Modelling of damage processes in biocomposites, fibre-reinforced composites and hybrid composites 9780081022894, 1331331331, 0081022891

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Modelling of Damage Processes in Biocomposites, Fibre-Reinforced 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 Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-08-102289-4 For information on all Woodhead publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Matthew Deans Acquisition Editor: Gwen Jones Editorial Project Manager: Thomas Van Der Ploeg Production Project Manager: Poulouse Joseph Designer: Mark Rogers Typeset by TNQ Technologies

Dedicated to Parents of Dr. Mohamed Thariq Sufaidah Binti Mohd Musa-Mother Haji Hameed Sultan Bin Mohamed Sulaiman-Father

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Contents

List of contributors About the editors Preface 1

2

3

4

Finite element modeling of natural fiber-based 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 fiber-based hybrid composites 1.5 Conclusions References The effects of cut-out on thin-walled 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 Kamarul-Azhar 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 Kamarul-Azhar 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, Mohammed-Ouadi 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 Kamarul-Azhar 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 Kamarul-Azhar Kamarudin, Mohd Khir Mohd Nor, Al Emran Ismail, Iskandar Abdul hamid and Ahmad Sufian Abdullah 7.1 Introduction 7.2 Material properties of carbon fiber-reinforced 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

8

9

10

Investigation of damage processes of a microencapsulated self-healing mechanism in glass fiber-reinforced polymers J. Lilly Mercy and S. Prakash 8.1 Introduction 8.2 Chemistry of capsule-based self-healing 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 self-healing agent on mechanical properties 8.8 Study of the effect of self-healing agent on dynamic mechanical properties 8.9 Microstructural analysis 8.10 Discussion and conclusion References Finite element analysis of natural fiber-reinforced 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 fiber-reinforced polymer composites 9.5 Finite element analysis of natural fiber and natural fiber-reinforced 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

ix

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

x

11

12

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 Mohammed-Ouadi Bensalah Rabat, Morocco Nabil Bouhfid Morocco

Mohammed V-Rabat University, Faculty of Science,

Mohammed V-Rabat 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 V-Rabat 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

Kamarul-Azhar 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 Co-Curricular, 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

xiii

Marya Raji Mohammed V-Rabat 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/fire-retardant, 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 peer-reviewed 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 high-impact international peer-reviewed 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 fiber-reinforced 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 H-index is 40 (Google Scholar) and 35 (Scopus).

xvi

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, fire-retardant materials, natural fiber-reinforced 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, fiber-reinforced 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, fiber-reinforced 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 eco-friendly 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 up-to date techniques for aerospace, automotive, and construction applications. This book covers topics such as finite element modeling of natural fiber-based 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 self-healing mechanisms in glass fiber-reinforced polymers, element analysis of natural fiber-reinforced 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 fiber-based hybrid composites

1

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, Fibre-Reinforced Composites and Hybrid Composites https://doi.org/10.1016/B978-0-08-102289-4.00001-1 Copyright © 2019 Elsevier Ltd. All rights reserved.

2

1.1.1

Modelling of Damage Processes in Biocomposites, Fibre-Reinforced Composites

Natural fibers (or cellulose-based 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 cost-effective 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, plant-based fiber-reinforced 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 water-soluble 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 fiber-based 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 fiber-reinforced 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 water-soluble 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,

4

Modelling of Damage Processes in Biocomposites, Fibre-Reinforced 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 fiber-reinforced 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 fiber-reinforced composites are

Finite element modeling of natural fiber-based hybrid composites

5

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 fiber-reinforced composite can almost emulate the properties of glass-fiber-reinforced 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. High-performance 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].

6

Modelling of Damage Processes in Biocomposites, Fibre-Reinforced 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 fiber-based 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

Fiber-reinforced hybrid composites of various geometries are designed and used as structural components in many weight-sensitive and high-performance 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), mesh-free method, etc. have been utilized in the past to evaluate the desired responses by incorporating them into real-life 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, first-order shear deformation theory, or higher-order shear deformation theory (HSDT) mid-plane 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 mid-plane kinematics and Greene Lagrange strain terms. They analyzed the deformation and frequency characteristics

8

Modelling of Damage Processes in Biocomposites, Fibre-Reinforced 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 higher-order 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 mid-plane deformation behavior of fiber-reinforced laminated composite panels under uniform pressure. So, the displacements at any point for an arbitrary laminated panel with respect to the mid-plane 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 Mid-plane

Z x, 1 N

K

2 1

X Zk-1 Zk

Figure 1.2 Geometry and stacking sequence of a hybrid composite plate.

Finite element modeling of natural fiber-based 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 mid-plane 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 higher-order terms of Taylor series expansion defined at the mid-plane 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 mid-plane 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

mid-plane strain term that is the function of the x and y coordinates and the linear thickness coordinate matrices. The superscripts 0e3 in the individual mid-plane strain terms stand for the extension, bending, curvature, and higher-order terms, respectively.

10

Modelling of Damage Processes in Biocomposites, Fibre-Reinforced 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, Fibre-Reinforced 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 mid-plane 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 fiber-based 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 fiber-based hybrid composites

In this section, the finite element solutions of natural fiber-based hybrid composite plates are examined for different materials (volume fractions), number of layers, and geometrical (side-to-thickness ratio, a/h and side-to-length ratio, a/b) and support conditions. A homemade customized code in an FEM framework via a nine-noded isoparametric element based on the HSDT mid-plane 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, single-layered hybrid composite/double-layered composite (0 /90 )/three-layered 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, Fibre-Reinforced 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 single-layered, double-layered (0 /90 ), and three-layered (0 /90 /0 ) hybrid composites. It is observed that the central deflection is enhanced with the increase in side-to-thickness 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 side-to-length 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 side-to-length 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 fiber-based hybrid composites

15

Table 1.1 Central deflection (in mm) of a hybrid composite plate under uniform pressure for different side-to-thickness ratios Side-to-thickness 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 side-to-length ratios Side-to-length 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, Fibre-Reinforced 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 fiber-based composite structures. However, the modeling and analysis of natural fiber-based hybrid composites are still unexplored. To bridge this gap, natural fiber-based hybrid composites are modeled micromechanically and solved computationally using the higher-order 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 nine-noded isoparametric Lagrangian element with eighty-one 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 side-tolength ratio, and degrades with the increase in side-to-thickness ratio.

