This book was produced as part of the efforts funded by the European Research Coun-cil to better understand and characterise the nature of these composites in alabora-tory setting， and to resolve some of the numerous outstanding challenges and issues as viewed from mathematical and numerical perspectives. It therefore serves as a sum-mary of this project, which encapsulates three distinct topics in experimentation and seven facets with a basis in mathematics, numerical and computational methods. The core of the content is the peer-reviewed research articles generated by my colleagues, listed on the inside cover of the book, and ourselves. As we have primarily focussed on topics in magnetostatics, there is an added bonus that the content given herein can often be immediately(or with little effort) reformulated with electrostatics in mind.
Although Paul has been involved in the research of magneto-active polymer com-posites for many years, this had been a new topic for me when I initially took up this project. One of the goals that I hoped to achieve in terms of content transmission is to put forward, in as transparent a manner as possible, all of the details that I would have liked to have first known, and gotten a clearview of, when starting this line of work. As we were formulating the foundational work for some of the mathematically inclined sections, I came to realise that I had taken for granted how some of the fundamentals and key mathematical concepts surrounding magnetostatic s theory had been devel-oped and resolved.

Preface VII

Acknowledgment—IX

Acronyms XVII

Notation XIX

Table of Units XXIII

1 Introduction 1

2 Fabrication of magneto-active polymer composites 5

2.1 A general discussion on MAP fabrication 5

2.2 Characterisable MAP: Fromlabtofab 10

3 Experimental apparatus and testing procedure 15

3.1 Parallel-plate rotational rheometer 16

      3.1.1 Stress-controlled deformation 18

      3.1.2 Magneto-rheological device 21

      3.1.3 Influence of rotor geometry on magnetic field 23

      3.1.4 Caveats to consider when performing experiments 26

3.2 Experimental methodology 31

4 Magneto-mechanical characterisation of magneto-active polymer composites 35

4.1 Unfilled matrix (PDMS) 35

4.2 Microstructure developed in the cured filled MAP 37

4.3 Strain amplitude dependence of the composite MAP 39

      4.3.1 Response to mechanical loading 39

      4.3.2 Response to magneto-mechanical loading 42

      4.3.3 Influence of rotor geometries on the response of isotropic MAPs 45

      4.3.4 Modelling of isotropic MAPs at a fixed frequency 46

4.4 Frequency dependence of the composite MAP 47

      4.4.1 Response to mechanical loading 49

      4.4.2 Response to magneto-mechanical loading 52

5 Introduction to continuum magneto -mechanics 54

5.1 Continuum setting 54

      5.1.1 Continuum domain 54

      5.1.2 Kinematics 55

      5.1.3 Line, area and volume transformations 56

      5.1.4 Time derivatives 57

      5.1.5 Fundamentals of electromagnetism 58

5.2 Continuum theorems for materials with discontinuities 63

      5.2.1 Control volumes with surface discontinuities 64

      5.2.2 Control surfaces with line discontinuities 68

5.3 Governing equations 70

      5.3.1 Balance laws 72

      5.3.2 Spatial relationships 76

      5.3.3 Summary of balance laws 77

      5.3.4 Ponder o motive force and moment 82

      5.3.5 Formulations for magnetic potentials 84

      5.3.6 Weak formulation of conservation laws 86

      5.3.7 Variational formulations 88

5.4 Thermodynamics 93

      5.4.1 Firstlawofthermodynamics: Energybalance 94

      5.4.2 Secondlawofthermodynamics: Entropybalance 96

      5.4.3 Parameterisation of isothermal energy functions 98

6 General aspects of computational simulation of coupled problems 100

6.1 Finite element discretisation 100

      6.1.1 Displacement field 101

      6.1.2 Magnetic vector potential field 105

      6.1.3 Magnetic scalar potential field 107

      6.1.4 Summary of finite element implementation 109

      6.1.5 Tools automating the computation of finite element linearisation sand constitutive model tangent moduli 110

6.2 Evaluation of definite integrals 111

6.3 Solution of a time/load increment 113

      6.3.1 Solution to the time-independent non-linear problem using the Newton-Raphson method 114

      6.3.2 Formation and solution of the discrete system of linear equations 114

