Introduction to Process and Mechanical Modelling of Engineering Composites

506 Pages·2019·31.629 MB·English

by  Anthony Pickett

Introduction to Process and
Mechanical Modelling of
Engineering Composites
Part 1. Fundamentals
Anthony Pickett
IFB (Institute of Aircraft Design)
University of Stuttgart, Germany
Copyright © 2019 Anthony K. Pickett
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First Printing, 2018
Edition 2, April 2021
IFB (Institute of Aircraft Design)
University of Stuttgart
Pfaffenwaldring 31
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Germany
CONTENTS
Contents i
Preface ix
Acknowledgements xi
List of Symbols xi
I General introduction 1
1 Introduction to composites and analysis 3
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Composite materials . . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 FRP markets and trends. . . . . . . . . . . . . . . . . . . . 6
1.2.3 Versatility and properties of FRP composites . . . . . . . . 7
1.3 Fibres and resins . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.1 Fibre types and properties . . . . . . . . . . . . . . . . . . . 11
1.3.2 Resin types and properties . . . . . . . . . . . . . . . . . . 19
1.3.2.1 Thermoset resins . . . . . . . . . . . . . . . . . . . 20
1.3.2.2 Thermoplastic resins . . . . . . . . . . . . . . . . . 22
1.4 Fabrics and preforms . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.4.1 2D fabrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.4.2 2D braids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.4.3 Non-crimp fabrics . . . . . . . . . . . . . . . . . . . . . . . 26
1.5 Yarn architecture versus performance . . . . . . . . . . . . . . . . . 27
i
CONTENTS
1.6 Manufacturing methods . . . . . . . . . . . . . . . . . . . . . . . . 28
1.6.1 Manufacture of thermoset composites . . . . . . . . . . . . 29
1.6.1.1 Pre-preg composites . . . . . . . . . . . . . . . . . 29
1.6.1.2 Liquid composites moulding . . . . . . . . . . . . 31
1.6.1.3 Comparing pre-preg and LCM manufacture . . . . 33
1.6.2 Thermoforming of thermoplastics . . . . . . . . . . . . . . . 34
1.6.3 Other manufacturing methods . . . . . . . . . . . . . . . . 35
1.7 Introduction to composites terminology . . . . . . . . . . . . . . . 36
1.7.1 Heterogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.7.2 Dimensions of scale . . . . . . . . . . . . . . . . . . . . . . . 37
1.7.3 Micro- and Macromechanics . . . . . . . . . . . . . . . . . . 38
1.7.4 Isotropy, orthotropy and anisotropy . . . . . . . . . . . . . 39
1.8 A case study: metals versus composites . . . . . . . . . . . . . . . 41
1.8.1 Fibre properties and orientations versus performance . . . . 41
1.9 Composites analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 46
1.9.1 The finite element method . . . . . . . . . . . . . . . . . . . 47
1.9.2 Process analysis . . . . . . . . . . . . . . . . . . . . . . . . 49
1.9.3 Mechanical and field analysis . . . . . . . . . . . . . . . . . 52
1.9.4 Software products . . . . . . . . . . . . . . . . . . . . . . . 53
1.9.5 Chaining of analyses . . . . . . . . . . . . . . . . . . . . . . 54
1.9.6 Strategies for a virtual product design . . . . . . . . . . . . 55
1.10 Structure of the book chapters . . . . . . . . . . . . . . . . . . . . 57
II Classical analysis methods 59
2 Micro- and mesomechanics 61
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.2 Some notations and properties . . . . . . . . . . . . . . . . . . . . 62
2.2.1 Lamina notations . . . . . . . . . . . . . . . . . . . . . . . . 62
2.2.2 Volume fractions . . . . . . . . . . . . . . . . . . . . . . . . 62
2.2.3 Constituent fibre and resin properties . . . . . . . . . . . . 64
2.3 Representative volume and homogenisation . . . . . . . . . . . . . 64
2.4 Mechanisms of fibre-matrix load transfer . . . . . . . . . . . . . . . 68
