Continuum Mechanics Via Problems and Exercises Downloaded from

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1 Bibliography Continuum Mechanics Via Problems and Exercises Downloaded from [1] Batchelor, G. K., An introduction to fluid dynamics, University Press, Cambridge, [2] Bowen, R. M., Introduction to continuum mechanics for engineeers, Plenum Press, New York, [3] Brekovskikh, L. M. and Goncharov, V. V., Mechanics of continua and wave dynamics, Springer-Verlag, Berlin; New York, [4] Chung, T. J., Continuum mechanics, Prentice Hall, Englewood Cliffs, N. J., [5] Ciarlet, P., Mathematical elasticity, North Holland, New York, [6] Cook, H., editor, Continua: with the Houston problem book, M. Dekker, New York, [7] Davis, J. L., Introduction to dynamics of continuous media, Macmillan, New York, [8] Ericksen, J., Introduction to the thermodynamics of solids, Cnapman & Hall, London; New York, [9] Gennes, P. G. d and Prost, J., The physics of liquid crystals, Clarendon Press, Oxford University Press, Oxford, New York, [10] Gere, J. M. and Timoshenko, S. P., Mechanics of materials, PWS-KENT Pub. Co., Boston, 3 Edition, [11] Giles, R. V., Evett, J. B., and Liu, C, Schaum's outline of theory and problems of fluid mechanics and hydraulics, McGraw-Hill, New York, 3 Edition, [12] Green, A. E. and Adkins, J. E., Large elastic deformations, Clarendon Press, Oxford,

2 260 BIBLIOGRAPHY [13] Groot, S. and Mazur, P., Non-equilibrium Thermodynamics, Interscience Pub. Co, New York, [14] Gurtin, M. E., An introduction to continuum mechanics, Academic Press, New York, [15] Hunt, J. N., Incompressible fluid dynamics, Wiley, New York, Continuum Mechanics Via Problems and Exercises Downloaded from [16] Hunter, S. C., Mechanics of continuous media, E. Horwood ; Halsted Press, Chichester ; New York, 2 Edition, [17] Kachanov, L. M., Foundations of the theory of plasticity, North-Holland Pub. Co., Amsterdam, [18] Kochin, N. E., Kibel', I. A., and Rose, N. V., Theoretical hydrodynamics, Interscience Pub. Co., New York, 1964,1965. [19] Kubo, R., Thermodynamics. An advanced course with problems and solutions, Northholland Pub. Co., Amsterdam, [20] Kulikovskiy A. G., Lyubimov G. A., Magneto-Hydrodynamics, Addison-Wesley Reading, Massachusetts, [21] Lai, M., Rubin, D., and Krempl, E., Introduction to continuum mechanics, Pergamon Press, Oxford; New York, 3 Edition, [22] Lamb, H., Hydrodynamics, University Press, Cambridge; New York, [23] Landau, L. D. and Lifshits, E. M., Mechanics, Pergamon Press, Oxford; New York, 2 Edition, [24] Landau, L. D. and Lifshits, E. M., Fluid mechanics, Pergamon Press, Oxford; New York, 3 Edition, [25] Landau, L. D. and Lifshits, E. M., Electrodynamics of continous media, Pergamon Press, Oxford; New York, 2 Edition, [26] Landau, L. D. and Lifshits, E. M., Theory of elasticity, Pergamon Press, Oxford; New York, 3 Edition, [27] Loitsyanskii, L. G. ( Mechanics of liquids and gases, Pergamon Press, [28] Marsden, J. and Hughes, T., Mathematical foundations of elasticity, Prentice- Hall, Englewood Cliffs, N.J., 1983.

