Auxetic Materials and Structures (eBook)

Dr. T.C. Lim earned his PhD in the area of material mechanics in 2001, and thereafter pioneered auxetic solids research in Asia. In addition to being Asia’s leading researcher in the area of auxetic solids, Dr. Lim is also a pioneer and one of the world leaders in the area of semi-auxetics. Dr. Lim has been serving as a scientific committee member in auxetic conferences and workshops since 2009. In addition, he has been serving as a journal reviewer for auxetic-related papers since 2004.

Preface 7
Contents 8
About the Author 14
1 Introduction 15
Abstract 15
1.1 Definition of Poisson's Ratio 15
1.2 History of Poisson's Ratio 16
1.3 Definition of Auxetic Materials 17
1.4 History of Negative Poisson's Ratio 19
1.5 Naturally Occurring Auxetic Materials 20
1.6 Auxetic Foams 21
1.7 Auxetic Yarns and Textiles 46
1.8 Auxetic Liquid Crystalline Polymers 51
1.9 Other Topics 52
References 54
2 Micromechanical Models for Auxetic Materials 58
Abstract 58
2.1 Introduction 58
2.2 Re-entrant Open-Cell Microstructure 58
2.3 Nodule Fibril Microstructure---Hinging, Flexure and Stretching Modes of Fibrils 61
2.4 Generalized 3D Tethered-Nodule Model 71
2.5 Rotating Squares and Rectangles Models 75
2.6 Rotating Triangles Models 85
2.7 Tetrahedral Framework Structure 89
2.8 Hard Cyclic Hexamers Model 92
2.9 Missing Rib Models 94
2.10 Chiral and Anti-chiral Lattice Models 101
2.11 Interlocking Hexagons Model 110
2.12 Egg Rack Structure 116
References 116
3 Elasticity of Auxetic Solids 119
Abstract 119
3.1 Constitutive Relationships 119
3.2 Bounds in Poisson's Ratio for Isotropic Solids 125
3.3 Constitutive Relationships for Isotropic Solids 129
3.4 Moduli Relations 132
3.5 Density-Modulus Relation in Auxetic Foams 134
3.6 Large Elastic Deformation of Auxetic Solids 137
3.7 Anisotropic Auxetic Solids 140
3.8 Elastoplasticity of Auxetic Solids 154
3.9 Viscoelasticity of Auxetic Solids 156
References 156
4 Stress Concentration, Fracture and Damage in Auxetic Materials 158
Abstract 158
4.1 Introduction 158
4.2 Stress Concentration in Auxetic Solids with Cavities 159
4.3 Stress Concentration in Auxetic Solids with Rigid Inclusions 160
4.4 Stress Concentration in Auxetic Plates 162
4.5 Stress Concentration in Auxetic Rods 163
4.6 Fracture Characteristics of Auxetic Solids 167
4.7 Stress and Displacement Fields Around Notches in Auxetic Solids 169
4.8 Mode I Dimensionless Displacement Fields 172
4.9 Mode II Dimensionless Displacement Fields 173
4.10 Mode III Dimensionless Displacement Field 176
4.11 Damage in Auxetic Solids 177
4.12 Fatigue in Auxetic Materials 178
References 179
5 Contact and Indentation Mechanics of Auxetic Materials 181
Abstract 181
5.1 Introduction 181
5.2 Line Contact on Auxetic Materials 181
5.3 Point Contact on Auxetic Materials 190
5.4 Effect of Indenter Shape on Auxetic Materials 195
5.5 Contact Between Auxetic Spheres 201
5.6 Contact Deformation in Auxetic Composites 205
5.7 Indentation of Auxetic Foams 207
References 209
6 Auxetic Beams 210
Abstract 210
6.1 Stretching of Auxetic Bars 210
6.2 Cantilever Bending of Auxetic Beams with Circular Cross Sections 212
6.3 Cantilever Bending of Auxetic Beams with Rectangular Cross Sections 214
6.4 Cantilever Bending of Auxetic Beams with Narrow Rectangular Cross Sections 215
6.5 Cantilever Bending of Auxetic Beams with Wide Rectangular Cross Sections 215
6.6 Cantilever Bending of Auxetic Beams with Regular Rectangular Cross Sections 216
6.7 Uniformly Loaded Auxetic Beams with Narrow Rectangular Cross Sections 219
6.8 Torsion of Auxetic Rods 220
6.9 Remarks on Auxetic Rods with Circular Cross Sections 222
References 224
7 Auxetic Solids in Polar and Spherical Coordinates 225
Abstract 225
7.1 Introduction 225
7.2 Thick-Walled Auxetic Cylinders 226
7.3 Rotating Thin Auxetic Disks 229
7.4 Rotating Thick Auxetic Disks 232
7.5 Thick-Walled Auxetic Spheres 234
References 238
8 Thin Auxetic Plates and Shells 239
Abstract 239
8.1 Introduction 240
8.2 Flexural Rigidity of Auxetic Plates 240
8.3 Circular Auxetic Plates 249
8.4 Rectangular Auxetic Plates 267
8.5 Auxetic Plates on Auxetic Foundation 283
8.6 In-Plane Compression of Constrained Auxetic Plate 291
8.7 Spherical Auxetic Shells 293
A.0. Appendix A 298
A.0. Appendix B 299
References 299
9 Thermal Stresses in Auxetic Solids 301
Abstract 301
9.1 Introduction 301
9.2 General Thermoelasticity of Auxetic Solids 302
9.3 Thermoelasticity of 3D Auxetics with Complete Geometrical Constraints 306
