Dynamic Deformation, Damage and Fracture in Composite Materials and Structures [2 ed.] 0128239794, 9780128239797

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

Woodhead Publishing Series in Composites Science and Engineering

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures Second Edition

Edited by

Vadim Silberschmidt Professor, Chair of Mechanics of Materials, Loughborough University, Loughborough, United Kingdom

An imprint of Elsevier

Woodhead Publishing is an imprint of Elsevier 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, OX5 1GB, United Kingdom Copyright © 2023 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. ISBN: 978-0-12-823979-7 (print) ISBN: 978-0-12-823980-3 (online) For information on all Woodhead publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Matthew Deans Acquisitions Editor: Dennis McGonagle Editorial Project Manager: Zsereena Rose Mampusti Production Project Manager: Surya Narayanan Jayachandran Cover Designer: Mark Rogers Typeset by STRAIVE, India

Contributors

M. Ahmadi Amirkabir University of Technology, Tehran, Iran Yehia Bahei-El-Din Centre for Advanced Materials, The British University in Egypt, El-Shorouk City, Egypt A. Banerjee University of Auckland, Auckland, New Zealand Konstantinos P. Baxevanakis Wolfson School of Mechanical, Electrical and Manufacturing Engineering, Loughborough University, Loughborough, Leicestershire, United Kingdom Christophe Bouvet Institut Clement Ader, Universite de Toulouse, CNRS UMR 5312—INSA—ISAE-SUPAERO—IMT Mines Albi—UPS, Toulouse, France  J. Brechou Ecole nationale superieure des ingenieurs en arts chimiques et technologiques (ENSIACET), Toulouse, France Juan Camilo Calle Universidad de los Andes, Bogota, Colombia A. Chanda University of Auckland, Auckland, New Zealand Tianyu Chen Department of Aeronautical and Automotive Engineering, Loughborough University, Loughborough, Leicestershire, United Kingdom; Key Laboratory of Microgravity (National Microgravity Laboratory), Institute of Mechanics, Chinese Academy of Sciences, Beijing, China Zhong Chen School of Materials Science & Engineering, Nanyang Technological University, Singapore, Singapore J.D. Coe Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, United States Laurence A. Coles Wolfson School of Mechanical, Electrical and Manufacturing Engineering, Loughborough University, Loughborough, Leicestershire, United Kingdom R. Das University of Auckland, Auckland, New Zealand

xii

Contributors

D.M. Dattelbaum Shock and Detonation Physics, Los Alamos National Laboratory, Los Alamos, NM, United States M. Fotouhi Delft University of Technology, Delft, The Netherlands Christopher M. Harvey Department of Aeronautical and Automotive Engineering, Loughborough University, Loughborough, Leicestershire, United Kingdom; School of Mechanical and Equipment Engineering, Hebei University of Engineering, Handan, China Paul J. Hazell School of Engineering and Information Technology, The University of New South Wales, Canberra, ACT, Australia Md Niamul Islam Wolfson School of Mechanical, Electrical and Manufacturing Engineering, Loughborough University, Loughborough, Leicestershire, United Kingdom Young W. Kwon Department of Mechanical & Aerospace Engineering, Naval Postgraduate School, Monterey, CA, United States Genevieve S. Langdon Blast Impact and Survivability Research Unit (BISRU), Department of Mechanical Engineering, University of Cape Town, Cape Town, South Africa; Department of Civil and Structural Engineering, University of Sheffield, Sheffield, United Kingdom Alejandro Maranon Universidad de los Andes, Bogota, Colombia G. Minak Alma Mater Studiorum – University of Bologna, Forlı`, Italy N.K. Naik Aerospace Engineering Department, Indian Institute of Technology Bombay, Mumbai, India Keiji Ogi Graduate School of Science and Engineering, Ehime University, Matsuyama, Ehime, Japan Juan Pablo Casas-Rodriguez Universidad de los Andes, Bogota, Colombia R. Panciroli Niccolo Cusano University, Rome, Italy Vaibhav A. Phadnis 3M UK Plc, Bracknell, United Kingdom Christophe Pinna Department of Mechanical Engineering, The University of Sheffield, Sheffield, United Kingdom Samuel Rivallant Institut Clement Ader, Universite de Toulouse, CNRS UMR 5312—INSA—ISAE-SUPAERO—IMT Mines Albi—UPS, Toulouse, France

Contributors

xiii

Vicent Robinson Universidad de los Andes, Bogota, Colombia Anish Roy Wolfson School of Mechanical, Electrical and Manufacturing Engineering, Loughborough University, Loughborough, Leicestershire, United Kingdom Serag Salem Centre for Advanced Materials, The British University in Egypt, ElShorouk City, Egypt Andreas Schiffer Department of Aeronautics, Imperial College London, London, United Kingdom Mostafa Shazly Centre for Advanced Materials, The British University in Egypt, ElShorouk City, Egypt Yu Shi Physical, Mathematical and Engineering Sciences, University of Chester, Chester, United Kingdom Vadim V. Silberschmidt Wolfson School of Mechanical, Electrical and Manufacturing Engineering, Loughborough University, Loughborough, Leicestershire, United Kingdom Gregory M. Sinclair Blast Impact and Survivability Research Unit (BISRU), Department of Mechanical Engineering, University of Cape Town, Cape Town, South Africa Constantinos Soutis Aerospace Research Institute, The University of Manchester, Manchester, United Kingdom Vito L. Tagarielli Department of Aeronautics, Imperial College London, London, United Kingdom Himayat Ullah Wolfson School of Mechanical, Electrical and Manufacturing Engineering, Loughborough University, Loughborough, Leicestershire, United Kingdom Chris J. von Klemperer Department of Mechanical Engineering, University of Cape Town, Cape Town, South Africa Chen Wang Polymer Technology Group, Singapore Institute of Manufacturing Technology, Singapore, Singapore Hongxu Wang School of Engineering and Information Technology, The University of New South Wales, Canberra, ACT, Australia

xiv

Contributors

Simon Wang Department of Aeronautical and Automotive Engineering, Loughborough University, Loughborough, Leicestershire, United Kingdom; School of Mechanical and Equipment Engineering, Hebei University of Engineering, Handan, China Shigeki Yashiro Department of Aeronautics and Astronautics, Kyushu University, Fukuoka, Japan

Introduction

1

Vadim V. Silberschmidt Wolfson School of Mechanical, Electrical and Manufacturing Engineering, Loughborough University, Loughborough, Leicestershire, United Kingdom

Since the publication of the first edition of this book in 2016, the introduction of composites into various products and structures, already prominent, additionally accelerated. This has been driven by significant improvements in manufacturing processes, development of new composite systems as well as a net-zero environmental agenda. The last factor encourages the manufacturers to continue their lightweighting efforts to improve fuel efficiency of various transportation systems. In many cases, this can be achieved only with the use of composites, thanks to their excellent specific mechanical characteristics such as stiffness and strength. An additional drive related to net-zero policies is increased demands for sustainable materials, making the use of thermoplastic composites in various aerospace, automotive, marine, energy, and defence applications even more important because of their better recyclability. A direct consequence of more green policies is a considerable increase in the use of wind turbines—both onand off-shore—with their blades produced with laminates and composite-containing sandwich elements. It also has become clear that introduction of 3D-printed composites could be a game-changer for various applications that need complex shapes and architectured materials not easily achievable with traditional manufacturing methods employing lay-up techniques or braided composites. Most of the composite applications during their life in service are exposed to various dynamic loading—impacts, ballistic events, or shocks—that last from microseconds to milliseconds but can cause severe damage or even full failure due to their high intensity supplemented in many cases by spatial localisation. Such transient events together with inherent microstructural heterogeneity of composites and a contrast of their constituents’ mechanical properties result in a complex spatiotemporal evolution of multiple damage modes in such materials, a principal feature distinguishing them from most other types of structural materials. These scenarios are additionally complicated by interaction between these modes and stress waves as well as their effect on local and global deformation processes as a result of stiffness reduction. All these features and factors necessitate considerably more advanced—as compared with quasi-static loading regimes—methods of in situ characterisation of microstructural changes and damage evolution during such short events, mechanical testing under dynamic loading conditions, and numerical simulations of transient problems. As an example, consider the effect of specimen fixture on results of experimental measurements. Here, an interaction between the local deformation processes with a global response of the system ‘specimen/fixture’ depends considerably on the loading velocity. The increase of the latter changes this response, making it much more Dynamic Deformation, Damage and Fracture in Composite Materials and Structures. https://doi.org/10.1016/B978-0-12-823979-7.00001-6 Copyright © 2023 Elsevier Ltd. All rights reserved.

2

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

localised: beyond a specific level, the system’s behaviour is effectively insensitive to the character of fixing since the ‘information’ (stress wave) cannot reach it and return to the impact locus before the end of projectile-target interaction (e.g., perforation). Thus, in many cases, a direct assessment of dynamic mechanical properties of composites (especially related to their multiple damage modes) is impossible. As a result, a dynamic interaction between a loaded specimen and a loading device should be analysed to calculate the material parameters, employing various assumptions. Another example is in the area of numerical simulations. Here, not only extremely small time increments are required for an appropriate description of a very short event but also rather fine elements are needed for analysis of highly localised effects of damage growth and fracture propagation. Additionally, in many practically important cases, a transition to new equations of state for the loaded material is required (a typical example is the behaviour beyond the Hugoniot elastic limit). This book covers many important aspects related to problems of dynamic deformation, damage, and fracture. Obviously, it cannot cover all the problems; still, contributions by leading researchers from many countries provide a detailed, multifaceted overview of the field. This monograph, as its first edition, focuses mostly on laminates, with fibre-reinforced composites in its centre. Two main reasons underpin this: a broad use of laminates in various applications and their significantly more complex damage and failure scenarios compared to those in particulate composites. An important addition to the second edition is a review of dynamic behaviour of 3D-printed composites. This is an emerging area, with a rapid research development, thanks to many potential benefits, with its ability to manufacture complex shapes being the main one. In these materials, even a (relatively) simple matter of quantification of their quasistatic mechanical parameters using standard specimens is becoming controversial since they are more related to structures with discrete elements than continuous materials. Generally, the book has a balanced coverage of all the main aspects related to dynamic deformation, damage, and fracture of composites: from microstructural analysis to mechanical testing at various loading rates to relevant analytical models and numerical schemes. In many chapters, the starting point of analysis is linked to quasistatic loading, used as a reference for results obtained at low and high loading rates. Two main dynamic loading regimes—ballistic impact and explosion—are analysed in detail due to obvious differences in global responses and resultant processes of initiation and propagation of damage and failure. Some more specific cases of these regimes are included, for instance, underwater blasts, impacts with fragmenting projectiles, or multiple impacts. The book provides a comprehensive overview of the state-of-the-art testing, microstructural analysis, analytical approaches, and numerical simulations of the most common types of layered composite materials exposed to application-relevant dynamic loading conditions.

Damage tolerance of composite structures under low-velocity impact

2

Christophe Bouvet and Samuel Rivallant Institut Cl
ement Ader, Universit
e de Toulouse, CNRS UMR 5312—INSA— ISAE-SUPAERO—IMT Mines Albi—UPS, Toulouse, France

2.1

Introduction

Composite materials are increasingly used in industry due to their high performance/ mass ratio. This is especially true in aeronautical and aerospace fields due to the crucial importance of mass criterion. This high performance/mass ratio is due to the use of materials with high-specific mechanical properties, such as carbon, glass, Kevlar, or Zylon fibres [1]. Nevertheless, these types of material have the major drawback of being fragile and particularly sensitive to impact. This impact sensitivity leads to overdimensioning, and thus, a reduction in the potential gain to ensure their residual strength after impact. This sensitivity is also associated with a repair complexity of impact damage. Indeed, the repairs are often complex and repair methods are often inappropriate and still need to be tested on large structures subjected to impact, such as composite airplane fuselages (Fig. 2.1), for example, the Boeing 787 or the Airbus A350 fuselage. It is therefore essential, in the current context of aviation safety, to prove that these composite structures are able to sustain loads even with damages, regardless of the damage caused – impact, manufacturing errors, or scratches during maintenance operations, manufacturing, or service. This is the concept of damage tolerance. The damage tolerance concept [3,4] was introduced in the 1970s for civil aircraft structures, and these requirements are expressed by European certification JAR 25.571 [5]: “the damage tolerance evaluation of the structure is intended to ensure that should serious fatigue, corrosion or accidental damage occur within the operational life of the airplane, the remaining structure can withstand reasonable loads without failure or excessive structural deformation until the damage is detected” or by US certification FAR 25.571 [6]: “an evaluation of the strength, detail design, and fabrication must show that catastrophic failure due to fatigue, corrosion, manufacturing defects, or accidental damage, will be avoided throughout the operational life of the airplane”. The study of damage tolerance can therefore be defined as the study of the behaviour of a structure damaged by fatigue stress, corrosion, or accidental damage. The damage tolerance concept consists of verifying that the structure is able to sustain acceptable loads, without a break and significant deformation, until the damage is detected. Dynamic Deformation, Damage and Fracture in Composite Materials and Structures. https://doi.org/10.1016/B978-0-12-823979-7.00002-8 Copyright © 2023 Elsevier Ltd. All rights reserved.

4

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

Fig. 2.1 Visible impact damage on the outer surface of a fuselage [2].

50 mm 2.2

Principles of damage tolerance

The design of aeronautical structures is a particularly complex area, and the structure must sustain many load cases and damage configurations. Nevertheless, this can be summed up succinctly: l

l

The structure must withstand, statically, the limit load (LL) without damage (or plasticity for metals). Indeed, no damage or permanent deformation is permitted in service, that is to say, for realistic loads, i.e. less than or equal to the LL. Practically, a realistic event is defined as an event whose probability of occurrence is 10 5 per flight hour. The structure must withstand the ultimate load (UL) without catastrophic failure. The structure must remain whole (but damage is allowed) for loads that are very improbable, i.e. less than or equal to the UL. Practically, a very improbable event is defined as an event whose probability of occurrence is 10 9 per flight hour.

Nevertheless, this concept does not account for damage. But in reality, the damage is obviously inevitable and can grow in service. Indeed, structure loading in service is repetitive and induces fatigue solicitations. The growth of fatigue damage can therefore lead to a residual strength lower than the UL. Therefore, damage tolerance seeks to show that: l

l

Even with damage, the residual strength of the structure must remain higher than the LL. Any damage leading to residual strength below the UL must be quickly detected and repaired (and the repair should restore a strength higher than the UL).

The damage tolerance of metallic materials is based on this concept (Fig. 2.2A). Indeed, damage growth in metallic structures is relatively slow and often well controlled. It is then possible to determine inspection intervals to ensure that damage does not grow too much before being detected and therefore does not lead to a gap below the UL, which is very important. This is the slow-growth approach. At the same time, it is necessary to prove that the residual strength is always higher than the LL.

Damage tolerance of composite structures

5

Metallic materials: slow-growth approach Residual strength

UL

Interval below UL

LL Critical damage

Detectable damage Inspection interval Detection and repair of damage

Damage occurrence

Time

(A) Composite materials: no-growth approach Residual strength

Residual strength

Unacceptable damage

UL

Interval below UL

LL

Acceptable damages

UL

Interval below UL

LL Damage 1

Damage occurrence

Detection and repair of the damage

Time

Damage occurrence

Damage 2

Detection and repair of the damage

Time

(B) Fig. 2.2 General principle of damage growth and repair of metallic (A) and composite (B) materials.

This case is illustrated in Fig. 2.2A. In this figure, a schematic representation of the residual strength of a structure versus time is plotted. At first, this residual strength is constant and obviously higher than UL until the occurrence of damage (usually a crack initiating at the edge of a hole). From there, the damage grows and residual strength decreases until reaching (if damage is not detected) the LL; this damage is called critical damage. The goal was obviously to decrease the time spent below the UL. Finally, once the damage is detected, it should be repaired and should restore residual strength higher than the UL. This concept of damage tolerance is usually not relevant for composite structures. Indeed, composite materials are almost insensitive to fatigue. This is particularly true for composites with carbon fibres, and slightly less for composites with glass fibre [7,8]. At the same time, as most of the primary aircraft structures are manufactured from carbon fibres, the primary aircraft structures are almost insensitive to fatigue. This insensitivity to fatigue of composite materials usually leads to no growth of damage in service. Then, it becomes impossible, as in the case of metallic structures, to define maintenance intervals based on the concept of slow growth (Fig. 2.2). Furthermore, composite materials being very sensitive to impact, impact damage can reduce the residual strength below the UL. However, the requirements direct (AC 20-107B [6]) that composite structures must not lead to less security than metallic structures. So it should be proved that the time spent below the UL of a composite structure (Fig. 2.2B) is less critical than for a metallic structure (Fig. 2.2A). It is

6

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

obviously necessary to take into account both the time spent below the UL and the gap between the residual strength and the UL. Clearly, damage leading to residual strength just below the UL (Damage 2, Fig. 2.2B) may be present longer than damage leading to a residual strength just above the UL (Damage 1, Fig. 2.2B). This is the no-growth approach. Not to mention that in addition to this, it must be proved that the residual strength remains, in any case, above the LL (except in special cases to be discussed later). Impact damage tolerance of composite structures depends therefore on two factors: l

l

The loss of residual strength due to impact. Impact is characterised by the mass of the impactor, the shape of the impactor, the impact velocity, and the impact energy level. Obviously, the higher the impact energy level (for a given impactor), the more damage is important, and therefore, the loss of residual strength is important (Fig. 2.4). In practice, the strength loss can reach 50%–75% of the strength without impact [9,10]. The detectability of the impact (Fig. 2.3). Once again, the higher the impact energy level (for a given impactor), the more damage is important, and therefore, the more damage is detectable. Moreover, for composite laminates of standard thickness (a few millimetres), the damage is detectable on the impacted side before being visible on the nonimpacted side. But at the same time, access to the nonimpacted side is usually impossible (inside the wing, the fuselage, the wingbox, etc.), so it is only once the impact is visible on the impacted side that it is considered detectable (Fig. 2.4).

This curve of the residual strain in compression versus impact energy (Fig. 2.4) is particularly important in the context of impact damage tolerance. At first, it should be noted that this is the compression loading that is mainly considered for justification of impact damage tolerance, because this is the characteristic most affected by impact [9–12]. This is because the compressive strength is often due to buckling (this is true for laminates not too thick, typically for a thickness of several millimetres). Nevertheless, we must not forget that the fundamental parameter of impact damage tolerance is the detectability of the damage, not the impact energy. It is therefore interesting to plot the residual strength versus permanent indentation left by the impact (Fig. 2.5). This graph is then used to bring up the three areas of the design of a composite structure to impact damage tolerance: Fig. 2.3 25 J impact of a carbon–PPS woven laminate plate of 2 mm thickness with a 16-mmdiameter hemispherical impactor.

Impact point

100 mm

Impacted side

Nonimpacted side

Damage tolerance of composite structures

7

Impact

Compressive failure strain Hrcomp –1.3%

Compressive residual strain (%)

Impact –0.6 Impact –0.4

–0.2 Damage Undetectable detectable damage nonimpacted side

Damage detectable impacted and nonimpacted side

Impact energy Fig. 2.4 Compressive residual strain after impact of a UD carbon–epoxy laminate plate typical of the aeronautical field.

Impact

Residual strength after impact

Undetectable damage

Detectable damage

Obviously detectable damage

Static requirements Impact

UL Impact

Damage tolerance LL Flight loading

Permanent indentation

Fig. 2.5 Residual strength after impact versus permanent indentation left by impact.

8 l

l

l

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

The area of undetectable damage: the structure has to withstand the UL. The area of detectable damage during maintenance inspection: the structure has to withstand the LL. In this case, an inspection procedure must be defined in order to detect the damage as early as possible. Once detected, the damage must be repaired and, once repaired, has to withstand the UL. The area of immediately detectable damage: the structure must withstand loads compatible with continued safe flight. This type of damage deals with very improbable impact or damage readily detectable by the flight crew. Once detected, the damage must be repaired (or the area changed), and once repaired, the structure has to withstand the UL.

2.3

The different damage types

Classifying the different types of damage of an aeronautical structure as complex as an aircraft is a delicate operation. However, there is an official classification made by US aviation authorities [6] via an advisory circular (AC) dealing specifically with composite aircraft structures (AC 20-107B [6]). The damages are then classified into five categories according to their criticality and their detectability, from the smallest to largest (Fig. 2.6). Category 1

Allowable damage that may go undetected by scheduled or directed field inspection or allowable manufacturing defects. Structural substantiation for Category 1 damage includes demonstration of a reliable service life while retaining UL capability. Some examples of Category 1 damage include barely visible impact damage (BVID) and CAT 1

CAT 2

CAT 3

CAT 4

CAT 5

Total damage on inner side of structure

Visible damage on outside surface

Residual sustainable load

UL

LL 0.85*LL 0.70*LL Damage not detectable

Damage detectable

Allowable damage limit (ADL) / detectability threshold Static requirement

DT domain

UL: Ultimate load LL: Limit load DT: Damage tolerance LDC: Large damage capability DSD: Discrete source damage

Damage readily detectable

Damage immediately detectable

Max. readily detectable damage (max. RDD)

Critical damage threshold (CDT) LDC

Very large damages --> Out of scope of A/C design limits Max. discrete source damage (max. DSD)

Damage size

DSD range

Fig. 2.6 Impact damage tolerance and the different damage categories [2].

Damage tolerance of composite structures

9

allowable defects caused in manufacturing or service (e.g. small delamination, porosity, small scratches, gouges, and minor environmental damage) which have substantiation data showing ultimate load is retained for the life of an aircraft structure. Category 2

Damage that can be reliably detected by scheduled or directed field inspections performed at specified intervals. Structural substantiation for Category 2 damage includes the demonstration of a reliable inspection method and interval while retaining loads above LL capability. The residual strength for a given Category 2 damage may depend on the chosen inspection interval and method of inspection. Some examples of Category 2 damage include visible impact damage (VID), deep gouges or scratches, manufacturing mistakes not evident in the factory, detectable delamination or debonding, and major local heat or environmental degradation that will sustain sufficient residual strength until found. This type of damage should not grow or, if slow or arrested growth occurs, the level of residual strength retained for the inspection interval is sufficiently above LL capability. Category 3

Damage that can be reliably detected within a few flights of occurrence by operations or ramp maintenance personnel without special skills in composite inspection. Such damage must be in a location such that it is obvious by clearly visible evidence or cause other indications of potential damage that becomes obvious in a short time interval because of loss of the part form, fit, or function. Both indications of significant damage warrant an expanded inspection to identify the full extent of damage to the part and surrounding structural areas. In practice, structural design features may be needed to provide sufficient large damage capability to ensure the limit or near LL is maintained with easily detectable Category 3 damage. Structural substantiation for Category 3 damage includes the demonstration of a reliable and quick detection, while retaining limit or near LL capability. The primary difference between Category 2 and 3 damages is the demonstration of large damage capability at limit or near LL for the latter after a regular interval of time which is much shorter than with the former. Some examples of Category 3 damage include Large VID or other obvious damage that will be caught during walk-around inspection or during the normal course of operations (e.g. fuel leaks, system malfunctions, or cabin noise). Category 4

Discrete source damage from a known incident such that flight manoeuvres are limited. Structural substantiation for Category 4 damage includes a demonstration of residual strength for loads specified in the regulations. Some examples of Category 4 damage include rotor bursts, bird strikes (as specified in the regulations), tire bursts, and severe in-flight hail. Category 5

Severe damage created by anomalous ground or flight events, which is not covered by design criteria or structural substantiation procedures. This damage is in the current

10

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

guidance to ensure that the engineers responsible for composite aircraft structure design and the US Federal Aviation Administration (FAA) work with maintenance organisations in making operations personnel aware of possible damage from Category 5 events and the essential need for immediate reporting to responsible maintenance personnel. Some examples of Category 5 damage include severe service vehicle collisions with aircraft, anomalous flight overload conditions, abnormally hard landings, maintenance jacking errors, loss of aircraft parts in flight, and possible subsequent high-energy, wide-area (blunt) impact with adjacent structure. Of course, this classification is not sufficient to precisely understand the damage developing in a composite structure during impact loading. This damage is very complex and depends on impact conditions, composite material, draping sequence, etc. The case of low-velocity/low-energy impact is very important, because it is representative of tools dropped during manufacturing or maintenance operations. Moreover, this type of damage is very penalising, because it often induces undetectable damage, and then, the structure must sustain the UL during all of the aircraft’s life. Thus, it is important to understand the damage induced by this type of impact to better understand the problem of impact damage tolerance.

2.4

Impact damage

A low-velocity/low-energy impact on a unidirectional (UD) composite laminate induces three types of damage: matrix cracks, fibre fractures, and delamination (Fig. 2.7) [13–16]. The first damage to appear is conventionally matrix cracking. When this damage grows, delamination quickly occurs. An interaction between these two damage phenomena is clearly visible during the impact tests (Fig. 2.7). This interaction is crucial to explain the very original morphology of the delaminated interfaces in the thickness of the plate: a C-scan investigation shows delamination as twin triangles (Fig. 2.11) [13,17], with the size growing from the impacted side to the nonimpacted side. This formation of twin triangles is illustrated by Renault [17] on a simple stacking sequence [ 45, 0, 45] (Fig. 2.8). The impact damage begins with the development of matrix cracks in the impact zone below the impactor. These matrix cracks grow up during the loading in the fibre direction. Therefore, in each ply, a strip of fibres and resin disjoins and slides in the normal direction of this ply. This disjointed strip creates an interlaminar zone of tension stress between two consecutive plies and can induce in this zone the formation of a delamination (Fig. 2.8A). The zone of tension stress, limited by the disjointed strips of the two adjacent plies, has a triangular shape with the size growing from the impacted side to the nonimpacted side. Fig. 2.8B illustrates the interlaminar zones of tension stress between the 45°/0° and 0°/45° plies. This scenario enables us to illustrate some originalities of the impact damage morphology, such as delamination aligned with the fibres of the lower ply or the generally conical shape of delamination. However, some questions remain subject to debate. In order to study a real impact damage in detail, postmortem microscopic sections can be observed (Fig. 2.10). This 25 J impact was performed on a UD laminate

Damage tolerance of composite structures

(A)

11

(C)

Matrix cracking

Longitudinal cracking Direction of solicitation

0 90 0

Delamination

Matrix cracking

90 0 90 Delamination initiation

Delamination initiation

Transverse cracking

(B)

Fibers failure

10-17-2003 07:42:35 SUPAERO/LMS COMPO

100 Pm HV:20kV Tilt:15° WD:15 nm SS:6 G:350 Entrez ici la description de l'échantillon

Delamination propagation

Delamination propagation

(a) Cracking + saturation of lower ply => delamination

(b) Bending cracking => delamination of the ply nonimpacted side

10-17-2003 07:50:01 SUPAERO/LMS (952x680)

COMPO

100 Pm HV:20kV Tilt:15° WD:15 nm SS:6 G:450 Entrez ici la description de l'échantillon

(952x680)

(D)

5 mm

Impact energy = 4.1 J

Rayons X A

Fiber fracture in 90° layer

Impact axis

A 0° direction

Impact Delaminations 0° 90° 0° 90° 0° 90°2 0° 90° 0° 90° 0°

2 mm

Section A-A

Tension matrix cracks

Shear matrix cracks and delaminations

Shear matrix cracks and delaminations

Fig. 2.7 (A, B, and D) Impact damage in UD composite. (C) Diagram of interaction between intra- and interlaminar damage. (A) From C. Bouvet, N. Hongkarnjanakul, S. Rivallant, J.J. Barrau, Chapter 8: Discrete impact modelling of inter- and intra-laminar failure in composites, in: Dynamic Failure of Composite and Sandwich Structures,Springer, Dordrecht, Heidelberg: New York, London, 2012, pp. 339–392, with kind permission from Springer Science and Business Media. (B) From S. Petit, Contribution a` l’
etude de l’influence d’une protection thermique sur la tol
erance aux dommages des structures composites des lanceurs (the`se de doctorat), Universit
e de Toulouse, 2005. (C) From F.K. Chang, H.Y. Choi, H.S.Wang, Damage of laminated composites due to low velocity impact, in: 31st AIAA/ASME/ASCE/AHS/ASC, Structures, Struct. Dyn. and Mater. Conf., Long Beach, CA, April 2–4, 1990, pp. 930–940. (D) From F. Aymerich, P. Priolo, Characterization of fracture modes in stiched and unstiched cross-ply laminates subjected to low-velocity impact and compression after impact loading, Int. J. Imp. Eng. 35 (2008) 591–560.

Impacted side Ply n°3 (45°)

Ply n°2 (0°) Propagation direction Impact zone

Impact zone

Ply n°3 (45°) Ply n°2 (0°) Ply n°1 (–45°)

45°

A

B

0° –45°

Impact zone

Intralaminar matrix cracks: creation of disjointed strips Ply n°2 (0°)

A

Propagation direction

Impact zone

Disjointed strip

Section A-A

Nonimpacted side

(A) Fig. 2.8 Mechanism of delamination formation proposed by Renault [17].

45° 0° -45° 45° Zones of interlaminar 0° -45° tension stress

0° -45°

Section B-B

Ply n°1 (-45°)

Impact zone 45°

B

Interlaminar tension stress zone of triangular shape

(B)

Damage tolerance of composite structures

13

carbon–epoxy T700/M21 with the draping sequence [02, 452, 902, 452]S, and 4.16 mm thickness [18]. The impact has been achieved with the AITM 1-0010 standardisation [19] with a 100  150 mm2 plate simply supported on a 75  125 mm2 window (Fig. 2.9). The impactor consists of a 2-kg mass with a hemispherical tip 16 mm in diameter impacting the plate perpendicularly with an initial speed of 5 m/s (hence the impact kinetic energy of 25 J). Two cuts were then performed after impact at 0° and 90° (Fig. 2.10). Seven plies are observed in these cuts, or more exactly seven plies grouped with same orientation [02, 452, 902, 454, 902, 452, 02]. In these photographs (Fig. 2.10), the different damage types mentioned previously – matrix cracking, delamination, and fibre breakage – are observed. It can be noticed Impact velocity 150 mm

I 16 mm

100 mm

Zone damaged during impact 4 mm

75 mm

125 mm

Fig. 2.9 Impact test with AITM 1-0010 standardisation [19] of a laminate composite plate and typical damage zone.

90° cut

Impactor

90°

Matrix cracks

0°

Fiber failures

Impacteur

Delaminations

0° cut

2 mm

Debris

Open delaminations

Fig. 2.10 Damage of a carbon–epoxy laminate composite plate with 150  100  4.16 mm3 dimensions impacted at 25 J [18].

14

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

that delaminations are present at the junction between two consecutive plies of different orientations. This is due to the significant interlaminar shear stress in these areas [9,12]. Important matrix cracks are present in the area under the impactor, mainly at 45° (Fig. 2.10), indicating a significant proportion of out-of-plane shear stress in this area. A lot of junctions between matrix cracking and delamination are also observed, which confirms the interaction between these two phenomena, mentioned earlier. The 0° cut shows a very significant delamination of the first interface nonimpacted side between the plies at 0° and 45°, still wide open, despite the release of the impact force. A C-scan of this plate obviously shows that this interface is the most delaminated interface during impact (Fig. 2.11) [18]. These C-scans (Fig. 2.11) show that all interfaces are delaminated and the damage is more extended nonimpacted side. It is also observed that delaminations are oriented in the fibre direction of the lower ply of the considered interface. This is because the

50 mm

25 J impact 6: 45/0

90°

45°

Impacted side 0 mm

0° -45° 2 mm

1: 0/45 2: 45/90 3: 90/-45

4: -45/90 5: 90/45

4.16 mm Nonimpacted side

Impacted side

(A) 50 mm

25 J impact

90°

45°

Nonimpacted side 0 mm

0° -45° 2 mm

1: 0/45 2: 45/90 3: 90/-45

Nonimpacted side

4: -45/90

4.16 mm Impacted side

(B) Fig. 2.11 Delaminated interfaces observed with C-scan after 25 J impact: (A) impacted side and (B) nonimpacted side.

Damage tolerance of composite structures

15

greatest stress is in the fibre direction, which generates interlaminar shear stress and thus a spread of delamination in this direction. Then, the delamination creation scenario can be divided into two stages: l

l

First, matrix cracks are created under the impactor, mainly due to high out-of-plane shear stress in this area. Once these cracks reach the lower interface, they create an opening force in the interface and cause its damage initiation (Fig. 2.12A). This damage initiation is mainly due to an opening mode of fracture failure (mode I). In practice, the problem is more complex and a significant proportion of shearing or tearing mode (mode II or III) exists. Once created, delamination grows due to interlaminar out-of-plane shear stress (Fig. 2.12B). This shear stress being greater in the fibre direction of the lower ply, delamination preferentially propagates in this direction.

The central cone heavily damaged by matrix cracking under the impactor is strongly related to permanent indentation. Indeed, it is when the cone separates (obviously partly) from the rest of the plate and puts down that the permanent indentation appears. This point will be explained in more detail subsequently.

0° 90° 0° 90° 0°

Impactor

Central cone heavily damaged

Delamination initiation in opening mode (I)

(A) 0° 90° 0° 90° 0°

Impactor

Delamination growth in shearing mode (II) High out-of-plane shear stress

(B)

Fig. 2.12 Typical scenario of delamination growth during impact loading: (A) initiation and (B) propagation.

16

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

2.5

Damage detectability

The aeronautical field relies heavily on visual inspection methods to determine damages in composite structures. Four levels of inspections are defined (Fig. 2.13) [4,20–24]: l

l

l

l

Walk-around: This is a visual inspection at a relatively large distance for the detection of perforations and large indentation surfaces or fibre breakage. This inspection is performed daily. The minimum size of detectable damage typically by this type of inspection is called ‘large visual impact damage’ (large VID). General visual inspection: This is a close inspection of relatively large surfaces, belonging to the internal or external structure for indicators of impact damage or other abnormalities of the structure. Adequate and appropriate lighting may be required. The inspection support tools (such as mirrors) and surface cleaning may also be required. The minimum size of detectable damage typically by this type of inspection is called ‘minor visual impact damage’ (Minor VID). Detailed visual inspection: This is a detailed inspection of relatively near locations to be inspected belonging to the internal or external structure, for the presence of impact injuries or other abnormalities of the structure. As for the general inspection, adequate and appropriate lighting may be required. More sophisticated support tools (such as magnifying glasses) and surface cleaning may also be required. The minimum size of detectable damage typically by this type of inspection is called barely visible impact damage (BVID). Special detailed inspection: This is an inspection for specific locations for invisible damages. Nondestructive inspection techniques (such as ultrasound or X-ray) may be used. This type of inspection is mainly carried out during production, or around a damaged area identified with a coarser inspection.

Then, the detectability of damage is defined using a threshold depending on inspection type: BVID, Minor VID, and Large VID. These thresholds deal generally with the dent depth left by the impact on the structure. In the aeronautic and aerospace fields, it has been shown that a permanent indentation between 0.25 and 0.5 mm is detectable during a detailed visual inspection with a probability greater than 90%. Tropis et al. [24]

BVID

Indentation: 0.5 mm

Minor VID

Large VID

Indentation: 2 mm or Perforation: I 20 mm

Perforation: I 50 mm

Walk around General visual inspection Detailed visual inspection Special detailed inspection

Fig. 2.13 Damage size depending on inspection type.

Damage size

Damage tolerance of composite structures

17

have shown that for a 0.5-mm indentation, this probability reaches 99% with a 95% confidence interval. Thus, according to the controller’s experience, it is possible to say with 95% confidence [4,21] that: l

l

A dent depth of 0.2–0.23 mm is detectable at 2 m distance. A dent depth of 0.1 mm is detectable during detailed inspection.

Then, it can be agreed that a 0.5-mm indentation (BVID) is detectable using a detailed visual inspection and a 2-mm indentation (Minor VID) is detectable using a general visual inspection (Figs 2.13 and 2.14). However, the use of the permanent indentation to quantify the extent of damage also has disadvantages. The dent depth depends on many impact parameters, such as the geometry of the impactor, and may not be a good indication of the form and the extent of internal damage. Thomas [21] has thus demonstrated that the dent depth can decrease over time as a result of fatigue and humidity due to viscoelasticity (Fig. 2.15). In some cases, the initial dent depth obtained just after impact is three times greater than at the end of life. Detectability probability 100 % 99 %

Detailed visual inspection

0.5 mm

General visual detection

Walk around

2 mm

Permanent indentation

Fig. 2.14 Detectability probability of permanent indentation greater than a given permanent indentation. Material relaxation

Fatigue End life dent depth of 0.3 mm (BVID)

Initial dent depth of 1 mm Wet aging

Thermal loading

Fig. 2.15 Evolution of dent depth versus time depending on external conditions [16].

18

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

Moreover, the decrease of the dent depth versus time is also dependent on the material. Thus, using the permanent indentation as the detection of damage during maintenance requires that BVID sizing tests are carried out on samples with end-of-life dent depth (after relaxation, fatigue, etc.) greater than the detectability threshold mentioned. In the absence of these parameters, a dent depth of 1 mm just after the impact must be considered so that it is detected after relaxation [16,21,23]. The Minor and Large VID thresholds are similarly defined: it should be demonstrated to the certification authorities that damage of this size can be detected during such inspection type with sufficient probability. This is not the standard that imposes these thresholds, but the aircraft manufacturer in cooperation with the airline company. Typically, these thresholds are (Fig. 2.13) about: l

l

l

0.3–0.5 mm of permanent indentation for BVID. 2 mm of permanent indentation and/or a perforation of about 20 mm diameter for Minor VID. 50-mm-diameter perforation for Large VID.

These thresholds are only indicative and may vary by airline companies and aircraft manufacturers. The permanent indentation is thus a crucial element to design a composite structure using impact damage tolerance. To investigate this permanent indentation, many experimental tests have been performed in the literature [9,16,25,26]. Impact tests are generally performed according to AITM 1-0010 standard [19]. This standard consists of a 100  150 mm2 plate about 4 mm thick simply supported on a 75  125 mm2 window impacted with a hemispherical 16-mm-diameter impactor. The typical evolution of the permanent indentation versus impact energy level is plotted in Fig. 2.16. This curve generally has three parts: l

l

The first part deals with the low-impact energy level. Impact damage consists of small extended-matrix cracking and delamination (Fig. 2.17A). During unloading (rebound of the impactor), shear matrix cracking and delamination remain partially open (Fig. 2.19), leading to a relatively small permanent indentation, generally less than BVID. In matrix cracks, resin and fibre debris, created during impact, block (Fig. 2.18) and prevent their closure. In the case of delamination, debris generally consists of cusps (Fig. 2.19C and D) [26,27], typical of a shearing fracture mode (mode II). Other phenomena, such as “plasticity” of the resin or friction in the cracks, preventing their reclosure, also contribute to the creation of the permanent indentation. When the impact energy level increases, then fibre fractures, mainly located under the impactor (Fig. 2.10), lead to a faster increase of the permanent indentation (Fig. 2.16). These fibre failures are due to compression loading under the impactor and to traction loading at mid-thickness or in the lower part of the plate. Nevertheless, the lowest ply generally does not break in fibre failure because the important delamination in this area tends to unload it. The main fibre failures are generally located between the mid-thickness and the nonimpacted side of the plate. These fibre failures, due to tensile (and partially shear) loading, make it possible to detach the highly damaged central cone from the rest of the plate (Fig. 2.17B) and induce the creation of a larger permanent indentation. It is in this area that the BVID is usually reached (Fig. 2.16).

Damage tolerance of composite structures

19

Impact

Matrix cracking + Delamination

Matrix cracking + Delamination + Fiber failures

Permanent indentation

Impact

Impact

Important fiber failures

Perforation

BVID

Impact energy Fig. 2.16 Schematic evolution of permanent indentation versus impact energy level.

0° 90° 0° 90° 0°

Impactor

Central cone heavily damaged

Impactor

Compressive fiber failure

Central cone heavily damages

(A)

0° 90° 0° 90° 0°

Tensile fiber failure

(B)

Fig. 2.17 Schematic scenario of damage-inducing permanent indentation during impact: (A) before and (B) after the important fibre failures.

Impactor

Debris

0° 90° 0°

Open delaminations

Fig. 2.18 Schematic scenario of creation of permanent indentation.

20

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

(A)

Impactor

Matrix crack 100 Pm

02 902 02

Delaminations 2 mm

(B) 100 Pm

100 Pm

(C)

(D)

Cusps

Fig. 2.19 Blocking of debris (A) after impact, (B) in matrix cracks, and (C and D) in delaminations [26].

l

The development of these fibre breakages obviously has a detrimental effect on the residual strength of the structure after impact; but at the same time, it makes it possible to create a large permanent indentation and thus to detect impact damage. Thus to increase the impact damage tolerance of a composite structure, it is necessary, on the one hand, to prevent the fibre failure to avoid excessive decrease of the residual strength after impact and, on the other hand, to promote the fibre failure to improve detectability of impact damage. This is the complexity of impact damage tolerance. It is in this part that we find the most critical cases of impact damage tolerance: the impacts producing a permanent indentation slightly lower than BVID and therefore undetectable by visual inspection. The structure has to sustain the UL throughout the aircraft life with this damage (Fig. 2.6). The last part of the curve deals with the large energy level near perforation (Fig. 2.16). An asymptote of the indentation is observed even if it is difficult to speak of permanent indentation in the presence of a perforation. This part is paradoxically less dangerous than the last case, because the impact is easily detectable and only the LL is required (Fig. 2.6).

The scenario of the creation of the permanent indentation is very complex and many works are still necessary to better understand it [26]. Nevertheless, this scenario is fundamental to optimise a composite structure using impact damage tolerance. It is indeed necessary, on the one hand, to promote the creation of the permanent indentation to improve the detection of impact damage and thus promote damage and, on the other hand, to reduce damage to avoid decreasing the residual strength after impact of the structure. This is the paradox and complexity of impact damage tolerance.

2.6

Residual strength after impact

In order to better understand impact damage tolerance, it is important to better understand the residual strength after impact and, in particular, the failure scenario leading to final failure during the experimental test of compression after impact (CAI). Indeed,

Damage tolerance of composite structures

21

the compression characteristic is generally the most affected by impact with the loss of characteristic up to 75% (Fig. 2.4), due to buckling of the delaminated plies. At the same time, the tensile characteristics are generally relatively unaffected by impact [9]. Thus, an experimental test of CAI has been standardised to evaluate the residual strength after impact of a composite laminate; this standard is obviously shared with the impact (AITM 1-0010 [19]). The 100  150  4 mm3 laminate plate previously mentioned is thus impacted using this standard (Fig. 2.15) and subjected to compression loading. However, to avoid premature buckling of the plate, a system of antibuckling knives is placed on both sides of the specimen (Fig. 2.20). These knives are located 10 mm from the free edge and have a low area of contact with the plate in order not to impede its movement in the loading direction. During a CAI test, two damage phenomena leading to failure are observed [9,11,28]: l

l

The first phenomenon is the buckling of the sublaminates delaminated during impact (Fig. 2.21). These sublaminates buckle in compression, at the same time induce growth of impact delaminations, and thus increase the buckling. Generally, the delamination grows in the direction perpendicular to the loading direction. This phenomenon is usually catastrophic and quickly leads to the rupture of the specimen. This buckling is much earlier than the plate being deformed from its original planar shape, that is to say the permanent indentation is important. The second damage phenomenon is the growth of a compressive fibre failure of the plies oriented in the loading direction. Moreover, laminates subjected to compression loading are generally oriented in the loading direction, that is to say, more plies are oriented in the loading direction compared to the other directions. Thus, a compression crack can be created near the heavily damaged area during impact, and this crack can grow during loading (Figs 2.10 and 2.21). Furthermore, this phenomenon is generally invisible from the outside, because the outside plies are often oriented at 45° to increase the buckling strength.

Now look at the laminate, previously mentioned, unidirectional carbon–epoxy T700/ M21, with draping sequence [02, 452, 902, 452]S, 4.16 mm total thickness, and 25 J impacted after CAI failure (Fig. 2.22). The compressive fibre failure is clearly seen on the impacted side. A section perpendicular to the crack makes it possible to observe a lot of delaminations developing during the impact and breaking during the compression. Kink bands, characteristic of compressive fibre failure, are observed on the upper ply corresponding to the impacted side [29,30].

150 mm CAI damage 100 mm

Compressive loading

4 mm Antibuckling system

Zone damaged during impact

Fig. 2.20 CAI test with the AITM 1-0010 standard of laminate composite plate and typical damage.

22

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

A

A-A cut

Impact permanent indentation

Delamination growth during CAI

Impact compressive fiber failure Delamination buckling during CAI

Growth of compressive fiber failure during CAI

Impact delamination

CAI loading A

Fig. 2.21 Typical scenario of damage and failure of CAI test.

To conclude this discussion of the residual strength after impact, we may recall that to optimise the CAI of a composite laminate, it is necessary to minimise impact damage in order to delay buckling of delaminated interfaces and propagation of the compressive fibre failure. The paradox is that to be able to clearly detect impact, it is necessary to promote the permanent indentation and therefore the impact damage.

2.7

Impact threat

As mentioned previously, the UL is required for all undetectable damage, that is to say, damage whose permanent indentation is below the BVID (Figs 2.5 and 2.6). In fact, the problem is more complicated. Indeed, this is true for a thin structure (Fig. 2.23), but not for a thick structure. For a thick structure, impact energy level necessary to reach BVID can be so high that it becomes extremely improbable. Thus, two thresholds of impact energy are defined (Fig. 2.23) [4,22–24]: l

l

The realistic impact energy level corresponding to a realistic event, i.e. whose probability of occurrence is 10 5 per flight hour. The improbable impact energy level corresponding to an extremely improbable event, i.e. whose probability of occurrence is 10 9 per flight hour.

Damage tolerance of composite structures

Compressive fiber failure

23

Impact indentation

Kink-bands

Delaminations

Y-Y cut 150 mm

Fiber failure

200 μm

[02, 452, 902, -452]s

500 μm

Y-Y cut

0° 45° 90°

Fig. 2.22 Postmortem observation after CAI.

Damage detectability (permanent indentation)

D

Thin plate

LL

Large VID B

Detectable impact damage: k.LL required

LL

UL

BVID

C A Realistic and undetectable impact damage: UL required

Readily detectable and/or improbable impact damage: Particular case (required load LL)

UL Realistic energy (10-5/fh)

Unrealistic and undetectable impact damage: k.LL required

Thick plate

Improbable energy (10-9/fh)

Impact energy

Fig. 2.23 Classification of the impact threats depending on damage detectability and impact energy level, and corresponding required load.

24

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

Practically these energies depend on the concerned area of the aircraft. A very exposed area has obviously greater impact energy thresholds than a less exposed area. It is therefore difficult to give values, but in a case of structure loaded mainly to falling tools during maintenance and manufacturing, the realistic impact energy may reach 30–50 J, while the improbable energy may reach 100 J. Between these two values of realistic and improbable energy threshold, we may assume that the probability of occurrence of an impact energy higher than a given energy corresponds to a log-linear law. Obviously, this law is only an approximation and reality is always more complex. In reality, it is necessary, to perform a statistical study of the impact threats of each aircraft area, to define the thresholds of realistic and improbable impact energy, and to check whether, between these two thresholds, the log-linear law reflects the reality. However, knowing the impact energy (or its probability) is not enough to define the threat of a given impact; it is also necessary to know whether it is possible to detect it and therefore to know the permanent indentation left by this impact. Then, several areas can be defined according to the damage detectability and to the impact energy, to be able to sort the impact threats and then to define the required loads (Fig. 2.23) [2,4,22–24,31,32]: l

l

The area of undetectable and realistic impact damage (area A). This area corresponds to events of realistic impact energy leaving a permanent indentation smaller than the BVID. In this case, the structure must withstand the UL, throughout its life. This zone corresponds to damage of Category 1 described by the standard. The limit of this area is usually the most restrictive for aircraft composite structures. Indeed, a structure with impact slightly below the BVID can be heavily damaged and yet must withstand the UL. Demonstration of damage no-growth should also be done. The area of detectable impact damage whose energy is less than the improbable energy (area B). This area corresponds to impacts leaving a permanent indentation higher than the BVID and lower than the Large VID. This type of damage can be detected during a visual inspection and must withstand at least the LL. This zone corresponds to damage of Category 2 described by the standard. It is also necessary to prove that the damage can be detected within a reasonable period of time. Indeed, if the time interval is too long, the structure can pass too much time under the UL and safety rules are not respected (Fig. 2.2B). Then, two types of approach can be adopted for design demonstration: a deterministic or a probabilistic approach. The deterministic approach requires that the structure withstands k.UL (with k > 1) until its detection, regardless of the impact energy level and the permanent indentation (obviously between the BVID and the Large VID); it is the type of approach that is followed by Boeing [21,22,27]. Then, it remains to determine k to avoid spending too much time under the UL. Remember that the k relationship between the UL and LL is generally between 1.1 and 1.5. Another approach, probabilistic, is to define the design loads such that the probability of encountering a damage reducing the residual strength below the design load is extremely improbable, i.e. with a probability lower than 10 9 per flight hour (CS 25.1309) [5]. This type of approach is used by Airbus [4,23,24]. This defines a k.UL load requirement (k > 1) with k close to 1 if the damage is close to the BVID, and k – most importantly – close to that of the UL if the damage is close to the Large VID. The coefficient k therefore depends on the size of the damage and the impact energy level, unlike the deterministic approach, where it is constant.

Damage tolerance of composite structures l

l

25

The area of unrealistic and undetectable impact damage (area C). This zone corresponds to the impacts leaving a permanent indentation less than the BVID with an energy level between the realistic and the improbable energy thresholds. As before, a deterministic or probabilistic approach can be adopted. In the case of the deterministic approach, the damage being undetectable, the UL is required, regardless of the impact energy level, obviously lower than the improbable energy threshold. In the case of the probabilistic approach, the required load k.UL depends on the probability of occurrence of the impact energy, such as the probability of encountering damage, reducing the residual strength below the required load, which remains extremely improbable. If this energy is close to the improbable energy threshold, k is close to 1; and if this energy is close to the realistic energy, k is close to that of the UL. This area is part of the Category 1 described by the standard (see Section 2.3), although it is not directly explained. The area of extremely improbable impact energy and/or readily detectable damage (area D in Fig. 2.23). This area corresponds to an impact energy level higher than the improbable energy threshold and/or impacts leaving a permanent indentation higher than the Large VID. In this case, the impacts are easily and quickly detected, and the structure can withstand loads lower than the LL. This area corresponds to damage of Categories 3, 4, and 5 described by the standard (see Section 2.3 and Fig. 2.6). This area is subject to specific design processes developed in collaboration between the manufacturer and the airline company.

2.8

Conclusions

Composite materials are very sensitive to impact; impact damage tolerance is therefore an important design case for aircraft composite structures. The complexity of this design case is that it is necessary to consider, on the one hand, the impact damage, or more precisely the loss of residual strength due to the impact, and, on the other hand, the impact detectability, or more precisely the permanent indentation left by the impact. So in order to justify a composite structure’s impact damage tolerance, the approach: l

l

Evaluates the residual strength and the permanent indentation versus the impact energy level of the considered composite structure. Shows that the conditions of impact damage tolerance defined by the four categories of damage threats (A, B, C, and D in Fig. 2.23) are satisfied.

In particular, for undetectable damage, the structure must withstand the UL, and for a detectable damage, the structure must withstand loads between the LL and UL. This approach is summarised in Fig. 2.23 showing the required loads, depending on the damage detectability and the impact energy level. The most complicated part of this approach is obviously to determine the residual strength and the permanent indentation depending on the impact energy level, because it requires a large number of experimental tests and can be long and expensive. An alternative, of course, is to determine the data using a numerical model. We can then distinguish two approaches:

26 l

l

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

The numerical model is able to directly determine the residual strength and the permanent residual indentation [33] versus impact energy. It is the simplest case, because it is possible to numerically and easily optimise the composite structure to impact damage tolerance. This type of approach is the main approach used at the present time in the aeronautical field. The main drawback of this type of approach is that it is limited to the types of impact identified by the numerical model. The numerical model is able to completely simulate the impact damage, in particular with the permanent indentation, and the residual strength after impact. The main advantage of this type of approach is to be valid for a large range of impact and residual loads. But this type of model can be very complex and unreliable. In fact, the complete numerical simulation of impact damage and CAI is still a challenge and is the subject of much research [34–36].

The problem is more complex if the objective is to numerically optimise a composite structure to impact damage tolerance. In this case, only the load is known, and the objective is to determine the composite structure as light and/or as cheap as possible, reaching the conditions of impact damage tolerance (as well as other cases of required loading). Again, numerical models would be of great utility but are limited in their range of validity. Finally, the problem is further complicated by the paradoxical character of the two main criteria of impact damage tolerance: residual strength and permanent indentation. It is usually necessary to increase the thickness of a composite plate to increase its residual strength; but at the same time, it is usually necessary to reduce its thickness to increase the detectability of the impact damage. There may be cases where an increase of the plate thickness can paradoxically decrease its resistance to impact damage tolerance, for example, by making the damage undetectable.

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eriaux Composites, Hermes/Lavoisier, 2005. [2] A.J. Fawcett, G.D. Oakes, Boeing Commercial Airplanes, Boeing composite airframe damage tolerance and service experience, in: Presented at the FAA Composite Damage Tolerance and Maintenance Workshop held in Chicago, July 19–21, 2006. [3] M. Calomfirescu, F. Daoud, T. P€uhlhofer, A new look into structural design philosophies for aerostructures with advanced optimization methods and tools, in: IV European Conference on Computational Mechanics, Paris, France, 2010. [4] J. Rouchon, Certification of large airplane composite structures, recent progress and new trends in compliance philosophy, in: 17th ICAS Congress, Stockholm, 1990. [5] Joint Airworthiness Requirements 25 (JAR25), Part 1 Requirements, Part 2 Acceptable Means of Compliance and Interpretations, for Composite Structures: JAR25 § 25.603 and ACJ 25.603. [6] Federal Aviation Administration 25 (FAR25), Advisory Circular 25.571: Damage Tolerance and Fatigue Evaluation of Structure, 2011. [7] N. Alif, L.A. Carlsson, Failure mechanisms of woven carbon and glass composites, in: Composite Materials, Fatigue and Fracture, vol. 6, ASTM International, 1997, pp. 471–493. ASTM STP 1285. [8] M. Bizeul, Contribution a` l’
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Damage tolerance of composite structures

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[10] M. De Freitas, L. Reis, Failure mechanisms on composite specimens subjected to compression after impact, Compos. Struct. 42 (1998) 365–373. [11] ASTM D7137/D7137M, Standard Test Method for Compressive Residual Strength Properties of Damaged Polymer Matrix Composite Plates, 2005. [12] J.M. Berthelot, Mat
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ees par un impact basse vitesse-applications aux structures a
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ements finis associ
ee, Rapport interne EADS CCR, 1994. [18] C. Bouvet, N. Hongkarnjanakul, S. Rivallant, J.J. Barrau, Chapter 8: Discrete impact modelling of inter- and intra-laminar failure in composites, in: Dynamic Failure of Composite and Sandwich Structures, Springer, Dordrecht, Heidelberg: New York, London, 2012, pp. 339–392. [19] AITM1-0010, Airbus Test Method: Determination of Compression Strength after Impact, Blagnac, 2005. [20] H.A. Kinnison, Aviation Maintenance Management, McGraw-Hill, New York, 2004. [21] M. Thomas, Study of the evolution of the dent depth due to impact on carbon/epoxy laminates, consequences on impact damage visibility and on in service inspection requirements for civil aircraft composite structures, in: MIL-HDBK 17 Meeting, Monterey, CA, 1994. [22] US MIL-HDBK-17, Composite Materials Handbooks, Department of Defense of United States of America, 1997. [23] J. Rouchon, Fatigue and damage tolerance aspects for composite aircraft structures, in: Proceedings of the ICAF Symposium, ICAF-DOC-2051. Delft, The Netherlands, 1995. [24] A. Tropis, M. Thomas, J.L. Bounie, P. Lafon, Certification of the composite outer wing of the ATR72, J. Aerosp. Eng. Proc. Inst. Mech. Eng. Part G 209 (1994) 327–339. [25] A.E. Abi, C. Bouvet, S. Rivallant, B. Broll, J.J. Barrau, Experimental analysis of damage creation and permanent indentation on highly oriented plates, Compos. Sci. Technol. 69 (7–8) (2009) 1238–1245. [26] N. Hongkarnjanakul, S. Rivallant, C. Bouvet, A. Miranda, Permanent indentation characterization for low-velocity impact modelling using three-point bending test, J. Compos. Mater. 48–20 (2014) 2441–2454. [27] A.J. Smiley, R.B. Pipes, Rate sensitivity of mode II interlaminar fracture toughness in graphite/epoxy and graphite/PEEK, Compos. Sci. Technol. 29 (1987) 1–15. [28] S. Rivallant, C. Bouvet, A.E. Abi, B. Broll, J.J. Barrau, Experimental analysis of CFRP laminates subjected to compression after impact: the role of impact-induced cracks in failure, Compos. Struct. 111 (2014) 147–157.

