CONFIGURATIONAL FORCES AS BASIC CONCEPTS OF CONTINUUM PHYSICS
CONTENTS 1 INTRODUCTION A BACKGROUND B VARIATIONAL DE?NITION OF CON?GURATIONAL FORCES C INTERFACIAL ENERGY A FURTHER ARGUMENT FOR A CON?GURATIONAL FORCE BALANCE D CON?GURATIONAL FORCES AS BASIC OBJECTS E THE NATURE OF CON?GURATIONAL FORCES F CON?GURATIONAL STRESS AND RESIDUAL STRESS INTERNAL CON?GURATIONAL FORCES G CON?GURATIONAL FORCES AND INDETERMINACY H SCOPE OF THE BOOK I ON OPERATIONAL DE?NITIONS AND MATHEMATICS J GENERAL NOTATION TENSOR ANALYSIS J1 ON DIRECT NOTATION J2 VECTORS AND TENSORS FIELDS J3 THIRD-ORDER TENSORS (3-TENSORS) THE OPERATION T J4 FUNCTIONS OF TENSORS A CON?GURATIONAL FORCES WITHIN A CLASSICAL CONTEXT 2 KINEMATICS A REFERENCE BODY MATERIAL POINTS MOTIONS B MATERIAL AND SPATIAL VECTORS THE SETS ESPACE AND EMATTER C MATERIAL AND SPATIAL OBSERVERS D CONSISTENCY REQUIREMENT OBJECTIVE ?ELDS 3 STANDARD FORCES WORKING A FORCES B WORKING STANDARD FORCE AND MOMENT BALANCES AS CONSEQUENCES OF INVARIANCE UNDER CHANGES IN SPATIAL OBSERVER 4 MIGRATING CONTROL VOLUMES STATIONARY AND TIME-DEPENDENT CHANGES IN REFERENCE CON?GURATION A MIGRATING CONTROL VOLUMES P P(T) VELOCITY ?ELDS FOR ?P(T) AND ?¯P(T) B CHANGE IN REFERENCE CON?GURATION B1 STATIONARY CHANGE IN REFERENCE CON?GURATION B2 TIME-DEPENDENT CHANGE IN REFERENCE CON?GURATION 5 CON?GURATIONAL FORCES A CON?GURATIONAL FORCES B WORKING REVISITED C CON?GURATIONAL FORCE BALANCE AS A CONSEQUENCE OF INVARIANCE UNDER CHANGES IN MATERIAL OBSERVER D INVARIANCE UNDER CHANGES IN VELOCITY ?ELD FOR ?P(T) CON?GURATIONAL STRESS RELATION E INVARIANCE UNDER TIME-DEPENDENT CHANGES IN REFERENCE EXTERNAL AND INTERNAL FORCE RELATIONS F STANDARD AND CON?GURATIONAL FORMS OF THE WORKING POWER BALANCE 6 THERMODYNAMICS RELATION BETWEEN BULK TENSION AND ENERGY ESHELBY IDENTITY A MECHANICAL VERSION OF THE SECOND LAW B ESHELBY RELATION AS A CONSEQUENCE OF THE SECOND LAW C THERMOMECHANICAL THEORY D FLUIDS CURRENT CON?GURATION AS REFERENCE 7 INERTIA AND KINETIC ENERGY ALTERNATIVE VERSIONS OF THE SECOND LAW A INERTIA AND KINETIC ENERGY B ALTERNATIVE FORMS OF THE SECOND LAW C PSEUDOMOMENTUM D LYAPUNOV RELATIONS 8 CHANGE IN REFERENCE CON?GURATION A TRANSFORMATION LAWS FOR FREE ENERGY AND STANDARD FORCE B TRANSFORMATION LAWS FOR CON?GURATIONAL FORCE 9 ELASTIC AND THERMOELASTIC MATERIALS A MECHANICAL THEORY A1 BASIC EQUATIONS A2 CONSTITUTIVE THEORY B THERMOMECHANICAL THEORY B1 BASIC EQUATIONS B2 CONSTITUTIVE THEORY B THE USE OF CON?GURATIONAL FORCES TO CHARACTERIZE