Tianyou Fan. Mathematical Theory of Elasticity of Quasicrystals and Its Applications

Transcription

1 Tianyou Fan Mathematical Theory of Elasticity of Quasicrystals and Its Applications

2 Tianyou Fan Mathematical Theory of Elasticity of Quasicrystals and Its Applications With 82 figures

3 Author Tianyou Fan Department of Physics, School of Science Beijing Institue of Technology Beijing , China ISBN Science Press Beijing ISBN e-isbn Springer Heidelberg Dordrecht London New York Library of Congress Control Number: Science Press Beijing and Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: Frido Steinen-Broo, EStudio Calamar, Spain Printed on acid-free paper Springer is part of Springer Science+Business Media (

4 Preface This monograph is devoted to the development of a mathematical theory of elasticity of quasicrystals and its applications. Some results on elastodynamics and plasticity of quasicrystals are also included to document preliminary advances in recent years. The first quasicrystal was observed in 1982 and reported in November At that time several physical and mathematical theories for quasicrystal study already existed. Soon after the discovery, the theory of elasticity of quasicrystals was put forward. Based on Landau-Anderson symmetry-breaking, a new elementary excitation - the phason - was introduced in addition to the well known phonon. The phason concept was suggested in the 1960 s in incommensurate phase theory. Group theory and discrete geometry e.g. the Penrose tiling provide further explanations to quasicrystals and their elasticity from the standpoint of algebra and geometry. The phonon and phason elementary excitations form the basis of the theory of elasticity of this new solid phase. Many theoretical (condensed matter) physicists have spent a great deal of effort on constructing the fundamental physical framework of the theory of elasticity of quasicrystals. Applications of group theory and group representation theory further enhance the physical basis of the description. On the basis of the physical framework and by extending the methodology of mathematical physics and classical elasticity, the mathematical theory of elasticity of quasicrystals has been developed. Recent studies on the elasto-/hydro-dynamics and the plasticity of quasicrystals have made preliminary but significant progress. As regards the dynamics, there are various viewpoints in the quasicrystal community, which reveal the unusual characteristics of phason dynamics. Yet the effect of the phason degrees of freedom on plastic deformation is not well understood, and the basic plastic properties of the material are virtually unknown. Because of many unsolved critical issues, the study of quasicrystals has attracted many researchers. The contex of the last few chapters in this book is a probe of the fascinating research area. As this book is focused on the mathematical theory of elasticity of quasicrystals, it does not include in-depth discussions on the physics of the phason degrees of freedom and the physical nature of the phason variables. These research subjects are important to the quasicrystal study. I sincerely thank the National Natural Science Foundation of China and the Alexander von Humboldt Foundation of Germany for their support over the years. I also thank Professors Fanghua Li (Institute of Physics in the Chinese Academy of

5 vi Preface Science), Longan Ying (School of Mathematics of Peking University), Dihua Ding (Department of Physics, Wuhan University), Jiann-Quo Tarn (Cheng Kung University / Zhejiang University), Weiqiu Chen (Zhejiang University), Qingping Sun (Hong Kong Science and Technology University), U. Messerschmidt (Max Planck Institut fuer Mikrostruktur Physik, Halle), H.-R. Trebin (Institut fuer Theoretische und Angewandte Physik, Universitaet Stuttgart), and K. Edagawa (University of Tokyo) for their helpful discussions. Last but not least, I thank my co-workers and former and present students, especially Professor Xianfang Li (Central South University), for their help and contributions. Tianyou Fan January 2010, Beijing

6 Preface Chapter 1 Crystals Periodicity of crystal structure, crystal cell Three-dimensional lattice types Symmetry and point groups Reciprocal lattice Appendix of Chapter 1: Some basic concepts...6 References...10 Chapter 2 Framework of the classical theory of elasticity Review on some basic concepts Basic assumptions of theory of elasticity Displacement and deformation Stress analysis and equations of motion Generalized Hooke s law Elastodynamics, wave motion Summary...23 References...24 Chapter 3 Quasicrystal and its properties Discovery of quasicrystal Structure and symmetry of quasicrystals A brief introduction on physical properties of quasicrystals One-, two- and three-dimensional quasicrystals Two-dimensional quasicrystals and planar quasicrystals...30 References...30 Chapter 4 The physical basis of elasticity of quasicrystals Physical basis of elasticity of quasicrystals Deformation tensors Stress tensors and the equations of motion Free energy and elastic constants Generalized Hooke s law Boundary conditions and initial conditions A brief introduction on relevant material constants of quasicrystals Summary and mathematical solvability of boundary value or initial-

7 viii boundary value problem Appendix of Chapter 4: Description on physical basis of elasticity of quasicrystals based on the Landau density wave theory...46 References...50 Chapter 5 Elasticity theory of one-dimensional quasicrystals and simplification Elasticity of hexagonal quasicrystals Decomposition of the problem into plane and anti-plane problems Elasticity of monoclinic quasicrystals Elasticity of orthorhombic quasicrystals Tetragonal quasicrystals The space elasticity of hexagonal quasicrystals Other results of elasticity of one-dimensional quasicrystals References...65 Chapter 6 Elasticity of two-dimensional quasicrystals and simplification Basic equations of plane elasticity of two-dimensional quasicrystals: point groups 5m and 10mm in five- and ten-fold symmetries Simplification of the basic equation set: displacement potential function method Simplification of the basic equations set: stress potential function method Plane elasticity of point group 5, 5 pentagonal and point group 10, 10 decagonal quasicrystals Plane elasticity of point group 12mm of dodecagonal quasicrystals Plane elasticity of point group 8mm of octagonal quasicrystals, displacement potential Stress potential of point group 5, 5 pentagonal and point group 10, 10 decagonal quasicrystals Stress potential of point group 8mm octagonal quasicrystals Engineering and mathematical elasticity of quasicrystals...98 References Chapter 7 Application I: Some dislocation and interface problems and solutions in one- and two-dimensional quasicrystals Dislocations in one-dimensional hexagonal quasicrystals Dislocations in quasicrystals with point groups 5m and 10mm symmetries Dislocations in quasicrystals with point groups 5, 5 five-fold and 10,