9.4 Thermoelasticity of Plates with Temperature Variation Along Thickness 306
9.5 Thermoelasticity of Beams with Temperature Variation Along the Beam Thickness 308
9.6 Dimensionless Thermal Stresses for Auxetic Plates and Shells 310
9.7 Thermal Stresses for Auxetic Plates and Shells 316
9.8 Summary on Thermal Stresses in Auxetic Plates and Shells 324
9.9 Thermal Conductivity in Multi--re-entrant Honeycomb Structures 326
References 328
10 Elastic Stability of Auxetic Solids 329
Abstract 329
10.1 Introduction 329
10.2 Buckling of Auxetic Columns 330
10.3 Buckling of Rectangular Auxetic Plates 332
10.4 Buckling of Circular Auxetic Plates 336
10.5 Buckling of Cylindrical Auxetic Shells 340
10.6 Buckling of Spherical Auxetic Shells 344
10.7 Recent Advances on Instability in Relation to Auxetic Materials and Structures 346
References 352
11 Vibration of Auxetic Solids 353
Abstract 353
11.1 Introduction 353
11.2 Vibration of Circular Auxetic Plates 354
11.3 Vibration of Rectangular Auxetic Plates 358
11.4 Vibration of Cylindrical Auxetic Shells 366
11.5 Vibration of Spherical Auxetic Shells 369
11.6 Advanced Topics on Vibration and Acoustics of Auxetic Solids and Structures 370
References 372
12 Wave Propagation in Auxetic Solids 374
Abstract 374
12.1 Introduction 374
12.2 Longitudinal Waves in Prismatic Auxetic Bars 376
12.3 Plane Waves of Dilatation in Auxetic Solids 377
12.4 Plane Waves of Distortion in Auxetic Solids 378
12.5 Rayleigh Waves in Auxetic Solids 380
12.6 Non-dimensionalization of Wave Velocities 381
12.7 Advanced Topics on Wave Motion in Auxetic Solids 387
References 389
13 Wave Transmission and Reflection Involving Auxetic Solids 391
Abstract 391
13.1 Introduction 391
13.2 Analysis 393
13.3 Longitudinal Wave (1D Stress State or 3D Strain State) 395
13.4 Longitudinal Wave (Width-Constrained Plates) 396
13.5 Plane Waves of Dilatation (1D Strain State or 3D Stress State) 396
13.6 Torsional Waves 397
13.7 Rayleigh Waves 398
13.8 Non-dimensionalization of Transmitted and Reflected Stresses 398
13.9 Dimensionless Transmitted Stress in Longitudinal Waves (1D Stress State) 400
13.10 Dimensionless Transmitted Stress in Longitudinal Waves (Constrained-Width Plates) 401
13.11 Dimensionless Transmitted Stress in Plane Waves of Dilatation 403
13.12 Dimensionless Transmitted Stress in Torsional Waves 405
13.13 Dimensionless Transmitted Stress in Rayleigh Waves 407
13.14 Summary on Stress Wave Transmission Involving Auxetic Solids 409
References 410
14 Longitudinal Waves in Auxetic Solids 411
Abstract 411
14.1 Introduction 411
14.2 Review of Elementary Analysis 413
14.3 Density Correction 414
14.4 Lateral Inertia 415
14.5 Density Correction and Lateral Inertia 417
14.6 Analogy with Plane Waves of Dilatation 422
14.7 Lateral Inertia in Auxetic Love Rods 425
14.8 Lateral Inertia and Density Correction in Auxetic Love Rods 428
References 431
15 Shear Deformation in Auxetic Solids 432
Abstract 432
15.1 Introduction 432
15.2 Laterally-Loaded Thick Auxetic Beams 433
15.3 Shear Correction Factors for Isotropic Plates Within 22121 2264 v 2264 0.5 441
15.4 Laterally-Loaded Thick Circular Auxetic Plates 445
15.5 Laterally-Loaded Thick Polygonal Auxetic Plates 449
15.6 Laterally-Loaded Thick Rectangular Auxetic Plates 452
15.7 Buckling of Thick Auxetic Columns 458
15.8 Buckling of Thick Auxetic Plates 464
15.9 Vibration of Thick Auxetic Plates 472
References 477
16 Simple Semi-auxetic Solids 479
Abstract 479
16.1 Introduction 479
16.2 Elastic Properties of a Directional Semi-auxetic Solid 480
16.3 Kinematical Studies on Rotation-Based Semi-auxetics 486
16.4 Analysis of Semi-auxetic Yarns 493
16.5 Processing of Semi-auxetic Yarns 502
16.6 Functionally-Graded Semi-auxetic Beams 507
16.7 Semi-auxetic Rods 512
16.8 Semi-auxetic Sandwich Plates 519
16.9 Mixed Auxeticity of Semi-auxetic Sandwich Structures 525
References 535
17 Semi-auxetic Laminates and Auxetic Composites 537
Abstract 537
17.1 Introduction 537
17.2 Semi-auxetic Unidirectional Fiber Composites 538
17.3 Out-of-Plane Modulus of Semi-auxetic Laminates 540
17.4 In-plane Modulus of Semi-auxetic Laminates 548
17.5 Further Counter-Intuitive Modulus from Semi-auxetic Laminates 554
17.6 Comparison Between In-Plane and Out-of-Plane Modulus of Semi-auxetic Laminates 561
17.7 Semi-auxetic and Alternating Positive and Negative Thermal Expansion Laminates 562
17.8 Auxetic Composites 573
References 583
Index 585