THE THEORY OF COMPOSITES
CONTENTS INTRODUCTION WHAT ARE COMPOSITES, AND WHY STUDY THEM? WHAT MAKES COMPOSITES USEFUL? THE EFFECTIVE TENSORS OF COMPOSITES HOMOGENIZATION FROM AN INTUITIVE VIEWPOINT PERIODIC HOMOGENIZATION HOMOGENIZATION IN RANDOM MEDIA HOMOGENIZATION IN THE SETTINGS OF G-, H-, AND -CONVERGENCE REFERENCES SOME EQUATIONS OF INTEREST AND NUMERICAL APPROACHES TO SOLVING THEM THE CONDUCTIVITY AND RELATED EQUATIONS MAGNETOTRANSPORT AND CONVECTION ENHANCED DIFFUSION THE ELASTICITY EQUATIONS THERMOELECTRIC, PIEZOELECTRIC, AND SIMILAR COUPLED EQUATIONS THERMOELASTICITY AND POROELASTICITY PYROELECTRIC EQUATIONS AND THEIR RELATION TO CONDUCTIVITY AND MAGNETOTRANS- PORT EQUATIONS IN ?BROUS COMPOSITES THE EQUIVALENCE BETWEEN ELASTICITY IN ?BROUS COMPOSITES AND TWO-DIMENSIONAL PIEZOELECTRICITY AND THERMOELASTICITY NUMERICAL METHODS FOR ?NDING EFFECTIVE TENSORS DUALITY TRANSFORMATIONS IN TWO-DIMENSIONAL MEDIA DUALITY TRANSFORMATIONS FOR CONDUCTIVITY PHASE INTERCHANGE IDENTITIES FOR TWO-PHASE MEDIA THE CONDUCTIVITY OF TWO-DIMENSIONAL POLYCRYSTALS DUALITY TRANSFORMATIONS FOR PYROELECTRICITY DUALITY TRANSFORMATIONS FOR ELASTICITY DUALITY TRANSFORMATIONS FOR OTHER ELASTIC MEDIA THE EFFECTIVE SHEAR MODULUS OF INCOMPRESSIBLE TWO-DIMENSIONAL POLYCRYS- TALS AND SYMMETRIC MATERIALS TRANSLATIONS AND EQUIVALENT MEDIA TRANSLATIONS APPLIED TO CONDUCTIVITY A FORMULA FOR THE HALL COEF?CIENT IN TWO-DIMENSIONAL POLYCRYSTALS A FORMULA FOR THE HALL COEF?CIENT IN TWO-PHASE, TWO-DIMENSIONAL MEDIA† INHOMOGENEOUS TRANSLATIONS FOR THREE-DIMENSIONAL CONDUCTIVITY TRANSLATIONS FOR ELASTICITY A PROOF THAT THE YOUNG’S MODULUS OF A METAL PLATE WITH HOLES DOES NOT DEPEND ON THE POISSON’S RATIO OF THE METAL THE ELASTIC MODULI OF CERTAIN TWO-DIMENSIONAL POLYCRYSTALS AND SYMMETRIC MATERIALS SOME MICROSTRUCTURE-INDEPENDENT EXACT RELATIONS THE UNIFORM ?ELD ARGUMENT THE BULK MODULUS OF POLYCRYSTALS WITH CUBIC SYMMETRY THE ELASTIC MODULI OF A COMPOSITE WITH A CONSTANT SHEAR MODULUS THE THERMAL EXPANSION TENSOR AND CONSTANT OF SPECI?C HEAT IN A COMPOSITE OF TWO ISOTROPIC PHASES THE EXTENSION TO NONLINEAR THERMAL EXPANSION THE THERMAL EXPANSION TENSOR AND SPECI?C HEAT IN COMPOSITES OF TWO ANISOTROPIC PHASES EXACT THERMOELASTIC RELATIONS FOR POLYCRYSTALS THE EFFECTIVE POROELASTIC MODULI OF TWO-PHASE MEDIA THE ELASTIC MODULI OF TWO-PHASE ?BROUS COMPOSITES EXACT RELATIONS FOR PYROELECTRIC, CONDUCTIVITY, AND