Forming of Laminates
Proefschrift
Ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus prof.dr.ir. J.T. Fokkema, voorzitter van het College voor Promoties,
in het openbaar te verdedigen op donderdag 1 juli 2004 om 15:30 uur door Tjarko Wouter DE JONG
ingenieur in de luchtvaart en ruimtevaart geboren te Krimpen aan de Lek
Dit proefschrift is goedgekeurd door de promoteren: Prof.dr.ir. M.J.L. van Tooren
Prof.dr.ir. L.B. Vogelesang
Samenstelling promotiecommissie: Rector Magnificus, voorzitter
Prof.dr.ir. M.J.L. van Tooren, Technische Universiteit Delft, promotor Prof.ir. L.B. Vogelesang, Technische Universiteit Delft, promotor Prof.dr. D.F. Adams, University of Wyoming
Prof.dr.ir. H. Huétink, Universiteit Twente
Dr.ir. G.H.J.J. Roebroeks, Fibre Metal Laminates centre of competence Ir. J. Sinke, Technische Universiteit Delft
Prof.ir. A. Beukers, Technische Universiteit Delft, reservelid
Ir. J. Sinke heeft als begeleider in belangrijke mate aan de totstandkoming van het proefschrift bijgedragen.
This research was carried out under project number MP97017 in the framework of the Strategic Research Programe of the Netherlands Institute for Metals Research in the Netherlands (www.nimr.nl).
Published and distributed by: DUP Science DUP Science is an imprint of
Delft University Press P.O. Box 98 2600 MG Delft The Netherlands Telephone: +31 15 27 85 678 Telefax: +31 15 27 85 706 E-mail: info@library.tudelft.nl ISBN 90-407-2506-3
Keywords: Fibre-Metal-Laminates, Formability, Interlaminar shear Copyright © 2004 by Tjarko de Jong
All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the publisher:
Delft University Press Printed in the Netherlands
TABLE OF CONTENTS
List of Symbols_________________________________________________________ xi 1. Introduction ______________________________________________________ 1 1.1. Laminates ... 1 1.1.1. Fibre-Metal-Laminates... 2 1.1.2. Metal-Polymer-Sandwiches ... 4 1.1.3. Composite Laminates... 4 1.2. Laminate Application... 5 1.3. Forming... 6 1.4. Objectives... 8 1.5. Research Approach ... 10 1.6. Outline... 10 2. Laminate Behaviour _______________________________________________ 13 2.1. Laminate Composition... 132.2. Typical Laminate Failures ... 15
2.2.1. Constituent Failure ... 15
2.3. Consistency Failure... 18
2.3.1. Delamination Buckling ... 18
2.3.2. Interlaminar Shear Failure... 19
2.3.3. Edge Delamination. ... 20 2.4. Internal Stresses ... 21 2.5. Interfaces... 22 2.6. Edge Effects ... 24 3. Laminate Deformation _____________________________________________ 27 3.1. Approach... 27 3.1.1. Deformation-Based Calculations ... 28
3.1.2. Basic Deformation Modes ... 28
3.1.3. Laminate Deformation Functions... 29
3.2. Classical Laminate Theory... 30
3.3. Reissner-Mindlin Plate Theory ... 31
3.4. First Order Shear Deformation Theory... 33
3.5. Higher Order Shear Deformation Theories... 34
3.6. Continuity for Layerwise Theories ... 35
3.9. Double Cross Section Transverse Shear Calculation... 40
3.10. Parabolic Layerwise Shear with DCS ... 43
3.11. Parabolic Layerwise with Global Shear... 46
3.12. Implemented Models ... 46
4. Design Tool Development __________________________________________ 49 4.1. Requirements ... 49 4.2. Structure ... 51 4.3. Components ... 52 4.3.1. Materials... 52 4.3.2. Laminates... 53 4.3.3. Deformation ... 54 4.3.4. Process... 54 4.4. Object-Orientated Design ... 55 4.5. Inter-Level Functions... 55 4.5.1. Material Component ... 56 4.5.2. Laminate Component... 61 4.5.3. Deformation Component... 61 4.5.4. Process Component ... 62 4.6. Programming Language... 63
5. Design Tool Components___________________________________________ 65 5.1. Stress and Strain Tensors ... 65
5.2. Material Models ... 67
5.2.1. Elastic Material Model ... 67
5.2.2. Anisotropic Material Model... 68
5.2.3. Orthotropic Material Model ... 69
5.2.4. Elastic Perfect-Plastic Material Model ... 71
5.2.5. Elastic Strain-Hardening Material Model... 73
5.2.6. Flow Theory Material Model... 74
5.3. Laminate Component... 76
5.4. Deformation Models ... 76
5.4.1. Reissner-Mindlin Plate Theory... 77
5.4.2. Constant Layerwise Shear with Parabolic Shear ... 80
5.4.3. Double Cross Section Plate Theory... 81
5.4.4. Double Cross Section Parabolic Layerwise Shear... 82
5.4.5. Double Cross Section, Parabolic Layerwise with Global Shear83 5.4.6. Maximum Interface Shear-Stress... 84
5.5. Process Models. ... 87
5.5.1. Tensile Test, Bending Radius... 87
5.5.2. Engineering Properties... 88
5.5.3. Forming Limit Diagram - Sub Sample ... 90
5.5.4. Forming Limit Diagram - Strain Path... 92
5.5.5. Strain Path Failure Limits. ... 92
5.5.6. Equal Interface Shear-Stress Lines (Vm-Plot)... 93
5.5.7. Bending Wizard... 94
5.5.8. Custom Process Models... 95
6. Model Behaviour _________________________________________________ 97
6.1. Reference Laminate ... 97
6.2. Without Transverse Shear... 99
6.2.1. In-Plane Uniaxial Stretching ... 99
6.2.2. Pure Bending ... 101
6.2.3. Model Comparison ... 103
6.3. Transverse Shear ... 103
6.3.1. Cross Section ... 104
6.3.2. Transverse Stress-Strain Curve. ... 108
6.4. Interactions- Failure Limits... 109
6.4.1. Biaxial Stretching ... 109
6.4.2. Stretch - Bending ... 110
6.4.3. Double Curvature ... 111
6.5. Interlaminar Shear Strength Failure... 112
6.5.1. Constant Layerwise Shear Method ... 114
6.5.2. Double Cross Section Method ... 117
6.6. Influences Of Laminate Composition ... 118
6.6.1. Influence Of Layers... 118
6.6.2. Homogenous Laminates... 120
6.6.3. Yield Point... 122
6.6.4. Interface Shear Strength ... 125
6.7. Double Cross Section Iterating. ... 127
6.7.1. Maximum Interface Shear Stress. ... 128
6.7.2. Accuracy of Parabolic Layerwise Shear Distribution... 129
6.8. Thickness Change / Neutral Fibre Position... 130
7. Test Method and Results __________________________________________ 133 7.1. Three-Point Test Method ... 134
7.1.1. Laminate Test Direction ... 134
7.1.2. Force-Displacement Curve... 135
7.1.3. Test Speed ... 136
7.2. Stiffness from Three Point-Bending Tests... 138
7.2.1. Initial Results ... 139
7.2.2. Sandwich Shear... 141
7.2.3. Beam Shear ... 142
7.2.4. Curve Fitting... 143
7.2.5. Influence Of Stiffness Components ... 144
7.2.6. Results For Different Laminates... 146
7.2.7. Results For Different Rotation Angles... 150
7.3. Iosipescu Out-Of-Plane Stiffness Tests ... 150
7.3.1. Monolithic Aluminium ... 151
7.3.2. Glare 3-5/4-0.4 L ... 152
7.3.3. Conclusion ... 153
