Modelling of Engineering Materials

A brief review of simplistic and linear models used to characterize the mechanical response of materials is presented. This is followed by a description of models that characterize the nonlinear response of solids and fluids from first principles. Emphasis is given to popular models that characterize the nonlinear response of materials.

The book also presents case studies of materials, where a comprehensive discussion of material characterization, experimental techniques and constitutive model development, is presented. Common principles that govern material response of both solids and fluids within a unified framework are outlined. Mechanical response in the presence of non-mechanical fields such as thermal and electrical fields applied to special materials such as shape memory materials and piezoelectric materials is also explained within the same framework.

C. Lakshmana Rao is a Professor in the Department of Applied Mechanics at Indina Institute of Technology (IIT) Madras, India. He obtained his B.Tech and MS from IIT Madras and Sc.D. from Massachusetts Institute of Technology. He teaches courses on continuum mechanics, numerical modeling, fracture mechanics and mechanics of materials. His research interests include modeling of failure in brittle materials, buckling control using smart materials and ballistic impact mechanics.

Abhijit P. Deshpande is a Professor in the Department of Chemical Engineering at IIT Madras, India. He obtained his B.Tech from IIT Bombay, M.S. from the University of Pittsburgh and Ph.D. from the University of Washington. He teaches courses on continuum mechanics, polymer rheology, momentum transfer and thermodynamics. His research interests include polymer rheology, flow visualization, ionically conducting polymers and polymeric composites.