COHERENT PHASE INTERFACES 10 INTERFACE KINEMATICS 11 INTERFACE FORCES SECOND LAW A INTERFACE FORCES B WORKING C STANDARD AND CON?GURATIONAL FORCE BALANCES AT THE INTERFACE D INVARIANCE UNDER CHANGES IN VELOCITY ?ELD FOR S (T) NORMAL CON?GURATIONAL BALANCE E POWER BALANCE INTERNAL WORKING F SECOND LAW INTERNAL DISSIPATION INEQUALITY FOR THE INTERFACE G LOCALIZATIONS USING A PILLBOX ARGUMENT 12 INERTIA BASIC EQUATIONS FOR THE INTERFACE A RELATIVE KINETIC ENERGY B DETERMINATION OF BS AND ES C STANDARD AND CON?GURATIONAL BALANCES WITH INERTIA D CONSTITUTIVE EQUATION FOR THE INTERFACE E SUMMARY OF BASIC EQUATIONS F GLOBAL ENERGY INEQUALITY LYAPUNOV RELATIONS C AN EQUIVALENT FORMULATION OF THE THEORY IN?NITESIMAL DEFORMATIONS 13 FORMULATION WITHIN A CLASSICAL CONTEXT A BACKGROUND REASON FOR AN ALTERNATIVE FORMULATION IN TERMS OF DISPLACEMENTS B FINITE DEFORMATIONS MODI?ED ESHELBY RELATION C IN?NITESIMAL DEFORMATIONS 14 COHERENT PHASE INTERFACES A GENERAL THEORY B IN?NITESIMAL THEORY WITH LINEAR STRESS-STRAIN RELATIONS IN BULK D EVOLVING INTERFACES NEGLECTING BULK BEHAVIOR 15 EVOLVING SURFACES A SURFACES A1 BACKGROUND SUPER?CIAL STRESS A2 SUPER?CIAL TENSOR ?ELDS B SMOOTHLY EVOLVING SURFACES B1 TIME DERIVATIVE FOLLOWING S NORMAL TIME DERIVATIVE B2 VELOCITY ?ELDS FOR THE BOUNDARY CURVE ?G OF A SMOOTHLY EVOLVING SUBSURFACE OF S TRANSPORT THEOREM B3 TRANSFORMATION LAWS 16 CON?GURATIONAL FORCE SYSTEM WORKING A CON?GURATIONAL FORCES WORKING B CON?GURATIONAL FORCE BALANCE AS A CONSEQUENCE OF INVARIANCE UNDER CHANGES IN MATERIAL OBSERVER C INVARIANCE UNDER CHANGES IN VELOCITY ?ELDS SURFACE TENSION SURFACE SHEAR D NORMAL FORCE BALANCE INTRINSIC FORM FOR THE WORKING 17 SECOND LAW 18 CONSTITUTIVE EQUATIONS A FUNCTIONS OF ORIENTATION B CONSTITUTIVE EQUATIONS C EVOLUTION EQUATION FOR THE INTERFACE 19 TWO-DIMENSIONAL THEORY A KINEMATICS B CON?GURATIONAL FORCES WORKING SECOND LAW C CONSTITUTIVE THEORY D EVOLUTION EQUATION FOR THE INTERFACE E CORNERS F ANGLE-CONVEXITY THE FRANK DIAGRAM G CONVEXITY OF THE INTERFACIAL ENERGY AND EVOLUTION OF THE INTERFACE E COHERENT PHASE INTERFACES WITH INTERFACIAL ENERGY AND DEFORMATION 20 THEORY NEGLECTING STANDARD INTERFACIAL STRESS A STANDARD AND CON?GURATIONAL FORCES WORKING B POWER BALANCE INTERNAL WORKING C SECOND LAW C1 SECOND LAW INTERFACIAL DISSIPATION INEQUALITY C2 DERIVATION OF THE INTERFACIAL DISSIPATION INEQUALITY USING A PILLBOX ARGUMENT D CONSTITUTIVE EQUATIONS E CONSTRUCTION OF THE PROCESS USED IN RESTRICTING THE CONSTITUTIVE