7.4. Edge Effect on Specimen Stiffness... 153
7.5. Strength from Bending Tests ... 156
7.5.1. VM-Plots ... 156
7.5.4. VM Failure Limits For Different Laminates... 159
7.5.5. Results For Different Rotation Angles... 161
7.5.6. Glass-Epoxy Composite... 163
7.6. Edge Effect on Specimen Failure... 165
8. Discussion _____________________________________________________ 169 8.1. Test Results outside Design Tool... 169
8.1.1. Flange Length ... 169
8.1.2. Delamination Buckling ... 170
8.2. Stiffness Calculations & Measurements ... 172
8.2.1. Thickness Differences ... 172
8.2.2. Bending Stiffness... 174
8.2.3. Shear Stiffness... 174
8.3. Bending Strength... 175
8.4. Interlaminar Shear Strength ... 176
8.4.1. Tmax Determination for Cl-Gp... 178
8.4.2. Tmax Determination for Double Cross-Section ... 181
8.4.3. Comparison between Model Results... 184
8.5. Explanation of Differences in VM... 184
8.5.1. Different Loads At Failure Point ... 184
8.5.2. Incorrect Shear Stress Distribution Model... 185
8.5.3. Yielding Of the Matrix ... 186
8.6. Improving the Predictions... 186
8.6.1. Force Introduction... 186
8.6.2. Composite Plasticity. ... 186
8.6.3. Failure Criterion for Cl-Gp... 186
8.6.4. Corner Points for Dcs... 188
8.6.5. Relevancy for Production ... 190
9. Conclusions ____________________________________________________ 193 9.1. Deformation Model... 193
9.2. Fibre Metal Laminate Behaviour ... 194
9.3. Design Tool Usability ... 195
9.4. Recommendations... 195
Appendix A: Fitting Of Third Order Function ________________________________ 201 Appendix B: Forming Program ___________________________________________ 203 B.1. Introduction... 203
B.2. Installation... 204
B.3. Starting the Program ... 204
B.4. Outline Of Tasks ... 206
B.5. Files... 208
B.5.1. Library Functions ... 208
B.6. Material and Laminate Definition Tools... 209
B.6.1. Material Properties... 209
B.6.2. New Material ... 210
B.6.3. Material Editors... 211
B.6.5. Use Of Materials... 213
B.6.6. Laminate Selector ... 213
B.6.7. Laminate Editor ... 214
B.7. Deformation Level Functions ... 216
B.7.1. Classic Plate Theory... 218
B.7.2. Constant Layerwise - Parabolic Global Shear Strain Deformation Theory... 219
B.7.3. Double Cross Section Deformation Model... 220
B.7.4. Double Cross Section - Iterating Deformation Model... 220
B.7.5. Double Cross Section With Global Parabolic Deformation .. 221
B.8. Process Level Laminate Functions ... 222
B.8.1. Interactive Graphical Plots ... 222
B.8.2. Stress-Strain Curve ... 223
B.8.3. Bending Moment - Bending Radius Curve ... 223
B.8.4. Bending Radius - Product Radius Curve... 224
B.8.5. Stretching ... 225 B.8.6. Stretch Bending... 225 B.8.7. Double Curvature ... 226 B.8.8. VM-Curve... 226 B.8.9. Laminate Overview ... 227 B.8.10. Bending Wizard... 228 B.9. Other Functions... 230
Appendix C: Forming Script Language _____________________________________ 231 C.1. Introduction... 231
C.1.1. Data Types ... 232
C.1.2. Flow Control Statements ... 233
C.1.3. Math Module... 233
C.2. Forming Module ... 234
C.2.1. Deformation Model Functions... 234
C.2.2. Laminate Functions ... 235 C.2.3. Chart Functions ... 236 C.3. Example Programs ... 236 C.3.1. Plot A Circle ... 236 C.3.2. Laminate Information ... 237 C.3.3. Stress-Strain Curve ... 237 C.3.4. Corner Points... 237
Appendix D: Programmers Guide _________________________________________ 241 D.1. Program Language ... 241 D.1.1. Data Types ... 241 D.1.2. Form Files... 242 D.1.3. Streaming ... 242 D.1.4. Components ... 242 D.2. Units ... 242 D.2.1. Main ... 243 D.2.2. Supporting Units ... 243
D.2.4. Material Models... 244
D.2.5. Material Level Interface Functions ... 244
D.2.6. Laminate Models... 245
D.2.7. Laminate Level Interface Functions ... 245
D.2.8. Direct Low-Level Calculations... 246
D.2.9. Generalized Low-Level Calculation ... 246
D.2.10. High Level / Process Level ... 247
D.3. File Format... 247
D.4. Adding Material Models ... 248
D.4.1. Creating The Unit And Class... 248
D.4.2. Specifying Parameters ... 248
D.4.3. Strain Stress Function... 249
D.4.4. Integrating in to the Program... 250
D.4.5. Adding A Failure Mechanism ... 250
D.5. Using the Calculation Units ... 251
D.5.1. Crosscalc ... 251
D.5.2. Cross_Trans_Calc ... 252
D.6. Adding Process Simulations ... 253 Appendix E Test Measurements___________________________________________ 255 Appendix F: Out-of-Plane Stiffness ________________________________________ 261 Appendix G: Lay-Up of Glare Laminates ___________________________________ 265 Acknowledgements ____________________________________________________ 269 About The Author______________________________________________________ 269 Summary_____________________________________________________________ 271 Samenvatting _________________________________________________________ 273
LIST OF SYMBOLS
A,B,D terms in laminate stiffness matrix
a,b,c parameters of specimen stiffness fitting function
C Voce constant
Cij stiffness component (in ij direction)
Cijk stiffness component (in ij direction for Layer k) C stiffness matrix
d width of laminate
E Young’s modulus
Ex stiffness in fibre direction,
Ey stiffness perpendicular to the fibres eij strain deviator component
F force
Ffibre strength of a layer in fibre direction
Fmatrix strength of a layer perpendicular to fibre direction Fshear shear strength of a layer
G shear modulus
Gxy in-plane shear stiffness Gyz. out of plane shear stiffness I moment of interia
k indication of a layer in the laminate
l length
L support span
Li support span of Specimen i M moment per unit width N normal force per unit width n strain-hardening exponent P force applied by the test machine
Q distance between support and the edge of the specimen R residual difference (error)
Ri residual difference (error) of Specimen i sij stress deviator component
S compliance matrix t thickness of laminate tk thickness of layer k
u deformation (function) in x-direction
unk layerwise coefficients in deformation function (nth coefficient of the kth layer)
v deformation (function) in y-direction
vk deformation (function) in y-direction of Layer k V transverse shear force per unit width
w deformation (function) in z-direction
wk deformation (function) in z-direction of Layer k wbending deformation in z-direction due to bending wgbl global deformation in z-direction
wshear deformation in z-direction due to shear w0,w1,.. coefficients in deformation function
x,y,z directions in othonormal co-ordinate system.
