EBOOK composite materials design and applications third edition (daniel gay) Thiết kế vật liệu composite và ứng dụng phiên bản thứ ba

Design and
and Applications
Applications
design

COMPOSITE materials
MATERIALS
Composite

Materials Science/Mechanical Engineering
Materials Science/Mechanical Engineering

T h i r d E d i T i o n
T H I R D E D I T I O N

Composite
COMPOSITEmaterials
MATERIALS
design
and
Applications
Design
and
Applications

“This book covers the topics related to the mechanics of composite matopics
to the mechanics
of composite
terials“This
in abook
very covers

simplethe
way.
... itrelated
is addressed
to graduate
and under-materialsstudents
in a very
... It is addressed
graduate
graduate
as simple
well asway.
to practical
engineers to
who
want toand
en- undergraduate
students
as
well
as
to
practical
engineers
who
want
hance their knowledge and learn the guidelines of the use of composite to
enhance
their book
knowledge

learn
the guidelines
of the use a
of good
composmaterials.
... This
is...a and
good
classroom
material...[and]
ite
materials.
...
This
book
is...good
classroom
material...[and]
a good
reference.”
reference.”
—Dr. Pierre Rahme, University of Notre Dame, Indiana, USA
—Dr. Pierre Rahme, University of Notre Dame, Indiana, USA
Considered to have contributed greatly to the pre-sizing of composite
structures,
Composite
Materials:
Design
andtoApplications
is aofpopular

Considered
to have
contributed
greatly
the pre-sizing
composite
reference
book
for
designers
of
heavily
loaded
composite
parts.
Fully
structures, Composite Materials: Design and Applications is a popular
updated
to
mirror
the
exponential
growth
and
development
of
composreference book for designers of heavily loaded composite parts. Fully
ites, this
English-language
Third Edition:

updated
to mirror the exponential
growth and development of composites,
this
English-language
Third
Edition:
• Contains all-new coverage of nanocomposites and biocomposites
• Contains
all-new
coverage ofprocesses
nanocomposites
and biocomposites
• reflects
the latest
manufacturing
and applications
in the
•
Reflects
the
latest
manufacturing
processes
and
applications
in the
aerospace, automotive, naval, wind turbine, and sporting goods
aerospace, automotive, naval, wind turbine, and sporting goods
industries

industries
• Provides a design method to define composite multilayered plates
• Provides a design method to define composite multilayered plates
underunder
loading,
along
withwith
all numerical
information
needed
for for
loading,
along
all numerical
information
needed
implementation
implementation
• Proposes
original
studystudy
of composite
beams
of any
section
shapes
• Proposes
original
of composite
beams

of any
section
shapes
and thick-laminated
composite
plates,
leading
to technical
formulaand thick-laminated
composite
plates,
leading
to technical
formulathat not
are found
not found
in literature
the literature
tions tions
that are
in the
• Features
numerous
examples
of the
pre-sizing
of composite
parts,
• Features
numerous

examples
of the
pre-sizing
of composite
parts,
processed
from
industrial
cases
and
reworked
to
highlight
key
processed from industrial cases and reworked to highlight key in- information
formation
• Includes test cases for the validation of computer software using
• includes test cases for the validation of computer software using
finite elements
finite elements
Consisting of three main parts, plus a fourth on applications, Composite
Consisting of three main parts, plus a fourth on applications, Composite
Materials: Design and Applications, Third Edition features a technical
Materials: Design and Applications, Third Edition features a technical
level that rises in difficulty as the text progresses, yet each part still can
level that rises in difficulty as the text progresses, yet each part still can
be explored
independently.
While
the heart

of the
book,
devoted
to the
be explored
independently.
While
the heart
of the
book,
devoted
to the
methodical
pre-design
of structural
parts,
retains
its original
character,
methodical
pre-design
of structural
parts,
retains
its original
character,
the
contents
have
been

significantly
rewritten,
restructured,
and
expandthe contents have been significantly rewritten, restructured, and expanded
to
better
illustrate
the
types
of
challenges
encountered
in
modern
ed to better illustrate the types of challenges encountered in modern
engineering
practice.
engineering practice.
6000 Broken Sound Parkway, NW
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TT h
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Composite
COMPOSITE
materials
MATERIALS
design
Applications

Design and Applications

Gay
Gay
TThHi rI RdD
I OnN
EEdDi TI Ti o

K19063
K19063

ISBN: 978-1-4665-8487-7

90000
9 781466 584877

Daniel Gay
daniel
Gay


T h i r d

E d i T i o n

Composite
materials
design and Applications

This page intentionally left blank


T h i r d

E d i T i o n

Composite
materials
design and Applications

daniel Gay

Boca Raton London New York

CRC Press is an imprint of the
Taylor & Francis Group, an informa business


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Contents
Preface.................................................................................................................................xix
Acknowledgments...............................................................................................................xxi
Author.............................................................................................................................. xxiii

Section I  PRINCIPLES OF CONSTRUCTION
  1 Composite Materials: Interest and Physical Properties.................................................3
1.1

1.2

1.3
1.4

1.5
1.6

What Is a Composite Material?................................................................................ 3
1.1.1 Broad Definition......................................................................................... 3
1.1.2 Main Features.............................................................................................. 4
Fibers and Matrices.................................................................................................. 4
1.2.1 Fibers........................................................................................................... 4
1.2.1.1 Definition.................................................................................... 4
1.2.1.2 Principal Fiber Materials............................................................. 5
1.2.1.3 Relative Importance of Different Fibers in Applications.............. 6
1.2.2 Materials for Matrices.................................................................................. 7
What Can Be Made Using Composite Materials?.................................................... 7
A Typical Example of Interest................................................................................... 9
Some Examples of Classical Design Replaced by Composite Solutions....................10
Main Physical Properties.........................................................................................10

