INTRODUCTION TO CONTINUUM MECHANICS
TABLE OF CONTENTS CHAPTER INTRODUCTION WHAT IS CONTINUUM MECHANICS? CHAPTER TENSORS PART A: INDICIAL NOTATION SUMMATION CONVENTION, DUMMY INDICES FREE INDICES THE KRONECKER DELTA THE PERMUTATION SYMBOL INDICIAL NOTATION MANIPULATIONS PROBLEMS FOR PART A PART B: TENSORS TENSOR: A LINEAR TRANSFORMATION COMPONENTS OF A TENSOR COMPONENTS OF A TRANSFORMED VECTOR SUM OF TENSORS 210 PRODUCT OF TWO TENSORS 211 TRANSPOSE OF A TENSOR 212 DYADIC PRODUCT OF VECTORS 213 TRACE OF A TENSOR 214 IDENTITY TENSOR AND TENSOR INVERSE 215 ORTHOGONAL TENSORS 216 TRANSFORMATION MATRIX BETWEEN TWO RECTANGULAR CARTESIAN COORDINATE SYSTEMS 217 TRANSFORMATION LAW FOR CARTESIAN COMPONENTS OF A VECTOR 218 TRANSFORMATION LAW FOR CARTESIAN COMPONENTS OF A TENSOR 219 DEFINING TENSOR BY TRANSFORMATION LAWS 220 SYMMETRIC AND ANTISYMMETRIC TENSORS 221 THE DUAL VECTOR OF AN ANTISYMMETRIC TENSOR 222 EIGENVALUES AND EIGENVECTORS OF A TENSOR 223 PRINCIPAL VALUES AND PRINCIPAL DIRECTIONS OF REAL SYMMETRIC TENSORS 224 MATRIX OF A TENSOR WITH RESPECT TO PRINCIPAL DIRECTIONS 225 PRINCIPAL SCALAR INVARIANTS OF A TENSOR PROBLEMS FOR PART B PART C: TENSOR CALCULUS 226 TENSOR-VALUED FUNCTIONS OF A SCALAR 227 SCALAR FIELD AND GRADIENT OF A SCALAR FUNCTION 228 VECTOR FIELD AND GRADIENT OF A VECTOR FUNCTION 229 DIVERGENCE OF A VECTOR FIELD AND DIVERGENCE OF A TENSOR FIELD 230 CURL OF A VECTOR FIELD 231 LAPLACIAN OF A SCALAR FIELD 232 LAPLACIAN OF A VECTOR FIELD PROBLEMS FOR PART C PART D: CURVILINEAR COORDINATES 233 POLAR COORDINATES 234 CYLINDRICAL COORDINATES 235 SPHERICAL COORDINATES PROBLEMS FOR PART D KINEMATICS OF A CONTINUUM DESCRIPTION OF MOTIONS OF A CONTINUUM MATERIAL DESCRIPTION AND SPATIAL DESCRIPTION MATERIAL DERIVATIVE ACCELERATION OF A PARTICLE DISPLACEMENT FIELD KINEMATIC EQUATION FOR RIGID BODY MOTION INFINITESIMAL DEFORMATION GEOMETRICAL MEANING OF THE COMPONENTS OF THE INFINITESIMAL STRAIN TENSOR PRINCIPAL STRAIN 310 DILATATION 311 THE INFINITESIMAL ROTATION TENSOR 312 TIME RATE OF CHANGE OF A MATERIAL ELEMENT 313 THE RATE OF DEFORMATION TENSOR 314 THE SPIN TENSOR AND THE ANGULAR VELOCITY VECTOR 315 EQUATION OF CONSERVATION OF MASS 316 COMPATIBILITY CONDITIONS FOR INFINITESIMAL STRAIN COMPONENTS 317 COMPATIBILITY CONDITION FOR RATE OF DEFORMATION COMPONENTS 318 DEFORMATION GRADIENT 319 LOCAL RIGID BODY MOTION 320 FINITE DEFORMATION 321 POLAR DECOMPOSITION THEOREM 322 CALCULATION OF STRETCH AND ROTATION TENSORS FROM THE DEFORMATION GRADIENT 323 RIGHT CAUCHY-GREEN DEFORMATION TENSOR 324 LAGRANGIAN STRAIN TENSOR TABLE OF CONTENTS 325 LEFT CAUCHY-GREEN DEFORMATION TENSOR 326 EULERIAN STRAIN TENSOR 327 CHANGE OF AREA DUE TO DEFORMATION 328 CHANGE OF VOLUME DUE TO DEFORMATION 329 COMPONENTS OF DEFORMATION TENSORS IN OTHER COORDINATES 330 CURRENT