2.5 Short fibre composites . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.5.1 Kelly-Tyson model for failure and critical fibre length . . . 71
2.5.2 The Cox model for elastic stress distribution and stiffness . 73
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CONTENTS
2.6 Micromechanical models for stiffness . . . . . . . . . . . . . . . . . 73
2.6.1 Mechanics of materials . . . . . . . . . . . . . . . . . . . . . 74
2.6.2 Voigt (parallel) model for longitudinal modulus E . . . . . 74
1
2.6.3 Reuss (series) model for transverse modulus E . . . . . . . 76
2
2.6.4 Comparing models, and their limitations . . . . . . . . . . . 77
2.6.5 Law of mixture formulae for in-plane G , ν and ν . . . 78
12 12 21
2.6.6 Hopkins-Chamis model for E and G . . . . . . . . . . . . 79
2 12
2.6.7 Halpin-Tsai relations for E and G . . . . . . . . . . . . . 80
2 12
2.6.8 Comparing in-plane elastic properties G , ν and ν . . . 81
12 12 21
2.6.9 Transverse elastic properties G , G , ν and ν . . . . . 82
13 23 13 23
2.6.10 Comparing micromechanics and FE models for E . . . . . 84
2
2.7 Micromechanical models for failure . . . . . . . . . . . . . . . . . . 84
2.7.1 Notations and failure modes . . . . . . . . . . . . . . . . . . 86
2.7.2 Longitudinal tensile failure F . . . . . . . . . . . . . . . . 88
1t
2.7.3 Transverse tensile failure F . . . . . . . . . . . . . . . . . 89
2t
2.7.4 Longitudinal compression failure F . . . . . . . . . . . . . 93
1c
2.7.5 Transverse compressive failure F . . . . . . . . . . . . . . 95
2c
2.7.6 In-plane shear failure F . . . . . . . . . . . . . . . . . . . . 96
6
2.7.7 Transverse shear failure F and F . . . . . . . . . . . . . . 97
4 5
2.8 Other micromechanical laws . . . . . . . . . . . . . . . . . . . . . . 98
2.8.1 Conduction coefficients . . . . . . . . . . . . . . . . . . . . 98
2.8.2 Hygrothermal behaviour . . . . . . . . . . . . . . . . . . . . 99
2.8.3 Thermal expansion coefficients . . . . . . . . . . . . . . . . 99
2.8.4 Thermal (and curing) stresses . . . . . . . . . . . . . . . . . 101
2.8.5 Moisture expansion coefficients and stresses . . . . . . . . . 102
2.9 Textile composites analysis . . . . . . . . . . . . . . . . . . . . . . 103
2.9.1 Orientation averaging method . . . . . . . . . . . . . . . . . 104
2.9.2 Limitations of mechanics of materials models . . . . . . . . 111
2.10 Eshelby-type inclusion methods . . . . . . . . . . . . . . . . . . . . 112
2.11 Mosaic method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
2.12 Finite element and voxel methods . . . . . . . . . . . . . . . . . . . 114
2.13 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
3 Ply mechanics and laminate analysis 117
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
3.2 Laminate notations and sign conventions . . . . . . . . . . . . . . . 118
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3.3 Stress and strain definitions . . . . . . . . . . . . . . . . . . . . . . 120
3.4 Ply mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
3.4.1 Fibre frame stiffness . . . . . . . . . . . . . . . . . . . . . . 122
3.4.2 Coordinate transformations . . . . . . . . . . . . . . . . . . 123
3.4.3 Stress transformations . . . . . . . . . . . . . . . . . . . . 125
3.4.4 Strain transformations . . . . . . . . . . . . . . . . . . . . . 126
3.4.5 Coordinate frame stiffness . . . . . . . . . . . . . . . . . . . 129
3.4.6 Engineering constants for a ply . . . . . . . . . . . . . . . . 131
3.5 Classical laminate theory . . . . . . . . . . . . . . . . . . . . . . . 133
3.5.1 Laminate extensional stiffness [A] and compliance [a]=[A]−1 136
3.5.2 Laminate coupling [B] and bending stiffness [D] matrices . 143
3.6 Laminate lay-up types and their properties . . . . . . . . . . . . . 157
3.7 Laminate engineering constants . . . . . . . . . . . . . . . . . . . . 159