  2 Manufacturing Processes............................................................................................17
2.1

2.2

Molding Processes...................................................................................................17
2.1.1 Contact Molding........................................................................................17
2.1.2 Compression Molding................................................................................18
2.1.3 Vacuum Molding........................................................................................18
2.1.4 Resin Injection Molding.............................................................................19
2.1.5 Injection Molding with Prepreg................................................................. 20
2.1.6 Foam Injection Molding............................................................................ 20
2.1.7 Molding of Hollow Axisymmetric Components........................................ 20

Other Forming Processes........................................................................................ 22
2.2.1 Sheet Forming........................................................................................... 22
2.2.2 Profile Forming......................................................................................... 23
2.2.3 Forming by Stamping................................................................................ 23
v


vi  ◾  Contents

2.3

2.2.4 Preforming by Three-Dimensional Assembly............................................. 24
2.2.4.1 Example: Carbon/Carbon......................................................... 24
2.2.4.2 Example: Silicon/Silicon............................................................ 24
2.2.5 Automated Tape Laying and Fiber Placement............................................ 24
2.2.5.1 Necessity of Automation............................................................ 24
2.2.5.2 Example..................................................................................... 24
2.2.5.3 Example......................................................................................25
2.2.5.4 Example: Robots and Software for AFP—Automatic
Fiber Placement Coriolis Composites (FRA)..............................25
Practical Considerations on Manufacturing Processes............................................ 26
2.3.1 Acronyms.................................................................................................. 26
2.3.2 Cost Comparison...................................................................................... 27

  3 Ply Properties..............................................................................................................29
3.1

3.2

3.3

3.4

3.5

3.6
3.7
3.8

Isotropy and Anisotropy......................................................................................... 29
3.1.1 Isotropic Materials......................................................................................31
3.1.2 Anisotropic Material.................................................................................. 32
Characteristics of the Reinforcement–Matrix Mixture............................................33
3.2.1 Fiber Mass Fraction................................................................................... 34
3.2.2 Fiber Volume Fraction............................................................................... 34
3.2.3 Mass Density of a Ply.................................................................................35
3.2.4 Ply Thickness..............................................................................................35
Unidirectional Ply.................................................................................................. 36
3.3.1 Elastic Modulus......................................................................................... 36
3.3.2 Ultimate Strength of a Ply......................................................................... 38
3.3.3 Examples................................................................................................... 39
3.3.4 Examples of High-Performance Unidirectional Plies..................................41
Woven Ply...............................................................................................................41
3.4.1 Forms of Woven Fabrics.............................................................................41
3.4.2 Elastic Modulus of Fabric Layer................................................................ 42
3.4.3 Examples of Balanced Fabric/Epoxy.......................................................... 43
Mats and Reinforced Matrices.................................................................................45
3.5.1 Mats...........................................................................................................45
3.5.2 Example: A Summary of Glass/Epoxy Layers.............................................45
3.5.3 Microspherical Fillers.................................................................................45

3.5.4 Other Classical Reinforcements................................................................. 48
Multidimensional Fabrics....................................................................................... 49
3.6.1 Example: A Four-Dimensional Architecture of Carbon Reinforcement..... 49
3.6.2 Example: Three-Dimensional Carbon/Carbon Components..................... 50
Metal Matrix Composites....................................................................................... 50
3.7.1 Some Examples.......................................................................................... 50
3.7.2 Unidirectional Fibers/Aluminum Matrix....................................................52
Biocomposite Materials...........................................................................................53
3.8.1 Natural Plant Fibers...................................................................................53
3.8.1.1 Natural Fibers.............................................................................53
3.8.1.2 Pros............................................................................................53


Contents  ◾  vii

3.8.1.3 Cons...........................................................................................53
3.8.1.4 Examples................................................................................... 54
3.8.2 Natural Vegetable Fiber–Reinforced Composites...................................... 54
3.8.2.1 Mechanical Properties............................................................... 54
3.8.2.2 Biodegradable Matrices............................................................. 54
3.8.3 Manufacturing Processes........................................................................... 56
3.8.3.1 With Thermosetting Resins....................................................... 56
3.8.3.2 With Thermoplastic Resins.........................................................57
3.9 Nanocomposite Materials........................................................................................57
3.9.1 Nanoreinforcement.....................................................................................57
3.9.1.1 Nanoreinforcement Shapes.........................................................57
3.9.1.2 Properties of Nanoreinforcements.............................................. 58
3.9.2 Nanocomposite Material............................................................................61
3.9.3 Mechanical Applications........................................................................... 62
3.9.3.1 Improvement in Mechanical Properties..................................... 62

3.9.3.2 Further Examples of Nonmechanical Applications.................... 64
3.9.4 Manufacturing of Nanocomposite Materials............................................. 64
3.10 Tests....................................................................................................................... 66

  4 Sandwich Structures....................................................................................................69
4.1

4.2

4.3

4.4

4.5

What Is a Sandwich Structure?............................................................................... 69
4.1.1 Their Properties Are Surprising.................................................................. 69
4.1.2 Constituent Materials................................................................................ 70
Simplified Flexure................................................................................................... 71
4.2.1 Stress......................................................................................................... 71
4.2.2 Displacements........................................................................................... 72
4.2.2.1 Contributions of bending moment M and of shear force T .... 72
4.2.2.2 Example: A Cantilever Sandwich Structure............................... 73
Some Special Features of Sandwich Structures........................................................74
4.3.1 Comparison of Mass for the Same Flexural Rigidity 〈EI〉..........................74
4.3.2 Deterioration by Buckling of Sandwich Structures.....................................74
4.3.2.1 Global Buckling........................................................................ 75
4.3.2.2 Local Buckling of the Skins....................................................... 75
4.3.3 Other Types of Damage.............................................................................76
Manufacturing and Design Problems......................................................................76