Finite element modeling of natural fiber-based 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 fibre-reinforced 1-pentene/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.Spohr-Verlag, 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] Karger-Kocsis 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 fiber-reinforced rubber seed oil-based 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, Fibre-Reinforced Composites

[21] Sreekala MS, George J, Kumaran MG, Thomas S. The mechanical performance of hybrid phenol-formaldehyde-based 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 hygro-thermo-mechanical loading. Steel Compos Struct 2015;19(4):1011e33. [27] Mahapatra TR, Kar VR, Panda SK. Nonlinear flexural analysis of laminated composite panel under hygro-thermo-mechanical 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 higher-order equivalent single-layer theory. J Sandw Struct Mater 2017:1e24. [30] Sharma N, Mahapatra TR, Panda SK. Vibro-acoustic 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 higher-order finite-boundary 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 bio-fibre and bio-shell particle reinforced epoxy bio-composites. J Reinf Plast Comp 2015;34(13):1075e89.

The effects of cut-out on thin-walled 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 load-carrying capacity of the web reduces. The presence of shear buckle, which involves out-of-plane deflections on the web, is accompanied by changes in the stress distribution within the cross-section. In beam design, openings are frequently used to reduce the structural component weight and also to provide entrance for inspection and maintenance services. The cut-out 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 cut-out 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 thin-walled 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 cut-outs and subjected to in-plane shear. The cases considered in this study are: 1. Square plates with varying dimensions of central circular cut-outs. 2. Rectangular plates with varying aspect ratios and multiple central circular cut-outs.

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, Fibre-Reinforced Composites and Hybrid Composites https://doi.org/10.1016/B978-0-08-102289-4.00002-3 Copyright © 2019 Elsevier Ltd. All rights reserved.

20

Modelling of Damage Processes in Biocomposites, Fibre-Reinforced 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 general-purpose 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 cut-out on thin-walled 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 7075-T6. Aluminum 7075-T6 has good strength among the aluminum alloys, good durability, and low fatigue crack development. The basic properties of aluminum 7075-T6 are as tabulated in Table 2.1. The plates are simply supported on all edges with web length, a, depth, b, and cut-out diameter, d. From Fig. 2.3(a), displacement in the x-direction at the lower corner of the plate is restrained from any movement in the x-direction. Edge displacement in the y-direction at the lower edge of the plate is set to zero to avoid Table 2.1 Properties of aluminum 7075-T6 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, Fibre-Reinforced Composites

(a)

(b)

(c)

Figure 2.3 Boundary condition (BC) applied to plate models. (a) BC fixed at the x-direction. (b) BC fixed at the y-direction. (c) BC fixed at the z-direction.

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 z-direction. 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 first-order element has only a corner or end node, whereas the second-order element also includes mid-side nodes. The displacements within the element are linearly and quadratically interpolated for the firstorder and second-order elements, respectively. For thin-walled structures, the most

The effects of cut-out on thin-walled 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, Fibre-Reinforced 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 cut-out ratio d/b increases. This indicates that the strength of the plates decreases when the size of the cut-out increases. The trends of shear buckling coefficient for square perforated plates with different cut-out 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 cut-outs 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 cut-outs for plate ratios a/b of 2, 3, and 4 are 2, 3, and 4 cut-outs, respectively.

The effects of cut-out on thin-walled plates

25

Table 2.3 Shear buckling behavior of perforated square plates (a/b ¼ 1) Plate length, a (mm)

Plate width, b (mm)

Cut-out ratio of d/b

Critical shear load, scr (MPa)

Shear buckling coefficient, kv

S-1-100a

100

100

e

61.23

9.25

S-1-10d

100

100

0.10

55.86

8.44

S-1-20d

100

100

0.20

45.48

6.87

S-1-30d

100

100

0.30

35.23

5.32

S-1-40d

100

100

0.40

26.48

4.00

S-1-50d

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 cut-out ratio d/b with the shear buckling coefficient of the rectangular plates with multiple cut-outs. As shown in the figures, the shear buckling coefficients of the perforated rectangular plates are dropping as the cut-out 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 cut-outs (a/b ¼ 2).

Modelling of Damage Processes in Biocomposites, Fibre-Reinforced 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 cut-outs (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 cut-outs (a/b ¼ 4).

is a big difference in shear buckling coefficient between the nonperforated rectangular plate and the perforated rectangular plates (multiple cut-outs) as the cut-out ratio d/b increases. In addition, it is observed that the shear buckling coefficient for the perforated rectangular plates with multiple cut-outs is much lower when compared with the perforated rectangular plates with single cut-out. This indicates that the quantity of cut-out has an effect on the strength of the plates. Hence increased quantity of cut-outs 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 cut-out ratio d/b, plate ratio a/b, and the quantity of cut-outs assigned to the plate structures.

Modeling of crushing mechanisms of hybrid metal/fiber composite cylindrical tubes

3

Al Emran Ismail, Kamarul-Azhar 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 thin-walled 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 fiber-reinforced 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 quasi-static 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 fiber-reinforced 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

Modelling of Damage Processes in Biocomposites, Fibre-Reinforced Composites and Hybrid Composites https://doi.org/10.1016/B978-0-08-102289-4.00003-5 Copyright © 2019 Elsevier Ltd. All rights reserved.

