Acceleration 80–81, 84

Ampère-Maxwell law 

Eulerian form 269–271, 273–274, 275–277

Lagrangian form 269–271, 275–277

Angular momentum 116–117, 139–140, 256–257

Archimedes' principle 202–203

Area elements 112, 112f

Axial vector 43–44

Balance of angular momentum 117, 133–135

for electrodynamics 

Eulerian form 256–257, 269–271

Lagrangian form 256–257, 275–277

for thermomechanics 

Eulerian form 117–118, 131–132

Lagrangian form 140, 141–142, 144–145

Balance of linear momentum 117, 132–133

for electrodynamics 

Eulerian form 252–256, 269–271

Lagrangian form 252–256, 275–277

for thermomechanics 

Eulerian form 117–118, 131–132

Lagrangian form 140, 141–142, 144–145

Blatz-Ko model 194

Body 75, 76f, 78–80

Body force 

electromagnetic 252–256, 280–282

mechanical 116–117, 139–140, 252–256

Cartesian components 

curl 56

divergence 56, 58–59

dyadic product 34

fourth-order tensor 47–48

gradient 56–58, 66–67

identity tensor 26, 32–33

indicial notation 25

product of two tensors 26, 33–34

second-order tensor 31–32

second-order tensor, transpose 26, 33

summation convention 12–31

third-order tensor 47–48

Cauchy deformation tensor 89–91

Cauchy-Schwarz inequality 11, 12

Cauchy stress tensor 124, 125–128

Cayley-Hamilton theorem 45–46, 192–193

Chain rule 52–53

Charge 

bound 260–262, 263–264

density, referential and spatial 260–262

free 260–262, 263–264

total 260–262, 263–264

Christoffel symbols 65–66, 67–68

Chu model 279

Clausius-Duhem inequality 148–149, 150–153, 175, 216–217, 259–260, 284–285

Eulerian integral form 148–151, 259–260, 269

Lagrangian integral form 148–151, 259–260, 275

restrictions imposed on constitutive equations 148–149

nonlinear elastic materials 183–186

thermo-electro-magneto-elastic materials 285–291

viscous fluids 207–208

Configuration 

initial 75, 78–80, 79f

present 75, 76f, 78–80, 79f

reference 75, 76f, 78–80, 79f

Conservation laws 117

angular momentum  See (Balance of angular momentum)

energy  See (First law of thermodynamics)

linear momentum  See (Balance of linear momentum)