MAGNETOTRANSPORT EQUATIONS THE BULK MODULUS OF A SUSPENSION OF ELASTIC PARTICLES IN A ?UID EXACT RELATIONS FOR COUPLED EQUATIONS THE COVARIANCE PROPERTY OF THE EFFECTIVE TENSOR THE REDUCTION TO UNCOUPLED EQUATIONS FOR TWO-PHASE COMPOSITES WITH ISOTROPIC PHASES TRANSLATIONS FOR COUPLED EQUATIONS ELASTICITY AS A SPECIAL CASE OF COUPLED ?ELD EQUATIONS EQUIVALENT COUPLED ?ELD PROBLEMS IN TWO DIMENSIONS THE TWO-DIMENSIONAL EQUATIONS AS A SYSTEM OF ?RST-ORDER PARTIAL DIFFERENTIAL EQUATIONS THE COVARIANCE PROPERTY OF THE FUNDAMENTAL MATRIX LINKING SPECIAL CLASSES OF ANTIPLANE AND PLANAR ELASTICITY PROBLEMS EXPRESSING THE ?ELDS IN EACH PHASE IN TERMS OF ANALYTIC FUNCTIONS† ASSEMBLAGES OF SPHERES, ELLIPSOIDS, AND OTHER NEUTRAL INCLUSIONS THE COATED SPHERE ASSEMBLAGE MULTICOATED SPHERE ASSEMBLAGES A PHASE INTERCHANGE IDENTITY AND INEQUALITY ASSEMBLAGES OF SPHERES WITH VARYING RADIAL AND TANGENTIAL CONDUCTIVITY THE CONDUCTIVITY OF SCHULGASSER’S SPHERE ASSEMBLAGE THE CONDUCTIVITY OF AN ASSEMBLAGE OF SPHERES WITH AN ISOTROPIC CORE AND POLYCRYSTALLINE COATING ASSEMBLAGES OF ELLIPSOIDS AND THEIR ASSOCIATED RICATTI EQUATIONS† THE CONDUCTIVITY OF AN ASSEMBLAGE OF COATED ELLIPSOIDS† A SOLUTION OF THE ELASTICITY EQUATIONS IN THE COATED ELLIPSOID ASSEMBLAGE† EXPRESSIONS FOR THE DEPOLARIZATION FACTORS† NEUTRAL COATED INCLUSIONS TRICKS FOR GENERATING OTHER EXACTLY SOLVABLE MICROGEOMETRIES MODIFYING THE MATERIAL MODULI SO THE ?ELD IS NOT DISTURBED ASSEMBLAGES OF COATED SPHERES AND COATED ELLIPSOIDS WITH ANISOTROPIC CORES MAKING AN AF?NE COORDINATE TRANSFORMATION THE CONDUCTIVITY OF AN ASSEMBLAGE OF COATED ELLIPSOIDS WITH AN ANISOTROPIC CORE AND COATING MAKING A CURVILINEAR COORDINATE TRANSFORMATION† QUASICONFORMAL MAPPINGS GENERATING MICROGEOMETRIES FROM ?ELDS LAMINATE MATERIALS THE HISTORY OF LAMINATES AND WHY THEY ARE IMPORTANT ELEMENTARY LAMINATION FORMULAS LAMINATION FORMULAS WHEN THE DIRECTION OF LAMINATION IS ARBITRARY TARTAR’S LAMINATION FORMULA FOR TWO-PHASE SIMPLE AND COATED LAMINATES LAMINATION FORMULAS FOR ELASTICITY, THERMOELASTICITY, THERMOELECTRICITY, AND PIEZOELECTRICITY THE LAMINATION FORMULA FOR A COATED LAMINATE WITH ANISOTROPIC COATING AND ANISOTROPIC CORE REFERENCE TRANSFORMATIONS EXPLICIT FORMULAS FOR THE CONDUCTIVITY AND ELASTICITY TENSORS OF A COATED LAMINATE ORDINARY DIFFERENTIAL LAMINATES† PARTIAL DIFFERENTIAL LAMINATES† APPROXIMATIONS AND ASYMPTOTIC FORMULAS