Composites and Metamaterials

Rod Lakes

    R. S. Lakes, Composites and Metamaterials, World Scientific, July (2020). ISBN 978 981 121 636 7
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Chapter 1 introduces structure property relations and kinds of materials.
Chapter 2 introduces basic composite structures and their elastic properties, also upper and lower bounds on properties.
Chapter 3 presents governing equations for anisotropic elasticity, and the role of material symmetry in relation to properties.
Chapter 4 introduces coupled fields in which materials can respond to multiple field variables such as those associated with elastic, electric and thermal response.
Chapter 5 presents specific materials with particulate, fibrous, and platelet inclusions and their properties.
Chapter 6 introduces cellular solids: honeycomb, foam, and lattices including structural hierarchy as well as materials with cellular structure that attain negative or extreme physical properties.
Chapter 7 presents materials of biological origin and the role of structural hierarchy in biological materials.
Chapter 8 discusses the role of the size scale of structural heterogeneity in analysis and design.
Chapter 9 presents viscoelastic response of heterogeneous solids: creep, relaxation, energy dissipation, wave attenuation and rate dependence.
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Outline

chapter 1 Introduction 1
1.1 Heterogeneous materials 1
1.1.1 Overview 1
1.1.2 Classification and terminology 2
1.1.3 Assumptions about the material 3
1.1.4 Materials vs. structures 4
1.2 Outline 4
1.3 Role of density 5
1.3.1 Modulus and density 6
1.3.2 Strength and density 7
1.3.3 Soft materials 7

chapter 2 Structures, properties, bounds 9
2.1 Introduction 9
2.2 Bounds on properties 10
2.2.1 Bounds on elastic constants of a homogeneous solid 10
Bounds on elastic moduli 10
Bounds on Poisson's ratio 11
2.2.2 Bounds on heat capacity of a homogeneous solid 12
2.2.3 Bounds on composite elastic properties 13
Voigt bound 13
Reuss bound 13
Hashin-Shtrikman bounds 13
2.2.4 Bounds on composite dielectric constant 15
2.3 Attaining the bounds on properties 15
2.3.1 Voigt composite 16
2.3.2 Reuss composite 17
2.3.3 Laminates: dielectric constant 17
2.3.4 Laminates: structural hierarchy 18
2.3.5 Attaining the Hashin-Shtrikman bounds: spheres 20
2.3.6 Attaining the Hashin-Shtrikman bounds: laminates 21
2.4 Inclusion shape: dilute concentration 22
2.4.1 Spherical inclusions 22
2.4.2 Fiber inclusions 22
2.4.3 Platelet inclusions 22
2.5 Exceeding bounds 23
2.5.1 Negative structural stiffness and extreme damping 23
2.5.2 Extreme nonlinear energy dissipation 25
2.5.3 Phase transformations 25
2.5.4 Negative and extreme moduli: stored energy 27
Biological tissue negative properties 28
2.5.5 Negative and extreme moduli: energy flux 28
2.5.6 Negative heat capacity 28
2.5.7 Negative capacitance 29
2.6 Summary 29

chapter 3 Symmetry and anisotropy 34
3.1 Introduction and Rationale 34
3.2 Tensors 34
3.3 Elastic properties 35
3.3.1 Hooke's law 35
3.3.2 Reduced notation: matrix form 36
3.3.3 Symmetry classes 36
3.3.4 Quasicrystals 37
3.3.5 Modulus matrices and symmetry 38
3.3.6 Isotropy 40
Elastic constants 40
Modulus matrix for isotropic solid 40
Modulus C_ 1111 via elementary method 42
3.3.7 Physical interpretation: elastic modulus 42
3.3.8 Physical interpretation: elastic compliance 44
3.3.9 Physical interpretation: experiment 45
3.3.10 How to show the effect of symmetry 46
3.3.11 Neumann's principle 46
3.4 Stress concentration: anisotropy 47
3.5 Chirality 47
3.6 Dielectric and optical properties 49
3.7 Materials, symmetry and structure 49
3.7.1 Examples of materials 49
3.7.2 Poisson's ratio in materials with structure 50
3.7.3 Quasicrystal elasticity 51
3.8 Summary 51

