate, – = 3 b a – 4
a – 8
a – 4
a – 4
a – 8
b – 2 d
d
d
d
b – 2 a Rectangular plate, – = 4 b
Figure 2.2 Typical rectangular web panel (multiple openings).
measures half of the spacing, a/2, from the side edge and the distance between the openings is the spacing, a.
2.2.2
Material model and boundary conditions
The material is aluminum 7075T6. Aluminum 7075T6 has good strength among the aluminum alloys, good durability, and low fatigue crack development. The basic properties of aluminum 7075T6 are as tabulated in Table 2.1. The plates are simply supported on all edges with web length, a, depth, b, and cutout diameter, d. From Fig. 2.3(a), displacement in the xdirection at the lower corner of the plate is restrained from any movement in the xdirection. Edge displacement in the ydirection at the lower edge of the plate is set to zero to avoid Table 2.1 Properties of aluminum 7075T6 Type of properties
Value
Unit
Poisson ratio
0.33
e
Young’s modulus
71,700
MPa
Density
0.00281
g/mm3
Source: ASM Material Data Sheet, 2001.
22
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
(a)
(b)
(c)
Figure 2.3 Boundary condition (BC) applied to plate models. (a) BC ﬁxed at the xdirection. (b) BC ﬁxed at the ydirection. (c) BC ﬁxed at the zdirection.
any movement in that direction as shown in Fig. 2.3(b). Displacement on each side of the plate edges as illustrated in Fig. 2.3(c) is constrained from movement in the zdirection. The shear force applied to the plate model is deﬁned by assigning shell edge load on each edge of the plate model as shown in Fig. 2.4. In addition, the amount of shear load applied is 1 N/mm.
2.2.3
Element selection
Element selection is important because the elements represent the real structure in ﬁnite element analysis. Elements are interconnected by nodes that identify the locations where the displacements are computed. The ﬁrstorder element has only a corner or end node, whereas the secondorder element also includes midside nodes. The displacements within the element are linearly and quadratically interpolated for the ﬁrstorder and secondorder elements, respectively. For thinwalled structures, the most
The effects of cutout on thinwalled plates
23
Figure 2.4 The shear load applied to the plate models.
appropriate elements to be used are shell elements. ABAQUS provides a number of different types of shell elements in standard triangular and quadrilateral forms. S3 and S3R are the triangular elements and S4, S4R, S8R, S4R5, and S9R5 are the quadrilateral elements. Apparently, quadrilateral shell elements are suitable for analyzing the plate structures. As a result, the S4 quadrilateral shell element has been used to formulate the ﬁnite element models. The element has four connecting nodes and each of its nodes has six degrees of freedom, these being 3 translational and three rotational degrees of freedom.
2.2.4
Convergence test
A convergence investigation has been performed to ascertain a suitable size of ﬁnite element across the plate that leads to a solution with reasonable accuracy. While a number of cases having different lengths, shape factors, and number of ﬁnite elements have been considered, it is adequate to include the discussion of only two cases here as all cases result in a similar conclusion as far as the convergence study is concerned. A different number of meshes were used in this procedure with input variables: width of the plate ¼ 100 mm, thickness ¼ 1 mm, modulus of elasticity ¼ 71,700 N/mm2, and Poisson’s ratio, m ¼ 0.33. By using the shear buckling equation, the values of k obtained by ﬁnite element analysis for a square plate and rectangular plate are given in Table 2.2. It is seen that the solution converges to the exact values with errors ranging from 1% to 2%. The number of elements needed to obtain acceptable results with sufﬁcient accuracy was determined in this study.
24
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
Table 2.2 Convergence study using the ﬁnite element method Number of elements used
k
Percentage error
Square plate (b/a ¼ 1.0)
Number of elements used
k
Percentage error
Rectangular plate (b/a ¼ 0.33)
Exact value
9.34
e
Exact value
5.9
e
15,625 nos.
9.25
0.96
30,000 nos.
5.82
1.36
Rectangular plate (b/a ¼ 0.67)
Rectangular plate (b/a ¼ 0.25)
Exact value
7.10
e
Exact value
5.7
e
25,350 nos.
7.03
0.99
25,600 nos.
5.61
1.58
Exact value
6.60
e
31,250 nos.
6.52
1.21
Rectangular plate (b/a ¼ 0.5)
2.3 2.3.1
Analysis of results Shear buckling behavior of perforated square plates
In this section, the shear buckling behavior of perforated square plates is evaluated. The size of the square plate is 100 mm, and the thickness of the plate is 1 mm. Plates with ﬁve diameters are modeled, i.e., 10 , 20 , 30 , 40, and 50 mm and evaluated with a nonperforated square plate by comparing the critical shear load, scr , and shear buckling coefﬁcient for each plate. The parameters and results predicted by using the ﬁnite element method for each type of perforated square plate and nonperforated square plate are summarized in Table 2.3. Based on Table 2.3, the results show that critical shear load, scr , and the shear buckling coefﬁcient for the perforated square plates are reduced when the cutout ratio d/b increases. This indicates that the strength of the plates decreases when the size of the cutout increases. The trends of shear buckling coefﬁcient for square perforated plates with different cutout ratios d/b are presented in Fig. 2.5.
2.3.2
Shear buckling behavior of rectangular perforated plates
This section provides a discussion of the results obtained for perforated rectangular plates when subjected to shear loads. Analysis of perforated rectangular plates with multiple cutouts with plate ratios a/b of 2, 3, and 4 were attempted. The width and thickness of the perforated rectangular plates is kept constant throughout the analysis at 100 and 1 mm, respectively. The increasing value of plate ratio a/b depends on the length of the plates, i.e., 200, 300, and 400 mm. The quantity of cutouts for plate ratios a/b of 2, 3, and 4 are 2, 3, and 4 cutouts, respectively.
The effects of cutout on thinwalled plates
25
Table 2.3 Shear buckling behavior of perforated square plates (a/b ¼ 1) Plate length, a (mm)
Plate width, b (mm)
Cutout ratio of d/b
Critical shear load, scr (MPa)
Shear buckling coefﬁcient, kv
S1100a
100
100
e
61.23
9.25
S110d
100
100
0.10
55.86
8.44
S120d
100
100
0.20
45.48
6.87
S130d
100
100
0.30
35.23
5.32
S140d
100
100
0.40
26.48
4.00
S150d
100
100
0.50
19.55
2.95
Shear buckling coefficient
Types of plate
Shear buckling coefficient of square perforated plate with different hole diameter 11.0 10.0 Nonperforated square plate,kv= 9.34 9.0 8.0 7.0 6.0 5.0 4.0 3.0 2.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 Hole ratio of d/b
Figure 2.5 Shear buckling behavior of perforated square plates (a/b ¼ 1).
Shear buckling coefficient
Figs. 2.6e2.8 present the relationship between cutout ratio d/b with the shear buckling coefﬁcient of the rectangular plates with multiple cutouts. As shown in the ﬁgures, the shear buckling coefﬁcients of the perforated rectangular plates are dropping as the cutout ratio d/b increases for plate ratios a/b of 2, 3, and 4. There
8.5
Shear buckling coefficient of perforated plate with single hole and multiple holes (plate ration of a/b = 2)
7.5 6.5
Plate without hole, kv = 6.60 Plate with single hole Plate with multiple hole
Nonperforated rectanglar plate with a/b plate ratio of 2, kv = 6.60
5.5 4.5 3.5 2.5 1.5 0.00
0.05
0.10
0.15
0.20 0.25 0.30 Hole ratio of d/b
0.35
0.40
0.45
0.50
Figure 2.6 Shear buckling behavior of rectangular plates with multiple cutouts (a/b ¼ 2).
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
Shear buckling coefficient
26
Shear buckling coefficient of perforated plate with single hole and multiple holes (plate ration of a/b = 3) 7.5 Plate without hole, kv =5.90 6.5
Plate with single hole Plate with multiple hole
Nonperforated rectanglar plate with a/b plate ratio of 3, kv = 5.90
5.5 4.5 3.5 2.5 1.5 0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
Hole ratio of d/b
Figure 2.7 Shear buckling behavior of rectangular plates with multiple cutouts (a/b ¼ 3).
Shear buckling coefficient
Shear buckling coefficient of perforated plate with single hole and multiple holes (plate ration of a/b = 4) 7.5 6.5 5.5
Plate without hole, kv = 5.70 Plate with single hole Plate with multiple hole
Nonperforated rectanglar plate with a/b plate ratio of 4, kv = 5.70
4.5 3.5 2.5 1.5 0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
Hole ratio of d/b
Figure 2.8 Shear buckling behavior of rectangular plates with multiple cutouts (a/b ¼ 4).
is a big difference in shear buckling coefﬁcient between the nonperforated rectangular plate and the perforated rectangular plates (multiple cutouts) as the cutout ratio d/b increases. In addition, it is observed that the shear buckling coefﬁcient for the perforated rectangular plates with multiple cutouts is much lower when compared with the perforated rectangular plates with single cutout. This indicates that the quantity of cutout has an effect on the strength of the plates. Hence increased quantity of cutouts means less shear strength in plates. Based on the analysis conducted in this section, it can be summarized that the shear buckling coefﬁcient is inﬂuenced by the cutout ratio d/b, plate ratio a/b, and the quantity of cutouts assigned to the plate structures.
Modeling of crushing mechanisms of hybrid metal/ﬁber composite cylindrical tubes
3
Al Emran Ismail, KamarulAzhar Kamarudin Crashworthiness and Collisions Research Group (COLORED), Mechanical Failure Prevention and Reliability Research Centre (MPROVE), Faculty of Mechanical and Manufacturing Engineering, Universiti Tun Hussein Onn Malaysia, Batu Pahat, Johor, Malaysia
3.1
Introduction
The research area of crashworthiness deals with the enhancement of structural integrity to protect the occupants, especially during impact or accident events. Research in this area started in the 1970s and a summary can be found in the book authored by Lu and Yu [1]. There are a tremendous number of works that are summarized in Ref. [2] that discuss thinwalled structures as energy absorbers. To enhance the performance of energy absorption, some researchers wrapped the metallic components in composites [3]. In this work, metal tubes with different sizes are wrapped with different orientated ﬁberreinforced composites before the composites are impacted. Comparisons are also made between quasistatic and dynamic loading and it was found that speciﬁc energy absorptions under dynamic loading were always greater than under quasistatic loading. Crushing behavior can be characterized by analyzing the force versus displacement responses. The peak force, Pmax, is the maximum force observed on the curves and it is also indicated on the initiation of the ﬁrst fold. Subsequently, the crushing force experiences a sudden drop before it ﬂuctuates. These ﬂuctuations are the result of folding formation and each peak force corresponding to the respective folding process. The area under the curve represents the capability of energy absorption. It is known that the responses of the forceedisplacement curve are controlled by several factors [4], the most important of which is geometry. To increase the capability of energy absorption, composite materials are used to wrap over the surface of tubes. Jung et al. [5] investigated the energy absorption capability of aluminum/glass ﬁberreinforced plastic hybrid tubes experimentally. It was found that when the tube was wrapped with 0 /90 it showed better performance of energy absorption when compared with other orientations. It is estimated that energy absorption capability improved 1.29 times relative to the unwrapped tubes. Kalhor and Scott [3] numerically investigated the crashworthiness behavior of hybrid
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
aluminum/carbon ﬁberreinforced polymer tube under axial compression impact. It is suggested that hybridization the metallic tubes with composite produced the potential of the structures as a lightweight structure. It is also found that the speciﬁc energy absorption can be increased by 37% compared with virgin aluminum tube. Shin et al. [6] also investigated the crushing behavior of aluminum tube wrapped with glass ﬁberreinforced polymer, but a square tube is used instead of a circular tube. They then experimentally compared the crushing performance of hybrid and nonhybrid tubes. In the case of a tube under axial compression, the hybrid tubes with the 90 degree ply orientation showed better energy absorption capability compared with other types. ElHage et al. [7] numerically characterized the square aluminum composite tubes under quasistatic axial compression. The overwrap angles of the composites varied between 30 and 90 with respect to the tube axis. It was also found that the hybrid tubes offered a signiﬁcant increment in energy absorption than the aluminum tube. Research work [8e12] that discusses the role of natural ﬁber aspects on crushing performance can be found. Based on this literature, the lack of research has inspired the numerical study of hybrid tubes under oblique compression. In this work, ANSYS ﬁnite element software is used to model and solve problems related to oblique compression. Two parameters are involved: ﬁber orientations and oblique angles. Metallic tubes and composite laminates are modeled with a shell element and a proper contact algorithm is used to ensure that the ﬁnite element model is in the right condition. The hybrid tubes are positioned between two rigid plates and they are quasistatically compressed to obtain their forceedisplacement curves. Then, crashworthiness behavior is extracted and analyzed.
3.2
Research methodology
The ANSYS ﬁnite element program is used to model and analyze problems. A basic geometry of the hybrid composite is as shown in Fig. 3.1. The diameter of tubes, D, is 58 mm, the length, L, is 170 mm, and the thickness, t, is 1 mm. Composite thickness is related to the number of layers. The metallic tube is made of steel and the mechanical properties of steel are listed in Table 3.1. The composite is used to wrap on the outer surface of the steel tube. Two types of ﬁbers are used: carbon and eglass ﬁbers. Table 3.2 shows the mechanical strength of these composites, while Fig. 3.2 reveals the stressestrain curves for steel. In modeling hybrid composites, the ANSYS ﬁnite element program is used. Both steel tubes and composite material are modeled with shell element or SHELL163 Explicit Thin Structural Element and the formula is based on the BelytschkoTsay formulation, where reduced integration is used to decrease the computation time. A schematic diagram of the hybrid composite is shown in Fig. 3.1. In this work, a proper contact algorithm is used since it determines the accuracy of the simulation.
Modeling of crushing mechanisms of hybrid metal/ﬁber composite cylindrical tubes
29
(a) Compactor Hybride tube
Steel wall Composite layers
(b)
Gap
Steel Composite
External layer
Internal layer
Figure 3.1 (a) Basic components in constructing the hybrid composites and (b) enlarged area on the wall crosssections. Table 3.1 Mechanical properties of steel tube Density
7.82 3 10e6 kg/mm3
Modulus of elasticity
207.2 GPa
Yield stress
235 MPa
Poisson’s ratio
0.33
An eroding single surface relationship is used on the whole contact surface to identify the selfcontact or to contact with other bodies. Automatic surfacetosurface is used to identify the contact surface between the rigid plate and the hybrid composites. The frictional force between steel and composite is assigned with 0.2 and the contact algorithm
30
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
Table 3.2 Mechanical properties of carbon/epoxy and eglass/epoxy composites (Huang et al., 2012) Carbon/epoxy
Eglass/epoxy
Density, r (g/cm3)
1.53
1.80
Longitudinal modulus, Ea (GPa)
135
30.9
Transverse modulus, Eb (GPa)
9.12
8.3
Shear modulus, Gab (GPa)
5.67
2.8
Poisson’s ratio, n
0.021
0.0866
Longitudinal tensile strength, Xt (MPa)
2326
798
Longitudinal compressive strength, Xc (MPa)
1236
480
Transverse tensile strength, Yt (MPa)
51
40
Transverse compressive strength, Yc (MPa)
209
140
Shear strength, Sc (MPa)
87.9
70
1000
Stress (MPa)
800 600 400 200 0
0
0.05
0.1
0.15
0.2
0.25
0.3
Strain
Figure 3.2 Stressestrain curve for the steel tube.
between these layers used contact tiebreak node only considering that the shearing force is 100 N. Three thicknesses of composite layers are used: 0.5, 0.8, and 1.0 mm. In this work, three layers of composites are wrapped around the steel tubes producing 1.5, 2.4, and 3.0 mm of total composite thickness wrapped around the tubes. Once hybrid composite modeling is completed, it is positioned between two ﬂat rigid plates as in Fig. 3.1(a). Two loading conditions are selected: one is where the tubes are compressed axially and the other is where the tubes are compressed
Modeling of crushing mechanisms of hybrid metal/ﬁber composite cylindrical tubes
31
Force
θ°
Figure 3.3 Oblique compression during simulation.
(a)
y
(b)
a
y
a b
x
b
x
Figure 3.4 (a) Circular and (b) elliptical sections of tubes.
obliquely using different inclined angles such as 0, 5, 10, 15, and 20 degrees, as in Fig. 3.3. To study the effect of geometry on the crushing performance, different aspect ratios, a/b, are used such as 0.5, 0.8, 1.0, 1.2, and 1.8, as in Fig. 3.4. Each layer of composite is wrapped around the tubes three times using different angles such as 45, 60, and 90 degrees. The program is written using ANSYS parametric design language. Before the model is used further, it is essential to compare the model with the existing model, as shown in Fig. 3.5 obtained from Huang et al. (2012). It is revealed that both results are almost similar and the model in this work can be used for further analysis, while Fig. 3.6 shows the corresponding crushing mechanisms of the steel tubes under quasistatic compression. Fig. 3.7 shows the schematic diagram of the forceedisplacement curve of a tubular section under axial compression. It is composed of three regions: (1) linear elastic deformation, (2) plateau stress, and (3) densiﬁcation stage. The tube reached the maximum force, Pmax, after experiencing an elastic deformation. After that the Pmax dropped suddenly due to the formation ﬁrst of localized plastic deformation. Consequently, similar localized lobe formation occurred creating the plateau stage, which is determined by the mean force, Pmean. The densiﬁcation stage started when the tube wall is plastically crushed and there is no more wall to fold. Energy absorption
32
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites 200
Current work
Huang et al. (2012)
Force (kN)
150
100
50
0
0
20
40 60 Displacement (mm)
80
100
Figure 3.5 Comparison between the responses of forceedisplacement curves of current and existing results.
(a) (b)
(c)
Figure 3.6 Comparison between current and existing models (Huang et al., 2012). (a) Initial, (b) ﬁnal conditions, and (c) aerial view. 200 Peak force, Pmax
Force (kN)
160 120
Mean force, Fm
80 40 0 0
20
40 60 Displacement (mm)
Figure 3.7 Typical shape of forceedisplacement curves.
80
100
Modeling of crushing mechanisms of hybrid metal/ﬁber composite cylindrical tubes
33
performance is determined by calculating the area under the forceedisplacement curve, as in Eq. (3.1): Z E¼
Pds
(3.1)
where P is an applied force and ds is an elemental distance.
3.3 3.3.1
Results and discussion Effect of oblique compression angles on the forceedisplacement curves for empty tubes
Fig. 3.8 reveals the responses of forceedisplacement curves for empty steel tubes under axial and oblique quasistatic compressions. It is revealed that the tube under axial compression produced higher peak force compared with other tubes loaded obliquely. It is also found that higher oblique compression angles resulted in lower peak force. This is because higher oblique angles are capable of producing higher bending moments, thus reducing the peak forces. Based on this observation, higher ﬂuctuating forces occur during progressive collapse, while under oblique compression the forces are almost insigniﬁcant force ﬂuctuations. This is probably due to unsymmetrical
120
Angles 0 5 10 15 20
100
Force (kN)
80 60 40 20 0
0
10
20
30
40
50
60
70
80
90
Displacement (mm)
Figure 3.8 Forceedisplacement responses of empty tubes under axial and oblique compressions.
100
34
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
loading where only one side is under compression, while another side is under tension, thus creating small localized lobes or plastic deformations.
3.3.2
Effect of oblique angles on the speciﬁc energy absorption capability
Fig. 3.9 shows the inﬂuence of oblique angles on the speciﬁc energy absorption performance for circularshaped hybrid tubes. For all cases, increased composite thickness reduces the performance of speciﬁc energy absorption. This is because thicker composite layers produce heavier hybrid tubes, thus affecting the crushing performance in terms of speciﬁc energy absorption. The introduction of oblique compression angles seems to reduce the speciﬁc energy absorption capability. This is due to the sharp contact point between the hybrid tube and the rigid ﬂattened plate, thus producing a higher stress concentration region. This area accelerates the damaging processes and therefore reduces strength and produces lower peak forces.
3.3.3
Effect of tube aspect ratios on the crushing performance
Fig. 3.10(a) shows the effect of tube aspect ratios on speciﬁc energy absorption capability. Changing the aspect ratios seemed to affect the crushing performance. It is observed that for a/b < 1.0, the speciﬁc energy absorption performance is slightly higher compared with the tube with a/b ¼ 1.0. However, when a/b > 1.0 the crushing performance is slightly reduced compared with the tube with a/b ¼ 1.0. It is also observed that for a/b < 0.5 and a/b > 1.5, the effect of a/b on the speciﬁc energy absorption is insigniﬁcant. This can be related to the change of moment of inertia. For the circular section, the moment of inertia is almost uniform for all axes. However, when the aspect ratio changed, there was a higher tendency for the tube to plastically deform
Specific energy absorption (J/kg)
25 0.5
0.8
1
20 15 10 5 0
0
5
10 15 Oblique angles (θ°)
20
Figure 3.9 Effect of oblique compression angles on speciﬁc energy absorption.
Modeling of crushing mechanisms of hybrid metal/ﬁber composite cylindrical tubes
Specific energy absorption (J/kg)
(a)
Torce ratios, Favg / Fmm
(b)
35
35 30 25 20 15 10 5 0
0.2
0.5
0.8
1.0
1.2
1.5
1.8
Elliptical ratios, a/b 0.5 0.4 0.3 0.2 0.1 0
0.2
0.5
0.8 1.0 1.2 Elliptical ratios, a/b
1.5
1.8
Figure 3.10 Effect of elliptical ratios on (a) speciﬁc energy absorption and (b) force ratio.
in the direction of least moment of inertia, therefore reducing the tube strength to support the force under compression. Fig. 3.10(b) reveals the effect of elliptical ratios on the force ratio. It is observed that for a/b between 0.5 and 1.0, it is capable of producing a higher force ratio. For this range of force ratio, a higher peak force can be obtained as compared with the mean force. On the other hand, these elliptical ratios are also capable of preventing catastrophic failure.
3.3.4
Effect of materials, ﬁber orientations, and oblique compression angles on the crushing performance
Fig. 3.11 shows the effect of oblique compression angles on the speciﬁc energy absorptions when different ﬁber orientations are used. In this work, a ﬁber orientation of 90 degrees is aligned with the axis of the tubes. Based on numerical works, 45 and 60 degree ﬁber orientations produced slightly higher energy absorption
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
Specific energy absorption (J/kg)
36
25 45
20
60
90
15 10 5 0
0
5
10 15 Oblique angles (θ°)
20
Figure 3.11 Effect of oblique compression angles on the speciﬁc energy absorption when different ﬁber orientations are used.
capabilities compared with the tubes containing 90 degree ﬁber orientation. This is because when the ﬁber is aligned circumferentially around the tube, it is capable of restraining the localized wall deformation and therefore increasing the performance of energy absorptions. Fig. 3.12 shows the inﬂuence of ﬁber orientations on force ratios. It is deﬁned as the ratio between the peak and mean forces. A lower force ratio indicates that the tube failed catastrophically. As predicted in Fig. 3.11, 90 degree ﬁber orientations are unable to resist wall deformation signiﬁcantly as compared with other orientations. The insigniﬁcant effect of 90 degree ﬁber orientation in increasing the force ratio can also be seen where the force ratio has no strong effect when different oblique compression angles are used. This indicates that 90 degree ﬁber orientations must be reinforced with the orientations, especially 0 degree orientations.
0.39 45
Force ratio
0.37
60
90
0.35 0.33 0.31 0.29
0
5
10 15 Oblique angles (θ°)
Figure 3.12 Effect of oblique angles on the force ratio.
20
Specific energy absorption (J/kg)
Modeling of crushing mechanisms of hybrid metal/ﬁber composite cylindrical tubes
37
30 Ekaca/Epoksi
Karbon/Epoksi
20
10
0
0
5
10 15 Oblique angles (θ°)
20
Figure 3.13 Effect of oblique angles on the speciﬁc energy absorption.
Fig. 3.13 shows the effect of oblique compression angles on the speciﬁc energy absorption for two different materials. There are two distinct material responses when different oblique compression angles are used. For eglass ﬁberreinforced composites, the speciﬁc energy absorption performance is reduced. However, it is increased when carbon ﬁberreinforced composites are used where there is no signiﬁcant effect on the energy absorption capability when oblique angles are increased. This is probably related to the different values of strain to failure. Carbon ﬁber has a higher straintofailure value compared with eglass ﬁber. Thus it is capable of resisting circumferential wall deformation and therefore increasing the crashworthiness performance. Fig. 3.14 indicates the inﬂuence of different ﬁbers on the force ratios when compressed obliquely. It is revealed that as the oblique angles increased the force ratios slightly decreased. However, for carbon ﬁberreinforced composites, the force ratios slightly increased when oblique compression angles are increased. This can be related to better mechanical performance of carbon compared with glass ﬁbers.
0.5 Ekaca/Epoksi
Karbon/Epoksi
Force ratio
0.4 0.3 0.2 0.1 0.0
0
5
10 15 Oblique angles (θ°)
20
Figure 3.14 Effect of oblique angles on the force ration between two materials.
38
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
Figs. 3.15 and 3.16 show the effect of different ﬁbers on the crushing mechanisms of carbon and glass ﬁberreinforced composites, respectively. The differences among these ﬁbers are the value of strain to failure. It is known that maximum strain to failure for carbon is higher than eglass ﬁbers and therefore the capability of carbon ﬁber to resist wall deformation is better than glass ﬁber. Comparatively, glass ﬁberreinforced composite experiences severe ﬁber dispersion compared with carbon ﬁber, thus reducing the capability to absorb the crushing energy.
(a)
(b)
(c)
Figure 3.15 Failure mechanisms of (a) an asconstructed model, (b) side, and (c) aerial crushed hybrid composite tubes (carbon ﬁberreinforced composites).
(a)
(c) (b)
Figure 3.16 Failure mechanisms of (a) an asconstructed model, (b) side, and (c) aerial crushed hybrid composite tubes (glass ﬁberreinforced composites).
Modeling of crushing mechanisms of hybrid metal/ﬁber composite cylindrical tubes
3.4
39
Conclusion
Based on ﬁnite element analysis, several conclusions can be drawn: 1. The introduction of oblique compression angles reduced the crashworthiness performance compared with axial compressions. However, the force ﬂuctuations were signiﬁcantly reduced at the plateau stage. 2. Changing circular to elliptical shapes using elliptical ratios between 0.5 and 0.8 improved the performance of speciﬁc energy absorptions. 3. Utilizing carbon instead of glass ﬁbers was able to improve the crushing performance even when oblique compression angles were used.
Acknowledgments The authors acknowledge the Universiti Tun Hussein Onn Malaysia for sponsoring this project.
References [1] Lu G, Yu T. Energy absorption of structures and materials. New York: CRC Press; 2003. [2] Song HW, Wan ZM, Xie ZM, Du XW. Axial impact behavior and energy absorption efﬁciency of composite wrapped metal tubes. J Impact Eng 2000;24:385e401. [3] Kalhor R, Case SW. The effect of FRP thickness on energy absorption of metalFRP square tubes subjected to axial compressive loading. Compos Struct 2015;130:44e50. [4] Baroutajia A, Sajjiab M, Olabic AG. On the crashworthiness performance of thinwalled energy absorbers: recent advances and future developments. ThinWalled Struct 2017;118: 137e63. [5] Jung DW, Kim HJ, Choi NS. AluminumeGFRP hybrid square tube beam reinforced by a thin composite skin layer. Composites A 2009;40:1558e65. [6] Shin KC, Lee JJ, Kim KH, Song MC, Huh JS. Axial crush and bending collapse of an aluminum/GFRP hybrid square tube and its energy absorption capability. Compos Struct 2002;57:279e87. [7] ElHage H, Mallick PK, Zamani N. A numerical study on the quasistatic axial crush characteristics of square aluminumecomposite hybrid tubes. Compos Struct 2006;73: 505e14. [8] Huang MY, Tai YS, Hu HT. Numerical study on hybrid tubes subjected to static and dynamic loading. Appl Compos Mater 2012;19(1):1e19. [9] Ismail AE, Che Abdul Aziz MA. Tensile strength of woven yarn kenaf ﬁber reinforced polyester composites. J Mech Eng Sci 2015;8:1695e704. [10] Ismail AE, Sahrom MF. Lateral crushing energy absorption of cylindrical kenaf ﬁber reinforced composites. Int J Appl Eng Res 2015;10(8):19277e88. [11] Roslan MN, Ismail AE, Hashim MY, Zainulabidin MH, Khalid SNA. Modelling analysis on mechanical damage of kenaf reinforced composite plates under oblique impact loadings. Appl Mech Mater 2014;465e466:1324e8. [12] Ismail AE, Mohd Tobi AL. Axial energy absorption of woven kenaf ﬁbre reinforced composites. ARPN J Eng Appl Sci 2016;11(14):8668e72.
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Roles of layers and ﬁber orientations on the mechanical durability of hybrid composites
4
Muhammad Eka Novianta, Al Emran Ismail, KamarulAzhar Kamarudin Crashworthiness and Collisions Research Group (COLORED), Mechanical Failure Prevention and Reliability Research Centre (MPROVE), Faculty of Mechanical and Manufacturing Engineering, Universiti Tun Hussein Onn Malaysia, Batu Pahat, Johor, Malaysia
4.1
Introduction
Nowadays, composite materials are widely used, especially in aerospace industries, since composites offer a better strengthtoweight ratio. In automotive sectors, one of the key factors for choosing composite materials is their ability to absorb impact energy better than metallic components. Composites can be classiﬁed as shown in Fig. 4.1. In this work, continuous hybrid ﬁber is selected to take advantage of composite materials and other types of material, for example, metal, as shown in Fig. 4.2. Sinmazcelik et al. [1] reviewed ﬁber metal laminates over the past few decades. Fiber metal laminates are hybrid composite structures based on thin sheets of metal alloys and plies of ﬁberreinforced polymeric materials [2]. Fiber/metal composite technology combines the advantages of metallic materials and ﬁberreinforced matrix systems. Metals are, for instance, isotropic, have a high bearing strength and impact resistance, and are easy to repair, while full composites have excellent fatigue characteristics and high strength and stiffness. The fatigue and corrosion characteristics of metals, and the low bearing strength, impact resistance, and reparability of composites, can be overcome by combination [3,4]. Carrillo and Cantwell [5] studied the mechanical properties of a thermoplastic ﬁber/metal laminate. A modiﬁed polypropylene ﬁlm is used to bond aluminum with selfreinforced polypropylene. Single cantilever beam and tensile and impact tests revealed that there is some improvement in mechanical performance. Frangopol et al. [6] investigated the reliability of a composite laminate plate by using TsaieWu failure criterion. Fiber orientation, layer thickness, and number have a signiﬁcant effect on the reliability of ﬁberreinforced composite plates. Mortazavian et al. [7] experimentally and analytically investigated the anisotropy effects on tensile properties of two short glass ﬁberreinforced thermoplastics. Laminate analogy and modiﬁed TsaieHill criteria provided satisfactory predictions of elastic modulus and tensile strength. The ﬂexural properties of bidirectional hybrid epoxy
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites and Hybrid Composites https://doi.org/10.1016/B9780081022894.000047 Copyright © 2019 Elsevier Ltd. All rights reserved.
42
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
Composites
Fibre reinforced
Particle reinforced
Continuous (aligned)
Continuous (aligned)
Structural
Discontinuous (aligned)
Discontinuous (aligned)
Continuous (aligned)
Discontinuous (aligned)
Figure 4.1 Classiﬁcation of composite materials.
Composite Metal Composite Figure 4.2 Hybrid ﬁber continuous.
composites reinforced with glass and carbon ﬁber were investigated experimentally, using ﬁnite element analyses, by Dong et al. [8]. The failure analysis of composite laminates subjected to tension, shear, and compression were investigated by numerical analyses and experimentally [9,10]. High strain rate outofplane properties of an aramid fabricreinforced polyamide composite have been investigated experimentally. The inﬂuence of strain rate on the tensile and compression properties of glass, carbon, and aramidreinforced epoxy/polyamide composites have been studied experimentally and theoretically in [10,11]. Eksi and Genel [12] compared the mechanical properties of unidirectional and woven carbon, glass, and aramid ﬁberreinforced epoxy composites. Three mechanical tests were conducted such as tensile, compression, and shear tests. It was found that unidirectional carbon ﬁber showed better unidirectional ﬁber. The mechanical properties of aramid ﬁberreinforced composite were higher than those of glass and carbon ﬁber, when the woven types of ﬁbers were considered. Abdul Rahim et al. [13] developed a conceptual design of hybrid composite with a metal liner for underwater application. Metal liner was wrapped with eglass (roving and fabric) ﬁbers using different ﬁber orientations. Both analytical and ﬁnite element results
Roles of layers and ﬁber orientations on the mechanical durability of hybrid composites
43
indicated that the structure could withstand the desired pressure. Several other related works can be found elsewhere [14e18]. This chapter investigates the performance of hybrid composite plates subjected to tension stress. Composite layers are attached to both sides of a metallic plate. Three important parameters are used: number of layers, ﬁber orientations, and surrounding temperatures. Hybrid plates are modeled and solved using the ANSYS ﬁnite element program. It is an important task to examine the effect of variables on structural integrity and durability.
4.2
Methodology
In this work, the ANSYS WORKBENCH ﬁnite element program is used to model and solve problems related to hybrid laminated composites. The model is based on the geometry recommended by ASME E8 as shown in Fig. 4.3, while Table 4.1 lists the dimensions of such a model. This model is typically used to characterize the tensile behavior of metallic materials. It is then laminated with the composite materials to analyze their responses under similar loading. Steel material is used
20
12.58
210
102.50
Figure 4.3 Modeling for specimens using the ASTM E8 standard.
Table 4.1 Size model of specimens Gauge length, G
50 mm
Width, W
12.5 mm
Thickness, T
3 mm
Fillet radius, R
65 mm
Length of reduced section, A
57 mm
Length of grip section, B
50 mm
Width of grip section, C
20 mm
Length of specimen, L
210 mm
44
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
as a basic body, while a carbon/epoxy composite is used to attach to the surfaces of the steel plate. Tables 4.2 and 4.3 provide a list of mechanical properties of material used. There are several important variables used in this work such as number of plies, orientations, and material combinations. On the other hand, the solutions to numerical work are also solved at 125 and 250 C. Table 4.4 lists the variables involved in this work. Data from Tables 4.2 and 4.4 are implemented in ANSYS as shown in Fig. 4.4, while Fig. 4.5 reveals the model created using SOLIDWORK. Fig. 4.6 shows the construction of composite materials considering number of layers and orientations. Fig. 4.7(a) presents the meshed model, while Fig. 4.7(b) indicates an example of a hybrid metal/composite used in this work. The model is stressed axially while different temperatures are applied on the outer surfaces to investigate their effect on mechanical performance. Before the current model is further used, it is paramount to have validation. Fig. 4.8 shows the comparison between the results obtained from ANSYS and theory. It is revealed that the results agree well and can be used for further analysis. Fig. 4.9 shows previous work to investigate mechanical performance under different temperatures (Nguyen et al., 2014). It is indicated that higher temperatures are capable of degrading material integrity and therefore temperatures of 125 and 250 C are used.
4.3
Results and discussion
Fig. 4.10 shows the stress distributions of hybrid composites where a layer of composite is ﬁrmly attached to both sides of steel plate. Different composite orientations are used to study their effect on mechanical strength at a temperature of 125 C under Table 4.2 Mechanical properties of steel Modulus of elasticity, E
200 GPa
Poisson’s ratio, n
0.3
Thermal coefﬁcient, a
11.7 10 6/ C
Density, r
7.85 kg/m3
Table 4.3 Mechanical properties of carbon/epoxy Fiber mass per unit area
Cured ply thickness
Fiber volume fraction
Initial elastic modulus
Tensile strain to failure
Compressive strain to failure
21.2 g/m2
28.9 mm
41%
230 GPa
1.5
e
Materials
Twoply orientations
Fourply orientations
Sixply orientations
Temperature (8C)
Metal specimen
e
e
e
125
Composite specimens
[0/0]; [45/e45]; [90/90]
[0/45/e45/0]; [0/45/45/0]; [0/e45/e45/0]; [0/90/90/0]; [90/45/e45/90]; [90/45/45/90]; [90/e45/e45/90]; [45/e45/e45/45]
[0/90/45/45/90/0]; [0/90/45/e45/90/0]; [0/90/e45/e45/90/0]
125
Hybrid specimens
[0/0]; [45/e45]; [90/90]
[0/45/e45/0]; [0/45/45/0]; [0/e45/e45/0]; [0/90/90/0]; [90/45/e45/90]; [90/45/45/90]; [90/e45/e45/90]; [45/e45/e45/45]
[0/90/45/45/90/0]; [0/90/45/e45/90/0]; [0/90/e45/e45/90/0]
125
Metal specimen
e
e
e
250
Roles of layers and ﬁber orientations on the mechanical durability of hybrid composites
Table 4.4 Variables involved during numerical work
Continued
45
46
Table 4.4 Variables involved during numerical workdcont’d Twoply orientations
Fourply orientations
Sixply orientations
Temperature (8C)
Composite specimens
[0/0]; [45/e45]; [90/90]
[0/45/e45/0]; [0/45/45/0]; [0/e45/e45/0]; [0/90/90/0]; [90/45/e45/90]; [90/45/45/90]; [90/e45/e45/90]; [45/e45/e45/45]
[0/90/45/45/90/0]; [0/90/45/e45/90/0]; [0/90/e45/e45/90/0]
250
Hybrid specimens
[0/0]; [45/e45]; [90/90]
[0/45/e45/0]; [0/45/45/0]; [0/e45/e45/0]; [0/90/90/0]; [90/45/e45/90]; [90/45/45/90]; [90/e45/e45/90]; [45/e45/e45/45]
[0/90/45/45/90/0]; [0/90/45/e45/90/0]; [0/90/e45/e45/90/0]
250
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
Materials
Roles of layers and ﬁber orientations on the mechanical durability of hybrid composites
47
Figure 4.4 Insertions of mechanical properties of steel and carbon/epoxy.
Figure 4.5 CAD modeling of tension sample.
tension stress. Fig. 4.10(a) shows the stress distribution for [0/0] composite laminated on the surface of steel, while Fig. 4.10(b) reveals similar behavior although [90/90] ﬁber orientations are used. Fig. 4.10(c) indicates the stress condition when [45/e45] ﬁber orientations are attached to steel surfaces. Based on observation, it is found that the stress distributions for [45/e45] ﬁber orientation are more uniform and capable of eliminating the spot of maximum stresses, which then leads to increases in strength and durability. Fig. 4.11 shows the stressestrain behavior of the hybrid composites when they are stressed at different levels of loading. It is revealed that the hybrid composite with [0/0] ﬁber orientation is capable of withstanding higher stresses compared with other types of materials. However, the elongation capability of this composite is the lowest. Even though the composite is under axial stress, it is not the only stress; shear stress can be
48
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
Figure 4.6 Formation of composite laminates.
(a)
Y
X 0.00
70.00 (mm)
35.00 52.50
17.50
(b)
z x 0.000
20.000 (mm)
10.000 5.000
15.000
Figure 4.7 Finite element model: (a) aerial and (b) side views.
True stress (Mpa)
Roles of layers and ﬁber orientations on the mechanical durability of hybrid composites
49
1000 100 10 1 0.00E+00
5.00E–04
1.00E–03 1.50E–03 True strain (mm)
ANSYS
2.00E–03
2.50E–03
Theory
Figure 4.8 Numerical validation of the present model between results obtained using ﬁnite element analysis and theoretical expression.
True stress, MPa
400
25°C 100°C
300
150°C 200°C
200
100
0 0.0
0.2
0.4
0.6
True strain, µ
Figure 4.9 Stressestrain responses.
induced especially on the surfaces. This combination of stresses mainly degrades the composite and therefore reduces its performance. Compared with other composites ([90/90] and [45/e45]), the capability to elongate is higher since there are ﬁbers oriented at speciﬁc angles. These ﬁbers are capable of resisting steel deformation and therefore increasing the material’s integrity. As a result, higher elongation can be observed. Fig. 4.12 shows the stress distributions of hybrid plates under tension stress. Four plies of composite layers are used with two plies for each side. According to the observations, composites containing 0 degree ﬁber orientation such as in Fig. 4.12(a)e(d) experience higher stresses. In this work, 0 degree ﬁber orientation is deﬁned perpendicular to the loading’s axis where these kinds of ﬁbers are unable to assist mechanical deformation along the axis. On the other hand, Fig. 4.12(e), (g), and (h) reveal that when these 0 degree ﬁber orientations are aligned to a certain degree, for example, 45 degree, the maximum stress can be reduced signiﬁcantly. Similar stress behavior can be observed for a higher number of composite layers, as shown in Fig. 4.13. Fig. 4.13 shows the stress versus strain curves of different materials under axial tension stress. In general, the effect of ﬁber orientations on the stressestrain responses
50
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
(a)
(b)
(c)
Figure 4.10 Tensile responses of hybrid composites (two plies) at 125 C: (a) [0/0], (b) [90/90], and (c) [45/e45].
100000
True stress (MPa)
10000 1000 100 10 1 0
0.05
Hybrid 0/0 125°C
0.1
0.15
0.2 0.25 True strain
Hybrid 45/–45 125°C
0.3
0.35
0.4
Hybrid 90/90 125°C
Figure 4.11 Stressestrain diagram of twoply hybrid composites.
Roles of layers and ﬁber orientations on the mechanical durability of hybrid composites
51
(a)
(b)
(c)
(d)
(e)
Figure 4.12 Tensile responses of hybrid composites (four plies) at 125 C: (a) [0/45/e45/0], (b) [0/45/45/0], (c) [90/45/e45/90], (d) [0/90/90/0], (e) [90/e45/e45/90], (f) [0/e45/e45/0], (g) [90/e45/e45/90], and (h) [45/e45/e45/45].
52
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
(f)
(g)
(h)
Figure 4.12 cont'd.
seems insigniﬁcant. However, for certain conﬁgurations, for example [90/45/e45/90] and [90/e45/e45/90] ﬁber orientations, the hybrid composites are capable of elongating further compared with other types of composites. On the other hand, these composites are a lack of supporting higher axial stresses compared with others. For higher numbers of composite layers, there is an insigniﬁcant effect on the stressestrain curves when different ﬁber orientations are used. This is because when a higher number of composite thicknesses are used, the strength is almost similar to the metallic material. In addition, the change in ﬁber orientations does not strongly affect the overall strength of the hybrid composites as shown in Fig. 4.15. Fig. 4.16 compares the displacement performance when different ﬁber orientations are used. It is revealed that lower displacement can be obtained by introducing the 0 degree ﬁber orientations into the conﬁgurations. This is because 0 degree is deﬁned as the perpendicular axis relative to the axis of loading. In this case, these ﬁbers are unable to resist deformation effectively. On the other hand, when the ﬁbers are aligned
100000
True stress (MPa)
10000 1000 100 10 1 0
0.05
0.1
0.15
0.2 0.25 True strain
0.3
0.35
0.4
Hybrid 0/45/–45/0 125°C
Hybrid 0/45/45/0 125°C
Hybrid 90/45/–45/90 125°C
Hybrid 0/90/90/0 125°C
Hybrid 90/45/45/90 125°C
Hybrid 0/–45/–45/0 125°C
Hybrid 90/–45/–45/90 125°C
Hybrid 45/–45/–45/45 125°C
Figure 4.13 Stressestrain responses of fourply hybrid composites at 125 C.
(a)
(b)
(c)
Figure 4.14 Tensile responses of hybrid composites (four plies) at 125 C: (a) [0/90/45/e45/90/0], (b) [0/90/45/45/90/0], and (c) [0/90/e45/e45/90/0].
54
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites 100000
True stress (MPa)
10000
1000
100
10
1 0
0.2
0.4
0.6 0.8 True strain (mm)
Hybrid 0/90/45/–45/90/0 125°C
1
1.2
Hybrid 0/90/45/45/90/0 125°C
Hybrid 0/90/–45/–45/90/0 125°C
Figure 4.15 Stressestrain responses of sixply hybrid composites at 125 C.
180
Displacement (mm)
160 140 120
100 80 60
40 20
Composite 125°C
/9 0/ 0 –4 5/ –4 5/ 90 0 /– 45 /– 45 /9 45 0 /– 45 /– 45 /4 5
0
90
0/
/4 5/ 45 90
0/ 90 /9
5/
0 5/
/4 5/ –4
90
0/ 45 /4
0/ 4
5/
–4 5/ 0
0
Model and temperature of specimens Composite 250°C Hybrid 125°C Hybrid 250°C
Figure 4.16 Comparison of the effect of ﬁber orientations on displacement under tension force at different temperatures.
Roles of layers and ﬁber orientations on the mechanical durability of hybrid composites
55
16 14 Deformation (mm)
12 10 8 6 4 2 0 Composite 125°C Composite 250°C
0/90/45/45/90/0
Hybrid 125°C
0/90/ – 45/ – 45/90/0
Hybrid 250°C
0/90/45/ – 45/90/0
Figure 4.17 Comparison of the effect of surrounding temperatures on displacement under tension force for different ﬁber orientations.
using 45 or 90 degrees direction, the composites experience longer displacements. Fibers with 90 degree direction are parallel to the axis of loading. These ﬁbers restrain the composites and therefore produce higher maximum displacements. Fig. 4.17 compares the effect of surrounding temperatures on the maximum displacements. There are six plies containing different ﬁber orientations. According to the results, it is indicated that for a higher number of layers (six plies), there is an insigniﬁcant effect on the displacements when ﬁber orientations are varied. This is because when higher numbers of layers are used, for example six, the strength of such composites is as strong as steel or probably even stronger. At this level of thickness, surrounding temperatures insigniﬁcantly affect displacement as long as the maximum temperatures for these composites are not reached.
4.4
Conclusion
In this work, the ANSYS ﬁnite element program was used to model and solve the problems related to hybrid ﬁber/metal composites under tension stress at two different temperatures. Three types of layers were used: two, four, and six layers and each layer had different orientations. Based on numerical work, several conclusions can be made: 1. It is observed that by attaching the ﬁbers oriented parallel to the axis of loading, it is capable of producing a better surface stress distribution. 2. Fiber reinforcements play an important role in strengthening the metallic plates; however, this strongly depends on the composite thicknesses. If the number of layers is greater than six, the effect of the metallic plate is diminished even at higher temperatures. 3. Fiber orientations also play a great role where the presence of 45 and 90 degree ﬁber alignments are capable of enhancing the integrity of hybrid composites.