POLARIZABILITY OF A DIELECTRIC INCLUSION DIELECTRIC CONSTANT OF A DILUTE SUSPENSION OF INCLUSIONS TO THE ?RST ORDER IN THE VOLUME FRACTION DIELECTRIC CONSTANT OF A SUSPENSION OF WELL-SEPARATED SPHERES TO THE SECOND ORDER IN THE VOLUME FRACTION THE MAXWELL APPROXIMATION FORMULA THE EFFECTIVE MEDIUM APPROXIMATION FOR THE DIELECTRIC CONSTANT OF AN AGGREGATE WITH SPHERICAL GRAINS AVERAGE ?ELD APPROXIMATIONS† THE DIFFERENTIAL SCHEME FOR THE EFFECTIVE CONDUCTIVITY OF A SUSPENSION OF SPHERES THE EFFECTIVE MEDIUM APPROXIMATION AS THE ATTRACTOR OF A DIFFERENTIAL SCHEME APPROXIMATION FORMULAS FOR EFFECTIVE ELASTIC MODULI 10 10 ASYMPTOTIC APPROXIMATION FORMULAS 10 11 CRITICAL EXPONENTS AND UNIVERSALITY WAVE PROPAGATION IN THE QUASISTATIC LIMIT ELECTROMAGNETIC WAVE PROPAGATION IN THE QUASISTATIC LIMIT ELECTROMAGNETIC SIGNALS CAN PROPAGATE FASTER IN A COMPOSITE THAN IN THE CONSTITUENT PHASES ELASTIC WAVE PROPAGATION IN THE QUASISTATIC LIMIT THE CORRESPONDENCE PRINCIPLE AND THE ATTENUATION OF SOUND IN A BUBBLY ?UID TRANSFORMATION TO REAL EQUATIONS CORRESPONDENCE WITH THERMOELECTRICITY IN TWO DIMENSIONS RESONANCE AND LOCALIZED RESONANCE IN COMPOSITES† REFORMULATING THE PROBLEM OF ?NDING EFFECTIVE TENSORS RESOLVING A PERIODIC ?ELD INTO ITS THREE COMPONENT ?ELDS: THE -OPERATORS A WIDER CLASS OF PARTIAL DIFFERENTIAL EQUATIONS WITH ASSOCIATED EFFECTIVE TENSORS† A RELATED -OPERATOR THE EQUATION SATIS?ED BY THE POLARIZATION ?ELD THE EFFECTIVE TENSOR OF DILUTE SUSPENSIONS OF ALIGNED ELLIPSOIDS EXPRESSIONS FOR THE ACTION OF THE -OPERATORS IN REAL SPACE A FRAMEWORK FOR DE?NING EFFECTIVE TENSORS IN A MORE GENERAL CONTEXT VARIOUS SOLUTIONS FOR THE ?ELDS AND EFFECTIVE TENSOR THE DUALITY PRINCIPLE 12 10 THE EFFECTIVE TENSOR OF THE ADJOINT EQUATION 12 11 MAGNETOTRANSPORT AND ITS EQUIVALENCE TO THERMOELECTRICITY IN TWO DIMENSIONS VARIATIONAL PRINCIPLES AND INEQUALITIES CLASSICAL VARIATIONAL PRINCIPLES AND INEQUALITIES MONOTONICITY OF THE EFFECTIVE TENSOR NULL LAGRANGIANS VARIATIONAL PRINCIPLES FOR PROBLEMS WITH A COMPLEX OR OTHER NON-SELF-ADJOINT TENSOR HASHIN-SHTRIKMAN VARIATIONAL PRINCIPLES AND INEQUALITIES RELATION BETWEEN THE HASHIN-SHTRIKMAN AND CLASSICAL VARIATIONAL INEQUALITIES† VARIATIONAL INEQUALITIES FOR NONLINEAR MEDIA SERIES EXPANSIONS FOR THE ?ELDS AND EFFECTIVE TENSORS EXPANDING THE FORMULAS FOR THE EFFECTIVE TENSORS AND ?ELDS IN POWER SERIES THE SERIES EXPANSION IN A COMPOSITE