mass 

Eulerian form 117–118, 131–132, 222, 269–271

Lagrangian form 140, 144–145, 251–252, 275–277

Constitutive equations 216–217

in electrodynamics 

definition 283–284

elastic solids, large deformation 284–285

elastic solids, small deformation 296–299

restrictions on 285–286, 293–294

in mechanics 

definition 158

inviscid fluids 205

Newtonian fluids 200–201

nonlinear elastic solids 194–195

restrictions on 159–161

in thermomechanics 

definition 175–178

inviscid fluids 212

Newtonian fluids 209–211

nonlinear elastic solids 191–193

restrictions on 178–179

Constitutive limit 

incompressibility 227–229

isentropic incompressibility 235–236

isothermal incompressibility 234–235

Constraint equations 

incompressibility (absolute) 220–222, 237–240

isentropic incompressibility 232–233

isothermal incompressibility 229–232

motion-entropy, general 217–218

motion-temperature, general 216–217

Constraint response 216–217, 220, 221, 230–231, 238

Continuum 3

Couette flow 226

Curl 52–53

Current 

conductive (free) 260–262, 264–265

density 260–262, 264–265

magnetization 264–265

polarization 260–262, 264–265

Curvilinear coordinates 

contravariant basis vectors 62–64, 62f

coordinate surfaces 59, 60f

covariant basis vectors 60, 61f

cylindrical polar coordinates 48–50, 60

physical components 63–67

spatial derivatives 

Christoffel symbols 65–68

tensor divergence 70–72

vector gradient and divergence 66–67, 68–72

spherical coordinates 48–50, 60

Cylindrical polar coordinates 48–50, 60

Deformation gradient 101–103

polar decomposition 91, 92–93, 92f

Density 

in present configuration 115–116

in reference configuration 115–116

Dependent variable 175–178, 283–284

Description, of a quantity 

material 75–78

referential (Lagrangian) 75–78

spatial (Eulerian) 75–78

Determinant 36

Dilatation 89–90, 110–111

Direct notation 5

Discrete 3

Displacement 

Cartesian component form 83

material description 77–78

referential description 77–78

spatial description 78

Divergence 52–53, 56, 80–83

Divergence theorem 53, 55–56, 276

Dyadic product 22, 34, 44–46

Effective electromagnetic fields 

Chu model 279, 279t

definition 261

Lorentz model 278, 279t

Minkowski model 278, 279t, 280

statistical model 278–279, 279t

Eigenvalues 44, 46

Eigenvectors 44, 97

Elastic solids, nonlinear 

compressible 

restrictions on constitutive equations, 

imposed by the second law 183–186

imposed by invariance and angular momentum 187–189

imposed by material symmetry (isotropy) 191–193

strain energy models 194–195, 241–243

incompressible 237–240

constitutive equations 239–240

strain energy models 241–243

Electric displacement, referential and spatial 263–264

Electric field, referential and spatial 262

Electric flux 263–265

Electric permittivity 263

Electrodynamics, continuum 

constitutive equations  See (Thermo-electro-magneto-elastic materials)

coupling terms, electromagnetically induced 280–282

effective electromagnetic fields 261

fundamental laws 250–271

Ampère-Maxwell law 264–265

balance of angular momentum 256–257

conservation of electric charge 260–262

conservation of mass 251–252

Faraday's law 262

first law of thermodynamics 257–259

Gauss's law, 

electricity 263–264

magnetism 262–263

balance of linear momentum 251–252

second law of thermodynamics 259–260

Maxwell stress tensor 281, 282–283

notation and nomenclature 250–251, 251f

spatial and referential quantities, relations between 265–268

thermo-electro-magneto-mechanical process 283–284, 299–300, 300f, 301t

total stress tensor 281

Electromagnetic energy 257–259, 280–282

Electromotive force 262

Energy theorem 

for electrodynamics 

Eulerian form 269–271

Lagrangian form 275–277

for mechanics 

definition 130–131

Eulerian form 130–131, 135–136, 269

Lagrangian form 143–144, 276

Enthalpy 218–219, 315–316

Entropy 148–151, 259–260

Equations of motion, mechanical.  See Conservation laws See also mass; Balance of linear momentum; Balance of angular momentum

Equations of state 218–219, 314

Euclidean space 

definition 5, 7

properties 7–10

inner product space 9, 11–12

metric space 9

normed space 9

vector space 7–9

Euler-Almansi strain tensor.  See Eulerian strain tensor

Eulerian strain tensor 97, 99

Faraday's law 

Eulerian form 262, 269–271

Lagrangian form 262, 275–277

Finger deformation tensor 97, 101–103

First law of thermodynamics 

for electrodynamics 

Eulerian form 269

Lagrangian form 275

for thermomechanics 

Eulerian form 120, 153

Lagrangian form 141, 144–145

Fluid mechanics.  See Inviscid fluids See also Newtonian fluids; Viscous fluids

Fourth-order tensors 48

Fundamental laws.  Electrodynamics, Mechanics, and Thermomechanics

Gauss's law 

electricity 263–264, 269, 275

magnetism 262–263, 269, 275

Gibbs free energy 218–219, 316–317

Gradient 52–53, 81–82

Green's deformation tensor 88–90

Green-Lagrange strain tensor.  See Lagrangian strain tensor

Heat flux vector 

referential (Lagrangian) 142, 237

spatial (Eulerian) 124–125, 128–130

Helmholtz free energy 177, 185, 189

Hyperelastic material 181–182, 183–186

Ideal fluids.  See Inviscid fluids See also mechanical theory

Identity tensor 12–13, 26, 32–33

Incompressibility.  Elastic solids, incompressible theory and Newtonian fluids, incompressible theory

