Mathematical Theory Elasticity Generalhb: Mathematical Theory of Elasticity and Generalized Dynamics of Quasicrystals and Its Applications

This book gives a detailed description on mathematical theory of elasticity and generalized dynamics of solid quasicrystals and its applications.The Chinese edition of the book Mathematical Theory of Elasticity of Quasicrystals and Its Applications was published by the Beijing Institute of Technology Press in 1999, written by Prof Tian-You Fan. In this English edition of the book, the phonon-phason dynamics, defect dynamics and hydrodynamics of solid quasicrystals are included, so the scope of the book is beyond elasticity. Hence, the title in this edition is changed to Mathematical Theory of Elasticity and Generalized Dynamics of Quasicrystals and Its Applications. This book is the first and only monograph in the scope of quasicrystals since first published in 1999 in China and worldwide. In this edition, the two-dimensional quasicrystals of second kind, soft-matter quasicrystals and photonic bade-gap and application of photonic quasicrystals are added.This book combines the mechanical and physical behavior of quasicrystals and mathematical physics, which may help graduate students and researchers in the fields of new materials, condensed matter physics, applied mathematics and engineering science. Contents Preface Authors 1. Crystals 1.1 Periodicity of Crystal Structure, Crystal Cell 1.2 Three-dimensional Lattice Types 1.3 Symmetry and Point Groups 1.4 Reciprocal Lattice 1.5 Appendix: Some Basic Concepts 1.5.1 Concept of phonon 1.5.2 Incommensurate crystals 1.5.3 Glassy structure 1.5.4 Mathematical aspect of group 1.5.4.1 Mathematical definition of group 1.5.4.2 The linear representation of group References 2. Framework of Crystal Elasticity 2.1 Review of Some Basic Concepts 2.1.1 Vector 2.1.2 Coordinate frame 2.1.3 Coordinate transformation 2.1.4 Tensor 2.1.5 Algebraic operations of te 2.2 Basic Assumptions of Theory of Elasticity 2.3 Displacement and Deformation 2.4 Stress Analysis 2.5 Generalized Hooke’s Law 2.6 Elastodynamics, Wave Motion 2.7 Summary References 3. Solid Quasicrystals and Their Properties 3.1 Discovery of Solid Quasicrystal 3.2 Structure and Symmetry of Quasicrystals 3.3 A Brief Introduction of the Physical Properties of Solid Quasicrystals 3.4 One-, Two- and Three-dimensional Quasicrystals 3.5 Two-dimensional Quasicrystals and Planar Quasicrystals 3.6 The First and Second Kind of Two-dimensional Quasicrystals References 4. The Physical Basis of Elasticity of Solid Quasicrystals 4.1 Physical Basis of Elasticity of Quasicrystals 4.2 Deformation Tensors 4.3 Stress Tensors and Equations of Motion 4.4 Free Energy and Elastic Constants 4.5 Generalized Hooke’s Law 4.6 Boundary Conditions and Initial Conditions 4.7 A Brief Introduction on Relevant Material Constants of Quasicrystals 4.8 Summary and Mathematical Solvability of Boundary Value or Initial-boundary Value Problem 4.9 Appendix A: Description on Physical Basis of Elasticity of Quasicrystals Based on the Landau Density Wave Theory References 5. Elasticity Theory of One-dimensional Quasicrystals and Simplification 5.1 Elasticity of Hexagonal Quasicrystals 5.2 Decomposition of the Elasticity into a Superposition of Plane and Anti-plane Elasticity 5.3 Elasticity of Monoclinic Quasicrystals 5.4 Elasticity of Orthorhombic Quasicrystals 5.5 Tetragonal Quasicrystals 5.6 The Space Elasticity of Hexagonal