3 BIBLIOGRAPHY 261 [29] Mase, G. E., Schaum's outline of theory and problems of continuum mechanics, McGraw-Hill, New York, [30] Maugin, G. A., The thermomechamcs of plasticity and fracture, Cambridge University Press, New York, Continuum Mechanics Via Problems and Exercises Downloaded from [31] McConnell, A. J., Applications of tensor analysis, Dover Publications, New York, [32] Milne-Thompson, L. M., Theoretical hydrodynamics, Macmill&n, New York, 5 Edition, [33] Muller, I., Thermodynamics, Pitman, Boston, [34] Ogden, R., Non linear elastic deformations, Ellis Horwood ; Halsted Press, Chichester; New York, [35] Prager, W., Introduction to mechanics of continua, Ginn, Boston, [36] Schouten, J. A., Tensor analysis for physicists, Clarendon Press, Oxford, 2 Edition, [37] Sedov, L. I., Stmilartty and dimensional methods m mechanics, Academic Press, [38] Sedov, L. I., Introduction to the mechanics of a continuous medium, Addison- Welsley, Reading, [39] Sedov, L. I., Two-dimensional problems m hydrodynamics and aerodynamics, Interscience Pub. Co., New York, [40] Sedov, L. I., A course in continuum mechanics, volume 1-4, Wolters-Noordhoff, Groningen, [41] Smoller, J., Shock waves and reaction-diffusion equations, Springer-Verlag, New York, [42] Sokolnikoff, I. S., Mathematical theory of elasticity, McGraw-Hill, New York, 2 Edition, 1956 [43] Sokolnikoff, I. S., Tensor analysis, theory and applications to geometry and mechanics of continua, Wiley, New York, [44] Spenser, A. J. M., Continuum mechanics, Longman, London; New York, 1980.

4 262 BIBLIOGRAPHY [45] Timoshenko, S. P. and Goodier, J. N., Theory of elasticity, Mcgrawhill, New York, 3 Edition, [46] Truesdell, C, A First Course in Rational Continuum Mechanics, Academic Press, Boston, 2nd Edition, Continuum Mechanics Via Problems and Exercises Downloaded from [47] Valliappan, S., Continuum mechanics fundamentals, A. A. Balkema, Rotterdam, [48] Vekstein, G. E., Physics of continuous media: a collection of problems with solutions for physics students, Adam Hilger, Bristol, Philadelphia, [49] Whitham, G. B., Linear and nonlinear waves, Wiley, New York, 1974.

5 Index Continuum Mechanics Via Problems and Exercises Downloaded from A acceleration components in concomitant coordinate system 63 acceleration in the Eulerian description 4 active loading 218 additional energy 86, 104 additional energy input 95, 103 adiabatic process 87 airfoil 168 Airy stress function 65, 196, 197 Alfven waves 241 Almansi strain tensor 27, 28, 32 angular momentum conservation law 49,77 angular momentum equations 51, 77 apparent mass 140 Archimedes force 122 Archimedes' law 120, 122 associated law in plasticity 216 averaging 209 axisymmetric potential motions 131 B barotropic motion 66, 136, 163 basic units in dimensionality theory 245 beam 190 bending moment 190 Bernoulli integral 66, 70, 95, 136, 177 Biot-Savart formula 133 Blasius problem 151 Blasius-Chaplygin formulae 139, 141 body couple 77, 78, 104, 207 body force 63, 82, 180 boundary conditions for a viscous fluid 119 boundary conditions for an elastic solid 181 boundary conditions for an ideal fluid 118 boundary conditions on a free surface 119 boundary layer 142, 151, 152, 209 boundary layer equations 151 Boussinesq equation 160 Boussinesq turbulent hypothesis 154 bulk modulus 185 C cantilever 191 Carnot cycle 95, 98 Cauchy stress tensor 56, 63, 183 Cauchy-Lagrange integral 67, 136, 165 Cauchy-Poisson wave problem 156 cavitation 137 cavitation number 253 characteristic form of a differential equation 169 characteristics of differential equations 169 charged particle 232 Christoffel symbols 16 circulation of velocity 129, 133 Clapeyron equation 100 class of systems of dimensional units 246 Clausius inequality 89 coefficient of bulk viscosity