28

Modelling of Damage Processes in Biocomposites, Fibre-Reinforced Composites

aluminum/carbon fiber-reinforced 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. El-Hage 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 e-glass 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 SHELL-163 Explicit Thin Structural Element and the formula is based on the Belytschko-Tsay 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 cross-sections. 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 self-contact or to contact with other bodies. Automatic surface-to-surface 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, Fibre-Reinforced Composites

Table 3.2 Mechanical properties of carbon/epoxy and e-glass/epoxy composites (Huang et al., 2012) Carbon/epoxy

E-glass/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, Fibre-Reinforced 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, Fibre-Reinforced 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 circular-shaped 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, Fibre-Reinforced 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 E-kaca/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 e-glass fiber-reinforced composites, the specific energy absorption performance is reduced. However, it is increased when carbon fiber-reinforced 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 strain-tofailure value compared with e-glass 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 fiber-reinforced 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 E-kaca/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, Fibre-Reinforced Composites

Figs. 3.15 and 3.16 show the effect of different fibers on the crushing mechanisms of carbon and glass fiber-reinforced 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 e-glass fibers and therefore the capability of carbon fiber to resist wall deformation is better than glass fiber. Comparatively, glass fiber-reinforced 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 as-constructed model, (b) side, and (c) aerial crushed hybrid composite tubes (carbon fiber-reinforced composites).

(a)

(c) (b)

Figure 3.16 Failure mechanisms of (a) an as-constructed model, (b) side, and (c) aerial crushed hybrid composite tubes (glass fiber-reinforced 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 H-W, Wan Z-M, Xie Z-M, Du X-W. 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 metal-FRP square tubes subjected to axial compressive loading. Compos Struct 2015;130:44e50. [4] Baroutajia A, Sajjiab M, Olabic A-G. On the crashworthiness performance of thin-walled energy absorbers: recent advances and future developments. Thin-Walled Struct 2017;118: 137e63. [5] Jung D-W, Kim H-J, Choi N-S. 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] El-Hage H, Mallick PK, Zamani N. A numerical study on the quasi-static 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, Kamarul-Azhar 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 strength-to-weight 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 fiber-reinforced polymeric materials [2]. Fiber/metal composite technology combines the advantages of metallic materials and fiber-reinforced 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 self-reinforced 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 fiber-reinforced composite plates. Mortazavian et al. [7] experimentally and analytically investigated the anisotropy effects on tensile properties of two short glass fiber-reinforced 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, Fibre-Reinforced Composites and Hybrid Composites https://doi.org/10.1016/B978-0-08-102289-4.00004-7 Copyright © 2019 Elsevier Ltd. All rights reserved.

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Modelling of Damage Processes in Biocomposites, Fibre-Reinforced 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 out-of-plane properties of an aramid fabric-reinforced polyamide composite have been investigated experimentally. The influence of strain rate on the tensile and compression properties of glass, carbon, and aramid-reinforced 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 fiber-reinforced 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 fiber-reinforced 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 e-glass (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

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Modelling of Damage Processes in Biocomposites, Fibre-Reinforced 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

Two-ply orientations

Four-ply orientations

Six-ply 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 Two-ply orientations

Four-ply orientations

Six-ply 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, Fibre-Reinforced 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, Fibre-Reinforced 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, Fibre-Reinforced 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 two-ply 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, Fibre-Reinforced 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 four-ply 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, Fibre-Reinforced 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 six-ply 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.

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Modelling of Damage Processes in Biocomposites, Fibre-Reinforced 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 titanium-based 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 fibre-metal 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 e-glass 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 multi-axial 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 out-of-plane 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 , Mohammed-Ouadi Bensalah 1 , Rachid Bouhfid 2 , Abou el kacem Qaiss 2 1 Mohammed V-Rabat 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 scaling-up 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, Fibre-Reinforced Composites and Hybrid Composites https://doi.org/10.1016/B978-0-08-102289-4.00005-9 Copyright © 2019 Elsevier Ltd. All rights reserved.

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Modelling of Damage Processes in Biocomposites, Fibre-Reinforced 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 modulus-to-weight 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 cross-section [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 Fiber-reinforced composite.

60

Modelling of Damage Processes in Biocomposites, Fibre-Reinforced 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, filler-reinforced 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, Fibre-Reinforced 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, Fibre-Reinforced 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, Fibre-Reinforced 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, Fibre-Reinforced 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, Fibre-Reinforced 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].

One-point bounds The one-point 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, Fibre-Reinforced 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 (iso-strain) and (b) the Reuss (iso-stress) 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.

Two-point 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 two-phase 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 rule-of-mixture Investigations show that the obtained results by the rule-of-mixture 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, Fibre-Reinforced Composites

120 Voigt-reuss-hill

Stiffness (GPa)

100 80

Hashin-shtrikman (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 rule-of-mixtures 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.

76

Modelling of Damage Processes in Biocomposites, Fibre-Reinforced 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, Fibre-Reinforced 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, Fibre-Reinforced 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 Tanaka-solving 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:

82

Modelling of Damage Processes in Biocomposites, Fibre-Reinforced 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, Fibre-Reinforced 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 problem-solving 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 user-friendly 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 object-oriented 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, Fibre-Reinforced 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]. Fiber-hybrid 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 fiber-hybrid composites can be defined, see Fig. 5.12: • • •

Interlayer: also called layer-by-layer hybrids having different types of fibers in different layers, where each layer only has a single fiber type [126]. Intralayer: also called yarn-by-yarn hybrids having both types of fibers in a single layer; the layers can be stacked in different configurations [126]. Intrayarn: also called fiber-by-fiber 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 fiber-hybrid 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 rule-of-mixture, 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.

88

Modelling of Damage Processes in Biocomposites, Fibre-Reinforced 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 cross-sectional 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 fiber-reinforced composites

Hybrid particulate matrix fiber-reinforced 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 fiber-reinforced 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, Fibre-Reinforced 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 process-induced 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 all-carbon fiber composite [132]. HE, high elongation; LE, low elongation.