EQUATIONS F BASIC EQUATIONS WITH INERTIAL EXTERNAL FORCES F1 STANDARD AND CON?GURATIONAL BALANCES F2 SUMMARY OF BASIC EQUATIONS G GLOBAL ENERGY INEQUALITY LYAPUNOV RELATIONS 21 GENERAL THEORY WITH STANDARD AND CON?GURATIONAL STRESS WITHIN THE INTERFACE A KINEMATICS TANGENTIAL DEFORMATION GRADIENT B STANDARD AND CON?GURATIONAL FORCES WORKING C POWER BALANCE INTERNAL WORKING D SECOND LAW INTERFACIAL DISSIPATION INEQUALITY E CONSTITUTIVE EQUATIONS G LYAPUNOV RELATIONS 22 TWO-DIMENSIONAL THEORY WITH STANDARD AND CON?GURATIONAL STRESS WITHIN THE INTERFACE B FORCES WORKING C POWER BALANCE INTERNAL WORKING SECOND LAW E EVOLUTION EQUATIONS FOR THE INTERFACE F SOLIDI?CATION 23 SOLIDI?CATION THE STEFAN CONDITION AS A CONSEQUENCE OF THE CON?GURATIONAL FORCE BALANCE A SINGLE-PHASE THEORY B THE CLASSICAL TWO-PHASE THEORY REVISITED THE STEFAN CONDITION AS A CONSEQUENCE OF THE CON?GURATIONAL BALANCE 24 SOLIDI?CATION WITH INTERFACIAL ENERGY AND ENTROPY B APPROXIMATE THEORY THE GIBBS-THOMSON CONDITION AS A CONSEQUENCE OF THE CON?GURATIONAL BALANCE C FREE-BOUNDARY PROBLEMS FOR THE APPROXIMATE THEORY GROWTH THEOREMS C1 THE QUASILINEAR AND QUASISTATIC PROBLEMS C2 GROWTH THEOREMS G FRACTURE 25 CRACKED BODIES A SMOOTH CRACKS CONTROL VOLUMES B DERIVATIVES FOLLOWING THE TIP TIP INTEGRALS TRANSPORT THEOREMS 26 MOTIONS 27 FORCES WORKING C STANDARD AND CON?GURATIONAL FORCE BALANCES D INERTIAL FORCES KINETIC ENERGY 28 THE SECOND LAW A STATEMENT OF THE SECOND LAW B THE SECOND LAW APPLIED TO CRACK CONTROL VOLUMES C THE SECOND LAW APPLIED TO TIP CONTROL VOLUMES STANDARD FORM OF THE SECOND LAW D TIP TRACTION ENERGY RELEASE RATE DRIVING FORCE E THE STANDARD MOMENTUM CONDITION 29 BASIC RESULTS FOR THE CRACK TIP 30 CONSTITUTIVE THEORY FOR GROWING CRACKS A CONSTITUTIVE RELATIONS AT THE TIP B THE GRIF?TH-IRWIN FUNCTION C CONSTITUTIVELY ISOTROPIC CRACK TIPS TIPS WITH CONSTANT MOBILITY 31 KINKING AND CURVING OF CRACKS MAXIMUM DISSIPATION CRITERION A CRITERION FOR CRACK INITIATION KINK ANGLE B MAXIMUM DISSIPATION CRITERION FOR CRACK PROPAGATION 32 FRACTURE IN THREE SPACE DIMENSIONS (RESULTS) H TWO-DIMENSIONAL THEORY OF CORNERS AND JUNCTIONS NEGLECTING INERTIA 33 PRELIMINARIES TRANSPORT THEOREMS A TERMINOLOGY B TRANSPORT THEOREMS B1 BULK ?ELDS B2 INTERFACIAL ?ELDS 34 THERMOMECHANICAL THEORY OF JUNCTIONS AND CORNERS A MOTIONS B NOTATION C FORCES WORKING D SECOND LAW E BASIC RESULTS FOR THE JUNCTION F WEAK SINGULARITY CONDITIONS NONEXISTENCE OF CORNERS G CONSTITUTIVE EQUATIONS H FINAL JUNCTION CONDITIONS