Preface ...................................................... xiii
Acknowledgments ................................................ xv
Author ....................................................... xvii
1 Introduction ............................................... 1
1.1 Basic Definitions .......................................... 1
1.2 Polymeric Materials ........................................ 5
1.2.1 Basic Concepts ...................................... 5
1.2.2 Addition versus Condensation Polymers ............... 7
1.2.3 Molecular Structure ................................. 8
1.2.4 Thermoplastic versus Thermoset Polymers ............ 10
1.2.5 Amorphous versus Semicrystalline Thermoplastics .... 11
1.2.6 A-, B-, and C-Staged Thermosets .................... 12
1.2.7 The Glass Transition Temperature ................... 12
1.3 Fibrous Materials ......................................... 14
1.3.1 Glass Fibers ....................................... 15
1.3.2 Aramid Fibers ...................................... 16
1.3.3 Graphite and Carbon Fibers ......................... 17
1.3.4 Polyethylene Fibers ................................ 20
1.4 Commercially Available Forms .............................. 21
1.4.1 Discontinuous Fibers ............................... 21
1.4.2 Roving Spools ...................................... 22
1.4.3 Woven Fabrics ...................................... 22
1.4.4 Braided Fabrics .................................... 25
1.4.5 Pre-Impregnated Products or "Prepreg" .............. 27
1.5 Manufacturing Processes ................................... 29
1.5.1 Layup Techniques ................................... 29
1.5.2 Autoclave Process Cycles ........................... 31
1.5.3 Filament Winding ................................... 34
1.5.4 Pultrusion ......................................... 35
1.5.5 Resin Transfer Molding ............................. 37
1.6 Scope of This Book ........................................ 37
References ................................................ 39
2 Review of Force, Stress, and Strain Tensors ............... 41
2.1 The Force Vector .......................................... 41
2.2 Transformation of a Force Vector .......................... 43
2.3 Normal Forces, Shear Forces, and Free-Body Diagrams ....... 50
2.4 Definition of Stress ...................................... 52
2.5 The Stress Tensor ......................................... 54
2.6 Transformation of the Stress Tensor ....................... 59
2.7 Principal Stresses ........................................ 67
2.8 Plane Stress .............................................. 71
2.9 Definition of Strain ...................................... 79
2.10 The Strain Tensor ......................................... 83
2.11 Transformation of the Strain Tensor ....................... 85
2.12 Principal Strains ......................................... 90
2.13 Strains within a Plane Perpendicular to a Principal
Strain Direction .......................................... 93
2.14 Relating Strains to Displacement Fields .................. 100
2.15 Computer Programs 3DROTATE and 2DROTATE .................. 104
Homework Problems ........................................ 105
References ............................................... 109
3 Material Properties ...................................... 111
3.1 Material Properties of Anisotropic versus Isotropic
Materials ................................................ 111
3.2 Material Properties That Relate Stress to Strain ......... 118
3.2.1 Uniaxial Tests .................................... 119
3.2.2 Pure Shear Tests .................................. 122
3.2.3 Specialization to Orthotropic and Transversely
Isotropic Composites .............................. 126
3.3 Material Properties Relating Temperature to Strain ....... 133
3.3.1 Specialization to Orthotropic and Transversely
Isotropic Composites .............................. 135
3.4 Material Properties Relating Moisture Content to Strain .. 136
3.4.1 Specialization to Orthotropic and Transversely
Isotropic Composites .............................. 137
3.5 Material Properties Relating Stress or Strain to
Failure .................................................. 138
3.6 Predicting Elastic Composite Properties Based on
Constituents: The Rule of Mixtures ....................... 141
Homework Problems ........................................ 148
References ............................................... 152
4 Elastic Response of Anisotropic Materials ................ 155
4.1 Strains Induced by Stress: Anisotropic Materials ......... 155
4.2 Strains Induced by Stress: Orthotropic and Transversely
Isotropic Materials ...................................... 162
4.3 Strains Induced by a Change in Temperature or Moisture
Content .................................................. 172
4.4 Strains Induced by Combined Effects of Stress,
Temperature, and Moisture ................................ 173
Homework Problems ........................................ 176
5 Unidirectional Composite Laminates Subject to Plane
Stress ................................................... 179
5.1 Unidirectional Composites Referenced to the Principal
Material Coordinate System ............................... 179
5.2 Unidirectional Composites Referenced to an Arbitrary