91

92

Modelling of Damage Processes in Biocomposites, Fibre-Reinforced 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 “micro-meso-macro principle” defined by Hashin [143]. The RVE is defined as the minimum volume of a laboratory-scale specimen, such that the results obtained from this specimen can still be regarded as representative of a continuum (Van Mier) [144]. Ostoja-Starzewski 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 Ostoja-Starzewski’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 cross-section 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

Object-oriented modeling

Object-oriented 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).

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[142] Drugan WJ, Willis JR. A micromechanics-based 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 three-dimensional textile composites: a mesh objective multi-scale 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

Kamarul-Azhar 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

Fiber-reinforced 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 high-velocity 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 hemispherical-shaped 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 high-velocity impact with existing pretension on structures. Garcia

Modelling of Damage Processes in Biocomposites, Fibre-Reinforced Composites and Hybrid Composites https://doi.org/10.1016/B978-0-08-102289-4.00006-0 Copyright © 2019 Elsevier Ltd. All rights reserved.

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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 three-dimensional 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

Computationally efficient modeling of woven composites under uniaxial stress

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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 fiber-reinforced 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, Fibre-Reinforced 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 z-direction was allowed and there was no rotation during impact.

Computationally efficient modeling of woven composites under uniaxial stress

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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, Fibre-Reinforced 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 z-direction. 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.E-3)

0.10

0.15(x1.E-3)

(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 hemispherical-shaped 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 fiber-reinforced 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.

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Modelling of Damage Processes in Biocomposites, Fibre-Reinforced 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 flat-nosed 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 flat-shaped projectile damage of shear but preferably tensile failure using a hemispherical-shaped nose projectile shown in an

Computationally efficient modeling of woven composites under uniaxial stress

111

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 fiber-reinforced plastic target using a flat-nosed 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 flat-nosed 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 fiber-reinforced 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.

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Modelling of Damage Processes in Biocomposites, Fibre-Reinforced 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 fiber-reinforced 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 hemispherical-shaped 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,

Computationally efficient modeling of woven composites under uniaxial stress

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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 S2-glass fiber/toughened epoxy composite beams. Compos Sci Technol 2009;69:965e82. [3] Garcia-Avila M, Portanova M, Rabiei A. Ballistic performance of composite metal foams. Compos Struct 2015;125:202e11. [4] Garcia-Castillo S, Sanchez-Saez S, Lopez-Puente 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/10.1016/0266-3538(95) 00141-7. [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 3-D orthogonal woven composite. Compos Part B Eng 2009;40:552e60. [11] Choi HY, Chang F-K. A model for predicting damage in graphite/epoxy laminated composites resulting from low-velocity 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 low-velocity impact on composite laminates. Compos Struct 2014;108:161e71. [17] Lapczyk I, Hurtado JA. Progressive damage modeling in fiber-reinforced materials. Compos Part A Appl Sci Manuf 2007;38:2333e41. [18] Kamarudin KA, Ismail AE. Modelling high velocity impact on aluminium alloy 7075-T6 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

Kamarul-Azhar 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 stiffness-to-weight 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 nose-shaped projectiles. Besides having signs of local displacement, the target impacted by a flat-nosed 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, Fibre-Reinforced Composites and Hybrid Composites https://doi.org/10.1016/B978-0-08-102289-4.00007-2 Copyright © 2019 Elsevier Ltd. All rights reserved.

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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 gas-pressurized 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 low-velocity 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 in-plane load, especially those related to high-velocity impact. Ballistic limit prediction was mostly determined in high-velocity impact studies. In 2006, together with nonpretension targets, Garcia et al. [12] also investigated the ballistic limit of glass-reinforced 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 high-velocity 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 fiber-reinforced 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 low-velocity 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 composite-reinforced 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, Fibre-Reinforced Composites

Material properties of carbon fiber-reinforced 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 fiber-reinforced 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 four-nodes 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, Fibre-Reinforced 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, Fibre-Reinforced 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 z-direction 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 24-ply composite laminate carbon fiber-reinforced 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 hemispherical-shaped 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, Fibre-Reinforced 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 fiber-reinforced 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 fiber-reinforced 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 fiber-reinforced plastic target impacted by hemispherical projectiles.

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Modelling of Damage Processes in Biocomposites, Fibre-Reinforced 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 hemispherical-shaped 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 hemispherical-nosed 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 fiber-reinforced 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, Fibre-Reinforced 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 fiber-reinforced 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 fiber-reinforced 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 high-speed 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, Fibre-Reinforced 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 fiber-reinforced 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 fiber-reinforced 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 hemispherical-shaped 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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Modelling of Damage Processes in Biocomposites, Fibre-Reinforced Composites

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 7075-T6 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 2024-T3 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 3-D 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, Toso-PentecOte 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 low-velocity 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] García-Castillo S, Sanchez-Saez S, Barbero E, Navarro C. Response of pre-loaded laminate composite plates subject to high velocity impact. J Phys IV 2006;134:1257e63. [13] Garcia-Castillo S, Sanchez-Saez S, Lopez-Puente 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] Garcia-Castillo S, Navarro C, Barbero E. Damage in preloaded glass/vinylester composite panels subjected to high-velocity 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 fiber-reinforced 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 low-velocity impact on composite laminates. Compos Struct 2014;108:161e71. ˇ

Investigation of damage processes of a microencapsulated self-healing mechanism in glass fiber-reinforced 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, fiber-reinforced 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 out-of-plane 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 “damage-tolerant” materials for a stipulated service life. As an alternative approach to conventional repair, self-healing 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 self-healing, three basic methods were identified: (1) capsule based, (2) vascular, and (3) intrinsic self-healing [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 capsule-based self-healing materials

In capsule-based self-healing, 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, Fibre-Reinforced Composites and Hybrid Composites https://doi.org/10.1016/B978-0-08-102289-4.00008-4 Copyright © 2019 Elsevier Ltd. All rights reserved.