TO SECOND ORDER THERMOELASTIC COMPOSITES FOR WHICH THE THIRD AND HIGHER ORDER TERMS IN THE EXPANSION VANISH A LARGE CLASS OF EXACTLY SOLVABLE MATERIALS WITH COMPLEX MODULI† REDUCING THE DIMENSIONALITY OF THE PROBLEM† CONVERGENCE OF THE EXPANSIONS AND THE EXISTENCE AND UNIQUENESS OF THE ?ELDS AND EFFECTIVE TENSORS CONVERGENCE WHEN L IS NOT SELF-ADJOINT† EXTENDING THE DOMAIN OF CONVERGENCE† A SERIES WITH A FASTER CONVERGENCE RATE 14 10 A RELATED SERIES THAT CONVERGES QUICKLY 14 11 NUMERICAL COMPUTATION OF THE ?ELDS AND EFFECTIVE TENSOR USING SERIES EXPANSIONS CORRELATION FUNCTIONS AND HOW THEY ENTER SERIES EXPANSIONS† EXPRESSING THE THIRD-ORDER TERM OF THE SERIES EXPANSION IN TERMS OF CORRELA- TION FUNCTIONS THE TERMS IN THE SERIES EXPANSION FOR RANDOM MEDIA CORRELATION FUNCTIONS FOR PENETRABLE SPHERES CORRELATION FUNCTIONS FOR CELL MATERIALS REDUCED CORRELATION FUNCTIONS EXPANSIONS FOR TWO-PHASE RANDOM COMPOSITES WITH GEOMETRIC ISOTROPY SERIES EXPANSIONS FOR CELL MATERIALS WITH GEOMETRIC ISOTROPY OTHER PERTURBATION SOLUTIONS EFFECT OF A SMALL VARIATION IN THE MATERIAL MODULI APPLICATION TO WEAKLY COUPLED EQUATIONS OF THERMOELECTRICITY OR PIEZOELEC- TRICITY APPLICATION TO COMPUTING THE EFFECTIVE HALL COEF?CIENT THE VARIANCE OF THE ELECTRIC ?ELD IN A TWO-PHASE CONDUCTING COMPOSITE BOUNDS ON THE CONDUCTIVITY TENSOR OF A COMPOSITE OF TWO ISOTROPIC PHASES THE CHANGE IN THE EFFECTIVE TENSOR DUE TO A SHIFT IN THE PHASE BOUNDARY† PERTURBING THE LAMINATION DIRECTIONS IN A MULTIPLE-RANK LAMINATE† THE GENERAL THEORY OF EXACT RELATIONS AND LINKS BETWEEN EFFECTIVE TENSORS LINKS BETWEEN EFFECTIVE TENSORS AS EXACT RELATIONS: THE IDEA OF EMBEDDING NECESSARY CONDITIONS FOR AN EXACT RELATION SUF?CIENT CONDITIONS FOR AN EXACT RELATION AN EXACT FORMULA FOR THE SHEAR MODULUS OF CERTAIN THREE-DIMENSIONAL POLYCRYSTALS MORE EXACT RELATIONS FOR COUPLED EQUATIONS† EXACT RELATIONS WITH LIMITED STATISTICAL INFORMATION ADDITIONAL NECESSARY CONDITIONS FOR AN EXACT RELATION† ANALYTIC PROPERTIES ANALYTICITY OF THE EFFECTIVE DIELECTRIC CONSTANT OF TWO-PHASE MEDIA ANALYTICITY OF THE EFFECTIVE TENSOR FOR PROBLEMS INVOLVING MANY EIGENVALUES INTEGRAL REPRESENTATIONS FOR THE EFFECTIVE TENSOR FOR PROBLEMS INVOLVING TWO EIGENVALUES THE CORRESPONDENCE BETWEEN ENERGY FUNCTIONS AND MICROGEOMETRIES THE CORRESPONDENCE BETWEEN EFFECTIVE CONDUCTIVITY FUNCTIONS AND MICROGE- OMETRIES IN TWO DIMENSIONS† INTEGRAL