3.8 Principles of design with laminate analysis . . . . . . . . . . . . . . 162
3.9 Transverse shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
3.10 Hygrothermal behaviour of laminates . . . . . . . . . . . . . . . . . 174
3.10.1 Hygrothermal laminate stress resultants . . . . . . . . . . . 174
3.11 Limitations of CLT and alternative methods . . . . . . . . . . . . . 176
3.12 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
III Finite element analysis methods 179
4 Fundamentals of finite element analysis 181
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
4.2 Principles of the finite element method . . . . . . . . . . . . . . . . 182
4.3 Finite element types and their applications . . . . . . . . . . . . . 185
4.4 Strain-displacement relations . . . . . . . . . . . . . . . . . . . . . 188
4.5 Element stiffness formulation . . . . . . . . . . . . . . . . . . . . . 190
4.5.1 Bar element stiffness - direct method . . . . . . . . . . . . . 190
4.5.2 Element stiffness – formal derivations . . . . . . . . . . . . 191
4.5.3 Bar element stiffness – via minimisation of total potential
energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
4.6 Assembly, boundary conditions and solution . . . . . . . . . . . . . 195
4.7 Displacement and stiffness transformations . . . . . . . . . . . . . 198
4.8 Two-dimensional constant strain triangle element . . . . . . . . . . 203
4.9 Other 2D and 3D higher order continuum elements . . . . . . . . . 211
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4.10 Beam and shell elements . . . . . . . . . . . . . . . . . . . . . . . . 213
4.11 Characteristic matrices for field analysis . . . . . . . . . . . . . . . 214
4.12 Miscellaneous topics . . . . . . . . . . . . . . . . . . . . . . . . . . 217
4.12.1 Isoparametric elements . . . . . . . . . . . . . . . . . . . . . 217
4.12.2 Gaussian integration . . . . . . . . . . . . . . . . . . . . . . 220
4.13 Element assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
4.14 Boundary conditions and rigid body modes . . . . . . . . . . . . . 225
4.15 Solution assumptions and validation . . . . . . . . . . . . . . . . . 226
4.16 Typical linear FE composites analysis . . . . . . . . . . . . . . . . 228
4.17 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
5 Topics in advanced analysis 231
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
5.2 Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
5.3 Geometric nonlinear implicit analysis . . . . . . . . . . . . . . . . . 234
5.3.1 Derivation of element stiffnesses [k ], [k ] and [k ] . . . . . 235
L σ T
5.3.2 Large displacement matrix [B ] for stiffness [k ] . . . . . . 235
L L
5.3.3 Stress stiffness matrix [k ] . . . . . . . . . . . . . . . . . . . 237
σ
5.3.4 Geometric nonlinear analysis using [K ], [K ] and [K ] . . 244
O L σ
5.3.5 Newton-Raphson iterative solution . . . . . . . . . . . . . . 245
5.4 Explicit FE analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 248
5.5 Comparison of implicit and explicit methods . . . . . . . . . . . . 254
5.6 Nonlinear material behaviour . . . . . . . . . . . . . . . . . . . . . 255
5.7 Classical and FE analysis of plate bending . . . . . . . . . . . . . . 259
5.7.1 Kirchhoff and Mindlin deformation hypotheses . . . . . . . 260
5.7.2 Classical analysis of plate bending . . . . . . . . . . . . . . 261
5.7.3 A composite plate bending finite element . . . . . . . . . . 264
5.7.4 Shell finite elements . . . . . . . . . . . . . . . . . . . . . . 268
5.8 Buckling analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
5.8.1 Classical buckling analysis . . . . . . . . . . . . . . . . . . . 273
5.8.2 Finite element linear eigenvalue buckling analysis . . . . . . 276
5.9 Free frequency analysis . . . . . . . . . . . . . . . . . . . . . . . . . 280