6 coefficient of linear thermal expansion 93, 186 coefficient of shear viscosity 118 coefficient of thermal expansion 186 combustion 175 compatibility conditions for strain components 33 complex potential 129 composite materials 209 concomitant coordinate system 33 condition of adhesion 119 condition of impermeability 118 condition of incompressibility 124 conditions across a discontinuity surface 51 conditions at infinity 119, 125 conditions of evolutionarity 112 conservation laws 49, 69 constitutive equations 80, 91, 207 constitutive equations in couple elastisity 207 constitutive relations for Maxwell viscoelastic solid 223 constitutive relations for the elasticplastic body 217 constitutive relations for Voigt viscoelastic solid 223 constitutive relations of a rigid perfectly plastic solid 225 contact discontinuity 111 continuity equation 50, 52, 118, 180 continuity equation in cylindrical coordinates 53 continuity equation in spherical coordinates 53 contravariant components of a vector 13 contravariant transformation rule 14 control surface method 69 control volume method coordinate system, its local basis 13 coordinates, Eulerian 4 coordinates, Lagrangian 27, 114 Coriolis force 162 Couette flow 96, 146 couple elasticity 79, 206 couple stress tensor 77, 78, 207 couple stress vector 77 covariant components of a vector 13 covariant derivative 16, 32 covariant transformation rule 14 creep 223 critical velocity 177 curl of a vector field 25 cylindrical coordinates 18 cylindrical coordinates, metric tensor 18 D d'alembert paradox 140 Debye length 238 deformation plasticity theory 220 determining equations (see also constitutive equations) 91 determining parameters in the dimensionality theory 245 detonation 175 detonation adiabat 176 diffusion 103 diffusion of vorticity 147 dimension formula 246 dimensional quantities 246 dimensionality theory 245 dimensionally independent quantities 247, 248 dimensionless quantities 246 Dirichlet problem 125 discontinuity of electromagnetic field 233 discontinuity surface 114, 233 discrete structure 209

7 dispersing waves 157 dispersion of waves 157, 203, 209 dispersion relation 156, 203, 210, 213 dissipation function 105 dissipation inequality 89, 102 dissipation of mechanical energy 143 distortion tensor 184 divergence of a vector field 25 divergence of a velocity vector 31 Doppler effect 166 double shear 36 drag coefficient 255 drag force 74, 144, 149, 255 dynamic conditions across a tangential discontinuity 111 dynamic conditions across a weak discontinuity surface 109, 110 E eigenvalues of a tensor 10, 59 eigenvectors of a tensor 10 elastic medium 101, 179 elastic strains in plastic solid 216 elastic-plastic solid 216 elasticity theory equations in Lagrangian coordinates 180 electric charge 231, 232, 234 electric conductivity 237, 241 electric conductivity tensor 238 electric current 232 electric field 237, 241 electrohydrodynamics 241 electrohydrodynamics, system of equations 241 electromagnetic energy 235 electromagnetic field 231 electromagnetic force 235 electromagnetic momentum 235 electromagnetic stresses 235 energy at rest in Relativity theory 230 energy conservation law 50, energy dissipation rate 143 energy equation 51, 86, 104 energy-momentum tensor 231 enthalpy 91, 94, 98 entropy 87 entropy equation 51, 88 entropy increase in a shock wave 172 entropy of a linear elastic body 186 entropy of a linear elastic solid 98 entropy of a perfect gas 98 entropy production 87, 88, 105, 238 entropy production in a viscous fluid 90 entropy supply 87 entropy variation law 50, 87 equation of internal angular momentum 78 equation of internal energy 85, 86, 118, 207, 208 equation of magnetic induction 238, 239 equation of state 91, 118, 180 equations of angular momentum 77 equations of equilibrium 63 equations of equilibrium for fluid 119 equations of motion 63, 208 equations of motion for an ideal fluid (Euler equations) 66, 117 equations of motion in Lagrangian variables 183 equations of shallow water 158 equilibrium equations 63, 64, 182 Euler equations 66, 117 Euler equations in cylindrical coordinates 67 Euler equations in spherical coordinates 67 Eulerian description of motion 4 evolutionarity conditions 172 evolutionary discontinuities 172, 176 evolutionary shock wave 172