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Modelling of Damage Processes in Biocomposites, Fibre-Reinforced Composites

Catalyst Microcapsule Crack

Healing agent

Polymerized healing agent

Figure 8.1 Mechanism of self-healing (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 self-healing in a capsule-based healing system. The capsule-based self-healing process is the forerunner, which is also simple in design, fabrication, and application. The first capsule-based self-healing 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 self-healing in a capsule-based 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 real-time usage, there were other allied factors such as basic mechanical properties, the response of the self-healing system during the manufacturing cycle, etc. To get an in-depth understanding of the basic static and dynamic mechanical properties of a self-healing composite, tests were conducted according to ASTM standards. The makeup of the self-healing glass fiber-reinforced plastic (GFRP) was decided by the percentage of healing

Investigation of damage processes of a microencapsulated self-healing 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 spherical-sized 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.

136

Modelling of Damage Processes in Biocomposites, Fibre-Reinforced 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 self-healing GFRP

Self-healing GFRP plates were fabricated through layer-by-layer stacking of glass fiber mats impregnated with epoxy resin, microcapsules, and Grubb’s catalyst. GFRP with a self-healing system comprised DCPD-filled 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 self-healing GFRP.

Investigation of damage processes of a microencapsulated self-healing mechanism in glass

(a)

(c)

137

(b)

(d)

Figure 8.4 Fabrication of self-healing GFRP plates: (a) microcapsules and Grubbs catalyst mixed in epoxy resin; (b) hand lay-up process; (c) resin mixed with capsules poured on the fiber mat; (d) self-healing GFRP panel.

8.5

Experimental plan

As the focus of this work is on finding out the effect of adding self-healing 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 Sigma-Aldrich 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, Fibre-Reinforced 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 self-healing mechanism in glass

8.6.1

139

Static mechanical properties

The common static mechanical properties of a fiber-reinforced 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 (three-point bend test)dASTM: D790 In-plane shear test (double-notched 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) in-plane 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)

In-plane 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, Fibre-Reinforced Composites

Input parameters

Investigation of damage processes of a microencapsulated self-healing 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 D-4065-01 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 self-healing 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, Fibre-Reinforced Composites

Figure 8.7 Specimen size 20 mm  10 mm subjected to ASTM-D-4065-01-dual cantilever bending.

8.7 8.7.1

Study of the effect of a self-healing 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 self-healing 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 self-healing GFRP. This could be because the compressive load is mainly borne by the glass reinforcement in the self-healing GFRP material and hence is independent of the self-healing 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 self-healing GFRP was purely yielded due to fiber breakage. However, the compressive strength of the self-healing 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 self-healing 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, Fibre-Reinforced 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 in-plane shear strength

The in-plane 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 in-plane 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 self-healing 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 self-healing 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 high-temperature viscous region [13]. The self-healing 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 self-healing mechanism in glass

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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 in-plane shear strength.

146

Modelling of Damage Processes in Biocomposites, Fibre-Reinforced 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 self-healing 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 self-healing 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 self-healing 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 Cole-cole 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, Fibre-Reinforced 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 in-plane 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 in-plane shear specimen.

Investigation of damage processes of a microencapsulated self-healing 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.

150

Modelling of Damage Processes in Biocomposites, Fibre-Reinforced Composites

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 self-healing GFRP composite.

References [1] Blaiszik BJ, Kramer SLB, Olugebefola SC, Moore JS, Sottos NR, White SR. Self-healing 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. In-situ poly (urea formaldehyde) microencapsulation of dicyclopentadiene. J Microencapsul 2003;20(No. 6):719e30. [4] Brown EN, Sottos NR, White SR. Fracture testing of a self-healing 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 self-healing 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 fiber-reinforced 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 load-bearing 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 thermoset-based 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 fiber-reinforced composites. Synthetic fiberreinforced composites incorporate carbon fiber-reinforced plastic, glass fiberreinforced plastic, Kevlar fiber-reinforced plastic, and hybrid fiber-reinforced plastic. Natural fiber-reinforced 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 nonload-bearing 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, Fibre-Reinforced Composites and Hybrid Composites https://doi.org/10.1016/B978-0-08-102289-4.00009-6 Copyright © 2019 Elsevier Ltd. All rights reserved.

154

Modelling of Damage Processes in Biocomposites, Fibre-Reinforced Composites

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 fiber-reinforced 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.

Finite element analysis of natural fiber-reinforced polymer composites

155

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, LS-PrePost, 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 real-time 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, LS-DYNA, 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 higher-order 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 fiber-reinforced polymer composites 1

157

ANSYS

NODAL SOLUTION

R16.0

STEP=1 SUB=1 TIME-1 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, LS-PrePost, 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 well-known 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, Fibre-Reinforced 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 in-plane 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 three-point or four-point 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 fiber-reinforced 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 step-by-step 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 coupled-field 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.

160

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Modelling of Damage Processes in Biocomposites, Fibre-Reinforced 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, in-plane 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 X-axis 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 fiber-reinforced 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 fiber-reinforced polymer composites

Natural fiber-based polymer composites are the most promising alternative to synthetic fiber-reinforced polymer composites. An increase in awareness of eco-friendly 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 homogenization-based 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 cross-section 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 X-ray 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 strain-hardening 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 strain-induced 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 cross-section of the natural fiber and the tensile properties. They modeled and evaluated the tensile behavior of hemp fiber with different elliptical cross-sections. 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 cross-section gave more accurate results than a circular cross-section [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 draw-in 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 fiber-reinforced polymer composites

9.5.2.1

Thermal analysis of natural fiber-reinforced polymer composites

Natural fiber-based polymer composites are widely used in many industries due to their light weight, eco-friendliness, 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 fiber-based polymer composites. This section will focus on finite element modeling and analysis of complex structures of natural fiber-based 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 fiber-reinforced polymer composite.

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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 energy-efficient 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 so-called 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 n-sided 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 fiber-reinforced polymer composite

Micromechanics models for stiffness prediction The following models were used to evaluate the stiffness of the short natural fiber composites [24]: • • • • • •

MorieTanaka-type model. HalpineTsai equation and its extensions. Dilute model. Self-consistent model. Bounding model. Shear lag model.