Quasicrystals 5.7 Other Results of Elasticity of One-dimensional Quasicrystals References 6. Elasticity Theory of Two-dimensional Solid Quasicrystals of First Kind and Simplification 6.1 Basic Equations of Plane Elasticity in Two-dimensional Quasicrystals: Point Groups 5m and 10mm in Five- and 10-fold Symmetries 6.2 Simplification of the Basic Equation Set: Displacement Potential Function Method 6.3 Simplification of Basic Equations Set: Stress Potential Function Method 6.4 Plane Elasticity of Point Group 5,5 and 10, 10 Pentagonal and Decagonal Quasicrystals 6.5 Plane Elasticity of Point Group 12mm of Dodecagonal Quasicrystals 6.6 Plane Elasticity of Point Group 8mm of Octagonal Quasicrystals, Displacement Potential 6.7 Stress Potential of the Plane Field of Point Group 5, 5 Pentagonal and Point Group 10, 10 Decagonal Quasicrystals 6.8 Stress Potential of Point Group 8mm Octagonal Quasicrystals 6.9 The Three-dimensional Elasticity and the Field Equations of the First Kind of Two-dimensional Sold Quasicrystals 6.10 Other Discussions References 7. Application I — Some Dislocation and Interface Problems and Solutions in One-dimensional and First Kind of Two-dimensional Quasicrystals 7.1 Dislocations in One-dimensional Hexagonal Quasicrystals 7.2 Dislocations in Quasicrystals with Point Groups 5m and 10mm Symmetries 7.3 Dislocations in Quasicrystals with Point Groups 5, 5 fold and 10, 10 fold Symmetries 7.4 Dislocations in Quasicrystals with Eight fold Symmetry 7.4.1 Fourier transform method 7.4.2 Complex analysis method 7.5 Dislocations in Dodecagon Quasicrystals 7.6 Interface between Quasicrystal and Crystal 7.7 Dislocation Pile Up, Dislocation Group and Plastic Zone 7.8 Discussions and Conclusions References 8. Application II — Solutions of Notch and Crack Problems of One- and Two-dimensional Quasicrystals 8.1 Crack Problem and Solution of One-dimensional Quasicrystals 8.1.1 Griffith crack 8.1.2 Brittle fracture theory 8.2 Crack Problem in Finite-sized One-dimensional Quasicrystals 8.2.1 Cracked quasicrystal strip with finite height 8.2.2 Finite strip with two cracks 8.3 Griffith Crack Problems in Point Groups 5m and 10mm Quasicrystal Based on Displacement Potential Function Method 8.4 Stress Potential Function Formulation and Complex Analysis Method for Solving Notch/Crack Problem of Quasicrystals of Point Groups 5, 5 and 10, 10 8.4.1 Complex analysis method 8.4.2 The complex representation of stresses and displacements 8.4.3 Elliptic notch problem 8.4.4 Elastic field caused by a Griffith crack 8.5 Solutions of Crack/Notch Problems in Two-dimensional Octagonal Quasicrystals 8.6 Approximate Analytic Solutions of Notch/Crack of Two-dimensional Quasicrystals with Five- and 10-fold Symmetries 8.7 Cracked Strip with Finite Height of Two-dimensional Quasicrystals with 5- and 10-fold Symmetries and Exact Analytic Solution 8.8 Exact Analytic Solution of Single Edge Crack in a Finite Width Specimen of a Two-dimensional Quasicrystal of 10-fold Symmetry 8.9 Perturbation Solution of Three-dimensional Elliptic Disk Crack in One-dimensional Hexagonal Quasicrystals 8.10 Other Crack Problems in One- and Two-dimensional Quasicrystals 8.11 Plastic Zone Around Crack Tip 8.12 Appendix A: Some Derivations in Secition 8.1 8.13 Appendix B: Some Further Derivation of Solution in Section 8.9 References 9. Theory of Elasticity of Three-dimensional Quasicrystals and Their Applications 9.1 Basic Equations of Elasticity of Icosahedral