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

[29] M.J. Laffan, S.T. Pinho, P. Robinson, Measurement of the fracture toughness associated with the longitudinal fibre compressive failure mode of laminated composites, Compos. Part A Appl. Sci. Manuf. 43 (2012) 1930–1938. [30] I.M. Daniel, O. Ishai, Engineering Mechanics of Composite Materials, Editions Oxford University Press, New York, 1994. [31] H.P. Kan, Enhanced Reliability Prediction Methodology for Impact Damaged Composite Structures, US Department of Transportation, FAA, Office of Aviation Research, 1998. Report No. DOT/FAA/AR-97/79. [32] R.C. Alderliesten, Damage tolerance of bonded aircraft structures, Int. J. Fatigue 31 (6) (2008) 1024–1030. [33] V. Lopresto, G. Caprino, Damage mechanisms and energy absorption in composite laminates under low velocity impact loads, in: Dynamic Failure of Composite and Sandwich Structures, Springer, Dordrecht, Heidelberg: New York, London, 2012 (Chapter 6). [34] S. Rivallant, C. Bouvet, N. Hongkarnjanakul, Failure analysis of CFRP laminates subjected to compression after impact: FE simulation using discrete interface elements, Compos. Part A 55 (2013) 83–93. [35] W. Tan, B.G. Falzon, L.N.S. Chiu, M. Price, Predicting low velocity impact damage and compression-after-impact behaviour of composite laminates, Compos. Part A Appl. Sci. Manuf. 71 (2015) 212–226. [36] E.V. Gonza´lez, P. Maimı´, P.P. Camanho, A. Turon, J.A. Mayugo, Simulation of dropweight impact and compression after impact tests on composite laminates, Compos. Struct. 94 (2012) 3364–3378.

Dynamic interfacial fracture

3

Tianyu Chena,b, Christopher M. Harveya,c, Simon Wanga,c, and Vadim V. Silberschmidtd a Department of Aeronautical and Automotive Engineering, Loughborough University, Loughborough, Leicestershire, United Kingdom, bKey Laboratory of Microgravity (National Microgravity Laboratory), Institute of Mechanics, Chinese Academy of Sciences, Beijing, China, cSchool of Mechanical and Equipment Engineering, Hebei University of Engineering, Handan, China, dWolfson School of Mechanical, Electrical and Manufacturing Engineering, Loughborough University, Loughborough, Leicestershire, United Kingdom

3.1

Introduction

The dynamics of interfacial fracture significantly affects the fracture behaviour including crack initiation, propagation and arrest, and material properties such as dynamic fracture toughness. There are therefore strong motivations for analytical modelling of dynamic interfacial fracture to achieve a mechanical understanding or to facilitate postprocessing of experimental fracture data for the assessment of material properties, vital for maintaining structural integrity and preventing dynamic failure. Early investigations into dynamic interfacial fracture did not fully consider the dynamic regime in their analytical models, assuming instead a quasistatic motion. This led to contradictory findings reported in the literature concerning rate effects on fracture toughness. Aliyu and Daniel [1], for instance, conducted tests using double cantilever beams (DCBs) to measure the mode I fracture toughness of AS4/ 3501–6 epoxy/carbon fibre composites at loading rates between 8.5
106 and 8.5
103 m s1 and concluded that fracture toughness increased with increasing loading rates. Mall et al. [2], however, measured the fracture toughness of PEEK/carbon fibre composite at loading rates between 8.5
106 and 1.67
102 m s1 and found the opposite, with fracture toughness decreasing with increasing loading rates. At first thought, material properties might seem a possible explanation since PEEK as a thermoplastic is relatively ductile compared to brittle epoxy, with the materials having different rate dependencies. This is not always the case, however, as when the same material AS4/3501–6 epoxy/carbon fibre composite was tested in [3] at loading rates between 4.2
106 and 6.7
101 m s1, the fracture toughness was found to decrease with increasing opening rates, opposite to the conclusion in [1] for the same material. Comprehensive reviews on the loading effect on fracture toughness can be found in [4–6], but no conclusions concerning rate effects on it could be drawn. Experimental studies for the rate effects on mode II fracture toughness can be found in [7–14] and in a comprehensive review in [15], but—just as for the mode I investigations—there were no unanimous conclusions regarding fracture toughness rate effects. Dynamic Deformation, Damage and Fracture in Composite Materials and Structures. https://doi.org/10.1016/B978-0-12-823979-7.00004-1 Copyright © 2023 Elsevier Ltd. All rights reserved.

30

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

In general, a conventional analytical approach to modelling dynamic interfacial fracture is based on [16]. The kinetic energy is considered significant for the ‘crack driving force’ or energy release rate (ERR) and so included in a global energy balance approach, following Griffith, but based on quasistatic motion without any consideration of vibration. This conventional analytical approach, however, only provides a ‘smoothed’ dynamic ERR. Moreover, the kinetic energy contribution to the dynamic ERR is simply a baseline shift from its quasistatic component. Surprisingly, experimental data postprocessed using this approach appear to show that the dynamic effect under high loading rates is negligible [3,7,14,17]. This contradicts other experimental and numerical investigations, demonstrating that the dynamic ERR is, in general, oscillating. Meanwhile, in the absence of an analytical modelling capability to properly address the dynamic effect, hybrid experimental numerical approaches were also used. Recently, the authors have developed a complete analytical framework for dynamic interfacial fracture by including structural vibration and wave propagation. The resulting solutions show excellent agreement with experimental results and numerical simulations. The framework encompasses newly developed techniques for stationary and propagating cracks, quasistatic motion and vibration, mode I and mode II fractures, local and global approaches, and a variety of derived solutions for certain configurations and also includes rigid and elastic interfaces (although not reported here). It is referred to as a ‘framework’ since the collection of techniques will in general allow the application to new configurations and fracture mechanics phenomena, for example mixed-mode fracture. In this chapter, following a review of the conventional analytical approach for dynamic interfacial in Section 3.2, the authors’ recent analytical models and solutions for dynamic interfacial fracture are described in Sections 3.3 to 3.5. Mode I stationary cracks, mode I propagating cracks, and mode II stationary cracks are covered, respectively. Conclusions are given in Section 3.6.

3.2

Conventional analytical approach to modelling dynamic interfacial fracture

The conventional analytical approach to modelling dynamic interfacial fracture accounts for kinetic energy in the global energy balance. Fundamental structures such as those shown in Fig. 3.1 are typically considered, including double cantilever beams (DCBs), end-loaded split (ELS), and end-notched flexure (ENF) specimens. The applied loading velocity is extrapolated along with the specimen and then used to determine the kinetic energy. The ERR is subsequently determined as Gglobal ¼ d(Wext  U  K)/dA0 where Wext is the external work done, U is the strain energy, K is the kinetic energy, and A0 is the crack area. This global approach to determining the ERR dates back to Mott [16], a direct extension of Griffith’s approach by including the kinetic energy. There is, however, another local approach [18] based on a crack-tip energy flux integral. The ERR is

Dynamic interfacial fracture

a)

31 w0

b)

w0

a

L

a

h

h

h

h L0

w0

c)

w0

a

L

h h 2L

Fig. 3.1 Specimen configurations: (A) DCB; (B) ELS; (C) ENF.

determined as Glocal ¼ FðΩÞ=A_ 0 , where F(Ω) Ðis the energy  flux integral  into a small _ 1 =2 ds. contour Ω surrounding the crack tip: FðΩÞ ¼ Ω σ ij nj u_i + σ ij εij + ρu_i u_i an

3.2.1 Mode I fracture 3.2.1.1 Stationary cracks in DCBs Smiley and Pipes [3] were the first to account for kinetic energy and derive an expression for the dynamic ERR of a DCB under opening displacement of constant rate. They used their solution to postprocess experimental data to measure the interlaminar fracture toughness of epoxy/carbon-fibre and PEEK/carbon-fibre composites and to investigate the effect of loading rate on fracture toughness. Their analytical model incorporates the kinetic energy of quasistatic motion of the DCB arms by extrapolating the applied constant opening velocity v at the free end along the length of the DCB arm. This extrapolation gave the distribution of the applied constant opening velocity along one DCB arm as  w_ ðxÞ ¼ v

 3x2 x3 ,  2a2 2a3

(3.1)

where w(x) is the deflection of the DCB arm with an effective boundary condition (see Fig. 3.3B), w_ ðxÞ is the transverse velocity of the DCB arm, a is the crack length, and x is the coordinate along the length of the arm from the crack tip towards the free end with x ¼ 0 at the crack tip. The kinetic energy for one DCB arm is then calculated as 1 K ¼ ρA 2

ða 0

½w_ ðxÞ2 dx ¼

33 ρAv2 , 280

(3.2)

32

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

where ρ is the material density and A is the cross-sectional area. The contribution to the total ERR from the kinetic energy of the quasistatic motion is then 2dK/(b da) ¼  33ρAv2/(140b), where a factor of two was introduced due to the symmetry of the DCB and b is the beam width. The total dynamic ERR by the global approach is therefore G¼

P2 a2 33ρAv2  , bEI 140b

(3.3)

where P is the external load at the location of the applied opening displacement, E is the Young’s modulus, and I is the second moment of cross-sectional area. Smiley and Pipes used this analytical solution to determine the dynamic ERR in their experiments but found that the kinetic energy contribution of 33ρAv2/(140b) was negligible. Blackman et al. [17], using the same technique, derived an equivalent expression for the dynamic ERR of the DCBs under constant opening rate displacement as G¼

9EIv2 t2 33ρAv2 :  140b ba4

(3.4)

Eqs (3.3) and (3.4) are theoretically equivalent, but they imply a difference of what experimental data should be recorded in DCB tests to determine the fracture toughness (or critical ERR). In Smiley and Pipes’ work, the critical load Pc for crack initiation was supposed to be recorded and used in Eq. (3.3), whereas Blackman et al. [17] proposed that the time for crack initiation should be employed instead. This argument came from experimental observations of considerable oscillation in the externally applied force at high loading rates, and so this load could not be accurately recorded for crack initiation. Smiley and Pipes’ approach was nevertheless suitable for their own study [3] since the maximum opening rate was 0.67 m s-1 whereas in Blackman et al.’s study it was 15 m s-1.

3.2.1.2 Propagating cracks in DCBs Based on the same theoretical principle of including the kinetic energy of the quasistatic motion in the global energy balance, Blackman et al. [17] proposed a dynamic ERR solution for a propagating crack in DCBs. The kinetic energy was calculated from the transverse velocity, but this time including a contribution from crack propagation, that is, w_ ðxÞ ¼

dw dw + a_ , dt da

(3.5)

_ where a_ is the crack propagation speed and adw=da is the contribution from the crack propagation.

Dynamic interfacial fracture

33

This method seems plausible, but it depends on how the crack propagation speed a_ is determined. Blackman et al. [17] assumed the continuous crack propagation with G ¼ Gc at all times after initiation and constant fracture toughness. According to the quasistatic solution, these assumptions give a crack propagation speed of a_ ¼ a=ð2tÞ. Based on this crack propagation speed, the total kinetic energy (with a factor of two due to symmetry) is 1 K ¼ 2  ρA 2

ða  0

 dw dw 2 111 + a_ ρAav2 : dx ¼ dt da 280

(3.6)

Then, using the global approach, the total dynamic ERR for a propagating crack is G¼

9EIv2 t2 111ρAv2 :  280b ba4

(3.7)

Comparing Eqs (3.4) and (3.7) demonstrates that the ERR of a steadily propagating crack is less than that of a stationary crack due to the higher kinetic energy of the former. This method to determine the ERR of a propagating crack has some intrinsic shortcomings. One is that fracture toughness can be rate-dependent under high loading rates, and another is that experimental observations of DCB tests at high loading rates show that crack propagation is not continuous but, in fact, nonsmooth with a mixture of stable growth, fast unstable growth and arrest. This crack propagation behaviour is referred to as a ‘stick-slip’ propagation and was widely observed in many studies [8,19–24]. Together, this means that the crack propagation speed will not in general follow a_ ¼ a=ð2tÞ. That said, however, the kinetic energy contribution in the second term of Eq. (3.7) is small compared to the strain energy contribution (first term), and so the method could still be applicable for the low opening rates. In any case, the conventional approach to modelling dynamic interfacial fracture cannot capture stick-slip propagation and instead provides a ‘smoothed’ dynamic ERR.

3.2.2 Mode II fracture Dynamic mode II interfacial fracture, compared to mode I, received less research attention. The most common configurations to study mode II fracture in the quasistatic loading regime are ENF [25] and ELS [26] specimens. Other configurations can also be used, for instance, centre notch flexural (CNF) specimens [27,28]. The conventional analytical approach to study the dynamic effect in mode II fracture is to include the kinetic energy of the extrapolated quasistatic motion, similar to as described for mode I fracture. In this way, ENF [7], ELS [14], and CNF [12] configurations were considered. A crack growth in the ENF configuration is unstable and, thus, it can only be used to investigate crack initiation. For ELS specimens, as reported in [14], the dynamic effect was negligible, contributing less than 1% to the measured fracture toughness value. It was therefore assumed in [14] that the dynamic

34

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

contribution was also negligible for propagating cracks in ELS specimens, and so no analytical model for propagating cracks was developed.

3.2.2.1 Stationary cracks in ENF specimens Smiley and Pipes [7] conducted tests using ENF specimens to study the rate sensitivity of mode II interlaminar fracture toughness. To support this, they developed the following analytical solution for the dynamic ERR accounting for the kinetic energy contribution for the ENF specimen with the crack length ratio of a/L ¼ 0.5 (Fig. 3.1C): G¼

3P2 a2 ρAv2 + 0:078 : 64bEI b

(3.8)

Eq. (3.8) was still based on the global approach, with the kinetic energy determined by extrapolating the externally applied constant velocity v from the crosshead speed of the test machine. The kinetic energy contribution to the fracture toughness measured in their experiments using Eq. (3.8) was, however, negligible [7]. It contributed less than 0.01% for applied velocities between 4.6
106 and 9.2
102 m s1. Kusaka et al. [10] conducted a mode II fracture test using Hopkinson bars. They measured the kinetic energy term in Eq. (3.8) as ranging from 0.2% to 3% and concluded that the dynamic effect could be ignored. Cantwell [29] came to the same conclusion as the second term in Eq. (3.8) contributed less than 0.05% in their experiments with loading rates up to 3 m s1.

3.2.2.2 Stationary cracks in ELS specimens The conventional dynamic ERR solution for the ELS specimen was given by [14]. The same analytical approach was applied, but the kinetic energy contribution due to the extrapolated quasistatic motion from the applied constant loading velocity was derived as a function of crack length ratio η, where η ¼ a/(a + L) (Fig. 3.1B). The total ERR was derived as 108EIa2 v2 t2 G¼  2 b L3 "+ 3L2 a + 3La2 + 4a3  # ρAv2 η2 297 9 2 9 4 27 5 459 7 : + 9η  η + η + η + η   b ð1 + 3η3 Þ3 70 2 5 10 70

(3.9)

The kinetic energy contribution to the ERR—the last term in Eq. (3.9)—is still very small compared to the ERR component due to the quasistatic component, the first term in Eq. (3.9). As reported in [14], for the epoxy/carbon-fibre composite specimen under 5 m s-1 loading rate and with crack length ratio η ¼ 0.6, the kinetic energy contribution from the second term in Eq. (3.9) was only 5 N m-1, while the quasistatic component from the first term in Eq. (3.9) was 800 N m-1. It was therefore concluded in [14] that the kinetic energy contribution could be ignored.

Dynamic interfacial fracture

35

3.2.2.3 Stationary cracks in CNF specimens Maikuma et al. [12] conducted experiments using the CNF specimen (Fig. 3.2) under impact loading to study the dynamic mode II fracture behaviour of CFRPs and developed an analytical solution for the ERR to postprocess the experimental data. The dynamic ERR solution for the CNF specimen with the crack length ratio a/ L ¼ 0.5 is G¼

3P2 a2 ρAv2  0:633 , 256bEI b

(3.10)

where P is determined by the impact load history and v is the impact velocity. The latter ranged in their experiments from 1.25 m s-1 to 3.00 m s-1, and within this range, they found the kinetic energy contribution from the second term in Eq. (3.10) to be less than 1%.

3.2.3 Experimental–numerical hybrid method The experimental–numerical hybrid method is usually used when there are no available analytical solutions to postprocess the experimental data. Experimentally measured parameters, such as external force, crack length vs time curve and crack propagation speed, are incorporated into numerical models, and the desired fracture parameters, e.g. ERR and fracture toughness, are calculated by finite element method (FEM) simulations. Nishioka and Atluri [30] introduced an experimentally obtained crack length time curve for a wedge-loaded DCB into finite element models with their developed singular elements to determine the dynamic stress intensity factor (SIF). Guo and Sun [31] incorporated a similar curve for a DCB into a finite element model and developed a scheme of sequentially releasing nodes to calculate the dynamic ERR as G¼

1 ðΔALLSE + ΔALLKEÞ, Δa

(3.11)

where ALLSE is the total elastic strain energy and ALLKE is the total kinetic energy, both directly obtained in FEM simulations. Tsai et al. [32] followed the same approach to study mode II fracture in ENF specimens. They used the virtual crack closure technique (VCCT) to calculate the dynamic ERR and obtain results, which were demonstrated to be equivalent to Eq. (3.11). Kumar and Kishore [33] studied DCBs under

2a

2L

Fig. 3.2 Configuration of CNF test specimen.

36

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

impact loading and used the experimentally measured deflection and crack propagation speed to calculate the J-integral to determine the fracture toughness. Liu et al. [34] conducted DCB tests under impact using Hopkinson bars and incorporated the experimental data into a finite element model with the VCCT to determine the fracture toughness. Subsequently, Liu et al. [35] also used a cohesive-zone model (CZM) to postprocess experimental data to determine the fracture toughness.

3.3

Dynamic mode I interfacial fracture for stationary crack

3.3.1 Theoretical development with vibration To study the dynamic effect on mode I ERR, equal and opposite time-dependent displacement w0(t) are applied to the free-ends of DCB arms (Fig. 3.3A). The widely used method for analysing DCB considers the crack region of the DCB arm and assumes an effective boundary condition for the crack tip (Fig. 3.3B). This condition does not allow the crack tip to rotate and is discussed in Section 3.3.4. The DCB arm is assumed to be thin, where h ≪ a, and a classical idea of Euler-Bernoulli beam applies; the displacement is small, and no longitudinal forces are developed. In addition, no interfacial contact is assumed between two DCB arms, and the DCB arm with the effective boundary condition (Fig. 3.3B) can vibrate freely.

3.3.1.1 Dynamic response of thin beam For the DCB arms under constant opening rate v with the corresponding displacement w0(t) ¼ vt, the dynamic transverse response (deflection) of the DCB arm as shown in Fig. 3.3B can be derived by introducing a shifting function following Grant’s method [36], the transverse deflection of the beam is of the form wðx, tÞ ¼ wfv ðx, tÞ + FðxÞvt,

(3.12)

where wfv(x, t) is the free-vibration component and F(x) is the shifting function. Note that the product of applied displacement w0(t) ¼ vt and shifting function F(x) a)

b)

w0 ( t ) = vt

z

w0 ( t ) = vt

h

x

h

( 0,0 )

a

( a,0 )

w0 ( t ) = vt

Fig. 3.3 (A) Symmetric double cantilever beam; (B) effective boundary condition.

Dynamic interfacial fracture

37

represents the quasistatic motion of the DCB arm, and, therefore, the total dynamic response is deemed to be a combination of free vibration and quasistatic motion. The derivation of free vibration component and shifting function are given in [37]: rffiffiffiffiffiffi ∞   ρA X Λi 1 3 3 2 ϕ ðxÞ sin ðωi tÞ +  3 x + 2 x vt wðx, tÞ ¼ va EI i¼1 λ3i i 2a 2a " rffiffiffiffiffiffi  # ∞ ρA X Λi 1 3 3 2 2 ϕ ðxÞsin ðωi tÞ +  3 x + 2 x t , ¼v a EI i¼1 λ3i i 2a 2a 2

(3.13)

where λi is the solution for frequency equation tanh(λi)  tan (λi) ¼ 0 and pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi Λi ¼ ð1Þi σ 2i + 1 + σ 2i  1 with σ i ¼ [cosh(λi)  cos (λi)]/[sinh(λi)  sin (λi)]. The solutions for λi, σ i, and Λi are given in Table 3.1; ϕi(x) is the ith mode shape given as ϕi(x) ¼ [cosh(βix)  cos (βix)]  σ i[sinh(βix)  sin (βix)] and ωi is the ith mode pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi natural frequency given as ωi ¼ λ2i a2 EI=ðρAÞ with λi ¼ βia. The product of shifting function and applied velocity vF(x) ¼ v[ x3/(2a3) + 3x2/(2a2)] is also the quasistatic motion of the DCB arm and it has the same expression with the solution for quasistatic motion Eq. (3.1) in the conventional analytical scheme. Another notion should be noticed is that Λi comes from the initial modal velocity for pffiffiffiffiffiffiffiffi T_ i ð0Þ ¼ v ρAaΛi =λi , which is a coupling between the normal mode and the applied velocity via shifting function. Note that rather than only accounting for quasistatic motion in the conventional analytical scheme in Eq. (3.1), the dynamic response now in Eq. (3.13) also considers the free vibration due to the applied time-dependent displacement, and the significance is that the transverse motion now is a combination of local free vibration and quasistatic motion; their interaction or coupling should be also considered to study dynamic fracture. Another notion is that the total transverse deflection is proportional to the applied constant opening rate v, and terms in the bracket in Eq. (3.13) are an inherent property of a given DCB configuration.

Table 3.1 Modal parameters for fixed pinned beam in free vibration. Mode number

λi

σi

Λi

1 2 3 4 5 i>5

3.92660231 7.06858275 10.21017612 13.35176878 16.49336143 (4i + 1)π/4

1.000777304 1.000001445 1.000000000 1.000000000 1.000000000 1.0

1.375327127 1.415914585 pffiffiffi  2 pffiffiffi 2 pffiffiffi  2 pffiffiffi ð1Þi 2

38

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

3.3.1.2 Dynamic energy release rate and amplitude divergence The total transverse deflection in Eq. (3.13) is therefore used to calculate the strain and kinetic energies of the vibrating DCB arm under the applied time-dependent displacement as required to determine the ERR by the global approach similar to the conventional analytical scheme. (1) Strain and kinetic energies

Ð The strain energy of one DCB arm in Fig. 3.3B is U ¼ a0 M2(x, t)dx/(2EI), where M(x, t) ¼ EIw(2)(x, t), which is the internal bending moment. The strain energy of the vibrating DCB arm with a constant opening rate displacement at its free end by combining the total deflection Eq. (3.13) is therefore ð i2 h i2

1 a h ð2Þ ð2Þ ð2Þ ð2Þ dx EIwfv ðx, tÞ + 2EIwfv ðx, tÞF ðxÞvt + F ðxÞvt U¼ 2EI 0 (3.14) ∞ X 1 Λ2i 3EIv2 t2 2 2 sin ðωi tÞ + : ¼ ρAav 2 2a3 2 i¼1 λi The kinetic energy of one DCB arm in Fig. 3.3B is K ¼ ρA therefore ða  2  1 w_ fv ðx, tÞ + 2w_ fv ðx, tÞFðxÞv + v2 F2 ðxÞ dx K ¼ ρA 2 0

Ða 0

½w_ ðx, tÞ2 dx=2, and it is

∞ ∞ X X 1 Λ2i Λ2i 33 2 2 ρAav2 : cos ð ω t Þ  ρAav cos ðωi tÞ + ¼ ρAav2 t 2 2 2 280 λ λ i¼1 i i¼1 i

(3.15)

(2) Dynamic energy release rate

The dynamic ERR of the DCB shown in Fig. 3.3A (i.e. comprising two single arms in Fig. 3.3B with equal and opposite displacements) is now obtained using the global approach Gglobal ¼ d(Wext  U  K)/dA0, accounting for a global energy balance together with Eqs (3.14) and (3.15) (under displacement control Wext ¼ 0 during the crack growth), which gives 1d ðWext  U  K Þ b da 9 8 9EIv2 t2 > > > > > > > > a4 > > > > > > > > > > > > p ffiffiffiffiffiffiffiffiffiffi ffi < ∞ ∞ ∞ 2 2= 2 X X X 1 4 ρAEI v t Λ Λ 2 2 2 i i : ¼ + Λi sin ðωi tÞ + 2ρAv cos ðωi tÞ  ρAv 2 2> b> a2 > i¼1 i¼1 λi i¼1 λi > > > > > > > > > > > > > > > > > ; :  33 ρAv2 140 (3.16)

G¼

Dynamic interfacial fracture

39

Note that the natural frequency ωi is a function of crack length a with λi ¼ βia and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ωi ¼ λ2i a2 EI=ðρAÞ, and this was taken into account when differentiating the corresponding energy terms to derive dynamic ERR in Eq. (3.16). In Eq. (3.16), the ERR components with their sources are: (1) the first term is the contribution of the strain energy of the quasistatic motion denoted GU st ; (2) the second and third terms are due to the kinetic energy of coupling between the local vibration and the quasistatic motion, and the fourth term is due to the strain and kinetic energies of the local vibration, and therefore, these three terms are grouped together and denoted Gvib since they are vibration-related ERR component; (3) the last term is the ERR component due to the kinetic energy of the quasistatic motion and denoted GK st . Thus, the total dynamic ERR now can be written accordingly as G ¼ Gst + Gvib K ¼ GU st + Gst + Gvib ,

(3.17)

K where Gst is the ERR component due to quasistatic motion for Gst ¼ GU st + Gst . Note K that Gst is also the kinetic energy contribution in the conventional analytical scheme and Gst has the same expression as the kinetic energy contribution in conventional dynamic interfacial fracture; therefore, actually, in the conventional analytical studies, only the ERR due to quasistatic motion was derived by neglecting the vibration-related ERR component Gvib. In addition, the total dynamic effect is then Gdyn ¼ Gvib + GK st .

(3) ERR divergence and ERR with first vibration mode accuracy

The total dynamic ERR in Eq. (3.16) is based on the global approach accounting for kinetic energy contribution in the global energy balance via Gglobal ¼ d(Wext  U  K)/dA0, which was used extensively in linear elastic fracture mechanics (LEFM) and the conventional analytical scheme for dynamic interfacial fracture; thus, it was supposed to be able to predict the ERR accurately. But a close examination of the ERR derived in Eq. (3.16), particular for the ERR component due to vibration Gvib, reveals that the ERR determined in Eq. (3.16) provides a nonphysical and nonmechanical solution since the amplitude of the ERR component due to vibration Gvib shows a divergent fashion with adding more vibration modes, and its amplitude is with no bound. This divergence in the ERR amplitude including more vibration modes comes from P pffiffiffiffiffiffiffiffiffiffiffi 2 the term of 4 ρAEI v2 ta2 ∞ i¼1 Λi sin ðωi tÞ in Gvib. For the ith vibration mode, the amplitude of this ERR component is proportional to Λ2i . As Table 3.1 shows, the value pffiffiffi of Λi can be approximately taken as Λi  ð1Þi 2, so Λ2i  2, and this leads to the phenomenon that by adding more vibration modes the amplitude of Gvib keeps increasing. This phenomenon of divergence is also shown in Fig. 3.8 (Section 3.3.3.2). But the verification demonstrates that the dynamic ERR with only the first vibration mode can capture the oscillation in the FEM results and seems to be the envelope; and therefore, the dynamic ERR with the first vibration mode accuracy is proposed as

40

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

8 9EIv2 t2 33 > >  ρAv2 > 140 1 < a4 G¼ pffiffiffiffiffiffiffiffiffiffiffi b> 4 ρAEI v2 t 2 Λ2 Λ2 > > + Λ1 sin ðω1 tÞ + 2ρAv2 21 cos ðω1 tÞ  ρAv2 21 : 2 a λ1 λ1

9 > > > = > > > ;

:

(3.18)

Note that the first vibration mode has the lowest natural frequency, and, therefore, Eq. (3.18) is approximately applicable when the duration of the external load is long for the low-frequency vibration mode to be significant. (4) Dynamic factor

Based on the derived dynamic ERR in Eq. (3.18), a dynamic factor can be defined to investigate the dynamic effects. In the derived dynamic ERR with the first vibration mode accuracy in Eq. (3.18), the first term is the ERR component due to the strain energy of the quasistatic motion GU st , which is also the strain ERR in the quasistatic loading regime without any dynamic effect, and accordingly, the remaining terms can be grouped together as the dynamic component Gdyn, so that the total dynamic ERR is G ¼ GU st + Gdyn. Therefore, the dynamic factor fdyn is defined by fdyn ¼ Gdyn/GU st . A characteristic time can be pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi introduced to study the property of fdyn, that is, τ ¼ ta2 EI=ðρAÞ. Then, the dynamic factor can be written as a function of this characteristic time: Gdyn GU st    11 1 4 1 2  2  1 Λ21 1  ¼ Λ1 sin λ1 t + 2 2 2cos λ21 t  1  : 9τ 9 λ1 τ 420 τ2

fdyn ¼

(3.19)

Based on the above definition, the total dynamic ERR is then given by G ¼ GU st (1 + fdyn). The ERR is proportional to the static ERR with the dynamic factor, which is determined by the characteristic time only. Note that the characteristic time is an inherent and universal property of the DCB with given material properties and structural configuration. And thus, the dynamic factor defined by Eq. (3.19) is also inherent and universal for DCBs. As Eq. (3.19) suggests, the dynamic factor fdyn attenuates significantly with respect to the characteristic time τ with oscillation, and it is plotted against the characteristic time as the solid line in Fig. 3.4 together with the dynamic factor in the conventional analytical scheme represented by the dashed line. The conventional dynamic factor is derived from Eq. (3.4) and related to the characteristic time τ, which is found to be fdyn ¼ 11τ2/420, and this is also the last term of Eq. (3.19). As expected, the dynamic factor due to structural vibration oscillates, while the dynamic factor defined by the conventional analytical scheme is an inverse function of time without any oscillation; both dynamic factors decay with time. The oscillating dynamic factor is less than  1 during parts of the first two vibration periods, leading to the negative ERR. This finding is consistent with conventional

Dynamic interfacial fracture

41

Fig. 3.4 Dynamic factor vs characteristic time.

analytical investigations by Smiley and Pipes [3] and Blackman et al. [17], although they reported infinite negative ERR at t ¼ 0 or τ ¼ 0, whereas thanks to the dynamic ERR with the first vibration mode accuracy in Eq. (3.19), the ERR is always finite. In the literature, a negative ERR was also witnessed in the test for DCB under impact load [33] before crack initiation. According to the energy consideration, the negative ERR impedes crack propagation [38] because in this case the crack growth increases the potential energy rather than decreasing it [39]. At τ ¼ 0, the dynamic effect is at its maximum, and the limit of fdyn at τ ¼ 0 is lim fdyn ðτÞ  9:721:

τ!0

(3.20)

The dynamic factor, however, decays to 1.0 < fdyn < 1.0 after one characteristic time period. It then continues to drop steadily. After around 10 characteristic time periods, the dynamic factor reduces to 0.1 < fdyn < 0.1, which can be regarded as insignificant. Note that this dynamic factor is independent of the applied opening rate.

3.3.2 Theoretical development with wave propagation 3.3.2.1 ERR divergence and energy flux In Section 3.3.1, the ERR determined with the global approach including a kinetic energy contribution, that is, by Gglobal ¼ d(Wext  U  K)/dA0, is shown to be divergent in the ERR amplitude as more vibration modes are added. This is contrary to the physical and mechanical reality that unbounded ERR would lead to immediate rupture for any applied loading rate. This phenomenon of divergent ERR therefore implies that the global approach is not viable to study the dynamic fracture mechanics with beams or 1D waveguides in the context of wave propagation. This is an interesting finding since the global approach or global energy balance proposed by Griffith

42

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

[40] has long been used to study the fracture behaviour in the quasistatic loading regime of LEFM. Moreover, the strain ERR derived from the global energy balance is equivalent to the crack-tip-related magnitudes of SIF and J-integral. Smiley and Pipes [3] and Blackman et al. [17] extended the approach to include kinetic energy in the global energy balance, as proposed by Mott [16], to study the interfacial dynamic fracture. They did not, however, include vibration and, thus, they did not encounter this divergent ERR issue since their ERR is ‘smooth’ and not oscillating. Recall the alternative definition of dynamic ERR, based on a crack tip energy flux integral, that is, G ¼ FðΩÞ=A_ 0 , which is a local approach. If this local approach provides the correct dynamic ERR, as demonstrated in this section, and the global approach provides an unbounded nonphysical ERR since the amplitude is divergent as more vibration modes are added, then the following inequality can be written: 1 F ð ΩÞ 1 d ðWext  U  K Þ: < b a_ b da

(3.21)

This inequality shows that the actual amount of energy flowing into the crack tip (the left-hand side) is less than the amount of energy that can be potentially dissipated from the system (the right-hand side). In dynamics, beams are 1D waveguides and have a dispersive property, that is, flexural waves with higher frequencies travel faster [41,42]. The global approach for the global energy balance needs to be reconsidered in consideration of this since it treats the energy term from each vibration mode (flexural wave) indiscriminately by reckoning that they can be dissipated simultaneously. This, however, is not the case when looking at the crack tip for an infinitesimal time interval. The energy supplied to the crack tip by each vibration mode (flexural wave) is highly dependent on the speed of the flexural wave. A further analytical theory to address this issue with consideration for dispersion of wave propagation is developed, and it is called the dispersion-corrected global approach.

3.3.2.2 Dynamic energy release rate Combining Eqs (3.14) and (3.15), the total mechanical energy Π of the half DCB at a given time t is Π¼U+K ¼

∞ ∞ X 3EIv2 t2 33ρAav2 ρAav2 X Λ2i Λ2i + +  ρAav2 cos ðωi tÞ: 2 2 3 2a 280 2 i¼1 λi i¼1 λi

(3.22)

Note that the first and second terms in Eq. (3.22) are the strain and kinetic energies, respectively, due to the quasistatic motion; and the last two terms are the energy due to vibration. To aid in the following analytical development, these contributions are denK oted ΠU st , Πst , and Πvib, respectively.

Dynamic interfacial fracture

43

(1) ERR components due to quasistatic motion

Since the quasistatic motion is not dispersive, the ERR component due to the strain U energy of the quasistatic motion can be directly derived using GU st ¼ dΠst /dA0, and so does the ERR component due to the kinetic energy of the quasistatic motion K with GK st ¼ dΠst /dA0. These give the same solutions as the global approach provides K 2 2 4 2 in Eq. (3.4), where GU st ¼ 9EIv t /(ba ) and Gst ¼ 33ρAv /(140b) (a factor of 2 is applied due to the symmetry). (2) ERR component due to vibration

The ERR component due to vibration Gvib cannot be determined as  dΠvib/dA0, which leads to divergence as discussed before. Considering the definition in the local approach with G ¼ FðΩÞ=A_ 0 , where F(Ω) is the energy flux into a contour around the crack tip, and, accordingly, the ERR component due to vibration Gvib is calculated as Gvib ¼

Fvib ðΩÞ , A_ 0

(3.23)

where Fvib(Ω) is the energy flux due to vibration through the contour Ω shown in Fig. 3.5 where ε < < a. The energy flux due to vibration through the contour Ω can be calculated as Fvib ðΩÞ ¼ Evib C1p ,

(3.24)

where Evib is the total energy density due to vibration and C1p is the phase speed of the first-mode flexural wave since the first-mode flexural wave modulates all the other waves with higher frequencies [41,42], and, therefore, the average speed of total energy flux is C1p. To determine the total energy density due to vibration Evib, a small region 0 < x < ε in front of the crack tip is considered (the energy density in the small region  ε < x < 0 behind the crack tip is zero due to the effective boundary condition). The sign of the spatial distribution (represented by the normal modes) of the free vibration in this region is proportional to Λi, which alternates with vibration mode numbers

ε

vt

Ω h h

2ε

vt a

Fig. 3.5 Crack tip contour Ω for DCB to determine ERR component due to vibration.

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

pffiffiffi since Λi  ð1Þi 2 (Table 3.1). Note that for each vibration mode or flexural wave, its free vibration component has contributions from both the space and time domains. The contribution from the space domain is the spatial distribution of free vibration, which is proportional to Λi; the contribution from the time domain oscillates with pffiffiffi sin(ωit). Therefore, Λi  ð1Þi 2 means that the contribution of flexural waves with odd mode numbers tends to close the crack and decrease the total energy density in the space domain, while those with even mode numbers tend to open the crack and increase the total energy density. The total energy density due to vibration Evib in the contour is therefore Evib ¼

∞ X

ð1Þi Eivib ,

(3.25)

i¼1

where Eivib is the energy density due to the ith mode flexural wave (ith vibration mode). pffiffiffiffiffiffiffiffi Note that Λi comes from the initial modal velocity for T_ i ð0Þ ¼ v ρAaΛi =λi , which is the coupling of the normal mode Wi(x) (representing the free vibration) and the quasistatic motion vF(x) along the beam. The physical interpretation of Eq. (3.25) is therefore that when the spatial velocity of free vibration of one flexural wave, represented by the normal mode, is in the same direction as the applied opening velocity, this flexural wave opens the crack and increases the energy density in the contour in the space domain. Likewise, when the spatial velocity of free vibration of one flexural wave, represented by the normal mode, is in the opposite direction to the applied opening velocity, this flexural wave closes the crack and decreases the energy density in the contour in the space domain. The energy flux of the ith mode flexural wave Fivib(Ω) is Fivib(Ω) ¼ EivibCig, where i Cg is the wave’s group speed since the energy of a wave propagates at its group speed [43]. Now, combining Eqs (3.24) and (3.25) gives Fvib ðΩÞ ¼

∞ X

ð1Þi Fivib ðΩÞfi ,

(3.26)

i¼1

where fi ¼ C1p/Cig. The ith mode flexural wave energy flux is Fivib(Ω) ¼  dΠivib/dt, where Πivib ¼ ρAav2Λ2i /λ2i  2ρAav2Λ2i /λ2i cos (ωit) from Eq. (3.22) (but with a factor of 2 applied since Eq. (3.22) is for a half DCB). Substituting these results into Eq. (3.26) and combining with Eq. (3.23) gives the ERR component due to vibration as pffiffiffiffiffiffiffiffiffiffiffi ∞ 4 ρAEI v2 t X ð1Þi Λ2i fi sin ðωi tÞ Gvib ¼ 2 ba i¼1 ∞ ∞ 2ρAv2 X Λ2 ρAv2 X Λ2 + ð1Þi 2i fi cos ðωi tÞ  ð1Þi 2i fi : b i¼1 b i¼1 λi λi

(3.27)

Dynamic interfacial fracture

45

(3) Correction factor for dispersion

In deriving the ERR component due to vibration, a factor fi is introduced in Eqs (3.26) and (3.27), which is used to address the dispersive property of beams as highly dispersive waveguides. This factor is for accurate assessment of the amount of energy supplied to the crack tip to determine the ERR based on the global approach. Note that the global approach accounting for the global energy balance overestimates the energy supplied to the crack tip, and fi corrects this by considering the dispersive property of wave propagation. Therefore, this factor is called the correction factor for dispersion. pffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The phase speed of the first-mode flexural wave is C1p ¼ ω1 4 EI=ðρAÞ, the group pffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi speed of the ith mode flexural wave is Cig ¼ 2 ωi 4 EI=ðρAÞ [41–43]; the relationship pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi between the natural frequencies and frequency parameters λi is ωi ¼ λ2i a2 EI=ðρAÞ, so the correction factor for dispersion can be written as C1p

1 fi ¼ i ¼ Cg 2

rffiffiffiffiffiffi ω1 λ 1 ¼ : ωi 2λi

(3.28)

The value for the correction factor for dispersion fi is presented in Table 3.2 for different vibration mode numbers. The correction factor for dispersion fi decreases with increasing mode number as Table 3.2 shows. Also, note that the amplitude of each mode for the ERR component due to vibration Gvib is proportional to fi as shown in Eq. (3.27). This allows the amplitude ratio to be studied; for instance, the amplitude of the fifth vibration mode is only 12% of that for the first mode. This indicates that the dominant mode is the first, which partially justifies the approximation for the ERR with first-mode accuracy in Section 3.3.1.2; in addition, higher modes (i.e. i > 5) are not significant and can be reckoned as noise since their amplitudes are smaller and frequencies are higher than for the first five modes. Another important aspect of this correction factor for dispersion is that it characterises the energy transmission ability of beams as waveguides to generate the ERR, and it is an inherent property of beams with a given set of boundary conditions as Eq. (3.28) suggests. The effective boundary assumption to study the DCB gives the fixed pinned boundary conditions, and the correction factor for dispersion fi is dimensionless and universal for DCBs with its value given in Table 3.2.

Table 3.2 Correction factor for dispersion for stationary crack. Mode number

1

2

3

4

5

i>5

Correction factor fi

0.5

0.27775

0.19229

0.14704

0.11904

2:49975 ð4i + 1Þ

46

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

(4) Total dynamic ERR

By combining results for each ERR component and the correction factor for dispersion, for a stationary crack in DCBs, the total dynamic ERR is therefore: Gglobal stationary ¼

pffiffiffiffiffiffiffiffiffiffiffi ∞ 9EIv2 t2 33ρAv2 4 ρAEI v2 t X +  ð1Þi Λ2i fi sin ðωi tÞ 140b ba4 ba2 i¼1 ∞ ∞ 2 2ρAv2 X ρAv2 X Λ2 i Λi ð1Þ 2 fi cos ðωi tÞ  ð1Þi 2i fi : + b i¼1 b i¼1 λi λi

(3.29)

Note that this ERR expression was derived based on the global approach accounting for the global energy balance along with the correction for the dispersive properties of beams as waveguides; it is therefore referred to as dispersion-corrected global ERR, and this approach is called dispersion-corrected global approach to distinguish it from the global approach.

3.3.2.3 Simplified dynamic ERR with vibrational deflection (1) Local approach with vibrational deflection

For a DCB under quasistatic loads, the ERR can be calculated as G ¼ M2(a, t)/(bEI), where M(a, t) ¼ EIw(2)(a, t) is the crack tip bending moment. The static ERR is a local quantity related to the crack tip since it is only a function of the crack tip bending moment. Similarly, for a crack propagating in a DCB under dynamic loads, Freund [18] derived the ERR using the crack tip energy flux integral as Glocal propagation ¼

  a_ 2 M2 ða, tÞ 1 2 , bEI C0

(3.30)

pffiffiffiffiffiffiffiffi where C0 is the longitudinal wave speed with C0 ¼ E=ρ. The dynamic ERR of a propagating crack is only a function of the crack propagation speed a_ and the crack tip bending moment. Since the ERR under quasistatic loads and the ERR of a propagating crack under dynamic loads can both be determined by the crack tip local quantities, it is an interesting question whether the dynamic ERR of stationary crack can be determined by setting a_ ¼ 0 in Eq. (3.30) and using the vibrational deflection in Eq. (3.13) to determine M(a, t). This would give a simplified solution for dynamic ERR for a stationary crack compared to Eq. (3.29). It should be recognised that Eq. (3.13) is a vibrational solution, which assumes adequate time for all the flexural waves to form standing waves. Eq. (3.30), however, considers the actual values of the crack tip quantities at a given time. This means that the stationary dynamic ERR calculated with Eqs (3.13) and (3.30) together can only become accurate after a certain period of time after t ¼ 0, at least after the establishment of all the standing waves. Furthermore, the calculated ERR is overestimated

Dynamic interfacial fracture

47

during this initial period since the calculation assumes that the energies of all flexural waves are immediately available at the crack tip, when, in fact, flexural waves need time to travel along the beam (with lower frequency ones travelling more slowly due to the dispersive property of flexural waves in 1D waveguides). By comparison, the dispersion-corrected global ERR in Eq. (3.29) accounts for wave dispersion (by considering the energy flux into a contour around the crack tip), even though it is also based on the vibrational deflection of Eq. (3.13). By combining Eqs (3.13) and (3.30) and setting a_ ¼ 0, the simplified dynamic ERR for a stationary crack of a DCB under dynamic loads is Glocal stationary

pffiffiffiffiffiffiffiffiffiffiffi ∞ 9EIv2 t2 12 ρAEI v2 t X Λi ¼ + sin ðωi tÞ 4 2 λ ba ba i¼1 i " #2 ∞ 4ρAv2 X Λi sin ðωi tÞ : + b λ i¼1 i

(3.31)

This dynamic ERR is referred to as local ERR since it is determined by the crack tip bending moment only and derived with the local approach but with the assumption that vibrational deflection is applicable. It consists of three components: the first term is the ERR component due to the quasistatic motion, the second term is the ERR component due to the coupling between the local vibration and the quasistatic motion, and the third term is the ERR component due to the local vibration. Note that Eq. (3.31) overestimates the ERR for a short period of time in the beginning as discussed since it assumes that all the flexural waves are immediately available at the crack tip using the vibrational deflection when, actually, flexural waves need time to travel, and ones with lower frequencies travel slower due to dispersion. (2) Equivalence of local and dispersion-corrected global approaches

Now consider the equivalence between the simplified ERR with the local approach in Eq. (3.31) and dispersion-corrected global ERR in Eq. (3.29), that is, under which condition Eq. (3.31) is applicable and potentially equivalent to Eq. (3.29). To achieve this, global a relative ERR difference between Glocal stationary and Gstationary is defined as  global Glocal stationary  Gstationary ΔG ¼ : GU GU st st

(3.32)

It is the difference between the ERRs from the two methods, divided by GU st (the ERR component due to strain energy of quasistatic motion, or, equivalently, the static ERR 2 2 4 without any dynamic effect). Both methods give the same GU st ¼ 9EIv t /(ba ), which makes it an appropriate choice as the normalisation factor when defining the relative ERR difference. To study the characteristics of this relative difference, a relative time scale n is also pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi defined: n ¼ t=τ0 ¼ λ1 a2 t1 EI=ðρAÞ. It is the time t divided by the time taken for the first-mode flexural wave to travel the crack length a. The phase speed of the first-

48

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mode flexural wave is C1p ¼ λ1 a1 EI=ðρAÞ, and the time for this wave to travel from the excitation point (i.e. the free end of the beam) to the crack tip is τ0 ¼ a/C1p. Based on Eqs (3.29), (3.31), and (3.32) as well as the definition for relative time scale n above, the relative ERR difference becomes    ∞ ΔG 2 λ1 X 6Λi n i 2 λ1 sin λi ¼  ð1Þ Λi λi 9 n i¼1 λi λ1 GU st   ∞ 2 1 λ21 X n i λ1 Λi + ð 1 Þ 1  2 cos λ i 18 n2 i¼1 λ1 λ3i "   #2 ∞ 4 λ21 X Λi n 11 λ21 + 2 sin λi + : 9 n i¼1 λi λ1 420 n2

(3.33)

Eq. (3.33) is dimensionless and universal for DCBs. Evolution of the relative ERR difference ΔG/GU st with the relative time scale n is shown in Fig. 3.6A based on the first ten mode flexural waves (or vibration modes). Fig. 3.6B presents the same data, but with the y axis transformed to enhance low-amplitude variations of the relative ERR difference for both positive and negative values. Apparently, the relative ERR difference is mainly positive. This indicates that the local ERR is larger than the dispersion-corrected global ERR, consistent with the previous discussion. The most significant overestimation happens in the range 0 < n < 5; for n > 5, it rapidly decreases to zero. For n > 12, the relative ERR difference is approximately 0.7 % < ΔG/GU st < 1.9%. The dispersion-corrected global and local ERR solutions can, therefore, be considered equivalent for predicting the dynamic ERR provided that enough time has passed, with 0  n  12 being the transition period for the vibrational solution to be applicable. Note that when n ¼ 2, the firstmode flexural wave returns to the free end, with all the standing waves established (also the establishment of vibration). The local ERR solution, however, still needs time to ‘even out’ the additional ‘artificial’ energy in comparison to the dispersioncorrected global ERR solution, explaining the duration of the transition period beyond n ¼ 12.

Fig. 3.6 Evolution of relative ERR difference between local and global methods with n.

Dynamic interfacial fracture

49

Another important aspect of this equivalence is that it provides some insight into whether the global and local approaches give an equivalent assessment of the dynamic ERR. Generally, in dispersive waveguides, as demonstrated in this section, the global approach using Gglobal ¼ d(Wext  U  K)/dA0 does not provide a physically and mechanically sound definition of the dynamic ERR. In dispersive waveguides, the dynamic ERR should be defined with the dispersion-corrected global approach instead, accounting for dispersion of the waveguides. It is demonstrated in this section that the dynamic ERR defined by the dispersion-corrected global approach is equivalent to that defined by the local approach. (3) Simplified dynamic ERR for stationary crack

P One further simplification can be made to Eq. (3.31) for its third term: since [ ∞ i¼1 Λi/ P 2 λi sin (ωit)]2  [ ∞ i¼1 Λi/λi] , with the value of Λi and λi given in Table 3.1, this third term of Eq. (3.31) can be no larger than 0.2295ρAv2/b. This term is therefore small in comparison to the second term of Eq. (3.31) when n > 12, in which case the dynamic ERR further simplifies to Glocal stationary ¼

pffiffiffiffiffiffiffiffiffiffiffi ∞ 9EIv2 t2 12 ρAEI v2 t X Λi + sin ðωi tÞ: 4 2 λ ba ba i¼1 i

(3.34)

Eq. (3.34) provides the simplified dynamic ERR based on the local bending moment and vibrational deflection. When n > 12, it is equivalent to Eq. (3.29), but for an accurate calculation of ERR when n < 12, it is advisable to use Eq. (3.29) instead, which accounts for wave dispersion.

3.3.3 Numerical verification 3.3.3.1 Finite element model and verification case To verify the analytical solutions of ERR for a stationary crack with vibration in Section 3.3.1 and with wave propagation in Section 3.3.2, the symmetric DCB (Fig. 3.7) was considered; its width is 1 mm. An isotropic elastic material was used with the Young’s modulus of 10 GPa, the Poisson’s ratio of 0.3, and the density of 103 kg m3. A 2D finite element model was built employing plane stress elements (CPS4R) in Abaqus/Explicit, which includes the inertia effects; uniform mesh size of 0.1 mm was v = 1 m s-1

h = 2 mm h = 2 mm

L = 20 mm

a = 60 mm

v = 1 m s-1

Fig. 3.7 Geometry of DCB for numerical verification.

50

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

selected after convergence study and, in total, 32,000 elements were used. All the viscous parameters were set to zero to avoid unnecessary damping. The VCCT was used to determine the dynamic ERR numerically. The uncracked region was formed by sharing nodes of two DCB arms, and no contact was modelled.

3.3.3.2 Verification for developed theory with vibration As discussed in Section 3.3.1, the ERR derived directly from the global approach with consideration of vibration accounting for the global energy balance shows a divergent amplitude, and only the ERR with the first vibration mode accuracy can be used; this result is compared to the FEM result in Fig. 3.8A. The results based on the developed analytical theory with vibration and with the first vibration mode accuracy are in good agreement with the results from the numerical simulation: the analytical results capture the amplitude and frequency of ERR variation predicted by the FEM. The analytical theory is slightly out of phase with the FEM, which is due to the difference in boundary conditions: the finite element model simulates a full DCB, whereas the developed theory models the effective boundary condition shown in Fig. 3.3B. It is worth noting that the ERR with dynamic effects oscillates about a mean value of GU st (the ERR component due to strain energy of quasistatic motion or static ERR without any dynamic effect). In Eq. (3.19), the dynamic factor decays quickly with time, but in Fig. 3.8A, the oscillating amplitude actually increases with time. This indicates that the dominant contribution to this vibration amplitude is the increasing GU st . When the first two vibration modes are included in the analytical result as shown in Fig. 3.8B, the amplitude of ERR oscillation begins to diverge as predicted.

Fig. 3.8 Comparison of ERR from results of FEM simulation (grey line) and from the developed analytical theory (black line) using global approach accounting for vibration with first (A) and first two (B) vibration modes.

Dynamic interfacial fracture

51

3.3.3.3 Verification for developed theory with wave propagation The theory developed based on the dispersion-corrected global approach in Eq. (3.29) is compared against the results from FEM simulation in Fig. 3.9 for the first vibration mode (a) and first five vibration modes (b). Apparently, the divergence of ERR amplitude as more vibration modes are added is resolved by accounting for dispersion (Fig. 3.9B); this contrasts with Fig. 3.8B, calculated using the conventional global approach. By adding more vibration modes, the ERR from the analytical theory approaches the FEM simulation results, as expected. Recall that the ERR amplitude from each vibration mode is proportional to the correction factor for dispersion fi given in Table 3.2. Based on this, the first five modes were reckoned sufficient to predict the amplitude of the ERR. This is confirmed by Fig. 3.9B; adding more vibration modes does not significantly alter the ERR although may provide more details of the ERR. By examining Fig. 3.9, a slight phase difference between the analytical and FEM simulation results is observed. In addition, the analytical results are slightly higher than the FEM simulation results. Both these discrepancies are due to the effective boundary condition. This can be resolved either by using an effective crack length for crack tip rotation compensation, as per Section 3.3.4, or by treating the intact section of the beam as resting on an elastic foundation. The latter is not covered here, but full details are available in [44].

3.3.4 Crack tip rotation compensation for stationary crack When using the effective boundary condition, the dynamic ERR result is slightly overestimated, and the frequency is not accurately predicted; crack tip rotation compensation can be used to resolve this. In this section, the viability of making the required compensation by using an additional crack length Δ is investigated. This Δ is determined in the quasistatic regime following the modified compliance calibration (MCC) method from ASTM D5528 [45], originally from [46]. The viability is assessed by

Fig. 3.9 Comparison of ERR from results of FEM simulation (grey line) and from the developed analytical theory (black line) using the dispersion-corrected global approach with first (A) and first five (B) vibration modes.

52

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

comparing the dynamic analytical solution using the effective crack length against FEM results. Following the quasistatic method to determine Δ, the external force acting on the DCB at various crack lengths under constant opening rate is measured. Quasistatic FEM simulation is used (using Abaqus/Standard, where the inertia of the DCB is not accounted for), and the compliances are calculated for each crack length. Then, according to the MCC method, the cube root of compliance is plotted against the corresponding crack length. The results for the verification case in Section 3.3.3.1 are shown in Fig. 3.10, and linear regression is used to find the additional crack length corresponding to zero compliance: Δ  1.34 mm. There are two methods of implementing this additional crack length Δ in ASTM D5528: one method increases the crack length by Δ to give an effective crack length of aeff ¼ a + Δ, while the other method uses an effective flexural modulus Eeff ¼ a3E/(a + Δ)3. Considering the verification case in Section 3.3.3.1 again, for which Δ  1.34 mm, both methods are tried, first replacing a in Eq. (3.29) with effective crack length aeff and then replacing E in Eq. (3.29) with effective flexural modulus Eeff. The ERR results for these two methods are compared against the results of FEM simulation in Fig. 3.11. Apparently, employing the effective crack length aeff is more accurate than the use of the effective flexural modulus Eeff. For the former, there is still a very small difference between the analytical solution and the results from FEM simulation. Note that the two methods are proposed in ASTM D5528 and deemed equivalent for quasistatic loads, but as shown, they are not equivalent for a dynamic fracture. Including the additional crack length Δ for crack tip rotation compensation also allows accurate calculation of the ERR contribution from each vibration mode. The analytical results based on the effective crack length aeff for the first one, two, three, four, five, ten, 15, and 20 vibration modes (or flexural waves) are shown in Fig. 3.12A–H, respectively. As more vibration modes are included, the analytical solution becomes increasingly close to the FEM results in terms of the overall

Fig. 3.10 Regression analysis of cube root of compliance with respect to crack length from quasistatic FEM simulation.