4.4.1 Example of Core Material: Honeycomb.....................................................76
4.4.2 Shaping Processes...................................................................................... 77
4.4.2.1 Machining................................................................................. 77
4.4.2.2 Deformation.............................................................................. 77
4.4.2.3 Some Other Considerations....................................................... 77
4.4.3 Inserts and Attachment Fittings................................................................ 78
4.4.4 Repair of Laminated Facings..................................................................... 79
Nondestructive Inspection...................................................................................... 80
4.5.1 Main Nondestructive Inspection Methods................................................ 80
4.5.2 Acoustic Emission Testing..........................................................................81


viii  ◾  Contents

  5 Conception: Design and Drawing...............................................................................85
5.1

5.2

5.3

5.4

Drawing a Composite Part......................................................................................85
5.1.1 Specific Properties......................................................................................85
5.1.2 Guide Values of Presizing.......................................................................... 86
5.1.2.1 Material Characteristics............................................................. 86
5.1.2.2 Design Factors........................................................................... 88
Laminate................................................................................................................ 88
5.2.1 Unidirectional Layers and Fabrics............................................................. 88

5.2.1.1 Unidirectional Layer.................................................................. 88
5.2.1.2 Fabrics....................................................................................... 89
5.2.2 Correct Ply Orientation............................................................................. 89
5.2.3 Laminate Drawing Code........................................................................... 90
5.2.3.1 Standard Orientations............................................................... 90
5.2.3.2 Laminate Middle Plane............................................................. 90
5.2.3.3 Description of the Stacking Order............................................. 93
5.2.3.4 Midplane Symmetry.................................................................. 93
5.2.3.5 Specific Case of Balanced Fabrics.............................................. 94
5.2.3.6 Technical Minimum.................................................................. 95
5.2.4 Arrangement of Plies................................................................................. 96
5.2.4.1 Proportion and Number of Plies................................................ 96
5.2.4.2 Example of Pictorial Representation.......................................... 97
5.2.4.3 Case of Sandwich Structure....................................................... 97
Failure of Laminates............................................................................................... 98
5.3.1 Damages.................................................................................................... 98
5.3.1.1 Types of Failure......................................................................... 98
5.3.1.2 Note: Classical Maximum Stress Criterion Shows
Its Limits........................................................................... 99
5.3.2 Most Frequently Used Criterion: Tsai–Hill Failure Criterion.................. 100
5.3.2.1  Tsai–Hill Number.................................................................... 100
5.3.2.2 Notes........................................................................................101
5.3.2.3 How to Determine the Stress Components σℓ, σt, and τℓt
in Each Ply...............................................................................101
Presizing of the Laminate......................................................................................102
5.4.1 Modulus of Elasticity—Deformation of a Laminate.................................102
5.4.1.1 Varying Proportions of Plies.....................................................102
5.4.1.2 Example of Using Tables..........................................................103
5.4.2 Case of Simple Loading............................................................................103
5.4.3 Complex Loading Case: Approximative Proportions According

to Orientations.........................................................................................109
5.4.3.1 When the Normal and Tangential (Shear) Loads Are
Applied Simultaneously............................................................109
5.4.3.2 Example....................................................................................114
5.4.3.3 Note.........................................................................................117
5.4.4 Complex Loading Case: Optimum Composition of a Laminate............... 119
5.4.4.1 Optimum Laminate................................................................. 119
5.4.4.2 Example................................................................................... 122


Contents  ◾  ix

5.4.4.3 Example....................................................................................125
5.4.4.4 Notes....................................................................................... 126
5.4.5 Notes for Practical Use Concerning Laminates....................................... 127
5.4.5.1 Specific Aspects for the Design of Laminates........................... 127
5.4.5.2 Delaminations......................................................................... 128
5.4.5.3 Why Is Fatigue Resistance So Good?........................................129
5.4.5.4 Laminated Tubes......................................................................133

  6 Conception: Fastening and Joining...........................................................................135
6.1

6.2

6.3

Riveting and Bolting.............................................................................................135
6.1.1 Local Loss of Strength..............................................................................135
6.1.1.1 Knock-Down Factor.................................................................135

6.1.1.2 Causes of Hole Degradation.................................................... 136
6.1.2 Main Failure Modes in Bolted Joints of Composite Materials..................138
6.1.3 Sizing of the Joint.....................................................................................138
6.1.3.1 Recommended Values...............................................................138
6.1.3.2 Evaluation of Magnified Stress Values......................................140
6.1.4 Riveting....................................................................................................140
6.1.5 Bolting......................................................................................................141
6.1.5.1 Example of Bolted Joint............................................................141
6.1.5.2 Tightening of the bolt..............................................................143
Bonding.................................................................................................................143
6.2.1 Adhesives Used.........................................................................................143
6.2.2 Geometry of the Bonded Joints................................................................145
6.2.3 Sizing of the Bonding Surface Area..........................................................146
6.2.3.1 Strength of adhesive.................................................................146
6.2.3.2 Design......................................................................................147
6.2.3.3 Stress in Bonded Areas.............................................................148
6.2.3.4 Example of single-lap adhesive joint......................................150
6.2.4 Case of Bonded Joint with Cylindrical Geometry.....................................150
6.2.4.1 Bonded Circular Flange............................................................150
6.2.4.2 Tubes Fitted and Bonded into One Another.............................150
6.2.5 Examples of Bonding................................................................................150
6.2.5.1 Laminates.................................................................................150
Inserts....................................................................................................................152
6.3.1 Case of Sandwich Parts.............................................................................152
6.3.2 Case of Parts under Uniaxial Loads..........................................................154

