This book provides a unified theory of nonlinear electro-magnetomechanical interactions of soft materials capable of large elastic deformations. The authors include an overview of the basic principles of the classical theory of electromagnetism from the fundamental notions of point charges and magnetic dipoles through to distributions of charge and current in a non-deformable continuum, time-dependent electromagnetic fields and Maxwell’s equations. They summarize the basic ingredients of continuum mechanics that are required to account for the deformability of material and present nonlinear constitutive frameworks for electroelastic and magnetoelastic interactions in a highly deformable material. The equations contained in the book are used to formulate and solve a variety of representative boundary-value problems for both nonlinear electroelasticity and magnetoelasticity

We first became interested in the interaction between mechanical deformation and magnetic or electric fields just over 10 years ago. This interest was in part motivated by the development of elastomeric materials capable of large deformations that can be generated by the application of an electric or magnetic field. These were beginning to be used in various devices, ranging from activators and sensors to vibration and damping controls. Modelling of the behaviour of such materials requires an adequate theory of nonlinear deformations that can also accommodate electric or magnetic fields in the constitutive description. Although there were nonlinear theories in the literature based on continuum mechanics, we didn't find that they lent themselves easily to applications, in particular to the formulation of boundary-value problems. We therefore decided to seek an approach that could simplify the structure of the constitutive equations and, as a result, the governing equations of equilibrium and motion. We feel that this aim has been achieved in the last few years, and the subject is nowata point where it would be useful to write a connected account of this recent work, and this monograph is the result.
After an introductory chapter, we begin with a chapter (Chap. 2) that summarizes the necessary background from electrostatics, magnetostatic sand electrodynamics, aimed primarily at those who have not previously been exposed to much of this theory. This is followed by Chap. 3, which summarizes the essential ingredients of continuum mechanics and nonlinear elasticity theory, partly for the benefit of those who have not attended a course of continuum mechanics. For those with relevant backgrounds the material in these first two chapters will be familiar, but they provide the basic theory and notations that are required in order to merge these distinct subject areas and to derive a coupled nonlinear theory of electro elastic interactions and, separately, of magneto elastic interactions. In each case the theory is applied to simple representative problems to illustrate the influences of the electric and magnetic fields on the elastic behaviour of materials in the finite deformation regime. We also include a chapter on variational approaches to both electro elasticity and magneto elasticity. We then provide a discussion of the(linearized) incremental equations superimposed on an underlying configuration consisting of a finite deformation in the presence of either an electric field or a magnetic field. This is used, first in the electro elastic case, to evaluate the stability of the underlying configuration for some simple body geometries, and, for magneto elastic materials, the magneto acoustic approximation is adopted in order to study the propagation of magneto elastic homogeneous plane waves and surface waves.
In the course of writing we have obtained a significant number of new results, which are included here but otherwise unpublished. We have also tried to unify the notation, and therefore much of the notation in the later chapters differs from that in our various papers.
We have been very much helped by the encouragement of colleagues and friends who have been very positive about this project and by those researchers who have taken onboard our approach and developed it in different directions.
This monograph is aimed at researchers and graduate students whose interests are at the interface of electromagnetism and continuum mechanics, whether they are mathematicians, engineers or physicists. It requires some familiarity with vector and tensor calculus and some basic knowledge of electromagnetic theory and continuum mechanics.
Medford, MA, USA Luis Dorfmann
Glasgow, UKRay W. Ogden

