Introduction to Continuum Mechanics 3 Episode 15 pptx

Aris, R., Vectors, Tensors and the Basic Equations of Fluid Mechanics, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1962.. Noll, Viscometric Flows of Non-Newtonian Fluids, Springer

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CHAPTER 4

CHAPTERS

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CHAPTER 6

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CHAPTER 7

1 Aris, R., Vectors, Tensors and the Basic Equations of Fluid Mechanics, Prentice-Hall,

Inc., Englewood Cliffs, New Jersey, 1962

2 Astarita, G and G Marrucci, Principles of Non-Newtonian Fluid Mechanics,

McGraw-Hill Book Company (UK) limited, Maiden Head, England, 1974

3 Bird, R.B., R.C Armstrong, O Hassger, Dynamics of Polymeric Liquids, Vol 1: Fluid Mechanics, Wiley & Sons, New York, 1977.

4 Coleman, B.D, Markowitz, H and W Noll, Viscometric Flows of Non-Newtonian Fluids,

Springer-Verlag, New York, 1966

5 Eringen, A.C., Mechanics ofContinua, Wiley, New York, 1967.

6 Ferry, J.D., Viscoelastic Properties of Polymers, Wiley, 2nd edition, New York 1970.

7 Fung, Y.C., First Course in Continuum Mechanics, Prentice Hall, Englewood Cliffs,

New Jersey, 1977

8 Fung, Y.C., Foundation of Solid Mechanics, Prentice Hall, Englewood Cliffs, 1965.

9 Green, A.E., and W Zerna, Theoretical Elasticity, Oxford University Press, Fair Lawn,

New Jersey, 1954

10 Leigh, D.C., Nonlinear Continuum Mechanics, McGraw-Hill, New York, 1978

11 Malvern, L.E., Introduction to the Mechanics of a Continuous Medium, Prentice Hall,

Englewood Cliffs, New Jersey, 1969

12 Schlichting, H., Boundary Layer Theory, 7th edition, McGraw-Hill, New York, 1979.

13 Showater, W.R., Mechanics of Non-Newtonian Fluids, Pergamon Press, UK/USA, 1978.

550

References 551

14 Sokolnikoff, I.S Mathematical Theory of Elasticity, 2nd ed., McGraw-Hill, New York,

1956,

15 Sokolnikoff, I S., Tensor Analysis: Theory and Applications, John Wiley & sons, Inc.,

New York, 1951

16 Timoshenko, S.P and Goodier, J.N., Theory of Elasticity, 3rd ed., McGraw-Hill,

New York, 1970

17 Tniesdell, C, The Elements of Continuum Mechanics, Springer-Verlag, Inc., New York,

1966

18 Truesdell, C., and W Noll, The Non-linear Field Theories of Mechanics, Springer-Verlag, New York, 1992

19 Yih, C.S Fluid Mechanics, a Concise Introduction to the Theory, McGraw Hill, 1969,

West River Press, 1988

Acceleration of a particle Cauchy's stress principle, 173-174

in cylindrical coordinates, 88 Cayley-Hamilton theorem, 323

in rectangular coordinates, 87 Channel flow, 371-372,523

in spherical coordinates, 89 Characteristic equation, 39

Acoustic wave, 404 Choked flow, 417

Airy's stress function, 276,282 Co-rotational derivatives, 508

Anisotropic elastic solid, 219,293 Compatibility conditions

monoclinic, 299,312 for finite deformation, 144

orthotropic, 301,311 for infinitesimal strain, 114

plane of material symmetry, 296 for rate of deformation, 119

transversely isotropic, 303,308 Complex shear modulus, 470

Antisymmetric tensor, 35 Compliance matrix, 294

Apparent viscosity, 513,515-516 Compressible flow

Axial vector, 36,94 converging nozzle, 414

converging-diverging nozzle, 416 Barotropic flow, 409 one-dimensional, 412

Bernoulli's equations, 392 Compressible Newtonian fluid, 401 BKZ fluid, 503 Compressive stresses, 177

Body force, 187 Conjugate pairs, 207

Boundary layer, 399 Conservation of mass, 112,147,349,437 Bulk modulus, 220,228 Continuum mechanics, 79

Bulk viscosity, 358 Contraction of indices, 9

Control volume, 433-434 Cauchy stress tensor, 202,319,321 Convected Maxwell fluid, 512

Cauchy stress vector, 174 Conversion of elastic constants, 230 Cauchy's equations of motion, 189 Corotational Jeffrey fluid, 514

552

index 553

Couette flow, 380,389,526 area change, 145

Creep experiment, 465 isotropic elastic material, 322

Creep function, 466 volume change, 146

Current configuration as reference con- Finite deformation tensor, 121,128,134, figuration, 476 136,138,141,151,153,155-156,206,318-321

in other coordinates, 149 Deformation gradient, 120,126,317 Finite elastic deformation