x,y,z co-ordinate system axis, x,y are in-plane, z-out of plane. zk location of interface between Layer k-1 and Layer k
Greek
δij Kronecker delta
ε
strain tensorε1 in-plane main strain ε2 in-plane main strain
ε3 out of plane component of in-plane main strains εe equivalent strain
εfail failure strain
εi strain component (vector notation) εij strain component (tensor notation) εkk sum of the diagonal strain component
εx strain in x direction (primary in-plane strain) εx0 global strain in x direction
εy strain in y direction (secondary in-plane strain) εy0 global strain in y direction
εz strain in z direction (thickness strain) ν Poisson;s ratio
κ curvature
dλ proportionality factor between effective stress and effective strain rate.
λ failure multiplier
φ rotation of the cross section
γxy shear strain in xy direction (in-plane shear) γxy0 global strain in xy direction (in-plane shear)
γxz shear strain in xz direction (primary transverse shear) γyz shear strain in yz direction (secondary transverse shear)
ϕ proportionality factor between effective stress and effective strain. κgbl global curvature (in x-direction)
κx curvature in x-direction κxy in-plane shear curvature κy curvature in y-direction νxy in-plane Poisson’s ratio νyz out of plane Poisson’s ratio
σ
stress tensor σ0 yield stress σe equivalent stressσi stress component (vector notation) σij stress component (tensor notation) σii sum of the diagonal stress components τ shear stress
τmax maximum shear stress at an interface τxzk shear stress in xz direction of Layer k ψ tilting function
1. INTRODUCTION
Structural engineers are constantly looking for ways to reduce the weight of their designs. This weight is determined by material properties, structural con-cepts and manufacturing methods. In this introduction we will initially focus on the material properties and the structural concepts. These two are often hard to distinguish1. Especially for so-called laminates, engineers use the term material for the structural concept. The manufacturing methods can only be discussed after the material properties and the structural concepts have been discussed.
1.1. Laminates
Laminates are stacked plies of materials. The individual plies can be made of different materials. For structures that are loaded in a plane stress situation, the use of laminates allows an optimization and tuning of the material properties. In the history of manmade structures, different reasons have led to applications in the form of laminates. Below, the advantages of using laminates for specific material groups are highlighted.
Wood has anisotropic properties, along its fibres the strength is much higher than perpendicular to its fibres. For wooden beams this can be an advantageous property, but for plates it is a handicap. The anisotropy can be reduced by using thin layers at different orientations bonded together [1]. This is plywood and was already used by the ancient Egyptians. When engineering knowledge in-creased, the anisotropy of the plywood was made to match the needed structural performance. The “De Havilland Mosquito” was a prime example of a wood structure; five types of wood were used to get the best structural performance [2]. Another benefit of plywood is that it is possible to create plates with a larger size then the original tree-trunk [3]. A plywood structure can therefore have fewer joints.
Metal laminates have been used to improve durability by increasing the fracture toughness. Steel-iron laminates were used as armour in the fifteenth century [1]. Later on, metal laminates were also used for corrosion protection in silver-plated sheets and clad aluminium sheets.
After the discovery of the very good properties of glass in fibrous form, other fibres where developed. These materials could only be used in structures by making bundles. Although bundles can be used directly they are commonly combined into fabrics. These fabrics are impregnated with a matrix, then cured to become a laminate. Before curing, the material can be easily draped and formed on a mould; this gives a very large freedom of shape. A more recent de-velopment is the use of a thermoplastic matrix material. This allows the lami-nate to be formable under elevated temperature, and weldable [4].
In the last decades a start has been made with hybrid laminates, i.e. metal plies and composite plies blended or metal plies and polymer plies. At the moment, in the class of the metal-polymer-laminates (Figure 1.1) there exist two concepts of structural importance: the Fibre-Laminates (FML) and the Metal-Polymer-Sandwiches (MPS). These materials will be studied now in more de-tail.
1.1.1. Fibre-Metal-Laminates
Fibre-Metal-Laminates originate from metal bonding. In the 1950’s metal bonding was developed in the aerospace industry to avoid inter rivet buckling of thin sheets. Although riveting was perceived as an industrial process, the bonded structure showed higher (compression) strength [5].
In 1954, the well-known accidents with the De Havilland Comet occurred and metal fatigue became a serious issue in the aerospace industry. During fatigue testing at Fokker and the NLR, it was discovered that the laminated metal sheets had a favourable resistance to fatigue and in addition showed high damage tol-erance.
In the 1970’s there were experiments in which a metal structure was reinforced by composites; this inspired researchers at Fokker to experiment with fibres added to their laminated metals. The observed two or three times slower crack growth was however not spectacular in view of the costs of the laminate.
Metal-Polymer Sandwiches Laminates
Metal Polymer Laminates
Fibre-Metal Laminates
Composite Laminates ...
Glare Hylite
... Arall ... ...
At Delft University of Technology it was shown that thinner sheets (0.3-0.5mm) improved the fatigue crack growth resistance even more (10-1000 times) and even could result in a stop in crack growth. During the 1980's the material was further developed. The first standardised family of FML’s used aramid fibres and was called Arall. During the testing of Arall it was found that the aramid fibres did not perform well when the laminate was subjected to compressive loads. This led to the choice of S2-Glass fibres embed-ded in epoxy between 2024 sheets: the Glare family. At the same time other beneficial prop-erties of the laminates became apparent such as the impact strength, residual strength and its fire resistance properties.
In the 1990's the research was focused to make Glare ready for a major application. Especially the splicing concept (Figure 1.3) was important to reduce the costs of the finished product. With splices the Glare panels can be much larger than the size of the metal sheets. Also a lot of effort was put into quality assurance of the laminates. Studies at Airbus showed that Glare could lead to a 25% weight reduction [6]. In the second half of the 1990's Airbus showed sincere interest in applying Glare on the A3XX that was under development at that time. Finally this inter-est resulted in the commitment of Airbus to apply Glare in the fuselage of the A380, (as the A3XX was renamed).
This does not mean that the fundamental research in Delft stopped. The concept of Fibre-Metal-Laminates is still under development and several fundamental questions still need to be answered. Each new application of Glare gives new questions that have to be solved.
The co-operation between the people that develop the Glare material and the people that design a fuselage led to interesting insights. For example, Glare 4 was developed with the internal pressurisation of the fuselage in mind. The in-ternal pressure in a cylinder leads to circumferential stresses in the skin which have twice the magnitude of the longitudinal stresses. Therefore Glare 4 has twice as many fibres in the circumferential direction as in the longitudinal di-rection. However, in the actual fuselage the bending may be dominant, which Figure 1.2: Glare
circumferential direction. The orientation of the metal layers is fixed during production. This demonstrates the flexibility of the Glare concept.
