Mathematical Theory of Elasticity of Quasicrystals and Its Applications

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

Author: Tianyou Fan

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Tianyou Fan

Mathematical Theory of Elasticity of Quasicrystals and Its Applications

Tianyou Fan

Mathematical Theory of Elasticity of Quasicrystals and Its Applications With 82 figures

Author Tianyou Fan Department of Physics, School of Science Beijing Institue of Technology Beijing 100081, China

ISBN 978-7-03-025669-0 Science Press Beijing ISBN 978-3-642-14642-8 e-ISBN 978-3-642-14643-5 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2010931505 © 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 (www.springer.com)

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 1984. 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

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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

Contents Preface Chapter 1 Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Periodicity of crystal structure, crystal cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Three-dimensional lattice types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.3 Symmetry and point groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Reciprocal lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 Appendix of Chapter 1: Some basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Chapter 2 Framework of the classical theory of elasticity . . . . . . . . . . . . . 13 2.1 Review on some basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Basic assumptions of theory of elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 Displacement and deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4 Stress analysis and equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.5 Generalized Hooke’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.6 Elastodynamics, wave motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Chapter 3 Quasicrystal and its properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.1 Discovery of quasicrystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 Structure and symmetry of quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.3 A brief introduction on physical properties of quasicrystals . . . . . . . . . . . . . 29 3.4 One-, two- and three-dimensional quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.5 Two-dimensional quasicrystals and planar quasicrystals . . . . . . . . . . . . . . . . 30 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Chapter 4 The physical basis of elasticity of quasicrystals . . . . . . . . . . . . . 35 4.1 Physical basis of elasticity of quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.2 Deformation tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.3 Stress tensors and the equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.4 Free energy and elastic constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.5 Generalized Hooke’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.6 Boundary conditions and initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.7 A brief introduction on relevant material constants of quasicrystals . . . . . 43 4.8 Summary and mathematical solvability of boundary value or initial-

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boundary value problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.9 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.1 Elasticity of hexagonal quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.2 Decomposition of the problem into plane and anti-plane problems . . . . . . 56 5.3 Elasticity of monoclinic quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.4 Elasticity of orthorhombic quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.5 Tetragonal quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5.6 The space elasticity of hexagonal quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.7 Other results of elasticity of one-dimensional quasicrystals . . . . . . . . . . . . . 65 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Chapter 6 Elasticity of two-dimensional quasicrystals and simpliﬁcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 6.1 Basic equations of plane elasticity of two-dimensional quasicrystals: point groups 5m and 10mm in ﬁve- and ten-fold symmetries . . . . . . . . . . . 71 6.2 Simpliﬁcation of the basic equation set: displacement potential function method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 6.3 Simpliﬁcation of the basic equations set: stress potential function method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 6.4 Plane elasticity of point group 5, 5 pentagonal and point group 10, 10 decagonal quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6.5 Plane elasticity of point group 12mm of dodecagonal quasicrystals . . . . . 85 6.6 Plane elasticity of point group 8mm of octagonal quasicrystals, displacement potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 6.7 Stress potential of point group 5, 5 pentagonal and point group 10, 10 decagonal quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 6.8 Stress potential of point group 8mm octagonal quasicrystals . . . . . . . . . . . . 95 6.9 Engineering and mathematical elasticity of quasicrystals . . . . . . . . . . . . . . . 98 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Chapter 7 Application I: Some dislocation and interface problems and solutions in one- and two-dimensional quasicrystals . . . . 103 7.1 Dislocations in one-dimensional hexagonal quasicrystals . . . . . . . . . . . . . . . 104 7.2 Dislocations in quasicrystals with point groups 5m and 10mm symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 7.3 Dislocations in quasicrystals with point groups 5, ¯5 ﬁve-fold and 10,

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10 ten-fold symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 7.4 Dislocations in quasicrystals with eight-fold symmetry . . . . . . . . . . . . . . . . 117 7.5 Dislocations in dodecagonal quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 7.6 Interface between quasicrystal and crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 7.7 Conclusion and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Chapter 8 Application II: Solutions of notch and crack problems of one-and two-dimensional quasicrystals . . . . . . . . . . . . . . . . . 127 8.1 Crack problem and solution of one-dimensional quasicrystals . . . . . . . . . . 128 8.2 Crack problem in ﬁnite-sized one-dimensional quasicrystals . . . . . . . . . . . 134 8.3 Griﬃth crack problems in point groups 5m and 10mm quasicrystals based on displacement potential function method . . . . . . . . . . . . . . . . . . . . . 139 8.4 Stress potential function formulation and complex variable function method for solving notch and crack problems of quasicrystals of point groups 5, ¯5 and 10, 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 8.5 Solutions of crack/notch problems of two-dimensional octagonal quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 8.6 Other solutions of crack problems in one-and two-dimensional quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 8.7 Appendix of Chapter 8: Derivation of solution of Section 8.1 . . . . . . . . . . 154 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 Chapter 9

Theory of elasticity of three-dimensional quasicrystals and its applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 9.1 Basic equations of elasticity of icosahedral quasicrystals . . . . . . . . . . . . . . . 160

9.2 Anti-plane elasticity of icosahedral quasicrystals and problem of interface between quasicrystal and crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 9.3 Phonon-phason decoupled plane elasticity of icosahedral quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 9.4 Phonon-phason coupled plane elasticity of icosahedral quasicrystals— displacement potential formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 9.5 Phonon-phason coupled plane elasticity of icosahedral quasicrystals— stress potential formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 9.6 A straight dislocation in an icosahedral quasicrystal . . . . . . . . . . . . . . . . . . . 173 9.7 An elliptic notch/Griﬃth crack in an icosahedral quasicrystal . . . . . . . . . 178 9.8 Elasticity of cubic quasicrystals—the anti-plane and axisymmetric deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Chapter 10 Dynamics of elasticity and defects of quasicrystals . . . . . . 191 10.1 Elastodynamics of quasicrystals followed the Bak’s argument . . . . . . . . 192 10.2 Elastodynamics of anti-plane elasticity for some quasicrystals . . . . . . . . 192 10.3 Moving screw dislocation in anti-plane elasticity . . . . . . . . . . . . . . . . . . . . . 194 10.4 Mode III moving Griﬃth crack in anti-plane elasticity . . . . . . . . . . . . . . . 197 10.5 Elasto-/hydro-dynamics of quasicrystals and approximate analytic solution for moving screw dislocation in anti-plane elasticity . . . . . . . . . 199 10.6 Elasto-/hydro-dynamics and solutions of two-dimensional decagonal quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 10.7 Elasto-/hydro-dynamics and applications to fracture dynamics of icosahedral quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 10.8 Appendix of Chapter 10: The detail of ﬁnite diﬀerence scheme . . . . . . . 221 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 Chapter 11 Complex variable function method for elasticity of quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 11.1 Harmonic and quasi-biharmonic equations in anti-plane elasticity of one-dimensional quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 11.2 Biharmonic equations in plane elasticity of point group 12mm twodimensional quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 11.3 The complex variable function method of quadruple harmonic equations and applications in two-dimensional quasicrystals . . . . . . . . . . 231 11.4 Complex variable function method for sextuple harmonic equation and applications to icosahedral quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . 243 11.5 Complex analysis and solution of quadruple quasiharmonic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 11.6 Conclusion and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 Chapter 12 Variational principle of elasticity of quasicrystals, numerical analysis and applications . . . . . . . . . . . . . . . . . . . . . . . 257 12.1 Basic relations of plane elasticity of two-dimensional quasicrystals . . . . 258 12.2 Generalized variational principle for static elasticity of quasicrystals . . . . 259 12.3 Finite element method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 12.4 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 Chapter 13 Some mathematical principles on solutions of elasticity of quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 13.1 Uniqueness of solution of elasticity of quasicrystals . . . . . . . . . . . . . . . . . . . 273

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13.2 Generalized Lax-Milgram theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 13.3 Matrix expression of elasticity of three-dimensional quasicrystals . . . . . 278 13.4 The weak solution of boundary value problem of elasticity of quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 13.5 The uniqueness of weak solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 13.6 Conclusion and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 Chapter 14 Nonlinear behaviour of quasicrystals . . . . . . . . . . . . . . . . . . . . . . 289 14.1 Macroscopic behaviour of plastic deformation of quasicrystals . . . . . . . . 290 14.2 Possible scheme of plastic constitutive equations . . . . . . . . . . . . . . . . . . . . . 292 14.3 Nonlinear elasticity and its formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 14.4 Nonlinear solutions based on simple models . . . . . . . . . . . . . . . . . . . . . . . . . . 295 14.5 Nonlinear analysis based on the generalized Eshelby theory . . . . . . . . . . 301 14.6 Nonlinear analysis based on the dislocation model . . . . . . . . . . . . . . . . . . . 305 14.7 Conclusion and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 14.8 Appendix of Chapter 14: Some mathematical details . . . . . . . . . . . . . . . . . 309 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 Chapter 15 Fracture theory of quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 15.1 Linear fracture theory of quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 15.2 Measurement of G IC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 15.3 Nonlinear fracture mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 15.4 Dynamic fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 15.5 Measurement of fracture toughness and relevant mechanical parameters of quasicrystalline material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 Chapter 16 Remarkable conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 Major Appendix: On some mathematical materials . . . . . . . . . . . . . . . . . . . . 333 Appendix I Outline of complex variable functions and some additional calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 A.I.1 Complex functions, analytic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 A.I.2 Cauchy’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 A.I.3 Poles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 A.I.4 Residue theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 A.I.5 Analytic extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 A.I.6 Conformal mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 A.I.7 Additional derivation of solution (8.2-19) . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 A.I.8 Additional derivation of solution (11.3-53) . . . . . . . . . . . . . . . . . . . . . . . . . . 342

Contents

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

A.I.10

On the calculation of integral (9.2-14) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

Appendix II Dual integral equations and some additional calculations . . . . . 348 A.II.1 Dual integral equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 A.II.2

Additional derivation on the solution of dual integral equations

(8.3-8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 A.II.3

Additional derivation on the solution of dual integral equations

(9.8-8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

Chapter 1 Crystals This book discusses mainly elasticity and defects of quasicrystals, however quasicrystals have inherent connection with crystals. This chapter provides the basic knowledge on crystals which may be beneﬁcial to study quasicrystals and relevant topics.

1.1

Periodicity of crystal structure, crystal cell

Based on X-ray diﬀraction patterns, it is known that a crystal consists of particles (i.e., collections of ions, atoms and molecules) which are arranged regularly in space. The arrangement is a repetition of the smallest unit, called a unit cell, resulting in the periodicity of a complete crystal. The frame of the periodic arrangement of centres of gravity of particles is called a lattice. Thus, the properties of corresponding points of diﬀerent cells in a crystal are the same. The positions of these points can be deﬁned by the radius vectors r and r in the coordinate frame e1 , e2 , e3 , and a, b and c are three non-mutually co-linear vectors, respectively (the general concept on vector refering to Chapter 2). Hence we have r = r + la + mb + nc, (1.1-1) in which a, b and c are the basic translational vectors describing the particle arrangement in a complete crystal, and l, m and n are arbitrary integers. If the physical properties are described by function f (r), the above invariance can be expressed mathematically as f (r ) = f (r + la + mb + nc) = f (r). (1.1-2) This is called the translational symmetry or long-range translational order of a crystal, because the symmetry is realized by the operation of translation. Formula (1.1-1) represents a kind of translational transform, while (1.1-2) shows that the lattice is invariant under transformation (1.1-1). The collection of all translational transform remaining lattice invariant constitutes the translational group.

1.2

Three-dimensional lattice types

Cells of lattice may be described by a parallel hexahedron having lengths of its three sides a, b and c and angles α, β and γ between the sides. According to the relationship of length of sides and angles there are seven diﬀerent forms observed for the cells, which form seven crystal systems given in Table 1.2-1. T. Fan, Mathematical Theory of Elasticity of Quasicrystals and Its Applications © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2011

2

Chapter 1 Table 1.2-1

Crystals

Crystals and the relationship of the length of sides and angles

Crystal system Triclinic Monoclinic Orthorhombic Rhombohedral Tetragonal Hexagonal Cubic

Characters of cell a = b = c, α = β = γ a = b = c, α = γ = 90◦ = β a = b = c, α = β = γ = 90◦ a = b = c, α = β = γ = 90◦ a = b = c, α = β = γ = 90◦ a = b = c, α = β = 90◦ , γ = 120◦ a = b = c, α = β = γ = 90◦

Among each crystal system there are certain classes of crystals that are classiﬁed based on the conﬁguration such that whether the face centre or body centre contains lattice point. For example, the cubic system can be classiﬁed as three classes: the simple cubic, body centre cubic and face centre cubic. According to this classiﬁcation, the seven crystal systems contain 14 diﬀerent lattice types, called Bravais cells as shown in Fig. 1.2-1.

Fig. 1.2-1

The 14 Crystal cells of three-dimension

(a) Simple triclinic, (b) Simple monoclinic, (c) Button centre monoclinic, (d) Simple orthorhombic, (e) Button centre orthorhombic, (f) Body centre orthorhombic, (g) Face centre orthorhombic, (h) Hexagonal, (i) Rhombohedral, (j) Simple tetragonal, (k) Body centre tetragonal, (l) Simple cubic, (m) Body centre cubic, (n) Face centre cubic.

Apart from the above mentioned 14 Bravais cells with three-dimensional lattices, there are 5 Bravais cells of two-dimensional lattice, we do not go any further.

1.3

1.3

Symmetry and point groups

3

Symmetry and point groups

In Section 1.1 we have discussed the translational symmetry of crystals. Here we point out that the symmetry reveals invariance of crystals under translational transformation T = la + mb + nc. (1.3-1) Formula (1.3-1) is referred to as an operation of symmetry, which is a translational operation. Apart from this, there are rotation operation and reﬂection (or mapping) operation, they belong to so-called point operation. A brief introduction on the rotation operation and orientational symmetry of crystals is given below. By rotating about an axis through a lattice, the crystal always returns to the original state since the rotational angles are 2π/1, 2π/2, 2π/3, 2π/4 and 2π/6 or integer times of these values. This is the orientational symmetry or the long-range orientational order of a crystal. Because of the constraint of translational symmetry, the orientational symmetry holds for n = 1, 2, 3, 4 and 6 only, which is neither equal to 5 nor more greater than 6 where n is the denominator of 2π/n (e.g. a molecule can have ﬁve-fold rotation symmetry, but a crystal cannot have this symmetry because the cells either overlap or have gaps between them when n = 5, Fig. 1.3-1 is an Fig.1.3-1 There is no ﬁve-fold rotational symmetry in crystals example). The fact constitutes the following fundamental law of crystallography: Law of symmetry of crystals Under rotation operation, n-fold symmetry axis is marked by n. Due to the constraint of translational symmetry, there are axes n = 1, 2, 3, 4 and 6 exist only, neither 5 nor number greater than 6 exists. In contrast to translational symmetry, rotation is a point symmetry. Other point symmetries are: plane of symmetry, the corresponding operation is mapping, marked by m; centre of symmetry, the corresponding operation is inversion, marked by I; rotation-inversion axis, the corresponding operation is composition of rotation and inversion, when the inversion after rotation 2π/n, marked by n ¯. For crystals, the point operation consists of eight independent ones only, i.e., 1, 2, 3, 4, 6,

¯ I = (I),

m = ¯2, ¯4,

(1.3-2)

which are the basic symmetric elements of point symmetry. The rotation operation is also denoted by Cn , n = 1, 2, 3, 4, 6. The mapping operation can also be expressed by σ. The horizontal mapping by mh and Sh , the vertical one by Sv .

4

Chapter 1

Crystals

The mapping-rotation is a composite operation, denoted by Sn , which can be understood as Sn = Cn σh = σh Cn . The inversion can be understood as I = S2 = C2 σh = σh C2 . Another composite operation—rotation-inversion n ¯ is related to Sn , e.g. ¯1 = S2 = I, ¯2 = S1 = σ, ¯3 = S6 , ¯4 = S4 , ¯6 = S3 . So that (1.3-2) can be redescribed as C1 , C2 , C3 , C4 , C6 , I, σ, S4 .

(1.3-3)

The collection of each symmetric operation among these eight basic operations constitutes a point group, the collection of their composition forms 32 point groups listed in Table 1.3-1. Table 1.3-1 Sign Cn I σ(m) Cnh Cnν Dn Dnh Dnd Sn T Th Td O

32 Point groups of crystals

Meaning of sign Having n-fold axis Symmetry central Mapping Having n-fold axis and horizontal symmetry plane Having n-fold axis and vertical symmetry plane Having n-fold axis and n 2-fold axes, they are perpendicular to each other Meaning of h is the same as before d means in Dn there is a symmetry plane dividing the angle between two 2-fold axes Having n-fold mapping-rotation axis Having four 3-fold axes and three 2-fold axes Meaning of h is the same as before Meaning of d is the same as previous Having three 4-fold axes which perpendicular to each other and six 2-fold axes and four 3-fold axes

Point group C1 , C2 , C3 , C4 , C6 I(i) σ(m)

Number 5 1 1

C2h , C3h , C4h , C6h

4

C2ν , C3ν , C4ν , C6ν

4

D2 , D3 , D4 , D6

4

D2h , D3h , D4h , D6h

4

D2d , D3d

2

S4 , S6 = C3i

2

T

1

Th

1

Td

1

O, Oh

2

Note: T = C3 D2 means the composition between operations C3 and D2 , where suﬃxes 3 denotes a 3-fold axis. O = C3 C4 C2 means the composition between operations C3 , C4 and C2 , where 3 represents a 3-fold axis, 2 represents a 2-fold axis.

1.4

Reciprocal lattice

5

The concept and sign of point groups will be used in the subsequent chapters.

1.4

Reciprocal lattice

The concept of the reciprocal lattice will be concerned in the subsequent chapters, here is a brief introduction. Assume that there is relation between the base vectors a1 , a2 and a3 of a lattice (L) and the base vectors b1 , b2 and b3 for another lattice (LR ) bi · aj = δij =

1, i = j, 0, i = j,

i, j = 1, 2, 3,

(1.4-1)

such that the lattice with the base vectors {b1 , b2 , b3 } is the reciprocal lattice LR of crystal lattice L with the base vectors {a1 , a2 , a3 }. Between bi and aj there exist the relationship b1 =

a2 × a3 , Ω

b2 =

a3 × a1 , Ω

b3 =

a1 × a2 , Ω

(1.4-2)

where Ω = a1 · (a2 × a3 ) is the volume of lattice cell. Denote Ω ∗ = b1 · (b2 × b3 ), then Ω∗ =

1 . Ω

The position of any point in the reciprocal lattice can be expressed by G = h1 b1 + h2 b2 + h3 b3

(1.4-3)

in which G called the reciprocal vector and h1 , h2 , h3 = ±1, ±2, · · · Points in the lattice can be described by a1 , a2 , a3 as well as by b1 , b2 , b3 . In a similar fashion the concept of the reciprocal lattice can be extended to the higher dimensional space, e.g. six-dimensional space, which will be used in the discussion in Chapter 4. The brief introduction above provides a background for reading the subsequent text of the book. Further information on the crystals, the diﬀraction theory, and the point groups can be found in books[1, 2]. We will recall the concepts in the following text.

6

1.5

Chapter 1

Crystals

Appendix of Chapter 1: Some basic concepts

Some basic concepts will be described in the following chapters, with which most of physicists are familiar. For the readers of non-physicists, a simple introduction is provided as follows; the details can be found in the relevant references. 1.5.1

Concept of phonon

In general, the course of crystallography does not contain the contents given in this section. Because the discussion here is dependent on quasicrystals, especially with the elasticity of quasicrystals, we have to introduce some of the simplest relevant arguments. In 1900, Planck put forward the quantum theory. Soon after Einstein developed the theory and explained the photo-electric eﬀect, which leads to the photon concept. Einstein also studied the speciﬁc heat cv of crystals by using the Planck quantum theory. There are some unsatisfactory points in the work of Einstein although his formula explained the phenomenon of cv = 0 at T = 0, where T denotes the absolute temperature. To improve Einstein s work, Debye[3] and Born et al. [4,5] applied the quantum theory to study the speciﬁc heat arising from lattice vibration in 1912 and 1913 respectively, and got a great success. The theoretical prediction is in excellent agreement to the experimental results, at least for the atom crystals. The propagation of the lattice vibration is called the lattice wave. Under the long-wavelength approximation, the lattice vibration can be seen as continuum elastic vibration, i.e., the lattice wave can be approximately seen as the continuum elastic wave. The motion is a mechanical motion, but Debye and Born assumed that the energy can be quantized based on Planck’s hypothesis. With the elastic wave approximation and quantization, Debye and Born successfully explained the speciﬁc heat of crystals at low temperature, and the theoretical prediction is consistent with the experimental results in all range of temperature, at least for the atomic crystals. The quanta of the elastic vibration, or the smallest unit of energy of the elastic wave is named phonon, because the elastic wave is one of acoustic waves. Unlike photon, the phonon is not an elementary particle, but in the sense of quantization, the phonon presents natural similarity to that of photon and other elementary particles, thus can be named quasi-particle. The concept created by Debye and Born opened the study on lattice dynamics, an important branch of solid state physics. Yet according to the view point at present, the Debye and Born theory on solid belongs to a phenomenological theory, though they used the classical quantum theory. Landau[6] further developed the phenomenological theory and put forward the concept of elementary excitation. According to his concept, photon and phonon etc.

1.5

Appendix of Chapter 1: Some basic concepts

7

belong to elementary excitations. In general one elementary excitation corresponds to a certain ﬁeld, e.g. photon corresponds to electro-magnetic wave, phonon corresponds to elastic wave, etc. The phonon concept is further extended by Born[5] ; he pointed out the phonon theory given by Debye corresponds to the vibration as molecule as a whole body, which is called phonetic frequency vibration modes, or the phonetic branch of phonon. In this case the physical quantity phonon describing displacement ﬁeld deviated from the equilibrium position of particles (atoms, or ions, or molecules) at lattice is also called as phonon ﬁeld, or brieﬂy phonon. Macroscopically it is the displacement vector u of elastic body (if the crystal is regarded as an elastic body). But Born emphasized that there is another vibration in crystals with compound lattice, i.e., the relative vibration between atoms within a molecule, which is called photonic frequency vibration modes, or photonic branch of phonon. For this branch the phonon cannot be understood simply as macroscopic displacement ﬁeld. But our discussion here is conﬁned to the frame work of continuum medium, with no concern with the photonic branch, so the phonon ﬁeld is the macro-displacement ﬁeld in the consideration. In many physical systems (classical or quantum systems) the motion presents the discrete spectrum (the energy spectrum or frequency spectrum, which corresponds to the discrete spectrum of an eigen-value problem of a certain operator from the mathematical point of view), the lowest energy (frequency) level state is called ground state, and that beyond the ground state is called excited state. The so-called elementary excitation induces a transfer from the ground state to the state with the smallest non-zero energy (or frequency). Strictly speaking, it should be named lowest energy (or frequency) elementary excitation. The solid state physics was intensively developed in 1960∼70’s, then evolved into the condensed matter physics. The condensed matter physics is not only extending the scope of solid state physics by considering the liquid state and micro-powder structure, but also developing the basic concepts and principles. Modern condensed matter physics is established as a result of the construction of its paradigm, in which the symmetry-breaking is in a central place, which was contributed by Landau[6] and Anderson[7] and other scientists. Considering the importance of the concept and principle of symmetry-breaking in development of elasticity of quasicrystals, we give a brief discussion here. It is well known that for a system with a constant volume, the equilibrium state thermodynamically requires the free energy of the system F = E − TS

(1.5-1)

be minimum, in which E is the internal energy, S is the entropy and T is the absolute temperature.

8

Chapter 1

Crystals

Landau proposed the so-called second order phase transition theory by introducing a macroscopic order parameter η to describe (order-disorder) phase transition, i.e., assuming that the free energy can be expanded as a power series of η, F (η, T ) = F0 (T ) + A(T )η 2 + B(T )η 4 + · · · ,

(1.5-2)

in which, according to the requirement of the stability condition of phase transition (i.e., the variational condition δF = 0 or ∂F/∂η = 0), the coeﬃcients of odd terms should be taken to zero, and B(T ) > 0. At high temperature the system is in a disorder state, so A(T ) > 0, too; as the temperature reduces A(T ) changes its sign; at the critic temperature TC there exists A(TC ) = 0. The simplest choice in satisfying these conditions is A(T ) = α(T − TC ),

B(T ) = B(TC ),

(1.5-3)

in which α is a constant. Without concerning concrete micro-mechanism, the Landau theory has the merits of simplicity and generality; it can be used to many systems and has achieved successes, especially for the study of superconductivity, liquid crystals,high energy physics (To the author’s understanding, the quasicrystals study is another area that has been achieved following the line of the symmetry-breaking principle). Applying the above principle to periodic crystals we have[7] F =

1 α(|G|)(T − TC (G))η 2 + higher-order terms, 2

(1.5-4)

where the constant α is related to the reciprocal vector G (the concepts on the reciprocal vector and the reciprocal lattice, refer to Section 1.4). Further, Anderson[7] showed that for crystals if the density of periodic crystals can be expressed by Fourier series (the expansion exists due to the periodicity of the structure in threedimensional lattice or the reciprocal lattice) ρ(r) =

G∈LR

ρG exp{iG · r} =

|ρG | exp{−iΦG + iG · r},

(1.5-5)

G∈LR

where G is a reciprocal vector, and LR the reciprocal lattice in three-dimensional space, ρG is a complex number, ρG = |ρG | e−iΦG

(1.5-6)

with the amplitude |ρG | and the phase angle ΦG , due to ρ(r) being real, |ρG | = |ρ−G | and ΦG = −ΦG , the order parameter is η = |ρG | .

(1.5-7)

1.5

Appendix of Chapter 1: Some basic concepts

9

Anderson pointed out that for crystals the phase angle ΦG contains the phonon u for crystal, i.e., ΦG = G · u,

(1.5-8)

in which both G and u are in three-dimensional physical space. If we consider only the phonetic branch of the phonon, then u can be understood as the displacement ﬁeld. So the displacement ﬁeld in periodic crystals can be understood as the phonon ﬁeld from the Landau symmetry-breaking hypothesis, though it possesses an intuitive physical meaning under the approximation of the long-wavelength (refer to Chapter 2). The description based on the symmetry-breaking, physical quantity u is connected to the reciprocal vector G and reciprocal lattice LR of crystals, so it presents more profound insight (a result of symmetry-breaking) than that of the intuitive description of the displacement ﬁeld u, though the explanation is still based on phenomenological theory (because the Landau theory is a phenomenological theory), rather than on the ﬁrst-principles. The concept of phonon is originated from Debye[3] and Born[4,5] , which describes the mechanical vibration of lattice mass points (atoms, or ions , or molecules) deviating from their equilibrium position. The propagation of the vibration leads to the lattice wave, and the motion can be quantized, the quanta is the phonon. This is an elementary excitation in condensed matter. The symmetry-breaking leads to appearance of new order phase (e.g. crystals), new order parameter (e.g. wave amplitude of mass density wave), the new elementary excitation (e.g. phonon), and the new conservation law (e.g. the crystal symmetry law given in Section 1.3). The above description on the phonon from the symmetry-breaking point of view helps us in understanding in-depth the physical nature of the phonon. Following this scheme, some elementary excitations (e.g. phason) temporarily without intuition meaning can also be explained by the Landau theory, and one can further ﬁnd out their physical meaning from the point of view of symmetry rather than from intuition, because for some complex phenomena, the simple intuition cannot give a complete and correct explanation. We recall that some of the elementary excitations are related to broken symmetry or symmetry-breaking, e.g. a liquid behaves arbitrary translational symmetry and arbitrary orientational symmetry, then the periodicity of crystals (i.e., the lattice) breaks the translational symmetry and the orientational symmetry of liquid, the phonon is elementary excitation, resulting from the symmetry-breaking. In this case we may say the quasicrystal is a result of symmetry-breaking of crystal, which will be discussed in Chapter 4. If description on phason concept is unnecessary, this section can be omitted. The reader can skip this section if not interested in.

10

1.5.2

Chapter 1

Crystals

Incommensurate crystals

In this book we do not discuss incommensurate phases, but quasicrystals are related with so-called incommensurate structure, we mention it in brief. Since 1960’s, incommensurate crystals have been studied by many physicists, see e.g. [8]. The incommensurate phase means that it is plus an additional incommensurate modulate at the basic lattice, in which the modulated ones may be displacements or compounds of atoms or arrangement of spin etc. As an example, if a modulated displacement λ is plus to a lattice with period a, and if λ/a is an rational number, the crystal becomes a super-structure with long period (which is the integer times of a); and if λ/a is an irrational number, the crystal becomes an incommensurate structure. In this case along the modulate direction the periodicity is lost. The modulation can be one-dimensional, e.g. Na2 CO3 , NaNO2 , etc, or two-dimensional, e.g. TaSe2 , quartz, etc, or three-dimensional, e.g. Fe1−x O, etc. In the incommensurate phases, the modulation is only a “perturbation” of another period of the basic lattice, the diﬀraction pattern of the basic lattice holds, i.e., the crystallography symmetry holds, so one calls the structure being the incommensurate crystals. It is noticed that there are new degrees of freedom in the phases, be named the phasons. Here the phason modes present long-wavelength propagation similar to that of the phonon modes. In Chapter 4 we shall study that the phason modes in the quasicrystals present quite diﬀerent nature, i.e., the motion of atoms exhibits discontinuous jumps rather than the long-wavelength propagation. In addition, in the quasicrystals there is non-crystallographic orientational symmetry, which is essentially diﬀerent from that of incommensurate crystals. 1.5.3

Aphomus(glassy structure)

The crystals are solid with long-range order due to regular atom arrangement. In contrast, there is a solid without order, but it has short-range order in scale within the atom size. This material is the aphomus, as a branch of the condensed matter physics.

References [1] Kittel C.Introduction to the Solid State Physics. New York: John Wiley & Sons, Inc, 1976 [2] Wybourne B G. Classical Group Theory for Physicists. New York: John Wiley & Sons, Inc, 1974 [3] Debye P. Die Eigentuemlichkeit der speziﬁschen Waermen bei tiefen Temperaturen. Arch de Gen´eve, 1912, 33(4): 256–258

References

11

[4] Born M, von K´ arm´ an Th. Zur Theorie der speziﬁschen Waermen, Physikalische Zeitschrift. 1913, 14(1): 15–19 [5] Born M, Huang K. Dynamic Theory of Crystal Lattices, Oxford: Clarendon Press, 1954 [6] Landau L D, Lifshitz M E. Theoretical Physics V: Statistical Physics. 3rd ed. Oxford: Pergamon Press, 1980 [7] Anderson P W. Basic Notions of Condensed Matter Physics. Menlo Park: BenjaminCummings, 1984 [8] Blinc B and Lavanyuk AP. Imcommensurate Phases in Dielectrics, I. II. Amsterdan: North Holand, 1986

Chapter 2 Framework of the classical theory of elasticity As the knowledge of crystals is a beneﬁt for understanding quasicrystals, it is worthwhile having a concise review of the classical theory of elasticity before learning the elasticity of quaicrystals. Here is a brief description to the theory. The detailed material for this theory can be found in many monographs and textbooks, e.g. Landau and Lifshitz[1] . Though the discussion here is limited within the framework of continuum medium mechanics, there are still connections to physical nature of the elasticity of crystals reﬂected by phonon concept (discussed in Section 1.5). The readers are advised to refer to the relevant chapters and sections of monographs of Born and Huang[2] and Anderson[3] which would help us understanding the phonon concept so the phason concept and elasticity of quasicrystals, which will be presented in the following chapters. The practice shows that it would be hard to understand phason concept and the phason elasticity if we limited our knowledge only within the classical continuum medium and complete intuition. For simplicity, the tensor algebra will be used in the text.

2.1 2.1.1

Review on some basic concepts Vector

A quantity with both magnitude and direction is named vector, denoted by a, and a = |a| represents its magnitude. The scalar product is deﬁned as a · b = ab cos(a, b) of two vectors a and b. The vector product is deﬁned as a × b = n ab sin (a, b), in which n is the unit vector perpendicular to both a and b, so |n| = 1. A more general deﬁnition on vector is given later. 2.1.2

Coordinate frame

To describe the vector and the tensor, it is convenient to introduce the coordinate frame. We will consider the orthogonal frame. Assume that e1 , e2 and e3 are three unit vectors and mutually perpendicular, i.e., e1 · e2 = 0, e2 · e3 = 0, e3 · e1 = 0 T. Fan, Mathematical Theory of Elasticity of Quasicrystals and Its Applications © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2011

14

Chapter 2

Framework of the classical theory of elasticity

and e3 = e1 × e2 , e2 = e3 × e1 , e1 = e2 × e3 , then they are the base vectors of an orthogonal coordinate frame, and often called base vectors brieﬂy. In the orthogonal coordinate frame e1 , e2 , e3 , any vector a can be expressed by a = a1 e1 + a2 e2 + a3 e3 or a = (a1 , a2 , a3 ). 2.1.3

(2.1-1)

Coordinate transformation

Considering another orthogonal frame e1 , e2 , e3 which can be expressed in terms of e1 , e2 , e3 . From (2.1-1), we have e1 = c11 e1 + c12 e2 + c13 e3 , e2 = c21 e1 + c22 e2 + c23 e3 , e3 = c31 e1 + c32 e2 + c33 e3 ,

(2.1-2)

where c11 , c12 , · · · , c33 are some scalar constants. The relation (2.1-2) is the coordinate transformation, which can be expressed in the matrix form ⎡ ⎤ ⎡ ⎤ e1 e1 ⎢ ⎥ ⎢ ⎥ (2.1-3) ⎣ e2 ⎦ = [C] ⎣ e2 ⎦ , e3 e3 where

⎡

c11 ⎢ c [C] = ⎣ 21 c31

c12 c22 c32

⎤ c13 c23 ⎥ ⎦, c33

which is an orthogonal matrix, consequently [C]T = [C]−1 ,

(2.1-4)

here notation “T” marks transpose operation, and “−1” the inversion operation. It follows, ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ e1 e1 e1 ⎢ e ⎥ T ⎢ e ⎥ −1 ⎢ e ⎥ (2.1-5) ⎣ 2 ⎦ = [C] ⎣ 2 ⎦ = [C] ⎣ 2 ⎦ . e3 e3 e3 Based on (2.1-1), in the frame e1 , e2 , e3 , a = a1 e1 + a2 e2 + a3 e3 Substituting (2.1-5) into (2.1-1) yields a = (c11 a1 + c12 a2 + c13 a3 )e1 + (c21 a1 + c22 a2 + c23 a3 )e2

(2.1-6)

2.1

Review on some basic concepts

15

+ (c31 a1 + c32 a2 + c33 a3 )e3 .

(2.1-7)

It follows that by the comparison between (2.1-6) and (2.1-7) a1 = c11 a1 + c12 a2 + c13 a3 , a2 = c21 a1 + c22 a2 + c23 a3 , a3 = c31 a1 + c32 a2 + c33 a3 ,

(2.1-8)

or in the matrix expression, i.e., ⎡

⎤ ⎡ ⎤ a1 a1 ⎣ a2 ⎦ = [C] ⎣ a2 ⎦ . a3 a3

(2.1-8 )

Whatever (2.1-8) or (2.1-8 ), there is ai =

3

cij aj = cij aj .

(2.1-9)

j=1

The summation sign in the right-handside of (2.1-9) is omitted, when the repeated indexes in cij aj represent summing. Henceforth, the summation convention will be used throughout. A set of number (a1 , a2 , a3 ) satisfying the relation (2.1-9) under the linear transformation (2.1-2) is a vector regardless its physical meaning. This is an algebraic deﬁnition of vector; it is more general than saying that the vector has both magnitude and direction. 2.1.4

Tensor

Let us deﬁne nine numbers in the orthogonal frame e1 , e2 , e3 as A: ⎡ ⎤ A11 A12 A13 A = ⎣ A21 A22 A23 ⎦ , A31 A32 A33

(2.1-10)

in which the components satisfy the relation Akl =

3

cki clj Aij = cki clj Aij

(2.1-11)

i,j=1

under the linear transformation, then A is a tensor of rank 2, where cij are given by (2.1-3), and the summation sign is omitted in the right-handside of (2.1-11). It is evident that the concept of tensor is an extension of that of vector. According to the deﬁnition Aij represents a tensor where i=1, 2, 3, j=1, 2, 3. It is understood that it represents a component with the indexes i and j of the tensor.

16

Chapter 2

2.1.5

Framework of the classical theory of elasticity

Algebraic operation of tensor

1) Unit tensor

I = δij =

0, i = j, 1, i = j,

(2.1-12)

which is named the Kronecker delta conventionally. 2) Transpose of tensor ⎡

A11 AT = ⎣ A12 A13

A21 A22 A23

⎤ A31 A32 ⎦ . A33

(2.1-13)

3) Algebraic sum of a tensors A ± B = Aij ± Bij .

(2.1-14)

4) Product of a tensor by a scalar mA = mAij .

(2.1-15)

AB = Aij Bkl .

(2.1-16)

5) Product of tensors

Other operations about tensors will be provided where required.

2.2

Basic assumptions of theory of elasticity

The theory of linear elasticity is a branch of continuum mechanics, it follows the basic assumptions thereof, whereas there are 1) Continuity In the theory one assumes that the medium is ﬁlled the full space that it occupies, this means the medium is continuous. Connected with this, the ﬁeld variables concerning the medium are continuous and diﬀerentiable functions of coordinates. 2) Homogeneity Physical constants describing the medium are independent of coordinates, so the medium is homogeneous. 3) Small deformation Assume that displacements ui small and ∂ui /∂xj much less than unit. Due to small deformation, the boundary conditions are expressed with reference to the boundaries before deformation though those boundaries have undergone deformation. This makes the problems to be linearized and simpliﬁes the solution procedure.

2.3

2.3

Displacement and deformation

17

Displacement and deformation

That elastic body exhibits deformation is connected to the relative displacement between points in it. So we ﬁrst look for the displacement ﬁeld. Consider a region R in an elastic body, refer to Fig. 2.3-1, it turns into another region R after deformation. The point O with radius vector r, before deformation, becomes point O with radius vector r after deformation, and u is the displacement vector of point O during the deformation process (see Fig.2.3-1) i.e.,

Fig. 2.3-1

Displacement of a point in an elastic body

r = r + u

(2.3-1)

u = r − r = xi − xi .

(2.3-1 )

or In Fig. 2.3-1, e1 , e2 , e3 depicts any orthogonal coordinate system, especially we use the rectilinear coordinate system (x1 , x2 , x3 ) or (x, y, z). Assume that O1 in R is a point near the point O, the radius vector connecting them is dr = dxi . The point O1 becomes point O1 in R after deformation. The radius vector connecting points O1 and point O1 is dr = dxi = dxi + dui . The displacement of point O1 is u , thus u = u + du, (2.3-2) i.e.,

dui = ui − ui

and dui =

∂ui dxj . ∂xj

(2.3-3) (2.3-4)

(2.3-4) expresses the Taylor expansion at point O and takes the ﬁrst order term only. Under the small deformation assumption, this reaches a very high accuracy. It denotes

18

Chapter 2

Framework of the classical theory of elasticity

∂ui = εij + ωij , ∂xj in which εij =

1 2

1 ωij = 2

∂ui ∂uj + ∂xj ∂xi ∂uj ∂ui − ∂xj ∂xi

(2.3-5) ,

(2.3-6)

,

(2.3-7)

here εij is a symmetric tensor εij = εji

(2.3-8)

and called the strain tensor, while ωij an asymmetric tensor, which has only three independent components ∂uz 1 ∂uy − , Ωx = ωyz = 2 ∂z ∂y 1 ∂uz ∂ux Ωy = ωzx = (2.3-9) − , 2 ∂x ∂z 1 ∂ux ∂uy − . Ωz = ωxy = 2 ∂y ∂x The physical meaning of εij describes the volume and shape change of a cell, and that of ωij the rigid-body rotation, which is independent of deformation. Henceforth, it suﬃces to consider εij . The components ε11 , ε22 and ε33 (if denote x = x1 , y = x2 , z = x3 , then we have εxx , εyy and εzz ) represent normal strains, describing the volume change of a cell, while ε32 = ε23 , ε13 = ε31 and ε12 = ε21 (or εyz = εzy , εzx = εxz and εxy = εyx ) represent shear strains, describing the shape change of a cell.

2.4

Stress analysis and equations of motion

The stress in the internal forces per unit area due to deformation denoted it by σij , it is zero if there is no deformation. When the body is in static equilibrium, the equilibrium equations can be derived from the law of momentum conservation as follows: ∂σij + fj = 0, (2.4-1) ∂xi which holds for any inﬁnitesimal volume element of the body, σij represents the components of the stress tensor as mentioned above, and suﬃx j the acting direction of the component, i the direction of outward normal vector of the surface element on which the stress component exerted, fi the body force density vector. Among all, the components of σij , σxx , σyy and σzz are normal to the surface elements which they exerted, and σyz , σzy , σzx , σxz , σxy and σyx are along the tangent directions

2.5

Generalized Hooke’s law

19

of the surface elements, the former are called normal stresses, and the latter shear stresses. According to the angular momentum conservation, one ﬁnds that σij = σji ,

(2.4-2)

this means the stress tensor is a symmetric tensor, and (2.4-2) is named the shear stress mutual equal law. The external surface forces density (tractions) Ti subjected to the surface of a body should be balanced with the internal stresses, this leads to σij nj = Ti ,

(2.4-3)

where nj is the unit vector along the outward normal to the surface element. People also call Ti area force density. Equation (2.4-3) describes the stress boundary conditions which play a very important role for elasticity.

2.5

Generalized Hooke’s law

Between the stresses σij and the strains εij , there exists a certain relationship depended upon the material behaviour of the body. Hereafter we consider only the linear elastic behaviour of materials, and the state without initial stresses. In the case, the classical experimental law——Hooke’s law can be generalized as σij =

∂U = Cijkl εkl , ∂εij

(2.5-1)

in which U denotes the free energy, or the strain energy density, i.e., U =F =

1 Cijkl εij εkl 2

(2.5-2)

and Cijkl is the elastic constant tensor, consisting of 81 components. Due to the symmetry of σij and εij , each of them has 6 independent components only, such that the independent components of Cijkl reduce to 36. Formula (2.5-2) shows that U is a quadratic function of εij , considering the symmetry of εij , then we have Cijkl = Cklij ,

(2.5-3)

so the independent components 36 reduce to 21. The relation (2.5-1) with 21 independent elastic constants is named generalized Hooke’s law. The generalized Hooke’s law describes anisotropic elastic bodies including crystals. Stress and strain tensors can also be expressed by corresponding vectors with

20

Chapter 2

Framework of the classical theory of elasticity

6 independent elements, then can be denoted by the corresponding elastic constants matrix [Bijkl ], ⎤ ⎡ ⎤⎡ ⎤ ⎡ B1122 B1133 B1123 B1131 B1112 B1111 ε11 σ11 ⎢ ⎥ ⎢ σ22 ⎥ ⎢ B2222 B2233 B2223 B2231 B2212 ⎥ ⎥ ⎢ ⎥ ⎢ ε22 ⎥ ⎢ ⎥ ⎢ ⎢ σ33 ⎥ ⎢ B3333 B3323 B3331 B3312 ⎥ ⎢ ε33 ⎥ ⎥ ⎢ ⎥. ⎢ ⎢ ⎥ ⎢ σ23 ⎥ = ⎢ B2323 B2331 B2312 ⎥ ⎥ ⎢ ⎥ ⎢ ε23 ⎥ ⎢ ⎦ ⎣ ⎣ σ31 ⎦ ⎣ (symmetry) B3131 B3112 ε31 ⎦ σ12 B1212 ε12 (2.5-4) Applying formula (2.5-4) to crystals, between the elements Cijkl (or Bijkl ) there are some relations by considering certain symmetry of the crystals, so that, the resulting number of the elastic constants for certain individual crystal systems may be less than 21. In the following we give a brief discussion on the argument. 1) Triclinic system (classes 1 or C1 and Ci ) The triclinic symmetry does not add any restrictions to the components of tensor Cijkl (or Bijkl in (2.5-4)), however, appropriate choice of the coordinate system enables us to reduce the number of non-zero independent elastic constants. Because the orientation of the coordinate system is determined by three rotation angles, this provides three conditions to restrict some components in Cijkl (or Bijkl in (2.5-4)); for example. one can take three of them to be zero, such that, the triclinic crystal system has 18 components of elastic moduli. 2) Monoclinic system (classes Cs , C2 and C2h ) In the class Cs , there is plane of symmetry, we take it as x3 = 0(z = 0) in coordinate system {e1 , e2 , e3 }. Making a coordinate transformation with this plane of symmetry one can obtain a new coordinate system {e1 , e2 , e3 } . Between these two coordinate systems, there are relations e1 = e1 ,

e2 = e2 ,

e3 = −e3 .

(2.5-5)

This operation is the reﬂection or mapping. In addition, we know that between σij in e1 , e2 , e3 and σij in e1 , e2 , e3 , there are (refer to Section 2.1) σkl = αkj αli σji ,

(2.5-6)

in which αij are the coeﬃcients of linear transformation, i.e., ei = αij ej .

(2.5-7)

Under the transformation (2.5-5), there are α11 = 1, α22 = 1, α33 = −1,

others = 0.

(2.5-8)

2.5

Generalized Hooke’s law

21

Therefore, under the transformation, for Cijkl in (2.5-1) (or Bijkl in (2.5-4)) whose suﬃxes containing 3 with an odd number of times (1 or 3) will change sign, while the others will remain invariant. Considering the symmetry of the crystal, however, the physical properties including Cijkl (or Bijkl in (2.5-4)) should remain unchanged under symmetric operation (including the reﬂection). So it is obvious that all components with an odd number of suﬃxes 3 must vanish, i.e., B1123 = B1131 = B2223 = B2231 = B3323 = B3331 = B2312 = B3112 = 0.

(2.5-9)

Consequently, there are only 13 independent elastic constants. A similar discussion can be done for the classes C2 and C2h . 3) Orthorhombic system (classes C2v , D2 and D2h ) This crystal system has a macroscopic correspondence, i.e., the orthotropic materials, in which there exist two planes of symmetry perpendicular to each other. Let us take x3 = 0 and x1 = 0 as the planes. If on reﬂection in plane x3 = 0, it is just the case for monoclinic system mentioned above. Subsequently consider the mapping in plane x1 = 0, between the new and old coordinate systems, there is the relation such as ⎤⎡ ⎤ ⎡ ⎤ ⎡ −1 0 0 e1 e1 ⎣ e2 ⎦ = ⎣ 0 1 0 ⎦ ⎣ e2 ⎦ . e3 e3 0 0 1 By a similar description to that for monoclinic system, one ﬁnds that B1112 = B2212 = B3312 = B2331 = 0.

(2.5-10)

Collecting to (2.5-9), the system contains 9 independent elastic constants. 4) Tetragonal system (classes C4v , D2d , D4 and D4h ) This crystal system has 4 axes of symmetry. Similar to previous discussion, that independent elastic moduli are B1111 ,

B3333 ,

B1122 ,

B1212 ,

B1133 ,

B1313 .

The total number of them is six. 5) Rhombohedral system (classes C3v , 3 or C3 , D3 , D3d and S6 ) In this system there is a third-order axis of symmetry (or three-fold symmetric axis). We can take axis of symmetry as the axis e3 , after a lengthy derivation that six independent elastic constants are as follows B3333 ,

Bξηξη ,

Bξξηη ,

Bξη33 ,

Bξ3η3 ,

with ξ = x1 + ix2 ,

η = x1 − ix2 .

Bξξξ3

22

Chapter 2

Framework of the classical theory of elasticity

The moduli can also be written in conventional version as B3333 ,

B1212 ,

B1122 ,

B1323 ,

B1113 .

6) Hexagonal system (class C6 ) The crystal system has a macroscopic correspondence—the transversely isotropic material, whose elasticity presents fundamental importance to elasticity of one- and two-dimensional quasicrystals. There is a sixth-order axis of symmetry (or say six-fold symmetric axis) in the system. Take this axis as x3 -axis, and use the coordinate substitution ξ = x1 + ix2 , η = x1 − ix2 . In a rotation with angle 2π/6 about the x3 -axis, the new coordinates ξ and η are transformed by ξ → ξei2π/6 , η → ηe−i2π/6 . Then one can see that only those components Cijkl (or Bijkl in (2.5-4)) do not vanish which have the same number of suﬃxes ξ and η. These are B3333 ,

Bξηξη ,

Bξξηη ,

Bξη33 ,

Bξ3ηξ ,

or in conventional expressions, C1111 = C2222 , C3333 ,

C2323 = C3131 , C1122 ,

C1133 = C2233 , C1212 ,

in which 2C1212 = C1111 −C1122 , so the number of independent elastic constants is 5. 7) Cubic system (clases T, O) For this system there are 3 four-fold symmetric axes, in which there is tetragonal symmetry. If taking the four-fold symmetric axis of the tetragonal symmetry in the x3 -direction, the number of independent components of Cijkl (or Bijkl in (2.5-4)) are B1111 , B1122 , B1212 . 8) Isotropic body In this case there are two elastic moduli, e.g. the Young’s modulus and Poisson’s ratio, E, ν, respectively, or the Lam´e constants λ=

νE , (1 + ν)(1 − 2ν)

μ=

E , 2(1 + ν)

(2.5-11)

or the bulk modulus of compression and shear modulus K=

E E ,μ = = G. 3(1 − 2ν) 2(1 + ν)

In this case the generalized Hooke’s law presents very simple form, i.e., σij = 2μεij + λεkk δij ,

(2.5-12)

where εkk = ε11 + ε22 + ε33 = εxx + εyy + εzz , δij is the unit tensor. An equivalent form of (2.5-12) is 1+ν ν σij − σkk δij , (2.5-13) εij = E E in which σkk = σ11 + σ22 + σ33 = σxx + σyy + σzz .

2.6

2.6

Elastodynamics, wave motion

23

Elastodynamics, wave motion

When the inertia eﬀect is considered in (2.4-1), then it becomes ∂σij ∂ 2 ui + fi = ρ 2 , ∂xj ∂t

(2.6-1)

where ρ is the mass density of the material. Considering an isotropic medium and omitting body forces, from (2.6-1), (2.3-6) and (2.5-12) the equations of wave motion are obtained as (c21 − c22 ) where c1 and c2 deﬁned by

c1 =

∂ 2 ui ∂ 2 uj ∂ 2 uj + c22 = , ∂xi ∂xj ∂x2i ∂t2

λ + 2μ ρ

12 ,

12 μ c2 = , ρ

(2.6-2)

(2.6-3)

which are speeds of elastic longitudinal and transverse waves respectively. If put u = ∇φ + ∇ × ψ,

(2.6-4)

then (2.6-2) can be reduced to ∇2 φ =

1 ∂2φ , c21 ∂t2

∇2 ψ =

1 ∂2ψ , c22 ∂t2

where φ is the scalar potential, and ψ the vector potential, and ∇2 =

(2.6-5) ∂2 ∂2 + 2+ 2 ∂x ∂y

∂2 , (2.6-5) are typical wave equations of mathematical physics. To solve the prob∂z 2 lem apart from the boundary conditions one needs initial conditions, i.e., ui (xi , 0) = ui0 (xi ), u˙ i (xi , 0) = u˙ i0 (xi ),

2.7

xi ∈ Ω .

Summary

The classical theory of elasticity is concluded to solve the following initial-boundary value problem 1 ∂ui ∂uj , εij = + 2 ∂xj ∂xi ∂σij ∂ 2 uj = ρ 2 − fj , ∂xi ∂t

t > 0, xi ∈ Ω ,

σij = Cijkl εkl , ui (xj , 0) = ui0 (xj ) xj ∈ Ω (Initial condition) u˙ i (xj , 0) = u˙ i0 (xj )

24

Chapter 2

Framework of the classical theory of elasticity

σij nj = Ti , t > 0, xj ∈ Γt ui = u ¯i , t > 0, xj ∈ Γu

(Boundary condition)

¯i are known functions, Ω denotes the region of mawhere ui0 (xj ), u˙ i0 (xj ), Ti and u terials we studied, Γt and Γu are parts of boundary Γ on which the tractions and ∂ 2 uj = 0, the prob∂t2 lem reduces to a static problem as pure boundary value problem, there are no initial conditions at all. displacements are prescribed respectively and Γ = Γt + Γu . If

References [1] Landau L D, Lifshitz E M. Theoretical Physics V: Theory of Elasticity. Oxford: Pergamon Press, 1986 [2] Born M, Huang K. Dynamic Theory of Crystal Lattices. Oxford: Clarendon Press, 1954 [3] Anderson P W. Basic Notions of Condensed Matter Physics. Menlo Park: BenjaminCummings, 1984

Chapter 3 Quasicrystal and its properties The observation of quasicrystal in 1982 was a dramatic event and is an important discovery in the development history of physics. This has given rise to a great deal of attention of researchers to the unusual properties of the new structure of solid. Due to the thermodynamic stability of the most quasicrystals produced to date, which become a novel material. It is necessary to give a brief discussion on the structural characters and physical properties of quasicrystals for understanding their elasticity and defects which are the main focus of the following chapters.

3.1

Discovery of quasicrystal∗

The ﬁrst observation of quasicrystal was done in April 1982 while D. Shechtman was working in the Bureau of Standards in USA as a guest scholar. He observed from the electronic microscopy that a rapid cooled Al-Mn alloy exhibits ﬁve-fold orientational symmetry by the diﬀraction patterns with bright diﬀraction spots and sharp Bragg reﬂections, as shown in Fig. 3.1-1. Because the ﬁve-fold orientational symmetry is of contradiction with the basic law of symmetry of crystals (refer to Chapter 1), the result could not be understood within the ﬁrst two year since the discovery. I. Blech in Israel a colleague of Shechtman, gave him a powerful support, explaining that it might be an icosahedral glass. They drafted a paper concerning the experimental results and submitted to a journal, but was rejected. Then they submitted it to another journal, which could not be published too. J. W. Cahn, the hosting scientist in the Bureau of Standards recommended streamlining the paper; leaving out details of the model and experiment, and limiting it solely the experimental ﬁndings. After consulting with D. Gratias, a mathematical crystallographer at the Centre National de la Recherche Scientiﬁque in France, the group submitted an abbreviated article to Physical Review Letters (PRL) in October 1984, more than two years after Shechtman’s initial experiment. The article was published several weeks later. This is the Ref. [1]. After four weeks, Levine and Steinhardt[2] published their work by PRL to intro∗ This is referred to article PRL Top 10: #8 of APS.

T. Fan, Mathematical Theory of Elasticity of Quasicrystals and Its Applications © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2011

26

Chapter 3

Quasicrystal and its properties

duce ordered structure with quasiperiodicity and called the novel alloy as “quasicrystal” formally, in which their theoretical (computed) diﬀraction pattern in excellent agreement with that of the experimental observation. Soon after, other groups, for example, Ye et al[3] , Zhang et al[4] etc. also found similar structure of ﬁve-fold symmetry and icosahedral quasicrystal in Ni-V and Ni-Ti alloys. The icosahedral quasicrystals are one of three-dimensional quasicrystals, in which the atomic arrangement is quasiperiodic in three directions. Another three-dimensional quasicrystal is cubic quasicrystal observed by Fung et al[5] later.

Fig. 3.1-1

The patterns of diﬀraction of icosahedral quasicrystal

(a) The ﬁve-fold symmetry, (b) the stereographic structure of the quasicrystal.

Successively two-dimensional quasicrystals were observed. Here, the atomic arrangement is quasiperiodic along two directions and periodic along the third direction, which is just the directions of ﬁve-, eight-, ten- and twelve-fold symmetrical axes of the two-dimensional quasicrystals observed to date. Such that one ﬁnds four kinds of two-dimensional quasicrystals with ﬁve-, eight-, ten- and twelve-fold rotation symmetries which are also called pentagonal, octagonal, decagonal and dodecagonal quasicrystals, respectively (see Bendersky[6] , Chatopadihyay et al[7] , Fung et al[8] , Urban et al[9] , Wang et al[10] , Li et al[11] ). There is another class of quasicrystals, the one-dimensional quasicrystals, in which the atomic arrangement is quasiperiodic along one axis, and periodic along the plane perpendicular to the axis (see Merdin et al[12] , Hu et al[13] ,Fung et al[14] , Teranchdi et al[15] , Chen et al[16] , and Yang et al[17] ). The discovery of this novel matter with long-range order and non-crystallographic symmetry changes the traditional concept of classifying solids into two classes: crystal and noncrystal, and gives a strong impact on traditional crystallography, and brings new physical idea into the matter structure and symmetry.

