546 Answers CHAPTER 4 CHAPTERS Answers 547 548 Answers CHAPTER 6 Answers 549 CHAPTER 7 References 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. Index 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 [...]... stretch,-shear vibration, 251 Torricelli's formula, 39 4 Torsion of a circular cylinder, 258 ,33 1 Torsion of a noncircular cylinder, 266 Transformation laws of tensors, 30 ,32 of vectors, 28 Transformation matrix, 26 Transversely isotropic elastic solid, 30 3 ,30 8 Turbulent flow, 37 0 Two point components, 155 -156 for deformation gradient, 151 Two point components for relative deformation gradient, 4 83 Uniaxial stress,... Stagnation enthalpy, 402 Stagnation pressure, 410 Steady and unsteady flow, 37 0 Stiffness matrix, 294 Storage modulus, 471 Stored energy function, 222 Strain energy function, 222,2 93- 294 Strain tensor (infinitesimal), 98 Streakline, 36 8 Streamline, 36 6 Stress boundary condition, 192 Stress concentration, 287 Stress power, 2 03 Stress relaxation experiment, 466 Stress tensor (Cauchy), 174 components of,... Uniaxial stress, 228 Unit elongation, 99, 137 Unit step function, 467 Unsteady flow, 37 0 Vibration of an infinite plate, 251 Viscoelastic fluid linear, 464 nonlinear, 476 Viscometric flow, 516 Viscometric functions, 522 Viscosity, 35 7 Viscosity function, 500 Viscous stress tensor, 35 6 Vorticity tensor, 112 Vorticity transport equation, 39 6 Vorticity vector, 38 7 Young's modulus, 218,228 ... stresses, 177 symmetry of, 178 tangential stresses, 177 Stress vector, 1 73 Stresses in viscometric flow, 520 Stretch, 95,122 Stretch tensor, 124,126,128 Stretching, 109 Summation convention, 3 Surface tractions, 192 Symmetric tensor, 35 Tanner and Simmons network model, 501 Tensile stresses, 177 Tensors definition of, 11 inverse of, 23 product of, 18 sum of, 17 trace of, 22 transpose of, 20 Thick-walled...5S6 index Shear modulus, 220,228 Shear strain, 100 Shear stress function, 506,5 13, 522 Shear wave, 242 Shearings, 110 Simple bending, 269 Simple extension, 254 Simple shear stress state, 229 Simple shearing motion, 82 Simply-connected region, 116 Snell's law, 251 Spatial coordinates, 84 Spatial description, 83 Speed of sound, 406 Spherical pressure vessel, 291 Spin tensor, 108, 111 St Venant's . material, 32 2 Creep function, 466 volume change, 146 Current configuration as reference con- Finite deformation tensor, 121,128, 134 , figuration, 476 136 , 138 ,141 ,151 ,1 53, 155 -156 ,206 ,31 8 -32 1 in . 508 Dissipation functions, 38 3 First Piola Kirchhoff stress tensor, 202 Divergence theorem, 430 Flow Dual vector, 36 ,94 channel flow, 37 2,5 23 Dummy index, 3 Couette, 38 0 ,38 9,526 Dyadic . Continuum mechanics, 79 Bulk viscosity, 35 8 Contraction of indices, 9 Control volume, 433 - 434 Cauchy stress tensor, 202 ,31 9 ,32 1 Convected Maxwell fluid, 512 Cauchy stress vector,