Dynamic interfacial fracture

53

Fig. 3.11 Implementation of Δ for the compensation of crack tip rotation: (A) by effective crack length aeff; (B) by effective flexural modulus Eeff.

magnitude of ERR as well as the frequencies. It also demonstrates that the first five vibration modes (Fig. 3.12E) are adequate to capture the dynamic ERR accurately. Note that the additional crack length Δ cannot be determined using compliance calibration and results of DCB tests under dynamic loads because the compliance measured in such experiments is not accurate due to oscillation of the external loads. Parallel quasistatic experiments or FEM simulations must therefore be conducted to derive the additional crack length Δ to use this method for experiments under dynamic loads.

3.4

Propagation of dynamic mode I interfacial crack

3.4.1 Rate dependency of fracture toughness In Section 3.3, the dynamic ERR of a stationary crack was derived based on the developed analytical framework. The equation of motion was solved with a time-dependent boundary condition, and the global approach including kinetic energy with correction for dispersion was used, forming the dispersion-corrected global approach. According to the fracture criterion, the crack begins to propagate when the dynamic ERR reaches the crack’s initiation toughness. Then, while the crack propagates, the dynamic ERR equals or exceeds the crack propagation toughness. These criteria are, however, complicated by the fact that dynamic fracture toughness is not, in general, constant. It is generally believed that dynamic fracture toughness is dependent on crack propagation speed based on experimental observations and results, that is, Gc ¼ f ða_ Þ. In early results from experiments with a low crack propagation speed, fracture toughness was shown to be approximately constant, but with increasing crack propagation speed, the dynamic fracture toughness increased exponentially after a limiting value of a_ L . A comprehensive review of these early experiments is given in [47], and it was concluded that the relationship between the dynamic fracture toughness and the crack propagation speed might be a material property. Zhou et al. [48] reported their experimental results for the dynamic fracture toughness of PMMA vs crack propagation speed and fitted the respective relation as

54

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

Fig. 3.12 Dynamic ERR for stationary crack vs time results from developed theory with effective crack length (black line) and from FEM (grey line) with increasing numbers of vibration modes.

Dynamic interfacial fracture

 a_ L , Gc ða_ Þ ¼ G0 log a_ L  a_

55



(3.35)

where G0 and a_ L are the fitting parameters. The microstructures of failure for various crack propagation speeds were also examined to explain the relationship between Gc and a_ together with the failure mechanisms. Based on the literature discussed, the fracture toughness for slow crack propagation can be considered approximately constant, but for fast crack propagation, the fracture toughness depends on the crack propagation speed or rate. Therefore, here, for mode I crack propagation, two scenarios of constant and rate-dependent fracture toughness are investigated separately, with analytical solutions developed for each.

3.4.2 Theoretical development 3.4.2.1 Analytical solution for constant fracture toughness For a steadily propagating crack, consider the fracture criterion of G  Gc ¼ 0, and its total derivative at a given time t, and the corresponding crack length a is ∂ðG  Gc Þ ∂ðG  Gc Þ da + dt ¼ 0: ∂a ∂t

(3.36)

Rearranging Eq. (3.36), the crack propagation speed at time t is a_ ¼

da ∂ðG  Gc Þ=∂t : ¼ dt ∂ðG  Gc Þ=∂a

(3.37)

Assuming that the contribution to strain and kinetic energies due to the crack propagation speed is small, the ERR solution for the stationary crack can be used to study the crack propagation speed derived in Eq. (3.37). Since the analytical ERR solution with the first mode accuracy in Eq. (3.18) predicts the envelope of ERR compared to results from FEM simulation, it can be used to obtain the crack propagation speed in Eq. (3.37) approximately. Evaluating respective terms in Eq. (3.37): ∂ Gc/∂ a and ∂ Gc/∂ t are found to be zero due to the constant Gc, and terms ∂ G/∂ a and ∂ G/∂ t are derived from Eq. (3.18), giving the approximate crack propagation speed as a_ ¼ 

∂G=∂t a ¼ : ∂G=∂a 2t

(3.38)

For oscillating ERR in Eq. (3.18), the crack propagation speed does not oscillate as Eq. (3.38) shows since all of the oscillatory terms in ∂ G/∂ t and ∂ G/∂ a cancel out. Note that Eq. (3.38) only applies under the steady and slow crack propagation without contact. Physically, it means that during crack propagation, the time oscillation of the ERR is balanced by the gradient of ERR.

56

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

The crack propagation speed in Eq. (3.38) is the same as that obtained for a DCB under quasistatic loads. Nevertheless, it was derived using the theory developed for dynamic interfacial fracture and is therefore also valid for dynamic crack propagation under the stated assumptions and limitations. Integrating Eq. (3.38) gives the solution for a crack length vs time curve as pffiffiffiffiffiffiffiffiffiffi a  a0 ¼ A1 t  t0 ,

(3.39)

where a0 is the initial crack length, t0 is the time for crack initiation. The crack initiation time can be determined using the condition for the stationary crack of G ¼ Gc, and, therefore, the coefficient A1 is determined accordingly by the crack initiation condition. Note that Eq. (3.39) has an interesting implication that the crack propagation speed is independent of fracture toughness once it starts to propagate and crack extension follows a fixed pattern as Eq. (3.39) indicates. But the coefficient A1 is dependent on fracture toughness Gc. Eqs (3.38) and (3.39) are only applicable to brittle materials with a moderate material density and a constant fracture toughens Gc. If the material density is high, the inertial effect can cause the crack surfaces to close, causing crack arrests. This theory for constant Gc, however, cannot predict the crack arrest phenomenon for two reasons: (1) The condition used in deriving Eqs (3.38) and (3.39) is that G ¼ Gc at all times after crack initiation, meaning that the crack must always propagate. (2) The theory does not consider contact between crack surfaces, and, furthermore, the interpenetration of crack surfaces gives nonzero ERR. For materials with high density, Eq. (3.39) can still accurately predict the slope of the crack length-time curve.

3.4.2.2 Analytical solution for rate-dependent fracture toughness (1) Problem description

For a DCB of a material with a rate-dependent fracture toughness Gc ða_ Þ, that is, the fracture toughness depends on the crack propagation speed, the fracture criterion is _ tÞ  Gc ða_ Þ ¼ 0, where the dynamic ERR is treated as a function of crack length Gða, a, a, crack propagation speed a_ and time t. Considering the total derivative of this fracture criterion and differentiating it with regard to t give _ tÞ _ tÞ _ tÞ dGc ða_ Þ ∂Gða, a, ∂Gða, a, ∂Gða, a, a_ + a€+  a€¼ 0, ∂a ∂a_ ∂t d a_

(3.40)

where a€ is the acceleration of crack propagation. Note that the last term in Eq. (3.40) is the contribution of dynamic interfacial fracture toughness, and it is neither always known nor with mature constitutive models, that is, for different materials they can take different forms and expressions. Therefore, solving Eq. (3.40) analytically to derive a solution for the crack propagation speed and then the crack length curve is generally infeasible in contrast to the case with constant fracture toughness in Section 3.4.2.1. Rather, the theory developed hereby is to provide analytical solutions to determine the dynamic ERR and then

Dynamic interfacial fracture

57

to measure and study the dynamic interfacial fracture toughness and fracture behaviours accordingly. To achieve this, the assumption is made that crack length a is a given parameter, which can be measured in experiments. And since the crack propagates, Freund’s formula in Eq. (3.30) can be applied to determine the dynamic ERR; however, it depends on the crack tip bending moment. This requires solving partial differential equations with a moving boundary condition to determine the deflection at the crack tip and then the crack tip bending moment as the crack propagates. This cannot be achieved by rigorous mathematics due to the unknown expression for crack length a(t) [49–51]; however, a general analytical engineering solution is developed, which can be applied for a given crack length curve, for example, from experimental results. (2) Theoretical derivation

Consider the DCB in Fig. 3.13A and one DCB arm with effective boundary in Fig. 3.13B with prescribed coordinate, so that the crack can propagate in the x positive direction [52]. The deflection of the DCB arm in Fig. 3.13B is obtained by coordinate transformation of the solution in Eq. (3.13) as rffiffiffiffiffiffi ∞  3  ρA X Λi x 3x ϕ ða  xÞsin ðωi tÞ +  + 1 vt: wðx, tÞ ¼ va 2a3 2a EI i¼1 λ3i i 2

(3.41)

For a propagating crack of instantaneous length a1 by assuming the deflection given in Eq. (3.41), the bending moment at the crack tip is M(a1, t) ¼ EIw(2)(a1, t). Then, by combining Eqs (3.30) and (3.41) and employing the simplification described in Section 3.3.2.3, the dynamic ERR is Glocal propagation

  9EIv2 t2 a_ 21 ¼ 1 2 ba4pffiffiffiffiffiffiffiffiffiffiffi C 0  ∞ 12 ρAEI v2 t Λi a_ 21 X 1  sin ðωi tÞ: + 2 2 C0 i¼1 λi ba1

a)

(3.42)

b)

z

vt

vt h h

x

( 0,0 ) vt

( a,0 )

a

Fig. 3.13 DCB configuration for crack propagation: (A) symmetric DCB; (B) effective and prescribed boundary conditions.

58

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

The first term is due to the strain energy of quasistatic motion and its expression is the same for any given crack length, and, therefore, does not need to be adjusted for dispersion. The second term, however, is related to vibration and wave propagation, and so does need to be adjusted to account for dispersion and the Doppler effect [53]. By combining Eqs (3.23) (Gvib ¼ Fvib ðΩÞ=A_ 0 ) and (3.42) for a crack length a1, the vibration energy flux into a contour around the crack tip (see Fig. 3.14) is F1 ðΩÞ ¼ a_ 1 b

∞ X

Givib ða1 Þ,

(3.43)

i¼1

   pffiffiffiffiffiffiffiffiffiffiffi  where Givib ða1 Þ ¼ 12 ρAEI v2 t C20  a_ 21 Λi sin ðωi tÞ= λi ba21 C20 . Now consider crack propagation from a1 to a2 over the time interval Δt. Ahead of the initial crack tip position at x ¼ a1, a new beam section of length (a2  a1) is formed as shown in Fig. 3.14, with the total energy related to vibration (or wave propagation) supplied to this new beam section being F1(Ω)Δt. Following the same arguments and techniques developed for the consideration of dispersion in Section 3.3.2.2, the energy flux due to F1(Ω)Δt at the new crack-tip position at x ¼ a2 is F2 ðΩÞ ¼ a_ 1 b

∞ X

fi 0 Givib ða1 Þ,

(3.44)

i¼1

where fi0 is the correction factor that accounts for the dispersion of flexural waves in the new beam section of (a2  a1); it is a function of the new beam section’s boundary conditions only. By combining Eqs (3.23) (Gvib ¼ Fvib ðΩÞ=A_ 0 ) and (3.44), the ERR component due to vibration at the new crack length a2 is therefore pffiffiffiffiffiffiffiffiffiffiffi   ∞ F2 ðΩÞ 12 ρAEI v2 t a_ 1 Λi 0 a_ 21 X ¼ 1 2 f sin ðωi tÞ: (3.45) Gvib ða2 Þ ¼ 2 ba_ 2 a_ 2 C0 i¼1 λi i ba1 The ratio fi0 or correction factor for dispersion for propagating crack is determined in the same way as fi in Section 3.3.2.2. Note that the boundary conditions for the new beam section of (a2  a1), formed by crack propagation, are different to the boundary conditions for the stationary crack in Section 3.3.2.2; so, fi0 must be different to fi. If

Fig. 3.14 Crack propagation from a1 to a2 over time interval Δt.

Dynamic interfacial fracture

59

(a2  a1) is small, then a_ 1  a_ 2 and the boundary conditions for the new beam section can be taken as approximately fixed-fixed, where the deflections and slopes at both x ¼ a1 and x ¼ a2 are all zero. From Section 3.3.2.2, fi0 is still derived as fi0 ¼ λ10 / (2λi0 ), but now λi0 is determined from the frequency equation of a beam with the fixed-fixed boundary condition, that is, cos(λi0 ) cosh (λi0 )  1 ¼ 0. The value of λi0 and the corresponding correction factor for dispersion for propagating crack is given in Table 3.3. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The frequency of the ith mode flexural wave is ωi ¼ λ2i a2 EI=ðρAÞ. The excitation point is at the free end of the DCB arms, and these waves travel towards the crack tip; since the crack propagates, the frequency of each flexural wave that the crack tip observes needs to be modified due to the Doppler effect. The frequencies of flexural waves observed at the crack tip decrease with increasing crack propagation speeds. For a propagating crack, the actual frequency of the ith mode flexural wave observed at the crack tip is ! a_ 0 (3.46) ωi ¼ 1  i ωi : Cp Combining Eqs (3.42), (3.45) and (3.46), the total dynamic ERR for a propagating crack is   9EIv2 t2 a_ 2 local Gpropagation ¼ 1 2 C0 ba4 " ! # (3.47) pffiffiffiffiffiffiffiffiffiffiffi 2   ∞ 12 ρAEI v t Λi 0 a_ 2 X a_ + 1 2 fi sin 1  i ωi t : Cp C0 i¼1 λi ba2 (3) Limiting speed of crack propagation in DCB

For the first term of Eq. (3.47), which is the ERR component due to the strain energy of quasistatic motion, the crack propagation speed a_ should be no larger than pffiffiffiffiffiffiffiffi C0 ¼ E=ρ. This component of ERR would otherwise become negative, inhibiting crack propagation. For the second term of Eq. (3.47), which is the ERR component due to vibration, there is another constraint in addition to a_  C0 : the crack propagation speed a_ must be no greater than C1p, that is, a_  C1p . Vibration energy would

Table 3.3 Correction factor for dispersion for propagating crack. Mode number

1

2

3

4

5

i>5

Frequency solution λi0 Correction factor fi0

4.73004

7.85320

10.99561

14.13717

17.27876

0.5

0.30115

0.21509

0.16729

0.13687

ð2i + 1Þπ 2 1:50562 ð2i + 1Þ

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

otherwise not be supplied to the propagating crack tip (recall that C1p is the phase speed of the first-mode flexural wave, which carries all the other higher mode waves). This speed is λ1 C1p ¼ a

sffiffiffiffiffiffi sffiffiffi EI λ1 E ¼ pffiffiffiffiffi r , ρA ρ 12

(3.48)

where r ¼ h/a is the aspect ratio of the half DCB, and so the limiting speed of crack propagation in DCBs is proportional to r; λ1  3.9266 (Table 3.1). Freund [18] derived that for mode I fracture, the crack propagation speed cannot surpass the Rayleigh wave speed CR, where CR ¼ (0.862 + 1.14ν)CS/(1 + ν) approxpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi imately and the shear wave speed CS ¼ E=½2ρð1 + νÞ. The ratio of these two limiting speeds, C1p and CR, is C1p λ1 1+ν pffiffiffiffiffiffiffiffiffiffi r: ¼ pffiffiffi CR 6 ð0:862 + 1:14νÞ 1  ν

(3.49)

These two different limiting speeds, C1p and CR, are not in contradiction to each other: instead, C1p is a development of CR when applied to a DCB with a given aspect ratio. Freund’s original derivation of the limiting crack propagation speed as the Rayleigh wave speed CR was based on a crack in an infinite sheet. The crack provides a traction-free surface on a semiinfinite medium, where Rayleigh waves can form. If, however, additional structural constraints are included, for example, in the form of another traction-free surface imposed parallel to the existing one, the semiinfinite medium develops into a plate. For a thick plate, Rayleigh waves become Rayleigh-Lamb waves, which are the superposition of A0 (asymmetric mode) and S0 (symmetric mode) Lamb waves [54]. A0 Lamb waves resemble flexural waves and S0 Lamb waves resemble axial waves [43]. For a thin plate, A0 Lamb waves become the 2D counterpart of flexural waves in Euler-Bernoulli beams [42], and, if axial motion is absent, the influence of S0 Lamb waves can be ignored. For the increasingly constrained structure described above, as it develops from a semiinfinite sheet to an Euler-Bernoulli beam, Rayleigh waves develop into flexural waves in a beam, and, accordingly, the Rayleigh wave speed drops to the phase speed of flexural waves. This leads to a reduction in the limiting speed of crack propagation, as determined by Eq. (3.49). According to the assessment in Eq. (3.49), the limiting crack propagation speed decreases with the decreasing aspect ratio and the decreasing Poisson’s ratio as shown in Fig. 3.15. The limiting speed of a crack in a DCB, therefore, decreases with the increasing crack length. For the conventional DCB test employed to determine the mode I fracture toughness in CFRPs, the aspect ratio magnitude typically ranges from

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61

Fig. 3.15 Effect of aspect ratio on limiting crack propagation speed for various Poisson’s ratios.

0.01 to 0.1 [45], and so, from Eq. (3.49), the corresponding limiting speed of crack propagation is therefore in the range from 0.02CR to 0.25CR. There are currently relatively few experimental data on dynamic crack propagation available in the literature [6] for CFRPs in DCBs, at least partly due to the challenges such tests pose for an experimental setup design. Experimental data for high loading rates are available in [17,19], which used a servo-hydraulic test machine with a ‘lost motion device’ to achieve opening rates up to 15 m s1, and in [35], that employed an electromagnetic Hopkinson bar to obtain opening rates up to 30 m s1. The crack propagation speeds, measured in [17,19] under various loading rates for two kinds of CFRPs, PEEK/carbon-fibre and epoxy/carbon-fibre composites, are plotted in Fig. 3.16. The theoretical limiting speed of crack propagation, predicted by Eq. (3.48), is also plotted. The measured crack propagation speeds are well below the predicted limiting speed for all the opening rates up to 15 m s-1. Ref. [35] reports crack propagation speeds for 18 specimens of unidirectional epoxy/carbon-fibre composite, ranging from 108 m s1 to 253 m s1. For the opening rates of up to 30 m s1, the lowest predicted limiting crack propagation speed from Eq. (3.48) is 385 m s 1, and so the measured crack propagation speeds are also within the predicted limit. A further two points should be noted concerning Eq. (3.49). First, for aspect ratios in the range 0.4–0.5 when the specimen thickness and the crack length are of the same order, it might appear that the limiting crack propagating speed can reach CR as Rayleigh waves can form [55]. The Euler-Bernoulli beam assumption, however, requires that r ¼ h/a < 0.1. Second, when the aspect ratio approaches zero, it might appear that the limiting crack propagation speed should also approach zero, which would mean that crack cannot propagate. For zero aspect ratio, however, the structure instead behaves like a string and is unable to bear compression or bending loads. This should not be the case with real DCB configurations.

62

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

Fig. 3.16 Comparison of experimentally measured crack propagation speeds under various opening (loading) rates [17,19] against theoretically predicted limiting speed.

Dynamic interfacial fracture

63

3.4.3 Experimental verification Fracture toughness is generally rate-dependent as shown by the slip-stick nonsmooth propagation behaviour under high loading rates; this is also demonstrated by the experimental results in [17,19], which are used to verify the analytical solution for the rate-dependent fracture toughness developed in Section 3.4.2.2. There are relatively few experimental studies of DCBs under high loading rates in the literature [6]. Blackman et al. [17,19], however, performed a comprehensive series of experiments under high opening rates of up to 15 m s1. Two of these DCB experiments, with PEEK/carbon-fibre composite under opening rates of 6.5 m s1 and 10 m s1, were selected to verify the analytical solution developed in Section 3.4.2.2. For this composite, the longitudinal modulus is taken as 115 GPa, the Poisson’s ratio as 0.28 and the density as 1540 kg m3 [17,19]. The half DCB thickness is 1.5 mm, and the width is 20 mm. A plane strain condition is therefore assumed in the analytical calculations with the effective Young’s modulus taken as E/(1  ν2) ¼ 124.78 GPa. Before crack propagation, that is, for a stationary crack, the solution of Eq. (3.29) in Section 3.3.2.2 was used, which provides a continuous ERR with respect to time since the crack length does not increase. During crack propagation, Eq. (3.47) developed in Section 3.4.2.2 for rate-dependent fracture toughness was used to calculate the dynamic ERR. The first five vibration modes were considered in both cases, which Section 3.3.2.2 and Section 3.3.3.3 showed to be adequate. The original experimental data of crack length vs time curves in [17,19] include 51 data points at evenly spaced intervals for the test with the 6.5 m s1 opening rate and 31 points for the test with the 10 m s1 opening rate. A central difference calculation was therefore used to estimate the crack propagation speed a_ ðtn Þ ¼ ½aðtn + 1 Þ aðtn1 Þ=ðtn + 1  tn1 Þ since there are not enough data points to get an accurate estimation otherwise (further study for assessment of crack propagation speed is in Section 3.4.5). When calculating the ERR, the effective crack length of aeff ¼ a + Δ was used, where a is the actual crack length and Δ is the additional crack length aimed to compensate for crack tip rotation [17,19], which otherwise is not captured by EulerBernoulli beams with effective boundary condition. The corresponding values of Δ for each test are from [17,19]. The value of Δ for 6.5 m s1 opening rate is 4.4 mm, and that for 10 m s1 opening rate is calculated by linear interpolating for opening rates 6.5 m s1 and 14.9 m s1, giving Δ ¼ 5 mm.

3.4.3.1 Experimental verification for DCB under 6.5 m s1 loading rate The theoretical results of dynamic ERR for the DCB test with the 6.5 m s1 opening rate are plotted in Fig. 3.17. They are based on the experimentally observed crack length and propagation speeds (the crack length is also plotted in Fig. 3.17 with values on the secondary axis). Note that Ref. [56] simulates Blackman et al.’s experiments [17,19] using FEM with interfacial thick-level-set modelling (ITLSM), and in these simulations, the crack initiation time is higher than the actual crack initiation time

64

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

Fig. 3.17 Evolution of dynamic ERR and crack length for 6.5 m s1 loading rate based on experimentally observed crack propagation speed.

in the experiments. This is because ‘the lost motion device was set to allow a period of pretravels to ensure that the test was conducted at constant velocity’ in experiments [17,19], and so this pretravels period should be taken into consideration. The FEM results in [56] were therefore used to shift the initiation time in the experimental data. The experimental results in Fig. 3.17 are plotted on this shifted time scale in order to make a valid comparison with the theory. The crack initiation toughness (CIT) of 1400 N m1 and crack arrest toughness (CAT) of 670 N m1 are taken from the same reference [56], in which Liu et al. performed a parametric FEM study of different values of CIT and CAT, aiming to match the experimentally measured crack length vs time curves. These values are not required in the calculation of dynamic ERR and are only shown for comparison. Using the developed analytical solutions—Eq. (3.29) for stationary crack and Eq. (3.47) for propagating crack—and the experimentally observed crack length and propagation speed, the dynamic ERR was calculated for both stationary and propagating cracks. For the stationary crack, it is expected that the crack should not propagate until the ERR exceeds the CIT (1400 N m1). For the propagating crack, it is expected that the crack should propagate if the ERR exceeds the CAT (670 N m1) and should be arrested otherwise. Moreover, once the dynamic ERR drops below the CAT and crack propagation is arrested, it is expected that it reinitiates (i.e. continues propagating again) only once the dynamic ERR builds up and exceeds the CIT again. Apparently, the developed analytical theory is generally in excellent agreement with the experimental results (Fig. 3.17): (1) The crack initiates and begins to propagate once the dynamic ERR reaches the expected CIT (i.e. after 0.6 ms). (2) The dynamic ERR is equal to or greater than the CAT, while it propagates continuously between about 0.6 and 3.5 ms. Note that a small number of data points in this period are slightly below the CAT and that in the period between 2.4 and 2.7 ms (with three

Dynamic interfacial fracture

65

sample points), there appears to be a very short period of arrest. Nevertheless, the general tendency in the period from 0.6 to 3.5 ms is that of propagation with the dynamic ERR equal to or greater than the CAT. There are several considerations, which indicate that the general tendency is more important than close attention to singular or small groups of data points when interpreting these experimental results. In both cases, at least part of the cause is the relatively few experimental sample points, which prevented accurate estimation of the crack propagation speed (see Section 3.4.5). Also, concerning the short arrest period, the FEM simulation shows its absence (see Fig. 3.19). Generally, dynamic tests are well known to be probabilistic in nature [57], particularly where damage is concerned, and on small scales of time and space, whereas the developed theory is deterministic. For all these reasons, it is sensible to consider that the crack propagates during the short period between 2.4 and 2.7 ms with the dynamic ERR equal to or greater than the CAT (thus permitting continued propagation). (3) At about 3.5 ms, for the reasons described above, it is also within the probable error margin that the dynamic ERR drops below the CAT, and the propagation is arrested. (4) Once propagation is arrested, the dynamic ERR needs to exceed the CIT again in order to reinitiate. Indeed, there is a crack arrest period between 3.5 and about 4.8 ms, where the dynamic ERR is lower than the CIT. (5) At about 4.8 ms, the dynamic ERR reaches the CIT again and then the crack propagates until about 6 ms with the dynamic ERR within the limits set by the CIT and the CAT. (6) At 6 ms, the dynamic ERR drops below the CAT and the propagation arrests once more.

3.4.3.2 Experimental verification for DCB under 10 m s1 loading rate The theoretical results of dynamic ERR for the DCB test with the 10 m s1 opening rate are presented in Fig. 3.18. These results are based on the experimentally observed crack length and propagation speeds (the crack length is also plotted in Fig. 3.18 with

Fig. 3.18 Evolution of dynamic ERR and crack length for 10 m s1 loading rate based on the experimentally observed crack propagation speed.

66

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

values on the secondary axis). As previously explained, to account for the period of pretravels in the experiment, the experimental results are plotted on a shifted time scale, with the required amount of shift determined using the FEM results in [56]. The values of CIT and CAT were also determined in the same reference as 1300 N m1 and 300 N m1, respectively. These toughness values are not required in the calculation of dynamic ERR and are only shown for comparison. Fig. 3.18 again demonstrates an excellent agreement between the developed analytical theory and the experimental results: (1) The crack initiates and begins to propagate once the dynamic ERR reaches the expected CIT. (2) After crack initiation, there is a significant drop of dynamic ERR, which remains above the CAT until t ¼ 2.6 ms, when the propagation arrests. In both comparisons with the dynamic DCB experiments at different opening rates, the developed analytical theory predicts dynamic ERRs that are in line with the observed crack propagation behaviour. This is a strong confirmation that the developed analytical theory accurately predicts the values of CIT and CAT at times of crack initiation or reinitiation and arrest.

3.4.4 Numerical verification The experimental verification (Section 3.4.3) for the theory developed for rate-dependent fracture toughness of Eq. (3.47) can accurately predict the main features of fracture behaviour such as initiation, propagation, and arrest. More detailed comparisons between the developed analytical theory and experiment were not possible, however, since the dynamic ERR cannot be measured directly from experiments without postprocessing (for example, using the developed theory) and because of the insufficient number of sample points for accurate measurement of the crack propagation speed. Instead, the results from FEM analysis in [56], which simulated the same experiments in [17,19] considered in Section 3.4.3, were used to further verify the developed analytical theory. The FEM analysis by Liu et al. [56] used the ITLSM to simulate 3D DCB under different loading rates; full details are given in [56]. Following the same procedure described in Section 3.4.3 for experimental verification, the FEM results for crack length and crack propagation speed were used together with the developed analytical theory for rate-dependent fracture toughness to determine the analytical dynamic ERR, which could then be compared directly with the dynamic ERR of the respective FEM results.

3.4.4.1 Numerical verification for DCB under 6.5 m s1 loading rate The dynamic ERR calculated with developed analytical theory (based on the crack length vs time curve obtained from the FEM simulation as well as the crack propagation speed) and the FEM results for the 6.5 m s1 loading case are compared in Fig. 3.19. They are in excellent agreement for both stationary and propagating cracks. Note that the developed analytical theory predicts a slightly higher ERR in comparison to the FEM results. This is reasonable since the FEM model used a 3D formulation with

Dynamic interfacial fracture

67

Fig. 3.19 Evolution of dynamic ERR and crack length for 6.5 m s1 loading rate based on FEM results of crack propagation speed.

the anisotropic material properties of E11 ¼ 115 GPa and E22 ¼ 8 GPa, making this model less stiff than the analytical one. The discrepancy is not significant.

3.4.4.2 Numerical verification for DCB under 10 m s1 loading rate The dynamic ERR determined with the developed analytical theory (based on the evolution of the crack length calculated with the FEM result as well as the crack propagation speed) and the numerical result are compared in Fig. 3.20 for 10 m s1 loading case. They are in excellent agreement for both regimes of crack—stationary and propagation. The analytical theory developed for the material with rate-dependent fracture toughness is therefore verified and can accurately predict the dynamic ERR in

Fig. 3.20 Evolution of dynamic ERR and crack length for 10 m s1 loading rate based on FEM results of crack propagation speed.

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

DCB tests under high loading rate and also characterise the main feature of fracture behaviour such as crack initiation, propagation, and arrest.

3.4.5 Crack propagation speed assessment and dynamic ERR An accurate prediction of dynamic ERR requires an accurate estimation of the crack propagation speed. Due to the relatively small number of experimental sample points, the central difference method was used in Sections 3.4.3 and 3.4.4 to assess the crack propagation speed at a given time based on two crack length measurements to avoid sudden jumps in the crack propagation speed, so a_ ðtn Þ ¼ ½aðtn + 1 Þ aðtn1 Þ=ðtn + 1  tn1 Þ was used. Another two possible methods to assess the crack propagation speed include the backward difference method with a_ ðtn Þ ¼ ½aðtn Þ  aðtn1 Þ=ðtn  tn1 Þ and the forward difference method with a_ ðtn Þ ¼ ½aðtn + 1 Þ  aðtn Þ=ðtn + 1  tn Þ. Ideally, these three methods would give values of crack propagation speed in close agreement, but this may not be the case in real experiments under high loading rates when the actual number of sample points for the crack length is determined by the capability of the experimental setup, for example, high-speed cameras. For the experimental verification case in Section 3.4.3 (i.e. for PEEK/carbon-fibre composite under an opening rate of 6.5 m s1), the three methods of determining the crack propagation speed are examined. At t ¼ 3.065 ms, the crack propagation speed is 15.15 m s-1 by the central difference method, 26.81 m s1 by the backward difference method, and 6.14 m s1 by the forward difference method. The dynamic ERR predicted with these crack propagation speeds are 645.33 N m1, 809.01 N m1, and 625.19 N m1, respectively, and the mean and range of these dynamic ERR values is therefore 693.18  82.32 N m-1. Following this approach, the dynamic ERR for every sample point is calculated, and the mean and range results of dynamic ERR are shown in Fig. 3.21.

Fig. 3.21 Mean value of dynamic ERR and its range assessed by three methods of assessing crack propagation speed.

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Apparently, precise prediction of dynamic ERR based on the developed analytical solution of Eq. (3.47) in Section 3.4.2.2 depends on the accurate determination of the crack propagation speed. The mean value of the dynamic ERR from the three estimation methods of the crack propagation speed can capture the main crack propagation behaviours including its arrest and reinitiation. The CAT measured from the experimental results in this way is a range rather than the single value of 670 N m-1 determined by the FEM simulation. The crack reinitiation value at t ¼ 4.81 ms is not affected by estimation of the crack propagation speed, however, because it is zero before reinitiation. There is a particular point at t ¼ 2.46 ms, where the mean dynamic ERR drops significantly below the CAT value of 670 N m1 (the value determined by the FEM simulation). This one point is not considered problematic for several reasons: (i) the overall crack behaviour around this time is one of crack propagations (see Section 3.4.3); (ii) the range still allows for a dynamic ERR that exceeds the CAT; and, furthermore, (iii) there are other contributions to the experimental error besides estimation of the crack propagation speed, which is the only one considered here. At t ¼ 2.46 ms, the forward and backward differences for crack propagation speed give a lower limit of 0 m s1 and an upper limit of 53.31 m s1, respectively, and the central difference gives 24.12 m s1. To study the sensitivity of the dynamic ERR to the crack propagation speed, the dynamic ERR is calculated for this particular point at t ¼ 2.46 ms with the crack propagation speed ranging from 0 to 55 m s1 while keeping the other parameters the same; the results are shown in Fig. 3.22. The dynamic ERR ranges from 521.9 N m1 to 724.9 N m1. For crack propagation speeds between 10 m s1 and 35 m s1, the dynamic ERR is above the CAT value of 670 N m1. Note that the lower limit of the crack propagation speed at this point is 0, while the upper one is 53.31 m s1. The mean value gives the dynamic ERR above 670 N m1; this is as expected since the general crack behaviour around this point is propagation. In conclusion, application of the theoretical solution developed in this section clearly requires an accurate estimation of the crack propagation speed. For a relatively

Fig. 3.22 Dynamic ERR vs crack propagation speed for sample point t ¼ 2.46 ms.

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

small number of experimental sample points, however, the central difference method may offer a reasonable prediction of the dynamic ERR.

3.5

Dynamic mode II interfacial fracture

3.5.1 Introduction Laminated composite materials are widely used in aerospace, automotive, and naval applications to save weight. Their weak transverse properties, however, make them susceptible to damage from transverse loadings such as impact, which causes delamination and significant weakening of structural strength. Impact-induced delamination tends to be mode-II-dominant [58], and so it is important to understand the dynamic mode II delamination behaviour of structures in addition to the corresponding loading rate-dependent delamination toughness of the material. As pointed out in a recent review paper [15], which also provides a broad review of dynamic mode II delamination, there is therefore still a clear need for closed-form analytical solutions to study the dynamic mode II fracture behaviour, and that is what this section aims to provide. Analytical models of dynamic mode II delamination should include structural vibration and not just quasistatic motion as in conventional studies [7,12,14]. Under rapidly applied loads, the arms of the ELS specimen slide back and forth, generating the oscillating relative displacement and ERR, which these conventional studies do not capture. There is a disagreement among various experimental studies concerning the rate effects of dynamic mode II fracture toughness: positive rate effects in [11,59–61], negative in [7,12], and insignificant in [14,62–65]. This disagreement might be explained by structural vibration, not considered in the data reduction. Sections 3.3 and 3.4 show the capability of structural dynamics and vibration to accurately predict the dynamic ERR for mode I dynamic fracture in [37,52]. In this section, structural dynamics and vibration are used to model the dynamic ELS test and to derive the dynamic mode II ERR. It is demonstrated that this is not just a straightforward application of the existing analytical theory [37,44,52] but requires additional novel ideas and interpretation, including the handling of contact in the modelling; the representation of the dynamic effect in terms of a dynamic factor and a spatial factor to facilitate understanding and the insight regarding dominant vibration modes.

3.5.2 Theoretical development In this section, the dynamic mode II ERR of an ELS specimen is derived analytically by considering beam dynamics and structural vibration. Fig. 3.23 shows such an ELS specimen in its initial undeformed state, with three beam sections, ①, ②, and ③. A time-dependent downward displacement w0(t) ¼ vt is applied to the midplane of the free end of the beam section ②, where v is a constant displacement rate. The length of the beam section ① (the intact region) is L, while beam sections ② and ③ are above

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Fig. 3.23 Schematic of ELS specimen.

and below, respectively, the crack with length a; the total length of the ELS specimen is L0. In the conventional ELS specimen [26], beam sections ② and ③ have the same thickness h. The x axis is positive to the right, with the crack tip located at x ¼ L. The transverse deflections of beam sections ①, ②, and ③ are in the x - z plane and denoted as w1(x, t), w2(x, t), and w3(x, t), respectively. As beam ② deflects downwards according to the applied w0(t), the interfacial contact between beam sections ② and ③ also drives beam section ③ downwards. It is assumed that beam section ③ has the same vertical deflection as beam section ②, and so a pure mode II delamination is produced. Under further assumptions of h < < a and h < < L, the Euler-Bernoulli beam assumption is appropriate to derive the respective deflections and, thus, the ERR. In general, the ERR of a stationary crack under dynamic loading can be determined by crack tip bending moments by using a crack tip energy flux integral [18], and the dynamic ERR is

 1 M22 ðL, tÞ M32 ðL, tÞ M12 ðL, tÞ , +  G¼ 2bE I2 I3 I1

(3.50)

where M1(L, t), M2(L, t), and M3(L, t) are the crack tip bending moments of beam sec(2) tions ①, ②, and ③, respectively: M1(L, t) ¼ EI1w(2) 1 (L, t), M2(L, t) ¼ EI2w2 (L, t), and (2) M3(L, t) ¼ EI3w3 (L, t). The deflections of respective beam sections are derived in Section 3.5.2.1.

3.5.2.1 Dynamic response of ELS specimen Under the applied constant rate displacement of w0(t) ¼ vt, the dynamic transverse deflections of beam sections ①, ②, and ③ take the following forms by introducing shifting functions [36]: w1 ðx, tÞ ¼ w1fv ðx, tÞ + F1 ðxÞvt,

(3.51)

w2 ðx, tÞ ¼ w2fv ðx, tÞ + F2 ðxÞvt,

(3.52)

w3 ðx, tÞ ¼ w3fv ðx, tÞ + F3 ðxÞvt:

(3.53)

In Eqs (3.51) to (3.53), w1fv(x, t), w2fv(x, t), and w3fv(x, t) are the free vibration components of beam sections ①, ②, and ③, respectively; F1(x), F2(x), and F3(x) are the

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respective shifting functions. The effect of the shifting functions is to distribute the applied displacement of w0(t) ¼ vt along the three beam sections. Note that in order to satisfy the assumed contact condition that beam sections ② and ③ have the same deflection, that is, w2(x, t) ¼ w3(x, t), both their free vibration components and their shifting functions must be equal, that is,w2fv(x, t) ¼ w3fv(x, t) and F2(x) ¼ F3(x) by enforcing homogeneous conditions. Therefore, the deflection and shifting function of beam section ③ are replaced by those of beam section ②, which simplifies and facilitates the derivation process. The derivation of free vibration components and shifting functions are given in [66]: w1 ðx, tÞ ¼ w1fv ðx, tÞ + F1 ðxÞvt ( )  ∞ X Hi W1i ðxÞsin ðωi tÞ + F1 ðxÞt , ¼v  ωi i¼1

(3.54)

w2 ðx, tÞ ¼ w2fv ðx, tÞ + F2 ðxÞvt ( )  ∞ X Hi W2i ðxÞsin ðωi tÞ + F2 ðxÞt : ¼v  ωi i¼1

(3.55)

where W1i(x)Ðand W2i(x) are the ith normalÐ modes for beam sections ① and ②, respectively; Hi ¼ L0 4ρA1W1i(x)F1(x)/EI1dx + L+a L ρA2W2i(x)F2(x)/EI2dx, which represents the coupling of free vibration and the applied constant displacement rate v, that is, how the free vibration of the beam responds to the applied excitation. Note that Hi is independent of applied displacement rate v, and, thus, this is an inherent property representing the beam configuration; and x3 + 3ðL + aÞx2 , F 1 ðxÞ ¼  3 2 L + 3L2 a + 3La2 + 4a3 F 2 ðxÞ ¼

4ðx  L  aÞ3 + 3ðL2 + 2La + 4a2 Þðx  L  aÞ   + 1: 2 L3 + 3L2 a + 3La2 + 4a3

(3.56)

(3.57)

3.5.2.2 Dynamic energy release rate By combining Eqs (3.50), (3.54), and (3.55), the total dynamic ERR for the ELS specimen shown in Fig. 3.23 is obtained as 108EI 2 a2 v2 t2 G¼  2 b L3 + 3L2 a + 3La2 + 4a3 " #2 ∞ ∞ X 18EI 2 av2 t 3EI 2 v2 X    3 Λi sin ðωi tÞ + Λi sin ðωi tÞ , 4b b L + 3L2 a + 3La2 + 4a3 i¼1 i¼1 (3.58)

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where Λi ¼ [ Ci3 sinh (βia) + Ci4 sin (βia)](Ci3  Ci4)[ρA2/(EI2)]3/2. In Eq. (3.58), the first term is the ERR component due to the strain energy of quasistatic motion, which has the same value as the static ERR without any dynamic effect; the second term is the ERR component due to coupling between static motion and local vibration; and the last term is the ERR component due to local vibration. These three terms are cp loc denoted GU st , Gdyn, and Gdyn, respectively. It should be recognised that the total deflections in Eqs (3.54) and (3.55) as well as the total dynamic ERR in Eq. (3.58) are vibrational solutions, and as such, assume that sufficient time has passed for all the flexural waves to form standing waves. Section 3.3.2.3 derives the dynamic ERR of a double cantilever beam (DCB) employing the dispersion-corrected global approach and the local approach with vibrational solutions. It is shown that for a DCB the difference between the two approaches is essentially zero after 12 times the duration taken for the first-mode flexural wave to travel the crack length and already very small after just five times. Following this guide from Section 3.3.2.3, the vibrational approach used here is considered justified, as verified in Section 3.5.3. To establish the time required for all flexural waves to form standing waves in the ELS specimen, the phase speed of the first-mode flexural wave is considered since it travels the slowest. The time needed for this wave to travel from the free end of the ELS specimen to the crack tip is τ1 ¼ a/C1ap, where C1ap ¼ ω1/β1 is the phase speed of the first-mode flexural wave in beam section ②. Likewise, the time needed for this wave to travel from the crack tip to the fixed end is τ2 ¼ L/C1Lp, where C1Lp ¼ ω1/ α1 is the phase speed of the first-mode flexural wave in beam section ①. Note that pffiffiffi C1ap =C1Lp ¼ α1 =β1 ¼ 1= 2. Therefore, the time τ0 needed for this wave to travel from the free end to the fixed end of the ELS specimen is 1 τ0 ¼ τ1 + τ2 ¼ pffiffiffiffiffiffi ω1

rffiffiffiffiffiffiffiffi pffiffiffi 4 3ρ 2a + L : 2 Eh

(3.59)

There are two limiting cases, namely, a ! 0 and a ! L0, where L0 ¼ a + L is the constant total length of the ELS specimen. Therefore, the time needed for the first-mode flexural wave to travel from the free end to the fixed end is in the range τ0 ða ! 0Þ < τ0 < τ0 ða ! L0 Þ,

(3.60)

pffiffiffi where τ0(a ! 0) ¼ L0/C1Lp and τ0 ða ! L0 Þ ¼ L0 =C1ap ¼ 2L0 =C1Lp . Either of these two limiting cases represents a cantilever beam under constant loading rate, which is the configuration modelled in Section 3.3.2.3. Therefore, the multiple of 12 can be introduced as guided by the conclusion of Section 3.3.2.3. The minimum time for the developed dynamic ERR solution based on structural vibration to become applicable, or, equivalently, the minimum test time required to postprocess experimental results by this method, is therefore 12 t0 ¼ 12τ0 ¼ pffiffiffiffiffiffi ω1

rffiffiffiffiffiffiffiffi pffiffiffi 4 3ρ 2a + L , 2 Eh

(3.61)

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

Therefore, the maximum velocity that can be applied in a test is wcrit/t0, where wcrit is the critical displacement, at which crack initiates.

3.5.2.3 Dynamic factor Since the ERR component due to the strain energy of quasistatic motion has the same value as the ERR without any dynamic effect, the total dynamic effect contribution to the ERR is loc Gdyn ¼ Gcp dyn + Gdyn :

(3.62)

The ERR component due to local vibration Gloc dyn has a maximum value of 2 P∞ 3EI2v ( i¼1 Λi)/(4b), while the amplitude of the ERR component due to the coupling of vibration and quasistatic motion Gcp dyn increases with time t. Therefore, given sufloc ficient time, Gcp dyn is more significant than Gdyn, and, so, in order to better understand the dynamic effect, the total dynamic effect can be taken as Gdyn  Gcp dyn. A comparison of these ERR components for the verification case in Section 3.5.3.1 is shown in Fig. 3.24. Note that in Fig. 3.24, the ERR component due to the dynamic effect in [14] is also plotted; it is the solid horizontal line with a constant value of 6.914 N m1. This shows the limitation of the conventional approach of using only quasistatic motion to account for the total kinetic energy contribution to the ERR. This is the reason why the dynamic effect only accounts for 1% of the measured fracture toughness value in [14] and is described as negligible. Therefore, to accurately determine the dynamic mode II ERR of an ELS specimen, vibration must be considered. A total dynamic factor for mode II crack can be defined as cp

Gdyn Gdyn  U GU Gst st  3  ∞ L + 3L2 a + 3La2 + 4a3 X ¼ Λi sin ðωi tÞ: 6at i¼1

fdyn ¼

Fig. 3.24 Comparison of ERR components.

(3.63)

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Note that Gcp dyn for the ith vibration mode is proportional to parameter Λi. Typical values of Λi for the verification case in Section 3.5.3.1 are plotted in Fig. 3.25 based on the specified beam configuration and material properties. It can be seen that for the given configuration, the most significant contribution to the total dynamic ERR is from the first vibration mode, whereas the contribution to the total dynamic ERR from the second vibration mode is near zero since Λ2  0, as discussed in Section 3.5.2.4 and reflected by results presented in Fig. 3.30. From Eq. (3.63), the dynamic factor for the verification case in Section 3.5.3.1 is plotted in Fig. 3.26 for the first (dashed line) and the first four (solid line) vibration modes. The maximum value for the dynamic factor is found in the first cycle of the first vibration mode, and then it drops dramatically. The dynamic factor with the first four vibration modes oscillates around that of the first vibration mode since the first vibration mode makes the most significant contribution to the total ERR. This indicates the feasibility of using only the first vibration mode to quantify the dynamic ERR for this beam configuration; however, this is not always the case as shown in Section 3.5.2.5. The dynamic factor based on [14] is also plotted in Fig. 3.26 as the dotted line; it does not oscillate but decays very quickly to around zero.

3.5.2.4 Normal modes and crack-tip-loading condition In Section 3.5.2.3, the dynamic factor of the ith vibration mode is found to be proportional to parameter Λi. For the values of Λi shown in Fig. 3.25 (for the verification case in Section 3.5.3.1), it can be seen that Λ2  0, and, accordingly, the dynamic factor of the second vibration mode f2dyn  0 (the superscript 2 denoting the second vibration mode). This means that the second vibration mode does not contribute to the total dynamic ERR in this case. To understand this phenomenon, the normal mode and its slope of the first three vibration modes of the verification case in Section 3.5.3.1 were examined to investigate the crack-tip-loading condition. It is apparent that the slopes of the first and third normal modes (Fig. 3.27B and F) are not smooth at the crack tip (that is, there is a discontinuity of curvature), whereas it is smooth for the second normal mode (Fig. 3.27D). This demonstrates that the first Fig. 3.25 Typical value of parameter Λi.

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Fig. 3.26 Dynamic factor for verification case.

Fig. 3.27 Normalised normal modes and slopes for first three vibration modes.

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and third normal modes contribute to the bending moment at the crack tip and the ERR according to Eq. (3.50). (2) (2) For the second normal mode, however, W(2) 12 (L) ¼ W22 (L). Since EI1W12 (L) ¼ (2) (2) (2) 2EI2W22 (L)(continuity condition), W12 (L) ¼ W22 (L) ¼ 0. This means that, in this configuration, the second vibration mode does not contribute to the bending moment at the crack tip or to the ERR. The parameter Λi can therefore be interpreted as a description of the crack-tiploading condition. Note that Λi is a function of the ELS specimen’s configuration, and the second vibration mode does not always give Λi ¼ 0 for different configurations. A more general conclusion about the ith vibration modal contribution to the ERR is given in Section 3.5.2.5.

3.5.2.5 ith vibration modal contribution to ERR The dynamic factor of the ith vibration mode is 

i fdyn

 L3 + 3L2 a + 3La2 + 4a3 Λi sin ðωi tÞ, ¼ 6at

(3.64)

which is a sine function decreasing with time as it oscillates (superscript i denoting the vibration mode). It is seen that the dynamic effect comes from both the time and the space domains, where the former varies with time and the latter depends on structural properties. To facilitate understanding of the relationship between the beam configuration and the modal contributions of each vibration mode, a spatial factor fisp can be defined by isolating the structural properties from fidyn as fspi

  Λi L3 + 3L2 a + 3La2 + 4a3 ¼ , 6a

(3.65)

where fisp has units of seconds and superscript i denotes the vibration mode. Using the material properties from the verification case in Section 3.5.3.1 and maintaining the total length of the specimen as L0 ¼ L + a ¼ 60 mm, fisp was calculated for the first five vibration modes for crack length ratios η ¼ a/L0 in the range 0.05  η  0.95 and plotted in Fig. 3.28. Fig. 3.28A shows the spatial factors of each vibration mode oscillating around zero with the crack length ratio η. Clearly, the first vibration mode has the lowest frequency, and the frequency of fisp increases with increasing vibration mode numbers. It is noteworthy that there are some crack length ratios, for which fisp ¼ 0, meaning that the ith vibration mode does not contribute to the total ERR in this case. Furthermore, for the ith vibration mode, there are i ratios, which produce fisp ¼ 0. For example, the first vibration mode has only one crack length ratio (0.61, approximately) that provides f1sp ¼ 0, but the second vibration mode has two crack length ratios (0.34 and 0.78, approximately) that provide f2sp ¼ 0. Thus, for the verification case in Section 3.5.3.1 with the crack length ratio of 0.33, the second mode contribution to the total ERR is close to zero.

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Fig. 3.28 Spatial factor of ith vibration mode against crack-length ratio.

The absolute value of fisp for various crack length ratios is presented in Fig. 3.28B. The range of this ratio can be divided into four regions, according to the relative contribution of each mode, as shown. Note that since fisp is independent of the applied loading rate and an inherent property of an ELS specimen with the given crack length ratio η, this classification of regions is general for ELS specimens and is not just for this specific verification case. In Region I, where 0.05  η  0.15, the contribution of each vibration mode is significant, and so all the modes should be taken into account when determining the total dynamic ERR. In Region II, where 0.15  η  0.5, the contribution of the first vibration mode is dominant, and the total ERR can be approximated using only the first vibration mode. In Region III, where 0.5  η  0.7, the dominant vibration mode contributing to the total dynamic ERR is the second vibration mode. In Region IV, where 0.7  η  0.95, the dominant vibration mode contributing to the total dynamic ERR changes from the second vibration mode back to the first vibration mode. This classification of regions is important when designing specimens for high-loading-rate ELS tests to measure the dynamic mode II crack toughness of a given material. For example, it may be desirable to design ELS specimens in Region II (with crack length ratios in the range 0.15  η  0.5) since then only the first vibration mode would be dominant, simplifying the postprocessing of test data and reducing the amplitude of higher vibration modes, which could be helpful for data recording and regression. This classification of regions is also important to help understand the structural behaviour in the presence of a mode II crack; for instance, a structure could be designed to avoid certain vibration modes in the presence of certain crack length ratios. Alternatively, to reduce the total dynamic effect, crack length ratios in Region III are preferred due to smaller spatial factors of all vibration modes.

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3.5.3 Numerical verification 3.5.3.1 Numerical verification for isotropic bilayer composite Finite element simulations were used to verify the analytical theory developed in Section 3.5.2 for the total dynamic ERR. The geometry for the ELS specimen verification case is shown in Fig. 3.29. Isotropic elastic material properties were used with the Young’s modulus of 10 GPa, the Poisson’s ratio of 0.3, and the density of 103 kg m3. The applied constant displacement rate was 5 m s1. A 2D FEM model was built in Abaqus/Explicit using four-node plane-stress elements (CPS4R) with a uniform element size of 0.1 mm; the total number of elements was 24,000. The crack region was modelled using a contact algorithm. To eliminate any damping effect in the dynamic response, the viscosity parameters were set to zero. The total dynamic ERR from the FEM was calculated using the VCCT and compared to the analytical theory. Fig. 3.30 shows comparison of the mode II dynamic ERR from the analytical theory developed in Section 3.5.2 (solid black line) and from FEM simulation results (solid grey line) with various numbers of vibration modes. The dashed line represents the static component of the ERR. In addition, the dynamic ERR from [14] is also plotted as a dashed line, which almost overlaps with the static component or quasistatic solution with only a baseline upward shift of 6.914 N m1. The significant difference between the dynamic ERR from [14] and the developed analytical theory demonstrates the necessity of considering vibration for ELS specimen under dynamic loads. The developed analytical theory and FEM simulations are in excellent agreement up to and including the first four vibration modes. From the fifth vibration mode onwards, the oscillation amplitudes from the analytical solution are slightly larger than that from FEM simulations, suggesting that the assumption in the theory that beam sections ② and ③ share the same deflection is valid for the lower vibration modes, but not quite so accurate for higher-order ones. In addition, the Euler-Bernoulli beam theory was used, which overlooks shear and rotational inertia that might become significant for higher modes. Since the actual ERR amplitudes due to higher vibration modes are small, however (see Fig. 3.25), this discrepancy is not significant in estimating ERR if only the first few vibration modes are used, but the dominant vibration mode must be included as discussed in Section 3.5.2.5. z

Fig. 3.29 ELS geometry for FEM verification.

v = 5 m s-1 L = 40 mm

①

a = 20 mm ②

h = 2 mm

③

h = 2 mm

x

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Fig. 3.30 Dynamic ERR vs time results from developed theory (black line) and from FEM (grey line) with increasing numbers of vibration modes for isotropic bilayer composite.

The crack length ratio in this verification case is 0.33, and, according to the analysis in Sections 3.5.2.3, 3.5.2.4, and 3.5.2.5, the contribution of the second vibration mode to the total dynamic ERR is close to zero. This is also confirmed by comparing Fig. 3.30A and B, where adding the ERR contribution from the second vibration mode in the developed analytical theory does not alter the total ERR.

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3.5.3.2 Numerical verification for orthotropic fibre-reinforced composite In Section 3.5.3.1, the numerical verification demonstrates the agreement between the developed theory and FEM simulation for an isotropic bilayer composite. In this section, the developed theory is verified against a simulation of an orthotropic fibrereinforced composite material. To apply the developed theory, the conventional method, as in ASTM D5528 [45], is to use the longitudinal modulus of elasticity (if it is dominant and the aspect ratio L/h is high). The orthotropic material properties of unidirectional T800H/3900-2 carbon-fibrereinforced polymer, as given in Table 3.4, were taken from [67] and adopted for FEM simulation. The ELS dimensions used, in accordance with ISO 15114 [26], were L ¼ 40 mm, a ¼ 60 mm, and h ¼ 1.5 mm. The applied loading rate selected was 5 m s-1. The density of T800H/3900–2 was taken as 1.25
103 kg m-3 from the manufacturer’s data sheet. All other FEM settings were as described in Section 3.5.3.1. For the analytical solution, the modulus of elasticity in the fibre direction (E11 in Table 3.4) is used to derive the dynamic mode II ERR by Eq. (3.58). The comparison between the analytical solution and the FEM simulation results is shown in Fig. 3.31. Results of the analytical solution and the FEM simulation are generally in excellent agreement, although the analytical solution predicts slightly higher amplitudes, which is not significant. The crack length ratio for this ELS geometry is 0.6. According to Fig. 3.28 in Section 3.5.2.5, the dominant vibration mode should be the second one, with the first vibration mode not contributing to the total ERR since f1sp  0. This is confirmed in Fig. 3.31A, in which the analytical solution with the first vibration mode only almost overlaps the static component. Fig. 3.31B–F show that by adding more vibration modes, the analytical solution becomes increasingly closer to the FEM simulation results, and that, therefore, the developed theory is also applicable to orthotropic fibre-reinforced composite ELS specimens by the conventional method of using the longitudinal modulus of elasticity.

Table 3.4 Orthotropic material properties of unidirectional T800H/3900-2 graphite/epoxy. E11 5 154.72 GPa

E22 5 7.58 GPa

E33 5 7.58 GPa

G12 ¼ 4.27 GPa ν12 ¼ 0.32

G13 ¼ 4.27 GPa ν13 ¼ 0.32

G23 ¼ 2.88 GPa ν23 ¼ 0.32

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Fig. 3.31 Dynamic ERR vs time results from developed theory (black line) and from FEM (grey line) with increasing numbers of vibration modes for orthotropic fibre-reinforced composite.