3.2

Structure and symmetry of quasicrystals

27

The unusual structure of quasicrystals leads to a series of properties diﬀerent from those of crystals, which has created a great deal of attention by researchers in a range of ﬁelds, such as physics, crystallography, chemistry etc. A decade before the discovery, Penrose[18] put forward a mathematical model of quasicrystals, afterward called Penrose tiling, in which a tiling without an overlap or a gap in two diﬀerent rhombohedra can result in the quasiperiodic structure. After the discovery of quasicrystals, Penrose tiling has become a geometry tool of the new solid phase. As an example, a Penrose tiling for describing a two-dimensional ﬁve-fold symmetry is shown in Fig. 3.1-2, in which the local structure of the tiling is similar, this is called local isomorphism (LI). The quasiperiodic symmetry and local isomorphism of the Penrose tiling present signiﬁcance for describing the new solid phase—quasicrystals.The discovery of quasicrystal physically gave rise to the development of the Penrose geometry and the relevant discrete geometry, and the development of ergodic theory, group theory, Fourier analysis etc. Quasicrystals with thermodynamical stability are becoming a new class of functional and structural materials, which have many poFig. 3.1-2 Penrose tiling of tential engineering applications. They gentwo-dimensional quasicrystal with erate the study on physical and mechanical ﬁve-fold symmetry properties of the material. The elasticity and defects are of important topics among the mechanical behaviour of quasicrystals. These provide many new challenges as well as opportunities to the continuum mechanics.

3.2

Structure and symmetry of quasicrystals

The quasicrystals are diﬀerent from the periodic crystals. With a certain symmetry, it is one of kind of aperiodic crystals. The unusual characters of the quasicrystals are originated from their special atomic constitution. The character of this structure is explored by diﬀraction patterns. Just through these diﬀraction patterns people discovered diﬀerences between quasicrystals and crystals, so did the quasicrystals. Similar to other aperiodic crystals, quasiperiodicity induces new degrees of freedom, which can be explained as follows. In crystallography and solid state physics, the Miller indices (h, k, l)are often used to describe the structure of crystals. These indices can explain the spectra of diﬀraction patterns of all crystals. In Chapter 1 we

28

Chapter 3

Quasicrystal and its properties

mentioned that the number of base vectors for crystal N is identical to the number of the dimensions d of the crystal, i.e., N = d. However, because quasicrystals have quasiperiodic symmetry (including both or either quasiperiodic translational and orientational symmetries disallowed by the rule of crystallography), the Miller indices cannot be used and instead we need to employ six indices (n1 , n2 , n3 , n4 , n5 , n6 ). This feature implies, it is necessary to introduce higher dimensional (four-, or ﬁveor six-dimensional) spaces to characterize the symmetry of quasicrystals. This idea is identical to that of group theory, i.e., the quasiperiodic structure is periodic in higher dimensional space (four-, or ﬁve- or six-dimensional space). Quasicrystals in the real three-dimensional space (physical space) may be seen as a projection of a periodic lattice in the higher dimensional space (mathematical space). The projection of the periodic lattice at four-, ﬁve-, and six-dimensional space to the physical space generates one-, two- and three-dimensional quasicrystals, respectively. The six-dimensional space is denoted by E 6 , which consists of two sub-spaces, one is the physical space, called the parallel space and denoted by E3 , another is the comple3 , so that mentary space also called the vertical space and denoted by E⊥ 3 E 6 = E3 ⊕ E⊥ ,

(3.2-1)

where ⊕ denotes the direct sum. For one-, two- and three-dimensional quasicrystals, the number of the base vectors is N = 4, 5, 6; the number of the realistic dimension of the material (in physical space) is d = 3, so N > d, this is diﬀerent from that of crystals. The method of group theory is the most appropriate method to describe the symmetry of quasicrystals. The one-dimensional quasicrystals have 31 point groups, consisting of 6 quasicrystal systems and 10 Laue classes, in which the all the point groups are crystallographic point groups listed in Table 3.2-1 . Table 3.2-1

The systems, Laue classes and point groups of one-dimensional quasicrystals

System Triclinic

Laue class 1

Point group 1, 1

Monoclinic

2 3

2, mh , 2/mh 2h , m, 2h /m

Orthorhombic

4

2h 2h 2, mm2, 2h mmh , mmmh

Tetragonal

5 6

4, 4, 4/mh 42h 2h , 4mm, 4/mh , 4/mh mm

Rhombohedral

7 8

3, 3 32h , 3m, 3m

Hexagonal

9 10

6, 6, 6/mh 62h 2h , 6mm, 6m2h , 6/mh mm

3.3

A brief introduction on physical properties of quasicrystals

29

The two-dimensional quasicrystals have 57 point groups, in which 31 are crystallographic point groups listed in Table 3.2-1, and other 26 are non-crystallographic point groups listed in Table 3.2-2. The three-dimensional quasicrystals have 60 point groups. They are: (1) 32 crystallographic point groups and 28 non-crystallographic point groups, i.e., icosahedral point groups (235, m¯3¯5) and 26 point groups with 5-,8-,10-and12-fold symmetries (5, 5,52, 5m, 5m, and N ,N ,N/m, N 22, N mm, N m2 , N/mmm, N = 8, 10, 12), the latter have been listed in Table 3.2-2. Table 3.2-2

The systems, Laue classess and point groups of

non-crystallography of two-dimensional quasicrystals

3.3

System Pentagonal

Laue class 11 12

Point groups 5, 5 5m, 52, 5m

Decagonal

13 14

10, 10, 10/m 10mm, 1022, 10m2, 10/mmm

Octagonal

15 16

8, 8, 8/m 8mm, 822, 8m2, 8/mmm

Dodecagonal

17 18

12, 12, 12/m 12mm, 1222, 12m2, 12/mmm

A brief introduction on physical properties of quasicrystals

The unusual structure of quasicrystals leads to some new physical properties of the material. The mechanical behaviour of quasicrystals, especially, the distinct features of elasticity to those of crystals in particular, have aroused a great deal of research interest. These will be discussed in detail starting from Chapter 4. We do not talk about them any further here. The thermal properties of quasicrystals are a ﬁeld that attracts attention[19∼27] . The thermal conductivity of quasicrystals is lower than that of the conventional metals. Among the profound properties of quasicrystals being studied, the ﬁrst is the structural and elastic properties, the second is the properties of electricity of the material. The conductivity of electricity of quasicrystals is lower. The Hall eﬀect[28∼32] was well studied. The absolute number of the Hall coeﬃcient RH is two orders of magnitude greater than that of the conventional metals. In addition, the pressureresistance properties of quasicrystals have also been discussed, see, e.g. [33]. The light conductivity rate of quasicrystals is quite diﬀerent from that of the conventional metals, the singularity is of interest, see [34∼36]. Recently the study of

30

Chapter 3

Quasicrystal and its properties

quasicrystals photonic crystal study becomes a focus[37∼39] , with a trend of further development[40∼42] . The electronic structure of quasicrystals and relevant topics have also been concerned after the discovery[43∼45] . Due to lack of periodicity, the Bloch theorem and Brillouin zone concept cannot be used. By some simple models and through numerical computation, one can obtain results on the electronic energy spectra. Some results on wave functions obtained exhibit behaviour neither the in extending state nor in the localization state. For some quasicrystalline materials, e.g. Al-Cu-Li, Al-Fe etc, there are pseudogaps when energy is over the Fremi energy. Because the present book discusses elasticity of quasicrystals only, the other properties will not be dealt with. The above descrption provides a very simple introduction only.

3.4

One-, two- and three-dimensional quasicrystals

It is needed to recall the concept on one-, two- and three-dimensional quasicrystals. The one-dimensional quasicrystals are the ones in which the atom arrangement is quasiperiodic in one direction, and periodic in other two directions. The twodimensional quasicrystals belong to ones in which the atom arragement is quasiperiodic in two directions and periodic in the other one. The three-dimensional quasicrystals behave in such a manner that the arrangement presents quasiperiodicity in all three directions. There exist 200 quasicrystals observed to date, in which about 100 are icosahedral and about 70 decagonal quasicrystals, respectively, so these two classes of quasicrystal systems present major importance in the material.

3.5

Two-dimensional quasicrystals and planar quasicrystals

The two-dimensional quasicrystals and planar quasicrystlas are diﬀerent concepts. The two-dimensional quasicrystals have been introduced in the previous section, which represent a three-dimensional structure with two-dimensional quasiperiodic planes stacked along the third direction; in this direction the atom arrangement is periodic. While the planar quasicrystals belong to a two-dimensional structure within the plane the atom arrangement is quasiperiodic, and there is no third dimension.

References [1] Shechtman D, Blech I, Gratias D et al. Metallic phase with long-range orientational order and no translational symmetry. Phys Rev Lett, 1984, 53(20): 1951–1953

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[2] Levine D, Steinhardt P J.Quasicrystals: A new class of ordered structure. Phys Rev Lett, 1984, 53(26): 2477–2450 [3] Ye H Q, Wang D, Kuo K H. Five-fold symmetry in real and reciprocal space. Ultramicrossopy, 1985, 16(2): 273–277 [4] Zhang Z, Ye H Q, Kuo K H. A new icosahedral phase with m35 symmetry. Phil Mag A, 1985, 52(6): L49–L52 [5] Feng Y C, Lu G, Witers R I. An incommensurate structure with cubic point group symmetry in rapidly solidiﬁed V-Vi-Si alloy. J Phys Condens Matter, 1989, 1(23): 3695–3700 [6] Bendersky L. Quasicrystal with one-dimensional translational symmetry and a tenfold rotation axis. Phys Rev Lett, 1985, 55(14): 1461–1463 [7] Chattopadhyay K, Lele S, Thangarai N et al. Vacancy ordered phases and onedimensional quasiperiodicity. Acta Metall, 1987, 35(3): 727–733 [8] Fung K K, Yang C Y, Zhou Y Q et al. Icosahedrally related decagonal quasicrystal in rapidly cooled Al-14-at.%-Fe alloy. Phys Rev Lett, 1986, 56(19): 2060–2063 [9] Urban K, Mayer J, Rapp M et al. Studies on aperiodic crystals in Al-Mn and Al-V alloys by means of transmission electron microscopy. Journal de Physique Colloque, 1986, 47C(3): 465–475 [10] Wang N, Chen H, Kuo K H. Two-dimensional quasicrystal with eightfold rotational symmetry. Phys Rev Lett, 1987, 59(9): 1010–1013 [11] Li X Z, Guo K H. Decagonal quasicrystals with diﬀerent periodicities along the tenfold axis in rapidly solidiﬁed Al-Ni alloys. Phil Mag Lett, 1988, 58(3): 167–171 [12] Merdin R, Bajema K, Clarke R et al. Quasiperiodic GaAs-AlAs heterostructures. Phys Rev Lett, 1985, 55(17): 1768–1770 [13] Hu A, Tien C, Li X et al. X-ray diﬀraction pattern of quasiperiodic (Fibonacci) Nb-Cu superlattices. Phys Lett A, 1986, 119(6): 313–314 [14] Feng D, Hu A, Chen K J et al. Research on quasiperiodic superlattice. Mater Sci Forum, 1987, 22(24): 489–498 [15] Terauchi H, Noda Y, Kamigami K et al. X-Ray diﬀraction patterns of conﬁgurational Fibonacci lattices. J Phys Jpn, 1988, 57(7): 2416–2424 [16] Chen K J, Mao G M, Fend D et al. Quasiperiodic a-Si: H/a-SiNx: H multilayer structures. J Non-cryst Solids, 1987, 97(1): 341–344 [17] Yang W G, Wang R H, Gui J. Some new stable one-dimensional quasicrystals in Al65 Cu20 Fe10 Mn5 alloy. Phil Mag Lett, 1996, 74(5): 357–366 [18] He L X, Li X Z, Zhang Z et al. One-dimensional quasicrystal in rapid solidiﬁed alloys. Phys Rev Lett, 1988, 61(9): 1116–1118 [19] Biham O, Mukamel D, Shtrikman. Symmetry and stability of icosahedral and other quasicrystalline phases. Phys Rev Lett, 1986, 56(20): 2191–2194

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Chapter 3

Quasicrystal and its properties

[20] Scheater R J, Bendersky L A. Introduction to Quasicrystals. Jaric M, Boston V. MA: Academic Press, 1988 [21] Widom M, Deng D P, Henleg C L. Transfer-matrix analysis of a two-dimensional quasicrystal. Phys Rev Lett, 1989, 63(3): 310–313 [22] Yang W G, Ding D H, Wang R H, et al. Thermodynamics of equilibrium properties of quasicrystals. Z Phys. B, 100(3), 447–454 [23] Hu C Z, Yang W G, Wang R H et al. Quasicrystal symmetry and physical properties. Progress in Phys, 1997, 17(4): 345–367 [24] Fan T Y. A study on speciﬁc heat of one-dimensional hexagonal quasicrystal. J Phys: Condens Matter, 1999, 11(45): L513–L517 [25] Fan T Y, Mai Y W. Partition function and state equation of point group 12mm two-dimensional dodecagonal quasicrystals. Euro Phys J B, 2003, 31(2): 17–21 [26] Li C Y, Liu Y Y. Phason-strain inﬂuence on low-temperature speciﬁc heat of the decagonal Al-Ni-Co quasicrystal. Chin Phys Lett, 2001, 18(4): 570–573 [27] Li C Y, Liu Y Y. Low-temperature lattice excitation of icosahedral Al-Mn-Pd quasicrystals. Phys Rev B, 2001, 63(6): 064203 [28] Biggs B D, Li Y, Poon S J. Electronic properties of icosahedral, approximant, and amorphous phases of an Al-Cu-Fe alloy. Phys Rev B, 1991, 43(10): 8747–8750 [29] Lindqvist P, Berger C, Klein T et al. Role of Fe and sign reversal of the Hall coeﬃcient in quasicrystalline Al-Cu-Fe. Phys Rev B, 1993, 48(1): 630–633 [30] Klein T, Gozlen A, Berger C et al. Anomalous transport properties in pure Al-Cu-Fe icosahedral phases of high structural quality. Europhys Lett, 1990, 13(2): 129–134 [31] Pierce F S, Guo Q, Poon S J. Enhanced insulatorlike electron transport behavior of thermally tuned quasicrystalline states of Al-Pd-Re alloys. Phys Rev Lett, 1994, 73(16): 2220–2223 [32] Wang A J, Zhou X, Hu C Z et al. Properties of nonlinear elasticity of quasicrystals with ﬁve-fold symmetry. Wuhan University Journal Nat Sci, 2005, 51(5): 536–566 [33] Zhou X, Hu C Z, Gong P et al. Piezoresistance properties of quasicrystals. J Phys: Condens Matter, 2004, 16(30): 5419–5425 [34] Homes C C,Timusk T, Wu X et al. Optical conductivity of the stable icosahedral quasicrystal Al63.5 Cu24.5 Fe12 . Phy Rev Lett, 1991, 67(19): 2694–2696 [35] Burkov S E. Optical conductivity of icosahedral quasicrystals. J Phys: Condens Matter, 1992, 4(47): 9447–9458 [36] BasovD N, Timusk T, Barakat F et al. Anisotropic optical conductivity of decagonal quasicrystals. Phys Rev Lett, 1994, 72(12): 1937–1940 [37] Villa D, Enoch S, Tayeb G et al. Band gap formation and multiple scattering in photonic quasicrystals with a Penrose-type lattice. Phys Rev Lett, 2005, 94(18): 183903

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[38] Mayou D. Generalized Drude formula for the optical conductivity of quasicrystals. Phys Rev Lett, 2000, 85(6), 1290–1293 [39] Notomi M, Suzuk H, Tamamura T et al. Document lasing action due to the twodimensional quasiperiodicity of photonic quasicrystals with a Penrose lattice. Phys Rev Lett, 2004, 92(12): 123906 [40] Feng Z F, Zhang X D, Wang Y Q. Negative refraction and imaging using 12-foldsymmetry quasicrystals. Phys Rev Lett, 2005, 94(24): 247402 [41] Lifshitz R, Arie A, Bahabad A. Photonic quasicrystals for nonlinear optical frequency conversion. Phys Rev Lett, 2005, 95(13): 13390 [42] Marn W, Megens M. Experimental measurement of the photonic properties of icosahedral quasicrystals. Nature, 2005, 436(7053): 993–996 [43] Kohmoto M, Sutherland B. Electronic states on a Penrose lattice. Phys Rev Lett, 1986, 56(25): 2740–2743 [44] Sutherland B. Simple system with quasiperiodic dynamics: a spin in a magnetic ﬁeld. Phys Rev B, 1986, 34(8): 5208–5211 [45] Fujiwara T, Yokokawa T. Universal pseudogap at Fermi energy in quasicrystals. Phys Rev Lett, 1991, 66(3): 333–336

Chapter 4 The physical basis of elasticity of quasicrystals The physical background on elasticity of quasicrystals is quite diﬀerent from that of the classical elasticity, the discussion about this provides a basis of the subsequent contents of the book.

4.1

Physical basis of elasticity of quasicrystals

Quasicrystal has become a type of functional and structural materials, having potential engineering applications. As a material, quasicrystal is deformable under applied forces, thermal loads and certain internal eﬀects. The deformation of crystals has been discussed in Chapter 2. Questions arise as to what the characteristics in the deformation process of the quasicrystal are. How to describe mathematically the behaviour of the quasicrystal deformation? To answer these questions, it is necessary to consider the physical background of elasticity of quasicrystals. The study in this regard was conducted soon after discovery of the new solid phase. Because the quasicrystal is a new structure of solid, theoretical physicists have proposed various descriptions of its elasticity. The majority agrees that the Landau density wave theory (Refs[1-25]) is the physical basis of elasticity of the quasicrystals. We shall introduce the theory in Appendix of this chapter (Section 4.9). Essentially, the description suggested that there are two displacement ﬁelds u and w in a quasicrystal: the former is similar to that in crystals, named as the phonon ﬁeld, its macro-mechanical behaviour has been discussed in Chapter 2; the latter is a displacement ﬁeld, named as the phason ﬁeld. The total displacement ﬁeld in a quasicrystal is expressed by ¯ = u ⊕ u⊥ = u ⊕ w, u

(4.1-1)

where ⊕ denotes the direct sum; u is in the physical space, or the parallel space E3 ; 3 w is in the complement space, or the perpendicular space E⊥ , which is an internal space Furthermore, the two displacement vectors depend on the coordinate vector r T. Fan, Mathematical Theory of Elasticity of Quasicrystals and Its Applications © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2011

36

Chapter 4

The physical basis of elasticity of quasicrystals

in the physical space, u = u(r ), w = w(r )

(4.1-2)

For simplicity, the superscript of r will be removed hereafter. From the angle of mathematical theory of elasticity of quasicrystals and its technological applications the formulas (4.1-1) and (4.1-2) are suﬃcient for comprehending the following contents of the book. If readers are interested in more of the physical background on the phonon and phason ﬁelds in quasicrystals, we suggest that they could read the Appendix of this chapter (i.e., the Section 4.9). With basic formulas (4.1-1) and (4.1-2) and some fundamental conservation laws well known in physics, the macroscopic basis of the continuous medium model of elasticity of quasicrystals can be set up, in some extent, the discussion is an extension to that in Chapter 2, which will be done in the following sections.

4.2

Deformation tensors

In Chapter 2 we introduced that the deformation of phonon ﬁeld lies in the relative displacement (i.e., the rigid-body translation and rotation do not result in deformation), which is expressed by du = u − u. If we set up an orthogonal coordinate system (x1 , x2 , x3 ) or (x, y, z), then u = (u1 , u2 , u3 ) = (ux , uy , uz ) and ∂ui dui = dxj (4.2-1) ∂xj in which ∂ui /∂xj has the meaning of the gradient of vector u. It may also be denoted as ⎡ ⎤ ∂ux ∂ux ∂ux ⎢ ∂x ∂y ∂z ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ∂uy ∂uy ∂uy ⎥ ∂ui ⎢ ⎥ =⎢ (4.2-2) ∇u = ∂xj ∂y ∂z ⎥ ⎢ ∂x ⎥ ⎢ ⎥ ⎣ ∂uz ∂uz ∂uz ⎦ ∂x and

⎡

∂ux ⎢ ∂x ⎢ ⎢ ⎢ ∂uy ⎢ ⎢ ∂x ⎢ ⎢ ⎣ ∂uz ∂x

∂ux ∂y ∂uy ∂y ∂uz ∂y

∂y ⎤ ∂ux ∂z ⎥ ⎥ ⎥ ∂uy ⎥ ⎥ ∂z ⎥ ⎥ ⎥ ∂uz ⎦ ∂z

∂z

4.2

Deformation tensors

⎡

37

∂ux ∂x

⎢ ⎢ ⎢ ⎢ 1 ∂u x ⎢ + =⎢ ⎢ 2 ∂y ⎢ ⎢ ⎢ 1 ∂ux ⎣ + 2 ∂z ⎡

1 2

∂ux ∂uy + ∂y ∂x

1 2

∂ux ∂uz + ∂z ∂x

⎤

⎥ ⎥ ⎥ ∂uy ∂uy 1 ∂uy ∂uz ⎥ ⎥ + ⎥ ∂x ∂y 2 ∂z ∂y ⎥ ⎥ ⎥ 1 ∂uy ∂uz ∂uz ∂uz ⎥ ⎦ + ∂x 2 ∂z ∂y ∂z ∂ux 1 ∂uz ∂ux 1 ∂uy − − − 0 − ⎢ 2 ∂x ∂y 2 ∂x ∂z ⎢ ⎢ ⎢ 1 ∂u ∂uy 1 ∂uz ∂uy x ⎢ − − 0 − − +⎢ ∂x 2 ∂y ∂z ⎢ 2 ∂y ⎢ ⎢ ∂uz 1 ∂uy ∂uz ⎢ 1 ∂ux ⎣ − − − − 0 2 ∂z ∂x 2 ∂z ∂y ∂uj ∂ui 1 ∂uj 1 ∂ui + − − = εij + ωij , = 2 ∂xj ∂xi 2 ∂xi ∂xj 1 ∂ui ∂uj , εij = + 2 ∂xj ∂xi 1 ∂ui ∂uj ωij = − , 2 ∂xj ∂xi

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(4.2-3) (4.2-4)

which means the gradient of the phonon vector u can be decomposed into two parts εij and ωij , in which εij has contribution to the deformation energy, and ωij represents a kind of rigid-body rotations. We consider only εij , which is the phonon deformation tensor, or the strain tensor, it is a symmetric tensor: εij = εji . Similarly for the phason ﬁeld we have dwi = and

∂wi dxj ∂xj

⎡

∂wx ⎢ ∂x ⎢ ⎢ ∂wy ∂wi ⎢ ∇w = =⎢ ⎢ ∂x ∂xj ⎢ ⎢ ⎣ ∂wz ∂x

∂wx ∂y ∂wy ∂y ∂wz ∂y

(4.2-5) ⎤ ∂wx ∂z ⎥ ⎥ ⎥ ∂wy ⎥ ⎥, ∂z ⎥ ⎥ ⎥ ∂wz ⎦

(4.2-6)

∂z

though it can be decomposed into symmetric and asymmetric parts, all components of

∂wi contribute to the deformation of the quasicrystals, the phason deformation ∂xj

38

Chapter 4

The physical basis of elasticity of quasicrystals

tensor or the phason strain tensor is deﬁned by wij =

∂wi , ∂xj

(4.2-7)

which is asymmetric tensor wij = wji , and describes the local rearrangement of atoms in a cell. The diﬀerence between εij and wij given by (4.2-3) and (4.2-7) are originated from the physical properties of the phonon modes and the phason modes. This can also be explained by the group theory, in that, they follow diﬀerent irreducible representations for some symmetry transformations for most quasicrystal systems, except the three-dimensional cubic quasicrystal system. The detail about this is omitted here. For the three-dimensional cubic quasicrystals, the phason modes exibit the same behaviour as that of the phonon modes, which will be particularly discussed in Chapter 9.

4.3

Stress tensors and the equations of motion

The gradient of the displacement ﬁeld w ﬁgures out the local rearrangement of atoms in a cell in the quasicrystals. External forces are needed to drive the atoms through barriers when they make the local rearrangement in a cell. Such that, there is another kind of body forces and tractions apart from the conventional body forces f and tractions T for deformed quasicrystals, named as the generalized body forces (density) g and the generalized tractions ( the generalized area forces density) h. At ﬁrst, we consider the static case. Denoting the stress tensor corresponding to εij by σij , the phonon stress tensor, and that to wij by Hij , the phason stress tensor, we have the following equilibrium equations ⎧ ⎪ ∂σij + f = 0, ⎪ i ⎪ ⎨ ∂xj x, y, z ∈ Ω (4.3-1) ⎪ ⎪ ⎪ ∂Hij + gi = 0, ⎩ ∂xj based on the momentum conservation law. Apply the angular momentum conservation law to the phonon ﬁeld d ˙ r × ρudΩ = r × f dΩ + r × T dΓ dt Ω Ω Ω

(4.3-2)

by using the Gauss theorem, it follows that σij = σji ,

(4.3-3)

4.3

Stress tensors and the equations of motion

39

this indicates that the phonon stress tensor is symmetric, Since r and w(g, h) transform under diﬀerent representations of the point groups, more precisely that, the former transform like a vector, but latter does not, the product representation: r × w, r × g and r × h do not contain any vector representations. This implies that for the phason ﬁeld there is no equation analogous to (4.3-2), from which it follows that, generally Hij = Hji

(4.3-4)

except the case for three-dimensional cubic quasicrystals. In the dynamic case, the deformation process is rather complicated; there are diﬀerent arguments. Lubensky et al[5] claimed that the phonon modes and the phason modes are diﬀerent based on their role in six-dimensional hydrodynamics, the phonons are wave propagation while the phasons are diﬀusive with very large diﬀusive time. Physically the phason modes represent a relative motion of the constituent density waves. Dolinsek et al[22,23] further developed the idea by bringing in the concept of atom ﬂip or atom hopping for the phason dynamics. But according to Bak[1,2] , the phason describes particular structural disorders or structure ﬂuctuations in quasicrystals, and it can be formulated based on a six-dimensional space description. Since there are six continuous symmetries, there exist six hydrodynamic vibration modes. In the following we give a brief introduction on elastodynamics based on the Bak’s argument as well as argument of Lubensky et al. Ding et al[26] and Hu et al[16] derived that ⎧ ∂σij ∂ 2 ui ⎪ ⎪ + f = ρ , i ⎪ ⎨ ∂xj ∂t2 (x, y, z) ∈ Ω , t > 0 (4.3-5) ⎪ ⎪ ∂ 2 wi ∂Hij ⎪ ⎩ + gi = ρ 2 , ∂xj ∂t based on the law of momentum conservation. We believe that the derivation is carried out by following the Bak’s argument, in which ρ is the mass density of the quasicrystals. If following the argument of Lubensky et al, people cannot obtain (4.3-5), instead we have ⎧ ∂σij ∂ 2 ui ⎪ ⎪ + f = ρ , ⎪ i ⎨ ∂xj ∂t2 (x, y, z) ∈ Ω , t > 0, (4.3-6) ⎪ ∂w ∂H ⎪ ij i ⎪ ⎩ + gi = κ , ∂xj ∂t in which κ = 1/Γw , Γw is the kinetic coeﬃcient of the phason ﬁeld. The equations are given by Fan et al[27] , which are identical to those given by Lubensky et al[5] for

40

Chapter 4

The physical basis of elasticity of quasicrystals

the linear case. Lubensky et al gave their formulation based on the hydrodynamics principle, so equation (4.3-6) may be seen as elasto-/hydro-dynamic equation of quasicrystals. In particular, the second equation of (4.3-6) presents the dissipation feature of motion of the phason degrees in the dynamic process, it is irreversible thermodynamically. The elastodynamics of quasicrystals poses a great challenge, see e.g. [32, 33], we discuss it only in a quite narrow scope (only in Chapter 10), the inﬂuence of those distinguishing arguments in the circle of the quasicrystal study is limited to the range in Chapter 10. The content therein does not aﬀect the results and conclusions presented in other chapters.

4.4

Free energy and elastic constants

Consider the free energy or the strain energy density of a quasicrystal F (εij , wij ) whose general expression is diﬃcult to obtain. We take a Taylor expansion in the neighbourhood of εij = 0 and wij = 0, and remain up to the second order term, then 1 1 ∂2F ∂2F F (εij , wij ) = εij εkl + εij wkl 2 ∂εij ∂wkl 0 2 ∂εij ∂wkl 0 1 1 ∂2F ∂2F + wij wkl + wij εkl 2 ∂wij ∂wkl 0 2 ∂wij ∂εkl 0 1 1 1 1 = Cijkl εij εkl + Rijkl εij wkl + Kijkl wij wkl + Rijkl wij εkl 2 2 2 2 = Fu + Fw + Fuw ,

(4.4-1)

where Fu , Fw and Fuw denote the parts contributed by the phonon, the phason and the phonon-phason coupling, respectively, and ∂2F Cijkl = (4.4-2) ∂εij ∂εkl 0 is the phonon elastic constant tensor, discussed in Chapter 2 already, and Cijkl = Cklij = Cjikl = Cijlk ,

(4.4-3)

the tensor can be expressed by the symmetric matrix [C]9×9 . In (4.4-1) another elastic constant tensor is ∂2F Kijkl = , ∂wij ∂wkl 0

(4.4-4)

4.4

Free energy and elastic constants

41

3 in which the suﬃxes j, l belong to space E3 and i, k to space E⊥ , and

Kijkl = Kklij .

(4.4-5)

All components of Kijkl can also be expressed by the symmetric matrix [K]9×9 . In addition,

∂2F , Rijkl = ∂εij ∂wkl 0 ∂2F = Rijkl ∂wij ∂εkl 0

(4.4-6) (4.4-7)

are the elastic constants of the phonon-phason coupling. To be noted that, the 3 suﬃxes i, j, l belong to space E3 and k belongs to space E⊥ , and Rijkl = Rjikl ,

Rijkl = Rklij ,

Rklij = Rijkl ,

(4.4-8)

but Rijkl = Rklij ,

Rijkl = Rklij ,

(4.4-9)

of which all components can be expressed by the symmetric matrices [R]9×9 ,

[R ]9×9

and [R]T = [R ] ,

(4.4-10)

where T denotes the transpose operator. The composition of four matrices [C] , [K] , [R] and [R ] forms a 18 × 18 matrix ⎤ ⎡ [C] [R] [C] [R] ⎦. =⎣ (4.4-11) [C, K, R] = T [R ] [K] [K] [R] If the strain tensor is expressed by a row vector with 18 elements, i.e., ε11 , ε22 , ε33 , ε23 , ε31 , ε12 , ε32 , ε13 , ε21 , [εij , wij ] = , (4.4-12) w11 , w22 , w33 , w23 , w31 , w12 , w32 , w13 , w21 the transpose of which denotes the array vector, then the free energy (or strain energy density) may be expressed by ⎡ ⎤ [C] [R] 1 ⎦ [εij , wij ]T , (4.4-13) F = [εij , wij ] ⎣ T 2 [K] [R] which is identical to that given by (4.4-1)

42

4.5

Chapter 4

The physical basis of elasticity of quasicrystals

Generalized Hooke’s law

To apply theory of elasticity of quasicrytals, one must determine the displacement ﬁeld and stress ﬁeld, this requires that we need to set up relationship between the strains and the stresses, the relations are the generalized Hooke’s law of quasicrystalline material. From the free energy (4.4-1) or (4.4-13), we have σij =

∂F = Cijkl εkl + Rijkl wkl , ∂εij

(4.5-1)

∂F = Kijkl wkl + Rklij εkl , Hij = ∂wij or in the matrix form

where

σij Hij

=

4.6

σij Hij εij wij

[C] T [R]

[R] [K]

εij wij

,

(4.5-2)

= [σij , Hij ]T , (4.5-3) T

= [εij , wij ] .

Boundary conditions and initial conditions

The above general formulas give a description of the basic law of elasticity of quasicrystals, and provide a key to solve those problems in application for theoretical research and engineering practice, the formulas hold in any interior of the body, i.e., (x, y, z) ∈ Ω , where (x, y, z) denote the coordinates of any point of the interior, and Ω denotes the body. The formulas are concluded as some partial diﬀerential equations. To solve them, it is necessary to know the situation of the ﬁeld variables at the boundary Γ of Ω , without appropriate information at the boundary, the solution has no any physical meaning. According to practical case the boundary Γ consists of two parts Γt and Γu , i.e., Γ = Γt + Γu , at Γt the tractions are given and at Γu the displacements are prescribed. For the former case,

σij nj = Ti , (x, y, z) ∈ Γt , (4.6-1) Hij nj = hi , where nj represents the unit outward normal vector at any point at Γ , Ti and hi the traction and generalized traction vectors, which are given functions at the boundary. Formula (4.6-1) is called the stress boundary conditions. And for the latter case,

¯i , ui = u (4.6-2) (x, y, z) ∈ Γu , ¯i , wi = w

4.7

A brief introduction on relevant material constants of quasicrystals

43

where u ¯i and w ¯i are known functions at the boundary. Formula (4.6-2) is named the displacement boundary conditions. If Γ = Γt (i.e., Γu = 0), the problem for solving equations (4.2-3), (4.2-7), (4.3-1) and (4.4-14) under boundary conditions (4.6-1) is a traction boundary value problem. While Γ = Γu (i.e., Γt = 0) , the problem for solving equations (4.2-3), (4.2-7), (4.3-1) and (4.4-14) under boundary conditions (4.6.2) is a displacement boundary value problem. If Γ = Γu + Γt and both Γt = 0, Γu = 0, the problem for solving equations (4.2-3), (4.2-7), (4.3-1) and (4.4-14) under boundary conditions (4.6-1) and (4.6.2) is a mixed boundary value problem. For the dynamic problem, if taking the wave equations (4.3-5) together with equation (4.2-3), (4.2-7) and (4.4-14), besides boundary conditions (4.6-1) and (4.6-2), we must prescribe initial value conditions:

ui (x, y, z, 0) = ui0 (x, y, z), u˙ i (x, y, z, 0) = u˙ i0 (x, y, z), (x, y, z) ∈ Ω , (4.6-3) wi (x, y, z, 0) = wi0 (x, y, z), w˙ i (x, y, z, 0) = w˙ i0 (x, y, z), in which ui0 (x, y, z, 0), u˙ i0 (x, y, z, 0),wi0 (x, y, z, 0) and w˙ i0 (x, y, z, 0) are given functions and u˙ i =

∂ui etc. In this case the problem is called initial-boundary value ∂t

problem. If taking the wave equations coupling diﬀusion equations (4.3-6) together with (4.2-3) and (4.4-14), the initial value conditions are

ui (x, y, z, 0) = ui0 (x, y, z), u˙ i (x, y, z, 0) = u˙ i0 (x, y, z), (x, y, z) ∈ Ω , (4.6-4) wi (x, y, z, 0) = wi0 (x, y, z), this is also an initial-boundary value problem, but diﬀerent from the previous one.

4.7

A brief introduction on relevant material constants of quasicrystals

In above discussion we ﬁnd that the quasicrystals posses diﬀerent nature with those of the crystals. Connection with this, the material constants of the solid should be diﬀerent from those of the crystals and other conventional structural materials, in which the constants appearing in the above basic equations are interesting at least. We here give a brief introduction to help readers in conceptional point of view. The detailed introduction will be given in Chapters 6, 9 and 10. The measurement of material constants of quasicrystals is diﬃcult, but the experimental technique is in progress, especially in recent years. Due to the majority of icosahedral and decagonal quasicrystals in the material, the measured data are mainly for these two kinds of the solid phase.

44

Chapter 4

The physical basis of elasticity of quasicrystals

For icosahedral quasicrystals, the independent nonzero components of the phonon elastic constants Cij are only λ and μ, the phason elastic contants Kij only K1 and K2 , and the phonon-phason coupling elastic constants Rij only R. For the most important icosahedral Al-Pd-Mn quasicrystal, the measured data including the mass density and kinetic coeﬃcient of phason are[28,29] : 3

ρ = 5.1g/cm ,

λ = 74.9,

μ = 72.4GPa,

K1 = 72,

K2 = −37MPa,

R ≈ 0.01μ,

Γw = 4.8 × 10−10 cm3 · μs/g = 4.8 × 10−19 m3 · s/kg and two-dimensional quasicrystals, the independent nonzero components of the phonon elastic constants are only C11 , C33 , C44 , C12 , C13 , and C66 = (C11 − C12 )/2, the phason ones only K1 , K2 , K3 , and the phonon-phason coupling ones only R1 , R2 . For decagonal Al-Ni-Co quasicrystal, the measued data are[28] : 3

ρ = 4.186g/cm ,

C11 = 234.3,

C33 = 232.22,

C44 = 70.19,

C12 = 57.41,

C13 = 66.63(GPa), R1 = −1.1,

|R2 | < 0.2(GPa)

and there are no measured data for K1 , K2 (but we can use those obtained by the Monte-Carlo simulation), and Γw can approximately be taken the value of the icosahedral quasicrystal. In addition, the tensile strength σc = 450MPa for decagonal Al-Cu-Co quasicrystals before annealing, and σc = 550MPa after annealing. The hardness for decagonal Al-Cu-Co quasicrystals is 4.10GPa[30,31] , the fracture tough√ ness is 1.0 ∼ 1.2MPa m [30] . With these basic data, the computation for stress analysis for statics and dynamics can be undertaken.

4.8

Summary and mathematical solvability of boundary value or initial-boundary value problem

For a static equilibrium problem, the mathematical formulation is ∂Hij ∂σij + fi = 0, + hi = 0, xi ∈ Ω , ∂xj ∂xj 1 ∂ui ∂uj ∂wi εij = + , xi ∈ Ω , , wij = 2 ∂xj ∂xi ∂xj

σij = Cijkl εkl + Rijkl wkl , xi ∈ Ω , Hij = Kijkl wkl + Rklij εkl , σij nj = Ti , Hij nj = h,

xi ∈ Γt ,

(4.8-1)

(4.8-2) (4.8-3) (4.8-4)

4.8

Summary and mathematical solvability of boundary value or...

ui = u ¯i , wi = w ¯i ,

xi ∈ Γ u .

45

(4.8-5)

For a dynamic problem, based on the Bak’s argument, the mathematical formulation is ∂ 2 ui ∂Hij ∂ 2 wi ∂σij + fi = ρ 2 , + gi = ρ 2 , xi ∈ Ω , t > 0, (4.8-6) ∂xj ∂t ∂xj ∂t 1 εij = 2

∂uj ∂ui + ∂xj ∂xi

, wij =

∂wi , ∂xj

σij = Cijkl εkl + Rijkl wkl , Hij = Kijkl wkl + Rklij εkl , σij nj = Ti , Hij nj = h, ¯i , wi = w ¯i , ui = u

xi ∈ Ω , t > 0, xi ∈ Ω , t > 0,

xi ∈ Γt , t > 0,

(4.8-7)

(4.8-8) (4.8-9)

xi ∈ Γu , t > 0,

(4.8-10)

ui |t=0 = ui0 , u˙ i |t=0 = u˙ i0 , wi |t=0 = wi0 , w˙ i |t=0 = w˙ i0 ,

xi ∈ Ω .

(4.8-11)

For a dynamic problem, based on the argument of Lubensky et al, the mathematical formulation is ∂σij 1 ∂ 2 ui ∂Hij ∂wi ,κ = + fi = ρ 2 , + hi = κ , ∂xj ∂t ∂xj ∂t Γw εij =

1 2

∂ui ∂uj + ∂xj ∂xi

, wij =

∂wi , ∂xj

σij = Cijkl εkl + Rijkl wkl , Hij = Kijkl wkl + Rklij εkl , σij nj = Ti , Hij nj = h, ui = u ¯i , wi = w ¯i ,

xi ∈ Ω , t > 0,

xi ∈ Ω , t > 0, xi ∈ Ω , t > 0,

xi ∈ Γt , t > 0,

xi ∈ Γu , t > 0,

ui |t=0 = ui0 , u˙ i |t=0 = u˙ i0 , wi |t=0 = wi0 ,

(4.8-12)

(4.8-13)

(4.8-14) (4.8-15) (4.8-16)

xi ∈ Ω .

(4.8-17)

The solution satisfying all equations and corresponding initial conditions and boundary conditions is just the realistic solution of elasticity of quasicrystals mathematically and has physical meaning. The existence and uniqueness of solutions of elasticity of quasicrystals will further be discussed in Chapter 13.

46

4.9

Chapter 4

The physical basis of elasticity of quasicrystals

Appendix of Chapter 4: Description on physical basis of elasticity of quasicrystals based on the Landau density wave theory

In Section 4.1 we gave the formula (4.1-1) as the physical basis of elasticity of quasicrystals and did not discuss its profound physical source, because the discussion is concerned with quite complicated background, which is unnecessary for the beginner before reading Sections 4.2∼4.8. At this point, it is advantageous to study the physical background. The reader is advised to read Section 1.5 (the Appendix of Chapter 1) ﬁrst. Chapter 3 shows quasicrystals belong to subject of condensed matter physics rather than traditional solid state physics, though the former is evolved from the latter. In the development, the symmetry-breaking (or broken symmetry) forms the core concept and principle of the condensed matter physics. According to the understanding of physicists that the Landau density wave description on the elasticity of quasicrystals is a natural choice, though there are some other descriptions, e.g. the unit cell description based on the Penrose tiling. Now the diﬃculty lies in that the workers in other disciplines are not so familiar with the Landau theory and the relevant topics. For this reason we have introduced the Landau theory in the Appendix of Chapter 1 (i.e., Section 1.5, in which the concept on incommensurate crystals is also introduced in brief, which will be concerned though it is not related to the Landau theory) and the Penrose tiling in Section 3.1. These important physical and mathematical results can help us to understand the elasticity of quasicrystals. Immediately after the discovery of quasicrystals, Bak[1] published the theory of elasticity in which he used the three important results in physics and mathematics mentioned above, but the core is the Landau theory on elementary excitation and symmetry-breaking of condensed matter. Bak[1,2] pointed out too, ideally, one would like to explain the structure from ﬁrst-principles calculations taking into account the actual electronic properties of constituent atoms. Such a calculation is hardly possible to date, so he suggested that the Landau phenomenological theory[3] on structural transition can be used, i.e., the condensed phase is described by a symmetry-breaking order parameter which transforms as an irreducible representation of the symmetry group of a liquid with full translational and rotational symmetry. According to the Landau theory, the order parameter of quasicrystals is the density wave. For the density of the ordered, low-temperature d-dimensional quasicrystal can be expressed as a Fourier series by extended formula (1.5-5) (the expansion exists due to the periodicity in lattice or reciprocal lattice of higher dimensional space) ρ(r) = ρG exp{iG · r} = |ρG | exp{−iΦG + iG · r}, (4.9-1) G∈LR

G∈LR

4.9

Appendix of Chapter 4: Description on physical basis of elasticity...

47

where G is a reciprocal vector, and LR the reciprocal lattice (the concepts on the reciprocal vector and reciprocal lattice, referring to Chapter 1), ρG is a complex number (4.9-2) ρG = |ρG | e−iΦG with an amplitude |ρG | and phase angle ΦG , due to ρ(r) being real, |ρG | = |ρ−G | and ΦG = −Φ−G . There exists a set of N base vectors, {Gn }, so that each G ∈ LR can be written as mn Gn for integers mn . Furthermore N = kd, where k is the number of the mutually incommensurate vectors in the d-dimensional quasicrystal. In general k = 2. A convenient parametrization of the phase angle is given by Φn = Gn · u + G⊥ n · w,

(4.9-3)

in which u can be understood similar to the phonon like that in conventional crystals, while w can be understood the phason degrees of freedom in quasicrystals, which describe the local rearrangement of unit cell description based on the Penrose tiling. Both are functions of the position vector in the physical space only, where Gn is the reciprocal vector in the physical space E3 just mentioned and G⊥ n is the conjugate 3 . We can realize that the above mentioned Bak’s vector in the perpendicular space E⊥ hypothesis is a natural development of Anderson’s theory introduced in Section 1.5. Almost in the same time, Levine et al[4] , Lubensky et al [5∼8] , Kalugin et al[9] , Torian and Mermin[10] , Jaric[11] , Duneau and Katz[12] , Socolar et al[13] , Gahler and Phyner[14] carried out the study on elasticity of quasicrystals. Though the researchers studied the elasticity from diﬀerent descriptions, e.g. the unit-cell description based on the Penrose tiling is adopted too, but the density-wave description based on the Laudau phenomenological theory on symmetry-breaking of condensed matter has played the central role and been widely acknowledged. This means there are two elementary excitations of low-energy, phonon u and phason w for quasicrystals, in which vector u is in the parallel space E3 and vector w is in the perpendicular 3 , respectively. So the total displacement ﬁeld for quasicrystals is space E⊥ ¯ = u ⊕ u⊥ = u ⊕ w, u which is the formula (4.1-1), where ⊕ represents the direct sum. According to the argument of Bak etc, that u = u(r ),

w = w(r ),

i.e., u and w depend upon special radius vector r in parallel space E3 only, this is formula (4.1-2). For simplicity the superscript of r is removed in the presentation in Sections 4.2∼4.8.

48

Chapter 4

The physical basis of elasticity of quasicrystals

Even if introducing u and w by such a way the concept of phason is hard to be accepted by some readers. We would like to give some additional explanation in terms of projection concept as below. We previously said that the quasicrystal in three-dimensional space may be seen as a projection of periodic structure in higher dimensional space. For example, onedimensional quasicrystals in physical space may be seen as a projection of “periodic crystals” in four-dimensional space, while in the one-dimensional quasicrystals, the atom arrangement is quasiperiodic only in one direction, say z-axis direction, and periodic in other two directions. The atom arrangement quasiperiodic axis may seen as a projection of two-dimensional periodic crystal shown in Fig. 4.9-1 (a), in which dots form the two-dimensional, e.g. right square, crystal, and lines with slop of irrational numbers can correspond to quasiperiodic structures (by contrast, if the slope is rational number it corresponds to the periodic structure). For this purpose we can use the so-called Fibonacci sequence formed by a longer segment L and a shorter segment S according to the recurrence (whose geometry depiction is shown in Fig. 4.9-1 (b)) Fn+1 = Fn + Fn−1 and F0 F1 F2 F3 F4

: S, : L, : LS, : LSL, : LSLLS.

The Fibonacci sequence is ordered but not periodic. The geometric expression of the sequence can be shown in the axis E1 i.e., E in Fig. 4.9-1 (a), and E is the socalled parallel space, and that perpendicular to which is the so-called vertical space E2 , i.e., E⊥ . The Fibonacci sequence is a useful tool to describe the geometry of one-dimensional quasiperiodic structure, like that the Penrose tiling to describe the geometry of two- and three-dimensional quasicrystals. The Fig. 4.9-1 may help us to understand the internal-space E⊥ . For one-dimensional quasicrystals, the ﬁgure can give an explicit description, while for two- and three-dimensional quasicrystals there is no such an explicit graph. Since quasicrystals belong to one of incommensurate phases, and there are phason modes in the incommensurate crystals, denoted by w(r ), which may be understood as the corresponding new displacement ﬁeld, if people have knowledge on incommensurate phases, then they may easily understand the origin of phason modes in quasicrystal, though conventional incommensurate crystals are not certainly the actual quasicrystals.

4.9

Appendix of Chapter 4: Description on physical basis of elasticity...

Fig. 4.9-1

49

A geometrical representation for one-dimensional quasiperiodic structure

(a) A projection of two-dimensional crystal can generate a one-dimensional quasiperiodic structure, (b) The Fibonacci sequence

The phonon variables u(r ) appearing in the physical space E3 , vector u represents the displacement of lattice point deviated from its equilibrium position due to the vibration of the lattice. The propagation of this vibration is sound waves in solids. Though vibration is a mechnical motion, which can be quantized, the quantum of this motion is named phonon. So the u ﬁeld is called phonon ﬁeld in physical terminology. The gradient of the u ﬁeld characterizes the changes in volume and in shape of cells—this is identical to that in the classical elasticity (see, e.g. Chapter 2, and previous sections of this chapter). As mentioned before, the phason variables are substantively related with structural transitions of alloys, some of them can be observed from the characteristics of diﬀraction patterns. Lubensky et al[5,7] and Horn et al[15] discussed the connection between the phenomena and phason strain. These profound observations could not be discussed here, reader can refer to the review given by Hu et al[16] . This makes us know that the phason modes exist. The physical meaning of phason variables can be explained as a quantity to describe the local rearrangement of atoms in a cell. We know that the phase transition in crystalline materials is just induced by the atomic local rearrangement. The unit-cell description on quasicrystals mentioned above predicts that w describes the local arrangement of Penrose titling. These ﬁndings

50

Chapter 4

The physical basis of elasticity of quasicrystals

may help us to understand the meaning of the unusual ﬁeld variables. Afterward experimental investigations by neutron scattering[17∼20] , Moessbauer spectroscopy[21] , NMR (Nuclear Magnetic Resonance)[22,23] and speciﬁc heat measurements[24,25] , the concept of thermal-induced phason ﬂips has been suggested, this is identical to the diﬀusive essentiality of phasons. Note that the so-called diﬀusion here is quite diﬀerent from that in metallic periodic crystals (which mainly results from the presence of vacancies in the lattice, and the vacancies are not necessary for atomic motion in the quasicrystal structures). We will discuss this aspect in Chapter 10. It should be noted that vector u and vector w are diﬀerent in nature under certain symmetry operations. This can be explained by the group theory. The discussion is omitted here.

References [1] Bak P. Phenomenological theory of icosahedral incommensurate (quaisiperiodic) order, in Mn-Al alloys. Phys Rev Lett, 1985, 54(8): 1517–1519 [2] Bak P. Symmetry, stability and elastic properties of icosahedral incommensurate crystals. Phys Rev B, 1985, 32(9): 5764–5772 [3] Landau L D, Lifshitz E M. Theoretical Physics V: Statistical Physics. 3rd ed. New York: Pregamen Press, 1980 [4] Levine D, Lubensky T C, Ostlund S et al. Elasticity and dislocations in pentagonal and icosahedral quasicrystals. Phys Rev Lett, 1985, 54(8): 1520–1523 [5] Lubensky T C, Ramaswamy S, Nad Toner J. Hydrodynamics of icosahedral quasicrystals. Phys Rev B, 1985, 32(11): 7444–7452 [6] Lubensky T C, Ramaswamy S, Toner J. Dislocation motion in quasicrystals and implications for macroscopic properties. Phys Rev B, 1986, 33(11): 7715–7719 [7] Lubensky T C, Socolar J E S, Steinhardt P J et al. Distortion and peak broadening in quasicrystal diﬀraction patterns. Phys Rev Lett, 1986, 57(12): 1440–1443 [8] Lubensky T C. Introduction to Quasicrystals. Jaric M V. Boston: Academic Press, 1988 [9] Kalugin P A, Kitaev A, Levitov L S. 6-dimensional properties of Al0.86 Mn0.14 alloy. J Phys Lett, 1985, 46(13): 601–607 [10] Torian S M, Mermin D. Mean-ﬁeld theory of quasicrystalline order. Phys Rev Lett, 1985, 54(14): 1524–1527 [11] Jaric M V. Long-range icosahedral orientational order and quasicrystals. Phys Rev Lett, 1985, 55(6): 607–610 [12] Duneau M, Katz A. Quasiperiodic patterns, Phys Rev Lett, 1985, 54(25): 2688–2691 [13] Socolar J E S, Lubensky T C, Steinhardt P J. Phonons, phasons, and dislocations in quasicrystals. Phys Rev B, 1986, 34(5): 3345–3360

References

51

[14] Gahler F, Rhyner J. Equivalence of the generalised grid and projection methods for the construction of quasiperiodic tilings. J Phys A: Math Gen, 1986, 19(2): 267–277 [15] Horn P M, Melzfeldt W, Di Vincenzo D P et al. Systematics of disorder in quasiperiodic material. Phys Rev Lett, 1986, 57(12): 1444–1447 [16] Hu C Z, Wang R H, Ding D H. Symmetry groups, physical property tensors, elasticity and dislocations in quasicrystals. Rep Prog Phys, 2000, 63(1): 1–39 [17] Coddens G, Bellissent R, Calvayrac Y et al. Evidence for phason hopping in icosahedral Al-Fe-Cu quasicrystals. Europhys Lett, 1991, 16(3): 271–276 [18] Coddens G, Sturer W. Time-of-ﬂight neutron-scattering study of phason hopping in decagonal Al-Co-Ni quasicrystals. 1999, Phys Rev B, 1999, 60(1): 270–276 [19] Coddens G, Lyonnard S, Hennion B et al. Triple-axis neutron-scattering study of phason dynamics in Al-Mn-Pd quasicrystals. Phys Rev B 2000, 62(10): 6268–6295 [20] Coddens G, Lyonnard S, Calvayrac Y et al. Atomic (phason) hopping in perfect icosahedral quasicrystals Al70.3 Pd21.4 Mn8.3 by time-of-ﬂight quasielastic neutron scattering. Phys Rev B, 1996, 53(6): 3150–3160 [21] Coddens G, Lyonnard S, Sepilo B et al. Evidence for atomic hopping of Fe in perfectly icosahedral Al-Fe-Cu quasicrystals by57 Fe Moessbauer spectroscopy. J Phys, 1995, 5(7): 771–776 [22] Dolisek J, Ambrosini B, Vonlanthen P et al. Atomic motion in quasicrystalline Al70 Re8.6 Pd21.4 : A two-dimensional exchange NMR study. Phys Rev Lett, 1998, 81(17): 3671–3674 [23] Dolisek J, Apih T, Simsic M et al. Self-diﬀusion in icosahedral Al72.4 Pd20.5 Mn7.1 and phason percolation at low temperatures studied by

27

Al NMR. Phys Rev Lett, 1999,

82(3): 572–575 [24] Edagawa K, Kajiyama K. High temperature speciﬁc heat of Al-Pd-Mn and Al-Cu-Co quasicrystals, Mater Sci and Eng A, 2000, 294∼296(5): 646–649 [25] Edagawa K, Kajiyama K, Tamura R et al. High-temperature speciﬁc heat of quasicrystals and a crystal approximant. Mater Sci and Eng A, 2001, 312(1∼2): 293–298 [26] Ding D H, Yang W G, Hu C Z et al. Generalized elasticity theory of quasicrystals. Phys Rev B, 1993, 48(10): 7003–7010 [27] Fan T Y, Wang X F, Li W et al. Elasto-/hydro-dynamics of quasicrystals. Phil Mag, 2009, 89(6), 501–512 [28] Edagawa K, Takeuchi S. Elasticity, dislocations and their motion in quasicrystals, Dislocation in Solids. 2007, Chapter 76. ed. by Nabarro E R N, Hirth J P, 367–417 [29] Edagawa K, Giso Y. Experimental evaluation of phonon-phason coupling in icosahedral quasicrystals. Phil Mag, 2007, 87(1): 77–95 [30] Meng X M, Tong B Y, Wu Y K. Mechanical properties of quasicrystal Al65 Cu20 Co15 . Acta Metallurgica Sinica, 1994, 30(2): 61–64 (in Chinese)

52

Chapter 4

The physical basis of elasticity of quasicrystals

[31] Takeuchi S, Iwanhaga H, Shibuya T. Hardness of quasicrystals. Japanese J Appl Phys, 1991, 30(3): 561–562 [32] Francoual S, Levit F, de Boussieu M et al. Dynamics of phason ﬂuctuations in the i-Al-Pd-Mn quasicrystals. Phys Rev Lett, 2003, 91(22): 225501 [33] Coddens G. On the problem of the relation between phason elasticity and phason dynamics in quasicrystals. Eur Phys J B, 2006, 54(1): 37–65

Chapter 5 Elasticity theory of one-dimensional quasicrystals and simpliﬁcation As mentioned in Chapter 4, there exist one-, two- and three-dimensional quasicrystals. Each can be further divided into sub-classes by symmetry consideration. In the class of one-dimensional quasicrystals, the atom arrangement is quasiperiodic in one direction (say, z-direction) and is periodic in the plane perpendicular to the direction (xy-plane). Although the quasicrystal is one-dimensional, the structure is three-dimensional; it is generated in a three-dimensional body. One-dimensional quasicrystals can be regarded as a projection of periodic crystals in four-dimensional space to physical space. There are 4 non-zero displacements, ux , uy , uz and wz (wx = wy = 0). In this sense, the elasticity of one-dimensional quasicrystals is a four-dimensional problem.