CONFIGURATION AS THE REFERENCE CONFIGURATION APPENDIX 31: NECESSARY AND SUFFICIENT CONDITIONS FOR STRAIN COMPATIBILITY APPENDIX 32: POSITIVE DEFINITE SYMMETRIC TENSORS APPENDIX 33: THE POSITIVE DEFINITE ROOT OF U2 ¼ D PROBLEMS FOR CHAPTER STRESS AND INTEGRAL FORMULATIONS OF GENERAL PRINCIPLES STRESS VECTOR STRESS TENSOR COMPONENTS OF STRESS TENSOR SYMMETRY OF STRESS TENSOR: PRINCIPLE OF MOMENT OF MOMENTUM PRINCIPAL STRESSES MAXIMUM SHEARING STRESSES EQUATIONS OF MOTION: PRINCIPLE OF LINEAR MOMENTUM EQUATIONS OF MOTION IN CYLINDRICAL AND SPHERICAL COORDINATES BOUNDARY CONDITION FOR THE STRESS TENSOR 410 PIOLA KIRCHHOFF STRESS TENSORS 411 EQUATIONS OF MOTION WRITTEN WITH RESPECT TO THE REFERENCE CONFIGURATION 412 STRESS POWER 413 STRESS POWER IN TERMS OF THE PIOLA-KIRCHHOFF STRESS TENSORS 414 RATE OF HEAT FLOW INTO A DIFFERENTIAL ELEMENT BY CONDUCTION 415 ENERGY EQUATION 416 ENTROPY INEQUALITY 417 ENTROPY INEQUALITY IN TERMS OF THE HELMHOLTZ ENERGY FUNCTION 418 INTEGRAL FORMULATIONS OF THE GENERAL PRINCIPLES OF MECHANICS APPENDIX 41: DETERMINATION OF MAXIMUM SHEARING STRESS AND THE PLANES ON WHICH IT ACTS THE ELASTIC SOLID MECHANICAL PROPERTIES LINEARLY ELASTIC SOLID PART A: ISOTROPIC LINEARLY ELASTIC SOLID ISOTROPIC LINEARLY ELASTIC SOLID YOUNG’S MODULUS, POISSON’S RATIO, SHEAR MODULUS, AND BULK MODULUS EQUATIONS OF THE INFINITESIMAL THEORY OF ELASTICITY NAVIER EQUATIONS OF MOTION FOR ELASTIC MEDIUM NAVIER EQUATIONS IN CYLINDRICAL AND SPHERICAL COORDINATES PRINCIPLE OF SUPERPOSITION PLANE ELASTIC WAVES PLANE IRROTATIONAL WAVES 510 PLANE EQUIVOLUMINAL WAVES 511 REFLECTION OF PLANE ELASTIC WAVES 512 VIBRATION OF AN INFINITE PLATE SIMPLE EXTENSION, TORSION AND PURE BENDING 513 SIMPLE EXTENSION 514 TORSION OF A CIRCULAR CYLINDER 515 TORSION OF A NONCIRCULAR CYLINDER: ST VENANT’S PROBLEM 516 TORSION OF ELLIPTICAL BAR 517 PRANDTL’S FORMULATION OF THE TORSION PROBLEM 518 TORSION OF A RECTANGULAR BAR 519 PURE BENDING OF A BEAM PLANE STRESS AND PLANE STRAIN SOLUTIONS 520 PLANE STRAIN SOLUTIONS 521 RECTANGULAR BEAM BENT BY END COUPLES 522 PLANE STRESS PROBLEM 523 CANTILEVER BEAM WITH END LOAD 524 SIMPLY SUPPORTED BEAM UNDER UNIFORM LOAD 525 SLENDER BAR UNDER CONCENTRATED FORCES AND ST VENANT’S PRINCIPLE 526 CONVERSION FOR STRAINS BETWEEN PLANE STRAIN AND PLANE STRESS SOLUTIONS 527 TWO-DIMENSIONAL PROBLEMS IN POLAR COORDINATES 528 STRESS DISTRIBUTION SYMMETRICAL ABOUT AN AXIS 529 DISPLACEMENTS FOR SYMMETRICAL STRESS DISTRIBUTION IN PLANE STRESS SOLUTION 530 THICK-WALLED CIRCULAR CYLINDER UNDER INTERNAL AND EXTERNAL PRESSURE 531 PURE BENDING OF A CURVED BEAM 532 INITIAL STRESS IN A WELDED RING 533 AIRY STRESS FUNCTION ’ ¼ F(R) COS NY AND ’ ¼ F(R) SIN NY 534 STRESS CONCENTRATION DUE TO A SMALL CIRCULAR HOLE IN A PLATE UNDER TENSION 535 STRESS CONCENTRATION