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Assumptions and limitations •

•

The foregoing models are based on the assumptions that the cross-section of the fiber is either cylindrical or elliptical. However, the real cross-section 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 fiber-reinforced 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 fiber-reinforced 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 time-consuming finite element model, the so-called 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 fiber-reinforced 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

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shell element (four-noded multilayer). The damping coefficients were evaluated in longitudinal, transverse, and in-plane 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 fiber-reinforced polymer composites

The general failure mechanisms of fiber-reinforced 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 fiber-reinforced 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 fiber-based 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.

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• • • • • • • •

Modelling of Damage Processes in Biocomposites, Fibre-Reinforced 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 homogenization-based 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 fiber-reinforced 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 carbon-fabric-reinforced epoxy composites. J Eng Fabr Fibers 2015;10(1). [3] Standard A. D3039M-08. Standard test method for tensile properties of polymer matrix composite materials. Annu Book ASTM Stand 2008:1e13. [4] Standard A. D6856-03. Standard guide for testing fabric-reinforced textile composite materials. 2003.

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[5] Wang H, Qin Q-H, Xiao Y. Special n-sided Voronoi fiber/matrix elements for clustering thermal effect in natural-hemp-fiber-filled 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, Guicheret-Retel 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, Guicheret-Retel 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 non-linearity 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 G-Y. 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 X-J, Wang J-S. Interaction analysis of multiple coated fibers in cement composites by special n-sided interphase/fiber elements. Compos Sci Technol 2015;118: 117e26. [23] Wang H, Lei Y-P, Wang J-S, Qin Q-H, 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 short-fiber 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 fiber-reinforced 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 multi-scale 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 J-M. 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 Co-Curricular, 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, Fibre-Reinforced Composites and Hybrid Composites https://doi.org/10.1016/B978-0-08-102289-4.00010-2 Copyright © 2019 Elsevier Ltd. All rights reserved.

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Modelling of Damage Processes in Biocomposites, Fibre-Reinforced 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 fiber-reinforced 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 plane-stress 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 (A6XXX-T4) and Al-killed cold-rolled steel sheet SPCE [4].

10.2

Constitutive formulation

The use of composite materials in aerospace structures is on the increase mainly due to their strength-to-weight 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.

174

Modelling of Damage Processes in Biocomposites, Fibre-Reinforced Composites

In the literature, many researchers have studied the response of orthotropic materials at quasi-static 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 quasi-statically measured properties at different heattreated states. It can be observed that topics related to shock wave propagation in anisotropic materials in the isotropic solid-state physics and mechanics literature has received considerable attention [27e33]. The shock response of the aluminum alloy 7010-T6 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 AA6061-T6 may vary in accordance with the one-dimensional 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 AA6061-T6 [35]. In addition, it is also reported the spall strength of AA2024-T86 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 energy-based 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 plane-stress 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 three-dimensional 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 plane-stress 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

176

Modelling of Damage Processes in Biocomposites, Fibre-Reinforced 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 F
1 p . Subsequently, the pull-back 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 F
1 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)

178

Modelling of Damage Processes in Biocomposites, Fibre-Reinforced 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 $s$F
T 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 skew-symmetric part of the Mandel stress tensor Sa [55]. Numerous efforts are still required to establish the elastic domain and yield functions for the skew-symmetric 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 three-dimensional 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.

180

Modelling of Damage Processes in Biocomposites, Fibre-Reinforced 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 Mie-Gr€ 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 Mie-Gr€ 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 (U-shock 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 Mie-Gr€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.

182

Modelling of Damage Processes in Biocomposites, Fibre-Reinforced 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 fourth-order 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 strain-like internal variable for isotropic

184

Modelling of Damage Processes in Biocomposites, Fibre-Reinforced 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 F
T p $C$Fp
T _Ce ¼ F_ $C$F
1 þ F
T $C$F _
1 þ F
T $C$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

186

Modelling of Damage Processes in Biocomposites, Fibre-Reinforced 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 shock-loaded 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 three-dimensional 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 LLNL-DYNA3D 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 cm
g
ms 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 cross-section 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 one-dimensional 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 ms
1, 450 ms
1, and 895 ms
1. The material properties of each parts are shown in Table 10.1. The flyer is set as Aluminum 6082
T6. 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, Fibre-Reinforced 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 a
direction

sy

564 MPa

250 MPa

70 MPa

Tangent plastic modulus in a
direction

H

0.13 GPa

130 MPa

300 MPa

Pressure cutoff Density Hill’s parameters

pcut r R P Qbc Qba Qca


2.81 gcm
3 1 0.719 1 1 1

2.5 GPa 2.7 gcm
3







1.18 gcm
3






Gr€ uneisen parameters

C s1 s2 s3 G a

5200 ms
1 1.36 0.00 0.00 2.2 0.48

5240 ms
1 1.4 0.00 0.00 1.97 0.48

2180 ms
1 2.088
1.124 0.00 0.85 0.00

Grady parameters

r c0 εfail Kc B

2.81 gcm
3 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 ms
1 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 ms
1 impact.

test at 234 ms
1. 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 ms
1 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

190

Modelling of Damage Processes in Biocomposites, Fibre-Reinforced 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 ms
1 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 ms
1 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 ms
1 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 ms
1 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 ms
1 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 ms
1 impact.

velocity, smaller pull-back 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, Fibre-Reinforced 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 ms
1 (Transverse): Simulation Experiment

0.40 0.33

0.64 0.63

1.18 1.44

450 ms
1 (Longitudinal): Simulation Experiment

0.41 0.43

1.32 1.31

1.13 1.16

450 ms
1 (Transverse): Simulation Experiment

0.38 0.39

1.28 1.38

1.08 1.13

895 ms
1 (Longitudinal): Simulation Experiment

0.36 0.21

2.95 3.25

1.07 1.10

895 ms
1 (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 fiber-reinforced 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 through-thickness 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 quasi-orthotropic material in this analysis. Table 10.3 shows the woven composite plate properties. The properties are characterized from the layer macromechanical properties for the lay-up [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 ms
1. 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 surface-to-surface 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 kgm
3