Table 5.0-1

Systems, Laue classes and point groups of one-dimensional quasicrystals

Systems

Laue classes

Triclinic

1

Point groups ¯ 1 1

Monoclinic

2

2

mh

2/mh

m

2h /m

3

2h

Orthorhombic

4

2h mmh

Tetragonal

5

2h 2h 2 mm2 ¯ 4/mh 4 4 42h 2h 3 ¯ 3

¯ 42h m

4/mh mm

¯ 6m2h

6/mh mm

6

4mm

Rhombohedral

7

(or trigonal)

8

Hexgonal

9

32h 3m ¯ 3m ¯ 6/mh 6 6

10

62h 2h

6mm

mmmh

We here brieﬂy list the crystal systems and Laue classes of one-dimensional quasicrystals, in which the concept of point group must be concerned, and we do not concern with the concept of space group. T. Fan, Mathematical Theory of Elasticity of Quasicrystals and Its Applications © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2011

54

Chapter 5

Elasticity theory of one-dimensional quasicrystals and simpliﬁcation

In the following we discuss the elasticity of quasicrystals of certain types listed in Table 5.0-1.

5.1

Elasticity of hexagonal quasicrystals

As pointed out previously that for one-dimensional quasicrystals, there are phonon displacements ux , uy , uz and phason displacement wz (and wx = wy = 0), the corresponding strains are εxx =

εyz = εzy

1 = 2

∂ux ∂uy , εyy = , ∂x ∂y

wzx =

∂wz , ∂x

∂uz , ∂z

1 , εzx = εxz = 2 1 ∂ux ∂uy + , = 2 ∂y ∂x

∂uz ∂uy + ∂y ∂z εxy = εyx

εzz =

wzy =

∂wz , ∂y

wzz =

(5.1-1)

∂uz ∂ux + ∂x ∂z

,

∂wz ∂z

(5.1-2)

and other wij = 0. Formulas (5.1-1) and (5.1-2) hold for all one-dimensional quasicrystals. In this section we only discuss one-dimensional hexagonal quasicrytals. Let the strains given by (5.1-1) and (5.1-2) be expressed by the vector with 9 components, [ε11 , ε22 , ε33 , 2ε23 , 2ε31 , 2ε12 , w33 , w31 , w32 ]

(5.1-3)

[εxx , εyy , εzz , 2εyz , 2εzx , 2εxy , wzz , wzx , wzy ] ,

(5.1-4)

or

The corresponding stresses are [σxx , σyy , σzz , σyz , σzx , σxy , Hzz , Hzx , Hzy ] The elastic constant matrix is ⎡ C11 C12 ⎢ C12 C11 ⎢ ⎢ C13 C13 ⎢ ⎢ 0 0 ⎢ 0 0 [CKR] = ⎢ ⎢ ⎢ 0 0 ⎢ ⎢ R1 R1 ⎢ ⎣ 0 0 0 0

C13 C13 C33 0 0 0 R2 0 0

0 0 0 C44 0 0 0 0 R3

0 0 0 0 C44 0 0 R3 0

0 0 0 0 0 C66 0 0 0

R1 R1 R2 0 0 0 K1 0 0

0 0 0 0 R3 0 0 K2 0

(5.1-5)

0 0 0 R3 0 0 0 0 K2

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

5.1

Elasticity of hexagonal quasicrystals

55

where a short notation for the phonon elastic constant tensor is used, i.e., index 11→1, 22→2, 33→3, 23→4, 31→5, 12→6 and Cijkl is denoted as Cpq accordingly, C11 = C1111 = C2222 ,

C12 = C1122 ,

C33 = C3333 ,

C44 = C2323 = C3131 ,

C1111 − C1122 C11 − C12 = . 2 2 There are ﬁve independent phonon elastic constants, in addition K1 = K3333 , K2 = K3131 = K3232 , i.e., two independent phason elastic constants, and R1 = R1133 = R2233 , R2 = R3333 , R3 = R2332 = R3131 , i.e., three phonon-phason coupling elastic constants. The corresponding stress-strain relations are: ⎧ σxx = C11 εxx + C12 εyy + C13 εzz + R1 wzz , ⎪ ⎪ ⎪ ⎪ σyy = C12 εxx + C11 εyy + C13 εzz + R1 wzz , ⎪ ⎪ ⎪ ⎪ σzz = C13 εxx + C13 εyy + C33 εzz + R2 wzz , ⎪ ⎪ ⎪ ⎪ ⎨σyz = σzy = 2C44 εyz + R3 wzy , σzx = σxz = 2C44 εzx + R3 wzx , (5.1-6) ⎪ ⎪ ⎪σxy = σyx = 2C66 εxy , ⎪ ⎪ ⎪ ⎪Hzz = R1 (εxx + εyy ) + R2 εzz + K1 wzz , ⎪ ⎪ ⎪ ⎪Hzx = 2R3 εzx + K2 wzx , ⎪ ⎩ Hzy = 2R3 εyz + K2 wzy C13 = C1133 = C2233 ,

C66 =

and other Hij = 0. The equilibrium equations are: ⎧ ∂σ ∂σxy ∂σxz xx ⎪ + + = 0, ⎪ ⎪ ⎪ ∂x ∂y ∂z ⎪ ⎪ ⎪ ⎪ ∂σyy ∂σyz ∂σyx ⎪ ⎪ + + = 0, ⎨ ∂x ∂y ∂z ∂σzy ∂σzz ∂σzx ⎪ ⎪ ⎪ + + = 0, ⎪ ⎪ ∂x ∂y ∂z ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂Hzx + ∂Hzy + ∂Hzz = 0. ∂x ∂y ∂z

(5.1-7)

The above results are obtained by Wang et al[1] . The elastic equilibrium problem of one-dimensional hexagonal quasicrystals is more complicated than that of three-dimensional classical elasticity. Here there are 4 displacements, 9 strains and 9 stresses, which added up to 22 ﬁeld variables. The corresponding ﬁeld equations are also 22, consisting of 4 for equilibrium equations and 9 equations of deformation geometry and 9 stress-strain relations. We will present a rigorous treatment of the problem later on. In the following we give a simpliﬁed treatment.

56

5.2

Chapter 5

Elasticity theory of one-dimensional quasicrystals and simpliﬁcation

Decomposition of the problem into plane and anti-plane problems

If there is a straight dislocation or a Griﬃth crack along the direction of the atom quasiporiodic arrangement, the deformation is independent of the z-axis, say ∂ = 0, ∂z

(5.2-1)

Then ∂ui = 0, ∂z

i = 1, 2, 3,

∂wz = 0. ∂z

(5.2-2)

Hence, εzz = wzz = 0,

εyz = εzy =

1 ∂uz , 2 ∂y

∂σij = 0, ∂z

εzx = εxz =

∂Hij = 0. ∂z

The generalized Hooke’s law is simpliﬁed as ⎧ σxx = C11 εxx + C12 εyy , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ σyy = C12 εxx + C11 εyy , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ σxy = σyx = 2C66 εxy , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ σzz = C13 (εxx + εyy ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ σyz = σzy = 2C44 εyz + R3 wzy , ⎪ ⎪ ⎪ ⎪ σzx = σxz = 2C44 εzx + R3 wzx , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Hzz = R1 (εxx + εyy ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ H = 2R3 εzx + K2 wzx , ⎪ ⎪ zx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Hzy = 2R3 εyz + K2 wzy .

1 ∂uz , 2 ∂x

(5.2-3)

(5.2-4)

(5.2-5)

In the absence of the body force and generalized body force the equilibrium equations are ∂σxx ∂σxy ∂σyx ∂σyy ∂σzx ∂σzy + = 0, + = 0, + = 0, (5.2-6) ∂x ∂y ∂x ∂y ∂x ∂y ∂Hzx ∂Hzy + = 0. ∂x ∂y

(5.2-7)

5.2

Decomposition of the problem into plane and anti-plane problems

57

Equations(5.1-2), (5.1-3), (5.2-5)∼(5.2-7)deﬁne two uncoupled problems[2] . The ﬁrst of them is ⎧ σxx = C11 εxx + C12 εyy , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ σyy = C12 εxx + C11 εyy , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ σxy = (C11 − C12 )εxy , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ σzz = C13 (εxx + εyy ), (5.2-8) Hzz = R1 (εxx + εyy ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂σyx ∂σyy ∂σxx ∂σxy ⎪ ⎪ + = 0, + = 0, ⎪ ⎪ ∂x ∂y ∂x ∂y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ux ∂uy 1 ∂uy ∂ux ⎪ ⎩ εxx = , εyy = , εxy = + , ∂x ∂y 2 ∂x ∂y this is the classical plane elasticity of conventional hexagonal crystals. The second one is ⎧ σyz = σzy = 2C44 εyz + R3 wzy , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ σzx = σxz = 2C44 εzx + R3 wzx , ⎪ ⎪ ⎪ ⎪ Hzx = 2R3 εzx + K2 wzx , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Hzy = 2R3 εyz + K2 wzy , ⎪ ⎪ ⎨ (5.2-9) ∂Hzx ∂σzy ∂σzx ∂σzy ⎪ + = 0, + = 0, ⎪ ⎪ ⎪ ∂x ∂y ∂x ∂y ⎪ ⎪ ⎪ ⎪ ⎪ 1 ∂uz 1 ∂uz ⎪ ⎪ εzx = = εxz , εzy = = εyz , ⎪ ⎪ 2 ∂x 2 ∂y ⎪ ⎪ ⎪ ⎪ ⎪ ∂wz ∂wz ⎪ ⎪ ⎩ wzx = , wzy = , ∂x ∂y which is a phonon-phason coupling elasticity problem, involving only the displacements uz and wz . It is an anti-plane elasticity problem. The plane elasticity described by (5.2-8) has been studied, extensively using the stress function approach, e.g. it introduces σxx =

∂2U , ∂y2

σyy =

∂2U , ∂x2

σxy = −

∂2U , ∂x∂y

then equations (5.2-8) are reduced to solve ∇2 ∇2 U = 0. The problem is considerably discussed in crystal (or classical) elasticity, we do not consider it here.

58

Chapter 5

Elasticity theory of one-dimensional quasicrystals and simpliﬁcation

We are interested in the phonon-phason coupling anti-plane elasticity described by (5.2-9), which may bring new insight into the scope of elasticity of quasicrystals. Substituting the deformation geometry relations into the stress-strain relations, then into the equilibrium equations yields the ﬁnal governing equations such as

C44 ∇2 uz + R3 ∇2 wz = 0, R3 ∇2 uz + K2 ∇2 wz = 0,

(5.2-10)

because C44 K2 − R32 = 0, we have ∇2 uz = 0, where ∇2 =

∇2 wz = 0,

(5.2-11)

∂ ∂ + 2 , so uz and wz are harmonic functions. 2 ∂x ∂y

It is well known that the two-dimensional harmonic functions uz and wz can be a real part or an imaginary part of any analytic functions φ(t) and ψ(t) of complex √ variable t = x + iy, i = −1, respectively, i.e.,

uz (x, y) = Re φ(t),

(5.2-12)

wz (x, y) = Re ψ(t). In this version, equations (5.2-11) should be identically satisﬁed. Determination of φ(t) and ψ(t) depends upon appropriate boundary conditions, which will be discussed in detail in Chapters 7 and 8. The complex variable function method for solving elasticity of one-, two- and three-dimensional quasicrystals will be summarized fully in Chapter 11.

5.3

Elasticity of monoclinic quasicrystals

Decomposition procedure suggested by the author e.g. in Ref.[2,3] is applicable not only for hexagonal quasicrystals but also for other one-dimensional quasicrystals. Herein we discuss monoclinic quasicrystals. For this kind of one-dimensional quasicrystals, there are 25 non-zero elastic constants in total, namely, C1111 , C2222 , C3333 , C1122 , C1133 , C1112 , C2233 , C2212 , C3312 , C3232 , C3231 , C3131 , C1212 for the phonon ﬁeld, K3333 , K3131 , K3232 , K3132 for the phason ﬁeld and R1133 , R2233 , R3333 , R1233 , R2331 , R2332 , R3132 , R1233 for the phonon-phason coupling. The generalized Hooke’s law for the case is as follows[1] :

5.3

Elasticity of monoclinic quasicrystals

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

59

σxx = C11 εxx + C12 εyy + C13 εzz + 2C16 εxy + R1 wzz , σyy = C12 εxx + C22 εyy + C23 εzz + 2C26 εxy + R2 wzz , σzz = C13 εxx + C23 εyy + C33 εzz + 2C36 εxy + R3 wzz , σyz = σzy = 2C44 εyz + 2C45 εzx + R4 wzx + R5 wzy ,

σzx = σxz = 2C45 εyz + 2C55 εzx + R6 wzx + R7 wzy , ⎪ ⎪ ⎪ ⎪ σxy = σyx = C16 εxx + C26 εyy + C36 εzz + 2C66 εxy + R8 wzz , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Hzx = 2R4 εyz + 2R6 εzx + K1 wzx + K4 wzy , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Hzy = 2R5 εyz + 2R7 εzx + K4 wzx + K2 wzy , ⎪ ⎪ ⎪ ⎩ Hzz = R1 εxx + R2 εyy + R3 εzz + 2R8 εxy + K3 wzz ,

(5.3-1)

where the short notation is used for the phonon elastic constant tensor, such that the index 11→1,22→2,33→3, 23→4,31→5,12→6 and Cijkl is denoted as Cpq ; for the phason elastic constants, K3131 = K1 , K3232 = K2 , K3333 = K3 ,K3132 = K4 and for the phonon-phason coupling elastic constants, R1133 = R1 , R2233 = R2 , R3333 = R3 , R2331 = R4 , R2332 = R5 , R3131 = R6 , R3132 = R7 , R1233 = R8 . Under the assumption (5.2-1), the problem can be decomposed into two separate problems as follows: ⎧ σxx = C11 εxx + C12 εyy + 2C16 εxy , ⎪ ⎪ ⎪ ⎪ ⎨ σyy = C12 εxx + C22 εyy + 2C26 εxy , σxy = σyx = C16 εxx + C26 εyy + 2C66 εxy , (5.3-2) ⎪ ⎪ σ = C ε + C ε + 2C ε , ⎪ zz 13 xx 23 yy 36 xy ⎪ ⎩ Hzz = R1 εxx + R2 εyy + R3 εzz + 2R8 εxy , and

⎧ σyz = σzy = 2C44 εyz + 2C45 εzx + R4 wzx + R5 wzy , ⎪ ⎪ ⎨ σzx = σxz = 2C45 εyz + 2C55 εzx + R6 wzx + R7 wzy , Hzx = 2R4 εyz + 2R6 εzx + K1 wzx + K4 wzy , ⎪ ⎪ ⎩ Hzy = 2R5 εyz + 2R7 εzx + K4 wzx + K2 wzy

(5.3-3)

in which the problem described by equations (5.3-2) is plane elasticity of monocline crystals. By introducing the displacement potential G(x, y), ∂2 ∂2 ∂2 ux = C16 2 + C26 2 + (C12 + C66 ) G, ∂x ∂y ∂x∂y ∂2 ∂2 ∂2 G, uy = − C11 2 + C66 2 + 2C16 ∂x ∂y ∂x∂y the equations of elasticity reduce to ∂4 ∂4 ∂4 ∂4 ∂4 + c5 4 G = 0 c1 4 + c2 3 + c3 2 2 + c4 ∂x ∂x ∂y ∂x ∂y ∂x∂y3 ∂y

60

Chapter 5

Elasticity theory of one-dimensional quasicrystals and simpliﬁcation

with constants 2 − C11 C66 , c2 = 2(C16 C12 − C11 C26 ), c1 = C16 2 − 2C16 C26 + 2C12 C66 − C11 C22 , c3 = C12 2 c4 = 2(C26 C12 − C16 C22 ), c5 = C26 − C22 C66 .

Since this is classical elasticity and does not have a direct connection to phason elasticity of one-dimensional quasicrystal, we do not consider it further. We are interested in the problem described by equation (5.3-3), which is a phonon-phason coupling problem. Substituting the equations of deformation geometry into the stress-strain relations and then into the equilibrium equations, we obtain the governing equation: ∂4 ∂4 ∂4 ∂4 ∂4 (5.3-4) a1 4 + a2 3 + a3 2 2 + a4 + a5 4 F = 0, ∂x ∂x ∂y ∂x ∂y ∂x∂y 3 ∂y where

and

⎧ ∂2 ∂2 ∂2 ⎪ ⎪ u F, = R + R + (R + R ) 6 5 4 7 ⎪ ⎨ z ∂x2 ∂y 2 ∂x∂y ⎪ ⎪ ∂2 ∂2 ∂2 ⎪ ⎩ wz = − C55 + C + 2C F, 44 45 ∂x2 ∂y 2 ∂x∂y

⎧ ⎨ a1 = R62 − K1 C56 , a2 = 2(R6 (R4 + R7 ) − K1 C45 − K4 C55 ), a3 = 2R5 R6 + (R4 + R7 )2 − K1 C44 − K2 C55 − 4K4 C45 , ⎩ a4 = 2[R5 (R4 + R7 ) − K2 C45 − K4 C44 ], a5 = R52 − K2 C44 .

(5.3-5)

(5.3-6)

In the following, we consider only anti-plane elasticity of monocline quasicrystals, its solution has the complex representation F (x, y) = 2Re

2

Fk (zk ),

zk = x + μk y,

(5.3-7)

k=1

where Fk (zk ) are analytic functions of zk and μk = αk + iβk

(5.3-8)

are the distinct complex parameters to be determined by the characteristic equation a5 μ4 + a4 μ3 + a3 μ2 + a2 μ + a1 = 0

(5.3-9)

and μ1 = μ2 . If the roots of equation (5.3-9) are double roots, i.e., μ1 = μ2 , then F (x, y) = 2Re[F1 (z1 ) + z¯1 F2 (z1 )],

z1 = x + μ1 y.

(5.3-10)

5.4

Elasticity of orthorhombic quasicrystals

61

Substitution of formula (5.3-7) into (5.3-5) and then into (5.3-3) we arrive at the complex representation of the displacements and stresses as follows: ⎧ 2 ⎪ ⎪ ⎪ ⎪ = 2Re [R6 + (R4 + R7 )μk + R5 μ2k ]fk (zk ), u z ⎪ ⎪ ⎪ ⎪ k=1 ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ w = −2Re (C55 + 2C45 μk + C44 μ2k )fk (zk ), ⎪ z ⎪ ⎪ ⎪ k=1 ⎪ ⎪ ⎪ ⎪ ⎪ = σ σ zy yz ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ = 2Re [R6 C45 − R4 C55 + (R6 C44 − R4 C45 + R7 C45 − R5 C55 )μk ⎪ ⎪ ⎪ ⎪ k=1 ⎪ ⎪ ⎪ ⎪ +(R7 C44 − R5 C45 )μ2k ]fk (zk ), ⎪ ⎪ ⎪ ⎪ ⎨ σzx = σxz 2 ⎪ ⎪ ⎪ ⎪ = 2Re [R4 C55 − R6 C45 + (R5 C55 + R4 C45 − R6 C44 − R5 C55 )μk ⎪ ⎪ ⎪ ⎪ k=1 ⎪ ⎪ ⎪ ⎪ ⎪ +(R3 C45 − R7 C44 )μ2k ]μk fk (zk ), ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ = 2Re [(R7 + R5 μk )(R6 + R4 μk + R7 μk + R5 μ2k ) H ⎪ zx ⎪ ⎪ ⎪ k=1 ⎪ ⎪ ⎪ ⎪ ⎪ −(K4 + K2 μk )(C55 + 2C45 μk + C44 μ2k )]fk (zk ), ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ H = 2Re [(R6 + R4 μk )(R6 + R4 μk + R7 μk + R5 μ2k ) zy ⎪ ⎪ ⎪ ⎪ k=1 ⎪ ⎪ ⎩ −(K + K μ )(C + 2C μ + C μ2 )]f (z ), 4

2

2 k

∂zk2

55

45 k

44 k

k

k

(5.3-11)

Fk (zk ).

where fk (zk ) ≡ ∂ Fk (zk ) = Determination of analytic functions Fk (zk ) must be supplemented with the boundary conditions of speciﬁc problems, which will be given in Chapters 7 and 8.

5.4

Elasticity of orthorhombic quasicrystals

In Table 5.0-1, we know that orthorhombic quasicrystals contain the points 2h 2h 2, mm2, 2h mmh and mmmh ,which belong to one Laue class 4. Due to the increase of symmetric elements of this quasicrystal system in comparison with the monocline quasicrystal system, one has C16 = C26 = C36 = C45 = K4 = R4 = R7 = R8 = 0.

(5.4-1)

Thus, the total number of non-zero elastic constants in the case reduces to 17,i.e., C11 , C22 , C33 , C12 , C44 , C55 , C66 for the phonon ﬁeld, K1 , K2 , K3 for the phason ﬁelds and R1 , R2 , R3 , R5 , R6 for the phonon-phason coupling ﬁeld.

62

Chapter 5

Elasticity theory of one-dimensional quasicrystals and simpliﬁcation

With results (5.4-1), then (5.3-6) is simpliﬁed to a5 = R52 −K2 C66 (5.4-2) and a1 and a5 remain the same as in (5.3-6). Then, equation (5.3-4) reduces to ∂4 ∂4 ∂4 a1 4 + a3 2 2 + a5 4 F = 0, (5.4-3) ∂x ∂x ∂y ∂y

a2 = a4 = 0,

a1 = R62 −K1 C55 ,

a3 = 2R5 R6 −K1 C44 −K2 C55 ,

The solution takes the form ⎧ 2 ⎪ ⎪ ⎪ [R6 + R5 μ2k ]fk (zk ), uz = 2Re ⎪ ⎪ ⎪ ⎪ k=1 ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ = −2Re (C55 + C44 μ2k )fk (zk ), w z ⎪ ⎪ ⎪ ⎪ k=1 ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ σ = σ = 2Re (R6 C44 − R5 C55 )μk fk (zk ), ⎪ zy yz ⎪ ⎪ ⎨ k=1 2 ⎪ (R5 C55 − R6 C44 )μ2k fk (zk ), σzx = σxz = 2Re ⎪ ⎪ ⎪ ⎪ k=1 ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ Hzy = 2Re [R5 R6 − K2 C55 + (R52 − K2 C44 )μ2k ) ⎪ ⎪ ⎪ ⎪ k=1 ⎪ ⎪ ⎪ − (K 4 + K2 μk )]μk fk (zk ), ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ [R62 − K1 C55 + (R5 R6 − K1 C44 )μ2k ]fk (zk ). ⎪ ⎩ Hzx = 2Re

(5.4-4)

k=1

in which fk (zk ) represents analytic function of zk .

5.5

Tetragonal quasicrystals

From Table 5.0-1 we know that one-dimensional tetragonal quasicrystal have 7 point groups, in which groups ¯42h m, 4mm, 42h 2h and 4/mh mm belong to Laue class 6. Besides (5.4-1), we also have C11 = C22 ,

C13 = C23 ,

C44 = C55 ,

K1 = K2 ,

R1 = R2 ,

R5 = R6 . (5.5-1)

The total number of non-zero elastic constants reduces to 11. With (5.5-1) and (5.4-2), one can ﬁnd the complex representation of the solution for the anti-plane elasticity of quasicrystals of Laue class 6. The point groups 4 ,¯4 and 4/mh belong to Laue class 5, in which the solution of the anti-plane elasticity can also be derived in a similar manner. The governing equation of anti-plane elasticity for this kind of quasicrystals is ∇2 ∇2 F = 0.

(5.5-2)

5.6

5.6

The space elasticity of hexagonal quasicrystals

63

The space elasticity of hexagonal quasicrystals

The decomposition procedure proposed in Ref[2] has been developed to simplify the elasticity of various systems of one-dimensional quasicrystals, in the previous sections. The main feature of the procedure lies in decomposition a space (threedimensional) elasticity into plane elasticity and anti-plane elasticity for the quasicrystalline material under study. The merits will be shown in Chapters 7 and 8 and other chapters. In some cases, when the procedure is not applicable, we have to solve the problem of space elasticity. As an example, we discuss the solution of space elasticity of hexagonal quasicrystals here. Substituting (5.5-1) into (5.1-6), then into (5.1-7) we obtain the equilibrium equations expressed in terms of the displacements as follows ⎧ ∂2 ∂2 ∂2 ∂ 2 uy ⎪ ⎪ C + C + C + (C − C ) u ⎪ 11 66 44 x 11 66 ⎪ ⎪ ∂x2 ∂y 2 ∂z 2 ∂x∂y ⎪ ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ ∂ uz ∂ wz ⎪ ⎪ + (R1 + R3 ) = 0, +(C13 + C44 ) ⎪ ⎪ ∂x∂z ∂x∂z ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ 2 ux ∂2 ∂2 ∂2 ⎪ ⎪ (C11 − C66 ) + C + C + C uy ⎪ 66 11 44 ⎪ ∂x∂y ∂x2 ∂y2 ∂z 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ 2 uz ∂ 2 wz ⎪ ⎪ + (R1 + R3 ) = 0, +(C13 + C44 ) ⎪ ⎨ ∂y∂z ∂y∂z (5.6-1) 2 ⎪ ⎪ ∂2 ∂2 ∂2 ∂ ux ∂ 2 uy ⎪ ⎪ (C13 + C44 ) + + C44 2 + C44 2 + C33 2 uz ⎪ ⎪ ∂x∂z ∂y∂z ∂x ∂y ∂z ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂2 ∂2 ⎪ ∂2 ⎪ ⎪ + R3 + 2 + R2 2 wz = 0, ⎪ 2 ⎪ ∂x ∂y ∂z ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ∂ 2 ux ∂2 ∂ ∂ 2 uy ∂2 ⎪ ⎪ ⎪ + R ) + (R + + R + R uz 1 3 3 2 ⎪ ⎪ ∂x∂z ∂y∂z ∂x2 ∂y 2 ∂z 2 ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ∂ ∂2 ∂2 ⎪ ⎪ ⎩ + K wz = 0. + K2 + 1 ∂x2 ∂y 2 ∂z 2 If we introduce 4 displacement functions as ⎧ ∂ ∂F4 ∂ ∂F4 ⎪ ⎨ ux = (F1 + F2 + F3 ) − , uy = (F1 + F2 + F3 ) + , ∂x ∂y ∂y ∂x ⎪ ⎩ uz = ∂ (m1 F1 + m2 F2 + m3 F3 ), wz = ∂ (l1 F1 + l2 F2 + l3 F3 ) ∂z ∂z

(5.6-2)

and ∇2i Fi = 0,

i = 1, 2, 3, 4,

(5.6-3)

64

Chapter 5

Elasticity theory of one-dimensional quasicrystals and simpliﬁcation

∇2i =

2 ∂2 ∂2 2 ∂ + + γ , i ∂x2 ∂y 2 ∂z 2

i = 1, 2, 3, 4

(5.6-4)

with mi , li and γi deﬁned by C44 + (C13 + C44 )mi + (R1 + R3 )li C11 C33 mi + R2 li = C13 + C44 + C44 mi + R3 li R2 mi + K1 li C44 = = γi2 , i = 1, 2, 3, = γ42 , R1 + R3 + R3 mi + K2 li C66

(5.6-5)

consequently, equations (5.6-1) are identically satisﬁed. The ﬁnal governing equations (5.6-3) are three-dimensional harmonic equations. By substituting (5.6-2) into (5.1-1), (5.1-2), and then into (5.1-6), we express the stresses in terms of F1 , F2 , F3 and F4 as ∂2 ∂ 2 F4 ∂2 σxx = C11 2 + (C11 − 2C66 ) 2 (F1 + F2 + F3 ) − 2C66 ∂x ∂y ∂x∂y 2 2 ∂ ∂ + C13 2 (m1 F1 + m2 F2 + m3 F3 ) + R1 2 (l1 F1 + l2 F2 + l3 F3 ), ∂z ∂z ∂2 ∂ 2 F4 ∂2 σyy = (C11 − 2C66 ) 2 + C11 2 (F1 + F2 + F3 ) + 2C66 ∂x ∂y ∂x∂y ∂2 ∂2 + C13 2 (m1 F1 + m2 F2 + m3 F3 ) + R1 2 (l1 F1 + l2 F2 + l3 F3 ), ∂z ∂z ∂2 2 ∂2 2 2 σzz = −C13 2 (γ1 F1 + γ2 F2 + γ3 F3 ) + C33 2 (m1 F1 + m2 F2 + m3 F3 ) ∂z ∂z ∂2 + R2 2 (l1 F1 + l2 F2 + l3 F3 ), ∂z 2 ∂2 ∂2 ∂ σxy = σyx = 2C66 − (F1 + F2 + F3 ) + C66 F4 , ∂x∂y ∂x2 ∂y 2 ∂2 [(m1 + 1)F1 + (m2 + 1)F2 + (m3 + 1)F3 ] σyz = σzy = C44 ∂y∂z ∂ 2 F4 ∂2 + C44 + R3 (l1 F1 + l2 F2 + l3 F3 ), ∂x∂z ∂y∂z ∂2 [(m1 + 1)F1 + (m2 + 1)F2 + (m3 + 1)F3 ] σzx = σxz = C44 ∂x∂z ∂ 2 F4 ∂2 + R3 (l1 F1 + l2 F2 + l3 F3 ), − C44 ∂y∂z ∂x∂z ∂2 ∂2 Hzz = −R1 2 (γ12 F1 + γ22 F2 + γ32 F3 ) + R2 2 (m1 F1 + m2 F2 + m3 F3 ) ∂z ∂z ∂2 + K1 2 (l1 F1 + l2 F2 + l3 F3 ), ∂z

References

65

∂2 [(m1 + 1)F1 + (m2 + 1)F2 + (m3 + 1)F3 ] ∂x∂z ∂ 2 F4 ∂2 + K2 (l1 F1 + l2 F2 + l3 F3 ), − R3 ∂y∂z ∂x∂z ∂2 = R3 [(m1 + 1)F1 + (m2 + 1)F2 + (m3 + 1)F3 ] ∂y∂z ∂ 2 F4 ∂2 + K2 (l1 F1 + l2 F2 + l3 F3 ). + R3 ∂y∂z ∂y∂z

Hzx = R3

Hzy

Harmonic equations (5.6-3) can be solved under appropriate boundary conditions, which will be discussed in Chapter 8.

5.7

Other results of elasticity of one-dimensional quasicrystals

There are other results of elasticity of one-dimensional quasicrystals, e.g. Fan et al[8] on elasticity of one-dimensional quasicrystal-crystal coexisting phase (refer to Chapter 7 for a brief discussion), Chen et al[9] , Wang et al,[10] etc on the three-dimensional elasticity of hexagonal quasicrystals, they carried out considerable research in the area, and have got quite a lot of achievements. Due to the space limitation, the details of their work could not be handled here and reader can refer to the original literatures.

References [1] Wang R H, Yang W G, Hu C Z et al. Point and space groups and elastic behaviour of one-dimensional quasicrystals. J Phys Condens Matter, 1997, 9(11): 2411–2422 [2] Fan T Y. Mathematical theory of elasticity and defects of quasicrystals. Advances in Mechanics, 2000, 30(2): 161–174 (in Chinese) [3] Fan T Y, Mai Y W. Elasticity theory, fracture mechanics and some relevant thermal properties of quasicrystalline materials. Appl Mech Rev, 2004, 57(5): 325–344 [4] Liu G T, Fan T Y, Guo R P. Governing equations and general solutions of plane elasticity of one-dimensional quasicrystals. Int J Solid and Structures, 2004, 41(14): 3949–3959 [5] Liu G T. The complex variable function method of the elastic theory of quasicrystals and defects and auxiliary equation method for solving some nonlinear evalution equations, Dissertation. Beijing Institute of Technology, 2004 (in Chinese) [6] Peng Y Z, Fan T Y. Elastic theory of 1-D quasiperiodic stacking 2-D crystals. J. Phys: Condens Matter, 2000, 12(45): 9381–9387 [7] Peng Y Z. Study on elastic three-dimensional problems of cracks for quasicrystals, Dissertation. Beijing Institute of Technology, 2002 (in Chinese)

66

Chapter 5

Elasticity theory of one-dimensional quasicrystals and simpliﬁcation

[8] Fan T Y, Fan L, Xie L Y et al. Study on interface of quasicrystal-crystal. J. Phys.: Condens. Matter, 2009, submitted [9] Chen W Q, Ma Y L, Ding H J. On three-dimensional elastic problems of onedimensional hexagonal quasicrystal bodies. Mech Res Commun, 2004, 31(5): 633– 641 [10] Wang X. The general solution of one-dimensional hexagonal quasicrystal. Appl Math Mech, 2006, 33(4): 576–580

Chapter 6 Elasticity of two-dimensional quasicrystals and simpliﬁcation As has been shown in Chapter 5, in one-dimensional quasicrystals, elasticity can be decomposed into plane elasticity and anti-plane elasticity in case that the conﬁguration is independent of the quasiperiodic axis. In this case, the plane elasticity is a classical elasticity problem and its solution is well known, whereas the anti-plane elasticity is a problem concerned with the quasiperiodic structure, which is our main concern. This decomposition leads to great simpliﬁcations for the solution. We now consider elasticity of two-dimensional quasicrystals, which is mathematically much more complicated than that of one-dimensional quasicrystals. The decomposition procedure developed in Chapter 5 hints that the elasticity of twodimensional quasicrystals may somehow also be made of a decomposition for a wide range of applications. In this way the problem can be greatly simpliﬁed and it is helpful to solve the boundary value problems by analytic method. Two-dimensional quasicrystals so far observed cover four systems, i.e., those involving ﬁve-fold, eight-fold, ten-fold and twelve-fold symmetries, named the pentagonal, octagonal, decagonal and dodecagonal respectively, and among them there are diﬀerent Laue classes. The importance of two-dimensional quasicrystals is only less than that of the three-dimensional icosahedral quasicrystals. To date, among the almost 200 quasicrystals, there are 100 icosahedral quasicrystals, and 70 decagonal quasicrystals. These two kinds of quasicrystals constitute the majority of the material. The elasticity of icosahedral quasicrystals will be discussed in Chapter 9. We will, at ﬁrst, give a brief description of the point groups and Laue classes of the two-dimensional quasicrystals. There are 31 kinds of crystallographic point groups and 26 kinds of noncrystallographic point groups of the quasicrystals, the former will not be discussed here and we focus on the latter, which is further divided into eight Laue classes and four quasicrystal systems observed so far, as listed in Table 6.0-1. Like that in the one-dimensional quasicrystals, the phonon ﬁeld of two-dimensional quasicrystals is transversally isotropic. If we take the xy- plane (or x1 x2 - plane) as T. Fan, Mathematical Theory of Elasticity of Quasicrystals and Its Applications © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2011

68

Chapter 6

Elasticity of two-dimensional quasicrystals and simpliﬁcation

the quasiperiodic plane, and axis z (or x3 ) as the periodic axis, then xy- plane is the elasticity isotropic plane, within which the elastic constants are C1111 = C2222 = C11 , C1122 = C12 , C1212 = C1111 − C1122 = C11 − C12 = 2C66 .

Table 6.0-1

Quasicrystal systems, Laue classes and point groups of two-dimensional quasicrystal

Quasicrystal systems Pentagonal

Laue classes 11 12 13 14 15 16 17 18

Decagonal Octagonal Dodecagonal

Point groups 5, ¯ 5 5m, 52, ¯ 5m 10, 10, 10/m 10mm, 1022, 10m2, 10/mmm 8, ¯ 8, 8/m 8mm, 822, ¯ 8m2, 8/mmm 12, 12, 12/m 12mm, 1222, 12m2, 12/mmm

This shows that C66 is not independent. Other independent elastic constants are out of the plane, i.e., C2323 = C3131 = C44 , C1133 = C2233 = C13 , C3333 = C33 . which are listed in Table 6.0-2. Table 6.0-2 11 22 33 23 31 12

Phonon elastic constants in two-dimensional quasicrystals (Cijkl ) 11 C11 C12 C13 0 0 0

22 C12 C11 C13 0 0 0

33 C13 C13 C33 0 0 0

23 0 0 0 C44 0 0

31 0 0 0 0 C66 0

12 0 0 0 0 0 C66

The relevant phason elastic constants and phonon-phason coupling elastic constants are listed in Tables 6.0-3∼6.0-6. Table 6.0-3 11 22 23 12 13 21

11 K1 K2 K7 0 K6 0

Phason elastic constants for Laue class 11 (Kijkl ) 22 K2 K1 K7 0 K6 0

23 K7 K7 K4 K6 0 −K6

12 0 0 K6 K1 −K7 −K2

13 K6 K6 0 −K7 K4 K7

21 0 0 −K6 −K2 K7 K1

Chapter 6

Elasticity of two-dimensional quasicrystals and simpliﬁcation Table 6.0-4

69

Phonon-phason coupling elastic

constants for Laue class 11 (Rijkl )

XXX

XXXwij X

εij

X X

11 12 33 23 31 12

11

22

23

12

13

21

R1 −R1 0 R4 −R3 R2

R1 −R1 0 −R4 R3 R2

R6 −R6 0 0 0 −R5

R2 −R2 0 R3 R4 −R1

R5 −R5 0 0 0 R6

−R2 R2 0 R3 R4 R1

For Laue class 12: If 2//x1 , m⊥x1 : K6 = R2 = R3 = R6 = 0. If 2//x2 , m⊥x2 :K7 = R2 = R4 = R6 = 0. For Laue class 13: K6 = K7 = R3 = R4 = R5 = R6 = 0. For Laue class 14: K6 = K7 = R2 = R3 = R4 = R5 = R6 = 0.

Table 6.0-5

The phason elastic constants for Laue class 15 (Kijkl )

11 K1 K2 0 K5 0 K5

11 22 23 12 13 21

22 K2 K1 0 −K5 0 −K5

23 0 0 K4 0 0 0

12 K5 −K5 0 K 0 K3

13 0 0 0 0 K4 0

21 K5 −K5 0 K3 0 K

K = K1 + K2 + K3 .

Table 6.0-6

for Laue class 15 (Rijkl )

``` εij

The phonon-phason coupling elastic constants

``` wij ``` ` ` 11 22 33 23 31 12

11

22

23

12

13

21

R1 −R1 0 0 0 R2

R1 −R1 0 0 0 R2

0 0 0 0 0 0

R2 −R2 0 0 0 −R1

0 0 0 0 0 0

−R2 R2 0 0 0 R1

For Laue class 16: K6 = R2 = 0. For Laue class 17: the constants Kijkl are the same as table 6.0-5 and Rijkl = 0. For Laue class 18: the constants Kijkl are the same as those of Laue 16 and Rijkl = 0.

The experimental measurement of the material data for diﬀerent quasicrystals is essential in studying the elasticity. We here list some data for decagonal quasicrystals in Tables 6.0-7∼6.0-9.

70

Chapter 6

Table 6.0-7

Elasticity of two-dimensional quasicrystals and simpliﬁcation

Phonon elastic constants of decagonal quasicrystal (the unit of Cij , B, G is GPa) by experimental measurement[1]

Alloy Al-Ni-Co

C11 234.33

C33 232.22

C44 70.19

C12 57.41

C13 66.63

B 120.25

G 79.98

ν 0.228

in which B is the bulk modulus, G the shear modulus, and ν the Poisson’s ratio, respectively. The phason elastic constants, for a decagonal Al-Ni-Co quasicrystal, anisotropic diﬀuse scattering has been observed in synchrotron X-ray diﬀraction measurements[2] . It has been shown that the measurement can attributed the phason elastic constants, although no quantitative evaluation on K1 and K2 . The Monto-Carlo simulation was used to evaluate the phason elastic constants, e.g. given by Table 6.0-8[3] . Table 6.0-8

Phason elastic constants of an Al-Ni-Co decagonal quasicrystal by Monte-Carlo simulation[3] K1 /(1012 dyn/cm2 ) 1.22

Alloy Al-Ni-Co

K2 /(1012 dyn/cm2 ) 0.24

2

where 1GPa = 1010 dyn/cm . It should be noted that the values given by the MonteCarlo simulation are subjected to veriﬁcation. Recently the experimental measurement for phonon-phason coupling elastic constants for decagonal quasicrystals have been achieved, the results are listed in Table 6.0-9. Table 6.0-9

Coupling elastic constants for point group 10, 10, Al-Ni-Co decagonal quasicrystals[4]

Alloy Al-Ni-Co

R1 /GPa −1.1

|R2 |/GPa <0.2

Tables 6.0-2∼6.0-6 indicate that both the phonon elasticity of two-dimensional quasicrystals, and the phason and phonon-phason coupling elasticity are threedimensional, not two-dimensional. In general, they can not be reduced to twodimensional problems. In this case there are 29 ﬁeld variables and 29 ﬁeld equations, it is very diﬃcult to obtain their analytic solution. In practice there is often a case in which the conﬁguration of a system is uniform along a periodic axis, say the axis z, both physically and geometrically, so that the ﬁeld variables are free from the coordinate z, i.e., ∂ = 0. ∂z

(6.0-1)

6.1

Basic equations of plane elasticity of two-dimensional quasicrystals: point . . .

71

Under this condition, the elasticity of two-dimensional quasicrystals can be decomposed into plane elasticity of quasicrystal and anti-plane elasticity of crystal, where the latter is a pure phonon or classical elasticity, whose governing equation and boundary conditions are decoupled with the plane elasticity of quasiperiodic structure, which can be treated separately. Uniform conﬁguration along the z-axis, in physical and geometrical sense, represents a number of useful physical problems, for example, a straight dislocation line or a Griﬃth crack along the direction, and so on. This shows that the decomposition procedure presents important physical meaning. In what follows, we derive the ﬁnal governing equations for the plane and antiplane elasticity of the four diﬀerent systems, by introducing some displacement or stress potential functions. We will see that the ﬁeld equations are dramatically simpliﬁed, this is helpful to solve them by analytic method, Furthermore we use the Fourier analysis or complex variable function method to construct analytic solutions of some boundary value problems of the equations. But the mathematical methods and numerous exact solutions for diﬀerent boundary value problems will be performed in Chapters 7 and 8 in detail.

6.1

Basic equations of plane elasticity of two-dimensional quasicrystals: point groups 5m and 10mm in ﬁve- and tenfold symmetries

Following the introduction given in Chapter 1, it is understood that the sign 10 in the symbol 10mm represents a ten-fold rotation symmetry and that m signiﬁes a mirror symmetry. Thus, the notation 10mm therefore means a combined operation of one rotation symmetry and two mirror symmetries. For other signs the explanation is similar. The point groups 5m and 10mm quasicrystals are quasicrystals with ﬁve-fold and ten-fold symmetries repectively, whose plane elasticity has been studied earlier, see e.g. in reviews given by Fan and Mai[5] or by Hu et al[6] or by Bohsung et al[7] . The ﬁrst solution of dislocation of pentagonal quasicrystals was given by De and Pelcovis[8] , according to our understanding, it is a solution for point group 5m two-dimensional quasicrystal. The formulation and the solution method developed here and herein are diﬀerent from those given by Ref. [8]. The diﬀraction pattern and relevant Penrrose tiling are shown in Figs. 6.1-1 and 6.1-2. The Penrose tiling of pentagonal quasirystals has been shown in Chapter 3. Quasicrystals with these symmetries belong to the class of two-dimensional quasicrystals, i.e., the atomic arrangement is quasiperiodic in a plane, and periodic in the third direction. For clarity, the quasicrystals can be deﬁned as being generated

72

Fig. 6.1-1

Chapter 6

Elasticity of two-dimensional quasicrystals and simpliﬁcation

Diﬀraction pattern of two-dimensional quasicrystal with ten-fold symmetry

Fig. 6.1-2

Penrose tiling of ten-fold symmetry quasicrystals

by stacking from planar quasiperiodic structures of ﬁve-fold or ten-fold symmetry along the third symmetry axis. Here the quasiperiodic plane is the xy- plane, and the ﬁve-fold or ten-fold rotational axis is the z-axis, which is the axis of symmetry. Due to the presence of scaling of incommensurate length in the quasiperiodic plane, it leads to the additional degree of freedom that does not exist in conventional crystals, named the phason ﬁeld w. Although two-dimensional quasicrystal can be understood as an stacked planar quasiperiodic structure along the periodic symmetrical axis, yet it is threedimensional in elasticity, and is diﬀerent from the plane problem in classical elasticity. And it can be decomposed into plane and anti-plane elasticity in special cases. Here we consider only the plane elasticity, because the anti-plane elasticity is a classical one and independent of the phason variables.

6.1

Basic equations of plane elasticity of two-dimensional quasicrystals: point . . .

73

Tables 6.0-2∼6.0-6 listed all the elastic constants. Consider a plane in twodimensional quasicrystals, and assume that it is perpendicular to the periodic symmetrical axis (say, axis z). In this case, w = (wx , wy , 0),

u = (ux , uy , uz ),

so the strains are wzz = wzx = wxz = wzy = wyz = 0. Assumption (6.0-1) leads to εzz = εxz = εyz = 0, and Table 6.0-2 is simpliﬁed to the following Table 6.1-1 for the phonon elastic constants for the plane elasticity. Table 6.1-1

Phonon elastic constants for plane

elasticity of two-dimensional quasicrystals 11

11 C11

22 C12

12 0

21 0

22

C12

C11

0

0

12

0

0

C66

C66

21

0

0

C66

C66

The phonon elastic constants listed in Table 6.1-1 can be expressed in the fourthorder tensor Cijkl = Lδij δkl + M (δjk δjl + δil δjk ),

i, j, k, l = 1, 2,

(6.1-1)

C11 − C12 (6.1-2) = C66 . 2 There are only two independent phonon elastic constants. The data given by Table 6.1-1 and relation by equations (6.1-1), (6.1-2) hold for all two-dimensional quasicrystals of plane elasticity. In the plane elasticity for two-dimensional quasicrystals with point groups 5m, 52, 5m, 10mm, 1022, 10mm and 10/mmm, Table 6.0-3 is simpliﬁed to Table 6.1-2 below. L = C12 ,

Table 6.1-2

M=

Phason elastic constants for plane elasticity for point group 10mm quasicrystals

11 22 12 21

11 K1 K2 0 0

22 K2 K1 0 0

12 0 0 K1 −K2

21 0 0 −K2 K1

This means K1111 = K2222 = K2121 = K1 , K1122 = K2211 = −K2112 = −K1221 = K2

(6.1-3)

74

Chapter 6

Elasticity of two-dimensional quasicrystals and simpliﬁcation

and other Kijkl = 0, and the expression of them by tensor of four rank is Kijkl = K1 δik δjl + K2 (δij δkl − δil δjk ),

i, j, k, l = 1, 2.

(6.1-4)

Table 6.0-4 in the present case is simpliﬁed into Table 6.1-3 as follows. Table 6.1-3

hhh

Phonon-phason coupling elastic constants for plane elasticity

hhh h

wij

hhh

εij

11 22 12 21

hh h

11

22

12

21

R −R 0 0

R −R 0 0

0 0 −R −R

0 0 R R

This shows that R1111 = R1122 = −R2222 = R1221 = R2121 = −R1212 = −R2211 = −R2121 = R (6.1-5) or Rijkl = R(δi1 − δi2 )(δij δkl − δik δjl + δiljk ),

i, j, k, l = 1, 2.

(6.1-6)

One can ﬁnd that for plane elasticity point group 5m has the same elastic constants with point group 10mm, so they can be discussed in the same line. The deﬁnition of strain tensor are as given in Chapter 4, i.e. , 1 ∂ui ∂uj ∂wi εij = + . (6.1-7) , wij = 2 ∂xj ∂xi ∂xj In Chapter 4 it was seen that the stress, strain and elastic constant tensors can be expressed by matrices. The above mentioned elastic constants may be denoted by matrix[CKR]. For the present case, the strain vector deﬁned by (4.4-12) is simpliﬁed into (6.1-8) [εij wij ] = [ε11 ε22 ε12 ε21 w11 w22 w12 w21 ] and [CKR] is ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ [CKR] = ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

L + 2M L 0 0 R R 0 0

L L + 2M 0 0 −R −R 0 0

0 0 R R 0 0 −R −R M M 0 0 M M 0 0 0 0 K 1 K2 0 0 K2 K1 −R −R 0 0 R R 0 0

0 0 −R −R 0 0 K1 −K2

0 0 R R 0 0 −K2 K1

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

6.1

Basic equations of plane elasticity of two-dimensional quasicrystals: point . . .

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎢ ⎣

L + 2M

L L + 2M

0 0 M

0 0 M M

(symmetry)

R R 0 0 −R −R 0 0 0 0 −R R 0 0 −R R 0 0 K1 K2 0 0 K1 K1 −K2 K1

75

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ . (6.1-9) ⎥ ⎥ ⎥ ⎥ ⎦

The free energy (or strain energy density) is F =

1 1 Lεii εii + M εij εij + K1 wij wij + K2 (wxx wyy − wxy wxy ) 2 2 + R [(εxx − εyy )(wxx + wyy ) + 2εxy (wxy − wyx )]

(6.1-10)

By (6.1-9) and (4.5-3) or by (6.1-10) and (4.5-1), the generalized Hooke’s law for plane elasticity of point group 10mm quasicrystals of ten-fold symmetry is ⎧ σxx = L(εxx + εyy ) + 2M εxx + R(wxx + wyy ), ⎪ ⎪ ⎪ ⎪ ⎪ σyy = L(εxx + εyy ) + 2M εyy − R(wxx + wyy ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ σxy = σyx = 2M εxy + R(wyx − wxy ), Hxx = K1 wxx + K2 wyy + R(εxx − εyy ), ⎪ ⎪ ⎪ Hyy = K1 wyy + K2 wxx + R(εxx − εyy ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Hxy = K1 wxy − K2 wyx − 2Rεxy , ⎪ ⎪ ⎪ ⎩ Hyx = K1 wyx − K2 wxy + 2Rεxy .

(6.1-11)

In addition, for anti-plane elasticity there is

σxz = 2M εxz , σyz = 2M εyz .

(6.1-12)

Monograph [9] pointed ﬁrst out that (6.1-9)∼(6.1-11) hold for the plane elasticity for both point group 5m and point group 10mm, this is identical to the argument of Ref. [6]. Equations (6.1-11) are the physical basis of elasticity of point groups 5m and 10mm quasicrystals. The geometry (or kinetics) basis of the subject is equation (6.1-7). Another necessary basis comes from statics, i.e., ⎧ ∂σxx ∂σxy ∂σyx ∂σyy ⎪ ⎪ + = 0, ⎨ ∂x + ∂y = 0, ∂x ∂y ⎪ ∂Hyx ∂Hyy ∂Hxx ∂Hxy ⎪ ⎩ + = 0, + = 0. ∂x ∂y ∂x ∂y

(6.1-13)

76

Chapter 6

Elasticity of two-dimensional quasicrystals and simpliﬁcation

In addition, for the anti-plane elasticity there is ∂σzx ∂σzy + = 0. ∂x ∂y

(6.1-14)

Here the body force density is omitted. From (6.1-7),(6.1-11) and (6.1-13) ,we found that there are 18 ﬁeld variables, i.e., four displacements ux , uy , wx , wy , seven strains εxx , εyy , εxy = εyx , wxx , wyy , wxy , wyx and seven stresses σxx , σyy , σxy = σyx , Hxx , Hyy , Hxy , Hyx . And the number of corresponding ﬁeld equations is also 18, including four equations of statics, seven stress-strain relations and seven equations of deformation geometry. The number of ﬁeld equations is equal to that of ﬁeld variables. This means that the mathematical presentation of the problem is consistent and solvable under appropriate boundary conditions. In addition, (6.1-7), (6.1-12) and (6.1-14) give the description of anti-plane pure classical (phonon) elasticity. De and Pelcovits[10] solved the equations for dislocation problem by using the Green function method and iteration procedure. Ding et al[11] solved the similar problem in terms of Fourier transforms and the Green function method. Taking diﬀerent way from either De and Pelcovits or Ding et al, we at ﬁrst reduce the 18 equations into a single partial diﬀerential of higher order by introducing some displacement potentials or stress potentials, which enable us to solve the problem by the Fourier transform method and complex variable function method. In the following we derive the ﬁnal governing equations and fundamental solutions for various quasicrystal systems. And the applications of the theory and methodology will be given in detail in Chapters 7 and 8 for dislocation and crack problems of quasicrystals, respectively.

6.2

Simpliﬁcation of the basic equation set: displacement potential function method

The number of basic equations as given in the previous section is too large and it is very hard to solve them directly. In mathematical physics, a conventional procedure is to reduce the number of ﬁeld equations. In the classical theory of elasticity, there is a way by introducing the so-called displacement or stress potential functions to realize the purpose, and the procedure is called the displacement potential method or stress potential method, respectively. In this section we eliminate the stress and strain components in the basic equations, and obtain some equilibrium equations expressed by displacement components, thus the strain compatibility is automatically satisﬁed. Furthermore, by introducing a displacement potential function, the ﬁnal governing equations become a single quadruple harmonic equation concerning the displacement potential function, thus this is the so-called displacement potential method. In the next section, we will introduce the stress potential method, there

6.2

Simpliﬁcation of the basic equation set: displacement potential function method 77

the governing equation becomes a single quadruple harmonic equation concerning the stress potential function that would be the so-called stress potential method. In this way the huge number of equations involving elasticity can be simpliﬁed very much. Substituting (6.1-7) into (6.1-11) and then into (6.1-12), we obtain 2 ⎧ ∂ ∂ 2 wy ∂ wx ∂wx ⎪ 2 ⎪ u + (L + M ) + 2 ∇ · u + R − = 0, M ∇ x ⎪ ⎪ ∂x ∂x2 ∂x∂y ∂y 2 ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ∂ ∂ 2 wx ∂ wy ∂ 2 wy ⎪ 2 ⎪ u + (L + M ) − 2 ∇ · u + R − = 0, M ∇ y ⎪ ⎨ ∂y ∂x2 ∂x∂y ∂y 2 2 ⎪ ⎪ ∂ 2 ux ∂ ux ∂ 2 uy ⎪ 2 ⎪ − = 0, K ∇ w + R − 2 1 x ⎪ ⎪ ∂x2 ∂x∂y ∂y 2 ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ∂ uy ∂ 2 ux ∂ 2 uy ⎪ ⎩ K1 ∇2 wy + R + 2 − =0 ∂x2 ∂x∂y ∂y2

(6.2-1)

in which ∇2 =

∂2 ∂2 + 2, 2 ∂x ∂y

∇·u=

∂ux ∂uy + . ∂x ∂y

Equations (6.2-1) are actually the displacement equilibrium equations of the plane elasticity of point group 5m with ﬁve-fold symmetry or point group 10mm with ten-fold symmetry, as mentioned in the previous section. Here there are only 4 displacement components ux , uy , wx , wy , i.e., the number of ﬁeld variables are only 4, and the number of order of equations is 8. Observation for the ﬁrst two equations of (6.2-1) suggests that by introducing the unknown functions ϕ(x, y) and ψ(x, y) as follows: ⎧ ∂2ϕ ∂2ψ ∂2ψ ⎪ ⎪ ux = (L + M ) + M 2 + (L + 2M ) 2 , ⎪ ⎪ ∂x∂y ∂x ∂y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂2ϕ ∂2ϕ ∂2ψ ⎪ ⎪ ⎪ u = − (L + 2M ) 2 + M 2 + (L + M ) , ⎪ ⎨ y ∂x ∂y ∂x∂y 2 ⎪ ⎪ M (L + 2M ) ∂ ϕ ∂2ψ ∂2ψ ⎪ ⎪ ⎪ − wx = − 2 + , ⎪ ⎪ R ∂x∂y ∂x2 ∂y2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ M (L + 2M ) ∂ 2 ϕ ∂ 2 ϕ ∂ 2ψ ⎪ ⎪ ⎩ wy = − − 2 . R ∂x2 ∂y 2 ∂x∂y

(6.2-2)

then the ﬁrst two equations are automatically satisﬁed already. Substituting (6.2-2)

78

Chapter 6

Elasticity of two-dimensional quasicrystals and simpliﬁcation

into the last two equations in (6.2-1), there follows the equations ⎧ ∂2 ∂2 ∂2ϕ ⎪ ⎪ + αΠ ψ = 0, + βΠ ) − βΠ (αΠ ⎪ 1 2 1 2 ⎨ ∂x∂y ∂y2 ∂x2 ⎪ ∂2 ∂2 ∂2ψ ⎪ ⎪ = 0. ⎩ αΠ2 2 − βΠ1 2 ϕ + (αΠ2 + βΠ1 ) ∂x ∂y ∂x∂y where Π1 = 3

∂2 ∂2 − 2, 2 ∂x ∂y

Π2 = 3

⎧ ⎨ α = R(L + 2M ) − ωK1 , ⎩ ω = M (L + 2M ) . R

∂2 ∂2 − 2, 2 ∂y ∂x

(6.2-3)

(6.2-4)

β = RM − ωK1 , (6.2-5)

Equation (6.2-3) is much simpler compared with (6.2-1), still it can be simpliﬁed by letting ∂2F ∂2 ∂2 . (6.2-6) ϕ = βΠ2 2 − αΠ1 2 F, ψ = (αΠ1 + βΠ2 ) ∂x ∂y ∂x∂y in which F (x, y) is any function, then the ﬁrst equation of (6.2-3) is satisﬁed. Substituting (6.2-6) into the second one of equations (6.2-3), after manipulation, it reduces to ∇2 ∇2 ∇2 ∇2 F = 0.