56
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
References [1] Sinmazcelik T, Avcu E, Ozgur Bora M, Coban O. A review: ﬁbre metal laminates, background, bonding type and applied test methods. Mater Design 2011;32:3671e85. [2] Cortes P, Cantwell WJ. The prediction of tensile failure in titaniumbased thermoplastic ﬁbreemetal laminates. Compos Sci Technol 2006;66(13):2306e16. [3] Alderliesten RC, Benedictus R. Fibre/metal composite technology for future primary aircraft structures. In: 48th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference 15th; April 23e26, 2007. Honolulu: Hawaii; 2007. p. 1e12. [4] Chang PY, Yeh PC, Yang JM. Fatigue crack initiation in hybrid boron/glass/aluminium ﬁbre metal laminates. Mater Sci Eng A 2008;496(1e2):273e80. [5] Carrilo JG, Cantwell WJ. Mechanical properties of a novel ﬁbremetal laminate based on a polypropylene composite. Mech Mater 2009;41(7):828e38. [6] Frangopol DM, Recek S. Reliability of ﬁbre reinforced composite laminate plates. Probab Eng Mech 2003;18(2):119e37. [7] Mortazavian S, Fatemi A. Effect of ﬁbre orientations and anisotropy on tensile strength and elastic modulus of short ﬁbre reinforced polymers composites. Compos B Eng 2015;72: 116e29. [8] Dong C, Davies IJ. Flexural strength of bidirectional hybrid epoxy composites reinforced by eglass and T700S carbon ﬁbres. Compos B Eng 2015;72:65e71. [9] Hu H, Lin W, Tu F. Failure analysis of ﬁbre reinforced composite laminates subjected to biaxial loads. Compos B Eng 2015;83:153e65. [10] Tan JLY, Deshpande VS, Fleck NA. Failure mechanisms of a notched CFRP laminate under multiaxial loading. Compos A Appl Sci Manuf 2015;77:56e66. [11] Naresh K, Shankar K, Rao BS, Velmurugan R. Effect of high strain rate on glass/carbon/ hybrid ﬁbre reinforced epoxy laminated composites. Compos B Eng 2016;100:125e35. [12] Qian X, Wang H, Zhang D, Wen G. High strain rate outofplane compression properties of aramid fabric reinforced polyamide. Composite 2016;53:314e22. [13] Eksi S, Genel K. Comparison of mechanical properties of unidirectional and woven carbon, glass and aramid ﬁber reinforced epoxy composites. Acta Phys Pol A 2017;132:879e82. [14] Khairul Izman AR, Othman AR, Mohd Rizal A. Pressure hull development using hybrid composite with liner concept. Indian J Geomar Sci 2011;40(2):207e13. [15] Ismail AE. Oblique crushing performances of kenaf aluminum composite tubes. Int J Appl Eng Res 2016;11(8):5997e6002. [16] Ismail AE, Ibrahim MN, Jamian S, Kamarudin KA, Awang MK, Nor MKM. Oblique crushing performances of hybrid woven Kenaf ﬁbre reinforced aluminium hollow cylinder. MATEC Web Conf 2017;108:01006. [17] Ismail AE, Noranai Z, Nor NHM, Tobi ALM, Ahmad MH. Effect of hybridized ﬁber wrapped around the aluminum tubes on the crushing performances. IOP Conf Ser Mater Sci Eng 2017;160(1):012019. [18] Roslan MN, Ismail AE, Hashim MY, Zainulabidin MH, Khalid SNA. Modelling analysis on mechanical damage of kenaf reinforced composite plates under oblique impact loadings. Appl Mech Mater 2014;465e466:1324e8.
Further reading [1] Ismail AE, Zainulabidin MH, Roslan MN, Mohd Tobi AL, Muhd Nor NH. Effect of velocity on the impact resistance of woven jute ﬁbre reinforced composites. Appl Mech Mater 2014; 465e466:1277e81.
Numerical modeling of hybrid composite materials
5
Nabil Bouhﬁd 1 , Marya Raji 1, 2 , Radouane Boujmal 1 , Hamid Essabir 2 , MohammedOuadi Bensalah 1 , Rachid Bouhﬁd 2 , Abou el kacem Qaiss 2 1 Mohammed VRabat University, Faculty of Science, Rabat, Morocco; 2Moroccan Foundation for Advanced Science, Innovation and Research (MAScIR), Institute of Nanomaterials and Nanotechnology (NANOTECH), Laboratory of Polymer Processing, Rabat, Morocco
5.1
Introduction
Composite materials are the result of a combination of at least two phases where the reinforcement element and the matrix are integrated to improve the properties of the composites. The use of composite materials is slowly emerging from the realm of advanced materials [1e3], allowing them to invade more and more space in both academic and industrial ﬁelds such as automotive, wind energy, aeronautics, civil applications, etc. [4,5]. These kinds of materials are replacing conventional materials due to their interesting performance such as improved mechanical, thermal, and electrical properties and also to offset the high price of the matrices [6e9]. Several studies have shown that the composite materials ﬁlled by natural or synthetic loading provide several advantages over other materials such as good durability, high corrosion resistance, and low density [10e12]. However, the key culprits to the lack of their structural properties were the manufacturing approach [9], shaping, and mainly the state of interphase links [13]. There are also numerous factors that have a direct impact on the mechanical behavior of composite materials, such as active mechanisms of various constitutive elements [14], for example: volumetric fraction [15e17], morphology [18,19], distribution [20], dispersion [21,22], and the state of interfaces and contents dispositions [23,24]. Accordingly, these microscopic elements are the determining factors in predicting the composite material properties and are used to explain the properties of the composite materials at the macroscopic level. However, there are many cases in which the experimental results are unable to explain certain phenomena observed at the macroscopic scale [25]. In fact, the properties of the composite are closely linked to its internal structures, which induce a high heterogeneity of the microstructure, so it is necessary to look more deeply into scalingup approaches that establish the transition between the local heterogeneity state and the global homogeneity state. In general, the heterogeneity problem of composite materials at the microscopic level makes it difﬁcult to move toward a homogeneous global level where the behavior of the material can be measured [26]. The passage through the micro to the macroscale can only take place through rough models and satisfactory calculation tools. The best
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites and Hybrid Composites https://doi.org/10.1016/B9780081022894.000059 Copyright © 2019 Elsevier Ltd. All rights reserved.
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
knowledge of component (matrix and ﬁllers) performance can, through homogenization methods, predict the new material’s properties with acceptable precision [27]. This chapter presents a thorough review of many homogenization methods that pretend to give a more or less close representation of the real mechanical behaviors of composites. First, the classiﬁcation of composite materials and ﬁller types is presented. Then, the renowned analytical models, including the Voigt and Reuss method, Hirsh’s method, Tsai’s model, and other analytical models, are discussed. Finally, the mechanical properties of hybrid composites are evaluated using proposed models.
5.2
Classiﬁcation of materials and ﬁller types
A composite material or simply composite, from the Latin composĭtus, is a complex material made of an assembly of two or more distinct phases (matrix phase and dispersed phase) [2], and having properties signiﬁcantly different from those of any of the constituents [28]. The primary phase is the matrix called the continuous phase, which is more ductile and less hard [29]. The secondary phase, called the dispersed phase, is the reinforcing phase embedded within the matrix, generally stronger than the primary phase, and responsible for enhancing one or more properties of the matrix [30e32]. The main advantages of composite materials in structural applications are as follows [33]: • • •
A composite is a multiphase material consisting of two or more physically distinct components and mechanically separable materials [34]. Composites are made by mixing the components in a controlled way to achieve a uniform dispersion of the constituents leading to optimum properties [35]. Composite mechanical properties are superior to the properties of each individual component and in some cases uniquely different from the properties of their constituents [36].
Composite materials have been widely used in practical applications because of their many advantages: high strength and modulustoweight ratio, easy processability, and cost effectiveness. Furthermore, they are ﬂexible and have a structured design [37].
5.2.1
Matrix
The choice of the matrix is driven by the composite’s industrial application depending on many factors such as resistance to atmospheric agents, operating environment, mechanical properties, and cost [38]. The matrix plays two key roles in composite materials: ﬁrst, transferring the load to the reinforcement and second, protecting the reinforcement against chemical attacks and adverse environmental effects [39]. The matrix may also serve as a barrier to crack propagation. Composites can be grouped into three categories depending upon the matrix: • • •
Organic matrix composites (OMC) [40]. Ceramic matrix composites (CMC) [41]. Metal matrix composites (MMC) [42,43].
While the two last composite types (MMC) and (CMC) are suited for industrial applications in which hardness, thermal stability, high temperature, or corrosion resistance are critical, their use is limited by the high manufacturing costs. The most
Numerical modeling of hybrid composite materials
59
widely used composites are those based on a polymeric matrix because they are often used as lightweight metal replacements and have the advantages of being stiff, strong, of low density, and simple to manufacture with relatively low manufacturing costs [44]. Considering the huge range of potential polymer matrix materials that can be combined with a number of different reinforcement types, which themselves can be arranged in various architectures, the range of OMCs becomes apparent. Two distinct categories of polymers are generally considered: thermoplastics and thermosetting [45]. The difference between them depends on a curing or crosslinking reaction that either does or does not occur during the molding process.
5.2.2
Reinforcements
Reinforcements are strong materials with a particular morphology that is incorporated into the matrix to improve a composite’s physical properties [46]. The different reinforcements used in composites have different properties and so affect the properties of the composite in different ways [47]. Consequently, the properties of composites are a function of the properties of this dispersed phase, its relative amounts, and its morphology, which mean that the composites can be classiﬁed according to their types of reinforcement [48,49]. They are thus divided into four categories.
5.2.2.1
Fiber reinforcements
Reinforcements of this type of composite are in the form of ﬁbers. The length of the ﬁbers is much greater than the dimensions of their crosssection [50,51]. According to their applications, the ﬁbers either take the length of the piece (continuous ﬁbers) [52,53] or are cut into short lengths (short ﬁbers) [54,55], see Fig. 5.1.
5.2.2.2
Particulate reinforcements
This reinforcement system is considered as a particulates if their dimensions are approximately equal in all directions and small in front of the other dimensions of the material [21,56,57]. The hard particles are randomly dispersed in the less rigid matrix (Fig. 5.2).
Figure 5.1 Fiberreinforced composite.
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
Figure 5.2 Particulate composite.
5.2.2.3
Flake reinforcements
Flakes have a very small dimension compared to the other dimensions. The dispersion of this type of reinforcement is generally random [58]. However, the ﬂakes can be arranged in parallel so as to have more uniform properties in the plane, see Fig. 5.3.
5.2.2.4
Hybrid composite materials (more than one element of distinct reinforcements)
Several studies have shown that the combination of two or more types of reinforcement in the same matrix makes it possible to improve the mechanical and thermal properties of the composite [59e61]. It may be a combination of the reinforcing
Figure 5.3 Flake composite.
Numerical modeling of hybrid composite materials
61
elements of different shapes and/or characteristics, as in the case of hybrid composite reinforced with clay particles and natural ﬁbers [62e64]. In principle, it appears that to have a valuable hybrid composite material the reinforcement elements must not exceed two types. These materials can bring new functionalities and can therefore meet technical and/or economic requirements in a more efﬁcient way than conventional composite materials [65].
5.3
Various modeling techniques of composite mechanical properties
5.3.1
Goal
Modeling is used to understand the behavior of the material from the properties of the constituents, which then allows structures or constructions to be designed based on this material. The model must be able to predict a material’s reactions against external stresses before they are synthesized [66].
5.3.2
Mechanics of composites
To predict the performance of composite materials, characteristic mechanical properties such as Young’s modulus, Poisson’s ratio, shear modulus, failure strain, or stress need to be deﬁned for the selected composite [67]. However, ﬁllerreinforced composites are inhomogeneous and nonisotropic. This is why additional levels of complexity are introduced and the analysis of the mechanics of composites is generally studied at two levels. The ﬁrst level is the micromechanics level, in which the interaction between two or more distinct materials must be examined on a microscopic scale as a result of this difference, often in the form of a representative volume element (RVE) [27,68]. In the micromechanical analysis, stiffness, strength, thermal, and moisture expansion coefﬁcients of a lamina are found using the individual properties of constituents (ﬁller and matrix). The second level is the macromechanical level, in which the response of the two constituents in the composite is considered as homogeneous material on a macroscopic scale. Stresses, strains, and deﬂections are determined most often with orthogonal or anisotropic elasticity [69,70].
5.3.2.1
Micromechanical analysis
In the micromechanical approach, the composite materials are considered as the combination of numerous materials and derive mechanical properties based on the homogenization procedures using the individual properties of each component [66,71e73]. This special approach is valuable to study the interaction of constituent materials at the microscopic level and to optimize the design parameters of composites associated with constituent structures, such as the ﬁller volume fraction, ﬁller arrangement, and also perhaps ﬁller distribution to meet the target properties of composites, generally conducted by the use of a mathematical model to determinate the basic elastic
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
properties of the composite [74e76]. Longitudinal and transverse modulus, Poisson’s ratio, and shear modulus are given, respectively, by the following relations: E1 ¼ Vf E1f þ Vm Em E1f Em Vf Em þ Vm E2f
E2 ¼
v12 ¼ Vf v12f þ Vm vm G12 ¼
G12f Gm Vf Gm þ Vm G12f
(5.1)
where the subscripts 1 and 2 and f and m denote the longitudinal and transverse directions and ﬁller and matrix properties, respectively, and Vf and Vm represent ﬁller and matrix volume fractions, respectively. The average property of the composite materials is then given by the rule of mixtures approximation [68,77,78]: Discrete:
Pc ¼ Vf Pf þ Vm Pm þ / ¼ SVi Pi Z . Pc ¼ PðxÞdV
Continuum:
5.3.2.2
(5.2)
Macromechanical analysis
Homogenization approaches assume simplifying expectations and deﬁne the laws linking macroscopic properties to local properties. Primarily, a representative elementary volume (REV also known as RVE) must be deﬁned. It is a homogeneous ﬁctitious medium having on average a behavior identical to the set observed on the microscopic scale [66,79,80]. The REV representation consists of deﬁning the different supposed homogeneous phases, and in particular their shape, orientation, and distribution. In the context of the study of homogeneous and elastic linear material, stress s and strain ε tensors are connected by a linear law known as Hooke’s law: (
s ¼ C: ε ε ¼ S: s
( 5
sij ¼ Cijkl $εkl εij ¼ Sijkl $skl
(5.3)
where C : S ¼ I is the identity tensor. The tensors C and S are symmetric. In fact, according to the equation of equilibrium and the theory of the deformation gradient, the tensors of the stresses and the deformations possess index symmetry:
Numerical modeling of hybrid composite materials
63
sij ¼ sji ; εij ¼ εji (
Cijkl ¼ Cjikl ¼ Cijlk ¼ Cjilk Sijkl ¼ Sjikl ¼ Sijlk ¼ Sjilk
:
(5.4)
Hooke’s law is equivalent to the existence of a potential state l4 from which stress is derived [79]: 2l4 ¼ ε: C : ε l
v4 ¼ s; vε
sij ¼ l
v4 vεij
2l4 ¼ Cpqrs εpq εrs 2sij ¼ Cpqrs ðdip djq εrs þ εpq dir djs Þ ¼ Cijrs εrs þ Cpqij εpq sij ¼ Cjikl εkl :
(5.5)
Since this is a potential, then: v2 l4 v2 l4 ¼ vεij vεkp vεkp vεij 0Cijkl ¼ Cklij (
Sijkl ¼ Sklij ij; kl ˛f11; 22; 33; 23; 31; 12g
.
(5.6)
This is the relation of the great symmetry of the tensor C. Finally, the tensor C has 21 independent components. This is the most general case of an anisotropic material that has no symmetry and whose response of the REV subjected to a stress depends on the direction in which this stress is applied. The tensor is then written in the form of a matrix with 34 ¼ 81 terms, which is not very convenient.
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
Voigt proposed, as for the tensors of the second order, a regrouping of the indices and taking into account their symmetry:
2
C1111
6 6 C2211 6 6 6C 6 3311 C¼6 6 6 C2311 6 6 6 C3111 4 C1211
3
C1122
C1133
C1123
C1131
C1112
C2222
C2233
C2223
C2231
C2212 7
C3322
C3333
C3323
C3331
C2322
C2333
C2323
C2331
C3122
C3133
C3123
C3131
C1222
C1233
C1223
C1231
7 7 7 7 C3312 7 7 7. 7 C2312 7 7 7 C3112 7 7 5 C1212
(5.7)
This allows us to write the law of elasticity, with s and ε represented by vectors following Voigt’s convention:
s11 C1111 s22 C2211 s33 C3311 ¼ s23 C2311 s C 31 3111 s C 12
1211
C1122
C1133
C1123
C1131
C2222
C2233
C2223
C2231
C3322
C3333
C3323
C3331
C2322
C2333
C2323
C2331
C3122
C3133
C3123
C3131
C1222
C1233
C1223
C1231
C1112 ε11 C2212 ε22 C3312 ε33 . C2312 ε23 C3112 ε31 C ε 1212
(5.8)
12
The coefﬁcient 2 before the components ε is introduced to compensate for the fact that these components appear twice in the summation:
s1 C11 s2 C12 s3 C13 ¼ s4 C14 s C 5 15 s C 6
16
C12
C13
C14
C15
C22
C32
C42
C25
C23
C33
C43
C35
C24
C34
C44
C45
C52
C35
C45
C55
C62
C36
C46
C56
C16 ε1 C26 ε2 C36 ε3 . C46 2ε4 C56 2ε5 C 2ε 66
(5.9)
6
When the material has rotational invariance around a particular axis, it is called transverse isotropic. A medium is said to be transversely isotropic with an axis Xi if it is isotropic in all planes orthogonal to Xi.
Numerical modeling of hybrid composite materials
65
If the axis of the ﬁbers corresponds to the axis X1 (the case of the folds of a composite material with parallel ﬁbers embedded in a matrix), in the composite stiffness matrix, there remain only ﬁve independent coefﬁcients:
C11 C12 C13 0 0 0
C12
C13
0
0
C22
C32
0
0
C23
C33
0
0
0
0
ðC22 C23 Þ=2
0
0
0
0
C66
0
0
0
0
0 0 . 0 0 C66 0
(5.10)
In the case of an isotropic material where the response to a stress remains unchanged in all directions, the elastic properties are identical and the stiffness matrix contains only two independent coefﬁcients:
2
C11
6 6 C12 6 6 6C 6 12 C¼6 6 60 6 6 60 4 0
3
C12
C12
0
0
0
C11
C12
0
0
0
C12
C11
0
0
0
0
0
ðC12 C12 Þ=2
0
0
0
0
0
ðC12 C12 Þ=2
0
0
0
0
0
ðC12 C12 Þ=2 (5.11)
7 7 7 7 7 7 7 7. 7 7 7 7 7 7 5
These coefﬁcients are expressed as a function of Lame’s constants as follows: C12 ¼ l ðC11 C12 Þ ¼ m. 2
(5.12)
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
The coefﬁcients E and n are directly accessible during a tensile test: E¼
mð3l þ 2mÞ ; ðl þ mÞ
v¼
l . 2ðl þ mÞ
(5.13)
Representation This step consists of repeating the number of phases of the composite material, characterizing the behavior of each phase that is taken as a homogeneous continuous medium (stiffness coefﬁcients, nature of interaction between phases, etc.), and ﬁnally giving an idea on the morphology of the phases (shape, distribution, etc.). The representative elementary volume considered is of a material with a heterogeneous linear elastic structure, which occupies a domain U of boundary vU.
Location This consists in establishing the link between microscopic and macroscopic characteristics. We then formulate either a situation with homogeneous constraints or a situation with homogeneous deformations. Situation with homogeneous deformation To ﬁnd the relationship between the macroscopic and microscopic characteristic quantities, we ﬁrst have to deﬁne the loading on the domain U of the REV from the macroscopic quantities: vy vU. vx The displacement condition applied to vU is: uðxÞ ¼ E$x where E is the homogeneous macroscopic deformation and uðxÞ is the displacement at the microscopic scale. See Fig. 5.4. To deﬁne a homogenized behavior, forces of volume and acceleration are supposed to be inﬁnitely small, which allows the following equation to be written:
u(x) = E.x ∂Ω
x ε(u) Ω
Figure 5.4 Situation with homogeneous deformation.
Numerical modeling of hybrid composite materials
(
uðxÞ ¼ E$xðvUÞ divsðxÞ ¼ 0ðUÞ
.
67
(5.14)
Taking into account Eq. (5.14), and the relationship between the displacement and deformation ս and Gauss’s theorem, we have: 1 2
Z U
1 ui;j þ uj;i dU ¼ 2
Z ðui nj þ uj ni ÞdS vU
Z Z 1 ¼ Eik xk nj dS þ Ejk xk ni dS 2 vU vU ¼
Z Z 1 xk;j vU þ Ejk xk;i vU ¼ jUjEij Eik 2 vU vU
0hεi ¼
1 jUj
Z U
ε x dU ¼ E. H
(5.15)
(5.16)
i where hi represent the average operator on U, սij is the gradient of displacement du duj , and ni is the component of the normal vector of the surface. Eq. (5.5) reﬂects that the macroscopic homogeneous deformations E applied to the REV contour are equal to the average of local deformations in the REV. P We deﬁne the macroscopic constraints by the following relation: 1 S ¼ hsi ¼ jUj
Z U
sðxÞdU.
(5.17)
The problem with localization is linear elasticity since it is considered here that the linear elastic materials: 8 > > > > < sðxÞ ¼ CðxÞ: εðxÞ divsðxÞ ¼ 0 > > > > : uðxÞ ¼ E$x
ðUÞ ðUÞ .
(5.18)
ðvUÞ
The microscopic deformation ﬁeld εðxÞ is a linear function of the tensor of homogeneous macroscopic deformation (E) since we are dealing with a linear elastic material: εðxÞ ¼ A: E
(5.19)
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
where A denotes the localization tensor, which has the following characteristics: • • •
The tensor A has an average equal to the identity tensor A ¼ I A is minor symmetric Aijkl ¼ Ajikl A is not major symmetric Aijkl s Aklij
Situation with homogeneous stress In the case of homogeneous system, the solicitation are expressed as follows (Fig. 5.5): 8 < sðxÞ$nðxÞ ¼ SðxÞ$nðxÞ : divsðxÞ ¼ 0
ðUÞ ðvUÞ
.
(5.20)
Considering this last equation and Gauss’s theorem, we show: Z U
Z sij dU ¼
Z
Z ðsik xj Þ;k dU ¼
U
ðsik xj nk ÞdS ¼ Sik vU
xj nk dS vU
Z ¼ Sik
U
xj;k dU ¼ Sik jUjdij ¼ jUjSik djk ¼ jUjSij .
(5.21)
This makes it possible to conclude that the macroscopic homogeneous stresses applied to the contour of REV are equal to the average of the local stresses in REV. We then have: 1 0hsi ¼ jUj
Z U
sðxÞdU ¼ S.
(5.22)
We deﬁne the macroscopic deformations E by the following relation: E ¼ hεi ¼
1 jUj
Z U
εðxÞdU.
(5.23)
n ∂Ω
(x) Σ.(n)
σ(x)
Ω
Figure 5.5 Situation with homogeneous stress.
Numerical modeling of hybrid composite materials
69
Since we have a homogeneous elastic material problem, it allows us to write: 8 > > > > < sðxÞ ¼ CðxÞ: εðxÞ divsðxÞ ¼ 0 > > > > : sðxÞ:n ¼ S:n
ðUÞ ðUÞ .
(5.24)
ðvUÞ
The microscopic stress ﬁeld sðxÞ is a linear function of the tensor of homogeneous macroscopic stress (S) since we are dealing with the case of a linear elastic material: sðxÞ ¼ B : S
(5.25)
where B denotes the localization tensor, which has the following characteristics: • • •
The tensor B has an average equal to the identity tensor B ¼ I B is minor symmetric Bijkl ¼ Bjikl B is not major symmetric Bijkl s Bklij
Homogenization For any statically admissible stress ﬁeld sðxÞ and any deformation ﬁeld εðxÞ that is admissible, we have the relation: hsðxÞ: εðxÞi ¼ hsðxÞi: E.
(5.26)
We deduce for the contour condition compatible with a homogeneous stress that: hsðxÞ: εðxÞi ¼ S: E.
(5.27)
And it is known that the potential energy in the heterogeneous material is preserved by a change of scale: Z U
ðsðxÞ: εðxÞÞdU ¼ sðxÞ: εðxÞU ¼ S: ε:U ¼ E : S:U
(5.28)
provided that the macroscopic deformation tensor is deﬁned by: hεi ¼ S.
(5.29)
Conversely, in the dual problem, the conservation of the energy between the scales is veriﬁed provided that we take for macroscopic stress the average stress of volume: s ¼ S.
(5.30)
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
This shows that the deﬁnition of the macroscopic stress and strain from average volume is compatible with the conservation of energy and vice versa. Consider an REV of the composite material of volume V and boundary S, containing a volume of n phases of elementary volume Ui. The phase matrix of volume ðD Si Ui Þ will be designated by the index 0 (Fig. 5.6). The application of one of the boundary conditions allows the determination of one or the other of the dual responses; in other words, if the boundary conditions are of the stress type, then it is the deformation that must be calculated. Thus the elastic tensor of the equivalent homogeneous material can be established: S ¼ C c $E
(5.31)
where Cc is the elastic tensor of composite material. The problems related to composite materials are deﬁned at the structural level where the materials are homogeneous by phase, and then the REV model can be used to express the equivalent characteristic tensors as a function of the volume fraction f of each phase: S ¼ Sni¼0 f i si ¼ Sni¼0 f i Ci εi
(5.32)
E ¼ Sni¼0 f i εi .
To distinguish the matrix phase with index m, the previous equations can be written as follows: S ¼ f m C m εm þ Sni¼1 f i C i εi
(5.33)
E ¼ f m εm þ Sni¼1 f i εi .
S
Ωn REV
Ω2 Ω1
V Figure 5.6 Representative elementary volume (REV) with n phases.
Numerical modeling of hybrid composite materials
71
This allows us to write: S ¼ C m E þ Sni¼1 f i Ci Cm εi
(5.34)
for a given phase i, and by using Eq. (5.3), which connects the stress and the deformation by means of the stiffness and ﬂexibility tensors, Eq. (5.31) can be rewritten as follows: (
S ¼ f m sm þ Sni¼1 f i si E ¼ f m Sm sm þ Sni¼1 f i Si si
.
(5.35)
By combining these two equations we have: E ¼ Sm S þ Sni¼1 f i Si Sm si .
(5.36)
By introducing Eqs. (5.19), (5.25), and (5.27) into Eqs. (5.34) and (5.36) we get: 8 < C c ¼ C m þ Sni¼1 f i C i C m Ai . : Sc ¼ Sm þ S n f i Si Sm B i i¼1
(5.37)
Determination of the tensors of stiffness Cc or of ﬂexibility Sc of the composite material depends on the determination of the location tensors Ai and Bi relative to the different phases of the composite. All the modeling approaches use the same approach mentioned previously, but they differ in the determination of the localization tensors.
5.3.2.3
Bounding models
Bounding techniques to isotropic composite materials in the presence of constituents having nondeﬁnite moduli have proved to be powerful and robust tools in practical applications because the bounds can yield useful estimates for rigorous upper and lower limits on the mechanical properties of linear elastic multiphase materials given the composition [81,82]. These types of models generally concentrate on the prediction of the upper and lower bounds of the module rather than determining formulae for the exact calculation of the module, which cannot go any higher than an upper bound and vice versa for a lower bound [83].
Onepoint bounds The onepoint bound models allow a framing of the effective tensor of the rigidity of the composite materials based on the semicrystalline polymer and between two extreme limits: upper bound (Voigt) [62,81,84] and lower bound (Reuss) [49,62,85]. These two classical bounds are the simplest approaches used to compute the effective elastic properties because they require few data at the microstructure
72
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
level; the volume fraction fi of the amorphous phase and stiffness tensors are sometimes sufﬁcient. The Voigt model assumes the uniformity of strain and the Reuss model assumes uniformity of stress in composite materials. Voigt model The Voigt model assumes that the deformation in each phase εi is identical to the applied macroscopic deformation E [12,86]: εi ¼ E.
(5.38)
This means that, according to Eq. (5.19), the localization tensor Ai ¼ I, and from Eq. (5.37) we can write: C c ¼ C 0 þ Sni¼1 f i C i C 0 .
(5.39)
In the case of a composite with a single reinforcement element, the stiffness tensor is expressed as follows: Cc ¼ C0 1 f i þ f i Ci .
(5.40)
Then the stress tensor: sc ¼ si f i þ 1 f i s0
(5.41)
where sc designates the stress of the composite, s0 designates the stress in the matrix, si designates the stress in phase i, and fi designates the volume fraction of phase i. Reuss model This model is the complement of the Voigt model, see Fig. 5.7. The model of Reuss assumes that stress is constant, which means [49,51,87]: si ¼ S.
(5.42)
(a)
σ
σ
(b)
σ
σ
Figure 5.7 Schematic of (a) the Voigt (isostrain) and (b) the Reuss (isostress) limit.
Numerical modeling of hybrid composite materials
73
According to Eq. (5.19), the localization tensor Bi ¼ I, and from Eq. (5.37) we can write: Sc ¼ S0 þ Sni¼1 f i Si S0 ¼ Sni¼0 f i Si .
(5.43)
In the case of a composite with a single reinforcement element, the ﬂexibility tensor is expressed as follows: Sc ¼ S0 þ f i Si S0 ¼ S0 1 f i þ f i Si .
(5.44)
Then the deformation tensor: E c ¼ εi f i þ 1 f i ε0 .
(5.45)
VoigteReusseHill average A common simplistic approach to estimating the effective moduli is to calculate the average of the Voigt and Reuss bounds [85]. The Voigte ReusseHill averages are deﬁned as the arithmetic mean of the Voigt and Reuss average, and often provide reasonable estimators of the effective constants once the elastic constants of anisotropic composites are known [85]. The VoigteReusseHill estimates are frequently employed as semiempirical tools.
Twopoint bounds Hashin and Shtrikman model Hashin and Shtrikman optimal bounds are the tightest constraints possible and can be determined without a detailed description of the microstructure of a twophase material (the size and shape of phases of the composite and their spatial distribution) [88,89]. The Hashin and Shtrikman model is the most popular bound (Fig. 5.8), as it only requires the volume fractions of the phases to calculate the rigorous upper and lower bounds of elasticity tensor of any class of composite material (namely, isotropic microstructures) [76,90]: h i1 RðHSþÞ ¼ RI þ cM ðRM RI Þ1 þ cI S I R1 I h RðHSÞ ¼ RM þ cI ðRI RM Þ1 þ cM S
5.3.2.4
1 M RM
i1
.
(5.46)
Semiempirical models
Semiempirical relations are also called semiphysical relations and are mostly based on parameters that have physical signiﬁcance [40,91]. Fitting parameters or the correcting factors are involved in these relations for easy design procedure [76,92].
Modiﬁed ruleofmixture Investigations show that the obtained results by the ruleofmixture model do not agree well with experimental and ﬁnite element data in the case of random ﬁller composites
74
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
120 Voigtreusshill
Stiffness (GPa)
100 80
Hashinshtrikman (upper and lower)
Voigt
60 40 Reuss
20 0
0
0.2
0.4
0.6
0.8
1
Volume fraction
Figure 5.8 Comparison of the bounds on overall composite stiffness.
because it does not take into account the ﬁller orientation and ﬁller/matrix interaction effects [87,93,94]. For this reason, Curtis and coworkers modiﬁed the ruleofmixtures into the following form: 1 hf $V f hm $V m ¼ f þ E22 Em E22
(5.47)
where factors hf, hm are calculated as:
hf ¼
f E22 $V f þ
f E22 $V f þ Em $V m
h hf ¼
h i f 1 vf12 $vf21 $Em þ vm $vf21 $E11 $V m
i f f 1 vm2 $E11 1 vm$v12 $Em $V f þ Em $V m f E22 $V f þ E m $V m
Vf
þ
h0 $V m Gm
Gf 1 ¼ 12 G12 V f þ h0 $V m
with 0 < h0 < 1, but it is preferable to take h0 equal to 0.6.
(5.48)
Numerical modeling of hybrid composite materials
75
HalpineTsai’s method Equations expressing the HalpineTsai model make it possible to predict, in a simple and semiempirical manner, the moduli of a composite reinforced by short aligned ﬁbers [95]:
1 þ znf i 1 nf i
1 þ zn0 f i sc ¼ s0 1 n0 f i E c ¼ ε0
(5.49)
where:
εi ! 1 0 takes account of the modulus and the strength of the reinforcements; n ¼ εi ε þz ε0 si ! 1 0 n0 ¼ s i takes account of the modulus and the resistance of the matrix. s þz s0 For rigid inclusions n ¼ 1. For a homogeneous material n ¼ 0. For voids n ¼ 1z. Ec is the transverse laminate modulus. The coefﬁcient z is an adjustable parameter generally of the order of unity that depends on the measure of reinforcement of the composite material (the geometry of the ﬁbers, their arrangement, and the loading condition), given to the matrix by the presence of the ﬁbers. Generally, z must be determined experimentally, which presents a weakness of the HalpineTsai equations. Theoretically, z can vary between 0 and N. It is easy to see that for z ¼ 0, we get the typical result of the Reuss model representing the inferior limit: Ec ¼
εi ε0 . ε0 f i þ ð1 f i Þεi
(5.50)
In contrast, for z ¼ N, we have the following relationship, which is the typical result of the parallel type and model that represents the superior limit: E c ¼ εi f i þ 1 f i ε0
(5.51)
z represents a measure of the reinforcement given to the matrix by the elements of reinforcement; the contribution of these elements to the structural reinforcement of the matrix increases with the increase in the parameter z.
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
In the procedure for homogenization, the resolution of localization problem calls for different approaches that allow an estimation of material behavior or the determination of extreme limits of this behavior narrower than the limits of Voigt and Reuss.
Hirsch’s method The Hirsch model is a combination of the two previous models (Voigt and Reuss) and introduces the adjustable parameter (x), which determines the transfer of stresses between the reinforcement elements and the matrix [49,96]. This parameter is, however, considered to be mainly determined by the orientation of the ﬁbers, the length of the ﬁbers, and the concentration of the stresses at the ends of the ﬁbers [94]. Young’s modulus and composite stress are determined by the following equations: E c ¼ x εi f i þ 1 f i ε0 þ ð1 xÞ sc ¼ x εi si þ 1 f i s0 þ ð1 xÞ
5.3.2.5
εi ε0 ε0 f i þ ð1 f i Þεi
f i s0
si s0 . þ ð1 f i Þsi
(5.52)
(5.53)
Homogenization models
Analytical homogenization models are based on the separation of scales between the overall mechanical response of the composite material, belonging to the macroscopic scale, and the mechanical behavior of the heterogeneities, belonging to the microscopic scale information (e.g., constituent properties, volume fraction, shape, orientation, etc.) [97]. The purpose of homogenization models is to derive the relationship between stress and strain at the macroscopic scale (S and E) and those at the microscopic scale (s(x) and ε(x)) to predict the effective mechanical properties of heterogeneous materials [27,93].
TsaiePagano’s method Based on the orthotropic elasticity theory, this model introduces a term multiplier 4 that is a function of the type of arrangement of the reinforcements and allows the determination of the modulus of a short ﬁber isotropic composite in the plane [7,98,99]:
1 þ znf i ; 1 nf i
(5.54)
1 þ zn0 f i s ¼s 1 n0 f i
(5.55)
E c ¼ ε0
c
0
Numerical modeling of hybrid composite materials
77
1 4max 0 4¼1þ V 42max
εi ! 1 0 where z ¼ K 1 and K ¼ 1 þ 2ldn ¼ ε i ε þz ε0 For a square arrangement of the ﬁbers: 4max ¼ 0:785. For a random arrangement of the ﬁbers: 4max ¼ 0:82. Shape factor of the ﬁber: dl . Considering the theory of orthotropic elasticity and for the cases of composites with short ﬁbers of random distribution [100], Tsai and Pagano propose the following formulation: 3 5 Ec ¼ E k þ E t. 8 8
(5.56)
By referring to the models of Voigt and Reuss, in the case of a unidirectional composite, this equation can be written: 3 c 5 c Ec ¼ EVoigt þ EReuss . 8 8
(5.57)
Eshelby’s method The Eshelby method is the point of reference for many micromechanical models. The idea is based on the resolution of an elementary conﬁguration; it is a solution of the inclusion problem in isotropic solids in a ﬁrst time then Eshelby also pointed out that to obtain explicit expressions analogous to those for heterogeneity composites [101,102]. This method can be done in three steps. The ﬁrst step considers the inclusion immersed in the matrix. The inclusion is deﬁned such that its mechanical characteristics are identical to those of the matrix [103]. For the second step, this time it will be a heterogeneity immersed in the matrix. In this case, the heterogeneity possesses mechanical characteristics different from those of the matrix [68]. The last step will be to extend the previous case to composites, where the matrix contains several heterogeneities. Inclusion problem The inclusion undergoes a deformation called free of constraint. This deformation εL would not cause stress if it were applied to the same material, taken separately from its reference medium D. However, in the reference medium the matrix disturbs this perturbation εL , which induces a stress and deformation ﬁeld at any point in the domain D (Fig. 5.5). Indeed, determination of the expression of these ﬁelds is the major object of this kind of study. To do this, two methods can be proposed: one analytic and the other physical based on a logical sequence of events (Fig. 5.9). Intuitive approach Eshelby proposed an approach based on a sequence of logical and physical steps. These steps are shown schematically in Fig. 5.10.
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
Inclusion Cm CI , Matrix
Figure 5.9 Inclusion problem.
The ﬁrst step is the ﬁctitious division of inclusion. This one came out of its reference medium. The second step that is used for expressing the free stress strain e*. The third step is to replace the inclusion in its reference medium. To do this, it is necessary to apply to the latter a force enabling it to restore its initial size. The last step is the relaxation of this force once the inclusion is repositioned in its original environment. A deformation ﬁeld is created in the medium and in the inclusion. Equivalent inclusion method For the equivalent inclusion method, the inclusion has different mechanical characteristics from those of the reference medium. Then, the inclusion will be called heterogeneity. In other words, an elastic heterogeneity of module cI is embedded in a matrix of elastic moduli cm . In the vicinity of heterogeneity, the macroscopically applied stress is perturbed by the presence of heterogeneity. On the other hand, on a macroscopic scale, far from the heterogeneity, the perturbation is negligible. S and E are, respectively, the constraint applied to inﬁnity and the corresponding deformation, which are linked by the following relation: S ¼ C c E.
(5.58)
ε =εL
ε =0
ε = εI
Figure 5.10 The different steps for resolution.
Numerical modeling of hybrid composite materials
79
The perturbations of the stresses and deformations are noted, respectively, by spt and εpt , then in the neighborhood of the heterogeneity and in each of the two domains (inclusion and matrix), the stresses and the total deformations can be written: S þ spt
(5.59)
E þ εpt
(5.60)
which, taking into account Hooke’s law, will allow us to write: In the heterogeneity
S þ spt ¼ CI : ðE þ εpt Þ
In the matrix
S þ spt ¼ C m : ðE þ εpt Þ
.
(5.61)
To return to the previous case, the equivalent inclusion method assimilates the perturbation of constraints, linked to the presence of heterogeneity, by that caused by an inclusion. If we now consider that the whole domain D is homogeneous, the stiffness tensor, the heterogeneity then becomes an inclusion and, as in the previous case, undergoes a free strain deformation εL called transformation deformation. Then, Eq. (5.52) can be written: In the inclusion S þ spt ¼ C m : E þ εpt εL In the matrix
S þ spt ¼ Cm : ðE þ εpt Þ
.
(5.62)
For the two problems of inclusion and heterogeneity to be equivalent, the expression of the given constraints (Eqs. 5.61 and 5.62) must be identical: CI : ðE þ εpt Þ ¼ Cm : E þ εpt εL .
(5.63)
The problem of the inclusion undergoing a free strain deformation makes it possible to connect the Eshelby tensor to the perturbation distortion: εpt ¼ SEsh : εL
(5.64)
where SEsh is the Eshelby tensor connecting the deformation εpt , resulting from the accommodation of the deformation between the matrix and the inclusion to the free deformation εL . This allows expression of the transformation strain, after replacement in Eq. (5.63): εL ¼
1 I C C m E. C m C I SEsh Cm
(5.65)
The deformations in the heterogeneity can be obtained as a function of the transformation deformation Eq. (5.15) knowing that: εI ¼ E þ εpt and εpt ¼ SEsh : εL .
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
Then the deformations in the heterogeneity are: εI ¼ E þ εpt ¼ E þ SEsh εL 1 I εI ¼ E þ SEsh C m CI SEsh C m C Cm E h 1 I i εI ¼ I þ SEsh C m C I SEsh Cm C C m E.
(5.66)
h i1 This gives εI ¼ I þ SEsh ðCm Þ1 ðC I C m Þ E h i1 εI ¼ I þ SEsh ðCm Þ1 ðC I C m Þ E. So we can determine stresses in inclusion, knowing that: sI ¼ S þ spt ¼ C I εI ¼ C I ðE þ εpt Þ.
(5.67)
This allows us to write: h i1 sI ¼ C I I þ SEsh ðC m Þ1 CI Cm E h
s ¼C IþS I
I
Esh
I m 1
ðC Þ
C C
m
i1
(5.68) m 1
ðC Þ
S.
Then, every reinforcement deformation is given by Eq. (5.66). In other words, the reinforcement is considered to be embedded in a medium having the properties of the matrix alone and that this reinforcement does not undergo interactions with the others. It should be noted that this approach concerns only composite materials with a moderate reinforcement ratio. For this approach, deformations and stresses in the reinforcements are given by relations Eqs. (5.66) and (5.17). They are functions of macroscopic constraints and deformations applied to inﬁnity. Hence ions can deﬁne the localization tensors in deformations and stresses, respectively, AI and BI : h i1 AI ¼ I þ SEsh ðC m Þ1 CI Cm h i1 m 1 BI ¼ C I I þ SEsh ðC m Þ1 CI Cm ðC Þ .
(5.69)
The equivalent elastic properties of the composite material can then be determined if the expression of the locating tensors is known. The calculation can be done by replacing these localization tensors in the relations (Eq. 5.30): CC ¼ Cm þ
n X h I i1 m 1 C Cm f i C I C m I þ SEsh I ðC Þ I¼1
Numerical modeling of hybrid composite materials
SC ¼ Sm þ
n X h I i1 m 1 m 1 m f i SI Sm CI I þ SEsh ðC Þ C ðC Þ . C I
81
(5.70)
I¼1
Each reinforcement is called an Eshelby tensor. Indeed, an Eshelby tensor is a function of the geometrical nature of the reinforcement, its shape, and also its close environment. This is why, in determining the mechanical properties of the composite, Eshelby tensors are directly involved in the calculation.
MorieTanaka model To predict the effective properties of composites given by an explicit formula for the stiffness matrix, the MorieTanaka approach takes into account the interaction between the inclusions [69,93]. For this, the mean of deformation in the inclusions is estimated by that which is established in an inclusion of the same form and of the same elastic characteristics, immersed in a matrix subjected to inﬁnity at an average deformation E [87,101,104,105]. This model treats the elastic behavior of composite materials reinforced by heterogeneities of different shapes, orientations, and characteristics. In addition, the Morie Tanakasolving approach is not directly based on the research of localization tensors Ai and Bi [106]. The resolution process is based on research of the macroscopic strain E when a macroscopic stress ﬁeld is applied. Thus the elasticity tensor can be deduced by the relation: S ¼ C C $E.
The starting point is the reference medium (matrix) without reinforcements or loads. When the matrix is subjected to a stress S, a deformation E is created therein: S ¼ C m $E.
(5.71)
The second step consists in introducing the reinforcing elements into the reference medium, the n reinforcements (phases) in the matrix. Then, by the presence of these reinforcements, a constraint of perturbation is added to the initial constraint. In the matrix, the constraint ﬁeld is then expressed: sm ¼ S þ e s ¼ C m ðEm þ eεÞ εm ¼ ðEm þ eεÞ
(5.72)
where s and ε are the averages of the stress and deformation ﬁelds generated by the presence of the reinforcements. The third step consists in expressing the mean ﬁelds of deformation and stress in the reinforcement i:
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
si ¼ S þ e s þ sipt ¼ C i E m þ eε þ εipt εi ¼ E m þ eε þ εipt
(5.73)
where sipt and εipt are the constraints and disturbance deformations. Finally, to determine the strain ﬁelds εi , the method of equivalent inclusion is used. The reinforcement is replaced by an inclusion, with the same mechanical properties as the matrix, but undergoing a transformation deformation εi . Then Eq. (5.73) can be written as follows: si ¼ S þ e s þ sipt ¼ Ci Em þ eε þ εipt ¼ C m Em þ eε þ εipt εi
(5.74)
i With : εipt ¼ SEsh i ε
(5.75)
is the Eshelby tensor. where SEsh i From Eqs. (5.74) and (5.75) it is possible to determine εi : εi ¼ Li ðE m þ eεÞ With :
Li ¼
(5.76)
h
i1 m C m C i SEsh Ci Cm . C i
(5.77)
The homogenizing relation Eq. (5.33) is used. This means that the average of the stresses in all the phases taken on the representative elementary volume is equal to the applied macroscopic stress. This can be written considering Eqs. (5.73) and (5.75): S ¼ f m sm þ
n X
f i si ¼ f m Cm ðEm þ eεÞ þ
i¼1
n X
f i C m E m þ eε þ εipt εi .
i¼1
(5.78) Eq. (5.78) is simpliﬁed by using Eqs. (5.71) and (5.75), which allows it to express eε as follows: eε ¼
n n X X f i εipt εi ¼ f i SEsh I εi . i i¼1
(5.79)
i¼1
Note that f m þ
n P
f i ¼1.
i¼1
Taking into account Eqs. (5.77) and (5.79) we can write: "
n X eε ¼ I þ f i SEsh I Li i i¼1
#1 "
# n X f i SEsh I Li E m . i i¼1
(5.80)
Numerical modeling of hybrid composite materials
83
The homogenizing relation Eq. (5.33) is used. The average of the deformations in all the phases taken on the representative elementary volume is equal to the macroscopic deformation E. Taking into account Eqs. (5.33) and (5.79) we can write: E ¼ f m ðE m þ eεÞ þ
n n X X f i E m þ eε þ εipt ¼ E m þ f i εi . i¼1
(5.81)
i¼1
By combining Eq. (5.80), (5.81), and (5.77) we can write: "
n X
E¼ Iþ
!" i i
f L
n X f i SEsh I Li Iþ i
i¼1
#1 # Em .
(5.82)
i¼1
According to the Hooke’s law (Eq. 5.71), the elasticity tensor of the homogeneous equivalent material can then be determined: " C ¼C c
m
Iþ
n X i¼1
!" i i
f L
n X f i SEsh I Li Iþ i
#1 #1 .
(5.83)
i¼1
The elasticity tensor of the homogeneous equivalent material is a function of the microstructural parameters of the material, the volume fraction, the mechanical characteristics of the matrix, and the reinforcements and their geometry. To treat the case of composites with randomly oriented reinforcements, the orientations are discretized in N families. Each of these N families of reinforcements has a particular orientation; they are therefore considered as N different phases.
5.4
Numerical modeling of the mechanical behavior of composite material
Numerical models are widely used in polymer composite applications as an effective means for investigating and predicting their mechanical properties through the development of powerful analysis software and computing devices [107]. There are various numerical techniques for solving practical engineering problems analytically with a reasonable degree of accuracy such as the ﬁnite elements method (FEM) [26,69], the ﬁnite difference method (FDM) [108,109], and the boundary element method (BEM) [110e112] in continuous mechanics, or the discrete element method as a discontinuous model [113,114] (Fig. 5.11). The numerical modeling method is the most ﬂexible tool for the veriﬁcation of the results calculated by other methods and for also studying the effect of changes in design and input parameters. In the next paragraphs, we will detail various numerical models in continuous mechanics.