Independent variable 175–178, 216, 219, 283–284

Indicial notation 25, 42–43

Inner product space 9, 11–12

Internal constraints.  See Constraint equations

Internal energy 119, 218–219, 257–259

Invariance under superposed rigid body motions 

application to inviscid fluids 205

application to nonlinear elastic solids 187–189

application to thermo-electro-magneto-elastic solids 293–294

application to viscous fluids 198–200, 207–208

comparision with material frame indifference 305

requirements on constitutive equations 171–174, 179, 293–294

Invariants 45–46, 191–193, 194–195, 237

Inverse, of a tensor 39–41

Inviscid fluids 

mechanical theory 205

ideal fluids 205–206

thermomechanical theory 

perfect fluids 212–214

Irrotational motions 105

Isochoric motions 110–113, 220

Kinetic energy 119, 257–259

Kronecker delta 25

Lagrange multiplier 221, 231, 239

Lagrangian strain tensor 97, 99

Left Cauchy-Green deformation tensor.  See Finger deformation tensor

Legendre transformation 177, 219, 288, 289t

Levinson-Burgess model 194–195

Linear momentum 116–117, 139–140, 252–256

Localization theorem 122–123

Lorentz model 278

Magnetic field, referential and spatial 264–265

Magnetic flux 262–263

Magnetic flux density, referential and spatial 262

Magnetization, referential and spatial 264–265

Magnetic permeability 264

Mass 115–116, 139–140, 251–252

Material derivative 80, 81

Material line 109–110

Material point 109, 109f

Material surface 109–110

Material symmetry 

basic principles 171–174

isotropy considerations, nonlinear elastic solids 191–193

Material volume 110

Matrix representation, of a tensor 26–33, 34

Maxwell's equations.  See Electrodynamics, continuum

Maxwell stress 281, 282–283

Mechanics 

constitutive equations  (Constitutive equations in mechanics)

inviscid fluids 205

nonlinear elastic materials 181–182

viscous fluids 

Navier-Stokes equations 202–204

Newtonian fluids 200–203

fundamental laws  See (Equations of motion, mechanical)