REPRESENTATIONS FOR PROBLEMS INVOLVING MORE THAN TWO EIGENVALUES: THE TRAJECTORY METHOD THE LACK OF UNIQUENESS IN THE CHOICE OF INTEGRAL KERNEL: CONSTRAINTS ON THE MEASURE† INTEGRAL REPRESENTATIONS FOR A BROADER CLASS OF COMPOSITE PROBLEMS† Y-TENSORS THE Y-TENSOR IN TWO-PHASE COMPOSITES THE Y-TENSOR IN MULTIPHASE COMPOSITES A FORMULA FOR THE EFFECTIVE THERMOELASTIC TENSOR IN TERMS OF THE ELASTICITY Y-TENSOR THE HILBERT SPACE SETTING FOR THE Y-TENSOR PROBLEM† THE Y-TENSOR POLARIZATION PROBLEM† VARIATIONAL INEQUALITIES AND PRINCIPLES FOR Y-TENSORS† Y-TENSORS AND EFFECTIVE TENSORS IN ELECTRICAL CIRCUITS† THE INCIDENCE MATRIX AND THE ?ELDS OF POTENTIAL DROPS AND CURRENTS THE SUBDIVISION OF BONDS IN AN ELECTRICAL CIRCUIT THE Y-TENSOR OF THE ELECTRICAL CIRCUIT THE EFFECTIVE TENSOR OF THE PASSIVE NETWORK THE INTERPRETATION OF THE SUBSPACE U (1) THE RELATION BETWEEN THE EFFECTIVE TENSOR AND THE Y-TENSOR IN AN ELECTRICAL CIRCUIT BOUNDS ON THE PROPERTIES OF COMPOSITES WHY ARE BOUNDS USEFUL? WHAT ARE BOUNDS? THE ROLE OF BOUNDS IN STRUCTURAL OPTIMIZATION: A MODEL PROBLEM CLASSICAL VARIATIONAL PRINCIPLE BOUNDS MULTIPHASE CONDUCTING COMPOSITES ATTAINING ENERGY BOUNDS OPTIMAL BOUNDS ON THE CONDUCTIVITY OF ISOTROPIC POLYCRYSTALS OPTIMAL BOUNDS ON THE BULK MODULUS OF ISOTROPIC POLYCRYSTALS THE COMPLETE CHARACTERIZATION OF THE SET G F UE0 FOR N-PHASE COMPOSITES AND POLYCRYSTALS THE G-CLOSURE IN TWO DIMENSIONS OF AN ARBITRARY SET OF CONDUCTING MATERIALS BOUNDS ON COMPLEX EFFECTIVE TENSORS† BOUNDS FROM THE HASHIN-SHTRIKMAN VARIATIONAL INEQUALITIES BOUNDS ON THE EFFECTIVE CONDUCTIVITY OF AN ISOTROPIC COMPOSITE OF N ISOTROPIC PHASES OPTIMAL BOUNDS ON THE EFFECTIVE CONDUCTIVITY OF AN ANISOTROPIC COMPOSITE OF TWO ISOTROPIC PHASES BOUNDS FOR TWO-PHASE, WELL-ORDERED MATERIALS BOUNDS ON THE ENERGY THAT INVOLVE ONLY THE VOLUME FRACTIONS BOUNDS ON THE EFFECTIVE TENSOR THAT INVOLVE ONLY THE VOLUME FRACTIONS BOUNDS FOR TWO-PHASE COMPOSITES WITH NON-WELL-ORDERED TENSORS† BOUNDING THE COMPLEX EFFECTIVE MODULI OF AN ISOTROPIC COMPOSITE OF TWO ISOTROPIC PHASES† USING QUASICONFORMAL MAPPINGS TO OBTAIN BOUNDS OPTIMAL TWO-DIMENSIONAL MICROGEOMETRIES: REDUCTION TO A DIRICHLET PROBLEM† 23 10 BOUNDS FOR CELL POLYCRYSTALS BOUNDS USING THE COMPENSATED COMPACTNESS OR TRANSLATION METHOD THE TRANSLATION BOUND AND COMPARISON BOUND UPPER BOUNDS ON THE BULK MODULUS OF TWO-PHASE COMPOSITES AND POLYCRYS- TALS IN TWO