84
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
Numerical methods Continuous models Boundary element method
Finite difference method
Discontinuous models
Finite elements method
Discrete element method
Figure 5.11 Flow chart of numerical techniques.
5.4.1
Finite difference method
In the literature survey, the FDM is a method of choice because it is easy to formulate and can solve partial differential equations effortlessly [109]. However, it is generally suitable for modeling problems with simple geometries (regular) due to the regularity of the grid structure, but it is really popular for solving partial differential equations of heat transfer, ﬂuid ﬂow, mass, and momentum transfer numerically. ! 1 vui vuj εij ¼ þ . (5.84) 2 vxj vxi Hooke’s law (stressestrain, constitutive equations): sij ¼
2my dij εkk þ 2mεij 12y
(5.85)
where m is the shear modulus, ʋ is Poisson’s ratio, and dij is the Kronecker delta function whose properties are: ( 0; i s j . (5.86) dij ¼ 1; i ¼ j Differential equations of stress: vsij þ fi ¼ 0 vxj where fi is the body force vector.
(5.87)
Numerical modeling of hybrid composite materials
5.4.2
85
Finite element method
Nowadays, the FEM, which involves elements throughout the volume, is used to replace all other numerical methods in most engineering problemsolving scenarios in part because of the augmented freedom one has in terms of reinforcement geometry, distribution, volume fraction of constituent phases, interface, and mechanical properties, and in part due to advances in computational power and userfriendly codes [115]. Finite element analyses is a numerical solution to problems with complex boundary shapes that can be expressed in mathematical language in the form of partial derivative equation systems to produce reliable results within the body of the structure [26]. With FEM, the problem under consideration is divided into a number of small regions of limited sizes known as the meshing operation [116]: X m
½km ½dm ¼ ½F
(5.88)
where m represents the element number, [k]m is the element stiffness matrix, [d]m is the element displacement vector, and [F] is the vector containing all the external forces. In ﬁnite element modeling, there are three different approaches: multiscale RVE [117] modeling, unit cell modeling [88], and objectoriented modeling [88], see Section 5.5.2.
5.4.3
Boundary element method
The BEM has been shown to be well suited in the structural simulation and characterization of composite bodies for several periods [118]. Unlike other existing numerical methods, this approach is a continuum mechanics method that involves solving boundary integral equations for the evaluation of stress and strain concentration, fracture mechanics, and contact analysis due to its high resolution for materials in complex stress states on the surface and easy modiﬁcation of geometry, as well as physical interpretation and simple implementation [110]. BEM uses elements only along the surface or boundary of the problem under consideration from micro to macroscale [119]. For the initial stress sij the approach is related to the ﬂowing displacement integral equation: cij ðx0 Þuj ðx0 Þ ¼
Z
Z Uij ðx0 ; xÞtj ðxÞdGðxÞ Tij ðx0 ; xÞuj ðxÞdGðxÞ G G Z þ Eikj ðx0 ; XÞsrjk ðXÞdUðXÞ U
(5.89)
where cij is a constant, which depends on the position of the collocation point, uj and tj are components of displacements and tractions, respectively, Uij, Tij, Ejki are fundamental solutions of elastostatics, x0 is the collocation point, X is a boundary point, and dU(X) is a domain point. Unlike the elastostatic case, Eq. (5.89) contains the domain plastic term, which depends on the unknown plastic stress srjk . To obtain the stress ﬁelds in the
86
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
domain the stress integral equation is used. For the initial stress approach the equation is: 0
sij ðx Þ ¼
Z
0
Z
Uijk ðx ; xÞtk ðxÞdGðxÞ Tijk ðx0 ; xÞuk ðxÞdGðxÞ G Z r 0 þ Eijkl ðx ; XÞskl ðXÞdUðXÞ þ Fijkl srkl ðx0 Þ G
U
(5.90)
where Uijk, Tijk, Eijkl, and Fijkl are other fundamental solutions.
5.5
Numerical modeling of hybrid composite materials
The mechanical behavior of hybrid composite materials is studied using many numerical simulation methods, which is difﬁcult to task because of the complexity of their internal structure and interactions between constituents that consist of n phases operating together (n > 2) [120e122]. The hybridization approach was initially developed to reduce the cost of some of the ﬁbers in the 1960s and to explore their mechanical properties [123,124].
5.5.1
Fiber hybridization
Fiber hybridization is one of the few strategies that can lead to improved composite properties [62,125]. Fiberhybrid composites can be deﬁned as a composite material that contains more than one of type of ﬁber and/or matrix system. According to the conﬁgurations of both ﬁber types, three essential forms of ﬁberhybrid composites can be deﬁned, see Fig. 5.12: • • •
Interlayer: also called layerbylayer hybrids having different types of ﬁbers in different layers, where each layer only has a single ﬁber type [126]. Intralayer: also called yarnbyyarn hybrids having both types of ﬁbers in a single layer; the layers can be stacked in different conﬁgurations [126]. Intrayarn: also called ﬁberbyﬁber hybrids having both ﬁber types in a single tow; this type of conﬁguration leads to a better dispersion of both ﬁber types [126].
This section focuses on studying the effects of hybridization on the mechanical properties of composites. Starting with the tensile modulus of ﬁberhybrid composites,
Figure 5.12 Hybrid conﬁgurations: (a) interlayer, (b) intralayer, and (c) intrayarn conﬁgurations.
Numerical modeling of hybrid composite materials
87
which can be precisely predicted by a linear ruleofmixture, according to many researchers [127]: E ¼ Ef 1 Vf 1 þ Ef 2 Vf 2 þ Em Vm
(5.91)
where Ef1 and Ef2 are the longitudinal tensile moduli of both ﬁbers, Em is the matrix elastic modulus, and V are the volume fractions of the respective components. For example, the typical stressestrain diagram of a hybrid composite shows two distinct peaks (Fig. 5.13), which are associated with failure of the low and high elongation ﬁbers, respectively. In 1972, Hayashi was the ﬁrst to report a remarkable effect of sandwiching a carbon ﬁber layer in between glass ﬁber layers. The tensile failure strain of the carbon ﬁber layer increased by 40%, which meant that the ﬁrst peak shifted from εc to ε0c [128]. This increase is called the “hybrid effect”: Hybrid effect ¼
ε0c εc . εc
(5.92)
With regard to the failure strain, the hybrid effect is expected to occur. In 1977, Zweben was the ﬁrst author to extend shear lag models for unidirectional composites to hybrid composites with the intention of predicting the hybrid effect for failure strain [129,130]. Zweben derived an analytical expression for the strain concentrations and ineffective length in both packings, with alternating low elongation (LE) [131] and high elongation (HE) [132,133] ﬁbers, as illustrated in Fig. 5.14. The strain concentration factor k was deﬁned as the ratio of the strain in a ﬁber next to a single broken ﬁber over the applied strain that depends on r, which is the ratio of normalized stiffness of both ﬁbers: r¼
ELE $ALE EHE $AHE
(5.93)
LE failure Stress
LE
co
mp
os
ite
Hybrid composite
HE
co
m
p
it os
e
Positive hybrid effect
ε c ε ′c
Strain
Figure 5.13 Schematic stressestrain diagram of a hybrid composite and its two reference composites. HE, high elongation; LE, low elongation.
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
HE fibre
LE fibre
Figure 5.14 Schematic representation of 1D alternating packing ﬁbers (low elongation [LE] and high elongation [HE]).
where ELE and EHE are the Young’s modulus of the LE and HE ﬁbers and ALE and AHE are the crosssectional areas of both ﬁber types. The ineffective length d for the hybrid composite can be determined as: d¼F
ELE SLE dm Gm tm
1=2 (5.94)
where dm and tm are the width and thickness of the matrix region between the ﬁbers, Gm is the matrix shear modulus, and F is a factor that solely depends on r. Combining Eqs. (5.93) and (5.94) with the Weibull distributions for ﬁber strength yields (Eq. 5.95) for the hybrid effect Rhyb:
Rhyb
εh;c ¼ ¼ εLE;c
sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ" q #1=2m εHE;f dh $ kh 1 $ εLE;f 2$d$ðkq 1Þ
(5.95)
where εLE and εHE are the mean failure strains of the LE and HE ﬁbers at the considered gauge length and q is the Weibull shape parameter of the ﬁbers, which was considered to be equal in both ﬁber types. The Zweben model can be applied not only to unidirectional hybrid composites but also to more complex situations of multidirectional hybrids. In 1984 Fukuda modiﬁed the model developed by Zweben for understanding the statistical effect in hybrid composites, according to three other intrinsic shortcomings encountered in that model; however, he still used the same basic geometry [129]. First, Fukuda’s model revealed that the failure strain of the high elongation ﬁber is not a
Numerical modeling of hybrid composite materials
89
realistic criterion for hybrid composites. Second, in hybrid composites the strain concentration factor depends only on the ratio r of the normalized stiffnesses of the two ﬁber types. Finally, the Zweben model predicts stress concentration factors smaller than Hedgepeth’s solution. The latter argument may not be a valid one as mentioned by Fukuda [134]. Addressing these three shortcomings, Fukuda obtained Eq. (5.96) for the enhancement of the LE composite failure strain: "
Rhyb
q #1=2m dh $ kh 1 ¼ . 2$d$ðkq 1Þ
(5.96)
As seen, the Fukuda equation is similar to Eq. (5.95), but with two signiﬁcant differences. First, the ratio of the failure strains of the two ﬁbers is no longer included in this model. This would mean that the failure strain of HE ﬁbers does not affect the hybrid effect. Second, stress concentrations and ineffective lengths have been calculated more precisely. All of the reported hybrid effects models were found in unidirectional composites, which are even more difﬁcult to test than multidirectional composites. Table 5.1 summarizes some of the numerical models of ﬁber hybrid composite materials.
5.5.2
Hybrid particulate ﬁberreinforced composites
Hybrid particulate matrix ﬁberreinforced composites consisting of an isotropic polymer matrix reinforced by both particles and short or long ﬁbers offer the potential for simple functional grading to tailor mechanical properties that are critically determined by the shape, volume fractions, size, and interactions between the ﬁbers and particles [76]. The mechanical properties of the hybrid composites are required for their engineering applications. In the last few decades, the use of numerical micromechanical modeling as an RVE [135,136] or a representative unit cell (RUC) [137] or objectoriented modeling of the composite microstructure appears to be well suited to describe the mechanical behavior of hybrid particulate matrix ﬁberreinforced composites [138], as seen in Fig. 5.15.
5.5.2.1
Representative volume element
RVE [135,139] is a statistical representation of typical material properties that connect between the macroscopic properties of materials with the properties of the microscopic constituents and microscopic structures of the materials [79]. Nowadays, the RVE is widely used to describe the mechanics of macroscopic structures of heterogeneous materials and plays a central role when predicting their effective properties [27]. All deﬁnitions reveal that the RVE should contain enough information on the microstructure and should be sufﬁciently smaller than the macroscopic structural
90
Table 5.1 Overview of the hybrid effect of failure strain References
Year
Fibers
Conﬁguration
UD/ MD
Vf ratio
Hybrid effect (%)
Hayashi
1972
Carbon/glass
Interlayer
UD
25/75
þ45
Bunsell & Harris
1974
Carbon/glass
Interlayer
UD
33/67 to 50/50
þ42 to þ84
Very low failure strain for LE ﬁbers measured, and short gauge lengths used (50 mm)
Perry & Adams
1975
Carbon/glass Carbon/Kevlar
Interlayer
UD/ MD
86/14 82/18
þ12 þ14
HE ﬁber was under 45 degrees
Aveston & Sillwood
1976
Carbon/glass
Interlayer
UD
10/90
þ116
Ultimate failure strain was used to calculate the hybrid effect
Phillips
1976
Carbon/glass
Interlayer
UD
20/80 25/75 33/67 50/50
þ20 þ17 þ13 þ2
Zweben
1977
Carbon/Kevlar
Interlayer
UD/ MD
50/50 50/50
þ4 þ32
Manders & Badeer
1981
Carbon/glass
Interlayer
UD
5/95 to 50/50 6/94 to 0.4/99.6
42 to þ16 þ30 to þ52
0.4/99.6 was achieved by a carbon tow in between HE layers
Chamis
1981
Carbon/glass Carbon/Kevlar
Interlayer Interlayer
UD UD
70/30 to 90/10 70/30 to 90/10
42 to þ16 66 to þ10
High fractions of LE ﬁber
Remarks
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
1991
Carbon/ polyethylene
Interlayer
20/80 to 80/20
Yerramalli &Waas
2003
Carbon/glass
Interlayer
You Yj
2007
Carbon/glass
Interlayer Intrayarn
UD UD
47/53 47/53
þ9 to þ27 þ14 to þ33
Taketa
2010
Carbon/ polypropylene
Interlayer
MD
31/69 to 60/40
þ7 to þ18
Pandya
2011
Carbon/glass
Interlayer
MD
45/55
þ36 to þ90
Ultimate failure strain was used to calculate the hybrid effect
Diao
2012
Carbon/carbon
Intrayarn
UD
34/66
8
Carbon ﬁbers with different failure strains. Negative effect attributed to processinduced damage
Zhang
2012
Carbon/glass
Interlayer
MD
25/75 to 50/50
þ10 to þ32
Jagannath & Harish
2015
Carbon/glass
Interlayer
MD
0/60 to 60/0
þ35 to þ68
30/70 Ultimate failure strain was used to calculate the hybrid effect
Numerical modeling of hybrid composite materials
UD
þ5 to þ12
Peijs
The column UD/MD indicates whether the composites were unidirectional (UD) or multidirectional (MD). The hybrid effect is calculated as the relative failure strain enhancement of the carbon ﬁbers in the hybrid composites compared to their failure strain in an allcarbon ﬁber composite [132]. HE, high elongation; LE, low elongation.
91
92
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
Figure 5.15 Generated preform microstructural models for (a) a hybrid composite with short ﬁbers and particles, (b) a hybrid composite with continuous ﬁbers and particles, and (c) ﬁberreinforced laminates with particles.
dimensions [79]. Scientists have deﬁned the RVE as a sample of a heterogeneous material as follows: •
•
•
•
• •
The deﬁnition of RVE given by Hill [140]: RVE is “a sample that is structurally entirely typical of the whole mixture, and which contains a sufﬁcient number of inclusions for the effective global modules to be effectively independent of the surface tensile and displacement values, provided that these values are macroscopically uniform.” The deﬁnition of RVE postulated by Trusov and Keller [141]: An RVE is the minimum size of the material volume, which contains enough statistical mechanisms of deformation processes. The increase in this volume should not lead to changes of evolution equations for ﬁeld values, describing these mechanisms. The deﬁnition of RVE postulated by Drugan and Willis that should be sufﬁciently larger than the microstructural size and the RVE is the smallest material volume element of the composite for which the usual spatially constant macroscopic constituting “global modulus” is sufﬁciently accurate models to represent the average constitutive response [142]. The RVE is a model of the material to be used to determine the corresponding effective properties for the homogenized macroscopic model. The RVE must be large enough to contain sufﬁcient information on the microstructure to be representative; however, it should be much smaller than the macroscopic body. This is known as the “micromesomacro principle” deﬁned by Hashin [143]. The RVE is deﬁned as the minimum volume of a laboratoryscale specimen, such that the results obtained from this specimen can still be regarded as representative of a continuum (Van Mier) [144]. OstojaStarzewski introduced a deﬁnition of RVE based on: (1) statistical homogeneity and ergodicity of the material; these two properties assure that the RVE is statistically representative of the macroresponse, and (2) some scale L of the material domain, sufﬁciently large relative to the microscale “d” (inclusion size) so as to ensure the independence of boundary conditions [117]. The RVE from OstojaStarzewski’s point of view is clearly deﬁned in two situations only: (1) the unit cell in a periodic microstructure, and (2) the volume containing a very large (mathematically inﬁnite) set of microscale elements (e.g., grains), possessing statistically homogeneous and ergodic properties (Evesque) [117,145].
All these deﬁnitions bring together the prediction of the hybrid particulate ﬁberreinforced composite’s mechanical properties, see Fig. 5.15.
Numerical modeling of hybrid composite materials
5.5.2.2
93
Representative unit cell
The RUC approach involves some oversimplifying assumption: the hybrid composite crosssection is perfectly periodic with stilling the building block of the composite that simpliﬁes the mathematics for numerical modeling and for a computationally efﬁcient method [137,146,147]. Most researchers assume that the RUC is the periodic RVE because it consists of a relatively large size (usually in micrometers) and contains a signiﬁcant number of ﬁllers (usually in tens to hundreds or more); both approaches are sometimes used interchangeably [146].
5.5.2.3
Objectoriented modeling
Objectoriented modeling developed by the National Institute of Standards and Technology is a relatively new approach of modeling and simulation that incorporates microstructure images such as scanning electron microscopy micrographs into ﬁnite element grids to accurately predict the overall mechanical properties of highly variable and irregular angular structures of ﬁllers, using approximation of simple geometrical particles that capture the complex morphology, size, and spatial distribution of the reinforcement [88,138].
5.6
Conclusion
Hybrid composite materials are increasingly utilized in many engineering applications because they offer a number of enhanced properties and various advantages over traditional composite materials. The mechanical properties of hybrid composites consist of n (n > 2) jointly working phases, which are very important. For this reason, the modeling of the mechanical properties of hybrid composites as mentioned previously is done by using a linear coupling of numerical simulation models. However, the mechanical behavior of hybrid composites depends not only on the character of a matrix and reinforcements but also on properties of the interface between these components and the matrix, which must be taken into consideration in the numerical modeling of the mechanical properties. Furthermore, the effect of environmental aging should be taken into account for numerical modeling of hybrid composite materials.
Acknowledgments This work was supported by MAScIR (Moroccan Foundation for Advanced Science, Innovation and Research), as well as MESRSFC and CNRST Morocco (Grant No. 1970/15).
References [1] Essabir H, Raji M, Essassi EM, Rodrigue D, Bouhﬁd R, Qaiss A el kacem. Morphological, thermal, mechanical, electrical and magnetic properties of ABS/PA6/
94
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
[2]
[3]
[4]
[5] [6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14] [15]
[16] [17]
SBR blends with Fe3O4 nanoparticles. J Mater Sci Mater Electron 2017;28(22): 17120e30. Raji M, El M, Mekhzoum M, Qaiss A. el K, Bouhﬁd R. Nanoclay modiﬁcation and functionalization for nanocomposites development: effect on the structural, morphological, mechanical and rheological properties. Nanoclay Reinf Polym Compo 2016:1e34. Raji M, Essabir H, Essassi EM, Rodrigue D, Bouhﬁd R, Qaiss A el K. Morphological, thermal, mechanical, and rheological properties of high density polyethylene reinforced with illite clay. Polym Polym Compos 2016;16(2):101e13. Sana A, Laaziz A, Raji M, Hilali E, Essabir H, Rodrigue D, et al. Biocomposites based on polylactic acid and argan nut shell: production and properties. Int J Biol Macromol 2017;104:30e42. Abdellaoui H, Bensalah H, Raji M, Rodrigue D, Bouhﬁd R. Laminated epoxy biocomposites based on clay and jute ﬁbers. J Bionic Eng 2017;14(2):379e89. El Mechtali FZ, Essabir H, Nekhlaoui S, Bensalah MO, Jawaid M, Bouhﬁd R, et al. Mechanical and thermal properties of polypropylene reinforced with almond shells particles: impact of chemical treatments. J Bionic Eng 2015;12(3). Essabir H, Nekhlaoui S, Bensalah MO, Rodrigue D, Bouhﬁd R, Qaiss AEK. Phosphogypsum waste used as reinforcing ﬁllers in polypropylene based composites: structural, mechanical and thermal properties. J Polym Environ 2016;25(3):658e66. Essabir H, Boujmal R, Bensalah MO, Rodrigue D, Bouhﬁd R, Qaiss AEK. Mechanical and thermal properties of hybrid composites: oilpalm ﬁber/clay reinforced high density polyethylene. Mech Mater 2016;98:36e43. Raji M, Essabir H, Bouhﬁd R, Qaiss A el kacem. Impact of chemical treatment and the manufacturing process on mechanical, thermal, and rheological properties of natural ﬁbersbased composites. In: Handbook of composites from renewable materials. Hoboken, NJ, USA: John Wiley & Sons, Inc.; 2017. p. 225e52. Boujmal R, Kakou CA, Nekhlaoui S, Essabir H, Bensalah MO, Rodrigue D, et al. Alfa ﬁbers/clay hybrid composites based on polypropylene. J Thermoplast Compos Mater 2017;7. Essabir H, Raji M, Bouhﬁd R, Qaiss A el K. Nanoclay and natural ﬁbers based hybrid composites: mechanical, morphological, thermal and rheological properties. Nanoclay Reinf Polym Compos 2016:29e49. Bensalah H, Gueraoui K, Essabir H, Rodrigue D, Bouhﬁd R, Qaiss AEK. Mechanical, thermal, and rheological properties of polypropylene hybrid composites based clay and graphite. J Compos Mater 2017. https://doi.org/10.1177/0021998317690597. Prasad A, Fotou G, Li S. The effect of polymer hardness, pore size, and porosity on the performance of thermoplastic polyurethanebased chemical mechanical polishing pads. J Mater Res 2013;28(17):2380e93. Hutchinson JM. Physical aging of polymers. Prog Polym Sci 1995;20(4):703e60. Esposito Corcione C, Prinari P, Cannoletta D, Mensitieri G. Maffezzoli a. Synthesis and characterization of claynanocomposite solventbased polyurethane adhesives. Int J Adhes Adhes 2008;28(3):91e100. Issa B, Obaidat IM, Albiss BA, Haik Y. Magnetic nanoparticles: surface effects and properties related to biomedicine applications. Int J Mol Sci 2013;14(11):21266e305. Universiti S, Universiti Z. Stiffness prediction of hybrid Kenaf/glass ﬁber reinforced polypropylene composites using rule of mixtures (ROM) and rule of hybrid mixtures (RoHM) stiffness prediction of hybrid Kenaf/glass ﬁber reinforced polypropylene composites using rule of. 2013.
Numerical modeling of hybrid composite materials
95
[18] Gamona G, Evona P, Rigala L. Twinscrew extrusion impact on natural ﬁbre morphology and material properties in poly(lactic acid) based biocomposites. Ind Crops Prod 2013;46: 173e85. [19] Babrekar HA, Kulkarni NV, Jog JP, Mathe VL, Bhoraskar SV. Inﬂuence of ﬁller size and morphology in controlling the thermal emissivity of aluminium/polymer composites for space applications. Mater Sci Eng B SolidState Mater Adv Technol 2010;168(1):40e4. [20] Wang B. Dispersion of cellulose nanoﬁbers in biopolymer based dispersion of cellulose nanoﬁbers in biopolymer based nanocomposites. 2008. 157. [21] Supov a M, Martynkova GS, Barabaszova K. Effect of nanoﬁllers dispersion in polymer matrices: a review. Sci Adv Mater 2011;3(1):1e25. [22] Geethamma VG, Joseph R, Thomas S. Short coir ﬁberreinforced naturalrubber composites e effects of ﬁber length, orientation, and alkali treatment. J Appl Polym Sci 1995; 55(4):583e94. [23] Rohini R, Katti P, Bose S. Tailoring the interface in graphene/thermoset polymer composites: a critical review. Polymer (UK) 2015;70:A17e34. [24] Tang LC, Wan YJ, Yan D, Pei YB, Zhao L, Li YB, et al. The effect of graphene dispersion on the mechanical properties of graphene/epoxy composites. Carbon NY 2013;60:16e27. [25] Kakou CA, Arrakhiz FZ, Trokourey A, Bouhﬁd R, Qaiss A, Rodrigue D. Inﬂuence of coupling agent content on the properties of high density polyethylene composites reinforced with oil palm ﬁbers. Mater Des June 2014;63:641e9 [cited 2014 Jul 11]. [26] Xu J, Lomov SV, Verpoest I, Daggumati S, Van Paepegem W, Degrieck J. A progressive damage model of textile composites on mesoscale using ﬁnite element method: fatigue damage analysis. Comput Struct 2015;152:96e112. [27] Wang Y, Huang Z. A review of analytical micromechanics models on composite elastoplastic behaviour. Procedia Eng 2017;173:1283e90. [28] Ashik KP, Sharma RS. A review on mechanical properties of natural ﬁber reinforced hybrid polymer composites. J Miner Mater Charact Eng 2015;3:420e6. September. [29] Zuhri MYM, Sapuan SM, Ismail N. Oil palm ﬁbre reinforced polymer composites: a review. Prog Rubber Plast Recycl Technol 2009;25(4):233e46. [30] Beyou E, Akbar S, Chaumont P, Cassagnau P. Polymer nanocomposites containing functionalised multiwalled carbon nanoTubes: a particular attention to polyoleﬁn based materials. Synth Appl Carbon Nanotub Their Compos 2013:77e115. [31] Ku H, Wang H, Pattarachaiyakoop N, Trada M. A review on the tensile properties of natural ﬁber reinforced polymer composites. Compos Part B Eng 2011;42(4):856e73. [32] Haque Z, Yadav S, Kumar S. Mechanical behaviour of COIR/GLASS ﬁber reinforced EPOXY based composites. 2016. p. 568e72. [33] Dragan K. Ageing of composites in the rotorcraft industry. Ageing Compos 2008: 311e25. [34] Bledzki A. Composites reinforced with cellulose based ﬁbres. In: Progress in polymer science; 1999. p. 221e74. [35] Tejado A, Pen C, Labidi J, Echeverria JM, Mondragon I. Physicochemical characterization of lignins from different sources for use in phenol e formaldehyde resin synthesis. Bioresour Technol 2007;98:1655e63. [36] Jose JP, Malhotra SK, Thomas S, Joseph K, Goda K, Sreekala MS. Advances in polymer composites: macro and microcomposites e state of the art, new challenges, and opportunities. In: polymer Composites First Edition. In: Thomas S, Joseph K, Malhotra SK, editors. Koichi Goda, and Meyyarappallil Sadasivan Sreekala; 2012. p. 1e16.
96
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
[37] Thakur VK, Thakur MK, Michael R. Kessler. Handbook of composites from renewable materials. Vol 1 Struct Chem 2017. 575 p. [38] Hussain F, Hojjati M, Okamoto M, Gorga RE. Review article: polymermatrix nanocomposites, processing, manufacturing, and application: an Overview. J Compos Mater 2006;40(17):1511e75. [39] Yu L, Dean K, Li L. Polymer blends and composites from renewable resources. Prog Polym Sci 2006;31(6):576e602. [40] Dahlen C, Springer GS. Delamination growth in composites under cyclic loads. J Compos Mater May 27, 1994;28(8):732e81 [cited 2017 Sep 27]. [41] Xia Z, Li L. Adv Ceram Matrix Compos 2014:1e709. [42] Surappa MK. Aluminium matrix composites: challenges and opportunities. Sadhana 2003;28(1e2):319e34. [43] Cox J, Luong D, Shunmugasamy V, Gupta N, Strbik O, Cho K. Dynamic and thermal properties of aluminum alloy A356/silicon carbide hollow particle syntactic foams. Metals (Basel) 2014;4(4):530e48. [44] Composites P, Properties M, Performance T. 4Polymermatrix composites: mechanical properties and thermal performance. In: Carbon composites. 2nd ed. Elsevier Inc.; 2017. 218e255 p. [45] Mohammed L, Ansari MNM, Pua G, Jawaid M, Islam MS. A review on natural ﬁber reinforced polymer composite and its applications. Int J Polym Sci. 2015;2015:1e15. [46] Verma D, Gope PC, Shandilya A, Gupta A, Maheshwari MK. Coir ﬁbre reinforcement and application in polymer composites: a review. J Mater Environ Sci 2013;4(2):263e76. [47] Abdellaoui H, Bouhﬁd R, Qaiss A el K. Clay, natural ﬁbers and thermoset resin based hybrid composites: preparation, characterization and mechanical properties. Nanoclay Reinf Polym Compos 2016:225e46. [48] Taj S, Munawar MA, Khan S. Natural ﬁberreinforced polymer composites. Proc Pakistan Acad Sci 2007:129e44. [49] Essabir H, Bensalah MO, Rodrigue D, Bouhﬁd R, Qaiss AEK. Biocomposites based on Argan nut shell and a polymer matrix: effect of ﬁller content and coupling agent. Carbohydr Polym 2016;143:70e83. [50] Chen T, Liu W, Qiu R. Mechanical properties and water absorption of hemp ﬁbersreinforced unsaturated polyester composites: effect of ﬁber surface treatment with a heterofunctionl monomer. BioResources 2013;8(2):2780e91. [51] Essabir H, Elkhaoulani A, Benmoussa K, Bouhﬁd R, Arrakhiz FZ, Qaiss A. Dynamic mechanical thermal behavior analysis of doum ﬁbers reinforced polypropylene composites. Mater Des October 2013;51:780e8 [cited 2014 Jul 11]. [52] Zampaloni M, Pourboghrat F, Yankovich SA, Rodgers BN, Moore J, Drzal LT, et al. Kenaf natural ﬁber reinforced polypropylene composites: a discussion on manufacturing problems and solutions. Compos A Appl Sci Manuf 2007;38(6):1569e80. [53] Arrakhiz FZ, El Achaby M, Benmoussa K, Bouhﬁd R, Essassi EM, Qaiss A. Evaluation of mechanical and thermal properties of Pine cone ﬁbers reinforced compatibilized polypropylene. Mater Des 2012;40:528e35. [54] Li Z, Zhang Y, Zhou X. Short ﬁber reinforced geopolymer composites manufactured by extrusion. J Mater Civ Eng 2005;17(6):624e31. [55] Curvelo AAS, De Carvalho AJF, Agnelli JAM. Thermoplastic starchcellulosic ﬁbers composites: preliminary results. Carbohydr Polym 2001;45(2):183e8. [56] Paul DR, Robeson LM. Polymer nanotechnology: nanocomposites. Polymer (Guildf) 2008;49(15):3187e204.
Numerical modeling of hybrid composite materials
97
[57] Elsevier F, Mech JNF, Engineering P. Talcthermoplastic compounds: particle orientation in ﬂow and rheological properties. 1996. p. 62. [58] Song SH, Park KH, Kim BH, Choi YW, Jun GH, Lee DJ, et al. Enhanced thermal conductivity of epoxygraphene composites by using nonoxidized graphene ﬂakes with noncovalent functionalization. Adv Mater 2013;25(5):732e7. [59] Ghasemi I, Kord B. Longterm water absorption behaviour of polypropylene/wood ﬂour/ organoclay hybrid nanocomposite. Iran Polym J 2009;18(9):683e91. [60] Jacob M, Thomas S, Varughese KT. Mechanical properties of sisal/oil palm hybrid ﬁber reinforced natural rubber composites. Compos Sci Technol 2004;64(7e8):955e65. [61] Zhang J, Chaisombat K, He S, Wang CH. Hybrid composite laminates reinforced with glass/carbon woven fabrics for lightweight load bearing structures. Mater Des 2012;36: 75e80. [62] Essabir H, Bensalah MO, Rodrigue D, Bouhﬁd R, Qaiss A. Structural, mechanical and thermal properties of biobased hybrid composites from waste coir residues: ﬁbers and shell particles. Mech Mater 2016;93:134e44. [63] Thwe MM, Liao K. Durability of bambooglass ﬁber reinforced polymer matrix hybrid composites. Compos Sci Technol 2003;63(3e4):375e87. [64] Abdul Khalil HPS, Jawaid M, Abur Bakar A. Woven Hybrid composites: water absorption and thicjness swelling behaviours. BioResources 2011;6(2):1043e52. [65] Thakur VK, Thakur MK, Pappu A. Hybrid polymer composite materials properties and characterisation. 2017. 420 p. [66] Ahzi S, Bahlouli N, Makradi A, Belouettar S. Mechanics of Materials and Structures composite modeling for the effective elastic. J Mech Mater Struct 2007;2:1e21. [67] Cocchieri E, Almeida R, José S, Paulo S. A review on the development and properties of continuous ﬁber/epoxy/aluminum hybrid composites for aircraft structures 2. Prod Metal Laminate Hybrid Compos 2006;9(3):247e56. [68] Valavala PK, Odegard GM. Modeling techniques for determination of mechanical properties of polymer nanocomposites. Rev Adv Mater Sci 2005;9(1):34e44. [69] Toledo MWE, Nallim LG, Luccioni BM. A micromacromechanical approach for composite laminates. Mech Mater 2008;40(11):885e906. [70] Carvelli V, Lomov SV. Fatigue of textile composites. 2015. 511 p. [71] Eichhorn SJ, Dufresne A, Aranguren M, Marcovich NE, Capadona JR, Rowan SJ, et al. Review: current international research into cellulose nanoﬁbres and nanocomposites. J Mater Sci 2010;45(1):1e33. [72] Dai G, Mishnaevsky L. Fatigue of hybrid glass/carbon composites: 3D computational studies. Compos Sci Technol 2014;94:71e9. [73] Ertas AH, Sonmez FO. Design optimization of spotwelded plates for maximum fatigue life. J Compos Mater 2011;47(4):413e23. [74] Asaad M. Damage accumulation in hybrid woven fabric composites. 2002. December. [75] Post N.L., Case S.W., Lesko J.J. Modeling the Residual Strength Distribution of Structural GFRP Composite Materials Subjected to Constant and Variable Amplitude TensionTension Fatigue Loading Modeling the Residual Strength Distribution of Structural GFRP Composite Materials Subjected to. 2005; (October 2014). [76] Annis DS, Mosher DF, Roberts DD. Micromechanics and structural response of functionally graded, particulatematrix, ﬁberreinforced composites. NIH Public Access 2009; 27(4):339e51. [77] Horrocks AR, Anand SC. Handbook of technical textiles. 2nd Edition. Technical Textile Applications; 2016. 442 p. [78] Horrocks AR, Anand SC. Technical textile applications. 2016. 442 p.
98
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
[79] Gitman IM. Representative volumes and multiscale modelling of quasibrittle materials. Master in Science thesis. Perm State Technical University; 2006. 127 p. [80] Gibson RF. Principles of composite material mechanics. 2012. 659 p. [81] Pinfold MK. Composite mechanical properties for use in structural analysis. 1995. p. 275. [82] Fu S, Feng X, Lauke B, Mai Y. Effects of particle size, particle/matrix interface adhesion and particle loading on mechanical properties of particulate e polymer composites. Compos Part B 2008;39:933e61. [83] Haddad S, Mokdad L, Youcef S. Bounding models families for performance evaluation in composite web services. J Comput Sci 2013;4(4):232e41. [84] Liu B, Feng X, Zhang SM. The effective Young’s modulus of composites beyond the Voigt estimation due to the Poisson effect. Compos Sci Technol 2009;69(13):2198e204. [85] Watt JP, Davies GF, O’Connell RJ. The Elastic Properties of Composite Materials. Rev Geophys Space Phys 1976;14:541e63. [86] Ramanathan T, Abdala AA, Stankovich S, Dikin DA, HerreraAlonso M, Piner RD, et al. Functionalized graphene sheets for polymer nanocomposites. Nat Nanotechnol 2008; 3(6):327e31. [87] Younes R, Hallal A, Fardoun F, Hajj F. Comparative review study on elastic properties modeling for unidirectional composite materials. Compos Their Prop 2012. https:// doi.org/10.5772/50362. [88] Hu H, Onyebueke L, Abatan A. Characterizing and modeling mechanical properties of nanocomposites review and evaluation. J Miner Mater Charact Eng 2010;9(4):275e319. [89] Kochmann DM, Milton GW. Rigorous bounds on the effective moduli of composites and inhomogeneous bodies with negativestiffness phases. J Mech Phys Solids 2014;71(1): 46e63. [90] Enikolopyan NS, Stalnova IO. Filled polymers : mechanical properties and processability. Adv Polym Sci 1990;96:1e67. [91] Ramakrishna S, Lim TC, Inai R, Fujihara K. Modiﬁed HalpinTsai equation for clayreinforced polymer nanoﬁber. Mech Adv Mater Struct 2006;13(1):77e81. [92] Alavinasab A. Nonlocal theory and ﬁnite element modeling of nanocomposites. 2009 (August):116. [93] Younes R, Hallal A, Fardoun F, Chehade FH. Comparative review study on elastic properties modeling for unidirectional composite materials. Compos Their Prop 2012: 391e408. [94] Sreekumar PA, Joseph K, Unnikrishnan G, Thomas S. A comparative study on mechanical properties of sisalleaf ﬁbrereinforced polyester composites prepared by resin transfer and compression moulding techniques. Compos Sci Technol 2007;67(3e4): 453e61. [95] Patterson W, Force A. HalpinTsai Eq Rev 1976;16(5). [96] Nekhlaoui S, Essabir H, Kunal D, Sonakshi M, Bensalah MO, Bouhﬁd R, et al. Comparative study for the Talc and two kinds of Moroccan clay as reinforcements in polypropyleneSEBSgMA matrix. Polym Compos 2014;10(1e10). [97] Muc A, Jamroz M. Homogenization models for carbon nanotubes. Mech Compos Mater 2004;40(2):101e6. [98] Essabir H, Bensalah MO, Rodrigue D, Bouhﬁd R, Qaiss AEK. A comparison between bio and mineral calcium carbonate on the properties of polypropylene composites. Constr Build Mater 2017;134:549e55. [99] Essabir H, Achaby ME, Hilali EM, Bouhﬁd R, Qaiss A. Morphological, structural, thermal and tensile properties of high density polyethylene composites reinforced with treated argan nut shell particles. J Bionic Eng 2015;12(1):129e41.
Numerical modeling of hybrid composite materials
99
[100] Ku H, Wang H, Pattarachaiyakoop N, Trada M. A review on the tensile properties of natural ﬁber reinforced polymer composites. Compos Part B Eng 2011;42(4):856e73. [101] McCraryDennis MCL, Okoli OI. A review of multiscale composite manufacturing and challenges. J Reinf Plast Compos 2012;31(24):1687e711. [102] Jiang Y. Micromechanics constitutive model for predicting the stressestrain relations of particle toughened bulk metallic glass matrix composites. Intermetallics 2017;90: 147e51. [103] Seçkin Erden KH. Fiber technology for ﬁberreinforced composites. Composites Science and Engineering Woodhead Publishing is an imprint of Elsevier; 2017. 320 p. [104] Irving PE, Soutis C. Polymer composites in the aerospace industry. 2015. 257 p. [105] Younes R., Hallal A., Fardoun F., Chehade F.H. World’s largest science, technology & medicine open access book publisher comparative review study on elastic properties modeling for unidirectional composite Materials. [106] Brooks R. Materials property modelling and design of short ﬁbre composites. Flowinduced alignment in composite materials. Woodhead Publishing Ltd.; 293e323 p. [107] Petrů M, Novak O. Measurement and numerical modeling of mechanical properties of polyurethane foams. Asp Polyurethanes 2017;3:73e109. [108] Wennberg D, Stichel S, Wennhage P. Finite difference adaptation of the decomposition of layered composite structures on irregular grid. J Compos Mater 2014;48(20):2427e39. [109] G€unay E. Micromechanical analysis of polymer ﬁber composites micromechanical analysis of polymer ﬁber composites under tensile loading by ﬁnite element method under tensile loading by ﬁnite element method. In: Perusal of the ﬁnite element method; 2017. p. 89e114. [110] Carrer JAM, Costa VL. Boundary element method formulations for the solution of the scalar wave equation in onedimensional problems. J Brazilian Soc Mech Sci Eng 2015; 37(3):959e71. [111] Liu YJ, Nishimura N, Otani Y, Takahashi T, Chen XL, Munakata H. A fast boundary element method for the analysis of ﬁberreinforced composites based on a rigidinclusion model. J Appl Mech 2005;72(1):115. [112] Pan L, Adams DO, Rizzo FJ. Boundary element analysis for composite materials and a library of Green’s functions. Comput Struct 1998;66(5):685e93. [113] Wang Y, Xu D, Tsui KY. Discrete element modeling of contact creep and aging in sand. 2008. p. 1407e11. September. [114] Avrami E, Guillaud H, Hardy M. Terra literature review an Overview of research in earthen architecture conservation. 2008. 21e31 p. [115] Jiang X, Song J, Qiang X, Kolstein H, Bijlaard F. Moisture absorption/desorption effects on ﬂexural property of Glassﬁberreinforced polyester laminates: threepoint bending test and coupled hygromechanical ﬁnite element analysis. Polymers (Basel) 2016;8(8). [116] Riva E., Nicoletto G. Modeling and prediction of the mechanical properties of woven laminates by the ﬁnite element method. In: Fracture and damage of composites to. 2005. p. 105e125. [117] OstojaStarzewski M. Microstructural randomness versus representative volume element in thermomechanics. J Appl Mech 2002;69(1):25. [118] Sadd MH. Numerical ﬁnite and boundary element methods. In: Elasticity; 2014. p. 505e29. [119] Gun H, Kose G. Prediction of longitudinal modulus of aligned discontinuous ﬁberreinforced composites using boundary element method. Sci Eng Compos Mater 2014; 21(2):219e21.
100
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
[120] Fiore V, Valenza A, Di Bella G. Mechanical behavior of carbon/ﬂax hybrid composites for structural applications. J Compos Mater 2012;46(17):2089e96. [121] Mohammed FHL, Ansari MNM, Pua G, Jawaid M, Islam MS. A Review on Natural Fiber Reinforced Polymer Composite and Its Applications. Int J Polym Sci 2015. https:// doi.org/10.1155/2015/243947. [122] Nunna S, Chandra PR, Shrivastava S, Jalan A. A review on mechanical behavior of natural ﬁber based hybrid composites. J Reinf Plast Compos 2012;31(11):759e69. [123] Gupta G, Gupta A, Dhanola A, Raturi A. Mechanical behavior of glass ﬁber polyester hybrid composite ﬁlled with natural ﬁllers. IOP Conf Ser Mater Sci Eng 2016;149:12091. [124] Banerjee S, Sankar BV. Composites : Part B Mechanical properties of hybrid composites using ﬁnite element method based micromechanics. Compos Part B 2014;58:318e27. [125] Thwe MM, Liao K. Effects of environmental aging on the mechanical properties of bambooglass ﬁber reinforced polymer matrix hybrid composites. Compos Part A Appl Sci Manuf 2002;33(1):43e52. [126] Salit MS, Jawaid M, Yusoff NB, Hoque ME. Manufacturing of natural ﬁbre reinforced polymer composites e Google Livres. 2015 [cited 2018 Jan 22]. 395 p. [127] Mingchao W, Zuoguang Z, Zhijie S. The hybrid model and mechanical properties of hybrid composites reinforced with different diameter ﬁbers. J Reinf Plast Compos 2008; 28(3):257e64. [128] Hayashi T. On the improvement of mechanical properties of composites by hybrid composition. Proc 8th Intl Reinf Plast Conf 1972:149e52. [129] Fukuda H. An advanced theory of the strength of hybrid composites. J Mater Sci 1984; 19(3):974e82. [130] Swolfs Y, Verpoest I, Gorbatikh L. Maximising the hybrid effect in unidirectional hybrid composites. Mater Des 2016;93:39e45. [131] Swolfs Y, Verpoest I, Gorbatikh L. Tensile failure of hybrid composites: measuring, predicting and understanding. IOP Conf Ser Mater Sci Eng 2016;139(1). [132] Swolfs Y, Gorbatikh L, Verpoest I. Fibre hybridisation in polymer composites . A Review 1 Introduction 2014;67:181e200. [133] Goodship V, Middleton B, Cherrington R. Design and manufacture of plastic components for multifunctionality. Design Manuf Plastic Compon Multifunct 2016:103e70. [134] Fukunaga H, Chou TW, Fukuda H. Strength of intermingled hybrid composites. J Reinf Plast Compos 1984;3:145e60. [135] Gitman IM, Askes H, Sluys LJ, Valls OL. The concept of Representative Volume for elastic, hardening and softening materials. Adv Probl Mech 2004:180e4. [136] Praud F, Chatzigeorgiou G, Chemisky Y, Meraghni F. Hybrid micromechanicalphenomenological modelling of anisotropic damage and anelasticity induced by microcracks in unidirectional composites. Compos Struct 2017;182:223e36. [137] Oakeshott JL, Iannucci L, Robinson P. Development of a representative unit cell model for biaxial NCF composites. J Compos Mater 2007;41(7):801e35. [138] Gholap S, Panchagade DR, Patil V. Continuum modeling techniques to determine mechanical properties of nanocomposites. Int J Mod Eng Res. 2014;4(1):9e15. [139] Ehsan Mohammadpour MA. Modeling the tensile stressstrain response of carbon nanotube/polypropylene nanocomposites using nonlinear representative volume element. Mater Des 2014;1(1). [140] Hill R. Elastic properties of reinforced solids: some theoreticel principles. J Mech Phys Solids 1963;11:357e72. [141] Trusov PV, Keller IE. The theory of constitutive relations. Part I. Perm In Russian: Perm State Technical University; 1997.
Numerical modeling of hybrid composite materials
101
[142] Drugan WJ, Willis JR. A micromechanicsbased nonlocal constitutive equa tion and estimates of representative volume element size for elastic composites. J Mech Phys Solids 1996;44(4):497e524. [143] Hashin Z. Analysis of composite materials e a survey. J Appl Mechan 1983;50:481e505. [144] van Mier JGM. Reality behind ﬁctitious crack? In: n VC Li, Leung CKY, Willam KJ, Billington SL, editors. Fifth international conference on fracture mechanics of concrete and concrete structures; 2004. p. 11e30. [145] Evesque P. Fluctuations, correlation and representative elementary volume (REV) in granular materials. June 15, 2005. [146] Patel DK, Waas AM. Damage and failure modelling of hybrid threedimensional textile composites: a mesh objective multiscale approach. Phil Trans R Soc A 2016;374(2071). [147] Dubrovski PD. Woven Fabric Engineering. 2010. 436 p.
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Computationally efﬁcient modeling of woven composites under uniaxial stress
6
KamarulAzhar Kamarudin 1 , Al Emran Ismail 1 , Iskandar Abdul Hamid 2 , Ahmad Suﬁan Abdullah 3 1 Crashworthiness and Collisions Research Group (COLORED), Mechanical Failure Prevention and Reliability Research Centre (MPROVE), Faculty of Mechanical and Manufacturing Engineering, Universiti Tun Hussein Onn Malaysia, Batu Pahat, Johor, Malaysia; 2Crash Reconstruction Unit, Vehicle Safety & Biomechanics Research Centre, Malaysian Institute of Road Safety Research, Kajang, Malaysia; 3ARTeC, Faculty of Mechanical Engineering, Universiti Teknologi MARA, Permatang Pauh, Malaysia
6.1
Introduction
Fiberreinforced polymer (FRP) composite materials have been used for automotive crash tubes and armor designs because of their superior mechanical properties such as high speciﬁc stiffness and strength, and other properties such as corrosion and fatigue resistance. Unfortunately, they are very vulnerable to impact damage, which can signiﬁcantly reduce the stiffness, strength, and delamination of the structure. The penetration of targets by projectiles at high impact velocity has been investigated analytically during the last few decades and experimentally for more than two centuries [1]. Sevkat et al. presented the damage behavior of FRP plates subjected to highvelocity impact by comparing both the simulation and the experimental data. Aside from having good agreement on the postimpact damage pattern between both results, the target impacted by the hemisphericalshaped projectile also resulted in delamination of the FRP [2]. Matias et al. studied the impact behavior of composite metal foam when struck by armor projectiles. The composite metal foam had a thickness of approximately 25 mm with and without backplates of aluminum or FRP. The results showed that composite metal foams absorbed up to 70% of the total kinetic energy of the projectile and stopped both types of projectiles with less depth of penetration and backplate deformation. It was also shown that numerical simulation was validated using a mesh sensitivity technique using course, medium, and ﬁne mesh. Close agreement was found between experimental and corresponding ﬁnite element (FE) results [3]. To date, a large number of studies have focused on impact but less attention has been given to highvelocity impact with existing pretension on structures. Garcia
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites and Hybrid Composites https://doi.org/10.1016/B9780081022894.000060 Copyright © 2019 Elsevier Ltd. All rights reserved.