Quasicrystals 9.2 Anti-plane Elasticity of Icosahedral Quasicrystals and Problem of Interface of Quasicrystal–Crystal 9.3 Phonon–Phason Decoupled Plane Elasticity of Icosahedral Quasicrystals 9.4 Phonon–Phason Coupled Plane Elasticity of Icosahedral Quasicrystals — Displacement Potential Formulation 9.5 Phonon–Phason Coupled Plane Elasticity of Icosahedral Quasicrystals — Stress Potential Formulation 9.6 A Straight Dislocation in an Icosahedral Quasicrystal 9.7 Application of Displacement Potential to Crack Problem of Icosahedral Quasicrystal 9.8 An Elliptic Notch/Griffith Crack in an Icosahedral Quasicrystal 9.8.1 The complex representation of stresses and displacements 9.8.2 Elliptic notch problem 9.8.3 Brief summary 9.9 Elasticity of Cubic Quasicrystals — The Anti-plane and Axisymmetric Deformation References 10. Phonon–Phason Dynamics and Defect Dynamics of Solid Quasicrystals 10.1 Acoustics of Quasicrystals Followed Bak’s Argument 10.2 Acoustics of Anti-plane Elasticity for Some Quasicrystals 10.3 Moving Screw Dislocation in Anti-plane Elasticity 10.4 Mode III Moving Griffith Crack in Anti-plane Elasticity 10.5 Two-dimensional Phonon–Phason Dynamics, Fundamental Solution 10.6 Phonon–Phason Dynamics and Solutions of Two-dimensional Decagonal Quasicrystals 10.6.1 The mathematical formalism of dynamic crack problems of decagonal quasicrystals 10.6.2 Examination on the physical model 10.6.3 Testing the Scheme and the Computer Program 10.6.3.1 Stability of the scheme 10.6.3.2 Accuracy test 10.6.3.3 Influence of mesh size (space step) 10.6.4 Results of dynamic initiation of crack growth 10.6.5 Results of the fast crack propagation 10.7 Phonon–Phason Dynamics and Applications to Fracture Dynamics of Icosahedral Quasicrystals 10.7.1 Basic Equations, Boundary and Initial Conditions 10.7.2 Some results 10.7.3 Conclusion and discussion 10.8 Appendix A — The Detail of Finite Difference Scheme References 11. Complex Analysis Method for Elasticity of Quasicrystals 11.1 Harmonic and Biharmonic in Anti-plane Elasticity of One-dimensional Quasicrystals 11.2 Biharmonic Equations in Plane Elasticity of Point Group 12mm Two-dimensional Quasicrystals 11.3 The Complex Analysis of Quadruple Harmonic Equations and Applications in Two-dimensional Quasicrystals 11.3.1 Complex representation of solution of the governing equation 11.3.2 Complex representation of the stresses and displacements 11.3.3 The complex representation of boundary conditions 11.3.4 Structure of complex potentials 11.3.4.1 Arbitrariness in the definition of the complex potentials 11.3.4.2 General formulas for finite multiply connected regions 11.3.4.3 Case of infinite regions 11.3.5 Conformal mapping 11.3.6 Reduction of the boundary value problem to function equations 11.3.7 Solution of the function equations 11.3.8 Example 1: Elliptic Notch/Crack Problem and Solution 11.3.9 Example 2: Infinite plane with an elliptic hole subjected to a tension at infinity 11.3.10 Example 3: Infinite plane with an elliptic hole subjected to a distributed pressure at a part of surface of the hole 11.4 Complex Analysis for Sextuple Harmonic Equation and Applications to Three-dimensional Icosahedral Quasicrystals 11.4.1 The complex representation of stresses and displacements 11.4.2 The complex representation of boundary conditions 11.4.3 Structure of complex potentials 11.4.3.1 The arbitrariness of the complex potentials 11.4.3.2 General formulas for finite multiply connected