DUE TO A SMALL CIRCULAR HOLE IN A PLATE UNDER PURE SHEAR 536 SIMPLE RADIAL DISTRIBUTION OF STRESSES IN A WEDGE LOADED AT THE APEX 537 CONCENTRATED LINE LOAD ON A 2-D HALF-SPACE: THE FLAMONT PROBLEM ELASTOSTATIC PROBLEMS SOLVED WITH POTENTIAL FUNCTIONS 538 FUNDAMENTAL POTENTIAL FUNCTIONS FOR ELASTOSTATIC PROBLEMS 539 KELVIN PROBLEM: CONCENTRATED FORCE AT THE INTERIOR OF AN INFINITE ELASTIC SPACE 540 BOUSSINESQ PROBLEM: NORMAL CONCENTRATED LOAD ON AN ELASTIC HALF-SPACE 541 DISTRIBUTIVE NORMAL LOAD ON THE SURFACE OF AN ELASTIC HALF-SPACE 542 HOLLOW SPHERE SUBJECTED TO UNIFORM INTERNAL AND EXTERNAL PRESSURE TABLE OF CONTENTS 543 SPHERICAL HOLE IN A TENSILE FIELD 544 INDENTATION BY A RIGID FLAT-ENDED SMOOTH INDENTER ON AN ELASTIC HALF-SPACE 545 INDENTATION BY A SMOOTH RIGID SPHERE ON AN ELASTIC HALF-SPACE APPENDIX 5A1: SOLUTION OF THE INTEGRAL EQUATION IN SECTION 545 PROBLEMS FOR CHAPTER 5, PART A PART B: ANISOTROPIC LINEARLY ELASTIC SOLID 546 CONSTITUTIVE EQUATIONS FOR AN ANISOTROPIC LINEARLY ELASTIC SOLID 547 PLANE OF MATERIAL SYMMETRY 548 CONSTITUTIVE EQUATION FOR A MONOCLINIC LINEARLY ELASTIC SOLID 549 CONSTITUTIVE EQUATION FOR AN ORTHOTROPIC LINEARLY ELASTIC SOLID 550 CONSTITUTIVE EQUATION FOR A TRANSVERSELY ISOTROPIC LINEARLY ELASTIC MATERIAL 551 CONSTITUTIVE EQUATION FOR AN ISOTROPIC LINEARLY ELASTIC SOLID 552 ENGINEERING CONSTANTS FOR AN ISOTROPIC LINEARLY ELASTIC SOLID 553 ENGINEERING CONSTANTS FOR A TRANSVERSELY ISOTROPIC LINEARLY ELASTIC SOLID 554 ENGINEERING CONSTANTS FOR AN ORTHOTROPIC LINEARLY ELASTIC SOLID 555 ENGINEERING CONSTANTS FOR A MONOCLINIC LINEARLY ELASTIC SOLID PART C: ISOTROPIC ELASTIC SOLID UNDER LARGE DEFORMATION 556 CHANGE OF FRAME 557 CONSTITUTIVE EQUATION FOR AN ELASTIC MEDIUM UNDER LARGE DEFORMATION 558 CONSTITUTIVE EQUATION FOR AN ISOTROPIC ELASTIC MEDIUM 559 SIMPLE EXTENSION OF AN INCOMPRESSIBLE ISOTROPIC ELASTIC SOLID 560 SIMPLE SHEAR OF AN INCOMPRESSIBLE ISOTROPIC ELASTIC RECTANGULAR BLOCK 561 BENDING OF AN INCOMPRESSIBLE ISOTROPIC RECTANGULAR BAR 562 TORSION AND TENSION OF AN INCOMPRESSIBLE ISOTROPIC SOLID CYLINDER APPENDIX 5C1: REPRESENTATION OF ISOTROPIC TENSOR-VALUED FUNCTIONS NEWTONIAN VISCOUS FLUID FLUIDS COMPRESSIBLE AND INCOMPRESSIBLE FLUIDS EQUATIONS OF HYDROSTATICS NEWTONIAN FLUIDS INTERPRETATION OF L AND M INCOMPRESSIBLE NEWTONIAN FLUID NAVIER-STOKES EQUATIONS FOR INCOMPRESSIBLE FLUIDS NAVIER-STOKES EQUATIONS FOR INCOMPRESSIBLE FLUIDS IN CYLINDRICAL AND SPHERICAL COORDINATES BOUNDARY CONDITIONS 610 STREAMLINE, PATHLINE, STEADY, UNSTEADY, LAMINAR, AND TURBULENT FLOW 611 PLANE COUETTE FLOW TABLE OF CONTENTS 612 PLANE POISEUILLE FLOW 613 HAGEN-POISEUILLE FLOW 614 PLANE COUETTE FLOW OF TWO LAYERS OF INCOMPRESSIBLE VISCOUS FLUIDS 615 COUETTE FLOW 616 FLOW NEAR AN OSCILLATING PLANE 617 DISSIPATION FUNCTIONS