  7 Composite Materials and Aerospace Construction...................................................155
7.1

Aircraft.................................................................................................................. 155

7.1.1 Composite Components in Aircraft.......................................................... 155
7.1.2 Allocation of Composites Depending on Their Nature.............................156
7.1.2.1 Glass/Epoxy, Kevlar/Epoxy......................................................156
7.1.2.2 Carbon/Epoxy..........................................................................157
7.1.2.3 Boron/Epoxy............................................................................157
7.1.2.4 Honeycombs.............................................................................157
7.1.3 Few Comments........................................................................................158


x  ◾  Contents

7.2

7.3

7.1.4 Specific Aspects of Structural Strength.....................................................158
7.1.5 Large Transport Aircraft...........................................................................159
7.1.5.1 Example....................................................................................159
7.1.5.2 How to Determine the Benefits................................................159
7.1.5.3 Example: Civil Transport Aircraft A380-800,
Airbus (EUR) ...................................................................... 161
7.1.5.4 Example: Civil Transport Aircraft B 787-800,
Boeing (USA).....................................................................161
7.1.5.5 Example: Civil Transport Aircraft A350-900,
Airbus (EUR)................................................................... 163
7.1.6 Regional Aircraft and Business Jets..........................................................165
7.1.6.1 Example: Regional Aircraft ATR 72-600, EADS (EUR),
Alenia (ITA) ............................................................................165
7.1.6.2 Example: Business Aircraft Falcon, Dassault Aviation (FRA)....... 165
7.1.6.3 Example: Cargo Aircraft WK2 and Suborbital Space Plane

SST2, Scaled Composites (USA)–Virgin Group (UK).............166
7.1.7 Light Aircraft...........................................................................................168
7.1.7.1 Trends......................................................................................168
7.1.7.2 Aircraft with Tractor Propeller.................................................168
7.1.7.3 Aircraft with Pusher Propeller..................................................169
7.1.7.4 Modern Glider Planes...............................................................170
7.1.8 Fighter Aircraft.........................................................................................170
7.1.9 Architecture and Manufacture of Composite Aircraft Parts.....................171
7.1.9.1 Sandwich Design......................................................................171
7.1.9.2 Rib-Stiffened Panels.................................................................173
7.1.10 Braking Systems.......................................................................................178
Helicopters............................................................................................................179
7.2.1 Situation...................................................................................................179
7.2.2 Composite Areas.......................................................................................180
7.2.2.1 Example: Helicopter EC 145 T2, Airbus-Helicopter (EUR).....180
7.2.2.2 Example: Helicopter X4, Thales–Safran (FRA),
Airbus- Helicopter (EUR)���������������������������������������������������������180
7.2.3 Blades.......................................................................................................181
7.2.3.1 Design of a Main Rotor Blade..................................................181
7.2.3.2 Advantages...............................................................................181
7.2.3.3 Consequences...........................................................................181
7.2.4 Rotor Hub................................................................................................183
7.2.4.1 Example: Rotor Hub Starflex, Eurocopter (FRA–GER)...........183
7.2.4.2 Example: Rotor Hub Spheriflex, Eurocopter (FRA–GER).......184
7.2.5 Other Working Composite Parts..............................................................184
Airplane Propellers................................................................................................186
7.3.1 Propellers for Conventional Aerodynamics...............................................186
7.3.1.1 Example: Propeller Blade, Hamilton Sundstrand (USA)–
Ratier Figeac (FRA)�����������������������������������������������������������������186
7.3.1.2 Example: Airplane with Tilt Rotors, V-22 Osprey Bell

Boeing (USA) and Dowty Propellers (UK)�����������������������������187
7.3.2 High-Speed Propellers..............................................................................188


Contents  ◾  xi

7.4

7.5

Aircraft Reaction Engine.......................................................................................190
7.4.1 Employed Materials..................................................................................190
7.4.2 Refractory Composites.............................................................................191
7.4.2.1 Specific Features.......................................................................191
7.4.2.2 Fibers........................................................................................191
7.4.2.3 Matrices....................................................................................192
7.4.2.4 Applications..............................................................................192
7.4.2.5 Example: Jet Engine Leap®, CFM International,
General Electric (USA)–SNECMA (FRA) �����������������������������193
Space Applications.................................................................................................194
7.5.1 Satellites...................................................................................................194
7.5.2 Propellant Tanks and Pressure Vessels......................................................195
7.5.3 Nozzles.....................................................................................................196
7.5.4 Other Composite Components for Space Application..............................198
7.5.4.1 For Engines..............................................................................198
7.5.4.2 For Thermal Protection.............................................................198
7.5.4.3 For Energy Storage.................................................................. 200