Differential type constitutive equations bending of a bar, 327

incompressible fluids, 503 extension of incompressible solid, 324 Dilatation, 105,220 simple shear of an isotropic material, 325 Dilatational wave, 240 torsion-tension, 331

Displacement field, 92 First coefficient of viscosity, 357

Displacement gradient, 95 First Jaumann derivative, 508

Dissipation functions, 383 First Piola Kirchhoff stress tensor, 202

Divergence theorem, 430 Flow

Dual vector, 36,94 channel flow, 372,523

Dummy index, 3 Couette, 380,389,526

Dyadic product, 21 Hagen-Poiseuille, 374

irrotational, 390 Eigenvalues of a tensor, 38 oscillating plate, 381

Eigenvectors of a tensor, 38 parallel, 361

Einstein's summation convention, 4 plane Couette flow, 371

Elastic constants plane Couette of two layers, 377

table of, 231 simple shearing, 82

Elastic medium under large deformation, 319 uni-directional, 361

Elasticity, 217 Fluid flow

Elasticity tensor, 221 boundary conditions, 365

components of, 225 Fluid pressure, 357

Energy equation, 208,402 Fluids

Newtonian fluid, 384 definition of, 348

Enthalpy, 402 Frame

Entropy inequality, 209 change of frame, 314,317,496

Equations of hydrostatics, 350 frame-indifferent quantities, 315

Equations of motion, 187 principle of material frame indifference, 319

in cylindrical coordinates, 190 Free index, 5

in reference configuration, 201

in spherical coordinates, 190 Gauss's theorem, 431

Equilibrium equations, 189 Generalized linear Maxwell fluid

Equivoluminal wave, 242 continuous spectrum, 474

Euler's equation of motion, 391 discrete relaxation spectra, 471

Eulerian description, 84 integral form, 473

Eulerian strain tensor, 141,319 Global principle, 427

Extra stress, 464 Green's deformation tensor, 129

Green's theorem, 427 Finger deformation tensor, 138

Finite deformation, 121

554 Index

Hagen-Poiseuille flow, 374 Lame's constants, 226

History of relative deformation tensor, 486 Laminar flow, 370

Homogeneous media, 219 Left Cauchy-Green tensor, 138,151,155-Hookean elastic solid 156,318,321

linear, 220 Linear anisotropic elastic solid, 293 nonlinear, 322 Linear elastic solid, 220

Hugoniot equation, 413 Linear isotropic elastic solid, 225,306-307 Hydrostatic pressure, 349 Linear Maxwell fluid, 464,469,475 Hydrostatic stress, 179,230 Linear transformation, 11

Linearly viscous fluid, 356 Identity tensor, 23 Local principle, 427

Incompressible elastic material, 232 Longitudinal wave, 239

Incompressible material, 113,147 Loss modulus, 471

Incompressible Newtonian fluid, 359

Incompressible simple fluid, 497 Mach number, 411

Indeterminate pressure, 359 Material coordinates, 80, 83

Infinitesimal deformations, 94 Material derivative, 85

Infinitesimal rotation tensor, 106 Material description, 83

Infinitesimal strain tensor, 98 Material volume, 433

Inhomogeneous media, 219 Maximum shearing stress, 182

Integral type constitutive equation Maxwell element, 464

linear, 473 Mean normal compressive stress, 357 nonlinear, 498,503 Memory function, 475

Irrotational flow Modulus of elasticity, 218,228

as solution of Navier-Stokes equation, 394 Monoclinic elastic solid, 299-300,312 inviscid compressible fluid, 408 Moving control volume, 449

inviscid fluid, 391 Moving frames of reference, 447

Irrotational wave, 240

Isentropic pressure density relation, 406 Navier's equations

Isochoric condition, 324 cartesian coordinates, 235

Isotropic elastic solid, 219,225,306 cylindrical coordinates, 236

Isotropic function, 322,502 spherical coordinates, 236

Isotropic function(al), 497 Navier-Stokes equations

Isotropic tensor, 225 cylindrical coordinates, 364

incompressible fluid, 360 Jaumann derivative of stress, 507 spherical coordinates, 365

Newtonian fluid, 355 Kelvin's problem, 190 Non-Newtonian fluid, 462

Kinematic equations of motion, 80 Normal strains, 100

Kinematic viscosity, 396 Normal stress differences, 505-506 Kronecker delta, 6 Normal stress functions, 500,514-516,522

Nth Jaumann derivative, 508 Lagrange multiplier, 184

Lagrange stress tensor, 196 Objective quantities, 315

Lagrangian description, 84 Objective rate of stress, 506

Lagrangian strain tensor, 134,136,206,319 Objective scalar, vector, tensor, 316

index 555

Oldroyd 3-constant fluid, 515 Quotient rule, 34

Oldroyd 4-constant fluid, 516

Oldroyd fluid A, 515 Rate of change of a material element, 106 Oldroyd lower convected derivative, 508 Rate of deformation tensor, 108