FOR NEWTONIAN FLUIDS 618 ENERGY EQUATION FOR A NEWTONIAN FLUID 619 VORTICITY VECTOR 620 IRROTATIONAL FLOW 621 IRROTATIONAL FLOW OF AN INVISCID INCOMPRESSIBLE FLUID OF HOMOGENEOUS DENSITY 622 IRROTATIONAL FLOWS AS SOLUTIONS OF NAVIER-STOKES EQUATION 623 VORTICITY TRANSPORT EQUATION FOR INCOMPRESSIBLE VISCOUS FLUID WITH A CONSTANT DENSITY 624 CONCEPT OF A BOUNDARY LAYER 625 COMPRESSIBLE NEWTONIAN FLUID 626 ENERGY EQUATION IN TERMS OF ENTHALPY 627 ACOUSTIC WAVE 628 IRROTATIONAL, BAROTROPIC FLOWS OF AN INVISCID COMPRESSIBLE FLUID 629 ONE-DIMENSIONAL FLOW OF A COMPRESSIBLE FLUID 630 STEADY FLOW OF A COMPRESSIBLE FLUID EXITING A LARGE TANK THROUGH A NOZZLE 631 STEADY LAMINAR FLOW OF A NEWTONIAN FLUID IN A THIN ELASTIC TUBE: AN APPLICATION TO PRESSURE-FLOW RELATION IN A PULMONARY BLOOD VESSEL THE REYNOLDS TRANSPORT THEOREM AND APPLICATIONS GREEN’S THEOREM DIVERGENCE THEOREM INTEGRALS OVER A CONTROL VOLUME AND INTEGRALS OVER A MATERIAL VOLUME THE REYNOLDS TRANSPORT THEOREM THE PRINCIPLE OF CONSERVATION OF MASS THE PRINCIPLE OF LINEAR MOMENTUM MOVING FRAMES A CONTROL VOLUME FIXED WITH RESPECT TO A MOVING FRAME THE PRINCIPLE OF MOMENT OF MOMENTUM 710 THE PRINCIPLE OF CONSERVATION OF ENERGY 711 THE ENTROPY INEQUALITY: THE SECOND LAW OF THERMODYNAMICS NON-NEWTONIAN FLUIDS PART A: LINEAR VISCOELASTIC FLUID LINEAR MAXWELL FLUID A GENERALIZED LINEAR MAXWELL FLUID WITH DISCRETE RELAXATION SPECTRA INTEGRAL FORM OF THE LINEAR MAXWELL FLUID AND OF THE GENERALIZED LINEAR MAXWELL FLUID WITH DISCRETE RELAXATION SPECTRA A GENERALIZED LINEAR MAXWELL FLUID WITH A CONTINUOUS RELAXATION SPECTRUM COMPUTATION OF RELAXATION SPECTRUM AND RELAXATION FUNCTION PART B: NONLINEAR VISCOELASTIC FLUID CURRENT CONFIGURATION AS REFERENCE CONFIGURATION RELATIVE DEFORMATION GRADIENT RELATIVE DEFORMATION TENSORS CALCULATIONS OF THE RELATIVE DEFORMATION TENSOR 810 HISTORY OF THE RELATIVE DEFORMATION TENSOR AND RIVLIN-ERICKSEN TENSORS 811 RIVLIN-ERICKSEN TENSORS IN TERMS OF VELOCITY GRADIENT: THE RECURSIVE FORMULA 812 RELATION BETWEEN VELOCITY GRADIENT AND DEFORMATION GRADIENT 813 TRANSFORMATION LAW FOR THE RELATIVE DEFORMATION TENSORS UNDER A CHANGE OF FRAME 814 TRANSFORMATION LAW FOR RIVLIN-ERICKSEN TENSORS UNDER A CHANGE OF FRAME 815 INCOMPRESSIBLE SIMPLE FLUID 816 SPECIAL SINGLE INTEGRAL-TYPE NONLINEAR CONSTITUTIVE EQUATIONS 817 GENERAL SINGLE INTEGRAL-TYPE NONLINEAR CONSTITUTIVE EQUATIONS 818 DIFFERENTIAL-TYPE CONSTITUTIVE EQUATIONS FOR INCOMPRESSIBLE FLUIDS 819 OBJECTIVE RATE OF STRESS 820 RATE-TYPE CONSTITUTIVE EQUATIONS PART C: VISCOMETRIC FLOW OF AN INCOMPRESSIBLE SIMPLE FLUID 821 VISCOMETRIC FLOW 822 STRESSES IN VISCOMETRIC FLOW OF AN INCOMPRESSIBLE SIMPLE FLUID 823 CHANNEL FLOW 824 COUETTE FLOW APPENDIX 81: GRADIENT OF SECOND-ORDER TENSOR FOR ORTHOGONAL COORDINATES REFERENCES