(6.2-7)

This is the ﬁnal governing equation of the plane elasticity of point groups 5m and 10mm quasicrystals. We call F (x, y) as a displacement potential function, or simply displacement potential. The equation (6.2-7) is a quadruple harmonic equation, the order of which is much higher than that in the classical elasticity, where that is biharmonic equation. All of displacement and stress components can be expressed by potential function F (x, y) as follows: ⎧ ∂2 ⎪ 2 ⎪ ⎪ ⎨ ux = [M αΠ1 + (L + 2M )βΠ2 ] ∂x∂y ∇ F, ⎪ ∂2 ∂2 ⎪ ⎪ ⎩ uy = M αΠ1 2 − (L + 2M )βΠ2 2 ∇2 F, ∂y ∂x ⎧ ∂2 ⎪ ⎪ ⎪ ⎨ wx = ω(α − β)Π1 Π2 ∂x∂y F, 2 2 ⎪ ⎪ 2 ∂ 2 ∂ ⎪ ⎩ wy = −ω αΠ1 2 + βΠ2 2 F, ∂y ∂x

(6.2-8)

(6.2-9)

6.3

Simpliﬁcation of the basic equations set: stress potential function method

⎧ ∂3 2 ⎪ ⎪ σ = 2M (L + M )αΠ ∇ F, xx 1 ⎪ ⎪ ∂y 3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∂3 σyy = 2M (L + M )αΠ1 2 ∇2 F, ⎪ ∂x ∂y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 3 ⎪ ⎪ ⎩ σxy = σyx = −2M (L + M )αΠ1 ∂ ∇2 F, ∂x∂y 2

79

(6.2-10)

⎧ ∂ 2 2 2 ∂ ∂2 ∂2 ⎪ 2 2 ⎪ Hxx = αβ ∇ ∇ ∇ F + ω(K1 − K2 ) αΠ1 2 + βΠ2 2 F, ⎪ ⎪ ∂y ∂y ∂y ∂x ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂3 ∂ ⎪ ⎪ ⎪ Hyy = αβ ∇2 ∇2 ∇2 F − ω(K1 − K2 )(α − β)Π1 Π2 2 F, ⎪ ⎨ ∂y ∂x ∂y ⎪ 2 2 ⎪ ∂ 2 2 2 ∂ ⎪ 2 ∂ 2 ∂ ⎪ = −αβ ∇ ∇ F − ω(K − K ) + βΠ ∇ αΠ F, H ⎪ xy 1 2 1 2 ⎪ ⎪ ∂x ∂x ∂y 2 ∂x2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 3 ⎪ ⎪ ⎩ Hyx = αβ ∂ ∇2 ∇2 ∇2 F + ω(K1 − K2 )(α − β)Π1 Π2 ∂ F. ∂x ∂x∂y 2 (6.2-11) [12] have suggested the approach, and the practice shows that it is very Li and Fan eﬀective. In the next two chapters many applications of the approach will be given. In addition (6.1-12) and (6.1-14) yield the ﬁnal governing equation for anti-plane elasticity ∇2 uz = 0.

(6.2-12)

It is obvious that the anti-plane problem is decoupled with the plane one.

6.3

Simpliﬁcation of the basic equations set: stress potential function method

The stress potential method has been widely used in classical theory of elasticity. We have extended it to elasticity of quasicrystals [13], [14]. From (6.1-7) the strain compatibility equations ⎧ 2 ∂ 2 εxy ∂ εxx ∂ 2 εyy ⎪ ⎪ ⎪ ⎨ ∂y 2 + ∂x2 = 2 ∂x∂y , ⎪ ⎪ ⎪ ⎩ ∂wxy = ∂wxx , ∂x ∂y

(6.3-1) ∂wyx ∂wyy = . ∂y ∂x

The strain components εij and wij can be expressed by the stress components σij

80

Chapter 6

Elasticity of two-dimensional quasicrystals and simpliﬁcation

and Hij , i.e., ⎧ 1 1 ⎪ ⎪ (σxx + σyy ) + [(K1 + K2 )(σxx + σyy ) − 2R(Hxx + Hyy )], εxx = ⎪ ⎪ 4(L + M ) 4C ⎪ ⎪ ⎪ ⎪ ⎪ 1 1 ⎪ ⎪ (σxx + σyy ) − [(K1 + K2 )(σxx + σyy ) − 2R(Hxx + Hyy )], ⎪ εyy = ⎪ 4(L + M ) 4C ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ εxy = εyx = [(K1 + K2 )σxy − R(Hxy + Hyx )], ⎪ ⎪ C ⎪ ⎪ ⎪ ⎨ 1 1 (Hxx − Hyy ) + [M (Hxx + Hyy ) − R(σxx − σyy )], wxx = 2(K − K ) 2C 1 2 ⎪ ⎪ ⎪ ⎪ 1 1 ⎪ ⎪ ⎪ wyy = − (Hxx − Hyy ) + [M (Hxx + Hyy ) − R(σxx − σyy )], ⎪ ⎪ 2(K − K ) 2C 1 2 ⎪ ⎪ ⎪ ⎪ ⎪ 1 (R2 − M K1 )Hxy + (R2 − M K2 )Hyx ⎪ ⎪ = − Rσ , ⎪ w xy xy ⎪ ⎪ C K1 − K 2 ⎪ ⎪ ⎪ ⎪ ⎪ 1 (R2 − M K1 )Hxy + (R2 − M K2 )Hyx ⎪ ⎪ ⎩ wyx = −Rσxy − . C K1 − K 2 (6.3-2) with C = M (K1 + K2 ) − 2R2 . (6.3-3) If substituting (6.3-2) into (6.3-1), one can ﬁnd the strain compatibility equations expressed by stress components (because they are too long to be listed, but we will give them in Section 6.9). In addition, there are equilibrium equations ⎧ ∂σyx ∂σyy ∂σxx ∂σxy ⎪ ⎪ + = 0, ⎨ ∂x + ∂y = 0, ∂x ∂y (6.3-4) ⎪ ∂Hxx ∂Hxy ∂Hyx ∂Hyy ⎪ ⎩ + = 0, + = 0. ∂x ∂y ∂x ∂y So that we have 7 equations in total in which there are 3 compatibility equations expressed by stress components referring to (6.9-2a)∼(6.9.2c) and 4 equilibrium equations, and the number of the unknown functions is also 7,i.e., σxx , σyy , σxy = σyx , Hxx , Hyy , Hxy , Hyx . The equation set is closed and solvable. If introducing the stress functions ϕ, ψ1 and ψ2 , such as ⎧ ∂2ϕ ∂2ϕ ∂2ϕ ⎪ ⎪ = , σ = , σ = σ = − σ , xx yy xy yx ⎪ 2 2 ⎨ ∂y ∂x ∂x∂y (6.3-5) ∂ψ1 ∂ψ1 ∂ψ2 ∂ψ2 ⎪ ⎪ , H , H , H . H = = − = − = ⎪ xx xy yx yy ⎩ ∂y ∂x ∂y ∂x then the equilibrium equations (6.3-4) are automatically satisﬁed. Substituting (6.3-5) into deformation compatibility equations expressed by stress components

6.4

Plane elasticity of point group 5, 5 pentagonal and point group 10, 10 . . .

81

yields ⎧ K1 + K 2 2 2 R ∂ ∂ 1 ⎪ 2 2 ⎪ ∇ ∇ ϕ+ ∇ ∇ ϕ+ Π1 ψ1 − Π2 ψ2 = 0, ⎪ ⎪ ⎪ 2C(L + M ) 2C C ∂y ∂x ⎪ ⎪ ⎪ ⎨ C ∂ + M ∇2 ψ1 + R Π1 ϕ = 0, ⎪ K1 − K2 ∂y ⎪ ⎪ ⎪ ⎪ ⎪ ∂ C ⎪ ⎪ + M ∇2 ψ2 − R Π2 ϕ = 0. ⎩ K1 − K2 ∂x (6.3-6) in which Π1 and Π2 are deﬁned by (6.2-4), C is given by (6.3-3). By now, the numbers of equations and unknown functions have been reduced to 3. Now introducing a new unknown function G(x, y), such as ⎧ ⎪ ⎪ ϕ = D ∂ Π1 ∇2 G, ⎪ ⎪ ⎪ ∂y ⎪ ⎪ ⎪ ⎪ 1 ⎪ 2 2 2 2 ⎪ ⎪ ⎨ ψ1 = − R (M K1 − R2 )[(L + 2M )(K1 − K2 ) − 2R ]∇ ∇ ∇ G (6.3-7) ∂2 ⎪ ⎪ ⎪ + (L + M )(K1 − K2 )R Π1 Π2 G, ⎪ ⎪ ∂x∂y ⎪ ⎪ ⎪ ⎪ 2 ⎪ ∂ ⎪ ⎪ Π1 Π2 G. ⎩ ψ2 = (L + M )(K1 − K2 )R ∂x∂y If ∇2 ∇2 ∇2 ∇2 G = 0,

(6.3-8)

then equations (6.3-6) are satisﬁed, in which D = 2(M K1 − R2 )(L + M ).

(6.3-9)

At the same time the equation (6.2-12) holds for anti-plane elasticity too.

6.4

Plane elasticity of point group 5, 5 pentagonal and point group 10, 10 decagonal quasicrystals

For plane elasticity, the ﬁve-fold and ten-fold symmetries quasicrystals of point groups 5, 5 and 10, 10 are diﬀerent in elasticity with that of point groups 5m and 10mm, the diﬀerence lies in only the phonon-phason coupling elastic constants, in which the former has two coupling elastic constants R1 and R2 rather than one constant R. Thus we have Rijkl = R1 (δi1 − δi2 )(δij δkl − δik δjl + δil δjk ) + R2 [(1 − δij )δkl + δij (δi1 − δi2 )(δk1 δl2 − δk2 δl1 )],

i, j, k, l = 1, 2. (6.4-1)

82

Chapter 6

Elasticity of two-dimensional quasicrystals and simpliﬁcation

Apart from this, the phonon and phason elastic constants of point group 5,5 and point group 10, 10 quasicrystals are the same as those of point groups 5m and 10mm. The corresponding elastic constant matrix is ⎤ ⎡ R2 −R2 L + 2M L 0 0 R1 R1 ⎥ ⎢ ⎢ L L + 2M 0 0 −R1 −R1 −R2 R2 ⎥ ⎥ ⎢ ⎥ ⎢ 0 0 M M R2 R2 −R1 R1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 0 M M R2 R2 −R1 R1 ⎥ ⎢ ⎥ ⎢ [CKR] = ⎢ ⎥ R1 −R1 R2 R2 K1 K2 0 0 ⎥ ⎢ ⎥ ⎢ ⎢ R1 −R1 R2 R2 K2 K1 0 0 ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ −R2 −R1 −R1 0 0 K1 −K2 ⎥ R2 ⎦ ⎣ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

−R2 L + 2M

R2

R1

R1

0

0

−K2

L

0

0

R1

L + 2M

0

0

−R1

−R1

−R2

M

M

R2

R2

−R1

M

R2

R2

−R1

K1

K2

0

K1

0

(symmetry)

R1

R2

K1

K1 −R2

⎤

⎥ R2 ⎥ ⎥ ⎥ R1 ⎥ ⎥ ⎥ R1 ⎥ ⎥. ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ ⎥ −K2 ⎥ ⎦ K1 (6.4-2)

By this elastic constant matrix, the stress-strain relation can be written as ⎧ σxx = L(εxx + εyy ) + 2M εxx + R1 (wxx + wyy ) + R2 (wxy − wyx ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ σyy = L(εxx + εyy ) + 2M εyy − R1 (wxx + wyy ) − R2 (wxy − wyx ), ⎪ ⎪ ⎪ ⎪ ⎪ σ = σyx = 2M εxy + R1 (wyx − wxy ) + R2 (wxx + wyy ), ⎪ ⎪ ⎨ xy Hxx = K1 wxx + K2 wyy + R1 (εxx − εyy ) + 2R2 εxy , ⎪ ⎪ ⎪ ⎪ Hyy = K1 wyy + K2 wxx + R1 (εxx − εyy ) + 2R2 εxy , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Hxy = K1 wxy − K2 wyx − 2R1 εxy + R2 (εxx − εyy ), ⎪ ⎪ ⎪ ⎩ Hyx = K1 wyx − K2 wxy + 2R1 εxy − R2 (εxx − εyy ).

(6.4-3)

In addition, the stresses σij and Hij satisfy the same equilibrium equations as (6.1-13). Substituting (6.1-7) into (6.4-3), then into (6.1-12) leads to the equilibrium equa-

6.4

Plane elasticity of point group 5, 5 pentagonal and point group 10, 10 . . .

83

tions expressed by displacements as ⎧ 2 ∂ ∂ 2 wy ∂ wx ∂ 2 wx ⎪ 2 ⎪ u + (L + M ) + 2 ∇ · u + R − M ∇ ⎪ x 1 ⎪ ⎪ ∂x ∂x2 ∂x∂y ∂y 2 ⎪ ⎪ ⎪ ⎪ ⎪ ∂ 2 wx ∂ 2 wy ∂ 2 wy ⎪ ⎪ −2 − = 0, −R2 ⎪ ⎪ 2 ⎪ ∂x ∂x∂y ∂y 2 ⎪ ⎪ ⎪ 2 ⎪ ⎪ ∂ wy ∂ 2 wy ∂ ∂ 2 wx ⎪ 2 ⎪ ⎪ ⎨ M ∇ uy + (L + M ) ∂y ∇ · u + R1 ∂x2 − 2 ∂x∂y − ∂y 2 2 ⎪ ∂ 2 wy ∂ wx ∂ 2 wx ⎪ ⎪ +R + 2 − = 0, ⎪ 2 ⎪ ⎪ ∂x2 ∂x∂y ∂y2 ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ ∂ 2 uy ∂ 2 ux ∂ ux ∂ uy ∂ 2 ux ∂ 2 uy ⎪ 2 ⎪ ⎪ ⎪ K1 ∇ wx + R1 ∂x2 − 2 ∂x∂y − ∂y 2 + R2 ∂x2 + 2 ∂x∂y − ∂y2 = 0, ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ ∂ 2 ux ∂ 2 uy ∂ uy ∂ ux ∂ 2 uy ∂ 2 ux ⎪ ⎪ + 2 − 2 − − − R = 0. ⎩ K1 ∇2 wy + R1 2 ∂x2 ∂x∂y ∂y 2 ∂x2 ∂x∂y ∂y 2 (6.4-4) The equations are similar to those of (6.2-1), the deﬁnitions on operators ∇2 and ∇· are the same there. It is obvious that equations (6.4-4) are more complex than those of (6.2-1). We now simplify the equations in terms of the displacement potential function method. Introducing displacement potentials ϕ(x, y) and ψ(x, y) as below: ⎧ ∂2ϕ ∂2ψ ∂2ψ ⎪ ⎪ ⎪ + M 2 + (L + 2M ) 2 , ux = (L + M ) ⎪ ⎪ ∂x∂y ∂x ∂y ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ ∂ ϕ ∂ ϕ ∂2ψ ⎪ ⎪ u = − (L + 2M ) + M + (L + M ) , ⎪ y ⎪ ⎪ ∂x2 ∂y 2 ∂x∂y ⎪ ⎪

2 ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ wx = −ω 2R1 ∂ − R2 ∂ − ∂ ϕ ⎪ ⎨ ∂x∂y ∂x2 ∂y 2 (6.4-5) 2 ⎪ ∂2 ∂ ∂2 ⎪ ⎪ − ψ , + 2R + R ⎪ 1 2 ⎪ ∂x2 ∂y2 ∂x∂y ⎪ ⎪ ⎪ ⎪

⎪ 2 2 ⎪ ∂ ∂2 ∂ ⎪ ⎪ ϕ + 2R2 wy = ω R 1 − ⎪ ⎪ ∂x2 ∂y 2 ∂x∂y ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ∂2 ∂2 ∂ ⎪ ⎪ − 2R − − R ψ . ⎩ 1 2 ∂x∂y ∂x2 ∂y 2 in which ω=

M (L + 2M ) , R2

R2 = R12 + R22 .

(6.4-6)

The functions ϕ(x, y) and ψ(x, y) deﬁned by (6.4-4) automatically satisfy the ﬁrst two of equations (6.4-4), and the substitution of formulas (6.4-6) into the second two

84

Chapter 6

Elasticity of two-dimensional quasicrystals and simpliﬁcation

of equations (6.4-4) results in ⎧ ∂ ∂ ∂ ∂ ⎪ ⎪ ⎨ (L + 2M )c2 ∂x Λ1 ϕ + M c1 ∂y Λ2 ϕ + (L + 2M )c2 ∂y Λ1 ψ − M c1 ∂x Λ2 ψ = 0, ⎪ ∂ ∂ ∂ ∂ ⎪ ⎩ (L + 2M )c2 Λ2 ϕ − M c1 Λ1 ϕ + (L + 2M )c2 Λ2 ψ + M c1 Λ1 ψ = 0 ∂x ∂y ∂y ∂x (6.4-7) with ⎧ ∂ ∂ ⎪ ⎪ ⎨ Λ1 = R1 ∂y Π1 + R2 ∂x Π2 , (6.4-8) ⎪ ∂ ∂ ⎪ ⎩ Λ2 = R1 Π2 − R2 Π1 , ∂x ∂y c1 = (L + 2M )K1 − R2 ,

c2 = M K1 − R2 .

If we introduce a new function F (x, y) by means of ⎧ ∂ ∂ ⎪ ⎪ ⎨ ϕ = −(L + 2M )c2 R ∂y Λ1 F + M c1 R ∂x Λ2 F, ⎪ ∂ ∂ ⎪ ⎩ ψ = (L + 2M )c2 R Λ1 F + M c1 R Λ2 F. ∂x ∂y

(6.4-9)

(6.4-10)

Then the ﬁrst one of equations (6.4-7) is identically satisﬁed, and from the second one we ﬁnd that (6.4-11) ∇2 ∇2 ∇2 ∇2 F = 0. The deﬁnitions of operators ∇2 and ∇· are the same as before. All components of displacement vectors and stress tensors can be expressed by the potential function F (x, y) as ∂ ∂ ux = R c2 Λ1 + c1 Λ2 ∇2 F, (6.4-12a) ∂x ∂y ∂ ∂ (6.4-12b) uy = R c2 Λ1 − c1 Λ2 ∇2 F, ∂y ∂x wx = −c0 Λ1 Λ2 F,

(6.4-12c)

wy = −R−1 [c2 (L + 2M )Λ21 + c1 M Λ22 ]F,

(6.4-12d)

σxx = 2c0 c2

∂2 Λ1 ∇2 F, ∂y 2

(6.4-12e)

σyy = 2c0 c2

∂2 Λ1 ∇2 F, ∂x2

(6.4-12f)

σxy = σyx = −2c0 c2

∂2 Λ1 ∇2 F, ∂x∂y

(6.4-12g)

6.5

Plane elasticity of point group 12mm of dodecagonal quasicrystals

∂ ∂ 2 2 2 ∇ ∇ ∇ F + R−1 K0 [c2 (L + 2M )Λ21 + c1 M Λ22 ]F, ∂y ∂y

(6.4-12h)

∂ ∂ 2 2 2 ∇ ∇ ∇ F − R−1 K0 [c2 (L + 2M )Λ21 + c1 M Λ22 ]F, ∂x ∂x

(6.4-12i)

Hxx = −c1 c2 R

Hxy = c1 c2 R

85

Hyx = −c1 c2 R

∂ ∂ 2 2 2 ∇ ∇ ∇ F − c0 K0 Λ1 Λ2 F, ∂x ∂y

(6.4-12j)

Hyy = −c1 c2 R

∂ 2 2 2 ∂ ∇ ∇ ∇ F + c0 K0 Λ1 Λ2 F ∂y ∂x

(6.4-12k)

with c0 = R(L + M ),

K0 = K1 − K2 .

(6.4-13)

The results were given by Li and Fan[15] and Li et al[16] . Recently, Li and Fan[14] have derived the ﬁnal governing equation of elasticity of the same point groups through the stress potential method, the resulting equation is also quadruple harmonic equation, of course the unknown function is the stress potential. Application will be shown in Chapter 8 for solving notch problem of point group 5, 5 and point group 10, 10 two-dimensional quasicrystals. The frequent appearance of the quadruple harmonic equations in (6.2-7), (6.3-8) and (6.4-11), suggests that this kind of equations is very important in theory and practice.

6.5

Plane elasticity of point group 12mm of dodecagonal quasicrystals

Point group 12mm dodecagonal quasicrystals are two-dimensional ones, the Penrose tiling and the diﬀraction pattern are shown in Figs. 6.5-1 and 6.5-2 respectively. If taking z-axis as periodic arrangement direction, and supposing that the ﬁeld variables independent of the coordinate z, then the elasticity problem can be decomposed into plane elasticity and anti-plane elasticity. As mentioned in previous sections, the quasiperiodic plane is taken as xy-plane, so the z-axis represents the twelve-fold symmetry axis. As in other two-dimensional quasicrystals, the quasiperiodic plane is an elastic isotropic plane. There are two non-zero independent elastic constants Cijke in the plane, i.e., L and M , L = C12 ,

M=

C11 − C12 = C66 . 2

(6.5-1)

86

Chapter 6

Fig.6.5-1

Elasticity of two-dimensional quasicrystals and simpliﬁcation

Penrose tiling of dodecagonal quasicrystal

Here we have non-zero Kijkl as K1111 = K2222 = K1 , K1122 = K2211 = K2 , K1221 = K2112 = K3 ,

K2121 = K1212 = K1 + K2 + K3

(6.5-2)

and others are zero. The results can also be expressed as Kijkl = (K1 − K2 − K3 )(δik − δil ) + K2 δij δkl + K3 δil δjk + 2(K2 + K3 )(δi1 δj2 δk1 δl2 + δi2 δj1 δk2 δl1 ), i, j, k, l = 1, 2.

(6.5-3)

In addition, the phonon and phason are decoupled, i.e., Rijkl = 0.

Fig.6.5-2

(6.5-4)

Diﬀraction pattern of quasicrystal of twelve-fold symmetry, in the condition of the phason strain ﬁeld being uniformly

6.5

Plane elasticity of point group 12mm of dodecagonal quasicrystals

87

The elastic constant matrix is ⎡

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ [CKR] = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎤

L + 2M

L

0

0

0

0

0

0

L

L + 2M

0

0

0

0

0

0

0

0

M M

0

0

0

0

0

0

M M

0

0

0

0

0

0

0

0

K1 K 2

0

0

0

0

0

0

K2 K1

0

0

0

0

0

0

0

0

K1 + K2 + K3

K3

0

0

0

0

0

0

K3

K1 + K2 + K3

L + 2M

L L + 2M

0 0 M

(symmetry)

0 0 M M

0 0 0 0 0 0 0 0 K1 K2 K1

0 0 0 0 0 0 K1 + K2 + K3

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎤ 0 ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥. ⎥ 0 ⎥ ⎥ 0 ⎥ ⎦ K3 K 1 + K2 + K3

(6.5-5) The relevant free energy (or strain energy density) is F =

where εij

1 1 1 2 2 L(∇ · u)2 + M εij εij + K1 wij wij + K2 (wyx + wxy + 2wxx wyy ) 2 2 2 1 (6.5-6) + K3 (wyx + wxy )2 , 2 and wij denote the strain tensors 1 ∂ui ∂uj ∂wi + . (6.5-7) , wij = εij = 2 ∂xj ∂xi ∂xj

From (4.5-1) in Chapter 4 and (6.5-5) or (6.5-6), the generalized Hooke’s law for plane elasticity of dodecagonal quasicrystals is ⎧ σxx = L(εxx + εyy ) + 2M εxx , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ σyy = L(εxx + εyy ) + 2M εyy , ⎪ ⎪ ⎪ ⎪ ⎪ σ = σyx = 2M εxy , ⎪ ⎪ ⎨ xy Hxx = K1 wxx + K2 wyy , (6.5-8) ⎪ ⎪ ⎪ ⎪ Hyy = K1 wyy + K2 wxx , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Hxy = (K1 + K2 + K3 )wxy + K3 wyx , ⎪ ⎪ ⎪ ⎩ Hyx = (K1 + K2 + K3 )wyx + K3 wxy

88

Chapter 6

Elasticity of two-dimensional quasicrystals and simpliﬁcation

and there are the equilibrium equations in the absence of the body force ⎧ ∂σyx ∂σyy ∂σxx ∂σxy ⎪ ⎪ + = 0, ⎪ ⎨ ∂x + ∂y = 0, ∂x ∂y ⎪ ∂Hxx ∂Hxy ⎪ ⎪ + = 0, ⎩ ∂x ∂y

∂Hyx ∂Hyy + = 0. ∂x ∂y

(6.5-9)

Eliminating the stress and strain components from (6.5-7)∼(6.5-9), we obtain the equilibrium equations expressed by displacement components as below: ⎧ ∂ ⎪ ⎪ M ∇2 ux + (L + M ) ∇ · u = 0, ⎪ ⎪ ∂x ⎪ ⎪ ⎪ ⎪ ⎪ ∂ ⎪ ⎪ M ∇2 uy + (L + M ) ∇ · u = 0, ⎪ ⎪ ⎨ ∂y ⎪ ∂ ∂wx ⎪ 2 ⎪ + K ∇ w + (K + K ) 1 x 2 3 ⎪ ⎪ ∂y ∂y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ ∂wx ⎪ ⎩ K1 ∇2 wy + (K2 + K3 ) + ∂x ∂y

∂wy ∂x ∂wy ∂x

(6.5-10) = 0,

= 0.

If deﬁning two displacement potentials F (x, y) and G(x, y), such as ⎧ ∂2F ⎪ ⎪ = (L + M ) , u ⎪ x ⎪ ⎪ ∂x∂y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂2F ∂2F ⎪ ⎪ u = −(L + 2M ) − M , ⎪ y ⎨ ∂x2 ∂y 2 ⎪ ⎪ ∂2G ⎪ ⎪ = (K + K ) , w x 2 3 ⎪ ⎪ ∂x∂y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂2G ∂2G ⎪ ⎪ ⎩ wy = −K1 2 − (K1 + K2 + K3 ) 2 , ∂x ∂y

(6.5-11)

then equations (6.5-10) will be reduced to ∇2 ∇2 F = 0,

∇2 ∇2 G = 0,

(6.5-12)

which is found in Ref.[12]. One can see that the problem is concluded to solve two biharmonic equations, and the theory and method studying this kind of equations are well developed in the theory of classical elasticity, which can be used in studying elasticity of quasicrystals. In this respect, the most systematic method is the complex variable function method. If assume ϕ1 (z), ψ1 (z), π1 (z) and χ1 (z) are analytic functions of complex variable

6.6

Plane elasticity of point group 8mm of octagonal quasicrystals, displacement . . . 89

z = x + iy(i =

√ −1), then ⎧ ⎪ ⎪ ⎪ F (x, y) = Re z¯ϕ1 (z) + ψ1 (z)dz , ⎨ ⎪ ⎪ ⎪ ⎩ G(x, y) = Re z¯π1 (z) + χ1 (z)dz .

(6.5-13)

where z¯ = x − iy and Re denotes the real part of a complex number. Theory of analytic functions is a powerful tool to solve the boundary value problems of harmonic, biharmonic and multiple harmonic equations, the work on quadruple and sextuple harmonic equations is developed by the study of elasticity of quasicrystals, the details will be given in the subsequent chapters. As a complete description for the new development of the method some detailed summarization will be displayed in Chapter 11.

6.6

Plane elasticity of point group 8mm of octagonal quasicrystals, displacement potential

Octagonal quasicrystals belong to two-dimensional quasicrystals, the Penrose tiling of its quasiperiodic plane is shown in Fig.6.6-1. The stacking of the planes along the third direction perpendicular them will result in the quasicrystals.

Fig.6.6-1

Penrose tiling of eight-fold symmetry quasicrystal

In the following considering only a simple case of elasticity of the material, i.e., all of the ﬁeld variables are independent of the axis along which the atom arrangement is periodic. In the case, the elasticity can be decomposed into plane elasticity and anti-plane elasticity. We focus on the solution for the plane elasticity. Within the quasiperiodic plane, the phonon elasticity is isotropic, so Cijkl are the same of those discussed in previous sections. The phason elasticity here is anisotropic, but Kijkl are the same with those of point group 12mm quasicrystals. Between phonon and

90

Chapter 6

Elasticity of two-dimensional quasicrystals and simpliﬁcation

phason ﬁelds there is coupling, the corresponding elastic constants Rijkl are the same given by (6.1-6), i.e., Rijkl = R(δi1 − δi2 )(δij δkl − δik δjl + δil δjk ),

i, j, k, l = 1, 2.

(6.6-1)

such that we have the elastic constants matrix ⎡

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ [CKR] = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

L + 2M

L

0

0

L

L + 2M

0

0

0

0

M

M

0

0

0

M

M

R

−R

0

R

−R

0

0

0

0

0

L+2M

R

0

0

0

0

−R

R

0

0

−R

R

0

K1

K2

0

0

0

K2

K1

0

0

0

0

K1 +K2 +K3

K3

0

0

K3

K1 +K2 +K3

R

R

R

⎤

0

−R −R

R

−R −R

L

0

0

L+2M

0

0

M

M

0

M

0 K1

(symmetry)

R

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

0

0

0

0

0

−R

R

0

−R

R

K2

0

0

K1

0

0

K1 +K2 +K3

K3

−R −R

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

K1 +K2 +K3

(6.6-2) The corresponding free energy is 1 1 1 2 2 + wyx + 2wxx wyy ) F = L(∇ · u)2 + M εij εij + K1 wij wij + K2 (wxy 2 2 2 1 + K3 (wxy + wyx )2 + R[(εxx − εyy )(wxx + wyy ) + 2εxy (wyx − wxy )] (6.6-3) 2 where εij and wij are deﬁned as before. From (6.6-3) and (4.5-1) the generalized Hooke’s law can be expressed as ⎧ σxx = L(εxx + εyy ) + 2M εxx + R(wxx + wyy ), ⎪ ⎪ ⎪ ⎪ ⎪ σyy = L(εxx + εyy ) + 2M εyy − R(wxx + wyy ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ σxy = σyx = 2M εxy + R(wyx − wxy ), ⎨ Hxx = K1 wxx + K2 wyy + R(εxx − εyy ), (6.6-4) ⎪ ⎪ ⎪ ⎪ Hyy = K1 wyy + K2 wxx + R(εxx − εyy ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Hxy = (K1 + K2 + K3 )wxy + K3 wyx − 2Rεxy , ⎪ ⎪ ⎪ ⎩ Hyx = (K1 + K2 + K3 )wyx + K3 wxy + 2Rεxy .

6.6

Plane elasticity of point group 8mm of octagonal quasicrystals, displacement . . . 91

For brevity, stress equilibrium equations are not listed. Through a similar procedure, equilibrium equations given by displacement components are as follows: 2 ⎧ ∂ ∂ 2 wy ∂ wx ∂ 2 wx ⎪ 2 ⎪ +2 − = 0, M ∇ ux + (L + M ) ∇ · u + R ⎪ ⎪ ∂x ∂x2 ∂x∂y ∂y 2 ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ∂ wy ∂ 2 wy ∂ ∂ 2 wx ⎪ 2 ⎪ = 0, ⎪ ⎨ M ∇ uy + (L + M ) ∂y ∇ · u + R ∂x2 − 2 ∂x∂y − ∂y 2 2 2 ⎪ ⎪ ∂ wx ∂ 2 wy ∂ 2 uy ∂ ux ∂ 2 ux ⎪ 2 ⎪ K1 ∇ wx + (K2 + K3 ) + −2 +R − = 0, ⎪ ⎪ ∂y 2 ∂x∂y ∂x2 ∂x∂y ∂y2 ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ ∂ wx ∂ 2 ux ∂ uy ∂ 2 wy ∂ 2 uy ⎪ 2 ⎩ K1 ∇ wy + (K2 + K3 ) +2 + − +R = 0. ∂x∂y ∂x2 ∂x2 ∂x∂y ∂y 2 (6.6-5) It is evident that, if K2 + K3 = 0, the equations will be reduced to (6.2-1). In fact, K2 + K3 = 0, so the equations are more complex than those given in the previous sections. But the ﬁnal governing equation for the present case presents more interesting in mathematical physics, we can see immediately. At ﬁrst we introduce two auxiliary function ϕ(x, y) and ψ(x, y) in such way ⎧ ∂2ϕ ∂2ψ ∂2ψ ⎪ ⎪ u = (L + M ) + (L + 2M ) , + M ⎪ x ⎪ ∂x∂y ∂x2 ∂y 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂2ϕ ∂2ϕ ∂2ψ ⎪ ⎪ , u = − (L + 2M ) 2 + M 2 + (L + M ) ⎪ ⎨ y ∂x ∂y ∂x∂y 2 ⎪ ⎪ ∂2ψ ∂2ψ ∂ ϕ ⎪ ⎪ wx = −ω 2 + , − ⎪ ⎪ ∂x∂y ∂x2 ∂y 2 ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ∂ ϕ ∂2ϕ ∂2ψ ⎪ ⎩ wy = ω − 2 −2 . ∂x2 ∂y ∂x∂y

(6.6-6)

where ω = M (L + M )/R, so that (6.6-5) is simpliﬁed as ⎧ ∂2 ∂2 ∂2ϕ ⎪ ⎪ ⎪ ⎨ (γΠ1 + δΠ2 ) ∂x∂y + αΠ1 ∂y 2 − βΠ2 ∂x2 ψ = 0, ⎪ ∂2ψ ∂2 ∂2 ⎪ ⎪ + αΠ2 2 − βΠ1 2 ϕ = 0, ⎩ (γΠ2 + δΠ1 ) ∂x∂y ∂y ∂x in which

(6.6-7)

92

Chapter 6

Elasticity of two-dimensional quasicrystals and simpliﬁcation

⎧ ∂2 ∂2 ∂2 ∂2 ⎪ ⎪ Π = 3 − , Π = 3 − , 1 2 ⎪ ⎪ ∂x2 ∂y 2 ∂y2 ∂x2 ⎪ ⎪ ⎨ α = R(L + 2M ) − ω(K1 + K2 + K3 ), ⎪ ⎪ ⎪ β = RM − ωK1 , δ = RM − ω(K1 + K2 + K3 ), ⎪ ⎪ ⎪ ⎩ γ = R(L + 2M ) − ωK1 .

(6.6-8)

and ω is given above. At last the displacement potential F (x, y) is introduced ⎧ ∂2 ∂2 ⎪ ⎪ ⎪ ϕ = βΠ F, − αΠ 2 1 ⎨ ∂x2 ∂y2 (6.6-9) ⎪ ∂2F ⎪ ⎪ ⎩ ψ = (γΠ1 + δΠ2 ) ∂x∂y and (6.6-7) reduces to a single equation as below: (∇2 ∇2 ∇2 ∇2 − 4ε∇2 ∇2 Λ2 Λ2 + 4εΛ2 Λ2 Λ2 Λ2 )F = 0, where

⎧ ∂2 ∂2 ∂2 ∂2 ⎪ 2 2 ⎪ ⎪ ∇ = + , Λ = − , ⎨ ∂x2 ∂y 2 ∂x2 ∂y 2 ⎪ R2 (L + M )(K2 + K3 ) ⎪ ⎪ . ⎩ ε= [M (K1 + K2 + K3 ) − R2 ][(L + 2M )K1 − R2 ]

(6.6-10)

(6.6-11)

It is obvious that if K2 + K3 = 0, then ε = 0, equation (6.6-10) will be reduced to (6.2-7). If F (x, y) is the solution of (6.6-10), substituting it into (6.6-9) then into (6.6-6), one can obtain the displacement ﬁeld ⎧ ∂2 ⎪ ⎪ ⎪ u = [M αΠ1 + (L + 2M )βΠ2 ] + 4ω(K2 + K3 ) ⎪ ⎪ x ∂x∂y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂2 ∂2 ⎪ ⎪ × M 2 + (L + 2M ) 2 Λ2 F, ⎪ ⎪ ⎪ ∂x ∂y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂2 ∂2 ⎨ uy = M αΠ1 2 − (L + 2M )βΠ2 2 ∇2 F (6.6-12) ∂y ∂x ⎪ ⎪ ∂4 ⎪ 2 ⎪ ⎪ −4ω(K2 + K3 )(L + M ) 2 2 Λ F, ⎪ ⎪ ∂x ∂y ⎪ ⎪ ⎪ ⎪ 2 ⎪ ∂ ⎪ ⎪ [ω(α − β)Π1 Π2 − 4ω 2 (K2 + K3 )Λ2 Λ2 ]F, wx = ⎪ ⎪ ⎪ ∂x∂y ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ ∂4 ⎪ 2 ⎩ wy = −ω αΠ12 ∂ + βΠ22 ∂ (K + K ) Λ2 F. F − 8ω 2 3 ∂y 2 ∂x2 ∂x2 ∂y 2

6.6

Plane elasticity of point group 8mm of octagonal quasicrystals, displacement . . . 93

Similarly, the stress ﬁeld has the following expressions: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

σxx = 2M (L + M )αΠ1

∂3 2 ∂5 ∇ F + 8M ω(L + M )(K2 + K3 ) 2 3 Λ2 F, 3 ∂y ∂x ∂y

σyy = 2M (L + M )αΠ1

∂3 ∂5 ∇2 F + 8M ω(L + M )(K2 + K3 ) 4 Λ2 F, 2 ∂x ∂y ∂x ∂y

∂3 ∂5 2 ∇ F −8M ω(L+M )(K +K ) Λ2 F, 2 3 ∂x∂y 2 ∂x3 ∂y 2

∂2 ∂ ∂3 2 M αΠ1 Λ + 2(L + 2M )βΠ2 2 ∇2 + 4Rω(K2 + K3 ) 2 Hxx = R ∂y ∂x ∂x ∂y 2 2 3 ∂ ∂ ∂ M 2 + (2L + 5M ) 2 Λ2 + K1 ω 2 [(α − β)Π1 Π2 − 4ω(K2 + K3 ) ∂x ∂y ∂x ∂y ∂3 2 ∂3 ∂5 2 2 2 2 Λ Λ ] − K2 ω α 3 Π1 + β 2 Π2 + 8ω(K2 + K3 ) 2 3 Λ F, ∂y ∂x ∂y ∂x ∂y

∂2 ∂ M αΠ1 Λ2 + 2(L + 2M )βΠ2 2 ∇2 Hyy = R ∂y ∂x 3 2 ∂ ∂2 ∂ + 4Rω(K2 + K3 ) 2 M 2 + (2L + 5M ) 2 Λ2 ∂x ∂y ∂x ∂y

σxy = σyx = −2M (L + M )αΠ1

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Hxy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Hyx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

∂3 [(α − β)Π1 Π2 − 4ω(K2 + K3 )Λ2 Λ2 ] ∂x2 ∂y ∂3 ∂3 ∂5 − K1 ω α 3 Π12 + β 2 Π22 + 8ω(K2 + K3 ) 2 3 Λ2 F, ∂y ∂x ∂y ∂x ∂y

∂2 ∂ 2M α 2 Π1 − (L + 2M )βΠ2 Λ2 ∇2 = −R ∂x ∂y ∂3 + 4Rω(L + 2M )(K2 + K3 ) Λ2 Λ2 ∂x∂y 2 ∂3 + (K1 + K2 + K3 )ω [(α − β)Π1 Π2 − 4ω(K2 + K3 )Λ2 Λ2 ] ∂x∂y 2 ∂3 ∂3 2 ∂5 2 2 Π + β 3 Π2 + 8ω(K2 + K3 ) 3 2 Λ − K3 ω α F, ∂x∂y2 1 ∂x ∂x ∂y

∂ ∂2 = R 2M α 2 Π1 − (L + 2M )βΠ2 Λ2 ∇2 ∂x ∂y ∂3 − 4Rω(L + 2M )(K2 + K3 ) Λ2 Λ2 ∂x∂y 2 ∂3 + K3 ω [(α − β)Π1 Π2 − 4ω(K2 + K3 )Λ2 Λ2 ] − (K1 + K2 + K3 )ω ∂x∂y 2 ∂3 2 ∂5 ∂3 2 2 F. Π + β Π + 8ω(K + K ) Λ α 2 3 ∂x∂y 2 1 ∂x3 2 ∂x3 ∂y 2 (6.6-13) + K2 ω

94

Chapter 6

Elasticity of two-dimensional quasicrystals and simpliﬁcation

A part of above results was reported by Refs. [12], [15] and [17].

6.7

Stress potential of point group 5, 5 pentagonal and point group 10, 10 decagonal quasicrystals

In Section 4 we discussed the displacement potential of plane elasticity of point group 5, 5 pentagonal and point group 10, 10 decagonal quasicrystals. But the stress potential for the quasicrystals is also beneﬁcial, which will be introduced in this section. From the basic formulas listed in Section 6.4, if we exclude the displacements then there are the deformation compatibility equations ∂ 2 εyy ∂ 2 εxy ∂ 2 εxx + = 2 , ∂y 2 ∂x2 ∂x∂y

∂wxy ∂wxx = , ∂x ∂y

∂wyx ∂wyy = . ∂y ∂x

(6.7-1)

If introducing stress potential functions φ(x, y), ψ1 (x, y) and ψ2 (x, y), such as ∂2φ ∂2φ ∂2φ , , σyy = , σxy = σyx = − 2 2 ∂y ∂x ∂x∂y ∂ψ1 ∂ψ1 ∂ψ2 ∂ψ2 = , Hxy = − , Hyx = − , Hyy = , ∂y ∂x ∂y ∂x

σxx = Hxx

(6.7-2)

then equilibrium equations ∂σij /∂xj = 0 and ∂Hij /∂xj = 0 will be automatically satisﬁed. Based on the generalized Hooke’s law (6.4-3), all strain components can be expressed by relevant stress components: ⎧ 1 1 ⎪ ⎪ (σ + σyy ) + [(K1 + K2 )(σxx − σyy ) ε = ⎪ ⎪ xx 4(L + M ) xx 4c ⎪ ⎪ ⎪ ⎪ − 2R1 (Hxx + Hyy ) − 2R2 (Hxy − Hyx )], ⎪ ⎪ ⎪ ⎪ ⎪ 1 1 ⎪ ⎪ εyy = (σxx + σyy ) − [(K1 + K2 )(σxx − σyy ) ⎪ ⎪ 4(L + M ) 4c ⎪ ⎪ ⎪ ⎪ ⎪ − 2R (H + H ) − 2R 1 xx yy 2 (Hxy − H)], ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ε = εyx = [(K1 + K2 )σxy − R2 (Hxx + Hyy ) + R1 (Hxy − Hyx )], ⎪ ⎪ ⎨ xy 2c 1 1 (Hxx − Hyy ) + [M (Hxx + Hyy ) − R1 (σxx − σyy ) − 2R2 σxy ], wxx = ⎪ ⎪ 2(K − K ) 2c ⎪ 1 2 ⎪ ⎪ ⎪ 1 1 ⎪ ⎪ ⎪ wyy = − (Hxx − Hyy ) + [M (Hxx + Hyy ) ⎪ ⎪ 2(K1 − K2 ) 2c ⎪ ⎪ ⎪ ⎪ (σ − σ ) − 2R σ ], − R ⎪ 1 xx yy 2 xy ⎪ ⎪ ⎪ ⎪ 1 1 M ⎪ ⎪ wxy = [−R2 (σxx −σyy )+2R1 σxy ]+ (Hxy +Hyx )+ (Hxy −Hyx ), ⎪ ⎪ 2c 2(K1 −K2 ) 2c ⎪ ⎪ ⎪ ⎪ 1 1 M ⎪ ⎪ ⎩wyx = [R2 (σxx −σyy)−2R1 σxy]+ (Hxy +Hyx )− (Hxy −Hyx ), 2c 2(K1 −K2 ) 2c (6.7-3)

6.8

Stress potential of point group 8mm octagonal quasicrystals

95

in which c = M (K1 + K2 ) − 2(R12 + R22 ).

(6.7-4)

So the deformation compatibility equations (6.7-2) can be rewritten by the stresses σij , Hij , then by employing (6.4-6), one has ⎧ R1 ∂ ∂ K1 + K2 1 ⎪ 2 2 ⎪ ∇ φ + ψ − ψ + ∇ Π Π ⎪ 1 1 2 2 ⎪ ⎪ 2(L + M ) 2c c ∂y ∂x ⎪ ⎪ ⎪ ⎪ ⎪ ∂ R2 ∂ ⎪ ⎪ Π2 ψ1 + Π1 ψ2 = 0, ⎪ ⎨ + c ∂x ∂y (6.7-5) ⎪ c ∂ ∂ ⎪ 2 ⎪ + M ∇ ψ1 + R1 Π1 φ + R2 Π2 φ = 0, ⎪ ⎪ ⎪ K1 − K2 ∂y ∂x ⎪ ⎪ ⎪ ⎪ ⎪ ∂ ∂ c ⎪ ⎪ + M ∇2 ψ2 − R1 Π2 φ + R2 Π1 φ = 0, ⎩ K1 − K2 ∂x ∂y where ∇2 =

∂2 ∂2 + 2, 2 ∂x ∂y

Π1 = 3

∂2 ∂2 − 2, 2 ∂x ∂y

Π2 = 3

∂2 ∂2 − 2. 2 ∂y ∂x

The equations (6.7-5) will be satisﬁed when we choose a new function G, which is called the stress function, such that φ = c1 ∇2 ∇2 G, ∂ ∂ ψ1 = − R1 Π1 + R2 Π2 ∇2 G, ∂y ∂x ∂ ∂ ψ2 = R1 Π2 − R2 Π1 ∇2 G, ∂x ∂y in which

(6.7-6)

c +M K1 − K 2

(6.7-7)

∇2 ∇2 ∇2 ∇2 G = 0.

(6.7-8)

c1 = and

So the ﬁnal governing equation based on the stress potential is the same as that based on the displacement potential.

6.8

Stress potential of point group 8mm octagonal quasicrystals

The ﬁnal governing equation of plane elasticity of point group 8mm octagonal quassicrystals was given in Section 6 by displacement potential, similarly we can also give a derivation by stress potential.

96

Chapter 6

Elasticity of two-dimensional quasicrystals and simpliﬁcation

The strain compatibility equations and the deﬁnition on stress potential are the same as those given by (6.7-1) and (6.7-2), and the strain-stress relations are as follows:

1 1 (K1 + K2 )(L + 2M ) − 2R2 σxx εxx = (L + M )c 4 1 1 − (K1 + K2 )L + 2R2 σyy − R(L + M )(Hxx + Hyy ) , (6.8-1a) 4 2

1 1 1 − (K1 + K2 )L + 2R2 σxx + [(K1 + K2 )(L + 2M ) εyy = (L + M )c 4 4 1 −2R2 ]σyy + R(L + M )(Hxx + Hyy ) , (6.8-1b) 2 1 (6.8-1c) εxy = εyx = [(K1 + K2 )σxy + R(Hxy − Hyx )], 2c 1 1 wxx = R(K1 − K2 )(σyy − σxx ) (K1 − K2 )c 2 + (K1 M − R2 )Hxx − (K2 M − R2 )Hyy , (6.8-1d) 1 1 R(K1 − K2 )(σyy − σxx ) wyy = (K1 − K2 )c 2 − (K2 M − R2 )Hxx + (K1 M − R2 )Hyy , (6.8-1e) 1 {R(K1 + K2 + 2K3 )σxy (K1 + K2 + 2K3 )c + (K1 + K2 + K3 )M − R2 Hxy − (K3 M + R2 )Hyx , 1 = {−R(K1 + K2 + 2K3 )σxy (K1 + K2 + 2K3 )c −(K3 M + R2 )Hxy + (K1 + K2 + K3 )M − R2 Hyx ,

wxy =

wyx

(6.8-1f)

(6.8-1g)

in which c = M (K1 + K2 ) − 2R2 . If introducing stress potential functions φ(x, y), ψ1 (x, y) and ψ2 (x, y) such as ∂2φ , ∂y2 ∂ψ1 = , ∂y

∂ 2φ ∂2φ , σ = σ = − , xy yx ∂x2 ∂x∂y ∂ψ1 ∂ψ2 ∂ψ2 =− , Hyx = − , Hyy = . ∂x ∂y ∂x

σxx =

σyy =

Hxx

Hxy

(6.8-2)

then equilibrium equations (6.5-9) (or ∂σij /∂xj = 0, ∂Hij /∂xj = 0) will be automatically satisﬁed. Substituting the stress formulas (6.8-2) into the strain-stress relation (6.8-1) then

6.8

Stress potential of point group 8mm octagonal quasicrystals

into the deformation compatibility equations (6.7-1), then we have ⎧ ∂ ∂ ⎪ ⎪ c1 ∇2 ∇2 φ + 2R(L + M ) Π1 ψ1 − 2R(L + M ) Π2 ψ2 = 0, ⎪ ⎪ ∂y ∂x ⎪ ⎪ ⎪ ⎨ 1 ∂ 2 2 2 ∂ ψ2 ∂ ψ1 ∂ ψ1 − c4 = 0, − c3 − R Π1 φ + c2 2 ⎪ 2 ∂y ∂x∂y ∂x ∂y 2 ⎪ ⎪ ⎪ ⎪ 2 2 2 ⎪ ⎪ ⎩ 1 R ∂ Π2 φ + c2 ∂ ψ1 − c4 ∂ ψ2 − c3 ∂ ψ2 = 0, 2 ∂x ∂x∂y ∂x2 ∂y 2

97

(6.8-3)

where ⎧ ∂2 ∂2 ∂2 ∂2 ∂2 ∂2 ⎪ 2 ⎪ = + , Π = 3 − , Π = 3 − , ∇ 1 2 ⎪ ⎪ ∂x2 ∂y 2 ∂x2 ∂y2 ∂y2 ∂x2 ⎪ ⎪ ⎪ ⎪ ⎨ K3 M + R2 K2 M + R2 + , c1 = (K1 + K 2 )(L + 2M ) − 2R2 , c2 = ⎪ K1 + K 2 + 2K3 K1 − K 2 ⎪ ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ ⎩ c3 = (K1 + K 2 + K3 )M − R , c4 = K1 M + R . K1 + K 2 + 2K3 K1 − K 2 (6.8-4) ∂ The manipulation of the second equation of (6.8-3) by × Π2 + the third equation ∂x ∂ of (6.8-3) by× Π1 , adding them together, leads to ∂y ∂3 ∂3 ∂3 Π − c Π − c Π c2 1 3 2 4 2 ψ1 ∂x∂y2 ∂x3 ∂x∂y 2 =

c4

∂3 ∂3 ∂3 + c Π − c Π Π 1 3 1 2 2 ψ2 . ∂x2 ∂y ∂y 3 ∂x2 ∂y

There exists a function A(x, y) such that ⎧ ∂3 ∂3 ∂3 ⎪ ⎪ ψ = c + c Π − c Π Π 4 1 3 1 2 2 A, ⎪ ⎨ 1 ∂x2 ∂y ∂y 3 ∂x2 ∂y ⎪ ⎪ ∂3 ∂3 ∂3 ⎪ ⎩ ψ2 = c2 Π − c Π − c Π 1 3 2 4 2 A. ∂x∂y 2 ∂x3 ∂x∂y2

(6.8-5)

(6.8-6)

Substituting (6.8-6) into the third equation of (6.8-3), we arrive at the function G(x, y) satisfying the relations such as φ = −c3 c4 ∇2 ∇2 G, A=

1 RG. 2

(6.8-7)

In the derivation the relation c2 = c4 − c3 has been used. The stress potentials φ, ψ1 , ψ2 can be expressed by the new function G(x, y), i.e.,

98

Chapter 6

Elasticity of two-dimensional quasicrystals and simpliﬁcation

⎧ φ = −c3 c4 ∇2 ∇2 G, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂3 ∂3 ∂3 1 ⎨ ψ1 = R c4 2 Π1 + c3 3 Π1 − c2 2 Π2 G, 2 ∂x ∂y ∂y ∂x ∂y ⎪ ⎪ ⎪ 3 3 ⎪ ∂ ∂ ∂3 1 ⎪ ⎪ Π1 − c3 3 Π2 − c4 Π2 G. ⎩ ψ2 = R c2 2 ∂x∂y 2 ∂x ∂x∂y 2

(6.8-8)

Substituting (6.8-8) into the ﬁrst equation of (6.8-3) yields − c1 c3 c4 ∇2 ∇2 ∇2 ∇2 G + R2 (L + M ) c4 + c3

∂4 2 ∂4 2 G = 0. Π + c Π 4 2 2 ∂y 4 ∂x2 ∂y 2

∂4 ∂4 2 ∂4 2 Π + c Π − 2c Π1 Π2 3 2 ∂x2 ∂y 2 1 ∂y 4 1 ∂x2 ∂y 2 (6.8-9)

Considering the following relations ! ∂4 1 = ∇2 ∇2 − Λ2 Λ2 , Π1 Π2 = ∇2 ∇2 − 4Λ2 Λ2 , ∂x2 ∂y 2 4 ∂2 ∂2 ∂2 ∂2 ∇2 = + 2 , Λ2 = − 2, 2 2 ∂x ∂y ∂x ∂y

c2 = c4 − c3 ,

equation (6.8-9) can be simpliﬁed as ∇2 ∇2 ∇2 ∇2 G − 4ε∇2 ∇2 Λ2 Λ2 G + 4εΛ2 Λ2 Λ2 Λ2 G = 0

(6.8-10)

with R2 (L + M )(K2 + K3 ) R2 (L + M )(c3 − c4 ) = . −c1 c3 + R2 (L + M )c3 [(K1 + K2 + K3 )M − R2 ][K1 (L + 2M ) − R2 ] (6.8-11) The ﬁnal governing equation in this case is exactly in agreement to that given by the displacement potential formulation, discussed in Section 6. ε=

6.9

Engineering and mathematical elasticity of quasicrystals

The complicated structure of quasicrystals leads to tremendous complexity of their elasticity equations. Even though, the solution can also be carried out, and fruitful results are found. This formulation is eﬀective not only for linear elasticity of quasicrystals but also for nonlinear material response of the solids if using some simple physical models[18,19] . Those studies belong to mathematical theory of elasticity of quasicrystals. In the subsequent chapters we can realize that the diﬃculty for analytic solutions does not only lie in the complexity of the equations but also the boundary conditions. In some cases the boundary conditions can be simpliﬁed, useful approximate

6.9

Engineering and mathematical elasticity of quasicrystals

99

solutions may be constructed with relative ease. For this purpose we consider an example. From Section 6.3, we have deformation compatibility equations ⎧ 2 ∂ 2 εxy ∂ εxx ∂ 2 εyy ⎪ ⎪ ⎨ ∂y 2 + ∂x2 = 2 ∂x∂y , ⎪ ⎪ ⎩ ∂wxy = ∂wxx , ∂wyx = ∂wyy . ∂x ∂y ∂y ∂x

(6.9-1)

Then the substitution of (6.3-2) into (6.9-1) yields the compatibility equations expressed by σij and Hij L+M ∇ (σxx + σyy ) − C 2

∂2 ∂2 − ∂x2 ∂y2

× [(K1 + K2 )(σxx − σyy ) − 2R(Hxx + Hyy )] L + M ∂2 [(K1 + K2 )σxy − R(Hxy + Hyx )] C ∂x∂y (R2 − M K1 )Hxy + (R2 − M K2 )Hyx 1 ∂ Rσxy − C ∂x K1 − K 2 1 1 ∂ ∂ = (Hxx −Hyy )+ [M (Hxx +Hyy )−R(σxx −σyy )] 2(K1 −K2 ) ∂y 2C ∂y (R2 − M K1 )Hxy + (R2 − M K2 )Hyx 1 ∂ −Rσxy − C ∂y K1 − K 2 −1 1 ∂ ∂ = (Hxx −Hyy )+ [M (Hxx +Hyy )−R(σxx −σyy )], 2(K1 −K2 ) ∂x 2C ∂x =8

(6.9-2a)

(6.9-2b)

(6.9-2c)

in which C, M, L, K1 , K2 and R are material constants given in Section 6.3. These three equations are combined with the equilibrium equations ⎧ ∂σxx ∂σxy ∂σyx ∂σyy ⎪ ⎪ + = 0, ⎨ ∂x + ∂y = 0, ∂x ∂y ⎪ ⎪ ∂Hxx + ∂Hxy = 0, ∂Hyx + ∂Hyy = 0 ⎩ ∂x ∂y ∂x ∂y

(6.9-3)

and this provides a basis for solving the problem. A very meaningful example is pure bending of beam, shown in Fig.6.9-1. For this sample, at the upper and lower surfaces of the beam there are following boundary conditions σyy = 0, σyx = 0, (6.9-4) Hyy = 0, Hyx = 0,

100

Chapter 6

Fig.6.9-1

Elasticity of two-dimensional quasicrystals and simpliﬁcation

Beam of quasicrystal of ten-fold symmetry under pure bending

in addition there are so-called St. Venant boundary conditions at the end sections ⎧ h/2 h/2 ⎪ ⎪ σxx dy = 0, yσxx dy = Mz , ⎪ ⎪ ⎪ −h/2 ⎪ ⎪ −h/2 ⎪ ⎪ h/2 ⎨ h/2 (6.9-5) Hxx dy = 0, yHxx dy = Lz , ⎪ −h/2 −h/2 ⎪ ⎪ ⎪ ⎪ h/2 h/2 ⎪ ⎪ ⎪ ⎪ σxy dy = 0, Hyx dy = 0. ⎩ −h/2

−h/2

where Mz and Lz represent the resultant moments of σxx and Hxx . The direction of vector of the moments is z. Boundary conditions (6.9-5) are relaxation boundary conditions, this gives some ﬂexibility for solution. At ﬁrst we assume that the value of Lz momentarily be undetermined, and assume further σxx = A1 y,

σyy = σxy = σyx = 0,

Hyy = Hxy = Hyx = 0,

Hxx = f (y)

and A1 and f (y) are to be determined. Substituting σxx = A1 y into (6.9-5), we ﬁnd that 2Mz (6.9-6) A1 = 3 h so that Mz σxx = y, (6.9-7) I where I = 1 · h3 /12 represents the inertia moment of the transverse section with height h and width 1. It can be veriﬁed that the above results have satisﬁed equations (6.7-3) and (6.7-2). Substituting (6.9-7) into (6.9-2b) yields Hxx =

2RMz y Rσxx , = M 1 1 M + I + 2(K1 − K2 ) 2C K1 − K2 C

(6.9-8)

References

so the equation is satisﬁed. So that we have the solution ⎧ Mz ⎪ ⎪ ⎨ σxx = I y, σyy = σxy = σyx = 0, 2RMz y ⎪ ⎪ ⎩ Hxx = , Hyy = Hxy = Hyx = 0. I[1/(K1 − K2 ) + M/C]

101

(6.9-9)

At last, combining (6.9-9) and (6.9-5) determines the value of Lz . Here we utilize the physical character of the sample to solve the problem easily. This procedure is quite eﬀective for some practical problems, and needs not so hard mathematics. This treatment is similar to that in the engineering elasticity of conventional structural materials, namely there is engineering elasticity of quasicrystals. But this monograph mainly discusses mathematical elasticity of quasicrystals on problems with topological or metric defects such as dislocations, cracks concerning stress singularities, or with local discontinuity of physical quantities such as in contact, shock wave problems. The complexity of boundary conditions renders the above engineering approximate treatment ineﬀective, we must develop systematic and direct analysis methods. In the classical elasticity, such systematic and direct methods were developed by Muskhelishvili[20] in 1930’s and Sneddon[21] in 1940’s and present important eﬀects promoting elasticity and relevant disciplines of science and engineering. By extending the methods in the classical elasticity to quasicrystal elasticity we will display in detail the application in Chapters 7, 8 for one-and two-dimensional quasicrystals, in Chapter 9 for three-dimensional quasicrystals, respectively.