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
et al. [4] ran an experiment to look for glass/polyester woven plates under the inﬂuence of pretensions at 31% of material ultimate tensile strength. Under biaxial pretension, the ballistic limit showed a difference of 10% with the existence of pretensions. They also found that the ballistic limit was found to be contributed by the ﬁnal yarn and not by sample stretching. Using numerical simulations to predict the performance of a target has also been done by many researchers [5e7]. From an engineering point of view, there is considerable interest in the development of numerical simulation for the penetration and perforation of plates. Numerical simulations have been successful in predicting the response of targets to impact with an acceptable computing time [8,9]. Many researchers have applied mesh size sensitivity studies to model impacts on a structure using the FE method. The reason for using a mesh sensitivity technique is to decide the best element size, which might contribute to the most accurate prediction and utilize minimum processing time. Studies have shown that elements for isotropic material are shaped like a cube [5], while elements for orthotropic material have a cuboid geometry shape [6,10]. The element shape was described by the aspect ratio (AR) of length to thickness. After running a mesh sensitivity analysis for ﬁne and coarse meshes, Choi and Chang showed that a fairly good result for force and delamination in the composite laminate can be a best ﬁt by using a relatively coarse mesh size of 6.33 mm 6.25 mm 0.54 mm [11]. The parameters (force and delamination) were found converged as the element increased toward ﬁne meshes. A similar investigation was performed by Chan et al. [12], for various element sizes in comparison with the target stoppage time in the experiments, which showed that an AR of 13.3 was chosen based on the optimum simulation time. Mikkor et al. studied the effect of mesh size on the damage prediction in plain weave carbon ﬁber composites due to impact. They found that to predict damage behavior, which is similar to the experimental results, the impact velocity required for the simulation should be much lower than the experimental velocity [6]. This chapter deals with simple techniques that are used to investigate by simulation the penetration and perforation of carbon ﬁber reinforced plastic (CFRP) laminates by rigid projectiles using two shapes of projectiles, i.e., ﬂat and hemispherical. Earlier, experiments were carried out using the same method and results were compared with those obtained from threedimensional numerical simulation in this study. The approach is based on the sensitivity technique using both types of projectiles. Due to the elasticeplastic behavior of the CFRP, its relationship with ballistic limit results is presented and discussed.
6.2
Material
As an input into the FE simulations, mechanical properties are taken from the experimental tests combined with previous studies using similar CFRP material
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[13,14]. The composite plate was made of 24 layers or unidirectional carbon ﬁbers in an epoxy resin of a [0,90]12 layup. The mechanical properties are summarized in Table 6.1. Abaqus/Explicit was used to simulate the impact scenario shown in Fig. 6.1, which also presents the mesh pattern used in the study. The size of the rectangular CFRP target was 100 mm 45 mm 3 mm. An adaptive mesh was applied to the models when the impact area, which is located in the central area as shown in Fig. 6.1, consisted of the most reﬁned mesh. Moving further from the center of the impact area, the elements became less dense. The purpose was to reduce the computational time for the simulations. A ﬁx of 12 elements through the plate thickness was used for the entire model. Since the critical area was considered to be only in the middle of the plate, element behavior outside the impact area was not critically analyzed.
Table 6.1 Material properties of unidirectional carbon ﬁberreinforced plastic Description
Symbol
Value
Young’s modulus in ﬁber direction 1 (GPa)
E11
145
Young’s modulus in ﬁber direction 2 (GPa)
E22
11
Young’s modulus in ﬁber direction 3 (GPa)
E33
11
Poisson’s ratio
v12
0.3
Poisson’s ratio
v13
0.3
Poisson’s ratio
v23
0.45
Shear modulus, 1e2 plane (GPa)
G12
4.5
Shear modulus, 1e3 plane (GPa)
G13
4.5
Shear modulus, 2e3 plane (GPa)
G23
2.5
Tensile failure stress in ﬁber direction 1 (MPa)
X1T
1620
Compression failure stress in ﬁber direction 1 (MPa)
X1C
1200
Tensile failure stress in transverse matrix direction 2 (MPa)
X2T
55
Compression failure stress in transverse matrix direction 2 (MPa)
X2C
250
Tensile failure stress in transverse matrix direction 3 (MPa)
X3T
55
Compression failure stress in transverse matrix direction 3 (MPa)
X3C
250
Shear strength, 1e2 plane (MPa)
S12
120
Shear strength, 1e3 plane (MPa)
S13
137
Shear strength, 2e3 plane (MPa)
S23
90
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
Impact area (high dense area)
Less dense mesh area
Figure 6.1 Model of target plate and projectile.
6.3 6.3.1
Finite element modeling Finite element model
The Abaqus/Explicit FE software package with progressive failure model for composite materials was used for modeling impact. The available Hashin’s damage model in Abaqus/Explicit was used to cooperate with the element by using a continuum shell element. The target was modeled with a deformable shell element (SC8R), using the “reduce” integration method, while the projectile was modeled as an analytical rigid body using a bilinear quadrilateral four nodes element (R3D4). The target experienced two conditions of nonpretension and pretension when the projectile impacted the center of the target at high velocity.
6.3.2
Boundary conditions and pretension technique
Two different boundary conditions were used to model nonpretension and pretension in Abaqus/Explicit. The methods that were used involved applying the required boundary conditions to both parallel sides of the plate, but keeping the other two sides free, as shown in Fig. 6.2. The ultimate strength for CFRP was taken as 705 MPa; hence the pretension values calculated for 10%, 30%, and 50% of the ultimate strength were 70.5, 211.5, and 352.5 MPa, respectively. In the Abaqus/Explicit simulation, the projectile nose tip was assigned with a reference point and a boundary condition was applied such that only translation movement in the zdirection was allowed and there was no rotation during impact.
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Figure 6.2 (a) Fixed boundary conditions. (b) Displacement boundary conditions.
For nonpretension, ﬁxed boundary conditions were applied to both ends, as shown in Fig. 6.2(a). For pretension conditions, axial displacements were applied to the two ends of the plate by using the amplitude function, which is already built into the Abaqus software to create pretension stress in the target. The amplitude function began from 0 and increased to 1 linearly, and then remained constant, as shown in Fig. 6.3. A time range was given for the force to pull both ends and create the required pretension stresses in the target. The pretension remained constant throughout the test from projectile impact until it stopped. The average stress was calculated based on the pretension value. The ultimate strength
1.4 1.2
Amplitude 0 1 1
1 Amplitude
Time (ms) 0 0.0625 0.2
0.8 0.6 0.4 0.2 0
0
0.05
Figure 6.3 Amplitude used for displacement in pretension.
0.1 Time (ms)
0.15
0.2
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
for CFRP was taken as 705 MPa; hence the pretension values calculated for 10%, 30%, and 50% of the ultimate strength were 70.5, 211.5, and 352.5 MPa, respectively. The displacements required to generate these pretensions were calculated based on the extension stiffness of the plate and are shown in Fig. 6.4. Two pretension values were used, which were 10% and 50% pretension. In addition, the boundary condition for the projectile was assigned at the nose tip, with the reference point moving only in the zdirection. The projectile was also assumed to have no rotation during impact. The initial position of the projectile was such that the projectile would only travel to impact the target after the pretension had been applied to the target and reached a steady state.
Pretension percentage (x1.E6) 50.
10%
Stress
40.
30.
20.
10.
0. 0.00
0.10
0.20
0.30 Time
0.40
0.50 0.60(x1.E3)
0.10
0.15(x1.E3)
(x1.E9) 0.25
50%
Stress
0.20
0.15
0.10
0.05
0.00 0.00
0.05
Time
Figure 6.4 Pretension using the amplitude function.
Computationally efﬁcient modeling of woven composites under uniaxial stress
6.3.3
109
Interaction in modeling
The algorithm used for contact and interaction in this chapter was the contact pair algorithm. To model this interaction, surface interaction was selected between the CFRP plate and the projectile. In Abaqus, the surface of the projectile was selected as the master surface, while the surface made of nodes of the CFRP plate was selected as a slave surface. Interaction properties for Abaqus/Explicit were determined by two criteria: tangential behavior and normal behavior. Tangential behavior used a friction coefﬁcient of 0.3, which was also used in other studies [12,15,16], and a “penalty” contact was deﬁned using friction formulation. In normal behavior, the hard contact was chosen under pressure overclosure, with the help of separation of elements after contact. Failure energies for ﬁber tension and compression were taken as 12.5 kJ/m2, while failure energies for matrix tension and compression were taken to be 1 kJ/m2 [17].
6.4
Mesh sensitivity analysis
A mesh sensitivity study was performed on a 3 mm thick CFRP panel. The plate was modeled by 12 continuum shell elements in the plate thickness, which represents 24 layers of lamina in the structure. Due to limitations of the material damage model in predicting tensile failure, the analysis only focused on a hemisphericalshaped projectile. The mesh studies involved dimension variables and ARs of the elements in the impacted area. The element has the same size in length and width and this size was varied, while the thickness of each element remained at 0.25 mm. Six element sizes are shown in Table 6.2. The elements constructed by these unique six meshes were in a cuboid shape, which differs from the isotropic cubic elements employed for the aluminum alloy panel using a solid element [18]. Table 6.2 Mesh sensitivity results for carbon ﬁberreinforced plastic impacted by a hemispherical projectile 0% pretension (vi [ 135 m/s)
10% pretension (vi [ 135 m/s)
30% pretension (vi [ 135 m/s)
50% pretension (vi [ 135 m/s)
Element size (mm3)
vr(m/s)
vr(m/s)
vr(m/s)
vr(m/s)
0.55 0.55 0.25
80
e
e
e
0.625 0.625 0.25
85
100
92
115
0.714 0.714 0.25
85
100
92
115
1 1 0.25
82
105
100
115
1.67 1.67 0.25
91
105
107
118
2.5 2.5 0.25
107
e
100
e
The (e) sign indicates no value taken.
110
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites (b)
135
0
0.5
1.5
200
2
2.5
3
Element width length
(c)
Residual velocity(m/s)
1
Residual velocity(m/s)
200 180 160 140 120 100 80 60 40 20 0
200 180 160 140 120 100 80 60 40 20 0
135 0
0.5
1
1.5
200 2
2.5
Element width length
200 180 160 140 120 100 80 60 40 20 0
135 0
0.5
3
1
200
1.5
2
2.5
3
Element width length
(d)
Residual velocity(m/s)
Residual velocity(m/s)
(a)
200 180 160 140 120 100 80 60 40 20 0
135 0
0.5
200
1 1.5 2 2.5 Element width length
3
Figure 6.5 Mesh sensitivity of (a) 0% pretension and (b) 30% pretension (c) 30% pretension and (d) 50% pretension.
To analyze the sensitivity of the mesh, the impact of a projectile with a mass of 3 g on the CFRP panel was simulated. Initial impact velocity used was at 135 m/s for all mesh sizes in prediction of the residual velocities. Pretensions of 0%, 10%, 30%, and 50% were applied to the CFRP plate. Fig. 6.5 shows the effect of mesh size on the predicted residual velocity of the projectile after impacting the panel under various pretension levels. The results from an impact velocity of 200 m/s were also included. The reason for having two impact velocities for the mesh sensitivity study was to validate the convergence behavior at a velocity of 135 m/s. It can be seen that, as the element got smaller, the residual velocity tended to become constant. Due to this convergence, the element size 1 mm 1 mm 0.25 mm was selected for further analysis.
6.4.1
Mesh sensitivity study impact of a ﬂat projectile
Mesh sensitivity studies were also run using a ﬂatnosed projectile on the target plate. Unfortunately, the results were not convincing due to unstable residual velocity values as shown in Fig. 6.6. The ﬂuctuation graph instead of converging (as in Fig. 6.6) showed that the element was not prone to ﬂatshaped projectile damage of shear but preferably tensile failure using a hemisphericalshaped nose projectile shown in an
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Residual velocity (m/s)
120 100 80 60
Fluctuated graph instead of convergence to show the sensitivity of the element.
40 20 0
0
0.5
1
1.5
2
2.5
Element width length (mm)
Figure 6.6 Mesh sensitivity of 0% pretension on a carbon ﬁberreinforced plastic target using a ﬂatnosed projectile.
earlier section. As also mentioned by Fan et al. [19], convergence during mesh sensitivity was needed to provide good prediction for simulation. Due to this limitation, the prediction for ballistic limits using a ﬂatnosed projectile used the analytical method, which is not explained in this chapter.
6.4.2
Ballistic limit prediction of hemispherical projectile using FE simulation
From the mesh sensitivity results in Table 6.2, an element size of 1 mm 1 mm 0.25 mm was chosen to model the CFRP plate and to run the impact simulation to determine the relevant ballistic limit. Four different pretensions, 0%, 10%, 30%, and 50%, of the overall material strength were applied to the panel, which was then impacted by a hemispherical projectile. Simulations were performed at a series of impact velocities and the residual velocities were determined. The results are shown in Table 6.3. Table 6.3 Simulation results for a carbon ﬁberreinforced plastic target under hemispherical projectile impact Impact velocity, vi (m/s) 200
135
130
120
110
100
95
90
80
60
62
46
0
Residual velocity, vr (m/s)
Pretension (%) 0
170
82
e
e
37
0
e
10
172
100
81
e
47
32
0
30
173
100
e
81
e
50
0
50
183
115
e
104
e
71
e
The (e) sign indicates no value taken.
112
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
200
Residual velocity(m/s)
180 160
Hem mis0%Sim
140
Hem mis10%Sim
120
Hem mis30%Sim Hem mis50%Sim
100 80 60 40 20 0 0
20
40 0
60
80 100 120 140 Impact velocity (m/s)
160
180
200
220
Figure 6.7 Parametric ﬁnite element simulation of carbon ﬁberreinforced plastic impacted by a hemispherical projectile.
The results of the impact velocity simulation in Table 6.3 are plotted in Fig. 6.7, showing the comparisons between results at 0%, 10%, 30%, and 50% pretensions. The residual velocity increased parabolically near to the ballistic limit, and linearly increased with the increase in impact velocity. The ballistic limits were estimated from the simulation results where the projectile failed to perforate the sample. Ballistic limit reductions for nonpretension to 10% pretension, 10%e30% pretension, and 30%e50% pretension were found to be 5%, 5.2%, and 33%, respectively, and the total reduction in the ballistic limit between a target with no pretension (0%) and 50% pretension was approximately 40%.
6.5
Conclusions
Abaqus/Explicit was used to develop a reliable FE model to simulate the impact behavior of a CFRP target, which was given various pretensions by a hemisphericalshaped nose projectile. The initial work on mesh sensitivity determined the optimum size of the continuum shell element as 1 mm 1 mm 0.25 mm. The FE model was further veriﬁed by comparing its ballistic limit and residual velocity for 0% and 30% pretension to the results from an experiment. The difference in ballistic limit between simulation and experiment for 0% and 30% pretension was 6.7% and 4.0%, respectively. Based on the results and discussion, the pretension on the target plate signiﬁcantly affected the ballistic limit, residual velocity of the projectile, impact force, the perforation process during impact, and the mode of failure. Total reduction of ballistic limit from no pretension to 50% pretension was approximately 40%. As the main failure mechanisms of the perforation were tensile failure, the pretension or uniaxial loading,
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which was in the direction of the stress that caused the tensile failure, will amplify the stress caused by the projectile impact followed by easier perforation. This failure mechanism explained the results in terms of residual velocity and the displacement of the projectile after perforation. The higher the pretension, the further the displacement of the projectile and the faster the projectile will perforate because it perforates easier. This failure mechanism also explains the increment of impact force as the pretension decreases.
Acknowledgments This study was supported by the Ministry of Higher Education Malaysia and Universiti Tun Hussein Onn Malaysia.
References [1] Wen HM. Predicting the penetration and perforation of FRP laminates struck normally by projectiles with different nose shapes. Compos Struct 2000;49:321e9. [2] Sevkat E, Liaw B, Delale F, Raju BB. A combined experimental and numerical approach to study ballistic impact response of S2glass ﬁber/toughened epoxy composite beams. Compos Sci Technol 2009;69:965e82. [3] GarciaAvila M, Portanova M, Rabiei A. Ballistic performance of composite metal foams. Compos Struct 2015;125:202e11. [4] GarciaCastillo S, SanchezSaez S, LopezPuente J, Barbero E, Navarro C. Impact behaviour of preloaded glass/polyester woven plates. Compos Sci Technol 2009;69: 711e7. [5] Iqbal MA, Gupta G, Diwakar A, Gupta NK. Effect of projectile nose shape on the ballistic resistance of ductile targets. Eur J Mech A Solids 2010;29:683e94. https://doi.org/10. 1016/j.euromechsol.2010.02.002. [6] Mikkor KM, Thomson RS, Herszberg I, Weller T, Mouritz AP. Finite element modelling of impact on preloaded composite panels. Compos Struct 2006;75:501e13. [7] Sun CT, Vaidya RS. Prediction of composite properties from a representative volume element. Compos Sci Technol 1996;56:171e9. https://doi.org/10.1016/02663538(95) 001417. [8] Jiying F, Zhongwei G, Westley JCNO. Modeling perforation in glass ﬁber reinforced composites subjected to low velocity impact loading. Polym Polym Compos 2008;16: 101e13. [9] Mohd Nor MK, Vignjevic R, Campbell J. Modelling of shockwave propagation in orthotropic materials. Appl Mech Mater 2013;315:557e61. [10] Sun B, Liu Y, Gu B. A unit cell approach of ﬁnite element calculation of ballistic impact damage of 3D orthogonal woven composite. Compos Part B Eng 2009;40:552e60. [11] Choi HY, Chang FK. A model for predicting damage in graphite/epoxy laminated composites resulting from lowvelocity point impact. J Compos Mater 1992;26:2134e69. [12] Chan S, Fawaz Z, Behdinan K, Amid R. Ballistic limit prediction using a numerical model with progressive damage capability. Compos Struct 2007;77:466e74.
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[13] Kamarudin KA. Ballistic response of aluminium alloy and Cfrp panels with pretension. University of Manchester; 2015. [14] ABAQUS standard user’s manual. ABAQUS user’s manual. 2010. [15] Rebouillant S. Surface treated aramid ﬁbers and a process for making them. 5,520,705. 1996. [16] Feng D, Aymerich F. Finite element modelling of damage induced by lowvelocity impact on composite laminates. Compos Struct 2014;108:161e71. [17] Lapczyk I, Hurtado JA. Progressive damage modeling in ﬁberreinforced materials. Compos Part A Appl Sci Manuf 2007;38:2333e41. [18] Kamarudin KA, Ismail AE. Modelling high velocity impact on aluminium alloy 7075T6 under axial pretension. Appl Mech Mater 2014;629:498e502. [19] Fan J, Guan Z, Cantwell WJ. Structural behaviour of ﬁbre metal laminates subjected to a low velocity impact. Sci China Phys Mech Astron 2011;54:1168e77.
Progressive damage modeling of synthetic ﬁber polymer composites under ballistic impact
7
KamarulAzhar Kamarudin 1 , Mohd Khir Mohd Nor 1 , Al Emran Ismail 1 , Iskandar Abdul Hamid 2 , Ahmad Suﬁan Abdullah 3 1 Crashworthiness and Collisions Research Group (COLORED), Mechanical Failure Prevention and Reliability Research Centre (MPROVE), Faculty of Mechanical and Manufacturing Engineering, Universiti Tun Hussein Onn Malaysia, Batu Pahat, Johor, Malaysia; 2Crash Reconstruction Unit, Vehicle Safety & Biomechanics Research Centre, Malaysian Institute of Road Safety Research, Kajang, Malaysia; 3ARTeC, Faculty of Mechanical Engineering, Universiti Teknologi MARA, Permatang Pauh, Malaysia
7.1
Introduction
Composite structures offer many advantages compared to conventional materials, especially where high strength and stiffnesstoweight ratio are concerned [1]. Thus composites have been used widely in many applications such as marine components, bicycle parts, petrochemicals, and protective gadgets. However, they are relatively sensitive to brittle behavior when loaded under static or fatigue conditions, which leads to damage and loss of stiffness. A number of studies have been conducted to evaluate the damage behavior of target plates under various impact velocities [2,3]. The damage characteristics were compared under the inﬂuence of different projectile nose geometries, composite ﬁber orientations, and target properties. Iqbal et al. [4], compared the deformation of the targets by different noseshaped projectiles. Besides having signs of local displacement, the target impacted by a ﬂatnosed projectile resulted in minimal displacement compared to hemispherical projectiles. However, hemispherical projectiles showed the highest displacement during global plastic deformation. Studies related to impacts on targets have mostly focused on impacts without pretension [5e8]. At low and high impact velocity, a structure may perform differently when subjected to nonpretension and pretension acting on its body. Pressurized vessels are an example of a pretension structure but using gas or air as the pressure medium. Pressure
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites and Hybrid Composites https://doi.org/10.1016/B9780081022894.000072 Copyright © 2019 Elsevier Ltd. All rights reserved.
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
creates structural expansion from inside the wall. Due to pretension, impact by an external object could lead to catastrophic failure and the creation of debris could create more damage. In 1997, Lambert and Schneider [9] investigated gaspressurized vessels using impact caused by a hypervelocity projectile. The projectile and pressure vessel were made of aluminum, impacted at normal trajectory with a constant velocity of 7000 m/s. Due to the impact, its kinetic energy had increased and the pressure vessel experienced burst at the front and rear. A number of tests were performed where front and rear bursts were separated by a boundary that depended on the stress level on the vessel wall. Under lowvelocity impact, Kelkar et al. [10] found that a higher pretension had resulted in a larger impact force, whereas the damaged area was also increased. This ﬁnding was similar to Chiu et al. [11], who concluded that the peak force and damage area will increase together with pretension. There are very few studies on the impact behavior of composite laminates under inplane load, especially those related to highvelocity impact. Ballistic limit prediction was mostly determined in highvelocity impact studies. In 2006, together with nonpretension targets, Garcia et al. [12] also investigated the ballistic limit of glassreinforced plastics (GRPs) under the effect of uniaxial and biaxial pretension. The pretension on the target was 114 MPa, which was approximately 26% of the ultimate strength of the target. The ballistic limit was found to be up to 4% higher between nonpretension and uniaxial pretension and approximately 6% higher between uniaxial and biaxial pretension. The experimental ballistic limits were also compared with the analytical model. The results did not agree with the analytical ﬁndings, whereby the ballistic limit was reduced as the pretension increased. They claimed that the difference in the ballistic limit between the experimental and analytical ﬁndings was due to the static parameter used in the equations instead of dynamic parameters. Besides that, the researchers did not mention the ﬁber volume fraction for the sample target, which made it inconclusive and questionable (ﬁve ﬁber layers resulted in 3.19 mm, which was too thick, hence the sample would be rich in resin). However, different damage patterns were reported. Garcia et al. observed that the damage grew linearly as the impact energy was increased, until it reached the ballistic limit. At velocities higher than the ballistic limit, the damage area was seen to decrease. The damage area from the images was found to be related to the delamination of the impacted specimens. In 2009, Garcia et al. [13] performed an experiment using a similar glass/polyester woven plate under pretension at 31% of the ultimate tensile strength. The tests focused on biaxial pretension, and were compared with nonpretension tests. With the existence of pretension on the target, the ballistic limit was found to be higher than without pretension. At velocities below the ballistic limit, the kinetic energy of the projectile decreased due to the inﬂuence of secondary yarn deformation and plate delamination.
Progressive damage modeling of synthetic ﬁber polymer composites under ballistic impact
117
However, at velocities higher than the ballistic limit, the formation of cones at the back side of the target plate became the main contributor to the main energy absorption mechanism [14]. The latest research paper by Garcia et al. was done in 2013 [15] on highvelocity impact with target pretension, focusing on impact damage in a wide interval of impact velocities between 90 and 360 m/s. Material types of GRP with different ﬁber arrangements were used in the study. There were three pretension boundary conditions used in the experiment: nonpretension, uniaxial, and biaxial. The pretensions applied to the uniaxial and biaxial target were approximately 166 and 122 MPa, which were 38% and 27% of their ultimate strength, respectively. The ballistic limit for the biaxial pretension was the lowest compared to uniaxial (10% difference) and nonpretension targets, which had the highest limit (12% difference between uniaxial and nonpretension). The ballistic limit on each plate showed reduction due to the existence of pretension. The pretension values for both uniaxial and biaxial were dissimilar after comparison (27% and 38%). In another major study, Mikkor et al. [16] investigated the impact damage at low, medium, and high velocities on carbon ﬁberreinforced plastic (CFRP) targets under pretension. The simulation results were compared with the experimental results of targets without pretension and with axial pretension. They found that the damage for lowvelocity impacts was internal and invisible, but caused a reduction in overall strength. It was also found that, at certain impact velocities, the model experienced catastrophic failure. Catastrophic failure was shown in the model by experiencing damage and in this case it split the part into two, which was in the direction perpendicular to the axial loading. Similar to Garcia et al. [12], the damage area was found to increase with increasing impact velocity. Most catastrophic damage occurs below the ballistic limit. With a minimum of 50 kN of applied pretension load, the stress was measured at approximately 294 MPa, which was 42% of the material ultimate strength; this pretension was found to be too high to avoid any catastrophic failure. At certain loads, the pretension applied was above 50% of material ultimate strength, whereby the maximum pretension load applied ranged from 98 kN to 544 MPa (78% of ultimate strength). Until recently, there have not been many studies that have reported damage and ballistic limit effects by the pretension on the target of carbon compositereinforced plastic. This chapter provides numerical simulation to measure the ballistic limit and damage behavior of CFRP and the results will be compared with the results of experimental tests done onsite. The pretension applied to the target used amplitude operation from the Abaqus/CAE package with impact velocity similar to the experimental test. The applied pretensions in the axial direction were at 10%, 30%, and 50% from the CFRP ultimate strength.
118
7.2
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
Material properties of carbon ﬁberreinforced plastic
The composite plate is made up of 24 layers of unidirectional carbon ﬁbers in an epoxy resin of a [0,90]12 layup. The mechanical properties as an input into the ﬁnite element (FE) simulations were taken from the experimental tests and previous studies using similar CFRP material [17]. The mechanical properties are summarized in Table 7.1. Failure energies for ﬁber tension and compression were taken as 12.5 kJ/m2, while failure energies for matrix tension and compression were taken to be 1 kJ/m2 [18].
Table 7.1 Material properties of unidirectional carbon ﬁberreinforced plastic Description
Symbol
Value
Young’s modulus in ﬁber direction 1 (GPa)
E11
145
Young’s modulus in ﬁber direction 2 (GPa)
E22
11
Young’s modulus in ﬁber direction 3 (GPa)
E33
11
Poisson’s ratio
v12
0.3
Poisson’s ratio
v13
0.3
Poisson’s ratio
v23
0.45
Shear modulus, 1e2 plane (GPa)
G12
4.5
Shear modulus, 1e3 plane (GPa)
G13
4.5
Shear modulus, 2e3 plane (GPa)
G23
2.5
Tensile failure stress in ﬁber direction 1 (MPa)
X1T
1620
Compression failure stress in ﬁber direction 1 (MPa)
X1C
1200
Tensile failure stress in transverse matrix direction 2 (MPa)
X2T
55
Compression failure stress in transverse matrix direction 2 (MPa)
X2C
250
Tensile failure stress in transverse matrix direction 3 (MPa)
X3T
55
Compression failure stress in transverse matrix direction 3 (MPa)
X3C
250
Shear strength, 1e2 plane (MPa)
S12
120
Shear strength, 1e3 plane (MPa)
S13
137
Shear strength, 2e3 plane (MPa)
S23
90
Progressive damage modeling of synthetic ﬁber polymer composites under ballistic impact
7.3
119
Finite element modeling using the continuum shell element
In Abaqus/Explicit, the only element that is able to use Hashin’s damage model is the continuum shell element. The target was modeled with a deformable shell element (SC8R) using the “reduce” integration method, while the projectile was modeled as an analytical rigid body using a bilinear quadrilateral fournodes element (R3D4). The target experienced two conditions of nonpretension and pretension with the projectile impacting the middle of the target at high velocity.
7.3.1
Progressive damage modeling
Damage initiation in the composite is based on Hashin’s failure criteria, using four different damage mechanisms: ﬁber tension, ﬁber compression, matrix tension, and matrix compression. The failure criteria are adapted from Abaqus [17] for damage initiations and are expressed as: Fibre tension : ðs11 0Þ
s11 X1T
Ff T ¼
2
s12 þa S12
2 ¼ 1;
Fibre compression : ðs11 < 0Þ FfC ¼
s11 X1C
2 ¼ 1;
Matrix tension : ðs22 0Þ FmT ¼
s22 X2T
2 þ
s12 S12
2 ¼ 1;
Matrix compression : ðs22 < 0Þ " # 2 s22 2 X2C 2 s22 s12 þ 1 ¼ 1; FmC ¼ þ 2S13 2S13 X2C S12
(7.1)
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
where s11,s22, and s12 are the longitudinal, transverse, and shear stresses in the lamina, X1T and X1C denote tensile and compression strength in the ﬁber direction, X2T and X2C denote tensile and compression strength in the transverse direction, and S12 and S13 denote the longitudinal and transverse shear strength. The coefﬁcient a ¼ 0 was used as the contributor of shear stress to the ﬁber tensile damage initiation in the present work. Based on the brittle behavior of the CFRP, the material was linearly elastic before damage initiation. Damage evolution occurred just after damage initiation [17]. The general form of constitutive laws for orthotropic elastic materials is computed as: s ¼ Cd ε
(7.2)
where Cd is the elasticity matrix given as: 2 Cd ¼
ð1 df ÞE11
16 6 ð1 df Þð1 dm Þv12 E22 D4 0
ð1 df Þð1 dm Þv21 E11
0
ð1 dm ÞE22
0
0
3 7 7 5
ð1 ds ÞG12 D (7.3)
where D ¼ 1 (1 df) (1 dm)v12v21 and df, dm, and ds reﬂect the current state of ﬁber damage, matrix damage, and shear damage, respectively. E11 and E22 are Young’s modulus in the ﬁber and transverse directions, respectively, while G12 is the shear modulus and v12 and v21 are Poisson’s ratios of the laminate. The damage variables, df, dm, and ds used in Eq. (7.3) were derived from the damt , and d c , which were related to the four failure modes (ﬁber age variables dft , dfc , dm f tensile, ﬁber compression, matrix tensile, matrix compression) as follows [17]: df ¼
dm ¼
8 t < df if s11 0 : dc if s < 0 11 f 8 t < dm if s22 0
(7.4)
: d c if s < 0 22 m t c ds ¼ 1 1 dft 1 dfc 1 dm 1 dm The damage variables were determined by considering a bilinear equivalent stresse displacement relationship. Element usage in simulation becomes an important factor, because most results depend on element shape, size, and dimensions. The constitutive law in Abaqus is
Progressive damage modeling of synthetic ﬁber polymer composites under ballistic impact
121
expressed in terms of stressedisplacement relations. The equivalent displacements and stresses for the four damage modes are deﬁned as follows [17]: Fibre tension : ðs11 0Þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ d feqT ¼ Lc hε11 i2 þ aε212 hs11 ihε11 i þ as12 ε12 . d fteq Lc
sfeqT ¼
Fibre compression : ðs11 < 0Þ c d fC eq ¼ L hε11 i
sfC eq ¼
hs11 ihε11 i . d fteq Lc
Matrix tension : ðs22 0Þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ c ¼ L dmT hε22 i2 þ ε212 eq smT eq ¼
(7.5)
hs22 ihε22 i þ s12 ε12 . c dmt eq L
Matrix compression : ðs22 < 0Þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ c ¼ L dmC hε22 i2 þ ε212 eq smC eq ¼
hs22 ihε22 i þ s12 ε12 . c dmc eq L
where Lc is a characteristic length, which is based on the element geometry. In the present analysis, a shell element was employed to model the composite laminate, and Lc was calculated to be the square root of the surface area of the shell element. After damage initiation (i.e., deq d0eq ), the damage variable for a particular mode was given by the following expression [17]: d feq deq d0eq ¼1 d¼ deq d feq d0eq
(7.6)
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
where d0eq is the equivalent displacement at which the initiation criterion for the mode is met and d feq is the displacement at which the material is completely damaged in this failure mode. The value d0eq for the various failure modes depends on the elastic stiffness and the strength parameters speciﬁed. In Abaqus, it is necessary to assign each failure mode to the energy dissipated due to failure, Gc. Therefore the values of d feq for the various modes depend on the respective Gc values.
7.3.2
Finite element model
Abaqus/Explicit was used to simulate the impact scenario shown in Fig. 7.1, which also presents the mesh pattern used in the study. The size of the rectangular CFRP target was 100 mm 45 mm 3 mm. An adaptive mesh was applied to the models when the impact area, which was located in the central area, consisted of the most reﬁned mesh. Moving further from the center of the impact area, the elements became less dense. The purpose was to reduce the computational time for the simulations. A ﬁx of 12 elements through the plate thickness was used for the entire model. Since the critical area was considered to be only in the middle of the plate, element behavior outside the impact area was not critically analyzed.
7.3.3
Boundary conditions and pretension technique
Two different boundary conditions were used to model nonpretension and pretension in Abaqus/Explicit. The methods that were used involved applying the required boundary conditions to both parallel sides of the plate, but keeping the other two sides free, as shown in Fig. 7.2. The ultimate strength for CFRP was taken as 705 MPa; hence the pretension values calculated for 10%, 30%, and 50% of the ultimate strength were 70.5, 211.5, and 352.5 MPa, respectively. The projectile nose tip was assigned with a reference point and a boundary condition was applied such that only translation movement in the zdirection was allowed and there was no rotation during impact.
7.3.4
Interaction in modeling
The algorithm used for contact and interaction in this chapter was the contact pair algorithm. To model this interaction, surface interaction was selected between the Impacttarget Loading direction
Figure 7.1 Target mesh used for simulation.
Progressive damage modeling of synthetic ﬁber polymer composites under ballistic impact
123
24 ply (0°, 90°) Loading direction
Impactor direction
Loading direction
Figure 7.2 Boundary conditions of uniaxial pretension on bidirectional 24ply composite laminate carbon ﬁberreinforced plastic.
CFRP plate and the projectile. In Abaqus, the surface of the projectile was selected as the master surface, while the surface made of nodes of the CFRP plate was selected as a slave surface. Interaction properties for Abaqus/Explicit were determined by two criteria: tangential behavior and normal behavior. Tangential behavior used a friction coefﬁcient of 0.3, which was used in other studies [19,20], and a “penalty” contact was deﬁned using friction formulation. In normal behavior, the hard contact was chosen under pressure overclosure, with the help of separation of elements after contact.
7.3.5
Mesh sensitivity analysis
A mesh sensitivity study was performed on a 3 mm thick CFRP panel. The plate was modeled by 12 continuum shell elements in the plate thickness, which represents 24 layers of lamina in the structure. Due to limitations of the material damage model in predicting tensile failure, the analysis only focused on a hemisphericalshaped projectile. The mesh studies involved dimension variables and aspect ratios of the elements in the impacted area. The element had the same size in length and width and this size was varied, while the thickness of each element remained at 0.25 mm. Six element sizes were used in implementing the technique. The elements constructed by these unique six meshes were in a cuboid shape, which differed from the isotropic cubic elements employed for the aluminum alloy panel [3]. To analyze the sensitivity of the mesh, the impact of a projectile with a mass of 3 g on the CFRP panel was simulated. The predicted residual velocities were obtained from using various mesh sizes when the initial impact velocity was taken at 135 and 200 m/s. The reason for having two impact velocities for the mesh sensitivity study was due to the unclear convergence behavior at a speciﬁc velocity. Pretensions of 0%, 10%, 30%, and 50% were applied to the CFRP plate. Fig. 7.3 shows the effect of mesh size on the predicted residual velocity of the projectile after impacting the panel under the 30% pretension level. It could be seen that as the element became smaller the residual velocity tended to become constant.
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
200
Residual velocity (m/s)
180 160 140 120 100 80 60 40
135
20 0 0
0.5
1
200
1.5
2
2.5
3
Element width length
Figure 7.3 Mesh sensitivity of 30% pretension.
7.4 7.4.1
Results and discussion Ballistic limit prediction of hemispherical projectile using ﬁnite element simulation
From the mesh sensitivity results in Fig. 7.3, an element size of 1 mm 1 mm 0.25 mm was chosen to model the CFRP plate and to run the impact simulation to determine the relevant ballistic limit (refer to Table 7.2). Four different pretensions, 0%, 10%, 30%, and 50%, of the overall material strength were applied to the panel, which was then impacted by a hemispherical projectile. Simulations were performed at a series of impact velocities and the residual velocities were determined. The results are shown in Table 7.2. The results of the impact velocity simulation in Table 7.2 were plotted in Fig. 7.4, showing the comparisons between results at 0% and 30% pretensions. The residual velocity increased parabolically near to the ballistic limit, and linearly increased
Table 7.2 Simulation results for a carbon ﬁberreinforced plastic target under hemispherical projectile impact Impact velocity, vi (m/s) 200
135
130
120
110
100
95
90
80
60
62
46
0
Residual velocity, vr (m/s)
Pretension (%) 0
170
82
e
e
37
0
e
10
172
100
81
e
47
32
0
30
173
100
e
81
e
50
0
50
183
115
e
104
e
71
e
Progressive damage modeling of synthetic ﬁber polymer composites under ballistic impact
125
200
Residual velocity (m/s)
180 160
Hemis0%sim
140
Hemis10%sim
120
Hemis30%sim Hemis50%sim
100 80 60 40 20 0
0
20
40
60
80 100 120 140 160 Impact velocity (m/s)
180
200
220
Figure 7.4 Parametric ﬁnite element simulation of carbon ﬁberreinforced plastic impacted by a hemispherical projectile.
with the increase in impact velocity. The ballistic limits were estimated from the simulation results where the projectile failed to perforate the sample. Ballistic limit reductions for nonpretension to 10% pretension, 10%e30% pretension, and 30%e50% pretension were found to be 5%, 5.2%, and 33%, respectively, and the total reduction in the ballistic limit between a target with no pretension (0%) and a target with 50% pretension was approximately 40%.
7.4.2
Ballistic limit ﬁnite element against experimental results
Fig. 7.5 shows the predicted residual velocity of a hemispherical projectile impacting on plates under the inﬂuence of 0% and 30% pretension, in comparison with 200 Hemis0%sim
Residual velocity (m/s)
180
Hemis30%sim
160
Hemis 0% exp
140
Hemis 30% exp
120 100 80 60 40 20 0 0
20
40
60
80 100 120 140 Impact velocity (m/s)
160
180
200
Figure 7.5 Experimental and simulation results of residual velocity versus impact velocity for a carbon ﬁberreinforced plastic target impacted by hemispherical projectiles.
126
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
experimental results. The comparison showed a fairly good agreement between simulation and experiments, where both lines were seen to align with each other. For a target without pretension (0%), there was a difference of only 6.7% between the simulated and experimental ballistic limit, under impact from a hemispherical projectile. Similarly, in targets with 30% pretension, a small difference of 4% in the simulated ballistic limit was found in comparison with the experimental results. From the graphs, it is clear that the existence of pretension does contribute to a reduction in ballistic limit for a CFRP target. Due to limitations of the material model, this analysis only presented ballistic results for a hemispherical projectile.
7.4.3
Failure modes of a CFRP target after impact
Fig. 7.6 shows the sequence of damage events in a 3 mm thick CFRP target impacted by a hemisphericalshaped projectile at a velocity of 110 m/s, a velocity just above the ballistic limit. This study showed the effect of pretension on failure modes in the target. It was observed that due to the hemisphericalnosed geometry, the projectile caused a tensile failure scenario when the material was pushed sideways during perforation and caused thinning in the contact area. The increase of pretension helped the projectile to perforate more easily, which contributed to a reduction in the ballistic limit velocity. This could be seen at 100 ms, where the projectile moved further as the pretension increased, which showed that it perforated more easily during penetration. Impact using a hemispherical projectile also contributed to the formation of petalling near to the impact target area.
7.4.4
Damage assessment from simulation and experiment
The images provided both front and rear target views of damage under various conditions of pretension. It was interesting to note that the FE result showed important damage phenomena with “logarithmic strain” images, which were similar to the experimental results. With the target under pretension, there were more obvious strain variations that could be seen in the FE target simulation. Fig. 7.7 presents damage to the nonpretension CFRP target at below and higher than the ballistic limit. Experimental observation of the front view image (a) corresponded to FE simulation image (b). Due to impact velocities below the ballistic limit, an indentation occurred (image b), which showed similarities in comparison with the experimental result in image (a). Strain images were seen concentrated at the impact and damage area. Images (c) and (d) show the rear view of targets (a and b) with a vertical high strain formation pattern (image d), which shifted toward a crack pattern in the experiment target. When impact velocity was above the ballistic limit, as expected, the front view images (e and f) showed perforation. Fibers were broken due to the tensile failure caused by the hemispherical projectile. However, image (g) shows broken ﬁbers on the back, close to the perforation hole after the hemispherical projectile went through. Fibers were not shown in the simulation of image (f ) due to the element deletion technique that was used by the FE software. Images (g and h) present a rear image of the target plates (e and f). The ﬁber damage in (g) shows petalling from the impact,
Progressive damage modeling of synthetic ﬁber polymer composites under ballistic impact
Pretension
25 µs
50 µs
127
100 µs
0%
10%
30%
50%
Figure 7.6 Failure sequence of a hemispherical projectile impacted on various pretension carbon ﬁberreinforced plastic targets.
while image (h) simulates the vertical high strain formation, which is similar to the ﬁber damage pattern in (g). Fig. 7.8 compares the damage by the hemispherical projectile when pretension in the plate was increased to 30%. The images show similar damage behavior in the plates from the front and rear views of experimental and FE simulation. Comparison between
128
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
Partial (0% pretension)
(a)
(b) Indentation
Front view
(c)
(d) Crack formation
Rear view
Limit/complete (0% pretension)
(e)
(f)
s Circle shape perforation
Front view
(g)
(h)
Crack formation
Rear view
Figure 7.7 Hemispherical projectile impact on 0% pretension carbon ﬁberreinforced plastic targets.
Progressive damage modeling of synthetic ﬁber polymer composites under ballistic impact
129
Partial (30% pretension)
(i)
(j)
Indentation
Front view
(l)
(k) Rear view
Vertical High strain area
Crack
Limit/complete (30% pretension)
(m)
(n)
Front view
(o)
(p)
Damage vertical formation
Rear view
Figure 7.8 Hemispherical projectile impact on 30% pretension carbon ﬁberreinforced plastic targets.
images (k and l) shows a similarity in ﬁber cracking with high strain formation in the FE simulation. An experiment panel in images (m and o) did not fail catastrophically upon impact. From observation of the highspeed camera during the experiment, the panel will ﬁrst experience cracking together with a few vibrations (bouncing forward and backward) before catastrophic failure occurs. However, in the FE simulation, higher strain formation could be seen from the rear view in images (n and p), which might have contributed to ﬁber cracking and catastrophic failure.
130
7.4.5
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
Impact force
Four predicted impact forceetime curves for hemispherical projectiles impacting on plates with various pretensions are plotted in Fig. 7.9. The impact velocity for all four cases was taken at 135 m/s. At 0% pretension, the force achieved the highest value compared to other pretension values. As pretension in the target was increased, the graph showed a reduction in the peak load. The highest reduction was experienced from 50% target pretension; the reduction from 0% to 50% are pretension was found to be approximately 25%. The graph indicates that pretension in the target reduced the time taken for the projectile to perforate. In the early impact stages, the graph showed an early step in load before reaching its peak. The phenomenon is stated as Region 1 in Fig. 7.9. Similar phenomena were noted by Sun et al., who concluded that this was due to the target bounce caused by the impact from the projectile [6]. As the pretension increased, the ﬁber became more stretched and this reduced its bounce. The various pretensions applied to the CFRP target gave the characteristic shape to the velocity proﬁle, as shown in Fig. 7.10. An increase in projectile movement was seen for the projectile to perforate as the pretension of the target was increased. Although each curve has a very similar pattern, there was still a small difference. As pretension was increased from 0% to 10%, the velocity showed a reduction (higher resistance). However, as the pretension was increased to 30% and 50%, the target’s resistance showed a reduction, which allowed for easier perforation. Pretension made the ﬁber more stretched and increased its resistance to the projectile. However, pretension also increased the stress in the ﬁber and made its failure easier, thus this Region 1
6
0%
Region 2
10%
5
30% 50%
Force (kN)
4
3
2
1
0
0
10
20
30
40
50
60
Time (µs)
Figure 7.9 Graph force versus time on various pretension carbon ﬁberreinforced plastic targets.
Progressive damage modeling of synthetic ﬁber polymer composites under ballistic impact
131
120 110
Vel 0%
100
Vel 10% Vel 30%
Velocity (m/s)
90
Vel 50%
80 70 60 50 40 30 20 10 0 0
100
200
300
400
Time (µs)
Figure 7.10 Velocity versus displacement of a hemispherical projectile impacted on various pretension carbon ﬁberreinforced plastic targets.
reduced the resistance of the plate to the projectile. It is believed that ﬁber stretch was dominant when pretension was low, and high pretension in the plate contributed more to ﬁber failure.
7.5
Conclusions
An Abaqus/Explicit FE was used to simulate the impact behavior of a CFRP target from a hemisphericalshaped nose projectile. Pretension was applied to the composite plate and damage behavior on the target plate was investigated. It is important to run a mesh sensitivity study to determine an optimum element size for ballistic limit analysis. The predicted amount of ballistic limit by the specimen was found to agree well with the experimental data over the complete range of projectile velocity. The model also captured the failure mode, form indentation to petalling, and a perforation process at higher velocities. It was shown that toward the investigated velocity range the contribution to damage failure modes was highly signiﬁcant to the experimental results even without using cohesive interaction effects on the elements.
Acknowledgments The authors wish to convey sincere gratitude to Universiti Tun Hussein Onn Malaysia and the Ministry of Higher Education Malaysia for providing ﬁnancial assistance during the preparation of this work.