The same co-operation led to the invention of the splice concept. The splices could not only be used to create a laminate of a larger size than the sheets it is composed of, splices could also be used to insert local reinforcements in the laminate. The local reinforcement means that these laminates must be consid-ered a structure instead of an advanced material.
1.1.2. Metal-Polymer-Sandwiches
Metal-Polymer-Sandwiches (MPS) are laminates consisting of two layers of metal with a polymer layer in between (Figure 1.4). They have a very low weight compared with metal sheets with the same bending stiffness. The low density core material increases the specific mo-ments of inertia significantly with only a small weight penalty. Corus, Alusuisse and others have developed these sandwiches. The Corus sand-wiches are named Hylite [7] and consist of AA5182 skins with a thickness of 0.2 mm and a polypropene core available in several thicknesses. The skin material can be delivered in a soft an-nealed or a full hard condition. The Alusuisse sandwich has a similar composition [8] and is marketed as wall-panel while the Corus sandwich is marketed for automotive applications. The smooth outer surface and the high specific bending stiffness makes the sandwich ideal for horizontal body panels like the roof, bonnet and trunk.
1.1.3. Composite Laminates
Composite laminates consist of layers of fibrous material that are impregnated by a matrix material that bonds the fibres together. Most composites use some kind of polymer for the matrix material although there are metal matrix com-posites. The matrix allows loads to be transferred from one fibre (layer) to the other thus providing out-of-plane stiffness.
For structural applications two fibres are common: carbon fibre and glass fibre. For each fibre, different types are available with different properties. The most used matrix materials are thermosetting polymers (polyester and epoxy). Re-cently thermoplastic matrix composites were introduced for aerospace applica-tions. Composites will not be investigated in this research since they are not cold formable. They are, however, almost always an alternative for products that are made with Metal-Polymer Laminates and a large part of the analysis origi-nates from composite analysis.
1.2. Laminate
application
In order to understand the requirements on Metal-Polymer Laminates, it is im-portant to understand the type of structure in which they are used. The nature of laminates makes them mainly useful for thin-walled structures. The dominant structural concepts for thin-wall structures are discrete stiffened shells and sandwiches. Thin-walled structures must have some provision to withstand out-of-plane loads and compressive loads. Thin walls by themselves cannot support these loads.
For discrete stiffened shells (Figure 1.5a), structural elements as stiffeners and frames provide the out-of-plane stiffness which prevents buckling. For the sandwich concept (Figure 1.5b) out-of-plane loads can be sustained by differen-tial in-plane loads in the two facings. The low-density core provides plane shear stiffness. Due to the large cross-sectional area of the core the out-of-plane shear stresses are relatively low. The sandwich concept has a ‘clean’ look; however this might change when other structures are attached to the sandwich. In-plane loads must be transferred to both facings and out-of-plane loads must be transferred as a shear load into the core. This might require inserts or a tem-porary deviation of the sandwich concept.
The two structural concepts are not bound to a specific material. The Dassault Atlantique uses a sandwich with metal facings with a honeycomb core while the Extra 400 uses carbon facings with a foam core. The Airbus A380 uses the stiff-ened shell concept for the carbon empennage as well as (weldable) aluminium in the lower fuselage and Glare panels in the upper fuselage.
Although laminates in general are not bound to a specific structural concept, the two major Polymer-Laminates, Fibre-Laminates and Metal-Polymer-Sandwiches, were designed with a different structural concept in mind. The Metal-Polymer-Sandwiches are, as their name says, sandwiches with their high specific bending stiffness. The FML’s were developed for their high in-plane properties (especially their fatigue strength and damage tolerance), with the conventional stiffened shell structure in mind.
A
B
frame stiffner skin core facing insertBoth laminates have more or less the same thickness in their normal composi-tions, between one and five millimetres. For the FML’s this is only the skin thickness while it is the structure (sandwich) thickness for the MPS’s. It can therefore be noticed that the FML-structures are likely to be larger than the MPS structures.
Both laminates, as is typical for all new materials, have suffered from a hesita-tion to use the laminates due to lack of experience. Only recently has this changed. In the future it is very likely that the use of laminates in general will increase since most products with a significant structure are in a consolidation stage. Therefore needed improvements must come mostly from optimising ex-isting structures and less from new concepts2. A more optimised structure can mean more laminates, since laminates provide a mixture of existing material properties and can be fine-tuned for their application. There have also been im-provements in the analysis methods, which make it possible to optimize such designs.
Another difference between Glare and Hylite is the intended market. Glare has been developed for aircraft structures and Hylite for automotive applications. Both laminates were optimised for their own application. A new application may require new instances of FML or MPS. Such an approach where the “mate-rial” is designed for the application will become more common with the im-proved control in the production of formerly bulk products as materials were. The main disadvantages of laminates are the production cost and the joining techniques. The joining of laminates does not give any problem; the problem lies in the fact that welding outperforms other joining techniques and laminates are in general not weldable. The cost is difficult to analyse since the only valid comparison is by the price of the complete structure including the production cost for the intended production rate.
1.3. Forming
The two types of laminates (FML’s and MPS’s) have different applications; however, the problems involving manufacturing have, on a fundamental level, similarities. During manufacturing there are roughly two steps: the creation of parts and assembly of the parts into the structure. In this research project we will focus on forming, which is one manner to create parts. Common sheet metal forming processes in the aerospace industry are (Figure 1.6): bending, stretching and rubber pad forming.
Forming processes are those processes in which a semi-finished product like a flat sheet or a straight profile is deformed permanently into a specific shape. The forming processes are common processes in metal workshops. The fact that Metal-Polymer-Laminates contain metal layers indicates a potential that the laminates can be formed in a manner similar to that of monolithic metal sheets. Forming is not the only method to create products from laminates; it is also pos-sible to deform first the layers and than make a laminate from the individual plies. In general the plies of a laminate are very flexible; therefore a mould is used in the laminating phase (Figure 1.7). Of course flat laminates are created using a flat mould; thus the difference in the laminating process with a product-dependent mould is not that large.
However, forming of laminates allows for different shapes then the lay-up proc-ess because the deformation principle is different. With forming of Fibre-Metal-Laminates the (cured) prepreg layers are deformed elastically and the metal lay-ers are deformed plastically. The obtainable deformation by forming is often limited by the amount of strain that the glass fibres can take or by the strength of the interfaces between the layers. The lay-up approach uses the flexibility of the prepreg before curing while the metal layers are only deformed elastically. The lay-up process is limited by the deformation of the metal layers at the rela-tively low pressure during curing.
The two shaping methods are more or less complementary to each other. In the aerospace industry, forming is the method used for structural elements like
stiff-aa bb cc
Figure 1.6: Forming processes. a: Bending b: Stretching c: Rubber pad forming.
ply preparation
laminating forming
laminating to shape
product
When the product has special requirements it is possible to use a combination of the two technologies. Some structural elements can be made by forming thin laminates and then bonding (with prepreg or adhesive only) the formed parts to become a thick laminate. Pre-forming of the thin metal sheets before laminating is too difficult and expensive to be cost effective.