      6.3.3 Special considerations when using the magnetic scalar potential formulation 116

7 Constitutive modelling 118

7.1 Preliminaries to magneto-mechanical energy functions 120

      7.1.1 Volumetric-isochoric split 120

      7.1.2 Invariants for isotropic media 121

      7.1.3 Transverse isotropy 122

7.2 Visco magneto -viscoelasticity 123

7.3 Transverse isotropy and particle chain dispersion 133

8 Phenomenological modelling of the curing process 142

8.1 A continuum framework for the curing of polymers 145

      8.1.1 Curing in visco magneto-viscoelastic materials 147

      8.1.2 Curing with field-sensitive shrinkage effects 149

9 Homogenisation 157

9.1 First-order homogenisation of magneto-coupled materials 159

      9.1.1 Hill's condition on the equality of micro-and macro-scale virtual powers 162

      9.1.2 Consistent boundary conditions arising from the equality of virtual power 164

      9.1.3 Computation of consistent tangent moduli for the macro-scale problem 174

      9.1.4 Homogenisation of curing using the Mori-Tanaka method 178

9.2 The stochastic finite element method 184

      9.2.1 Extension into stochastic dimensions 185

      9.2.2 Defining material discontinuities using level set functions 188

      9.2.3 Basis function selection 189

10 Modelling and computational simulation at the micro-scale 200

10.1 Single particle representative volume element problem 200

      10.1.1 Solution accuracy and finite element discretisation 201

      10.1.2 Computation of magnetic forces and torques 206

10.2 Micro-structural studies of MAP compositions 211

      10.2.1 Full resolution simulation of a prototype magnetostatic microstructural model 212

      10.2.2 Influence of microstructural organisation in a representative volume element on the response characteristics of a prototype magneto elastic material 215

11 Modelling and computational simulation at the macro-scale 224

11.1 Macro-to micro-scale transition using the FE² approach 224

11.2 lmm ersion of magnetic bodies in freespace 228

      11.2.1 Mesh motion in the freespace 229

11.3 Mixed variational approach for quasi-incompressible media 238

12 Further reading 251

A Identities 255

A.1 Operation identities 255

A.2 Generic differential identities 257

A.3 Differentialandrateidentities: Continuummechanics 262

B Calculus 265

B.1 Microscopic theorems 265

B.2 Continuum theorems 266

      B.2.1 Materials without discontinuities 268

      B.2.2 Materials with discontinuities 271

B.3 Continuum identities 275

C Derivations and proofs 277

C.1 Fundamentals of electromagnetism 277

      C.1.1 Electrostatics 277

      C.1.2 Magnetostatics 278

C.2 Continuum mechanics for magneto elasticity 279

C.3 Stress tensors 283

      C.3.1 Definitions 283

      C.3.2 Divergences 285

      C.3.3 Jumps 286

C.4 Ponderomotive forces and tractions 287

      C.4.1 Definitions of the Lorentz forces 287

C.5 Thermomechanical and electromagnetic balance laws 290

      C.5.1 Derivation of spatial statement of Maxwell's equations in a non-relativistic Eulerian reference frame 290

      C.5.2 Derivation of spatial statement of mechanical conservation laws (for magnetostatic systems) 294

      C.5.3 Derivation of additional balance and jump identities for potentials 298

      C.5.4 Transformation of conservation laws to their referential description 299

      C.5.5 Weak formulation of quasi-static balance of linear momentum 306

C.6 Legendre transformations 307

C.7 Thermodynamics 313

      C.7.1 Work performed by the Lorentz forces 313

      C.7.2 Boundary contributions to external power 315

      C.7.3 Combined contribution of external mechanical and electromagnetic powers 316

C.8 Homogenisation 317

      C.8.1 Relationship between macroscopic and microscopic field quantities 317

      C.8.2 The Hill-Mandel condition 319

      C.8.3 Algorithmically consistent tangent moduli 321

C.9 Constitutive modelling 325

      C.9.1 Free energy function derivatives 325

      C.9.2 Invariants 325

      C.9.3 Invariant derivatives 326

      C.9.4 Freespace stored energy function derivatives 327

      C.9.5 Volumetric/devia toric split of free energy function 328

C.10 Time integrators for rate-dependent materials described by internal variables 329

Bibliography 333

Image reproduction 367

Index 369

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