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References [1] Abdullah AS, Yamin AFM, Ghafar H. Finite element modelling of composite hollowcore slabs. Proc Annu Conf Can Soc Civ Eng 2016;3:33e40. [2] Shi Y, Swait T, Soutis C. Modelling damage evolution in composite laminates subjected to low velocity impact. Compos Struct 2012;94:2902e13. [3] Kamarudin KA, Ismail AE. Modelling high velocity impact on aluminium alloy 7075T6 under axial pretension. Appl Mech Mater 2014;629:498e502. [4] Iqbal MA, Gupta G, Diwakar A, Gupta NK. Effect of projectile nose shape on the ballistic resistance of ductile targets. Eur J Mech A Solids 2010;29:683e94. [5] Abdullah AS, Kuntjoro W, Yamin AFM. Finite element modelling of aluminum alloy 2024T3 under transverse impact loading. AIP Conf Proc 2017;50005:1e6. [6] Sun B, Liu Y, Gu B. A unit cell approach of ﬁnite element calculation of ballistic impact damage of 3D orthogonal woven composite. Compos Part B Eng 2009;40:552e60. [7] Mohd Norihan I, Ahmad Zaidi AM, Siswanto WA. Numerical study of 2024 T3 aluminum plates subjected to impact and perforation. J Mech Sci Technol 2014;28:4475e82. [8] Heimbs S, Bergmann T, Schueler D, TosoPentecOte N. High velocity impact on preloaded composite plates. Compos Struct 2014;111:158e68. [9] Lambert M, Schneider E. Hypervelocity impact on gas ﬁlled pressure vessel. Int J Impact Eng 1997;20:491e8. [10] Kelkar A, Sankar J, Rajeev K, Aschenbrenner R, Schoeppner G. Analysis of tensile preloaded composites subjected to lowvelocity impact loads. In: 39th AIAA/ASME/ ASCE/AHS/ASC Struct. Struct. Dyn. Mater. Conf. Exhib., American Institute of Aeronautics and Astronautics; 1998. [11] Chiu ST, Liou YY, Chang YC, Ong C long. Low velocity impact behavior of prestressed composite laminates. Mater Chem Phys 1997;47:268e72. [12] GarcíaCastillo S, SanchezSaez S, Barbero E, Navarro C. Response of preloaded laminate composite plates subject to high velocity impact. J Phys IV 2006;134:1257e63. [13] GarciaCastillo S, SanchezSaez S, LopezPuente J, Barbero E, Navarro C. Impact behaviour of preloaded glass/polyester woven plates. Compos Sci Technol 2009;69: 711e7. [14] Ulven C, Vaidya UK, Hosur MV. Effect of projectile shape during ballistic perforation of VARTM carbon/epoxy composite panels. Compos Struct 2003;61:143e50. [15] GarciaCastillo S, Navarro C, Barbero E. Damage in preloaded glass/vinylester composite panels subjected to highvelocity impacts. Mech Res Commun 2014;55:66e71. [16] Mikkor KM, Thomson RS, Herszberg I, Weller T, Mouritz AP. Finite element modelling of impact on preloaded composite panels. Compos Struct 2006;75:501e13. [17] ABAQUS standard user’s manual. ABAQUS user’s manual. 2010. [18] Lapczyk I, Hurtado JA. Progressive damage modeling in ﬁberreinforced materials. Compos Part A Appl Sci Manuf 2007;38:2333e41. [19] Chan S, Fawaz Z, Behdinan K, Amid R. Ballistic limit prediction using a numerical model with progressive damage capability. Compos Struct 2007;77:466e74. [20] Feng D, Aymerich F. Finite element modelling of damage induced by lowvelocity impact on composite laminates. Compos Struct 2014;108:161e71. ˇ
Investigation of damage processes of a microencapsulated selfhealing mechanism in glass ﬁberreinforced polymers
8
J. Lilly Mercy, S. Prakash School of Mechanical Engineering, Sathyabama Institute of Science and Technology, Chennai, India
8.1
Introduction
High stiffness, high strength, and light weight are the major features of composite materials when they compete with traditional metals and alloys. Of all the varieties of composites in use, ﬁberreinforced plastics (FRPs) take on an important role in the composite industry because of their ease of manufacture and variety of applications. Energy absorption during the stress of FRP leads to plastic deformation, whereas the outofplane loading leads to damage, expressed as intraply matrix cracks or interply delamination. Traditional repair techniques for polymeric composites are difﬁcult and costly and hence engineers are on the lookout for “damagetolerant” materials for a stipulated service life. As an alternative approach to conventional repair, selfhealing mechanisms came to light and gave rise to various strategies and models. The possibility of a material healing on its own without any external intervention became the crux of research. Of the many processes of selfhealing, three basic methods were identiﬁed: (1) capsule based, (2) vascular, and (3) intrinsic selfhealing [1]. These three basic methods developed into many variations. Many researchers, by varying the different healing agents, the method of delivering healing agents, and the rate of healing efﬁciency and fracture toughness, analyzed the triggers needed to activate the selfhealing system.
8.2
Chemistry of capsulebased selfhealing materials
In capsulebased selfhealing, the healing agent is held inside a microcapsule or nanocapsule and it ﬂows out when the capsule breaks when a crack propagates through the composite. This healing agent ﬁlls the gap and cures to become a hard and tough material through the polymerization process with the help of a catalyst trigger. The viscosity of the healing agent, its shelf life inside the embedded capsule,
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites and Hybrid Composites https://doi.org/10.1016/B9780081022894.000084 Copyright © 2019 Elsevier Ltd. All rights reserved.
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
Catalyst Microcapsule Crack
Healing agent
Polymerized healing agent
Figure 8.1 Mechanism of selfhealing (White et al., 2001).
the toughness of the cured or polymerized healing agent, the stiffness of the capsule wall, and the availability of catalyst or hardener near the damaged area are factors that determine the success of selfhealing in a capsulebased healing system. The capsulebased selfhealing process is the forerunner, which is also simple in design, fabrication, and application. The ﬁrst capsulebased selfhealing system was described by White et al. [2], who inculcated microcapsules made of ureaformaldehyde (UF) shell containing dicyclopentadiene (DCPD) in monomer form in an epoxy resin matrix. Fig. 8.1 shows the mechanism of selfhealing in a capsulebased composite.
8.3
Background to the study
The focus of the entire research work over the past decade was to ﬁnd an efﬁcient healing agent that would give better healing efﬁciency and resistance to future cracks. However, when the material was put into realtime usage, there were other allied factors such as basic mechanical properties, the response of the selfhealing system during the manufacturing cycle, etc. To get an indepth understanding of the basic static and dynamic mechanical properties of a selfhealing composite, tests were conducted according to ASTM standards. The makeup of the selfhealing glass ﬁberreinforced plastic (GFRP) was decided by the percentage of healing
Investigation of damage processes of a microencapsulated selfhealing mechanism in glass
135
agent included, the method of delivery of the healing agent, the matrix and reinforcement material, etc. Hence these were taken as parameters and varied to fabricate specimens of different percentages of constituents and tested for their mechanical properties.
8.4 8.4.1
Fabrication process Fabricating microcapsules
The procedure for making microcapsules was adopted from Brown et al. [3]. The agitation rates during the procedure decided the size of the microcapsules obtained from the process. The thickness of the microcapsule was affected by a high amount of ammonium chloride or resorcinol, whose initial pH was very low, etc. Hence the polymerization process was done with extreme care (Fig. 8.2). Capsules were prepared at three different agitation rates of 300e400, 600e700 and 800e900 rpm. The capsules obtained with these three different agitation rates were analyzed for shape uniformity. Uniform sphericalsized capsules of DCPD with a UF shell wall were obtained at all the agitation rates. No capsules were found to be sticking to each other and dry surface capsules were observed. The yield of the microcapsules was achieved by the ratio of the mass of the capsules obtained from preparation with the mass of DCPD plus UF. At an agitation of
Figure 8.2 Microcapsules of varied diameters obtained through in situ polymerization.
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
200 μm
EHT = 5.00kV WD = 8.6 mm
Signal A = SE2 Mag = 115 X
Date :16 oct 2014 Time :9:44:33
Figure 8.3 Surface of the microcapsule.
300e400 rpm, the capsule yield was around 68% and at 600e700 rpm it was 70%, whereas at an agitation of 800e900 rpm, the yield was around 80%. The average diameter of the capsule sample obtained at 300e400 rpm was around 900 mm, at 600e700 rpm it was 580 mm, and at 800e900 rpm it was 280 mm. The surface morphology of the capsules was studied using scanning electron microscopy (SEM). The outer surface of the shell seemed to have a rough surface texture as shown in Fig. 8.3. The capsule is made of smaller particle blocks of around 5e20 mm, which are compactly packed. The precipitation of higher molecular weight prepolymer and its aggregation and deposition at the capsule surface result in a rough outer layer of UF shell [4].
8.4.2
Fabricating selfhealing GFRP
Selfhealing GFRP plates were fabricated through layerbylayer stacking of glass ﬁber mats impregnated with epoxy resin, microcapsules, and Grubb’s catalyst. GFRP with a selfhealing system comprised DCPDﬁlled microcapsules and Grubb’s catalyst mixed with epoxy resin and hardener and hand laid with glass ﬁber mats. To maintain the isotropic nature of the fabricated composite, the mats were alternately arranged in different orientations, say 0/30/45/60/90 degrees. The panel was built to a thickness of 3 mm. The fabricated panel was kept under compression at 80 C under a pressure of 13 MPa and cured for 6 h. Fig. 8.4 shows the sequence of fabrication of selfhealing GFRP.
Investigation of damage processes of a microencapsulated selfhealing mechanism in glass
(a)
(c)
137
(b)
(d)
Figure 8.4 Fabrication of selfhealing GFRP plates: (a) microcapsules and Grubbs catalyst mixed in epoxy resin; (b) hand layup process; (c) resin mixed with capsules poured on the ﬁber mat; (d) selfhealing GFRP panel.
8.5
Experimental plan
As the focus of this work is on ﬁnding out the effect of adding selfhealing microcapsules to the composite and its mechanical properties, samples were fabricated by adding microcapsules of different sizes and concentrations. The concentration of the catalyst was also considered. These three factors along with catalyst size were found to inﬂuence the fracture toughness and healing efﬁciency of the selfhealing epoxy composite as reported by Brown et al. [4]. Because the catalyst was procured from SigmaAldrich at a speciﬁc particle size, the size of the catalyst could not be included as a factor in this research work. Table 8.1 shows the factors and
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
Table 8.1 Factors and levels considered for fabrication of samples Levels Factors
1
2
3
Microcapsule size (mm) 50 mm
300
600
900
Microcapsule concentration (%wt)
5
15
25
Catalyst concentration (%wt)
0.5
1.5
2.5
levels considered for experiments. The levels for microcapsule concentration and size were decided based on the previous investigations by Brown et al. [4] and Joseph et al. [6] where maximum healing efﬁciency and fracture toughness were observed. The levels of microcapsule size were based on the size of microcapsules harvested during the in situ polymerization process of microcapsules by altering the agitation rpm. More than 25% of microcapsules, when mixed in the epoxy resin, increased their viscosity and hence the levels of microcapsule concentration were set below 25%. A catalyst concentration of 2.5% was found to be the optimum concentration for healing efﬁciency and hence the concentration levels were set based on that amount. When the factors inﬂuencing the output responses were increased, the process became costly and it was difﬁcult to conduct experiments with a combination of all levels in each factor. Hence the design of experiments was formulated, which is a systematic method to ﬁnd the interrelationship between factors and the effect of each input factor on the output responses. When compared with the classical design of experiments, the one proposed by Taguchi was found to be robust, produced reproducible results, and hence there was no random error [5]. Therefore Taguchi’s orthogonal array was used in designing the experiments in this study. To design an experiment with three factors and three levels, an L9 or L27 array was suggested. Because an L27 array was a full factorial array involving a combination of all input factors in all its levels, to minimize the experimental work and cost, an L9 orthogonal array was chosen.
8.6
Testing
Materials behave differently under different loading conditions. Stress occurs on a material when a load is applied at rest and is different when a load applied in motion. A new material needs to be studied for both its static and dynamic aspects before it can be put into a speciﬁc application.
Investigation of damage processes of a microencapsulated selfhealing mechanism in glass
8.6.1
139
Static mechanical properties
The common static mechanical properties of a ﬁberreinforced composite include ultimate tensile strength, yield strength, elastic modulus, ultimate rupture stress and strain, fatigue and creep behavior, delamination between adjacent layers of the composite, shearing between adjacent layers, impact strength, etc. To study all these aspects, four major static mechanical tests were chosen for the study: 1. 2. 3. 4.
Tensile testdASTM: D3039 Compression testdASTM: D695 Flexural test (threepoint bend test)dASTM: D790 Inplane shear test (doublenotched shear)dASTM: D3846
The nine samples obtained by the addition of various concentrations and sizes of microcapsules and catalyst were tested for all four mechanical properties (Fig. 8.5). The strengths acquired through different testing methods are tabulated in Table 8.2.
8.6.2
Dynamic mechanical properties
Dynamic mechanical properties are those properties that are obtained by subjecting the composite to movement and variation. The three major parameters that are used in the assessment of dynamic mechanical properties are: (1) storage modulus, a measure of maximum energy stored in the material, (2) loss modulus, a measure of energy dissipated as heat, and (3) damping factor, a ratio of loss modulus to storage modulus [7]. The dynamic mechanical properties are dependent on the type of ﬁber, its orientation, volume of ﬁbers, ﬁllers, impact modiﬁers, coupling agents, mode of testing, etc.
(a)
(b)
(c)
(d)
Figure 8.5 Test specimens after static mechanical testing: (a) tensile test specimen; (b) compressive test specimen; (c) ﬂexural test specimen; (d) inplane shear test specimen.
140
Table 8.2 Experimental results of static mechanical properties Output response (MPa)
Experiment no.
Microcapsule size (mm) ±50 (A)
Microcapsule concentration (%wt) (B)
Catalyst concentration (%wt) (C)
Tensile strength (MPa)
Compressive strength (MPa)
Flexural strength (MPa)
Inplane shear stress (MPa)
1
300
5
0.5
308.2
1393
1988
56
2
300
15
1.5
312.8
1393
1972
64
3
300
25
2.5
315.6
1391
1962
69
4
600
5
1.5
314.6
1390
1905
60
5
600
15
2.5
313.9
1390
1896
65
6
600
25
0.5
316.2
1389
1893
75
7
900
5
2.5
322.4
1391
1882
63
8
900
15
0.5
320.2
1388
1853
67
9
900
25
1.5
323.6
1388
1879
80
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
Input parameters
Investigation of damage processes of a microencapsulated selfhealing mechanism in glass
141
[7e10]. The glass transition temperature (Tg), where the material moves from a hard to a rubbery state, can be found using the curves of loss modulus (E00 ) and damping factor (tand) [8]. Dynamic mechanical analysis was conducted using a Dynamic Mechanical Spectrometer DMS6100, Japan, having a force range of 7.8 N. A specimen of size 20 mm 10 mm was cut and the measurement was done according to ASTM D406501 through dual cantilever bending in synthetic oscillation mode to measure elastic modulus transformation at multiple frequencies. The temperature was varied from 50 to 220 C at the constant rate of 2 C/min, and tested for frequencies of 0.5, 1, 2, 5, and 10 Hz. The synthetic wave oscillation mode of the spectrometer varied between ﬁve different frequencies. The selfhealing GFRP sample was subjected to dynamic frequency scan to obtain the material ﬂow elasticity or stiffness as a function of frequency. Sampling frequency with simultaneous temperature scans is the right way to test polymers, due to their viscoelastic nature, which turns the material soft at a temperature inﬂuencing the dynamic modulus of the material. Figs. 8.6 and 8.7 show the dynamic mechanical spectrometer used for the study.
Figure 8.6 Dynamic mechanical spectrometer DMS6100.
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
Figure 8.7 Specimen size 20 mm 10 mm subjected to ASTMD406501dual cantilever bending.
8.7 8.7.1
Study of the effect of a selfhealing agent on mechanical properties Effect of factors on tensile strength
From Table 8.1, it can be observed that the tensile strength is at the maximum when the microcapsule size and concentration are high and the catalyst concentration is low. The highest microcapsule size ensures more storage of the healing agent DCPD being delivered to the crack area and high microcapsule concentration enables proximity of the capsule near the crack [4]. The tensile strength being high at high microcapsule size and concentration seems also to be favorable for the healing process. It is evident from the experimental results that the bonding between the capsules with the matrix and glass ﬁber reinforcement seems to be stronger than the bonding between mere epoxy matrix and glass ﬁber. This can be attributed to the microcapsules settling in the interstitial spaces of the glass ﬁber mat, adding up to the surface area of bonding with the ﬁber. It has been reported that the virgin fracture properties in a polymer composite were enhanced by the addition of microcapsules and catalyst [4]. The catalyst concentration being low in percentage did not show much deviation in tensile strength due to its addition.
Investigation of damage processes of a microencapsulated selfhealing mechanism in glass
8.7.2
143
Effect of factors on compressive strength
Out of the three measured outputs, compressive strength did not show signiﬁcant deviation in response to the variation of microcapsule size, concentration, and catalyst concentration in selfhealing GFRP. This could be because the compressive load is mainly borne by the glass reinforcement in the selfhealing GFRP material and hence is independent of the selfhealing microcapsules and catalyst. Moreover, the Young’s modulus of the capsule was reported to be around 3.7 GPa [11], irrespective of the size of capsule, and hence the selfhealing GFRP was purely yielded due to ﬁber breakage. However, the compressive strength of the selfhealing GFRP was observed to be roughly four times that of tensile strength. For thin specimens, failure due to compressive load occurs due to the buckling of the reinforced ﬁbers [12]. As the glass ﬁber is reinforced in mat form and placed at alternate orientations, buckling does not seem to occur. As it can be observed from Table 8.1, the compressive strength ranges from 1389 to 1393 MPa, which is a very minor deviation. Hence it can be concluded that the presence of microcapsules does not make any difference to the compressive strength of the selfhealing GFRP.
8.7.3
Effect of factors on ﬂexural strength
The ﬂexural strength values observed at different combinations of the L9 array gave results contrary to the tensile strength, which was higher at lower microcapsule size, concentration, and catalyst concentration as shown in Figs. 8.8 and 8.9. This can be attributed to a large area separation of the adjacent ﬁber layers by the microcapsules, where the epoxy settles down to ﬁll the gap during curing. When the capsule size is smaller, it just ﬁlls the interstices and layer separation is not so marked. Out of the three parameters, it can be found that microcapsule size contributes more toward ﬂexural strength. Effect of factors on tensile strength 324
Tensile strength
322 320 318 316 314 312 310 308 306
1
2
3
Microcapsule size
312.2
314.9
322.1
Microcapsule concentration
315.1
315.6
318.5
Catalyst concentration
314.9
317
317.3
Figure 8.8 Effect of factors on tensile strength.
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
Compressive strength
Effect of factors on compressive strength 1392.5 1392 1391.5 1391 1390.5 1390 1389.5 1389 1388.5 1388 1387.5
Microcapsule size Microcapsule concentration Catalyst concentration
1392
1
2 1390
3 1389
1391
1390
1389
1390
1390
1391
Figure 8.9 Effect of factors on compressive strength.
8.7.4
Effect of factors on inplane shear strength
The inplane shear stress was found to be highly dependent on microcapsule concentration followed by microcapsule size, and the combined effect of an increase in microcapsule concentration and size also increased the inplane shear strength to a maximum of 42%. Catalyst concentration does not seem to have any signiﬁcant effect on response as it only triggers the polymerization reaction whether it is available in large or small quantities. However, the closer proximity of the catalyst to the microcapsule matters for the polymerization reaction to occur. The capsules seem to bond well with the resin and ﬁber to ﬁll the interstices and so they exhibit high shear strength (Figs. 8.10 and 8.11).
8.8
Study of the effect of selfhealing agent on dynamic mechanical properties
The storage modulus curves for different frequencies and temperatures are shown in Fig. 8.12. The ﬁgure shows a decrease in storage modulus (E0 ) with the decrease in frequency until selfhealing GFRP reaches its glass transition phase after which the material turns viscous and the modulus decreases substantially. Storage modulus curve can be segmented into three different regions: lowtemperature glassy region, steep descent in modulus region, and hightemperature viscous region [13]. The selfhealing GFRP has a steep decrease in E0 between 70 and 110 C. Less frequency means the material has more time to respond to the temperature as time and frequency are inversely proportional and so the modulus value decreases [14]. Loss modulus (E00 ) is the measure of the energy dissipated as heat when the material turns viscous. Fig. 8.13 shows the loss modulus curves as a function of temperature at varied frequencies. The peak in loss modulus curve indicates the Tg region, which varies across different frequencies.
Investigation of damage processes of a microencapsulated selfhealing mechanism in glass
145
Effect of factors on flexural strength
Flexural strength
2000 1980 1960 1940 1920 1900 1880 1860 1840 1820 1800 Microcapsule size
1 1974
1898
2
3 1871
Microcapsule concentration
1925
1907
1911
Catalyst concentration
1911
1919
1913
Figure 8.10 Effect of factors on ﬂexural strength.
The Tg represented through the loss modulus curve was found to increase with an increase in frequency because low frequencies allow the material to ﬂow for a long time, thus decreasing the loss modulus. The Tg values varied between 85 and 110 C for different frequency values. Until the material reaches the rubbery state the dissipation of heat is higher and after that the material takes in heat in its viscous nature and the curve drops down. The interfacial adhesion between the microcapsules, matrix, and glass ﬁbers of selfhealing GFRP can be well understood using the damping curve shown in Fig. 8.14. The peak of the damping curve denotes Tg of the composite material. Damping Effect of factors on inplane shear strength
Implane shear strength
80 75 70 65 60 55 50 Microccapsule size Microccapsule concentration Catalyst concentration
63
1
2 66.67
3 70
59.67
65.33
74.67
66
68
65.67
Figure 8.11 Effect of factors on inplane shear strength.
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites 3500 0.5 Hz
1 Hz
Storage modulus (E') in MPa
3000
2 Hz 5 Hz
2500
10 Hz
2000
1500 1000 500
0 60
80
100
120
140 160 180 Temperature (°C)
200
220
240
Figure 8.12 Storage modulus curve.
peak is affected by the bonding between the individual materials in the composite, curing conditions, nature of the ﬁllers, reinforcements, etc. [8]. It was found that the tand values increased with increased frequencies. The Tg value from the tand curve ranged across 105e130 C for different frequencies. The ColeeCole plot is constructed with log E0 versus log E00 values and the semicircular nature of the curve indicates that the material is homogeneous [15]. To describe the viscoelasticity of a polymer, single relaxation peaks are not enough and hence the ColeeCole plot is considered as it expresses the dielectric data [7]. The ColeeCole plot of selfhealing GFRP is shown in Fig. 8.15. The
Shift peaks of loss modulus
350
Loss modulus (E") IN MPa
0.5 Hz 1 Hz
300
2 Hz 5 Hz 10 Hz
250 200
150
100 50 0 60
80
100
120
140
160
Temperature (°C)
Figure 8.13 Loss modulus curves.
180
200
220
240
Investigation of damage processes of a microencapsulated selfhealing mechanism in glass
147
0.35 0.5 Hz 1 Hz 2 Hz 5Hz 10 Hz
0.3 0.25 Tan δ
0.2 0.15 0.1 0.05 0 60
80
100
120
140 160 Temperature (°C)
180
200
220
240
Figure 8.14 Tan d curve.
nonsemicircular curve shows that the selfhealing GFRP is heterogeneous and indicates good ﬁberemicrocapsuleematrix adhesion. Until the glass transition value, the material behaves like elastic and hence frequency shows little deviation in the ColeeCole plot, whereas after the material turns viscous, frequency seems to have no effect.
8.9
Microstructural analysis
The specimens after conducting various tests were subjected to microstructural analysis. Microstructural analysis was performed through by SEM. Fig. 8.16 shows the Colecole plot for various frequencies
9
Log (E")
8.5
8
7.5 0.5 Hz
7
1 Hz 2 Hz 5 Hz
6.5 8
10Hz
8.2
Figure 8.15 ColeeCole plot.
8.4
8.6
8.8 Log (E')
9
9.2
9.4
9.6
148
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
100 μm
EHT = 20.00kV
signal A = SE2 Mag = 351 X
Date :15 may 2015 Time :14:57:06
Figure 8.16 Scanning electron microscopy image of the surface of resin with microcapsules.
uniform distribution of epoxy resin and the capsules embedded in the matrix. The random distribution of the capsules was observed. Fig. 8.17 shows the edge of the inplane shear specimen, where the groove was cut. Due to shear the capsules were pulled apart leaving empty holes in the resin portion. Fig. 8.18 also shows the holes left behind by the capsules due to the shearing action. It could also be observed that a broken empty sheared capsule still remains attached to the resin surface. Fig. 8.19 shows the capsule embedded in the matrix layer between the ﬁber layers, which becomes the site for crack initiation. Fig. 8.20 shows the capsule with polymerized DCPD. The inner wall of the capsule is smooth.
20 μm
EHT = 20.00kV wd = 7.2 mm
signal A = SE2 Mag = 524 X
Date :15 may 2015 Time :14:58:44
Figure 8.17 Scanning electron microscopy image at the edge of a groove in an inplane shear specimen.
Investigation of damage processes of a microencapsulated selfhealing mechanism in glass
100 μm
EHT = 20.00kV WD = 6.8 mm
signal A = SE2 Mag = 115 X
149
Date :15 may 2015 Time :15:00:42
Figure 8.18 Scanning electron microscopy image showing a sheared microcapsule.
8.10
Discussion and conclusion
It has been observed that the tensile strength increased with higher concentration and size of microcapsules with lesser catalyst concentration. On the contrary, higher ﬂexural strength was achieved at lesser microcapsule concentration, size, and catalyst concentration, whereas compressive strength did not show a signiﬁcant rise by the variation of the input parameters. It is evident that the capsules bonding with the ﬁber layers need to be strong enough to hold the ﬁbers together, which resulted in an
100 μm
EHT = 20.00kV WD = 7.4 mm
signal A = SE2 Mag = 115 X
Date :15 may 2015 Time :15:01:34
Figure 8.19 Scanning electron microscopy image showing a capsule embedded in a matrix layer.
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100 μm
EHT = 10.00kV WD = 25.2 mm
signal A = SE2 Mag = 346 X
Date :9 nov 2016 Time :15:39:59
Figure 8.20 Scanning electron microscopy image of a capsule with polymerized dicyclopentadiene.
increase in tensile strength. The decrease in ﬂexural strength by the increase in microcapsule size and concentration is because larger capsules separate the ﬁber layers to a large distance, giving way for the matrix to yield during the bending load. Dynamic mechanical properties such as storage modulus, loss modulus, and damping factor were tested and analyzed for different frequencies for a constant temperature increase. Viscoelastic behavior was observed clearly from the data obtained and the Tg values obtained from the loss modulus curve were correlated with the Tg values obtained from the tan d curve. It was observed that the increase in frequency increased the storage modulus, loss modulus, and damping factor because frequency is a function of time. Finally, the irregular, nonsemicircular ColeeCole plot conﬁrms the heterogeneity of the selfhealing GFRP composite.
References [1] Blaiszik BJ, Kramer SLB, Olugebefola SC, Moore JS, Sottos NR, White SR. Selfhealing polymers and composites. Annu Rev Mater Res 2010;40:179e211. [2] White SR, Sottos NR, Geubelle PH, Moore JS, Kessler MR, Sriram SR, Brown EN, Viswanathan S. Autonomic healing of polymer composites. Nature 2001;409:794e7. [3] Brown EN, Kessler MR, Sottos NR, White SR. Insitu poly (urea formaldehyde) microencapsulation of dicyclopentadiene. J Microencapsul 2003;20(No. 6):719e30. [4] Brown EN, Sottos NR, White SR. Fracture testing of a selfhealing polymer composite. Exp Mech 2002;42(4):372e9. [5] Taguchi G, Yokoyama Y. Taguchi methods: design of experiments, vol. 4. The University of Michigan, ASI Press; 1993. [6] Joseph DR, Nancy RS, White SR. Effect of microcapsule size on the performance of selfhealing polymers. Polymer 2007;48:3520e9.
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[7] Mandal S, Alam S. Dynamic Mechanical Analysis and morphological studies of Glass/ bamboo ﬁbre reinforced unsaturated polyester resin based hybrid composites. J Appl Polym Sci 2012;125:E382e7. [8] Manoharan S, Suresha B, Ramadoss G, Bharath B. Effect of short ﬁbre reinforcement on Mechanical properties of Hybrid phenolic composites. Journal of Materials; 2014. https:// doi.org/10.1155/2014/478549. [9] Poletto M, Zeni M, Zattera AJ. Dynamic Mechanical Analysis of recycled polystyrene composites reinforced with wood ﬂour. J Appl Polym Sci 2012;125:935e42. [10] Tajvidi M, Falk RH, C.Hermanson J. Effect of natural ﬁbres on thermal and mechanical properties of natural ﬁbre polypropylene composites studied by dynamic mechanical analysis. J Appl Polym Sci 2006;101:4341e9. [11] Keller MW, Sottos NR. Mechanical properties of microcapsules used in a selfhealing polymer. Exp Mech 2006;46:725e33. [12] Mallick PK. Fiber reinforced composites. Materials, Manufacturing and Design, CRC Press; 2007. [13] Pothan LA, Thomas S, Groeninckx G. The role of ﬁbre/matrix interactions on the dynamic mechanical properties of chemically modiﬁed banana ﬁbre/polyester composites. Compos A Appl Sci Manuf 2006;37(No. 9):1260e9. [14] Menard KP. Dynamic mechanical analysis: a Practical introduction. CRC Press; 1999. [15] Idicula M, Malhotra SK, Joseph K, Thomas S. Dynamic mechanical analysis of randomly oriented intimately mixed short banana/sisal hybrid ﬁbre reinforced polyester composites. Compos Sci Technol 2005;65(No. 7e8):1077e87. https://doi.org/10.1016/j.compscitech. 2004.10.023.
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Finite element analysis of natural ﬁberreinforced polymer composites
9
J. Naveen 1 , Mohammad Jawaid 2 , A. Vasanthanathan 3 , M. Chandrasekar 4 1 Department of Mechanical and Manufacturing Engineering, Universiti Putra Malaysia, Serdang, Malaysia; 2Laboratory of Biocomposite Technology, Institute of Tropical Forestry and Forest Products (INTROP), Universiti Putra Malaysia, Serdang, Malaysia; 3Department of Mechanical Engineering, Mepco Schlenk Engineering College, Sivakasi, India; 4 Department of Aerospace Engineering, Universiti Putra Malaysia, Serdang, Malaysia
9.1
Introduction
The term “composite” is attributed to the combination of two or more constituents. The three constituents of a composite (Fig. 9.1) are reinforcement, matrix, and interface. The reinforcement is the loadbearing member of the composite, while the matrix is the binding medium for the reinforcement. Interface is the common contact surface between reinforcement and matrix. According to the type of matrix medium, composites are categorized as polymer matrix composites (PMCs), metal matrix composites, ceramic matrix composites, and carbon/carbon composites. PMCs predominantly use thermosetbased matrices such as polyester, epoxy, and phenolic resins due to their good properties. PMCs are principally used in the aircraft and spacecraft industries due to their high speciﬁc strength, speciﬁc stiffness, light weight, high fatigue resistance, and high corrosion resistance. From the standpoint of types of ﬁbers, composites are classiﬁed as synthetic ﬁberreinforced composites and natural ﬁberreinforced composites. Synthetic ﬁberreinforced composites incorporate carbon ﬁberreinforced plastic, glass ﬁberreinforced plastic, Kevlar ﬁberreinforced plastic, and hybrid ﬁberreinforced plastic. Natural ﬁberreinforced composites implicate composites with rice husk, coir, cotton, sugarcane bagasse, jute, sisal, wool, hemp, etc., as reinforcements, which are shown in Fig. 9.2. Synthetic ﬁbers are ﬁbers produced by a manufacturing process, while natural ﬁbers are available readily on the earth. The noteworthy application of natural ﬁberreinforced polymer composites (NFRPCs) is in nonloadbearing applications in the case of automobile and aircraft structures. The potential use of natural ﬁbers in the automotive industry comprises the production of interior panels, door panels, body panels, etc. The aircraft industry also takes advantage of natural ﬁbers through their usage in secondary structures, namely interiors, panels, etc. In short, natural ﬁberreinforced composites have been gaining much attention in various applications because of their biodegradability, low material cost, availability, and recyclability.
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites and Hybrid Composites https://doi.org/10.1016/B9780081022894.000096 Copyright © 2019 Elsevier Ltd. All rights reserved.
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Matri
Reinforcement Interphase
Figure 9.1 Constituents of polymer composites.
Rice husk
Coir
Cotton
Sugarcane bagasse
Jute
Sisal
Wool
Hemp
Figure 9.2 Natural ﬁbers.
Exact solutions for simple geometries such as beams, columns, plates, and shells made from natural ﬁberreinforced composites can be found in the literature. For complex structures and shapes used in aircraft assemblies, as well as for the defects and damage caused by bird strike and other mechanical loads, it is not feasible to attain exact solutions wherein only approximate solutions are possible. Approximate solutions are acceptable solutions, which are near to the exact solution. There are many approximate methods for solving complex problems, namely the RayleigheRitz method, Galerkin’s method, ﬁnite element method (FEM), ﬁnite difference method, and ﬁnite volume method. Among the various approximate methods, FEM [1] is the most popular among the engineering communities because, “Any structure with complex shape, material, boundary conditions, loading and material model could be easily solved by FEM.” Through ﬁnite element analysis (FEA), virtual experiments of the actual complex phenomenon could be conducted and viewed in a graphical user interface environment. Engineers conduct many iterations using FEA to optimize results and increase accuracy, which could reduce the down time in product development, enhance its lifetime, and account for uncertainties. Fig. 9.3 represents an overview of the process steps in FEA.
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FEA
Preprocessor Defining the problem
Solution Solving
Defining geometry Assigning loads,constraints Specifying element type and solving Defining material properties Creating meshes and nodes
Postprocessor Viewing the results Lists of nodal displacements Deflection plots Stress contour diagrams Animation
Figure 9.3 Overview of ﬁnite element analysis (FEA).
9.2 9.2.1
Basic steps in ﬁnite element analysis Preprocessing
The principal objectives of preprocessing include 1D, 2D, or 3D modeling of the problem, assigning suitable material models, elements, meshing, material properties, and applying proper structural boundary conditions or thermal boundary conditions and loads such as thermal loads, structural loads, electrical loads, or magnetic loads depending on the application requirement. Commercial software packages are available for carrying out preprocessing modules, namely AUTOCAD, SOLIDWORKS, CATIA, CREO, SOLID EGDE, ANSYS, LSPrePost, OptiStruct, COMSOL, etc. For complex geometries such as automobile car body design, train car body design, and aircraft fuselage design it is essential to use dedicated software for meshing the complex geometries to capture the realtime response of the material to the applied loads. HYPERMESH is the frequently used meshing software used by the ﬁnite element communities. Usually, preprocessing works in the front end of any analysis software. Fig. 9.4 highlights the preprocessing of an aircraft wing box model.
ANSYS R15.0
Figure 9.4 Preprocessing of an aircraft wing box model.
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9.2.2
Solving
In ﬁnite element computations, all the parameters such as load, stiffness matrix, nodal displacements, element stresses, and element strains are stored in matrix form. In the preprocessor, after meshing has been successfully performed, the element stiffness matrix, global stiffness matrix, and global force vectors are formed in the back end of the analysis software. Solving usually represents a dedicated solver to solve the following linear equation of an element as well as a global system. Eq. (9.1) represents the governing equation of an element, while Eq. (9.2) is the governing equation of the global system in any ﬁnite element formulation: ff g ¼ ½k fug
(9.1)
where {f} is the elemental force vector, [k] is the elemental stiffness matrix, and {u} are the elemental nodal displacements: fFg ¼ ½K fUg
(9.2)
where {F} is the global force vector, [K] is the global stiffness matrix, and {U} are the global nodal displacements. On the back end of any commercial FEA software package the foregoing governing equation of the global system must be solved. If proper meshing, loading, and boundary conditions are not applied, then there are chances of error in solving the ﬁnite element equations. Basically, the ﬁnite element equations are solved based on the Gaussian elimination method on the back end of any FEA software. The typical ﬁnite element solvers used over the years are ANSYS, LSDYNA, etc. In FEA computations, the computation time depends on the solver, which in turn depends on the number of elements and number of nodes. Hence a ﬁnite element analyst has to take care of the number of elements while meshing so as to decrease the solving time. The solving time also greatly depends on the type of element, namely linear element and higherorder elements. The higherorder elements would consume more computation time compared to the linear elements. Only after successfully solving any ﬁnite element model can the ﬁnite element outcomes, namely displacements, temperatures, pressures, or velocities, be estimated. If there is any error in solving, then the entire meshing should be cleared and reworked with reference to a new mesh pattern before solving it again.
9.2.3
Postprocessing
The primary objective of postprocessing is to display the FEA results, namely displacements (for structural problems), temperatures (for thermal analysis), pressures/ velocities (for ﬂuid analysis), stresses, and strains. The postprocessors in FEA also have the capability of generating the results in terms of tables, graphs, plots, and animation. Through the FEA postprocessor, the response of a structure under any loading, i.e., static, impact, thermal, fatigue, torque, etc., could be studied in detail before the product development stages. Commercially available software packages
Finite element analysis of natural ﬁberreinforced polymer composites 1
157
ANSYS
NODAL SOLUTION
R16.0
STEP=1 SUB=1 TIME1 USUM (AVG) RSYS=0 DMX =.301484 SMX =.301484
FEB 16 2016 14:04:46 Y Z
X
MX
0
.033498
.066997
.100495
.133993
.167491
.20099
.234488
.267986
.301484
Figure 9.5 Postprocessing of an aircraft wing box model.
are available for carrying out postprocessing capabilities, e.g., ANSYS, LSPrePost, OptiStruct, etc. Fig. 9.5 shows the postprocessing results of an aircraft wing box model wherein the displacements are found to be maximum at the center.
9.2.4
Convergence in ﬁnite element analysis
It is mandatory to carry out convergence analysis for every ﬁnite element computation. Convergence is the art of selecting an optimal number of nodes for the given problem. It is a wellknown fact that ﬁne meshes exhibit more accurate results than the coarse mesh counterpart. Even though ﬁne mesh gives good results, it is not that the entire domain would be discretized completely by ﬁne meshes. Discretization of any structure proceeds from the coarse mesh toward the ﬁne mesh. While linearly increasing the number of elements, at one point the solution converges with the exact solution, analytical solution, or experimental solution. At that point, convergence occurs and the respective number of elements is termed “optimal number of elements.” The number of elements selected based on the convergence study yields good results with greater accuracy and takes the least computation time. If the number of elements is not identiﬁed based on the convergence study, then the obtained FEA results exhibit greater error and are also time consuming. Also, for better approximation, free mesh and mapped mesh have to be used while meshing complex engineering structures.
9.3
Finite element analysis of polymer matrix composites
For product development using PMCs, it is essential to carry out subsequent FEA for the virtual simulation of the product under various loading environments. The FEA of PMCs could be performed either by using FEA software packages or coding using MATLAB, C language, etc.
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
Signiﬁcance of material characterization of composites in ﬁnite element analysis
For any ﬁnite element computation, it is essential to feed the experimental material properties of the composite material into the ﬁnite element model so that the virtual simulation of the composite product behaves similar to the actual model. The basic material properties of composites would be experimentally estimated through material or mechanical characterization [2]. Fig. 9.6 highlights the various American Standards for Testing and Materials (ASTM) standards used for determination of the material properties of composite materials through experiments. The material properties of a composite material would be estimated in accordance with the ASTM [3,4]. ASTM: D3039 emphasizes the estimation of material properties, namely unidirectional tensile strength, elastic modulus, and Poisson’s ratio. For the testing of a composite specimen using ASTM: D3039, the uniaxial tensile load is applied over the composite test coupon along the longitudinal direction. For instance, while preparing ASTM D3039 (Fig. 9.6a) composite test coupons, ﬁbers are arranged along 0/90 degrees. Strain gauges needs to be pasted across the longitudinal and transverse directions for measuring Poisson’s ratio. The inplane shear strength and shear modulus could also be experimentally estimated via conducting tensile testing over an ASTM: D3518 composite test coupon. For the fabrication of an ASTM: D3518 (Fig. 9.6b) test coupon, ﬁbers are arranged along þ45/e45 degrees. ASTM D790 (Fig. 9.6c) is used for the experimental determination of ﬂexural strength of a composite specimen under a threepoint or fourpoint bending load. The ASTM standards just
(a)
25 250
ASTM D3039
(b) 250.00 45° 25.00 45°
ASTM D3518/ D3518M
(c) 125.00
12.70
ASTM D790/ D790M Figure 9.6 American Standards for Testing and Materials (ASTM) standards for testing material properties of composites: (a) ASTM: D3039; (b) ASTM: D3518/D3518M; (c) ASTM: D790/D790M.
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listed were intended only for the uniaxial material characterization of composites, which is the simpliﬁcation of complex loading in a real structure. Tensile properties could be determined using a universal testing machine (UTM) with grippers for rigidly ﬁxing the composite test coupons, and the ideal failure has to occur nearly at the center of the specimen or in the gauge length for good accuracy of the results. Using a computer interface with the UTM, the stressestrain curve and loadedisplacement curve for the composite material would be plotted. The load curve and stressestrain curve of the tensile testing would be fed into the ﬁnite element model.
9.4
An overview of ﬁnite element analysis of natural ﬁberreinforced polymer composites
For the development of products using NFRPCs, it is essential to carry out subsequent FEA.
9.4.1
Finite element modeling
The FEA of the NFRPC structure under static and structural loading has to be performed according to the following stepbystep procedure.
9.4.1.1
Analysis type
The selection of analysis type is the very ﬁrst step in any FEA work. The various analysis types are structural analysis, thermal analysis, ﬂuid analysis, heat transfer analysis, electromagnetic analysis, buckling analysis, electrical analysis, and multiphysics analysis (coupled ﬁeld). Depending on the nature of loading, the type of analysis has to be selected. Only based on the analysis type, the associated elements choices would be activated in any FEA software. Whenever any two ﬁelds have to be coupled, the coupledﬁeld analysis has to be selected in FEA software.
9.4.1.2
Part modeling
After assuming the proper analysis type, part modeling of the structure has to be performed using any preprocessor. While modeling any structure, the analysis simpliﬁcations assumed for the structure may be taken into account. Using plane problems, i.e., plane stress, plane strain, and axisymmetry, the 3D shape of the structure is reduced to a 2D shape to reduce computation time and for simplicity. For example, a composite plate with small thickness can be modeled as 2D using plane stress and composite plate with larger thickness can be modeled as 2D using plane strain. Since the thickness is negligible, it could be taken as a 2D structure. Any axisymmetric structures such as a cylindrical, conical, or spherical shell can be modeled as 2D. Also, using the various symmetries, e.g., planar symmetry, cyclic symmetry, or repetitive symmetry, the shape of the complex structure can be simpliﬁed at the part modelling stage itself for the sake of reducing computation time.
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9.4.1.3
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
Material modeling
For the material modeling of NFRPCs, an orthotropic material model is used because the composite is an orthotropic material, unlike steel, which is an isotropic material. An orthotropic material model assumes elastic modulus in the X, Y, and Z directions separately. Also, the orthotropic material model exhibits lines of symmetry.
9.4.1.4
Material properties
Experimentally observed material properties of the NFRPC materials have to be incorporated into the ﬁnite element model, which are usually the prerequisite material properties, namely unidirectional elastic modulus, ultimate tensile strength, Poisson’s ratio, inplane shear strength, and shear modulus. Analytical models to compute the bidirectional material properties from the experimentally observed unidirectional material properties can be found in the literature. Prediction of the bidirectional material properties is a tedious and highly complex process, which requires advanced computers to run the simulation.
9.4.1.5
Meshing
Element choice and size decide the accuracy of the ﬁnite element model. Initially, the ﬁnite element model of the NFRPC product is discretized with coarse mesh and then the mesh reﬁnement process is carried out based upon the experimental observations. For example, in the case of the impact analysis shown in Fig. 9.7, for an impact velocity of 3 m/s, a mesh size of 2 mm is optimum. It can also be observed that the accuracy of the results is not altered by reducing the size of the mesh further. In Fig. 9.7, the Yaxis indicates the negative shell compression heights, while the Xaxis indicates the mesh size.
4
3.5
Mesh size (mm) 3 2.5
2
1.5 10
14 FEA
16
Exp
18 20 22
Y displacement (mm)
12
24 26
Figure 9.7 Convergence study. Exp, Experiment; FEA, ﬁnite element analysis.
Finite element analysis of natural ﬁberreinforced polymer composites
9.4.1.6
161
Boundary conditions
To solve the ﬁnite element model accurately, suitable boundary conditions, i.e., structural boundary conditions and thermal boundary conditions, need to be incorporated. It should be noted that all the boundary conditions need to be applied to the nodes.
9.4.1.7
Loading
Before solving the ﬁnite element model, appropriate loadings have to be applied along the nodes. The loads may be structural load, thermal load, pressure load, buckling load, or fatigue load as per the physics of the given NFRPC problem. Once proper loading is incorporated into the ﬁnite element model, the database ﬁle is ready for solving.
9.4.1.8
Solving
After all the preprocessing works of the FEA of the NFRPC, the database ﬁle should be solved using a ﬁnite element solver.
9.4.1.9
Extraction of results
After successful solving of the ﬁnite element model using the postprocessor, the extraction results, e.g., displacements, stresses, strains, temperature, pressure, and velocity, should be performed, through which the virtual simulation of the actual NFRPC product is possible.
9.4.1.10 Outcome of ﬁnite element analysis results The outcome of the FEA would certainly give ample recommendation for redesign of or further experiments in the NFRPC.
9.5
Finite element analysis of natural ﬁber and natural ﬁberreinforced polymer composites
Natural ﬁberbased polymer composites are the most promising alternative to synthetic ﬁberreinforced polymer composites. An increase in awareness of ecofriendly material makes it imperative to utilize natural ﬁber as a potential reinforcement in polymer composites. Natural ﬁbers are abundantly available, biodegradable, and recyclable, which makes them accepted in the automotive, aircraft, and construction industries. Many researchers have experimentally studied the mechanical, thermal, stiffness, vibration, and tribological properties of NFRPCs. The main drawback of experimentation is prolonged time, high cost, inaccuracy, machine error, and human error. Nowadays, most biocomposite researchers are moving toward computational methods to model the NFRPC to simulate the mechanical and thermal properties. Finite element modeling is widely used by many researchers to model the natural ﬁbers and NFRPC [5]. Natural ﬁbers have complicated microstructures and their
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properties mainly depend on their origin, plant variety, weather, and soil condition. FEA is an effective tool to evaluate the effect of reinforcement, volume fraction, aspect ratio, and orientation of natural ﬁber on the mechanical and thermal behavior of NFRPCs. Generally, natural ﬁber contains cellulose, hemicellulose, lignin, and pectin. The macroscale mechanical properties mainly depend on their chemical composition and microstructure [6]. Due to the complicated structures in different length scales it is essential to use a homogenized computational method to evaluate the relationship between micro and macrostructural behavior. The representative volume element (RVE) method is a most efﬁcient homogenizationbased multiscale ﬁnite element model and represents the relevant features of natural ﬁber and NFRPC in a uniform microstructure.