If the product can be produced both by laminating and by forming, then a cost benefit for forming can be expected. Flat laminates are easier to produce by the lay-up process. Flat laminates need a flat plate as a mould instead of a more ex-pensive shaped mould that is product-specific. Stacking laminates directly on top of each other allows more laminates in the autoclave at the same time com-pared to a curved mould. The simpler (faster) lay-up reduces the lay-up costs and the expensive autoclave cycle is shared by a larger amount of laminates. Another benefit of forming is that laminate production can be separated from part manufacturing. One is able to buy the laminates with no need for laminat-ing equipment, clean-room environments and the technical know-how. More-over, the forming equipment is identical to that used for sheet metal.
1.4. Objectives
When products are formed with Metal-Polymer-Laminates there is a large inter-action of the laminate composition and the production method on the produci-bility of the design. This interaction
be-tween the design, material and production method is larger for the Metal-Polymer-Laminates than for the more traditional materials. This is a result of the additional failure modes of a laminate and the influ-ence of laminate composition parameters on these failure modes.
The composition of the laminate and the shape of the product are design driven. For the sake of producability it might be
un-avoidable to change either the shape of the product or the composition of the laminate. On the other hand, a different process parameter can prevent a failure during production. This strong relationship underlies the interaction between material, design and production (see Figure 1.8).
The research project described in this report aims for the development of a tool to support the designer with the production of his design with laminates. This includes the development of one or more models inside the design tool for the laminate behaviour. The tool must be able to predict the formability properties, like failure limits and required forces, of different laminates. The tool should be capable of treating not only existing laminates but also laminates that will be developed in the future. This must result in a fast and flexible design tool for the evaluation of the forming properties of these laminates (Figure 1.9).
For monolithic materials, especially metals, there are a large number of existing forming tools, knowledge and software. This research therefore aims at those properties for which the forming properties of laminates differ from monolithic materials. Those differences are more prominent at specific locations in larger products, since the difference between the forming of laminates and monolithic sheets lies mostly in the differences in failure mechanisms. The principle behind the force-displacement relations (including plasticity) is the same for laminates and monolithic sheets. The design tool will therefore focus on local phenomena within a larger product.
The number of layers and the function of each layer are not predetermined, an idealised approach possible for sandwiches. Forming processes use out-of-plane deformations; the stacking order of the layers is thus important. Therefore a rule of mixtures (metal volume fraction) approach cannot be used. In addition, the plastic behaviour needed for forming typically involves non-linear calculations. It is therefore inevitable that any formability prediction model will require nu-merical solution methods. It is preferable that these nunu-merical methods are in the form of a computer program that can be used by the end user. This program has to be a tool in which the relations between the laminate, the process and the product are incorporated.
Since the subject of forming is very broad and Metal-Polymer-Laminates are relatively new, some limitations are necessary. Forming processes involving elevated temperature, strain path dependency or strain rate dependency, for ex-ample, are not taken into account.
1.5. Research
Approach
Three approaches can be identified for the development of a formability predic-tion tool:
1. Building such a tool using empirical relations based on test results. With this method large numbers of tests should be performed to find the required re-lations. It is not guaranteed that the found relations will lead toward a fun-damental understanding of the forming behaviour, and it could be hard to predict the behaviour of new laminates by extrapolation.
2. A method using detailed finite element analyses of the laminate. This method would probably lead to a close fit between experiments and calcula-tions, although the amount of needed verification experiments is not known. Although it would certainly show the internal behaviour in the laminate, it would not lead to a flexible design tool.
3. Building a tool based on an analytical model (numerically solved). This would be the most difficult approach, and it would not be certain whether the model would be able to predict the actual behaviour. However, when this method works, it will be by far the most flexible.
Each approach has its noted advantages and disadvantages. However, the pros-pects of the last method are the best. The first method is more or less close to the current practice of obtaining the forming-properties of a laminate on a trial and error basis, extending previous experience [9]. It would be hard to extend this approach to achieve a predictive capability. Therefore it was decided not to use the empirical relations approach.
The primary decision was whether to use a finite element approach or an ana-lytical approach. The anaana-lytical approach was chosen for this research on the basis of the following arguments:
• It would be relatively easy to evolve the model into a design tool, without special requirements on hardware or software.
• It would lead to a better understanding of the parameters involved, since there is less of a "black-box" involved when everything has to be specified in the model.
• An analytical approach would have the best guarantees for an accurate pre-diction for laminates other than the validated laminates.
• It would need less calculation (due to the analytical foundation) and there-fore would give results faster.
1.6. Outline
The text of the dissertation will follow the development of the tool and the models inside it. The text starts with the background information, which is fol-lowed by construction of the forming prediction models and the structure of the design tool, and finishes with the verification of the developed models.
The first piece of background information that is needed before the model can be built is the differences between monolithic sheets and laminates. The model should focus on the ways in which the laminates differ, since otherwise the ex-isting prediction tools can be used. The main differences between monolithic sheets and laminates are the layered structure, the different failure modes and the boundary conditions at the interfaces. This is described in Chapter 2. The next information that is needed is the deformation model. The deformation model gives the relation between the external deformation and the internal strains and between the internal stresses and the external forces acting on the laminate. It must use the interface conditions as found in chapter 2 and it must be capable at calculating the factors involved in the failure prediction. The de-formation models are described in Chapter 3.
Although a deformation model is the core of a forming prediction tool, it needs more components to be useable. Since it is desirable to compare different mod-els with each other, the computer program becomes a framework for forming prediction models. The structure of this framework is the topic of Chapter 4, together with other items related to the implementation of the models in a com-puter program. When the structure of the program is known, it is time to de-velop the components that will fit into that structure. Chapter 5 describes the material model components, the laminate model, the calculation models and the process models that fit into the designed framework.
The interpretation of the model results starts in Chapter 6 with the rationalisa-tion of the model results. It must be possible to relate the output of the model to its input. However the real verification must be done on the basis of these re-sults. Chapter 7 describes the test methods used and the test results, and in Chapter 8 the model predictions are compared to these test results.
2. LAMINATE
BEHAVIOUR
Most forming methods used in aerospace engineering for thin plate structures are developed for monolithic sheet materials. To apply, and if necessary modify, these methods for Metal-Polymer-Laminates, a proper understanding of the de-formation behaviour of laminates is needed.
The major difference between a laminate and a monolithic material is of course the layered structure. This implies that a laminate can delaminate, a mechanism by which the integrity of the laminate is lost. Consequently a laminates have more failure mechanisms than monolithic materials. Implicitly related to layered structure is the fact that most laminates consist of different materials. These dif-ferent materials can have difdif-ferent failure modes. But equally important is that the differences in mechanical behaviour result in complex internal stress and strain distributions. The interface between two layers has special boundary con-ditions and the edge of the laminate has special stress redistributions.