  8 Composite Materials for Various Applications..........................................................203
8.1

8.2

8.3
8.4

Comparative Importance of Composites in Applications...................................... 203
8.1.1 Relative Importance in terms of Mass and Market Value........................ 204
8.1.2 Mass of Composites Implemented According to the
Geographical Area............................................................................. 205
8.1.3 Average Prices.......................................................................................... 205
Composite Materials and Automotive Industry.................................................... 206
8.2.1 Introduction............................................................................................ 206
8.2.1.1 Example: Golf Model, Volkswagen (GER).............................. 206
8.2.1.2 Relative Weight Importance of Materials................................ 207
8.2.2 Composite Parts...................................................................................... 208
8.2.2.1 Brief Reminder........................................................................ 208
8.2.2.2 Current Functional Design...................................................... 208
8.2.2.3 Notable Composite Components..............................................210
8.2.2.4 Notes........................................................................................212
8.2.2.5 Use of Natural Fibers................................................................213
8.2.3 Research and Development......................................................................214
8.2.3.1 Structure..................................................................................215
8.2.3.2 Mechanical Parts......................................................................215
8.2.4 Motor Racing...........................................................................................216
Wind Turbines......................................................................................................217
8.3.1 Components.............................................................................................217
8.3.2 Manufacturing Processes..........................................................................218
Composites and Shipbuilding................................................................................219
8.4.1 Competition.............................................................................................219

8.4.1.1 Example: Ocean-Going Maxi-Trimaran.................................. 220
8.4.1.2 Example: Single Scull.............................................................. 222
8.4.1.3 Example: Surfboard................................................................. 223
8.4.2 Vessels..................................................................................................... 223


xii  ◾  Contents

8.5

8.6

Sports and Leisure................................................................................................ 223
8.5.1 Skis.......................................................................................................... 223
8.5.1.1 Equipment of a Skier............................................................... 223
8.5.1.2 Main Components of a Ski...................................................... 224
8.5.2 Bicycles.................................................................................................... 225
8.5.2.1 Machine.................................................................................. 226
8.5.2.2 Other Specific Equipments...................................................... 226
8.5.3 Tennis Rackets........................................................................................ 226
Diverse Applications............................................................................................. 226
8.6.1 Pressure Gas Bottle.................................................................................. 226
8.6.2 Bogie Frame............................................................................................ 227
8.6.3 Tubes for Offshore Installations............................................................... 227
8.6.4 Biomechanical Applications..................................................................... 228
8.6.5 Cable Car................................................................................................ 229

Section II  Mechanical Behavior of Laminated Materials
   9 Anisotropic Elastic Medium......................................................................................233
9.1

9.2
9.3

Some Reminders....................................................................................................233
9.1.1 Continuum Mechanics.............................................................................233
9.1.2 Number of Distinct φijkℓ Terms............................................................... 234
Orthotropic Material............................................................................................ 236
Transversely Isotropic Material............................................................................. 236

  10 Elastic Constants of Unidirectional Composites.......................................................239
10.1
10.2
10.3
10.4
10.5

Longitudinal Modulus Eℓ..................................................................................... 239
Poisson Coefficient................................................................................................241
Transverse Modulus Et......................................................................................... 242
Shear Modulus Gℓt ............................................................................................. 244
Thermoelastic Properties........................................................................................245
10.5.1 Isotropic Material: Recall.........................................................................245
10.5.2 Case of Unidirectional Composite........................................................... 246
10.5.2.1 Coefficient of Thermal Expansion along the Direction ℓ......... 246
10.5.2.2 Coefficient of Thermal Expansion along the Transverse
Direction t................................................................................247
10.5.3 Thermomechanical Behavior of a Unidirectional Layer........................... 248

  11 Elastic Constants of a Ply in Any Direction..............................................................249

11.1 Flexibility Coefficients...........................................................................................249
11.2 Stiffness Coefficients..............................................................................................255
11.3 Case of Thermomechanical Loading......................................................................257
11.3.1 Flexibility Coefficients..............................................................................257
11.3.2 Stiffness Coefficients.................................................................................259

  12 Mechanical Behavior of Thin Laminated Plates.......................................................263

12.1 Laminate with Midplane Symmetry..................................................................... 263
12.1.1 Membrane Behavior................................................................................ 263
12.1.1.1 Loadings.................................................................................. 263
12.1.1.2 Displacement Field.................................................................. 264


Contents  ◾  xiii

12.1.2 Apparent Elastic Moduli of the Laminate.................................................267
12.1.3 Consequence: Practical Determination of a Laminate Subject
to Membrane Loading..............................................................................267
12.1.3.1 Givens of the Problem..............................................................267
12.1.3.2 Principle of Calculation........................................................... 268
12.1.3.3 Calculation Procedure............................................................. 269
12.1.4 Flexure Behavior...................................................................................... 272
12.1.4.1 Displacement Field.................................................................. 272
12.1.4.2 Loadings.................................................................................. 273
12.1.4.3 Notes........................................................................................275
12.1.5 Consequence: Practical Determination of a Laminate Subject to Flexure...... 278
12.1.6 Simplified Calculation for Bending......................................................... 278
12.1.6.1 Apparent Failure Strength in Bending..................................... 278
12.1.6.2 Apparent Flexure Modulus...................................................... 279

12.1.7 Thermomechanical Loading Case............................................................ 280
12.1.7.1 Membrane Behavior................................................................ 280
12.1.7.2 Behavior under Bending.......................................................... 283
12.2 Laminate without Midplane Symmetry................................................................ 283
12.2.1 Coupled Membrane–Flexure Behavior.................................................... 283
12.2.2 Case of Thermomechanical Loading........................................................ 285

Section III Justifications, Composite Beams, and Thick
Laminated Plates
  13 Elastic Coefficients....................................................................................................289