3.6

Conclusions

The conventional analytical approach to modelling dynamic interfacial fracture is a global energy balance approach. In this approach, the kinetic energy of quasistatic motion alone is considered without any vibration, and, as a consequence, conflicting results are obtained concerning the rate effects on fracture toughness. In addition, the approach fails to capture the vibration behaviour of the structure during dynamic fracture, which, as experiments and FEM simulations demonstrate, can be significant. Although using an experimental-numerical hybrid approach can capture the physics, it requires extensive computational resources, brings with it a degree of uncertainty around best practice for reliable and accurate simulations, is restricted to discrete

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cases, and may not reveal much of the underlying mechanics for understanding. A new and complete analytical framework, which accounts for structural vibration and wave propagation, has therefore been developed by the authors to study the dynamic interfacial fracture. This analytical framework encompasses newly developed techniques for stationary and propagating cracks, quasistatic motion and vibration, mode I and mode II fractures, local and global approaches, and a variety of derived solutions for certain configurations and (although not reported here) also includes rigid and elastic interfaces. It is referred to as a framework since the collection of techniques, in general, allows applications to new configurations and fracture-mechanics phenomena, for example, mixed-mode fracture. The first solutions of the authors’ analytical framework were developed for dynamic mode I interfacial fracture in DCBs, which are the most fundamental engineering structure for the study of the fracture. An effective boundary condition was assumed at the crack tip for one DCB arm, and the dynamic deflection of the DCB arm was solved by vibration analysis for time-dependent boundary conditions. The derived dynamic deflection was then used to determine the strain and kinetic energies to define the dynamic ERR by the global energy balance approach. The results from this approach, however, displayed nonphysical behaviour, with a divergent ERR amplitude as more vibration modes were included, and the derived dynamic ERR was only accurate for the first vibration mode. Further examination revealed that the global approach overlooked the wave propagation properties of beams as dispersive waveguides and assumed all the flexural waves arrived at the crack tip simultaneously. To address this, the dispersive property of beams was included to study energy flux into the crack tip at a given time, and a correction factor for dispersion was introduced to modify the global approach. This new approach is named the dispersion-corrected global approach. Dynamic ERR results from the dispersioncorrected global approach showed good agreement with results from FEM simulations but were slightly out of phase due to the effective boundary condition constraining crack tip rotation. This can be compensated for by using an effective crack length (also known as the modified compliance calibration (MCC) method, which is widely used in the quasistatic loading regime). After introducing an effective crack length, the theoretical results fell into an even better and overall excellent agreement with the FEM results. The investigation also demonstrated the equivalence of the dispersioncorrected global approach and the local approach (that is, by using the crack tip bending moment) in determining the dynamic ERR. The developed theory was then further extended to study mode I crack propagation under dynamic loads. For a material with constant fracture toughness, the first-order accurate dynamic ERR was used to derive the crack propagation speed and the crack length vs time curve by enforcing G ¼ Gc at all times after initiation. This analytical solution, however, could not predict crack arrest phenomenon. Then, for materials with rate-dependent fracture toughness, the dynamic ERR of a propagating crack was derived based on the dispersion-corrected global approach, an additional assumption regarding energy conservation, correction of frequency due to the Doppler effect, and the experimentally observed crack length vs time curve. The resulting analytical

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

solution for propagating cracks was verified against the experimental data and the FEM simulation results, showing excellent agreement. For the best agreement, the effective crack length was still used but had to be determined by data regression of the experimentally observed crack length as the MCC method could not be used. The analytical solution also allowed the limiting speed of crack propagation in DCBs to be studied; it was found to be a function of the Poisson’s ratio and the DCB’s aspect ratio r (r ¼ h/a, the ratio between the DCB’s arm thickness and the crack length). For conventional DCBs with aspect ratios between 0.01 and 0.1, the limiting crack propagation speed is between 0.02CR and 0.25CR where CR is the Rayleigh wave speed. For dynamic mode II interfacial fracture, the ELS test configuration was investigated using the local approach. The derived analytical solution shows that the contribution to the total ERR from different vibration modes is dependent on the crack length ratio, which is the ratio of the crack length to the total length of the ELS specimen. A dynamic factor was defined as the ratio of the dynamic contribution to the ERR to the static contribution, and, out of the dynamic factor, a spatial factor was then defined by isolating the structural properties from the time domain. This revealed that each crack length ratio has a dominant vibration mode based on the spatial factor and that the crack-tip-loading condition is linked to the spatial factor. These findings are relevant for designing ELS specimens to avoid unwanted high-amplitude oscillation from higher-order vibration modes, for designing structures to avoid certain vibration modes in the presence of a certain crack length or reducing the total dynamic effect on the ERR of a crack in a structure. Overall, all the analytical solutions are readily applicable for various applications, including for the design of engineering structures and dynamic fracture tests, postprocessing experimental results from such tests, measuring dynamic fracture toughness in modes I and II and verifying FEM implementations. The developed analytical framework also provides fundamental understanding and a collection of techniques and methods for general application to other fracture-mechanics phenomena and configurations.

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[7] A.J. Smiley, R.B. Pipes, Rate sensitivity of mode II interlaminar fracture toughness in graphite/epoxy and graphite/PEEK composite materials, Compos. Sci. Technol. 29 (1) (1987) 1–15. [8] K. Friedrich, R. Walter, L.A. Carlsson, A.J. Smiley, J.W. Gillespie, Mechanisms for rate effects on interlaminar fracture toughness of carbon/epoxy and carbon/PEEK composites, J. Mater. Sci. 24 (9) (1989) 3387–3398. [9] K. Kageyama, I. Kimpara, Delamination failures in polymer composites, Mater. Sci. Eng. A 143 (1–2) (1991) 167–174. [10] T. Kusaka, Y. Yamauchi, T. Kurokawa, Effects of strain rate on mode II interlaminar fracture toughness in carbon-fibre/epoxy laminated composites, J. Phys. 4 (8) (1994) 671–676. [11] L. Berger, W.J. Cantwell, Temperature and loading rate effects in the mode II interlaminar fracture behavior of carbon fiber reinforced PEEK, Polym. Compos. 22 (2) (2001) 271–281. [12] H. Maikuma, J.W. Gillespie, D.J. Wilkins, Mode II interlaminar fracture of the center notch flexural specimen under impact loading, J. Compos. Mater. 24 (2) (1990) 124–149. [13] P. Compston, P.Y.B. Jar, P. Davies, Matrix effect on the static and dynamic interlaminar fracture toughness of glass-fibre marine composites, Compos. Part B Eng. 29 (4) (1998) 505–516. [14] B.R.K. Blackman, et al., The failure of fibre composites and adhesively bonded fibre composites under high rates of test: part III mixed-mode I/II and mode II loadings, J. Mater. Sci. 31 (17) (1996) 4467–4477. [15] M. May, H. Channammagari, P. Hahn, High-rate mode II fracture toughness testing of polymer matrix composites—a review, Compos. Part A Appl. Sci. Manuf. 137 (2020), 106019. [16] N.F. Mott, Fracture of metals: theoretical considerations, Engineering 265 (1948) 16–18. [17] B.R.K. Blackman, A.J. Kinloch, Y. Wang, J.G. Williams, The failure of fibre composites and adhesively bonded fibre composites under high rates of test: part II mode I loading— dynamic effects, J. Mater. Sci. 31 (17) (1996) 4451–4466. [18] L.B. Freund, Dynamic Fracture Mechanics, Cambridge University Press, 1990. [19] B.R.K. Blackman, et al., The failure of fibre composites and adhesively bonded fibre composites under high rates of test—part I mode I loading-experimental studies, J. Mater. Sci. 30 (23) (1995) 5885–5900. [20] M.F. Kanninen, C. Popelar, P.C. Gehlen, Dynamic analysis of crack propagation and arrest in the double-cantilever-beam specimen, in: ASTM Special Technical Publication, 1977, pp. 19–38. no. 627. [21] M.F. Kanninen, A dynamic analysis of unstable crack propagation and arrest in the DCB test specimen, Int. J. Fract. 10 (3) (1974) 415–430. [22] P. Davidson, A.M. Waas, C.S. Yerramalli, Experimental determination of validated, critical interfacial modes I and II energy release rates in a composite sandwich panel, Compos. Struct. 94 (2) (2012) 477–483. [23] P. Davidson, A.M. Waas, Nonsmooth mode I fracture of fibre-reinforced composites: an experimental, numerical and analytical study, Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 370 (2012) 1942–1965. [24] C. Sun, M.D. Thouless, A.M. Waas, J.A. Schroeder, P.D. Zavattieri, Ductile-brittle transitions in the fracture of plastically-deforming, adhesively-bonded structures. Part I: experimental studies, Int. J. Solids Struct. 45 (10) (2008) 3059–3073. [25] ASTM D7905, Standard Test Method for Determination of the Mode II Interlaminar Fracture Toughness of Unidirectional Fiber-Reinforced Polymer Matrix Composites, ASTM International, 2014.

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[26] ISO 15114:2014, Fibre-Reinforced Plastic Composites—Determination of the Mode II Fracture Resistance for Unidirectionally Reinforced Materials Using the Calibrated End-Loaded Split (C-ELS) Test and an Effective Crack Length Approach, Int. Stand. Organ., Geneva, Switz., 2014. [27] H. Maikuma, J.W. Gillespie, J.M. Whitney, Analysis and experimental characterization of the center notch flexural test specimen for mode II interlaminar fracture, J. Compos. Mater. 23 (8) (1989) 756–786. [28] J. Ratcliffe, W.J. Cantwell, Center notch flexure sandwich geometry for characterizing skin-core adhesion in thin-skinned sandwich structures, J. Reinf. Plast. Compos. 20 (11) (2001) 945–970. [29] W.J. Cantwell, The influence of loading rate on the mode II interlaminar fracture toughness of composite materials, J. Compos. Mater. 31 (14) (1997) 1364–1380. [30] T. Nishioka, S.N. Atluri, Numerical analysis of dynamic crack propagation: generation and prediction studies, Eng. Fract. Mech. 16 (3) (1982) 303–332. [31] C. Guo, C.T. Sun, Dynamic mode-I crack-propagation in a carbon/epoxy composite, Compos. Sci. Technol. 58 (9) (1998) 1405–1410. [32] J.L. Tsai, C. Guo, C.T. Sun, Dynamic delamination fracture toughness in unidirectional polymeric composites, Compos. Sci. Technol. 61 (2001) 87–94. [33] P. Kumar, N.N. Kishore, Initiation and propagation toughness of delamination crack under an impact load, J. Mech. Phys. Solids 46 (10) (1998) 1773–1787. [34] H. Liu, H. Nie, C. Zhang, Y. Li, Loading rate dependency of mode I interlaminar fracture toughness for unidirectional composite laminates, Compos. Sci. Technol. 167 (2018) 215– 223. [35] H. Liu, X. Meng, H. Zhang, H. Nie, C. Zhang, Y. Li, The dynamic crack propagation behavior of mode I interlaminar crack in unidirectional carbon/epoxy composites, Eng. Fract. Mech. 215 (2019) 65–82. [36] D.A. Grant, Beam vibrations with time-dependent boundary conditions, J. Sound Vib. 89 (4) (1983) 519–522. [37] T. Chen, C.M. Harvey, S. Wang, V.V. Silberschmidt, Dynamic interfacial fracture of a double cantilever beam, Eng. Fract. Mech. 225 (2020) 1–9. [38] S.B. Park, C.T. Sun, Effect of electric field on fracture of piezoelectric ceramics, Int. J. Fract. 70 (3) (1995) 203–216. [39] V. Govorukha, M. Kamlah, V. Loboda, Y. Lapusta, Fracture Mechanics of Piezoelectric Solids with Interface Cracks, Springer, 2017. [40] A.A. Griffith, The phenomena of rupture and flow in solids, Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 221 (1921) 163–198. [41] K.F. Graff, Wave Motion in Elastic Solids, Oxford University Press, 1991. [42] S. Gopalakrishnan, Wave Propagation in Materials and Structures, CRC Press, 2017. [43] V. Giurgiutiu, Structural Health Monitoring with Piezoelectric Wafer Active Sensors, Elsevier, 2007. [44] T. Chen, C.M. Harvey, S. Wang, V.V. Silberschmidt, Dynamic delamination on elastic interface, Compos. Struct. 234 (2020), 111670. [45] ASTM D5528, Standard Test Method for Mode I Interlaminar Fracture Toughness of Unidirectional Fiber-Reinforced Polymer Matrix Composites, ASTM International, 2014. [46] S. Hashemi, A.J. Kinloch, J.G. Williams, Corrections needed in double-cantilever beam tests for assessing the interlaminar failure of fibre-composites, J. Mater. Sci. Lett. 8 (2) (1989) 125–129. [47] J.W. Dally, W.L. Fourney, G.R. Irwin, On the uniqueness of the stress intensity factor— crack velocity relationship, Int. J. Fract. 27 (3–4) (1985) 159–168.

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[48] F. Zhou, J.F. Molinari, T. Shioya, A rate-dependent cohesive model for simulating dynamic crack propagation in brittle materials, Eng. Fract. Mech. 72 (9) (2005) 1383–1410. [49] N.L. Balazs, On the solution of the wave equation with moving boundaries, J. Math. Anal. Appl. 3 (1961) 472–484. [50] L. Gaffour, Analytical method for solving the one-dimensional wave equation with moving boundary—abstract, J. Electromagn. Waves Appl. 20 (1998) 63–73. [51] B. Pelloni, D.A. Pinotsis, Moving boundary value problems for the wave equation, J. Comput. Appl. Math. 234 (2010) 1685–1691. [52] T. Chen, C.M. Harvey, S. Wang, V.V. Silberschmidt, Delamination propagation under high loading rate, Compos. Struct. 253 (2020), 112734. [53] I.C. Christov, C.I. Christov, On mechanical waves and Doppler shifts from moving boundaries, Math. Methods Appl. Sci. 40 (12) (2017) 4481–4492. [54] C. Hakoda, C.J. Lissenden, Comparison of quasi-Rayleigh waves and Rayleigh waves, and clarifying the cut-off frequency of quasi-Rayleigh waves, Ultrasonics 92 (2019) 50–56. [55] I.A. Viktorov, Rayleigh and Lamb Waves, 1967. [56] Y. Liu, F.P. van der Meer, L.J. Sluys, Cohesive zone and interfacial thick level set modeling of the dynamic double cantilever beam test of composite laminate, Theor. Appl. Fract. Mech. 96 (2018) 617–630. [57] D.R. Curran, L. Seaman, D.A. Shockey, Dynamic failure of solids, Phys. Rep. 147 (5–6) (1987) 253–388. [58] G.A.O. Davies, X. Zhang, Impact damage prediction in carbon composite structures, Int. J. Impact Eng. 16 (1) (1995) 149–170. [59] J.J.M. Machado, E.A.S. Marques, R.D.S.G. Campilho, L.F.M. da Silva, Mode II fracture toughness of CFRP as a function of temperature and strain rate, Compos. Part B Eng. 114 (2017) 311–318. [60] R.J. Brambleby, L.A. Louca, S.E. Mouring, Influence of loading rate on the mode II fracture toughness of vinyl ester GRP, Compos. Part A Appl. Sci. Manuf. 93 (2017) 153–162. [61] W.J. Cantwell, Loading rate effects in the mode II fracture of carbon fibre poly-etherether-ketone composites, J. Mater. Sci. Lett. 15 (7) (1996) 639–641. [62] H. Zabala, L. Aretxabaleta, G. Castillo, J. Aurrekoetxea, Dynamic 4 ENF test for a strain rate dependent mode II interlaminar fracture toughness characterization of unidirectional carbon fibre epoxy composites, Polym. Test. 55 (2016) 212–218. [63] P. Compston, P.Y.B. Jar, P.J. Burchill, K. Takahashi, Effect of matrix toughness and loading rate on the mode-II interlaminar fracture toughness of glass-fibre/vinyl-ester composites, Compos. Sci. Technol. 61 (2) (2001) 321–333. [64] X.F. Wu, Y.A. Dzenis, Determination of dynamic delamination toughness of a graphitefiber/epoxy composite using hopkinson pressure bar, Polym. Compos. 26 (2) (2005) 165–180. [65] M. Colin de Verdiere, A.A. Skordos, M. May, A.C. Walton, Influence of loading rate on the delamination response of untufted and tufted carbon epoxy non crimp fabric composites: mode I, Eng. Fract. Mech. 96 (2012) 11–25. [66] T. Chen, C.M. Harvey, S. Wang, V.V. Silberschmidt, Theory of dynamic mode-II delamination in end-loaded split tests, Compos. Part C Open Access 3 (2020), 100055. [67] B.D. Davidson, R.D. Bialaszewski, S.S. Sainath, A non-classical, energy release rate based approach for predicting delamination growth in graphite reinforced laminated polymeric composites, Compos. Sci. Technol. 66 (10) (2006) 1479–1496.

Low-velocity impact of composite laminates: Damage evolution

4

Yu Shia, Christophe Pinnab, and Constantinos Soutisc a Physical, Mathematical and Engineering Sciences, University of Chester, Chester, United Kingdom, bDepartment of Mechanical Engineering, The University of Sheffield, Sheffield, United Kingdom, cAerospace Research Institute, The University of Manchester, Manchester, United Kingdom

4.1

Introduction

Composite laminated structures usually show a complex damage pattern that develops internally and becomes difficult to detect [1–3]. Failure of the composite laminates generally includes both intra- and interlaminar damage mechanisms. Intralaminar failure occurs within a single lamina in the form of fibre tensile and compressive breakage, matrix tensile and compressive damage as well as debonding between fibres and matrix (splitting, matrix cracking). Interlaminar failure is represented by delamination that occurs between neighbouring plies, which is commonly observed during impact events [4]. Intra- and interlaminar damage modes may also interact with each other during damage evolution. For instance, transverse matrix cracking/splitting parallel to the fibres’ direction have been recognised as the first damage modes of composite laminates due to their resin-dominated characteristic [5]. These damage modes lead to the degradation of both the stiffness and strength of composites but usually do not cause direct failure of the composite (for the composite to fail, fibres have to break). When cracks in the matrix have propagated to the ply interface, delamination develops due to stress concentration generated at the interface, followed eventually by fibre breakage when fibres have become the only load-carrying constituent [6,7]. Barely visible impact damage (BVID) under relatively low-impact energy has become a critical issue (difficult to detect) and limitation in the design and fabrication of aircraft structural components [8]. It is therefore essential to investigate and understand the various damage modes and their interaction during the failure process of composite laminates in order to maximise the resistance and tolerance to damage of these materials under low-velocity impact by selecting the optimal lay-up configuration. Finite element (FE) techniques have been widely used to predict the complex internal damage mechanisms of composite structures subjected to impact loading. Such simulations minimise risks prior to implementation, prevent excessive waste of material and expensive testing, and reduce the manufacturing time at an early stage of the design process. Numerical simulation methods have been published to predict intraand interlaminar damage modes based on stress-based failure criteria or damage/fracture mechanics by developing the constitutive damage models for composites under Dynamic Deformation, Damage and Fracture in Composite Materials and Structures. https://doi.org/10.1016/B978-0-12-823979-7.00005-3 Copyright © 2023 Elsevier Ltd. All rights reserved.

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impact [9–22]. Even though these models effectively simulate the individual damage modes, they cannot predict damage evolution processes such as the propagation of matrix cracking or splitting and the complexity of the interaction of the various damage modes that characterises the failure of composite laminates under impact loading. This chapter therefore starts by briefly presenting the damage theories for initiation and evolution. The most efficient damage laws for the individual damage modes are first identified, and a damage model is then developed for the prediction of impactinduced damage evolution using the FE code Abaqus/Explicit [16]. An approach based on cohesive zone elements (CZEs) located within each lamina along the fibre direction between neighbouring elements is then introduced to capture the damage evolution process (matrix cracking/splitting). Results are discussed in terms of improved accuracy for the prediction of damage development during impact [17,18]. Nondestructive evaluation (NDE) using X-ray radiography is also presented to experimentally detect internal damage modes in composite materials and validate the developed numerical model.

4.2

Composite damage criteria

4.2.1 Background Damage in composites can be modelled using both failure criteria methods and damage mechanics approaches. The failure criteria approaches are generally classified into two categories: (1) noninteractive such as maximum stress and maximum strain criteria and (2) interactive such as Tsai-Wu and Tsai-Hill failure criteria. Maximum stress and maximum strain criteria are the most simple and straightforward approaches to numerically predict failure when the stress or strain in the material coordinate system is greater than a critical value, which can be expressed in Eqs. (4.1)–(4.4) as
f ¼ max

σ 11 σ 22 σ 12 j, j j, j j j X Y S12

 (4.1)

where σ 11  0 ) X ¼ XT ;σ 11 < 0 ) X ¼ XC σ 22  0 ) Y ¼ Y T ;σ 22 < 0 ) Y ¼ Y C

(4.2)

or
f ¼ max

ε11 ε22 γ j 0 j , j 0 j , j 12 j ε11 ε22 γ 012



where 0C ε11  0 ) ε11 ¼ ε0T 11 ;ε11 < 0 ) ε11 ¼ ε11

(4.3)

Low-velocity impact of composite laminates: Damage evolution 0C ε22  0 ) ε22 ¼ ε0T 22 ;ε22 < 0 ) ε22 ¼ ε22

91

(4.4)

In the aforementioned equations, σ 11, σ 22, ε11, and ε22 are the tensile or compressive stress and strain in the axial (1) and transverse (2) directions, respectively. σ 12 and γ 12 are the in-plane shear stress and strain. XT, XC, YT, and YC represent tensile and compressive strengths for failure prediction in their respective directions. S12 and γ 012 denote the in-plane shear strength and failure strain, respectively. An equivalent maximum strain criterion is shown in Eqs.(4.3) and (4.4), where ε011 and ε022 represent the failure strain in fibre and transverse directions, respectively. The maximum stress/maximum strain criteria are relatively simple to apply and understand due to direct comparison with failure strength or strain values. However, the failure modes within composites generally do not occur independently but rather interact with each other during damage evolution. Therefore, the interactive criteria taking into account the interaction of the stresses in the axial and transverse directions have been developed. This type of failure criteria based on the equivalent stress or strain is usually used to describe the failure envelope. The corresponding second-order polynomial equation for plane stresses is given in Eq. (4.5) f ¼ F11 σ 211 + F22 σ 222 + F66 σ 212 + 2F12 σ 11 σ 22 + F1 σ 11 + F2 σ 22 + F6 σ 12

(4.5)

where σ 11, σ 22, and σ 12 are axial and in-plane shear stresses, respectively. Different coefficients Fii and Fj determine the specific failure criteria derived. For instance, the Tsai-Hill criterion [23] defines these coefficients using tensile and compressive strengths as shown in Eq. (4.6): F11 ¼

1 1 1 1 , F22 ¼ 2 ,F12 ¼  2 ,F66 ¼ 2 X2 Y 2X S12

(4.6)

F1 ¼ 0, F2 ¼ 0,F6 ¼ 0 where X, Y, and S12 in Eq. (4.6) have the same definition as in Eq. (4.1). For the Tsai-Wu [24] criterion, the coefficients are as follows: F11 ¼ F1 ¼

1 XT X C

, F22 ¼

1 YTYC

,F66 ¼

1 S212

(4.7)

1 1 1 1  C , F2 ¼ T  C , F6 ¼ 0 T X X Y Y

with the coefficient F12 defined as F12 F12 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T X XC Y T Y C

(4.8)

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

F12 in the aforementioned equation ranges from 1 to 1 and can be obtained by fitting the equibiaxial experimental data. Hoffman [25] derived the same coefficients as in Eq. (4.7) but defined F12 as F12 ¼ 

1 2XT XC

(4.9)

Although the failure criteria approaches can be relatively simple and straightforward to predict the interactive damage within the composite structure, the damage mechanisms for different modes cannot be clearly characterised. It is especially hard to simulate the damage evolution process through numerical methods unless these criteria are applied at the ply level. Therefore, a progressive damage method has been developed to model the failure of composites, and this widely used method is mainly studied in this work. The prediction methods of damage onset and evolution will be introduced in the following sections.

4.2.2 Damage initiation criteria Hashin proposed a ply-by-ply failure criterion for unidirectional composites to separately model four distinct failure modes associated with fibre damage in tension and compression and matrix tensile and compressive failure [26,27]. Hashin’s criteria have been widely used in the industry, even though they cannot predict the matrix compression damage mode accurately. They can be expressed as follows: Fibre tension (σ 11  0) Fft ¼




σ 2 11 XT

+κ

σ 12 S12

2 ¼1

(4.10)

Fibre compression (σ 11 < 0) Ffc ¼

σ 2 11

XC

¼1

(4.11)

Matrix tension (σ 22  0) σ 2
σ 2 22 12 ¼1 Fmt ¼ T + Y S12

(4.12)

Matrix compression (σ 22 < 0)


σ 22 Fmc ¼ 2S23

# 2 "
C 2
 Y σ 22 σ 12 2 + 1 C + ¼1 2S23 Y 2S12

(4.13)

Low-velocity impact of composite laminates: Damage evolution

93

In the aforementioned equations, σ ij (i, j ¼ 1, 2, 3) are the stress components defined in the material coordinate system. XT and XC denote the fibre tensile and compressive strengths, YT and YC are the tensile and compressive strengths of the matrix, and Sij (i, j ¼ 1, 2, 3) denotes the longitudinal and transverse shear strengths of the composite, respectively. The coefficient κ in Eq. (4.10) accounts for the contribution of shear stress to fibre tensile failure and generally ranges between 0 and 1. Hashin’s quadratic failure criteria (could also be formulated as linear stress functions) have been proved to give an accurate prediction of individual damage modes, except for matrix compression damage, since fracture may occur at an angle through the ply thickness. Puck and Schurmann [28] developed a damage model for transverse compression. They proposed to use the failure criteria of Mohr [29] instead of the yield criterion of von Mises, which is normally applied (both, of course, were developed for isotropic homogeneous materials). Puck’s damage criterion for compression damage mode can be expressed as Eq. (4.14)


σ NT Fmc ¼ A S23 + μNT σ NN

2
2 σ NL + ¼1 S12 + μNL σ NN

(4.14)

In Eq. (4.14), σ ij (i, j ¼ L, T, N) are the stresses σ ij (i, j ¼ 1, 2, 3) rotated to the fracture plane, by reference to the axes  shown in Fig. 4.1 σ NN ¼ σ 2 m2 + σ 3 1  m2 + 2τ23 mn (4.15)   σ NT ¼ σ 2 mn + σ 3 mn + τ23 2m2  1

(4.16)

  σ TT ¼ σ 2 1  m2 + σ 3 m2  2τ23 mn

(4.17)

σ NL ¼ τ12 m + τ13 n

(4.18)

σ LT ¼ τ12 n + τ13 m

(4.19) Fig. 4.1 Fracture plane for matrix compressive failure relative to the material coordinate system. Ref. [16].

3

σ TN

α

σ NN 2

σ LN T

1

L

N

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

SA 23 ¼

YC 1  sin φ 2 cos φ

φ ¼ 2α  90°

(4.20) (4.21)

where m ¼ cos α and n ¼ sin α in Eqs. (4.15)–(4.19). SA 23 is the transverse shear strength along the fracture plane, which can be determined by the transverse compression strength YC and the angle of the fracture plane as shown in Eqs. (4.20) and (4.21). The key concept to Puck’s failure criteria is to determine the fracture plane by calculating the angle, α (see Fig. 4.1). The friction coefficients μNT and μNL in Eq. (4.14) can be defined based on the material friction angle (see Eq. (4.21)), φ, and material properties by reference to the Mohr failure criteria μNT ¼ tan φ ¼ tan ð2α  90°Þ μNL ¼

μNT S12 SA 23

(4.22) (4.23)

Generally, according to experimental observation, the matrix compression damage occurs along a fracture plane oriented at α ¼ 53° with respect to the through-thickness direction [28]. However, this angle of the fracture plane was determined under uniaxial compressive loading, and therefore, a multidirectional load such as impact could lead to various values of the fracture angle. To predict this fracture angle for different load states, the angle of the fracture plane should be a variable that can be numerically determined using an FE simulation. Sun et al. [30] also developed a criterion to predict the matrix compressive damage mode of composites based on an appropriate modification of Hashin’s criteria Fmc ¼

2 σ 2
σ 22 12 + ¼1 YC S12  ςσ 22

(4.24)

where ς is a constant determined experimentally and generally regarded as an internal material friction parameter (uncertainty in measuring and validity of these parameters still remains). Camanho et al. [31] developed a failure criterion (LaRC03 as shown in Fig. 4.2) based on continuum damage mechanics (CDM) and compared the failure envelope plotted in the plane (transverse stress σ 22, in-plane shear stress τ12) with various failure criteria (Fig. 4.2). It can be clearly seen from Fig. 4.2 that all damage criteria, compared with Worldwide Failure Exercise (WWFE) test results, give an accurate prediction for the tensile damage mode, except the maximum stress criterion. The maximum stress criterion only defines the failure along individual directions of the material and does not describe the interaction of the stress components. Therefore, it cannot give a satisfactory prediction of the failure of composites, especially when damage is matrix dominated.

Low-velocity impact of composite laminates: Damage evolution

95

τ 12, MPa

WWFE test ’02

100

Puck ’02 LaRC03 #1

Sun ’96

Max. stress 50

Hashin ’73 Hashin ’80

Hashin

LaRC03 #2

–150

–100

–50

0

σ 22 , MPa

50

Fig. 4.2 Failure envelope under different failure criteria for composites [31].

Pinho et al. [32,33] found that the interactive quadratic failure criterion expressed as Eq. (4.25) can accurately predict the tensile transverse matrix cracking based on the WWFE experimental measurement [34], without the need for any additional derived or experimental measured parameters, unlike other criteria such as LaRC03 σ 2
σ 2
σ 2 22 12 23 + ¼1 Fmt ¼ T + Y S12 S23

(4.25)

From Fig. 4.2, Hashin’s criterion was shown to be the least accurate for the prediction of the matrix compression damage mode. Sun’s criterion [30] and continuum damage criteria LaRC03 developed by Camanho [31] slightly improve the prediction accuracy, while Puck’s envelope is the most accurate. Therefore, in this work, Hashin’s criteria for damage initiation have been used for both fibre tensile and compressive damage modes; the interactive quadratic failure criterion Eq. (4.25) is applied to simulate tensile transverse matrix cracking, while Puck’s law is used to capture the onset of matrix compression damage.

4.2.3 Damage evolution criteria In general, carbon fibre/epoxy laminates show brittle properties as damage progresses. To predict the damage behaviour of composites, the damage evolution law is generally defined to degrade the material stiffness after different damage initiation modes have been satisfied. The easiest way to define the damage progression is to apply a degradation parameter directly associated with the individual failure modes to simulate the softening effect due to damage. Tita et al. [9] reduced the stiffness by using appropriate factors with respect to the various failure modes observed experimentally. For instance, the transverse Young’s modulus E22 and the plane Poisson’s ratio ν12 were directly reduced to 0 to represent complete damage in their work. Although this

96

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

method is straightforward, the determination of degradation parameters needs experimental measurements. Moreover, progressive damage evolution could not be simulated, as stiffness values were directly set to 0 after the onset of damage, which is unlikely to represent the physical damage process of composite structures. Therefore, it is a critical challenge to develop an approach to model progressive damage propagation with respect to the individual failure mechanisms. Chang and Chang [35–37] partially defined the degradation law. Considering the matrix failure, they proposed a degradation law to reduce the moduli E11 and G12 based on a decaying exponential function (Eq. 4.26), but other moduli were reduced abruptly to 0 once the damage was initiated:

A Ed11 ¼ E11 exp  H A0

A Gd12 ¼ G12 exp  H A0

(4.26)

where A is the area of the damage zone and A0 is the area of the interaction zone of the fibre failure from Chang and Chang [36,37]. H is a factor to control the degradation of the material stiffness. Progressive damage evolution laws such as that from Eq. (4.26) are obviously more realistic than an abrupt reduction of stiffness during the damage process, and therefore complete progressive damage evolution laws have been developed based on CDM, initiated by the work of Kachanov [38] and Rabotnov [39]. The stress–strain model in the numerical analysis process can exhibit strain-softening behaviour. Following this concept, Matzenmiller, Lubliner, and Taylor [40] developed a damage model called the MLT model for the nonlinear analysis of composite laminates. They constructed the damage model using damage variables with respect to the individual failure modes in the material principal directions. The model assumes that each unidirectional lamina in the composite acts as a continuum irrespective of the damage state. Damage growth is controlled based on Weibull distribution. The postdamage softening behaviour of the composite can be predicted by an exponential function

0 m  1 E ε (4.27) d ¼ 1  exp me X where E0 is the modulus of the material, ε is the strain related to the progressive damage at different time steps, e is Napier’s constant, X is the tensile or compressive strength with regard to the different damage modes in the different loading directions, and m is the strain-softening parameter during damage progression. In general, a high value of m could result in brittle failure of the material, while a low value of m indicates a ductile failure response with a high amount of absorbed energy. The effect of m is shown in Fig. 4.3. Modelling damage growth in composite laminates using the strain-softening parameter m in the MLT model has been shown to be an effective approach [12,41,42]. The appropriate value of m is usually related to the mesh size and loading

Low-velocity impact of composite laminates: Damage evolution

97

2500

Stress (MPa)

2000

m=1

1500 m=2

m = 20 1000 m = 10 m=5

500 m = 50 0 0.00

0.01

0.02

0.03

0.04 0.05 0.06 Strain (mm/mm)

0.07

0.08

0.09

0.10

Fig. 4.3 Effect of variations in the exponent, m, on the longitudinal stress–strain behaviour predicted by the MLT model for a [45/90/45/0]3S T800H/3900–2 CFRP plate [41].

conditions and it is crucial to select an optimum value of m for each failure mode. The value of m for various damage modes can be determined using uniaxial tensile or compressive tests, and the set of identified m values is thus applied to model the complex failure process of composites. Obviously, the m value has a strong effect on the prediction of damage progression in composites. An inappropriate value of m could give rise to unrealistic simulations of the damage growth process in brittle composites [41], with, for instance, a relatively low value of m predicting a ductile behaviour as shown in Fig. 4.3. An energy-based damage mechanics approach was then developed to model the progressive failure in the composite laminates [13,43,44]. The damage model implemented into an FE model effectively defined the damage variable for degradation by using the strains at damage onset and at complete damage and accurately captured damage progression for composite laminates [14,15]. This approach, which physically explains damage development through the energy consumed by the various damage modes, will be briefly introduced in the following sections.

4.2.3.1 Tensile failure modes The damage variables for tensile damage modes along the fibre and transverse directions can be expressed as t ¼ d1,2


 εft1,2 ε0t 1,2 1  ε1,2 εft1,2  ε0t 1,2

(4.28)

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

where the subscripts 1 and 2 denote the fibre and transverse directions, respectively; ε0t 1,2 is the strain when the damage initiation condition is fulfilled. Due to the irreversibility of the damage process, the strain calculated at each time step is updated in comparison with the strain at damage initiation ε1,2 ¼ max (ε1,2, ε0t 1,2) in Eq. (4.28). In order to avoid a zero or even negative energy absorption, the complete failure strain is defined to be greater than the initial failure strain εft1,2 > ε0t 1, 2 . The failure initiation strain can be given by the following equation ε0t 1,2 ¼

σ T1,2 E1,2

(4.29)

For tensile failure in fibres, σ T is the tensile strength XT, while YT is used for the matrix tensile failure mode. εft1,2 represents the maximum strain for complete failure in Eq. (4.28) and the stress is thus reduced to 0 at this point with a corresponding damage variable value of 1. εft can be derived from the fracture energy GT1,2 for the individual failure modes, the failure strength of the material, and the characteristic length as follows εft1,2 ¼

2GT1,2 σ T lc

(4.30)

In the aforementioned equation, lc is the characteristic length that can maintain an energy release rate per unit area of crack constant and also keep the predicted results independent of the mesh size in an FE model. Several methods have been reported in the literature to evaluate lc [45–47]. Bazǎnt and Oh [45] proposed the following relation for a square element lc ¼

pffiffiffiffiffiffiffi AIP |θ|  45° cos θ

(4.31)

where AIP is the area of the element with one integration point and θ is the angle between the mesh line and the crack direction. This method seems useful to solve the issue of dependency of the solution on the mesh size. However, this approach is not suitable for multidirectional composite laminates for which the crack growth direction varies between layers. In order to overcome this limitation, an average characteristic length was proposed by Oliver for predicting crack propagation [46]. Moreover, Olivier proposed a method to evaluate the characteristic length based on the mesh discretisation taking into account the orientation of the crack. Pinho [47] proposed a method to define the characteristic length by reference to the element volume V and the fracture area A lc ¼

V A

(4.32)

Low-velocity impact of composite laminates: Damage evolution

(a)

3

99

(b)

F

Fracture plane area A = L1L3

3

L3

L3 2

Fracture plane area A = L2L3

2

F

F

L1

1

F

L1

1

L2

L2

Fig. 4.4 Schematics of fibre and matrix tensile damage with fracture plane in unidirectional composite laminate. (A) Fibre tensile damage mode. (B) Matrix tensile damage mode.

For the fibre and matrix tensile damage modes, the fracture plane is expected to be perpendicular to the load directions, as shown in Fig. 4.4. Therefore, the characteristic length can simply be found using Eq. (4.32). For instance, for the fibre tensile damage mode,the fracture plane area  is A ¼ L2L3 and the characteristic length l∗ is thus equal

to L1 lc ¼ VA ¼ LL1 L2 L2 L3 3 ¼ L1 ; similarly, the characteristic length is defined as L2 for the

matrix tensile damage mode shown in Fig. 4.4B. However, when the fracture plane is unknown, as in the case of the matrix compression damage mode, for instance, it can be more complicated to compute the characteristic length in the material model. An accepted way to correctly define the characteristic length is to perform a transformation of the nodal coordinates at the integration point of each element in the material subroutine, as shown in Fig. 4.5. Firstly, the element dimensions are computed in a single-lamina coordinate system defined in relation to the fibre direction, which is illustrated in Fig. 4.5A with the transformation from global coordinate (1,2,3) to material axes

(a)

(b) Lc (α )

La 2

a

c

b

L2

ω

α

Lc Fracture plane

L1

1

Lb

b

Fig. 4.5 Transformation of the material and fracture planes for definition of the characteristic length in a single element. (A) Transformation from global coordinate to local material coordinate system. (B) Transformation from material coordinate axes to the fracture plane.

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

(a,b,c). The fracture plane (see Fig. 4.5B) is then transformed from the material axes, and the characteristic length for the matrix compressive damage mode is derived using Eq. (4.33) l¼

L1 L2 L3 La LcðαÞ

(4.33)

 L L2 1 , sin where La in Fig. 4.5A can be defined as La ¼ min cosω ω , Lb can then be derived as Lb ¼ LL1 La 2 , and Lc is equal to the out of plane length L3. Similarly, Lc(α) is obtained by  L c Lb LcðαÞ ¼ min cos α , sin α as shown in Fig. 4.5B.

4.2.3.2 Fibre compressive failure mode Similarly, the compressive damage variable in the fibres is expressed as d1c


 εfc ε0c 1 1 ¼ fc 1 ε1 ε1  ε0c 1

(4.34)

The strains at the onset of failure and for complete failure can be obtained using Eqs. (4.35) and (4.36), respectively ε0c 1 ¼

XC E1

(4.35)

εfc 1 ¼

2GC1 X C lc

(4.36)

4.2.3.3 Matrix compressive failure mode As presented in Section 4.2.2, the damage criterion proposed by Puck and Schurmann [28] was applied in this work for the compression failure mode. A fracture plane was defined and the nominal stress and material strength were hence transformed to the fracture plane to predict the failure initiation and evolution according to the following equation d2c ¼


 εfc ε0c mat mat 1  0c εmat εfc mat  εmat

(4.37)

In the aforementioned equation, εmat is the strain updated at each time step that has pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi been transformed to the fracture plane and can be defined as εmat ¼ ε2NT + ε2NL . εij (i, j ¼ T, N, L) is the strain εij (i, j ¼ 1, 2, 3) rotated to the fracture plane, which can be expressed as

Low-velocity impact of composite laminates: Damage evolution

101

  εNT ¼ ε2 cos α sin α + ε3 cos α sin α + γ 23 2cos 2 α  1

(4.38)

εNL ¼ γ 12 cos α + γ 13 sin α

(4.39)

ε0c mat is the strain at the onset of failure, and it can be numerically obtained once the damage initiation criterion for compression has been met. α is the angle of the fracture plane as defined in Fig. 4.1. As mentioned previously, due to the different load states in composite laminates, the angle of the fracture plane can be accurately known through experiments or obtained from the FE program by selecting an optimal value in the range 90 °  α 90° that maximises the compressive damage initiation criterion Fmc. The strain for complete failure is derived in terms of the fracture energy for matrix compression damage mode GC2 , the damage initiation stress σ 0c mat , and the characteristic length lc . εfc mat ¼

2GC2 σ 0c mat lc

(4.40)

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi It is worth noting that the computed stresses σ mat ¼ σ 2NT + σ 2NL with the stresses defined in the fracture plane can be obtained by transforming the stresses from the global coordinate system using the transformation matrix T(α) in Eq. (4.15).

4.2.4 Nonlinear shear failure mode According to experimental observations, composite laminates generally exhibit nonlinear and irreversible shear behaviour. In general, there are two different approaches to predict the nonlinear shear behaviour of composite laminates: plasticity and progressive damage with degradation of the stiffness of laminates. Several studies have been published to model the nonlinear shear behaviour of composite laminates using continuum theories of plasticity [48], continuum damage mechanics [49], or a combination of these two methods [50]. A damage-modelling approach used to predict the nonlinear shear failure initiation and evolution in composites was proposed by Donadon et al. [14]. A polynomial cubic stress–strain relationship was used to predict the nonlinear shear behaviour τij ¼ c1 γ ij + c2 γ 2ij + c3 γ 3ij

(4.41)

where the subscripts i and j refer to coordinate directions; c1, c2, and c3 are determined by fitting experimental stress–strain curves. The theoretical model introduced earlier has been proved effective in simulating the nonlinear shear behaviour of composite laminates. However, due to its strong reliance on experimental measurements, alternative approaches have been developed for better predictive capability. Soutis et al. [51,52] developed a theoretical approach to predict the nonlinear shear response of composite systems and good agreement with

102

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

experimental results has been obtained. The nonlinear shear stress–strain relation is expressed as τij ¼ Si, j 1  exp

G0i, j γ Si, j

!! ,i, j ¼ 1,2,3

(4.42)

This analytical expression includes the ultimate shear strength Si,j and the elastic shear modulus G0i,j, both of which are easily measured composite materials constants. Moreover, this equation can be easily implemented into an FE program to model the nonlinear shear behaviour of composites. The total strain in this nonlinear shear model can be decomposed into two parts: elastic strain γ ei,j and inelastic strain γ in i,j. The inelastic strain can be defined as e γ in i, j ¼ γ i, j  γ i, j ¼ γ i, j 

τij G0i, j

(4.43)

where τij is the nonlinear shear stress corresponding to the different shear planes; G0i, j is the original shear modulus which is a material constant. The criterion for shear failure initiation is expressed in terms of the nonlinear shear stress and maximum shear strength FSij ¼

|τi, j | ¼1 Si, j

(4.44)

where τi,j is the shear stress for a given shear plane and Si,j indicates the relative ultimate shear strength on that plane. The damage evolution law for the nonlinear shear modes is expressed by the damage variable di,j [53] di, j ¼ 1 

 γ i, j,0  γ in i, j,0 1 + λ2 ð2λ  3Þ in γ i, j  γ i, j, 0

(4.45)

with λ¼

γ i, j  γ i, j,0  2γ in i, j,0 f γ i, j,0  γ in i, j,0  γ i, j

(4.46)

where γ fi, j is the shear strain at complete failure and is related to the shear fracture toughness (which varies with the mode of fracture) GS, shear strengths of the material Si,j, and the characteristic length lc γ fi, j ¼

2GS Si, j lc

(4.47)

Low-velocity impact of composite laminates: Damage evolution

103

γ in i, j,0 denotes the inelastic strain for shear damage initiation. When experimental results are not available, GS can be assumed to be equal to the intralaminar matrix compression fracture toughness GC2 for unidirectional plies of composite laminates [53].

4.3

Damage prediction of composites under low-velocity impact

In this section, impact-induced damage for composites is mainly simulated by implementing the damage model introduced in the aforementioned section into the finite element code Abaqus/Explicit 6.10 through the Vumat subroutine to predict the type and extent of individual intralaminar damage modes by referring to the work of Shi et al. [16]. Impact tests with different impact energies are also performed for cross-ply laminates for the validation of the numerical modelling. Moreover, in order to improve the prediction accuracy and simulate the propagation of matrix cracking, an extension of the model is proposed with cohesive elements inserted between adjacent elements along the fibre direction within the individual plies to model splitting and transverse matrix cracks resulting from impact loading. Prediction from the extended model is then compared to nondestructive measurements of damage development using X-ray radiography [17].

4.3.1 Impact tests Composite laminates, 2 mm thick, were fabricated from carbon fibre–epoxy resin prepreg with a stacking sequence of [0/90]2S for all impact tests. The prepreg was made from continuous unidirectional high-tensile-strength carbon fibres (Tenax HTS40 12K 300) impregnated with Cycom 977–2 epoxy resin, which is a typical high-temperature curing aerospace grade system [16]. A drop weight impact tower was used for the test program with a constant drop height of 0.75 m and an impactor’s mass equal to 1, 1.5, and 2 kg resulting in impact energies of 7.35, 11.03, and 14.7 J, respectively [16]. A force ring sensor PCB 203B SN 2205 with a sensitivity of 0.0562 mV/N [16] was embedded in the impactor’s head and connected to the dynamic signal analyser SigLab to record the dynamic loading history. The acceleration of the impactor is thus derived by dividing the measured force by the impact mass. Additional required variables such as velocity, displacement, and impact energy can be obtained based on this calculated acceleration. For instance, if the mass of the impactor is m and the initial velocity is v0, then the impact energy can be easily calculated as follows E¼

mv20 2

The absorbed energy of the impacted samples can thus be derived as

(4.48)

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

Eabsorbed ¼ E 

mv2i ðtÞ mv20 mv2i ðtÞ  ¼ 2 2 2

(4.49)

where vi(t) is the velocity of the impactor at time t during impact, which can be obtained as Zt 1 Ft (4.50) vi ðtÞ ¼ v0  m 0

where F in the aforementioned equation is the force measured during the impact test.

4.3.2 Modelling impact-induced damage using damage criteria methods The algorithm for the developed damage model implemented through the user-defined subroutine Vumat is shown in Fig. 4.6 to simulate the damage evolution in composites under various impact energy levels. A 3D FE model of an impact test, including both impactor and a circular composite plate with appropriate boundary conditions, was built and is shown in Fig. 4.7. The 2-mm thick laminate consisted of eight plies with a ply thickness of 0.25 mm and a stacking sequence [0/90]2S. All the nodes at the edge of the plate were fixed in all directions (x, y, z) to simulate the experimental clamped conditions, and the impactor was modelled as a rigid body. The impact events were simulated

Hashin & Puck

σ

σn

ε

= C εn

No

Yes Damage initiation

dft Damage variables

dfc dmt

Damage evolution

σn

= d C εn

dmc dlt

Fig. 4.6 Algorithm for the damage criteria implemented into the Vumat subroutine of Abaqus/ Explicit [17].

Low-velocity impact of composite laminates: Damage evolution

105

ϕ = 75 mm ϕ = 15 mm

Impactor: Fixed except z axis initial velocity along z 2 mm

Contact between surfaces of impactor and top laminate surface z x

Fixed edge

y

Fig. 4.7 FE model with boundary conditions for the numerical simulation of the impact event [16].

for the three energy values mentioned earlier and a prescribed initial velocity of 3.83 m/s was assigned to the impactor. For the circular composite plate, each ply was meshed with eight-node linear brick elements (C3D8R). The computing time was reduced by introducing different mesh size/density in different regions of the FE model, with 1 mm  1 mm elements in the impacted zone and a progressively coarser mesh away from the impact zone (Fig. 4.7) where damage was not expected to occur, in agreement with experimental observations; size element and mesh density affect stress results and hence damage simulation. At each ply interface, COH3D8 cohesive elements were inserted to model delamination initiation and growth. The detailed material properties are listed in Table 4.1 [16]. Contact between the impactor and impacted plate, and between each ply of the laminate, was defined by the general contact algorithm within Abaqus/Explicit. This algorithm generated contact forces based on the penalty enforcement contact method. A tangential interaction was defined between the surfaces of adjacent layers using a Table 4.1 Material properties of the carbon fibre–epoxy unidirectional laminate [16]. Density (kg/m3) Orthotropic properties Strength (MPa) In-plane fracture toughness (kJ/m2)

1600 E01 ¼ 153 GPa; E02 ¼ E03 ¼ 10.3 GPa; ν12 ¼ ν13 ¼ 0.3; ν23 ¼ 0.4; G012 ¼ G013 ¼ 6 GPa; G023 ¼ 3.7 GPa T X ¼ 2537; XC ¼ 1580; YT ¼ 82; YC ¼ 236; S12 ¼ 90; S23 ¼ 40 GT1C ¼ 91.6; GC1C ¼ 79.9; GT2C ¼ 0.22; GC2C ¼ 2; GS ¼ 2

106

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

Coulomb friction model, τ ¼ μp, where the friction coefficient, μ, relates the shear stress τ to the contact pressure p. The friction coefficient between contacted layers is defined as a function of fibre orientation. For a 0°/0° interface, the value of μ ¼ 0.2 was reported, while a value of 0.8 was suggested for the interface between neighbouring 90° plies [54–56]. Thus, in the present analysis, an average friction coefficient of 0.5 was used for the 0°/90° interface of the cross-ply laminate [16]. In addition, a friction coefficient value of μ ¼ 0.3 was applied between the surface of the metal punches or supports and the outer surface of composite plate [16]. It is worth noting that these assumed μ coefficients have an effect on predicted absorbed energy values and they should therefore be ideally measured for the particular system investigated. Nondestructive evaluation (NDE) of the extent of damage in the impacted composite panels was carried out using penetrant-enhanced X-ray radiography. Results show transverse ply cracking and almost peanut-shaped delamination for an impact energy of 7.35 J, as illustrated in Fig. 4.8A. The predicted overall damage is shown for comparison in Fig. 4.8B, where all delamination is represented in rainbow colour (while in print in different shades of grey); red (dark in the centre of the image) regions indicate that the material has failed completely while blue (dark in the outer part of the maps) regions indicate that no damage has taken place. In between (lighter grey areas), the material has come close to failure but the failure criterion has not yet been reached. The delamination region shown in Fig. 4.8A is larger than that predicted, which indicates that the damage model slightly underestimated the energy absorbed during impact. Similar radiographs and predicted damage patterns are shown in Fig. 4.8C–F for the other two impact cases (11.03 and 14.7 J). As expected, the amount of damage introduced is more extensive as the impact energy increases. The delaminated area at each interface was individually simulated by cohesive zone elements and the delamination was found to be driven by the direction of the lower ply. For this research, the predicted delamination was therefore effectively validated through the peanut shape (Fig. 4.8) also seen in the experimental results for the crossply laminate. Matrix tensile failure and shear damage are also important damage mechanisms that absorb energy during impact, and the prediction from the model is shown in Fig. 4.9A–F. In these images, matrix tensile failure and in-plane shear damage of each ply are superimposed. The matrix tensile failure in Fig. 4.9A is initially small and becomes larger under increased impact energy (Fig. 4.9C and E). In-plane shear damage in each ply is superimposed in Fig. 4.9B and develops at approximately 45° with respect to the 0° fibre direction. A larger amount of shear damage is introduced for higher-impact energy levels (Fig. 4.9D and F). This damage might lead to a small permanent indentation after the impact event. Moreover, fibre breakage in Fig. 4.9G was predicted by the damage model when the applied load resulted in local stresses exceeding the tensile strength of the fibres, such as in the case of the 14.7-J impact. Fibre breakage was also observed experimentally as highlighted in Fig. 4.8E (darker regions corresponding to the zinc iodide solution completely penetrating the damaged plate).

Low-velocity impact of composite laminates: Damage evolution

(a)

107

(b) Matrix cracking

90°

90°

Delamination 10 mm

0°

(c)

0°

10 mm

(d) Delamination

90°

90° Matrix cracking

10 mm

0°

(e)

Matrix cracking

0°

10 mm

(f)

Splitting

90°

Fibre breakage Delamination 0°

10 mm

90° 0°

10 mm

Fig. 4.8 X-ray radiographs and superposition of numerically predicted delamination area. (Aand B) Impact energy of 7.35 J. (C and D) Impact energy of 11.03 J. (E and F) Impact energy of 14.7 J [16].

4.3.3 Modelling impact-induced matrix cracking and splitting using cohesive zone elements The type, location, and extent of damage were predicted by the proposed damage model in Section 4.3.1 and the overall damage area was in good agreement with observations and X-ray radiography images. However, the results also showed that the model slightly under-predicted the energy absorbed by the laminate during the 7.35-J impact, although the discrepancy kept decreasing with increasing impact

108

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

(a)

(b)

90°

90° 10 mm

0°

(c)

0°

10 mm

0°

10 mm

0°

10 mm

(d)

90°

90° 10 mm

0°

(e)

(f)

90°

90° 10 mm

0°

(g) Fibre breakage

90° 0°

10 mm

Fig. 4.9 Numerically predicted damage in the form of matrix tensile failure, (A, C, and E) and nonlinear shear damage (B, D, and F). (A and B) Impact energy of 7.35 J. (C and D) Impact energy of 11.03 J. (E and F) Impact energy of 14.7 J. (G) Predicted fibre breakage for impact energy of 14.7 J [16].

Low-velocity impact of composite laminates: Damage evolution

109

energy. This discrepancy could be attributed to the material properties variability, specimen quality/imperfections introduced during the manufacturing process, and/ or inaccuracies in the assumed value of the friction coefficient between the projectile and the plate or between individual plies. But it is most likely due to the lack of successful simulation of splitting, which can be experimentally observed in the bottom 0° ply of the cross-ply laminate, as typically shown in Fig. 4.8E. Therefore, in this section, the same impact events for cross-ply laminates have been modelled by combining together the damage criteria and the cohesive elements approach to respectively predict the different damage modes in order to improve the accuracy of the simulation [17]. The 3D FE model was built with a full size of 100  100  2 mm3 for the composite plate, clamps with a circular hole in the centre, and impactor, as shown in Fig. 4.10. The full model was built to allow for the simulation of a long splitting as observed experimentally, or transverse cracks, which can be influenced by the clamping conditions. The composite plate was meshed using eight-node linear brick elements with zero-thickness cohesive elements inserted within the individual plies to model matrix cracking and splitting as well as at the interface of adjacent plies to simulate delamination (see Fig. 4.10). Both clamps with the hole of 75 mm in diameter were represented by shell elements and fixed in all directions to constrain the composite plate while the impactor was constrained in all degrees of freedom (DOFs) except in the vertical impact direction. Contact between the impactor and the impacted plate, between both clamps and the plate, and between internally adjacent plies of the laminate, was simulated using the general contact algorithm. In order to reduce the computing time, the model was meshed with a refined element size of 2 mm  2 mm  0.25 mm in the impact zone while a relatively coarse mesh was applied in the area away from the impact zone. The refined size was determined based on the experimental observation of minimum splitting size in the nonimpacted 0° ply (see Fig. 4.14A).

z y

x

Delamination

Matrix cracking

Fig. 4.10 A full FE model for impact simulation with defined planes of matrix cracking and delamination by using interface cohesive elements [17].

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

Table 4.2 Material parameters used for the interface cohesive elements [17].