8 explosion at a point 68, 173, 257 extension, uniaxial 34 F fading memory 223 first law of thermodynamics 50, 85, 86 flow around a circular cylinder 130 flow around a sphere 150 formulae of Blasius-Chaplygin 139 four-dimensional metric 230 four-dimensional velocity 230 Fourier heat conduction law 87, 181 free energy 91, 98, 181, 209 free energy of a linear elastic solid 98 Froude number 125 frozen electric charge 242 frozen magnetic field 239 G gas dynamics, system of equations 164 Gibbs identity 90, 105, 180 Gibbs thermodynamic potential 91, 98 Green identity 127 Green strain tensor 27, 28, 32 Gromeka-Lamb equations 135 group speed 157, 203, 210 gyromagnetic coefficients 78 H Hamilton-Cayley theorem 44 hardening 215, 217 hardening parameter 217 harmonic function 125 harmonic wave 156, 203 Hartman number 240 Hartman problem 241 heat capacity 87 heat conduction equation 95 heat flux vector 86, 93 Helmholtz equation 133, 147 Hill's spherical vortex 135 homogeneous fluid Hooke's law 67, 93, 179, 181, 183 Hugoniot adiabat 111, 175 hydraulic jump 159 hydrostatics 119 hypothesis of plane sections 191 I ideal compressible fluid, constitutive relationships 92 ideal fluid 89, 117 ideal incompressible fluid, constitutive relationships 91 incompressibility condition 52, 124 inertial coordinate systems 227 inhomogeneous fluid 120 instability of Kelvin-Helmholtz 161 internal angular momentum 78, 104, 207 internal energy 85, 98, 104, 162, 181, 207 internal orientation 78 invariants of a tensor 12 inviscid fluid 117 irreversible process 87, 100 isothermal process 87 isotropic compression 185 isotropy 43 J Jaumann derivative 80, 82 Joukovski formula 74, 139 Joukovski-Chaplygin postulate 130 Joule heat 238, 239 jump conditions 109, 110 jump conditions for an ideal fluid 111 K Kelvin circulation theorem 125, 133, 164 Kelvin wave 162 kinematic conditions across a discontinuity surface 109

9 kinematic conditions across a tangential discontinuity 111 kinematic conditions across a weak discontinuity surface 109, 110 kinematical viscosity coefficient 142 kinetic energy theorem 68, 85, 86 kinetic relationships 90, 91 Korteweg-de Vries equation 160 Kronecker symbols 9 Kutta condition 130 L Lagrange interpolation polynomial 46 Lagrange's theorem 165 Lagrangian coordinates 3 Lagrangian description of motion 3 Lame coefficients 68, 93, 182 Lame equations 68, 182 laminar boundary layer 151 laminar motion 153 Laplace equation 125 Laplace operator 67 Larmor frequency 234 Levi-Civita tensor 23, 31 light velocity 228 limit load 222 linear couple elasticity 209, 210 linear elastic medium 67, 181 linearly viscous fluid 67, 92, 103 liquid crystal 78 logarithmic velocity distribution in turbulent flow 155 long waves 158 longitudinal wave 203, 210 Lorentz coordinate system 227 Lorentz force 231, 239 Lorentz transformations 228 M Mach number 125, 163 magnetic field 82, 237, magnetic fluid 108 magnetic induction 231 magnetic Reynolds number 239 magnetic viscosity coefficient 238 magnetizable media 78 magnetization 82, 233 magnetized liquid crystal 81 magnetohydrodynamics 237 magnetohydrodynamics, system of equations 237 mass at rest in Relativity theory 230 mass conservation law 49, 52 mathematical models of continua 91 Maxwell energy-momentum tensor 234 Maxwell equations 232, 233 Maxwell equations, integral form 232 Maxwell visco-elastic solid 223 method of conformal mappings 129 method of Huntley 248 method of mirror reflections 129 method of singular points 129 metric tensor 17 microstructure 209 Minkowski space 227 Mises yield criterion 216 model of the atmosphere 123 momentum conservation law 49 momentum equations 50, 63 monochromatic wave 165 motion description, Eulerian 4 motion description, Lagrangian 3 motion law 3, 114 Murnaghan medium 205 N Navier-Stokes equations 67, 118, 142 Navier-Stokes model 118 nematic liquid crystal 78, 104 Neumann problem 125 neutral axis of bent beam 190, 191 neutral loading 218

10 Newtonian fluid 92, 118 non-newtonian fluids 118 normality rule in plasticity 216 O Ohm's law 238, 242 one-dimensional unsteady motion 166, 169, 203 Onsager principle 90, 103, 105 Orr-Sommerfeld equation 148 orthotropic medium 183, 187 orthotropy 47 P paradox of d'alembert 140 perfect gas 91, 92, 101, 123 perfect plasticity 215 phase speed 156, 157, 203 physical components 20 Pi-theorem (II-theorem) 69, 247 Piola-Kirchhoff stress tensor 56, 62, 63, 183 planar potential flows 128 planar source 130 plane problem of elasticty theory 196 plane strain state 196 plane stress state 196 plane waves 166 plastic flow theory 215, 216 plastic strains 216 plasticity 215 point vortex 130, 133, 177 Poiseuille flow 146 Poisson adiabat 94 Poisson coefficient 182 Poisson equation 124 polarization 233 polarizing media 78 polytropic atmosphere 124 potential body force 66 potential flow potential motion 53, 125 Poynting vector 235, 240 Prandtl boundary layer equations 151 Prandtl turbulent hypothesis 154 Prandtl-Reuss equations 219 pressure coefficient 252 pressure function 66, 136 principal axes of a tensor 59 principal components of a tensor 59 principal coordinate system of a tensor 59 problem of Cauchy-Poisson 156 problem of Dirichlet 125 problem of Neumann 125 progressive wave 156 propeller, ideal 112 proper coordinate system 229 proper length in relativity theory 229 proper time in relativity theory 229 propulsive force of a rocket engine 75 pure bending 190, 191 Q quasi-one-dimensional flow 178 R Rayleigh waves 204 resonance phenomenon 159 reversible process 87, 89 Reynolds equations 154 Reynolds number 142 Riemann invariants 159, 169 Riemann wave 159, 170, 206 rigid-plastic solids 217 Rossby wave 162 rotational flow 133 rotor of a vector field 25 S second law of thermodynamics 50, 87, 88 self-similar problem 69

