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 ixNotations xiiiChapter 1 : Introduction 11.1 Introduction to material modelling 11.2 Complexity of material response in engineering 21.3 Classification of modelling of material response 51.3.1 Empirical models 61.3.2 Micromechanical models 71.3.3 Phenomenological models 81.4 Limitations of the continuum hypothesis 91.5 Focus of this book 10Chapter 2 : Preliminary Concepts 132.1 Introduction 132.2 Coordinate frame and system 132.3 Tensors 142.3.1 Tensors of different orders 152.3.2 Notations for tensors 172.4 Derivative operators 22Summary 25Exercise 25Chapter 3 : Continuum Mechanics Concepts 293.1 Introduction 293.2 Kinematics 303.2.1 Transformations 343.2.1.1 Transformation of line elements 343.2.1.2 Transformation of volume elements 353.2.1.3 Transformation of area elements 363.2.2 Important types of motions 373.2.2.1 Isochoric deformations 383.2.2.2 Rigid body motion 393.2.2.3 Homogeneous deformations 403.2.3 Decomposition of deformation gradient 403.2.3.1 Polar decomposition theorem 403.2.3.2 Stretches 423.2.4 Strain measures 423.2.4.1 Displacements 433.2.4.2 Infinitismal strains 443.2.5 Motions 443.2.5.1 Velocity gradient 453.2.6 Relative deformation gradient 483.2.7 Time derivatives viewed from different coordinates 493.2.7.1 Co-rotational derivatives 503.2.7.2 Convected derivatives 523.3 Balance laws 553.3.1 Transport theorem 563.3.2 Balance of mass 573.3.3 Balance of linear momentum 583.3.4 Balance of angular momentum 623.3.5 Work energy identity 633.3.6 Thermodynamic principles 653.3.6.1 First law of thermodynamics 653.3.6.2 Second law of thermodynamics 673.3.6.3 Alternate energy measures in thermodynamics 683.3.7 Referential description of balance laws 703.3.7.1 Relations between variables in deformed and undeformed configurations 703.3.7.2 Statement of the balance laws in reference configuration 723.3.8 Indeterminate nature of the balance laws 733.3.9 A note on multiphase and multi-component materials 743.3.9.1 Chemical potential 753.4 Constitutive relations 753.4.1 Transformations 763.4.1.1 Euclidean transformations 763.4.1.2 Galilean transformations 773.4.2 Objectivity of mathematical quantities 773.4.3 Invariance of motions and balance equations 793.4.4 Invariance of constitutive relations 793.4.4.1 Frame invariance in a thermoelastic material 813.4.4.2 Constitutive relations for thermoelastic materials 823.4.4.3 Frame invariance and constitutive relations for a thermoviscous fluid 853.4.5 Frame invariance of derivatives 87Summary 89Exercise 90Chapter 4 : Linear Mechanical Models of Material Deformation 954.1 Introduction 954.2 Linear elastic solid models 964.2.1 Small strain assumption of linear elasticity 984.2.2 Classes of elastic constants 984.2.2.1 General anisotropic linear elastic solid 994.2.2.2 Materials with single plane of elastic symmetry 1004.2.2.3 Materials with two planes of elastic symmetry 1004.2.2.4 Materials with symmetry about an axis of rotation 1014.2.2.5 Isotropic materials 1024.3 Linear viscous fluid models 1034.3.1 General anisotropic viscous fluid 1044.3.2 Isotropic viscous fluid 1054.4 Viscoelastic models 1064.4.1 Useful definitions for description of viscoelastic behaviour 1074.4.1.1 Creep compliance and relaxation modulus 1074.4.1.2 Phase lag, storage modulus and loss modulus 1074.4.2 Simplistic models of viscoelasticity 1104.4.2.1 Maxwell model 1114.4.2.2 Kelvin-Voigt model 1184.4.2.3 Mechanical analogs for viscoelastic models 1194.4.3 Time temperature superposition 121Summary 122Exercise 122Chapter 5: Non-linear Models for Fluids 