APPLIED SOLID MECHANICS

By Z. HIZII

CONTENTS MODELLING SOLIDS INTRODUCTION HOOKE’S LAW LAGRANGIAN AND EULERIAN COORDINATES STRAIN STRESS CONSERVATION OF MOMENTUM LINEAR ELASTICITY THE INCOMPRESSIBILITY APPROXIMATION ENERGY BOUNDARY CONDITIONS AND WELL-POSEDNESS COORDINATE SYSTEMS 1 11 1 CARTESIAN COORDINATES 1 11 2 CYLINDRICAL POLAR COORDINATES 1 11 3 SPHERICAL POLAR COORDINATES EXERCISES LINEAR ELASTOSTATICS LINEAR DISPLACEMENTS ISOTROPIC EXPANSION SIMPLE SHEAR UNIAXIAL STRETCHING BIAXIAL STRAIN GENERAL LINEAR DISPLACEMENT ANTIPLANE STRAIN TORSION MULTIPLY-CONNECTED DOMAINS PLANE STRAIN DE?NITION THE AIRY STRESS FUNCTION BOUNDARY CONDITIONS PLANE STRAIN IN A DISC PLANE STRAIN IN AN ANNULUS PLANE STRAIN IN A RECTANGLE PLANE STRAIN IN A SEMI-IN?NITE STRIP PLANE STRAIN IN A HALF-SPACE PLANE STRAIN WITH A BODY FORCE COMPATIBILITY GENERALISED STRESS FUNCTIONS GENERAL OBSERVATIONS PLANE STRAIN REVISITED PLANE STRESS AXISYMMETRIC GEOMETRY THE GALERKIN REPRESENTATION PAPKOVICH–NEUBER POTENTIALS MAXWELL AND MORERA POTENTIALS SINGULAR SOLUTIONS IN ELASTOSTATICS THE DELTA FUNCTION POINT AND LINE FORCES THE GREEN’S TENSOR POINT INCOMPATIBILITY LINEAR ELASTODYNAMICS NORMAL MODES AND PLANE WAVES NORMAL MODES WAVES IN THE FREQUENCY DOMAIN SCATTERING P-WAVES AND S-WAVES MODE CONVERSION IN PLANE STRAIN LOVE WAVES RAYLEIGH WAVES DYNAMIC STRESS FUNCTIONS WAVES IN CYLINDERS AND SPHERES WAVES IN A CIRCULAR CYLINDER WAVES IN A SPHERE INITIAL-VALUE PROBLEMS SOLUTIONS IN THE TIME DOMAIN FUNDAMENTAL SOLUTIONS CHARACTERISTICS MOVING SINGULARITIES CONCLUDING REMARKS APPROXIMATE THEORIES LONGITUDINAL DISPLACEMENT OF A BAR TRANSVERSE DISPLACEMENTS OF A STRING TRANSVERSE DISPLACEMENTS OF A BEAM DERIVATION OF THE BEAM EQUATION COMPRESSION OF A BEAM WAVES ON A BEAM LINEAR ROD THEORY LINEAR PLATE THEORY DERIVATION OF THE PLATE EQUATION SIMPLE SOLUTIONS OF THE PLATE EQUATION AN INVERSE PLATE PROBLEM MORE GENERAL IN-PLANE STRESSES VON K´ARM´AN PLATE THEORY ASSUMPTIONS UNDERLYING THE THEORY THE STRAIN COMPONENTS THE VON K´ARM´AN EQUATIONS WEAKLY CURVED SHELL THEORY STRAIN IN A WEAKLY CURVED SHELL LINEARISED EQUATIONS FOR A WEAKLY CURVED SHELL SOLUTIONS FOR A THIN SHELL NONLINEAR BEAM THEORY DERIVATION OF THE MODEL EXAMPLE: DE?ECTION OF A DIVING BOARD WEAKLY NONLINEAR THEORY AND BUCKLING NONLINEAR ROD THEORY GEOMETRICALLY NONLINEAR WAVE PROPAGATION 4 11 1 NONLINEARITY AND SOLITONS 4 11 2 GRAVITY-TORSIONAL WAVES 4 11 3 TRAVELLING WAVES ON A BEAM 4 11 4 WEAKLY NONLINEAR WAVES ON A BEAM NONLINEAR ELASTICITY STRESS AND STRAIN REVISITED DEFORMATION AND STRAIN THE PIOLA–KIRCHHO?STRESS TENSORS THE MOMENTUM EQUATION EXAMPLE: ONE-DIMENSIONAL NONLINEAR ELASTICITY THE CONSTITUTIVE RELATION POLAR DECOMPOSITION STRAIN INVARIANTS FRAME INDI?ERENCE AND ISOTROPY THE ENERGY EQUATION HYPERELASTICITY INCOMPRESSIBILITY EXAMPLES OF CONSTITUTIVE RELATIONS EXAMPLES PRINCIPAL STRESSES AND STRAINS BIAXIAL LOADING OF A SQUARE MEMBRANE BLOWING UP A BALLOON CAVITATION CONCLUSION ASYMPTOTIC ANALYSIS THE LINEAR PLATE EQUATION NONDIMENSIONALISATION AND SCALING DIMENSIONLESS EQUATIONS LEADING-ORDER EQUATIONS BOUNDARY CONDITIONS AND ST VENANT’S PRINCIPLE BOUNDARY LAYER SCALINGS EQUATIONS AND BOUNDARY CONDITIONS SOLVABILITY CONDITIONS ASYMPTOTIC EXPANSIONS THE VON K´ARM´AN PLATE EQUATIONS AND WEAKLY CURVED SHELLS BACKGROUND SCALINGS EQUATIONS FOR A WEAKLY CURVED SHELL THE EULER–BERNOULLI PLATE EQUATIONS ASYMPTOTIC STRUCTURE OF THE SOLUTION LONGITUDINAL STRETCHING OF A PLATE THE LINEAR ROD EQUATIONS LINEAR SHELL THEORY GEOMETRY OF THE SHELL CONCLUSIONS FRACTURE AND CONTACT STATIC BRITTLE FRACTURE PHYSICAL BACKGROUND MODE III CRACKS MATHEMATICAL METHODOLOGIES FOR CRACK PROBLEMS MODE II CRACKS MODE I CRACKS DYNAMIC FRACTURE CONTACT CONTACT OF ELASTIC STRINGS OTHER THIN SOLIDS SMOOTH CONTACT IN PLANE STRAIN PLASTICITY MODELS FOR GRANULAR MATERIAL STATIC BEHAVIOUR GRANULAR ?OW DISLOCATION THEORY PERFECT PLASTICITY THEORY FOR METALS TORSION PROBLEMS THREE-DIMENSIONAL YIELD CONDITIONS PLASTIC ?OW SIMULTANEOUS ELASTICITY AND PLASTICITY MORE GENERAL THEORIES VISCOELASTICITY SPRINGS AND DASHPOTS THREE-DIMENSIONAL LINEAR VISCOELASTICITY LARGE-STRAIN VISCOELASTICITY THERMOELASTICITY COMPOSITE MATERIALS AND HOMOGENISATION ONE-DIMENSIONAL HOMOGENISATION TWO-DIMENSIONAL HOMOGENISATION THREE-DIMENSIONAL HOMOGENISATION POROELASTICITY ANISOTROPY