DIMENSIONS ALLOWING QUASICONVEX TRANSLATIONS A LOWER BOUND ON THE EFFECTIVE BULK MODULUS OF A THREE-DIMENSIONAL, TWO- PHASE COMPOSITE USING THE IDEA OF EMBEDDING TO EXTEND THE TRANSLATION METHOD THE TRANSLATION BOUNDS AS A COROLLARY OF THE COMPARISON BOUNDS† EMBEDDING IN A HIGHER ORDER TENSORIAL PROBLEM: A LOWER BOUND ON THE CONDUCTIVITY TENSOR OF A POLYCRYSTAL A GEOMETRIC CHARACTERIZATION OF TRANSLATIONS† 24 10 TRANSLATION BOUNDS ON THE Y-TENSOR 24 11 DERIVING THE TRACE BOUNDS† 24 12 MIXED BOUNDS 24 13 VOLUME FRACTION INDEPENDENT BOUNDS ON THE CONDUCTIVITY OF A MIXTURE OF TWO ISOTROPIC PHASES 24 14 BOUNDS CORRELATING DIFFERENT EFFECTIVE TENSORS CHOOSING THE TRANSLATIONS AND ?NDING MICROGEOMETRIES THAT ATTAIN THE BOUNDS† OTHER DERIVATIONS OF THE TRANSLATION BOUNDS AND THEIR EXTENSION TO NONLINEAR PROBLEMS EXTREMAL TRANSLATIONS ATTAINABILITY CRITERIA FOR THE COMPARISON BOUNDS ISOTROPIC POLYCRYSTALS WITH MINIMUM CONDUCTIVITY CONSTRUCTED FROM A FULLY ANISOTROPIC CRYSTAL ATTAINABILITY CRITERIA FOR THE TRANSLATION BOUNDS ATTAINABILITY CRITERIA FOR THE HASHIN-SHTRIKMAN-HILL BOUNDS ON THE CONDUC- TIVITY AND BULK MODULUS A GENERAL PROCEDURE FOR ?NDING TRANSLATIONS THAT GENERATE OPTIMAL BOUNDS ON SUMS OF ENERGIES TRANSLATIONS FOR THREE-DIMENSIONAL ELASTICITY BOUNDS INCORPORATING THREE-POINT CORRELATION FUNCTIONS† A BRIEF HISTORY OF BOUNDS INCORPORATING CORRELATION FUNCTIONS THREE-POINT BOUNDS ON THE CONDUCTIVITY OF A TWO-PHASE MIXTURE THREE-POINT BOUNDS ON THE ELASTIC MODULI OF A TWO-PHASE MIXTURE CORRELATION FUNCTION INDEPENDENT ELASTICITY BOUNDS: IMPROVING THE HASHIN- SHTRIKMAN-HILL-WALPOLE BOUNDS USING THE TRANSLATION METHOD TO IMPROVE THE THIRD-ORDER BOUNDS THIRD-ORDER BOUNDS FROM CROSS-PROPERTY BOUNDS GENERAL THIRD-ORDER BOUNDS FOR A TWO-PHASE COMPOSITE THIRD-ORDER BOUNDS FOR TWO-PHASE COMPOSITES WITH GEOMETRICAL ISOTROPY BOUNDS USING THE ANALYTIC METHOD A BRIEF HISTORY OF BOUNDS DERIVED USING THE ANALYTIC METHOD A TOPOLOGICAL CLASSI?CATION OF RATIONAL CONDUCTIVITY FUNCTIONS BOUNDS THAT INCORPORATE A SEQUENCE OF SERIES EXPANSION COEF?CIENTS RELATION BETWEEN THE BOUNDS AND PAD´E APPROXIMANTS BOUNDS INCORPORATING KNOWN REAL OR COMPLEX VALUES OF THE FUNCTION AND SERIES EXPANSION COEF?CIENTS NUMERICAL COMPUTATION OF THE BOUNDS† BOUNDS FOR TWO-DIMENSIONAL ISOTROPIC COMPOSITES† BOUNDS FOR SYMMETRIC MATERIALS† REDUCING THE SET OF INDEPENDENT BOUNDS 27 10 PROVING ELEMENTARY