13.1 Elastic Coefficients for an Orthotropic Material................................................... 289
13.1.1 Reminders............................................................................................... 289
13.1.2 Elastic Behavior Equation in Orthotropic Axes....................................... 290
13.2 Elastic Coefficients for a Transverse Isotropic Material......................................... 292
13.2.1 Elastic Behavior Equation........................................................................ 292
13.2.2 Rotation about an Orthotropic Transverse Axis...................................... 295
13.2.2.1 Problem................................................................................... 295
13.2.2.2 Technical Form....................................................................... 300
13.3 Case of a Ply......................................................................................................... 302

  14 Damage in Composite Parts: Failure Criteria...........................................................303

14.1 Damage in Composite Parts................................................................................. 303
14.1.1 Industrial Emphasis of the Problem......................................................... 303
14.1.1.1 Causes of Damage................................................................... 303
14.1.1.2 Diversity of Composite Parts................................................... 304
14.1.2 Influence of Manufacturing Process........................................................ 304
14.1.2.1 Example: Injected Part with Short Fibers................................. 305
14.1.2.2 Example: Parts with Pronounced Curvatures.......................... 305

14.1.3 Typical Area and Singularities in a Same Part......................................... 305
14.1.4 Degradation Process within the Typical Area.......................................... 306
14.1.4.1 Example: Composite Short Fiber Plate.................................... 306
14.1.4.2 Example: Laminate Consisting of Unidirectional Plies............ 307


xiv  ◾  Contents

14.2 Form of a Failure Criterion....................................................................................310
14.2.1 Features of a Failure Criterion..................................................................310
14.2.1.1 Failure Criterion Is a Design Tool.............................................310
14.2.1.2 Many Criteria...........................................................................310
14.2.2 General Form of a Failure Criterion.........................................................310
14.2.2.1 Development of a Criterion......................................................310
14.2.2.2 Case of an Orthotropic Material...............................................311
14.2.3 Linear Failure Criterion............................................................................312
14.2.3.1 Example: Plane State of Stress in an Orthotropic Material.......312
14.2.3.2 Example: Maximum Stress Failure Criterion............................313
14.2.3.3 Note: Maximum Eligible Strain Criterion................................313
14.2.4 Quadratic Failure Criterion......................................................................314
14.2.4.1 General Form...........................................................................314
14.2.4.2 Specific Case of Plane Stress.....................................................314
14.2.4.3 Note: Simplified Form for the Quadratic Criterion.................. 315
14.3 Tsai–Hill Failure Criterion....................................................................................316
14.3.1 Isotropic Material: The von Mises Criterion.............................................316
14.3.1.1 Material Is Elastic and Isotropic...............................................316
14.3.1.2 Notes........................................................................................318
14.3.2 Orthotropic Material: Tsai–Hill Criterion............................................... 320
14.3.2.1 Notes....................................................................................... 320
14.3.2.2 Case of a Transversely Isotropic Material..................................321

14.3.2.3 Case of Unidirectional Ply under In-Plane Loading................ 323
14.3.3 Evolution of Strength Properties of a Unidirectional Ply Depending
on the Direction of Solicitation���������������������������������������������������������������324
14.3.3.1 Tensile and Compressive Strength............................................324
14.3.3.2 Shear Strength..........................................................................325

  15 Bending of Composite Beams of Any Section Shape.................................................327

15.1 B
 ending of Beams with Isotropic Phases and Plane of Symmetry......................... 328
15.1.1 Degrees of Freedom..................................................................................329
15.1.1.1 Equivalent Stiffnesses...............................................................329
15.1.1.2 Longitudinal Displacement......................................................329
15.1.1.3 Rotation of the Section.............................................................329
15.1.1.4 Elastic Center.......................................................................... 330
15.1.1.5 Transverse Displacement along y Direction............................. 330
15.1.1.6 Transverse Displacement along z Direction..............................331
15.1.2 Perfect Bonding between the Phases.........................................................332
15.1.2.1 Displacements..........................................................................332
15.1.2.2 Strains......................................................................................332
15.1.2.3 Stress........................................................................................333
15.1.3 Equilibrium Relationships........................................................................333
15.1.3.1 Longitudinal Equilibrium........................................................333
15.1.3.2 Transverse Equilibrium........................................................... 334
15.1.3.3 Moment Equilibrium...............................................................335


Contents  ◾  xv

15.1.4 Constitutive Equations............................................................................ 336

15.1.5 Technical Formulation.............................................................................337
15.1.5.1 Assumptions.............................................................................337
15.1.5.2 Expression of Normal Stress.....................................................337
15.1.5.3 Expression of Shear Stress.........................................................338
15.1.5.4 Shear Coefficient for the Section.............................................. 340
15.1.6 Energy Interpretation.............................................................................. 342
15.1.6.1 Energy Due to Normal Stress σxx. ........................................... 342
15.1.6.2 Energy Due to Shear Stress τ................................................... 343
15.1.7 Extension to the Dynamic Case.............................................................. 344
15.2 Case of Beams of Any Cross Section (Asymmetric).............................................. 346
15.2.1 Technical Formulation............................................................................ 347
15.2.2 Notes........................................................................................................351

  16 Torsion of Composite Beams of Any Section Shape..................................................353

16.1 Uniform Torsion....................................................................................................353
16.1.1 Torsional Degree of Freedom....................................................................354
16.1.2 Constitutive Equation..............................................................................354
16.1.3 Determination of Φ(y, z)..........................................................................355
16.1.3.1 Local Equilibrium....................................................................355
16.1.3.2 External Boundary Condition..................................................356
16.1.3.3 Internal Boundary Conditions..................................................356
16.1.3.4 Uniqueness of Function Φ........................................................356
16.1.4 Energy Interpretation...............................................................................357
16.2 Location of the Torsion Center..............................................................................358
16.2.1 Coordinates in Principal Axes..................................................................358
16.2.2 Summary of Results.................................................................................359
16.2.3 Flexion–Torsion Coupling........................................................................361