This chapter first defines the terminology used to describe the composition of the laminate, then the different failure modes of laminates. These failure modes must be implemented in the design tool. The chapter continues with a descrip-tion of the boundary condidescrip-tions at the interface, and a short descripdescrip-tion of the free-edge effect found in laminates.
2.1. Laminate
composition
A laminate consists of a number of stacked plies, the layers. The properties of a laminate are determined by the properties of the individual layers (plies) and by the stacking order. The stacking sequence determines how the properties of the individual plies are reflected in the properties of the laminate as a whole. Lami-nates can have a symmetric or an asymmetric stacking sequence; the symmetry is with respect to the central plane of the laminate.
Each ply is made of a material. In a Metal-Polymer-Laminate there is at least one ply made from a metal (steel, aluminium, titanium, etc.). A polymer layer can consist of a pure polymer (ABS, polypropylene, epoxy, etc.) or a more complex material such as a fibre-reinforced polymer where the layer itself con-sists of multiple materials. The properties of continuous fibre reinforced poly-mers strongly depend on the orientation of the material, and they are ortho-tropic. Other materials in laminates may also show anisotropy. Therefore the
Each ply can be characterised by its position within the laminate, material type, orientation and thickness. In practise the layer thickness cannot have an arbi-trary value. For each material there are only certain thickness available; this de-pends on the manufacturer.
Laminates are not just a collection of plies, the plies must be mechanically con-nected to each other. Often some kind of adhesive is used to connect layers to-gether. A layer in a laminate can also act as the adhesive, or a major component in a layer can act as the adhesive. In Glare the matrix in the glass fibre layer bonds with the aluminium layers. The adhesive connection is however not a re-quirement for laminates; other methods are also be possible such as (ultrasonic) welding and brazing.
For the further discussion it is useful to define a co-ordinate system for the laminates. The x-y plane will be the plane in which the laminate lies. In the laminates under study some layers are orthotropic. Therefore it is important to define the orientation of each ply with respect of the global orientation of the laminate. The z-axis is perpendicular to the laminate; therefore all rotations are with respect to the z-axis (Figure 2.1). It is customary to use as the laminate x-axis either the dominant direction of the structure the laminate is used in (-45°/+45° for shear webs instead of 0°/90° for composites) or the direction of important or visible layers (the rolling direction of the aluminium sheets for Glare).
Since the outside layers of most metal-polymer-laminates are metallic layers, the metal designation for the sheet orientation is commonly used for the expla-nation of the test direction. The rolling direction of those metal layers, the L-direction, is used as the x-direction for the laminate orientation. The LT-direction is perpendicular to the L-LT-direction and thus concurs with a rotation of 90°. For the Glare laminates the orientation of the fibre layers is always defined with respect the rolling direction of the aluminium layers. (All the aluminium layers have the same orientation.)
Layer angle Laminate angle L am in ate th ic k n es s La y er th ic kne ss
Z
Z
X
X
processY
Y
process 55 44 33 22 11Y
Y
laminateX
X
laminateThe laminate itself can also be rotated; the analysis will be done in the co-ordinate system of the process. In the process co-co-ordinate system, the primary deformation direction is along the x-axis.
2.2.
Typical laminate failures
The formability of laminates is limited by failure. The failure can take place in one of the constituents (constituent failure) or at the interface between the plies (delamination failure). For both cases a number of different failure modes can occur. In order to get a good formability prediction, the different failure modes have to be identified, the mechanism that causes the failure has to be determined and failure criteria have to be found. Whether or not a certain phenomenon oc-curring during forming should be considered as a failure mode depends on the application of the laminate. Sometimes a forming-induced surface roughness could be regarded as a failure while for some other product certain types of “damages” could be allowed. The eventual forming prediction tool must allow the user to select the failure modes to be considered and the limit settings of the related parameters. In this thesis a laminate will be called failed if some sort of crack occurs in the laminate.
The formability limits are closely related to the failure limits of laminates under mechanical loads. The classification of the failure modes will be done on a practical basis, not on the fundamental cause of the failure. This means that the determination of the different failure modes will be done based on unmagnified visible characteristics.
2.2.1. Constituent failure
The term constituent failure is used for intra ply failure. This is the normal fail-ure type for monolithic sheet (metal, polymer or homogenous composite). It oc-curs when the strain in a sheet exceeds the limit strain for that material. The constituent failure is, of course, different for different constituents. Since the constituent materials are not limited in this research project it is impossible to provide a complete list of all the possible constituent failure modes. This section gives examples of known constituent failures in laminates; these failure modes must be implemented in the design tool. Other constituent failure modes may exist but are not simple to test. When the material models are implemented in the design tool (§5.2), the different constituent failure modes of that material are combined into a single (multi-mode) failure criterion.
Metallic plies
For a metallic layer there is only one failure mode in practice: tension (Figure 2.2). In a laminate, the nor-mal location of this failure is at the outer metal layer. The fracture is identical to the fracture for mono-lithic metal sheet. However, in a laminate the fracture does not extend over the whole thickness. It is possi-ble to create laminates where this failure occurs inside the laminate. Usually this type of fracture is ac-companied by a loud sound. When a layer is cracked, the load of that layer must be transferred to the other lay-ers through the interface between the cracked layer and the other layers. This load distribution can invoke delaminations.
Composite plies
The composite layer in a laminate shows more than one failure mode. One of these is the failure of the matrix. This occurs when a laminate is bent with the bendline parallel to the fibre direction (Figure 2.3). The tensile failure strain in that direction is lower that the failure strain of the surrounding metal layers. This failure type cannot be observed from the outside. Due to the already very limited contribution of the matrix to the me-chanical properties of the laminate, it is very likely that the influence of this failure on the mechanical properties of the laminate can be neglected, as long as no delaminations are induced. On the other hand, when durability
has to be considered this failure may not be tolerated, since moisture could enter deep into the laminate through the cracks.
Another likely constituent failure mode in a composite ply is the failure of the fibres in compression (Figure 2.4), kinks. At a higher magnification it is clear that the failure is induced by the buckling of individual fibres. The buckled fi-bres force the outside metal layer into a rim, which is visible from the outside. Compared to delamination buckling (§2.3.1), the rim of the compression failure is smaller, rougher and positioned near the centre of the bending zone.
Figure 2.2: Cracked metal layer of a v-bend specimen
In a composite ply there is also the possibility of the failure of fibres in tension (Figure 2.5). Since the fibres are able to take a major part of the overall tensile load on the laminate, it is likely that after the failure of a fibre layer the whole laminate will fail violently. The huge amount of elastic energy released in the specimen “shatters” the laminate. The fracture area is large, with fibres sticking out of the laminate. Therefore it is unlikely that that this type of failure will re-main unnoticed.
Composite materials have more failure modes; most of these more exotic failure modes have to do with displacements of fibres within the matrix [4].
Figure 2.4: Fibres failed in compression.
2.3. Consistency
failure
Consistency failures are unique for laminates; with these failures the coherence of the laminate is lost. The individual layers are undamaged or damaged due to secondary effects. Consistency failure can also be a secondary effect of con-stituent failure.