1 Introduction 1

References 5

2 Electromagnetic Theory 9

2.1 Electrostatics 9

      2.1.1 Preliminary Remarks 9

      2.1.2 The Electric Field 10

      2.1.3 The Lorentz Law of Force 10

      2.1.4 Coulomb's Law 11

      2.1.5 Charge Conservation 12

      2.1.6 Units 14

      2.1.7 The Field of a Static Charge Distribution 14

      2.1.8 Gauss's Theorem 16

      2.1.9 The Field of a Dipole 18

      2.1.10 The Force and Couple on a Dipole in an Electric Field 20

2.2 Magnetostatics 21

      2.2.1 Preliminary Remarks 21

      2.2.2 The Biot-Savart Law and the Vector Potential 23

      2.2.3 Scalar Magnetic Potential 24

      2.2.4 Ampere's Circuital Law 27

      2.2.5 Force and Couple on a Dipole in a Magnetic Field 28

2.3 Faraday'sLaw of Induction 30

      2.3.1 Preliminary Remarks 30

      2.3.2 Electromotive Force 31

      2.3.3 Flux of a Magnetic Field Through a Moving Circuit 31

      2.3.4 Faraday's Law 34

2.4 Maxwell's Equations 36

      2.4.1 The Full Set of Maxwell's Equations 36

      2.4.2 Polarization and Magnetization in Materials 39

2.5 Boundary Conditions 43

      2.5.1 Boundary Conditions for E and D 43

      2.5.2 Boundary Conditions for B and H 45

References 46

3 Nonlinear Elasticity Background 47

3.1 Continuum Kinematics 47

      3.1.1 Deformation of a Solid Continuum 47

      3.1.2 Spatial Derivatives of Field Variables 49

      3.1.3 The Deformation Gradient 50

      3.1.4 Motion of a Continuum 55

      3.1.5 Integral Theorems 58

      3.1.6 Transport Formulas 60

3.2 Mechanical Balance Equations 61

      3.2.1 Mass Conservation 61

      3.2.2 Forces and Momenta 61

      3.2.3 Euler's Laws of Motion 62

      3.2.4 Energy Balance 65

3.3 Constitutive Equations for Hyperelastic Solids 67

      3.3.1 Hyper elasticity 67

      3.3.2 Objectivity 68

      3.3.3 Material Symmetry 70

      3.3.4 Incompressible Materials 76

3.4 Bouni dary-Value Problems 81

      3.4.1 Extension and Inflation of a Thick-Walled Tube 83

      3.4.2 The Azimuthal Shear Problem 86

      3.4.3 The Axial Shear Problem 89

      References 90

4 Nonlinear Electro elastic Interactions 91

4.1 Preliminaries 91

4.2 Equilibrium and Stress 92

4.3 Energy Balance and Constitutive Laws 96

4.4 Lagrangian Formulations 99

      4.4.1 Electrostatic Equations and Boundary Conditions 99

      4.4.2 Equilibrium Equation and Traction Boundary Condition 101

      4.4.3 Constitutive Equations 101

      4.4.4 Incompressible Materials 104

      4.4.5 Material Symmetry Considerations 105

4.5 Specialization to Linear Elasticity 108

      4.5.1 Relation to Linear Piezo electricity 109

      4.5.2 General Linear Theory 111

References 112

5 Electro elastic Boundary-Value Problems 113

5.1 Governing Equations 113

5.2 Homogeneous Deformations 115

      5.2.1 Pure Homogeneous Strain 116

      5.2.2 Simple Shear 118

5.3 Non-homogeneous Deformations 120

      5.3.1 Extension and Inflation of a Tube 120

      5.3.2 Helical Shear of a Circular Cylinder 128

      5.3.3 Inflation of aSpherical Shell 132

      References 136

6 Nonlinear Magneto elastic Interactions 137

6.1 Preliminaries 137

6.2 Equilibrium and Stress 138

      6.2.1 Magnetic Forces and Couples 138

      6.2.2 Mechanical Equilibrium 142

6.3 Constitutive Equations 144

      6.3.1 Eulerian Formulations 144

6.4 Lagrangian Formulations 148

      6.4.1 Magnetostatic Equations and Boundary Conditions 148

      6.4.2 Equilibrium Equation and Traction Boundary Condition 149

      6.4.3 Constitutive Equations 149

      6.4.4 Incompressible Materials 152

      6.4.5 Material Symmetry Considerations 152

6.5 Linear Magneto elasticity and Piezo magnetism 154

References 155

7 Magnetoelastic Boundary-Value Problems 157

7.1 Preliminaries 157

7.2 Governing Equations 158

7.3 Homogeneous Deformations 160

      7.3.1 Pure Homogeneous Strain 160

      7.3.2 Illustration 162

      7.3.3 Simple Shear 164

7.4 Application to Circular Cylindrical Geometry 167

      7.4.1 Extension and Inflation of a Tube 168

      7.4.2 Helical Shear of a Tube 173

References 179

8 Variational Formulations in Electro elasticity and Magnetoelasticity 181

8.1 Variational Formulations in Electroelasticity 181

      8.1.1 Formulation in Terms of the Electrostatic Scalar Potential 182

      8.1.2 Formulation in Terms of the Electrostatic Vector Potential 186

8.2 Variational Formulations in Magneto elasticity 190

      8.2.1 Formulation in Terms of the Magnetostatic Scalar Potential 191

      8.2.2 Formulation in Terms of the Magnetostatic Vector Potential 194

References 196

9 Incremental Equations 199

9.1 Continuum Electrodynamics 199

      9.1.1 Maxwell's Equations 200

      9.1.2 Equations of Motion 202

9.2 Incremental Equations 203

      9.3 Incremental Deformations 206

      9.3.1 Incremental Electro elasticity 206

      9.3.2 Incremental Magneto elasticity 217

      9.3.3 Incremental Boundary Conditions 218

      9.3.4 Constitutive Relations 220

References 229

10 Electro elastic Stability 231

10.1 Preliminary Remarks 231

10.2 Governing Equations 232

      10.2.1 Incremental Formulation Based on Ω 233

      10.2.2 Incremental Formulation Based on Ω* 234

10.3 The Elecectroelastic Half-Space 234

      10.3.1 Exterior Electric Field 234

      10.3.2 Incremental Fields and Equations 236

      10.3.3 Exterior Equations 237

      10.3.4 Boundary Conditions 240

      10.3.5 Solution 240

      10.3.6 An Electro elastic Neo-Hooke an Solid 342

10.4 An Electro elastic Plate 245

      10.4.1 Solution 247

      10.4.2 Application to the Neo-Hooke an Electro elastic Solid 249

10.5 An Electroelastic Plate with Electrodes 251

      10.5.1 Incremental Equations 253

      10.5.2 Incremental Traction Boundary Conditions 254

      10.5.3 Incremental Electric Boundary Conditions 254

References 258

11 Magneto elastic Wave Propagation 261

11.1 Preliminaries 261

11.2 The Quasimagnetostatic Approximation 262

11.3 Incremental Homogeneous Plane Waves 264

      11.3.1 Isotropic Magneto elasticity 265

      11.3.2 Example: A Prototype Magneto elastic Solid 266

      11.3.3 Two-Dimensional Specialization 267

11.4 Surface Waves 271

      11.4.1 The Case B1= 0 272

      11.4.2 The Case B2= 0 279

      11.4.3 Love-Type Surface Waves 282

References 297

A Basic Vector and Tensor Operations 299

A.1 Vector and Tensor Identities 299

A.2 Differential Operations on Scalar, Vector and Tensor Fields 300

      A.2.1 Rectangular Cartesian Coordinates 301

      A.2.2 Cylindrical Polar Coordinates 302

      A.2.3 Spherical Polar Coordinates 304

A.3 Integral Theorems 306

Index 309

A. Luis Dorfmann and Raymond W. Ogden are giants in this field and in the field of Nonlinear Elasticity. Dr. Dorfmann is a faculty member at Tufts University and Dr. Ogden is a faculty member at the University of Glasgow.

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