chapter 4 Coupled fields 54
4.1 Introduction: piezoelectricity, thermoelasticity 54
4.2 Piezoelectric properties 55
4.2.1 Piezoelectric properties and symmetry 55
Piezoelectricity: cubic symmetry 56
Piezoelectricity: isotropic solids 57
4.2.2 Piezoelectric materials 57
Voltage sensitivity 59
4.2.3 Strongly piezoelectric materials 61
4.2.4 Lead free piezoelectric materials 62
4.2.5 Experimental piezoelectric measurement 62
4.2.6 Electrostriction 62
4.2.7 Pyroelectric materials 62
4.2.8 Applications of piezoelectric and pyroelectric solids 63
4.3 Thermal expansion 63
4.3.1 Thermoelasticity, symmetry, causes 63
4.3.2 Thermal expansion anisotropy 64
4.3.3 Small or negative thermal expansion 65
4.3.4 Composite thermal expansion bounds 66
4.3.5 Applications and thermal expansion 66
4.3.6 Piezocaloric and related effects 66
Stress analysis via thermography 67
4.4 Fluid-solid composites 67
4.4.1 Constitutive equations 67
4.4.2 Experimental determination of constants 68
Undrained compliance 69
Waves 69
4.4.3 Applications: geology and geological engineering 69
4.4.4 Foams 70
4.4.5 Streaming potentials 70
4.4.6 Vascular materials 70
Healing and cooling 70
Laser cooling 71
4.5 Hall effect 72
4.6 Reciprocity 72
4.6.1 Non-reciprocal and extreme materials 73
4.7 Slow and fast processes 73
4.7.1 Overview 73
4.7.2 Isothermal and adiabatic moduli 74
4.7.3 Short and open circuit moduli 75
Electrical conductivity 75
4.7.4 Fluid-solid composites 75
4.8 Artificial muscles 77
4.9 Artificial tentacles 78
4.10 Energy harvesting 78
4.11 Other coupled fields 79
4.12 Summary 79

chapter 5 Particles, fibers, platelets 87
5.1 Introduction: structure 87
5.2 Particulate polymer matrix solids 88
5.2.1 Dental composites 88
5.2.2 Asphalt 90
5.2.3 Toughened polymers 90
5.2.4 Filled polymers; tire rubber; nano-fillers 91
5.2.5 Self healing polymers 93
5.3 Fibrous polymer matrix solids 93
5.3.1 Why fibers? 93
5.3.2 Unidirectional fibrous composites 94
Stress concentration factor 94
5.3.3 Laminates 94
Laminate analysis 95
Quasi-isotropic laminates 97
Cross-ply laminates; stacking sequence 97
5.3.4 Nano-tubes as fibers 99
5.3.5 Effects of moisture 100
5.3.6 Damage 100
5.3.7 Making fibrous composites 101
5.4 Platelet reinforcement 101
5.5 Metal matrix composites 102
5.5.1 Particulate metal matrix composite stiffness and strength 102
5.5.2 Nano-size particle inclusions in metal 103
5.5.3 Fiber inclusions in metal 103
5.6 Composites with renewable constituents 103
5.7 Thermoelastic composites 104
5.7.1 Thermal expansion, Voigt 104
5.7.2 Unidirectional composites: thermal expansion 104
5.7.3 Thermal benders 105
5.8 Piezoelectric composites 106
5.8.1 Piezoelectric composite structure and rationale 106
5.8.2 Piezoelectric Voigt composite 108
5.8.3 Piezoelectric benders 109
5.8.4 Piezoelectric composite uses and fabrication 109
5.9 In situ composites 110
5.10 Summary 111

chapter 6 Cellular solids and lattices 116
6.1 Introduction 116
6.2 Tessellations 116
6.3 Honeycomb 118
6.3.1 Honeycomb modulus 119
In plane modulus: scaling 119
In plane modulus: exact 120
Out of plane modulus 121
6.3.2 Honeycomb Poisson's ratio 121
In-plane Poisson's ratio 121
Out of plane Poisson's ratio 122
6.3.3 Honeycomb strength 122
In plane: elastic buckling 122
Out of plane: elastic buckling 123
Out of plane: plastic buckling 123
Out of plane: yield 123
6.3.4 Square cell honeycombs 124
6.3.5 Hierarchical solids: honeycombs 125
6.3.6 Making honeycomb 125
6.4 Foams 126
6.4.1 Foam elastic modulus 126
6.4.2 Poisson's ratio of foams 128
6.4.3 Nonlinearity and strength of foams 128
6.4.4 Toughness of foams 129
6.4.5 Dense foams and syntactic foams 130
6.4.6 Making foams 131
6.5 Lattices 131
6.5.1 Truss lattices: ribs 132
6.5.2 Continuous rib lattices 134
6.5.3 Lattice property bounds 135
6.5.4 Plate lattices 137
6.5.5 Surface lattices 138
6.5.6 Hierarchical lattices 140
6.5.7 Making lattices 142
6.6 Poisson's ratio tuning 143
6.6.1 Poisson's ratio in anisotropic materials 143
6.6.2 Poisson's tuning ratio in foams: negative or extreme 144
6.6.3 Poisson's ratio tuning in hinged structures: negative or extreme 146
6.6.4 Poisson's ratio tuning in lattices: negative or extreme 148
Applications, Poisson's ratio 151
6.7 Tuning coupled fields 151
6.7.1 Tuning thermal expansion: negative or extreme 151
6.7.2 Piezoelectric lattices 153
6.7.3 Tuning the Hall effect 153
6.8 Control of waves 154
6.8.1 Role of resonance 154
6.8.2 Tuning refraction of waves. Negative index. 155
Quasicrystal lattices 156
6.8.3 Electromagnetic lattices; cloaking 156
6.8.4 Acoustic lattices; cloaking 157
6.9 Applications of cellular solids 158
6.9.1 Applications of foam and honeycomb 158
6.9.2 Applications of lattices 158
6.10 Summary 159