BOUNDS USING THE METHOD OF VARIATION OF POLES AND RESIDUES 27 11 PROVING THE BOUNDS USING THE METHOD OF VARIATION OF POLES AND ZEROS FRACTIONAL LINEAR TRANSFORMATIONS AS A TOOL FOR GENERATING BOUNDS† ELIMINATING THE CONSTRAINTS IMPOSED BY KNOWN SERIES EXPANSION COEF?CIENTS ELIMINATING THE CONSTRAINTS IMPOSED BY KNOWN REAL VALUES OF THE FUNCTION AN ALTERNATIVE APPROACH THAT TREATS THE COMPONENTS ON A SYMMETRIC BASIS THE EXTENSION OF THE FRACTIONAL LINEAR TRANSFORMATIONS TO MATRIX-VALUED ANALYTIC FUNCTIONS THE ?ELD EQUATION RECURSION METHOD† ASSOCIATIONS BETWEEN OPERATIONS ON ANALYTIC FUNCTIONS AND OPERATIONS ON SUBSPACE COLLECTIONS HINTS OF A DEEPER CONNECTION BETWEEN ANALYTIC FUNCTIONS AND SUBSPACE COLLECTIONS THE ?ELD EQUATION RECURSION METHOD FOR TWO-PHASE COMPOSITES REPRESENTING THE OPERATORS AS IN?NITE-DIMENSIONAL MATRICES THE ?ELD EQUATION RECURSION METHOD FOR MULTIPHASE COMPOSITES WITH ISOTROPIC COMPONENTS BOUNDS ON THE ENERGY FUNCTION OF A THREE-PHASE CONDUCTING COMPOSITE PROPERTIES OF THE G-CLOSURE AND EXTREMAL FAMILIES OF COMPOSITES AN EQUIVALENCE BETWEEN G-CLOSURE PROBLEMS WITH AND WITHOUT PRESCRIBED VOLUME FRACTIONS STABILITY UNDER LAMINATION AND THE CONVEXITY PROPERTIES OF THE G-CLOSURE CHARACTERIZING THE G-CLOSURE THROUGH MINIMUMS OF SUMS OF ENERGIES AND COMPLEMENTARY ENERGIES CHARACTERIZATION OF THE G-CLOSURE BY SINGLE ENERGY MINIMIZATIONS EXTREMAL FAMILIES OF COMPOSITES FOR ELASTICITY: PROVING THAT ANY POSITIVE- DE?NITE TENSOR CAN BE REALIZED AS THE EFFECTIVE ELASTICITY TENSOR OF A COMPOSITE† AN EXTREMAL FAMILY OF UNIMODE MATERIALS FOR TWO-DIMENSIONAL ELASTICITY† AN EXTREMAL FAMILY OF BIMODE MATERIALS FOR TWO-DIMENSIONAL ELASTICITY† EXTREMAL MATERIALS FOR THREE-DIMENSIONAL ELASTICITY THE BOUNDING OF EFFECTIVE MODULI AS A QUASICONVEXI?CATION PROBLEM QUASICONVEXI?CATION PROBLEMS IN ELASTICITY THEORY THE INDEPENDENCE OF THE QUASICONVEXI?ED FUNCTION ON THE SHAPE AND SIZE OF THE REGION REPLACING THE AF?NE BOUNDARY CONDITIONS WITH PERIODIC BOUNDARY CONDITIONS THE EQUIVALENCE OF BOUNDING THE ENERGY OF MULTIPHASE LINEAR COMPOSITES AND QUASICONVEXI?CATION THE LINK BETWEEN THE LAMINATION CLOSURE AND -CONVEXI?CATION QUASICONVEX HULLS AND RANK-1 CONVEX HULLS LAMINATE ?ELDS BUILT FROM RANK-1 INCOMPATIBLE MATRICES EXAMPLE OF A RANK-1 FUNCTION THAT IS NOT QUASICONVEX† A COMPOSITE WITH AN ELASTICITY TENSOR THAT CANNOT BE MIMICKED BY A MULTIPLE-RANK LAMINATE MATERIAL†