Interface stiffness (GPa/mm) Interlaminar strength (MPa) Interlaminar fracture toughness (J/m2)

Mode I

Mode II

Mode III

106 62.3 280

106 92.3 790

106 92.3 790

The material properties used for the composite were consistent with those reported in Table 4.1 while the properties defined for cohesive zone elements have been listed in Table 4.2, including interface stiffness, strength, and fracture toughness for the simulation of the damage evolution process. It is worth noting that since cohesive elements with zero thickness were used within each ply and at the interface of adjacent plies, an interface stiffness value of 106 GPa/mm was used for the simulation [17,18]. The impact force–time curves for different energies applied are shown in Fig. 4.11A–C including the experimental measurement, the new numerical prediction with cohesive elements inserted for the simulation of matrix cracking and splitting, as well as the previous numerical prediction using damage criteria only. In Fig. 4.11A, it can be seen that the force–time curve corresponding to the prediction with inserted cohesive elements for an impact energy of 7.35 J shows a closer match with experimental results from the initial contact to the maximum force in comparison with the numerical prediction obtained using damage criteria only. For both numerical models, though, the force takes a longer time to degrade to 0 during the rebound phase compared to measurements. However, the model including splitting clearly shows a better match with the experimental data. Similar observations can be made for the increased impact energy of 11.03 J (Fig. 4.11B), while the accuracy of FE predictions including the effect of matrix cracking and splitting is considerably improved for the impact energy of 14.7 J (Fig. 4.11C). The impact energy–time histories are presented in Fig. 4.12A–C for the three impact energies. At low energy levels, the FE analysis without cohesive elements is under-predicted by 10%–14% of the energy absorbed, although the prediction is improved at the higher impact level of 14.7 J, for which delamination and fibre breakage may dominate the damage process. The absorbed energy for each energy level is better captured when matrix cracking and splitting are included in the energy calculations. The final absorbed energy values obtained from the experiments and both numerical models are presented in Table 4.3. The difference between predicted and measured values for the applied impact energy of 7.35 J drops from almost 14% to less than 5.2% when intralaminar cracking is taken into account. The predicted damage extent from the improved numerical model is compared in Fig. 4.13 to both experimental results using X-ray radiography and previous simulation results obtained using the damage criteria only. The typical example of the composite panel under the impact energy of 14.7 J was studied since more internal damage modes were clearly visible experimentally for this particular case, as shown in

Low-velocity impact of composite laminates: Damage evolution

111

(a) 3500 Experiment Simulation with splitting Simulation without splitting

3000

Force (N)

2500 2000 1500 1000 500 0 0

1

2

(b) 4000

3 4 Time (ms)

5

6

Experiment Simulation with splitting Simulation without splitting

Force (N)

3000

2000

1000

0 0

1

2

(c) 5000

3 4 Time (ms)

5

6

Experiment Simulation with splitting Simulation without splitting

Force (N)

4000 3000 2000 1000 0 0

1

2

3 4 Time (ms)

5

6

Fig. 4.11 Impact force–time curves for both experimental measurements and numerical predictions. (A) Impact energy of 7.35 J. (B) Impact energy of 11.03 J. (C) Impact energy of 14.7 J [17].

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(a) 8

Experiment Simulation with splitting Simulation without splitting

Energy (J)

6

4

2

0 0

1

2

3 4 5 6 Time (ms) Experiment Simulation with splitting Simulation without splitting

1

2

3 4 Time (ms)

(b) 12 10

Energy (J)

8 6 4 2 0 0

5

6

(c) 15 12

Energy (J)

9 Experiment Simulation with splitting Simulation without splitting

6 3 0

0

1

2

3 4 Time (ms)

5

6

Fig. 4.12 Impact energy–time curves for both experimental measurements and numerical predictions. (A) Impact energy of 7.35 J. (B) Impact energy of 11.03 J. (C) Impact energy of 14.7 J [17].

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Table 4.3 Experimental and numerical values of absorbed impact energy [17]. Absorbed energy Impact energy (J)

Test (J)

Earlier model (J)

Model with splitting (J)

Difference (%)a

7.35 11.03 14.7

5.5 7.1 9.52

4.49 6.02 9.08

5.12 6.98 9.88

5.17 (13.74)b 1.08 (9.79)b 2.45 (2.99)b

a b

Between experimental and FE results. () Difference when intralaminar cracking is ignored.

(a)

(b)

90°

90° 0°

10 mm

10 mm

0°

(c)

10 mm

Fig. 4.13 Experimentally measured and numerically predicted damage extent for the impact energy of 14.7 J. (A) X-ray radiograph. (B) Numerical model without matrix cracking and splitting. (C) Extended model [17].

Fig. 4.13A. A large area of delamination is observed together with extensive matrix cracks at the impact location as well as obvious splitting along the 0° fibre direction in the experiment. Fig. 4.13B shows the predicted overall damage from the earlier FE model while Fig. 4.13C illustrates the predicted delamination area simulated using cohesive elements.

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It is clearly seen in Fig. 4.13B that the predicted damage area agrees well with the measured delamination area but the damage mode of splitting is not explicitly shown. This can explain why the absorbed energy induced by the impact is always under-predicted, especially for lower impact energies, since matrix cracking and splitting provide an additional contribution to the energy absorbed by the composite panel. The cohesive-element-based numerical model more accurately captures the splitting and delamination patterns, as illustrated in Fig. 4.13C. The black lines in Fig. 4.13C represent the splitting mode in the bottom 0° ply that experiences maximum out-of-plane deflection due to bending. Splitting developed in the bottom 0° ply of the [0/90]2S laminate is clearly shown in Fig. 4.14 for the impact event of 14.7 J. The numerically simulated splitting gives a damage pattern similar to that observed in the experiment. The mixed-mode effect of large normal tensile and shear stresses drove the formation of splitting. The fractured material strips created by splitting measure 76 mm in length and 9 mm in width experimentally, while the numerical model predicts strips 72 mm long and 10 mm wide, which shows reasonable agreement. Regions in the ply where splitting started to grow but is not yet fully formed can be captured by the FE model, while this is not visible in the physical experiment. Also, cracks may close once the load is removed, therefore becoming undetectable by X-ray radiography. The FE analysis in this new model therefore provides very valuable insight into the full damage process by effectively capturing matrix cracking development during the whole impact event from initial contact to complete

(a)

y x

10 mm

(b)

y x

10 mm

Fig. 4.14 Splitting in the bottom 0° ply of [0/90]2S laminate under the impact energy of 14.7 J. (A) Experimental result. (B) Simulation [17].

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115

(a)

(b) Bending crack

(c)

(d)

Delamination

Fig. 4.15 Evolution process of matrix cracking in a cross-ply composite laminate subjected to impact energy of 7.35 J. (A) Initial contact. (B) Complete transfer of impact energy from impactor to laminate. (C) Rebounding phase. (D) End of impact event. Ref. [17].

rebound, as shown in Fig. 4.15. In this example, the impact energy of 7.35 J was selected to simulate the damage evolution process in the cross-ply laminate since 90° matrix cracking is the main expected damage mode together with crackinduced delamination (the impact energy is too low to cause fibre breakage). Matrix cracking is initially formed in the bottom 90° ply with few cracks found in the middle 90° plies (Fig. 4.15A). As the impactor further contacts the composite laminate, the surface indentation is formed with a bending crack appearing on the top 0° surface ply due to concentrated compressive load, while complete cracks through the thickness of the 90° layers are developed accompanied by delamination at the 0/90 interfaces (Fig. 4.10B). As the impact event progresses, the growth

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rate of cracks in the middle and bottom 90° plies is found to decrease, while the projectile rebound occurred (Fig. 4.10C). The simulation shows that cracks start to close once the complete rebound is reached (Fig. 4.10D). Extensive delaminations are shown in Fig. 4.10D, which are similar to those detected experimentally by X-ray radiography.

4.4

Conclusions

In this chapter, the numerical prediction of impact-induced damage evolution for a cross-ply [0/90]2S composite laminate has been investigated for three different impact energy levels. A stress-based failure criterion was introduced to predict damage initiation and evolution in the form of intra- and interlaminar damage modes using energy-based criteria. The damage model was implemented via a user subroutine into the commercially available finite element code Abaqus/Explicit. Moreover, delamination was simulated in the model by inserting cohesive element layers between all plies with appropriate traction laws and damage initiation and evolution criteria; damage development was compared favourably with the experimental results and observations. Although the damage model slightly under-predicted the energy absorbed by the laminate at lower impact energy levels, the discrepancy was significantly reduced for the highest impact energy level, for which a very good agreement was found between the experimental and numerically predicted delaminated areas. The discrepancy between experimental measurement and numerical prediction is most likely due to the splitting damage mode, which was not considered in the simulation, but was observed experimentally using X-ray radiography. Therefore, in order to further improve the prediction accuracy, the numerical model was extended by using cohesive elements inserted along the fibre direction between neighbouring elements within individual lamina for the cross-ply laminate [0/90]2S. The matrix cracking/splitting damage mode can be successfully captured using cohesive elements with appropriate material properties. The inclusion of intralaminar cracking improved the accuracy of the FE prediction, especially for the lower impact energy levels, reducing the difference with experimental results. Results therefore demonstrated the significant contribution of the splitting damage mode for low-energy impact. The extended FE model also demonstrated an improved predictive capability of the extent and pattern of damage in comparison with X-ray radiography measurements. The splitting damage mode at the bottom ply of the [0/90]2S laminate for the highest impact energy level was indeed predicted, giving confidence that the assumed stiffness and strength properties of the cohesive elements are appropriate for the selected lay-up and carbon fibre–epoxy system. Moreover, in order to understand the full history of the crack evolution process in composites, the evolution of matrix cracking was simulated as the main damage mode, accompanied by delamination using cohesive elements for the lowest-impact energy level. The model showed regions in the ply where resin cracks initiated, propagated, and then closed during rebounding. Such cracks are difficult to detect using X-ray radiography or any other nondestructive detection technique, which therefore underestimate the severity and extent of internal damage.

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The developed physically based damage model has shown its capability in predicting the full damage evolution process in cross-ply composite laminates under low-velocity impact. Future developments with the model could include the simulation of the residual (postimpact) strength properties and fatigue life of such laminates and other multidirectional lay-ups.

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[19] Y. Shi, C. Pinna, C. Soutis, Impact damage characteristics of carbon fibre metal laminates: experiments and simulation, Appl. Compos. Mater. 27 (5) (2020) 511–531. [20] Y. Shi, C. Soutis, Modelling low velocity impact induced damage in composite laminates, Mech. Adv. Mater. Mod. Process. 3 (2017) 1–12. [21] Y. Shi, C. Soutis, Modelling transverse matrix cracking and splitting of cross-ply composite laminates under four point bending, Theor. Appl. Fract. Mech. 83 (2016) 73–81. [22] Y. Shi, C. Pinna, C. Soutis, Interface cohesive elements to model matrix crack evolution in composite laminates, Appl. Compos. Mater. 21 (2014) 57–70. [23] S.W. Tsai, Strength Characterisation of Composite Material, Tech Rep NASA CR-224, National Aeronautics and Space Agency, 1965. [24] S.W. Tsai, E.M. Wu, A general theory of strength for anisotropic materials, J. Compos. Mater. 5 (1971) 58–80. [25] O. Hoffman, The brittle strength of orthotropic materials, J. Compos. Mater. 1 (1967) 200–206. [26] Z. Hashin, A. Rotem, A fatigue failure criterion for fiber-reinforced materials, J. Compos. Mater. 7 (1973) 448–464. [27] Z. Hashin, Failure criteria for uni-directional fibre composites, J. Appl. Mech. 47 (1) (1980) 329–334. [28] A. Puck, H. Schurmann, Failure analysis of FRP laminates by means of physically based phenomenological models, Compos. Sci. Technol. 58 (10) (1998) 1045–1067. [29] J. Salencon, Handbook of Continuum Mechanics: General Concepts, Thermoelasticity, Springer, New York, 2001. [30] C.T. Sun, B.J. Quinn, D.W. Oplinger, Comparative evaluation of failure analysis methods for composite laminates, 1996. DOT/FAA/AR-95/109. [31] C.G. Davila, P.P. Camanho, C.A. Rose, Failure criteria for FRP laminates, J. Compos. Mater. 39 (2005) 323–345. [32] S.T. Pinho, P. Robinson, L. Iannucci, Physically-based failure models and criteria for laminated fibre reinforced composites with emphasis on fibre kinking: part I: development, Compos. Part A 37 (1) (2006) 63–73. [33] S.T. Pinho, P. Robinson, L. Iannucci, Physically-based failure models and criteria for laminated fibre reinforced composites with emphasis on fibre kinking: part II: FE implementation, Compos. Part A 37 (5) (2006) 766–777. [34] P.D. Soden, M.J. Hinton, A.S. Kaddour, Biaxial test results for strength and deformation of a range of e-glass and carbon fibre reinforced composite laminates: failure exercise benchmark data, Compos. Sci. Technol. 62 (2002) 1489–1514. [35] F.K. Chang, K.Y. Chang, Post-failure analysis of bolted composite joints in tension or shear-out mode failure, J. Compos. Mater. 21 (9) (1987) 809–833. [36] F.K. Chang, K.Y. Chang, A progressive damage model for laminated composites containing stress concentrations, J. Compos. Mater. 21 (9) (1987) 834–855. [37] K.Y. Chang, S. Liu, F.K. Chang, Damage tolerance of laminated composites containing an open hole and subjected to tensile loadings, J. Compos. Mater. 25 (3) (1991) 274–301. [38] L.M. Kachanov, On the Creep Rupture Time, vol. 8, Izv SSSR Otd Tekhn Nauk, 1958, pp. 26–31. [39] Y.N. Rabotnov, On the equations of state for creep, in: Progress in Applied Mechanics, Prager Anniversary Volume Macmillan, New York, 1963. [40] A. Matzenmiller, J. Lubliner, R.L. Taylor, A constitutive model for anisotropic damage in fiber-composites, Mech. Mater. 20 (2) (1995) 125–152. [41] K.V. Williams, R. Vaziri, Application of a damage mechanics model for predicting the impact response of composite materials, Comput. Struct. 79 (2001) 997–1011.

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[42] A. Tabiei, S.B. Aminjikarai, A strain-rate dependent micro-mechanical model with progressive post-failure behaviour for predicting impact response of unidirectional composite laminates, Compos. Struct. 88 (2009) 65–82. [43] L. Iannucci, M.L. Willows, An energy based damage mechanics approach to modelling impact onto woven composite materials: part II. Experimental and numerical results, Compos. Part A 38 (2007) 540–554. [44] L. Iannucci, J. Ankersen, An energy based damage model for thin laminated composites, Compos. Sci. Technol. 66 (2006) 934–951. [45] Z.P. Bazǎnt, B.H. Oh, Crack band theory for fracture of concrete, Mater. Struct. 16 (1983) 155–177. [46] J. Olivier, A consistent characteristic length for smeared cracking models, Int. J. Numer. Methods Eng. 28 (1989) 461–474. [47] S.T. Pinho, Modelling Failure of Laminated Composites Using Physically-Based Failure Models (Ph.D. thesis), Imperial College London, 2005. [48] S.A. Khan, S. Huang, Continuum Theory of Plasticity, John Wiley and Sons, New York, 1995. [49] J. Lemaitre, J.L. Chaboche, Mechanics of Solid Materials, Cambridge University Press, Cambridge (UK), 1990. [50] R. Danesi, B. Luccioni, S. Oller, Coupled plastic-damaged model, Comput. Methods Appl. Mech. Eng. 129 (1–2) (1996) 81–89. [51] P. Berbinau, C. Soutis, I.A. Guz, Compressive failure of 0o unidirectional carbon-fibrereinforced plastic (CFRP) laminates by fibre microbuckling, Compos. Sci. Technol. 59 (1999) 1451–1455. [52] P. Berbinau, C. Soutis, P. Gouta, P.T. Curtis, Effect of off-axis ply orientation on 0o fibre microbuckling, Compos. Part A Appl. Sci. Manuf. 30 (1999) 1197–1207. [53] M.V. Donadon, S.F.M. de Almeida, M.A. Arbelo, A.R. de Faria, A three-dimensional ply failure model for composite structures, Int. J. Aerosp. Eng. 2009 (2009) 1–22. [54] N. Sung, N. Suh, Effect of fiber orientation on friction and wear of fiber reinforced polymeric composites, Wear 53 (1979) 129–141. [55] J. Schon, Coefficient of friction of composite delamination surfaces, Wear 237 (2000) 77–89. [56] Q. Bing, C.T. Sun, Effect of transverse normal stress on mode II fracture toughness in fiber composites, in: 16th International Conference on Composite Materials (ICCM16), Kyoto, Japan, 2007.

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R. Pancirolia, M. Ahmadib, M. Fotouhic, and G. Minakd a Niccolo Cusano University, Rome, Italy, bAmirkabir University of Technology, Tehran, Iran, c Delft University of Technology, Delft, The Netherlands, dAlma Mater Studiorum – University of Bologna, Forlı`, Italy

5.1

Low-velocity impact on thin and thick laminates

The impact resistance of laminated composite structures has received much attention over the last decades and continues to play an important role within the research community due to the heterogeneous, anisotropic, and brittle behaviour of the material. Depending on the ratio between the impactor speed and the structural dynamic response, impacts are usually divided into two categories: lowvelocity impact and high-velocity impact [1]. In low-velocity impact, the structure has more time to respond to the impactor and consequently more energy can be absorbed elastically. During their service life, composite laminates are likely to experience low-velocity impacts. The impact energy is absorbed by the composite structure in the form of elastic energy and internal damage. Different damage mechanisms relate to low-velocity impacts, spanning fibre breakage, matrix cracking, global or local buckling, and delamination [1–4]. Delamination is a particularly challenging damage mechanisms as it is often invisible to the naked eye but extremely detrimental in terms of residual resistance of the structure to membrane and flexural loads. Several factors – such as the material properties, projectile characteristics, layup and stitching, preload, environmental conditions, and curvature of the laminate – may affect the impact damage mechanics. In addition, the thickness of the whole laminate and of each lamina may change the induced damage mechanisms. A thin-ply laminate benefits from lower interlaminar stresses and less free-edge effects; however, there are more interlaminar boundaries in this type of laminate, which result in greater chance of delamination, more resin-rich zones and lower fibre volume fraction in a constant volume. The difference between damage mechanisms that may appear in low-velocity impact on the thin and thick plies and laminates is shown in Figs. 5.1 and 5.2. In thick laminates, high localised contact stresses cause matrix cracks on the impacted surface of the laminate. Damage progresses downward and makes a pinetree pattern. For thin laminates, due to bending stresses, matrix cracking starts in the lowest layer, and intra-ply cracks and interface delaminations propagate from

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V

V Delamination

M

Fibre compressive failure

t (total)

M

Fibre tensile failure Thick-ply laminate

Thin-ply laminate

Fig. 5.1 Different types of damage mechanisms by low-velocity impact in thin and thick laminates.

V

V

Thin laminate

M

Reversed pine tree

Thick laminate

M

Pine tree

Fig. 5.2 Impact-induced damage pattern in thick and thin laminates [4].

the lowest surface up toward the impacted surface, giving a reverse pine-tree pattern [4]. The impact behaviour of laminated composite materials has been investigated for many years but the role of in-plane stresses has not been considered in the majority of the studies, which for the most concentrate on plane laminates under simply supported boundary conditions, and with minor interest curved structures [3]. In real life, low-velocity impacts are expected to occur during the life of the structure, which is subjected to the design load [5]. Geometry and preload are known to have an effect on the impact response of metallic structures [6, 7] and similar influence is expected on composite structures as well. In addition, there is less study in the case of curved laminates under impact, whereas there is huge application of curved laminates in engineering structures. Therefore, the first objective of the following sections is to investigate the effect of preload on laminated composite materials under impact while the second is to understand the effect of curvature on the impact response.

Low-velocity impact on preloaded and curved laminates

5.2

123

Low-velocity impact on thin and thick laminates under preload (tension/compression)

In the operational environment, a composite structure may experience different loading conditions, which might lead to substantially different damage mechanisms compared to unload structures. As a result, it is important to understand the preloading effect on the impact behaviour of laminated composite materials. In the following section, the previously reported research on this area is reviewed.

5.2.1 Uniaxial preloading Some researchers have studied uniaxial tension and compression preloading prior to impact on laminated composites to evaluate the effect on the induced damage [8–14]. The schematic experimental setup for the uniaxially preloaded test is shown in Fig. 5.3, which contains the loading fixture and an instrumented drop weight tower. In this way, it is possible to apply different kinds of pure compression, pure tension, and pure shear load on the specimens prior to the impact test. Sankar et al. [8] investigated the effect of initial tensile stress on the low-velocity impact behaviour of graphite–epoxy laminates. Their results confirm that also in the case of preloaded structures, there is a critical velocity for the initiation of impact damage, and it is observed that in the prestressed samples such velocity threshold value is lower, but beyond a certain impact, velocity tensile prestresses may reduce the rise of delamination damage. The effect of uniaxial pretension and precompression loadings on the impacted composite laminates was studied by Chiu et al. [9]. It was found that both pretension and precompression have the effect of decreasing the impact resistance. The impactinduced damage occurs in a larger area for the specimens subjected to pretension. Pretension increases the flexural stiffness of a specimen while precompression decreases it; as a result, the maximum force is larger in the samples subjected to pretension and it is lower in the samples subjected to precompression, if compared to stress-free specimens. The damage size of a pretensioned specimen is higher than in a specimen

Drop weight impactor Load control

Specimen

Fig. 5.3 Schematic experimental setup for the uniaxial preloading [8].

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

without preload. Precompressed specimens bend into large deflections under the impact load and the damage size increases as a result of the delamination due to local buckling. Tensile preloaded thin and thick laminated composites subjected to low-velocity impact loads were investigated by Kelkar et al. [10]. The results showed that the impact energy intensities per ply basis, which are needed to initiate internal damage, remain approximately constant varying the laminate thickness. In addition, it was observed that for thicker laminates, higher amounts of energy per ply are required to cause back-face damage. Their results indicate that for laminates under tensile preloads, the incipient damage occurs much earlier. Heimbs et al. [11] performed experiments and numerical simulations on fully clamped plain laminates with and without a uniaxial compressive prestrain, indicating an increased deflection for the preloaded composite plates, which lead to a higher extent of material damage compared to the unloaded plates and observed that delamination onset and propagation is the main energy absorption mechanism. Fig. 5.4 illustrates C-scan results of the impacted laminates. It can be seen that by applying the same amount of energy levels per ply, the damage areas in the thin laminates are smaller than those in the thick laminates. Another interesting report in this chapter was that the impact-induced failure (A)

(B)

16-ply 1.5″

3.5″

6.5″

Induced damage area

Induced damage area

Drop height

1 0.8 0.6 0.4 0.2 0

32-ply 2.5

1.2

0

800

1600

3.5″

Drop height

6″

11″

2 1.5 1 0.5 0

2400

0

800

Preload (μs)

1600

2400

Preload (μs)

(C)

64-ply

Induced damage area

12 Drop height

10

10.375″

23.187″

36″

8 6 4 2 0

0

800

1600

2400

Preload (μs)

Fig. 5.4 Impact-induced damage in specimens: (A) 16-ply, (B) 32-ply, and (C) 64-ply laminates [9].

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125

mechanisms are different in thin and thick laminates, and for the thick laminates transverse shear effects are the main reason for failure. Further experimental results indicated that the duration of the impact event in thick laminates is lower than for thin laminates. Such behaviour should be ascribed to the increased stiffness of thicker laminates. It is follows that by increasing the preload, the impact load increases and the impact duration decreases. Heimbs et al. [11] studied the low-velocity impact behaviour of carbon-fibrereinforced plastic (CFRP) materials under a compressive preload. They found that there is an increase in the deflection and energy absorption in the samples prestressed with a load equal to 80% of the buckling load. An increase of the size of the delaminations occurring within individual plies is the main reason for the increase in the absorbed energy. Zhang et al. [12] studied the effect of compression preload on compression after impact (CAI) strength. They showed that if preload approaches the initial buckling value, the CAI strength benefits from the presence of the preload. In this case, the plate stiffness decreases and the impact-induced peak force is reduced. On the other hand, if preload approaches the CAI strength, the induced delamination can propagate catastrophically during the impact, but at a preload value below the CAI strength. The effects of preloading on damage and failure mechanisms of laminated composite materials were investigated by Pickett et al. [15]. Results of this study show obvious changes of the observed failure modes. By increasing the preload, the laminate impact tolerance is reduced, and during impact the superimposed loading induces major intra-ply shear and fibre failure damages. The C-scans show that almost 50% more delamination emerges in comparison with unloaded panels. The operating temperature is a further parameter which can affect impact response of composite structures. Fang et al. [16, 17] presented a study on CFRP wires under varying tensile preload and operating temperature, showing that low temperatures facilitate the generation and propagation of impact damages; that the energy dissipating capacity of CFRP wires increases as temperature increases; that the maximum impact force, transverse deflection at fracture, and maximum wire tension of the CFRP wire increase with temperature. The ballistic resistance of uniaxially loaded composite laminates have been presented in Zhikharev et al. [18], showing that tensile preloading decreases the amount or energy that can be transferred to the composite over a wide range of projectile velocities. The higher the initial stresses is, the lower the energy the composite will absorb.

5.2.2 Biaxial preloading In practical applications, laminated composite structures may be subjected to a complex stress state that could not be reproduced by a uniaxial preloading condition. To better simulate the realistic situation of actual structures, some studies investigated biaxially preloaded laminated composites subjected to low-velocity impact [19–22]. For applying a biaxial preload, the specimens are subjected to loading by means of specially designed test setups that enable biaxial loading tests in different stress/strain quadrants. An example of a setup used by Mitrevski et al. [21] is shown in Fig. 5.5.

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

Fig. 5.5 Impact test rig [19].

Whittingham et al. [19] investigated impact tests for two levels of impact energies, i.e. 6 and 10 J, in carbon-fibre laminated plates subjected to uniaxial and biaxial tension, pure shear, and null preloading condition. They observed that at the low impact energy level (6 J), peak load, absorbed energy, and penetration/perforation depth are essentially independent of the nature and amount of the prestress, but that the preloads became more significant at the higher levels of impact energy (10 J). Robb et al. [20] published a comprehensive paper in this field, which includes a wide range of preloading. They showed that size, shape, and orientation of the damage area depend on the nature and magnitude of the prestress. Major effects of the prestrain are only observed in the samples subjected to the highest prestrain situations. The results of specimens under the highest prestrain condition (600 μm) are shown in Fig. 5.6 and Table 5.1. The impacted specimens subjected to shear loading have the largest increase in the impact-induced damage area. Mitrevski et al. [21] applied two levels of biaxial pretension (500 and 1000 με) on thin glass fibre-reinforced polyester laminates. The results showed that as a result of the stiffening effect of the biaxial tension, the deflection, and contact duration decrease with increasing preload. There is also significant increase in the indentation depth produced by the conical impactor. On the other hand, absorbed energy, damage

Impact force (N)

Impact force (N)

5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0 –500 0

2

4 Time (ms)

6

8

5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0 –500 0

5000 4500

4000 3500

4000 3500

Impact force (N)

Impact force (N)

5000 4500

3000 2500 2000 1500 1000

2

4 Time (ms)

6

8

4

Impact force (N)

8

2000 1500 1000 500 0 –500 0

2

4

6

8

Time (ms)

Biaxial compression

Impact force (N) 2

6

3000 2500

Uniaxial tension 5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0 –500 0

4 Time (ms)

Uniaxial compression

No prestress

500 0 –500 0

2

6

8

5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0 –500 0

2

4

6

Time (ms)

Time (ms)

Biaxial tension

Biaxial tension/compression

8

Fig. 5.6 Typical force/time traces at
6000 micro-strain loading [19]. Table 5.1 Damage indices evaluated at
600 με [20].

Absorbed energy Peak impact force Damage area Peak indentationa a

Uniaxial Free tension

Uniaxial Biaxial compression tension

Biaxial Biaxial tension/ compression compression

1

1.15

1.38

0.93

1.47

1.55

1

1.03

0.87

1.1

0.91

0.67

1 1

1.24 1.12

1.17 1.68

1.18 0.88

1.02 0.98

2.64 2.81

Only one specimen scanned to obtain indentation results.

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

area, and peak force are only slightly affected by the magnitude of the applied preload. Khalili et al. [22] found similar results in the impact response of uniaxial and biaxial tensile preloaded reinforced graphite–epoxy composites. They found that the contact duration and deflection are strongly influenced by the preload, but the peak contact force is marginally affected. Pickett et al. [15] investigated the effects of preloading on damage and failure of composite materials when subjected to low-velocity impact. Results indicate that preloading diminishes the composite impact tolerance and alters the occurred failure modes and may lead to earlier catastrophic failure of the laminate. The role of in-plane biaxial preload on the ballistic response of laminates is presented in Wang et al. [23] through experimental results and numerical simulations. Results show a positive effect on the delamination resistance of laminated plate. On the other hand, the interlaminar stresses introduced by the biaxial in-plane tensile preloads tend to result in the weakening effect on the delamination resistance. Therefore, the influence of the biaxial in-plane tensile preloads on the delamination resistance of laminated plate is considered as the competition between the increased bending stiffness and the interlaminar stresses-induced weakening effect. Highvelocity impacts on biaxially preloaded composite structures have been also investigated in Moallemzadeh et al. [24]. Results confirm that prestress phenomena can significantly change the behaviour of composite plates under high-velocity impact, with any kind of preload generally reducing the impact resistance. Similar experiments, but on nanoreinforced specimens, have been reported in Moallemzadeh et al. [25]. Apart of the improvement in impact resistance introduced by the nanoclay modification, results show that any kind of preload (tension, compression, or combined tension/compression) reduces the ballistic performance and energy absorption capacity.

5.3

Analytical and numerical solutions

Simple analytical models for the prediction of the impact dynamics and the delamination threshold load have been developed in the past for plain laminates under no preload [4]. Such simplified analytical models have been found to be sufficiently accurate to study impact on curved laminates, provided that the flexural modulus of the laminate is adjusted to account for the increased stiffness introduced by the curvature [26], but often fail when preload is introduced. Several analytical and numerical investigations were carried out by various researchers to examine the effect of preload on the impact response of laminated composites [27–33]. Sun and Chattopadhyay [27] studied the low-velocity impact behaviour of cross-ply laminate plates subjected to initial biaxial stress. They found that a higher initial tensile stress raises the maximum contact force but on the other hand decreases the deflection and with it the contact time. They also found that pretension stresses lower the energy absorbed by the plate during the impact. Khalili et al. [22] utilized a similar method to study the uniaxial and biaxial tensile prestress effect on the impact response of reinforced graphite–epoxy plates, reporting similar results. Sun and Chen [30] used finite element (FE) modelling to study the effect of

Low-velocity impact on preloaded and curved laminates

129

three preloading cases, i.e. the zero, biaxial tension/tension, and biaxial compression/compression loading conditions. Results show that pretensile stress reduces the contact time and increases the contact force, whereby opposite results were found in the case of compressive stresses. It is also reported that a precompression stress may enhance the dynamic response. Mikkor et al. [31] used FE modelling to predict the behaviour of preloaded carbon–epoxy panels subjected to a range of lowenergy level impacts. The results show that there is a critical impact energy beyond which the panel fails catastrophically, and that by increasing the preload the critical energy level decreases. It was observed that at lower preloads, with increasing impact velocity, catastrophic failure is precipitated by reduction in residual strength and increase in damage size. At higher preloads, in contrast, increasing the impact velocity does not seem to affect considerably either residual strength or damage size before catastrophic failure. Generally speaking, the residual strength and damage size were affected in regions close to the critical velocity, but in other regions they are found to be independent of the preload. Ghelli and Minak [33] presented a comprehensive study on the mechanical behaviour of in-plane preloaded laminated composite plates subjected to low-velocity impact. Their study focused on both uniaxial and biaxial tensile and compressive preloading conditions. Results indicate that the impact-induced peak stresses are higher in the case of tensile preloads compared with the stress-free specimen, and that such effect increases with the laminate thickness. The most significant influence of compressive preloads is observed at medium span-to-thickness ratios for preloads comparable with the buckling load, while negligible or even beneficial effects are found in other cases. Zhang et al. [34] developed a nonlinear FE model to study the effect of a tensile preload on fibre metal laminates impacted by a hemispherical nose projectile at high velocities. Their result show how appropriate tensile preload can decrease the delamination area due to the reduced bending deflection, while excessive tensile preload results in more evident bending deformation and more serious delamination damage under ballistic impact.

5.4

Low-velocity impact on curved laminates

Curved laminates are widely utilised in engineering applications such as aircraft wings or pressure vessels. Radius and type of curvature are important structural parameters affecting the low-velocity impact behaviour of the laminated composite. The aim of this section is to review the literature [26, 35–46] on the effect of these parameters on the impact behaviour of thin and thick laminates. The dynamic behaviour and impact-induced damage of curved laminated composites were studied by Kim et al. [35]. He found that increasing the curvature of a cylindrical composite shell causes higher contact force. In addition, the delaminated areas of the cylindrical panels are greater when compared with flat plates under the same impact velocity. A schematic representation of the investigated cylindrical composite shell is shown in Fig. 5.7. The boundary conditions of the specimens were simply supported. The effects of boundary conditions were also found to be important to the impact response of the curved laminates [36]. The results showed that curved

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

Impactor

+

r

Fig. 5.7 Schematic representation of the cylindrical composite shell [35].

laminates behave more flexibly in simply supported edges, and consequently the deflection increases, whereas clamped edges make the laminate much stiffer, largely affecting the impact dynamics. Ambur and Starnes [37] investigated the behaviour of thin cylindrically curved, quasiisotropic graphite–epoxy laminates with various thicknesses and radii subjected to low-velocity impact. Results showed that in cylindrically curved plates subjected to a given level of impact energy the contact force is a function of plate thickness and curvature. As curvature decreases, the magnitude of the maximum contact force initially decreases, to eventually show an increasing trend, as illustrated in Fig. 5.8. Furthermore, the results for flat and curved 8- and 16-ply thick plates showed that the CAI residual strength is a function of the plate radius. The residual strength of already impacted flat and curved laminates, with barely visible impact damage, is

Maximum contact force (lb)

1200 1000 Experiment–drop weight

800

Analysis–drop weight

600

Analysis–airgun

400 200 0 0

50

100

150

200

Plate radius (in)

Fig. 5.8 Contact force results for 16-ply thick, quasiisotropic plates impacted by dropped weight and airgun-propelled impactors with 1.5 ft-lbs of impact energy [36].

Low-velocity impact on preloaded and curved laminates

131

approximately 3% and 15% lower than the residual strength for the same undamaged plates, respectively. In higher speeds, the impact response of the plate is localised and the plate behaviour is not affected by the curvature. Experimental, numerical, and analytical investigations were done by Kistler and Waas [38, 39] to understand the influence of panel curvature and thickness under low-velocity impact. Results show that increasing the thickness results in a decrease of the deflection during the impact, and that curvature effects become decreasingly important. They found that stiffer specimens produce higher impact forces, shorter contact duration times and smaller deflections. It is also mentioned that for a nonlinear curved plate under low-velocity impact the bending and membrane effects are more important than the effect of inertia, and should be taken into account. Other researchers also studied the influence of thickness, radius of curvature, and stacking sequence on damage behaviour of composite shells [40, 41]. Results showed that as the laminate thickness changes, the impact velocity threshold, which is required for initiation of the damage, is affected marginally. On the other hand, maximum damage size is altered significantly. The damage size depends on the total dynamic deformation, and impact-induced displacement is larger in thinner shells. Therefore, as the thickness of the composite shell increases, for a specific impact velocity the damage area decreases (see Fig. 5.9). As the shell flattens, the maximum damage size increases. This is due to higher stiffness of shells with smaller radii of curvature. The location of the largest damage through the laminate thickness depends on its radius of curvature. In flat composite plates, the maximum damage size occurs at the bottom layer but in curved shells, it occurs at the top layer. Preload and curvature effects on the impact response of cylindrical laminated composite shells have also been investigated numerically in Yokoyama et al. [42]. The investigated layups are listed in Table 5.2. Some of the obtained results are illustrated in Figs. 5.10–5.12. The numerical results show that the damage size on pressurised composite laminates subjected to impact loading is significantly affected by thickness, curvature, and preload variables. The results show that as the thickness, curvature, and preload increase, the dissipated energy decreases. Minak and coauthors [26, 43] also investigated the effect of preload on the residual strength and impact resistance of low-velocity-impacted CFRP tubes subjected to torque. The testing machine and the pendulum utilised to perform the impact are illustrated in Fig. 5.13. Results showed that the residual torsional strength of an impacted specimens is lower than half the undamaged tubes strength. It is also found that an impact under torsional preload might lead to a sudden collapse of the structure even for a low-energy impact. Such behaviour is ascribed to an increase of delaminated area with the torsional preload, to the point that the tube collapses due to local buckling phenomena under the impact load. As the preload increases, the residual torsional strength and the plies’ critical buckling load decrease, even if the preload is lower than the load required to cause the first matrix crack. The effect of preloading on the impact response of curved laminates was also investigated by Saghafi et al. [44]. They studied low-velocity impacts on glass–epoxy laminated composites for three different initial impact energies and preloads under two

132

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

(A)

4 (03/903/03) (03/903/03/903/03) (03/903/03/903/03/903/03)

Damage size (cm)

3

2

1

0

(B)

0

2

4

8 V (m/s)

10

12

14

16

6 R = 4.775 cm R = 9.459 cm R = Infinity

5 Damage size (cm)

6

4 3 2 1 0

0

2

4

6 V (m/s)

8

10

12

Fig. 5.9 Impact velocity threshold and maximum damage size for composite shell with different (A) thicknesses and (B) radii of curvature [36].

different boundary conditions. The experimental setup for this study is shown in Fig. 5.14. Due to the bending stress distribution, the panel curvature increases and the upper and lower surfaces of the laminate are put under tensile and compression stresses, respectively. Their results showed that preload and curvature are effective parameters affecting the low-velocity impact behaviour of the specimens and make significant changes on the impact parameters such as absorbed energy, damaged area,

Low-velocity impact on preloaded and curved laminates

133

Table 5.2 Simulation scenarios [42]. Simulation R1T R1T R1T R1T R2T R2T R2T R2T R3T R3T R3T R3T R4T R4T R4T R4T

1P0 1P1 2P0 2P1 1P0 1P1 2P0 2P1 1P0 1P1 2P0 2P1 1P0 1P1 2P0 2P1

Radius of Pressure load curvature (mm) Thickness (mm) (atm) 100

2.29 4.56

125

2.29 4.56

200

2.29 4.56

inf

2.29 4.56

0.0 0.5 0.0 0.5 0.0 0.5 0.0 0.5 0.0 0.5 0.0 0.5 0.0 0.5 0.0 0.5

maximum load, and maximum indentation. It was found that in the upper surface tension stress increases the propagation of matrix cracks, while in the lower surface compression stress reduces it. The damaged area is found to increase with the preload, but the total absorbed energy decreases, which is due to an increase in stiffness caused by the larger curvature of the laminate. Similar results have been reported in Asokan et al. [47], where it has been shown that the load carrying capacity is higher in convex laminates, alongside a reduction of the damage in the impacted surface area. Impacts on curved GFRP specimens have been also performed in Gemi et al. [45]. Results reported therein indicate that in the case of rather compliant structures displacement determines the formation of damages, especially in the form of delamination.

5.5

Conclusions

Preload conditions and curvature are important parameters affecting the low-velocity impact behaviour of composite structures. The severity of damage, the damage tolerance, and the residual strength of laminated composites are largely influenced by these parameters and must be correctly accounted for if transversal impacts are expected. The most interesting results of the analysis conducted in this work are summarised in Table 5.3 and are explained in the following.

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

(A)

Curvature effects

Dissipated energy (J)

9.0 R1T 2P0 R2T 2P0 R3T 2P0 R4T 2P0

7.5 6.0 4.5 3.0 1.5 0.0

0

4

8

12

16

20

24

10

12

Impact energy (J)

Dissipated energy (J)

(B)

Curvature effects 9.0 R1T 1P0 R2T 1P0 R3T 1P0 R4T 1P0

7.5 6.0 4.5 3.0 1.5 0.0

0

2

4

6

8

Impact energy (J) Fig. 5.10 Impact energy versus dissipated energy curves for (A) 4.56 mm and (B) 2.29 mm, without pressure loading [42].

l

l

Tensile preload increases the rigidity and flexural stiffness of laminates, often leading to an increase in the induced damage of the specimens. As a result, pretension increases the contact force whereas reducing contact time and deflection, if compared to stress-free structures. Delamination is found to have a minor effect on the residual strength of impacted composite structures subjected to tensile load. Compressive preload decreases the structure flexural stiffness and the maximum force is lowered in comparison with specimens without preload. This trend is reversed for damage size of precompressed specimens. The additional complexity of plate buckling may arise in damaged structures subjected to compressive preloading, and delamination is the most detrimental

Low-velocity impact on preloaded and curved laminates

(A)

135

Curvature effects

Dissipated energy (J)

9.0 R1T 2P1 R2T 2P1 R3T 2P1 R4T 2P1

7.5 6.0 4.5 3.0 1.5 0.0

0

4

8

12

16

20

24

Impact energy (J)

Dissipated energy (J)

(B)

Curvature effects 9.0 R1T 1P1 R2T 1P1 R3T 1P1 R4T 1P1

7.5 6.0 4.5 3.0 1.5 0.0

0

2

4

6

8

10

12

Impact energy (J) Fig. 5.11 Impact energy versus dissipated energy curves for (A) 4.56 mm and (B) 2.29 mm, with pressure loading (P ¼ 0.5 atm) [42].

l

l

damage mechanism that affects the structural damage resistance. Consequently, the damage size of precompressed specimens increases due to local buckling due to delamination. The laminates subjected to preshear loading have the largest increase in damage area compared with unstressed laminates. The impact failure mechanisms and the intensity of the induced damage are different for preloaded laminates of different thicknesses. The damage areas in thin laminates are smaller than thick ones, and transverse shear effects are found to be the main reason for failure in thick laminates.

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

(A)

Thickness effects

Dissipated energy (J)

8.0 R4T 1P0 R4T 2P0

6.0

4.0 2.0

0.0

0

2

4

6

8

10

12

14

16

18

20

22

24

18

20

22

24

Impact energy (J)

(B)

Thickness effects

Dissipated energy (J)

8.0 R4T 1P1 R4T 2P1

6.0 4.0 2.0 0.0

0

2

4

6

8

10

12

14

16

Impact energy (J) Fig. 5.12 Impact energy versus dissipated energy curves for the plates (A) without and (B) with pressure for different thicknesses [42].

l

l

As the radius of the plate increases (decreasing the curvature), the magnitude of the maximum contact force decreases at first and then increases. The delaminated areas of cylindrical panels are greater when compared with flat plates under similar impact velocity. For a curved plate, the larger the curvature, the larger the amount of dissipated energy during the impact. Moreover, as the plate thickness increases, the amount of dissipated energy decreases.

From the above review, it can be concluded that more attention is required in the design and maintenance of preloaded and curved laminated composite structures.

Fig. 5.13 Torque machine and impact pendulum utilised in Minak et al. [26, 43].

20 mm

(A)

3.1

168 mm

mm

(B)

P-3550 strain indicator

(C)

Hydraulic cylinder

Load cell

Fig. 5.14 (A) Schematic picture of the specimen. (B) Experimental setup. (C) Schematic drawing of the fixture [44].

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

Table 5.3 Effect of preload on low-velocity impact response of laminates. Preload types

Tensile

Compression

Shear

Flexural stiffness Damage area Contact force Contact time Deflection Detrimental damage

" " " # " Delamination

# " # " " Delam. + ply buckling

¼ "" ¼ " / Delam. + ply buckling

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[34] C. Zhang, Q. Zhu, Y. Wang, P. Ma, Finite element simulation of tensile preload effects on high velocity impact behavior of fiber metal laminates, Appl. Compos. Mater. 27 (3) (2020) 251–268. [35] S.J. Kim, N.S. Goo, T.W. Kim, Effect of curvature on the dynamic response and impactinduced damage in composite laminates, Compos. Sci. Technol. 57 (7) (1997) 763–773. [36] K.S. Krishnamurthy, P. Mahajan, R.K. Mittal, A parametric study of the impact response and damage of laminated cylindrical composite shells, Compos. Sci. Technol. 61 (12) (2001) 1655–1669. [37] D. Ambur, J. Starnes Jr., Effect of curvature on the impact damage characteristics and residual strength of composite plates, in: 39th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference and Exhibit, American Institute of Aeronautics and Astronautics, Reston, VA, 1998. [38] L.S. Kistler, A.M. Waas, Experiment and analysis on the response of curved laminated composite panels subjected to low velocity impact, Int. J. Impact Eng. 21 (9) (1998) 711–736. [39] L.S. Kistler, A.M. Waas, On the response of curved laminated panels subjected to transverse impact loads, Int. J. Solids Struct. 36 (9) (1999) 1311–1327. [40] G.P. Zhao, C.D. Cho, Damage initiation and propagation in composite shells subjected to impact, Compos. Struct. 78 (1) (2007) 91–100. [41] S. Ganapathy, K.P. Rao, Failure analysis of laminated composite cylindrical/spherical shell panels subjected to low-velocity impact, Comput. Struct. 68 (6) (1998) 627–641. [42] N.O. Yokoyama, M.V. Donadon, S.F.M de Almeida, A numerical study on the impact resistance of composite shells using an energy based failure model, Compos. Struct. 93 (1) (2010) 142–152. [43] G. Minak, S. Abrate, D. Ghelli, R. Panciroli, A. Zucchelli, Low-velocity impact on carbon/ epoxy tubes subjected to torque—experimental results, analytical models and FEM analysis, Compos. Struct. 92 (3) (2010) 623–632. [44] H. Saghafi, G. Minak, A. Zucchelli, Effect of preload on the impact response of curved composite panels, Compos. Part B Eng. 60 (2014) 74–81. € Şahin, Experimental and statistical anal[45] L. Gemi, M. Kayrıcı, M. Uludag, D.S. Gemi, O.S. ysis of low velocity impact response of filament wound composite pipes, Compos. Part B Eng. 149 (March) (2018) 38–48. [46] I. Mokhtar, M.Y. Yahya, A.S. Abd Kader, S.A. Hassan, C. Santulli, Transverse impact response of filament wound basalt composite tubes, Compos. Part B Eng. 128 (2017) 134–145. [47] R. Asokan, H. Balaji, B. Rubanrajasekar, M. Dinesh, Effect of low velocity impact at prestressed condition on curved glass-carbon composites laminate, Int. J. Veh. Struct. Syst. 11 (4) (2019) 434–438.

High-velocity impact damage in CFRP laminates

6

Shigeki Yashiroa and Keiji Ogib a Department of Aeronautics and Astronautics, Kyushu University, Fukuoka, Japan, b Graduate School of Science and Engineering, Ehime University, Matsuyama, Ehime, Japan

6.1

Introduction

Advanced composite materials like carbon-fibre-reinforced plastics (CFRPs) have been increasingly used in various industrial applications, including automobiles, because they have higher specific strength/modulus and better fatigue properties than conventional metals. In the latest aircraft designs, CFRPs have been applied to primary load-bearing structures of airframes and the fan blades and fan cases of turbo-fan engines. One technical issue concerning these composite structures involves high-velocity impact of foreign objects such as small stones and birds. Despite their lightweight, projectiles have great kinetic energy, which will induce local but catastrophic failure of composite structures. When a fan blade breaks and flies apart (i.e. a fan blade out event), a composite fan case needs to bear the impact and to contain the broken blade. Therefore, the response of composite materials to projectiles with a velocity near the speed of sound in the air should be clarified in order to improve the reliability of composite structures in aerospace applications. Out-of-plane impacts frequently cause delamination in composite laminates. Since compressive strength degrades severely after low-velocity impacts like tool-drops [1–3], an enormous number of experiments (e.g. [4–7]) and analyses (e.g. [8–13]) have been reported on this issue. Typical parameters influencing the impact response of composite laminates include impact velocity, material, geometry, impactor characteristics, and environmental conditions [14]. The impact response of thin-ply composites [15–17] is a new topic in recent years. Among them, the impact velocity is an important parameter governing the damage pattern. Breen et al. [18] investigated the effect of impact velocity within a low-velocity range and pointed out that the response in the target became localised with increased velocity. The duration of contact is a key parameter controlling the impact response from the global deformation found in quasi-static indentation to the local deformation induced by elastic waves [14]. Hypervelocity impact, with a projectile velocity of several kilometres per second, represents the opposite end of impact problems. It is also a major issue in systems to shield spacecraft from debris and in armour applications. A number of studies have been reported on the response of composite materials to hypervelocity impact and the generation of a debris cloud, i.e., fragments of a target (e.g. [19–24]). A good review on hypervelocity impact can also be found in the literature [25]. Dynamic Deformation, Damage and Fracture in Composite Materials and Structures. https://doi.org/10.1016/B978-0-12-823979-7.00007-7 Copyright © 2023 Elsevier Ltd. All rights reserved.

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Studies on the impact response of composites in the intermediate velocity range have increased in the last decade because of its importance for aeronautical applications. In the intermediate velocity range, a projectile bounces back after creating a crater, or occasionally remains in the target without perforation, and thus the impact response is particularly complex. This makes it difficult to address high-velocity impact problems. Experiment-based studies have mainly discussed the ballistic limit energy and energy absorption during perforation. Early studies by Cantwell and Morton [26–28] characterised the response of CFRP laminates to low- and high-velocity impacts. They concluded that high-velocity impacts induced local deformation near the contact point, in sharp contrast to the global deformation caused by low-velocity impacts. This result is similar to that of Breen et al. [18], although the velocity range is different. They observed a conical shear-fracture zone beneath the impact point regardless of the stacking configuration and proposed a simple model to predict the ballistic limit energy. Recent studies on high-velocity impacts on composite materials are summarised in the latest review [29]; the major topic is still the ballistic limit energy and energy absorption during the perforation process for shield applications (e.g. [30]). The delamination area has also been a point of interest in the latest studies. Lo´pez-Puente et al. [31] investigated the dependence of the delamination area on the temperature. Tanabe et al. [32] deduced that laminates with low interlaminar shear strength would efficiently dissipate the kinetic energy of the projectile by extensive extension of delamination. Heimbs et al. [33] indicated that the deflection affects the extent of delamination by observing the impact damage in preloaded carbon/epoxy laminates. Lulu et al. [34] observed the unfavourable effect of the hygrothermal ageing (i.e. weakened fibre/matrix interface) on high-velocity impact resistance. Kristnama et al. [35] correlated the residual tensile strength of carbon/epoxy laminates after high-velocity oblique impact with the impact velocity and damage extent. Hazell et al. [36,37] experimentally demonstrated that the projected delamination area, i.e., the energy absorbed by damage extension, was constant against the projectile velocity in cases of great kinetic energy. These results may be interpreted as follows: nearhypervelocity impacts produce constant energy dissipation consisting of fragmentation in a conical zone beneath the impact point and delamination in the locally deformed area. However, within the transition range from low-velocity impact to high-velocity impact, the ratio of energy dissipation by fibre breaks to total absorbed energy will increase with increasing velocity, depending on the severity of local deformation and fragmentation. Since damage extension in composite materials is complex, the impact response should be clarified when designing and using composite structures. Characterising the extension process of high-velocity impact damage including delamination and fibre breaks is essential for understanding the mechanism of energy absorption and for quantitatively evaluating it. However, we can find only a few discussions on the generation and extension of such damage in the literature. Several finite-element analyses representing impact responses have been reported (e.g. [38–43]). They predicted the delamination area or damage area, which agreed reasonably with the experiment results. However, detailed discussions on the mechanism of damage extension have still not been provided.

High-velocity impact damage in CFRP laminates

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This chapter therefore seeks to characterise high-velocity impact damage in CFRP laminates and to clarify the mechanism of damage extension. Yashiro et al. [44,45] focused on damage states near perforation and performed a series of experiments and impact simulations. This chapter will quantitatively evaluate the high-velocity impact damage and discuss its extension mechanism based on detailed observations of near-perforation damage. We will also investigate the influence of the stacking sequence and interlayers on the damage states.

6.2

Experiments

6.2.1 Factors affecting high-velocity impact damage Many factors can affect high-velocity impact damage in composite laminates. The following lists typical factors: Test conditions: input energy, impact velocity, boundary condition, temperature. Projectile: mass, shape (sharpness of the tip), hardness. Materials: stiffness, strength and toughness, stacking configuration.

Among these, this chapter mainly focuses on the stacking configuration.

6.2.2 High-velocity impact test Fig. 6.1 is a schematic diagram of a high-speed impact testing machine with an electric-heat gun (Maruwa Electronic Incorporated). This machine consists of a control unit for the power source, a chamber in which a sphere is shot at a target, and acceleration sensors for velocity measurement. A pulse current at high voltage

Data logger

Acceleration sensor

Electric-heat gun

Electric power supply Specimen Sabot stopper

Power source control panel

Chamber

Fig. 6.1 High-velocity impact testing machine with an electric-heat gun.

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(3–6 kV) is applied to a piece of aluminium foil in the gun, and a sabot with a projectile is accelerated by high-pressure aluminium plasma. The sabot stops at the end of the gun, and the projectile moves forward by inertia. The velocity of the projectile is approximately controlled by the voltage applied to the aluminium foil and measured from the flight time, which is obtained by two acceleration sensors on the sabot stopper and support jig. This testing system provides a velocity range from 40 to 1500 m/s, depending on the projectile mass. In most of the experiments in this chapter, a steel ball with a diameter of 1.5 mm (14.2 mg mass) was used as the projectile. The target specimen was set in a square-frame jig with a square cavity (Fig. 6.2), being fixed rigidly on all four sides. Consequently, bending deformation due to impact load was possible. The experiment conditions will be stated when they differ from the previously mentioned.

(a)

(b)

50

55

Fig. 6.2 Fixture for the target specimen.

Spacer

Specimen

High-velocity impact damage in CFRP laminates

145

After impact tests, damage states on the front and back surfaces were observed by a stereomicroscope. Damage in the cross-section, including the impact point, was observed by an optical microscope. Here, the tested specimens were cut off on a separate line from the impact point not to affect impact-induced damage. A small piece was then embedded in cold mounting resin, and the cross-section was polished using fine abrasive papers. Moreover, delamination was observed by soft X-ray radiography.

6.2.3 Material This study used carbon-fibre-reinforced epoxy composites (T700S/#2521R, Toray Industries) and prepared four types of laminates: unidirectional (UD) [016], simple cross-ply (CP1) [04/904]S, cross-ply with many ply interfaces (CP2) [0/90]4S, and quasi-isotropic (QI) [0/45/90/
45]S2. The fibre direction in the top and bottom plies was defined as the 0° direction. These laminates, which were 1.6 mm in thickness, were manufactured by a vacuum hot-pressing machine and were cut into square specimens 55 mm on a side. The influence of toughened interlayers on damage extension is investigated further in the last part of this chapter. CFRP laminates with interlayers (T800S/#3900-2B, Toray Industries) were tested, and the experiment results for toughened laminates were compared with those for general-purpose CFRP laminates (T700S/#2500, Toray Industries). The material properties are listed in Table 6.1.

6.3

Experiment results

6.3.1 Unidirectional laminate Fig. 6.3 depicts the damage states on the front and back surfaces of the UD laminates. A crater was generated on the front surface and splits extended from its edge. Moreover, fibre breaks and additional short splits appeared inside the crater. The size and severity of the damage increased with increasing impact velocity, while the damage pattern did not change. A single matrix crack was generated on the back surface at a relatively low velocity (200 m/s), and fibre breaks beneath the impact point and multiple matrix cracks were observed at a high velocity (619 m/s). Table 6.1 Material properties of CFRP unidirectional laminates used in this study.

Longitudinal Young’s modulus E1 (GPa) Transverse Young’s modulus E2 (GPa) In-plane shear modulus G12 (GPa) In-plane Poisson’s ratio ν12

T700S/ #2521R

T700S/ #2500

T800S/#39002B

135

135

153

7.8

8.5

8.0

4.4 0.34

4.8 0.34

4.0 0.34

146

(a)

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

200 m/s

619 m/s

2 mm

2 mm

4 mm

4 mm

(b)

Fig. 6.3 Damage states on the (a) front and (b) back surfaces of UD laminate.

Fig. 6.4 depicts the damage states of the UD laminates beneath the impact point. Cone cracks, i.e., matrix cracks extending obliquely from the crater edge, were observed in the cross-section normal to the 0° direction (hereafter, the normal cross-section), and these oblique cracks were connected with each other by delamination (Fig. 6.4a). The number of delaminated interfaces increased with increasing velocity. Fig. 6.4b presents optical micrographs of the cross-section parallel to the 0° direction (hereafter, the parallel cross-section). Delamination was observed at some ply interfaces, and its opening displacement increased with increasing velocity. In addition, fibres in all of the plies broke just beneath the impact point. Fig. 6.5 depicts soft X-ray photographs of the UD laminates. A matrix crack appeared on the back surface at 315 m/s. When fibre breaks got through all of the plies at a high velocity (over 400 m/s), the bottom ply was peeled off as depicted in Figs 6.3b and 6.4b. In this case, elongated delamination (peeled region) was observed along with the matrix cracks.