Preface ix

Notations xiii

Chapter 1 : Introduction 1

1.1 Introduction to material modelling 1

1.2 Complexity of material response in engineering 2

1.3 Classification of modelling of material response 5

1.3.1 Empirical models 6

1.3.2 Micromechanical models 7

1.3.3 Phenomenological models 8

1.4 Limitations of the continuum hypothesis 9

1.5 Focus of this book 10

Chapter 2 : Preliminary Concepts 13

2.1 Introduction 13

2.2 Coordinate frame and system 13

2.3 Tensors 14

2.3.1 Tensors of different orders 15

2.3.2 Notations for tensors 17

2.4 Derivative operators 22

Summary 25

Exercise 25

Chapter 3 : Continuum Mechanics Concepts 29

3.1 Introduction 29

3.2 Kinematics 30

3.2.1 Transformations 34

3.2.1.1 Transformation of line elements 34

3.2.1.2 Transformation of volume elements 35

3.2.1.3 Transformation of area elements 36

3.2.2 Important types of motions 37

3.2.2.1 Isochoric deformations 38

3.2.2.2 Rigid body motion 39

3.2.2.3 Homogeneous deformations 40

3.2.3 Decomposition of deformation gradient 40

3.2.3.1 Polar decomposition theorem 40

3.2.3.2 Stretches 42

3.2.4 Strain measures 42

3.2.4.1 Displacements 43

3.2.4.2 Infinitismal strains 44

3.2.5 Motions 44

3.2.5.1 Velocity gradient 45

3.2.6 Relative deformation gradient 48

3.2.7 Time derivatives viewed from different coordinates 49

3.2.7.1 Co-rotational derivatives 50

3.2.7.2 Convected derivatives 52

3.3 Balance laws 55

3.3.1 Transport theorem 56

3.3.2 Balance of mass 57

3.3.3 Balance of linear momentum 58

3.3.4 Balance of angular momentum 62

3.3.5 Work energy identity 63

3.3.6 Thermodynamic principles 65

3.3.6.1 First law of thermodynamics 65

3.3.6.2 Second law of thermodynamics 67

3.3.6.3 Alternate energy measures in thermodynamics 68

3.3.7 Referential description of balance laws 70

3.3.7.1 Relations between variables in deformed and undeformed configurations 70

3.3.7.2 Statement of the balance laws in reference configuration 72

3.3.8 Indeterminate nature of the balance laws 73

3.3.9 A note on multiphase and multi-component materials 74

3.3.9.1 Chemical potential 75

3.4 Constitutive relations 75

3.4.1 Transformations 76

3.4.1.1 Euclidean transformations 76

3.4.1.2 Galilean transformations 77

3.4.2 Objectivity of mathematical quantities 77

3.4.3 Invariance of motions and balance equations 79

3.4.4 Invariance of constitutive relations 79

3.4.4.1 Frame invariance in a thermoelastic material 81

3.4.4.2 Constitutive relations for thermoelastic materials 82

3.4.4.3 Frame invariance and constitutive relations for a thermoviscous fluid 85

3.4.5 Frame invariance of derivatives 87

Summary 89

Exercise 90

Chapter 4 : Linear Mechanical Models of Material Deformation 95

4.1 Introduction 95

4.2 Linear elastic solid models 96

4.2.1 Small strain assumption of linear elasticity 98

4.2.2 Classes of elastic constants 98

4.2.2.1 General anisotropic linear elastic solid 99

4.2.2.2 Materials with single plane of elastic symmetry 100

4.2.2.3 Materials with two planes of elastic symmetry 100

4.2.2.4 Materials with symmetry about an axis of rotation 101

4.2.2.5 Isotropic materials 102

4.3 Linear viscous fluid models 103

4.3.1 General anisotropic viscous fluid 104

4.3.2 Isotropic viscous fluid 105

4.4 Viscoelastic models 106

4.4.1 Useful definitions for description of viscoelastic behaviour 107

4.4.1.1 Creep compliance and relaxation modulus 107

4.4.1.2 Phase lag, storage modulus and loss modulus 107

4.4.2 Simplistic models of viscoelasticity 110

4.4.2.1 Maxwell model 111

4.4.2.2 Kelvin-Voigt model 118

4.4.2.3 Mechanical analogs for viscoelastic models 119

4.4.3 Time temperature superposition 121

Summary 122

Exercise 122

Chapter 5: Non-linear Models for Fluids 125

5.1 Introduction 125

5.2 Non-linear response of fluids 126

5.2.1 Useful definitions for non-Newtonian fluids 126

5.2.1.1 Steady shear 127

5.2.1.2 Normal stresses 130

5.2.1.3 Material functions in extensional flow 130

5.2.2 Classification of different models 131

5.3 Non-linear viscous fluid models 132

5.3.1 Power law model 134

5.3.2 Cross model 134

5.4 Non-linear viscoelastic models 135

5.4.1 Differential-type viscoelastic models 135

5.4.2 Integral -type viscoelastic models 137

5.5 Case study: rheological behaviour of asphalt 138

5.5.1 Material description 138

5.5.2 Experimental methods 139

5.5.3 Constitutive models for asphalt 140

5.5.3.1 Non-linear viscous models 141

5.5.3.2 Linear viscoelastic models 141

5.5.3.3 Non-linear viscoelastic models 142

Summary 147

Exercise 147

Chapter 6 : Non-linear Models for Solids 149

6.1 Introduction 149

6.2 Non-linear elastic material response 149

6.2.1 Hyperelastic material models 151

6.2.2 Non-linear hyperelastic models for finite deformation 152

6.2.2.1 Network models of rubber elasticity 153

6.2.2.2 Mooney-Rivlin model for rubber elasticity 154

6.2.2.3 Ogden’s model for rubber elasticity 155

6.2.2.4 Non-linear hyperelastic models in infinitismal deformation 156

6.2.3 Cauchy elastic models 156

6.2.3.1 First order Cauchy elastic models 157

6.2.3.2 Second order Cauchy elastic models 158

6.2.4 Use of non-linear elastic models 158

6.3 Non-linear inelastic models 159

6.3.1 Hypo-elastic material models 160

6.4 Plasticity models 161

6.4.1 Typical response of a plastically deforming material 163

6.4.2 Models for monotonic plastic deformation 165

6.4.3 Models for incremental plastic deformation 170

6.4.4 Material response under cyclic loading 174

6.4.5 Generalized description of plasticity models 181

6.5 Case study of cyclic deformation of soft clayey soils 183

6.5.1 Material description 183

6.5.2 Experimental characterization 184

6.5.3 Constitutive model development for monotonic and cyclic behaviour 185

6.5.4 Comparison of model predictions with experimental results 187

Summary 189

Exercise 190

Chapter 7 : Coupled Field Response of Special Materials 193

7.1 Introduction 193

7.1.1 Field variables associated with coupled field interactions 194

7.2 Electromechanical fields 195

7.2.1 Basic definitions of variables associated with electric fields 195

7.2.2 Balance laws in electricity - Maxwell’s equations 196

7.2.3 Modifications to mechanical balance laws in the presence of electric fields 197

7.2.4 General constitutive relations associated with electromechanical fields 198

7.2.5 Linear constitutive relations associated with electromechanical fields 199

7.2.6 Biased piezoelectric (Tiersten’s) model 200

7.3 Thermomechanical fields 201

7.3.1 Response of shape memory materials 202

7.3.1.1 Response of shape memory alloys 202

7.3.1.2 Response of shape memory polymers 203

7.3.2 Microstructural changes in shape memory materials 204

7.3.2.1 Microstructural changes associated with shape memory alloys 205

7.3.2.2 Microstructural changes associated with shape memory polymers 206

7.3.3 Constitutive modelling of shape memory materials 208

7.3.3.1 Constitutive models for shape memory alloys 208

7.3.3.2 Constitutive models for shape memory polymers 209

Summary 210

Exercise 210

Chapter 8 : Concluding Remarks 213

8.1 Introduction 213

8.2 Features of models summarized in this book 214

8.3 Current approaches for constitutive modelling 215

8.4 Numerical simulation of system response using continuum models 218

8.5 Observations on system response 220

8.6 Challenges for the future 222

Summary 232

Exercise 232

Appendix 225

Bibliography 233

Index 235

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