region 11.4.4 Case of infinite regions 11.4.5 Conformal mapping and function equations at ζ-plane 11.4.6 Example: Elliptic notch problem and solution 11.5 Complex Analysis of Generalized Quadruple Harmonic Equation 11.6 Conclusion and Discussion 11.7 Appendix: Basic Formulas of Complex Analysis 11.7.1 Complex functions, analytic functions 11.7.2 Cauchy’s formula 11.7.3 Poles 11.7.4 Residual theorem 11.7.5 Analytic extension 11.7.6 Conformal mapping References 12. Variational Principle of Elasticity of Quasicrystals, Numerical Analysis and Applications 12.1 Review of Basic Relations of Elasticity of Icosahedral Quasicrystals 12.2 General Variational Principle for Static Elasticity of Quasicrystals 12.3 Finite Element Method for Elasticity of Icosahedral Quasicrystals 12.4 Numerical Results 12.4.1 Test example — An icosahedral Al–Pd–Mn quasicrystal bar subjected to uniaxial tension 12.4.2 Specimen of icosahedral Al–Pd–Mn quasicrystal with a crack under tension 12.5 Conclusion References 13. Some Mathematical Principles on Solutions of Elasticity of Solid Quasicrystals 13.1 Uniqueness of Solution of Elasticity of Quasicrystals 13.2 Generalized Lax–Milgram Theorem 13.3 Matrix Expression of Elasticity of Three-dimensional Quasicrystals 13.4 The Weak Solution of Boundary Value Problem of Elasticity of Quasicrystals 13.5 The Uniqueness of Weak Solution 13.6 Conclusion and Discussion References 14. Nonlinear Behaviour of Quasicrystals 14.1 Macroscopic Behaviour of Plastic Deformation of Quasicrystals 14.2 Possible Scheme of Plastic Constitutive Equations 14.3 Nonlinear Elasticity and Its Formulation 14.4 Nonlinear Solutions Based on Some Simple Models 14.4.1 Generalized Dugdale–Barenblatt model for anti-plane elasticity for some quasicrystals 14.4.2 Generalized Dugdale–Barenblatt model for plane elasticity of two-dimensional point groups 5m, 10mm and 5, 5, 10, 10 quasicrystals 14.4.3 Generalized Dugdale–Barenblatt model for plane elasticity of three-dimensional icosahedral quasicrystals 14.5 Nonlinear Analysis Based on the Generalized Eshelby Theory 14.5.1 Generalized Eshelby energy–momentum tensor and generalized Eshelby integral 14.5.2 Relation between crack tip opening displacement and the generalized Eshelby integral 14.5.3 Some further interpretation on application of E-integral to the nonlinear fracture analysis of quasicrystals 14.6 Nonlinear Analysis Based on the Dislocation Model 14.6.1 Screw dislocation pile-up for hexagonal or icosahedral or cubic quasicrystals 14.6.2 Edge dislocation pile-up for pentagonal or decagonal two-dimensional quasicrystals 14.6.3 Edge dislocation pile-up for three-dimensional icosahedral quasicrystals 14.7 Conclusion and Discussion 14.8 Appendix: Some Mathematical Details 14.8.1 Proof on path independency of E-integral 14.8.2 Proof on the equivalency of E-integral to energy release rate under linear elastic case for quasicrystals 14.8.3 On the evaluation of the critic value of E-integral References 15. Fracture Theory of Solid Quasicrystals 15.1 Linear Fracture Theory of Quasicrystals 15.2 Crack Extension Force Expressions of Standard Quasicrystal Samples and Related Testing Strategy for Determining Critical Value GIC 15.2.1 Characterization of GI and GIC of three-point bending quasicrystal samples 15.2.2 Characterization of GI and GIC of compact tension quasicrystal sample 15.3 Nonlinear Fracture Mechanics 15.4 Dynamic Fracture 15.5 Measurement of Fracture Toughness and Relevant