6.3.2 Simple cross-ply laminate Fig. 6.6 depicts the damage states on the front and back surfaces of the CP1 laminate. A crater was generated at the impact point and splits appeared from its edge. Fibre breaks were also observed at a higher velocity (618 m/s). A matrix crack was

High-velocity impact damage in CFRP laminates

(a)

147

200 m/s

430 m/s 0°

0°

1 mm

1 mm

(b) 0°

0°

1 mm 1 mm

Fig. 6.4 Damage states beneath the impact point of the UD laminate: (a) normal cross-section and (b) parallel cross-section [46].

(a)

0° direction

10 mm

(b)

10 mm

Fig. 6.5 Soft X-ray photographs of the impacted UD laminate at (a) 315 m/s and (b) 550 m/s.

148

(a)

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

186 m/s

618 m/s

2 mm

2 mm

(b)

2 mm

4 mm

Fig. 6.6 Damage states on the (a) front and (b) back surfaces of CP1 laminate.

observed on the back surface, and fibre breaks beneath the impact point, as well as ply peeling, were generated at a higher velocity. This damage pattern on the surfaces was the same as for the UD laminates (Fig. 6.3). Fig. 6.7 depicts the damage state beneath the impact point of the CP1 laminates. In the normal cross-section, a crater and accompanying matrix cracks appeared in the top 0° plies at a low velocity (186 m/s), but no damage was observed in the lower plies. The fibres in the middle 90° plies broke, and the crack opening distance of delamination was large at a high velocity (508 m/s). This crack opening induced a bump on the back surface. In the parallel cross-section, fibres broke at the crater edge and the middle 90° plies exhibited catastrophic failure with many matrix cracks and much delamination. Oblique cone cracks were also observed outside the ply-failure zone. The bottom 0° plies peeled off by connecting fibre breaks and matrix cracks to extensive delamination. Soft X-ray photographs of the tested CP1 laminates are presented in Fig. 6.8. A matrix crack and fan-shaped (or peanut-shaped) delamination were observed. It should be noted that this pattern was similar to the delamination caused by lowvelocity impact [47]. Elongated delamination along the matrix crack of the bottom ply was observed at a higher velocity (687 m/s).

High-velocity impact damage in CFRP laminates

(a)

149

186 m/s

508 m/s 0°

0°

0.5 mm

(b)

1 mm

217 m/s

600 m/s 0°

0°

0.5 mm

0.5 mm The bottom 0° plies peeled off due to extensive delamination.

Fig. 6.7 Damage states beneath the impact point of the CP1 laminate: (a) normal cross-section and (b) parallel cross-section.

(a)

0° direction

10 mm

(b)

10 mm Fig. 6.8 Soft X-ray photographs of the impacted CP1 laminate at (a) 186 m/s and (b) 687 m/s.

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

6.3.3 Cross-ply laminate with many ply interfaces Fig. 6.9 depicts the surface damage of the CP2 laminates. The damage pattern was similar to that of the CP1 laminates (Fig. 6.6), including a crater and splits on the front surface and a matrix crack on the back surface. At high velocity (666 m/s), more matrix cracks appeared on the back surface than in the CP1 laminate. This was caused by the thickness of the bottom ply. In contrast to the thickness of a ply in the CP2 laminate (0.1 mm), the bottom 0° plies in the CP1 laminate were thick (0.4 mm), so its damage state would thus be similar to the UD laminate. Fig. 6.10 depicts the damage state in the normal cross-section of the CP2 laminates. Fibre breaks, matrix cracks, and delamination were generated beneath the impact point at low velocity (140 m/s), but the damage extended only to a middle ply. Delamination spread from the tips of oblique matrix cracks. A ply-failure zone with the width of the crater was generated beneath the impact point, and delamination grew outside it at high velocity (500 m/s). It should be noted that delamination always existed below a transverse ply and that Yashiro et al. [44] observed the same delamination pattern in the parallel cross-section. This observation suggests that delamination was generated from the tip of a matrix crack in an upper ply. Fig. 6.11 presents soft X-ray photographs of the CP2 laminates. Fan-shaped delamination spread in both the 0° direction and the 90° direction. In particular, some

(a)

140 m/s

666 m/s

2 mm

2 mm

4 mm

4 mm

(b)

Fig. 6.9 Damage states on the (a) front and (b) back surfaces of CP2 laminate.

(a) 0°

0.5 mm

(b) 0°

0.5 mm

Fig. 6.10 Damage states in the normal cross-section of the CP2 laminate at (a) 140 m/s and (b) 500 m/s.

(a)

0° direction

10 mm

(b)

10 mm Fig. 6.11 Soft X-ray photographs of the impacted CP2 laminate at (a) 200 m/s and (b) 500 m/s.

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

shadows of delamination with different sizes were observed in one direction. The Xray photograph, along with the delamination opening observed in Fig. 6.10b, suggests that the delamination area increased when approaching the back surface.

6.3.4 Quasi-isotropic laminate Fig. 6.12 depicts the damage state on the surfaces of QI laminates, which had a crater and splits on the front surface and matrix cracks and fibre breaks on the back surface. The damage pattern on the back surface at high velocity (559 m/s) was similar to that of the CP2 laminate (Fig. 6.9b). The damage state in the parallel cross-section and a soft X-ray photograph are presented in Figs 6.13 and 6.14. A ply-failure zone was generated beneath the impact point, and fan-shaped delamination was observed at most of the ply interfaces around the ply-failure zone. This delamination pattern was similar to that generated by lowvelocity impact [48] except for the peeling off of the bottom 0° ply.

(a)

480 m/s

559 m/s

2 mm

2 mm

(b)

Barely visible damage

4 mm Fig. 6.12 Damage states on the (a) front and (b) back surfaces of QI laminate [44].

High-velocity impact damage in CFRP laminates

153

0°

0.5 mm Fig. 6.13 Damage state in the parallel cross-section of the QI laminate at an impact velocity of 530 m/s.

0° direction

10 mm Fig. 6.14 Soft X-ray photographs of the impacted QI laminate at a velocity of 870 m/s.

6.4

Discussion

The input energy, i.e., the initial kinetic energy of the projectile, is converted into three parts in an impact event: (1) strain energy stored in the target, (2) energy that generates damage, and (3) kinetic energy remaining in the rebounding projectile. The sum of the first two is the finally dissipated energy. Furthermore, the energy that generates damage is divided into energy release by matrix cracking, by delamination, and by fibre breaks. We will discuss how this energy dissipation changes with impact velocity and stacking configuration.

6.4.1 Mechanism of high-velocity impact damage Fig. 6.15 compares the high-velocity impact damage of the CP1 laminates with the low-velocity impact damage at two energy levels. The top of the tap was a half sphere with a diameter of 1.5 mm, the same as that of the projectile. In contrast to the highvelocity impact (Figs 6.8a and 6.15a), low-velocity impact induced only a matrix crack on the back surface in the low-energy case (0.33 J). However, at a high energy

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

(a)

(b)

10 mm

10 mm

Fig. 6.15 Comparison of damage states in simple CP1 laminates between high-velocity impact and low-velocity impact. (a) 217 m/s (b) 0.9 m/s, 0.33 J and (c) 687 m/s (d) 1.6 m/s, 3.35 J.

level (3.35 J), the delamination area generated by a low-velocity impact (202 mm2) exceeded that of a high-velocity impact (172 mm2), which included the peeling off of the bottom ply. In low-velocity impacts, all of the target plates can be deformed and the impact energy will be absorbed by elastic deformation and delamination growth. In contrast, deformation of the target will be localised in high-velocity cases, and therefore the impact energy will be absorbed by a ply-failure zone and delamination. However, the energy dissipated by fibre breaks will be much greater than the strain energy stored in the target in low-velocity cases. This is the reason for the smaller delamination area in the high-velocity impact than in the low-velocity cases. It should be noted that the impact energy in Fig. 6.15 is the input energy and not the absorbed energy. Additional tests will be required to confirm the previously mentioned discussion. Fig. 6.16 plots the delamination area of the four stacking configurations against the impact velocity. The delamination area increased gradually with increasing velocity up to 350 m/s then jumped sharply at velocities of 350–500 m/s. In all of the four laminations, fibre breaks appeared on the back surface in this velocity range. The delamination area became constant and independent of the velocity in the higher velocity range, regardless of the stacking configuration. High-velocity impact always generated a catastrophic failure zone beneath the crater, but observation of the crosssections revealed that their volume had an upper limit approximated by a cylinder with the width of the crater. Therefore, this result suggests that the energy absorbed by damage extension becomes constant with further increases in impact velocity. The estimated damage extension caused by high-velocity impact in a QI laminate is illustrated in Fig. 6.17. Damage is generated sequentially from the top ply to the bottom ply by indentation of the projectile and by propagation of the stress wave; major damage appears in the following order: crater, delamination at the first interface, failure of the second ply, and delamination at the second interface. The fact that the delamination patterns were similar to those caused by low-velocity impacts suggests

High-velocity impact damage in CFRP laminates

155

Fig. 6.16 Projected area of the internal damage versus the impact velocity. Some points have been added to Yashiro et al. [44].

Projected delamination area (mm2)

300

250

200

UD CP1 CP2 QI

150

100

50

0 0

200

400

600

800

1000

Impact velocity (m/s)

that the mechanism of delamination extension is the same as in the low-velocity cases. As observed in the CP2 laminate (Fig. 6.10), delamination is generated from the tip of a matrix crack in the upper ply and grows in a fan-shaped area between the matrix crack in the upper ply and one in the lower ply. The bottom ply is peeled off by connecting the delamination with fibre breaks beneath the impact point. The projected delamination pattern (Fig. 6.17f) agreed well with the soft X-ray photograph (Fig. 6.14).

6.4.2 Influence of the stacking sequence on damage severity The projected damage area on the front and back surfaces was measured to quantitatively evaluate the influence of the stacking configuration (Fig. 6.18). The projected damage area increased with increasing velocity on both surfaces, although the data varied widely. This trend was most prominent in the QI and CP2 laminates, followed by the CP1 laminate. However, the UD laminate had a lower damage area increase rate. This difference appeared significantly on the back surface. Assuming that damage on the back surface was mainly caused by bending deformation, the bending stiffness was calculated using classical lamination theory (Table 6.2). In the UD and CP1 laminates, the transverse bending stiffness D22 was much smaller than the longitudinal stiffness D11, and these laminates experienced inhomogeneous deformation states despite the axisymmetric loading. Bending in the transverse direction easily generates matrix cracks and decreases fibre breaks on the back surface. The CP2 and QI laminates have balanced stiffness components D11 and D22. Therefore, one-sided bending will diminish, and the effect of the local deformation beneath the impact point will become dominant in generating fibre breaks, which causes peeling off of the bottom ply. Therefore, the smaller stiffness difference (D11–D22) generates more fibre breaks on the back surface and enlarges the damage area.

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

(a)

(d)

Delamination

Matrix cracks

Crater

Matrix failure (Normal crosssection)

Splits

Matrix failure

Matrix cracks

Extensive fiber breaks (Parallel cross-section)

Matrix failure

Delamination

(b)

(e)

Delamination Matrix cracks (upper ply)

Delamination Matrix cracks

Matrix failure Matrix Fiber cracks breaks

Matrix cracks

Delamination

Matrix failure

(c)

Fiber breaks

Delamination

Ply peeling-off

(f)

Delamination

Fiber breaks

Matrix cracks

Delamination by peeling-off

Matrix failure

Fig. 6.17 Schematic diagram of high-velocity impact damage in a quasi-isotropic laminate. (a) Top 0°, (b) 45°, (c) 90°, (d)
45°, (e) bottom 0° ply, and (f) projected lamination.

6.4.3 Influence of toughened interlayers on damage severity CFRP quasi-isotropic laminates with toughened interlayers were prepared. The material used was high-strength carbon-fibre-reinforced epoxy with a toughened interlayer (T800S/#3900-2B, Toray Industries), and the stacking configuration was [0/
45/90/45]2S. Quasi-isotropic [02/
452/902/452]2S laminates made of generalpurpose CFRP (T700S/#2500, Toray Industries) were also prepared. These two

High-velocity impact damage in CFRP laminates

20

Projected damage area (mm2)

UD CP1 15

CP2

157

Fig. 6.18 Relationship between the projected damage area on the (a) front and (b) back surfaces and the impact velocity [44].

QI

10

5

0 0

200

400 600 800 1000 1200 1400 Impact velocity (m/s)

120

Projected damage area (mm2)

UD 100

CP1 CP2

80

QI

60

40

20

0 0

200

400 600 800 1000 1200 1400 Impact velocity (m/s)

laminates had the same thickness (3 mm) despite the different number of stacked plies. The target plates were set in the fixture jig (Fig. 6.2), and a 4 mm-thick steel plate with a hole (20 mm diameter) was inserted behind the specimen. The projectile steel ball had a diameter of 4.0 mm (260 mg mass). Fig. 6.19 depicts the damage state on the front surface. In a manner similar to the thin laminates (Fig. 6.12), both laminates suffered a crater, long splits from its edge, some short splits in the crater, and fibre breaks. Differences in the lengths of the major splits in the two material systems were negligible (Fig. 6.20a). However, more short

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

Table 6.2 Bending stiffness calculated using classical lamination theory. Stacking configuration

D11 (N ∙m)

D22 (N ∙m)

D12 (N ∙m)

UD [016] CP1 [04/904]S CP2 [0/90]4S QI [0/45/90/
45]S2

46.4 40.9 28.6 22.4

2.7 8.1 20.4 17.3

1.5 1.5 1.5 6.2

(a)

2 mm

(b)

2 mm Fig. 6.19 Magnified view of the front surfaces of the quasi-isotropic specimens with/without toughened interlayers. (a) Toughened, 265 m/s and (b) general-purpose laminate, 345 m/s.

High-velocity impact damage in CFRP laminates

(b)

20 T700S/#2500 T800S/#3900-2B 15

10

5

0 0

Projected damage area (mm2)

Average split length (mm)

(a)

159

40 T700S/#2500 T800S/#3900-2B 30

20

10

0

100 200 300 400 500 600 Impact velocity (m/s)

(c)

0

100 200 300 400 500 600 Impact velocity (m/s)

Maximum crack length (mm)

25 T700S/#2500 T800S/#3900-2B

20 15 10 5 0 0

100 200 300 400 500 600 Impact velocity (m/s)

Fig. 6.20 Influence of the toughened interlayers on the surface damage; (a) front surface, split length (b) front surface, damaged area, and (c) back surface, crack length.

splits were observed in the toughened laminates than in the general-purpose laminates, and these short splits enlarged the projected damage area on the front surface of the toughened laminates (Fig. 6.20b). A couple of matrix cracks, along with delamination, were generated on the back surface of the general-purpose laminates, and these cracks grew with increasing velocity (Fig. 6.20c). However, in the toughened laminates, no damage or a barely visible crack was observed within the tested velocity range. This result indicates tiny delamination in the lowest interface. The bending stiffness was evaluated as described in the previous section. Stiffness components D11 and D22 of the toughened laminate were 185 Nm and 116 Nm, which were slightly greater than those of the general-purpose laminate (D11 ¼ 156 Nm and D22 ¼ 105 Nm). Since the stiffness and transverse strength of the two material systems were almost identical, a simple discussion based on bending

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deformation cannot account for the clear difference seen in Fig. 6.20c. The toughened laminate has a critical energy release rate much greater than that of the generalpurpose laminate, and the interlayer suppresses delamination. Since delamination has difficulty growing, many splits are generated as depicted in Fig. 6.19a, and delamination extends from the crack tips to absorb impact energy. In addition, the energy dissipation by catastrophic ply failure in the toughened laminate is also greater than that of the general-purpose laminate because of the higher strength. Therefore, these two factors, i.e., the greater critical energy release rate and the higher strength, will diminish the ply-failure zone and prevent perforation.

6.5

Concluding remarks

In this chapter, we observed high-velocity impact damage in CFRP laminates and discussed the mechanism of damage extension. A catastrophic ply-failure zone was formed just beneath the impact point, and delamination extended around it. The ply-failure zone was approximated by a cylinder with the same width as the crater, which was close to the diameter of the projectile. Delamination spreads in a fanshaped area between two matrix cracks in the neighbouring two plies. This damage state was observed in all laminations. The delamination pattern was similar to that of a low-velocity impact, but the catastrophic ply failure and small delamination area are characteristics of high-velocity impact that accompany local deformation. The degree of anisotropy in the bending stiffness is an index of the severity of the impact damage. A greater difference in the two bending stiffness components induces one-sided bending deformation despite the axisymmetric loading. In this case, the surface damage area will be small, but the laminate will be perforated easily. In contrast, slight anisotropy in the bending stiffness will result in large delamination areas and many fibre breaks in the ply-failure zone. This condition absorbs greater energy than the large anisotropy case. In addition, toughened interlayers represent an effective approach to absorbing greater impact energy, similarly to low-velocity impacts. The major factors of energy absorption are delamination and fibre breaks beneath the impact point. Both the volume of the ply failure and the delamination area became constant at high velocity. This result supports the conventional knowledge that the absorbed energy is constant in hyper-velocity impact problems. If the energy absorbed by ply failure is calculated by the strain energy density multiplied by the cylinder volume and if the energy absorbed by delamination is calculated by the critical energy release rate multiplied by the delamination area, these two energy dissipations are in the same range. Therefore, both the lamina property and the interlayer property are important to prevent perforation. In conclusion, we present the following three rough guidelines on material choice for shielding applications such as fan blade-out containment: a material system that can store great strain energy (i.e. high-strength type CFRPs) along with good interlayer toughness will be appropriate, a stacking configuration with many ply interfaces should be employed, and the degree of anisotropy in the bending stiffness should be small.

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Dynamic damage in FRPs: From low to high velocity

7

Vaibhav A. Phadnisa, Anish Royb, and Vadim V. Silberschmidtb a 3M UK Plc, Bracknell, United Kingdom, bWolfson School of Mechanical, Electrical and Manufacturing Engineering, Loughborough University, Loughborough, Leicestershire, United Kingdom

7.1

Introduction

With an increasing use of fibre-reinforced polymer composites (FRPs) as primary load-carrying components in military vehicles, commercial aircraft, the shipping industry, infrastructure, and sports goods, understanding their impact behaviour is critical to designers and end-users. Because of their highly heterogeneous microstructure, FRPs usually demonstrate a multiplicity of damage mechanisms under varying impact conditions. A wealth of knowledge is available on the low-velocity impact response of composites, although with continuously emerging materials and structures, systematic structure-property-performance relationships that could provide guidelines on the dynamic impact behaviour of composites are rare. This chapter is focused on providing insight into the low- to high-velocity impact performance of FRPs, with emphasis on high-velocity impact. Two case studies— ballistic-impact performance of woven hybrid composite laminates and blast response of curved carbon-fibre-reinforced laminates—are discussed to elaborate this.

7.2

Impact response of composite materials

In general, composite structures often suffer impact loads under service conditions. Aerospace structures, for example, can receive impacts during maintenance operations, or during service caused, e.g., by hailstones or debris. Impact responses of materials can be broadly divided into four main categories: (1) low velocity (large mass), (2) intermediate velocity, (3) high velocity (small mass), and (4) hypervelocity. These regimes are shown in Fig. 7.1 and are briefly described below.

7.2.1 Low-velocity impact Low-velocity impact (LVI) can be treated as a quasistatic event, with the upper limit of velocity varying from 1 to 10 m/s, depending on target stiffness, material properties, and the mass and stiffness of impactor [2]. The response of a target material is controlled by an impactor/target mass ratio rather than impact velocity [1]. Here, the structural response of the target is important, since the contact duration is long Dynamic Deformation, Damage and Fracture in Composite Materials and Structures. https://doi.org/10.1016/B978-0-12-823979-7.00008-9 Copyright © 2023 Elsevier Ltd. All rights reserved.

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Fig. 7.1 Classification of impact regimes. (A) High-velocity impact, dominated by dilatational wave; very small impact time. (B) Intermediate-velocity response dominated by flexural and shear waves; short impact times. (C) Low-velocity impact, long impact times with quasistatic response [1].

enough for the entire structure to respond to the impact and, consequently, energy is absorbed elastically. Though a quasistatic formulation is considered as suitable for LVI, the scientific community has different opinions about this. Some researchers [3] classified low velocity up to 10 m/s, using test techniques such as Charpy, Izod, and drop-weight impact. A few other researchers (e.g. Ref. [4]) suggested that the type of impact should be classified based on the assessment of damage that occurred. A large group of researchers define an LVI event as one, in which a throughthickness stress wave plays no significant part in stress distribution. The failure modes in LVI depend strongly on specimen size, stiffness, and boundary conditions [5–8]. The impact energy in LVI is absorbed by a composite specimen primarily in the form of strain energy, in addition to that dissipated through various failure modes such as matrix cracking, fibre breakage, and delamination. A typical example is shown in Fig. 7.2.

7.2.2 Intermediate-velocity impact Intermediate-velocity impact (IVI) is considered to be between low- and high-velocity impacts. The range of impact velocities falling in this category is not clear [5,6]. Depending on the projectile mass, large deformations may occur in the IVI range, but it may differ from LVI in terms of a loading rate and momentum. Typically, IVI arises from events such as road-debris impact on automobiles, bullet impacts at lower-end velocities, hail impact, or even a baseball bat striking a ball.

7.2.3 High-velocity (ballistic) impact FRP composites are being extensively employed in ballistic armour applications. Thus, understanding penetration mechanisms and failure is important. Abrate [6] defines the high-velocity-impact (HVI) regime as the one with a ratio of velocities of an impactor and a transverse compression wave greater than the failure strain in that direction. This kind of response is dominated by the stress-wave propagation through the target’s thickness, with a structure not getting enough time to respond, leading to localised damage. Effects of boundary conditions are of low importance, since the impact event finishes before the initiated stress waves reach the boundary. Cantwell and Morton [3] found that such small-mass, high-velocity impacts are more detrimental to carbon-fibre-reinforced polymer (CFRP) laminates than low-velocity

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Fig. 7.2 (A) Delamination and fibre fracture at bottom layer under low-velocity impact event. (Material system CY225/E-glass fabric (Vf
60%), laminate nominal thickness 3 mm, diameter of hemispherical impactor 12.7 mm, mass 5.22 kg.) (B) A typical schematic illustration of different damage modes shown in 2A [9].

drop-weight impacts. They also provided a guideline for when an impact event can be considered high velocity. According to them, if the velocity of impact is higher than 10% of the wave speed in that material, it can be considered HVI. Following this definition, Abrate [6] mentioned that the range of HVI should be from 50 to 1000 m/s, depending on the impact system (i.e. impactor and target structure).

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

Damage mechanisms of FRPs under high-velocity impact

The mechanical response, and thereby damage mechanisms, of FRPs can differ considerably more under dynamic loading conditions than under quasistatic ones. Impact dynamics are characterised by two features that differentiate them from the more conventional disciplines of classical mechanics of rigid or deformable bodies under quasistatic conditions. The first is the importance of inertia effects, which must be included in all the governing equations based on the fundamental conservation laws of mechanics and physics. The second is the role of stress-wave propagation in the analysis of problems, recognising the fact that most impact events are transient phenomena, where steady-state conditions do not exist. This is discussed next to consider ballistic and blast responses of FRP composites—typical high-velocity impact events.

7.3.1 Air-blast response Thin, laminated composite structures are attractive for many lightweight applications such as military vehicles and civil infrastructure due to their durability, versatility, light weight, and high mechanical performance. One of the main technical challenges in their design is to determine the level of blast protection. Such structures often undergo large-deflection dynamic motion under high-pressure explosive blast loads and experience progressive material damage and even a structural collapse. In 2008, Tekalur et al. [10] conducted an experimental study of material behaviour and damage evolution for E-glass vinyl ester and carbon composites subjected to static and blast loads. Their findings in the case of blast loads were limited to qualitative descriptions of different damage scenarios for the two studied composite materials. Perhaps more importantly, they acknowledged that the response of composites to explosives and air blast was a complex phenomenon to implement in a laboratory setting and thus was rarely studied experimentally. As a result, the level of understanding of response of these materials to high loading rates, as would normally be observed in blast events, is not yet as well established as that under static conditions [11] due to inadequacy of experimental data. This typically results in composite structures being conservatively designed with large safety factors to ensure that damage will not occur. This inherent conservativeness leads to overdesigns, which do not afford full-weight savings possible with composites—hence, development of finite-element schemes capable of adequately modelling these events is necessary. Historically, two experimental methodologies have been used to impart shockloading conditions to structures: (1) explosives and (2) shock tubes [10–12]. Although the use of explosives offers an ease of use, there are associated deficiencies such as spherical-wave fronts and pressure signatures, which are often spatially complex and difficult to capture. Shock tubes offer the advantages of plane-wave fronts and wave parameters that are easily controlled and repeated. When composite materials are subjected to blast loading conditions, they may experience damage in the form of several distinct mechanisms occurring in the in-plane and through-thickness directions.

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In general terms, in blast events, the in-plane damage mechanisms consist of fibre breakage and matrix cracking, while the through-thickness damage is dominated by delamination of the plies. Experimental studies on shock loading of materials examined their response over a range of loading rates. Nurick et al. [13,14] studied the effects of boundary conditions on plates subjected to blast loading and identified distinct failure modes depending on the magnitude of impulse and stand-off. Tekalur et al. [10] investigated the effects of shock loading on both E-glass- and carbon-based laminates. This study used a shock tube to impart pure shock loading as well as a small-scale explosion tube to consider a shock load combined with effects of heat generated during combustion of explosive materials. Mourtiz [15,16] studied an effect of shock loading on flexural and fatigue properties of composite laminates subjected to underwater shock loading. These studies demonstrated that under relatively low impulsive loading, laminates sustained little damage (primarily matrix cracking) and their mechanical properties remained the same as those for undamaged laminates. However, once a critical loading threshold was exceeded, the panels experienced fibre breakage and the material’s strength significantly degraded. Mouritz [17] studied the effectiveness of adding a lightweight, through-thickness stitching to increase damage resistance of composites. LeBlanc et al. [18] studied effects of shock loading on 3D woven composite materials. Recently, there has been an increased interest in the study of the effect of shock loading on sandwich structures. These studies include the effects of shock and impactloading conditions [12,19–22]. A time history of a typical response of a sandwich composite under blast load obtained by Jackson and Shukla [19] is shown in Fig. 7.3.

Fig. 7.3 Blast response of sandwich composite made of E-glass woven laminate face sheet and SAN foam core [19].

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In general, it should be noted that this research area is still relatively new with a limited number of research studies.

7.3.2 Ballistic response Penetration of laminated composites is often evident when they are loaded under ballistic conditions. This failure mode occurs when the fibre failure reaches a critical extent enabling an impactor to completely penetrate a target material [6]. A threshold impact energy required to penetrate the specimen increases with an increase in its thickness. The major energy-absorbing mechanisms during penetration are shear-out (also called shear-plug), delamination, and elastic flexure. The shear-out (shear-plug) mechanism is briefly discussed here. In HVI, the impact energy is dissipated over a smaller region of a target composite material, giving rise to the damage mechanism known as shear-plug. Due to the high stresses created at the point of impact, the target material around the perimeter of the projectile is sheared and pushed forward, causing a hole, or plug, slightly larger than the projectile’s diameter, and its size increases as it penetrates the composite. A typical example depicting this is shown in Fig. 7.4. Next, two case studies—an air-blast response of hybrid composite laminates and ballistic-impact performance of woven FRPs—are discussed.

Fig. 7.4 A typical penetration process observed in S-glass-epoxy composite [23].

Dynamic damage in FRPs: From low to high velocity

7.4

171

Air-blast response of curved CFRP laminates

7.4.1 Introduction Controlled and accidental explosions or detonations cause dynamic loading of huge intensity on structures in the immediate vicinity of the event. Thus, it becomes imperative to critically assess blast resistance of structures that may not have been designed to resist explosions, such as crucial civilian as well as governmental and defence buildings and structures. Generally, composite structures need not be limited to flat geometries; several applications require curved shapes, such as composite shells used in submarine hulls [18,24]. The effect of curvature of such structures on their blastmitigation properties was the subject of interest of some studies [25,26]. Generally, blast experiments are rather complex to carry out due to difficulty in obtaining reliable output data. Hence, a robust and reliable numerical model can potentially be a valuable tool in the design of structures with improved blast resistance. In this section, the dynamic response of CFRP panels with quasi-isotropic properties and three different radii of curvature to blast loading is studied numerically. A finite-element (FE) model of blast loading of CFRP panels was developed in ABAQUS 6.11 [27] and its results were compared to experimental findings reported in Ref. [26].

7.4.2 Experimental procedure 7.4.2.1 Material and specimens Panels with three different radii of curvature (Fig. 7.5) were utilised in the experiments: infinite (i.e. flat; Panel A); 304.8 mm (Panel B); and 111.8 mm (Panel C). The specimens were fabricated using unidirectional AS4/3501-6 prepreg (fibre volume fraction of 60%) manufactured by the Hercules Corporation of Magna, Utah. The stacking sequence of this composite laminate—[0°/90°/+45°/45°]4s with 32 layers of unidirectional plies—was selected to provide quasi-isotropic effective properties; the specimens were 203 mm  203 mm  2 mm in size. For the curved panels, arc lengths of the curved edges corresponded to the plate length of 203 mm. The material properties of the studied laminate are listed in Table 7.1.

7.4.2.2 Shock-loading apparatus and loading conditions In experiments, a blast load can be imposed onto a structure using two different methods—either by a controlled detonation of explosives or with the use of shock tubes. The use of explosives is dangerous and produces spherical wave fronts and pressure signatures, which are spatially complex and difficult to measure. In contrast, a shock tube offers the advantage of planar-wave fronts so that the wave parameters may be easily controlled. Furthermore, such imposed loading conditions are easier to replicate in finite-element (FE) simulations. Thus, the shock-tube apparatus was the preferred choice in the application of the blast load in our experiments. The apparatus used in this study and locations of pressure transducers, recording pressure history of incident and reflected waves, are shown in Fig. 7.5B and C, respectively.

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

Fig. 7.5 (A) Specimen geometry. (B) Shock tube at University of Rhode Island used in experiments. (C) Muzzle section showing locations of transducers [28].

Details of the shock tube apparatus employed in experiments, and pressure profiles obtained for studied CFRP panels, can be found in Ref. [26].

7.4.3 Finite-element model A numerical simulation of deformation and damage processes in composite panels is a valuable tool, as it can significantly curtail the need to conduct expensive and laborious experiments on life-size specimens. Regarding this, a dynamic FE model of blast loading and response of curved carbon-epoxy panels was developed in the generalpurpose FE software package ABAQUS/Explicit [27]. Details of the FE modelling strategy, including the material modelling procedure, are discussed next.

7.4.3.1 Material model A user-defined damage model (VUMAT) with 3D continuum elements was developed and implemented to predict damage characteristics through the laminate’s thickness under the blast load. The model is able to characterise damage in a composite laminate by employing a stiffness-degradation concept with the help of an element-deletion approach based on the initiation and evolution of damage in the meshed domain [29].

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Table 7.1 Mechanical properties of AS4/3501–6 UD composite laminate [26,29,30]. Exx 5 147 GPa, Eyy 5 11.2 GPa, Exy 5 7 GPa

Elastic moduli Tensile strength in fibre direction, X1t Compressive strength in fibre direction, X1c Tensile strength in transverse direction, X2t Compressive strength in transverse direction, X2c In-plane shear strength, S12 Stiffness of cohesive zone elements Traction in normal direction, τn Traction in shear directions, τs ¼ τt Mode I fracture energy, GIC Mode II fracture energy, GIIC Power law coefficient, β

2004 MPa 1197 MPa 65 MPa 285 MPa 80 MPa 5  106, N/mm3 53 MPa 86 MPa 0.08 N/mm 0.55 N/mm 1.8 Average strain rates, ε_

Strain-rate-dependent properties

0.0001

1

400

2400

3800

Transverse modulus, E2 (GPa) Shear modulus, G12 (GPa) Transverse tensile strength, F2t (MPa) Transverse compressive strength, F2c (MPa)

11.2 7 65 285

12.9 8.2 80 345

14.5 9 90 390

[14.8] [9.3] [94] [405]

[14.96] [9.4] [95] [409]

Note: Numbers in brackets denote extrapolated values.

Another damage mode—interply delamination—was simulated using cohesive elements inserted between the adjacent plies of the laminate. The general contact algorithm in ABAQUS/Explicit was used to model contact conditions between the shock wave and the composite laminate, and between the laminates by defining appropriate contact-pair properties. The results of numerical simulations were evaluated using comparison with the experimental data.

Damage initiation Damage modelling in composites at a laminate level typically requires input of several parameters, including homogenised ply properties, interply strength and information about a laminate’s lay-up. Here, a layer-by-layer modelling strategy to capture failure in each ply [29] was adopted. This approach offers several advantages. First, full 3D stress states can be analysed. Typically, FE models of deformation in composites involve the use of 2D shell elements to represent composite plies; this does not allow for accurate representation of stresses through the composite’s thickness. Second, intraply and interply damages can be introduced discretely along with phenomenological models that account for a complex interaction between them.

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To model damage initiation and propagation in the composite laminate, the element-removal scheme of ABAQUS/Explicit was employed, that is, a finiteelement was removed from the mesh as a respective threshold level, primarily in the fibre direction, was attained by the instantaneous stress in this element. A combined approach, employing advantages of both linear-elastic fracture mechanics (LEFM) and damage mechanics used for delamination modelling, is also discussed in this section. Many criteria are available for modelling damage in fibre-reinforced polymer composites; each has its advantages and shortcomings. For example, Hashin’s criteria [31] possess the capability to differentiate between discrete damage modes of fibres and a polymer matrix material employing merely six input parameters that include ply strength, stiffness, and the Poisson’s ratio. Hence, they have been extensively used in industry for years, thanks to their simplicity; still, some studies indicate that they are limited in predicting damage in a brittle polymer matrix with acceptable accuracy [32,33]. In this regard, Puck’s criterion [34] was shown to provide a reasonably good estimate of damage in epoxy matrix both qualitatively and quantitatively; an extensive review on this is available in [32,33]. In our FE model, a combination of Hashin’s and Puck’s failure criteria was used to employ advantages of both schemes. Hashin’s criteria are used to estimate damage in carbon fibres, while damage in epoxy matrix is modelled using Puck’s criteria. The empirical formulation of these criteria is given next. Hashin’s criteria for failure in fibres. Fibre tensile failure (σ 11  0):


σ 11 X1t

2
2
2 σ 11 σ 11 + + ¼ 1, dft ¼ 1 S12 S13

(7.1)

Fibre compressive failure (σ 11 < 0):


σ 11 X1c

2 ¼ 1, dfc ¼ 1

(7.2)

Puck’s criteria for failure in epoxy matrix. Matrix failure: "


σ 11 2X1t

2


2 #
 ðσ 22 Þ2 σ 12 1 1 ¼ 1: + + + + σ 22 |X2t  X2c | S12 X2t X2c σ 22 + σ 33 > 0, dmt ¼ 1

(7.3)

σ 22 + σ 33 < 0,dmc ¼ 1 Here, σ 11, σ 22, σ 33, and σ 12 are the components of the stress tensor at an integration point of an element; dft, dfc, dmt, and dmc are the damage variables associated with failure modes in fibre tension, fibre compression, matrix tension, and matrix compression, respectively. X1t, X2t, and X2c are the tensile failure stress in the fibre direction, and the tensile failure and compressive failure stresses in the transverse direction,

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respectively, whereas S11, S12, and S13 are the shear failure stresses in 1–2, 2–3, and 1–3 planes, respectively. The mechanical properties of the CFRP laminate used in this FE analysis are listed in Table 7.1. Puck’s formulation was suitably modified to include the strain-rate dependence of the epoxy matrix at high strain rates that are observed in blast events. These modified equations (Eqs 7.4 and 7.5) were then implemented in a user-defined material model (VUMAT) in ABAQUS/Explicit.

Modelling rate dependency The polymer matrix material in a CFRP composite demonstrates strain-rate sensitivity at high strain rates (
103 s 1), which are typical for a blast event. This effect becomes significant, particularly for transverse directions, in cases where a polymer matrix is a primary load-bearing member [35–37]. Many test methods have been developed to facilitate the dynamic characterisation of composite materials at high deformation rates. Previous test studies highlighted the increase in stiffness and strength of composites with an increasing strain rate in matrix-dominated regions [35–38]. In some cases, explicit empirical relations were formulated to derive such material properties at corresponding strain rates [30,37]. The composite laminate used in our experiments—AS4/3501–6—was shown to exhibit a strain-hardening behaviour at high loading rates [30]. The response of AS4/3501-6 laminate at various biaxial stress states, e.g., combined transverse compression and shear, at strain rates varying from 104 to 400 s 1 was characterised. Stress-strain data at failure, initial moduli, and strength were also recorded. Empirical relationships between the matrix-dominated properties and strain rates under highstrain-rate deformation were as follows: l

For in-plane shear and transverse moduli:


  ε_ +1 : Eðε_ Þ ¼ Eðε_ 0 Þ me log ε_ 0

(7.4)

me ¼ 0:045, ε_ 0 ¼ 104 s1 l

For in-plane shear and transverse strength:


  ε_ +1 : Fðε_ Þ ¼ Fðε_ 0 Þ mf log ε_ 0

(7.5)

mf ¼ 0:057, ε_ 0 ¼ 104 s1 Here, Eðε_ Þ and Fðε_ Þ are the instantaneous in-plane moduli of elasticity and shear strength at a strain rate of ε_ , respectively; ε_ 0 ¼ 104 s1 is the reference strain rate, which corresponds to quasistatic loading, while me and mf are curve-fitting parameters. Eqs (7.4) and (7.5) form the basis of dynamic material properties used in our simulations. Average levels of maximum strain rates for the studied composite panels under blast loading were analysed, initially without specifying strain-rate-dependent properties,

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with our FE simulations. They were observed to be in the range of 2200–2400 s 1 for Panel A, 3500–3800 s 1 for Panel B, and 1500–1800 s 1 for Panel C. The matrixdominated properties reported by Daniel et al. [30] for a similar composite were suitably extrapolated to match these strain rates (Table 7.1) and incorporated in the failure criteria of our simulations. For this, Eqs (7.1) and (7.3) were modified as follows:


σ 11 S11

"


2

 σ 11 2 σ 11 2 + + ¼ 1, dft ¼ 1 S12 ðε_ Þ S13 ðε_ Þ

σ 11 2X1t

2


 #
 ðσ 22 Þ2 σ 12 2 1 1 + + ¼1 + + σ 22 |X2t ðε_ Þ  X2c ðε_ Þ| S12 ðε_ Þ X2t ðε_ Þ X2c ðε_ Þ σ 22 + σ 33 > 0, dmt ¼ 1

(7.6)

(7.7)

σ 22 + σ 33 < 0, dmc ¼ 1, where S12 ðε_ Þ and S13 ðε_ Þ are dynamic shear strengths in transverse directions (1–2 and 1–3), while X2t ðε_ Þ and X2c ðε_ Þ are dynamic tension and compression strengths in transverse direction at respective strain rates for the studied composite panels. The element-deletion approach used to remove the failed elements from the mesh was based on the magnitude of damage variables as calculated with Eqs (7.2), (7.6), and (7.7), applied to discrete damage modes in the modelled CFRP composite. The element was removed when the maximum damage condition was satisfied at its integration point. The damage parameter, d  max(dft,dfc,dmt,dmc), based on the formulations mentioned earlier, was calculated so that when d ¼ 1 (at the integration point of an element), the element was removed from the mesh and offered no subsequent resistance to deformation.

Delamination modelling Delamination at the interface of neighbouring plies of a laminate was modelled using cohesive-zone elements (CZEs) available in ABAQUS/Explicit. The elastic response of such an interface was modelled using stiffness of CZEs, calculated with an empirical formula suggested by Turon et al. [39]. Delamination initiation was modelled using a bilinear traction-separation law with a quadratic nominal-stress criterion [29], while its postdamage response was calculated using a power law [29,39] that accounted for mode mixity. The mechanical properties of cohesive elements used in our simulations are listed in Table 7.1.

7.4.3.2 Finite-element model set-up The 3D finite-element model developed in ABAQUS 6.11 consisted of a shock-tube wall and a CFRP panel (refer to Fig. 7.6A–C). The shock tube was modelled with shell elements with five integration points through its thickness. The elements of the wall had an edge length of 25.4 mm and shell thickness of 2 mm. The CFRP panels were modelled as a solid continuum with mechanical properties listed in Table 7.1.

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177

Fig. 7.6 (A) FE model set-up. (B) Boundary conditions, meshed CFRP Panel B and locations of cohesive zone elements (fibre orientation in a ply is shown with white dotted lines).

These panels were meshed with eight-node, one-integration-point hexahedral elements C3D8R with an element size of 1 mm along the length, while each ply was assigned one element through its thickness. There were a total of 1.7 million elements in this structural domain. The material’s coordinate system was assigned to the panel such that it captured a discrete orientation of each element accurately following the curvature. The schematic for meshed CFRP Panel C is shown in Fig. 7.6B. The boundary conditions employed in this model reflected the respective physical constraints due to specimen’s fixture applied in the experiments. All edges of the panels were

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

Fig. 7.7 Orientation of specimens.

fully constrained. All the degrees of freedom at the shock-tube wall were also fully constrained, since it was considered rigid in our simulations. The three panels in our simulations were positioned against the wall of the shock tube as shown in Fig. 7.7.

7.4.3.3 Fluid-structure coupling and shock-wave loading A fluid model in shock simulations consisted of the air inside and outside the shock tube as well as the air surrounding the plate (refer to Fig. 7.6A). The air outside the tube was modelled with an Eulerian formulation as a cuboid with a domain size of 400 mm 400 mm in the X–Y plane and 2000 mm along the tube’s axis. The model had 100 mm of air along the tube axis behind the plate to ensure that the plate remained in air during deformation caused by shock-wave loading. The air inside the tube was also modelled in the Eulerian domain with the element size of 3 mm. All the fluid elements were meshed with the Eulerian eight-node, one-integrationpoint hexahedral elements EC3D8R. The acoustic structural coupling between acoustic pressure of the fluid mesh and structural displacements of the CFRP panel was accomplished with a surface-based tie constraint at the common surface. The master-slave type of contact was established between the annular surface of the shock tube in contact with the CFRP panel and the top surface of the panel. The surface of the external fluid at the interface was designated as the master surface. The incident-wave front was assumed to be planar. For this planar wave, two reference points, namely the stand-off and source points, were defined (Fig. 7.6A). The relative positions of these two reference points were used to determine the direction of travel for the incident shock wave; the pressure history at the stand-off point was used to drive it. The ‘amplitude’ definition in ABAQUS/Explicit was employed to specify the shock load on the front surface of the CFRP panel using

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179

the pressure history data. The entire analysis was divided into two steps pertaining to the wave incidence and reflection, with appropriate magnitudes of average shockwave velocity and density used. Linear fluid mechanics was used for the entire model. The observed total pressure in the fluid was divided into two components: the incident wave itself, which was known, and a calculated wave field in the fluid due to reflections at the fluid boundaries and interactions with the solid.

7.4.4 Results and discussion The results obtained in these numerical studies are discussed next. The FE analysis was employed to compute the out-of-plane deflections of CFRP panels at the centre of their rear faces (as it was measured contactlessly in the tests), energy distributions during blast and damage in the panels.

7.4.4.1 Finite-element model validation The FE model allowed for observation of interaction of the shock front with the CFRP panels and their deformation under shock loading. The deflection, velocity, and strain data acquired employing the digital image correlation (DIC) technique [24–26] were used as a basis to validate the FE model. Initially, the flat panel (Panel A) had a uniform deflection within a central region of loading; this out-of-plane deflection decayed gradually towards the edges. When the radius of curvature was increased, the effective loading area changed its shape from circular to elliptical. For the curved panels, shock loading primarily acted upon the projected area unlike the circular area as in the case of the flat panel, which caused this change in a shape. During the early stages of shock loading, contours of out-of-plane deflection were not affected by the boundary conditions. Deflection of Panel A started as a circular region, which continued until 150 μs. This was a localised circular deflection contour, which had roughly the same diameter as that of the muzzle (at t ¼ 50 μs). At that moment, the boundary conditions started to affect the development of deflection contours in the panel. The stress waves generated in the specimen travelled outwards and were reflected by the boundary. This reflected stress wave caused the change in the shape of the deflection contours. The full-field deflection at the failure loading for these panels can be found in Ref. [26].

7.4.4.2 Modes of deflection in CFRP panels The FE analysis demonstrated that total deflection of the studied CFRP panels was the combined result of two deflection modes, namely the indentation and flexure modes. It was seen that all the panels started deflecting in the indentation mode initially. In the flat panel (Panel A), the global flexural mode quickly began dominating the deflection process. This was confirmed by the continuous nature of displacement contours that showed a monotonic increase in deflection from the edge of the specimen to its centre after t ¼ 200 μs. Deflection of Panel B continued in the indentation mode up to about 400 μs, after which it changed to the global flexural deflection mode. These deflection

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

contours show a continuous increase in deflection from the edge of the panel to its centre and the transition from elliptical contours back to the circular shape. In Panel C, the deflection was observed to be lower than that in Panels A and B, since only the central loading region was affected. In addition to the out-of-plane deflection, velocities and in-plane strain data were also extracted from the simulations with the developed FE model at the centre point of the back of the studied CFRP panels (Fig. 7.8A–C). Fig. 7.8A demonstrates that the deflection rate (35 m/s), for the initial 200 μs, was almost the same for all the three panels, though Panels A and B attained higher levels of deflection as compared to that of Panel C. This means that Panel C was stiffer than the other two panels since it sustained a higher pressure and had a lower deflection. Panels A and B showed similar trends up to 1000 μs. At this time, damage was observed to initiate in Panel B, which explains the rapid increase in its deflection. The lower out-of-plane deflection (Fig. 7.8B) and in-plane strain (Fig. 7.8C) in Panel C showed that this panel had higher flexural rigidity. Panel B exhibited higher in-plane shear strain, which led to its catastrophic failure.

Fig. 7.8 Experimental and numerical results. (A) Out-of-plane deflections. (B) Out-of-plane velocities. (C) In-plain strain.

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181

7.4.4.3 Damage in CFRP panels The observed variability of spatiotemporal evolution of deflections in the studied CFRP panels under blast load resulted in different types of realisation of damage modes (refer to Fig. 7.9). The damage development in Panel A at failure loading is shown in Fig. 7.9A. Apparently, fibre breakage was the dominant damage mode as observed from the results obtained with the FE model and confirmed from Ref. [26]. The initiated damaged regions were primarily located along the clamped edges, due to the constraints imposed, exposing the underlying fibres to the excessive tensile loading.

Fig. 7.9 Damage evolution in plates under blast loading. (A) Fibre breakage in front face of Panel A at failure load. (B) Panel B at failure load. (C) Panel C at threshold load. No global fracture is observed.

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

Fig. 7.10 Calculated response of Panel C at failure load. (A) Deflection. (B) In-plain strain.

The extent of damage was about the same at the clamped edges though varied along their thickness. In Panel B, a similar trend was observed, though on a larger scale. Fibre breakage was the governing damage mode, though a large-scale delamination was also detected. Damage initiated in the form of fibre breakage at the clamped boundaries and propagated towards the midregion, where Panel B failed (Fig. 7.9B). The damage in Panel C at the threshold loading is shown in Fig. 7.9C, where no fibre breakage or delamination was observed. Using the developed FE model, this panel was then exposed to a higher pressure of 8 MPa, where it failed catastrophically at around 1000 μs, following the similar damage pattern as observed in Panel B. The deflection and in-plane strain data at the centre point of the Panel C at this load (8 MPa) are presented in Fig. 7.10.

7.4.4.4 Energy distribution during blast The incident and remaining energies associated with the shock-loading intensities were analysed with the developed FE models. The energy lost was obtained by subtracting the remaining energy from the incident energy. The magnitudes of energies (incident, remaining, and lost) for all the three loading cases are shown in Fig. 7.11. Panel C was subjected to the highest intensity of shock loading, and so the incident energy was the highest for this panel. The energy remaining in the gas is identical with impact energy, as this is the actual energy that the panel experienced due to shock loading. Since Panel C reflected a major part of the incident shock energy, it was exposed to lower impact energy, whereas Panel B was subjected to the highest impact energy. The ratio of impact energy (remaining energy in the gas) to the incident energy was considered as an indication of the blast-mitigation ability of the studied CFRP panels. Panel C had the lowest ratio of impact to incident energy, which indicates its enhanced blast-mitigation capacity. Consequently, Panel B, with the highest ratio, demonstrates its poor blast-mitigation capacity. To clarify, the energy evolution analysis for all three panels was performed at the same loading pressure. The panels had different levels of energy-dissipation capacities. Again, Panel C had the lowest ratio of impact to incident shock energy, whereas Panel B had the highest.

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183

Fig. 7.11 Energy distribution in CFRP panels. (A, B) Panels A and B (failure load). (C) Panel C (threshold load).

7.5

Ballistic-impact response of hybrid woven FRPs

7.5.1 Introduction Woven fabric-based polymer-matrix composites (PMCs) are finding an increasing use in defence-related applications thanks to their high strength and stiffness, and their ability to produce structures with application-tailored shapes and mechanical properties. Additionally, employment of such PMCs leads to better energy absorption in ballistic-impact events, especially due to their balanced in-plane properties. Recent studies [40–43] showed that a hybrid composite structure—e.g., a combination of glass- and carbon-based epoxy composites—may further improve the energyabsorbing capacity of such composites. Thus, it is of great interest to the scientific community to understand the mechanical behaviour of these materials in highvelocity impact events. In ballistic-impact events, PMCs absorb the projectile’s kinetic energy by undergoing either elastic or permanent deformation. In the latter process, PMCs often exhibit different damage modes such as delamination, punching, and fibre breakage. Typical ballistic experiments involve a plethora of safety protocols and often their output is difficult to quantify in a reliable manner, given the short duration of such events. Thus, computer simulations are often employed as a virtual tool to aid design of structures for ballistic events.

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

Simulating a mechanical behaviour of a fabric-reinforced composite structure under ballistic impact is a challenging task. Unlike metallic components that can yield and dissipate energy by undergoing plastic deformation, these structures can only dissipate energy through various damage processes that usually degrade their stiffness. Hence, an advanced modelling tool that can adequately model such events is essential in the design process. However, due to the complexity involved in this process, most models attempt to provide an acceptable trade-off in performance analysis [29,44–46]. This section focuses on the development of an FE model of ballistic-impact response of wovenfabric-reinforced composites. The experimental studies are discussed first, followed by a brief description of the FE model. Its results and discussion are presented next.

7.5.2 Ballistic experiments A ballistic-impact test apparatus operated by a single-stage gas gun (Fig. 7.12) was used to carry out experimental studies. It consisted of a projectile-propelling mechanism, a chronograph for velocity measurement, a support stand to hold the specimen, a containment chamber, safety devices, and a strain-measuring facility. Compressed air was used as a propellant in the system. The main components of this propelling mechanism were a cylindrical barrel to guide the projectile, a quick-release valve to relieve the trapped air and a nitrogen gas-driven solenoid valve to operate this valve. The cylindrical barrel (through which a bullet was propelled) was 1.5 m long. Its inner diameter was chosen to suit a projectile used in this experiment. Velocity of the projectiles in the tests was varied up to 200 m/s by changing the air pressure in the cylinder. The experimental studies were carried out on flat specimens with dimensions 125 mm 125 mm with different thicknesses (refer to Table 7.2).

Fig. 7.12 Ballistic test set-up at IIT Bombay used in experiments. (A) Single-stage gas gun. (B) Typical test specimen in fixture.

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Table 7.2 Scheme of ballistic tests (target dimensions: 125 mm 125 mm; projectile diameter: 6.36 mm). S. no.

Projectile mass (g)

Projectile length (mm)

Target thickness (mm)

(A) G: Plain-weave E-glass epoxy 1 2 3 4

6.42 6.42 6.42 6.42

25.3 25.3 25.3 25.3

2.5 3.0 4.5 5.0

(B) C: 8H satin-weave T300 carbon/epoxy 1

6.42

25.3

3.0

25.3 25.3

3.0 3.0

(C) H4 and H5: hybrids 1 2

6.42 6.42

The specimen’s dimensions were chosen such that they can be accommodated into the fixture designed as an integral part of the ballistic-impact test apparatus. The mass of flat-end cylindrical projectiles made of hardened steel (and its diameter) was kept constant for all the tests. The experiments were carried out on at least four specimens for each impact condition to ensure repeatability. Four symmetric cross-ply woven fabric composites—plain-weave E-glass fabric/ epoxy, 8H satin-weave T300 carbon fabric/epoxy, and their hybrids—were studied. Specifications of tows/stands, fabrics, resin, and composites for plain-weave E-glass/ epoxy and 8H satin-weave T300 carbon/epoxy composites can be found in Ref. [41]. For simplicity, these composite laminates are designated as G, C, H4, and H5, respectively, from here onwards; the ply architecture of H4 and H5 is shown in Fig. 7.13.

7.5.3 Finite-element model A ply-level constitutive model was developed to analyse the mechanical response of the fabric-epoxy composites. This model was implemented as a material subroutine, VUMAT, in ABAQUS. Details of the constitutive model for fabric-reinforced composites and damage modelling framework used in this FE model are explained in the previous sections and thus not repeated here. Each ply was modelled as a homogeneous orthotropic elastic material with a potential to sustain progressive stiffness degradation due to fibre-matrix cracking and plastic deformation under shear loading. Delamination between the neighbouring plies was modelled using the cohesive-zoneelement (CZE) technique. The set-up of the developed FE model is shown in Fig. 7.14. Both the woven-fabric-reinforced plate and the bullet were modelled as 3D deformable solids. The dynamic explicit solver was used in simulations to account for the time-dependent loading and complex interaction between the target and the projectile.

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

Fig. 7.13 Architecture of hybrid composite laminates. (A) H4. (B) H5.

Fig. 7.14 FE model of ballistic impact. (A) Set-up. (B) Typical meshed specimen, thickness 3 mm.

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187

A 3-mm-thick symmetric cross-ply laminate was modelled; it consisted of five plies, each with a thickness of 0.32 mm for C and 0.28 mm for G. The local coordinate systems were defined to account for orientations of individual plies. In experiments, a cylindrical bullet of mass 6.42 g and length 25.3 mm impacted the target’s centre orthogonally. In simulations, this was achieved using a predefined velocity-boundary condition. The contacts between the bullet and the composite plate and all the contacted plies of the laminate were defined with the general contact algorithm available in ABAQUS/Explicit. This algorithm generated the contact forces based on the penalty-enforced contact method. A coefficient of friction, μ, was used to account for shear stress of the surface traction with contact pressure, p, and can be represented as τ ¼ μp. In this case, the frictional contact between the bullet and the composite laminate was modelled with a constant coefficient of friction of 0.3 [29]. The models required on average 11 h on 24 Intel quad-core processors with 48 GB RAM each to complete numerical analysis using the high-performance computing (HPC) facility available at Loughborough University.

7.5.4 Results and discussions The developed FE models of ballistic impact on the studied composite laminates were validated using an experimentally measured ballistic limit velocity (V50) and analytically calculated energy absorbed by the laminate. The ballistic limit velocity V50 was assessed for the same thickness of laminates, geometry, and mass of the impactor to provide a comparison of their relative performance. It should be noted that although the thickness of all the studied laminates was the same, the areal weight of the underlying fabric materials (E-glass plain-weave and carbon satin-weave) was different. Thus, a more viable comparison of their ballistic performance was provided in terms of V50 normalised with the weight of a unit area of the target. The fracture mechanisms in these laminates were also studied, and the contribution of these to absorption of incident impact energy is discussed.

7.5.4.1 V50 for same target thickness and per-unit areal density In the performed FE simulations, V50 was calculated at the reference point Rf tied to a bullet using an equation constraint to reduce the computational efforts. Fig. 7.15 presents the ballistic limit velocity V50 for G, C, H4, and H5 composites for the same thickness and areal density. It can be observed that the ballistic limit velocity, V50, had considerably different hierarchies for assessed criteria and laminates; thus, a selection of appropriate composite mostly depends on the application—i.e., if the working structure needs higher stiffness or better energy-absorption capacity.