References [1] Chernikov M A, Ott H R, Bianchi A et al. Elastic moduli of a single quasicrystal of decagonal Al-Ni-Co: evidence for transverse elastic isotropy. Phys Rev Lett, 1998, 80(2): 321–324 [2] Abe H, Taruma N, Le Bolloc’h D et al. Anomalous-X-ray scattering associated with short-range order in an Al70 Ni15 Co15 decagonal quasicrystal. Mater Sci and Eng A, 2000, 294∼296(12): 299–302 [3] Jeong H C, Steinhardt P J. Finite-temperature elasticity phase transition in decagonal quasicrystals. Phys Rev B, 1993, 48(13): 9394–9403 [4] Edagawa K. Phonon-phason coupling in decagonal quasicrystals. Phil Mag, 2007, 87(18∼21): 2789–2798 [5] Fan T Y, Mai Y W. Elasticity theory, fracture mechanics and relevant thermal properties of quasicrystalline materials. Appl Mech Rev, 2004, 57(5), 325–344 [6] Hu C Z, Yang W G, Wang R H et al. Symmetry and physical properties of quasicrystals. Prog Phys, 1997, 17(4): 345–374

102

Chapter 6

Elasticity of two-dimensional quasicrystals and simpliﬁcation

[7] Bohsung J, Trebin H R. Introduction to the Quasicrystal Mathematics. Jaric M V, New York: Academic Press, 1989 [8] De P, Pelcovis R A. Disclinations in pentagonal quasicrystals. Phys Rev B, 1987, 35 (17): 9604–9607 [9] Yang S H, Ding D H. Foundation of Dislocation Theory of Crystals. Vol. II. Beijing: Science Press, 1998 (in Chinese) [10] De P, Pelcovits R A. Linear elasticity of pentagonal quasicrystals. Phys Rev B, 1987, 36(13): 8609–8620 [11] Ding D H, Wang R H, Yang W G et al. General expressions for the elastic displacement ﬁelds induced by dislocations in quasicrystals. J Phys: Condens Matter, 1995, 7(28): 5423–5426 [12] Li X F, Fan T Y. New method for solving elasticity problems of some planar quasicrystals and solutions. Chin Phys Letter, 1998, 15(4): 278–280 [13] Guo Y C, Fan T Y. Mode II Griﬃth crack in decagonal quasicrystals. Appl Math Mech, English Edition, 2001, 22(10): 1311–1317 [14] Li L H, Fan T Y. Complex function method for solving notch problem of point 10 two-dimensional quasicrystal based on the stress potential function. J Phys: Condens Matter, 2006, 18(47): 10631–10641 [15] Li X F, Fan T Y, Sun Y F. A decagonal quasicrystal with a Griﬃth crack. Phil Mag A, 1999, 79(7): 1943–1952 [16] Li X F, Duan X Y, Fan T Y, et al. Elastic ﬁeld for a straight dislocation in a decagonal quasicryatal. J Phys: Condens Matter, 1999, 11(3): 703–711 [17] Zhou W M, Fan T Y. Plane elasticity of octagonal quasicrystals and solutions. Chin Phys, 2001, 10(8): 743–747 [18] Fan T Y, Trebin A R, Messerschmidt U et al. Plastic ﬂow coupled with a Griﬃth crack in some one- and two-dimensional quasicrystals. J Phys: Condens Matter, 2004, 16 (37): 5419–5429 [19] Fan T Y and Fan L. Plastic fracture of quasicrystals, Phil. Mag., 2008, 88(4), 321– 335 [20] Muskhelishvili N I. Some Basic Problems in the Mathematical Theory of Elasticity. Groringen, P Noordhoﬀ Ltd: 1953 [21] Sneddon I N. Fourier Transforms. New York: McGraw-Hill, 1951

Chapter 7 Application I−−Some dislocation and interface problems and solutions in oneand two-dimensional quasicrystals In Chapters 5 and 6, with the physical basis of quasicrystal elasticity based on the density wave model, we have performed some mathematical operations, by proper simpliﬁcation, to reduce the original problems to the boundary value problems of high-order partial diﬀerential equations, and to establish the standard solving procedure and the fundamental solutions. This work is the development of the boundary value problems in classical elasticity. Here, we need to pose a question: Do these mathematical operations contribute to solving the quasicrystal elasticity problems? This is answered only by practice. The following two chapters will provide the applications of these theories, including the solutions to some dislocation, crack and interface problems in one- and two-dimensional quasicrystals. The calculation results indicate that the mathematical operations discussed in Chapters 5 and 6 are powerful in solving these problems. As we know, almost all monographs of classical elasticity do not place their main focus upon dislocation and crack problems. These problems have been investigated in monographs relating to dislocation theory and fracture mechanics, respectively. As a monograph on theories of elasticity of quasicrystals, the current work is not going to focus entirely on dislocations and cracks in quasicrystals. Therefore, as an attempt to examine the theories developed in Chapters 5 and 6 and to show their applications in elasticity of quasicrystals, we present some calculation examples of certain realistic dislocation and crack problems in elasticity of quasicrystals. The methods developed in this work can be used for studying other problems in quasicrystals. Historically, soon after the discovery of quasicrystals, scientists proposed the possibility of existing dislocations in quasicrystals. De and Pelcovits[1,2] ﬁrst studied elastic ﬁeld around dislocations and disclinations in quasicrystals. Furthermore, Ding et al[3,4] conducted systematic investigation of this topic by using the Green T. Fan, Mathematical Theory of Elasticity of Quasicrystals and Its Applications © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2011

104

Chapter 7

Application I—Some dislocation and interface problems and . . .

function method. Now people have recognized that quasicrystal is a kind of ordered phase of quasiperiodic long-range. Similar to crystals with ordered phase of long-range, the breaking of long-range regularity takes place usually through topological defects, i.e., dislocations lead to the breaking of the long-range symmetry. As mentioned in Chapter 3, quasicrystals also possess the orientational symmetries incompatible with those permitted in crystal theory. In quasicrystals, another type of defect, disclinations, also exists simultaneously, which lead to breaking of the orientational symmetry in quasicrystals. In some cases, the crystalline phases often coexist with the quasicrystalline phases. So there is another kind of defects, the interface between quasicrystal and crystal. The existence of these defects including the stacking fault dramatically aﬀects the mechanical properties of quasicrystals. Therefore, it is important to study the elastic properties of quasicrystals with dislocations, disclinations, interfaces and stacking faults. As aforementioned, during the study of dislocations in quasicrystals, physicists have developed some mathematical methods, such as the Green function method. These important methods are available in relevant literature. Hereafter, by using the elementary methods developed in Chapters 5 and 6, complex variable function and Fourier transform, we study the displacement and stress ﬁelds around dislocations in one- and two-dimensional quasicrystals quantitatively. The three-dimensional dislocation and dynamic dislocation problems will be discussed in Chapters 9 and 10 respectively.

7.1

Dislocations in one-dimensional hexagonal quasicrystals

Following the sequence from simplicity to complexity, we ﬁrst study the dislocations in one-dimensional quasicrystals. A dislocation in a n-dimensional quasicrystal bears || || || the Burgers vector b|| ⊕b⊥ , where b|| =(b1 , b2 , · · · , bn ) is its Burgers vector of phonon || ⊥ ⊥ ﬁeld, and b⊥ =(b⊥ 1 ,b2 , · · · , bn ) the Burgers vector of phase ﬁeld. And b is located in the physical space E|| , while b⊥ in the supplementary space or the vertical space || || || E⊥ . For one-dimensional quasicrystals, b|| =(b1 , b2 , b3 ) and b⊥ =(0,0, b⊥ 3 ) since || || || || ⊥ ⊥ ⊥ ⊥ b1 = b2 =0. Therefore, b ⊕ b =(b1 , b2 , b3 , b3 ) here, which can be dealt with as || || || || || the superposition of (b1 , b2 ,0,0) and (0,0, b3 , b⊥ 3 ). The (b1 ,b2 ,0,0) corresponds to the blade dislocation in regular hexagonal crystals, whose elastic solution is available in common metal physics or dislocation monographs, (e.g. Refs.[7],[8]). The one corre|| spondence to the component (0,0,b3 , b⊥ 3 ) is the screw dislocation in one-dimensional quasicrystal. We solve the elastic ﬁeld induced by this dislocation in the following. In Section 5.2, we have obtained the governing equations for anti-plane strain

7.1

Dislocations in one-dimensional hexagonal quasicrystals

105

problems in one-dimensional hexagonal quasicrystal elasticity such as ∇2 uz = 0,

∇2 wz = 0.

(7.1-1)

The boundary conditions corresponding to the screw dislocation with the Burgers || vector (0, 0, b3 , b⊥ 3 ) are ⎧ " " ⎨ uz ""y=0+ − uz ""y=0− = b , 3 (7.1-2a) " " ⎩ wz "y=0+ − wz "y=0− = b⊥ 3 or

Γ

duz = b3 ,

Γ

dwz = b⊥ 3,

(7.1-2b)

where Γ indicates an arbitrary contour surrounding the dislocation core. The solution to the boundary value problem (7.1-1) and (7.1-2) is

uz =

y b3 arctan , 2π x

wz =

y b⊥ 3 arctan . 2π x

(7.1-3)

The stress components can be extracted according to the stress-strain relation (5.2-8) in Chapter 5 as ⎧ C44 b3 y y R3 b⊥ ⎪ 3 ⎪ ⎪ σ = σ = − − , xz zx ⎪ ⎪ 2π x2 + y 2 2π x2 + y 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ C44 b3 x x R3 b⊥ ⎪ 3 ⎪ = σ = − σ + , ⎪ zy ⎨ yz 2π x2 + y 2 2π x2 + y 2 (7.1-4) ⎪ ⊥ ⎪ b b K y y R 2 3 ⎪ 3 3 ⎪ Hzx = − − , ⎪ ⎪ 2π x2 + y 2 2π x2 + y 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ K b⊥ x x R3 b3 ⎪ ⎩ Hzy = − 2 3 + . 2π x2 + y 2 2π x2 + y 2 ||

The displacement and stress ﬁelds corresponding to the Burgers vector b|| ⊕ b⊥ =(b1 , || || b2 , b3 , b⊥ 3 ) can be obtained by superposing the above elastic ﬁeld on that of the regular hexagonal crystals. The elastic strain energy induced by the screw dislocation is 1 ∂uz ∂wz σzj dx1 dx2 W = + Hzj 2 Ω ∂xj ∂xj ∂uz ∂wz 1 R0 2π drdθ = r σzj + Hzj 2 r0 0 ∂xj ∂xj 2

⊥ = (C44 b3 + K2 b⊥2 3 + 2R3 b3 b3 )

R0 1 ln , 4π r0

(7.1-5)

106

Chapter 7

Application I—Some dislocation and interface problems and . . .

where r0 is the size of the dislocation core, R0 is the size of dislocation network or inclusion, which are available in theory of dislocations in regular crystals.

7.2

Dislocations in quasicrystals with point groups 5m and 10mm symmetries

Consider a dislocation in pentagonal point group 5m or decagonal point group 10mm || || || || ⊥ quasicrystals with the Burgers vector b|| ⊕ b⊥ =(b1 ,b2 ,b3 ,b⊥ 1 ,b2 ), where b is its ⊥ Burgers vector of phonon ﬁeld, and b is its Burgers vector of phase ﬁeld. Here, b⊥ 3 =0 due to wz =0. This dislocation problem can be dealt with by superposition of two dislocation || || || ⊥ problems of the Burgers vectors (b1 ,b2 ,0,b⊥ 1 ,b2 ) and (0,0,b3 ), respectively. On the dislocation line parallel to the direction of periodic arrangement (z-direction), the || || ⊥ elastic ﬁeld induced by the dislocation of the Burgers vector (b1 ,b2 ,0,b⊥ 1 ,b2 ) can be simpliﬁed as an in-plane elastic ﬁeld in quasicrystal elasticity, which can be analyzed by the methods discussed in Section 6.1 and Section 6.2. The elastic ﬁeld induced || by (0,0,b3 ) is governed by equation ∇2 uz =0, and its solution has been obtained in Section 7.1. Therefore, in the following two sections, we always assume the dislocation line parallel to the direction of atom periodic arrangement in quasicrystals, and we only study the elastic ﬁeld induced by the dislocation with the Burgers vector || || ⊥ (b1 ,b2 ,0,b⊥ 1 ,b2 ). To demonstrate the methodology concisely, we ﬁrst consider a special case of || (b1 ,0,0,b⊥ 1 ,0). During this procedure, in order to further simplify the mathematical || || ⊥ process, we ﬁrst analyze the case of b1 = 0 and b2 = b⊥ 1 = b2 = 0. In this case, we consider the problem in the upper half-space of y 0, and the problem has the following boundary conditions: ⎧ σij (x, y) → 0, Hij (x, y) → 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ σyy (x, 0) = 0,

#

x2 + y 2 → ∞,

wx (x, 0) = wy (x, 0) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ux (x, 0+ ) − uy (x, 0− ) = b or dux = b1 , 1

(7.2-1)

Γ

where Γ indicates an arbitrary contour surrounding the dislocation core. The dislocation problem is reduced to solve equation (6.2-7), i.e., the following equation ∇2 ∇2 ∇2 ∇2 F = 0 under boundary conditions (7.2-1).

7.2

Dislocations in quasicrystals with point groups 5m and 10mm symmetries

By introducing the Fourier transform ∞ ˆ F (x, y)eiξx dx, F (ξ, y) =

107

(7.2-2)

−∞

where ξ is the Fourier transform parameter, the above equation can be reduced into

d2 − ξ2 dy 2

4 Fˆ = 0.

(7.2-3)

This is an ordinary diﬀerential equation with constant coeﬃcients, and its general solution is Fˆ (ξ, y) = [A1 + B1 y + C1 y 2 + D1 y 3 ]e−|ξ|y + [A2 + B2 y + C2 y 2 + D2 y 3 ]e|ξ|y ,

(7.2-4)

where A1 , B1 , · · · , D2 are functions with respect to ξ to be determined by boundary conditions. We are going to use the Fourier transform to search for solutions in Chapters 7∼10. For convenience, we only consider the problem in the upper half-space (or the lower half-space) based on the symmetry or anti-symmetry of the problems. It should be cautious that here the symmetry or anti-symmetry is in the view of macroscopic continuum media, namely the even and odd characteristics of the displacement function F (x,y) or stress function G(x,y) with respect to x, but not in the scale of crystalline structures. Furthermore, during the use of the Fourier transform, we attempt to make the boundary condition homogeneous at inﬁnity (zero boundary condition). Therefore, we only consider the case in the upper half-space, and the formal solution (7.2-4) can be simpliﬁed as Fˆ (ξ, y) = [A1 + B1 y + C1 y 2 + D1 y 3 ]e−|ξ|y .

(7.2-5)

In the evident reason the suﬃx of A1 , B1 , C1 , D1 can be removed in the following. For conciseness in writing, introduce the following indicator: X = (A, B, C, D),

Y = (1, y, y 2 , y 3 )T ,

(7.2-6)

where the symbol T indicates the matrix transpose. Therefore, (7.2-5) can be expressed as Fˆ (ξ, y) = XY e−|ξ|y . (7.2-5 ) Since A, B, C, and D are arbitrary functions with respect to ξ, without loss of generality, (7.2-5) can be rewritten as

108

Chapter 7

Application I—Some dislocation and interface problems and . . .

Fˆ (ξ, y) = (4ξ 4 )−1 XY e−|ξ|y .

(7.2-7)

By performing the Fourier transform on the displacement expressions (6.2-8), (6.2-9) and stress expressions (6.2-10), (6.2-11) and using the notation (7.2-7), we have u ˆx (ξ, y) = iξ −1 X[2nξ 2 Y + (m − 5n) |ξ| Y − (2m − 5n)Y ]e−|ξ|y ,

(7.2-8a)

u ˆy (ξ, y) = |ξ|−1 X[2nξ 2 Y − (m + 5n) |ξ| Y + (2m + 5n)Y ]e−|ξ|y , (7.2-8b) w ˆx (ξ, y) = −iω(α − β)ξ −1 X[2n |ξ|3 Y − 12ξ 2 Y + 15 |ξ| Y − 10Y ]e−|ξ|y ,

(7.2-9a)

w ˆy (ξ, y) = −ω(α − β) |ξ|−1 X[4 |ξ|3 Y − 12ξ 2 Y + 15 |ξ| Y − (10 + e0 − e1 )Y ]e−|ξ|y ,

(7.2-9b)

σ ˆxx (ξ, y) = 2M α(L + M )X(−2ξ 2 Y + 8 |ξ| Y − 13Y )e−|ξ|y ,

(7.2-10a)

σ ˆyy (ξ, y) = 2M α(L + M )X(2ξ 2 Y − 4 |ξ| Y + 3Y )e−|ξ|y , σ ˆxy (ξ, y) = σ ˆyx (ξ, y) = i2M α(L + M )ξ

−1

2

|ξ| X(2ξ Y − 6 |ξ| Y

(7.2-10b)

+ δY )e−|ξ|y ,

(7.2-10c)

ˆ xx (ξ, y) = −ω(α − β)(K1 − K2 ) |ξ|−1 X[4 |ξ|3 Y H − 16ξ 2 Y + 27 |ξ| Y − (25 + e2 )Y ]e−|ξ|y

(7.2-11a)

ˆ yy (ξ, y) = −ω(α − β)(K1 − K2 )X[−4 |ξ|3 Y + 12ξ 2 Y − 15 |ξ| Y H + (10 − e1 )Y ]e−|ξ|y ,

(7.2-11b)

ˆ xy (ξ, y) = iω(α − β)(K1 − K2 )ξ −1 |ξ| X[−4 |ξ|3 Y + 12ξ 2 Y H − 15 |ξ| Y + (10 + e2 )Y ]e−|ξ|y ,

(7.2-11c) 3

ˆ yx (ξ, y) = −iω(α − β)(K1 − K2 )ξ −1 |ξ| X[−4 |ξ| Y + 16ξ 2 Y H − 27 |ξ| Y + (25 − e1 )Y ]e−|ξ|y , where ⎧ ⎨ m = M α + (L + 2M )β, n = M α − (L + 2M )β, 2αβ 2αβ α+β , e2 = + ⎩ e1 = ω(α − β)(K1 − K2 ) ω(α − β)(K1 − K2 ) α − β and α, β, and ω are given in (6.2-5). The inverse of Fourier transform reads ∞ 1 Fˆ (ξ, y)e−iξx dξ, F (x, y) = 2π −∞

(7.2-11d)

(7.2-12)

(7.2-13)

7.2

Dislocations in quasicrystals with point groups 5m and 10mm symmetries

109

Similarly, the inverses of displacements and stresses are obtained as follows: ∞ 1 u ˆj (ξ, y)e−iξx dξ, (7.2-14) uj (x, y) = 2π −∞ ∞ 1 wj (x, y) = w ˆj (ξ, y)e−iξx dξ, (7.2-15) 2π −∞ ∞ 1 σ ˆjk (ξ, y)e−iξx dξ, (7.2-16) σjk (x, y) = 2π −∞ ∞ 1 ˆ jk (ξ, y)e−iξx dξ. Hjk (x, y) = H (7.2-17) 2π −∞ In the above dislocation problem, ux (x,y) is an odd function with respect to x, therefore F (x, y) must be an odd function with respect to x according to the ﬁrst expression in (6.2-8). For an odd function, the Fourier transform (7.2-2) is rewritten as ∞ ˆ F (x, y) sin(ξx)dx (7.2-18) F (ξ, y) = 0

and its inversion is F (x, y) =

2 π

∞

Fˆ (ξ, y) sin(ξx)dξ.

(7.2-19)

0

If F (x,y) is an even function with respect to x, it has ∞ ˆ F (ξ, y) = F (x, y) cos(ξx)dx

(7.2-20)

0

and its inversion is

2 F (x, y) = π

∞

Fˆ (ξ, y) cos(ξx)dξ.

(7.2-21)

0

Thereby, the corresponding integrals relating uj , wj , σjk and Hjk can be simpliﬁed. So far, the displacement and stress components in forms of integrals have been obtained, which all include the unknown functions A(ξ), B(ξ), C(ξ) and D(ξ) to be determined by the second expressions in the boundary conditions (7.2-1) such that A=

9Jsgnξ , 4ξ 2

B = 2Jξ 2 ,

C=

Jsgnξ , 2ξ 2

D = 0,

(7.2-22)

where J is a constant to be determined. By using the last expression in (7.2-1), i.e., the dislocation condition, J is found to be

J=

b1 . 8(n − m)

(7.2-23)

Substitution of the above expressions into (7.2-8)∼(7.2-11) and then into the inversion formulas (7.2-14)∼(7.2-17) leads to the displacement and stress ﬁelds induced

110

Chapter 7

Application I—Some dislocation and interface problems and . . . ||

by dislocation with Burgers vector (b1 ,0,0,0) in quasicrystals of point group 10mm ten-fold symmetry as below y xy b1 (L + M )K1 , arctan + 2π x (L + M )K1 + (M K1 − R2 ) r2 r b1 (M K1 − R2 ) uy = − ln 2π (L + M )K1 + (M K1 − R2 ) a 2 (L + M )K1 y + , (L + M )K1 + (M K1 − R2 ) r2 $ % 3 b1 2x y (L + M )K1 , wx = 2π (L + M )K1 + (M K1 − R2 ) r4 $ % 2 2 (L + M )K1 2x y b1 , wy = 2π (L + M )K1 + (M K1 − R2 ) r4

ux =

(7.2-24b) (7.2-25a) (7.2-25b)

σxx = −A

(7.2-26a)

σyy

(7.2-26b)

σxy Hxx Hyy Hxy Hyx where r =

y(3x2 + y 2 ) , r4 2 2 y(x − y ) =A , r4 x(x2 − y 2 ) = σyx = A , r4 R(K1 − K2 ) x2 y(3x2 − y 2 ) = −A , M K1 − R 2 r6 2 R(K1 − K2 ) x y(3y2 − x2 ) , = −A M K1 − R 2 r6 2 2 R(K1 − K2 ) xy (3x − y 2 ) =A , M K1 − R 2 r6 3 2 R(K1 − K2 ) x (3y − x2 ) = −A , M K1 − R 2 r6

(7.2-24a)

(7.2-26c) (7.2-27a) (7.2-27b) (7.2-27c) (7.2-27d)

# x2 + y 2 , a is the size of dislocation core and $ A=

b1 π

%

(L + M )(M K1 − R2 ) . (L + M )K1 + (M K1 − R2 )

(7.2-28)

Considering L = C12 , and M =(C11 − C12 )/2=C66 in (6.1-1), and substituting them into (7.2-24a) and (7.2-25), we ﬁnd that our solutions exactly consistent with those given by Ding et al[4,7] . They used the Green function method. This examines the correction of our derivation. In the case of R=0, the above solution recovers the solution of dislocation in hexagonal crystals[8] . If the material is isotropic, therefore L = λ and M = μ.

7.2

Dislocations in quasicrystals with point groups 5m and 10mm symmetries

111

Substitution of L and M into (7.2-28) yields

A=

μb1 , 2π(1 − ν)

(7.2-29)

where λ and μ are the Lam´e constants, and ν is the Poisson’s ratio of phonon ﬁeld. In this case, ui and σij recover those in a regular isotropic crystal with an edge dislocation while wi =0 and Hij =0. || ⊥ Let us consider another special case of b⊥ 1 = 0 and b1 = b2 = b2 = 0. In this case, the boundary conditions can be described as # ⎧ σij (x, y) → 0, Hij (x, y) → 0, x2 + y 2 → ∞, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ Hyy (x, 0) = 0, (7.2-30) ux (x, 0) = uy (x, 0) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ wx (x, 0+ ) − wx (x, 0− ) = b⊥ dwx = b⊥ 1, 1 or Γ

where the meaning of Γ is aforementioned. By using the similar procedure, the displacement and stress ﬁelds can mined as c1 − c2 2xy 3 b⊥ 1 k0 xy − ux = , 2πc2 r2 2c1 r4 c1 − c2 y 2 (x2 − y 2 ) b⊥ xy 1 k0 uy = − 2 + , 2πc2 r 2c1 r4 y c k xy(3x2 − y 2 )(3y 2 − x2 ) b⊥ 0 0 wx = 1 arctan + , 2π x 2c1 c2 3r 6 b⊥ M c0 k0 y 2 (3x2 − y 2 )2 L + 2M r − wy = 1 1− ln + , 2π 2c1 2c2 a 2c1 c2 3r 6 ⊥ 2 c0 b1 k0 x y(3x2 − y 2 ) , σxx = − πc1 R r6 ⊥ 3 2 c0 b1 k0 y (3x − y 2 ) σyy = − , πc1 R r6 ⊥ c0 b1 k0 2xy 2 (y 2 − x2 ) σxy = σyx = , πc1 R r6 k0 b⊥ y 2x2 y(3x2 − y 2 )(3y 2 − x2 ) 1 Hxx = (e1 + e2 ) 2 − , 2πe1 r r8 ⊥ k 0 b1 y 2(x2 − y 2 ) (x2 − y 2 )(3x2 − y 2 )(3y 2 − x2 ) Hyy = − + , 2πe1 r4 r8 k0 b⊥ x 2xy2 (3x2 − y 2 )(3y 2 − x2 ) 1 Hxy = (e1 + e2 ) 2 + , 2πe1 r r8

be deter-

(7.2-31a) (7.2-31b) (7.2-31c) (7.2-31d) (7.2-32a) (7.2-32b) (7.2-32c) (7.2-33a) (7.2-33b) (7.2-33c)

112

Chapter 7

Hyx = −

k 0 b⊥ 1x 2πe1

Application I—Some dislocation and interface problems and . . .

2(x2 − y 2 ) (x2 − y 2 )(3x2 − y 2 )(3y 2 − x2 ) + , r4 r8

where e1 =

2c1 c2 , c0 k0

e2 =

c1 c2 c0 k0

c1 = (L + 2M )K2 − R2 ,

c c1 + 2 c1 c2

(7.2-33d)

,

c2 = M K2 − R2

and c0 , c1 ,c2 and k0 are given in Chapter 6, i.e., c0 = (L + 2M )R,

c1 = (L + 2M )K1 − R2 ,

c2 = M K1 − R2 ,

k0 = R(K1 − K2 ). ||

Superposition of the above two solutions yields the solution of dislocation (b1 , 0, || ⊥ b1 , 0). The solution of (0, b2 , 0, b⊥ 2 ) can be determined similarly. Their superposition || || ⊥ , yields the solution of (b1 , b2 , b⊥ 1 b2 ). Part of this work can be found in paper given by Li and Fan[5] . Readers may examine that the present solutions are identical to those given by Ding et al[4] using the method of Green functions (Note that L = C12 and M = (C11 − C12 )/2=C66 ).

7.3

Dislocations in quasicrystals with point groups 5, ¯ 5 ﬁvefold and 10, 10 ten-fold symmetries

Similar to those discussed in the proceeding section, as two-dimensional quasicrys|| || || ⊥ tals, the Burgers vector of the dislocation is b|| ⊕ b⊥ =(b1 ,b2 ,b3 ,b⊥ 1 , b2 ). Due to w3 =0 and our assumption that dislocation line is parallel to the direction of periodic arrangement (z-direction), the above Burgers vector can be regarded as the || ⊥ ⊥ superposition of (b1 ,b2 ,b⊥ 1 ,b2 ) and (0,0,b3 ). Under this condition, the ﬁeld variables do not vary with the change of the variable z. Therefore, the elastic ﬁeld corre|| || ⊥ sponding to (b1 ,b2 ,b⊥ 1 ,b2 ) can be determined by using the theory of plane elasticity of two-dimensional quasicrystals discussed in Section 6.4. The anti-plane elastic ﬁeld induced by (0,0,b⊥ 3 ) is very simple, and can be solved by using the method discussed in Section 7.1. From Section 6.4, we have obtained the governing equation of plane elastic ﬁeld of quasicrystals with point groups 5, ¯5 ﬁve-fold and 10, ¯1¯0 ten-fold symmetries: ∇2 ∇2 ∇2 ∇2 F = 0,

(7.3-1)

where F (x,y) is the displacement potential function, deﬁned in (6.4-10). By performing the Fourier transform of (7.2-2) and considering the case of upper half-space (y 0), its Fourier transform Fˆ (ξ, y) has the form of Fˆ (ξ, y) = (4ξ 4 R2 )−1 XY e−|ξ|y ,

(7.3-2)

7.3

Dislocations in quasicrystals with point groups 5, ¯ 5 ﬁve-fold and 10, 10 . . .

113

where X and Y have the same meanings as those in Section 7.2, ξ is the parameter of Fourier transform and (7.3-3) R2 = R12 + R22 . After some algebraic manipulation with (6.2-8)∼(6.2-11), the displacement components in the Fourier transform domain are ¯ 0 X[2nξ 2 Y + (m − 5n) |ξ| Y − (2m − 5n)Y ]e−|ξ|y , (7.3-4a) u ˆx (ξ, y) = iξ −1 R −1 ¯ 2 −|ξ|y u ˆy (ξ, y) = |ξ| R , (7.3-4b) 0 X[2nξ Y − (m + 5n) |ξ| Y + (2m + 5n)Y ]e 3 −1 ¯ 2 2 −|ξ|y w ˆx (ξ, y) = ic0 ξ R0 X[4 |ξ| Y − 12ξ Y + 15 |ξ| Y − 10Y ]e , (7.3-4c) −1

w ˆy (ξ, y) = c0 |ξ|

¯ 02 X[4 |ξ|3 Y − 12ξ 2 Y + 15 |ξ| Y − (10 + e0 R02 )Y ]e−|ξ|y , R (7.3-4d)

where

⎧ ⎪ ⎪ ⎨ m = c2 + c1 ,

−[C66 c1 + C11 c2 ] , Rc0 ¯ 0 = R1 − iR2 sgn ξ . R R

n = c2 − c1 ,

⎪ ⎪ ⎩ R0 = R1 + iR2 sgn ξ , R

e0 =

(7.3-5)

Similarly, the stress components in the Fourier transform domain are ¯ 0 X(−2ξ 2 Y + 8 |ξ| Y − 13Y )e−|ξ|y , σ ˆxx (ξ, y) = 2c0 c2 R−1 R ¯ 0 X(2ξ 2 Y − 4 |ξ| Y + 3Y )e−|ξ|y , σ ˆyy (ξ, y) = 2c0 c2 R−1 R σ ˆxy (ξ, y) = σ ˆyx (ξ, y) = i2c0 c2 R 2

−1

(7.3-6a) (7.3-6b)

¯ 0 (sgnξ) R

· X(2ξ Y − 6 |ξ| Y + 7Y )e−|ξ|y , ˆ xx (ξ, y) = c0 k0 R ¯ 02 X[4 |ξ|3 Y − 16ξ 2 Y + 27 |ξ| Y H

(7.3-6c)

− (25 + e2 R02 )Y ]e−|ξ|y , ˆ yy (ξ, y) = c0 k0 R ¯ 02 X[−4 |ξ|3 Y + 12ξ 2 Y − 15 |ξ| Y H

(7.3-6d)

+ (10 − e1 R02 )Y ]e−|ξ|y , ˆ xy (ξ, y) = ic0 k0 R ¯ 2 (sgnξ)X[−4 |ξ|3 Y + 12ξ 2 Y − 15 |ξ| Y H 0

(7.3-6e)

+ (10 + e2 R02 )Y ]e−|ξ|y , ˆ yx (ξ, y) = ic0 k0 R ¯ 02 (sgnξ)X[−4 |ξ|3 Y H

(7.3-6f)

+ 16ξ 2 Y − 27 |ξ| Y + (25 − e1 R02 )Y ]e−|ξ|y , ⎧ c1 c2 ⎪ ⎨ e1 = 2c1 c2 , e2 = c1 c2 , + c0 k0 R c0 k0 R c1 c2 ⎪ ⎩ c1 = (L + 2M )K2 − R2 , c2 = M K2 − R2 , where c0 and K0 are given in (6.4-13).

(7.3-6g)

(7.3-7)

114

Chapter 7

Application I—Some dislocation and interface problems and . . .

As introduced in the proceeding section, X includes four unknowns A(ξ), B(ξ), C(ξ) and D(ξ), which can be determined by boundary conditions. Once A(ξ), B(ξ), C(ξ) and D(ξ) are determined, ui , wi , σjk and Hjk can be determined by the Fourier inverse. || In the case of dislocation problem b|| ⊕b⊥ =(b1 ,0,b⊥ 1 ,0), we try to determine these functions and their Fourier inverses in the following. || In order to simplify the calculation, we ﬁrst consider the case of b1 = 0 and b⊥ 1 = 0. In this case the boundary conditions are # ⎧ σ (x, y) → 0, H (x, y) → 0, x2 + y 2 → ∞, ij ij ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ σyy (x, 0) = 0, ⎨ (7.3-8) wx (x, 0) = wy (x, 0) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ dux = b1 , duy = 0. Γ

Γ

In the Fourier transform domain, solutions (7.3-4), (7.3-6) and the ﬁrst three expressions in boundary conditions (7.3-8) yield A(ξ) =

9C(ξ) , 4

B(ξ) = 2C(ξ),

D(ξ) = 0,

and the dislocation condition leads to ⎧ ⎪ ⎪ dux = −4c1 R−1 (R1 ReC + R2 sgnξImC), ⎨ Γ ⎪ ⎪ ⎩ duy = 4c1 R−1 (R2 ReC − R1 sgnξImC),

(7.3-9)

(7.3-10)

Γ

where the indicator Re and Im stand for the real and imaginary parts of a complex, respectively. Finally, expressions (7.3-10) and (7.3-8) deﬁne

C(ξ) = ReC + iImC = −

R0 b1 (R1 + iR2 sgnξ)b1 =− , 4πRc1 4πc1

(7.3-11)

where R0 is given by (7.3-5), and c1 is given by (6.4-9), i.e., c1 = (L + 2M )K1 − R2 ,

R2 = R12 + R22 .

(7.3-12)

Therefore, y c − c xy b1 1 2 , arctan + 2π x c1 r2 r c1 − c2 b1 r y2 uy = − ln + ln + 2 , 2π a c1 a r

ux =

(7.3-13a) (7.3-13b)

7.3

Dislocations in quasicrystals with point groups 5, ¯ 5 ﬁve-fold and 10, 10 . . .

y 2 (3x2 + y 2 ) , r4 3 R2 2x y c 0 b1 R1 y 2 (3x2 + y 2 ) wy = + , 2πc1 R r4 R r4

c 0 b1 wx = 2πc1

R1 R

2x3 y + r4

R2 R

115

(7.3.13c) (7.3-13d)

where c0 is given by (6.4-13), and a indicates the radius of dislocation core. The corresponding stress components can be determined by (7.3-13) and generalized Hooke’s law (6.4-3): $ % c0 c2 b1 y(3x2 + y 2 ) σxx = − , (7.3-14a) πc1 R r4 % $ c0 c2 b1 y(x2 − y 2 ) σyy = , (7.3-14b) πc1 R r4 $ % c0 c2 b1 x(x2 − y 2 ) , (7.3-14c) σxy = σyx = πc1 R r4 3 2 R2 x (3y − x2 ) c0 k0 b1 R1 x2 y(3x2 − y 2 ) Hxx = − + , (7.3-14d) πc1 R r6 R r6 2 2 R1 x2 y(3x2 − y 2 ) R2 xy (3x − y 2 ) c0 k0 b1 , (7.3-14e) Hyy = − − πc1 R r6 R r6 2 R2 x y(3y 2 − x2 ) c0 k0 b1 R1 xy 2 (3x2 − y 2 ) Hxy = − + , (7.3-14f) πc1 R r6 R r6 2 R2 x y(3x2 − y 2 ) c0 k0 b1 R1 x3 (3y 2 − x2 ) + − . (7.3-14g) Hyx = − πc1 R r6 R r6 ||

0. In this case, the corresponding Now let us consider the case of b1 = 0 and b⊥ 1 = boundary conditions are ⎧ # ⎪ x2 + y 2 → ∞, σij (x, y) → 0, Hij (x, y) → 0, ⎪ ⎪ ⎪ ⎪ H (x, 0) = 0, ⎪ ⎨ yy (7.3-15) ux (x, 0) = uy (x, 0) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ dwx = b⊥ dwy = 0. ⎩ 1, Γ

Γ

By using the similar analysis, the corresponding displacement and stress ﬁelds can be determined as the following formulas (7.3-16) and (7.3-17),

c1 − c2 2xy 3 c1 b⊥ R1 xy 1 − ux = πc0 e1 R r2 c1 r4 R2 y 2 c1 − c2 y 2 (x2 − y 2 ) + + , R r2 c1 r4

(7.3-16a)

116

Chapter 7

uy =

c1 b⊥ 1 πc0 e1 +

wx =

−

Application I—Some dislocation and interface problems and . . .

c1 − c2 y 2 (x2 − y 2 ) R1 y 2 − R r2 c1 r4

c1 − c2 2xy 3 R2 xy + , R r2 c1 r4

(7.3-16b)

y R2 − R2 xy(3x2 − y 2 )(3y 2 − x2 ) b⊥ 1 1 2 arctan + 2π x e1 R2 3r 6 +

2R1 R2 e1 R2

y 2 (3x2 − y 2 )2 , 3r 6

(7.3-16c)

2 R1 − R22 y 2 (3x2 − y 2 )2 b⊥ r 1 e2 ln + wy = 2πe1 a R2 3r 6

σxx σyy σxy Hxx Hyy Hxy Hyx

2R1 R2 xy(3x2 − y 2 )(3y 2 − x2 ) − , e1 R2 3r 6 R1 x2 y(3x2 − y 2 ) R2 x3 (3y 2 − x2 ) 2c2 b⊥ 1 , =− + πe1 R R r6 R r6 2c2 b⊥ R1 y 3 (3x2 − y 2 ) R2 xy 2 (3y 2 − x2 ) 1 =− + , πe1 R R r6 R r6 2c2 b⊥ R1 xy 2 (3x2 − y 2 ) R2 x2 (3y 2 − x2 ) 1 = σyx = − + , πe1 R R r6 R r6 2 R1 −R22 K0 b ⊥ y 2R1 R2 1 = h21 (x, y)− h22 (x, y) , −(e1 +e2 ) 2 +x 2πe1 r R2 R2 2 2 K0 b ⊥ 2R1 R2 1 y R1 − R2 =− h (x, y) + h (x, y) , 22 21 2πe1 R2 R2 2 R1 −R22 K0 b ⊥ x 2R1 R2 1 (e1 +e2 ) 2 +y = h21 (x, y)− h22 (x, y) , 2πe1 r R2 R2 2 2 K0 b ⊥ 2R1 R2 1 x R 1 − R2 =− h (x, y) + h (x, y) , 22 21 2πe1 R2 R2

(7.3-16d)

(7.3-17a)

(7.3-17b)

(7.3-17c)

(7.3-17d)

(7.3-17e)

(7.3-17f)

(7.3-17g)

in which 2xy(3x2 − y 2 )(3y 2 − x2 ) , r8 2(x2 − y 2 ) (x2 − y 2 )(3x2 − y 2 )(3y 2 − x2 ) + . h22 (x, y) = r4 r8

h21 (x, y) =

||

(7.3-18)

The solution to dislocation problem (b1 ,0,b⊥ 1 ,0) can be determined by superpos-

7.4

Dislocations in quasicrystals with eight-fold symmetry

117

ing (7.3-13) and (7.3-14) onto (7.3-16) and (7.3-17). The solution to dislocation || problem (0,b2 ,0,b⊥ 2 ) can be determined in the similar way. As a result, the solu|| || ⊥ tion to dislocation problem with Burgers vector (b1 ,b2 ,b⊥ 1 ,b2 ) can be determined completely. The above work can be found in literature [6].

7.4

Dislocations in quasicrystals with eight-fold symmetry

The ﬁnal governing equation of elasticity of two-dimensional quasicrystals with eightfold symmetry can be expressed

in which

(∇2 ∇2 ∇2 ∇2 − 4ε∇2 ∇2 Λ2 Λ2 + 4εΛ2 Λ2 Λ2 Λ2 )F = 0,

(7.4-1)

⎧ ∂2 ∂2 ∂2 ∂2 ⎪ 2 2 ⎪ ⎪ ⎨ ∇ = ∂x2 + ∂y 2 , Λ = ∂x2 − ∂y 2 , ⎪ R2 (L + M )(K2 + K3 ) ⎪ ⎪ ⎩ ε= [M (K1 + K2 + K3 ) − R2 ][(L + 2M )K1 − R2 ]

(7.4-2)

(see (6.6-10),(6.6-11) for detail). Equation (7.4-1) is more complicated than those of (6.2-7) and (7.3-1), so the solution of which is also more complex than those discussed in the previous sections. Due to the space limitation we cannot list whole procedure of the solution, only give some main results of them in the following, in which the Fourier transform and complex variable function methods are respectively used. 7.4.1

Fourier transform method[9] ||

Consider dislocation problem b|| ⊕ b⊥ =(b1 ,0,b⊥ 1 ,0,0), we try to determine the displacement ﬁeld under action of the boundary conditions ⎧ # ⎪ x2 + y 2 → ∞, σij (x, y) → 0, Hij (x, y) → 0, ⎪ ⎪ ⎪ ⎨ σyy (x, 0) = 0, Hyy (x, 0) = 0, (7.4-3) ⎪ ⎪ ⎪ ⎪ dux = b1 , dwx = b⊥ ⎩ 1. Γ

Γ

By performing the Fourier transform to (7.4-1), it can be reduced to 2 2 4 2 4 d d d2 2 2 2 Fˆ = 0. −ξ − 4ε −ξ + 4ε +ξ dy 2 dy 2 dy 2

(7.4-4)

The eigen roots of equation (7.4-4) depend on the value of parameter ε, Zhou[9] gave a detailed discussion for the solutions corresponding to case (1): 0 < ε < 1 and

118

Chapter 7

Application I—Some dislocation and interface problems and . . .

case (2): ε < 0, but calculation is tremendous complex and lengthy which cannot be included here. For the case (1) the solution is ⎧ 2 2 ⎪ λ1 +λ22 y λ2 λ1 +λ22 y λ2 b1 1 ⎪ ⎪ arctan + − +arctan (x, y) = u ⎪ x ⎪ 2π 2 λ x λ1 λ1 x λ1 ⎪ ⎪ 1 ⎪ ⎪ ⎪ xy 2λ 2λ1 xy 3 ⎪ ⎪ +(F C + F D) arctan − arctan ⎪ 3 4 ⎪ x2 − (λ23 + λ24 )y 2 x2 − (λ21 + λ22 )y 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 x2 +2λ2 xy+(λ21 +λ22 )y 2 x2 +2λ4 xy+(λ23 +λ24 )y 2 ⎪ ⎪ F + +F6 ln 2 ⎪ 5 ln 2 ⎪ 4π x −2λ2 xy+(λ21 +λ22 )y 2 x −2λ4 xy+(λ23 +λ24 )y 2 ⎪ ⎪ ⎪ ⎪

⎪ ⎪ 1 λ2 2λ1 λ2 y 2 ⎪ ⎪ uy = − 2 arctan H1 arctan 2 ⎪ ⎪ ⎪ 2π x + (λ21 − λ22 )y 2 λ1 ⎪ ⎪ ⎪ ⎪ 2 ⎪ 2λ3 λ4 y λ4 ⎪ ⎪ +H2 arctan 2 ⎪ 2 − λ2 )y 2 − 2 arctan λ ⎪ ⎪ x + (λ 3 3 4 ⎪ ⎪ ⎪

⎪ 4 2 2 2 2 ⎪ 1 x + 2(λ1 − λ2 )x y ⎪ ⎪ + H3 ln 1 + ⎪ ⎪ ⎪ 4π (λ21 + λ22 )2 y 4 ⎪ ⎪ ⎪ ⎪ ⎪ x4 + 2(λ23 − λ24 )x2 y 2 ⎪ ⎪ , +H ln 1 + ⎪ 4 ⎪ ⎪ (λ23 + λ24 )2 y 4 ⎪ ⎪ ⎪

2 2 2 ⎪ ⎨ λ1 +λ2 y λ2 λ1 +λ22 y λ2 1 b⊥ 1 wx (x, y) = arctan + − +arctan 2π 2 λ1 x λ1 λ1 x λ1 ⎪ ⎪ ⎪ ⎪ ⎪ 2λ3 xy ⎪ ⎪ +(G C + G D) × arctan 2 ⎪ 3 4 ⎪ ⎪ x − (λ23 + λ24 )y 2 ⎪ ⎪ ⎪ ⎪ ⎪ 2λ1 xy ⎪ ⎪ − arctan 2 ⎪ 2 + λ2 )y 2 ⎪ ⎪ x − (λ ⎪ 1 2 ⎪ ⎪ ⎪ 2 2 2 2 ⎪ 1 +2λ xy+(λ x x2 + 2λ4 xy+(λ23 +λ24 )y 2 ⎪ 2 1 +λ2 )y ⎪ G5 ln 2 , + +G6 ln 2 ⎪ ⎪ ⎪ 4π x −2λ2 xy+(λ21 +λ22 )y 2 x −2λ4 xy+(λ23 + λ24 )y 2 ⎪ ⎪ ⎪

⎪ ⎪ 1 λ2 2λ1 λ2 y 2 ⎪ ⎪ w = − 2 arctan I arctan ⎪ y 1 ⎪ ⎪ 2π x2 + (λ21 − λ22 )y 2 λ1 ⎪ ⎪ ⎪ ⎪ 2 ⎪ 2λ3 λ4 y λ4 ⎪ ⎪ +I2 arctan 2 ⎪ ⎪ 2 − λ2 )y 2 − 2 arctan λ ⎪ x + (λ 3 ⎪ 3 4 ⎪ ⎪

⎪ 4 2 2 2 2 ⎪ 1 x + 2(λ1 − λ2 )x y ⎪ ⎪ + ⎪ I3 ln 1 + ⎪ ⎪ 4π (λ21 + λ22 )2 y 4 ⎪ ⎪ ⎪ ⎪ ⎪ x4 + 2(λ23 − λ24 )x2 y 2 ⎪ ⎪ ⎪ , +I4 ln 1 + ⎪ ⎩ (λ23 + λ24 )2 y 4 (7.4-5) in which F1 , · · · , F6 , G1 , · · · , G6 , H1 , · · · , H4 and I1 , · · · , I4 are some functions of λ1 , λ2 , λ3 and λ4 which are constants constituted from the original material constants M, L, K1 , K2 , K3 and R, the expressions are very complicated and lengthy, we have

7.4

Dislocations in quasicrystals with eight-fold symmetry

119

to omit them. By using the similar procedure, the solution for the case (2) can also be obtained. But the solving procedure is very tedious due to the complexity of the ﬁnal governing equation (7.4-1). We omit them for simplicity. 7.4.2

Complex variable function method

The equation (7.4-1) can also be solved by complex variable function method. For this purpose the equation can be rewritten as 8 ∂8 ∂8 ∂ + 4(1 − 4ε) + 2(3 + 16ε) ∂x8 ∂x6 ∂y 2 ∂x4 ∂y 4 ∂8 ∂8 (7.4-6) +4(1 − 4ε) 2 6 + 8 F = 0. ∂x ∂y ∂y The solution of equation (7.4-6) can be expressed in terms of four analytic functions Fk (zk ) of complex variable zk (k = 1, 2, 3, 4), i.e., F (x, y) = 2Re

4

Fk (zk ),

zk = x + μk y

(7.4-7)

k=1

and μk = αk + iβk (i = 1, 2, 3, 4) are complex parameters and determined by the roots of the following eigenvalue equation: μ8 + 4(1 − 4ε)μ6 + 2(3 + 16ε)μ4 + 4(1 − 4ε)μ2 + 1 = 0.

(7.4-8)

Based on the displacement expressions (6.6-12) and the dislocation condition dux = b1 , duy = b2 , dwx = b⊥ , dwy = b⊥ (7.4-9) 1 2, Γ

Γ

Γ

Γ

we can obtain the solution as follows: ux = 2Re

4

a1k fk (zk ),

uy = 2Re

k=1

wx = 2Re

4

4

a2k fk (zk ),

k=1

a3k fk (zk ),

wy = 2Re

k=1

4

a4k fk (zk ),

k=1

where fk (zk ) =

∂ 6 Fk (zk ) . ∂zk6

Some detail of this work can be found in Ref. [10].

(7.4-10)

120

7.5

Chapter 7

Application I—Some dislocation and interface problems and . . .

Dislocations in dodecagonal quasicrystals

In monograph [7] Ding et al give the solution in terms of the Green function method which has not been introduced in the present book, so the detail is omitted and only list the results, such as the phonon displacement ﬁeld ⎧ b1 L + M x 1 x2 x2 ⎪ ⎪ ⎪ ⎪ u1 = 2π arctan x + L + 2M r2 ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ L + M x22 M b2 r ⎪ ⎪ + + ln , ⎪ ⎪ ⎪ 2π L + 2M r0 L + 2M r2 ⎪ ⎨ (7.5-1) b1 L + M x21 M r ⎪ + u2 = − ln ⎪ ⎪ 2 ⎪ 2π L + 2M r0 L + 2M r ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ L + M x1 x2 b2 x2 ⎪ ⎪ + arctan − , ⎪ ⎪ 2π x1 L + 2M r2 ⎪ ⎪ ⎪ ⎩ u3 = 0. and the phason displacement ﬁeld ⎧ b⊥ x2 (K1 + K2 )(K2 + K3 ) x1 x2 ⎪ 1 ⎪ arctan − w1 = ⎪ ⎪ ⎪ 2π x1 2K1 (K1 + K2 + K3 ) r2 ⎪ ⎪ ⎪ ⎪ ⎪ b⊥ (K1 + K2 )(K2 + K3 ) x22 K2 (K1 + K2 + K3 ) − K1 K3 r ⎪ 2 ⎪ + + − ln , ⎪ ⎨ 4π K1 (K1 + K2 + K3 ) r0 K1 (K1 + K2 + K3 ) r2 ⎪ b⊥ (K1 + K2 )(K2 + K3 ) x21 K2 (K1 + K2 + K3 ) − K1 K3 r ⎪ 1 ⎪ ⎪ − ln ⎪ w2 = 4π ⎪ K1 (K1 + K2 + K3 ) r0 K1 (K1 + K2 + K3 ) r2 ⎪ ⎪ ⎪ ⎪ ⎪ (K1 + K2 )(K2 + K3 ) x1 x2 x2 b⊥ ⎪ ⎪ ⎩ . + 2 arctan + 2π x1 2K1 (K1 + K2 + K3 ) r2 (7.5-2) and the sign L, M are phonon elastic constants and others have the same meaning as before. Because the phonons and phasons are decoupled for dodecagonal quasicrystals, the coupling constant R = 0.

7.6

Interface between quasicrystal and crystal

In the previous sections we discussed dislocation problems in one- and two-dimensional quasicrystals, a series analytic solutions are obtained. In Chapter 8 we will discuss crack problems in the material. Apart from dislocations and cracks, interface is a kind of defects in quasicrystals too, which is of particular signiﬁcance for some physical processes. We know all quasicrystals observed to date are alloys. This type of alloys can be in crystalline phase, in quasicrystalline phase, or in crystal-quasicrystal coexisting phase. Li et al[16,17] observed the crystal-quasicrystal phase transition, and

7.6

Interface between quasicrystal and crystal

121

which is continuous transition. The transition process gives rise to interface between crystal and quasicrystal. Consequently, analysis of the interface problem of quasicrystals is of signiﬁcance. In this section we give a phenomenological study on elastic behaviour of the interface for one-dimensional quasicrystal-isotropic crystal. Refs.[15∼17] pointed out that the phase transition is induced by the phason strains. Such a problem is formidable. Here we focus only on the determination of the strains, and further studies will be given in Chapter 9 for icosahedral quasicrystalcubic crystal interface. Consider an orthorhombic quasicrystal at the upper half-plane (i.e., y > 0), the phonon-phason coupling problem is governed by (5.4-3), i.e., ∂4 ∂4 ∂4 (7.6-1) a1 4 + a3 2 2 + a5 4 F = 0, ∂x ∂x ∂y ∂y in which F (x, y) denotes the displacement potential, a1 , a3 , a5 the material constants composed of Cij , Ki and Ri deﬁned by (5.3-6). We assume that the crystal coexisting with the quasicrystal is located at the lower half-plane with thickness h(i.e., −h < y < 0). The plane y = 0 is the interface between the quasicrystal and crystal, as shown in Fig.7.6-1. For simplicity, suppose the crystal be an isotropic material, (c) characterizing by elastic constants Cij (E (c) , μ(c) ). At the interface there are the following boundary conditions:

Fig. 7.6-1

Coexisting phase of quasicrystal-crystal

y = 0, −∞ < x < ∞ :

σzy = τ f (x) + ku(x),

Hzy = 0,

(7.6-2)

in which f (x) is the distribution function of applied stress at the interface, u(x) = uz (x, 0) the value of displacement component of phonon ﬁeld at the interface, τ a constant shear stress and k a material constant to be taken as

122

Chapter 7

Application I—Some dislocation and interface problems and . . .

k=

μ(c) h

(7.6-3)

and where μ(c) and h the shear modulus and the thickness of the crystal. We further assume that the outer boundaries are stress free. Taking the Fourier transform ∞ F (x, y)eiξx dx (7.6-4) Fˆ (ξ, y) = −∞

to the equation (7.6-1) yields d4 d2 a5 4 + a3 ξ 2 2 + a1 ξ 4 Fˆ = 0. dy dy

(7.6-5)

If put solution of equation (7.6-5) being Fˆ (ξ, y) = e−λ|ξ|y , where λ is a parameter, and substituting it into (7.6-5) leads to a5 λ4 − a3 λ2 + a1 = 0 which has the roots λ1 , λ2 , λ3 , λ4 =

&

2a3 ±

& # # a23 − 4a1 a5 2a3 ± a23 − 4a1 a5 ,− . 2a5 2a5

(7.6-6)

(7.6-7)

There follows Fˆ (ξ, y) = Ae−λ1 |ξ|y + Be−λ2 |ξ|y + Ce−λ3 |ξ|y + De−λ4 |ξ|y . By considering the condition of stress free at y = ∞, then C = D = 0, i.e., Fˆ (ξ, y) = Ae−λ1 |ξ|y + Be−λ2 |ξ|y . According to Section 5.4, we have, e.g. ⎧ ∂3F ⎪ ⎪ , = (R C − R C ) σ ⎪ zy 6 44 5 55 ⎪ ⎪ ∂x2 ∂y ⎪ ⎪ ⎨ ∂2 ∂2 ∂ ∂ Hzy = − C55 2 + C44 2 + K2 F, K1 ⎪ ∂x ∂y ∂x ∂y ⎪ ⎪ ⎪ ⎪ ∂2 ∂2 ⎪ ⎪ ⎩ uz = R6 2 + R5 2 F. ∂x ∂y

(7.6-8)

(7.6-9)

The Fourier transforms of the stresses and displacement are ⎧ dFˆ ⎪ ⎪ ⎪ , σ ˆzy = −(R6 C44 − R5 C55 )ξ 2 ⎪ ⎪ dy ⎪ ⎪ ⎨ 2ˆ 3ˆ ˆ ˆ zy = −iC55 K1 |ξ| ξ 2 Fˆ + C55 K2 ξ 2 dF + iC44 K1 |ξ| d F − C44 K2 d F , H ⎪ dy dy2 dy 3 ⎪ ⎪ ⎪ 2 ⎪ d ⎪ ⎪ ⎩ u ˆz = −ξ 2 R6 + R5 2 Fˆ . dy (7.6-10)

7.6

Interface between quasicrystal and crystal

123

Substituting (7.6-8) into the second one of (7.6-10) then into the second one of (7.6-2) yields B = αA,

(7.6-11)

where α= c1 = C55 K1 ,

−λ1 c2 − λ31 c4 + i(c1 + λ21 c3 ) , λ2 c2 + λ32 c4 + i(−c1 − λ22 c3 )

c2 = C55 K2 ,

c3 = C44 K1 ,

c4 = C44 K2 .