9.5.1 9.5.1.1
Finite element models of different natural ﬁbers Macromechanical analysis
Many researchers proposed different methodologies to evaluate the mechanical and thermal properties of natural ﬁbers using a ﬁnite element model. In the case of micromechanical simulation the models were created by assuming uniform material properties in the crosssection and the gradient properties of the material in the culm axis. Silva et al. simulated the tensile, ﬂexural, and torsional behavior of bamboo stem with different models such as the homogeneous isotropic model, homogeneous orthotropic model, and the functionally graded material model [7]. Fu et al. developed a ﬁnite element model to investigate the shear capability of bamboo using brittle fracture mechanics with an assumption that bamboo has a gradient elastic modulus [8]. Chand et al. validated the tensile and bending behavior of bamboo using experimentation as well as simulation [9].
9.5.1.2
Micromechanical analysis
Palombini et al. studied the effect of microstructures on the mechanical properties of bamboo ﬁber. A 3D ﬁnite element model of bamboo was meshed on Xray microtomography. The elements chosen for parenchyma and sclerenchyma were tetrahedral elements. The experimental axial compressive strength of bamboo ﬁber was consistent with the simulated results. Nilsson et al. developed a 3D elastice plastic RVE model to evaluate the tensile properties of ﬂax and hemp ﬁbers with respect to ﬁber diameter, microstructure, and nonlinearity of the ﬁber. The cell layers were modeled with a truss element. While rotating the nodes of the truss element (cellulose), ﬁber dislocation could be achieved. The models were created with and without ﬁber dislocation. The mechanical properties of cellulose can be represented as linear elasticity, whereas for hemicellulose it is termed elastic plasticity. From the results it is clear that stiffness decreases while increasing the ﬁber diameter due to the maximum ﬁber dislocation angle. Also, the stressestrain relationship of hemicellulose developed by using a strainhardening model was almost similar to the experimental results [10].
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A 3D viscoelastic model can simulate the nonlinear tensile properties of natural ﬁbers accurately. Trivaudey et al. simulated the tensile behavior of hemp ﬁber using a 3D viscoelastic model with respect to microﬁbril angle, viscoelastic strain, and shear straininduced crystallization. The layers of hemp ﬁber were modeled as a thick hollow cylinder with helical orientation. Microﬁbril angle for the bulk region and dislocated regions were 11 and 30 degrees, respectively [11]. Del Masto et al. found the relationship between the crosssection of the natural ﬁber and the tensile properties. They modeled and evaluated the tensile behavior of hemp ﬁber with different elliptical crosssections. The result showed that the tensile properties have a strong inﬂuence on the degree of ellipticity. The microﬁbril angle and the viscoelastic properties played a vital role in the geometry of the natural ﬁber [12]. Thuault et al. created a ﬁnite element model of ﬂax ﬁber by using the following parameters: chemical composition, microﬁbril angle, and cell wall thickness. They explained the shape and characteristics of the tensile stressestrain curve with this ﬁnite element model. The ﬁber was modeled with three layers of cell wall. The microﬁbril angle for layers 1 and 3 was 45 degrees and for layer 2 it was 10 degrees. A quadratic 3D element was considered as a ﬁber element. The results show that the second cell wall thickness and the microﬁbril angle had a strong inﬂuence on the Young’s modulus of the ﬂax ﬁber [13]. Saaveedra Flores et al. created a 3D ﬁnite element model of palmetto wood using different RVE models at the nanoscale, microscale, and macroscale. The microﬁbril RVE model (nanoscale) comprised cellulose, hemicellulose, and lignin. The macroscale model contained a lot of macroﬁbers arranged periodically in the matrix. The microscale RVE model with thick cell wall structure was arranged inside the macroﬁbers, whereas the thin wall structure was arranged outside the macroﬁbers. These RVE models were effectively used to evaluate the mechanical properties of palmetto wood with respect to volume fraction, cell wall thickness, ﬁber/matrix porosity, microﬁbril angle, and crystallinity of cellulose. The results showed that the tensile modulus of bulk wood increases with an increase in cellulose crystallinity and volume fraction, whereas the tensile modulus decreases with an increase in ﬁber/matrix porosity.
9.5.1.3
Mesoscale representative volume element models
Inczyszyn developed an RVE model at the mesoscale to evaluate the mechanical properties of hemp ﬁber with different geometries. Accurate geometry of hemp ﬁbers has been developed by a digital imaging method. A 3D model of hemp ﬁber was developed and simulated the mechanical properties. From the experimental and simulation results a polygon crosssection gave more accurate results than a circular crosssection [14]. Tephany et al. proposed a new constitutive model to simulate tensile properties of woven ﬂax fabric under different loading conditions. A unit cell was developed using an isotropic shell element. The unit cell elastic properties were obtained from a uniaxial tensile test as well as a bias extension test. The forming model of ﬂax woven fabric was developed using the constitutive model. Simulation results showed that the tensile behavior of woven ﬂax fabric had a strong inﬂuence on the drawin and shear angle of the ﬂax fabric [15].
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The mechanical properties of the mesoscale RVE model were predicted based on the morphological studies of the natural ﬁber without considering the defects or damage that affect the mechanical properties of natural ﬁber. In the case of natural ﬁber bundles the mechanical properties were strongly inﬂuenced by the ﬁber trajectory and its geometry. However, these factors have not been investigated for ﬁnite element models [16].
9.5.2
Finite element analysis of natural ﬁberreinforced polymer composites
9.5.2.1
Thermal analysis of natural ﬁberreinforced polymer composites
Natural ﬁberbased polymer composites are widely used in many industries due to their light weight, ecofriendliness, low cost, and higher mechanical properties. Depending on the application, different ﬁber orientations were selected such as randomly oriented short ﬁber, long ﬁber, and woven arrangements. Fiber orientation, volume fraction, distribution, and aspect ratio signiﬁcantly affect the mechanical properties of the natural ﬁberbased polymer composites. This section will focus on ﬁnite element modeling and analysis of complex structures of natural ﬁberbased polymer composites. The thermal properties of NFRPC are strongly inﬂuenced by the microstructure and macrodistribution of natural ﬁber. In the case of conventional ﬁnite element modeling, researchers assumed that the structure was homogeneous and they did not consider the microstructure of natural ﬁber [17]. However, the microstructure of natural ﬁbers consists of a solid region and lumen, which strongly affects the thermal properties of NFRPC. To overcome this issue a 2D RVE model was created [18]. Fig. 9.8 shows the 2D model of the NFRPC. The effective thermal conductivity of NFRPC in the FEA can be calculated by using the following relation: k¼
ql ti t0
where k is the effective thermal conductivity, q is the heat ﬂux, ti and t0 are the temperatures at the boundaries, and l is the side wall length.
Natural fiber Lumen t0
ti
Matrix
Figure 9.8 2D model of the natural ﬁberreinforced polymer composite.
Finite element analysis of natural ﬁberreinforced polymer composites
165
Liu et al. developed a 2D ﬁnite element model of the NFRPC to evaluate the transverse thermal conductivity [19]. They found that the lumen size had a strong inﬂuence on the transverse thermal conductivity and they have validated their results with HasselmaneJohnson’s model [20]. Wang et al. developed a 2D ﬁnite element model to evaluate the thermal conductivity of the composites with respect to the volume fraction of natural ﬁber and lumen. From the results they concluded that the thermal conductivity was more strongly inﬂuenced by the volume fraction of the ﬁber than the lumen [21]. The lumen content in the natural ﬁber reduces the thermal conductivity of overall composites, which makes it ideal for use in cemented composites in green and energyefﬁcient buildings. Researchers have developed different mesoscale ﬁnite element models to evaluate the clustering effect of natural ﬁber on the thermal conductivity of cemented composites [22]. An RVE model of cemented composites ﬁlled with hemp ﬁber was created to investigate the factors that affect the thermal conductivity of the cemented composites. The following factors were considered for evaluation: volume fraction of the ﬁber, degree of ﬁber clustering, thermal conductivity of the ﬁber/matrix element, and random clustering. The results showed that the effective thermal conductivity of cemented composites was strongly affected by the thermal conductivity of the ﬁber/matrix element and the volume fraction of the ﬁber [23]. To evaluate the ﬁber clustering effect a special type of ﬁnite element was developed, the socalled Voronoi ﬁber/matrix element (n sided) [5]. This special element consists of two variable integral function ﬁelds such as nonconforming and conforming element temperature ﬁelds. This special type of element signiﬁcantly improved the accuracy and computational efﬁciency of the thermal analysis. To evaluate thermal interaction between the cemented matrix and the natural ﬁber, another special nsided interphase/ﬁber element was developed, which reduces the mesh effort and provides accurate results. The thermal properties of hybrid natural ﬁber (jute/banana)reinforced polymer composites both in the longitudinal as well as in the transverse direction were evaluated with a 3D RVE model. Thermal conductivity, diffusivity, and heat capacity decreased with ﬁber loading.
9.5.2.2
Mechanical analysis of natural ﬁberreinforced polymer composite
Micromechanics models for stiffness prediction The following models were used to evaluate the stiffness of the short natural ﬁber composites [24]: • • • • • •
MorieTanakatype model. HalpineTsai equation and its extensions. Dilute model. Selfconsistent model. Bounding model. Shear lag model.
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
Assumptions and limitations •
•
The foregoing models are based on the assumptions that the crosssection of the ﬁber is either cylindrical or elliptical. However, the real crosssection of each natural ﬁber varies in different lengths. Hence compared with these theoretical models, ﬁnite element RVE models provide more effective and accurate results. Moreover, the defects in the composites and the interface properties of the ﬁber/matrix element are not considered in these models.
9.5.2.3
Representative volume element modeling and analysis of natural ﬁberreinforced polymer composites
Different researchers use different ﬁnite element models to evaluate the mechanical properties of NFRPC such as the: • • • • •
Direct 3D RVE model of composites. Orientation averaging approach. Multiscale RVE model. 3D macroscale model. 2D macroscale model.
Randomly oriented short ﬂax ﬁberreinforced polypropylene composites were modeled using the 3D RVE method [25] with respect to the aspect ratio of the ﬁber, ﬁber defects, and ﬁber bundles. The ﬁber was modeled as a linear isotropic elastic material, whereas the matrix was modeled as a nonlinear plastic material. The defects and ﬁber bundles were modeled as a brittle material. Deformation started at the ﬁber endings and around the defects. Modniks and Andersons developed a simpliﬁed timeconsuming ﬁnite element model, the socalled orientation averaging approach, to evaluate the mechanical properties of the composites. Nonlinear deformation of the ﬂax/polypropylene composites was evaluated with the orientation averaging approach with different loading conditions as follows: transverse compression, transverse tension, axial tension, pure shear, and shear and equibiaxial tension. The simulated results showed good agreement with the experimental results [26]. Zhong et al. evaluated the damage prediction of unidirectional ﬂax/polypropylene composites with a multiscale RVE model. Initially, the microscale RVE model was created with respect to the damage to the ﬁber/matrix and stiffness degradation. Then macroscale simulation was performed to evaluate the bending and tensile behavior [27]. Reis et al. developed a 3D macroscale ﬁnite element model of laminated hemp/glass hybrid composites to evaluate the bending behavior of the laminated composites. Each layer of the composite was assumed as a linear, elastic, homogeneous, and orthotropic material. The stiffness and strength of the hybrid composites were compared with the glass ﬁberreinforced polymer composites [28]. Assarar et al. created a 2D macroscale ﬁnite element model to investigate the bending stiffness and damping behavior of hybrid laminated composites with different stacking sequences. They carried out the simulation using ABACUS software with a
Finite element analysis of natural ﬁberreinforced polymer composites
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shell element (fournoded multilayer). The damping coefﬁcients were evaluated in longitudinal, transverse, and inplane shear directions. From the results they concluded that the position of ﬂax layers in the hybrid composites plays a vital role in the bending stiffness and damping behavior of the laminated composites [29].
9.5.3
Failure modeling of natural ﬁberreinforced polymer composites
The general failure mechanisms of ﬁberreinforced polymer composites are as follows: • • • • • •
Fiber fracture. Interface failure. Fiber/matrix debonding. Fiber pullout. Matrix crack/failure. Delamination.
Most computational researchers consider ﬁber/matrix debonding failure. However, it is essential to consider all the failure modes to evaluate the properties accurately. Khaldi et al. developed a ﬁnite element model of alfa ﬁberreinforced polymer composites to study the crack initiation and propagation at the ﬁber/matrix interface [30]. The ﬁber was modeled as an elastic anisotropic material, whereas the matrix was a viscoelastic material. An energy method was adopted to evaluate microcrack initiation and propagation. The microcrack propagates perpendicular to the loading direction and the edge microcracks propagate more rapidly than the internal crack. Fatigue analysis of ﬂax ﬁberbased polymer composites has been performed with a 3D ﬁnite element model to evaluate the stress intensity factor aimed at creating a fatigue crack growth curve with respect to ﬁber direction and ﬁber volume fraction. The materials were assumed to be orthotropic and homogeneous. The highest fracture toughness was found when the ﬁber orientation was perpendicular to the loading direction. Energy analysis is adopted to describe the damage and crack propagation of the NFRPC. Three stages are often applied to evaluate the interfacial damage analysis of ﬁberreinforced polymer composites. Initially, a perfect bonding is applied to the boundaries of the ﬁber/matrix interface, and then a frictional contact is implemented on the ﬁber/ matrix interface to model the mechanical properties. Eventually, based on constitutive law the spring element is introduced to deﬁne the interphase.
9.6
Conclusion
The following conclusions are drawn: • •
Any natural ﬁber composite material with complex geometry and loading can be easily analyzed using RVE. FEM ﬁnds widespread application in aerospace, automotive, civil, and mechanical ﬁelds, and commercial products, electronic goods, etc.
168
• • • • • • • •
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
FEM is an integral part of the design of any products using natural ﬁbers. Convergence is an important criteria for selecting the optimal number of elements in FEA. The relationship between the microstructure of the natural ﬁber and its properties can be evaluated using multiscale ﬁnite element models. The representative volume element method is the most popular homogenizationbased multiscale constitutive method used in ﬁnite element modeling to evaluate the impact of microstructures on the mechanical and thermal properties of NFRPC. Formulation of an accurate and appropriate ﬁnite element model for the material will drastically reduce the design time and cost of experimentation. Challenges in the ﬁnite element modeling of natural ﬁbers deﬁne the ﬁber/matrix interface, 3D geometric modeling, and interfacial adhesion. For structural materials the ﬁnite element models should comprise thermal, mechanical, and dynamic performance. Progressive damage mechanics and analysis of NFRPC may be the most interesting research ﬁelds for the future.
Notations ASTM FEA FEM NFRPC PMC RVE UTM
American Standards for Testing and Materials Finite element analysis Finite element method Natural ﬁberreinforced polymer composites Polymer matrix composites Representative volume element Universal testing machine
Acknowledgments The authors would like to thank the Laboratory of Biocomposite Technology, INTROP, Universiti Putra Malaysia, and the Department of Mechanical Engineering, MEPCO SCHLENK Engineering College, Sivakasi, Tamil Nadu, India, for their continued support in preparing this chapter.
References [1] Ameen M. Boundary element analysis: theory and programming. CRC Press; 2001. [2] Arunachalam V, Nagaraj P. Correlation study of IR TNDT analysis with structural failure modes of carbonfabricreinforced epoxy composites. J Eng Fabr Fibers 2015;10(1). [3] Standard A. D3039M08. Standard test method for tensile properties of polymer matrix composite materials. Annu Book ASTM Stand 2008:1e13. [4] Standard A. D685603. Standard guide for testing fabricreinforced textile composite materials. 2003.
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[5] Wang H, Qin QH, Xiao Y. Special nsided Voronoi ﬁber/matrix elements for clustering thermal effect in naturalhempﬁberﬁlled cement composites. Int J Heat Mass Tran 2016; 92:228e35. [6] Flores EIS, Haldar S. Microemacro mechanical relations in Palmetto wood by numerical homogenisation. Compos Struct 2016;154:1e10. [7] Silva ECN, Walters MC, Paulino GH. Modeling bamboo as a functionally graded material: lessons for the analysis of affordable materials. J Mater Sci 2006;41(21):6991e7004. [8] Fu WS, Zhao ZR, Han W, Zhou JB, editors. Research on ﬁnite element model for parallel to bamboo culms axial shear. Applied mechanics and materials. Trans Tech Publ; 2014. [9] Chand N, Shukla M, Sharma MK. Analysis of mechanical behaviour of bamboo (Dendrocalamus strictus) by using FEM. J Nat Fibers 2008;5(2):127e37. [10] Nilsson T, Gustafsson PJ. Inﬂuence of dislocations and plasticity on the tensile behaviour of ﬂax and hemp ﬁbres. Compos Appl Sci Manuf 2007;38(7):1722e8. [11] Trivaudey F, Placet V, GuicheretRetel V, Boubakar ML. Nonlinear tensile behaviour of elementary hemp ﬁbres. Part II: modelling using an anisotropic viscoelastic constitutive law in a material rotating frame. Compos Appl Sci Manuf 2015;68:346e55. [12] Del Masto A, Trivaudey F, GuicheretRetel V, Placet V, Boubakar L. Nonlinear tensile behaviour of elementary hemp ﬁbres: a numerical investigation of the relationships between 3D geometry and tensile behaviour. J Mater Sci 2017;52(11):6591e610. [13] Thuault A, Bazin J, Eve S, Breard J, Gomina M. Numerical study of the inﬂuence of structural and mechanical parameters on the tensile mechanical behaviour of ﬂax ﬁbres. J Ind Textil 2014;44(1):22e39. [14] Ilczyszyn F, Cherouat A, Montay G, editors. Effect of hemp ﬁbre morphology on the mechanical properties of vegetal ﬁbre composite material. Advanced materials research. Trans Tech Publ; 2014. [15] Tephany C, Soulat D, Gillibert J, Ouagne P. Inﬂuence of the nonlinearity of fabric tensile behavior for preforming modeling of a woven ﬂax fabric. Textil Res J 2016;86(6):604e17. [16] Xiong X, Shen SZ, Hua L, Liu JZ, Li X, Wan X, et al. Finite element models of natural ﬁbers and their composites: a review. J Reinforc Plast Compos 2018;37(9):617e35. https://doi.org/10.1177/0731684418755552. [17] Muralidhar K. Equivalent conductivity of a heterogeneous medium. Int J Heat Mass Tran 1990;33(8):1759e66. [18] Zheng GY. Numerical investigation of characteristic of anisotropic thermal conductivity of natural ﬁber bundle with numbered lumens. Math Probl Eng 2014;2014. [19] Liu K, Takagi H, Yang Z. Evaluation of transverse thermal conductivity of Manila hemp ﬁber in solid region using theoretical method and ﬁnite element method. Mater Des 2011; 32(8e9):4586e9. [20] Hasselman D, Johnson LF. Effective thermal conductivity of composites with interfacial thermal barrier resistance. J Compos Mater 1987;21(6):508e15. [21] Wang H, Xiao Y, Qin Q. 2D hierarchical heat transfer computational model of natural ﬁber bundle reinforced composite. Sci Iran Trans B Mech Eng 2016;23(1):268. [22] Wang H, Zhao XJ, Wang JS. Interaction analysis of multiple coated ﬁbers in cement composites by special nsided interphase/ﬁber elements. Compos Sci Technol 2015;118: 117e26. [23] Wang H, Lei YP, Wang JS, Qin QH, Xiao Y. Theoretical and computational modeling of clustering effect on effective thermal conductivity of cement composites ﬁlled with natural hemp ﬁbers. J Compos Mater 2016;50(11):1509e21. [24] Tucker III CL, Liang E. Stiffness predictions for unidirectional shortﬁber composites: review and evaluation. Compos Sci Technol 1999;59(5):655e71.
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[25] Sliseris J, Yan L, Kasal B. Numerical modelling of ﬂax short ﬁbre reinforced and ﬂax ﬁbre fabric reinforced polymer composites. Compos B Eng 2016;89:143e54. [26] Modniks J, Andersons J. Modeling elastic properties of short ﬂax ﬁberreinforced composites by orientation averaging. Comput Mater Sci 2010;50(2):595e9. [27] Zhong Y, Kureemun U, Lee HP. Prediction of the mechanical behavior of ﬂax polypropylene composites based on multiscale ﬁnite element analysis. J Mater Sci 2017;52(9): 4957e67. [28] Reis P, Ferreira J, Antunes F, Costa J. Flexural behaviour of hybrid laminated composites. Compos Appl Sci Manuf 2007;38(6):1612e20. [29] Assarar M, Zouari W, Sabhi H, Ayad R, Berthelot JM. Evaluation of the damping of hybrid carboneﬂax reinforced composites. Compos Struct 2015;132:148e54. [30] Khaldi M, Vivet A, Bourmaud A, Sereir Z, Kada B. Damage analysis of composites reinforced with Alfa ﬁbers: viscoelastic behavior and debonding at the ﬁber/matrix interface. J Appl Polym Sci 2016;133(31).
Modeling shock waves and spall failure in composites as an orthotropic materials
10
Mohd Khir Mohd Nor 1,3 , N. Ma’at 1 , H.C. Sin 1 , M.S.A. Samad 2 1 Crashworthiness and Collisions Research Group (COLORED), Mechanical Failure Prevention and Reliability Research Centre (MPROVE), Faculty of Mechanical and Manufacturing Engineering, Universiti Tun Hussein Onn Malaysia, Batu Pahat, Johor, Malaysia; 2Department of Computer Aided Engineering, Vehicle Development and Engineering, Perusahaan Otomobil Nasional Sdn Bhd, Shah Alam, Selangor, Malaysia; 3 Centre for General Studies and CoCurricular, Universiti Tun Hussein Onn Malaysia, Batu Pahat, Johor, Malaysia
10.1
Introduction
There are various engineering applications related to composites developed over the past few decades. This trend has sparked massive attention on the requirement to closely model the behavior of such materials under dynamic loading. It is difﬁcult for engineers and the user of metal structures to ignore the realm of these topics. Lack of knowledge can impose limitations on the related engineering design. Research has been conducted to deal with these issues. However, it is generally agreed that there is much work needed to improve the prediction capability and the procedures involved in the characterization method of the required input parameters. Modeling materials behavior, which involves mathematical formulation, can be very complex due to the orientation changes of a material’s orthotropy [1]. It must be ﬁrst highlighted that the conventional stress tensor decomposition into isotropic and deviatoric parts is proven inappropriate to describe shock response of an orthotropic materials. Generally, the development of constitutive models for shock wave propagation can be divided into equation of state (EOS) and a strength model to describe the material response to uniform compression (change of volume) and shear deformation (change of shape), respectively. This separation between volumetric and deviatoric strain components is suitable for isotropic materials due to the colinearity of the principal axis of the stress and strain tensors. Conversely, this colinearity does not hold for orthotropic materials. Hence, the equivalent relationship is not suitable for orthotropic materials. A more general deﬁnition is needed if one assumes that pressure is the state of stress induced by an isotropic state of strain (uniform compression or expansion) [2]. This leads to a number
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites and Hybrid Composites https://doi.org/10.1016/B9780081022894.000102 Copyright © 2019 Elsevier Ltd. All rights reserved.
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
of possible deﬁnitions of pressure as a vector in the principal stress space. Vignjevic has proposed a new expression for generalized pressure to explore this statement further. First, the stress due to the isotropic component of strain (isotropic strain pressure) can be expressed as e ij ¼ cijkl dkl εss =3 ¼ cijkk εv Pj
(10.1)
e and jij where jij ¼ 0 c i s j, jij s 0 c i ¼ j, and εv ¼ εss/3. In above equation, P is deﬁned as
e ¼ εv P
sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 cijkk cijll jst jst
(10.2)
and jij ¼
cijkk εv Cijkk ¼ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ e 1 P cprkk cprll jst jst
(10.3)
In this formulation, the double contraction tensor jstjst must be speciﬁcally e and tensor jij. One can set jstjst ¼ 3. Bear determined to uniquely deﬁne P in mind that the tensor jij is characterized by the elastic stiffness properties. Further, Eqs. (10.1)e(10.3) can be written in Voigt notation as shown in Eqs. (10.4)e(10.6), respectively: 8 > > > > > > > > > > > > > < e P > > > > > > > > > > > > > :
9 j1 > > > > > j2 > > > > > > > j3 > =
2
6 6 6 6 6 6 6 ¼ 6 6 0 > > 6 > > 6 > > 6 0 > > 6 > > 6 > > 4 > ; 0
c11
c12
c13
0
0
0
c12
c22
c23
0
0
0
c13
c23
c33
0
0
0
0
0
0 c44
0
0
0
0
0
0
c55
0
0
0
0
0
0
c66
38 > 7> > 7> > 7> > 7> > > 7> > 7> 7< 7 7> 7> > 7> > 7> > 7> > 7> > 5> > :
9 εv > > > > > > εv > > > > > > εv > = 0 > > > > > > 0 > > > > > > > ; 0
(10.4)
Modeling shock waves and spall failure in composites as an orthotropic materials
173
e used to deﬁnes the magnitude of pressure can be expressed as The scalar P sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2 e ¼ ðc11 þ c12 þ c13 Þ þ ðc12 þ c22 þ c23 Þ þ ðc13 þ c23 þ c33 Þ εv ¼ 3Kj εv P 3 (10.5) jij, which is used to deﬁne the orientation of a new volumetric axis within the principal stress space and can be expressed as ci1 þ ci2 þ ci3 jðiiÞ ¼ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 ðc11 þ c12 þ c13 Þ þ ðc12 þ c22 þ c23 Þ2 þ ðc13 þ c23 þ c33 Þ2 3
(10.6)
Eventually, a newly generalized orthotropic pressure can be deﬁned as: e ¼ skl jkl P jsr jsr
(10.7)
In this formulation, jij becomes dij in the limit of isotropic materials. The above formulation has been examined against the experimental plate impact test data of carbon ﬁberreinforced epoxy where a good agreement is obtained [2]. In other works, this new stress tensor decomposition has been further evaluated using an orthotropic sheet metals under planestress conditions. In this work, the yield surface is assumed to be circular in a unique alignment [3]. A good agreement was obtained with respect to the experimental data for 6000 series aluminum alloy sheet (A6XXXT4) and Alkilled coldrolled steel sheet SPCE [4].
10.2
Constitutive formulation
The use of composite materials in aerospace structures is on the increase mainly due to their strengthtoweight ratio and low cost of manufacturing. Therefore, it is necessary to understand formation and propagation of shock waves in such materials for the development of constitutive models. This problem is complicated due to the anisotropic behavior and the mechanical properties dependency on loading rates and material orientation of composite materials.
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In the literature, many researchers have studied the response of orthotropic materials at quasistatic strain rates; see, for example, [3,5e7]. The behavior of such materials impacted with dynamic loading have been extensively investigated in [8e14]. Many have contributed to the study of anisotropic inﬂuence including shockwave propagation in the materials behavior [5,15e25]. The ﬁrst attempt to investigate a shock response was made by [26]. The ﬁndings showed that the Hugoniot Elastic Limit (HEL) and spall strengths for AA2024 followed similar trends as the quasistatically measured properties at different heattreated states. It can be observed that topics related to shock wave propagation in anisotropic materials in the isotropic solidstate physics and mechanics literature has received considerable attention [27e33]. The shock response of the aluminum alloy 7010T6 was investigated in [5] using a plate impact test. It is emphasized that an appropriate strength model and EOS must be adopted in addition, to the conservation laws to accurately describe the material’s nonlinear behavior and shockwave propagation in solids due to shock loading [24]. Butcher in his earlier work predicted the spall strength of AA6061T6 may vary in accordance with the onedimensional stress yield strength [34]. Later, it was found that there is no signiﬁcant effect on crack formation. With respect to spall response in orthotropic materials, Stevens and Tuler have emphasized that the degree of precompression had no effect on the spall strength of AA6061T6 [35]. In addition, it is also reported the spall strength of AA2024T86 decreased with increasing temperature [36]. The compressive input stress refers to the rising part of the shock described by the HEL of the material during shock loading. It is shown that whenever the shock reaches a free surface, it is reﬂected back as a release wave. This is, consequently, get the material back to ambient stress condition. The release waves can be arranged to meet in the middle of the target by manipulating the thickness of the specimen and ﬂyer resulting in a zone of net tension. It is found that a spall failure is developed when the tension exceeds the tensile strength of the material [5]. Normally in many simulation code, pressure cutoff and a maximum principal stress models are adopted to model spall failure due to their simplicity. In respect to the pressure cutoff model, spall is set to develop in the materials when the propagating pressure is less than the pressure cutoff value deﬁned by the user. Further, both the deviatoric stress tensor and the pressure are set to zero. Another spall failure model is an energybased failure model known as a Grady spall model. This model that assumes the spall is developed when the strain energy reaches a certain level using an appropriate mathematical modeling of fragmentation. Grady reports on results from dynamic compression and dynamic tension (spall) tests [37]. Grady failure model was ﬁrst introduced to predict spall in ductile metals in 1997 [38]. This model has shown an acceptable prediction of fracture and fragmentation of naturally fragmenting munitions in different materials and geometries [39]. De Vuyst in his works has used pressure cutoff, maximum principal stress and Grady spall models to model impact on water [40].
Modeling shock waves and spall failure in composites as an orthotropic materials
175
In the modeling of anisotropic yield surface, an anisotropic homogeneous yield function of degree two for orthotropic plastic response of rolled sheet is ﬁrst proposed by Hill [41]. This is a solid foundation to predict inelastic behavior in metals. This yield criterion shows ability to represent the full behavior of orthotropic materials by considering shear stress components in a planestress case. The yield function is limited to planar isotropy when the shear components are disappeared. This is a big advantage as a yield function [42,43]. Further, it should be emphasized that the yield criterion homogeneous characteristic is conserved by Hill’s effective stress formulation. Therefore, the convexity of the yield surface is maintained. In addition, the yield function parameters have a direct physical meaning. This a massive advantage for providing a simple formulation in a threedimensional case that allows for wide use in practice [44]. The yield function also requires a minimum number of mechanical parameters for characterization. Only three parameters are required for the case of planestress condition. In addition to this yield function, there are numerous other yield functions that have been constructed in the literature. The interested reader is directed to [24] for a comprehensive discussion on yield criteria for orthotropic materials.
10.2.1 Kinematics for ﬁnite strain deformation The multiplicative decomposition of the deformation gradient F is adopted in the proposed formulation in this work: F ¼ Fe Fp
(10.8)
where Fe and Fp represent thermoelastic part of the deformation and plastic part of the deformation (dislocation mechanics), respectively. An intermediate conﬁguration that corresponds to elastically unloaded material is introduced in this concept. This conﬁguration, also known as elastic reference conﬁguration, can be physically obtained through elastic unloading of the material. The additive decomposition of generalized strain measures that leads to spurious shear stresses is avoided in this work [45]. In addition, this formulation also shows incapability to track the evolution of material symmetry in such materials undergoing ﬁnite strain deformations [46]. Using Eq. (10.8), the elastic right CauchyeGreen tensor Ce and the elastic Greene Lagrange strain tensor Ee can be expressed as Ce ¼ FTe $Fe ;
1 1 Ee ¼ ðCe IÞ ¼ FTe $Fe I 2 2
(10.9)
The proposed constitutive model is developed in respect to the isoclinic conﬁguration to demonstrate strong material symmetries correlation within elastic and
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
plastic regimes as shown in [47]. This conﬁguration assumes that the principal directions of material elastic and plastic orthotropy coincide. This deﬁnition also simpliﬁes the numerical implementation by avoiding the explicit use of any corotational rate [48]. This is proven in recent works by [24,49,50]. For the sake of clarity, in this manuscript, ( ) is used upon each of kinematic and kinetic variables deﬁned with respect to the isoclinic conﬁguration. The orthotropy symmetric group w is characterized using the structural tensors Mii ji ¼ 1; 2; 3 [51]. These tensors are deﬁned as M1 ¼ n15n1, M2 ¼ n25n2, and M3 ¼ n35n3 where n1, n2, and n3 are unit vectors to describe an orthonormal frame of the material. Referring to Fig. 10.1, the structural tensors are pushed forward from an initial conﬁguration Uo to elastically unloaded conﬁguration Up as Mii ¼ Fp Mii F1 p . Subsequently, the pullback transformation of the structural tensors Mii from the b i can be perelastically unloaded conﬁguration Up to the isoclinic conﬁguration U formed by rotating back for plastically induced rigid body rotation due to plasticrelated deformations using the following transformation. ˇ
c M ii ¼ QTp Mii Qp
(10.10)
where Qp represents an orthonormal tensor that deﬁnes the rigid rotation. A triad of unit vectors for material symmetries representation is schematically shown by two orthogonal axes with arrows, as shown in Fig. 10.1. Both elastic stretching and rotation are embedded in the elastic part Fe of the deformation gradient F. Plastic part of F with b i is represented by Fp and F b p , respectively. The plastic rotation Rp respect to Up and U
Fp Fe
—
Ωp
RTp Ωt
Ω0 Fe Fp Ωi
Figure 10.1 Deﬁnition of isoclinic conﬁguration.
Modeling shock waves and spall failure in composites as an orthotropic materials
177
is assigned to Fp to ensure no rotation to the orthotropy principal axes (not inﬂuenced by plastic deformation) in the chosen conﬁguration. As shown in [52], the rotation and distortion for elastoplastic deformation contained in the elastic deformation gradient Fe. Alternatively, the plastic rotation Rp can be b e ¼ Fe Fd Rp by considering included in the deformation due to damage Fd to deﬁne F the elastic material parameters evolve due to damage. As emphasized in [49], the changes of material compliance are inﬂuenced by damage. Further, the elastic and plastic parts of the deformation gradient F are deﬁned in the isoclinic conﬁguration as: b F e ¼ Fe ; b F p ¼ RTp Fp ¼ RTp Rp Up ¼ Up
(10.11)
In the above formulation, plastic rotation Rp is used to describe an orthogonal rotation tensor developed by plastic deformation while Up refers to plastic right stretch b then can be formulated additively as: tensor. The total velocity gradient L :
:
:
:
1 1 1 1 b¼b be þ L bp L F$b F ¼b Fe$b Fe þ b Fe$b Fp$b Fp $b Fe ¼ L
(10.12)
b p ¼ 1 is ﬁnal adopted to ensure an incompressibility constraint for plastic det F deformation.
10.2.2 Stress tensor decomposition for composite materials First, the Mandel stress tensor S can be expressed as [53]: X
¼ C$S
(10.13)
where C and S refer to Right CauchyeGreen tensor and Second Piola Kirchhoff stress tensor, respectively. These tensors can be expressed in the intermediate conﬁguration Up as Ce ¼ FTe $Fe
(10.14)
T S ¼ F1 e $s$Fe
(10.15)
The above Kirchhoff stress tensor s is assumed symmetric in the current conﬁguration Ut as s ¼ J$s ¼ detðFÞ$s
(10.16)
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
where J is a volume ratio. Eqs. (10.14) and (10.15) can be substituted into Eq. (10.13) to deﬁne the Mandel stress tensor in the intermediate conﬁguration Up as X
T T ¼ FTe $s$FT e ¼ detðFÞ$Fe $s$Fe
(10.17)
This stress tensor that is common in the formulation of inelastic behavior of materials is adopted in this work [54]. Subsequently, the Mandel stress tensor can be expressed as X d
b T $s$ F b T ¼F e e
(10.18)
Further, we can express Eq. (10.17) as: X d
T T T T b T $s$ b b T $detðFÞ$s$ b ¼F Fe ¼ F F e ¼ detðFÞ$ b F e $s$ b Fe e e
(10.19)
Using sij ¼ Pdij þ Sij, the above equation can be expressed as X d
T
T
¼ detðFÞ$ b F e $s$ b Fe
T
T
¼ detðFÞ$ b F e $ðPd þ SÞ$ b Fe
(10.20)
Using the new generalized orthotropic pressure, Eq. (10.20) can be reexpressed as: X d
sj T T b $j $ b Fe ¼ detðFÞ$ F e $ S þ jj
(10.21)
This equation can be further extended as X d
T T T sj T Fe ¼ detðFÞ$ b F e $S$ b F e þ detðFÞ$ b F e $ $j$ b jj ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ ﬄ} 0 P P c ¼deviatoric c
(10.22)
p
b p represents the volumetric component (pressure) of the new Mandel stress where S tensor. Focusing on the deviatoric part of a full stress tensor, the deviatoric part of the new Mandel stress tensor can be deﬁned as X d0
sj T T Fe $j b ¼ detðFÞ$ b Fe s jj
(10.23)
Modeling shock waves and spall failure in composites as an orthotropic materials
179
The new deviatoric Mandel stress tensor deﬁned in the isoclinic conﬁguration Ui , eventually can be written as X d0
T T T sj T Fe ¼ detðFÞ$ b F e $s$ b F e detðFÞ$ b F e $ $j$ b jj ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} b S b Sp
(10.24)
T T ¼ detðFÞ$ b F e $S$ b Fe
It can be proven that the deviatoric component of the new Mandel stress tensor in the above equation is traceless. The existing experimental evidence shows that it is not easy to deduce sound data about a continuum elastic domain for the skewsymmetric part of the Mandel stress tensor Sa [55]. Numerous efforts are still required to establish the elastic domain and yield functions for the skewsymmetric part of the Mandel stress tensor, experimentally. Therefore, in this formulation, only the symmetric part of the Mandel stress is used. This option has been adopted by many researchers. See, for example, [56,57]. Further, in Section 10.6, the thermodynamics analysis is presented based on the second law in the form of ClausiusePlank (CP) inequality. It is necessary to introduce relevant conjugate variable pairs, starting with the stress power P to deﬁne the CP inequality. P ¼ s: L ¼ detðFÞ$s: L ¼ S: E_ ¼
X d0 b : Lp
(10.25)
The stress power is adopted to characterize the real mechanical power during dynamic process. The stress power can be represented by the product between work conjugate strain and stress measures. This part is discussed in Section 10.6.
10.2.2.1 Representation of orthotropic yield surface in the stress space The yield surface representation in the principal stress space is discussed in this section. First it should be noted that any arbitrary stress state can be decomposed into hydrostatic and deviatoric parts in the stress space. This isotropic decomposition is best presented in the principal stress space by a blue line as shown in Fig. 10.2. It can be seen in this ﬁgure their directions perpendicular to each other. Referring to the formulation of the new stress tensor decomposition, the representations of this decomposition jij in threedimensional space of the principal stress is shown by a purple line. It can be seen that this decomposition leads to a shift of the pressure vector away from the common. The alignment of the volumetric axis jij is not making the same angle with the principal stress directions. Using this deﬁnition, it can be deduced that any orthotropic materials will ﬁnd their own yield surface direction within the principal stress space.
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
σ1
ψ ij δ ij
σ3
σ2
Figure 10.2 j and d as a vector in a principal stress space.
10.2.3
Equation of state (EOS) for shock waves
An appropriate EOS is important for describing the anisotropic material response due to shock loadings. Therefore, the proposed formulation is combined with EOS in addition to the conservation laws to mathematically describe the material’s nonlinear behavior including shock wave propagation in solids due to shock loading. In the contemporary hydrocodes, the available EOSs are either of an analytical or a tabulated type. The very popular MieGr€ uneisen EOS extensively used for solid continua is used in this work [58,59]. This analytic EOS deﬁnes the pressure as a function of density r or speciﬁc volume and speciﬁc internal energy e as shown below P ¼ f ðr; eÞ ¼ pr ðvÞ þ
GðvÞ ðe er ðvÞÞ v
(10.26)
where v represents the speciﬁc volume while GðvÞ is the Gr€unesien gamma which can be deﬁned as vP GðvÞ ¼ v ve v
(10.27)
where G refers to a constant G ¼ G0 , or assumed that Gv00 ¼ Gv ¼ const alternatively. Functions pr and er are known functions of v on some reference curves. There are few possible reference curves for consideration such as the shock Hugoniot curve, the 0 K isotherm, etc. The form of the MieGr€ uneisen EOS that use the shock Hugoniot reference curve widely adopted for solid materials is deﬁned below
Modeling shock waves and spall failure in composites as an orthotropic materials
G P ¼ f ðr; eÞ ¼ pH $ þ 1 m þ rGe 2
181
(10.28)
where pH refers to Hugoniot pressure, m ¼ rr 1 is relative change of volume, G is 0 Gr€ uneisen parameter, r is density and e deﬁned as the speciﬁc internal energy. The RankineeHugoniot equations for the shock jump conditions can be characterized by deﬁning a relation between any pair of the r, P, e, up (the velocity of the particle directly behind the shock), and U (the velocity of shockwave that propagates through the medium). An empirical linear relationship between U and up for many liquids and most solids is shown below: U ¼ c þ Sup
(10.29)
where c is the intercept of the U up curve (Ushock velocity vs. up particle velocity curve), and S refers to the coefﬁcient of the U up curve slope. As deﬁned in [58], both the Hugoniot pressure and a shock velocity U can be expressed as a nonlinear function of particle velocity up as: U ¼ c þ S1 up þ S 2
u u 2 p p up up þ S3 U U
(10.30)
The Gr€ uneisen gamma for the undeformed materials can be expressed by G¼
g0 þ au 1þu
(10.31)
Later, the pressure deﬁned as a function of MieGr€uneisen EOS with cubic shock velocity can be expressed as
PEOS ¼ "
G G r0 c 2 m 1 þ 1 m m2 2 2 m2 m3 S3 1 ðS1 1Þm S2 mþ1 ðm þ 1Þ2
#2 þ ð1 þ mÞ$G$E
(10.32)
when m > 0 PEOS ¼ r0 c2 m þ ð1 þ mÞ$G$E when m < 0 where E is the internal energy per initial speciﬁc volume, S1, S2, S3 are the coefﬁcients deﬁned from the slope of the U up curve, g0 is the Gruneisen gamma for the undeformed material, a is the ﬁrst order volume correction to g0, c, S1, S2, S3, g0, a, r0 represent the material properties for EOS characterization.
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
In this work, the new stress tensor decomposition is combined with the MieGr€uneisen EOS using several modiﬁcations. Generally speaking, j is calculated using the material stiffness matrix C. The increment of deviatoric Mandel stress tensor c S0 is e ¼ PEOS the stress update then calculated using rate of deformation tensor D. Using P at time n þ 1 can be deﬁned as b 0nþ1 Pnþ1 j snþ1 ¼ S EOS
10.2.4
(10.33)
Elastic free energy function
A free strain energy function and a plastic level set function of orthotropic yield criterion is used to formulate the behavior of orthotropic materials in this work. As mentioned in the preceding section, the orientation of orthotropic symmetry group w is considered identical throughout plastic deformation. The Helmholtz free energy is then used to deﬁne the elastic orthotropy in terms of evolving structural tensors. The Helmholtz free energy can be decomposed into elastic and plastic components b i as follows in the isoclinic conﬁguration U b ¼J be E be þ J b pðisotÞ ðaÞ (10.34) J b e refers to the energy stored due to elastic deformations, deﬁned using be E J b e . The energy from isotropic plastic hardthe Elastic GreeneLagrange strain tensor E b pðisotÞ ðaÞ where a is adopted as an isotropic ening is represented by J hardening variable. Further, the elastic material response is assumed invariant b e; c be ¼ J be E under transformations of the material symmetry group w: J M 11 ; c M 22 ¼ b e QT ; Qc b e QE M 11 QT ; Qc M 22 QT , where Q is orthogonal rotation tensor. J
10.2.5
Orthotropic yield criterion
The dependence on plastic anisotropy is described by Hill’s yield function deﬁned in terms of the structural tensors c M ii i ¼ 1; 2; 3 of the material symmetry group w [41]. Using the hypothesis that shape of yield surface remained identical for a range of strain rates. Using Eq. (10.24), the yield function can be expressed as ! 0 X d bf ¼ bf ;a (10.35)
Modeling shock waves and spall failure in composites as an orthotropic materials
183
Again, a refers to isotropic hardening variable. The structural tensors c M ii is used to deﬁne the properties of symmetric orthotropy: X0 bf ¼ bf d ; c M ii ; a
(10.36)
The plastic anisotropy that is characterized by Hill’s anisotropy yield function then can be expressed as: rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ X0 X0 b : d bf ðaÞ ¼ 0 bf ¼ d : h
(10.37)
where b h is a fourthorder tensor used to represent the dependency of the proposed formulation on Hill’s yield criterion and structural tensors. bf ðaÞ deﬁnes the dependency of evolving ﬂow stress on isotropic hardening. The Hill’s effective stress b i as follows: then can be ﬁnalized in the isoclinic conﬁguration U vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 u 2 u 2 2 2 u 6 P0 P0 0 0 0 0 02 02 02 P P P P P P P u 6F c c þ G c z c x þ H c x c y þ 2L c yz þ 2M c zx þ 2N c xy 7 7 y z X 36 7 d0 u ¼u 7 u2 6 6 7 F þ G þ H u 4 5 t (10.38)
10.2.6 The evolution equations The second law of thermodynamics framework is adopted to model the evolution equation. The formulation can be expressed using the CP inequality as _ 0 D ¼ S: E_ J
(10.39)
_ is a rate of Helmholtz free energy function, and E_ can be written as where J 1 E_ ¼ C_ 2
(10.40)
Using Eq. (10.40), Eq. (10.39) can be expressed as follows: 1 _ 0 D ¼ S: C_ J 2
(10.41)
Assuming the Helmholtz free energy function can be formulated by the elastic CauchyeGreen strain tensor Ce, and the strainlike internal variable for isotropic
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
plastic hardening xh description, J ¼ J(Ce,xh), this function can be deﬁned into elastic and plastic parts as J ¼ Je ðCe Þ þ Jp ðxh Þ
(10.42)
The above equation can be further differentiated with respect to time to gives _ ¼ vJe : C_ e þ vJp $x_h J vCe vxh
(10.43)
Substituting Eq. (10.43) into Eq. (10.42) gives vJp _ 1 vJe _ D ¼ S: C_ : Ce þ $xh 0 2 vCe vxh
(10.44)
1 and its material time derivative Using Ce ¼ FT p $C$Fp T _Ce ¼ F_ $C$F1 þ FT $C$F _ 1 þ FT $C$F_ 1 , the Mandel stress tensor can be p p p p p p deﬁned as:
vJe T T vJe 1 S ¼ C$S ¼ Fp $Ce $Fp $ 2$Fp $ $F ¼ 2$Ce $ vCe p vCe
(10.45)
Eq. (10.45) can be further adopted to deﬁne the nonnegative of the internal dissipation as follows: D ¼ S: Lp
vJp _ $xh 0 vxh
(10.46)
In this work, the evolution equations for the plastic strain tensors are formulated using the principle of maximum plastic dissipation. The normality rule is used to deﬁne function of Lp and x_h as Lp ¼ l_
vf ; vS0
vf x_h ¼ l_ va
(10.47)
These equations satisfy the associative ﬂow rule and the expression for evolution equation. x_h is a work conjugate to a. The local dissipation inequality is then deﬁned as X0 b ¼d:L b p a: x_h 0 D
(10.48)
Modeling shock waves and spall failure in composites as an orthotropic materials
185
where b b p ¼ l_ v f ; L 0 P vc
vbf x_h ¼ l_ va
(10.49)
Only the symmetric part of the plastic velocity gradient is adopted in the proposed formulation to match with the chosen symmetric Mandel stress tensor that is thermodynamically conjugate to the plastic velocity gradient. This assumption holds rigorously for plastic isotropic but does not necessarily hold for plastic anisotropic cases. In the proposed formulation, the plastic spin that is assumed vanishes. by using the Mandel stress and setting the plastic spin to zero, Eq. (10.48) can be expressed as X0 b ¼ d : sym L b p a: x_h 0 D
(10.50)
The above equation is also equal to X0 b ¼d:D b p a: x_h 0 D
(10.51)
b p is further identiﬁed by applying the The evolution of the plastic deformation D yield function deﬁned in Eq. (10.37) into the ﬁrst part of Eq. (10.49): 0
P0 @b h$ c þ
c0 b P h$
b p ¼ l_ D 4
!T
P0 þb h $c þ T
P0 b h $c T
!T 1 A
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ P P0 c0 : b h: c
(10.52)
The inequality of dissipation energy in Eq. (10.51) then can be expressed as follows: 2 0
P0 6 @b h$ c þ 6 X0 6 b ¼ d $6l_ D 6 6 4
P0 b h$ c
!T
P0 þb h $c þ T
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ P0 P0 4 c : h: c
P0 b h $c T
! T 13 A7 7 7 7 a: x_h 7 7 5
0 (10.53) The CP inequality of the second law of thermodynamics is ﬁnally deﬁned below using the above identities
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
1 rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ X vf C d0 b X d0 b ¼ l_ B a: A 0 D : h: @ va 0
10.2.7
(10.54)
Grady failure model
The prediction of damage initiation and evolution in materials undergoing high velocity impacts is focusing on the materials behavior under high pressure and high strain rates. The application of this topic can be observed in various engineering applications such as blast loading and ballistic impact. Spallation is one failure mode used during high velocity impact. This phenomenon describes the material failure due to relations between two or more rarefaction waves [40]. It is observed that the impact on the outer surface is not powerful enough to cause penetration but still managed to create spalls formation at the inner surface. Spall failure can be regarded as the main failure mode in shockloaded metallic materials that is mainly inﬂuenced by materials microstructure. Spall damage nucleates, grows, and coalesces within the microstructure of the materials under shock loading and can be characterized using a threedimensional approach. Pressure cutoff and maximum principal stress are the simplest failure models. In this work, an energybased on the Grady failure spall model is adopted. This failure model assumes spall are developed when the strain energy reaches certain levels. Two mechanisms are investigated in this failure model: brittle and ductile fractures. The required energy refer to the critical fracture toughness for the brittle fracture model while also referring to the work required to reach failure strain for the case of ductile fracture model. This is then results in the following spall strengths: psðductileÞ ¼
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2rc20 sY εfail Ductile Failure
(10.55)
psðbrittleÞ ¼
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 3rc0 Kc2 ε_ Brittle Failure
(10.56)
where r ¼ density; c0 ¼ bulk sound speed; sY ¼ yield stress; εfail ¼ critical strain failure; Kc ¼ fracture toughness; and ε_ ¼ rate of volumetric dilatation. The spall stress is calculated for each cell including the local conditions in the cell at each cycle. This stress is then used as the local maximum principal stress failure criterion. It can be observed that there is a transition point between ductile and brittle spall for certain strain rates. This critical strain rate, εcrit is calculated from: ε_ crit
sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 8B20 ðsY εfail Þ3 ¼ 9rK 4c
(10.57)
where r ¼ density; sY ¼ yield stress; εfail ¼ critical strain failure; Kc ¼ fracture toughness; and B ¼ isotropic/kinematic hardening.