5.9.1 Classical frequency analysis . . . . . . . . . . . . . . . . . . 280
5.9.2 Finite element frequency analysis . . . . . . . . . . . . . . . 281
5.10 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
6 Draping, forming and consolidation 284
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6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
6.2 Forming processes and their analysis . . . . . . . . . . . . . . . . . 285
6.3 Fabric deformation mechanisms . . . . . . . . . . . . . . . . . . . . 287
6.4 Experimental methods for fabric properties . . . . . . . . . . . . . 289
6.4.1 Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
6.4.1.1 Picture frame shear test . . . . . . . . . . . . . . . 290
6.4.1.2 Bias extension shear test . . . . . . . . . . . . . . 293
6.4.2 Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
6.4.3 Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
6.4.4 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
6.5 Analytical models for fabrics and processing . . . . . . . . . . . . . 298
6.5.1 Kinematic models for dry fabrics . . . . . . . . . . . . . . . 299
6.5.2 Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
6.5.3 Consolidation . . . . . . . . . . . . . . . . . . . . . . . . . . 305
6.5.4 Intra-ply consolidation . . . . . . . . . . . . . . . . . . . . . 307
6.5.5 Inter-ply consolidation . . . . . . . . . . . . . . . . . . . . . 312
6.6 Geometric drape analysis . . . . . . . . . . . . . . . . . . . . . . . 314
6.7 Finite element draping simulation . . . . . . . . . . . . . . . . . . . 316
6.7.1 Constitutive model for dry fabric . . . . . . . . . . . . . . . 317
6.7.2 Coulomb sliding friction . . . . . . . . . . . . . . . . . . . . 319
6.8 Finite element thermoforming simulation. . . . . . . . . . . . . . . 321
6.8.1 Thermal properties for forming . . . . . . . . . . . . . . . . 322
6.8.2 Constitutive model for thermo-viscoelastic fabrics . . . . . 324
6.8.3 Viscous sliding friction . . . . . . . . . . . . . . . . . . . . . 326
6.9 Meso-scale finite element analysis of fabrics . . . . . . . . . . . . . 330
6.10 Other textile preforming processes . . . . . . . . . . . . . . . . . . 331
6.11 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
7 Composites infusion 335
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
7.1.1 Infusion manufacturing processes . . . . . . . . . . . . . . . 336
7.2 Darcy’s law for flow in a porous medium . . . . . . . . . . . . . . . 338
7.2.1 Darcy’s law for flow and velocity . . . . . . . . . . . . . . . 340
7.2.2 Darcy’s law for multi-dimensional flow . . . . . . . . . . . . 341
7.3 Measuring permeability . . . . . . . . . . . . . . . . . . . . . . . . 345
7.3.1 One-dimensional channel flow test . . . . . . . . . . . . . . 346
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7.3.2 Two-dimensional radial test . . . . . . . . . . . . . . . . . . 351
7.3.3 Three-dimensional through-thickness test . . . . . . . . . . 353
7.4 Analytical and empirical models for permeability . . . . . . . . . . 354
7.5 Measurement of fibre volume fraction and porosity . . . . . . . . . 357
7.6 Permeability and porosity of deformed fabrics . . . . . . . . . . . . 358
7.6.1 Fabric compaction . . . . . . . . . . . . . . . . . . . . . . . 359
7.6.2 Permeability and porosity in sheared fabrics . . . . . . . . . 361
7.6.3 Numerical methods to predict permeability . . . . . . . . . 364
7.7 Cure kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
7.8 Viscosity laws for infusion . . . . . . . . . . . . . . . . . . . . . . . 367
7.9 FE analysis of infusion . . . . . . . . . . . . . . . . . . . . . . . . . 370
7.9.1 Governing equations and a basic finite element . . . . . . . 371
7.9.2 Conforming finite elements and non-conservative mass flow 374
7.9.3 Non-conforming finite elements for mass conservation . . . 375