chapter 7 Biological material structural hierarchy 168
7.1 Introduction 168
7.2 Bone and teeth 168
7.2.1 Compact bone: stiffness and strength 168
7.2.2 Compact bone: piezoelectricity 173
7.2.3 Compact bone: adaptation 173
7.2.4 Compact bone: stress concentration 174
7.2.5 Spongy bone 174
7.2.6 Teeth 176
7.3 Ligament and tendon 176
7.4 Wood and other plant tissue 178
7.4.1 Wood structure, stiffness and strength 178
7.4.2 Wood piezoelectricity 181
7.5 Summary 181

chapter 8 Size of heterogeneity 186
8.1 Introduction 186
8.2 Stress concentrations 186
8.2.1 Experiment 186
8.2.2 Analysis: ad hoc criteria 187
8.2.3 Analysis: generalized elasticity 187
8.3 Size effects 188
8.3.1 Size effects: structural example 188
8.3.2 Size effects: continuum view 190
8.3.3 Size effects: experiment 190
Milli-scale 190
Nano-scale 192
Interatomic scale 193
8.4 Generalized continuum elasticity 193
8.4.1 Cosserat theory 193
8.4.2 Stress and strain fields: effect of microstructure 196
8.4.3 Physical causes 197
8.4.4 Homogenization analyses 197
8.4.5 Hinged structures 197
8.4.6 Chirality in elasticity 197
8.4.7 Other generalized continua 198
8.5 Flexo-electricity: gradient piezoelectricity 199
8.6 Surface and free edge effects 200
8.7 Summary 200

chapter 9 Viscoelastic composites 206
9.1 Viscoelastic properties: introduction 206
9.2 Viscoelastic functions 206
9.2.1 Creep 206
9.2.2 Relaxation 207
9.2.3 Response to sinusoidal input 208
9.2.4 Viscoelasticity of typical materials 209
9.3 Viscoelasticity of composites 209
9.3.1 Viscoelasticity of Voigt laminates 210
9.3.2 Stiffness-damping maps of extremal composites 211
9.3.3 Stiffness-damping map: inclusion shape 212
9.3.4 Bounds on viscoelastic properties 213
Bounds on moduli 213
Bounds for a composite 214
9.3.5 Waves in composites 214
9.3.6 Negative damping; acoustic amplification 215
9.3.7 Extreme viscoelastic composites: inclusion shape 216
9.3.8 Extreme viscoelastic composites: stored energy 216
9.3.9 Viscoelasticity of fibrous composites 216
9.3.10 Effect of temperature 217
9.3.11 Poisson's ratio of viscoelastic materials 217
9.3.12 Viscoelasticity of cellular solids 218
9.3.13 Viscoelasticity of bone 218
9.3.14 Viscoelasticity of tendon and ligament 218
9.3.15 Viscoelastic damping of metal matrix composites 219
9.4 Summary 219

A Appendix 224
A.1 Solved Problems 224
A.1.1 Transverse, fibrous 224
A.1.2 Physical meaning of C_ 1111 in applications 224
A.1.3 Adiabatic and isothermal compliance 225
A.1.4 Foam stiffness vs. density 226
A.1.5 Steel foam and solid polymer 226
A.1.6 Cardboard honeycomb strength 226
A.1.7 Poisson's ratio of honeycomb 227
A.1.8 Particle inclusion concentration 227
A.1.9 Multiple particle sizes 227
A.1.10 Spongy bone modulus 228
A.1.11 Sneaker sole design 228
A.1.12 Cubic lattice 229
A.1.13 Lattice with tubular ribs 229
A.1.14 Motion from piezoelectric disk 230
A.1.15 Voltage from piezoelectric disk 230
A.1.16 Piezoelectric bender 231
A.1.17 Laminate of steel and rubber 231

A.2 Problems and questions 232

B Symbols 235
B.1 Principal symbols and definitions 235

We have used this book for our course in multiphase and heterogeneous materials in the Department of Materials Science.