Coordinate System ........................................ 194
5.3 Calculating Transformed Properties Using Material
Invariants ............................................... 213
5.4 Effective Elastic Properties of a Unidirectional
Composite Laminate ....................................... 217
5.5 Failure of Unidirectional Composites Referenced to the
Principal Material Coordinate System ..................... 225
5.5.1 The Maximum Stress Failure Criterion .............. 227
5.5.2 The Tsai-Hill Failure Criterion ................... 228
5.5.3 The Tsai-Wu Failure Criterion ..................... 229
5.6 Failure of Unidirectional Composites Referenced to an
Arbitrary Coordinate System .............................. 233
5.6.1 Uniaxial Stress ................................... 233
5.6.1.1 Maximum Stress Criterion ................. 234
5.6.1.2 Tsai-Hill Criterion ...................... 236
5.6.1.3 Tsai-Wu Criterion ........................ 237
5.6.1.4 Comparison ............................... 239
5.6.2 Pure Shear Stress States .......................... 241
5.6.2.1 Maximum Stress Criterion ................. 242
5.6.2.2 Tsai-Hill Criterion ...................... 245
5.6.2.3 Tsai-Wu Criterion ........................ 246
5.6.2.4 Comparisons .............................. 247
5.7 Computer Programs UNIDIR and UNIFAIL ..................... 249
5.7.1 Program UNIDIR .................................... 250
5.7.2 Program UNIFAIL ................................... 251
Homework Problems ........................................ 251
References ............................................... 257
6 Thermomechanical Behavior of Multiangle Composite
Laminates ................................................ 259
6.1 Definition of a "Thin Plate" and Allowable Plate
Loadings ................................................. 259
6.2 Plate Deformations: The Kirchhoff Hypothesis ............. 264
6.3 Principal Curvatures ..................................... 269
6.4 Standard Methods of Describing Composite Laminates ....... 276
6.5 Calculating Ply Strains and Stresses ..................... 280
6.6 Classical Lamination Theory .............................. 294
6.6.1 Constant Environmental Conditions ................. 296
6.6.2 Including Changes in Environmental Conditions ..... 312
6.7 Simplifications due to Stacking Sequence ................. 326
6.7.1 Symmetric Laminates ............................... 329
6.7.2 Cross-Ply Laminates ............................... 332
6.7.3 Balanced Laminates ................................ 334
6.7.4 Balanced Angle-Ply Laminates ...................... 336
6.7.5 Quasi-Isotropic Laminates ......................... 337
6.8 Summary of CLT Calculations .............................. 339
6.8.1 A CLT Analysis When Loads Are Known ............... 340
6.8.2 A CLT Analysis When Midplane Strains and
Curvatures Are Known .............................. 341
6.9 Effective Properties of a Composite Laminate ............. 342
6.9.1 Effective Properties Relating Stress to Strain .... 343
6.9.1.1 Extensional Properties ................... 343
6.9.1.2 Flexural Properties ...................... 348
6.9.2 Effective Properties Relating Temperature or
Moisture Content to Strain ........................ 350
6.10 Transformation of the ABD Matrix ......................... 355
6.11 Computer Program CLT ..................................... 361
6.12 Comparing Classical Lamination Theory and Finite-
Element Analyses ......................................... 363
6.13 Free Edge Stresses ....................................... 375
6.13.1 The Origins of Free Edge Stresses ................ 375
6.13.2 Analytical and Numerical Studies of Free Edge
Stresses ......................................... 381
6.13.3 Typical Numerical Results ........................ 383
Homework Problems ........................................ 387
References ............................................... 394
7 Predicting Failure of a Multiangle Composite Laminate .... 397
7.1 Preliminary Discussion ................................... 397
7.2 Estimating Laminate Failure Strengths Using CLT .......... 401
7.2.1 Using CLT to Predict First-Ply Failure ............ 401
7.2.2 Predicting Last-Ply Failure ....................... 408
7.3 First-Ply Failure Envelopes .............................. 412
7.4 Computer Programs LAMFAIL and PROGDAM .................... 415
7.4.1 Program LAMFAIL ................................... 417
7.4.2 Program PROGDAM ................................... 418
Homework Problems ........................................ 421
References ............................................... 423
8 Composite Beams .......................................... 425
8.1 Preliminary Discussion ................................... 425
8.2 Comparing Classical Lamination Theory to Isotropic
Beam Theory .............................................. 426
8.3 Types of Composite Beams Considered ...................... 432
8.4 Effective Axial Rigidity of Rectangular Composite Beams .. 437
8.5 Effective Flexural Rigidities of Rectangular Composite
Beams .................................................... 440
8.5.1 Effective Flexural Rigidity of Rectangular
Composite Beams with Ply Interfaces Orthogonal
to the Plane of Loading ........................... 440
8.5.2 Effective Flexural Rigidity of Rectangular