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 Isotropic-Elastic-PlasticHydrodynamic material model named Material Type 10 (Mat10). The elastic material properties and the Mie-Gr€ 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 kgm
3 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

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Modelling of Damage Processes in Biocomposites, Fibre-Reinforced Composites

Table 10.5 Mie-Gr€uneisen EOS parameters of aluminum and composite Material Parameters Gr€ uneisen

Aluminum

Composite


1

3230 ms
1 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

2.5E-02

Z-stress (x 102 GPa)

2.0E-02 1.5E-02 1.0E-02

Front gauge mat92

5.0E-03 0.0E+00 0.5

1.0

1.5 2.0 Time (µs)

2.5

3.0

3.5

–5.0E-03

Figure 10.11 Stress comparison between the front gauge and Mat92.

Z-stress (x 102 GPa)

2.0E-02

1.5E-02

1.0E-02 Back gauge 5.0E-03

0.0E+00 2.0 –5.0E-03

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 hyperelastic-plastic 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 fiber-reinforced 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.

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Modelling of Damage Processes in Biocomposites, Fibre-Reinforced Composites

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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. Shock-wave compression of quartz. J Appl Phys 1962;33:922e37. [28] Zel’dovich YB, Raizer YP. Physics of shock waves and high-temperature 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. High-pressure 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 6061-T6 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 elasto-plasticity 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. Finite-strain 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.

198

Modelling of Damage Processes in Biocomposites, Fibre-Reinforced Composites

[50] Mohd Nor MK. Unique rotation tensor formulation to predict three-dimensional 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, crystal-plasticity and phase transitions. Springer; 2014. [53] Mandel J. Plasticité classiqueet viscoplastié. CISM lecture Notes. Wien: Springer-Verlag; 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 elasto-plasticity. Computational plasticity. Computational methods in applied sciences. Dordrecht: Springer; 2007. p. 13e36. [56] Vladimirov IN, Pietryga MP, Reese S. On the modelling of non-linear 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. UCRL-MA-106439. [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 strain-based constitutive formulation in LLNL-DYNA3D 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 fibre-epoxy composite. Comput Sci Technol 2007;67: 3253e60. [63] Vignjevic R, Millett JCF, Bourne NK, Meziere Y, Lukyanov A. The behaviour of a carbon-fibre 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 problem-solving 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 real-word situation, due to incomplete or non-obtainable 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, Fibre-Reinforced Composites and Hybrid Composites https://doi.org/10.1016/B978-0-08-102289-4.00011-4 Copyright © 2019 Elsevier Ltd. All rights reserved.

200

Modelling of Damage Processes in Biocomposites, Fibre-Reinforced 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 2-opt algorithm together with DEA in an iterative process to find the most efficient design, such that the 2-opt 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 Multi-Criteria 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 best-known 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.

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Modelling of Damage Processes in Biocomposites, Fibre-Reinforced 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 n-dimensional 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 negative-ideal 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 decision-making method. All the composite laminates are compared based on the TOPSIS method and ranking has been done. The decision matrix, normalization matrix, weight-normalized 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, Fibre-Reinforced 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, Fibre-Reinforced 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, Fibre-Reinforced 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 decision-making problems with fuzzy data. App Math Comput 2006;181:1544e51. [2] Athawale VM, Chakraborty S. Material selection using multi-criteria decision-making methods: a comparative study. Proc IMechE Part L J Mater Des App 2012;226:266e75. [3] Jiang J, Ying-Wu C, Da-Wei T, Yu-Wang 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 multi-criteria 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 multiple-criteria 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 weight-sensitive 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 higher-order 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 function-based 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. Hosseini-Hashemi 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 shear-deformation kinematics to examine the deflection and frequency responses of FGM

Modelling of Damage Processes in Biocomposites, Fibre-Reinforced Composites and Hybrid Composites https://doi.org/10.1016/B978-0-08-102289-4.00012-6 Copyright © 2019 Elsevier Ltd. All rights reserved.

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Modelling of Damage Processes in Biocomposites, Fibre-Reinforced 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 close-form solution to illustrate the bending responses of FGM flat/curved panels using the HSDT-based 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 higher-order 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 first-order shear deformation kinematic theory (FSDT). Haddadpour et al. [14] utilized KirchhoffeLove shell theory via the von KarmaneDonnell-type 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 kp-Ritz 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 eight-noded 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 power-law 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 power-law 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 power-law 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, Fibre-Reinforced 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 first-order 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 eight-noded 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, Fibre-Reinforced 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 eight-noded 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 power-law 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 power-law 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 power-law index values and reduces with increase in thickness ratios.

218

Modelling of Damage Processes in Biocomposites, Fibre-Reinforced 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 power-law 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 metal-rich surface of the FGM shell panel is under compression, as the top ceramic-rich surface is in tension. And the increase in power-law 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, Fibre-Reinforced 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, Fibre-Reinforced 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 power-law 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 power-law 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 power-law 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 power-law indices.