  17 Bending of Thick Composite Plates..........................................................................363

17.1 Preliminary Remarks............................................................................................ 363
17.1.1 Transverse Normal Stress σz.................................................................... 363
17.1.2 Transverse Shear Stress τxz and τyz........................................................... 364
17.1.3 Assumptions.............................................................................................365
17.2 Displacement Field................................................................................................367
17.3 Strains.................................................................................................................. 369
17.4 Constitutive Equations......................................................................................... 369
17.4.1 Membrane Behavior................................................................................ 369
17.4.2 Bending Behavior.....................................................................................370
17.4.3 Transverse Shear Behavior........................................................................372
17.4.3.1 Transverse Shear Resultant Qx..................................................372
17.4.3.2 Transverse Shear Resultant Qy..................................................373
17.5 Equilibrium Relationships.....................................................................................373
17.5.1 Transverse Equilibrium............................................................................373
17.5.2 Equilibrium in Bending............................................................................374


xvi  ◾  Contents

17.6 Technical Formulation for Bending.......................................................................374
17.6.1 Stress Due to Bending..............................................................................375
17.6.1.1 Plane Stress Values....................................................................375
17.6.1.2 Transverse Shear Stress Values..................................................376
17.6.2 Characterization of Warping Increments in Bending ηx and ηy................376
17.6.3 Particular Cases....................................................................................... 377
17.6.3.1 Orthotropic Homogeneous Plate............................................. 377
17.6.3.2 Cylindrical Bending about x- or y-Axis.....................................378
17.6.3.3 Multilayered Plate.....................................................................379
17.6.3.4 Consequences.......................................................................... 380

17.6.4 Warping Functions.................................................................................. 380
17.6.4.1 Boundary Conditions.............................................................. 380
17.6.4.2 Interfacial Continuity...............................................................381
17.6.4.3 Formulation of Warping Functions..........................................381
17.6.5 Consequences.......................................................................................... 382
17.6.5.1 Expression of Transverse Shear Stress...................................... 382
17.6.5.2 Transverse Shear Coefficients................................................... 382
17.6.6 Energy Interpretation.............................................................................. 384
17.7 Examples...............................................................................................................385
17.7.1 Orthotropic Homogeneous Plate..............................................................385
17.7.2 Sandwich Plate........................................................................................ 387
17.7.2.1 Case of Two Orthotropic Materials......................................... 387
17.7.2.2 Warping Functions.................................................................. 388
17.7.2.3 Transverse Shear Stress............................................................ 389
17.7.2.4 Transverse Shear Coefficients................................................... 389
17.7.3 Conclusion.............................................................................................. 390

Section IV Applications
  18 Applications Level 1..................................................................................................393
18.1 Simply Supported Sandwich Beam........................................................................393
18.2 Poisson Coefficient of a Unidirectional Layer....................................................... 396
18.3 Helicopter Blade................................................................................................... 397
18.4 Drive Shaft for Trucks.......................................................................................... 402
18.5 Flywheel in Carbon/Epoxy................................................................................... 408
18.6 Wing Tip Made of Carbon/Epoxy........................................................................410
18.7 Carbon Fiber Coated with Nickel......................................................................... 423
18.8 Tube Made of Glass/Epoxy under Pressure........................................................... 425
18.9 Filament-Wound Pressure Vessel: Winding Angle................................................ 428
18.10 Filament-Wound Pressure Vessel: Consideration of Openings in the Bottom Heads.......431
18.11Determination of Fiber Volume Fraction by Pyrolysis...........................................435

18.12Reversing Lever Made of Carbon/PEEK (Unidirectional and Short Fibers)......... 436
18.13Glass/Resin Telegraph Pole................................................................................... 439
18.14Unidirectional Layer of HR Carbon..................................................................... 443
18.15Manipulator Arm for a Space Shuttle................................................................... 444


Contents  ◾  xvii

  19 Applications Level 2..................................................................................................449
19.1 Sandwich Beam: Simplified Calculation of the Shear Coefficient......................... 449
19.2 Procedure for a Laminate Calculation Program.....................................................451
19.3 Kevlar/Epoxy Laminates: Stiffness in Terms of the Direction of Load..................455
19.4 Residual Thermal Stress Due to the Laminate Curing Process..............................459
19.5 Thermoelastic Behavior of a Glass/Polyester Tube................................................ 462
19.6 Creep of a Polymeric Tube Reinforced by Filament Wound under Thermal Stress...... 465
19.7 First-Ply Failure of a Laminate: Ultimate Strength................................................471
19.8 Optimum Laminate for Isotropic Plane Stress.......................................................475
19.9 Laminate Made of Identical Layers of Balanced Fabric..........................................481
19.10Carbon/Epoxy Wing Spar.................................................................................... 484
19.11 Elastic Constants of a Carbon/Epoxy Unidirectional Layer, Based on Tensile Test........491
19.12 Sailboat Hull in Glass/Polyester................................................................................492
19.13Balanced Fabric Ply: Determination of the In-Plane Shear Modulus.....................498
19.14Quasi-Isotropic Laminate..................................................................................... 499
19.15Pure Torsion of Orthotropic Plate........................................................................ 502
19.16Plate Made by Resin Transfer Molding ............................................................... 506
19.17Thermoelastic Behavior of a Balanced Fabric Ply...................................................512