8 ix 10 ten-fold symmetries Dislocations in quasicrystals with eight-fold symmetry Dislocations in dodecagonal quasicrystals Interface between quasicrystal and crystal Conclusion and discussion References Chapter 8 Application II: Solutions of notch and crack problems of one-and two-dimensional quasicrystals Crack problem and solution of one-dimensional quasicrystals Crack problem in finite-sized one-dimensional quasicrystals Griffith crack problems in point groups 5m and 10mm quasicrystals based on displacement potential function method Stress potential function formulation and complex variable function method for solving notch and crack problems of quasicrystals of point groups 5, 5 and 10, Solutions of crack/notch problems of two-dimensional octagonal quasicrystals Other solutions of crack problems in one-and two-dimensional quasicrystals Appendix of Chapter 8: Derivation of solution of Section References Chapter 9 Theory of elasticity of three-dimensional quasicrystals and its applications Basic equations of elasticity of icosahedral quasicrystals Anti-plane elasticity of icosahedral quasicrystals and problem of interface between quasicrystal and crystal Phonon-phason decoupled plane elasticity of icosahedral quasicrystals Phonon-phason coupled plane elasticity of icosahedral quasicrystals displacement potential formulation Phonon-phason coupled plane elasticity of icosahedral quasicrystals stress potential formulation A straight dislocation in an icosahedral quasicrystal An elliptic notch/griffith crack in an icosahedral quasicrystal Elasticity of cubic quasicrystals the anti-plane and axisymmetric deformation...185

9 x References Chapter 10 Dynamics of elasticity and defects of quasicrystals Elastodynamics of quasicrystals followed the Bak s argument Elastodynamics of anti-plane elasticity for some quasicrystals Moving screw dislocation in anti-plane elasticity Mode III moving Griffith crack in anti-plane elasticity Elasto-/hydro-dynamics of quasicrystals and approximate analytic solution for moving screw dislocation in anti-plane elasticity Elasto-/hydro-dynamics and solutions of two-dimensional decagonal quasicrystals Elasto-/hydro-dynamics and applications to fracture dynamics of icosahedral quasicrystals Appendix of Chapter 10: The detail of finite difference scheme References Chapter 11 Complex variable function method for elasticity of quasicrystals Harmonic and quasi-biharmonic equations in anti-plane elasticity of one-dimensional quasicrystals Biharmonic equations in plane elasticity of point group 12mm twodimensional quasicrystals The complex variable function method of quadruple harmonic equations and applications in two-dimensional quasicrystals Complex variable function method for sextuple harmonic equation and applications to icosahedral quasicrystals Complex analysis and solution of quadruple quasiharmonic equation Conclusion and discussion References Chapter 12 Variational principle of elasticity of quasicrystals, numerical analysis and applications Basic relations of plane elasticity of two-dimensional quasicrystals Generalized variational principle for static elasticity of quasicrystals Finite element method Numerical examples References Chapter 13 Some mathematical principles on solutions of elasticity of quasicrystals Uniqueness of solution of elasticity of quasicrystals...273

10 xi 13.2 Generalized Lax-Milgram theorem Matrix expression of elasticity of three-dimensional quasicrystals The weak solution of boundary value problem of elasticity of quasicrystals The uniqueness of weak solution Conclusion and discussion References Chapter 14 Nonlinear behaviour of quasicrystals Macroscopic behaviour of plastic deformation of quasicrystals Possible scheme of plastic constitutive equations Nonlinear elasticity and its formulation Nonlinear solutions based on simple models Nonlinear analysis based on the generalized Eshelby theory Nonlinear analysis based on the dislocation model Conclusion and discussion Appendix of Chapter 14: Some mathematical details References Chapter 15 Fracture theory of quasicrystals Linear fracture theory of quasicrystals Measurement of G IC Nonlinear fracture mechanics Dynamic fracture Measurement of fracture toughness and relevant mechanical parameters of quasicrystalline material References Chapter 16 Remarkable conclusion References Major Appendix: On some mathematical materials Appendix I Outline of complex variable functions and some additional calculations A.I.1 Complex functions, analytic functions A.I.2 Cauchy s formula A.I.3 Poles A.I.4 Residue theorem A.I.5 Analytic extension A.I.6 Conformal mapping A.I.7 Additional derivation of solution (8.2-19) A.I.8 Additional derivation of solution ( )...342

11 xii A.I.9 Detail of complex analysis of generalized cohesive force model for plane elasticity of two-dimensional point groups 5m, 10mm and 10, 10 quasicrystals A.I.10 On the calculation of integral (9.2-14) Appendix II Dual integral equations and some additional calculations A.II.1 Dual integral equations A.II.2 Additional derivation on the solution of dual integral equations (8.3-8) A.II.3 Additional derivation on the solution of dual integral equations (9.8-8) References Index...359