Modeling shock waves and spall failure in composites as an orthotropic materials
10.3
187
Results and analysis
The algorithms of the proposed constitutive model are written and implemented into the LLNLDYNA3D of Tun Hussein Onn University of Malaysia (UTHM)’s version and named Material Type 92 (Mat92). The work is purposefully performed to ensure a good book keeping of the algorithm to deal complex structure [24,60,61]. A cmgms unit system is used in each validation process.
10.3.1 Analysis on commercial aluminum alloy The Plate Impact test data published in [40] is ﬁrst used in this analysis using the ﬁnite element model depicted in Fig. 10.3. As can be seen in this ﬁgure, the model is divided into three main parts of rectangular bars. The crosssection is deﬁned symmetrically in the XY plane using 4x4 solid elements. The ﬂyer, test specimen, and Poly (methyl methacrylate) (PMMA) block are modeled with 25, 75, and 100 solid elements, respectively. In addition, the length for each PMMA block, test specimen, and ﬂyer are set as 12 mm 10 mm, and 2.5 mm, respectively. The model orientation is set parallel to Z axis (impact axis). The adopted mesh allows a onedimensional wave propagation along bars during the event of impact. The back of the Poly (methyl methacrylate) (PMMA) block is deﬁned using a nonreﬂecting boundary condition. A contact interface is used to deﬁne contact between the ﬂyer and the test specimen. A time history block is then embedded at the top of PMMA bar to measure the stress time histories during impact. In this test, the Z stress obtained at the top elements of the PMMA bar is compared against the experimental data recorded in the longitudinal and short transverse directions of the specimen. The impact velocity is set as 234 ms1, 450 ms1, and 895 ms1. The material properties of each parts are shown in Table 10.1. The ﬂyer is set as Aluminum 6082T6. PMMA blocks and the ﬂyer are characterized by Mat10.
PMMA
SPECIMEN Z
Y X
Figure 10.3 Plate impact test conﬁguration.
Flyer V
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
Table 10.1 Material properties for plate impact test analysis Materials Parameters
Al7010
Al6082
PMMA
Young’s modulus
Ea Eb Ec
70.6 GPa 71.1 GPa 70.6 GPa
Poisson’s ratio
vba vca vcb
0.342 0.342 0.342
Shear modulus
Gbc Gab Gac
26.31 GPa 26.48 GPa 26.48 GPa
26.8 GPa 26.8 GPa 26.8 GPa
2.3 GPa 2.3 GPa 2.3 GPa
Yield stress in adirection
sy
564 MPa
250 MPa
70 MPa
Tangent plastic modulus in adirection
H
0.13 GPa
130 MPa
300 MPa
Pressure cutoff Density Hill’s parameters
pcut r R P Qbc Qba Qca
2.81 gcm3 1 0.719 1 1 1
2.5 GPa 2.7 gcm3
1.18 gcm3
Gr€ uneisen parameters
C s1 s2 s3 G a
5200 ms1 1.36 0.00 0.00 2.2 0.48
5240 ms1 1.4 0.00 0.00 1.97 0.48
2180 ms1 2.088 1.124 0.00 0.85 0.00
Grady parameters
r c0 εfail Kc B
2.81 gcm3 0.52 0.5 0.0025 r/c20
The orthotropic material axes type 2 (AOPT 2) is used and set to a ¼ 0ax þ 0ay þ 1az and d ¼ 0dx þ 1dy þ 0dz. The results are shown in Figs. 10.4 until 10.9. Generally, the shape of pulse for both simulation and experiment are pretty close. The initial slope represents the HEL. It should be noted that the materials anisotropic level is described by a different HEL value. The numerical simulation also provides a good agreement in terms of Hugoniot stress levels to ensure the behavior prediction under extreme loading condition is good. Figs. 10.4 and 10.5 simulate the plate impact
Modeling shock waves and spall failure in composites as an orthotropic materials
189
0.80 0.70 Exp mat92
Stress (GPa)
0.60 0.50 0.40 0.30 0.20 0.10 0.00 2.5 –0.10
3.0
–0.20
3.5
4.0
4.5
5.0
Time (µs)
Figure 10.4 Longitudinal stress in the longitudinal direction at 234 ms1 impact.
0.8 0.7
Exp mat92
0.6
Stress (GPa)
0.5 0.4 0.3 0.2 0.1 0 0.1 0.2
2.5
3
3.5
4
4.5
5
Time (µs)
Figure 10.5 Longitudinal stress in the transverse direction at 234 ms1 impact.
test at 234 ms1. The tensile wave failure or spall is not developed at this impact velocity. This behavior is shown by the longitudinal stress reloading following the ﬁrst pulse. Referring to Figs. 10.6 and 10.7, a clear HEL can be observed in both traces when impacted with 450 ms1 impact velocity. The values of HEL are slightly difference, 0.41 GPa in the longitudinal and 0.38 GPa in the transverse directions. These values are reasonable compared to the experimental values which are 0.39 and 0.33 GPa in the longitudinal and transverse directions, respectively. A very good prediction is obtained for the Hugoniot stress levels. This is a good indication for the proposed
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
1.60 1.40
Stress (GPa)
1.20
Exp mat92
1.00 0.80 0.60 0.40 0.20
0.00 1.5 –0.20
2.0
2.5
3.0
3.5
4.0
Time (µs)
Figure 10.6 Longitudinal stress in the longitudinal direction at 450 ms1 impact.
1.60 1.40 Exp mat92
Stress (GPa)
1.20 1.00 0.80 0.60 0.40 0.20 0.00 1.5 –0.20
2.0
2.5
3.0
3.5
4.0
Time (µs)
Figure 10.7 Longitudinal stress in the transverse direction at 450 ms1 impact.
formulation to capture shockwaves in orthotropic materials. In addition, a clear spall is recorded in both traces: 0.31 and 0.42 GPa in the longitudinal and transverse directions, respectively. The numerical values are 0.21 GPa in the longitudinal and 0.10 GPa in the transverse directions which are smaller compared to the values obtained experimentally. It can be observed the constitutive model sensitive to the direction of impact as shown by the materials in the experimental work. This behavior can be clearly observed by the point where the spall starts. Figs. 10.8 and 10.9 describe the for 895 ms1 impact velocity. Generally, a good prediction is shown in terms of pulse shape and the Hugoniot stresses compared to the experimental data in both impact directions. As demonstrated in 450 ms1 impact
Modeling shock waves and spall failure in composites as an orthotropic materials
191
3.50 3.00 Exp mat92
Stress (GPa)
2.50 2.00 1.50 1.00 0.50 0.00 1.0 –0.50
1.5
2.0
2.5
3.0
3.5
Time (µs)
Figure 10.8 Longitudinal stress in the longitudinal direction at 895 ms1 impact. 3.50 3.00
Stress (GPa)
2.50 Exp mat92
2.00 1.50 1.00 0.50
0.00 1.0 –0.50
1.5
2.0
2.5
3.0
Time (µs)
Figure 10.9 Longitudinal stress in the transverse direction at 895 ms1 impact.
velocity, smaller pullback signals are predicted by the proposed constitutive model in both impact directions. Referring to the above analysis, it can be deduced that the proposed formulation of Mat92 is capable of providing a close prediction of the complex behavior of aluminum alloy AA7010. The results are satisfactory in terms of pulse shape, HEL and the EOS (Hugoniot stress level). HEL is lower in the transverse direction compared to the longitudinal direction. The Hugoniot stress levels prediction is closely captured. The width of the pulse is also good. Even though the Grady’s spall failure model of the proposed formulation shows a reasonable spall prediction, further development or a better spall failure model must be included to provide a very good prediction for a complex spall strength evolution at various impact velocity. The results of this validation analysis are summarized in Table 10.2.
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
Table 10.2 Comparison between Mat92 and plate impact test data Analysis Criteria HEL (GPa)
Hugoniot Stress Level (GPa)
Pulse (ms)
234 ms (Longitudinal): Simulation Experiment
0.40 0.39
0.64 0.65
1.15 1.65
234 ms1 (Transverse): Simulation Experiment
0.40 0.33
0.64 0.63
1.18 1.44
450 ms1 (Longitudinal): Simulation Experiment
0.41 0.43
1.32 1.31
1.13 1.16
450 ms1 (Transverse): Simulation Experiment
0.38 0.39
1.28 1.38
1.08 1.13
895 ms1 (Longitudinal): Simulation Experiment
0.36 0.21
2.95 3.25
1.07 1.10
895 ms1 (Transverse): Simulation Experiment
0.32 0.19
2.89 2.80
1.12 1.13
Impact Velocity/Direction 1
10.5.2
Validation against carbon ﬁberreinforced epoxy composites
The ﬁnal part of this validation process is an analysis against shockwave propagation data in composite target of woven carbon ﬁberseepoxy plies. The shockwave is propagating in the throughthickness direction of the composite. This is set by assigning the impact direction normal to the ﬁber direction. The ﬁnite element model of this analysis is depicted by Fig. 10.10. The carbon ﬁber composite target plate is set as a quasiorthotropic material in this analysis. Table 10.3 shows the woven composite plate properties. The properties are characterized from the layer macromechanical properties for the layup [0/90, 45] [62]. The initial material density r0 is 1500 kg/m3 and the longitudinal speed of sound is set as C0 ¼ 3020 ms1. First, it should be noted that a and b directions in the table refer to x and y coordinate axes. In this analysis, a surface metal cover is used for the target. The identical materials adopted to deﬁne both the ﬂyer and the metal cover. In addition, a thick PMMA is placed at the back of the model. The deformation of the model is set as a uniaxial strain under an adiabatic assumption. The identical boundary condition used in the analysis in the preceding section is adopted here. The ﬂyer target interface is deﬁned as a surfacetosurface contact algorithm.
Modeling shock waves and spall failure in composites as an orthotropic materials
193
Cover plate
Flyer Target PMMA
Figure 10.10 Plate impact test conﬁguration for carbon ﬁber/epoxy composite.
Table 10.3 Material properties for carbon ﬁber/epoxy composite Parameters
Composite
Young’s modulus in longitudinal direction
Ea Eb Ec
68.5 GPa 66.5 GPa 10.0 GPa
Poisson’s ratio
vba vca vcb
0.039 0.0044 0.045
Shear modulus
Gbc Gab Gac
3.57 GPa 4.57 GPa 3.57 GPa
Density
r
1500 kgm3
504 m/s impact velocity is assigned to the ﬂyer plate, as measured experimentally [62,63]. 30 elements are used to model the 5 mm thick of ﬂyer plate. Further, 142 elements are used for the test specimen and set parallel to the Z axis. In the experimental work, the front gauge is covered with 1 mm aluminum alloy plate while the rear gauge is covered by 12 mm of PMMA. The composite plate specimen is set 3.8 mm thick. The aluminum plate is modeled by the IsotropicElasticPlasticHydrodynamic material model named Material Type 10 (Mat10). The elastic material properties and the MieGr€ uneisen EOS parameters are given in Tables 10.4 and 10.5, respectively. The data used to deﬁne PMMA is taken from Table 10.1. The aluminum density is 2703 kgm3 The results of stress along the Z axis are shown in Figs. 10.11 and 10.12. The value is compared against the stress history obtained from the front gauge. The Table 10.4 Material properties of aluminum alloy (AA7010) Young’s modulus
Poisson’s ratio
Shear modulus
290.00 MPa
0.30
27.60 GPa
194
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
Table 10.5 MieGr€uneisen EOS parameters of aluminum and composite Material Parameters Gr€ uneisen
Aluminum
Composite
1
3230 ms1 0.92 0.00 0.00 0.84 0.50
5240 ms 1.4 0.00 0.00 1.97 0.48
C s1 s2 s3 G A
2.5E02
Zstress (x 102 GPa)
2.0E02 1.5E02 1.0E02
Front gauge mat92
5.0E03 0.0E+00 0.5
1.0
1.5 2.0 Time (µs)
2.5
3.0
3.5
–5.0E03
Figure 10.11 Stress comparison between the front gauge and Mat92.
Zstress (x 102 GPa)
2.0E02
1.5E02
1.0E02 Back gauge 5.0E03
0.0E+00 2.0 –5.0E03
mat92
2.5
3.0
3.5 4.0 Time (µs)
4.5
Figure 10.12 Stress comparison between the back gauge and Mat92.
5.0
Modeling shock waves and spall failure in composites as an orthotropic materials
195
stress in PMMA and the rear gauge is then compared. The stress values measured from the gauge and the numerical simulation are comparable since the shock front is planar and parallel to the composite plies. It can be observed the newly constitutive model is capable to provide a good prediction in terms of the stress value and the pulse length. The proposed constitutive model also correctly predicts the separation between the cover plate and the ﬂyer as both plates stay in contact after impact. It is important to note that a similar prediction capability is shown by the propose formulation in the analysis in the preceding section, i.e., separation of both plates after impact.
10.4
Conclusion
A new hyperelasticplastic constitutive model shockwave propagation and spall failure prediction in orthotropic materials is developed in this work. A new deﬁnition of Mandel stress tensor is introduced by a combination with a new generalized orthotropic pressure. The formulation is deﬁned within a multiplicative decomposition framework and combined with EOSs in the isoclinic conﬁguration. An isotropic hardening is used to control the evolution of orthotropic yield surface. The proposed formulation of a newly constitutive model is implemented into a Lawrence Livermore National Laboratory (LLNL)DYNA3D code of UTHM’s version and named as Material Type 92 (Mat92). The prediction capability of the proposed constitutive model is validated against the plate impact test data of aluminum alloy and carbon ﬁberreinforced epoxy composites where a good agreement is obtained. This is a good indication for a better prediction of the commercial orthotropic materials behavior under extreme conditions in various engineering applications.
Acknowledgments The authors wish to convey a sincere gratitude to Universiti Tun Hussein Onn Malaysia (UTHM) and the Ministry of Higher Education Malaysia (MOHE) for providing the ﬁnancial means to prepare and complete this work under Incentive Grant Scheme for Publication (IGSP), Vot U674, and Fundamental Research Grant Scheme (FRGS), Vot 1547, respectively.
References [1] Sitnikova E, Guan ZW, Schleyer GK, Cantwell WJ. Modelling of perforation failure in ﬁbre metal laminates subjected to high impulsive blast loading. Int J Solid Struct 2014;51: 3135e46. [2] Vignjevic R, Campbell J, Bourne NK, Djordjevic N. Modelling shock waves in orthotropic elastic materials. In: Conference on shock compression of condensed matter; 2007 June; Hawaii; 2007. [3] Mohd Nor MK, Vignjevic R, Campbell J. Planestress analysis of the new stress tensor decomposition. Appl Mech Mater 2013;315:635e9.
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
[4] Banabic D, Kuwabara T, Balan T, Comsa DS, Julean D. Nonquadratic yield criterion for orthotropic sheet metals under planestress conditions. J Mech Sci July 2003;45:797e811. [5] Vignjevic R, Bourne NK, Millett JCF, De Vuyst T. Effects of orientation on the strength of the aluminum alloy 7010T6 during shock loading: experiment and simulation. J Appl Phys 2002;92(8):4342e8. [6] Sinha S, Ghosh S. Modeling cyclic ratcheting based fatigue life of HSLA steels using crystal plasticity FEM simulations and experiments. Int J Fatig 2006;28(12):1690e704. [7] Mohd Nor MK, Mohamad Suhaimi I. Effects of temperature and strain rate on commercial aluminum alloy AA5083. Appl Mech Mater 2014;660:332e6. [8] Furnish MD, Chhabildas LC. Alumina strength degradation in the elastic regime. AIP Conf Proc 1998;429(1):501e4. [9] Minich R, Cazamias J, Kumar M, Schwartz A. Effect of microstructural length scales on spall behaviour of copper. Metall Mater Trans 2004;35(9):2663e73. [10] Colvin JD, Minich RW, Kalantar DH. A model for plasticity kinetics and its role in simulating the dynamic behaviour of Fe at high strain rates. Int J Plast 2009;25(4):603e11. [11] Kanel GI, Zaretsky EB, Rajendran AM, Razorenov SV, Savinykh AS, Paris V. Search for conditions of compressive fracture of hard brittle ceramics at impact loading. Int J Plast 2009;25(4):649e70. [12] Khan AS, Meredith CS. Thermomechanical response of Al 6061 with and without equal channel angular pressing (ECAP). Int J Plast 2010;26(2):189e203. [13] Zaretsky EB, Kanel GI. Plastic ﬂow in shockloaded silver at strain rates from 10[sup 4]s [sup  1] to 10[sup 7]s[sup  1] and temperatures from 296 K to 1233 K. J Appl Phys 2011; 110(7):073502. [14] Meredith CS, Khan AS. Texture evolution and anisotropy in the thermomechanical response of UFG Ti processed via equal channel angular pressing. Int J Plast 2012; 30e31:202e17. [15] Smallman RE. Modern physical metallurgy. 4th ed. London: Butterworths; 1985. [16] Gray GT, Bourne NK, Millett JCF. Shock response of tantalum: lateral stress and shear strength through the front. J Appl Phys 2003;94(10):6430e6. [17] Khan AS, Kazmi R, Farrokh B. Multiaxial and nonproportional loading responses, anisotropy and modeling of Tie6Ale4V titanium alloy over wide ranges of strain rates and temperatures. Int J Plast 2007;23(6):931e50. [18] Khan AS, Kazmi R, Farrokh B, Zupan M. Effect of oxygen content and microstructure on the thermomechanical response of three Tie6Ale4V alloys: experiments and modeling over a wide range of strainrates and temperatures. Int J Plast 2007;23(7):1105e25. [19] Nakamachi E, Tam NN, Morimoto H. Multiscale ﬁnite element analyses of sheet metals by using SEMEBSD measured crystallographic RVE models. Int J Plast 2007;23(3): 450e89. [20] Khan AS, Kazmi R, Pandey A, Stoughton T. Evolution of subsequent yield surfaces and elastic constants with ﬁnite plastic deformation. PartI: a very low work hardening aluminum alloy (Al6061T6511). Int J Plast 2009;25(9):1611e25. [21] Sitko M, Skoczen B, Wroblewski A. FCCBCC phase transformation in rectangular beams subjected to plastic straining at cryogenic temperatures. Int J Mech Sci 2010;52(7): 993e1007. [22] Mohd Nor MK, Vignjevic R, Campbell J. Modelling of shockwave propagation in orthotropic materials. Appl Mech Mater 2013;315:557e61. [23] Mohd Nor MK. The development of unique orthogonal rotation tensor algorithm in the llnldyna3d for orthotropic materials constitutive model. Aust J Basic Appl Sci 2015;9(37): 22e7.
Modeling shock waves and spall failure in composites as an orthotropic materials
197
[24] Mohd Nor MK. Modelling inelastic behaviour of orthotropic metals in a unique alignment of deviatoric plane within the stress space. Int J Non Lin Mech 2016;87:43e57. [25] Mohd Nor MK. Modeling of constitutive model to predict the deformation behaviour of commercial aluminum alloy AA7010 subjected to high velocity impacts. ARPN J Eng Appl Sci 2016;11(4):2349e53. [26] Rosenberg Z, Luttwak G, Yeshurun Y, Partom Y. Spall studies of differently treated 2024A1 specimens. J Appl Phys 1983;54(5):2147e52. [27] Wackerle J. Shockwave compression of quartz. J Appl Phys 1962;33:922e37. [28] Zel’dovich YB, Raizer YP. Physics of shock waves and hightemperature hydrodynamic phenomena, vols. 1 and 2. New York: Academic Press; 1966. [29] Davison L, Graham RA. Shock compression of solids. Phys Rep 1979;55(4):255e379. [30] Eliezer S, Ghatak A, Hora H, Teller E. An introduction to equations of state, theory and applications. Cambridge: Cambridge University Press; 1986. [31] Asay JR, Shahinpoor M. Highpressure shock compression of solids. New York: Springer; 1993. [32] Meyers MA. Dynamic behaviour of materials. New York: Wiley, Inc; 1994. [33] Drumheller DS. Introduction to wave propagation in nonlinear ﬂuids and solids. Cambridge (UK): Cambridge University Press; 1998. [34] Butcher BM. In Behaviour of dense media under high dynamic pressure. New York: Gordon and Breach; 1968. p. 245. [35] Stevens AL, Tuler FR. Effect of shock precompression on the dynamic fracture strength of 1020 steel and 6061T6 aluminum. J Appl Phys 1971;42(13):5665. [36] Schmidt RM, Davies FW, Lempriere BM. Temperature dependent spall threshold of four metal alloys. J Phys Chem Solid 1978;39(4):375e85. [37] Grady DE. The spall strength of condensed matter. J Mech Phys Solid 1988;36:353e84. [38] Grady DE, Kipp ME. Fragmentation properties of metals. Int J Impact Eng 1997;20(1e5): 293e308. [39] Wilson LT, Reedal DR, Kuhns LD, Grady DE, Kipp ME. Using a numerical fragmentation model to understand the fracture and fragmentation of naturally fragmenting munitions of differing materials and geometries. In: 19th International Symposium of ballistics; 7e11; Interlaken, Switzerland; 2001. [40] De Vuyst TA. Hydrocode modelling of water impact. Cranﬁeld University; 2003. [41] Hill R. A theory of the yielding and plastic ﬂow of anisotropic metals. Proc R Soc Ser A 1948;193:281e97. [42] Barlat F. Crystallographic texture, anisotropic yield surface and forming limits of sheet metals. Mat Sci Eng 1987;91:55. [43] Barlat F, Lian J. Plastic behaviour and stretchability of sheet metals. Part I: a yield function for orthotropic sheets under plane stress conditions. Int J Plast 1989:51e66. [44] Banabic D. Sheet metal forming processes. Heidelberg: Springer; 2010. [45] Itskov M. On the application of the additive decomposition of generalized strain measures in large strain plasticity. Mech Res Commun 2004;31:507e17. [46] Itskov M, Aksel N. A constitutive model for orthotropic elastoplasticity at large strains. Arch Appl Mech 2004;74:75e91. [47] Man C. On the correlation of elastic and plastic anisotropy in sheet metals. J Elast 1995; 39(2):165e73. [48] Aravas N. Finitestrain anisotropic plasticity and the plastic spin. Model Simulat Mater Sci 1994;2:483e504. [49] Vignjevic R, Djordjevic N, Panov V. Modelling of dynamic behaviour of orthotropic metals including damage and failure. Int J Plast 2012;38:47e85.
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Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
[50] Mohd Nor MK. Unique rotation tensor formulation to predict threedimensional deformation behaviour of aluminum alloy AA7010. Int J Mech Mechatron Eng 2016c;16(4): 70e5. [51] Boehler JP. On irreducible representations for isotropic scalar functions. Z Angew Math Mech 1977;57:323e7. [52] Schr€oder J, Hackl K. Plasticity and beyond: microstructures, crystalplasticity and phase transitions. Springer; 2014. [53] Mandel J. Plasticité classiqueet viscoplastié. CISM lecture Notes. Wien: SpringerVerlag; 1972. [54] Holzapfel GA. Nonlinear solid mechanics, A continuum approach for engineering. Chichester: John Wiley & Sons Ltd; 2007. [55] Montans FJ, Bathe KJ. Towards a model for large strain anisotropic elastoplasticity. Computational plasticity. Computational methods in applied sciences. Dordrecht: Springer; 2007. p. 13e36. [56] Vladimirov IN, Pietryga MP, Reese S. On the modelling of nonlinear kinematic hardening at ﬁnite strains with application to springback e comparison of time integration algorithms. Int J Numer Methods Eng 2008;75(1):1e28. [57] Reese S, Vladimirov IN. Anisotropic modelling of metals in forming processes. IUTAM symposium on theoretical computational and modelling aspects of inelastic media, 11; 2008. p. 175e84. [58] Steinberg DJ. Equation of state and strength properties of selected materials. Livermore (CA): Lawrence Livermore National Laboratory; 1991. Report No. UCRLMA106439. [59] Gruneisen E. The state of solid body. 1959. NASA R19542. [60] Mohd Nor MK, Ma’at N. Simpliﬁed approach to validate constitutive formulation of orthotropic materials undergoing ﬁnite strain deformation. J Eng Appl Sci 2016;11(10): 2146e54. [61] Mohd Nor MK, Ma’at N, Kamarudin KA, Ismail AE. Implementation of ﬁnite strainbased constitutive formulation in LLNLDYNA3D to predict shockwave propagation in commercial aluminum alloys AA7010. In: IOP Conf. series: materials science and engineering; 2016. p. 160. 012023. [62] Millett JCF, Bourne NK, Meziere YJE, Vignjevic R, Lukyanov A. The effect of orientation on the shock response of a carbon ﬁbreepoxy composite. Comput Sci Technol 2007;67: 3253e60. [63] Vignjevic R, Millett JCF, Bourne NK, Meziere Y, Lukyanov A. The behaviour of a carbonﬁbre epoxy composite under shock loading. In: Furnish MD, Elert ML, Russell TP, White CT, editors. Shock compression of condensed matter e 2005. Melville (New York): American Institute of Physics; 2006. p. 825e8.
Further reading [1] Maudlin PJ, Bingert JF, House JW, Chen SR. On the modeling of the Taylor cylinder impact test for orthotropic textured materials: experiments and simulations. Int J Plast 1999a;15(2):139e66. [2] Bronkhorst CA, Cerreta EK, Xue Q, Maudlin PJ, Mason TA, Gray III GT. An experimental and numerical study of the localization behaviour of tantalum and stainless steel. Int J Plast 2006;22(7):1304e35.
TOPSIS method for selection of best composite laminate
11
M.R. Sanjay 1, 2 , Mohammad Jawaid 3 , N.V.R. Naidu 2,4 , B. Yogesha 2,5 1 Department of Mechanical Engineering, Ramaiah Institute of Technology, Bengaluru, India; 2Visvesvaraya Technological University, Belagavi, India; 3Laboratory of Biocomposite Technology, Institute of Tropical Forestry and Forest Products (INTROP), Universiti Putra Malaysia, Serdang, Malaysia; 4Department of Industrial Engineering & Management, Ramaiah Institute of Technology, Bengaluru, India; 5Department of Mechanical Engineering, Malnad College of Engineering, Hassan, India
11.1
Introduction
Composite laminates have received considerable attention in aircraft industries due to their remarkable properties over conventional metallic materials, such as greater stiffness/weight ratio, high speciﬁc strength, and higher strength/weight ratio. However, selecting the best laminate from the available resources is a challenge. To overcome this issue, multiple criteria decision making (MCDM) has proved to be a solution. In MCDM, the best options from a set of alternatives are selected, each of which is evaluated against multiple criteria. Among various problemsolving techniques are the analytic hierarchy process, the analytic network process, elimination and choice translating reality), simple additive weighting, the simple multiattribute rating technique, and the Technique for Order Preference by Similarity to the Ideal Solution (TOPSIS). Among the multicriteria models for making complex decisions and multiple attribute models for the most preferable option, TOPSIS has been the preferred choice. In realword situation, due to incomplete or nonobtainable information, the data (attributes) are often not so deterministic, they are usually fuzzy/imprecise [1]. The TOPSIS method is used for the proper selection of machine tools, labor savings, improved product quality, and increased production rate with higher overall productivity. It is noted that the use of the TOPSIS method is quite capable and computationally easy to evaluate and selects the right machine tool from a given set of alternatives [2]. As a future possibility, a diffuse methodology based on TOPSIS can be developed to help decision makers make decisions in the presence of inaccurate and incomplete data. The TOPSIS method proposes a belief structure model to solve problems of the MCDM belief group. First, the MCDM problem belief group is structured as a decision matrix belief in the trials of each decision maker as the belief structure of the models is described, and then the probative reasoning approach is used for multitaker aggregation judgments of decisions. Subsequently, solutions of positive and negative ideals are deﬁned with the TOPSIS principle. To measure the separation of the ideal solutions, the concept and belief of the distance measurement algorithm, which can be used to compare the difference between the models of the belief structure, are Modelling of Damage Processes in Biocomposites, FibreReinforced Composites and Hybrid Composites https://doi.org/10.1016/B9780081022894.000114 Copyright © 2019 Elsevier Ltd. All rights reserved.
200
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
deﬁned. Finally, the relative closeness and the classiﬁcation index are calculated to classify the alternatives [3]. One of the approaches to solving the problem of disposal of facilities is data envelopment analysis (DEA). This approach proposes using a 2opt algorithm together with DEA in an iterative process to ﬁnd the most efﬁcient design, such that the 2opt algorithm in each stage with the previous scenario design generates design alternatives. The TOPSIS method is then used to test the DEA result by ranking the best response of each DEA iteration response [4]. Rohmatulloh and Winarni prepared a strategic training program related to the growing problems of inputs suggested by stakeholders. The problems identiﬁed required management to evaluate each of them to determine the scale of priorities of issues. Application of the TOPSIS method allows analysis of all issues related to the criteria guidelines. The analysis results showed that training programs with the support of important resources include training for reducing energy and mining inspectors [5]. Srikrishna et al. proposed a new procedure for selecting the best car among commercially available models for various speciﬁcations of the technical parameters and operationally considered style, lifetime, fuel economy, suspension, costs, etc. Some cars were considered with different attributes and the best car was selected using the TOPSIS technique [6]. A performance analysis was carried out for the performances of 12 family practice units on business were converted into a single score indicating the general performance level with the help of the TOPSIS method among the MultiCriteria Decision Making Techniques through the data of 8 criteria and such units were put into order in this regard and their performances were compared [7].
11.2
TOPSIS method
TOPOSIS is one of the bestknown methods for classical multiattribute decisionmaking techniques [8]. The underlying logic of TOPSIS is to deﬁne the ideal solution and the ideal negative solution. The TOPSIS method was ﬁrst proposed by Hwang and Yoon (1981) [9]. According to this technique, the best alternative would be to be close to the ideal positive solution and further away from the ideal negative solution [10]. The positive ideal solution consists of all the best values of achievable criteria, while the ideal negative solution is composed of all the values of criteria that are more difﬁcult to reach [11e14]. This technique can also be obtained through the gap between the ideal alternative and each alternative and the alternative classiﬁcation order so that it can be widely used in many ﬁelds. TOPSIS is used to add ratings and generate a general score to measure the performance of each alternative. The TOPSIS method offers a number of attributes or criteria in a systematic manner. Moreover, the advantages of the TOPSIS method are i) ability to identify the best alternative quickly, ii) simple and rationally comprehensive concept, iii) good computational efﬁciency, iv) ability to measure the relative performance of eachalternative in a simple mathematical form, v) large ﬂexibility in the deﬁnition of thechoice set, vi) a sound logic that represents the rationale of human choice and vii) asimple computational process that can be easily programmed into a spreadsheet [15e17].
TOPSIS method for selection of best composite laminate
11.3
201
Methodology adopted
Decide the criteria (properties) for the selection of the alternative (materials destined to be used for engineering applications). The criteria or attributes can be density, tensile strength, ﬂexural strength, interlaminar cut resistance, microhardness to impact resistance, and water absorption. Choose a set of alternative materials and measure the performance of each alternative to the attributes. Determine the importance of weights of the attributes using the variance method. Find the standard weighted matrix decision. Determine the ideal positive and negative solution. Calculate the segregation using the 3D Euclidean distance n. The separation of each alternative provides the ideal solution and ideal negative solution. Find the relative closeness to the ideal solution. The relative proximity of alternative Ai with respect to Aþ must be discovered. Finally as per the TOPSIS method. The various steps involved are as follows [2,5,8,15]: Step 1: Create a decision matrix for the ranking, and the problem of MCDM can be expressed in matrix format like: C1
C2 ..... Cn
A1 ⎡ x11 x12 ..... x1n ⎡ A ⎢ x21 x22 ..... x2n ⎢⎢ D= 2⎢ : ⎢ : : : : ⎢ Am ⎢⎣ xm1 xm2 ..... xmn ⎣⎢
(11.1)
where A1, A2, ., Am are the viable alternatives from which the decision makers have to choose, C1, C2, ., Cn are the criteria by which the alternative performance is measured, xij is the qualiﬁcation of the alternative Ai with respect to the criterion Cj, and wj is the weight of the criterion Cj. Step 2: Determine the normalized decision matrix, and the normalized value nij is obtained using the formula: xij nij ¼ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ m P x2ij
(11.2)
i1
where i ¼ 1, 2, ., m, j ¼ 1, 2, ., n. Step 3: Determine the weighted normalized decision matrix, and weighted normalized value vij is obtained using the formula: vij ¼ rij wj where wj is the relative weight of the jth criterion or attribute and
(11.3) n P j¼1
wj ¼ 1.
202
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
Step 4: Calculate the positive ideal solutions and negative ideal solutions, respectively: Aþ ¼
A ¼
n o ¼ vþ maxvij j˛Ub ; minvij j˛Uc j j ¼ 1; 2; .; n i
i
n o ¼ v minvij j˛Ub ; maxvij j˛Uc j ¼ 1; 2; .; n j i i
(11.4)
(11.5)
where Ub and Uc are the sets of beneﬁt criteria/attributes and cost criteria/attributes, respectively. Step 5: Determine the separation measure value using the ndimensional Euclidean distance method. The separation of each alternative from the ideal solution is given as: diþ
vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ uX 2 u m vij vþ ¼t ; j ¼ 1; 2; .; n. j
(11.6)
j¼1
Similarly, separation from the negativeideal solution is given as: vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ uX 2 u m vij v ; j ¼ 1; 2; .; n di ¼ t j
(11.7)
j¼1
Step 6: Determine the relative closeness to the ideal solution, and the relative closeness of the alternative Ai with respect to Aþ is obtained using the formula: clþ i ¼
diþ
di . þ di
(11.8)
Step 7: Finally, rank the preference order. A large value of closeness coefﬁcient clþ i indicates good performance of the alternative Ai. The best alternative is the one that has the greatest relative closeness to the ideal solution.
11.4
Results and discussion
By applying this TOPSIS method for selection of composite laminates for potential applications the selection is to ﬁnd the best laminate among nine composite laminates using a decisionmaking method. All the composite laminates are compared based on the TOPSIS method and ranking has been done. The decision matrix, normalization matrix, weightnormalized matrix, ideal positive and ideal negative solution, separation measure, relative closeness value, and ranking are shown in Tables 11.1e11.6, respectively.
ILSS (MPa)
Impact strength (J/m)
Microhardness (HV)
Water absorption (wt%)
309.315
18.288
1469.5
24.58
0.008
34.676
170.860
4.736
122.5
10.55
0.052
1.2127
45.185
190.083
6.331
151.9
13.51
0.051
S4
1.2233
42.954
176.994
5.071
134.4
11.57
0.055
S5
1.2221
47.439
181.916
5.837
171.5
15.89
0.049
S6
1.3611
85.447
198.444
8.549
792.4
20.13
0.016
S7
1.3512
101.348
232.824
11.305
897.4
22.27
0.009
S8
1.3581
88.512
218.194
9.39
860.1
21.97
0.011
S9
1.3548
129.183
235.531
12.557
1078.4
23.16
0.015
Laminates
Density (g/cc)
Tensile strength (MPa)
Flexural strength (MPa)
S1
1.4952
331.779
S2
1.2337
S3
TOPSIS method for selection of best composite laminate
Table 11.1 Decision matrix
ILSS, interlaminar cut resistance.
203
204
Table 11.2 Normalization matrix Tensile strength (MPa)
Flexural strength (MPa)
ILSS (MPa)
Impact strength (J/m)
Microhardness (HV)
Water absorption (wt%)
S1
0.378847
0.830715
0.4761
0.608423
0.622037
0.434206
0.074672
S2
0.312589
0.086822
0.262989
0.157562
0.051854
0.186366
0.485367
S3
0.307269
0.113135
0.292577
0.210626
0.064299
0.238654
0.476033
S4
0.309954
0.107549
0.27243
0.168707
0.056891
0.204384
0.513369
S5
0.30965
0.118779
0.280006
0.194191
0.072596
0.280697
0.457365
S6
0.344869
0.213944
0.305446
0.284416
0.335422
0.355596
0.149344
S7
0.342361
0.253757
0.358364
0.376106
0.379868
0.3934
0.084006
S8
0.344109
0.221618
0.335846
0.312395
0.364079
0.3881
0.102674
S9
0.343273
0.323451
0.362531
0.417758
0.456485
0.409121
0.14001
ILSS, interlaminar cut resistance.
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
Laminates
Density (g/cc)
Laminates
Density (g/cc)
Tensile strength (MPa)
Flexural strength (MPa)
ILSS (MPa)
Impact strength (J/m)
Microhardness (HV)
Water absorption (wt%)
S1
0.060616
0.132914
0.076176
0.097348
0.099526
0.069473
0.011947
S2
0.028133
0.007814
0.023669
0.014181
0.004667
0.016773
0.043683
S3
0.027654
0.010182
0.026332
0.018956
0.005787
0.021479
0.042843
S4
0.027896
0.009679
0.024519
0.015184
0.00512
0.018395
0.046203
S5
0.027869
0.01069
0.025201
0.017477
0.006534
0.025263
0.041163
S6
0.041384
0.025673
0.036654
0.03413
0.040251
0.042672
0.017921
S7
0.041083
0.030451
0.043004
0.045133
0.045584
0.047208
0.010081
S8
0.041293
0.026594
0.040301
0.037487
0.043689
0.046572
0.012321
S9
0.041193
0.038814
0.043504
0.050131
0.054778
0.049095
0.016801
TOPSIS method for selection of best composite laminate
Table 11.3 Weight normalized matrix
ILSS, interlaminar cut resistance.
205
206
Solution
Density (g/cc)
Tensile strength (MPa)
Flexural strength (MPa)
ILSS (MPa)
Impact strength (J/m)
Microhardness (HV)
Water absorption (wt%)
Aþ
0.060616
0.132914
0.076176
0.097348
0.099526
0.069473
0.010081
0.027654
0.007814
0.023669
0.014181
0.004667
0.016773
0.046203
A

ILSS, interlaminar cut resistance.
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
Table 11.4 Ideal positive and ideal negative solution
TOPSIS method for selection of best composite laminate
207
Table 11.5 Separation measure Laminates
SD
Se
S1
0
0.19541
S2
0.19533
0.00048
S3
0.18939
0.00768
S4
0.19287
0.00285
S5
0.18866
0.00985
S6
0.14718
0.05487
S7
0.13458
0.06801
S8
0.14209
0.06139
S9
0.12226
0.07991
Table 11.6 Relative closeness value and ranking Laminates
Closeness factor
Ranking
S1
1
1
S2
0.002446
9
S3
0.038947
7
S4
0.014543
8
S5
0.049638
6
S6
0.27158
5
S7
0.335693
3
S8
0.301716
4
S9
0.395276
2
Finally, the ranking of different composite laminates based on their properties is shown in Fig. 11.1. It has been observed that ranking of composite materials is as follows: Rank 1 (S1), Rank 2 (S9), Rank 3 (S7), Rank 4 (S8), Rank 5 (S6), Rank 6 (S5), Rank 7 (S3), Rank 8 (S4), and Rank 9 (S2).
208
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
TOPSIS method 1.2
Closeness factor
1 0.8 0.6 0.4
0.2 0 Laminates
S1
S2
S3
S4
S5
S6
S7
1
0.002
0.038
0.014
0.049
0.271
0.335
S8
S9
0.301 0.395
Figure 11.1 Ranking of the composite laminates.
11.5
Conclusion
From the results, it was observed that S6, S7, S8, and S9 obtained the relative closeness to the ideal solution and the values were 0.27, 0.33, 0.30, and 0.39, respectively. It was observed that S1 was selected as the best composite laminate among the all laminates, which had the best relative closeness value, and S9 was identiﬁed as the second best composite laminate. It was revealed that the TOPSIS method was considered as the best method to select the ideal laminate from the set of laminates.
References [1] Jahanshahloo GR, Hosseinzadeh Lotﬁ F, Izadikhah M. Extension of the TOPSIS method for decisionmaking problems with fuzzy data. App Math Comput 2006;181:1544e51. [2] Athawale VM, Chakraborty S. Material selection using multicriteria decisionmaking methods: a comparative study. Proc IMechE Part L J Mater Des App 2012;226:266e75. [3] Jiang J, YingWu C, DaWei T, YuWang C. TOPSIS with belief structure for group belief multiple criteria decision making. Inter J Auto Comp 2010;7:359e64. [4] Ghaseminejad A, Navidi H, Bashiri M. Using Data Envelopment Analysis and TOPSIS method for solving ﬂexible bay structure layout. Inter J Manag Sci Eng Manag 2011;6: 49e57. [5] Rohmatulloh, Winarni S. TOPSIS method for determining the priority of strategic training program. Inter J Adv Sci Eng Inform Tech 2014;4:31e4.
TOPSIS method for selection of best composite laminate
209
[6] Srikrishna S, Sreenivasulu Reddy A, Vani S. A new car selection in the market using TOPSIS technique. Inter J Eng Res Gen Sci 2014;2:177e81. [7] karaman E, kazan H. Performance evaluation in family physician: the application of topsis multicriteria decision making method. J Oper Res Stat Econ Manag Inf Syst 2015;3:1e2. [8] Purohit P, Ramachandran M. Selection of ﬂywheel material using multicriteria decision making fuzzy TOPSIS. Ind J Sci Tech 2015;8:1e5. [9] Kabir G, Ahsan Akhtar Hasin M. Comparative analysis of topsis and fuzzy topsis for the evaluation of travel website service quality. Inter J Q Res 2012;6:169e75. [10] Benitez JM, Martin JC, Roman C. Using fuzzy number for measuring quality of service in the hotel industry. Tour Manag 2007;28:544e5. [11] Ertugrul D, Karakasoglu N. Performance evaluation of Turkish cement ﬁrms with fuzzy analytic hierarchy process and TOPSIS methods. Exp Syst Appl Inter J 2009;36:702e5. [12] Wang YM, Elhag TMS. Fuzzy TOPSIS method based on alpha level sets with an application to bridge risk assessment. Exp Syst App Inter J 2006;31:309e19. [13] Wang TC, Chang TH. Application of TOPSIS in evaluating initial training aircraft under a fuzzy environment. Exp Syst App Inter 2007;33:870e80. [14] Wang YJ, Lee HS. Generalizing TOPSIS for fuzzy multiplecriteria group decisionmaking. Comp Math Appl 2007;53:1762e72. [15] Abdul Rahman NSF. A decision making support of the most efﬁcient Steaming speed for the liner business industry. Eur J Buss Manag 2012;4:37e9. [16] Olson DL. Comparisons of weights in TOPSIS models. Math Comp Model 2004;40: 721e7. [17] Shih HS, Shyur HJ, Lee ES. An extension of TOPSIS for group decision making. Math Comp Mod 2007;45:801e3.