224

Modelling of Damage Processes in Biocomposites, Fibre-Reinforced 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, Fibre-Reinforced 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, Fibre-Reinforced 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 power-law-based 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 power-law index because stiffness of the FGM shell panel degrades with the increment in power-law indices. The nondimensional central deflection increases with increase in curvature ratio and reduces with increase in thickness ratio. The bottom metal-rich surface of the FGM shell panel is always under compression and the top ceramic-rich surface is in tension for all cases considered. And the increase in power-law 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 shells-Theory 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 third-order 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. Three-dimensional vibration analysis of functionally graded plates under various boundary conditions. J Reinf Plastic Compos 2007;26(18):1847e63. [7] Hosseini-Hashemi S, Rokni Damavandi Taher H, Akhavan H, Omidi M. Free vibration of functionally graded rectangular plates using first-order 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 quasi-3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates. Compos Struct 2012;94:1814e25. [9] Thai H-T, Kim S-E. A simple higher-order 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 two-dimensional 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 higher-order 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. Thin-walled Struct 2007;45:591e9. [15] Santos H, Soares CMM, Mota Soares CA, Reddy JN. A semi-analytical 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 fiber-reinforced 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 fiber-reinforced composite, 41e43 Automobile engineering, 1 Automobile industry, 1 B Bamboo fiber macromechanical analysis, 162 micromechanical analysis, 162e163 Belytschko-Tsay formulation, 28e30 Boundary element method (BEM), 85e86 Bounding models one-point bound models classical bounds, 71e72 Reuss model, 72e73 Voigt model, 72 VoigteReusseHill average, 73 two-point bounds, 73 C Capsule-based self-healing healing agent, 133e134 mechanism, 134f microcapsule fabrication, 135e136 Carban fiber-reinforced 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 fiber-reinforced epoxy composites, shock waves modeling aluminum alloy (AA7010), 193, 193t material properties for, 192, 193t

232

Carbon fiber-reinforced epoxy composites, shock waves modeling (Continued) Mie-Gr€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 problem-solving 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, self-healing glass fiber-reinforced plastic (GFRP), 143 Constitutive formulation, shock waves modeling aluminum alloy 7010-T6, 174

Index

anisotropic homogeneous yield function, 175 compressive input stress, 174 dynamic loading, 174 elastic free energy function, 182 energy-based failure model, 174 equation of state (EOS) Mie-Gr€ 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 plastic-related deformations, 176e177 pull-back 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 AA6061-T6, 174 strength-to-weight 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 Eight-noded 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 Energy-based 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 Fiber-by-fiber hybrids, 86 Fiber clustering effect, 165 Fiber-hybrid composites, 86f Fiber/metal composite technology, 41e43 Fiber-reinforced plastics (FRPs), 133 Fiber-reinforced 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 fiber-reinforced 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 nine-noded 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 higher-order shear deformation theory (HSDT) mid-plane kinematics, 7e8 strainedisplacement relations, 9 Finite element model carban fiber-reinforced plastic (CFRP) composite. See Carban fiberreinforced plastic (CFRP) composite continuum shell element, 122 natural fiber-reinforced polymer composites (NFRPC). See Natural fiber-reinforced polymer composites (NFRPC) First-order 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 eight-noded serendipity shell element (SHELL281), 215 elemental stiffness matrix, 216 first-order 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 first-order shear deformation kinematic theory (FSDT), 211e212 higher-order shear deformation theory (HSDT), 211e212 micromechanical material modeling material profiles, 213f metal and ceramic materials, 214t power-law based Voigt’s micromechanical model, 212 volume fractions, 212 Young’s modulus, 213 G Glass-reinforced 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 Higher-order shear deformation theory (HSDT) mid-plane 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 fiber-hybrid composites, 86, 86f Fukuda’s model, 88e89 hybrid effect, 87 ineffective length, hybrid composite, 88 linear rule-of-mixture, 86e87 mechanics of composites, 84 strain concentration factor, 87e88 stressestrain diagram, 87, 87f Weibull distributions, 88 Zweben model, 88 hybrid particulate fiber-reinforced composites object-oriented 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 fiber-reinforced 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 In-plane shear strength, self-healing glass fiber-reinforced plastic (GFRP), 145f Intuitive approach, 77e78 Isoclinic configuration, 176f Isotropic strain pressure, 172 J Jute/banana-reinforced polymer composites, 165 K KarmaneDonnell-type strain, 211e212 KirchhoffeLove shell theory, 211e212 Kirchhoff stress tensor, 177e178 Kronecker delta function, 84 L Lame’s constants, 65 Layer-by-layer 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 fiber-reinforced composite, 41e43 continuous hybrid fiber, 41, 42f failure analysis, 41e43 fiber/metal composite technology, 41e43 fiber orientation effect, displacement, 54fe55f, 55 strain rate out-of-plane properties, 41e43 stressestrain response four-ply hybrid composites, 49e52, 53f six-ply hybrid composites, 54f two-ply hybrid composites, 47e49, 50f tensile responses four-ply hybrid composites, 49, 53f two-ply 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 rule-of-mixture 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 high-performance 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 Mie-Gr€uneisen EOS, 180e181 Modified rule-of-mixture 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 fiber-based hybrid composites finite element solutions clamped and simply supported (SCSC), 14 clamped condition (CCCC), 14 deflection behavior, 14, 15te16t density, 13 HSDT mid-plane 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 fiber-reinforced 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/banana-reinforced 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 Nine-noded quadrilateral isoparametric Lagrangian element, 11, 11f Nodal displacement vector, 12 O Object-oriented 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 Power-law 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 Self-healing glass fiber-reinforced 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 in-plane shear specimen, 148f sheared microcapsule, 149f self-healing agent ColeeCole plot, 146e147, 147f compressive strength, 143, 144f damping curve, 145e146, 147f flexural strength, 143 in-plane 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 rule-of-mixture model, 73e74 Shear moduli, 7 SHELL-163 Explicit Thin Structural Element, 28e30 Shock waves modeling carbon fiber-reinforced epoxy composites aluminum alloy (AA7010), 193, 193t material properties for, 192, 193t Mie-Gr€ 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, self-healing glass fiberreinforced plastic (GFRP), 142 Transverse modulus, 6 Transverse stiffeners, 20 TsaiePagano’s method, 76e77 Two-point 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 cut-out ratio d/b, 25e26 perforated square plates, 24, 25f, 25t rectangular perforated plates, 24e26, 25fe26f transverse stiffeners, 20 Weibull distributions, 88 Y Yarn-by-yarn hybrids, 86 Z Zweben model, 88