7.9.4 Flow and the control volume method . . . . . . . . . . . . . 377
7.9.5 Algorithm and examples for RTM FE analysis . . . . . . . 379
7.9.6 Theory, algorithm and examples of VIP analysis . . . . . . 383
7.10 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
8 Ply and laminate failure 389
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
8.2 Mechanical testing for stiffness and strength . . . . . . . . . . . . . 390
8.2.1 Individual fibre and matrix properties . . . . . . . . . . . . 391
8.2.2 Tension tests . . . . . . . . . . . . . . . . . . . . . . . . . . 392
8.2.3 Compression tests . . . . . . . . . . . . . . . . . . . . . . . 394
8.2.4 Shear tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 396
8.2.4.1 Intra-ply shear tests . . . . . . . . . . . . . . . . . 396
8.2.4.2 Inter-ply shear strength tests . . . . . . . . . . . . 398
8.2.5 Measurement of strains . . . . . . . . . . . . . . . . . . . . 399
8.2.6 Some closing remarks on failure measurements . . . . . . . 400
8.3 Macromechanical failure criteria . . . . . . . . . . . . . . . . . . . 401
8.3.1 Strength ratio . . . . . . . . . . . . . . . . . . . . . . . . . . 402
8.3.2 Lamina macromechanical failure criteria . . . . . . . . . . . 403
8.3.2.1 Non-interactivetheories: maximumstressandmax-
imum strain . . . . . . . . . . . . . . . . . . . . . 404
8.3.2.2 Interactive theories: Tsai-Hill and Tsai-Wu . . . . 406
8.3.2.3 Failure mode theories: Hashin and Sun . . . . . . 412
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8.3.2.4 Fracture plane theories: Puck and LaRC02 . . . . 414
8.3.3 Off-axis failure . . . . . . . . . . . . . . . . . . . . . . . . . 423
8.3.4 Laminate strength and analysis . . . . . . . . . . . . . . . . 424
8.4 Finite element failure analysis . . . . . . . . . . . . . . . . . . . . . 425
8.5 In-situ ply strength . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
8.6 Stiffness and strength of woven fabric composites . . . . . . . . . . 430
8.6.1 Stiffness and strength of orthogonal woven composites . . . 430
8.6.2 Stiffness of sheared fabric composites . . . . . . . . . . . . . 431
8.6.3 Failure of sheared fabric composites . . . . . . . . . . . . . 432
8.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433
9 Impact and crash 434
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434
9.2 Mechanical testing for impact . . . . . . . . . . . . . . . . . . . . . 435
9.2.1 Strain rates . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
9.2.2 Inter-laminar fracture toughness . . . . . . . . . . . . . . . 437
9.2.2.1 DCB test mode I . . . . . . . . . . . . . . . . . . . 438
9.2.2.2 ENF test mode II . . . . . . . . . . . . . . . . . . 440
9.2.2.3 MMB test mixed mode I/II . . . . . . . . . . . . . 441
9.2.2.4 Delamination strain rate testing . . . . . . . . . . 441
9.2.3 Damage tolerance . . . . . . . . . . . . . . . . . . . . . . . 442
9.3 Continuum damage mechanics . . . . . . . . . . . . . . . . . . . . 442
9.3.1 Ladev`eze’s ply damage model for IFF . . . . . . . . . . . . 445
9.3.2 A damage model for fibre failure . . . . . . . . . . . . . . . 452
9.3.3 Limitations of continuum damage. . . . . . . . . . . . . . . 454
9.4 Strain rate dependent failure and damage . . . . . . . . . . . . . . 455
9.5 Delamination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458
9.5.1 Delamination mechanisms . . . . . . . . . . . . . . . . . . . 458
9.5.2 Basic theory for delamination modelling . . . . . . . . . . . 460
9.6 Mesh dependency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465
9.7 Axial crash and fragmentation . . . . . . . . . . . . . . . . . . . . 467
9.8 Concluding comments to impact and crash . . . . . . . . . . . . . 470
9.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471
References 472
Index 486
viii