(7.6-12)

The Fourier transform of the phonon stress and displacement component at the interface are σ ˆzy (ξ, 0) = A(ξ) |ξ| (λ1 e−λ1 |ξ|y + αλ2 e−λ2 |ξ|y ), u ˆz (ξ, 0) = A(ξ)ξ 2 [(−R6 + λ21 R5 )e−λ1 |ξ|y + α(R6 − λ22 R5 )e−λ2 |ξ|y ].

(7.6-13)

The Fourier transform of the ﬁrst equation of (7.6-2) is uz (ξ, 0). σ ˆzy (ξ, 0) = τ fˆ(ξ) + kˆ

(7.6-14)

From (7.6-13) and (7.6-14), one determines the unknown function A(ξ) =

τ fˆ(ξ) . |ξ| (λ1 + αλ2 ) − kξ 2 [−R6 + λ21 R5 + α(R6 − λ21 R5 )]

(7.6-15)

Thus, the stress and displacement components for phonon and phason ﬁelds can be evaluated, e.g. ⎧ ∞ ∞ 1 1 ⎪ −iξx ⎪ σ ˆzy e dξ = A(ξ) |ξ|(λ1 e−λ1 |ξ|y + αλ2 e−λ2 |ξ|y )e−iξx dξ, ⎪ σzy = ⎪ ⎪ 2π −∞ 2π −∞ ⎪ ⎪ ⎪ ∞ ⎪ ⎪ 1 ⎪ ⎪ u = u ˆz e−iξx dξ ⎪ z ⎪ ⎪ 2π −∞ ⎪ ⎪ ⎪ ∞ ⎨ 1 A(ξ)ξ 2 [(−R6 + λ21 R5 )e−λ1 |ξ|y + α(R6 − λ22 R5 )e−λ2 |ξ|y ]e−iξx dξ, = ⎪ 2π −∞ ⎪ ⎪ ⎪ ∞ ⎪ ⎪ 1 ⎪ ⎪ w ˆz e−iξx dξ wz = ⎪ ⎪ 2π −∞ ⎪ ⎪ ⎪ ⎪ ∞ ⎪ ⎪ 1 ⎪ ⎪ A(ξ)ξ 2 [(−C55 +λ21 C66 )e−λ1 |ξ|y +α(C66 −λ22 C55 )e−λ2 |ξ|y ]e−iξx dξ. =− ⎩ 2π −∞ (7.6-16) The phason strain ﬁeld presents important eﬀect in the phase transition of crystal-quasicrystal, which is determined by the above solution such as

124

Chapter 7

Application I—Some dislocation and interface problems and . . .

⎧ ∞ ∂wz 1 ⎪ ⎪ = A(ξ)ξ 3 [(−C55 + λ21 C66 )e−λ1 |ξ|y = −i w ⎪ zx ⎪ ⎪ ∂x 2π −∞ ⎪ ⎪ ⎪ ⎪ ⎪ +α(C66 − λ22 C55 )e−λ2 |ξ|y ]e−iξx dξ, ⎪ ⎪ ⎨ ∂wz wzy = ⎪ ∂y ⎪ ∞ ⎪ ⎪ ⎪ 1 ⎪ ⎪ A(ξ)ξ 3 [λ1 |ξ| (−C55 + λ21 C66 )e−λ1 |ξ|y = ⎪ ⎪ 2π −∞ ⎪ ⎪ ⎪ ⎩ +λ |ξ| α(C − λ2 C )e−λ2 |ξ|y ]e−iξx dξ. 2 66 2 55

(7.6-17)

The solution varies from material constants Cij , Ki , Ri of quasicrystals, and the material constant μ(c) of crystals, applied stress τ and the size h of the crystals, so the results are interesting. Further discussion on interface will be given in Section 9.2 of Chapter 9.

7.7

Conclusion and discussion

The main object of this chapter lies in demonstration on eﬀect of the formulation in Chapters 5 and 6, it does not intend to fully explore the nature of dislocations and interfaces in quasicrystals. There are other work e.g. [11∼15,18,19] can be referenced. It has been found that singularity around the dislocation core occurs, for instance, stresses σij , Hij ∼ 1/r(r → 0), which is the direct result of broken symmetry due to appearance of the topological defect, where r is the distance measured from the dislocation core. This is similar to that of crystals, i.e., the broken symmetry also leads to the appearance of singularity in crystals. Physically, the high stress grade or the stress concentration around dislocation core results in plastic ﬂow in crystals as well as in quasicrystals. So dislocation and other defects in quasicrystals aﬀect the mechanical and physical properties of the material. The dislocation solutions present important application in studying plastic deformation and plastic fracture, as referring to [14] or Chapter 14 of this book. The discussion on interface here is in a primary version, but it may be helpful to study crystal-quasicrystal phase transition. However this is a very complicated problem, the knowledge in this respect is very limited so far. The dislocations and interfaces and the solutions in three-dimensional quasicrystals will be introduced in Chapter 9, and the dynamic dislocation problem can be referred in Chapter 10.

References [1] De P, Pelcovits R A. Linear elasticity theory of pentagonal quasicrystals. Phys Rev B, 1987, 35(16): 8609–8620 [2] De P, Pelcovits R A. Disclination in pentagonal quasicrystals. Phys Rev B, 1987, 36(17): 9304–9307

References

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[3] Ding D H, Wang R H, Yang W G et al. General expressions for the elastic displacement ﬁelds induced by dislocation in quasicrystals. J Phys Condens Matter, 1995, 7(28): 5423–5436 [4] Ding D H, Wang R H, Yang W G et al. Elasticity theory of straight dislocation in quasicrystals. Phil Mag Lett, 1995, 72(5): 353–359 [5] Li X F, Fan T Y. New method for solving elasticity problems of some planar quasicrystals and solutions. Chin Phys Lett, 1998, 15(4): 278–280 [6] Li X F, Duan X Y, Fan T Y et al. Elastic ﬁeld for a straight dislocation in a decagonal quasicrystal. J Phys: Condens Matter, 1999, 11(3): 703–711 [7] Yang S H, Ding D H. Fundamentals to Theory of Crystal Dislocations. Vol II. Beijing: Science Press, 1998 (in Chinese) [8] Firth J P, Lothe J. Theory of Dislocations. John Wiley, Sons. New York, 1982 [9] Zhou W M. Dislocation, crack and contact problems in two- and three-dimensional quasicrystals. Dissertation Beijing Institute of Technology, 2000 (in Chinese) [10] Li L H. Study on complex variable function method and exact analytic solutions of elasticity of quasicrystals. Dissertation. Beijing Institute of Technology, 2008 (in Chinese) [11] Fan T Y, Li X F, Sun Y F. A moving screw dislocation in a one-dimensional hexagonal quasicrystals. Acta Physica Sinica (Oversea Edition), 1999, 8(3): 288–295 [12] Li X F, Fan T Y. A straight dislocation in one-dimensional hexagonal quasicrystals. Phy Stat Sol (b), 1999, 212(1): 19–26 [13] Edagawa K. Dislocations in quasicrystals, Mater Sci Eng A, 2001, 309∼310(2): 528–538 [14] Fan T Y, Trebin H R, Messeschmidt U et al. Plastic ﬂow coupled with a crack in some one- and two-dimensional quasicrystals. J Phys: Condens Matter, 2004, 16(37): 5229–5240 [15] Hu C Z, Wang R H, Ding D H. Symmetry groups, physical property tensors, elasticity and dislocations in quasicrystals. Rep Prog Phys, 2000, 63(1): 1–39 [16] Li F H. In: Jacaman M J, Torres M. Crystal-Quasicrystal Transitions. Elsevier Sci Publ, 1988, 13–47 [17] Li F H, Teng C M, Huang Z R et al. In between crystalline and quasicrystalline states. Phil Mag Lett, 1988, 57 (1): 113–118 [18] Fan T Y, Xie L Y, Fan L et al. Study on interface of quasicrystal-crystal. Chin. Phys. B, submitted 2009 [19] Kordak M, Fluckider T, Kortan A R et al. Crystal-quasicrystal interface in Al-PdMn. Prog Surface Sci, 2004, 75(3∼8): 161–175

Chapter 8 Application II: Solutions of notch and crack problems of one-and two-dimensional quasicrystals Quasicrystals are potential materials to be developed for structural use, and their strength and toughness attract attention of researchers. Experimental observations[1,2] have shown that quasicrystals are brittle. With common experience of conventional structural materials, we know that failure of brittle materials is mainly related to the existence and growth of cracks. Chapter 7 indicated that dislocations have been observed in quasicrystals, and the accumulation of dislocations will eventually lead to cracking of the material. Now let us study crack problems in quasicrystals that have both theoretical and practical value in the view of applications in future. Chapters 5∼7 have discussed some elasticity and dislocation problems in oneand two-dimensional quasicrystals. It has shown that when the quasicrystal conﬁguration is independent of one coordinate, e.g. variable z, its elasticity problem can be decoupled into a plane problem and an anti-plane problem. In the case of onedimensional quasicrystals, if the z-direction accords with the quasiperiodic axis, the above plane problem belongs to classic elasticity problem, and the anti-plane problem is a coupling problem of phonon and phason ﬁelds. In the case of two-dimensional quasicrystals, if z-direction represents the periodic axis, the above plane problem is a coupling problem of phonon and phason ﬁelds, and the anti-plane problem belongs to a classical elasticity problem. Due to use of decomposition procedure, the resulting problem can be dramatically simpliﬁed. Chapters 5 and 6 have given their corresponding fundamental solutions, and Chapter 7 conducted the solutions of dislocations in detail. The present chapter is going to focus on crack problems, to continue using the above schemes, such as the fundamental solutions developed in Chapters 5 and 6, and the Fourier transform and complex variable function methods used in Chapter 7. But it emphasizes the complex function method, which will be developed in Sections 8.1, 8.2 and 8.4, this approach is powerful. Problems displayed in Sections 8.1 and 8.2 are relatively simpler, the detailed introduction T. Fan, Mathematical Theory of Elasticity of Quasicrystals and Its Applications © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2011

128

Chapter 8

Application II: Solutions of notch and crack problems of one-and...

may help reader understand and further handle the principle and technique of the complex potential method, though the representation does not beyond the classical Muskhelishvili[18] method, this is helpful to understand the solutions for more complicated problems displayed in Section 8.4 and in the next chapter. The further summary of the method will be introduced in Chapter 11, because the contents in Section 8.4 and Chapter 9 bring some new insights into the study and go beyond the Muskhelishvili method, the farther general discussion is necessary and may be beneﬁcial. Based on the common nature of exact solutions of diﬀerent static and dynamic cracks in diﬀerent quasicrystal systems for linear and nonlinear deformation (which are discussed in this chapter and Chapters 9, 10 and 14), the fracture theory of quasicrystalline material is suggested in Chapter 15, which can be seen as a development of fracture mechanics of conventional structural materials.

8.1 8.1.1

Crack problem and solution of one-dimensional quasicrystals Griﬃth crack

As shown in Fig. 8.1-1, assume a Griﬃth crack along the quasiperiodic axis (zdirection) of a one-dimensional hexangular quasicrystal. Under the action of ex(∞) (∞) ternal traction σyz = τ1 and/or Hzy = τ2 . The deformation induced is often termed longitudinal shearing. Obviously, the geometry of the crack is independent of variable z. In this case, all ﬁeld variables are independent of variable z. Therefore, based on the analysis in Chapter 5, this one-dimensional quasicrystal elasticity

Fig. 8.1-1

A Griﬃth crack subjected to a longitudinal shear where τ = τ1 corresponding to phonon ﬁeld, and τ = τ2 corresponding to phason ﬁeld

8.1

Crack problem and solution of one-dimensional quasicrystals

129

can be decomposed into a plane elasticity problem of regular crystal and an antiplane elasticity problem of the phonon-phason coupling ﬁeld. Plane elasticity problems of regular crystal have been studied extensively in classical elasticity, and its crack problems have also been studied in the classical fracture theory, see e.g. Ref. [3]. Therefore, we skip the discussion of crack problems in regular crystal. The antiplane elasticity problem of the phason-phonon coupling ﬁeld is described by using the following basic equations: ⎧ σyz = σzy = 2C44 εyz + R3 wzy ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ σzx = σxz = 2C44 εzx + R3 wzx (8.1-1) ⎪ Hzy = K2 wzy + 2R3 εzy ⎪ ⎪ ⎪ ⎪ ⎩ H = K w + 2R ε zx 2 zx 3 zx ⎧ 1 ∂uz ⎪ , εyz = εzy = ⎪ ⎪ ⎪ 2 ∂y ⎪ ⎪ ⎪ ⎪ ⎪ 1 ∂uz ⎪ ⎪ ⎨ εzx = εxz = 2 ∂x (8.1-2) ⎪ ∂wz ⎪ ⎪ , wzy = ⎪ ⎪ ⎪ ∂y ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ w = ∂wz zx ∂x ∂σzx ∂σzy ∂Hzx ∂Hzy + = 0, + =0 (8.1-3) ∂x ∂y ∂x ∂y The derivation in Chapter 5 shows that the above equations can be reduced to ∇2 uz = 0,

∇ 2 wz = 0

(8.1-4)

where ∇2 = ∂ 2 /∂x2 + ∂ 2 /∂y 2 . Figure 8.1-1 shows that the anti-plane deformation of a Griﬃth crack has the following boundary conditions: # x2 + y 2 → ∞ : σyz = τ1 , Hzy = τ2 , σzx = Hzx = 0 (8.1-5) y = 0, |x| < a : σyz = 0, Hzy = 0 where a is the half-length of the crack. Based on the results of linear elasticity analysis, if the quasicrystal is traction free at inﬁnity instead there are a traction σyz = −τ1 and a generalized traction Hzy = −τ2 at the crack surface, then the boundary conditions stand for # x2 + y 2 → ∞ : σyz = σzx = Hzx = Hzy = 0 (8.1-6) y = 0, |x| < a : σyz = −τ1 , Hzy = −τ2

130

Chapter 8

Application II: Solutions of notch and crack problems of one-and...

Note that the phason stress τ2 at the crack surface is assumed from point of view physically, though its measurement result has not been reported yet. For simplicity, we can assume τ2 =0, sometimes. In the following we are going to solve the boundary value problem of (8.1-4) and (8.1-5) ﬁrst. The complex variable function method will be used. To do so, we introduce the complex variable √ t = x + iy = reiθ , i = −1 (8.1-7) From equations (8.1-4), it is known that both uz (x, y) and wz (x, y) are harmonic functions that can be expressed in terms of the real part or the imaginary part of two arbitrary analytic functions φ1 (t) and ψ1 (t) of complex variable t in a region occupied by the quasicrystal. For simplicity we can call φ1 (t) and ψ1 (t) as complex potentials. Here, assume uz (x, y) = Reφ1 (t) (8.1-8) wz (x, y) = Reψ1 (t) in which the symbol Re indicates the real part of a complex number. It is well known that if a function F (t) is analytic, then dF ∂F = , ∂x dt

∂F idF = . ∂y dt

(a)

Furthermore, assume F (t) = P (x, y) + iQ(x, y) = ReF (t) + iImF (t),

(b)

where symbol Im denotes the imaginary part of a complex number, and P (x, y) and Q(x, y) represent the real and imaginary parts of F (t), respectively. Therefore, the Cauchy-Riemann relations of an analytic function is ∂P ∂Q = , ∂x ∂y

∂P ∂Q =− . ∂y ∂x

(c)

With the aid of relation (a), formula (8.1-8) and equations (8.1-1) and (8.1-2) lead to ⎧ ∂ ∂ ⎪ ⎪ σyz = σzy = C44 Reφ1 + R3 Reψ1 ⎪ ⎪ ∂y ∂y ⎪ ⎪ ⎪ ⎪ ⎪ ∂ ∂ ⎪ ⎪ ⎪ ⎨ σzx = σxz = C44 ∂x Reφ1 + R3 ∂x Reψ1 (8.1-9) ⎪ ∂ ∂ ⎪ ⎪ Hzx = K2 Reψ1 + R3 Reφ1 ⎪ ⎪ ∂x ∂x ⎪ ⎪ ⎪ ⎪ ⎪ ∂ ∂ ⎪ ⎪ ⎩ Hzy = K2 Reψ1 + R3 Reφ1 ∂y ∂y

8.1

Crack problem and solution of one-dimensional quasicrystals

131

Based on the Cauchy-Riemann relation (c), the above equation can be rewritten as

σzx − iσzy = C44 φ1 + R3 ψ1 Hzx − iHzy = K2 ψ1 + R3 φ1

where φ1 = dφ1 /dt, ψ1 = dψ1 /dt. According to formula (8.1-10), we have σyz = σzy = −Im(C44 φ1 + R3 ψ1 ) Hzy = −Im(K2 ψ1 + R3 φ1 )

(8.1-10)

(d)

Furthermore, for an arbitrary complex function F (t), its imaginary part is ImF (t) =

1 (F − F ) 2i

where F indicates the conjugate of F . Then the above formula (d) can be expressed as ⎧ 1 ⎪ ⎪ σyz = σzy = − [C44 (φ1 − φ1 ) + R3 (ψ1 − ψ1 )] + τ1 ⎨ 2i (8.1-11) ⎪ 1 ⎪ ⎩ Hzy = − [K2 (ψ1 − ψ1 ) + R3 (φ1 − φ1 )] + τ2 2i The expression given by (8.1-11) ensures the solution constructed in the following to automatically satisfy the boundary condition at inﬁnity, the solution is ⎧ ⎞ ⎛ & 2 ⎪ ⎪ τ − R τ ) ia(K t t ⎪ 2 1 3 2 ⎪ ⎝ − φ1 (t) = − 1⎠ ⎪ ⎪ ⎪ C44 K2 − R32 a a ⎨ (8.1-12) ⎛ ⎞ & ⎪ 2 ⎪ ⎪ t t τ − R τ ) ia(C 44 2 3 1 ⎝ ⎪ ⎪ − ψ1 (t) = − 1⎠ ⎪ ⎪ C44 K2 − R32 a a ⎩ whose derivation can be obtained in terms of the strict complex variable function method given in Section 8.7—Appendix of Chapter 8. From (8.1-12) ⎧ i(K2 τ1 − R3 τ2 ) t ⎪ ⎪ 1− √ ⎪ ⎨ φ1 (t) = C K − R2 t2 − a2 44 2 3 (8.1-13) ⎪ τ − R τ ) i(C t 44 2 3 1 ⎪ ⎪ 1− √ ⎩ ψ1 (t) = C44 K2 − R32 t2 − a2 and substitution of the above expressions into the ﬁrst expression in (8.1-11) yields t (8.1-14) σzx − iσzy = iτ1 − √ t2 − a2

132

Chapter 8

Application II: Solutions of notch and crack problems of one-and...

Separation of the real and imaginary parts of (8.1-14) leads to ⎧ τ1 r 1 1 ⎪ ⎪ ⎪ ⎨ σxz = σzx = (r1 r2 )1/2 sin θ − 2 θ1 − 2 θ2 ⎪ τ1 r 1 1 ⎪ ⎪ ⎩ σyz = σzy = cos θ − θ1 − θ2 (r1 r2 )1.2 2 2

(8.1-15)

where t = reiθ , or ⎧ # ⎪ r = x2 + y 2 , ⎪ ⎨

y ⎪ ⎪ , ⎩ θ = arctan x

r1 =

t − a = r1 eiθ1 , #

t + a = r2 eiθ2

(x − a)2 + y 2 ,

θ1 = arctan

y x−a

r2 =

#

(8.1-16)

(x + a)2 + y 2

,

θ2 = arctan

y x+a

(8.1-16 )

which can be shown in Fig. 8.1-2.

Fig. 8.1-2

Similarly

The coordinate system at the crack tip

t Hzx − iHzy = iτ2 − √ t2 − a2

For these stress components there are expressions similar to (8.1-15). As a result, ⎧ ⎨ √ τ1 x , |x| > a x2 − a2 σzy (x, 0) = ⎩ 0, |x| < a ⎧ ⎨ √ τ2 x , |x| > a x2 − a2 Hzy (x, 0) = ⎩ 0, |x| < a

(8.1-17)

(8.1-18)

(8.1-19)

The above two formulas indicate that at y = 0 and |x| < a : σzy = 0, Hzy = 0. Therefore, the solution given above also satisﬁes the boundary conditions at the crack surfaces.

8.1

Crack problem and solution of one-dimensional quasicrystals

133

# Formulas (8.1-14) and (8.1-17) also show that when x2 + y 2 → ∞, σyz = τ1 , σxz = 0 and Hzy = τ2 , Hzx = 0, namely, the solution given above satisﬁes boundary conditions at inﬁnity. 8.1.2

Brittle fracture theory

The above formulas show that stresses have singular characteristics near crack tips, for example ⎧ τ1 x + ⎪ ⎨ σzy (x, 0) = √x2 − a2 → ∞, x → a (8.1-20) ⎪ ⎩ Hzy (x, 0) = √ τ2 x → ∞, x → a+ x2 − a2 Deﬁne the mode III stress intensity factors of the phonon and phason ﬁelds such that: # || KIII = lim 2π(x − a)σzy (x, 0), + x→a

⊥ = lim KIII

x→a+

then ||

KIII =

#

2π(x − a)Hzy (x, 0),

√ πaτ1 ,

⊥ KIII =

√ πaτ2 ,

(8.1-21)

where suﬃx “III” stands for model III (longitudinal shearing mode) [3]. Now let us calculate the crack strain energy: a WIII = 2 (σzy ⊕ Hzy ) (uz ⊕ wz )dx 0 a [σzy (x, 0)uz (x, 0) + Hzy (x, 0)wz (x, 0)]dx =2

(8.1-22)

0

From (8.1-8) and (8.1-12), we have + ⎧ x 2 K2 τ1 − R3 τ2 ⎪ ⎪ u (x, 0) = Re(φ (t)) = a 1 − ⎪ z 1 t=x ⎨ C44 K2 − R32 a + ⎪ ⎪ x 2 C τ − R3 τ 1 ⎪ ⎩ wz (x, 0) = Re(ψ1 (t))t=x = a 44 2 1− 2 C44 K2 − R3 a

|x| < a (8.1-23) |x| < a

In addition, considering the equivalency of problems (8.1-5) and (8.1-6), we can take σyz (x, 0) = −τ1 and Hzy = −τ2 as |x| < a, then substitution of above results into (8.1-22) yields WIII =

K2 τ12 + C44 τ22 − 2R3 τ1 τ2 2 πa C44 K2 − R32

(8.1-24)

134

Chapter 8

Application II: Solutions of notch and crack problems of one-and...

From (8.1-24), we can determine the crack strain energy release rate (crack growth force) such that GIII =

K2 τ12 + C44 τ22 − 2R3 τ1 τ2 1 ∂WIII = πa 2 ∂a C44 K2 − R32 2

=

⊥2 ⊥ − 2R3 KIII KIII K2 KIII + C44 KIII 2 C44 K2 − R3

(8.1-25)

Clearly, the crack energy and energy release rate are related not only to the phonon but also to the phason and the phonon-phason coupling ﬁelds. If τ2 = 0, then

GIII =

K2 (KIII )2 C44 K2 − R32

(8.1-26)

Furthermore if R3 = 0, we have GIII =

πaτ12 , C44

||

or GIII =

(KIII )2 . C44

(8.1-27)

Since GIII comprehensively describes the coupling eﬀect of the phonon and phason ﬁelds and the stress states near crack tip, we recommend GIII = GIIIC

(8.1-28)

as the fracture criterion of quasicrystals under mode III deformation, where GIIIC is the critical (threshold) value of GIII , the mode III fracture toughness to be determined by testing. The stress intensity factor and the strain energy release rate are fundamental physical parameters, and constitute the basis of brittle fracture theory for both the conventional crystalline as well as the quasicrystalline materials.

8.2

Crack problem in ﬁnite-sized one-dimensional quasicrystals

The preceding section discussed the Griﬃth crack problems in one-dimensional quasicrystals, and obtained their exact solutions. In this section, we are going to discuss the solution of another kind of important crack problems, which go beyond the scope of the Griﬃth crack. In the preceding section, we assumed that the size of quasicrystal be much more larger than that of the defect, while in this case we consider the quasicrystal as an inﬁnite body. Therefore, this section is aimed to the study of crack problems in quasicrystals of ﬁnite size.

8.2

Crack problem in ﬁnite-sized one-dimensional quasicrystals

8.2.1

135

Cracked quasicrystal strip with ﬁnite height

As shown in Fig. 8.2-1, a one-dimensional hexangonal quasicrystal strip of height 2H has a semi-inﬁnite crack embedded at the mid plane with crack tip corresponding to coordinate origin. Crack surfaces near the crack tip, i.e., y = ±0 and −a < x < 0, are assumed under the action of uniform shear traction σzy = −τ1 , Hzy = 0, where a is a length used to simulate a ﬁnite-size crack. The upper and lower strip surfaces are assumed being traction free, therefore the boundary conditions are ⎧ y = ±H, −∞ < x < ∞ : σzy = 0, Hzy = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ x = ±∞, −H < y < H : σzx = 0, Hzx = 0, (8.2-1) ⎪ y = ±0, −∞ < x < −a : σzy = 0, Hzy = 0, ⎪ ⎪ ⎪ ⎪ ⎩ y = ±0, −a < x < 0 : σ = −τ , H = 0. zy 1 zy

Fig. 8.2-1

Finite strip of hexangonal quasicrystal with a crack

Formulas (8.1-1)∼(8.1-10) in the preceding section still hold in this section, and other notation and symbol are similar. Therefore, formula (8.1-11) should be modiﬁed as C44 (φ1 − φ1 ) + R3 (ψ1 − ψ1 ) = 2iτ1 f (x), (8.2-2) K2 (ψ1 − ψ1 ) + R3 (φ1 − φ1 ) = 0,

where f (x) =

0,

x < −a,

1,

−a < x < 0.

(8.2-3)

The conformal mapping 2 H 1+ζ , t = ω(ζ) = ln 1 + π 1−ζ

(8.2-4)

maps the domain in the t-plane onto the interior of the unit circle γ in the ζ-plane, ζ = ξ + iη. Therefore, crack tip t = 0 corresponds to ζ = −1, and t = −a accords to two points on the unit circle in the ζ-plane such that

136

Chapter 8

Application II: Solutions of notch and crack problems of one-and...

√ ⎧ −e−πa/H + 2i 1 − e−πa/H ⎪ ⎪ = , σ ⎪ ⎨ −a 2 − e−πa/H √ ⎪ −πa/H ⎪ − 2i 1 − e−πa/H ⎪ ⎩ σ−a = −e , 2 − e−πa/H

(8.2-5)

where σ = eiϕ = ζ||ζ|=1 denotes the value of ζ on γ. Equations in Section 8.1 and the Appendix of this chapter (i.e., Section 8.7) are useful. Nevertheless, the ﬁrst 2iω (σ)τ in (8.7-5) should be changed to 2if ω (σ)τ , and the ﬁrst term in (8.7-5) 2iτ 1 C44 2πi should be modiﬁed as 2iτ 1 C44 2πi

γ

f γ

ω (σ) dσ σ−ζ ω (σ) dσ. σ−ζ

Thus, equation (8.7-5) for the present problem becomes ⎧ R3 2iτ1 1 ω (σ) ⎪ ⎪ φ (ζ) + dσ, ψ (ζ) = f ⎨ C44 C44 2πi γ σ − ζ ⎪ R3 ⎪ ⎩ ψ (ζ) + φ (ζ) = 0, K2

(8.2-6)

where f is the function given by formula (8.2-3), which takes values between σ−a and σ−a in the ζ-plane. Integration of the right-handside of equation (8.2-6) leads to 1 ω (σ) 1 1 1+ζ f dσ = ln(σ − 1) − ln(σ − ζ) 2πi γ σ − ζ 2πτ 1 − ζ (1 − ζ)(1 + ζ 2 ) σ=σ−a 1 σ−i ζ 2 ln(1 + σ ) + ln − 2(1 + ζ 2 ) 2(1 − ζ 2 ) σ + i σ=σa ≡F (ζ)

(8.2-7)

where σ−a and σ−a are given by formula (8.2-5). Based on the above relation, (8.2-6) is further reduced to φ (ζ) =

K2 τ1 2iF (ζ), C44 K2 − R32

ψ = −

R3 τ1 2iF (ζ). C44 K2 − R32

(8.2-8)

Now let us calculate the stresses. During this process, the following relations will be used: φ1 (t) =

2iK2 τ1 φ (ζ) F (ζ) = , · ω (ζ) C44 K2 − R32 ω (ζ)

8.2

Crack problem in ﬁnite-sized one-dimensional quasicrystals

ψ1 (t) =

137

ψ (ζ) F (ζ) 2iR3 τ1 · = − . ω (ζ) C44 K2 − R32 ω (ζ)

Substitution of the above expressions into (8.1-10) leads to ⎧ ⎪ ⎨ σzx − iσzy = iτ1 F (ζ) , ω (ζ) ⎪ ⎩ H − iH = 0, zx

(8.2-9)

zy

which shows that the stress distribution is independent of the material constants. Substitution of (8.2-7) into (8.2-9) leads to the explicit expressions of stresses. The forms of F (ζ) and F (ζ) are relatively complex, while the expressions of stress in terms of variable ζ(= ξ + iη) are concise. In an attempt to invert them to the t-plane, inverse of transform (8.2-4) must be used, √ −(eπt/H − 2) ± 2i eπt/H − 2 −1 ζ = ω (t) = . (8.2-10) eπt/H − 2 Substitution of (8.2-10) into (8.2-9) yields to the ﬁnal expressions of σzx , σzy , Hzx and Hzx in the t-plane, which are very complex. Here we skip this procedure. Now let us calculate the stress intensity factors. According to (8.1-15) and (8.1-16), it is known that at the region such that r1 /a 1, ||

K θ1 σzx = − √ III sin , 2 2πr1

||

K θ1 σzy = √ III cos , 2 2πr1

therefore ||

σzx − iσzy

K = − √ III 2πr1

θ1 θ1 − i cos sin 2 2

K = − √ III , 2πt1

(8.2-11)

where t1 = r1 eiθ1 = x1 + iy1 . By using conformal mapping t = ω(ζ), we have t1 = ω(ζ1 ),

(8.2-12)

where ζ1 is the point in the ζ-plane corresponding to t1 . With relations (8.2-11), (8.2-12), and (8.2-9), we deﬁne

KIII = lim

ζ→−1

√

F (ζ) , πτ1 # ω (ζ)

⊥ KIII =0

(8.2-13)

where ζ = −1 is the point corresponding to the crack tip. Substitution of (8.2-4) and (8.2-7) into (8.2-13) yields

138

Chapter 8

Application II: Solutions of notch and crack problems of one-and...

KIII =

√ √ 2Hτ1 2eπa/H − 1 + 2eπa/H 1 − e−πa/H √ , ln 2π 2eπa/H − 1 − 2eπa/H 1 − e−πa/H

(8.2-14)

⊥ If we do not assume τ2 = 0 in (8.2-1), then the stress intensity factor KIII can be evaluated, and the expression is similar to (8.2-14), this is the extension of work given by Ref. [4] for classical elasticity to quasicrystal elasticity.

8.2.2

Finite strip with two cracks

The conﬁguration is shown in Fig. 8.2-2, there are the following boundary conditions ⎧ y = ±H, −∞ < x < ∞ : σzy = 0, Hzy = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ x = ±∞, −H < y < H : σzx = 0, Hzx = 0 ⎪ ⎪ ⎪ ⎨ y = ±0, −∞ < x < −a : σzy = 0, Hzy = 0; ⎪ ⎪ ⎪ ⎪ ⎪ −a < x < 0 : σzy = −τ1 , Hzy = −τ2 ; ⎪ ⎪ ⎪ ⎪ ⎩ L < x < ∞ : σzy = 0, Hzy = 0

(8.2-15)

The following conformal mapping function: 2

t = ω(ζ) =

H 1 + α (1 − ζ/1 + ζ) ln π 1 + βα (1 − ζ/1 + ζ)2

(8.2-16)

transforms the region at z-plane onto the interior of the unit circle γ at ζ-plane, in which 1 − e−πa/H , β = e−πL/H . (8.2-17) α= 1 − e−π(a+L)/H

Fig. 8.2-2

Two cracks in a strip

Then substitute (8.2-17) into (8.2-6), we can ﬁnd the solution φ (ζ) so the stress intensity factors such as

8.3

Griﬃth crack problems in point groups 5m and 10mm quasicrystals based...

⎧ # F (ζ) (0,0) ⎪ ⎪ KIII = lim −2 2πω(ζ)τ1 ⎪ ⎪ ζ→1+0 ω (ζ) ⎪ ⎪ ⎪ √ ⎪ √ √ ⎪ ⎪ 2Hτ1 1 + αβ 1+ α √ ⎪ ⎪ √ − β ln √ ln , = √ ⎪ ⎨ 1− α π 1−β 1 − αβ

# ⎪ F (ζ) ⎪ (L,0) ⎪ = lim −2 2π(L − ω(ζ))τ1 KIII ⎪ ⎪ ζ→−1−0 ⎪ ω (ζ) ⎪ ⎪ ⎪ √ √ √ ⎪ ⎪ 2Hτ1 √ 1 + αβ 1+ α ⎪ ⎪ √ √ √ ⎩ , = − ln β ln 1− α π 1−β 1 − αβ

139

(8.2-18)

in which φ (ζ) =

K2 τ1 2iF (ζ), C44 K2 − R32

ψ (ζ) = −

R3 τ1 2iF (ζ), C44 K2 − R32

√ √ 2H αA αβM F (ζ) = 2 − π (1 + ζ)2 + α(1 − ζ)2 (1 + ζ)2 + αβ(1 − ζ)2 2H iα(1 − β)(1 − ζ 2 ) i−ζ + 2 ln , 2 2 2 2 π [(1 + ζ) + α(1 − ζ) ][(1 + ζ) + αβ(1 − ζ) ] 1 − iζ √ √ (1 + α) 1 + αβ √ √ . A = ln , M = ln 1− α 1 − αβ

(8.2-19)

⊥ If we do not assume τ2 = 0 in (8.2-15), then the stress intensity factor KIII can be similarly evaluated, and the expression is similar to (8.2-18), this extends the study for the classical elasticity. The detail can be found in Refs. [5] and [6], some calculations on the function F (ζ) refer to Major Appendix of this book.

8.3

Griﬃth crack problems in point groups 5m and 10mm quasicrystals based on displacement potential function method

Literature [2] reported that Chinese materials scientists had started to characterize the fracture toughness of quasicrystals. Due to the lack of theoretical solutions to crack problems in quasicrystals, their experimental work was performed largely by indirectly measuring the fracture toughness. If the crack solution had been known then, the fracture toughness could have been determined by direct measurement, a much more simpler and accurate method. This section aims to solve the mode I Griﬃth crack in quasicrystal with ten-fold symmetry by using method of displacement functions. Therefore, the fundamental formulas in Section 6.2 and formulas using Fourier transform in Section 7.2 are the basis for this section. To save space, we do not plan to list those formulas in detail here, and interested readers may refer to the previous two chapters. In the next

140

Chapter 8

Application II: Solutions of notch and crack problems of one-and...

section, we are going to solve this problem using method of stress functions. On the one hand, this demonstrates the problem-solving procedure based on method of stress functions; on the other hand this is targeted to examine the results obtained by the method of displacement functions. For a correct solution, which can be examined using any available method. (∞) Consider a Griﬃth crack under the action of external traction, i.e., σyy = p, and the crack is assumed to penetrate the periodic axis(z-direction) of the quasicrystals, as shown in Fig. 8.3-1. Similar to the analysis in the preceding section, within the framework of Griﬃth’s theory, this problem can be replaced by an equivalent crack problem shown in Fig. 8.3-2. Furthermore, assume the external traction being independent of z, therefore the deformation of the quasicrystal is also independent of z, namely ∂ui ∂wi = 0, = 0, i = 1, 2, 3. (8.3-1) ∂z ∂z According to the analysis performed in Chapter 6, under this case, the twodimensional quasicrystal elasticity problem can be decoupled into a plane elasticity problem of phonon-phason coupling and an anti-plane pure elasticity problem. In this case, the latter only has a trivial solution under mode I external traction, which can be neglected. The plane elasticity problem of phonon-phason coupling with point groups 5m and 10mm has been studied in Section 6.2, and its ﬁnal governing equation is (8.3-2) ∇2 ∇2 ∇2 ∇2 F = 0. Here F (x, y) is the displacement potential function introduced in Section 6.2.

Fig. 8.3-1

Griﬃth crack along the periodic axis of quasicrystal and subjected to a tension

8.3

Griﬃth crack problems in point groups 5m and 10mm quasicrystals based...

Fig. 8.3-2

141

The same Griﬃth crack as Figure 8.3-1 with external traction acting on crack surfaces

As shown in Fig. 8.3-2, the Griﬃth crack is under the action of uniform traction at crack surfaces and without far-ﬁeld traction, i.e., σyy (x, 0) = −p, |x| < a. We modify this problem into the semi-plane problem, i.e., only study the case in the upper half-plane or the lower half-plane under the following conditions: ⎧ # ⎪ x2 + y 2 → ∞ : σij = Hij = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ y = 0, |x| < a : σyy = −p, σyx = 0, ⎪ ⎨ (8.3-3) Hyy = 0, Hyx = 0, ⎪ ⎪ ⎪ ⎪ y = 0, |x| > a : σyx = 0, Hyx = 0, ⎪ ⎪ ⎪ ⎪ ⎩ uy = 0, wy = 0. By performing the Fourier transform on equation (8.3-2) ∞ Fˆ (ξ, y) = F (x, y)eiξx dx.

(8.3-4)

−∞

which is therefore reduced to an ordinary diﬀerential equation such that 2 4 d 2 Fˆ (ξ, y) = 0. −ξ dy 2

(8.3-5)

If we choose the upper half-plane y > 0 for our study, the solution of the above equation is (8.3-6) Fˆ (ξ, y) = (4ξ 4 )−1 XY e−|ξ|y , where X = (A, B, C, D), Y = (1, y, y 2 , y 3 )T , and A, B, C, and D are arbitrary functions with respect to ξ to be determined according to boundary condition, “T”

142

Chapter 8

Application II: Solutions of notch and crack problems of one-and...

stands for the transpose of a matrix. Fourier transforms of the displacement and stress components can be expressed in term of Fˆ (ξ, y), i.e., X and Y as discussed in Section 7.2. Solution (8.2-6) has satisﬁed the boundary condition (8.2-3) at inﬁnity, and the left boundary condition in (8.2-3) results in ⎧ 21C(ξ)|ξ| − 3(32 − e2 )D(ξ) ⎪ ⎪ , ⎨ A(ξ) = 3 2 |ξ| ⎪ ⎪ ⎩ B(ξ) = 6C(ξ)|ξ| − 21D(ξ) ξ2 and the following set of dual integral equations: ∞ ⎧ 2 ⎪ ⎪ [C(ξ)ξ − 6D(ξ)] cos(ξx)dξ = −p, ⎪ ⎪ ⎪ ⎪ d11 0 ⎪ ⎪ ∞ ⎪ ⎪ ⎪ ⎪ ξ −1 [C(ξ)ξ − 6D(ξ)] cos(ξx)dξ = 0, ⎨ 0 ∞ ⎪ 2 ⎪ ⎪ D(ξ) cos(ξx)dξ = 0, ⎪ ⎪ d12 0 ⎪ ⎪ ⎪ ∞ ⎪ ⎪ ⎪ ⎪ ⎩ ξ −1 D(ξ) cos(ξx)dξ = 0,

(8.3-7)

0 < x < a, x > a, (8.3-8) 0 < x < a, x > a.

0

Here, e2 is given by the second relation in (7.2-12), i.e., e2 =

2αβ α−β + , ω(α − β)(K1 − K2 ) α + β

α and β are given in (6.2-5), i.e., α = R(L + 2M ) − ωK1 ,

β = RM − ωK1 ,

ω=

M (L + 2M ) , R

d11 and d22 are given as ⎧ nR ⎪ , d11 = ⎪ ⎪ ⎪ (4M/L + M )(L + 2M )(M K1 − R2 ) ⎪ ⎪ ⎪ ⎪ ⎨ nR2 d12 = , d0 M (L + 2M ) ⎪ ⎪ ⎪ ⎪ ⎪ d0 = −{(M K1 − R2 )[(L + 2M )(K1 + K2 ) − 2R2 ] ⎪ ⎪ ⎪ ⎩ −[(L + 2M )K1 − R2 ][M (K1 + K2 ) − 2R2 ]} and n is determined by (7.2-12), namely, n = M α − (L + 2M )β.

(8.3-9)

8.3

Griﬃth crack problems in point groups 5m and 10mm quasicrystals based...

143

The theory of dual integral equations is oﬀered in the Major Appendix of this monograph. Accordingly, solution to the set of equations (8.3-8) is 2C(ξ)ξ = d11 pαJ1 (aξ),

D(ξ) = 0,

(8.3-10)

where J1 (aξ) is the ﬁrst order Bessel function of the ﬁrst kind. So far, the unknown functions A(ξ), B(ξ), C(ξ), and D(ξ) have been determined completely. In the view of mathematics, this problem has been solved in the Fourier transform space. However, in the view of physics, we need to perform the Fourier inverse ∞ 1 (8.3-11) F (x, y) = Fˆ (ξ, y)e−iξx dξ 2π −∞ in order to express the ﬁeld variables in the physical space. Obviously, once F (x, y) is determined from the integral (8.3-11), uj , σjk , and Hjk can be determined by substituting F (x, y) into (8.3-8)∼(8.3-11). Alternatively, ˆ jk (ξ, y) can be determined by substituting X(ξ) u ˆj (ξ, y), w ˆj (ξ, y), σ ˆjk (ξ, y), and H and Y (ξ) into (8.3-8)∼(8.3-11), and then their Fourier inverses ﬁnally lead to uj , σjk , and Hjk . Luckily, the above integrals with Bessel function can be expressed explicitly using elementary functions. Nevertheless, their ﬁnal expressions in terms of variables x and y are extremely complex. However, the ﬁnal expressions appear more concise if using (r, θ), (r1 , θ1 ), and (r2 , θ2 ) to stand for the three polar coordinate systems with the crack center, left crack tip, and right crack tip as origins, respectively, as shown in Fig. 8.1-2, similar to (8.1-16), i.e.,

x = r cos θ = a + r1 cos θ1 = −a + r2 cos θ2 , y = r sin θ = r1 sin θ1 = r2 sin θ2 .

(8.3-12)

The following inﬁnite integrals involving Bessel functions are used for stress calculation: ⎧ ∞ ⎪ 1 z ⎪ −ξz ⎪ , J1 (aξ)e dξ = 1− ⎪ ⎪ 1/2 ⎪ 0 a (a2 − z 2 ) ⎪ ⎪ ⎪ ∞ ∞ ⎪ ⎪ ⎪ a ⎪ ⎪ ξJ ξJ1 (aξ)e−ξz dξ = , ⎪ 1 ⎨ 0 0 (a2 − z 2 )3/2 (8.3-13) ⎪ ∞ 3az ⎪ 2 −ξz ⎪ ⎪ ξ J1 (aξ)e dξ = , ⎪ ⎪ ⎪ 0 (a2 − z 2 )5/2 ⎪ ⎪ ⎪ ∞ ⎪ ⎪ 3a(4z 2 − a2 ) ⎪ 3 −ξz ⎪ ⎪ ξ J (aξ)e dξ = , 1 ⎩ 7/2 0 (a2 − z 2 ) where z = x + iy.

144

Chapter 8

Application II: Solutions of notch and crack problems of one-and...

After proper calculation, we have ⎧ ⎪ σxx = −p[1 + r(r1 r2 )−3/2 cos(θ − θ)] − pr(r1 r2 )−3/2 sin θ sin 3θ, ⎪ ⎪ ⎪ ⎪ ⎪ σyy = −p[1 − r(r1 r2 )−3/2 cos(θ − θ)] + pr(r1 r2 )−3/2 sin θ sin 3θ, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ σxy = σyx = pr(r1 r2 )−3/2 sin θ cos 3θ, ⎪ ⎨ Hxx = −4d21 pr(r1 r2 )−3/2 sin θ cos 3θ − 6d21 pr3 (r1 r2 )−5/2 sin2 θ cos(θ − 5θ), ⎪ ⎪ ⎪ ⎪ Hyy = −6d21 pr3 (r1 r2 )−5/2 sin2 θ cos(θ − 5θ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Hxy = 6d21 pr3 (r1 r2 )−5/2 sin2 θ sin(θ − 5θ), ⎪ ⎪ ⎪ ⎩ Hyx = 4d21 pr(r1 r2 )−3/2 sin θ cos 3θ + 6d21 pr3 (r1 r2 )−5/2 sin2 θ sin(θ − 5θ), (8.3-14a) in which r = r/a, r1 = r1 /a, r2 = r2 /a, θ = (θ1 + θ2 )/2, d21 =

R(K1 − K2 ) 4(M K1 − R2 )

(8.3-14b)

Similarly, displacement components uj and wj can be also expressed in terms of elementary functions. We list here only the component ⎧ |x| > a, ⎨ 0, √ 1 K1 p (8.3-15) uy (x, 0) = 2 2 + a − x , |x| < a, ⎩ 2 M K1 − R2 L + M From these results, we can ﬁnd the stress intensity factor # √ KI = lim+ 2π(x − a)σyy (x, 0) = πap.

(8.3-16)

x→a

and from (8.3-15) we have the crack strain energy a WI =2 (σyy (x, 0) ⊕ Hyy (x, 0))(uy (x, 0) ⊕ wy (x, 0))dx 0 1 K1 πa2 p2 + . = 4 L+M M K1 − R2 So there is the crack energy release rate 1 1 K1 1 ∂W1 = + (KI )2 . GI = 2 ∂a 4 L+M M K1 − R2

(8.3-17)

(8.3-18)

It is evident that the crack energy release rate depends on not only phonon elastic constants L(= C12 ), M (= (C11 − C12 )/2), but also phason elastic constant K1 and phonon-phason coupling elastic constant R. The further meaning and applications of these quantities will be discussed in detail in Chapter 15. This provides the basis of fracture mechanics of quasicrystalline materials. The detail on the work can be found in Ref [7].

8.4

8.4

Stress potential function formulation and complex variable function method...

145

Stress potential function formulation and complex variable function method for solving notch and crack problems of quasicrystals of point groups 5, ¯ 5 and 10, 10

The Fourier method cannot solve notch problems, while complex variable function method with conformal mapping can solve them. In this section we develop complex potential method for notch/crack problems for plane elasticity of point groups 5, ¯5 and 10, 10 quasicrystals. We will use the stress potential formulation[9] , of course, we can also use the displacement potential formulation given in Section 6.4. 8.4.1

Complex function method

From Section 6.7, we ﬁnd that, based on stress potential method the ﬁnal governing equation of plane elasticity of point group 10, 10 decagonal quasicrystals is ∇2 ∇2 ∇2 ∇2 G = 0.

(8.4-1)

The general solution of the equation (8.4-1) is 1 1 G = 2Re g1 (z) + z¯g2 (z) + z¯2 g3 (z) + z¯3 g4 (z) , 2 6

(8.4-2)

where gi (z)(i = 1, 2, 3, 4) are four analytic functions of a single complex variable z = x + iy = reiθ . The bar over the quantity denotes the complex conjugate hereinafter, i.e., z¯ = x − iy = re−iθ . We can see that the complex analysis here will be more complicated than that of Muskhelishvili method for plane elasticity of classical elasticity. 8.4.2

The complex representation of stresses and displacements

Substituting expression (8.4-2) into equation (6.7-6) then into equations (6.7-2) leads to ⎧ σxx = −32c1 Re[Ω (z) − 2g4 (z)], ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ σyy = 32c1 Re[Ω (z) + 2g4 (z)], ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ σxy = σyx = 32c1 ImΩ (z), ⎪ ⎪ ⎪ ⎨ Hxx = 32R1 Re[Θ (z) − Ω (z)) − 32R2 Im(Θ (z) − Ω (z)], (8.4-3) ⎪ ⎪ ⎪ ⎪ ⎪ Hxy = −32R1 Im[Θ (z) + Ω (z)) − 32R2 Re(Θ (z) + Ω (z)], ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Hyx = −32R1 Im[Θ (z) − Ω (z)) − 32R2 Re(Θ (z) − Ω (z)], ⎪ ⎪ ⎪ ⎪ ⎩ Hyy = −32R1 Re[Θ (z) + Ω (z)) + 32R2 Im(Θ (z) + Ω ],

146

where

Chapter 8

Application II: Solutions of notch and crack problems of one-and...

⎧ ⎨ Θ(z) = g (IV) (z) + z¯g (IV) (z) + 1 z¯2 g (IV) (z) 2 3 4 2 ⎩ (IV) (IV) Ω (z) = g3 (z) + z¯g4 (z)

(8.4-4)

in which the prime, two prime, three prime and superscript (IV) denote the ﬁrst to fourth order derivatives of gj (z)(j = 1, · · · , 4) to variable z, in addition Θ (z) = dΘ(z)/dz. We further derive the complex representations of displacement components of phonon and phason ﬁelds. The ﬁrst two equations of (6.7-3) can be rewritten as ⎧ K1 + K2 1 ⎪ ⎪ σyy − [R1 (Hxx + Hyy ) + R2 (Hxy − Hyx )], ⎨ εxx = c2 (σxx + σyy ) − 2c 2c ⎪ K 1 + K ⎪ 2 ⎩ εyy = c2 (σxx + σyy ) − 1 σxx + [R1 (Hxx + Hyy ) + R2 (Hxy − Hyx )], 2c 2c (8.4-5) where c + (L + M )(K1 + K2 ) c2 = . (8.4-6) 4(L + M )c Substituting equations (8.4-3) into (8.4-5) and by integration yield ux =128c1 c2 Reg4 (z) −

K1 + K2 ∂ φ 2c ∂x

32(R12 + R22 ) Re[g3 (z) + z¯g4 (z) − g4 (z)] + f1 (y), c K1 + K2 ∂ φ uy =128c1 c2 Img4 (z) − 2c ∂y 32(R12 + R22 ) − Im[g3 (z) + z¯g4 (z) + g4 (z)] + f2 (x). c With these results and other equations of equations (6.7-3) one ﬁnds that +

−

df1 (y) df2 (x) = . dy dx

This means these two functions must be constants which only give rigid-body displacements. Omitting the trial functions f1 (y), f2 (x), one obtains ux + iuy = 32(4c1 c2 − c3 − c1 c4 )g4 (z) − 32(c1 c4 − c3 )[g3 (z) + zg4 (z)],

(8.4-7)

where

R12 + R22 K1 + K 2 , c4 = . (8.4-8) c c Similarly, the complex representations of displacement components of phason ﬁelds can be expressed as follows: c3 =

wx + iwy =

32(R1 − iR2 ) Θ(z). K1 − K2

(8.4-9)

8.4

Stress potential function formulation and complex variable function method...

8.4.3

147

Elliptic notch problem

To illustrate the eﬀect of the stress potential and complex variable function method to the complicated stress boundary value problems of eight order partial diﬀerential equations given above, we here calculate the stress and displacement ﬁelds induced y2 x2 by an elliptic notch L : 2 + 2 = 1, see Fig. 8.4-1 (a), the edge of which is a b subjected to a uniform pressure p, this problem is equivalent to the case that the body is subjected to a tension at inﬁnity and the surface of the notch is stress free shown in Fig. 8.4-1(b) if α = π/2. The equivalency is proved in Subsection 11.3.9 of Chapter 11.

Fig. 8.4-1

(a) The elliptic notch in a decagonal quasicrystal subjected to a uniform

pressure and traction tree at inﬁnity; (b) An inﬁnite decagonal quasicrystal with an elliptic notch subjected to a tension

148

Chapter 8

Application II: Solutions of notch and crack problems of one-and...

For the problem shown by Fig. 8.4-1 (a) the boundary conditions can be expressed as follows: σxx cos(n, x) + σxy cos(n, y) = Tx ,

σxy cos(n, x) + σyy cos(n, y) = Ty ,

(x, y) ∈ L, (8.4-10)

(x, y) ∈ L, (8.4-11) where Tx = −p cos(n, x), Ty = −p cos(n, y) denote the components of surface traction, p is the magnitude of the pressure, hx and hy are generalized surface tractions, n represents the outward unit normal vector of any point of the boundary. But the measurement of generalized tractions has not been reported so far, for simplicity, we assume that hx = 0, hy = 0. From equations (8.4-3), (8.4-4) and (8.4-10), one has Hxx cos(n, x)+Hxy cos(n, y) = hx ,

g4 (z)

+

g3 (z)

+

zg4 (z)

i = 32c1

Hyx cos(n, x)+Hyy cos(n, y) = hy ,

(Tx + iTy )ds = −

1 pz, 32c1

z ∈ L.

(8.4-12)

Taking conjugate on both sides of equation (8.4-12) yields g4 (z) + g3 (z) + z¯g4 (z) = −

1 p¯ z, 32c1

z ∈ L.

(8.4-13)

From equations (8.4-3), (8.4-4) and (8.4-11), we have R1 ImΘ(z) + R2 ReΘ(z) = 0, −R1 ReΘ(z) + R2 ImΘ(z) = 0,

z ∈ L.

(8.4-14)

Multiplying the second formula of (8.4-14) by i and adding it to the ﬁrst, one obtains Θ(z) = 0,

z ∈ L.

(8.4-15)

Because the function g1 (z) does not appear in the displacement and stress formulas, boundary equations (8.4-12), (8.4-13) and (8.4-15) are enough for determining the unknown functions g2 (z), g3 (z) and g4 (z). However the calculation cannot be completed at the z-plane due to the complicity of the evaluation, we must use the conformal mapping 1 + mζ (8.4-16) z = ω(ζ) = R0 ζ to transform the region with ellipse at the z-plane onto the interior of the unit circle γ a−b a+b ,m= . at the ζ-plane, refer to Fig.8.4-2, where ζ = ξ+iη = ρeiϕ and R0 = 2 a+b

8.4

Stress potential function formulation and complex variable function method...

Fig. 8.4-2

149

Conformal mapping from the region at z-plane onto the interior of the unit circle γ at ζ-plane

For simplicity, we introduce the following new symbols (IV)

g2

(z) = F2 (z),

g3 (z) = F3 (z),

g4 (z) = F4 (z).

(8.4-17)

And we have Fj (z) = Fj (ω(ζ)) = Φj (ζ),

Fj (z) =

Φj (ζ) , ω (ζ)

(j = 1, · · · , 4).