1255.1 Introduction 1255.2 Non-linear response of fluids 1265.2.1 Useful definitions for non-Newtonian fluids 1265.2.1.1 Steady shear 1275.2.1.2 Normal stresses 1305.2.1.3 Material functions in extensional flow 1305.2.2 Classification of different models 1315.3 Non-linear viscous fluid models 1325.3.1 Power law model 1345.3.2 Cross model 1345.4 Non-linear viscoelastic models 1355.4.1 Differential-type viscoelastic models 1355.4.2 Integral -type viscoelastic models 1375.5 Case study: rheological behaviour of asphalt 1385.5.1 Material description 1385.5.2 Experimental methods 1395.5.3 Constitutive models for asphalt 1405.5.3.1 Non-linear viscous models 1415.5.3.2 Linear viscoelastic models 1415.5.3.3 Non-linear viscoelastic models 142Summary 147Exercise 147Chapter 6 : Non-linear Models for Solids 1496.1 Introduction 1496.2 Non-linear elastic material response 1496.2.1 Hyperelastic material models 1516.2.2 Non-linear hyperelastic models for finite deformation 1526.2.2.1 Network models of rubber elasticity 1536.2.2.2 Mooney-Rivlin model for rubber elasticity 1546.2.2.3 Ogden’s model for rubber elasticity 1556.2.2.4 Non-linear hyperelastic models in infinitismal deformation 1566.2.3 Cauchy elastic models 1566.2.3.1 First order Cauchy elastic models 1576.2.3.2 Second order Cauchy elastic models 1586.2.4 Use of non-linear elastic models 1586.3 Non-linear inelastic models 1596.3.1 Hypo-elastic material models 1606.4 Plasticity models 1616.4.1 Typical response of a plastically deforming material 1636.4.2 Models for monotonic plastic deformation 1656.4.3 Models for incremental plastic deformation 1706.4.4 Material response under cyclic loading 1746.4.5 Generalized description of plasticity models 1816.5 Case study of cyclic deformation of soft clayey soils 1836.5.1 Material description 1836.5.2 Experimental characterization 1846.5.3 Constitutive model development for monotonic and cyclic behaviour 1856.5.4 Comparison of model predictions with experimental results 187Summary 189Exercise 190Chapter 7 : Coupled Field Response of Special Materials 1937.1 Introduction 1937.1.1 Field variables associated with coupled field interactions 1947.2 Electromechanical fields 1957.2.1 Basic definitions of variables associated with electric fields 1957.2.2 Balance laws in electricity - Maxwell’s equations 1967.2.3 Modifications to mechanical balance laws in the presence of electric fields 1977.2.4 General constitutive relations associated with electromechanical fields 1987.2.5 Linear constitutive relations associated with electromechanical fields 1997.2.6 Biased piezoelectric (Tiersten’s) model 2007.3 Thermomechanical fields 2017.3.1 Response of shape memory materials 2027.3.1.1 Response of shape memory alloys 2027.3.1.2 Response of shape memory polymers 2037.3.2 Microstructural changes in shape memory materials 2047.3.2.1 Microstructural changes associated with shape memory alloys 2057.3.2.2 Microstructural changes associated with shape memory polymers 2067.3.3 Constitutive modelling of shape memory materials 2087.3.3.1 Constitutive models for shape memory alloys 2087.3.3.2 Constitutive models for shape memory polymers 209Summary 210Exercise 210Chapter 8 : Concluding Remarks 2138.1 Introduction 2138.2 Features of models summarized in this book 2148.3 Current approaches for constitutive modelling 2158.4 Numerical simulation of system response using continuum models 2188.5 Observations on system response 2208.6 Challenges for the future 222Summary 232Exercise 232Appendix 225Bibliography 233Index 235

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