Oldroyd upper convected derivative, 510 Rate of extension, 109

Orthogonal tensor, 24 Rate of heat flow, 207

Orthotropic elastic solid, 301,311 Rate of shear, 110

Rate type constitutive equations, 511 Particle in a continuum, 79 Recursive formulas

Pathline, 80,367 for Rivlin-Ericksen tensor, 491

Permutation symbol, 7 Reference configuration, 158

Phase angle, 471 Reference description, 84

Phase velocity, 239 Reference time, 79

Piezometric head, 362,374 Reflection of plane elastic waves, 248

Piola Kirchhoff stress tensor, 195,319 Refraction index, 250

first Piola Kirchhoff, 196,201 Relative deformation gradient, 159,477

second Piola Kirchhoff, 197,206,320 Relative deformation tensor, 478

Plane equivoluminal wave, 242 cylindrical coordinates, 482

Plane irrotational wave, 238 rectangular coordinates, 480

Plane of material symmetry, 296,299 spherical coordinates, 485

Plane Poiseuille flow, 372 transformation law in a change of frame, 494 Plane strain, 275 Relative Finger deformation tensor, 479

Plane strain in polar coordinates, 281 Relative left Cauchy-Green tensor, 159,479 Plane stress, 281 Relative left stretch tensor, 478

Poisson's ratio, 219,228 Relative Piola deformation tensor, 479

Polar decomposition theorem, 124,478 Relative right Cauchy-Green tensor, 159, Principal directions 479,499

strain, 105 Relative right stretch tensor, 478

tensor, 43 Relative rotation tensor, 478

Principal planes, of stress, 182 Relaxation function, 466

Principal Scalar invariants, 45 Reynolds number, 370

Principal strain, 105 Reynolds transport theorem, 435

Principal stresses, 182 Right Cauchy-Green tensor, 128,153, 155, Principal stretch, 122 318,320

Principal values, 43 Rigid body motion, 93

Principle of conservation of energy, 454 Rivlin's universal relation, 334

Principle of conservation of mass, 112,147, Rivlin-Ericksen fluid

349,437 incompressible of complexity n, 503

Principle of linear momentum, 187,440 Rivlin-Ericksen tensor, 486,488-490

Principle of material frame indifference, 319 in terms of velocity gradient, 491

Principle of moment of momentum, 178,451 transformation laws, 496

Principle of superposition, 238

Pure bending of a beam, 269 Second order fluid, 504

Pure bending of a curved beam, 285 Second Piola Kirchhoff stress tensor, 197, Pure stretch, 121 206,320

Second-order tensor, 11

5S6 index

Shear modulus, 220,228 Tanner and Simmons network model, 501 Shear strain, 100 Tensile stresses, 177

Shear stress function, 506,513,522 Tensors

Shear wave, 242 definition of, 11

Shearings, 110 inverse of, 23

Simple bending, 269 product of, 18

Simple extension, 254 sum of, 17

Simple shear stress state, 229 trace of, 22

Simple shearing motion, 82 transpose of, 20

Simply-connected region, 116 Thick-walled pressure vessel, 284

Snell's law, 251 Thickness stretch,-shear vibration, 251 Spatial coordinates, 84 Torricelli's formula, 394

Spatial description, 83 Torsion of a circular cylinder, 258,331 Speed of sound, 406 Torsion of a noncircular cylinder, 266 Spherical pressure vessel, 291 Transformation laws

Spin tensor, 108, 111 of tensors, 30,32

St Venant's principle, 256,262 of vectors, 28

Stagnation enthalpy, 402 Transformation matrix, 26

Stagnation pressure, 410 Transversely isotropic elastic solid, 303,308 Steady and unsteady flow, 370 Turbulent flow, 370

Stiffness matrix, 294 Two point components, 155-156

Storage modulus, 471 for deformation gradient, 151

Stored energy function, 222 Two point components

Strain energy function, 222,293-294 for relative deformation gradient, 483 Strain tensor (infinitesimal), 98

Streakline, 368 Uniaxial stress, 228

Streamline, 366 Unit elongation, 99,137

Stress boundary condition, 192 Unit step function, 467

Stress concentration, 287 Unsteady flow, 370

Stress power, 203

Stress relaxation experiment, 466 Vibration of an infinite plate, 251

Stress tensor (Cauchy), 174 Viscoelastic fluid

components of, 176 linear, 464

normal stresses, 177 nonlinear, 476

shearing stresses, 177 Viscometric flow, 516

symmetry of, 178 Viscometric functions, 522

tangential stresses, 177 Viscosity, 357

Stress vector, 173 Viscosity function, 500

Stresses in viscometric flow, 520 Viscous stress tensor, 356

Stretch, 95,122 Vorticity tensor, 112

Stretch tensor, 124,126,128 Vorticity transport equation, 396

Stretching, 109 Vorticity vector, 387

Summation convention, 3

Surface tractions, 192 Young's modulus, 218,228

Symmetric tensor, 35