(8.4-18)

Substituting (8.4-17) into (8.4-12), (8.4-13) and (8.4-15), then multiplying both sides 1 dσ , and integrating along the unit circle, we have of equations by 2πi σ − ζ 1 1 Φ4 (σ)dσ Φ3 (σ)dσ ω(σ) Φ4 (σ)dσ + + σ−ζ 2πi γ σ − ζ 2πi γ ω(σ) σ − ζ γ p 1 ω(σ)dσ , =− 32c1 2πi γ σ − ζ Φ4 (σ)dσ Φ3 (σ)dσ ω(σ) Φ4 (σ)dσ 1 1 1 + + 2πi γ σ − ζ 2πi γ σ − ζ 2πi γ ω(σ) σ − ζ ω(σ)dσ p 1 =− , 32c1 2πi γ σ − ζ 1 Φ2 (σ)dσ ω(σ) Φ3 (σ)dσ 1 + 2πi γ σ − ζ 2πi γ ω (σ) σ − ζ 2 2 ω(σ) Φ4 (σ)dσ ω(σ) ω (σ) Φ4 (σ)dσ 1 + − = 0, (8.4-19) 2πi γ [ω (σ)]2 σ − ζ [ω (σ)]3 σ−ζ γ 1 2πi

where σ = eiϕ (ρ = 1) represents the value of ζ at the unit circle.

150

Chapter 8

Application II: Solutions of notch and crack problems of one-and...

According to Cauchy’s integral formula and analytic extension of the complex variable function theory, similar to the calculations at Sections 8.1 and 8.2, from the ﬁrst and the second equation of (8.4-19), one obtains pR0 (1 + m2 )ζ , 32c1 mζ 2 − 1 pR0 Φ4 (ζ) = − mζ. 32c1 Φ3 (ζ) =

(8.4-20)

Substitution of ω(σ) σ2 + m = σ , ω (σ) mσ 2 − 1

2

ω(σ) ω (σ) 2σ(σ 2 + m)2 = − ω (σ)3 (mσ 2 − 1)3

and (8.4-20) into the third equation of (8.4-19) yields 1 2πi

γ

Φ2 (σ)dσ 1 + σ−ζ 2πi

σ γ

σ 2 + m Φ3 (σ)dσ 1 + mσ 2 − 1 σ − ζ 2πi

γ

σ(σ 2 + m)2 Φ4 (σ)dσ = 0. (mσ 2 − 1)3 σ − ζ

By Cauchy’s integral formula, we have 1 2πi 1 2πi 1 2πi

σ γ

γ

γ

Φ2 (σ)dσ = Φ2 (ζ), σ−ζ

σ 2 + m Φ3 (σ)dσ ζ2 + m = ζ Φ (ζ), mσ 2 − 1 σ − ζ mζ 2 − 1 3

ζ(ζ 2 + m)2 σ(σ 2 + m)2 Φ4 (σ)dσ = Φ (ζ). (mσ 2 − 1)3 σ − ζ (mζ 2 − 1)3 4

Substituting these equations into (8.4-19), one ﬁnds that at last Φ2 (ζ) =

pR0 ζ(ζ 2 + m)[(1 + m2 )(1 + mζ 2 ) − (ζ 2 + m)] . 32c1 (mζ 2 − 1)3

(8.4-21)

Utilizing the above mentioned results, the phonon and phason stresses can be determined at the ζ-plane. We here only give a simple example, i.e., along the edge of notch (ρ = 1), there are phonon stress components such as σϕϕ = p

1 − 3m2 + 2m cos 2ϕ , 1 + m2 − 2m cos 2ϕ

σρρ = −p,

σρϕ = σϕρ = 0,

which are identical to the well-known results of the classical elasticity theory.

8.4

8.4.4

Stress potential function formulation and complex variable function method...

151

Elastic ﬁeld caused by a Griﬃth crack

The solution of the Griﬃth crack subjected to a uniform pressure, has been observed by Li[10] in terms of the Fourier transform method, which can also be obtained from a the notch solution in corresponding to the case m = 1, R0 = of the present work. 2 For explicitness, we express the solution at z-plane. The inversion of transformation (8.4-16) is as m = 1, # 1 ζ = (z − z 2 − a2 ). (8.4-22) a From equations (8.4-20), (8.4-25) and (8.4-22), we have ⎧ z2 pa2 (IV) ⎪ ⎪ # g2 (z) = − , ⎪ ⎪ 128c1 (z 2 − a2 )3 ⎪ ⎪ ⎪ ⎨ a2 p √ , g3 (z) = − ⎪ 64c1 z 2 − a2 ⎪ ⎪ ⎪ ⎪ ⎪ p √ 2 ⎪ ⎩ ( z − a2 − z). g4 (z) = 64c1

(8.4-23)

So the stresses and the displacements can be expressed with complex variable z. Similar to (8.1-31), we introduce three pairs of polar coordinates (r, θ), (r1 , θ1 ) and (r2 , θ2 ) with the origin at crack center, at the right crack tip and at the left crack tip, i.e., z = reiθ , z − a = r1 eiθ1 , z + a = r2 eiθ2 respectively, the analytic expressions for the stress and displacement ﬁelds can be obtained. Moreover, the stress intensity factor and free energy of the crack and so on can be evaluated as the direct results of the solution. We here only list the stress intensity factor and energy release rate as below # √ KI = lim+ ( 2π(x − a)σyy (x, 0) = πap, x→a

∞ 1 ∂ [(σyy (x, 0) ⊕ Hyy (x, 0))(ux (x, 0) ⊕ wx (x, 0))]dx GI = 2 2 ∂a 0 =

L(K1 + K2 ) + 2(R12 + R22 ) 2 (KI ) , 8(L + M )c

(8.4-24)

where c = M (K1 + K2 ) − 2(R12 + R22 ) and L = C12 , M = (C11 − C12 )/2 = C66 . The crack energy release rate GI is dependent upon not only the phonon elastic constants L(= C12 ), M (= (C11 − C12 )/2), but also the phason elastic constants K1 , K2 , and the phonon-phason coupling elastic constants R1 , R2 , though we have assumed the generalized tractions hx = hy = 0. It is evident that the present solution covers the solution for point groups 5m, 10mm quasicrystals.

152

Chapter 8

Application II: Solutions of notch and crack problems of one-and...

The detail for some further principle of the complex function method will be discussed in-depth in Chapter 11. For the further implications and applications of the results to fracture mechanics of quasicrystals, refer to Chapter 15.

8.5

Solutions of crack/notch problems of two-dimensional octagonal quasicrystals

Zhou and Fan[11] and Zhou[12] obtained the solution of a Griﬃth crack in octagonal quasicrystals in terms of the Fourier transform and dual integral equations, the calculation is very complex and lengthy, which cannot be listed here. Li[13] gave solutions for a notch/Griﬃth crack problem in terms of complex variable function method based on the stress potential formulation, an outline of the algorithm was listed in Section 6.8. The ﬁnal governing equation of plane elasticity of octagonal quasicrystals of point group 8mm based on the stress potential formulation is the same of that of (6.6-12), but we here rewrite as 8 ∂8 ∂8 ∂8 ∂8 ∂ + 4(1 − 4ε) + 2(3 + 16ε) + 4(1 − 4ε) + G = 0, ∂x8 ∂x6 ∂y 2 ∂x4 ∂y4 ∂x2 ∂y6 ∂y 8 (8.5-1) in which G(x, y) is the stress potential function, and the material constant ε is the same as that given in Chapters 6 and 7. The complex representation of equation (8.5-1) is 4 Gk (zk ), zk = x + μk y, (8.5-2) G(x, y) = 2Re k=1

in which unknown functions Gk (zk ) are analytic functions of complex variable zk (k = 1, · · · , 4), to be determined, and μk = αk +iβk (k = 1, · · · , 4) are complex parameters and determined by the roots of the following eigenvalue equation: μ8 + 4(1 − 4ε)μ6 + 2(3 + 16ε)μ4 + 4(1 − 4ε)μ2 + 1 = 0.

(8.5-3)

The stresses can be expressed by functions Gk (zk ) such as σxx = −2c3 c4 Re

4

(μ2k + 2μ4k + μ6k )gk (zk ),

(8.5-4a)

k=1

σyy = −2c3 c4 Re

4

(1 + 2μ2k + μ4k )gk (zk ),

(8.5-4b)

k=1

σxy = σyx = 2c3 c4 Re

4 k=1

(μk + 2μ3k + μ5k )gk (zk ),

(8.5-4c)

8.6

Other solutions of crack problems in one- and two-dimensional quasicrystals

Hxx = RRe

4

[(4c4 − c3 )μ2k + 2(3c3 − 2c4 )μ4k − c3 μ6k )]gk (zk ),

153

(8.5-4d)

k=1

Hxy = −RRe

4

[(4c4 − c3 )μk + 2(3c3 − 2c4 )μ3k − c3 μ5k )]gk (zk ),

(8.5-4e)

k=1

Hyx = −RRe

4

[c3 μk + 2(c4 − 2c3 )μ3k − (4c4 − c3 )μ5k )]gk (zk ),

(8.5-4f)

[c3 + 2(c4 − 2c3 )μ2k − (4c4 − c3 )μ4k )]gk (zk ),

(8.5-4g)

k=1

Hyy = RRe

4 k=1

in which gk (zk ) =

∂ 6 Gk (zk ) , ∂zk6

gk (zk ) =

dgk (zk ) , dzk

(K1 + K2 + K3 )M − R2 K1 M − R 2 , c4 = . K1 + K2 + 2K3 K1 − K 2 We now consider an elliptic hole L : x2 /a2 + y 2 /b2 = 1, at which there are the boundary conditions c3 =

σxx cos(n, x)+σxy cos(n, y) = Tx ,

σxy cos(n, x)+σyy cos(n, y) = Ty ,

Hxx cos(n, x)+Hxy cos(n, y) = hx ,

Hyx cos(n, x)+Hyy cos(n, y) = hy ,

The complex variable zk can be rewritten as

zk = xk + iyk , xk = x + αk y, yk = βk y.

(x, y) ∈ L, (x, y) ∈ L. (8.5-5)

(8.5-6)

The second formula of equations (8.5-6) represents a coordinate transformation. Further taking conformal mapping, the complex potentials satisfying the boundary conditions can be determined. The detail is omitted here.

8.6

Other solutions of crack problems in one- and two-dimensional quasicrystals

Fan and Guo[14] constructed the solution of three-dimensional elliptic crack in hexagonal quasicrystals by using perturbation procedure and hydrodynamic analogy, this is the ﬁrst solution of elliptic crack in quasicrystalline materials, and the solution on penny-shaped crack of hexagonal quasicrystals given by Peng and Fan[15] is the special case of the solution of Fan and Guo[14] . Liu and Fan[16] and Liu[17] found the solution of Griﬃth crack of point group 10mm quasicrystals again by complex variable function-conformal mapping technique, Liu oﬀered a series crack solutions and solutions of interaction between cracks and dislocations for various one-dimensional quasicrystals.

154

Chapter 8

Application II: Solutions of notch and crack problems of one-and...

The characteristics of crack solutions mentioned here and those given in the previous sections provide a basis of fracture theory of quasicrystalline materials summarized in Chpater 15.

8.7

Appendix of Chapter 8: Derivation of solution of Section 8.1

In Section 8.1 we focused on the physical aspect of the problem, and omitted some mathematical details. Because the mathematical details are similar to that in Section 8.4, there is no need to repeat. But the strict derivation for complex displacement potentials (8.1-12) is signiﬁcant, which also provides a basis for the solutions in Section 8.2, we outline the derivation as follows. For simplicity we consider the boundary value problem (8.1-5). From (8.1-11) we take ⎧ 1 ⎪ ⎪ ⎨ σyz = σzy = − 2i [C44 (φ1 − φ1 ) + R3 (ψ1 − ψ1 )] + τ1 (8.7-1) ⎪ 1 ⎪ ⎩ Hzy = − [K2 (ψ1 − ψ1 ) + R3 (φ1 − φ1 )] + τ2 2i In the following we use L to represent the crack surface under consideration, and then the second expression of the boundary condition at L exhibited by (8.1-5) can be written as C44 (φ1 − φ1 ) + R3 (ψ1 − ψ1 ) − 2iτ1 = 0, t ∈ L (8.7-2) K2 (ψ1 − ψ1 ) + R3 (φ1 − φ1 ) − 2iτ2 = 0, t∈L By using the following conformal mapping, 1 a ζ+ z = ω(ζ) = 2 ζ

(8.7-3)

which maps the domain in the t-plane with a Griﬃth crack onto the interior of the unit circle γ in the ζ-plane (ζ = ξ + iη = ρeiϕ ), and the elliptic hole L onto a unit circle γ (similar to Fig. 8.4-2). On the unit circle γ, ζ = σ ≡ eiϕ , where ρ = 1. Furthermore, with the conformal mapping (8.7-3), the unknown functions φ1 (t) and ψ1 (t) and their corresponding derivatives can be expressed as φ1 (t) = φ1 [ω(ζ)] = φ(ζ), ψ1 (t) = ψ1 [ω(ζ)] = ψ(ζ) (8.7-4) φ1 (t) = φ (ζ)/ω (ζ), ψ1 (t) = ψ (ζ)/ω (ζ) Similar to the manipulation undertaken in Section 8.4 the boundary conditions

8.7

Appendix of Chapter 8: Derivation of solution of Section 8.1

(8.7-2) are reduced to the following function equations ⎧ 1 R3 1 φ (σ) ω (σ) ψ (σ) dσ 1 ⎪ ⎪ dσ − + dσ φ (σ) ⎪ ⎪ 2πi σ − ζ 2πi σ − ζ C 2πi ⎪ 44 γ γ ω (σ) γ σ−ζ ⎪ ⎪ ⎪ ⎪ ⎪ 2iτ1 1 R3 1 ω (σ) ω (σ) dσ ⎪ ⎪ ⎪ = dσ − ψ (σ) ⎪ ⎨ C44 2πi γ ω (σ) σ−ζ C44 2πi γ σ − ζ ⎪ 1 ψ (σ) ω (σ) φ (σ) dσ 1 R3 1 ⎪ ⎪ ψ (σ) dσ − + dσ ⎪ ⎪ (σ) ⎪ 2πi σ − ζ 2πi σ − ζ K 2πi σ −ζ 2 ω γ γ γ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ω (σ) ω (σ) dσ 2iτ2 1 R3 1 ⎪ ⎪ φ (σ) = dσ − ⎩ K2 2πi γ ω (σ) σ−ζ K2 2πi γ σ − ζ With the similar procedure that adopted in Section 8.4, we can ﬁnd ⎧ φ (σ) ω (σ) 1 dσ 1 ⎪ ⎪ dσ = φ (ζ), =0 φ (σ) ⎪ ⎪ ⎪ 2πi γ σ − ζ 2πi γ ω (σ) σ−ζ ⎪ ⎪ ⎪ ⎨ 1 ψ (σ) ω (σ) 1 dσ dσ = ψ (ζ), − =0 ψ (σ) 2πi γ σ − ζ 2πi γ ω (σ) σ−ζ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 a ω (σ) ⎪ ⎪ dσ = ⎩ 2πi σ−ζ 2

155

(8.7-5)

(a)

γ

so the solution of function equations (8.7-5) is φ (ζ) = ia

K 2 τ 1 − R3 τ 2 , C44 K2 − R32

C44 τ2 − R3 τ1 C44 K2 − R32

(8.7-6)

C44 τ2 − R3 τ1 ζ C44 K2 − R32

(8.7-7)

ψ (ζ) = ia

By integration from the solution (8.7-6) we have φ(ζ) = ia

K2 τ1 − R3 τ2 ζ, C44 K2 − R32

ψ(ζ) = ia

The single-valued inversion of conformal mapping & 2 t t ζ = ω −1 (t) = − −1 a a makes |z| = ∞ correspond to ζ = 0, and substituting it into (8.7-8) yields K 2 τ 1 − R3 τ 2 φ(ζ) = φ(ω −1 (t)) = φ1 (t) = ia ζ C44 K2 − R32 ⎞ ⎛ & 2 K2 τ1 − R3 τ2 ⎝ t t − 1⎠ − = ia C44 K2 − R32 a a ψ(ζ) = ψ(ω −1 (t)) = ψ1 (t) = ia

C44 τ2 − R3 τ1 ζ C44 K2 − R32

(8.7-8)

156

Chapter 8

Application II: Solutions of notch and crack problems and solutions...

⎞ & 2 C44 τ2 − R3 τ1 ⎝ t t = ia − 1⎠ − C44 K2 − R32 a a ⎛

these are complex displacement potentials (8.1-12).

References [1] Hu C Z, Yang W Z, Wang R H et al. Symmetry and physical properties of quasicrystals. Adv Phys, 1997, 17(4): 345–376 (in Chinese) [2] Meng X M, Dong B Y, Wu Y K. Mechanical property of quasi-crystal Al65 Cu20 Co15 . Acta Metal Sinica, 1994, 30(2): 61–64 (in Chinese) [3] Kanninen M F, Popelar C H, Advanced Fracture Mechanics. Cambridge: Cambridge University Press, 1985; Fan T Y. Foundation of Fracture Theory. Beijing: Science Press, 2003 (in Chinese) [4] Fan T Y. Exact analytic solutions of stationary and fast propagating cracks in a strip. Science in China, A, 1991, 34(5): 560–569 [5] Fan T Y. Mathematical Theory of Elasticity of Quasicrystals and Its Application. Beijing Institute of Technology Press. Beijing, 1999 (in Chinese) [6] Li L H, Fan T Y. Exact solutions of two semi-inﬁnite collinear cracks in a strip of one-dimensional hexagonal quasicrystal. Applied Mathematics and Computation, 2008, 196(1): 1–5 [7] Shen D W, Fan T Y. Exact solutions of two semi-inﬁnite collinear cracks in a strip. Eng Fracture Mech, 2003, 70(8): 813–822 [8] Li X F, Fan T Y, Sun Y F. A decagonal quasicrystal with a Griﬃth crack. Phil Mag A, 1999, 79(8): 1943–1952 [9] Li L H, Fan T Y. Complex function method for solving notch problem of point group 10, 10 two-dimensional quasicrystals based on the stress potential function. J Phys: Condens Matter, 2007, 18(47): 10631–10641 [10] Li X F. Defect problems and their analytic solutions of the theory of elasticity of quasicrystals. Dissertation. Beijing Institute of Technology, 1999 (in Chinese) [11] Zhou W M, Fan T Y. Plane elasticity problem of two-dimensional octagonal quasicrystal and crack problem. Chin Phys, 2001, 10(8): 743–747 [12] Zhou W M. Mathematical analysis of elasticity and defects of two- and three-dimensional quasicrystals . Dissertation. Beijing: Beijing Institute of Technology, 2000 (in Chinese) [13] Li L H. Study on complex variable function method and analytic solutions of elasticity of quasicrystals. Dissertation. Beijing: Beijing Institute of Technology, 2008 (in Chinese)

References

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[14] Fan T Y, Guo R P. Three-dimensional elliptic crack in one-dimensional hexagonal quasicrystals. Appl Math Mech, submitted, 2009 [15] Peng Y Z, Fan T Y. Elastic theory of 1D quasiperiodic stacking of 2D crystals. J Phys: Condens Matter, 2000, 12(45): 9381–9387 [16] Liu G T, Fan T Y. The complex method of the plane elasticity in 2D quasicrystals point group 10mm ten-fold rotation symmetry notch problems. Science in China E, 2003, 46(3): 326–336 [17] Liu G T. The complex variable function method of the elastic theory of quasicrystals and defects and auxiliary equation method for solving some nonlinear evolution equations. Dissertation. Beijing Institute of Technology, 2004 (in Chinese) [18] Muskhelishrili N I, Some Basic Problems of the Mathematical Theory of Elasticity, Groningen: Noordhoﬀ ltd, 1953

Chapter 9 Theory of elasticity of three-dimensional quasicrystals and its applications In Chapters 5∼8 we discussed the theories of elasticity of one- and two-dimensional quasicrystals and their applications. In this chapter the theory and applications of elasticity of three-dimensional quasicrystals will be dealt with. The three-dimensional quasicrystals include icosahedral quasicrystals and cubic quasicrystals. In all about 200 individual quasicrystals observed to date there are almost 100 icosahedral quasicrystals, so that they play the central role in this kind of solids. This suggests the major importance of elasticity of icosahedral quasicrystals in the study of mechanical behaviour of quasicrystalline material. There are some polyhedrons with the icosahedral symmetry, one among them is shown in Fig. 9.0-1, which consists of 20 right triangles and contains 12 ﬁve-fold symmetric axes A5, 20 three-fold symmetric axes A3 and 30 two-fold symmetric axes A2. One of the diﬀraction pattern is shown in Fig. 3.1-1 and the stereographic structure of one of icosahedral point groups is also depicted in Fig. 3.1-1. The elasticity of icosahedral quasicrystals was studied immediately after the discovery of the structure, which is the pioneering work of the ﬁeld. The outlook about this was ﬁgured out in the Chapter 4, in which the contribution of pioneers such as P. Bak etc was introduced. Afterward Ding et al[1] set up the physical framework of elasticity of icosahedral quasicrystals, they[2] Fig. 9.0-1 Icosahedral quasicrystal also summarized the basic relationship of elasticity of cubic quasicrystals. In terms of the Green function method, Yang et al[3] gave an approximate solution on dislocation for a special case, i.e., the phonon-phason decoupled plane elasticity of icosahedarl quasicrystal. In this chapter we mainly discuss the general theory of elasticity of icosahedral quasicrystals and the applications, in addition, those for cubic quasicrystals are also concerned. We focus on the T. Fan, Mathematical Theory of Elasticity of Quasicrystals and Its Applications © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2011

160

Chapter 9

Theory of elasticity of three-dimensional quasicrystals and its...

mathematical theory of the elasticity and the analytic solutions. Because of the large number of ﬁeld variables and ﬁeld equations involving elasticity of these two kinds of three-dimensional quasicrystals, the solution presents tremendous diﬃculty. We continue to develop the decomposition procedure adopted in the previous chapters, this can reduce the number of the ﬁeld variables and ﬁeld equations, and threedimensional elasticity can be simpliﬁed to two-dimensional elasticity to solve for some cases with important practical applications. The introducing of displacement potentials or stress potentials[4,5] can further simplify the problems. In the work some systematic and direct methods of mathematical physics and function theory have been developed, and a series of analytic solutions are constructed, which will be included in the chapter. Because the calculations are very complex, we would like to introduce them in detail as much as possible in order to facilitate comprehension of the text.

9.1

Basic equations of elasticity of icosahedral quasicrystals

The equations of deformation geometry are 1 ∂ui ∂uj + εij = , 2 ∂xj ∂xi

wij =

∂wi , ∂xj

(9.1-1)

which are similar in form to those given in previous chapters, but here ui and wi have 6 components, and εij and wij have 15 components in total. The equilibrium equations are as follows: ∂σij = 0, ∂xj

∂Hij = 0, ∂xj

(9.1-2)

which are also similar in form to those listed in previous chapters, however here adding σij and Hij gives 15 stress components. Between the stresses and strains there is the generalized Hooke’s law such as σij = Cijkl εkl + Rijkl wkl ,

Hij = Rklij εkl + Kijkl wkl ,

(9.1-3)

in which the phonon elastic constants are described by Cijkl = λδij δkl + μ(δik δjl + δil δjk ),

(9.1-4)

where λ and μ (= G in some references) the Lam´e constants. If the strain components are arranged as a vector according to the order [εij , wij ] = [ε11 ε22 ε33 ε23 ε31 ε12 w11 w22 w33 w23 w32 w12 w32 w13 w21 ]

(9.1-5 )

9.1

Basic equations of elasticity of icosahedral quasicrystals

161

and the stress components are also arranged according to the same order, i.e., [σij , Hij ] = [σ11 σ22 σ33 σ23 σ31 σ12 H11 H22 H33 H23 H12 H32 H13 H21 ]

(9.1-5 )

then phason and phonon-phason coupling elastic constants can be expressed by matrices of [K] and [R], ⎡

K1

0

0

0

K2

0

0

K2

0

⎤

⎥ ⎢ 0 K 0 0 −K2 0 0 K2 0 ⎥ ⎢ 1 ⎥ ⎢ ⎢ 0 0 0 0 0 0 0 ⎥ 0 K2 + K1 ⎥ ⎢ ⎥ ⎢ 0 K2 0 0 −K2 ⎥ 0 0 K1 − K 2 ⎢ 0 ⎥ ⎢ 0 0 K 1 − K2 0 0 0 0 ⎥ [K] = ⎢ ⎥, ⎢ K2 −K2 ⎥ ⎢ 0 K −K 0 0 0 0 0 K ⎥ ⎢ 2 1 2 ⎥ ⎢ ⎥ ⎢ 0 0 0 0 0 −K K − K 0 −K 2 1 2 2⎥ ⎢ ⎥ ⎢ 0 0 0 0 0 K 1 − K2 0 ⎦ ⎣ K2 K2 0 0 0 −K2 0 0 −K2 0 K1 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ [R] = R ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

1

1

−1 −1

1

0

0

0

0

1

1

0

0

0

0 −1

0

0

0

−2

0

0

0

0

0

0

0

0

0 −1 1

0

1

−1

0

0

1

0

0

0

0

0

−1 0 −1 0

0

0

0

0

0

0

0 −1 1

0

1

−1

0

0

1

0

0

0

0

0

−1 0 −1 0

0

0

0

⎤

0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ −1 ⎥ ⎥ 0 ⎥ ⎥. ⎥ 1 ⎥ ⎥ −1 ⎥ ⎥ ⎥ 0 ⎦ 1

(9.1-6)

Equations (9.1-1)∼(9.1-3) are basic equations of elasticity of icosahedral quasicrystals, there are 36 equations in total, and the number of the ﬁeld variables is also 36. It is consistent and solvable mathematically. Due to the huge number of the ﬁeld variables and ﬁeld equations, the mathematical solution is highly complex. One way to solve the elasticity problem is to reduce the number of the ﬁeld variables and ﬁeld equations above mentioned. For this purpose we can utilize the eliminating element method in classical mathematical physics. Based on the matrix expression of the generalized Hooke’s law of (4.5-3) in the Chapter 4, then one has the explicit relationship between stresses and strains as below:

162

Chapter 9

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

Theory of elasticity of three-dimensional quasicrystals and its...

σxx = λθ + 2μεxx + R(wxx + wyy + wzz + wxz ), σyy = λθ + 2μεyy − R(wxx + wyy − wzz + wxz ), σzz = λθ + 2μεyy − 2Rwzz , σyz = 2μεyz + R(wzy − wxy − wyx ) = σzy , σzx = 2μεzx + R(wxx − wyy − wzx ) = σxz , σxy = 2μεxy + R(wyx − wyz − wxy ) = σyx , Hxx = R(εxx − εyy + 2εzx ) + K1 wxx + K2 (wzx + wxz ), Hyy = R(εxx − εyy − 2εzx ) + K1 wyy + K2 (wxz − wzx ), Hzz = R(εxx + εyy − 2εzz ) + (K1 + K2 )wzz , Hyz = −2Rεxy + (K1 − K2 )wyz + K2 (wxy − wyx ), Hzx = 2Rεzx + (K1 − K2 )wzx + K2 (wxx − wyy ), Hxy = −2R(εyz + εxy ) + K1 wxy + K2 (wyz − wzy ), Hzy = 2Rεyz + (K1 − K2 )wzy − K2 (wxy + wyx ), Hxz = R(εxx − εyy ) + K2 (wxx + wyy ) + (K1 − K2 )wxz , Hyx = 2R(εxy − εyx ) + K1 wyx − K2 (wyz + wzy ),

(9.1-7)

where θ = εxx + εyy + εzz denotes the volume strain and εij and wij are deﬁned by (9.1-1). This explicit expression was ﬁrst given by Ding et al[1] . Substituting (9.1-7) into (9.1-2) yields one of forms of the ﬁnal governing equations— the ⎧ equilibrium equations in terms of displacements as follows: ∂ ⎪ ⎪ μ∇2 ux + (λ + μ) ∇ · u ⎪ ⎪ ⎪ ⎪ 2 ∂x 2 ⎪ ⎪ ∂ 2 wy ∂ wx ∂ wx ∂ 2 wx ∂ 2 wy ∂ 2 wz ⎪ ⎪ ⎪ + + 2 − − 2 + 2 = 0, + R ⎪ ⎪ ∂x2 ∂x∂z ∂y 2 ∂x∂y ∂y∂z ∂x∂z ⎪ ⎪ ⎪ ⎪ ∂ ⎪ ⎪ μ∇2 uy + (λ + μ) ∇ · u ⎪ ⎪ ∂y ⎪ ⎪ 2 ⎪ ⎪ ∂ 2 wx ∂ 2 wx ∂ wy ∂ 2 wy ∂ 2 wy ∂ 2 wz ⎪ ⎪ − −2 − −2 +2 = 0, +R ⎪ ⎪ ⎪ ∂x2 ∂x∂y ∂y 2 ∂z∂y ∂x∂z ∂y∂z ⎪ ⎪ ⎪ ∂ ⎪ ⎪ ⎪ μ∇2 uz + (λ + μ) ∇ · u ⎪ ⎪ ∂z ⎪ 2 ⎪ ⎪ ⎪ ∂ 2 wx ∂ 2 wz ∂ 2 wz ∂ wx ∂ 2 wx ∂ 2 wz ⎪ ⎪ − 2 + + − 2 − = 0, + R ⎪ 2 ⎨ ∂x ∂x∂y ∂y 2 ∂x2 ∂y 2 ∂z 2 (9.1-8) ∂ 2 wy ∂ 2 wz ∂ 2 wz ∂ 2 wx ∂ 2 wx ⎪ ⎪ +2 − − + K1 ∇2 wx + K2 2 ⎪ 2 2 2 ⎪ ∂x∂z ∂z ∂y∂z ∂x ∂y ⎪ ⎪ 2 ⎪ 2 2 2 2 2 ⎪ u u u u u uz ∂ 2 uz ∂ ∂ ∂ ∂ ∂ ∂ ⎪ x x x y y ⎪ +R +2 −2 − − −2 + = 0, ⎪ ⎪ ⎪ ∂x2 ∂x∂z ∂y 2 ∂x∂y ∂y∂z ∂x2 ∂y 2 ⎪ ⎪ ⎪ ⎪ ∂ 2 wy ∂ 2 wz ∂ 2 wy ∂ 2 wx ⎪ ⎪ −2 −2 − K1 ∇2 wy + K2 2 ⎪ ⎪ ∂y∂z ∂x∂z ∂x∂y ∂z 2 ⎪ ⎪ 2 ⎪ 2 2 2 ⎪ ⎪ ∂ uy ∂ ux ∂ ux ∂ uy ∂ 2 uy ∂ 2 uz ⎪ ⎪ +R −2 +2 + −2 −2 = 0, ⎪ ⎪ ∂x2 ∂y∂z ∂y 2 ∂x∂y ∂x∂z ∂x∂y ⎪ ⎪ ⎪ ⎪ ∂ 2 wy ∂ 2 wx ∂ 2 wx ∂ 2 wz ⎪ ⎪ ⎪ −2 − (K1 − K2 )∇2 wz + K2 2 + ⎪ 2 ⎪ ∂z ∂y∂x ∂x2 ∂y 2 ⎪ ⎪ ⎪ 2 2 2 2 ⎪ ∂ uz ∂ ux ∂ uz ∂ uz ⎪ ⎪ + = 0, +2 − +R ⎩ ∂x2 ∂x∂z ∂y 2 ∂z 2

9.1

Basic equations of elasticity of icosahedral quasicrystals

163

where

∂uz ∂2 ∂2 ∂2 ∂ux ∂uy + + . + + , ∇·u= ∂x2 ∂y2 ∂z 2 ∂x ∂y ∂z Equations (9.1-8) are 6 partial diﬀerential equations of second order on displacements ui and wi . So the number of the ﬁeld variables and ﬁeld equations is reduced already. But obtaining solution is still very diﬃcult, one of reasons is the boundary conditions for quasicrystals being much more complicated than those of the classical theory of elasticity. In the subsequent sections we will make a great eﬀort to solve some complex boundary value problems through diﬀerent approaches. It is obvious that the material constants of λ, μ, K1 , K2 and R are very important for stress analysis for diﬀerent icosahedral quasicrystals, which are experimentally measured through various methods (e.g. X-ray diﬀraction, neutron scattering etc) and listed by Tables 9.1-1∼9.1-3 respectively as follows: ∇2 =

Table 9.1-1

Phonon elastic constants of various icosahedral quasicrystals

Alloys Al-Li-Cu Al-Li-Cu Al-Cu-Fe Al-Cu-Fe-Ru Al-Pd-Mn Al-Pd-Mn Ti-Zr-Ni Cu-Yh Zn-Mg-Y

λ 30 30.4 59.1 48.4 74.9 74.2 85.5 35.28 33.0

μ(G) 35 40.9 68.1 57.9 72.4 70.4 38.3 25.28 46.5

B 53 57.7 104 87.0 123 121 111 52.13 64.0

ν 0.23 0.213 0.213 0.228 0.254 0.256 0.345 0.2913 0.208

Refs. [6] [7] [8] [8] [8] [9] [10] [11] [12]

In this table the measurement unit of λ, μ and B is GPa, and B = (3λ + 2μ)/3 represents the bulk modulus, and ν = λ/2(λ + μ) the Poisson’s ratio, respectively. Table 9.1-2 Alloys Al-Pd-Mn Al-Pd-Mn Al-Pd-Mn Zn-Mg-Sc

Table 9.1-3

Phason elastic constants of various icosahedral quasicrystals Source X-ray Neutron Neutron X-ray

Meas. Temp. R.T. R.T. 1043K R.T.

K1 /MPa 43 72 125 300

K2 /MPa −22 −37 −50 −45

Refs. [13] [13] [13] [14]

Phonon-phason coupling elastic constant of various icosahedral quasicrystals

Alloys Mg-Ga-Al-Zn Al-Cu-Fe

Source X-ray X-ray

R −0.04μ 0.004μ

Refs. [15] [15]

It is needed to point out that equations (9.1-8) are not the only form of ﬁnal governing equation of elasticity of icosahedral quasicrystals, there are other forms which will be discussed in Section 9.5.

164

Chapter 9

9.2

Theory of elasticity of three-dimensional quasicrystals and its...

Anti-plane elasticity of icosahedral quasicrystals and problem of interface between quasicrystal and crystal

People can ﬁnd equations (9.1-8) very complex, but they can be simpliﬁed for some meaningful cases physically. One of them is the so-called anti-plane case where the non-zero displacements are only uz and wz , and the other displacements vanish. In particular these two displacements and relevant strains and stresses are independent of the coordinate x3 (or z). If there is a Griﬃth crack along the axis z (see Fig. 9.2-1) or a straight dislocation line along the direction, etc., in addition, the applied external ﬁelds are independent of variable z, so ∂ ∂ = 0. (9.2-1) = ∂x3 ∂z

Fig. 9.2-1

One of conﬁguration of plane or anti-plane elasticity

Because there are only two components uz and wz , and others have vanished, the corresponding strains are only εyz = εzy =

1 ∂uz , 2 ∂y

εxz = εzx =

1 ∂uz , 2 ∂x

wzy =

∂wz , ∂y

wzx =

∂wz . ∂x

(9.2-2)

From the formulas listed in Section 9.1, the non-zero stress components are ⎧ σxz = σzx = 2μεxz + Rwzx , ⎪ ⎪ ⎪ ⎪ σ ⎪ yz = σzy = 2μεyz + Rwzy , ⎪ ⎪ ⎪ H ⎪ zx = (K1 − K2 )wzx + 2Rεxz , ⎪ ⎨ Hzy = (K1 − K2 )wzy + 2Rεyz , (9.2-3) H ⎪ xx = 2Rεxz + K2 wzx , ⎪ ⎪ ⎪ Hyy = −2Rεxz − K2 wzx , ⎪ ⎪ ⎪ ⎪ Hxy = −2Rεyz − K2 wzy , ⎪ ⎪ ⎩ Hyx = −2Rεyz − K2 wzy

9.2

Anti-plane elasticity of icosahedral quasicrystals and problem of interface...

and the equilibrium equations stand for ⎧ ∂σzx ∂σzy ⎪ ⎪ + = 0, ⎨ ∂x ∂y ∂Hxx ∂Hxy ⎪ ⎪ ⎩ + = 0, ∂x ∂y

∂Hzx ∂Hzy + = 0, ∂x ∂y ∂Hyx ∂Hyy + = 0. ∂x ∂y

165

(9.2-4)

Problem described by equations (9.2-2)∼(9.2-4) is anti-plane elasticity problem, and we have the ﬁnal governing equations ∇21 uz = 0,

∇21 wz = 0,

(9.2-5)

∂2 ∂2 + . ∂x2 ∂y 2 One can see that equations (9.2-5) are similar to (5.2-11), which can be solved using a procedure similar to that adopted in Chapters 5, 7 and 8. As an example of solution of anti-plane elasticity of icosahedral quasicrystals we discuss the interface problem between centre-body cubic crystals and icosahedral quasicrystals. The physical model is similar to that proposed in Section 7.6, i.e., the icosahedral quasicrystal is located in upper half-space y > 0, whose governing equations are listed above, while the centre-body cubic crystal lies in lower space y < 0 with ﬁnite thickness h (refer to Fig. 7.6-1) and governed by the following equation: where ∇21 =

∇2 u(c) z = 0,

(9.2-6)

with the following stress-strain relations (c) (c) = σyz = 2μ(c) ε(c) σzy yz , (c)

(c)

(c) (c) σzx = σxz = 2μ(c) ε(c) xz (c)

(c)

for crystals, in which εij = (∂ui /∂xj + ∂uj /∂xi )/2, μ(c) = C44 . After the Fourier transform, the solution of (9.2-5) is very easy to obtain such as ∞ ∞ 1 1 A(ξ)e−|ξ|y−iξx dξ, wz (x, y) = B(ξ)e−|ξ|y−iξx dξ uz (x, y) = 2π −∞ 2π −∞ (9.2-7) and the relevant stresses, e.g. ∞ ∞ 1 1 −|ξ|y−iξx σzy = −2μ |ξ|A(ξ)e dξ − R |ξ|B(ξ)e−|ξ|y−iξx dξ, 2π −∞ 2π −∞ ∞ ∞ 1 1 −|ξ|y−iξx Hzy = −R |ξ|A(ξ)e dξ − (K1 − K2 ) |ξ|B(ξ)e−|ξ|y−iξx dξ, 2π −∞ 2π −∞ (9.2-8) in which y > 0 and A(ξ) and B(ξ) are arbitrary functions to be determined.

166

Chapter 9

Theory of elasticity of three-dimensional quasicrystals and its...

According to the second condition of the boundary conditions at the interface y = 0, −∞ < x < ∞ :

σzy = τ f (x) + ku(x),

Hzy = 0,

(9.2-9)

where k=

μ(c) , h

(9.2-10)

then the relation between the two unknown functions is obtained B(ξ) = −

R A(ξ). K1 − K 2

(9.2-11)

From (9.2-7) and (9.2-8), we have

σ ˆ zy

R2 = − 2μ + K1 − K2

A(ξ) |ξ|

and from the ﬁrst one of conditions (9.2-9), one determines the unknown function A(ξ) = −

τ fˆ(ξ)

R + μ |ξ| + k K1 − K 2 2

(9.2-12)

so that

B(ξ) = (K1 − K2 )

Rτ fˆ(ξ) , R2 + μ |ξ| + k K1 − K 2

(9.2-13)

in which τ and k is deﬁned by (9.2-9) and (9.2-10) respectively. Thus the problem is solved. The phason strain ﬁeld can be determined as ⎧ ∞ |ξ| fˆ(ξ) 1 ⎪ ⎪ e−|ξ|y−iξx dξ, w (x, y) = −Rτ ⎪ zy ⎨ 2 2π −∞ (R − μ(K1 − K2 )) |ξ| + k(K1 − K2 ) ∞ ⎪ ⎪ ξ fˆ(ξ) 1 ⎪ ⎩ wzx (x, y) = iRτ e−|ξ|y−iξx dξ. 2 2π −∞ (R − μ(K1 − K2 )) |ξ| + k(K1 − K2 ) (9.2-14) Note that y > 0. The integrals in (9.2-14) can be evaluated by the residue theorem introduced in the Major Appendix of this book. For illustration, consider the ﬁrst example. Let that f (x) = 1, as −a/2 < x < a 2 a/2, and f (x) = 0, as x < −a/2 and x > a/2, so fˆ(ξ) = sin ξ , then from ξ 2 solution (9.2-14), we obtain

9.2

Anti-plane elasticity of icosahedral quasicrystals and problem of interface...

167

⎧ 1 μ(c) (K1 − K2 ) a a μ(c) R(K1 − K2 )τ ⎪ ⎪ (x, y) = sin w ⎪ zy ⎪ h [μ(K1 − K2 ) − R2 ]2 2 μ(K1 − K2 ) − R2 h ⎪ ⎪ ⎪ ⎪ (c) ⎪ ⎪ μ (K1 − K2 ) x μ(c) (K1 − K2 ) y ⎪ ⎪ cos × exp − ⎪ ⎨ μ(K1 − K2 ) − R2 h μ(K1 − K2 ) − R2 h ⎪ 1 μ(c) (K1 − K2 ) a a μ(c) R(K1 − K2 )τ ⎪ ⎪ ⎪ w (x, y) = sin zx ⎪ ⎪ h [μ(K1 − K2 ) − R2 ]2 2 μ(K1 − K2 ) − R2 h ⎪ ⎪ ⎪ (c) ⎪ ⎪ ⎪ μ (K1 − K2 ) x μ(c) (K1 − K2 ) y ⎪ ⎩ sin × exp − μ(K1 − K2 ) − R2 h μ(K1 − K2 ) − R2 h (9.2-15) (c) in which k = μ /h and the normalized expression has been used, i.e., x/h, y/h. Then consider the second example f (x) = δ(x), then the integrals (9.2-14) will be ⎧ ⎪ μ(c) R(K1 − K2 )τ ⎪ ⎪ (x, y) = w zy ⎪ ⎪ [μ(K1 − K2 ) − R2 ]2 ⎪ ⎪ ⎪ ⎪ (c) ⎪ ⎪ ⎪ μ(c) (K1 − K2 ) y μ (K1 − K2 ) x ⎪ ⎪ × exp − sin ⎪ ⎨ μ(K1 − K2 ) − R2 h μ(K1 − K2 ) − R2 h (9.2-16) ⎪ (c) ⎪ R(K1 − K2 )τ μ ⎪ ⎪ wzx (x, y) = ⎪ ⎪ [μ(K1 − K2 ) − R2 ]2 ⎪ ⎪ ⎪ ⎪ ⎪ (c) ⎪ ⎪ μ(c) (K1 − K2 ) y μ (K1 − K2 ) x ⎪ ⎪ × exp − cos ⎩ μ(K1 − K2 ) − R2 h μ(K1 − K2 ) − R2 h The detail of the evaluation is given in the Major Appendix of the book. The results are quite interesting. In the ﬁrst example, the phason strain ﬁeld is dominated by the elastic constants μ, K1 , K2 , R and μ(c) of quasicrystal and crystal, applied stress τ and geometry parameters a and h, while in the second example the geometry parameter is only h. For diﬀerent τ /μ, μ(c) /μ,a/h and given values of μ, K1 , K2 and R, one can ﬁnd a rich set of numerical results. The computation used the measured values of these quantities for Al-Pd-Mn icosahedral quasicrystals are provided by Tables 9.1-1, 9.1-2 and 9.1-3: μ = 72.4GPa, K1 = 125MPa, K2 = −50MPa, R = 0.04μ The numerical results show that the inﬂuence of the ratio μ(c) /μ of shear modulus of crystal and quasicrystal is very evident. In addition the inﬂuence of the applied stress τ /μ is also very important. While the inﬂuence of a/h is not evident for the ﬁrst example. Another feature of solution here is quite diﬀerent from that in Section 7.6 due to the diﬀerence of quasicrytalline systems. This work is given in the Ref. [24].

168

9.3

Chapter 9

Theory of elasticity of three-dimensional quasicrystals and its...

Phonon-phason decoupled plane elasticity of icosahedral quasicrystals

Yang et al[3] presented an approximate solution of a straight dislocation in icosahedral quasicrystals under the assumptions ∂ =0 ∂z

(9.3-1)

R=0

(9.3-2)

and

The conditions (9.3-1) and (9.3-2) result in a phonon-phason decoupled plane elasticity, in which there are εzz = 0,

wzz = wyz = wxz = 0.

(9.3-3)

Based on conditions (9.3-1) and (9.3-2), the ﬁnal governing equations (9.1-8) reduce to ⎧ ∂ ⎪ ⎪ μ∇21 ux + (λ + μ) ∇1 · u1 = 0, ⎪ ⎪ ∂x ⎪ ⎪ ⎪ ⎪ ⎪ ∂ ⎪ ⎪ ⎪ μ∇21 uy + (λ + μ) ∇1 · u1 = 0, ⎪ ⎪ ∂y ⎪ ⎪ ⎪ 2 ⎪ μ∇ u = 0, ⎪ 1 z ⎨ 2 ∂ 2 wz ∂ wz 2 (9.3-4) ⎪ K1 ∇1 wx + K2 ∂x2 − ∂y 2 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ 2 wz ⎪ ⎪ ⎪ = 0, K1 ∇21 wy − 2K2 ⎪ ⎪ ∂x∂y ⎪ ⎪ ⎪ 2 ⎪ ⎪ ∂ 2 wy ∂ wx ∂ 2 wy ⎪ 2 ⎪ −2 − = 0, ⎩ (K1 − K2 )∇1 wz + K2 ∂x2 ∂x∂y ∂y 2 where ∇21 =

∂2 ∂2 + 2, 2 ∂x ∂y

u1 = (ux , uy ),

∇1 · u1 =

∂ux ∂uy + . ∂x ∂y

Because the phonons and phasons are decoupled, the ﬁrst three equations of (9.34) are pure phonon equilibrium equations, in addition uz is independent of ux and uy , and the second three equations in (9.3-4) are pure phason equilibrium equations. Yang et al[3] solved the equations under the dislocation conditions dui = bi , dwi = b⊥ (9.3-5) i , Γ

Γ

9.4

Phonon-phason coupled plane elasticity of icosahedral quasicrystals...

169

where Γ represents a path enclosing the dislocation core. The authors used the Green function method to calculate. The results are ⎧ ⎪ b1 λ + μ x2 μ y λ + μ xy r b2 ⎪ ⎪ + arctan + ln + , ux = ⎪ ⎪ 2π x λ + 2μ r2 2π λ + 2μ r0 λ + 2μ r2 ⎪ ⎪ ⎪ ⎪ ⎪ b λ + μ y2 μ r y λ + μ xy b2 ⎪ ⎪ ⎪ uy = − 1 + ln arctan − + , ⎪ 2 ⎪ 2π λ + 2μ r0 λ + 2μ r 2π x λ + 2μ r2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ uz = b3 arctan y , ⎪ ⎪ ⎪ 2π x ⎪ ⎨ b⊥ xy y K 2 2xy3 1 − arctan + 2 wx = ⎪ 2π x 2K5 r4 r2 ⎪ ⎪ ⎪ ⎪ ⎪ b⊥ K 2 2x2 y 2 r b⊥ K2 xy ⎪ ⎪ − 2 2 ln + , + 3 ⎪ 4 ⎪ 4π K5 r0 r 2π K1 r2 ⎪ ⎪ ⊥ ⊥ ⎪ ⎪ ⎪ b⊥ K 2 y K22 xy 3 − x3 y r 2x2 y 2 b2 b3 K2 y 2 ⎪ ⎪ wy = 1 2 ln − , arctan + + − ⎪ ⎪ 4π K5 r0 r4 2π x 2K5 r4 2π K1 r2 ⎪ ⎪ ⎪ ⎪ 2 ⎪ K xy b⊥ b⊥ b⊥ y ⎪ 2 K1 K2 y 3 ⎩ wz = 1 K1 2 − + arctan , 2π K5 r2 2π K5 r2 2π x (9.3-6) in which K5 = K12 − K1 K2 − K22 . (9.3-7) The ﬁrst three of (9.3-6) are well known solution of pure phonon ﬁeld in the classical theory of dislocation, and the second three of (9.3-6) are new results for pure phason ﬁeld. Because Yang et al[3] ignored the coupling terms, the interaction between phonons and phasons could not be revealed.

9.4

Phonon-phason coupled plane elasticity of icosahedral quasicrystals—displacement potential formulation

In the previous section, Yang et al[3] introduced the assumption (9.3-2), which is not valid, and leads to loss a lot of information coming from the phonon-phason coupling. In studies of Fan and Guo[4] , Zhu and Fan[16] , Zhu, Fan and Guo[17] , and Li and Fan[5] considered the coupling eﬀects, i.e., R = 0 and obtained the complete theory for the plane elasticity of quasicrystals. In the study the assumption (9.2-1) or (9.3-1) still maintains, i.e., ∂ = 0. ∂z

(9.4-1)

In this case the three-dimensional elasticity can be reduced into a plane elasticity problem. From condition (9.3-1) directly we have

170

Chapter 9

Theory of elasticity of three-dimensional quasicrystals and its...

εzz = wzz = wxz = wyz = 0.

(9.4-2)

Thus the number of the ﬁeld variables and ﬁeld equations are reduced from 36 to 32. Though the reduction of the total number is not so much, the resulting equation system has been greatly simpliﬁed and with the following form: 2 ⎧ ∂ 2 wy ∂ wx ∂ ∂ 2 wy ⎪ 2 ⎪ ∇ − = 0, u + (λ + μ) · u + R + 2 μ∇ 1 1 ⎪ 1 x ⎪ ∂x ∂x2 ∂x∂y ∂y 2 ⎪ ⎪ ⎪ ⎪ ⎪ ∂ 2 wy ∂ ∂ 2 wx ∂ 2 wy ⎪ ⎪ μ∇21 uy + (λ + μ) ∇1 · u1 + R − 2 − = 0, ⎪ ⎪ ∂y ∂x2 ∂x∂y ∂y 2 ⎪ ⎪ ⎪ ⎪ ⎪ ∂ 2 wx ∂ 2 wy ∂ 2 wx ⎪ 2 ⎪ − 2 + ∇21 wz = 0, − ⎨ μ∇1 uz + R ∂x2 ∂x∂y ∂y 2 2 2 ⎪ ∂ 2 uy ∂ 2 ux ∂ 2 uz ∂ 2 uz ∂ wz ∂ 2 wz ∂ ux ⎪ 2 ⎪ ∇ w +K − −2 + − − +R = 0, K 1 1 x 2 ⎪ ⎪ ∂x2 ∂y 2 ∂x2 ∂x∂y ∂y 2 ∂x2 ∂y 2 ⎪ ⎪ ⎪ ⎪ ⎪ ∂ 2 wz ∂ 2 ux ∂ 2 uz ∂ 2 uy ∂ 2 uy ⎪ 2 ⎪ ∇ w − 2K + 2 − 2 + R − = 0, K 1 y 2 ⎪ 1 ⎪ ∂x∂y ∂x2 ∂x∂y ∂y 2 ∂x∂y ⎪ ⎪ ⎪ ⎪ ⎪ ∂ 2 wy ∂ 2 wx ∂ 2 wy ⎪ ⎩ (K1 − K2 )∇21 wz + K2 − 2 − + R∇21 uz = 0, ∂x2 ∂x∂y ∂y 2 (9.4-3) where ∇21 and ∇1 · u1 are the same of those in Section 9.3, but the suﬃx 1 of the two-dimensional Laplace operator will be omitted in the following for simplicity. The equation set is much simpler than that of (9.1-8) but still quite complicated. If we introduce a displacement potential F (x, y) such as ⎧ ∂2 ⎪ ⎪ ∇2 ∇2 [μαΠ1 + β(λ + 2μ)Π2 ]F u = R x ⎪ ⎪ ∂x∂y ⎪ ⎪ ⎪ ⎪ ⎪ ∂4 ∂2 ∂4 ∂4 ⎪ ⎪ Λ (3μ − λ) F R + 10(λ + μ) − (5λ + 9μ) +c 0 ⎪ ⎪ ∂x∂y ∂x4 ∂x2 ∂y 2 ∂y 4 ⎪ ⎪ ⎪ ⎪ ⎪ ∂2 ∂2 ⎪ 2 2 ⎪ μα F u = R∇ ∇ Π − β(λ + 2μ) Π y 1 2 ⎪ ⎪ ∂y 2 ∂x2 ⎪ ⎪ ⎪ ⎪ ⎪ ∂6 ∂6 ∂6 ∂6 ⎪ 2 ⎪ +c0 RΛ (λ + 2μ) 6 − 5(2λ + 3μ) 4 2 + 5λ 2 4 + μ 6 F, ⎨ ∂x ∂x ∂y ∂x ∂y ∂y 2 2 2 ⎪ ∂ ∂ ∂ ⎪ ⎪ (α − β)Λ2 Π1 Π2 + α 2 Π12 + β 2 Π22 F, uz = c1 ⎪ ⎪ ∂x∂y ∂y ∂x ⎪ ⎪ ⎪ ⎪ ⎪ ∂2 ⎪ ⎪ w = −ω ∇2 2c0 Λ2 ∇2 − (α − β)Π1 Π2 F x ⎪ ⎪ ∂x∂y ⎪ ⎪ ⎪ ⎪ ⎪ ∂2 2 ∂2 2 ⎪ 2 2 2 2 ⎪ wy = −ω∇ c0 Λ Λ ∇ + α 2 Π1 + β 2 Π2 F, ⎪ ⎪ ∂y ∂x ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ ∂ ∂ ∂2 ⎪ ⎩ wz = c2 (α − β)Λ2 Π1 Π2 + α 2 Π12 + β 2 Π22 F, ∂x∂y ∂y ∂x (9.4-4)

9.4

Phonon-phason coupled plane elasticity of icosahedral quasicrystals...

171

then the ﬁeld equations mentioned above will be satisﬁed if ∇2 ∇2 ∇2 ∇2 ∇2 ∇2 F (x, y) + ∇2 LF (x, y) = 0,

(9.4-5)

where ⎧ α = (λ + 2μ)R2 − ωK1 , β = μR2 − ωK1 , ω = μ(λ + 2μ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ μK22 +(K1 −3K2 )R2 (K1 −2K2 )Rω (K2 μ−R2 )ω ⎨ c0 = ω , c1 = , c2 = , 2 2 μ(K1 −K2 )−R μ(K1 −K2 )−R μ(K1 −K2 )−R2 ⎪ ⎪ ⎪ ⎪ ∂2 ∂2 ∂2 ∂2 ∂2 ∂2 ∂2 ∂2 ⎪ ⎪ ⎩ Π1 = 3 2 − 2 , Π2 = 3 2 − 2 , ∇2 = 2 + 2 , Λ2 = 2 − 2 , ∂x ∂y ∂y ∂x ∂x ∂y ∂x ∂y (9.4-6) in which the suﬃx 1 of the two-dimensional Laplace operator is omitted and α α ∂ 10 ∂ 10 ∂ 10 c0 − 10 11 − 10 − 10 + 5 4 − 5 L= 8 2 β ∂x β ∂x ∂y β ∂x6 ∂y4 ∂ 10 ∂ 10 α α α ∂ 10 . (9.4-7) + 10 10 − 11 − 5 5 − 4 − β ∂x4 ∂y6 β ∂x2 ∂y8 β ∂y 10 Assuming

R2 1 (9.4-8) μK1 (this is understandable, because the coupling eﬀect is weaker than that of phonon), then from equations (9.4-6) and (9.4-7), β → 1, α

∇2 L =

c0 2 2 2 2 2 2 ∇ ∇ ∇ ∇ ∇ ∇ , β

(9.4-9)

substituting (9.4-9) into (9.4-5), we ﬁnd that ∇2 ∇2 ∇2 ∇2 ∇2 ∇2 F (x, y) = 0.

(9.4-10)

This sextuple harmonic equation is the governing equation for plane elasticity of quasicrystals based on the displacement potential formulation. With the aid of the generalized Hooke’s law, the phonon and phason stress components can also be expressed in terms of potential function F (x, y) and these expressions are omitted here due to the limitation of the space. In other words, equation set (9.4-4) gives a fundamental solution in terms of F (x, y) for the plane elasticity problem of an icosahedral quasicrystal. Once the function F (x, y) satisfying equation (9.4-4) is determined for prescribed boundary conditions, the entire elastic ﬁeld of an icosahedral quasicrystal can be found from (9.4-4). This formulation has been reported brieﬂy by Fan and Guo[4] . In some extent it is a development of Li and Fan[18] for elasticity of two-dimensional quasicrystals. The application of the formulation and relevant solution will be given in Section 6.

172

Chapter 9

9.5

Theory of elasticity of three-dimensional quasicrystals and its...