7.5.4.2 Damage in composite panels Damage assessment in the studied composite laminates was carried out using the developed FE model. Damage patterns obtained at the front and back faces of the studied composite laminates are shown in Fig. 7.16A–D. Damage was visible on the either face of the composite panels and was measured (Fig. 7.17A and B) along both the

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

Fig. 7.15 Ballistic-impact velocities. (A) Same thickness. (B) Normalised with areal density for studied composite laminate (projectile mass 6.42 g, projectile length 25.3 mm).

Fig. 7.16 Damage patterns on front and back faces of composite panels. (A) 8H satin-weave T300 carbon/epoxy composites (V50 ¼ 82 m/s). (B) Plain-weave E-glass/epoxy composites (V50 ¼ 99.5 m/s). (C) Hybrid H4 (V50 ¼ 86 m/s). (D) Hybrid H5 (V50 ¼ 88 m/s). (Laminate thickness 3 mm.)

Dynamic damage in FRPs: From low to high velocity

189

Fig. 7.17 Damage size for studied composite laminates at their front (A) and back (B) faces at respective ballistic limit V50 (99.5 m/s for G; 82 m/s for C; 86 m/s for H4; 88 m/s for H5) (Laminate thickness 3 mm).

warp and fill directions from the centre of specimens, where they were impacted, and compared with those obtained from the FE analysis. The damage mechanism in G and C laminates was observed to be similar, although overall damage at the front and back face of G was more pronounced than that of C due to low stiffness of glass-fibre plies (Fig. 7.16A and B). Hybridisation of G and C laminates could possibly give rise to either improved or reduced material properties. In case of H4 (hybrid laminate with exterior G plies), the dominant damage processes were the same as found in C and G, namely tensile fracture of the back plies and crushing of the front plies under the impactor. The intraply cracks in the front and back plies grew upwards and downwards, respectively, leading to development of delamination cracks, and the final fracture took place by formation of a crack through the laminate thickness. However, the amount of damage at the back face of H4 laminate was far lower than in G and moderately less than in C under equivalent impact conditions (Fig. 7.16A and C). On the contrary, in case of H5 (the hybrid laminate with C plies at the exterior), apart from ply fracture at the back and crushing at the front, large deformations in the stiff carbon plies led to extensive delamination and intraply fracture that grew rapidly towards the back. Ply fracture due to crushing at the front also accelerated this process, resulting in greater damage on the back face of H5 (Fig. 7.16D).

7.5.4.3 Contribution of damage modes to energy absorption The trends of energy-absorption mechanisms in the studied laminates were quite similar to each other (refer to Fig. 7.18), with most of incident kinetic energy absorbed due to the deformation of secondary yarns, while energy absorption due to primary yarn/ fibre failure was relatively low and that due to matrix cracking and delamination was marginal. It may be explained by the highly transient nature of this event (a contact time between projectile and target
150 μs) and highly localised deformation of the target, where damage initiated in the form of fibre failure directly below the projectile and had hardly any time to diffuse through the laminate before the projectile penetrated through it.

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Fig. 7.18 Contribution of damage modes to kinetic-energy absorption. (A) Plain-weave E-glass-epoxy composite (V50 ¼ 99.5 m/s). (B) 8H satin-weave carbon-epoxy composite (V50 ¼ 82 m/s). (C) Hybrid composite H4 (V50 ¼ 86 m/s). (D) Hybrid composite H5 (V50 ¼ 88 m/s). (Laminate thickness 3 mm).

7.6

Conclusions

The mechanical behaviour of fibre-reinforced composites under low- to high-velocity impact is remarkably complex. For example, in FRPs—for a given fibre/resin system with exactly the same geometric and volumetric properties—the governing modes of failure may vary depending upon the rate of loading applied to them. In LVI, delamination and matrix cracks are the commonly observed modes of failure, while in HVI, fibre fracture is often found with complete penetration by impactor. This was demonstrated using two case studies—the blast response of curved CFRP laminates and the ballistic response of hybrid woven FRPs. The effect of curvature of composite panels on their blast-mitigation capacity was studied using the shock-tube apparatus. The performance of such panels under blast loading was characterised in terms of their out-of-plane deflection, in-plain stress as well as the damage and failure scenarios. A finite-element model was developed to simulate air-blast loading of curved CFRP panels and validated using the 3D DIC data coupled with high-speed photography. This model accurately accounted for the interaction between the shock wave, the curved panels, the shock tube, and the surrounding air. The following conclusions were drawn:

Dynamic damage in FRPs: From low to high velocity l

l

191

Two main deformation modes contributing to deflection of the studied panels were observed under shock loading: flexural and indentation. Flexural deformation decreased and indentation deformation increased as the radius of curvature was reduced. The indention mode was found to be more severe since it led to damage initiation in the panels. Fibre breakage was the dominant mode of damage observed in the studied panels at the failure loading, as confirmed in Ref. [26], and was captured reasonably accurately with the developed FE model.

In the second case study, ballistic-impact responses of four woven fabric composite laminates E, G, H4, and H5 were studied experimentally and using the developed finite-element model. The ballistic damage on the front and back faces of the studied laminates was observed. The effect of hybridisation on damage modes and their contribution to the energy-absorption capacity of the laminates were also discussed. The FE model predicted damage of the analysed laminates reasonably well and provided insight into their probable damage mechanisms. Some fundamental observations based on this study are listed: l

l

Improvements in the behaviour under impact by hybridisation of laminates were due to the higher strain-to-fracture of the E-glass–fibre plies located near the front and back laminate surfaces. These plies were able to sustain higher deformations before fracture and hindered propagation of damage to the inner plies from the broken plies on the front and back surfaces, increasing the maximum load-bearing capability of the composite. The hybridisation provided a reasonable trade-off between in-plain strength and failure strain that resulted in better ballistic-impact resistance properties compared to those of high-modulus fibre-reinforced composites. In addition, the presence of E-glass fibres helped to sustain higher deformations before laminate fracture by the percolation of a throughthickness crack, significantly improving the energy dissipated under impact.

Acknowledgements The authors acknowledge Dr. Puneet Kumar and Professor Arun Shukla for their valuable contribution in air-blast experiments performed at the University of Rhode Island, and Professor Niranjan Naik along with Mr. Kedar Pandya for contributing to experiments in ballistic damage study implemented at IIT Bombay.

References [1] R. Olsson, Mass criterion for wave controlled impact response of composite plate, Compos. Part A 31 (2000) 879–887. [2] P.O. Sjoblom, J.T. Hartness, T.M. Cordell, On low-velocity impact testing of composite materials, J. Compos. Mater. 22 (1998) 30–52. [3] W.J. Cantwell, J. Morton, The impact resistance of composite materials—a review, Compos. A: Appl. Sci. Manuf. 22 (5) (1991) 347–362. [4] D. Liu, L.E. Malvem, Matrix cracking in impacted glass/epoxy plates, J. Compos. Mater. 21 (1987) 594–609. [5] S. Abrate, Impact on laminated composite materials, Appl. Mech. Rev. 44 (4) (1991) 155–190.

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[6] S. Abrate, Impact Engineering of Composite Structures, CISM Courses and Lectures, CISM, Italy, 2011. [7] S. Abrate, Impact on Composite Structures, Cambridge University Press, Cambridge, UK, 1998. [8] M.O.W. Richardson, M.J. Wisheart, Review of low-velocity impact properties of composite materials, Compos. Part A 27A (1996) 1123–1131. [9] M. Aktas, C. Atas, B.M. Icten, R. Karakuzu, An experimental investigation of the impact response of composite laminates, Compos. Struct. 87 (2007) 307–313. [10] A.S. Tekalur, K. Shivakumar, A. Shukla, Mechanical behaviour and damage evolution in E-glass vinyl ester and carbon composites subjected to static and blast loads, Compos. Part B 39 (2008) 57–65. [11] J. LeBlanc, A. Shukla, Dynamic response of curved composite panels to underwater explosive loading: experimental and computational comparisons, Compos. Struct. 93 (2011) 3072–3081. [12] H. Arora, P. Hooper, J.P. Dear, Impact and blast resistance of glass fibre reinforced sandwich composite materials, in: Proceedings of IMPLAST, 2010. [13] G. Nurick, G. Shave, The deformation and tearing of thin square plates subjected to impulsive loads—an experimental study, Int. J. Impact Eng. 18 (1996) 99–116. [14] G. Nurick, M. Olson, J. Fagnan, A. Levin, Deformation and tearing of blast loaded stiffened square plates, Int. J. Impact Eng. 16 (1995) 273–291. [15] A.P. Mouritz, The effect of underwater explosion shock loading on the fatigue behaviour of GRP laminates, Composites 26 (1995) 3–9. [16] A.P. Mourtiz, The effect of underwater explosion shock loading on the flexural properties of GRP laminates, Int. J. Impact Eng. 18 (1996) 129–139. [17] A.P. Mouritz, Ballistic-impact and explosive blast resistance of stitched composites, Compos. Part B 32 (2001) 431–439. [18] J. LeBlanc, A. Shukla, C. Rousseau, A. Bogdanovich, Shock loading of three dimensional woven composite materials, Compos. Struct. 79 (2007) 344–355. [19] M. Jackson, A. Shukla, Performance of sandwich composites subjected to sequential impact and air blast loading, Compos. Part B 42 (2) (2011) 55–66. [20] P.M. Schubel, J. Luo, I. Daniel, Impact and post-impact behaviour of composite sandwich panels, Compos. A: Appl. Sci. Manuf. 38 (2007) 1051–1057. [21] N. DeNardo, M. Pinto, A. Shukla, Hydrostatic and shock-initiated instabilities in doublehull composite cylinders, J. Mech. Phys. Solids 120 (2018) 96–116, https://doi.org/ 10.1016/j.jmps.2017.10.020. [22] L. Yuan, R.C. Batra, Optimum first failure load design of one/two-core sandwich plates under blast loads, and their ultimate loads, Compos. Struct. 224 (2019), https://doi.org/ 10.1016/j.compstruct.2019.111022. [23] J.R. Xiao, B.A. Gama, J.W. Gillespie Jr., Progressive damage and delamination in plain weave S-2 glass/SC-15 composites under quasi-static punch-shear loading, Compos. Struct. 78 (2) (2007) 82–96. [24] S.A. Tekalur, A. Bogdanovich, A. Shukla, Shock loading response of sandwich panels with 3-D woven E-glass composite skins and stitched foam core, Compos. Sci. Technol. 69 (6) (2009) 736–753. [25] P. Kumar, J. LeBlanc, D.S. Stargel, A. Shukla, Effect of plate curvature on blast response of aluminium panels, Int. J. Impact Eng. 46 (2012) 74–85. [26] P. Kumar, D.S. Stargel, A. Shukla, Effect of plate curvature on blast response of carbon composite panels, Compos. Struct. 99 (2013) 19–30. [27] Hibbitt, Karlsson & Sorensen Inc, ABAQUS Version 6.11, 2011, Dassault System, USA.

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[28] V.A. Phadnis, P. Kumar, A. Shukla, A. Roy, V.V. Silberschmidt, Optimising curvature of carbon fibre-reinforced polymer composite panel for improved blast resistance: finiteelement analysis, Mater. Des. 57 (2014) 719–727. [29] V.A. Phadnis, F. Makhdum, A. Roy, V.V. Silberschmidt, Drilling in CFRP composites: experimental investigations and finite element implementation, Compos. Part A: Appl. Sci. Eng. 47 (2013) 41–51. [30] I.M. Daniel, B.T. Werner, J.S. Fenner, Strain-rate-dependent failure criteria for composites, Compos. Sci. Technol. 71 (3) (2011) 357–364. [31] Z. Hashin, Failure criteria for unidirectional fibre composites, J. Appl. Mech. 47 (1980) 321–329. [32] M.J. Hinton, A.S. Kaddour, P.D. Soden, A comparison of the predictive capabilities of current failure theories for composite laminates, judged against experimental evidence, Compos. Sci. Technol. 62 (12 13) (2002) 1725–1797. [33] C.R. Dandekar, Y.C. Shin, Modelling of machining of composite materials: a review, Int. J. Mach. Tools Manuf. 57 (2012) 102–121. [34] A. Puck, H. Sch€urmann, Failure analysis of FRP laminates by means of physically based phenomenological models, Compos. Sci. Technol. 58 (7) (1998) 1045–1067. [35] R.O. Ochola, K. Marcus, G.N. Nurick, T. Franz, Mechanical behaviour of glass and carbon fibre reinforced composites at varying strain rates, Compos. Struct. 63 (3) (2004) 455–467. [36] L. Raimondo, L. Iannucci, P. Robinson, P.T. Curtis, Modelling of strain rate effects on matrix dominated elastic and failure properties of unidirectional fibre-reinforced polymer–matrix composites, Compos. Sci. Technol. 72 (7) (2012) 819–827. [37] H. Koerber, J. Xavier, P. Camanho, High strain rate characterisation of unidirectional carbon-epoxy IM7-8552 in transverse compression and in-plane shear using digital image correlation, Mech. Mater. 42 (11) (2010) 1004–1019. [38] N.K. Naik, K.V. Rao, C. Veerraju, G. Ravikumar, Stress–strain behaviour of composites under high strain rate compression along thickness direction: effect of loading condition, Mater. Des. 31 (1) (2010) 396–401. [39] A. Turon, P.P. Camanho, J. Costa, J. Renart, Accurate simulation of delamination growth under mixed-mode loading using cohesive elements: definition of interlaminar strengths and elastic stiffness, Compos. Struct. 92 (8) (2010) 1857–1864. [40] N. Naik, A. Doshi, Ballistic-impact behaviour of thick composites: parametric studies, Compos. Struct. 82 (2008) 447–464. [41] K.S. Pandya, J.R. Pothnis, G. Ravikumar, N.K. Naik, Ballistic-impact behaviour of hybrid composites, Mater. Des. 44 (2013) 128–135. [42] M.M. Abedi, R.J. Nedoushan, W.-R. Yu, Enhanced compressive and energy absorption properties of braided lattice and polyurethane foam hybrid composites, Int. J. Mech. Sci. 207 (2021), https://doi.org/10.1016/j.ijmecsci.2021.106627. [43] L. Peng, M.T. Tan, X. Zhang, et al., Investigations of the ballistic response of hybrid composite laminated structures, Compos. Struct. (2021), https://doi.org/10.1016/j. compstruct.2021.115019. [44] S. Pinho, L. Iannucci, P. Robinson, Formulation and implementation of decohesion elements in an explicit finite element code, Compos. A: Appl. Sci. Manuf. 37 (5) (2006) 778–789. [45] M. Silva, Numerical simulation of ballistic-impact on composite laminates, Int. J. Impact Eng. 31 (2005) 289–306. [46] H. Ullah, A.R. Harland, V.V. Silberschmidt, Damage and fracture in carbon-fabric reinforced composites under impact bending, Compos. Struct. 101 (2013) 144–156.

The dynamic-loading response of carbon-fibre-filled polymer composites

8

D.M. Dattelbauma and J.D. Coeb Shock and Detonation Physics, Los Alamos National Laboratory, Los Alamos, NM, United States, bTheoretical Division, Los Alamos National Laboratory, Los Alamos, NM, United States

a

8.1

Introduction

8.1.1 Applications of carbon-fibre composites and dynamicloading conditions Carbon-fibre-loaded polymer composites have been used extensively in the automotive and aerospace industries, because their strength and fatigue properties per unit weight and production cost are superior to those of most metals and alloys [1]. This class of composites is now pervasive in a wide range of aircraft components, such as aircraft fuselages, automotive structural components, and sporting equipment (skis, bicycle components, etc.). Their mechanical response varies widely and can thus be tailored, with compositional features such as fibre content, degree of porosity, epoxy binder type, and polymer–filler adhesion, in addition to other microstructural details dependent on the fabrication method [2–5]. While epoxy binders are more common, phenolic binders are also attractive due to their dimensional stability as thermosetting resins [4–6]. For example, the most commonly encountered class of structural composites is comprised of
50–65% of chopped or filament-wound polyacrylonitrile (PAN)-based carbon fibres embedded in an Epon-type epoxy resin. Despite their widespread application, there are few reports detailing the impact (shock) and dynamic strength properties of carbon-fibre-filled composites, and most of the earlier works have focused on low shock pressures/low impact velocities (Table 8.1). In this chapter, we summarise the literature and our own recent experimental and theoretical results pertaining to the dynamic (shock) response of carbon-fibre-filled polymer composites. We will focus primarily on a summary of the shock response of  65% carbon-filled-polymer composites. In these materials, a high weight percentage of reinforcing carbon particles provides dimensional stability and mechanical strength, and the polymer binders included both Epon-based epoxies and other resins (phenolic, cyanate ester, modified epoxies). Phenyl groups incorporated into the backbone of resin molecules improve their thermal stability, and feature in both phenolic and cyanate ester resins. Features of the dynamic compressibility of several types of Dynamic Deformation, Damage and Fracture in Composite Materials and Structures. https://doi.org/10.1016/B978-0-12-823979-7.00009-0 Copyright © 2023 Elsevier Ltd. All rights reserved.

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Table 8.1 Shock/impact conditions and strain rates encountered in applications of carbonfibre composites, and laboratory methods used to reproduce those conditions.

Application Automotive structural components Aerospace (aircraft components, missile bodies) Munitions applications (munitions cases, structural components)

Shock or impact conditions

Strain rates in application

Low pressures Up to a few GPa Intermediate to high pressures 1–50 GPa Intermediate to high pressures 1–50 GPa

101–105 103–109

105–109

Laboratory methods Split-Hopkinson/Kolsky pressure bar, single-stage gas guns Single- and two-stage gas guns, laser-driven flyers, direct laser shock Single- and two-stage gas guns, laser-driven flyers, direct laser shock

carbon-fibre-based composites will be described, including quantitative details regarding observed anisotropy in dynamic response, dynamic yield strength (spall), and shock-driven dissociation. Measurement and interpretation of response to dynamic loading is vital for accurate simulation of shock propagation, used in part to determine the possibility of damage or failure. Many composites are fabricated based on filament-winding production methods, leading to highly anisotropic acoustic (and hence, shock-wave compression) properties. Existing Hugoniot data for high fillpercentage carbon-fibre–polymer composites have been limited largely to low shock stresses and impact velocities. In our own work, impact velocities were chosen to interrogate a variety of properties, from dynamic compressibility and anisotropy at low shock stresses to shock-driven transformations at high stresses. The focus of our research has been in the description of the hydrodynamic response of the composites, and thus focuses on the equation of state (EOS). The experimental results detailed below provide the broadest range of shock-wave compression data on chopped carbon-fibre composites reported to date. In addition to the thermodynamic response of these materials under impact conditions, we present particle-velocity wave profiles measured dynamically in situ. These provide additional information on the kinetics of chemical transformation under shock loading, and are (to our knowledge) the first profiles reported for polymeric materials within the mixed-phase region for solid polymers and composites.

8.1.2 Shock-wave compression concepts Shock waves produce discontinuous changes in material properties. As in any other context, however, such changes remain subject to conservation of mass, momentum, and energy. In addition to standard thermodynamic concepts such as pressure (P),

The dynamic-loading response of carbon-fibre-filled polymer composites

197

volume (V), density (ρ ≡ 1/V), and energy (E), we will consider shock velocity (U) and material (commonly referred to as ‘particle’) velocity (u) relative to some stationary reference frame. Properties of the unshocked material will be designated by a subscripted ‘0’, and those of the shocked material by a subscripted ‘H’. Specific (per mass) units will be employed throughout. For zero initial particle velocity,a the conservation laws are: V0 U , ¼ VH U  u PH ¼ P 0 +

Uu , V0

(8.1)

(8.2)

and
 1 2 Pu ¼ ρ0 U EH + u  E0 : 2

(8.3)

These are also known as the Rankine–Hugoniot relations. If the initial state is known, then 1 to 3 constitute a system of three equations in five unknowns. Substitution and elimination permits direct relation of any three variables, one of the most common (and useful) being elimination of the velocities from 3: 1 E ¼ E0 + PH ðVH  V0 Þ: 2

(8.4)

Perhaps more importantly, measurement of any two of the five quantities fully determines the state of the system. The locus of states accessible upon single-shock loading from a fixed origin is known as a Hugoniot. If that origin happens to be the ambient state, it is known as the principal Hugoniot. It is important to note that Hugoniots are not thermodynamic paths: one does not smoothly evolve a system along a Hugoniot as one would an isotherm in a diamond anvil cell experiment (for instance). Each point of the locus represents a discrete transition from the origin. The path actually followed in each of these transitions is the straight line linking the origin with the shocked state in the P  V plane, known as the Rayleigh line. It is found by eliminating u from Eqs (8.1), (8.2):
2 P H  P0 U ¼ V0  V H V0 a

(8.5)

Zero initial particle velocity relative to the laboratory frame is fairly typical in experimental contexts. It is also common to simplify the algebra by assuming zero initial pressure. While obviously an approximation, it is a very good one for most shock-wave experiments (pH
104 atm or greater) performed at or near ambient pressure (P
1 atm).

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The condition for shock stability is that the slope of this line be greater than that of the isentrope passing through the shocked state:


∂P  ∂V

 > SH

PH  P 0 V 0  VH

(8.6)

Pressure and particle velocity must be conserved across a shock interface. This fact gives rise to the very important interpretive procedure known as impedance matching.

8.1.3 Impedance matching States along the principal Hugoniot are readily determined using experimental techniques based on explosively driven, gas-gun-driven, or laser-based shock-wave compression [7]. The most common methods for the determination of the Hugoniot locus is by either shock-wave transmission (T) or reverse ballistic (front surface impact, FSI) experimental configurations. In these configurations, measurement of shock and/or particle velocities by optical velocimetry or wave arrival measurements (by pins, shorting wires, or optical velocimetry at interfaces) is used with the Rankine–Hugoniot relations Eqs (8.1)–(8.3) to determine P, ρ, Us, up, and specific internal energy, E. A simple experimental configuration for determining principal Hugoniot states is the ‘front surface impact’ (FSI) or reverse ballistic configuration in which a sample of interest is impacted into a window material, such as an oriented [100] single-crystal LiF, using smooth-bore gas or powder light gas guns. The particle velocity (uint) at the sample–window interface is measured using optical velocimetry techniques such as VISAR (velocity interferometry system for any reflector) [8] or PDV (photonic Doppler velocimetry) [9]. The measured velocities of a sample at the sample-LiF interface (uint) can then be used in concert with the measured projectile velocities (upr) to obtain final (P, up) states in the sample. For the composite experiments described below, we calculated final pressures in the initially unshocked (P0  0) LiF by substituting its Hugoniot (ρ0 ¼ 2.64 g/cm3, c0,LiF ¼ 5.15 mm/μs, sLiF ¼ 1.35) into the Rankine–Hugoniot relation for conservation of momentum, and then equating its particle velocity with that of the interface: PLiF ¼ ρ0,LiF Us,LiF up,LiF   ¼ ρ0, LiF c0,LiF + sLiF up,LiF up,LiF ¼ ρ0,LiF c0,LiF uint + ρ0,LiF sLiF u2int

:

(8.7)

Pressures in the composite were determined by impedance matching to the LiF: PLiF ¼ Pcomp ¼ P,

(8.8)

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199

and specification of the final shocked states was complete upon relating composite particle velocities to those measured for the projectile and at the interface: upr  uint ¼ up,comp ¼ up :

(8.9)

From this (P, up) combination, shock velocities Us were found by additional application of Eq. (8.7) to the composite only, and specific volumes V1 (or densities, ρ1) by conservation of mass: up V1 ρ 0 ¼ ¼1 : V0 ρ 1 Us

(8.10)

In Eq. (8.10) and for the remainder of the text, all quantities lacking a material designation are those of the sample (e.g. a binder or composite here). Similarly, impedance matching can be applied in a shock-transmission geometry, in which the shock velocity through a composite sample is determined by the measurement of the shock transit time (t) using optical velocimetry or other means (pins, shorting wires) to measure shock arrival times at the shock-wave input and exit interfaces, and prior measurement of the sample thickness. In practice, the experimental sample is often mounted on an impact baseplate or ‘driveplate’, which is impacted by the impactor/flyer plate, and simplifying measurement of shock-wave input. The flyer plate or impactor can be launched via explosively driven techniques or gas-gun-driven plate impact into the baseplate providing a well-defined shock-input condition. Typically, the impact condition is symmetric, e.g. the impactor and baseplate are the same material, and the impactor and baseplate are typically made from an EOS standard in which the Hugoniot is well characterised, to simplify the impedance-matching condition. In the experiments from our laboratory described below, a symmetric impact condition was created by launching an oxygen-free high-conductivity copper (OFHC-Cu) impactor into an OFHC-Cu baseplate using one of two two-stage light gas guns available at Los Alamos National Laboratory. The measured shock velocities were used in combination with measured projectile velocities and the OFHC-Cu Hugoniot (ρ0 ¼ 8.93 g/cm3, c0 ¼ 3.94 mm/μs, s ¼ 1.489) [10] to calculate up in the samples via the impedance-matching procedure described above. Shocked states in the sample were found from the intersection of its Rayleigh line (m ¼ ρ0Us) with the Hugoniot of the Cu projectile centred at the projectile velocity (u0 ¼ upr), assuming coincidence of the Cu Hugoniot and isentrope in this regime. The remaining Rankine–Hugoniot variables (ρ1, P1, and e1) followed from the conservation relations Eqs (8.7), (8.10).

8.1.4 General features of polymers and composites under shock-wave loading Extensive measurements of the shock adiabats of polymers and polymer composites were performed in the United States and Russia from the 1960s to the 1980s, motivated by the development of nuclear weapons [11,12]. The wealth of data produced

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

during that time still stands today as a benchmark of the shock properties of polymers over the large range of shock input conditions that were accessible by explosively driven flyer plate, and direct explosive-driven experimentation. Table 8.2 lists linear Rankine–Hugoniot relationships, Us ¼ c0 + sup, for common polymers used in engineering applications, including binders found in carbon-fibre composites, and related structural components. The summary in Table 8.2 is taken from experimental measurements from Los Alamos Scientific Laboratory (LASL) and summarised in both the LASL Shock Hugoniot Data compendium and a LASL report [11,13]. For many solid materials, in the absence of phase transitions and above the material’s elastic limit, shock Hugoniot data are linear in the shock velocity (Us)–particlevelocity (up) plane. In the Us–up plane, extrapolation of a linear Rankine–Hugoniot fit to up ¼ 0 ¼ c0, is often equivalent to the bulk sound velocity at ambient conditions for metals. However, three characteristic ‘features’ of the shock-wave compression response of polymers are worth noting. Firstly, for polymers, the linear extrapolation of the linear Rankine–Hugoniot fit to the y-intercept c0 always overestimates the bulk sound velocity by
200 to 500 m/s. Similar behaviour is observed in porous materials, such as foams and powders, porous composites such as plastic-bonded explosives (PBXs), and liquids, and is due to preferential compaction of free, network, and porous volume at low shock pressures. The low-pressure compaction phenomena give rise to increased shock heating compared with solid-density materials due to P △ V work (which can be estimated from the integral under the principal Hugoniot in the P–V plane). The difference between the bulk sound velocity and c0 from the linear Rankine–Hugoniot fit is illustrated in Fig. 8.1 for polyethylene. Furthermore, the bulk sound velocity is related to the ambient condition isentropic bulk modulus, Bs, by pffiffiffiffi cb ¼ ρBs . Inspection of bulk sound velocities in Table 8.2 provides a relative compar0 ison of the bulk compressibility of common polymers. The difference between c0,a, the intercept of the linear Rankine–Hugoniot fit of experimental Hugoniot data at low pressure, and cb further illustrates the relative contribution of free or network volume to the low-pressure shock compaction of the polymers. Secondly, at intermediate strain rates, such as those encountered in Hopkinson-bar or single-stage gas gun impact conditions, many polymers have been shown to exhibit viscoelastic responses [14]. Under shock loading, the viscoelastic response is indicated by rounding in the front of measured particle-velocity wave profiles, as described by Schuler and Nunziato, and more recently Clements [14]. Under lowvelocity impact conditions, polymers are first shocked to an ‘instantaneous’ state, which is followed by relaxation over 10s of ns to an equilibrium state. This relaxation is associated with an evolution to a softer, or more compressible response on the principal Hugoniot. Viscoelasticity in the wave profiles has been observed by our laboratory in shock-loaded PMMA, epoxy, PTFE, Kel-F 800, Kel-F 81, EPDM rubbers, and PDMS by the in situ measurement of particle-velocity wave profiles. The effect is also generally overdriven by
4 GPa, where there is convergence of the instantaneous and equilibrium Hugoniots. Note that the rounding in the profiles and increase in particle velocity with relaxation are expected to be observed in in situ measurements of the wave profiles, as well as measurements at a rear window

Table 8.2 Summary of Hugoniot data for selected polymers from LASL as summarised by Carter and Marsh. Polymer

ρ0 (g/cm3)

cb (km/s)

c0,a (mm/μs)

sa

c0,b (mm/μs)

sb

Pt (GPa), (Δ V/V)t (%)

Epoxy (Epon 828) Polymethyl methacrylate (PMMA) Polytetrafluoroethylene Polyethylene (linear) Polycarbonate Phenolic Polysulphone Polyvinylidene fluoride Polyurethane

1.192 1.186 2.151 0.954 1.196 1.385 1.235 1.767 1.265

2.264 2.227 1.139 2.166 1.933 2.442 1.976 1.853 2.068

2.69 2.59 1.68 2.86 2.33 2.98 2.35 2.58 2.54

1.51 1.52 1.79 1.57 1.57 1.39 1.55 1.58 1.57

2.88 2.90 2.08 3.27 2.06 2.05 1.58 2.98 2.25

1.35 1.33 1.62 1.43 1.39 1.55 1.51 1.39 1.47

23.1, 26.2, 41.6, 24.7, 20.0, 23.2, 18.5, 31.7, 21.7,

3.9 3.4 1.1 0.4 11.4 6.7 12.9 1.2 7.3

Initial densities and measured bulk sound velocities (cb) are also given. The linear Rankine–Hugoniot relationships, Us ¼ c0 + sup, from fitting experimental Hugoniot data are summarised below (a) and above (b) the threshold for shock-driven dissociation. Also listed are the approximate pressures and volume changes (cusp condition) associated with decomposition or reactant-to-product transition (t).

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

Polystyrene Polyethylene (Us + 1)

Shock velocity, Us (mm/µS)

14 12

n

10

cb = 2.17 km/s cint= 3.0 km/s

8 6 4

n

2

DV ~ 12%

0 0

1

2

3

4

5

Particle velocity, up (mm/Ps) Fig. 8.1 Shock Hugoniot data for polyethylene (red, offset byUs + 1) and polysulphone (blue) shown in the Us–up plane, illustrating two of the characteristics of polymers under shock-wave compression. At low pressures/particle velocities, there is preferential compression of network and free volume, resulting in an overestimate of the bulk sound speed by the intercept, c0, of the linear Rankine–Hugoniot fit to the Hugoniot data in this plane. The intercept and bulk sound velocities for polyethylene are given in the figure and in Table 8.2. The data in the figure are from S.P. Marsh (Ed.), LASL Shock Hugoniot Data. University of California Press, 1980. W.J. Carter, S.P. Marsh, Report No. LA-13006-MS (1995).

following shock propagation through a polymer sample. In front surface impact experiments, a decrease in particle velocity is expected to be observed with relaxation. Indeed, we have observed a measurable decrease in particle velocity at a polymer–LiF interface using VISAR in front surface impact experiments of a silica- and quartzfilled polydimethylsiloxane material at shock pressures up to 4 GPa. The last feature worth noting that pertains to the shock-wave compression response of polymers is their shock-driven chemical dissociation. Carter and Marsh (LASL) found, by studying a large number of polymers over a wide shock-pressure range, that polymers almost universally undergo a high-pressure transition marked by a change in slope in the Us  up plane between up
2 and 3 mm/μs associated with a ‘cusp’ in the principal Hugoniot at high pressures (typically between 20 and 30 GPa) [13]. The volume change associated with the transition is related to the polymer chain structure, with more extended backbone and side-chain groups resulting in larger volume collapses (>10%) across the transition. Shock Hugoniot data for polyethylene (red, offset by Us + 1) and polysulphone (blue) shown in the Us  up plane in Fig. 8.1 illustrate the ‘cusp’ in the Hugoniot above up
2 km/s. The volume change in polysulphone is large (
12%) and occurs above a shock input pressure of
18.5 GPa. For polyethylene, the volume change and magnitude of the nonlinearity is small, but still

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203

distinguished by a linear Rankine–Hugoniot fit to the low-pressure shock data below up ¼ 2 km/s. Table 8.2 gives the linear Rankine–Hugoniot relations from fits to experimental Hugoniot data below and above the change in slope in the Us  up plane for several polymers. The nonlinearity or ‘cusp’ is associated with a densification on the principal Hugoniot and is purported to be due to shock-driven dissociation and decomposition of polymers into small molecular products and carbon, similar to high explosives. From Table 8.2, it can be seen that the transition pressure is
20 to 30 GPa for most polymers. Furthermore, the magnitude of the volume change was found to be dependent on the backbone structure of the polymer, with larger backbone moieties giving rise to a larger volume decrease with the transition. For example, volume changes exceeding 10% were observed in polysulphone and polycarbonate, which both have benzene ring structures in the backbone chain. While the volume change is small in linear, non-cross-linked polymer networks, such as polyethylene, the transition is still measurable, even with only a few percent change in volume. In the case of densification reaction(s), shock inputs above the cusp condition will produce a first wave connecting the initial (P0,V0) to the first-wave state (P1,V1), and the input (P1) wave will slow down with propagation distance as it evolves [15]. A second wave then transforms the material to a higher-density state(s), with a rise time related to the transformation kinetics [15]. The rise time is related to the global shock-driven chemical dissociation rate. As shock stress is increased, the first-wave condition eventually becomes overdriven, and a single wave will once again be observed. Even then, particle-velocity profiles may reflect the timescale of transformation: for example, sluggish reactions may lead to measurable rounding in the wave front. Recently, we measured the first-wave profiles for carbon-fibre composites across the transition regime, and measured a two-wave structure, as indicated by a rounded wave structure, between 29 and 40 GPa [6]. At 29.1 GPa, the rise time of the second wave suggests that chemical reactions transform the composite to higher-density products over a period of roughly 45 ns. Such a rapid decomposition is perhaps unsurprising given our estimated shock temperature of roughly 1500 K at this condition. The global reaction rates for the dissociation/decomposition of polymers on the principal Hugoniot are similar to those measured in high explosives and simple molecules. Support of the densification transition being due to shock-driven decomposition or a reactant-to-product transition similar to high explosives comes from shock-wave recovery experiments. Morris et al. subjected polyethylene (PE) to a steady-state Mach compression disc and found that the polymer had irreversibly dissociated into amorphous carbon solid products [16]. Morris et al. also performed shock-recovery experiments on polytetrafluoroethylene (PTFE) to aid in the interpretation of the Hugoniot, and specifically the presence of cusps or nonlinearities at high pressures [17]. In the experimental apparatus, a PTFE cylinder was confined within a steel tube and a Mach disc was introduced into the centre of the sample. Recovery of the PTFE cylinder revealed a void in the centre of the cylinder filled with amorphous carbon, similar to PE. Analysis of the gas-phase products formed in the experiment found that CF4, C2F6, and other perfluorinated species were formed. More recently, real-time time-of-flight mass spectrometry experiments on polymers such as polydimethylsiloxane indicated that shock-wave-driven chain scission to form oligomeric species occurs in a similar

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pressure regime [18]. Persistent radicals have also been detected in shock-loaded and recovered samples of PMMA, Kapton, and Vespel, another indicator of shock-driven chain breaking [19]. Other works have examined changes in electrical conductivity of polymers under shock-wave loading, which may be related to the onset of chemical reaction. Measurements of the electrical conductivity of polymers under shock loading date to the late 1940s with work on PMMA [20]. Conductivity under both dynamic and static high pressures has been measured for PTFE, PE, polyvinylchloride, polyvinyl acetate, polychlorotrifluoroethylene, and polyethylene terephthalate. Graham and others have shown that several polymers exhibit shock-induced conductivity and shock-induced polarisation [19–22]. Shock-induced polarisation has been observed in polymers such as polymethyl methacrylate, polystyrene, nylon, epoxy, polyethylene, PTFE, polyvinyl chloride, and poly(pyromellitimide) (PPMI, or Vespel SP-1). Graham identified three regimes of shock-driven conductivity: an onset compression, a region of strong polarisation generation in which the polarisation ranges over three orders of magnitude, and a saturation region [22]. Graham also found that polymers with complex backbone structures exhibited the strongest shock-induced polarisation, and suggested that this may be due to an increased likelihood of shock-induced bond scission. There are a number of remaining scientific questions pertaining to the shock responses of polymers and polymer composites. These include dynamic strength and yielding phenomena, effects of shock loading on long-chain structures and crystalline versus amorphous domains, and the nature and extent of shock-driven chemical reactions on the principal Hugoniot and along other dynamic-loading pathways (offHugoniot, shock-ramp, isentropic compression). The effects of shock-driven dissociation on release pathways, and the pressure, temperature, and rate dependence of shock-driven chemical reaction pathways and final products have not been investigated to date. Similar to plastic-bonded explosives, the nature of shock-induced energy localisation in highly filled polymeric composites has also been difficult to discern, though energy localisation is expected to affect the bulk properties such as softening, yielding, dynamic strength, and the onset and extent of shock-driven chemical reactions.

8.2

Materials

In addition to summarising literature reports of the dynamic shock-wave response of carbon-fibre-filled polymer composites, we will describe experiments performed within our laboratory on three types of carbon-fibre-filled polymeric composites: a filament-wound carbon-fibre–epoxy composite with a high degree (16%) of porosity, a chopped carbon-fibre–phenolic composite, and a chopped carbon-fibre–cyanate ester composite. The compositions and densities of the three materials studied by our laboratory are given in Table 8.3. Shock Hugoniot data will be presented for the CFE in both through-thickness and in-fibre (0°) directions. For the chopped carbon-fibre composites, orientation of the fibres was moderate, and limited data are presented in the two orientations. All three composites contain carbon fibres based

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205

Table 8.3 Description of carbon-fibre-filled composites studied by our laboratory. Sample

Initial density (g/cm3)

Composition (by wt%)

Carbon-fibre–epoxy (CFE)

1.536 (Skeletal)

Carbon–phenolic (CP)

1.314 (Bulk) 1.555

Carbon–cyanate ester (CE)

1.555–1.556

60%–64% HexTow carbon fibres 36%–40% 55A epoxy 56.37% Carbon fibres 7.75% Graphite powder 35.0% Phenolic resin 10 GPa), as the dynamic yield strength in the fibre direction is overdriven. New plate impact experiments have recently been performed to quantify the Hugoniot-based equations-of-state for carbon-filled epoxy composites, using modern diagnostics to measure in situ and interface stress and particle-velocity wave profiles. In all of the studies, filament-wound carbon-fibre–epoxy composites were found to be highly anisotropic in their shock-compression responses, as expected based on the anisotropy in their ambient condition sound velocities, and other properties. In our work, the CFE studied was highly porous and contained a more ductile epoxy than related carbon-fibre–epoxy composites. These microstructural and chemical differences are manifested in its shock-compression response. Some notable features of the dynamic compression response of the CFE materials are that the shock compressibility of the composites in the transverse or through-thickness direction is similar to

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227

Fig. 8.17 Measured particle-velocity wave profiles from Alexander et al. [38] for experiments performed in the longitudinal (A) (in fibre, 0°) and transverse (B) directions. Note the quasielastic precursor wave observed in the fibre direction. Reprinted with permission from S. Alexander, C.T. Key, S.C. Schumaker, J. Appl. Phys. 114 (2013) 223515.

that of the epoxy resin. In the fibre direction, wave dispersion is observed with a quasielastic precursor propagating down the highly oriented fibres at velocities in excess of the longitudinal sound velocity in this orientation (for example, at velocities in excess of 10 km/s in our experiments). From elastic–plastic wave analysis in this orientation, the Hugoniot elastic limit (HEL) is estimated to be 3–4 GPa. Furthermore, in two different materials (Dattelbaum and Alexander), the unreacted Hugoniots of the CFE in the two orientations were found to converge above 10 GPa. The unreacted Hugoniot data for this class of composites from three different laboratories on three different composites compare well with one another. The reported Hugoniot data of a CFE with a lower initial density (ρ0 ¼ 1.50 g/cm3) linear Rankine–Hugoniot fit to up
0.9 km/s were reported to be Us ¼ 3.23 + 0.92up [34]. This linear fit (extrapolated to greater up) is shown in Fig. 8.13 overlaid with the

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

Fig. 8.18 Shock Hugoniot data for 62% and 68% carbon fibre CFE in the longitudinal (in fibre or 0°) and transverse (TT) directions shown in the shock stress versus particle-velocity plane from Alexander et al. [38] Note the HEL in the fibre direction near 3–4 GPa. The shock response of the composite in the TT direction is similar to that of the epoxy resin. Reprinted with permission from S. Alexander, C.T. Key, S.C. Schumaker, J. Appl. Phys. 114 (2013) 223515.

Hugoniot data of the CFE composite studied by our laboratory. Overall, the two nearfull density Hugoniots in the TT direction are quite similar, with our recent work extending the shock input pressure regime interrogated. The greater slope of the Rankine–Hugoniot fit for the composite studied by our laboratory may indicate that the pressure dependence of the bulk modulus is greater, e.g. it is more compressible. This may be a manifestation of the greater ductility of the 55A resin compared with more common epoxy resins, such as the Epon class of resins or, presumably, the Hexcel epoxy used in Millett’s formulation [34]. Preliminary experiments in our laboratory have also found that the spall strength of CFEs is weak (5 GPa. In the fibre direction, a Hugoniot elastic limit is observed near 3 GPa for two different CFEs.

Carbon-fibre–phenolic and carbon-fibre–cyanate ester composites Carbon-fibre–phenolic composites have featured prominently in aerospace applications. The dynamic response of a tape-wrapped carbon-fibre composite (TWCP) with an initial density of ρ0 ¼ 1.46 g/cm3 was previously studied by Wood et al. [4] The TWCP consisted of a carbon fibre weave impregnated with a phenolic resin, Durite SC-1008. Experimental target assemblies were constructed such that the shock

The dynamic-loading response of carbon-fibre-filled polymer composites

229

propagation direction was either parallel or perpendicular to the fibre weave direction. Using a 50-mm-diameter launch tube single-stage gas gun, impact velocities between 200 and 1000 m/s were achieved. Manganin gauges were used at multiple surfaces to measure the shock wave (in stress), and impedance matching using the impactor and impact condition was used to determine the remaining Hugoniot variables. In the experimental gauge records, oscillations were observed, which were purported to arise from shock reverberations within the carbon-fibre weave. Ten experiments with impact or flyer velocities ranging from 200 to 1000 m/s were used to define the principal Hugoniot. Note that the flyer in the experiments was either Al or Cu. The linear fit to the Hugoniot data in the Us  up plane for the TWCP is Us ¼ 3.69 + 0.59up. Note this can be compared with the Hugoniot-based EOS for the neat phenolic (SC-1008) resin reported by the same group: Us ¼ 2.49 + 3.79up. However, the deviation between the composite and resin was only found to exist below up < 0.9 mm/μs, and the two converged at greater particle velocities/pressures. The origin of the difference between the matrix and the composite at low shock stresses was due to curvature of the Hugoniot in the Us  up plane. Furthermore, the authors report a Hugoniot elastic limit for the TWCP of 1.39  0.19 GPa, which is lower than that estimated for the CFE composites in the fibre direction (
3–4 GPa) [4]. Recently, we compared the dynamic (shock) responses of two composites consisting of a chopped carbon-fibre-filled phenolic (CP) and a chopped carbon-fibre-filled cyanate ester (CE) [6]. A notable difference between this work and earlier experimental results is that the shock stresses obtained in the experiments were well in excess of the expected decomposition threshold for the polymer binder on its principal Hugoniot, as well as the graphite-to-diamond transition associated with shocked carbon. A summary of measured Hugoniot states is provided in Table 8.7, and those of both composites are plotted alongside historical data for a neat (unfilled) phenolic resin in Figs 8.19 and 8.20. At the time of their reporting, the historical data were taken to indicate that the neat phenolic binder dissociated chemically at first-shock pressures above 23.2 GPa (V/V0 ¼ 0.609), resulting in a 6.7% volume collapse. A linear fit below the cusp in the resin data of Fig. 8.19 gives Us ¼ 2.987 + 1.387up, with a y-intercept (cb) that overestimates the ambient bulk sound velocity (cb ¼ 2.173 km/s) by over 700 m/s. This deviation is a reflection of Hugoniot curvature at low particle velocities and is a well-known feature of polymers, liquids, and related materials. Based on their shock loci, the TT dynamic compressibilities of CP and CE were qualitatively similar to one another and to that of neat phenolic resin below about 25 GPa. Near-coincidence of Hugoniots for carbon-fibre–epoxy composites and those of their polymer binder (epoxy) alone has been observed previously. Based on Table 8.7 and Fig. 8.19, the lowest up to deviate from nominally linear behaviour (outside the curved region near the origin) was accessed in shot 2s-456 and corresponded to V/V0 ¼ 0.599. This volumetric compression ratio matches well the estimated threshold in the resin (listed above), even though corresponding shock stresses and densities differed due to differences in ρ0. Volume changes at the cusp near up  2.5 km/s were particularly similar in the case of CE and the neat resin, both of which were smaller than in CP. This feature is slightly curious given that the neat resin is that of CP, not of CE. We outline some possible explanations for this in the sections below.

Table 8.7 Summary of measured Hugoniot states for carbon phenolic (CP) and cyanate ester (CE) materials obtained by gas-gun-driven plate impact experiments. Shot#

Material

ρ0 (g/cm3)

Exp’t type

Proj. vel. (km/s)

69ts-10-09 69ts-10-14 69ts-11-04 2s-423 2s-435 2s-456 2s-572 69ts-12-04 1s-1574 69ts-11-02 69ts-11-03 69ts-11-06 2s-571 69ts-12-05 2s-744 2s-745 1s-1572 1s-1573

CP CP CP CP CP CP CP CP CP CE CE CE CE CE CE CE in fibre CE in fibre CE elastic CE plastic

1.554  0.001 1.556  0.001 1.556  0.001 1.556  0.001 1.550  0.001 1.554  0.001 1.550  0.001 1.558  0.002 1.554  0.002 1.555  0.001 1.556  0.001 1.556  0.001 1.555  0.001 1.530  0.001 1.556  0.005 1.556  0.005 1.556  0.005 1.556  0.005

T T T FSI FSI T FSI T T T T T FSI T T T T T

3.686  0.005 4.919  0.005 4.567  0.005 2.962  0.005 3.340  0.003 3.176  0.019 1.993  0.002 4.631  0.015 0.654  0.001 4.002  0.005 4.870  0.007 3.500  0.005 1.985  0.002 4.574  0.010 1.536  0.006 1.533  0.008 0.765  0.003 0.799  0.003

Us (km/s)

up (km/s)

P (GPa)

V (cm3/g)

6.818  0.020 7.614  0.135 7.314  0.126 5.795  0.010 6.171  0.010 6.462  0.046 4.720  0.100 7.495  0.039 3.217  0.004 6.886  0.112 7.993  0.151 6.591  0.102 4.798  0.100 7.363  0.059 4.818  0.108 4.588  0.020 3.657  0.020 6.472  0.040 3.806  0.020

2.977  0.004 3.947  0.012 3.752  0.010 1.940  0.007 2.188  0.008 2.576  0.016 1.363  0.005 3.715  0.012 0.575  0.001 3.238  0.009 3.875  0.014 2.838  0.008 1.350  0.008 3.689  0.009 1.286  0.007 1.293  0.003 0.662  0.003 0.056  0.001 0.681  0.003

31.5  0.1 46.8  0.7 42.7  0.6 17.6  0.1 20.9  0.1 25.9  0.2 10.0  0.1 43.4  0.2 2.87  0.01 34.7  0.5 48.2  0.7 29.1  0.4 10.1  0.1 41.6  0.3 9.64  0.19 9.23  0.05 3.77  0.03 0.57  0.01 4.30  0.03

0.3625  0.0054 0.3095  0.0046 0.3159  0.0047 0.4275  0.0064 0.4164  0.0062 0.3870  0.0058 0.4589  0.0070 0.3237  0.0022 0.5285  0.0009 0.3407  0.0051 0.3311  0.0049 0.3659  0.0055 0.4621  0.0070 0.3262  0.0033 0.4713  0.0050 0.4617  0.0020 0.5263  0.0020 0.6373  0.0020 0.5325  0.0020

The experimental geometries are denoted by T ¼ transmission (shock transit time measured) or FSI ¼ front surface impact (interface particle velocity measured). Experimental uncertainties in initial density, projectile velocity, shock-wave arrival time (accounting for projectile tilt), and initial sample thickness were propagated through the impedance match equations and are generally 0, see Fig. 9.12C), plate deceleration commences at velocity ζ, and flexural wave propagation is neglected, hence ζ ¼ R. In each phase, an axisymmetric, polynomial deflection profile is imposed, accounting for both shear and bending deflections, and satisfying compatibility: "  "     #    # Rr 2 Rr 3 Rr Rr 2  wðr, tÞ ¼ wB0 ðtÞ 3 2 + wS0 ðtÞ 2 : ζ ðtÞ ζðtÞ ζ ð tÞ ζ ðtÞ

(9.5)

Here, wB0(t) and wS0(t) are the centre point deflections due to bending and shear deformations, respectively. Introducing Eq. (9.5) a priori reduces the dimension of the problem to three degrees of freedoms (DOFs) in phase 1, namely central deflections due to bending and shear, wB0(t) and wS0(t), respectively, and wave-front position ζ(t). In phase 2, the response is described by only two DOFs, namely wB0(t) and wS0(t), as ζ ¼ R. The plate is assumed to be made from a symmetric and balanced laminate comprising n transversely isotropic composite laminas stacked at arbitrary orientations φk (k ¼ 1, 2, …, n). Introducing a reference system in cylindrical coordinates, (r, φ, z), for the laminate, the

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265

relationship between in-plane forces Ni (per unit width) and the corresponding inplane strains εi can be written as follows: ðNr Nt Nrt ÞT ¼ A  ð εr εt γ rt ÞT

(9.6)

where A denotes the in-plane stiffness matrix of the laminate as dictated by classical laminate theory. Likewise, for the bending and twisting moments Mi (per unit width), we write: ð Mr Mt Mrt ÞT ¼ D  ð κr κt κ rt ÞT

(9.7)

with D the bending stiffness matrix of the laminate and κi the bending/twisting curvatures. Assuming axisymmetric deformation (Eq. 9.5) and assuming that radial and tangential displacements of material points are negligible, that is j uj ≪ j w j, j vj ≪ j w j, the in-plane strains, bending curvatures, and transverse shear deformations can be written as follows: εr ¼

  1 wB0 + wS0 2 u ; εt ¼  0; γ rt ¼ 0; ζ 2 r

(9.8)

κr ¼

∂2 wB 1 ∂wB ; κ rt ¼ 0; ; κt ¼ ∂r 2 r ∂r

(9.9)

γ rz ¼

∂wS ; γ tz ¼ 0: ∂r

(9.10)

Underwater blast loading is modelled by prescribing a pressure-versus-time history on the fluid–structure interface; at any time t > 0, the pressure distribution at this interface is given by: pf ðr, tÞ ¼ 2p0 exp ðt=θÞ  ρw cw w_ ðr, tÞ:

(9.11)

It follows from Eq. (9.11) that the interface pressure pf is strongly affected by the transverse velocity field of the plate, w_ ðr, tÞ, and can drop to the value of the cavitation pressure of the fluid, pc  0, below which the fluid cannot sustain any further tensile loading and can take up arbitrarily large strains. The occurrence of the cavitation phenomenon introduces nonlinearities in the fluid response and renders the full analytical treatment of three-dimensional underwater blast problems impossible. Therefore, we deduce the pressure history at the fluid– structure interface from our previous one-dimensional analytical FSI model [7], accounting for the emergence and propagation of cavitation BF and CF and for the partial reflection of pressure waves at such fronts. The equations of motion of the system can be obtained by employing the principle of conservation of linear and angular momentum, utilising Eqs (9.5)–(9.11), as detailed in Ref. [24]. This results in a system

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

of three nonlinear ordinary differential equations (ODEs) in phase 1 and two ODEs in phase 2, which are integrated numerically by imposing the appropriate initial conditions. Once the equations of motion are obtained, dimensional analysis is performed and such equations are written in nondimensional form. This allows identifying the governing nondimensional parameters for this problem. These are listed as follows: h ¼ h=R; R ¼

pffiffiffi R ρ μ ρ cw p0 ; μ ¼ pffiffiffiffiffiffi ; α ¼ pwffiffiffiffiffiffi ; p0 ¼ ðnoteψ ¼ α=μÞ: E θ E Eρ θ Eρ

(9.12)

Here, h and R are thickness and radius of the circular plate, E and ρ are Young’s modulusc and density of the linear-elastic plate material, μ ¼ ρh represents the areal mass of the plate, and p0 and θ are peak pressure and decay time of the exponentially decaying blast wave. Finite element simulations conducted on elastic plates of different size confirmed that the scaling in Eq. (9.12) is adequate. Assuming that the relevant properties of the plate material, E and ρ, are homogeneous and independent of size, and focussing on the attention onto the elastic response of the plates, the problem under investigation can be regarded as scale-independent. It follows that the blast response of large naval components can be measured at laboratory scale by employing a scaled-down experimental setup, with a set of nondimensional parameters (Eq. 9.12) identical to that of full-size structures. For the case of full-scale naval components exposed to the threat of an explosion in water, typical ranges of the nondimensional parameters (Eq. 9.12) are as follows: 0:04 < h < 0:8; 0:001 < R < 2:5; 0:003 < μ < 0:4; 0:04 < α < 0:4; 5  105 < p0 < 0:025:

(9.13)

In our studies, the choice of specimen geometry, materials, and loading parameters were such to allow scaling of our small-scale experiments to real blast scenarios.

9.4.2 Analytical predictions and optimal design maps A full assessment of the fidelity of the model is reported in Refs. [21, 24]. In Fig. 9.16, we show an example of such fidelity, comparing measurements to analytical and FE predictions of the centre deflection-versus-time histories of two different composite plates made from glass-fibre-reinforced polymer (GFRP) and a vinyl ester matrix. Clearly, the analytical model is able to capture the peak deflection and the time response of the plates. We note that FE simulations predict a slightly different deflection response; a thorough investigation of this phenomenon revealed that this is due to the fact that FE simulations capture higher-order deflection modes subsequent to the flexural wave reaching the plate centre, which are not modelled analytically for simplicity. We have shown in Ref. [25] that it is possible to capture such higher-order modes by employing different shape functions; on the other hand, such higher-order c

For quasiisotropic composite plates, E is the equivalent in-plane stiffness deduced from the A matrix.

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(B)

(A)

2.4

2.4 Analytical predictions

Analytical predictions 2

1.6 Experiment 1.2

0.8

FEM

Centre deflection w0 (mm)

Centre deflection w0 (mm)

2

Experiment

1.6

1.2

0.8 FEM 0.4

0.4

0

0 0

0.1

0.2

Time (ms)

0.3

0.4

0

0.1

0.2

0.3

0.4

Time (ms)

Fig. 9.16 Measured centre deflection-versus-time histories w0(t) for two selected experiments performed on GFRP plates: (A) p0 ¼ 9.0 MPa, θ ¼ 0.12 ms; (B) p0 ¼ 7.0 MPa, θ ¼ 0.14 ms; analytical and FE predictions are included for comparison. The diameter of the plates was 25 m.

modes seem to be suppressed in the observed response, and we argue that this is due to a combination of material damping and viscous response of the water in contact with the structure. The good accuracy of the analytical predictions is further illustrated in Fig. 9.17, which shows a comparison of analytical and FE predictions over a wide range of applied impulses and plate geometries, for a chosen set of material properties. The validated analytical model, in combination with the findings of our experimental campaigns, can be used to draw conclusions on the relative blast performance of different material systems. In Fig. 9.18, we report analytical predictions of the maximum ¼ wmax normalised centre deflection, wmax 0 0 =R, as a function of the peak pressure of the incident blast wave, p0, for a given decay time, θ; the responses of CFRP–epoxy and GFRP–vinyl ester plates of equivalent mass are compared. In the experiments, it was observed that CFRP plates failed catastrophically at blast pressures on the order of p0 ¼ 9 MPa and corresponding normalised deflections of approximately wmax ¼ 0:125; on the other hand, the GFRP plates resisted pressures of 11 MPa with0 out failure (and corresponding normalised peak deflection of 0.2). We conclude that while GFRP plates deflect substantially more than CFRP plates of equivalent mass when subjected to a given blast, their higher ductility allows GFRP plates to outperform CFRP counterparts in blast resistance: glass-fibre composites will resist higher pressures than carbon-fibre composites, for a given areal mass and diameter. Strictly, this conclusion only applies when the active failure mechanism of the plates is that observed in our experiments (i.e. tensile tearing of the plates at the supports), and a more comprehensive experimental study is necessary in order to observe

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

Fig. 9.17 Comparisons between analytical and FE predictions of the as maximum centre deflection wmax 0 functions of the blast impulse per unit area I0 ¼ 2p0θ.