11 shallow water approximation 158 shear modulus 184 shock adiabat 175 shock wave 169, 171 similarity criteria 248 similarity of phenomena 248 simple extension 184 simple loading 220 simple shear 35, 36, 184 solitary wave (soliton) 160 sound propagation in mixture 168 sound wave 167 spatial coordinates 4 spatial source 177 special relativity theory 227 specific heat 87 speed of sound 163 speeds of characteristics 159 spherical coordinates 18 spherical coordinates, metric tensor 18 spherical tensor 12 St.Venant principle 187, 191 stability of a flow 161 stability of a steady flow of viscous fluid 142, 146, 148 stagnation parameters 177 standing wave 157 statics of fluids 119 steady motion 5 Stokes approximation 142, 148 Stokes equations 148 Stokes formula for drag force 150 Stokes theorem 133 strain rate tensor 30, 39 strain tensor of Almansi 27, 28, 33, 52 strain tensor of Green 27, 28, 33, 52 strain tensor, linearized 28 strain tensor, small 29, 181 stream function 129, 131 streamlines 5 stress concentration on a hole 198 stress function 65, 196, 197 stress surface 59 stress tensor 56, 63 stress tensor, its antisymmetric part 79, 105 stress vector 55 Strouhal number 125, 142 structure of a detonation wave 176 structure of a shock wave 172 structure of slow combustion front 176 substantive time derivative 4 summation convention 9 surface couple 77, 104 surface of strong discontinuity 109, 114 surface of weak discontinuity 109 surface tension 97, 254 symmetry group of a tensor field 44 symmetry group of tensors 43 system of dimensional units 245 T tangential discontinuity 111, 145 tensor 9 tensor deviator 12 tensor function 43, 81, 107 tensor of electromagnetic field 233 tensor of magnetic stresses 239 tensor product 9 tensor, its contravariant components 14 tensor, its covariant components 14 tensor, its invariants 10, 12, 52 tensor, its mixed components 15 tensor, its physical components 21 tensor, its principal axes 10 tensor, its principal components 10 thermal conductivity coefficient 87 thermoelastic medium, constitutive relationships 92 torsion 194 transverse isotropy

12 transverse wave 203, 210 transversely isotropic medium 183, 187 travelling wave 203 Tresca yield criterion 216, 217 turbulence 142 turbulent heat flux 155 turbulent motion 153 turbulent Reynolds stresses 154 twisting angle 194 U uncompensated heat 88, 105, 238 unloading 216, 218 V Van der Waals equation 101 velocity of light 231, 237 visco-elastic Maxwell solid 223 visco-elastic Voigt solid 223 visco-plastic solid 223 viscosity 223 viscosity coefficients 67, 92, 118 viscous compressible fluid, constitutive relationships 92 viscous fluid 89, 117 viscous incompressible fluid, constitutive relationships 92 viscous stress tensor 89 Voigt visco-elastic solid 223 vortex sheet 134 vorticity 147 vorticity vector 31, 105, 133 W wave equation 165, 203 wave packet 157 waves in stratified fluids 160 waves of small amplitudes 156 weak discontinuity 114 work of external and internal forces 68 work of internal couples 104 work of internal surface forces 68, 86, 93, 104 Y yield criterion 216 yield stress in pure shear 218 yield stress in pure tension 218 yield surface 216 Young's modulus 182 Z Zhukovskii formula