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Deformation characteristics of functionally graded composite panels using ﬁnite element approximation
12
V.R. Kar 1 , S.K. Panda 2 , P. Tripathy 2 , K. Jayakrishnan 3 , M. Rajesh 3 , A. Karakoti 3 , M. Manikandan 4 1 Department of Mechanical Engineering, NIT, Jamshedpur, India; 2Department of Mechanical Engineering, NIT, Rourkela, India; 3School of Mechanical Engineering, VIT, Vellore, India; 4Department of Mechanical Engineering, Amrita College of Engineering and Technology, Nagercoil, Tamil Nadu, India
12.1
Introduction
Advanced lightweight composites with high speciﬁc mechanical properties have been used effectively in aerospace and other weightsensitive industries. The major disadvantage of traditional composite material is sustaining their characteristics under the critical temperature environment. In general, metals are known for their excellent strength and toughness but have low thermal conductivity and antioxidant properties. The ceramics are exceptional in the thermal ﬁeld, whereas they are inferior under fatigue loading. Functionally graded materials (FGMs) are the distinct materials that amalgamate the superior properties of their constituents, i.e., metals/alloys and ceramics [1]. FGMs are novel kinds of advanced composites where material properties alter across the spatial direction smoothly, which relaxes the stress concentrations and delamination generally observed in layered structures [2]. In the past, many researchers investigated the ﬂexural behavior of FGM ﬂat/curved panels using different solution/kinematics schemes. Talha and Singh [3] examined the bending and free vibration responses of FGM plates using higherorder shear deformation theory (HSDT) via the ﬁnite element method. Ferreira et al. [4] investigated the deformation characteristics of FGM plates using the meshless collocation method via radial basis functions. Neves et al. [5] studied the static deformation analysis of FGM plates by the radial basis functionbased collocation method. Uymaz and Aydogdu [6] examined the vibrational behavior of FGM plates under various edge support cases using the small strain linear elasticity theory. HosseiniHashemi et al. [7] presented an analytical solution for free vibration analysis of moderately thick FGM rectangular plates for all six possible combinations of boundary conditions. Neves et al. [8] proposed a novel hyperbolic sine sheardeformation kinematics to examine the deﬂection and frequency responses of FGM
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites and Hybrid Composites https://doi.org/10.1016/B9780081022894.000126 Copyright © 2019 Elsevier Ltd. All rights reserved.
212
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
plates. Thai and Kim [9] proposed a new HSDT kinematics to obtain the ﬂexural and frequency characteristics of FGM plates. Oktem et al. [10] utilized the closeform solution to illustrate the bending responses of FGM ﬂat/curved panels using the HSDTbased generalized double Fourier series approach. Aragh and Hedayati et al. [11] employed the 2D generalized differential quadrature method to execute the free vibration and static analyses of FGM open cylindrical shells. Patel et al. [12] examined the free vibration characteristics of thick and thin FGM elliptical cylindrical shells employing finite element formulations based on higherorder theory. Tornabene and Viola [13] studied the free vibration behavior of FGM thick shells and panels of revolution using the generalized differential quadrature method incorporating the ﬁrstorder shear deformation kinematic theory (FSDT). Haddadpour et al. [14] utilized KirchhoffeLove shell theory via the von KarmaneDonnelltype strain to execute free vibration study of simply supported FGM cylindrical shells. Santos et al. [15] established a semianalytical axisymmetric model via ﬁnite element approximation and 3D linear elastic theory. Zhao et al. [16] obtained the ﬂexural and free vibration responses of an FGM cylindrical shell using the meshless kpRitz method and Sander’s FSDT kinematics. From these studies it is conﬁrmed that the deformation behavior of FGM curved panels is very limited. However, in the present study, the deformation characteristics of FGM ﬂat and curved (cylindrical and spherical) panels for different volume fractions, thickness ratios, curvature ratios, and boundary conditions are demonstrated and discussed. To achieve this, the ﬁnite element solutions are obtained in the ANSYS environment via the customized ANSYS parametric design language (APDL) code using eightnoded serendipity element (SHELL281) with 48 degrees of freedom.
12.2
Micromechanical material modeling
The overall material properties of the FGM are expected to be continuous along the spatial direction, and are obtained using a powerlaw based Voigt’s micromechanical model. The FGM properties P are dependent on the constituents’ material properties as well as the volume fractions, and are expressed as: P¼
k X
Pj Vfj
(12.1)
j¼1
where Pj and Vfj are the material property and volume fraction of the constituent material j, respectively. The volume fractions of all the constituent materials should add up to one, i.e.: k X j¼1
Vfj ¼ 1
(12.2)
Deformation characteristics of functionally graded composite panels using ﬁnite element
213
For a plate with a uniform thickness h and a reference surface at its middle surface, the volume fraction can be written as: Vf ¼
z 1 þ h 2
n (12.3)
where n is the powerlaw index, 0 n N. The variations of volume fraction of the ceramic phase through the dimensionless thickness are shown in Fig. 12.1 for different values of powerlaw index. For an FGM with two constituent materials, the Young’s modulus E and the mass density r can be expressed as: E ¼ ðEc Em Þ
z 1 þ h 2
z 1 r ¼ ðrc rm Þ þ h 2 v ¼ ðvc vm Þ
z 1 þ h 2
n þ Em
(12.4)
n þ rm
(12.5)
þ vm
(12.6)
n
Volume fraction of ceramics fraction
1.0
n = 0.2 n = 0.4 n = 0.5 n=1 n=2 n=5 n = 10
0.8
0.6
0.4
0.2
0.0 –0.6
–0.4
–0.2
0.0
0.2
0.4
0.6
Nondimensional thickness
Figure 12.1 Material proﬁles of a functionally graded material panel through the thickness.
214
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
Table 12.1 Mechanical properties of metal and ceramic materials Properties Materials
Young’s modulus E (GPa)
Poisson’s ratio n
Density r (kg/m3)
Aluminum (Al)
70
0.3
2707
Stainless steel (SUS304)
207.7877
0.31776
8166
Silicon nitride (Si3N4)
322.2715
0.24
2370
Zirconia (ZrO2)
151
0.3
3000
Alumina (Al2O3)
380
0.3
3000
From these equations, when z ¼ eh/2, E ¼ Em, r ¼ rm, and v ¼ vm; when z ¼ þh/2, E ¼ Ec, r ¼ rc, and v ¼ vm. The material properties alter smoothly from the metal phase (at the lower surface) to the ceramic phase (at the upper surface). The properties of the FGM constituents at the ambient temperature (300 K) are used as in Table 12.1.
12.3
Finite element approximations
A doubly curved FGM panel of uniform thickness h with a rectangular base of sides a and b is considered for the analysis as shown in Fig. 12.2. Rx and Ry are the principal radii of curvature of the shell panel along the x and y directions, respectively. The principal radii of curvature of spherical, cylindrical, and ﬂat panels can be presented as Rx ¼ Ry ¼ R, Rx ¼ R, Ry ¼N, and Rx ¼ Ry ¼N, respectively. The modeling of the doubly curved shell panel is governed by the ﬁrstorder shear deformation theory.
z y x Ceramic rich +h/2 –h/2 Rx
Ry
a
b
Metal rich
Figure 12.2 Geometrical derestriction of a functionally graded material curved panel.
Deformation characteristics of functionally graded composite panels using ﬁnite element
215
The displacements u, v, and w at any point along the x, y, and z directions, respectively, are given by: uðx; y; zÞ ¼ u0 ðx; yÞ þ zu1 ðx; yÞ vðx; y; zÞ ¼ v0 ðx; yÞ þ zv1 ðx; yÞ
(12.7)
wðx; y; zÞ ¼ w0 ðx; yÞ þ zw1 ðx; yÞ where u0, v0, and w0 are the midplane displacements and u1, v1, and w1 are the shear rotation terms. The present work utilizes an eightnoded serendipity shell element (SHELL281) with 48 degrees of freedom, which supports thin to moderately thick shell structures. The displacements are expressed in terms of shape functions (Ni): d¼
8 X
Ni di
(12.8)
i¼1
T where di ¼ u0i v0i w0i fxi fyi fzi . The shape functions for side nodes at i ¼ 1 to 4 and midside nodes at i ¼ 5 to 8, in xeh coordinates, can be expressed as: 1 N1 ¼ ð1 xÞð1 hÞð x h 1Þ; 4 1 N3 ¼ ð1 þ xÞð1 þ hÞðx þ h 1Þ; 4
1 N2 ¼ ð1 þ xÞð1 hÞðx h 1Þ 4 1 N4 ¼ ð1 xÞð1 þ hÞð x þ h 1Þ 4
N5 ¼
1 1 x2 ð1 hÞ; 2
1 N6 ¼ ð1 þ xÞ 1 h2 2
N7 ¼
1 1 x2 ð1 þ hÞ; 2
1 N8 ¼ ð1 xÞ 1 h2 2
(12.9)
The strain vector expressed in terms of nodal displacement vector is: fεg ¼ ½Bfdg
(12.10)
where [B] presents the product form of differential operators and approximation functions and {d} denotes the nodal displacement vector. The global stress tensor can be expressed in the following form as: fsg ¼ ½Dfεg
(12.11)
216
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
where
fsg ¼ f sx
T
sy
sz
sxy
syz
sxz gT
and
fεg ¼
εx
εy
εz
gxy
gyz gxz are the stress and strain vectors, respectively, and [D] is the rigidity matrix. The elemental stiffness matrix [K] can be written as: Z þ1 Z þ1 ½K ¼ ½BT ½D½BjJjdxdh (12.12) 1
1
where jJj is the determinant of the Jacobian matrix and [N] is the shape function matrix. The ﬁnal equilibrium equation for the ﬂexural behavior of an FGM curved panel subjected to uniform pressure P can be expressed as: ½Kfdg ¼ fPg
12.4
(12.13)
Results and discussions
The static behavior of functionally graded shell panels is analyzed by ANSYS APDL in conjunction with the Block Lanczos method. An eightnoded serendipity shell element (SHELL281) is used for the discretization of the developed model as deﬁned in ANSYS. The following sets of boundary conditions are used in the analysis: Clamped (CCCC): u0 ¼ v0 ¼ w0 ¼ u1 ¼ v1 ¼ w1 ¼ 0 at x ¼ 0, a and y ¼ 0, b. Simply supported (SSSS): v0 ¼ w0 ¼ v1 ¼ w1 ¼ 0 at x ¼ 0 and a. u0 ¼ w0 ¼ u1 ¼ w1 ¼ 0 at y ¼ 0 and b.
12.4.1
Convergence and validation tests
To check the efﬁcacy of the present ﬁnite element model, it is necessary to validate the model of FGM shell panels for the static analysis. A clamped FGM square plate with a/h ¼ 20 is considered for static analysis under uniform distributed load. Aluminum (Al) and alumina (Al2O3) are at the bottom and top surfaces of the panel, respectively. The dimensionless central deﬂections and load parameter are obtained by w ¼ w=h and po ¼ p/Emh4, respectively. For the convergence test, different mesh sizes are taken into consideration in the range of 2 2 to 10 10. It is observed that results converged at 8 8 mesh size as shown in Table 12.2. Additional comparison has been carried out to check the efﬁcacy of the present model for dimensionless central deﬂection and is tabulated in Table 12.3. The results are computed for a clamped FGM (Al/ZrO2) cylindrical panel with a/h ¼ 20 and R/a ¼ 5 for different powerlaw indices (n). It is observed that the results converged at 8 8 mesh size and the percentage differences between the present results and those reported in the literature are within 4%, which shows a good agreement.
Deformation characteristics of functionally graded composite panels using ﬁnite element
217
Table 12.2 Nondimensional central deﬂections ðw ¼ w=hÞ of a clamped square (Al/Al2O3) functionally graded material plate with a/h ¼ 20 Mesh size
n[0
n [ 0.2
n [ 0.5
n[1
n[2
n[5
n [ 10
22
0.00324
0.004
0.00502
0.0065
0.0083
0.00986
0.0108
44
0.0026
0.0032
0.004
0.00458
0.00636
0.00734
0.00778
66
0.00266
0.0033
0.00412
0.00486
0.00642
0.00752
0.00812
88
0.00266
0.0033
0.00412
0.00484
0.00644
0.00756
0.00816
10 10
0.00266
0.0033
0.00412
0.00484
0.00644
0.00756
0.00816
Ref. [3]
0.0026
0.0032
0.004
0.0047
0.0062
0.0074
0.0079
% difference
2.26
3.03
2.91
2.89
3.73
2.12
3.19
Table 12.3 Nondimensional central deﬂections ðw ¼ w=hÞ of a clamped square (Al/ZrO2) cylindrical panel with R/a ¼ 5 and a/h ¼ 20 Mesh size
n[0
n [ 0.2
n [ 0.5
n[1
n[2
n[5
22
0.01682
0.0191
0.02146
0.02396
0.02624
0.02846
44
0.01346
0.0152
0.01712
0.01912
0.02102
0.02294
66
0.01376
0.0156
0.01752
0.01957
0.02146
0.02336
88
0.01376
0.0156
0.01754
0.01958
0.02148
0.02336
10 10
0.01376
0.0156
0.01774
0.01958
0.02148
0.02336
Ref. [16]
0.01347
0.01516
0.01711
0.01915
0.02102
0.02289
% difference
2.11
2.82
3.55
2.20
2.14
2.01
12.4.2 Numerical illustrations Figs. 12.3e12.6 show the variation of nondimensional central deﬂection with curvature ratio for FGM (Si3N4/SUS304) cylindrical and spherical panels with a/h ¼ 10. The nondimensional central deﬂection w ¼ w=h and load parameter po ¼ p/Emh4 are used in the static problems, unless otherwise stated. It has been observed that as the curvature ratio and powerlaw index increase the nondimensional central deﬂection values also increase for all boundary conditions. Figs. 12.7e12.10 show the variation of nondimensional central deﬂection with thickness ratio for FGM (Si3N4/SUS304) cylindrical and spherical panels with R/ a ¼ 5. It has been observed that the nondimensional central deﬂection increases with increase in powerlaw index values and reduces with increase in thickness ratios.
218
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
Nondimensional center deflection
0.016
0.012
Ceramic n = 0.5 n=1 n=2 n = 10 Metal
0.008
0.004
0.000 0
10
20
30
40
50
Curvature ratio (R/a)
Figure 12.3 Nondimensional central deﬂection of a clamped functionally graded material (Si3N4/SUS304) cylindrical panel (a/h ¼ 10) with different curvature ratios.
Figs. 12.11e12.14 show the variation of nondimensional axial stress along with the thickness coordinate and powerlaw index for FGM (Si3N4/SUS304) shell panels with h2 a/h ¼ 10 and R/a ¼ 5. The nondimensional axial parameters are given as: sxx ¼ sxx pa 2 where p is the uniform pressure.
Nondimensional center deflection
0.016
0.012
Ceramic n = 0.5 n=1 n=2 n = 10 Metal
0.008
0.004
0.000 0
10
20
30
40
50
Curvature ratio (R/a)
Figure 12.4 Nondimensional central deﬂection of a clamped functionally graded material (Si3N4/SUS304) spherical panel (a/h ¼ 10) for various curvature ratios.
Deformation characteristics of functionally graded composite panels using ﬁnite element
219
Nondimensional center deflection
0.016
0.012
Ceramic n = 0.5 n=1 n=2 n = 10 Metal
0.008
0.004
0.000 0
10
20
30
40
50
Curvature ratio (R/a)
Figure 12.5 Nondimensional central deﬂection of a simply supported functionally graded material (Si3N4/SUS304) cylindrical panel (a/h ¼ 10) for various curvature ratios.
It has been observed that the bottom metalrich surface of the FGM shell panel is under compression, as the top ceramicrich surface is in tension. And the increase in powerlaw index increases the difference between the stress values, which are observed in all the cases considered.
Nondimensional center deflection
0.016
0.012
Ceramic n = 0.5 n=1 n=2 n = 10 Metal
0.008
0.004
0.000 0
10
20 30 Curvature ratio (R/a)
40
50
Figure 12.6 Nondimensional central deﬂection of a simply supported functionally graded material (Si3N4/SUS304) spherical panel (a/h ¼ 10) for various curvature ratios.
220
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
Nondimensional center deflection
0.024
0.020 Ceramic n = 0.5 n=1 n=2 n = 10 Metal
0.016
0.012
0.008
0.004
0
20
40
60
80
100
Thickness ratio (a/h)
Figure 12.7 Nondimensional central deﬂection of a clamped functionally graded material (Si3N4/SUS304) cylindrical panel (R/a ¼ 5) for various thickness ratios.
Figs. 12.15e12.18 show the variation of nondimensional axial stress along with the thickness coordinate and thickness ratio for FGM (Si3N4/SUS304) shell panels with n ¼ 2 and R/a ¼ 5. For all cases considered, the bottom surface is under compression as the top surface is in tension. The difference in stress magnitudes between top and bottom surfaces reduces as the thickness ratio increases.
Nondimensional center deflection
0.024 0.020
Ceramic n = 0.5 n=1 n=2 n = 10 Metal
0.016 0.012 0.008 0.004 0.000 0
20
40
60
80
100
Thickness ratio (a/h)
Figure 12.8 Nondimensional central deﬂection of a clamped functionally graded material (Si3N4/SUS304) spherical panel (R/a ¼ 5) for various thickness ratios.
Deformation characteristics of functionally graded composite panels using ﬁnite element
221
Nondimensional center deflection
0.024 Ceramic n = 0.5 n=1 n=2 n = 10 Metal
0.020
0.016
0.012
0.008
0.004
0.000 0
20
40
60
80
100
Thickness ratio (a/h)
Figure 12.9 Nondimensional central deﬂection of a simply supported functionally graded material (Si3N4/SUS304) cylindrical panel (R/a ¼ 5) for various thickness ratios.
Figs. 12.19e12.22 show the variation of nondimensional axial stress along with the thickness coordinate and curvature ratio for FGM (Si3N4/SUS304) shell panels with n ¼ 2 and a/h ¼ 10. For all cases considered, the bottom surface is under compression as the top surface is in tension. The difference in stress magnitudes between top and bottom surfaces increases as the curvature ratio increases.
Nondimensional center deflection
0.024 Ceramic n = 0.5 n=1 n=2 n = 10 Metal
0.020 0.016 0.012 0.008 0.004 0.000 0
20
40
60
80
100
Thickness ratio (a/h)
Figure 12.10 Nondimensional central deﬂection of a simply supported functionally graded material (Si3N4/SUS304) spherical panel (R/a ¼ 5) for various thickness ratios.
222
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
0.24
Nondimensional axial stress
0.18 0.12 0.06
n = 0.5 n=1 n=2 n = 10
0.00 –0.06 –0.12 –0.6
–0.4
–0.2 0.0 0.2 0.4 Nondimensional thickness
0.6
Figure 12.11 Nondimensional axial stress sxx of clamped functionally graded material (Si3N4/ SUS304) cylindrical panels (a/h ¼ 10 and R/a ¼ 5) at different powerlaw indices.
0.20
Nondimensional axial stress
0.15 0.10 0.05 n = 0.5 n=1 n=2 n = 10
0.00 –0.05 –0.10 –0.15 –0.6
–0.4
–0.2
0.0
0.2
0.4
0.6
Nondimensional thickness
Figure 12.12 Nondimensional axial stress sxx of clamped functionally graded material (Si3N4/ SUS304) spherical panels (a/h ¼ 10 and R/a ¼ 5) at different powerlaw indices.
Deformation characteristics of functionally graded composite panels using ﬁnite element
223
0.20
Nondimensional axial stress
0.15 0.10 0.05 n = 0.5 n=1 n=2 n = 10
0.00 –0.05 –0.10 –0.15 –0.6
–0.4
–0.2
0.0
0.2
0.4
0.6
Nondimensional thickness
Figure 12.13 Nondimensional axial stress sxx of simply supported functionally graded material (Si3N4/SUS304) cylindrical panels (a/h ¼ 10 and R/a ¼ 5) at different powerlaw indices.
0.20
Nondimensional axial stress
0.15 0.10 0.05
n = 0.5 n=1 n=2 n = 10
0.00 –0.05 –0.10 –0.15 –0.6
–0.4
–0.2
0.0
0.2
0.4
0.6
Nondimensional thickness
Figure 12.14 Nondimensional axial stress sxx of simply supported functionally graded material (Si3N4/SUS304) spherical panels (a/h ¼ 10 and R/a ¼ 5) at different powerlaw indices.
224
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
Nondimensional axial stress
0.20 0.15 0.10 0.05 a/h = 5 a/h = 10 a/h = 50 a/h = 100
0.00 –0.05 –0.10 –0.15 –0.6
–0.4
–0.2 0.0 0.2 Nondimensional thickness
0.4
0.6
Figure 12.15 Nondimensional axial stress sxx of clamped functionally graded material (Si3N4/ SUS304) cylindrical panels (n ¼ 2 and R/a ¼ 5) at different thickness ratios.
0.20
Nondimensional axial stress
0.15 0.10 0.05 0.00 a/h = 5 a/h = 10 a/h = 50 a/h = 100
–0.05 –0.10 –0.15 –0.6
–0.4
–0.2
0.0
0.2
0.4
0.6
Nondimensional thickness
Figure 12.16 Nondimensional axial stress sxx of clamped functionally graded material (Si3N4/ SUS304) spherical panels (n ¼ 2 and R/a ¼ 5) at different thickness ratios.
Deformation characteristics of functionally graded composite panels using ﬁnite element
225
0.20
Nondimensional axial stress
0.15 0.10 0.05 a/h = 5 a/h = 10 a/h = 50 a/h = 100
0.00 –0.05 –0.10 –0.15 –0.6
–0.4
–0.2 0.0 0.2 Nondimensional thickness
0.4
0.6
Figure 12.17 Nondimensional axial stress sxx of simply supported clamped functionally graded material (Si3N4/SUS304) cylindrical panels (n ¼ 2 and R/a ¼ 5) at different thickness ratios.
0.20
Nondimensional axial stress
0.15 0.10 0.05 0.00
a/h = 5 a/h = 10 a/h = 50 a/h = 100
–0.05 –0.10 –0.15 –0.6
–0.4
–0.2
0.0
0.2
0.4
0.6
Nondimensional thickness
Figure 12.18 Nondimensional axial stress sxx of simply supported functionally graded material (Si3N4/SUS304) spherical panels (n ¼ 2 and R/a ¼ 5) at different thickness ratios.
226
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
0.20
Nondimensional axial stress
0.15 0.10 0.05 R/a = 0.5 R/a = 1 R/a = 5 R/a = 10 R/a = 50 Plate
0.00 –0.05 –0.10 –0.15 –0.6
–0.4
–0.2
0.0
0.2
0.4
0.6
Nondimensional thickness
Figure 12.19 Nondimensional axial stress sxx of clamped functionally graded material (Si3N4/ SUS304) cylindrical panels (n ¼ 2 and a/h ¼ 10) at different curvature ratios.
0.20
Nondimensional axial stress
0.15 0.10 0.05 0.00 R/a = 0.5 R/a = 1 R/a = 5 R/a = 10 R/a = 50 Plate
–0.05 –0.10 –0.15 –0.6
–0.4
–0.2
0.0
0.2
0.4
0.6
Nondimensional thickness
Figure 12.20 Nondimensional axial stress sxx of clamped functionally graded material (Si3N4/ SUS304) spherical panels (n ¼ 2 and a/h ¼ 10) at different curvature ratios.
Deformation characteristics of functionally graded composite panels using ﬁnite element
227
Nondimensional axial stress
0.18
0.12
0.06 R/a = 0.5 R/a = 1 R/a = 5 R/a = 10 R/a = 50 Plate
0.00
–0.06
–0.12
–0.6
–0.4
–0.2
0.0
0.2
0.4
0.6
Nondimensional thickness
Figure 12.21 Nondimensional axial stress sxx of simply supported functionally graded material (Si3N4/SUS304) cylindrical panels (n ¼ 2 and a/h ¼ 10) at different curvature ratios.
0.20
Nondimensional axial stress
0.15 0.10 0.05 0.00
R/a = 0.5 R/a = 1 R/a = 5 R/a = 10 R/a = 50 Plate
–0.05 –0.10 –0.15 –0.6
–0.4
–0.2
0.0
0.2
0.4
0.6
Nondimensional thickness
Figure 12.22 Nondimensional axial stress sxx of simply supported functionally graded material (Si3N4/SUS304) spherical panels (n ¼ 2 and a/h ¼ 10) at different curvature ratios.
228
Modelling of Damage Processes in Biocomposites, FibreReinforced Composites
12.5
Conclusions
In this study, ﬂexural responses of FGM ﬂat and curved (cylindrical and spherical) panels are examined and presented. The overall material properties of FGM ﬂat/ curved panels are achieved through the powerlawbased Voigt’s micromechanical material model. Convergence and validation tests are executed to demonstrate the stability and exactness of the present ﬁnite element model. The results, computed via ANSYS APDL, are more robust and effective for different sets of parameters. The following points are revealed based on the parametric study of an FGM shell panel: • • •
•
The nondimensional central deﬂection increases with the increase in powerlaw index because stiffness of the FGM shell panel degrades with the increment in powerlaw indices. The nondimensional central deﬂection increases with increase in curvature ratio and reduces with increase in thickness ratio. The bottom metalrich surface of the FGM shell panel is always under compression and the top ceramicrich surface is in tension for all cases considered. And the increase in powerlaw index increases the difference between the stress values, which are observed in all the cases considered. The difference in stress magnitudes between top and bottom surfaces reduces as the thickness ratio increases, and increases as the curvature ratio increases.
References [1] Shen HS. Functionally graded material: nonlinear analysis of plates & shells. CRC press; 2009. [2] Reddy JN. Mechanics of laminated composite: plates and shellsTheory and analysis. 2nd ed. CRC press; 2003. [3] Talha M, Singh BN. Static response and free vibration analysis of FGM plates using higher order shear deformation theory. Appl Math Model 2010;34:3991e4011. [4] Ferreira AJM, Batra RC, Roque CMC, Qian LF, Martins PALS. Static analysis of functionally graded plates using thirdorder shear deformation theory and a meshless method. Compos Struct 2005;69:449e57. [5] Neves AMA, et al. Bending of FGM plates by a sinusoidal plate formulation and collocation with radial basis functions. Mech Res Commun 2011;38:368e71. [6] Uymaz B, Aydogdu M. Threedimensional vibration analysis of functionally graded plates under various boundary conditions. J Reinf Plastic Compos 2007;26(18):1847e63. [7] HosseiniHashemi S, Rokni Damavandi Taher H, Akhavan H, Omidi M. Free vibration of functionally graded rectangular plates using ﬁrstorder shear deformation plate theory. Appl Math Model 2010;34:1276e91. [8] Neves AMA, Ferreira AJM, Carrera E, Cinefra M, Roque CMC, Jorge RMN, Soares CMM. A quasi3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates. Compos Struct 2012;94:1814e25. [9] Thai HT, Kim SE. A simple higherorder shear deformation theory for bending and free vibration analysis of functionally graded plates. Compos Struct 2013;96:165e73.
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[10] Oktem AS, Mantari JL, Guedes Soares C. Static response of functionally graded plates and doubly ecurved shells based on a higher order shear deformation theory. Eur J Mech Solid 2012;36:163e72. [11] Sobhani Aragh B, Hedayati H. Static response and free vibration of twodimensional functionally graded metal/ceramic open cylindrical shells under various boundary conditions. Acta Mech 2012;223:309e30. [12] Patel BP, Gupta SS, Loknath MS, Kadu CP. Free vibration analysis of functionally graded elliptical cylindrical shells using higherorder theory. Compos Struct 2005;69:259e70. [13] Tornabene F, Viola E. Free vibration analysis of functionally graded panels and shells of revolution. Meccanica 2009;44:255e81. [14] Haddadpour H, Mahmoudkhani S, Navazi HM. Free vibration analysis of functionally graded cylindrical shells including thermal effects. Thinwalled Struct 2007;45:591e9. [15] Santos H, Soares CMM, Mota Soares CA, Reddy JN. A semianalytical ﬁnite element model for the analysis of cylindrical shells made of functionally graded materials. Compos Struct 2009;91:427e32. [16] Zhao X, Lee YY, Liew KM. Thermoelastic and vibration analysis of functionally graded cylindrical shells. Int J Mech Sci 2009;51:694e707.
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Index ‘Note: Page numbers followed by “f ” indicate ﬁgures, “t” indicate tables.’ A Abaqus/Explicit FE software package, 106 Aircraft wing box model postprocessing, 157f preprocessing, 155f Aluminum/glass ﬁberreinforced plastic hybrid tubes crushing behavior, 27e28 American Standards for Testing and Materials (ASTM) standards, 158e159, 158f Analytical model, 20f Anisotropic homogeneous yield function, 175 Aramid ﬁberreinforced composite, 41e43 Automobile engineering, 1 Automobile industry, 1 B Bamboo ﬁber macromechanical analysis, 162 micromechanical analysis, 162e163 BelytschkoTsay formulation, 28e30 Boundary element method (BEM), 85e86 Bounding models onepoint bound models classical bounds, 71e72 Reuss model, 72e73 Voigt model, 72 VoigteReusseHill average, 73 twopoint bounds, 73 C Capsulebased selfhealing healing agent, 133e134 mechanism, 134f microcapsule fabrication, 135e136 Carban ﬁberreinforced plastic (CFRP) composite
catastrophic failure, 117 continuum shell element. See Continuum shell element ﬁnite element modeling Abaqus/Explicit FE software package, 106 boundary conditions, 106e108, 107f interaction, 109 pretension, amplitude function, 108, 108f pretension technique, 106e108 target model, 106 hemispherical projectile, ballistic limit prediction experimental and simulation results, 125e126, 125f failure modes, 126, 127f impact forceetime curves, 130e131, 130fe131f nonpretension CFRP target, 126e127, 128f pretensions, 124, 124t, 127e129 velocity simulation, 124e125 material properties, 104e105, 105t mesh sensitivity analysis element sizes, 109, 109t ﬂat projectile, 110e111 hemispherical projectile, 109t, 111e112, 111t, 112f material damage model, 109 parametric ﬁnite element simulation, 112, 112f predicted residual velocity, 110 pretension, 110, 110f pretension applied, 117 target plate and projectile model, 105, 106f Carbon ﬁberreinforced epoxy composites, shock waves modeling aluminum alloy (AA7010), 193, 193t material properties for, 192, 193t
232
Carbon ﬁberreinforced epoxy composites, shock waves modeling (Continued) MieGr€uneisen EOS parameters, 194t plate impact test conﬁguration, 192, 193f stress comparison, 194f CauchyeGreen tensor, 175 Ceramic matrix composites (CMC), 58e59 ClausiusePlank (CP) inequality, 179 ColeeCole plot, 146e147, 147f Composite fabrication, 4e5 Composite laminates multiple criteria decision making (MCDM), 199e200 problemsolving techniques, 199e200 ranking, 208, 208f Technique for Order Preference by Similarity to the Ideal Solution (TOPSIS) advantages, 201 criteria selection, 201e202 data envelopment analysis (DEA), 199e200 decision matrix, 201, 203t diffuse methodology, 199e200 ideal negative solution, 200e202, 206t normalized decision matrix, 201e202, 204t performance analysis, 199e200 positive ideal solution, 200e202, 206t relative closeness, 202, 207t separation measure value, 202, 207t weighted normalized decision matrix, 202, 205t Composite materials and ﬁllers advantages, 4, 58 continuous phase, 4 cost and performance, 4 discontinuous phase, 4 matrix, 58e59 reinforcements, 58e59 ﬁber reinforcements, 59, 59f ﬂake reinforcements, 60, 60f hybrid composite materials, 60e61 particulate reinforcements, 59, 60f Compressive strength, selfhealing glass ﬁberreinforced plastic (GFRP), 143 Constitutive formulation, shock waves modeling aluminum alloy 7010T6, 174
Index
anisotropic homogeneous yield function, 175 compressive input stress, 174 dynamic loading, 174 elastic free energy function, 182 energybased failure model, 174 equation of state (EOS) MieGr€ uneisen EOS, 180e181 RankineeHugoniot equations, 180e181 evolution equations, 183e186 ﬁnite strain deformation CauchyeGreen tensor, 175 deformation gradient, 175 elastoplastic deformation, 177 isoclinic conﬁguration, 175e177, 176f plasticrelated deformations, 176e177 pullback transformation, 176e177 structural tensors, 176 total velocity gradient, 177 grady failure model, 186 Hugoniot Elastic Limit (HEL), 174 orthotropic yield criterion, 182e186 spall strength of AA6061T6, 174 strengthtoweight ratio, 173 stress tensor decomposition ClausiusePlank (CP) inequality, 179 deviatoric Mandel stress tensor, 179 generalized orthotropic pressure, 178 Kirchhoff stress tensor, 177e178 Mandel stress tensor, 177e178 orthotropic yield surface, 179 Continuous hybrid ﬁber, 41, 42f Continuous phase, composite materials, 4 Continuum shell element boundary conditions, 122, 123f ﬁnite element model, 122 mesh sensitivity analysis, 123, 124f pretension technique, 122 progressive damage modeling damage variables, 120e122 elasticity matrix, 120 equivalent displacements and stresses, 120e122 Hashin’s failure criteria, 119 laws for orthotropic elastic materials, 120 surface interaction, 122e123 Convergence, 23, 24t, 157
Index
D Damping coefﬁcients, 166e167 Damping factor, 139e141 Data envelopment analysis (DEA), 199e200 Decision matrix, 201, 203t Discontinuous phase, composite materials, 4 Discretization, 157 Displacement boundary conditions, 107f 2D macroscale ﬁnite element model, 166e167 3D macroscale ﬁnite element model, 166 E Eightnoded serendipity shell element (SHELL281), 215 Elastic free energy function, 182 Elasticity matrix, 120 Elastoplastic deformation, 177 Elemental force vector, 156 Elemental stiffness matrix, 216 Element selection, 22e23 Energybased failure model, 174 Eshelby method equivalent inclusion method Eshelby tensor, 80e81 free strain deformation, 79 heterogeneity, 78e81 inﬁnity and deformation, 78 localization tensors, 80 reinforcement deformation, 80 stresses and deformations, 79 transformation deformation, 79 transformation strain, 79 inclusion problem, 77, 78f intuitive approach, 77e78 steps involved in, 77 Eshelby tensor, 80e81 F Fiberbyﬁber hybrids, 86 Fiber clustering effect, 165 Fiberhybrid composites, 86f Fiber/metal composite technology, 41e43 Fiberreinforced plastics (FRPs), 133 Fiberreinforced polymer (FRP) composite automotive crash tubes and armor designs, 103
233
damage behavior, 103 glass/polyester woven plates, 103e104 mesh sensitivity analysis, 104 metal foam, 103 numerical simulation, 104 Finite element analysis (FEA) convergence, 157 natural ﬁberreinforced polymer composites. See Natural ﬁberreinforced polymer composites polymer matrix composites ASTM standards, 158e159 material characterization, 158e159 tensile testing, 159 universal testing machine (UTM), 159 postprocessing, 156e157, 157f preprocessing, 155, 155f solving, 156 Finite element formulations constitutive relations displacement vector, 12 ﬁber orientation angle, 10 ninenoded quadrilateral isoparametric Lagrangian element, 11, 11f nodal displacement vector, 12 shape functions, 11 strain energy expression, 12 total strain energy, 10 work done, 11e12 displacement ﬁeld, 8e9 governing equation, 13 higherorder shear deformation theory (HSDT) midplane kinematics, 7e8 strainedisplacement relations, 9 Finite element model carban ﬁberreinforced plastic (CFRP) composite. See Carban ﬁberreinforced plastic (CFRP) composite continuum shell element, 122 natural ﬁberreinforced polymer composites (NFRPC). See Natural ﬁberreinforced polymer composites (NFRPC) Firstorder shear deformation kinematic theory (FSDT), 211e212 Fixed boundary conditions, 107e108, 107f Fukuda’s model, 88e89
234
Functionally graded materials (FGMs) advanced composites, 211 ﬁnite element approximations eightnoded serendipity shell element (SHELL281), 215 elemental stiffness matrix, 216 ﬁrstorder shear deformation theory, 214e215 geometrical derestriction, 214e215, 214f global stress tensor, 215 nondimensional axial stress, 220e221, 226fe227f nondimensional central deﬂections, 217t, 218fe219f strain vector expression, 215 ﬁrstorder shear deformation kinematic theory (FSDT), 211e212 higherorder shear deformation theory (HSDT), 211e212 micromechanical material modeling material proﬁles, 213f metal and ceramic materials, 214t powerlaw based Voigt’s micromechanical model, 212 volume fractions, 212 Young’s modulus, 213 G Glassreinforced plastics (GRPs) ballistic limit, 116 material properties, 118, 118t Grady failure model, 186 Grady spall model, 174 H HalpineTsai equations, 6 HalpineTsai model, 75e76 Hashin and Shtrikman model, 73, 74f Hashin’s failure criteria, 119 Healing agent, 133e134 Helmholtz free energy, 182 Hemicellulose, 3e4 Higherorder shear deformation theory (HSDT) midplane kinematics, 7e8 Hill’s anisotropy yield function, 183 Hirsch model, 76 Homogeneous stress boundary conditions, 70 conservation of energy, 69
Index
contour condition, 69 elastic material problem, 69 equivalent characteristic tensors, 70 macroscopic, 68 matrix phase, 70 microscopic stress ﬁeld, 69 potential energy, 69 representative elementary volume (REV), 70f stiffness and ﬂexibility tensors, 71 Hooke’s law, 84 Hugoniot Elastic Limit (HEL), 174 Hugoniot stress levels, 190e191 Hybrid composite materials ﬁber hybridization 1D alternating packing ﬁbers, 88f failure strain, 84 ﬁberhybrid composites, 86, 86f Fukuda’s model, 88e89 hybrid effect, 87 ineffective length, hybrid composite, 88 linear ruleofmixture, 86e87 mechanics of composites, 84 strain concentration factor, 87e88 stressestrain diagram, 87, 87f Weibull distributions, 88 Zweben model, 88 hybrid particulate ﬁberreinforced composites objectoriented modeling, 93 representative unit cell, 93 representative volume element, 89e92 Hybrid composites, 4e5 Hybrid effect, 87 Hybridization, 5 Hybrid metal/ﬁber composite cylindrical tubes ANSYS ﬁnite element program composite layer thicknesses, 30e31 contact algorithm, 28e30 energy absorption performance, 31e33 forceedisplacement curve, 31e33, 32f loading conditions, 30e31 shell element, 28e30 components, 28, 29f crushing behavior aluminum/glass ﬁberreinforced plastic hybrid tubes, 27e28 energy absorption capability, 27e28
Index
ﬁber effect, 38, 38f ﬁber orientations, 36, 36f forceedisplacement curve, 27 force ratio, 37, 37f oblique compression angles, 33e36, 33f, 36f speciﬁc energy absorption capability, 34, 34f speciﬁc energy absorption, oblique angles, 37 tube aspect ratios, 34e35, 35f mechanical properties, 29te30t structural integrity, 27 HYPERMESH, 155 I Inequality of dissipation energy, 185 Inplane shear strength, selfhealing glass ﬁberreinforced plastic (GFRP), 145f Intuitive approach, 77e78 Isoclinic conﬁguration, 176f Isotropic strain pressure, 172 J Jute/bananareinforced polymer composites, 165 K KarmaneDonnelltype strain, 211e212 KirchhoffeLove shell theory, 211e212 Kirchhoff stress tensor, 177e178 Kronecker delta function, 84 L Lame’s constants, 65 Layerbylayer hybrids, 86 Lignin, 3e4 Longitudinal modulus, 6 Loss modulus, 139e141 M Mandel stress tensor, 177 Mechanical durability of hybrid composites aerospace industries, 41 ANSYS WORKBENCH ﬁnite element program CAD modeling of tension sample, 47f carbon/epoxy, 44t composite laminates, 48f
235
ﬁnite element model, 48f numerical validation, 49f size model of specimens, 43t specimens modeling, 43f steel material, 43e44, 44t stressestrain response, 44, 49f variables involved, 45te46t aramid ﬁberreinforced composite, 41e43 continuous hybrid ﬁber, 41, 42f failure analysis, 41e43 ﬁber/metal composite technology, 41e43 ﬁber orientation effect, displacement, 54fe55f, 55 strain rate outofplane properties, 41e43 stressestrain response fourply hybrid composites, 49e52, 53f sixply hybrid composites, 54f twoply hybrid composites, 47e49, 50f tensile responses fourply hybrid composites, 49, 53f twoply hybrid composites, 50f unidirectional carbon ﬁber, 41e43 Mechanics of composites boundary element method (BEM), 85e86 bounding models. See Bounding models ﬁnite difference method (FDM), 84 ﬁnite element method (FEM), 85 homogeneous deformation displacement condition, 66 forces of volume and acceleration, 66e67 linear elasticity, 67 macroscopic constraints, 67 microscopic deformation ﬁeld, 67e68 homogeneous stress, 68e69 homogenization models Eshelby method, 77e81 MorieTanaka model, 81e83 TsaiePagano’s method, 76e77 macromechanical analysis anisotropic material, 63 composite stiffness matrix, 65 homogenization approaches, 62 Hooke’s law, 63 index symmetry, 62e63 isotropic material, 65 Lame’s constants, 65 stress and strain tensors, 62 transverse isotropic, 64 Voigt’s convention, 64
236
Mechanics of composites (Continued) micromechanical analysis constituent material interaction, 61e62 longitudinal and transverse modulus, 61e62 Poisson’s ratio and shear modulus, 61e62 representative volume element (RVE), 61 rule of mixtures approximation, 62 semiempirical models HalpineTsai model, 75e76 Hirsch model, 76 modiﬁed ruleofmixture model, 73e74 Mesh sensitivity analysis, continuum shell element, 124f Mesoscale representative volume element models, 163e164 Metal matrix composites (MMC), 58e59 Microcapsules agitation rates, 135e136 size, 138 smaller particle blocks, 136 surface, 135e136, 136f thickness, 135, 135f Micromechanical material modeling highperformance composites, 5 hybrid system properties, 5 longitudinal modulus, 6 parameters, 5 Poisson’s ratio, 7 shear moduli, 7 transverse modulus, 6 volume fraction of ﬁllers, 6 MieGr€uneisen EOS, 180e181 Modiﬁed ruleofmixture model, 73e74 MorieTanaka model, 81e83 constraint of perturbation, 81 deformation in inclusions, 81 elasticity tensor, 81, 83 homogenizing relation, 83 mean ﬁelds of deformation and stress, 81e82 stress deformation, 81 transformation deformation, 82 Multiple criteria decision making (MCDM), 199e200 Multiscale RVE model, 166
Index
N Nanoﬁllers, 2e3 Natural ﬁberbased hybrid composites ﬁnite element solutions clamped and simply supported (SCSC), 14 clamped condition (CCCC), 14 deﬂection behavior, 14, 15te16t density, 13 HSDT midplane kinematics, 13 Poisson’s ratio, 13 shear modulus, 13 simply supported condition (SSSS), 14 volume fractions, 14 Young’s modulus, 13 ﬂexural responses, 16e17 Natural ﬁberreinforced polymer composites (NFRPC) failure modeling, 167 ﬁnite element modeling analysis type, 159 boundary conditions, 161 loading, 161 material modeling, 160 material properties, 160 meshing, 160, 160f outcomes, 161 part modeling, 159 solving, 161 mechanical analysis, 165e166 thermal analysis 2D ﬁnite element model, 164e165, 164f ﬁber clustering effect, 165 jute/bananareinforced polymer composites, 165 lumen content, 165 mesoscale ﬁnite element model, 165 thermal conductivity, 164f volume element modeling and analysis, 166e167 Natural ﬁbers advantages, 3 in automobile applications, 1 chemical composition, 3e4 classiﬁcation, 2, 2f ﬁnite element modeling macromechanical analysis, 162
Index
mesoscale representative volume element models, 163e164 micromechanical analysis, 162e163 hydrophilic nature, 3 mechanical properties, 3 nanoﬁllers, 2e3 plant ﬁbers, 2 resources, 1 Ninenoded quadrilateral isoparametric Lagrangian element, 11, 11f Nodal displacement vector, 12 O Objectoriented modeling, 93 “Optimal number of elements”, 157 Organic matrix composites (OMC), 58e59 Orientation averaging approach, 166 Orthotropic yield criterion, 182e186 P Perforated web panel, 20f Plant ﬁbers, 2 Plastic deformation, 185 Poisson’s ratio, 7 Powerlaw based Voigt’s micromechanical model, 212 Pressurized vessels, 115e116 R RankineeHugoniot equations, 180e181 Rectangular web panel, 20fe21f Representative elementary volume (REV), 62, 70f Representative unit cell, 93 Representative volume element (RVE), 89e92, 161e163 macroscale, 163 mesoscale, 163e164 microﬁbril, 163 microscale, 163 Reuss model, 72e73 S Selfhealing glass ﬁberreinforced plastic (GFRP) dynamic mechanical properties damping factor, 139e141 Dynamic Mechanical Spectrometer DMS6100, 141, 141f
237
loss modulus, 139e141 storage modulus, 139e141 experimental plan, 137e138 fabrication, 136, 137f, 138t healing agent, 134e135 microcapsules agitation rates, 135e136 size, 138 smaller particle blocks, 136 surface, 135e136, 136f thickness, 135, 135f microstructural analysis capsule with polymerized dicyclopentadiene, 150f epoxy resin and capsule distribution, 147e148 inplane shear specimen, 148f sheared microcapsule, 149f selfhealing agent ColeeCole plot, 146e147, 147f compressive strength, 143, 144f damping curve, 145e146, 147f ﬂexural strength, 143 inplane shear strength, 144, 145f loss modulus, 144e145, 146f storage modulus curve, 144, 146f tensile strength, 142, 143f static mechanical properties, 139, 140t viscoelastic behavior, 150 Semiempirical models HalpineTsai model, 75e76 Hirsch model, 76 modiﬁed ruleofmixture model, 73e74 Shear moduli, 7 SHELL163 Explicit Thin Structural Element, 28e30 Shock waves modeling carbon ﬁberreinforced epoxy composites aluminum alloy (AA7010), 193, 193t material properties for, 192, 193t MieGr€ uneisen EOS parameters, 194t plate impact test conﬁguration, 192, 193f stress comparison, 194f colinearity, 171e172 commercial aluminum alloy Hugoniot stress levels, 190e191 longitudinal stress, 189e190, 189f, 191f
238
Shock waves modeling (Continued) material properties, 188t numerical simulation, 188e189 orthotropic material axes type 2 (AOPT 2), 188 plate impact test conﬁguration, 187, 187f constitutive formulation. See Constitutive formulation, shock waves modeling constitutive models, 171 double contraction tensor, 172 generalized orthotropic pressure, 173 isotropic strain pressure, 172 magnitude of pressure, 173 Storage modulus, 139e141 Storage modulus curve, 144 Stressestrain response, 47e49, 49f Surface modiﬁcation, 4e5 T Tangential behavior, 122e123 Tensile responses, hybrid composites, 44e47, 49, 50fe51f Tensile strength, selfhealing glass ﬁberreinforced plastic (GFRP), 142 Transverse modulus, 6 Transverse stiffeners, 20 TsaiePagano’s method, 76e77 Twopoint bounds, 73 U Universal testing machine (UTM), 159
Index
V Voigt model, 72 VoigteReusseHill average, 73 Voronoi ﬁber/matrix element, 165 W Web buckling strength beam design, 19 boundary conditions, 19e20 ﬁnite element analysis analytical model, 20e21, 20f convergence test, 23, 24t element selection, 22e23 material model and boundary conditions, 21e22, 22fe23f perforated web panel, 20f rectangular web panel, 20fe21f square web panel, 20f shear buckle, 19 shear buckling behavior cutout ratio d/b, 25e26 perforated square plates, 24, 25f, 25t rectangular perforated plates, 24e26, 25fe26f transverse stiffeners, 20 Weibull distributions, 88 Y Yarnbyyarn hybrids, 86 Z Zweben model, 88