Composite Beams with Ply Interfaces Parallel to
the Plane of Loading .............................. 443
8.6 Effective Axial and Flexural Rigidities for Thin-Walled
Composite Beams .......................................... 449
8.7 Statically Determinate and Indeterminate Axially Loaded
Composite Beams .......................................... 467
8.8 Statically Determinate and Indeterminate Transversely
Loaded Composite Beams ................................... 472
8.9 Computer Program BEAM .................................... 487
Homework Problems ........................................ 488
References ............................................... 490
9 Stress Concentrations Near an Elliptical Hole ............ 491
9.1 Preliminary Discussion ................................... 491
9.2 Summary of the Savin Solution for an Anisotropic Plate
with Elliptical Hole ..................................... 492
9.3 Circular Holes in Unidirectional Laminates ............... 498
9.4 Elliptical Holes with an Aspect Ratio of Three
in Unidirectional Laminates .............................. 501
9.5 Circular Holes in Multiangle Laminates ................... 504
9.6 Computer Program HOLES ................................... 507
Homework Problems ........................................ 507
References ............................................... 508
10 The Governing Equations of Thin-Plate Theory ............. 509
10.1 Preliminary Discussion ................................... 509
10.2 Equations of Equilibrium for Symmetric Laminates ......... 515
10.2.1 Equations of Equilibrium Expressed in Terms of
Internal Stress and Moment Resultants,
Transverse Loading, and Out-of-Plane
Displacements ..................................... 516
10.2.2 Equations of Equilibrium Expressed in Terms of
the [ABD] Matrix, Transverse Loading, and
Midplane Displacement Fields ...................... 526
10.3 Boundary Conditions ...................................... 529
10.3.1 Geometric (Kinematic) Boundary Conditions ......... 530
10.3.2 Static (Natural) Boundary Conditions .............. 531
10.3.3 Combinations of Geometric and Static Boundary
Conditions ........................................ 535
10.3.3.1 Free Edge ................................ 537
10.3.3.2 Simply Supported Edges ................... 538
10.3.3.3 Clamped Edges ............................ 539
10.4 Representing Arbitrary Transverse Loads as
a Fourier Series .................................... 540
References ............................................... 546
11 Some Exact Solutions for Specially Orthotropic
Laminates ................................................ 547
11.1 Equations of Equilibrium for a Specially Orthotropic
Laminate ................................................. 547
11.2 In-Plane Displacement Fields in Specially Orthotropic
Laminates ................................................ 549
11.3 Specially Orthotropic Laminates Subject to Simple
Supports of Type S1 ...................................... 553
11.4 Specially Orthotropic Laminates Subject to Simple
Supports of Type S4 ...................................... 559
11.5 Specially Orthotropic Laminates with Two Simply
Supported Edges of Type SI and Two Edges of Type S2 ...... 566
11.6 The Navier Solution Applied to a Specially Orthotropic
Laminate Subject to Simple Supports of Type S4 ........... 572
11.7 Buckling of Rectangular Specially Orthotropic Laminates
Subject to Simple Supports of Type S4 .................... 575
11.8 Thermal Buckling of Rectangular Specially Orthotropic
Laminates Subject to Simple Supports of Type S1 .......... 585
11.9 Computer Program SPORTHO ................................. 589
References ............................................... 590
12 Some Approximate Solutions for Symmetric Laminates ....... 591
12.1 Preliminary Discussion ................................... 591
12.2 In-Plane Displacement Fields ............................. 598
12.3 Potential Energy in a Thin Composite Plate ............... 602
12.3.1 Evaluation of Strain Energy Component UI .......... 607
12.3.2 Evaluation of Strain Energy Component UII ......... 608
12.3.3 Evaluation of Strain Energy Component UIII ........ 612
12.3.4 Evaluation of Work Done by Transverse Loads ....... 617
12.4 Symmetric Composite Laminates Subject to Simple
Supports of Type S4 ...................................... 617
12.4.1 Deflections due to a Uniform Transverse Load ...... 618
12.4.2 Deflections due to a Sinusoidal Transverse Load ... 626
12.4.3 Deflections due to a Transverse Load Distributed
over an Interior Region ........................... 628
12.4.4 Deflections due to a Transverse Point Load ........ 630
12.5 Buckling of Symmetric Composite Plates Subject to
Simple Supports of Type S4 ............................... 631
12.6 Computer Program SYMM .................................... 635
References ............................................... 635
Appendix A: Experimental Methods Used to Measure In-Plane
Elastic Properties ....................................... 637
References ............................................... 641
Appendix B: Tables of Beam Deflections and Slopes ............. 643
Reference ................................................ 646
Index ......................................................... 647
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