  20 Applications Level 3..................................................................................................523
20.1 Cylindrical Bonding..............................................................................................523
20.2 Double-Lap Bonded Joint......................................................................................528

20.3 Composite Beam with Two Layers........................................................................533
20.4 Buckling of a Sandwich Beam...............................................................................537
20.5 Shear Due to Bending in a Sandwich Beam......................................................... 540
20.6 Shear Due to Bending in a Composite Box Beam................................................ 544
20.7 Torsion Center of a Composite U-Beam............................................................... 547
20.8 Shear Due to Bending in a Composite I-Beam..................................................... 549
20.9 Polymeric Column Reinforced by Filament-Wound Fiberglass..............................553
20.10Cylindrical Bending of a Thick Orthotropic Plate under Uniform Loading......... 563
20.11Bending of a Sandwich Plate................................................................................ 564
20.12Bending Vibration of a Sandwich Beam................................................................567

Appendix A: Stresses in the Plies of a Carbon/Epoxy Laminate Loaded in Its Plane........571
Appendix B: Buckling of Orthotropic Structures..............................................................585
Bibliography.......................................................................................................................595
Index���������������������������������������������������������������������������������������������������������������������������������599


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Preface
The developments in the field of composite materials since the last quarter of a century have made
this area popular due to the breadth and universality of applications.
The annual global growth rate of composites is 5%–6%, and tonnage, which was 8 million
tonnes in 2010, could rise to 10 million tonnes by 2015—a growth driven by advances in the
transportation and wind industries. The sector of composites is an area of business that is always
evolving.
The cost of composites is becoming increasingly competitive. For a quarter of a century, the
price of high-performance composites used in aerospace declined by more than half to compete
with sophisticated metal alloys. At the same time, the quality of semifinished products reached

remarkable levels. For example, the unidirectional prepreg tapes carbon/epoxy have their widths
defined within 0.2 mm, and their fiber volume content controlled within only a few fractions of a
percent, with obvious consequences for the evolution of the quality of parts.
The legislation on recyclability obligation also affects the composite activities. It leads to significant increases in research and development on topics concerning natural fibers and biodegradable polymers.
The growth in the use of composites has been aided by the development of modern design and
manufacturing methods for industrial components, which allow functional optimization based on
multiple technical and economic criteria. A good knowledge of what already exists helps develop
and use reliable numerical simulations for in-service behavior as well as for implementation during
the manufacturing.
The development of simulation tools is an important component of industrial development,
in general, and in composite domains, in particular. Without trying to replace testing, these tools
allow full exploitation of the experimental results in a much more complete manner, creating a
powerful synergy that saves time and cost.
This third edition has been updated to take into account this rapidly changing field as well as
the emergence and development of additional areas, such as those of bio- and nanocomposites. The
core of the book devoted to the methodical predesign of structural parts has been preserved. As in
previous editions, we have considered only a limited number of significant reinforcements and have
highlighted the specific features needed for predimensioning. This is, in fact, to limit the number
of performance tables accompanying the text. Other reinforcements not detailed in this book can
be readily adapted; the reader will find everything needed to use a spreadsheet in order to get the
desired results. He or she may also download a dedicated free utility as indicated in the book. The
chapters on composite beams of any cross-sectional shape and the chapter on laminated thick plates
still retain their original character, both with regard to the proposed method and to the results.
xix


xx  ◾  Preface

The book is structured into three levels of difficulty (even with regard to the applications).
The technical level becomes increasingly complicated from one section to the next. The first section corresponds to the undergraduate level, while the second and third sections correspond to the

graduate level. One can, however, work on each part independently.
Section IV, “Applications,” consists of 44 examples, including numerous cases of presizing of
composite parts, processed from industrial cases reworked so that the user can go directly to the
essentials. It also includes test cases for the validation of computer software using finite elements.
No other book in the literature covers the application domain in this way.
This book has had a run of two recent editions in English and five successive editions in French
(all of which are out of print). It is addressed to engineers and technicians in the field who deal with
problems of mechanical behavior that require designs, compositions, thicknesses, and fasteners to
be defined. It is addressed to teachers who want to structure a course on the subject, or simply
talk about composites. It is also addressed to students pursuing undergraduate and postgraduate
degrees and can help PhD students do an apprenticeship before moving on to specialized research.
This book does not focus on very detailed theoretical developments, which would not meet
the requirements of the targeted audience. In industry, there is little time for the consultation of
books, and the academic nature of initial training is often far from the daily concerns of the design
office. I have therefore adapted this presentation by taking into account readers who are always
in a hurry and who use the tools available to them or ones that they remember. The content of
this book is nevertheless anchored on solid scientific basis and will allow potential users to derive
maximum benefit from it.


Acknowledgments
I express my sincere thanks and gratitude to Dr. Stephane Gay, who wrote parts of the text and
reviewed and verified the appropriate use of technical terminology contained in this third edition,
especially in the field of aeronautics. I am also grateful to Pr. Suong Van Hoa, who kindly took
on the important task of translating the first edition of this book that I had originally written
in French.

xxi

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Author
Daniel Gay is a former student of the Ecole Normale Superieure de Cachan and served as a
­professor at the University Paul Sabatier Toulouse III. He led the Laboratory of Mechanical
Engineering of Toulouse, now the Clément Ader Institute, from its inception for over 15 years.
Dr. Gay has taught composite materials and structures at the undergraduate, graduate, and
postgraduate levels in many French schools and institutions (University of Toulouse III, IUT,
INSA, ENSICA, Supaero (ISAE), ENSTA, etc.). He is the author of numerous articles, scientific
publications, and industrial reports on the subject.

xxiii


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