The classification of the consistency failure modes will be based on the visible damage of the laminate and not on the fundamental cause of the failure. On a fundamental level there are only two mechanical failure drivers for these fail-ures: peel stress and shear stress. Peel stress is the stress perpendicular to the surface of an interface; shear stress is the stress along the interface (Figure 2.6). The reader may have experienced (with duct-tape for instance) that the shear strength is much higher than the peel strength.
Delamination in this context is a secondary process; the failure originates from a fracture surface that is induced by an exceedance of the peel or the shear strength, or a combination of both.
2.3.1. Delamination Buckling
Delamination buckling is the failure mode in which one or more layers of the laminate locally separate from the remainder of the laminate. The delaminated layer(s) was/were subjected to high compression loads. It is a failure mode that can occur during various forming processes.
This failure mechanism is not included in the analytical model since no failure criterion could be developed. At the start of the research there was only an as-sumption about the origin of this failure mode: the buckle is assumed to start from a peel load at the stress redistribution at a free edge [10]. The assumption that the delamination buckling is just a special Euler buckling case could also be
fracture surface
peel shear
Figure 2.6: Peel and shear stress induced failures.
valid. For both mechanisms no analytical failure criterion could be found. More insight in this failure mode is needed.
2.3.2. Interlaminar Shear failure
The interlaminar shear failure is a consistency failure mode in which an inter-face fails due to the lack of shear strength. This can be achieved by loading a specimen with a high out-of-plane (transverse) shear force. With microscopy the crack parallel to the surface of the layers in the laminate is hard to find (Figure 2.8), even under a higher magnification, since the crack is not opened after fail-ure.
During cantilever bending the transverse shear stresses are zero on the outer sur-faces and it is assumed that there is a maximum near the centre of the laminate, as engineering beam theory predicts. Thus it is assumed that the laminate fails first near the central layer and that other cracks are secondary.
Under a higher magnification (Figure 2.8) it can be seen that the interlaminar shear crack occurs just inside the composite layer between the most outside glass fibres and the matrix. This is also true if a fibre layer is the central layer in the laminate.
Figure 2.8: interlaminar shear failure.
Compressive loads on inside tensile loads on outside
Slippage after failure
induces bending deformation
P
P
P
P
Figure 2.9: the seagull shape is caused by a slippage between the compressive and tension side of a laminate.
Although the crack of the interlaminar shear failure does not open, it does not mean that this failure is hard to detect. After interlaminar shear failure each de-laminated part cannot have a residual longitudinal force, therefore the delami-nated parts have roughly the same length (Figure 2.9). The length differences from the centre curvature must be compensated by secondary curvature at the end of the delamination crack. The resulting shape is called a "seagull shape" (Figure 2.10) and this distinctive shape provides a simple identification of the interlaminar shear failure.
This same phenomenon has also been observed in metal-sandwiches, with the difference that the slippage is not caused by the failure of an interface but by shear yielding of the core material. This results in a similar "seagull " shape. Although for metal-sandwiches no crack has occurred, the extra curvature in the product is unwanted, unless a forming process uses this core shear yielding as an advantage. The core material of the Hylite sandwich whitens due to plastic strain when it is yielded.
2.3.3. Edge delamination.
Edge delamination is related to the interlaminar shear failure, the difference is that the interface does fail due to a lack of shear surface, not the lack of shear strength. The distinguishing feature of the edge delamination failure is that the failure is located in an unloaded part of the laminate.
Figure 2.10: The typical sea gull shape after an interlaminar shear failure.
When the loading of an interface in a laminate under shear loading is observed (Figure 2.12), then in the idealized case the loads on the interface will be con-stant throughout the loaded part of the laminate. It is however likely that the buildup of the shear stress on the interface is more gradual. Some of the load buildup will take place in the unloaded part of the laminate. However when there is an edge close to the force introduction, this buildup is not possible and the interface might fail.
This failure can be identified by that fact that an edge of the laminate has de-laminated (Figure 2.13) and a crack opens the edge.
2.4. Internal
stresses
A laminate differs from a monolithic sheet not only by the failure mechanisms exhibited. Most laminates are composed of plies of different materials (or of the same orthotropic material under different angles). This means that the constitu-tive relations (the relations between the stress and the strain) are not the same for all layers, and different constitutive relations mean that the stresses in the layers are different when the strains are equal (Figure 2.14).
idealized situation
smooth build-up
missing due
to edge effect
Figure 2.12: Shear load of an interface in a laminate under shear loading.
For forming there is always an out-of-plane deformation, otherwise the sheet remains flat. This out-of-plane deformation creates a non-constant strain distri-bution over the cross section. The stress distridistri-bution over the cross section of a laminate is therefore non-constant in the thickness direction for two reasons: non-constant strain and non-constant constitutive relations.
In order to effectively form a laminate, one or more plies in that laminate must consist of a material that will undergo a permanent deformation. Permanent de-formation behaviour means also a non-linear stress-strain relation. Half the dis-placement does not mean that the magnitude of forces acting on the laminate are half the magnitude of the original forces.
This different reaction to equal strains in not only true for mechanically applied strains, it is also true for thermal loads. It is common for laminates that the lay-ers are bonded together at an elevated temperature. At this elevated temperature there are no internal stresses in the laminate. When the laminate is cooled after curing the difference in thermal expansion coefficients between different con-stituents causes internal stresses in the laminate, while there are no outside loads.
2.5. Interfaces
A laminate is built from different materials that are mechanically connected (bonded) together, thus creating interfaces in the laminate. The interface, the surface with different materials on each side, causes special requirements for the analysis of laminate behaviour.
At an interface between two constituents there is a discontinuity in the constitu-tive equations. This paragraph describes which stress and strain components are continuous over the interface and which must be regarded as discontinuous.
soft
stiff
displacementFF
11FF
22Figure 2.15 shows the stresses acting on two small vol-umes on opposite sides of an interface. In order to keep equilibrium, the stresses acting on the shared interface surface must be equal for both volumes. These stresses are σz, τyz and τzx; the remaining stresses acting on the volumes (σx,σy, and τxy) may be discontinuous.
For the strains the same analysis could be done. How-ever, it is the deformation of the layers that must be compatible and not the strains.
The values of the deformation functions must be the same on both sides of the interface at any location on the interface. It is defined here that the interface is par-allel to the plane spanned by the x- and y-axis. Thus the values of the deformation functions at both sides of an interface must be equal for all x and y co-ordinates. This can only be true if the derivative of the deforma-tion funcdeforma-tions with respect to the x and y co-ordinates are equal on both sides of the interface.
For the normal strains this distinction has no influence but for the shear strains it does change the required continuity conditions. The Cauchy’s equations give the relation between small strains and the u, v, w deforma-tions: z u x w z w y w z v y v x v y u x u zx z yz y xy x ∂ ∂ + ∂ ∂ = ∂ ∂ = ∂ ∂ + ∂ ∂ = ∂ ∂ = ∂ ∂ + ∂ ∂ = ∂ ∂ =
γ
ε
γ
ε
γ
ε
From these equations it can be observed that εx, εy and γxy must be equal for both constituents; however εz may differ. For γyz and γzx it follows that they may be discontinuous. That discontinuity is caused entirely by ∂v/∂z and ∂u/∂z terms. This immediately shows that a straight cross section does not necessarily remain straight during deformation of a laminate.