Mechanical Parameters of Quasicrystalline Material 15.5.1 Fracture toughness 15.5.2 Tension strength References 16. Hydrodynamics of Solid Quasicrystals 16.1 Viscosity of Solid 16.2 Generalized Hydrodynamics of Solid Quasicrystals 16.3 Simplification of Plane Field Equations in Two-dimensional 5- and 10-fold Symmetrical Solid Quasicrystals 16.4 Numerical Solution 16.5 Conclusion and Discussion References 17. Two-dimensional Quasicrystals of the Second Kind 17.1 The Point Groups of the Second Kind of Two-dimensional Quasicrystals 17.2 Six-dimensional Embedding Space 17.3 Possible 18-fold Symmetry Solid Quasicrystals and Their Elasticity 17.4 Equation System of Elasticity of Quasicrystals of 18-fold Symmetry with Point Group 18mm 17.5 Possible 7-fold Symmetry Solid Quasicrystals and Their Elasticity 17.6 The Possible 9-fold Symmetrical Quasicrystals with Point Group 9m 17.7 Possible 14-fold Symmetry Solid Quasicrystals and Their Elasticity: The Possible 14-fold Symmetrical Quasicrystals with Point Group 14mm 17.8 Solution of Phason Field of Dislocation of Possible 9-fold Symmetry Solid Quasicrystals 17.9 Conclusion and Discussion References 18. Identification and Phase Transitions of Soft-matter Quasicrystals 18.1 Identification of Supramolecular Quasicrystals in Soft Matter 18.1.1 Basic notions and discovery of soft quasicrystals 18.1.2 12-fold dodecagonal quasicrystals in soft matters 18.1.3 10-fold axial decagonal DQC in soft matter 18.1.4 Soft quasicrystals with other symmetry 18.2 Phase Transitions and Formation Mechanisms of Soft Quasicrystals 18.2.1 Phase transitions of 12-fold DDQC 18.2.2 Phase transitions of DQC by giant molecules 18.2.3 Simulation of QCphase formation in soft matters 18.3 A Columnar 12-fold Quasicrystal with Thermodynamic Stability 18.4 Conclusion and Discussion References 19. Photonic Band Gap and Application of Two-dimensional Photonic Quasicrystals 19.1 Introduction 19.2 Design and Formation of Holographic PQCs 19.3 Band Gap of 8-fold PQCs 19.4 Band Gap of Multi-fold Complex PQCs 19.5 Fabrication of 10-fold Holographic PQCs 19.5.1 Material and writing system 19.5.2 Experimental results 19.6 PQC Sensor of Liquid Refractive Index 19.6.1 Design and optimization of the sensor 19.6.2 Performance analysis of the sensor 19.7 Conclusions References 20. Concluding Remarks and Perspectives References Major Appendix: On Some Mathematical Additional Materials Appendix I: Additional Calculations Related to Complex Analysis AI.1 Additional derivation of solution (8.2-19) AI.2 Additional derivation of solution (11.3-53) AI.3 Details of complex analysis of solution (14.4-7) of generalized cohesive force model for plane plasticity of two-dimensional point groups 5m, 10mm and 10, 10 quasicrystals AI.4 On the calculation of integral (9.2-14) AI.5 On the calculation of integral (8.8-9) References Appendix II: Dual Integral Equations and Some Additional Calculations AII.1 Dual Integral Equations AII.2 Additional Derivation on the Solution of Dual Integral Equations (8.3-8) and (9.7-4) AII.3 Additional Derivation on the Solution of Dual Integral Equations (9.8-8) References Appendix III: Poisson Brackets in Condensed Matter Physics, Lie Group and Lie Algebra and Their Applications AIII.1 Poisson Brackets in Condensed Matter Physics AIII.2 Other Relevant Formulas AIII.3 Derivation of Hydrodynamic Equations of Solid Quasicrystals AIII.4 Lie Group and Lie Algebra and Their Applications References Index