Metric space 9

Minkowski model 278, 280

Mooney-Rivlin model 

compressible 194–195

incompressible 241–243

Motion 75, 77, 78–80, 98–101

Navier-Stokes equations 

compressible 202–204

incompressible 219–224

neo-Hookean model 

compressible 194–195

incompressible 241–243

Newtonian fluids 

compressible 200–201, 209–211, 218–219

equations of state 209–211, 218–219, 314

mechanical theory 200–203

thermomechanical theory 209–211

incompressible 220–222, 227–229

thermal expansion, of 

definition 229

isentropic incompressibility 232–233, 235–236

isothermal incompressibility 229–232, 234–235

Nonlinear elasticity.  See Elastic solids, nonlinear

Normed space 9

Ogden model 

compressible 194–195

incompressible 242

Orthogonal 39, 41–43

Partial derivatives 48–53

Perfect fluids.  Inviscid fluids, thermomechanical theory

Permutation symbol 42–43

Piezoelectric materials 249, 302–303

Piola-Kirchhoff stress tensors 142, 145

Poiseuille flow 225f, 226

Polar decomposition 45, 46–47, 91–92

Polarization, referential and spatial 260–262

Position 

Cartesian component form 98–101

present 75–78

reference 75–78

Positive definite 39

Pressure 

constraint 221, 233, 238, 240

thermodynamic 200–201, 205, 209–211, 221–222, 230–231, 313–317

Product of two tensors 7–9, 16–17, 33–34

Product rule 80–83

Rate of deformation tensor 104–105

Real numbers 

definition 5, 6f

properties 5–7

Reference map 75–77

Residual dissipation inequality 286

Response functions.  See Constitutive equations

Right Cauchy-Green deformation tensor.  See Green's deformation tensor

Rotation tensors 88, 90, 91, 93–98, 100

Scalar product 35, 37, 38, 39, 40

Scalar triple product 43

Second law of thermodynamics.  See Clausius-Duhem inequality

Second-order tensor 12–17, 22, 26, 31–32

Skew tensor 17

Small neighborhood 85, 87f

Smart materials, modeling of.  See Electrodynamics, continuum

Spherical coordinates 48–50, 60

Stability, of thermodynamic equilibrium 7–10, 317–319

Statistical model 278–279

Steady laminar flow 219–224, 225f

Stokes's theorem 

Eulerian form 269

Lagrangian form 276

Strain energy 181–182, 183–186, 191–193, 194–195, 241–243

Strain-displacement relations 96–98, 296

Stress power 130–131

Stretch 

definition 88, 90–91

pure stretch 

in U 94–96, 100

in V 96, 100

tensors 88, 90–91

Summation convention 26

Superposed rigid body motions (SRBMs) 

definition 160–164, 160f

implications on invariance of constitutive equations  (Invariance under SRBMs)

transformations under a SRBM 

electromagnetic quantities, 

geometric and kinematic quantities 164–168

kinetic quantities 169–171

thermal quantities 178–180

Surface element 112–114

Symmetry, of a material.  See Material symmetry

Symmetric tensor 17, 43–44

Tangent basis vectors 60

Tensor algebra 12–13, 44–46

Cartesian components 24–31

Tensor calculus 47–48

Cartesian component form 56–59

Thermal expansion, of Newtonian fluids 

definition 229

isentropic incompressibility 232–233, 235–236

isothermal incompressibility 229–232, 234–235

Thermoelastic solids.  See Elastic solids, nonlinear

Thermo-electro-magneto-mechanical process 283–284, 299–300, 300f, 301t

Thermomechanical process 175–178

Thermoviscous fluids.  See Viscous fluids

Thermo-electro-magneto-elastic materials 

large-deformation theory 

angular momentum requirements 293

conjugate pairs 284–285

constitutive equations 286

fundamental laws  See (Electrodynamics, continuum)

invariance requirements 293, 294

Legendre transformation 288–291, 289t

reduced Clausius-Duhem inequality 284–285

residual dissipation inequality 286

small-deformation theory 

fundamental laws 295–296

linear constitutive equations 296–298, 297t

material symmetry, implications of 298–299, 299f

piezoelectric materials, specialization to 302–303

Thermomechanics 

constitutive modeling 

constitutive equations  (Constitutive equations in thermomechanics)

dependent variables 175

independent variables 175

thermomechanical process 176

fundamental laws 

balance of angular momentum  (Balance of angular momentum for thermomechanics)

balance of linear momentum  (Balance of linear momentum for thermomechanics)

conservation of mass  See (Conservation laws See also mass)

first law of thermodynamics  See (First law of thermodynamics)

Third-order tensor 47–48

Total stress tensor in electrodynamics 281

Trace 35

Traction 

referential (Lagrangian) 137–143, 252–256

spatial (Eulerian) 116–117, 124–125, 137–142, 252–256

Transport theorem 269, 271–273, 275

Transpose, of a tensor 17, 17f

Varga model 242

Vector product 42–43

Vector space 7–9

Velocity 80–81, 83, 104, 105

Velocity gradient 104–105

Viscosity 

bulk 209–211

dilatational 200–201, 205, 209–211

shear 200–201, 205, 209–211

Viscous fluids, compressible 

mechanical theory 

Navier-Stokes equations 202–204

Newtonian fluids 200–203

restrictions imposed by invariance 198–199

thermomechanical theory 

Newtonian fluids 209–211

restrictions imposed by invariance 207–208

restrictions imposed by the second law 207–208

Volume element 110–111, 111f, 113

Vorticity tensor 104, 105–107

Vorticity vector 104

Zero tensor 13