If the stress-based interface conditions are combined with the deformation-based interface conditions, it can be seen that for a transverse shear deformation on the sheet (γyz or γzx) with different shear stiffnesses of the constituents, the cross section cannot remain straight. The shear stresses must be equal at both sides of the interface, and thus the deformations must differ. This is an impor-tant conclusion to be used in the deformation models discussed in the next chapter.
Summarizing this, at the interface surface between Constituent A and Constitu-ent B the following relations must be valid:
σ
xσ
xσ
zσ
yσ
yτ
yzτ
yzτ
yzτ
yzτ
zxτ
zxτ
zxτ
zxτ
xyτ
xyτ
xyτ
xyFigure 2.15: Stresses on two infinitesimal small cubes (of different materials)
εxA=εxB σzA=σzB εyA=εyB τyzA=τyzB γxyA= γxyB τzxA=τzxB
This analysis shows that when a stress component must be continuous, the cor-responding the strain component can be discontinuous, and vice versa. It is im-portant that any description of the stress-strain distribution satisfies these re-quirements at an interface.
2.6. Edge
effects
There is another special phenomenon at the interface in a laminate and that is the stress redistribution at an edge of the laminate. This stress redistribution could lead to a peel force on the interface.
In a laminate the different constituents are mechanically connected to each other. This hinders each constituent in its
de-formation and thus introduces additional stresses in the laminate. However at a free (unloaded) surface all the stresses must be zero, even these additional stresses. There-fore a local stress redistribution is necessary. In what follows this effect will be shown for a uniaxial loading of a symmetric laminate. Under uniaxial tensile loading all the layers have the same tensile strain. Due to Pois-son’s ratio each layer also deforms perpen-dicular to the loading direction. The Pois-son’s ratios are different for the different layers; however, the contraction is the same for each layer. This means that for an infinite
plate under uniaxial loading there exist internal in-plane stresses perpendicular to the primary loading direction, although there is no resulting load in that di-rection. At the free edge these stresses must disappear, through the interfaces to the other layers. The interfaces lie on the outside of a layer and the Poisson-induced stresses work over the whole thickness of a layer (Figure 2.17). There-fore the stress redistribution includes a moment that can only be resolved by out-of-plane loads on the interface. If this out-of-plane load is a tensile load it could invoke the start of a laminate failure.
Figure 2.16: Peel stresses at an free edge.
This out of plane tensile load is the reason why the fatigue behaviour of a 0°/90°/90°/0° composite laminate differs from a 90°/0°/0°/90° composite lami-nate [11]. There are clues that a buckling failure can also have its origin at the laminate edge [12].
The laminate edge is a complex 3D-problem [14], especially for the case of a-symmetric laminates; it will therefore not be treated in the present study. The forming model will be limited to infinite laminates.
τyz
σz
3. LAMINATE
DEFORMATION
In this chapter we will discuss methods of obtaining “constitutive” equations for laminates. The main interest for forming analysis lies in the global behaviour of the laminate. The deformation (and the forces needed to obtain that deforma-tion) of the laminate is a global phenomenon while failure (as described in the first part of chapter 2) is caused by the local stress/strain state. This gives a need for “constitutive” equations that link external forces and deformations on the laminate to internal strains and stresses.
A laminate deformation model provides the “constitutive” equations on a lami-nate level (Figure 3.1), the relation between deformation of the lamilami-nate and forces/moments acting on the laminate. A
deformation model uses constitutive equa-tions of the laminate constituents and the boundary/interface conditions.
This chapter describes different laminate de-formation models, and their applicability for the prediction of the formability of polymer-laminates. The analysis of metal-polymer-laminates needs a plastic behaviour as well as a transverse shear behaviour. Many commonly used laminate deformation models do not include these behaviours, since they were developed for either com-posite laminates or thin-walled structures.
3.1. Approach
The analytical models will be based on the properties of the constituents and properties of the interfaces and not on experimental results of the whole lami-nate.
In order to use the properties of the constituents, the loads (either forces or dis-placements) on the laminate must be translated to the loads on the constituent in the laminate. The response of the constituents must then be transformed to the response of the laminate. This is the task of the laminate deformation model.
Figure 3.1: Interactions between global and local entities.
3.1.1. Deformation-based calculations
A laminate deformation model provides the relation between the external loads and displacements and the internal stresses and strains. Each of these four enti-ties can be used as the input parameters of the deformation models. One could use an assumed stress function and use this to calculate the strains and dis-placements. Timoshenko shows that this approach can be used to obtain beam deflections [13]. However, most forming problems are geometry driven and therefore deformation driven. This fact, in combination with the fact that most of the existing laminate deformation theories are based on displacements, made it attractive to choose displacements as the starting point.
The laminate deformation function takes an applied global deformation (§3.1.2) and transforms this, with a method that has yet to be determined, to the internal deformations inside the laminate, and thus the internal strain distribution, in such a way that it represents a good approximation of the actual deformations.
3.1.2. Basic deformation modes
The deformation at a global level has to be prescribed for the deformation model. Each deformation can be regarded as a combination of basic deformation modes, like bending, stretching, shearing, etc. The exact number of basic de-formation modes depends on the scale under consideration. At a larger scale there are more deformation modes. The
assumptions made within each defor-mation model determines the number and nature of that specific deformation model.
Although the applied deformation can be divided into a linear combination of the basic deformation modes, the re-sulting stress is not the linear combina-tion of the stresses resulting from the basic deformation modes. The desired plasticity during forming prevents that. Every combination of these basic de-formations must be regarded as a unique case
This does not mean that the deformations cannot be simplified in the initial analysis; the initial analysis needs simplifications. The first laminate deforma-tion models will ignore interlaminar shear. Later laminate deformadeforma-tion models start with a 2-dimensional implementation of the deformations.
An applied deformation of a piece of laminate results in forces that are needed to create the deformation. These forces will obey the sign convention of Figure 3.3. A common sign convention allows different deformation models to be compared with each other even when the initial deformations differ due to dif-ferent assumptions about the internal deformation.
3.1.3. Laminate deformation functions
The remainder of this chapter will discuss different laminate deformation func-tions and will hint at an appropriate solution method. Each deformation model uses an approach as seen in Figure 3.4. The simplistic deformation model of Figure 3.4 uses only one deformation parameter: the radius. The deformation parameter is translated into an internal strain distribution (Step 1). These inter-nal strains in combination with the constitutive relation for the constituents pro-vide the internal stress distribution (Step 2). The stresses can be integrated into the forces (per unit width) acting on the laminate (Step 3).
M
xM
yN
xN
xN
yN
yM
xV
xV
xV
yV
yM
yM
xyM
xyM
xyM
xyM
y x y zFigure 3.3: Sign convention for the forces (per unit width) acting on a laminate.
R Vx,Mx,Nx
