ELEMENTS OF NATURAL PHILOSOPHY: MECHANICS

By P. DIURT

CONTENTS INTRODUCTION BASIC CONCEPTS IN MICRO–MACRO MODELING HISTORICAL OVERVIEW OBJECTIVES OF THIS MONOGRAPH SOME BASICS OF THE MECHANICS OF SOLID CONTINUA KINEMATICS OF DEFORMATIONS DEFORMATION OF LINE ELEMENTS IN?NITESIMAL STRAIN MEASURES THE JACOBIAN OF THE DEFORMATION GRADIENT EQUILIBRIUM/ KINETICS OF SOLID CONTINUA POSTULATES ON VOLUME AND SURFACE QUANTITIES BALANCE LAW FORMULATIONS REFERENTIAL DESCRIPTIONS OF BALANCE LAWS THE FIRST LAW OF THERMODYNAMICS/AN ENERGY BALANCE THE SECOND LAW OF THERMODYNAMICS/A RESTRICTION LINEARLY ELASTIC CONSTITUTIVE EQUATIONS THE IN?NITESIMAL STRAIN CASE MATERIAL SYMMETRY MATERIAL CONSTANT INTERPRETATION CONSEQUENCES OF POSITIVE-DE?NITENESS HYPERELASTIC FINITE STRAIN MATERIAL LAWS BASIC REQUIREMENTS FOR FINITE STRAIN LAWS DETERMINATION OF MATERIAL CONSTANTS MODERATE STRAIN CONSTITUTIVE RELATIONS FUNDAMENTAL WEAK FORMULATIONS DIRECT WEAK FORMULATIONS AN EXAMPLE SOME RESTRICTIONS THE PRINCIPLE OF MINIMUM POTENTIAL ENERGY COMPLEMENTARY WEAK FORMS FUNDAMENTAL MICRO–MACRO CONCEPTS TESTING PROCEDURES THE AVERAGE STRAIN THEOREM THE AVERAGE STRESS THEOREM SATISFACTION OF HILL’S ENERGY CONDITION THE HILL-REUSS-VOIGT BOUNDS OBSERVATIONS CLASSICAL MICRO–MACRO MECHANICAL APPROXIMATIONS THE ASYMPTOTIC HASHIN-SHTRIKMAN BOUNDS THE CONCENTRATION TENSOR: MICRO?ELD BEHAVIOR THE ESHELBY RESULT DILUTE METHODS THE MORI-TANAKA METHOD FURTHER METHODS MICRO-GEOMETRICAL (MANUFACTURING) IDEALIZATIONS UPPER AND LOWER VARIATIONAL BOUNDS PROOF OF ENERGETIC ORDERING USES TO APPROXIMATE THE EFFECTIVE PROPERTY A BASIC FINITE ELEMENT IMPLEMENTATION FINITE ELEMENT METHOD IMPLEMENTATION FEM APPROXIMATION GLOBAL/LOCAL TRANSFORMATIONS DIFFERENTIAL PROPERTIES OF SHAPE FUNCTIONS DIFFERENTIATION IN THE REFERENTIAL COORDINATES POST PROCESSING ACCURACY OF THE FINITE ELEMENT METHOD LOCAL ADAPTIVE MESH RE?NEMENT A-POSTERIORI RECOVERY METHODS A-POSTERIORI RESIDUAL METHODS SOLUTION OF ALGEBRAIC EQUATIONS KRYLOV SEARCHES AND MINIMUM PRINCIPLES THE METHOD OF STEEPEST DESCENT THE CONJUGATE GRADIENT METHOD ACCELERATING COMPUTATIONS 5 10 REMARK ON PENALTY METHODS COMPUTATIONAL/STATISTICAL TESTING METHODS A BOUNDARY VALUE FORMULATION NUMERICAL DISCRETIZATION TOPOLOGICAL RESOLUTION ELEMENTALLY AVERAGED QUANTITIES ITERATIVE KRYLOV SOLVERS/ MICROSTRUCTURAL “CORRECTORS” OVERALL TESTING PROCESS: NUMERICAL EXAMPLES SUCCESSIVE SAMPLE ENLARGEMENT MULTIPLE SAMPLE TESTS A MINIMUM PRINCIPLE INTERPRETATION A PROOF BASED ON PARTITIONING RESULTS RELATION TO THE MATERIAL TESTS DEPENDENCY ON VOLUME FRACTION INCREASING THE NUMBER OF SAMPLES INCREASING SAMPLE SIZE VARIOUS EXTENSIONS AND FURTHER INTERPRETATIONS OF PARTITIONING PARTITIONING AND TRACTION TEST CASES ISOLATING THE SUBSAMPLING ERROR/NUMERICAL ERROR ORTHOGONALITY AN ERGODIC INTERPRETATION OF THE RESULTS STATISTICAL SHIFTING THEOREMS PARTITIONING AND ENSEMBLE AVERAGING PRIMAL PARTITIONING COMPLEMENTARY PARTITIONING HOMOGENIZED MATERIAL ORDERINGS EMBEDDED ORTHOGONAL MONOTONICITIES MOMENT BOUNDS ON POPULATION RESPONSES FIRST ORDER (AVERAGE) BOUNDS SECOND ORDER (STANDARD DEVIATION) BOUNDS THIRD ORDER (SKEWNESS) BOUNDS REMARKS DOMAIN DECOMPOSITION ANALOGIES AND EXTENSIONS BOUNDARY VALUE PROBLEM FORMULATIONS ERROR IN THE “BROKEN” PROBLEMS MULTISCALE PROXIMITY BOUNDS USES OF THE BOUNDS FOR DOMAIN DECOMPOSITION THE CONNECTION TO MATERIAL TESTING A “TOTAL” ORTHOGONAL SUM ITERATIVE EXTENSIONS METHOD I: GLOBAL/LOCAL CG ITERATIONS METHOD II: ITERATIVE EQUILIBRATION NONCONVEX–NONDERIVATIVE GENETIC MATERIAL DESIGN COMPUTATIONAL MATERIAL DESIGN CHARACTERISTIC OF SUCH OBJECTIVES NONCONVEXITY SIZE EFFECTS INTRODUCTION OF CONSTRAINTS FATIGUE-TYPE CONSTRAINTS CLASSICAL FATIGUE RELATIONS CONSTRUCTION OF A CONSTRAINT QUALITATIVE BEHAVIOR OF THE FATIGUE CONSTRAINTS NONCONVEX–NONDERIVATIVE GENETIC SEARCH NUMERICAL EXAMPLES SCOPE OF USE MODELING COUPLED MULTI?ELD PROCESSES 10 1 INTRODUCTION 10 2 A MODEL PROBLEM INVOLVING MULTI?ELD PROCESSES IN MULTIPHASE SOLIDS 10 3 CONSTITUTIVE ASSUMPTIONS 10 3 1 AN ENERGY BALANCE INCLUDING GROWTH 10 3 2 MASS TRANSFER AND REACTION-DIFFUSION MODELS 10 4 STAGGERED MULTI?ELD WEAK FORMULATIONS 10 4 1 A RECURSIVE ALGORITHM 10 4 2 CONVERGENCE AND CONTRACTION-MAPPING TIME STEPPING CONTROL 10 5 NUMERICAL EXPERIMENTS 10 6 CONCLUDING REMARKS CLOSING COMMENTS REFERENCES