0.12

Maximum central deflection w0 (m)

268

h = 10 mm θ = 0.15 ms

0.1

0.08

h/R = 0.02

0.06 h/R = 0.033

0.04

h/R = 0.05

0.02

h/R = 0.06 h/R = 0.1

0

4

0

4

1 10

2 10

4

3 10

Blast impulse I0 (Pa s)

0.25

GFRP h/R = 0.058 μ = 1.125 kg/m2

0.2

0.15 max

w0

CFRP failure 0.1

CFRP

0.05

h/R = 0.06 μ = 1.125 kg/m2 0 2

4

6

8

10

12

14

16

p (MPa) 0

Fig. 9.18 Analytical predictions of the normalised peak centre deflection wmax ¼ wmax 0 0 =R as a function of the peak shock-wave pressure p0 for CFRP and GFRP plates of equal areal mass, μ.

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Fig. 9.19 Design chart for circular isotropic plates subject to underwater blast loading with α ¼ 0:15 for a constrained normalised peak deflection of w0 ¼ 0:2; contours of nondimensional peak pressure p0 ¼ p0 =E (solid curves, underlined values) and areal pffiffiffiffiffiffi mass μ ¼ μ=ðθ EρÞ (dashed curves) are included. The path indicated by (h, R)max identifies designs that maximise the blast resistance, while the path indicated by (h, R)min denotes design of minimum blast resistance.

(h, R)

max

0.1

w = 0.2 0

0.09

0.007 0.006

Aspect ratio h

0.08

0.13

0.1

0.07 0.06

0.005

0.05 0.004

0.04 0.0033

0.003

0.03

(h, R)

min

0.0015

0.02

0.0025 0.002

0.067

0.0035 0.033

0.02

0.013

0.001

0.01 0

0.5

1

1.5

2

Nondimensional plate radius R

different failure modes (such as shear-off, delamination, etc.) by testing plates of substantially different aspect ratios. On the other hand, the information in Fig. 9.18 can be taken as a useful indication that GFRP might be preferable to CFRP for the construction of blast-resistant structures. Finally, we employ the analytical model in order to construct optimal design maps. An example of such a map is shown in Fig. 9.19. The map explores typical ranges of pffiffiffiffiffiffiffiffi plate aspect ratio h ¼ h=R and nondimensional radius R ¼ R ρ=E=θ, and is constructed with the constraint of a normalised peak deflection of w0 ¼ 0:2. The chart pffiffiffiffiffiffi includes contours of the nondimensional plate mass μ ¼ μ=ðθ EρÞ and blast pressure p0 ¼ p0 =E. We observe that the peak strain in the plate material scales with the nondimensional peak deflection; neglecting delamination, for a material which can safely sustain a normalised peak deflection of, for example w0 ¼ 0:2, the points in the chart correspond to structural designs that will respond elastically (therefore acceptable from a design point of view). The map includes a path, indicated by the full arrows and denoted by (h, R)max, which corresponds to designs of maximum blast resistance.d The chart in Fig. 9.19 allows designing plates of optimal blast resistance against the constraint of a given areal mass; similarly, it allows identifying designs of minimum mass against the constraint of a given blast resistance, which is very useful when selecting materials and geometries for composite plates that need to withstand a given peak pressure without failure. We note that this design chart is universal and allows the d

A similar path is also shown, indicated by the empty arrows and denoted by (h, R)min, which corresponds to designs of minimum blast resistance, for comparison.

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designer to conduct a preliminary, optimal dimensioning of blast-resistant composite plates without performing any calculation. Such designs can subsequently be verified by performing detailed FE analyses.

9.5

Conclusions

The elastic nature of fibre-reinforced polymer composites and the relatively high sonic speed of these materials allow composite structures to effectively resist dynamic loading, such as that experienced in blast and impact events. Slender composite structures perform particularly well when loaded by diffuse pressure, as in the case of blast in air or water. In the case of underwater blast, the dynamic structural response of composite plates is deeply influenced by FSI phenomena; motion of the plate reduces the pressure in the fluid, and the magnitude of such reduction is typically such that the pressure may decrease to 0, inducing water cavitation. BFs emerge from the point of first cavitation and propagate outwards at supersonic speed, expanding the region of cavitated water; such fronts can invert their direction of motion and become CFs, reducing the size of the cavitated region, if the pressure and velocity fields in the surrounding fluid are such to allow for it. The propagation of cavitation fronts is not independent from the structural response, as the latter affects the pressure and velocity field in the water; similarly, the motion of these fronts dramatically affects the structural response, as such fronts act as partially reflective interfaces for the pressure waves emanated by the structure. In order to understand this complex sequence of events, it is necessary to observe the phenomenon directly. In this chapter, we have described a recently developed apparatus that allowed, for the first time, simultaneous observation of structural motion and cavitation in the fluid during scaled-down underwater blast experiments; in addition, our apparatus allows investigating blast responses in deep water, allowing for initial water pressurisation prior to the blast loading. Such apparatus was used to investigate the underwater blast response of several structures, and the observations, with accompanying FE simulations, inspired the formulation of analytical predictive models that can be easily used in design. The main conclusions of our research are as follows: l

l

l

l

In loading of unsupported monolithic plates, cavitation always occurs at the fluid–structure interface; the presence of a support (either an elastic or plastic foundation, or a viscous support, like fluid at the back of the structure) locates the point of first cavitation at a finite distance from the structure, resulting in additional imparted impulse. The loading by underwater blast is less severe in deep water than in shallow water. Employing the sandwich construction results in a substantial reduction of the impulse compared to a monolithic structure of equivalent mass. Such reduction is more pronounced for water-backed structures than for air-backed structures, and again, the severity of blast loading on sandwich plates decreases with increasing water depth. Sandwich structures with a weaker core outperform those with strong cores. The underwater blast response of composite plates is governed by transient propagation of flexural waves; this induces a complex three-dimensional cavitation event, taking place at

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l

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the fluid–structure interface at the very early stages of the response; continued plate deflection results in a secondary cavitation event triggered at a finite distance from the structure; and such secondary cavitation evolves in a nearly one-dimensional manner. Both cavitation events affect substantially the imparted impulse to the composite plates and need to be captured by predictive models. Soft and relatively weak and heavy GFRP composite plates can outperform stiffer, lighter, and stronger CFRP structures in underwater blast events.

Acknowledgements We are grateful to professors Norman Fleck, Vikram Deshpande, Mike Ashby (University of Cambridge), John Hutchinson (Harvard), Tony Evans, Bob McMeeking (UCSB), and Haydn Wadley (University of Virginia) for their precious teaching and the inspiring conversations on the subject examined in this chapter.

References [1] R.H. Cole, Underwater Explosions, Princeton University Press, Princeton, NJ, USA, 1948. [2] M.M. Swisdak, Explosion Effects and Properties: Part II—Explosion Effects in Water, Dahlgren, VA, USA, Naval Surface Weapons Centre, 1978. [3] G.I. Taylor, The pressure and impulse of submarine explosion waves on plates, in: G.K. Batchelor (Ed.), The Scientific Papers of G.I. Taylor, vol. III, Cambridge University Press, Cambridge, UK, 1963, pp. 287–303. [4] E.H. Kennard, Cavitation in an elastic liquid, Phys. Rev. 63 (1943) 172–181. [5] V.S. Deshpande, N.A. Fleck, One-dimensional response of sandwich plates to underwater shock loading, J. Mech. Phys. Solids 53 (2005) 2347–2383. [6] Y. Liang, A.V. Spuskanyuk, S.E. Flores, D.R. Hayhurst, J.W. Hutchinson, R.M. McMeeking, A.G. Evans, The response of metallic sandwich panels to water blast, J. Appl. Mech. 71 (2007) 81–99. [7] A. Schiffer, V.L. Tagarielli, N. Petrinic, A.F.C. Cocks, The response of rigid plates to deep water blast: analytical models and finite element predictions, J. Appl. Mech. 79 (2012) 061014–061028. [8] V.S. Deshpande, A. Heaver, N.A. Fleck, An underwater shock simulator, Proc. R. Soc. A 462 (2006) 1021–1041. [9] H.D. Espinosa, S. Lee, N. Moldovan, A novel fluid structure interaction experiment to investigate deformation of structural elements subjected to impulsive loading, Exp. Mech. 46 (2006) 805–824. [10] F. Latourte, D. Gregoire, D. Zenkert, X. Wei, H.D. Espinosa, Failure mechanisms in composite panels subjected to underwater impulsive loads, J. Mech. Phys. Solids 59 (2011) 1623–1646. [11] L.F. Mori, S. Lee, Z.Y. Xue, A. Vaziri, D.T. Queheillalt, K.P. Dharmasena, H.N.G. Wadley, J.W. Hutchinson, H.D. Espinosa, Deformation and fracture modes of sandwich structures subjected to underwater impulsive loads, J. Mech. Mater. Struct. 2 (2007) 1981–2006. [12] L.F. Mori, D.T. Queheillalt, H.N.G. Wadley, H.D. Espinosa, Deformation and failure modes of I-core sandwich structures subjected to underwater impulsive loads, Exp. Mech. 49 (2009) 257–275.

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[13] J. LeBlanc, A. Shukla, Dynamic response and damage evolution in composite materials subjected to underwater explosive loading: an experimental and computation study, Compos. Struct. 92 (2010) 2421–2430. [14] H. Wadley, K. Dharmasena, Y. Chen, P. Dudt, D. Knight, R. Charette, K. Kiddy, Compressive response of multilayered pyramidal lattices during underwater shock loading, Int. J. Impact Eng. 35 (2008) 1102–1114. [15] A. Schiffer, V.L. Tagarielli, The response of rigid plates to blast in deep water: fluidstructure interaction experiments, Proc. R. Soc. A 468 (2012) 2807–2828. [16] S. Arezoo, V.L. Tagarielli, N. Petrinic, J.M. Reed, The mechanical response of Rohacell foams at different length scales, J. Mater. Sci. 46 (2011) 6863–6870. [17] S. Arezoo, V.L. Tagarielli, C.R. Siviour, N. Petrinic, Compressive deformation of Rohacell foams: effects of strain rate and temperature, Int. J. Impact Eng. 51 (2013) 50–57. [18] D.J. Korteweg, Uber die Fortpflanzungsgeschwindigkeit des Schalles in elastischen Rohren, Ann. Phys. 5 (1878) 525–542. [19] A. Schiffer, V.L. Tagarielli, One-dimensional response of sandwich plates to underwater blast: fluid-structure interaction experiments and simulations, Int. J. Impact Eng. 71 (2014) 34–49. [20] A. Schiffer, V.L. Tagarielli, The one-dimensional response of a water-filled double hull to underwater blast: experiments and simulations, Int. J. Impact Eng. 63 (2014) 177–187. [21] A. Schiffer, V.L. Tagarielli, The response of circular composite plates to underwater blast: experiments and modelling, J. Fluid Struct. 52 (2015) 130–144. [22] V.L. Tagarielli, V.S. Deshpande, N.A. Fleck, Prediction of the dynamic response of composite sandwich beams under shock loading, Int. J. Impact Eng. 37 (2010) 854–864. [23] V.L. Tagarielli, V.S. Deshpande, N.A. Fleck, The dynamic response of composite sandwich beams to transverse impact, Int. J. Solids Struct. 44 (2007) 2442–2457. [24] A. Schiffer, V.L. Tagarielli, The dynamic response of composite plates to underwater blast: theoretical and numerical modelling, Int. J. Impact Eng. 70 (2014) 1–13. [25] A. Schiffer, W.J. Cantwell, V.L. Tagarielli, An analytical model of the dynamic response of circular composite plates to high-velocity impact, Int. J. Impact Eng. 85 (2015) 67–82.

Dynamic loading on composite structures with fluid-structure interaction

10

Young W. Kwon Department of Mechanical & Aerospace Engineering, Naval Postgraduate School, Monterey, CA, United States

10.1

Introduction

Composite materials have been used more frequently for marine and offshore structures which are usually in contact with seawater. When a structure moves with water which may be located outside, inside, or both sides of the structure as sketched in Fig. 10.1, there is interaction between the fluid and the structure. If the deformation of the structure is not negligible, the fluid-structure interaction (FSI) becomes more significant. When the structure in contact with water is impacted or excited by loading, the FSI plays an important role [1–5]. This statement is particularly true for polymer composite structures because the densities of polymer composite materials are very comparable to that of water. As a result, the added mass effect resulting from FSI is critical to the polymer composite structures. This chapter presents both experimental and numerical studies of FSI with polymer composite structures. Both techniques complement each other. The study focused on impact loading as well as cyclic loading. In order to determine the effect of FSI, the numerical modelling and simulation as well as experimental tests were conducted in water and air, respectively. Then, their results were compared. Since natural frequencies and mode shapes play important roles in dynamic motion of composite structures, their characteristics were also compared in air and water, respectively.

(A)

structure

structure

water

water

(B)

structure

water

(C)

Fig. 10.1 Water locations relative to structure. (A) water outside structure, (B) water inside structure, and (C) water inside and outside structure. Dynamic Deformation, Damage and Fracture in Composite Materials and Structures. https://doi.org/10.1016/B978-0-12-823979-7.00011-9 Copyright © 2023 Elsevier Ltd. All rights reserved.

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10.2

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

Experimental study of impact on composite structures with FSI

Composite structures were fabricated and tested using an impact test machine inside an anechoic water tank in order to study the dynamic response and failure of the structures while they were in contact with water. In particular, the structural responses with and without water were compared to assess the effect of FSI on the structures. This section describes the experimental procedure for testing as well as the test results and discussion.

10.2.1 Description of experiment Composite panels were constructed for testing. Both laminated woven fabric composite panels and sandwich composite panels were fabricated using the vacuum assisted resin transfer moulding technique as shown in Fig. 10.2. The woven fabric materials were either glass or carbon fibres and the vinyl-ester resin. The core material was balsa wood. Their properties are listed in Table 10.1. Once the composite panels were fabricated, those were cut into the shapes of square plates and long rectangular strips. The square plates were clamped along all the boundaries, and the distance from one clamped edge to the opposite clamped edge was 0.3048 m such that the effective plate dimension was 0.3048 m by 0.3048 m. The thickness of the plate might be varied. Fig. 10.2 Resin flow during vacuum assisted resin transfer moulding process.

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Table 10.1 Material properties of composites. Material

Elastic modulus

Density (kg/m3)

E-glass fibre woven fabric composite Carbon fibre woven fabric composite Balsa wood

17.24 GPa 52.41 GPa 4.10 MPa

2020 1300 155

On the other hand, the long strips were 25.4 mm wide and clamped at two ends with the distance from the two clamped ends of 0.3048 m. In order to apply impact loading inside water, an impact test machine was designed and fabricated, and it was installed in an anechoic water tank which has a cubic shape of 3.048 m
3.048 m
3.048 m. Fig. 10.3 shows the impact test equipment before Fig. 10.3 Impact testing equipment just before being installed in anechoic water tank.

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

being installed into the anechoic water tank. In the figure, the water tank is covered by plywood for safety except for the opening for the impacting equipment. One of the features in the impact test equipment is that the free drop weight does not get into the water in order to minimise any disturbance to the water. Instead, the drop weight strikes the impact rod whose one end is above the water surface and the other end of the rod hits the composite sample under water. Because the initial distance between the tip of the impact rod and the composite sample is very short, usually a couple of millimetres, the movement of the impact rod inside the water is very small and its motion has a negligible effect on water. A load cell was attached to the end of the impact rod, which would strike the sample to measure the impact force. Fig. 10.4 shows a close-up view of the impact rod and a sandwich composite beam attached to the impact equipment. Strain gage rosettes were attached to every composite sample to be tested. Fig. 10.5 shows typical locations for strain gage rosettes on a square Fig. 10.4 Impact rod with a load cell just above a sandwich composite beam.

Fig. 10.5 Strain gage rosettes locations on square plate.

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277

composite plate. The locations for strain measurements were selected based on the available number of channels for data acquisition as well as symmetries of the plate and loading. Impact testing was conducted using the same impact mass and height while the water tank was filled with water or water was drained from the tank, respectively. In other words, the former test contains water with FSI (called the wet test) while the latter test does not include water (called the dry test). By comparing the two test results, the effect of FSI can be observed. In order to avoid any effect of water absorption, the composite samples were placed into the water only for a short period during the wet test. To further verify the water effect, the same composite sample was tested in dry, wet, and then dry conditions sequentially. Then, the two dry test results were compared to see whether there is any change in the responses. The test data show no difference between the two dry tests, one before and the other after the wet test. To study the dynamic response without damage, the same composite sample was used for both dry and wet conditions. However, to examine progressive damage, the same sample could not be used for both conditions because the sample had been already damaged from previous tests. In that case, one sample was used for the dry test and another sample was used for the wet test. In order to minimise the statistical difference from one sample to another, several samples were tested for each condition to determine their average behaviour. Those samples to be compared were fabricated from the same batch to maintain the consistency in the samples.

10.2.2 Experimental results and discussion The same impact mass and height were applied to an e-glass woven fabric composite plate while the plate was in air and water, respectively. Fig. 10.6 compares the plots for the impact force time history between the dry and wet tests. Two different impact heights were compared while the impact mass remained the same as 4.63 kg. Both heights resulted in greater impact peak forces for the wet tests than the dry tests. The difference in the peak forces between the wet and dry tests was larger for the greater impact height. This can be explained using FSI. When a plate is impacted while it is submerged in water, there is an effect of the added mass which is resulted from FSI. The added mass slows down the deflecting movement of the plate. For the composite plate, the density of the plate material is very comparable to that of water. As a result, the effect of added mass is quite significant. The slower movement in water yields a greater contact force between the impact rod and the plate. The effect of FSI is larger with the faster acceleration of the body resulting from the greater impact height. Dynamic response of the plate was measured using the strain gages as shown in Fig. 10.5. Strain measurements at some selected locations were compared between the dry and wet tests under the same impact height and mass. Fig. 10.7 shows the comparison of the strain responses at the gage location #1 which is located near the centre of the plate. As expected, there was a difference in the strain time histories between the dry and wet tests, and the difference became greater at later time. The next comparison of strains shown in Fig. 10.8 was made at the location #7 which is approximately in the

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3000 50.8 cm, Wet Test

2500

76.2 cm, Wet Test

2000

76.2 cm, Dry Test

Force (N)

50.8 cm, Dry Test

1500

1000

500 0

–500

0.005

0

0.01

0.015

0.02 0.025 Time (sec)

0.03

0.035

Fig. 10.6 Comparison of impact forces between dry and wet tests.

1.5 50.8 cm, Wet Test 76.2 cm, Wet Test 1

50.8 cm, Dry Test

Strain (millistrain)

76.2 cm, Dry Test 0.5

0

–0.5

–1

0

0.01

0.03 0.02 Time (sec)

Fig. 10.7 Comparison of strains at location #1 of Fig. 10.5.

0.04

0.05

0.04

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1.5 50.8 cm, Wet Test 76.2 cm, Wet Test 50.8 cm, Dry Test

1

Strain (millistrain)

76.2 cm, Dry Test

0.5

0

–0.5

–1

0

0.01

0.02 0.03 Time (sec)

0.04

0.05

Fig. 10.8 Comparison of strains at location #7 of Fig. 10.5.

middle along the diagonal direction between the plate centre and a corner. The figure shows a significant difference in the strain responses between the dry and wet tests. The wet test resulted in much greater strains than the dry test. Additionally, the peak strain was in tension for the wet tests while it was in compression for the dry test. In other words, the effect of FSI was so huge at that location. Figs 10.9 and 10.10

1 50.8 cm, Wet Test

Strain (millistrain)

76.2 cm, Wet Test 50.8 cm, Dry Test

0.5

76.2 cm, Dry Test

0

–0.5

0

0.01

0.02 0.03 Time (sec)

Fig. 10.9 Comparison of strains at location #12 of Fig. 10.5.

0.04

0.05

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures 1.5 50.8 cm, Wet Test

Strain (millistrain)

1

76.2 cm, Wet Test 50.8 cm, Dry Test 76.2 cm, Dry Test

0.5

0

–0.5

–1

0

0.01

0.02 0.03 Time (sec)

0.04

0.05

Fig. 10.10 Comparison of strains at location #15 of Fig. 10.5.

show the strains at the locations near the edge of the plate. Especially, Fig. 10.10 was a plot near a corner of the plate. As shown in the figures, the difference in strains between the dry and wet tests was much larger near the corner. The stain responses suggest that the effect of FSI is not uniform over the composite plate. The effect is greater near the corner than near the centre. If the FSI effect were uniform, the difference in strain responses between the dry and wet tests would be consistent for every location. This requires a further analysis to understand what results in such a large FSI effect around the plate corner. In order to provide some clues to this question, vibrational characteristics of the composite plate were studied in the later section. Similar impact tests were conducted for e-glass composite beams which were clamped at both ends. The beams were impacted at the centre, and strains were plotted and compared between the dry and wet tests. Fig. 10.11 shows the comparison of impact forces on the clamped beam between the dry and impact tests. As shown in the plate tests, the impact force was greater for the wet test under the same impact mass and height. Figs 10.12 and 10.13 compare strains near the centre and boundary of the clamped beam, respectively. The wet tests yielded higher strains than the dry tests at both locations. However, the difference in the strain between the dry and wet tests was larger near the centre than near the boundary. Additionally, strain measurements showed vibration of the beam after the impact force was absent. The vibrational frequency in the dry test was quite higher than that in the wet test. This confirms that the added mass effect due to FSI is significant for polymer composite structures. The magnitude of the peak strain for the wet test was 0.00152 m/m near the centre and 0.00144 m/m near the boundary while the dry test yielded 0.00134 m/m near the centre and 0.00136 near the boundary. In other words, the peak strain occurred near the centre of the beam for the wet test and near its boundary for the dry test. However,

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281

200 Wet Test Dry Test

Force (N)

150

100

50

0

–50 0

0.005

0.01 0.015 0.02 0.025 0.03 0.035 0.04 Time (sec)

0.045 0.05

Fig. 10.11 Comparison of impact forces on beam. 0.5

Strain (millistrain)

0

–0.5

–1

Wet test –1.5

–2

0

Dry test

0.01

0.02

0.03

0.04 0.05 Time (sec)

0.06

0.07

0.08

Fig. 10.12 Comparison of strain responses of beam near its centre.

the difference was not very large. This observation explains why the failure location changes from the dry test to the wet test as described later. Sandwich composite beams were tested under dry and wet conditions, respectively. The skins were made of glass fibre woven fabrics, and the core was balsa wood. The combination of the impact mass and height was selected such that the sandwich composite beams could fail. In this case, the same test sample could not be used for both

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1.5 Wet test Dry test

Strain (millistrain)

1

0.5

0

–0.5 0

0.01

0.02

0.03

0.04 0.05 Time (sec)

0.06

0.07

0.08

Fig. 10.13 Comparison of strain responses of beam near its boundary.

dry and wet tests. Therefore, about a dozen samples were tested with about a half of them in dry and the other half in wet conditions, respectively. The dry test showed five samples failed at the clamped boundary and one sample failed at the centre. On the other hand, the wet test resulted in five failures at the centre and two failures at the clamped boundary. In other words, the major failure location was shifted from the clamped boundary to the centre of the beams because of the effect of FSI. Figs 10.14 and 10.15 compare strains near the centre and boundary of the sandwich beams impacted in air and water, respectively. 30 Boundary

Strain (microstrain)

Centerline 20

10

0

–10 0

10

20

30 time (mSec)

40

50

60

Fig. 10.14 Plot of strains near centre and boundary of sandwich composite beam impacted in air.

Dynamic loading with fluid-structure interaction

283

10 Boundary

Strain (microstrain)

Centerline 5

0

–5

–10 0

10

20

30 time (mSec)

40

50

60

Fig. 10.15 Plot of strains near centre and boundary of sandwich composite beam impacted in water.

The next study compared progressive damage in composite plates as the drop height increased gradually for the same composite plate. For this test, a plate was impacted from an initially selected height. After each test, the composite plate was examined for any damage. Since an e-glass woven fabric composite plate was used, damage could be checked visually. The damage size and shape were recorded. Then, the second test was conducted using the same impact height followed by damage inspection. Now, the impact height was increased, and impact tests were conducted twice with the same impact height. This process continued with higher impact heights. The reason the impact tests were conducted twice using the same impact height was to determine whether damage progress was resulted from an increase in the impact height or not. The test result showed that there was no further damage growth after the second impact test. Therefore, the damage growth was due to the increase in the impact height, i.e., impact energy. The major damage was delamination in the composite plate, and the shape of the damage was close to a circular shape. As a result, the diameter of the damage size was measured against the impact height as plotted in Fig. 10.16, which shows that damage grows at a lower impact height for the wet test than for the dry test.

10.3

Numerical analysis of impact on composite structures with FSI

Computational modelling and simulations were conducted for impact tests on composite plates submerged in water. The structural solver and the fluid solver are executed in a staggered manner while the information is exchanged at the FSI interface between the two solvers. The fluid solver uses the fluid velocity at the interface while the structural solver uses the fluid pressure at the same interface.

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6 5

Dry Test

Delimination (cm)

Wet Test 4 3 2 1 0 0

0.4 0.5 0.3 Impact height (m)

0.2

0.1

0.8

0.7

0.6

Fig. 10.16 Plot of delamination growth in e-glass composite plate as a function of impact height.

10.3.1 Numerical modelling techniques Composite plates were generally modelled using the finite element method [6]. Both continuous and discontinuous Galerkin finite element techniques were used [7]. For FSI modelling, 3D solid-like plate/shell elements have an advantage because the elements have discrete nodes at the top and bottom surfaces of the plate/shell, respectively, and each node has three translational degrees of freedom [8] as sketched in Fig. 10.17. This makes it easier for the interface with the fluid domain. The 3D solid-like elements can also model each layer of the composite plate/shell discretely as necessary in order to model interlayer delamination [9]. w7

v7

7

u7

8 3 4 z

5

y x

1

Fig. 10.17 Three-dimensional solid-like plate finite element.

6 2

Dynamic loading with fluid-structure interaction

285

Additionally, the cellular automata technique was also applied for structural members [10]. The finite difference-based cellular automata technique is the simplest and quickest in terms of programming and computational time. As a result, the technique is described later for plate bending problems. The governing equation for plate bending is ρh


 ∂2 w ¼ r2 Dr2 w + q ∂t2

(10.1)

where w is the deflection of the plate, ρ is the mass density, h is the plate thickness, t denotes time, D is the plate rigidity, q is the applied pressure loading, and r2 is the Laplace operator. The governing equation is broken into two parts as follows: M ¼ Dr2 w ∂ w ρh 2 ¼ r2 M + q ∂t 2

(10.2)

in which, M ¼ (Mx + My)/(1 + ν). Here, Mx and My are the bending moments per unit length along x- and y-axis, respectively, and ν is Poisson’s ratio. Eq. (10.2) is expressed using the finite difference technique as follows: h    i Mit, j ¼ Di, j wti + 1, j + wti1, j  2wti, j =ðΔxÞ2 + wti, j + 1 + wti, j1  2wti, j =ðΔyÞ2 h    i t w€ti, j ¼ Mit + 1, j  Mi1 + 2Mit, j =ðΔxÞ2 + Mit, j + 1  Mit, j1 + 2Mit, j =ðΔyÞ2 + qi, j =ðρhÞi, j ,j w_ ti:j+ Δt ¼ w_ ti, j + w€ti, j ðΔtÞ wti,+j Δt ¼ wti, j + w_ ti,+j Δt ðΔtÞ (10.3)

where subscripts (i,j) denote the lattice point in the two dimensional domain of the plate;Δx, Δy, and Δt are the spacing in the x-, y-, and time axis; and the superimposed dot denotes the temporal derivative. These set of equations are solved for every lattice point repeatedly with an advance in time. The fluid domain was modelled using either the Navier-Stokes equation or the acoustic equation. If the fluid flow is important, the Navier-Stokes equation was selected for the fluid domain. Otherwise, the acoustic equation was used. The NavierStokes equation was solved using the lattice Boltzmann method while the acoustic equation was solved using the cellular automata technique [11]. In many cases, fluid motion as well as viscosity are not significant. The cellular automata technique has merits in terms of its very efficient computational cost as well as easy modelling of various boundary conditions such as nonreflected infinite boundaries. The linear acoustic equation is expressed as follows: ∂2 p ¼ c2 r2 p ∂t2

(10.4)

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

where p is the pressure and c is the speed of sound of the acoustic medium. The acoustic equation can be solved in two different ways using the cellular automata techniques. One way uses the following formulation: pti,+j Δt ¼

pti + 1, j + pti1, j + pti, j + 1 + pti, j1  2pti, j, k 2

(10.5)

for a 2D domain. The expression is applied to every other lattice in an alternative way. For example, Eq. (10.5) is applied to all black lattice points first in Fig. 10.18. Then, the same equation is applied to all white lattice points. This process repeats. The other equations for the cellular automata are as follows: p€ti, j ¼ c2i, j

h    i pti + 1, j + pti1, j  2pti, j =ðΔxÞ2 + pti, j + 1 + pti, j1  2pti, j =ðΔyÞ2 p_ ti:j+ Δt ¼ p_ ti, j + p€ti, j ðΔtÞ pti,+j Δt ¼ pti, j + p_ ti,+j Δt ðΔtÞ

ð10:6Þ

This expression is applied to all lattice points simultaneously. When Eqs (10.5) and (10.6) are compared, the former equation is computationally more efficient. However, because the former expression updates the value at every other lattice point, it is not easy to be coupled with the structural domain directly while the second expression does not have such a problem. Because it was not easy to couple the finite element structural domain to the first cellular automata expression directly, an intermediate neighbour of the finite element

Fig. 10.18 Lattice structure with alternating black and white grid points.

Dynamic loading with fluid-structure interaction

287

Structural finite element model

Finite element acoustic domain

Cellular automata acoustic domain

Fig. 10.19 Modelling of coupled structural and acoustic domain.

structural model was modelled using the finite element based acoustic model as sketched in Fig. 10.19. Some more detailed discussion is available in the reference [7]. The plate had 20
20
4 elements while the fluid domain had 60
60
60 elements. The top surface of the fluid domain was a free boundary while the rest of the surfaces were assumed nonreflecting boundaries. A numerical test was conducted for the same composite plate used for the previous experiment. Then, the strain response near the plate centre was compared between the experimental and numerical tests as shown in Fig. 10.20. The gage locations are sketched in Fig. 10.21. In general, they agreed well. Another strain comparison was made at the gage location #3 as shown in Fig. 10.22. At this location, both results are not favourably compared. As discussed at a later section, the effect of FSI was significant around the gage location #3. As a result, much more refined mesh as well as more careful modelling of FSI is required to improve the numerical solution. 0.2 Numerical Experimental

Milli-strain

0

–0.2 –0.4

–0.6

–0.8

0

0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 Time (s)

Fig. 10.20 Comparison of numerical and experimental strains for impact test in water at gage location #1 in Fig. 10.21.

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

Fig. 10.21 Strain gage locations for comparison of numerical and experimental data.

1

2

3 4

0.8 Numerical

Milli-strain

0.6

Experimental

0.4

0.2

0

–0.2

0

0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 Time (s)

Fig. 10.22 Comparison of numerical and experimental strains for impact test in water at gage location #3 in Fig. 10.21.

10.3.2 Composite failure modelling Failure of composite structures was modelled using a set of failure criteria based on a multiscale approach. That is, the failure criteria are based on the stresses and strains experienced in the fibre and matrix material, but not the homogenised composite material. To this end, a multiscale approach was used to determine those stresses and strains as sketched in Fig. 10.23. The multiscale approach consists of upscaling

Dynamic loading with fluid-structure interaction

289

Fig. 10.23 Multiscale analysis.

and downscaling processes as shown in the sketch. The upscaling process is to compute the effective composite material properties from the fibre and matrix material properties. The downscaling process is to compute the stresses and strains in the fibre and matrix materials from the composite level stresses and strains. The fibre stresses and strains do not mean those at an individual fibre. Instead, those represent the average stresses and strains of fibre bundles around the numerical integration points of the finite element analysis. In order to make the multiscale approach computationally efficient, both upscaling and downscaling processes must be efficient since they are used repeatedly as damage and failure evolves. To this end, analytical expressions were used for those processes, which were developed using a unit-cell model [11]. The failure criteria have three failure modes: fibre failure, matrix failure, and fibre/ matrix interface failure [12]. Delamination, fibre splitting, etc. are the examples of either matrix failure and/or fibre/matrix interface failure. As a result, there are three failure criteria. The criterion for the fibre failure is given as follows: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi      ffi ε1f

2

f + γ 12 f εfail

2

f + γ 13

2

1

(10.7)

where superscript f indicates the strains are those of the fibres, 1-axis is the fibre orif is the failure strain of fibres. The failure strain of the fibres is genentation, and εfail erally different between tensile and compressive loading because of fibre buckling under compression. The matrix failure depends on the matrix material. For a brittle resin with an isotropic material property, the maximum normal strain is selected as the matrix failure criterion. Finally, the fibre/matrix interface failure at the 1–2 plane is expressed as +2 pffiffiffiffi
!2 * τI + vf σ I2  σ I1 σ I2 + 1 I τfail σ Ifail

(10.8)

290

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

Fig. 10.24 Simplified impact model.

I where the superscript I indicates the interface stresses, τfail is the failure strength under I shear loading only, σ fail is the failure strength under normal loading only, hi is the Macaulay function which has a value for a positive σ 2I , and ν f is the fibre volume fraction. A very similar equation can be used for the interface failure at the 1–3 plane. A numerical impact model was analysed using the multiscale based failure criteria to show the effect of FSI on the fibre failure. An impact load was modelled using a spring and mass attached to a plate to be impacted as sketched in Fig. 10.24. The plate was considered to be in contact with water which was modelled as an acoustic medium. Another case was a dry plate. The fibre failure of the same clamped composite plate was compared in Fig. 10.25 between the two cases: with and without FSI. The results clearly showed the effect of FSI on the failure. The strain time history was also plotted in Figs 10.26 and 10.27 when the plate was dry or wet. The two figures showed that the effect of FSI was greater near the clamped edge than the centre of the plate. The gage locations are shown in Fig. 10.28.

10.4

Experimental study of vibration of composite structures in water

In this section, vibrational characteristics of composite structures were examined in water, and their dynamic behaviours were compared to those in air. First, experimental modal analysis was conducted for a composite beam while the beam was submerged in water or in air [13]. The beam was supported by two flexible strings at both ends to simulate the free-free boundary condition. Then, experimental modal analysis was conducted to determine natural frequencies and mode shapes. Then, modal curvatures were computed from the mode shapes by taking their second derivatives. For the modal analysis, 10 equally spaced accelerometers were attached to the beam to be tested. First, natural frequencies were measured for the beam which were submerged in water or in air, respectively. The comparison of the first three natural frequencies is shown in Table 10.2. There was an almost uniform reduction in all three

Dynamic loading with fluid-structure interaction

291

Fig. 10.25 Comparison of fibre failure (A) without and (B) with FSI.

frequencies resulting because of the added mass effect from FSI. The natural frequencies in water were close to 60% of those in air. The corresponding three mode shapes are plotted in Fig. 10.29. The overall mode shapes were almost the same for the beam whether it was in air or water. There were some differences in magnitudes locally along the beam. As the modal curvatures were compared as seen in Fig. 10.30, their differences were quite sizable at some locations along the beam. Modal curvatures were directly related to bending strains of the beam. As a result, comparison of modal curvatures suggests what would be the difference in the strain response due to FSI. The nest experiment was conducted to find out the free vibrational response of a composite beam. A composite beam was clamped at one end, and an initial force was applied at the other end of the beam. Then, the force was removed so that the beam could vibrate freely. The movement of the beam was measured using the digital image correlation technique and a high-speed camera. Fig. 10.31 shows the setup for the test.

292

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

Fig. 10.26 Normal strain responses of a dry plate of a simplified impact model at the gage locations shown in Fig. 10.28.

From the experimental measurement, natural frequencies were determined from the tip motion of the cantilever beam. When the beam was in air, the measured natural frequencies agreed well with the analytical solution with an error less than 1%. Then, the natural frequencies were compared between the beam in air and water, respectively. The frequency ratio of the wet beam to the dry beam was 0.3. In other words, the natural frequencies are reduced by 70% due to the added mass effect. The beam tested for this experiment was an e-glass composite. Since the e-glass composite is lighter than the carbon composite, the effect of FSI is expected to be greater for the e-glass composite beam. Additionally, the different boundary conditions also contributed to the difference between the e-glass and carbon composite beam test data. Figs 10.32 and 10.33 compare snapshots of free vibrational motions in air and water, respectively, of the cantilever beam at arbitrary time instances. The effect of FSI resulted in activation of higher frequency modes during the free vibrational motion even though the initially deformed shape was very close to the first mode

Dynamic loading with fluid-structure interaction

293

Fig. 10.27 Normal strain responses of a wet plate of a simplified impact model at the gage locations shown in Fig. 10.28.

shape. Figs 10.34 and 10.35 illustrate the vibrational motions in terms of the 3D perspective. The observation shows that the effect of FSI is very nonuniform along the beam, so that the strain response between the dry and wet beams will be different along the beam. This may also contribute to a change in failure locations due to the FSI effect.

10.5

Numerical analysis of vibration of composite structures in water

Numerical modal analysis can be substituted for the experimental modal analysis, if necessary. For example, in order to determine mode shapes of a composite plate accurately, a very large number of sensors must be attached to the plate. Many sensors introduce extra mass and stiffness to the structure, which can alter the response of

294

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

Fig. 10.28 Strain gage locations for the simplified impact model of a clamped plate.

Table 10.2 Comparison of natural frequencies of the free-free carbon composite beam obtained using experimental modal analysis. Mode shape

Frequency of wet beam (Hz)

Frequency of dry beam (Hz)

Frequency ratio of wet to dry beam

1 2 3

60.0 166 329

103 272 552

0.58 0.61 0.60

the structure itself. Additionally, the data acquisition system may be saturated with many sensor outputs. Some boundary conditions like a simply supported edge are not easy to be applied along all edges of a plate. As a result, the numerical modal analysis was undertaken in this section to complement the results in the previous section. The numerical modal analysis was conducted using the modelling technique described in Section 10.3. Then, impulse loading was applied to a selective location of the structure. The structural finite element model gives nodal displacement, velocity, and acceleration. Time dependent solutions of the nodal variables were transformed into the frequency domain using the fast Fourier transform technique. Then, natural frequencies and mode shapes were determined from the frequency domain solutions.

Dynamic loading with fluid-structure interaction First mode shape

1 Norm. displ.

295

Dry test Wet test

0.5 0 –0.5 0.1

0.2

0.3

0.4

Norm. displ.

0.8

0.9

1

Second mode shape

0.5 0

Dry test Wet test

–0.5 –1 0.1

0.2

0.3

0.4

0.8

0.9

1

0.8

0.9

1

Dry test Wet test

0

–0.5 0.1

0.5 0.6 0.7 Norm. beam length Third mode shape

0.5 Norm. displ.

0.5 0.6 0.7 Norm. beam length

0.2

0.3

0.4

0.5 0.6 0.7 Norm. beam length

Fig. 10.29 Comparison of mode shapes between dry and wet test for carbon composite beam with free ends.

First, a composite beam was studied using the numerical modal analysis [14]. The first example was a fibreglass composite beam with simply supported ends. There was no fluid around the beam so that the numerical results were compared to the analytical solutions. The first three natural frequencies were compared between the exact solutions and the numerical modal analysis results in Table 10.3. The numerical results were obtained using 40 beam elements. The comparison was good in overall. Figs 10.36 and 10.37 compare the first two mode shapes between the exact and numerical modal analysis results. They show an excellent agreement. These results proved that the numerical modal analysis provided acceptable results. The numerical modal analysis was applied to a clamped glass fibre composite beam to determine their mode shapes when the beam was inside water. Then, the mode shapes in air and water, respectively, were compared each other in order to determine the effect of FSI on the mode shapes of the beam, as shown in Figs 10.38 and 10.39. There was a very small difference between the two mode shapes in air and water as

296

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures First mode curvature Norm. curvature

0.3 0.2 0.1 Dry test Wet test

0 –0.1 0.1

0.2

0.3

0.4

0.7

0.8

0.9

1

0.8

0.9

1

0.8

0.9

1

Second mode curvature

1.5 Norm. curvature

0.5 0.6 Norm. beam length

1

Dry test Wet test

0.5 0 –0.5 0.1

0.2

0.3

0.4

Third mode curvature

3 Norm. curvature

0.5 0.6 0.7 Norm. beam length

Dry test Wet test

2 1 0 –1 0.1

0.2

0.3

0.4

0.5

0.6

0.7

Norm. beam length

Fig. 10.30 Comparison of modal curvatures between dry and wet test for carbon composite beam with free ends.

Fig. 10.31 Setup to measure free vibration of cantilever beam using digital image correlation technique.

0 t = 8s t = 12s

Amplitude (mm)

–0.05

–0.1

–0.15

–0.2 0

50

100

150

200

250

300

Distance from fixed end of beam (x 0.66503mm)

Fig. 10.32 Snapshot of free vibrational motion of cantilever beam in air at two time instances. 0.1 t = 6s t = 14s

Amplitude (mm)

0.05

0

–0.05

–0.1

0

50

100

150

200

250

Distance from fixed end of beam (X0.71843mm)

Fig. 10.33 Snapshot of free vibrational motion of cantilever beam in water at two time instances.

Fig. 10.34 Three-dimensional snapshot of free vibrational motion of cantilever beam in air.

298

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

Fig. 10.35 Three-dimensional snapshot of free vibrational motion of cantilever beam in water.

Table 10.3 Comparison of natural frequencies of simply supported composite beam obtained from the exact solutions and numerical modal analysis, respectively. Mode shape

Exact (Hz)

Num. modal analysis (Hz)

Error (%)

1 2 3

72 289 645

70 290 660

2.8 0.3 2.3

0 Exact

Norm. displ.

–0.2

Numerical

–0.4

–0.6

–0.8 –1

0

0.2

0.4 0.6 Norm. beam length

0.8

1

Fig. 10.36 Comparison of the first mode shape of a simply supported composite beam between the exact solution and numerical modal analysis solution.

demonstrated using the experimental modal analysis in the previous section. However, modal curvatures resulted in much sizable differences between the two cases. The next study was focused on a composite plate secured along all edges. As a numerical modal analysis, an impulse force was applied to the centre of the clamped plate. This was done purposely to simulate the previous impact testing. Snapshots of

Dynamic loading with fluid-structure interaction

299

1

Norm. displ.

0.5

0

Exact

–0.5

–1

Numerical

0

0.1

0.2

0.3

0.4 0.5 0.6 0.7 Norm. beam length

0.8

0.9

1

Fig. 10.37 Comparison of the second mode shape of a simply supported composite beam between the exact solution and numerical modal analysis solution.

0.02 In air In water

0

Norm. displ.

–0.2 –0.4 –0.6 –0.8 –1 –1.2 0

0.02

0.04

0.06 0.08 0.1 Norm. beam length

0.12

0.14

Fig. 10.38 Comparison of the first mode shape of a clamped composite beam in air and water, respectively.

vibrational motions of the composite plates are plotted in Figs 10.40 and 10.41. The former plot is for the plate in air while the latter plot is for the same plate in water. The overall shapes were similar each other, but a difference was noted around the corners of the plate. The contour plots clearly showed the difference. In order to understand why there was a difference around the corners, mode shapes were determined and compared between the two cases. Because the impulse force was

300

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

In air

1

In water

Norm. displ.

0.5

0

–0.5

–1 0.02

0

0.04

0.06 0.08 0.1 Norm. beam length

0.12

0.14

Fig. 10.39 Comparison of the second mode shape of a clamped composite beam in air and water, respectively.

x10–6 0 –0.5 –1 Plate deflection

–1.5 –2

–2.5 –3

–3.5 –4 0 0.2 0.4 0.6 0.8 Norm.plate length

1

1

0.8

0.6

0.4

Norm.plate width

Fig. 10.40 Snapshot of vibrational motion of a composite plate in air.

0.2

0

Dynamic loading with fluid-structure interaction

301

x 10–7

Plate displacement

2 1 0

–1

–2 0 0.2 0.4 0.6 0.8 Norm. Plate Width

1

0.2

0

0.4

0.6

Norm. Plate Length

Fig. 10.41 Snapshot of vibrational motion of a composite plate in water.

applied at the centre of the plate, any mode shape whose node was located at the plate centre would not be excited. The first mode shape looked like that in Fig. 10.40 regardless of whether the plate vibrated in air or water. However, the next mode shapes were quite distinguished themselves from each other. Figs 10.42 and 10.43 are the mode shapes in air and water, respectively. The mode shape in water had a quite different shape of deformation around the corner, which contributed to the vibrational motion as shown in Fig. 10.41. The numerical modal analysis showed that the mode shape could be changed locally because of FSI, which could also lead to the change in the failure location. Fig. 10.42 Mode shape of a composite plate in air.

1 0.5 0 –0.5 –1 1

1

0.5

0.5 0 0

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

1 0.5 0 –0.5 –1 1

1

0.5

0.5 0 0

Fig. 10.43 Mode shape of a composite plate in water.

10.6

Experimental study of cyclic loading on composite structures with FSI

This section studies the response of glass fibre composite beams subjected to cyclic loading while the beams are surrounded by water. The tests were conducted in terms of the three-point bending setup. The beams were simply supported, and cyclic loading was applied to the centre of the beams. Different frequencies of the cyclic loadings were applied, respectively. The same cyclic loading was applied to beam in air and water, respectively, to compare their responses. In particular, the number of cycles to failure was compared between the dry and wet tests. Before cyclic testing, static three-point bending tests were conducted to determine the failure load and the failure deflection. The cyclic loading was controlled by displacements. The maximum and minimum displacements of the cyclic tests were set at 75% and 25% levels of the failure deflection. Additionally, the bending stiffness was computed from the static testing, and natural frequencies were computed for the beams in air. The first natural frequency was around 350 Hz. With the effect of added mass, the frequency in water is expected to be lower. The frequency of applied cyclic loading was selected to avoid any resonance effect between the excitation and natural frequencies. The selected frequencies were 2, 5, and 10 Hz, which were much lower than the natural frequencies of the beams in air and water. Additionally, composite samples were examined whether there was moist absorption in the samples which could affect the test results. Most of the wet tests were conducted in water less than 1 h, sometime only around 10 min. As a result, there was not much time to adsorb moisture into the samples. Each sample was also weighed immediately before and after wet testing. There was no indication of weight change. As a result, the moisture was not considered to play a role in the test results. Because the cyclic tests were controlled by displacements, the forces were plotted as a function of time which is related to loading cycles. A typical force variation as a function of time is plotted in Fig. 10.44, which shows that the maximum and minimum

Dynamic loading with fluid-structure interaction

303

Force against time 1.4 Max Force Min Force

1.2 1

Force/kN

0.8 0.6 0.4 0.2 0 –0.2 0

200

400

600

800 Time/s

1000

1200

1400

1600

Fig. 10.44 Force variation during the cyclic loading on composite beam.

forces decrease as time (i.e. the number of loading cycle) increases until failure. Comparing the life cycles, i.e. the number of cycles to failure, between the dry and wet tests indicated that the FSI reduced the life cycles significantly. Table 10.4 shows the test results. The life cycles in air were about 50% greater than those in water for the 5 and 10 Hz loading. In other words, the FSI effect reduced the life cycles by approximately 35%. In order to understand the reason why there was such a reduction in life cycles, the forces were compared between the dry and wet tests. Fig. 10.45 compares how the forces varied during the cyclic loading at 5 Hz. The figure shows the initial several cycles for dry and wet tests, respectively. The comparison indicates that the maximum force is greater for the wet test because of the effect of FSI. Such an increase in the force with the same displacement influenced the life cycles. Table 10.4 Summary of ratios of fatigue cycles to failure in air to that in water. Life cycle ratio of air to water

Average Standard deviation

10 Hz

5 Hz

1.43 0.22

1.52 0.16

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Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

1.5

Force (N)

1

0.5

Wet Test Dry Test

0 0

0.1

0.2

0.3

0.6 0.4 0.5 Time (sec)

0.7

0.8

0.9

1

Fig. 10.45 Comparison of forces between dry and wet tests with cyclic loading at 5 Hz.

10.7

Numerical analysis of cyclic loading on composite structures with FSI

Computer modelling and simulation was conducted for cyclic loading on a composite beam as sketched in Fig. 10.46. This was to simulate a composite structure supporting rotating equipment while the structure was in contact with water on

F=F° sin(ωt)

k

Water

Fig. 10.46 Simplified computational model.

Composite beam

Dynamic loading with fluid-structure interaction

305

x10–6

3 2

Strain

1 0 –1 –2 Dry Test Wet Test

–3 –4

0

0.05

0.1

0.15 Time (sec)

0.2

0.25

0.3

Fig. 10.47 Comparison of bending strains at the centre of a clamped beam loaded at 10 Hz.

the other side. The equipment was modelled as a concentrated mass with a harmonic force. The beam was clamped at both ends. The same kind of modelling techniques described previously were used for FSI, and some of the numerical results are discussed later [15]. A harmonic force of 10 Hz was applied to a composite beam, and the bending strains were compared between the dry and wet tests under the same harmonic forcing function. To simplify the problem, a unit magnitude of force was applied. Fig. 10.47 compares the bending strain at the centre. The result also confirmed that the strain response was greater for the wet test, which could lead to reduction in the life cycle of the beam. In this example, the spring and mass of the equipment was adjusted so that the first natural frequency of the system was the same between the dry and wet test cases. Then, the excitation frequency was varied from 0 to 20 Hz. From each test, the maximum deflection was computed and divided by the static deflection, called deflection ratio which was compared between the dry and wet cases. As expected, the deflection ratio increased as the excitation frequency became closer to the natural frequency which was around 17 Hz as shown in Fig. 10.48. However, the wet case yielded larger deflection ratios consistently than the dry case. The difference was very small up to 3 Hz because the effect of FSI is small at those frequencies. This numerical result supported the experimental observation.

306

Dynamic Deformation, Damage and Fracture in Composite Materials and Structures

10 Dry Test Wet Test

Deflection Ratio

8 6

4 2 0 0

5

10 Frequency (Hz)

15

20

Fig. 10.48 Plot of deflection ratios against excitation frequencies.

10.8

Summary and conclusion

Both experimental and numerical studies were conducted to investigate the effect of FSI on polymer composite structures while they were subjected to dynamic impact loading or cyclic loading. Different test setups were designed and fabricated for each FSI study. In addition, diverse numerical models were developed for FSI, which used the finite element method, lattice Boltzmann method, and cellular automata technique. The test results showed that the effect of FSI was significant on those composite structures and could lead to premature failure when compared to the dry structures without FSI. Dynamic characteristics of the composite structures were also altered significantly because of the FSI effect. Both numerical and experimental studies complemented each other to confirm and support the findings. It is critical to include the FSI effect in design and analysis of composite structures in marine applications including water in order to provide necessary structural integrity to the structures.

References [1] Y.W. Kwon, Study of fluid effects on dynamics of composite structures, J. Press. Vessel. Technol. 133 (2011) 031301–031306. [2] Y.W. Kwon, Fluid-Structure Interaction of Composite Structures, Springer, Switzerland AG, 2020. 2020. [3] Y.W. Kwon, R.P. Conner, Low velocity impact on polymer composite plate in contact with water, Int. J. Multiphys. 6 (3) (2012) 179–197. [4] Y.W. Kwon, M.A. Violette, Damage initiation and growth in laminated polymer composite plates with fluid-structure interaction under impact loading, Int. J. Multiphys. 6 (1) (2012) 29–42.

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[5] Y.W. Kwon, M.A. Violette, R.D. McCrillis, J.M. Didoszak, Transient dynamic response and failure of sandwich composite structures under impact loading with fluid structure interaction, Appl. Compos. Mater. 19 (6) (2012) 921–940. 2012. [6] Y.W. Kwon, H.-C. Bang, Finite Element Method Using Matlab, second ed., CRC Press, Boca Raton, Florida, 2000. [7] L.E. Craugh, Y.W. Kwon, Coupled finite element and cellular automata methods for analysis of composite structures with fluid-structure interaction, Compos. Struct. 102 (2013) 124–137. [8] Y.W. Kwon, Analysis of laminated and sandwich composite structures using solid-like shell elements, Appl. Compos. Mater. 20 (4) (2013) 355–373. [9] Y.W. Kwon, L.E. Craugh, Progressive failure modeling in notched cross-ply fibrous composites, Appl. Compos. Mater. 8 (1) (2001) 63–74. [10] Y.W. Kwon, Finite difference based cellular automaton technique for structural and fluidstructure interaction applications, J. Press. Vessel. Technol. 139 (2017) 041301. [11] Y.W. Kwon, Multiscale and Multiphysics Modeling Techniques and Applications, CRC Press, Boca Raton, Florida, 2016. [12] Y.W. Kwon, J. Darcy, Failure criteria for fibrous composites based on multiscale modeling, Multiscale Multidiscip. Model. Exp. Des. 1 (1) (2018) 3–17. [13] Y.W. Kwon, E.M. Priest, J.H. Gordis, Investigation of vibrational characteristics of composite beams with fluid-structure interaction, Compos. Struct. 105 (2013) 269–278. [14] Y.W. Kwon, S.D. Plessas, Numerical modal analysis of composite structures coupled with water, Compos. Struct. 116 (2014) 325–335. [15] Y.W. Kwon, Dynamic responses of composite structures in contact with water while subjected to harmonic loads, Appl. Compos. Mater. 21 (2014) 227–245.

Shock response of polymer composites

11

Paul J. Hazell and Hongxu Wang School of Engineering and Information Technology, The University of New South Wales, Canberra, ACT, Australia

11.1

Shock propagation in composites

Much of what we understand with regards to shock waves in condensed matter evolved from work carried out during World War II and specifically the various nuclear weapons programmes. Although it was well known for well over 100 years that shock waves can be sustained in fluids, it was only in the 1940s that similar concepts were applied to solid materials [1]. In the UK, the subject was first introduced in the open literature in 1948 with a discussion on the shock propagation in steel and lead [2]. And indeed, there have many papers been published since on the shock propagation in solid homogeneous materials. Good reviews of the theory and the data are provided by Davison and Graham [3], Davison [4], and Meyers [5]. For composite materials, many of the papers on their shock compression started appearing in the late 1960s and early 1970s when interest began to emerge in this relatively new class of materials [6–11]. Needless to say, there has been an enormous proliferation of composite material use in recent years. This has been driven largely by their good specific properties. In particular, there has been a drive to introduce composite materials in the military and aerospace sectors, where their properties lead to weight savings [12–16]. Many of these structures are at risk of being subjected to explosions and high-velocity impacts where shock waves may arise. In particular, it is common-place these days to introduce polymer composites/ceramic armour targets [15,17–20]. However, not all impacts and explosions will result in the formation of a shock wave. In a continuum sense, a shock wave is a propagating discontinuity of density, temperature, and stress where these variables jump in value at the shock front. Shocks will form when the material is subjected to an intense compressive stimulus such that the atoms are forced into close proximity. For most military ballistic impact situations with bullet-shaped projectiles (