Phonon-phason coupled plane elasticity of icosahedral quasicrystals—stress potential formulation

In the previous section the displacement formulation exhibits its eﬀect, which reduces a very complicated partial diﬀerential equation set into a single partial diﬀerential equation with higher order (12th order), the latter will be easily to solve. At the meantime the stress potential formulation is eﬀective too. In this section we will introduce the formulation. To contrast the displacement potential formulation we now retain the stress components and eliminate the displacement components. From the deformation geometry equations (9.1-1) and considering (9.3-1) and (9.3-2) we obtain the deformation compatibility equations as follows: ⎧ 2 ∂εyz ∂εzx ∂ εxx ∂ 2 εyy ∂ 2 εxy ⎪ ⎪ , = , + = 2 ⎨ ∂y 2 2 ∂x ∂x∂y ∂x ∂y ⎪ ∂wxx ∂wyy ∂wyx ∂wzy ∂wzx ⎪ ∂wxy ⎩ = , = , = . ∂x ∂y ∂x ∂y ∂x ∂y

(9.5-1)

Thus the displacements are eliminated already. By combining (9.1-7) and (9.5-1) we obtain the deformation compatibility equations expressed by stress components, so that the strain components have been eliminated up to now ( those equations are too lengthy we here do not list them). So far one has the deformation compatibility equations expressed by stresses and equilibrium equations only. If we introduce stress potential functions ϕ1 (x, y), such as

with

ϕ2 (x, y),

ψ1 (x, y),

ψ2 (x, y),

ψ3 (x, y)

⎧ ∂ 2 ϕ1 ∂ 2 ϕ1 ∂ 2 ϕ1 ⎪ ⎪ , σ = − = , σxx = , σ xy yy ⎪ ⎪ ∂y 2 ∂x∂y ∂x2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ϕ2 ∂ϕ2 ⎪ ⎪ ⎪ ⎨ σzx = ∂y , σzy = − ∂x , ⎪ ⎪ ∂ψ1 ∂ψ1 ∂ψ2 ⎪ ⎪ ⎪ Hxx = ∂y , Hxy = − ∂x , Hyx = ∂y , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Hyy = − ∂ψ2 , Hzx = ∂ψ3 , Hzy = − ∂ψ3 , ∂x ∂y ∂x

(9.5-2)

9.6

A straight dislocation in an icosahedral quasicrystal

173

⎧ 2 ∂ ∂ ⎪ 2 ⎪ = c c R Π − Λ Π ϕ 2 ∇2 ∇2 G, ⎪ 1 2 3 2 1 ⎪ ⎪ ∂y ∂x2 ⎪ ⎪ ⎪ ⎪ ⎪ ϕ2 = −c3 c4 ∇2 ∇2 ∇2 ∇2 ∇2 G ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎨ ∂2 ∂ 2 2 ψ1 = c1 c2 R 2 2 2 Π1 Π2 − Λ Π1 ∇2 G + c2 c4 Λ2 ∇2 ∇2 ∇2 ∇2 G, ∂y ∂x ⎪ ⎪ 2 ⎪ ⎪ 2 ⎪ ∂2 ∂ ∂ ⎪ 2 2 2 ⎪ = c c R Π − Λ Π Π ψ 2 ∇2 ∇2 ∇2 ∇2 G, ⎪ 2 1 2 1 2 ∇ G − 2c2 c4 2 2 ⎪ ∂x∂y ∂x ∂x∂y ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎩ ψ3 = − K2 c3 c4 ∇2 ∇2 ∇2 ∇2 ∇2 G, R (9.5-3) then the equilibrium equations and the deformation compatibility equations will be identically satisﬁed if ∇2 ∇2 ∇2 ∇2 ∇2 ∇2 G = 0 (9.5-4) under the approximation R2 /K1 μ 1, which is the ﬁnal governing equation of plane elasticity of icosahedral quasicrystals, function G(x, y) is named the stress potential ⎧ in which R(2K2 − K1 )(μK1 + μK2 − 3R2 ) ⎪ ⎪ = , c 1 ⎪ ⎪ 2(μK1 − 2R2 ) ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ c2 = K2 (μK2 − R2 ) − R(2K2 − K1 ), ⎪ ⎪ ⎪ R ⎪ ⎪ ⎪ ⎪ ⎪ (μK2 − R2 )2 ⎪ 2 ⎪ = μ(K − K ) − R − , c ⎪ 3 1 2 ⎨ μK1 − 2R2 (9.5-5) 2 ⎪ − 2R 1 μK 1 ⎪ ⎪ c4 = c1 R + c3 K1 + , ⎪ ⎪ ⎪ 2 λ+μ ⎪ ⎪ ⎪ ⎪ ⎪ ∂2 ∂2 ∂2 ∂2 ⎪ ⎪ Π1 = 3 2 − 2 , Π2 = 3 2 − 2 , ⎪ ⎪ ∂x ∂y ∂y ∂x ⎪ ⎪ ⎪ ⎪ ⎪ 2 2 2 2 ⎪ ⎪ ⎩ ∇2 = ∂ + ∂ , Λ2 = ∂ − ∂ ∂x2 ∂y2 ∂x2 ∂y2 In derivation of (9.5-4) the approximation (9.4-8) is used at the last step. This work is given by Ref. [5], which may be seen as a development of the study for two-dimensional quasicrystals given by Guo and Fan (see e.g. Fan[19] or Guo and Fan[20] ).

9.6

A straight dislocation in an icosahedral quasicrystal

The formulations exhibited in the Sections 9.4 and 9.5 are meaningful, which have greatly simpliﬁed the complicated equations involving elasticity. Their applications will be addressed in this and subsequent sections, in which the Fourier analysis and complex variable function method play important roles.

174

Chapter 9

Theory of elasticity of three-dimensional quasicrystals and its...

We introduced the dislocation solution of Yang et al[3] in an icosahedral quasicrystal. The authors of Ref. [3] assumed that the coupling eﬀect between phonons and phasons is omitted, i.e., R = 0. In the case the phonon solution of the dislocation is the same to that of the classical isotropic elastic solution of an edge dislocation. And the phason solution of the problem is independent of the phonon ﬁeld. In this section we try to give a complete analysis of the problem in which the phonon-phason coupling eﬀect is taken into account. For a dislocation along x3 -axis (or z-axis) in an icosahedral quasicrystal with the ⊥ ⊥ core at the origin: the Burgers vector is denoted as b = b ⊕b⊥ = (b1 , b2 , b3 , b⊥ 1 , b2 , b3 ), where the dislocation conditions are duj = bj , dwj = b⊥ (9.6-1) j , Γ

Γ

in which x1 = x, x2 = y, x3 = z, and the integrals in (9.6-1) should be taken along the Burgers circuit surrounding the dislocation core in space E . By using the superposition principle, we here calculate ﬁrst the elastic ﬁeld for a special case, i.e., ⊥ ⊥ which corresponds to b1 = 0, b⊥ 1 = 0, b2 = b3 = 0 and b2 = b3 = 0. For simplicity we can solve a half-plane problem by considering symmetry and anti-symmetry of relevant ﬁeld variables, so there are the following boundary conditions including the dislocation condition: σyy (x, 0) = 0,

(9.6-2a)

σzy (x, 0) = 0,

(9.6-2b)

Hyy (x, 0) = 0,

(9.6-2c)

Hzy (x, 0) = 0, dux = b1 ,

(9.6-2d)

(9.6-2e)

Γ

Γ

dwx = b⊥ 1.

(9.6-2f)

In addition there are boundary conditions at inﬁnity: σij (x, y) → 0,

Hij (x, y) → 0,

# x2 + y 2 → ∞.

(9.6-3)

In the following we use the formulation of Section 9.4 to solve the above boundary value problem. Performing the Fourier transform to equation (9.4-10) and the above boundary conditions, we obtain the solution at the transformed domain, then taking

9.6

A straight dislocation in an icosahedral quasicrystal

175

inversion of the Fourier transform, we obtain the solution as follows: ⎧ xy xy 3 1 y ⎪ ⎪ = arctan + c b + c , ⎪ u x 12 2 13 4 ⎪ ⎪ 2π 1 x r r ⎪ ⎪ ⎪ ⎪ ⎪ y2 y 2 (y 2 − x2 ) 1 r ⎪ ⎪ + c22 2 + c23 −c21 ln , uy = ⎪ ⎪ ⎪ 2π r0 r 2r 4 ⎪ ⎪ ⎪ ⎪ ⎪ xy xy 3 1 y ⎪ ⎪ = arctan + c −c + c , u ⎪ z 31 32 33 ⎨ 2π x r2 r4 ⎪ 1 y xy xy 3 ⎪ ⊥ ⎪ = arctan + c b + c , w ⎪ x 42 2 43 4 ⎪ ⎪ 2π 1 x r r ⎪ ⎪ ⎪ ⎪ ⎪ y2 y 2 (y 2 − x2 ) 1 r ⎪ ⎪ −c51 ln , wy = + c52 2 + c53 ⎪ ⎪ ⎪ 2π r0 r 2r 4 ⎪ ⎪ ⎪ ⎪ ⎪ xy xy 3 1 y ⎪ ⎪ −c61 arctan + c62 2 + c63 4 , ⎩ wz = 2π x r r

(9.6-4)

in which r 2 = x2 + y2 , r0 is the radius of the dislocation core and cij are constants shown as follows: ⎧ 2 2 2 2 ⊥ ⎪ ⎪ ⎪ c12 = 2c0 (μ(2R +c0 μ)(λ +3λμ+2μ )b1 +R(−e(λ+μ)+2μc0 (λ+2μ) )b1 ) , ⎪ ⎪ −e(2e+μc0 (λ+2μ))+μc0 (λ+2μ)(e+2μc0 (λ + 2μ)) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2c0 R(λ + μ)(2Rμ(λ + μ)b1 + 2μc0 (λ + 2μ)b⊥ 1) ⎪ ⎪ , c13 = ⎪ ⎪ −e(2e + μc (λ + 2μ)) + μc (λ + 2μ)(e + 2μc (λ + 2μ)) ⎪ 0 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (2c20 μ3 (λ + 2μ) − 2e2 )b1 + 2c0 R(λ + 3μ)eb⊥ ⎪ 1 ⎪ c = , ⎪ 21 ⎪ ⎪ −e(2e + μc (λ + 2μ)) + μc (λ + 2μ)(e + 2μc (λ + 2μ)) 0 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2c0 (−μ2 (λ + μ)(−2R2 + c0 (λ + 2μ))b1 + R(−(λ + μ)e + 2c0 μ2 )b⊥ 1) ⎪ ⎪ = , c 22 ⎪ ⎪ −e(2e + μc0 (λ + 2μ)) + μc0 (λ + 2μ)(e + 2μc0 (λ + 2μ)) ⎪ ⎪ ⎨ 2c0 R(λ + μ)(2Rμ(λ + μ)b1 + 2c0 μ2 b⊥ 1) ⎪ ⎪ , c23 = ⎪ ⎪ −e(2e + μc (λ + 2μ)) + μc (λ + 2μ)(e + 2μc 0 0 0 (λ + 2μ)) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 3c1 e{2(c0 μ + 7e)μc0 (λ + 2μ)b1 + R(54c20 (λ2 + 3λμ + μ2 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −2(α − β)(e + μc0 (λ + 2μ)))b⊥ ⎪ 1} ⎪ = c , ⎪ 31 ⎪ ⎪ 4c R(−e(2e + μc (λ + 2μ)) + μc 0 0 0 (λ + 2μ)(e + 2μc0 (λ + 2μ)) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 3c1 e(2μ(−e + μc0 (λ + 2μ))b1 + R(−2e + 2μc0 (λ + 2μ))b⊥ 1) ⎪ ⎪ = , c ⎪ 32 ⎪ −e(2e + μc (λ + 2μ)) + μc (λ + 2μ)(e + 2μc (λ + 2μ)) ⎪ 0 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −3ec1 (2Rμ(λ + μ)b1 + 2μc0 (λ + 2μ)b⊥ ⎪ 1) ⎪ , ⎩ c33 = −e(2e + μc0 (λ + 2μ)) + μc0 (λ + 2μ)(e + 2μc0 (λ + 2μ))

176

Chapter 9

⎧ ⎪ ⎪ ⎪ c42 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ c43 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ c51 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ c52 ⎪ c53 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ c61 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ c62 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ c63

Theory of elasticity of three-dimensional quasicrystals and its...

=

−2e(2Rμ(λ + μ)b1 + 2μc0 (λ + 2μ)b⊥ 1) , −e(2e + μc0 (λ + 2μ)) + μc0 (λ + 2μ)(e + 2μc0 (λ + 2μ))

= 0,

{−4eμ2 c0 (λ + 2μ)b1 + R(2(λ + 2μ)(e + 0.5μc0 ) =− =−

+μ(2β 2 μ + 2c20 (λ + 2μ)2 + c0 (λ + 2μ)(−βμ + R2 (λ + μ)))b⊥ 1} , R(−e(2e + μc0 (λ + 2μ)) + μc0 (λ + 2μ)(e + 2μc0 (λ + 2μ)))

2e(2Rμ(λ + μ)b1 + 2μc0 (λ + 2μ)b⊥ 1) , −e(2e + μc0 (λ + 2μ)) + μc0 (λ + 2μ)(e + 2μc0 (λ + 2μ))

=0

3c2 e{(2(c0 μ + 7e)μc0 (λ + 2μ)b1 + R(54c20 (λ2 + 3λμ + μ2 ) =−

−2(α − β)(e + μc0 (λ + 2μ)))b⊥ 1 )} 4c0 R(−e(2e + μc0 (λ + 2μ))(e + 2μc0 (λ + 2μ)))

,

=

3ec2 (2μ(−e + μc0 (λ + 2μ))b1 + R(−2e + 2μc0 (λ + 2μ))b⊥ 1) , −e(2e + μc0 (λ + 2μ)) + μc0 (λ + 2μ)(e + 2μc0 (λ + 2μ))

=

−3ec2 (2Rμ(λ + μ)b1 + 2μc0 (λ + 2μ)b⊥ 1) . −e(2e + μc0 (λ + 2μ)) + μc0 (λ + 2μ)(e + 2μc0 (λ + 2μ))

(9.6-5)

2

with e = −(λ + μ)R . For the other two typical problems, in which the Burgers vector of the dislocation

⊥ is denoted by (0, b2 , 0, 0, b⊥ 2 , 0) and (0, 0, b3 , 0, 0, b3 ) respectively, a complete similar

consideration will yield similar results, which are omitted here. Alternatively, the (2)

(2)

(3)

(3)

expressions are denoted as uj , wj , uj , wj .

⊥ ⊥ Analytic expressions for elastic ﬁeld for a dislocation (b1 , b2 , b3 , b⊥ 1 , b2 , b3 ) in the icosahedral quasicrystal can be obtained by superposition of the corresponding ex ⊥ pressions for the elastic ﬁelds for (b1 , 0, 0, b⊥ 1 , 0, 0), (0, b2 , 0, 0, b2 , 0) and (0, 0, b3 , 0, 0, b⊥ 3 ), namely, (1)

(2)

(3)

uj = uj + uj + uj ,

(1)

(2)

(3)

wj = wj + wj + wj ,

i, j = 1, 2, 3.

(9.6-6)

We can see that the interaction among phonon-phonon, phason-phason and phononphason is very evident, so the solution (9.6-4) is quite diﬀerent from the solution given by Yang et al[3] (whose solution for phonon displacement ﬁeld is given by the ﬁrst three formulas of equations (9.3-6), and will be quoted again in the following, see formula (9.6-7)), where they took R = 0, i.e., they assumed the phonon and phason are decoupled, so the solution for phonon is the same as the classical solution for crystals. It is obvious that our solution given by (9.6-4) explores that the realistic

9.6

A straight dislocation in an icosahedral quasicrystal

177

case for quasicrystals is quite diﬀerent from that of crystal. To illustrate the coupling eﬀect we give some numerical results in Fig. 9.6-1 and Fig. 9.6-2 for the normalized

Fig. 9.6-1

The displacement u1 /b1 versus x for diﬀerent coupling elastic constants

Fig. 9.6-2

The displacement u1 /b1 versus y for diﬀerent coupling constants

displacement u1 /b1 versus x and y respectively, in which the results exhibit the inﬂuence of the coupling constant R; the coupling eﬀect is quite remarkable. In the calculation we take the data of elastic moduli as λ = 74.9,

μ = 72.9(GPa),

K1 = 72,

K2 = −73(MPa),

and the phonon-phason coupling elastic constant for three diﬀerent cases: i.e., R/μ = 0, R/μ = 0.04 and R/μ = 0.06, in which the ﬁrst one corresponds to decoupled case.

178

Chapter 9

Theory of elasticity of three-dimensional quasicrystals and its...

The ﬁgures show that the coupling eﬀect is very important, and the displacement is increasing with the growth of value of R. For icosahedral quasicrystals with the presence of a dislocation, there are ﬁve independent elastic constants. If R = 0, wi = 0, our solution is exactly reduced to the solution of dislocation of crystals, i.e., ⎧ b1 λ + μ x2 μ y λ + μ xy r b2 ⎪ ⎪ ⎪ u = + arctan + ln + , x ⎪ ⎪ 2π x λ + 2μ r2 2π λ + 2μ r0 λ + 2μ r2 ⎪ ⎪ ⎪ ⎪ ⎨ b1 λ + μ x2 μ r y λ + μ xy b2 + ln arctan − + , uy = − ⎪ ⎪ 2π λ + 2μ r0 λ + 2μ r2 2π x λ + 2μ r2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ u = b3 arctan y . z 2π x (9.6-7) The displacement potential function formulation establishes the basis for solving defects problem in icosahedral quasicrystals. The formulation is simpliﬁed the solution process. In the subsequent steps a systematic Fourier analysis is developed, which provides a constructive procedure to ﬁnd the analytic solution, it is eﬀective not only for dislocation problem, but also for more complicated mixed boundary value problems (e.g. crack problems refer to Ref. [25]). The present solution can be used as a fundamental solution for a dislocation in an icosahedral quasicrystal. Therefore, many elasticity problems in an icosahedral quasicrystal can be directly solved with the aid of this fundamental solution by superposition. This work has been published in Ref. [17].

9.7

An elliptic notch/Griﬃth crack in an icosahedral quasicrystal

The solution for notch problem of icosahedral quasicrystals is not available until 2006. The diﬃculty lies in that cannot be solved by the Fourier transform method. We must develop other methods, in which the complex variable function-conformal mapping method is particularly eﬀective. The results are given by Ref. [21], which may be seen as a development of the work of the same authors in Ref. [22]. In the present section, we consider an icosahedral quasicrystal with an elliptic notch along the z-axis. On the basis of the general solution obtained in Section 5, explicit expressions of stress and displacement components of phonon and phason ﬁelds in the quasicrystals are given. With the help of conformal mapping, analytic solution for elliptic notch problem of the quasicrystals is presented. The solution of the Griﬃth crack problem can be observed as a special case of the results, which

9.7

An elliptic notch/Griﬃth crack in an icosahedral quasicrystal

179

will be reduced to the well-known results in a conventional material if the phason ﬁeld is absent.

9.7.1

The complex representation of stresses and displacements

The solution of (9.5-4) can be expressed as G(x, y) = Re[g1 (z) + z¯g2 (z) + z¯2 g3 (z) + z¯3 g4 (z) + z¯4 g5 (z) + z¯5 g6 (z)],

(9.7-1)

where gi (z) are arbitrary analytic functions of z = x + iy, and the bar over complex variable or complex function denotes the complex conjugate. By equations (9.5-2)∼(9.5-4) and (9.7-1), the stresses can be expressed as follows: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

σxx + σyy = 48c2 c3 RImΓ (z), σyy − σxx + 2iσxy = 8ic2 c3 R(12Ψ (z) − Ω (z)), σzy − iσzx = −960c3 c4 f6 (z), σzz =

24λR c2 c3 ImΓ (z), (μ + λ)

Hxy − Hyx − i(Hxx + Hyy ) = −96c2 c5 Ψ (z) − 8c1 c2 RΩ (z), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Hyx − Hxy + i(Hxx − Hyy ) = −480c2 c5 f6 (z) − 4c1 c2 RΘ (z), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Hyz + iHxz = 48c2 c6 Γ (z) − 4c2 R2 (2K2 − K1 )Ω (z), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 24R2 ⎪ ⎩ Hzz = c2 c3 ImΓ (z), (μ + λ) where

⎧ R(2K2 − K1 )(μK1 + μK2 − 3R2 ) ⎪ ⎪ = c , ⎪ 1 ⎪ ⎪ 2(μK1 − 2R2 ) ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ c = K (μK2 − R2 ) − R(2K2 − K1 ), ⎪ ⎨ 2 R 2 ⎪ (μK2 − R2 )2 ⎪ ⎪ ⎪ , c3 = μ(K1 − K2 ) − R2 − ⎪ ⎪ μK1 − 2R2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 μK1 − 2R2 ⎪ ⎪ , ⎩ c4 = c1 R + c3 K1 + 2 λ+μ

(9.7-2)

180

and

Chapter 9

Theory of elasticity of three-dimensional quasicrystals and its...

⎧ z f6 (z), Ψ (z) = f5 (z) + 5¯ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Γ (z) = f4 (z) + 4¯ z f5 (z) + 10¯ z 2 f6 (z), ⎪ ⎪ ⎪ ⎪ ⎨ Ω (z) = f (z) + 3¯ z f4 (z) + 6¯ z 2 f5 (z) + 10¯ z 3 f6 (z), 3 ⎪ (IV ) ⎪ Θ(z) = f2 (z) + 2¯ ⎪ z f3 (z) + 3¯ z 2 f4 (z) + 4¯ z 3 f5 (z) + 5¯ z 4 f6 (z), ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ c5 = 2c4 − c1 R, c6 = (2K2 − K1 )R2 − 4c4 μK2 − R . ⎩ μK1 − 2R2

(9.7-3)

In the above expressions, the function g1 (z) does not appear, this implies that for stress boundary value problem in this formalism only ﬁve complex potentials g2 (z), g3 (z), g4 (z), g5 (z) and g6 (z) are needed, and one can take g1 (z) = 0. For simplicity, we have introduced the following new symbols: (9)

g2 (z) = f2 (z), (6)

g5 (z) = f5 (z),

(8)

g3 (z) = f3 (z),

(7)

g4 (z) = f4 (z),

(5)

(9.7-4)

g6 (z) = f6 (z),

(n)

where gi denotes the n-th derivative with the argument z, accordingly f1 (z) = 0. Similar to formulation in Chapter 8, the complex representations of displacement components can be written as follows (here we have omitted the rigid-body displacements) ⎧ 2c3 ⎪ ⎪ + c7 − 2c2 c7 RΩ (z), uy + iux = −6c2 R ⎪ ⎪ μ+λ ⎪ ⎪ ⎪ ⎪ ⎪ 4 ⎪ ⎪ ⎪ ⎪ uz = μ(K + K ) − 3R2 (240c10 Imf6 (z)) ⎪ 1 2 ⎪ ⎪ ⎪ ⎪ 2 ⎨ + c1 c2 R Im(Θ(z) − 2Ω (z) + 6Γ (z) − 24Ψ (z))), (9.7-5) ⎪ R ⎪ ⎪ + iw = − Ψ (z) − c Θ(z)), w (24c y x 9 8 ⎪ ⎪ c1 (μK1 − 2R2 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 4(μK2 − R2 ) ⎪ w = ⎪ (240c10 Imf6 (z)) z ⎪ ⎪ (K1 − 2K2 )R(μ(K1 + K2 ) − 3R2 ) ⎪ ⎪ ⎪ ⎩ + c1 c2 R2 Im(Θ(z) − 2Ω (z) + 6Γ (z) − 24Ψ (z))), in which

⎧ c3 K1 + 2c1 R ⎪ , c7 = ⎪ ⎪ ⎪ μK1 − 2R2 ⎪ ⎪ ⎪ ⎪ ⎨ c8 = c1 c2 R(μ(K1 − K2 ) − R2 ), ⎪ (μK2 − R2 )2 ⎪ ⎪ c = c + 2c c − c , ⎪ 9 8 2 4 3 ⎪ μK1 − 2R2 ⎪ ⎪ ⎪ ⎩ c10 = c1 c2 R2 − c4 (c2 R − c3 K1 ).

(9.7-6)

9.7

An elliptic notch/Griﬃth crack in an icosahedral quasicrystal

9.7.2

181

Elliptic notch problem

We consider an icosahedral quasicrystal solid with an elliptic notch, which penetrates through the medium along the z-axis direction,the edge of the elliptic notch is subject to a uniform pressure p, see Fig. 9.7-1. The boundary conditions of this problem can be expressed as follows: (x, y) ∈ L, (9.7-7) Hxx cos(n, x)+Hxy cos(n, y) = hx , Hyx cos(n, x)+Hyy cos(n, y) = hy , (x, y) ∈ L, (9.7-8) σzx cos(n, x) + σzy cos(n, y) = 0, Hzx cos(n, x) + Hzy cos(n, y) = 0, (x, y) ∈ L, (9.7-9) where σxx cos(n, x) + σxy cos(n, y) = Tx ,

cos(n, x) =

Fig. 9.7-1

dy , ds

cos(n, y) = −

σxy cos(n, x) + σyy cos(n, y) = Ty ,

dx , ds

Tx = −p cos(n, x),

Ty = −p cos(n, y), (9.7-7 )

An elliptic notch subject to an inner pressure in an icosahedral quasicrystal

Tx , Ty denote the components of surface traction, p is the magnitude of the pressure, hx , hy are components of the generalized surface traction, n is the outward x2 y 2 unit normal vector of any point of the boundary, L : 2 + 2 = 1 is the edge of the a b elliptic notch. Since the measurement of generalized traction has not been reported so far, for simplicity, we assume that hx = 0,

hy = 0.

Utilizing equations (9.7-2), (9.7-3), (9.7-7) and (9.7-7 ), one has z f5 (z) + 10¯ z 2 f6 (z)) − 4c2 c3 R[3(f4 (z) + 4¯

(9.7-8 )

182

Chapter 9

Theory of elasticity of three-dimensional quasicrystals and its...

− (f3 (z) + 3zf4 (z) + 6z 2 f5 (z) + 10z 3 f6 (z))] = (Tx + iTy )ds = ipz.

(9.7-10)

Taking complex conjugate on both sides of equation (9.7-10) yields − 4c2 c3 R[3(f4 (z) + 4zf5 (z) + 10z 2 f6 (z)) − (f3 (z) + 3¯ z f4 (z) + 6z 2 f5 (z) + 10¯ z 3 f6 (z))] = −ip¯ z.

(9.7-11)

From equations (9.7-2), (9.7-3), (9.7-8) and (9.7-8 ), we have 48c3 (2c4 − c1 R)ReΨ (z) + 2c1 c3 RReΘ(z) = 0, −48c3 (2c4 − c1 R)ImΨ (z) − 2c1 c3 RImΘ(z) = 0.

(9.7-12)

Multiplying the second formula of (9.1-12) by -i and adding it to the ﬁrst, we obtain 48c3 (2c4 − c1 R)Ψ (z) + 2c1 c3 RΘ(z) = 0.

(9.7-13)

By equations (9.7-2), (9.7-3) and (9.7-9), the following expression results in f6 (z) + f6 (z) = 0, z f6 (z)] + (2K2 − K1 )RRe[f4 (z) + 4¯ z f5 (z) 4c11 Re[f5 (z) + 5¯ + 10¯ z 2 f6 (z) + 20f6 (z)] = 0,

(9.7-14)

in which c11 = (2K2 − K1 )R −

4c4 (μK2 − R2 ) . (μK1 − 2R2 )R

(9.7-15)

However further calculation will be very diﬃcult on the z-plane owing to the complexity of the manipulation, we must employ conformal mapping 1 + mζ (9.7-16) z = ω(ζ) = R0 ζ to transform the region with the ellipse at the z-plane onto the interior of the unit circle γ at ζ-plane, in which R0 =

a+b , 2

m=

a−b . a+b

Let fi (z) = fi [ω(ζ)] = Φi (ζ),

i = 2, 3, · · · 6.

(9.7-17)

Substituting (9.7-16) into (9.7-10), (9.7-11), (9.7-13) and (9.7-14), then multiplying both sides of equations by dσ/[2πi(σ − ζ)](σ represents the value of ζ at the unit

9.7

An elliptic notch/Griﬃth crack in an icosahedral quasicrystal

183

circle) and integrating around the unit circle γ, by means of Cauchy’s integral formula and analytic extension of the complex variable function theory, we obtain (see the Major Appendix of this book for details) ⎧ R0 ipζ(ζ 2 + m)(m3 ζ 2 + 1) (2K2 − K1 )R0 ⎪ ⎪ + × Φ2 (ζ) = ⎪ ⎪ 2c2 c3 R (mζ 2 − 1)3 2c2 c3 c11 ⎪ ⎪ ⎪ ⎪ ⎪ pmζ 3 (ζ 2 + m)[m2 ζ 6 − (m3 + 4m)ζ 4 + (2m4 + 4m2 + 5)ζ 2 + m] ⎪ ⎪ ⎪ , ⎪ ⎪ (mζ 2 − 1)5 ⎪ ⎪ ⎪ ⎨ R0 ipζ(m2 + 1) (2K2 − K1 )R0 pmζ 3 (ζ 2 + m)(mζ 2 − m2 − 2) , Φ3 (ζ) = − ⎪ 4c2 c3 R (mζ 2 − 1) 12c2 c3 c11 (mζ 2 − 1)3 ⎪ ⎪ ⎪ ⎪ ⎪ R0 (2K2 − K1 )R0 pmζ(ζ 2 + m) ⎪ ⎪ ⎪ Φ4 (ζ) = − ipmζ − , ⎪ ⎪ 12c2 c3 R 2c2 c3 c11 (mζ 2 − 1) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Φ5 (ζ) = − (2K2 − K1 )R0 pmζ, Φ6 (ζ) = 0. 48c2 c3 c11 (9.7-18) The elliptic notch problem has been solved. The solution of the Griﬃth crack subjected to a uniform pressure can be obtained if put m = 1, R0 = a/2 in the above notch solution. The solution of crack can be expressed explicitly in the z-plane, for example, $ % ipa2 y z ia2 y 3(2K2 − K1 )R # +# −1 + σyy =Im ip √ 2c11 z 2 − a2 (z 2 − a2 )3 (z 2 − a2 )3 (2K2 − K1 )R ipy(2a4 − 3z z¯) (2K2 − K1 )R a2 pz(z z¯ − a2 ) # # − 2c11 4c11 (z 2 − a2 )5 (z 2 − a2 )5 a2 p¯ z (2K2 − K1 )R # , (9.7-19) + 2 4c11 (z − a2 )3 # 2c3 ip uy = − 6c2 R + c7 Re (z − z 2 − a2 ) μ+λ 24c2 c3 R # 2K2 − K1 z z¯ a2 + p √ −√ − z 2 − a2 24c2 c3 c11 z 2 − a2 z 2 − a2 ip a2 2K2 − K1 z z¯ √ − 2c2 c7 Re −√ − z¯ − ipy 8c2 c3 R 4c2 c3 c11 z 2 − a2 z 2 − a2 % $ z] a2 [(z z¯ − a2 ) + 2iy¯ 2K2 − K1 # p + 16c2 c3 c11 (z 2 − a2 )3 # a2 2K2 − K1 2z z¯ 2 2 + p √ −√ +2 z −a . (9.7-20) 16c2 c3 c11 z 2 − a2 z 2 − a2 +

From equations (9.7-19) and (9.7-20), the stress intensity factor and energy release rate can be evaluated as follows:

184

Chapter 9

Theory of elasticity of three-dimensional quasicrystals and its...

K1 =

√ πap,

a 1 ∂ 2 (σyy (x, 0) ⊕ H(x, 0))(uy (x, 0) ⊕ wy (x, 0))dx GI = 2 ∂a −a 1 c7 1 + (KI )2 , = 2 λ + μ c3

(9.7-21)

in which material constant c3 is given by (9.5-5) and c7 by (9.7-6), it is evident that the crack energy release depends upon not only phonon elastic constants λ, μ but also phason elastic constants K1 , K2 and phono-phason coupling elastic constant R, though we assumed phason tractions hx = hy = 0. 9.7.3

Brief summary

The notch problem can be solved only by complex variable function method, the solution includes that of Griﬃth crack problem naturally. Though the Fourier transform can solve the Griﬃth crack problem, referring to Zhu and Fan [25], it cannot solve the notch problem. Whatever solution for notch or crack here reveals the eﬀects of not only phonon but also phason and phonon-phason coupling. The numerical examples on crack opening displacement δ(x) = uy (x, +0) − uy (x, −0) and energy release rate GI for diﬀerent values of R/μ given by Ref. [25] are shown in Figs. 9.7-2 and 9.7-3 respectively. Both solutions given by the complex variable function method and the Fourier transform reduce to that of the classical theory when phason is absent, this is helpful to examine the present work.

Fig. 9.7-2

Inﬂuence of phason and phonon-phason coupling to the crack opening displacement

9.8

Elasticity of cubic quasicrystals—the anti-plane and axisymmetric deformation 185

Fig. 9.7-3

Inﬂuence of phason and phonon-phason coupling to the energy release rate

This work developed the previous work for elasticity of two dimensional quasicrystals of Fan and co-workers. The work is helpful to understand quantitatively the inﬂuence of elliptic notch and crack on the mechanical behaviour of icosahedral quasicrystals. The stress intensity factor and energy release rate are also obtained as the direct results of the solution, which are useful for fracture mechanics. The rigorous theory on the complex potential method will be summarized in Chapter 11, which can also be referred to Ref [26].

9.8

Elasticity of cubic quasicrystals—the anti-plane and axisymmetric deformation

Cubic quasicrystal is one of important three-dimensional quasicrystals. There are few analytic solutions availabe in the literature. Since the phasons in this case have the same irreducible representation with the phonons, the stress and strain tensors are symmetric. With this feature we consider two cases, one for anti-plane and another for axisymmetric elastic theory of cubic quasicrystal, and the latter can reveal the three-dimensional eﬀect of the elasticity. In addition, a penny-shaped crack problem under tensile loading in the material is investigated, and the exact analytic solution is obtained by using the Hankel transform and dual integral equations theory, and the stress intensity factor and the strain energy release rate are then determined, which provide some useful information for studying deformation and fracture of the quasicrystalline material. From the physical basis provided by Hu et al[2] , we ﬁrst discuss the anti-plane elasticity such as, i.e., the stress-strain relation

186

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Theory of elasticity of three-dimensional quasicrystals and its...

σ23 = 2C44 ε23 + R44 w23 , σ31 = 2C44 ε31 + R44 w31 , H23 = 2R44 ε23 + K44 w23 , H31 = 2R44 ε31 + K44 w31 , the deformation geometry ε23 =

1 ∂u3 , 2 ∂x2

ε31 =

1 ∂u3 , 2 ∂x1

w23 =

∂w3 , ∂x2

w31 =

∂w3 ∂x1

and the equilibrium equations ∂σ31 ∂σ32 + = 0, ∂x1 ∂x2

∂H31 ∂H32 + = 0. ∂x1 ∂x2

These equations are exactly similar to those of anti-plane elasticity of one-dimensional and icosahedral quasicrystals, so result in the ﬁnal governing equations ∇2 u3 = 0,

∇2 w3 = 0.

The solution can be derived from the relevant discussion in Chapters 5,7,8 and Section 9.2, so it need not be mentioned again. For the axisymmetric case Zhou and Fan[23] developed a displacement potential theory to reduce basic equations to a single partial diﬀerential equation with higher order in circular cylindrical coordinate system (r, θ, z), i.e., we assume ∂ =0 ∂θ and by the generalized Hooke’s law, ⎧ σrr = C11 εrr + C12 (εθθ + εzz ) + R11 wrr + R12 (wθθ + wzz ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ σθθ = C11 εθθ + C12 (εrr + εzz ) + R11 wθθ + R12 (wrr + wzz ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ σzz = C11 εzz + C12 (εrr + εθθ ) + R11 wzz + R12 (wθθ + wrr ), ⎪ ⎪ ⎪ ⎨ σzr = σrz = 2C44 εrz + 2R44 wrz , ⎪ Hzz ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Hrr ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Hθθ ⎪ ⎪ ⎪ ⎩ Hzr

= R11 εzz + R12 (εθθ + εrr ) + K11 wzz + K12 (wθθ + wrr ), = R11 εrr + R12 (εθθ + εzz ) + K11 wrr + K12 (wθθ + wzz ), = R11 εθθ + R12 (εrr + εzz ) + K11 wθθ + K12 (wrr + wzz ), = Hrz = 2R44 εrz + 2K44 wrz ,

and the equations of deformation geometry 1 ∂ui ∂uj 1 ∂wi ∂wj + + , wij = , εij = 2 ∂xj ∂xi 2 ∂xj ∂xi

(9.8-1)

(9.8-2)

9.8

Elasticity of cubic quasicrystals—the anti-plane and axisymmetric deformation 187

which are

⎧ ∂ur ur ∂uz ⎪ ⎪ , εθθ = , εzz = , εrr = ⎪ ⎪ ∂r r ∂z ⎪ ⎪ ⎪ ⎪ ⎪ 1 ∂ur ∂uz ⎪ ⎪ + , ⎨ εrz = εzr = 2 ∂z ∂r ∂wr wr ∂wz ⎪ ⎪ , wθθ = , wzz = , ⎪ wrr = ⎪ ⎪ ∂r r ∂z ⎪ ⎪ ⎪ ⎪ 1 ∂wr ∂wz ⎪ ⎪ + ⎩ wrz = wzr = 2 ∂z ∂r

(9.8-3)

and the equations of equilibrium ⎧ ∂σ ∂σrz σrr − σθθ rr ⎪ + + = 0, ⎪ ⎪ ⎪ ∂r ∂z r ⎪ ⎪ ⎪ ⎪ ⎪ ∂σzz σzr ∂σzr ⎪ ⎪ ⎨ ∂r + ∂z + r = 0, ⎪ ∂Hrz Hrr − Hθθ ∂Hrr ⎪ ⎪ + + = 0, ⎪ ⎪ ⎪ ∂r ∂z r ⎪ ⎪ ⎪ ⎪ ⎪ ∂Hzr ∂Hzz Hzr ⎩ + + = 0, ∂r ∂z r

(9.8-4)

If all the displacements and stresses can be expressed by a potential F (r, z) (the detail on deﬁnition of F (r, z) is given by [23]), which satisﬁes

2 2 6 2 4 ∂ ∂ ∂ 1 ∂ 1 ∂ ∂ ∂8 −b + +c + 8 2 6 2 ∂z ∂r r ∂r ∂z ∂r r ∂r ∂z 4 2 2 3 2 4 ∂ ∂ ∂ 1 ∂ 1 ∂ F = 0, −d + + e + ∂r2 r ∂r ∂z 2 ∂r2 r ∂r

(9.8-5)

then the equations (9.8-2)∼(9.8-4) are identically satisﬁed. As an application of above theory and method, the solutions of elastic ﬁeld of cubic quasicrystal with a penny-shaped crack are discussed in the following. Assume a penny-shaped crack with radius a in the centre of the cubic quasicrystal material, the size of the crack is much smaller than that of the solid, so that the size of the material can be considered as inﬁnite. At the inﬁnity, the quasicrystal material is subjected to a tension p in z-direction. The origin of coordinate system is at the centre of the crack (as shown in Fig. 9.8-1). From the symmetry of the problem, it is suﬃcient to study the upper half-space z > 0 or the lower half-space z < 0. In this case, for studying the upper half-space, the boundary conditions of the problem are described by

188

Chapter 9

Theory of elasticity of three-dimensional quasicrystals and its...

⎧ √ ⎪ r2 + z 2 → ∞ : σzz = p0 Hzz = 0, σrz = 0, Hrz = 0, ⎪ ⎨ z = 0, 0 r a, σzz = σrz = 0; Hzz = Hrz = 0, ⎪ ⎪ ⎩ z = 0, r > a : σrz = 0, uz = 0; Hrz = 0, wz = 0.

(9.8-6)

These boundary conditions can be replaced by ⎧ √ ⎪ r2 + z 2 → ∞ : σij = 0, Hij = 0, ⎪ ⎨ z = 0, 0 r a, σzz = −p, σrz = 0; Hzz = Hrz = 0, ⎪ ⎪ ⎩ z = 0, r > a : σrz = 0, uz = 0; Hrz = 0, wz = 0,

(9.8-6 )

which are equivalent to (9.8-6) in the sense of fracture mechanics, if p = p0 .

Fig. 9.8-1

Penny-shaped crack subject to a tension in a cubic quasicrystal

By taking the Hankel transform to equation (9.8-5) and boundary conditions (9.8-6 ), the solution at the transformed space is such as F¯ (ξ, z) = A1 e−λ1 ξz + A2 e−λ2 ξz + A3 e−λ3 ξz + A4 e−λ4 ξz ,

(9.8-7)

where Ai (i = 1, 2, 3, 4) are unknown functions of ξ to be determined and λi (i = 1, 2, 3, 4) are eigen roots obtained from the ordinary diﬀerential equation of F¯ (ξ, r). Accoding to the boundary conditions Ai (ξ) can be determined by solving the following dual integral equations

References

189

⎧ ∞ ⎪ ⎪ ξAi (ξ)J0 (ξr)dξ = Mi p0 , 0 < r < a, ⎨ 0 ∞ ⎪ ⎪ ⎩ Ai (ξ)J0 (ξr)dξ = 0, r > a

(9.8-8)

0

and i = 1, 2, 3, 4, in which Mi are some constants consisting of elastic moduli, J0 (ξr) the ﬁrst kind Bessel function of zero order. According to the theory of dual integral equations (refer to Major Appendix), we obtain the solution of dual integral equations (9.8-8) as follows: Ai (ξ) = 2a2 Mi p(2πaξ)−1/2 ξ −7 J3/2 (aξ),

(9.8-9)

in which J3/2 (aξ) is the ﬁrst kind Bessel function of 3/2-order (refer to the Major Appendix for the detailed calculation). After some calculation, the stress intensity factor KI , strain energy WI and strain energy release rate GI can be obtained as follows: KI =

2√ πap, π

WI = M p2 a3 ,

GI =

1 ∂WI 3M p2 a = , 2πa ∂a 2π

(9.8-10)

where M is the constant composed of the elastic constants which is quite lengthy so has not been included here.

References [1] Ding D H, Yang W G, Hu C Z et al. Generalized theory of elasticity of quasicrystals. Phys Rev B, 1993, 48(10): 7003–7010 [2] Hu C Z, Wang R H, Ding D H et al. Point groups and elastic properties of twodimensional quasicrystals. Acta Crystallog A, 1996, 52(2): 251–256 [3] Yang W G, Ding D H et al. Atomtic model of dislocation in icosahedral quasicrystals. Phil Mag A, 1998, 78(6): 1481–1497 [4] Fan T Y, Guo L H, Final governing equation of plane elasticity of icosahedral quasicrystals. Phys Lett A, 2005, 341(5): 235–239 [5] Li L H, Fan T Y. Final governing equation of plane elasticity of icosahedral quasicrystals– stress potential method. Chin Phys Lett, 2006, 24(9): 2519–2521 [6] Reynolds G A M, Golding B, Kortan A R et al. Isotropic elasticity of the Al-Cu-Li quasicrystal. Phys Rev B, 1990, 41(2): 1194–1195 [7] Spoor P S, Maynard J D, Kortan A R. Elastic isotropy and anisotropy in quasicrystalline and cubic Al-Cu-Li. Phys Rev Lett, 1995, 75(19): 3462–3465 [8] Tanaka K, Mitarai, Koiwa M. Elastic constants of Al-based icosahedral quasicrystals. Phil Mag A, 1996, 76(10): 1715–1723 [9] Duquesne J-Y, Perrin B. Elastic wave interaction in icosahedral AlPdMn. Physica B, 2002, 316–317: 317–320

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[10] Foster K, Leisure R G, Shaklee A et al. Elastic moduli of a Ti-Zr-Ni icosahedral quasicrystal and a 1/1 bcc crystal approximant. Phys Rev B, 1999, 59(17): 11132– 11135 [11] Schreuer J, Steurer W, Lograsso T A et al. Elastic properties of icosahedral iCd84 Yb16 and hexagonal h-Cd51 Yb14 . Phil Mag Lett, 2004, 84(10): 643–653 [12] Sterzel R, Hinkel C, Haas A et al. Ultrasonic measurements on FCI Zn-Mg-Y single crystals. Europhys Lett, 2000, 49(6): 742–747 [13] Letoublon A, de Boissieu M, Boudard M et al. Phason elastic constants of the icosahedral Al-Pd-Mn phase derived from diﬀuse scattering measurements. Phil Mag Lett, 2001, 81(4): 273–283 [14] de Boissieu M, Francoual S, Kaneko Y et al. Diﬀuse scattering and phason ﬂuctuations in the Zn-Mg-Sc icosahedral quasicrystal and its Zn-Sc periodic approximant. Phys Rev Lett, 2005, 95(10): 105503 [15] Edagawa K, So GI Y. Experimental evaluation of phonon-phason coupling in icosahedral quasicrystals. Phil Mag, 2007, 87(1): 77–95 [16] Zhu A Y, Fan T Y, Elastic ﬁeld of a mode II Griﬃth crack in icosahedral quasicrystals. Chinese Physics. B. 2007, 16(4): 1111–1118 [17] Zhu A Y, Fan T Y, Guo L H. A straight dislocation in an icosahedral quasicrystal. J Phys: Condens Matter, 2007, 19(23): 236216 [18] Li X F, Fan T Y. New method for solving elasticity problems of some planar quasicrystals. Chin Phys Lett, 1998, 15(4): 278–280 [19] Fan T Y, Mathematical Theory of Elasticity of Quasicrystals and Its Applications. Beijing: Beijing Institute of Technology Press, 1999 (in Chinese) [20] Guo Y C, Fan T Y, A mode-II Griﬃth crack in decagonal quasicrystals. Appl Math Mech, 2001, 22(11): 1311–1317 [21] Li L H, Fan T Y. Complex variable function method for solving Griﬃth crack in an icosahedral quasicrystal. Science in China G, 2008, 51(6): 773–780 [22] Li L H, Fan T Y. Complex function method for solving notch problem of point 10 twodimensional quasicrystal based on the stress potential function. J Phys: Condens, Matter, 2006, 18(47): 10631–10641 [23] Zhou W M, Fan T Y. Axisymmetric elasticity problem of cubic quasicrystal. Chinese Physics, 2000, 9(4): 294–303 [24] Fan T Y, Xie L Y, Fan L et al. Study on interface of quasicrystal-crystal. Chin. Phys. B, submitted, 2009 [25] Zhu A Y and Fan T Y. Elastic analysis of a Griﬃth crack in icosahedral Al-Pd-Mn quasicrystal, Int. J. Mod. Phys. B, 2009, 23(16): 3429–3444 [26] Fan T Y, Tang Z Y. The strict theory of complex varible function method of sextuple harmonic equation and applications, J. Math. Phys., 51(5), 053519, 2010

Chapter 10 Dynamics of elasticity and defects of quasicrystals The discussion in Chapters 5−9 is limited in the scope of elastostatics of quasicrystals. In this chapter we study the dynamic problems of the solid phase. Elastodynamics of quasicrystals is a topic with diﬀerent points of view. The focus of contradictions between diﬀerent scholar circles lies in the role of phason variables in the dynamic process. Lubensky et al[1] , Socolar and Lubensky[2] pointed out that the phonon ﬁeld u and the phason ﬁeld w play very diﬀerent roles in the hydrodynamics of quasicrystals, because w is insensitive to spatial translations, the phason modes represent the relative motion of the constituent density waves. They claimed that the phasons are diﬀusive, not oscillatory, and with very large diﬀusion times. On the other hand according to Bak[3,4] , the phason describes particular structure disorders or structure ﬂuctuations in quasicrystals, and it can be formulated based on a six-dimensional space description. Since there are six continuous symmetries, there exist six hydrodynamic vibration modes. Following this point of view u and w play similar roles in the dynamics. It is evident that the diﬀerence between the argument of Lubensky et al and Bak’s argument lies only in the dynamics. There is no diﬀerence between the arguments in statics (i.e., static equilibrium). Based on this reason in the discussions of the previous chapters, we need not distinguish the arguments of either Lubensky et al or Bak. Probably due to simpler mathematical formulation, many authors followed the Bak’s argument e.g.[5 ∼ 12] in dynamic study. In this chapter, we will present some results given in the references. These are Sections 10.1∼10.4 which constitute the part one of this chapter. In the meantime, we introduce some other results[13,14,27] which were carried out by following the argument of Lubensky et al. In this line of thinking it appears that elastodynamics and hydrodynamics are combined in some extent so it can be called elasto-/hydro-dynamics of quasicrystals. The discussion given in Sections 10.5∼10.7 constitutes the part two of this chapter. The results based on diﬀerent hypothesis are presented to provide readers for T. Fan, Mathematical Theory of Elasticity of Quasicrystals and Its Applications © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2011

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Dynamics of elasticity and defects of quasicrystals

their consideration and comparison. Though researchers believe that the hydrodynamics based on the argument of Lubensky et al is more fundamentally sound, the major shortcoming so far is lack of proper experimental data confirmation. Recently the research interest in the respect is growing up[13∼16] , but the most important accomplishment shall still be the quantitative results.

10.1

Elastodynamics of quasicrystals followed the Bak’s argument

Ding et al[5] ﬁrst discussed the elastodynamics of quasicrystals. The basic equations in deformation geometry and generalized Hooke’s law are the same as those of elastostatics, i.e., 1 ∂ui ∂uj ∂wi , wij = εij = + , (10.1-1) 2 ∂xj ∂xi ∂xj σij = Cijkl εkl + Rijkl wkl , Hij = Kijkl wkl + Rklij εkl .

(10.1-2)

They claimed that law of the momentum conservation holds for both phonons and phasons, namely for linear and small deformation case the equations of motion are ∂ 2 ui ∂σij =ρ 2 , ∂xj ∂t

∂Hij ∂ 2 wi =ρ 2 , ∂xj ∂t

(10.1-3)

where ρ denotes the average mass density of the material. This implies that they follow the Bak’s argument. After seven years, Hu et al[6] conﬁrmed the point of view again. In fact, the ﬁnal elastodynamic equations can be deduced by substituting (10.1-1) and (10.1-2) into (10.1-3). The mathematical structure of this theory is relatively simpler, the formulations are the extension to that of the classical elastodynamics, so many authors take this formulation to develop the elastodynamics of quasicrystals and give some applications in defect dynamics and thermodynamics. In the subsequent sections we will present examples of applications of the theory.

10.2

Elastodynamics of anti-plane elasticity for some quasicrystals

For three-dimensional icosahedral or cubic or one-dimensional hexagonal quasicrystals, in the anti-plane elasticity the basic equations have similar form. First we consider icosahedral quasicrystals, which have the constitutive relations as

10.2

Elastodynamics of anti-plane elasticity for some quasicrystals

⎧ ∂uz ∂wz ⎪ ⎪ σzy = σyz = μ +R , ⎪ ⎪ ∂y ∂y ⎪ ⎪ ⎪ ∂w ∂u ⎪ ⎨ σxz = σzx = μ z + R z , ∂x ∂x ∂uz ∂wz ⎪ ⎪ +R , Hzy = (K1 − K2 ) ⎪ ⎪ ⎪ ∂y ∂y ⎪ ⎪ ⎪ ⎩ H = (K − K ) ∂wz + R ∂uz . zx 1 2 ∂x ∂x

193

(10.2-1)

Substituting expressions (10.2-1) into equations of motion of (10.1-3) yields μ∇2 uz + R∇2 wz = ρ

∂ 2 uz , ∂t2

R∇2 uz + (K1 − K2 )∇2 wz = ρ

∂ 2 wz . ∂t2

(10.2-2)

If deﬁne displacement functions φ and ψ as[7] uz = αφ − Rψ,

wz = Rφ + αψ,

(10.2-3)

then equations (10.2-2) reduce to the standard wave equations ∇2 φ = where α=

1 ∂2φ , s21 ∂t2

∇2 ψ =

1 ∂ 2ψ , s22 ∂t2

# 1 [μ − (K1 − K2 ) + (μ − (K1 − K2 ))2 + 4R2 ] 2

and

+ sj =

εj , ρ

j = 1, 2,

(10.2-4)

(10.2-5)

(10.2-6)

# 1 [μ + (K1 − K2 ) ± (μ − (K1 − K2 ))2 + 4R2 ], 2 sj can be understood as the speeds of wave propagation in anti-plane deformation of the material. It is obvious that the wave speeds result from the phonon-phason coupling. If there is no coupling, i.e., R → 0, then & + μ (K1 − K2 ) s1 → , s2 → , (10.2-7) ρ ρ ε1,2 =

+ μ K1 − K2 represents the speed of transverse wave of the phonon ﬁeld and where ρ ρ represents the speed of the pure phason elastic wave, requiring K1 − K2 > 0. Substituting (10.2-3) into (10.2-1) the stresses can be expressed by φ and ψ +

194

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Dynamics of elasticity and defects of quasicrystals

⎧ ∂φ ∂ψ ⎪ ⎪ + R(α − μ) , σyz = σzy = (αμ + R2 ) ⎪ ⎪ ∂y ∂y ⎪ ⎪ ⎪ ∂φ ∂ψ ⎪ ⎪ + R(α − μ) , ⎨ σxz = σzx = (αμ + R2 ) ∂x ∂x (10.2-8) ∂φ ⎪ 2 ∂ψ ⎪ Hzy = R(α + (K1 − K2 )) + (α(K , − K ) − R ) ⎪ 1 2 ⎪ ⎪ ∂y ∂y ⎪ ⎪ ⎪ ∂φ ∂ψ ⎪ ⎩ Hzx = R3 (α + (K1 − K2 )) + (α(K1 − K2 ) − R2 ) . ∂x ∂x Formulas (10.2-3) and (10.2-8) give the expressions for displacements and stresses in terms of displacement functions φ and ψ, which satisfy the standard wave equations (10.2-4) for elastodynamics of anti-plane elasticity of three-dimensional icosahedral quasicrystals. The above discussion is valid for anti-plane elasticity of three-dimensional cubic quasicrystals or one-dimensional quasicrystals too. The diﬀerence between these quasicrystals is only the material constants. If μ, K1 − K2 and R are replaced by C44 , K44 and R44 (see Section 9.8) for cubic quasicrystals, or by C44 , K2 and R3 for one-dimensional hexagonal quasicrystals with the Laue classes 6/mh and 6/mh mm (see Sections 7.1 or 8.1) one can ﬁnd the similar equations. The solution of (10.2-4) can be done by using method for solving pure wave equations in classical mathematical physics.

10.3

Moving screw dislocation in anti-plane elasticity

Assume a straight screw dislocation line parallel to the axis which moves along one of axes, say, the x-axis in the plane perpendicular to the x-axis. For simplicity, considering the dislocation moves with constant velocity V . For the problem, a dislocation condition is assumed || duz = b3 , dwz = b⊥ (10.3-1) 3, Γ

Γ

||

i.e., we assume that the dislocation has the Burgers vector (0, 0, b3 , 0, b⊥ 3 ), and Γ denotes the Burgers circuit surrounding the core of the moving dislocation . Starting now we denote the ﬁxed coordinates as (x1 , x2 , t) and moving ones as (x, y). By introducing the Galilean transformation x = x1 − V t,

(10.3-2) 1 ∂2 wave equations (10.2-4) reduce to the Laplace equations i.e., ∇2 − 2 2 → ∇21 , s1 ∂t 2 2 2 ∂ 1 ∂ ∂ + ∇2 − 2 2 → ∇22 , ∇2 = s2 ∂t ∂x21 ∂x22 ∇21 φ = 0,

y = x2 ,

∇22 ψ = 0,

(10.3-3)

10.3

Moving screw dislocation in anti-plane elasticity

where

195

∂2 ∂2 ∂2 ∂2 + 2 , ∇21 = + 2, 2 2 ∂x ∂y1 ∂x ∂y2 , yj = βj y, βj = 1 − V 2 /s2j , j = 1, 2. ∇21 =

Let complex variable zj be zj = x + iyj ,

i=

√

−1,

(10.3-4a) (10.3-4b)

(10.3-5)

the solution of equations (10.3-3) is φ = ImF1 (z1 ),

ψ = ImF2 (z2 ),

(10.3-6)

where F1 (z1 ) and F2 (z2 ) are analytic functions of z1 and z2 respectively, and notation Im marks the imaginary part of a complex function. The boundary condition (10.3-1) determines the analytic functions as[7] φ(x, y1 ) =

A1 y1 arctan , 2π x

with constants

||

ψ(x, y1 ) =

A2 y2 arctan 2π x

(10.3-7a)

||

αb3 + Rb⊥ αb3 − Rb⊥ 3 3 , A = , (10.3-7b) 2 α 2 + R2 α2 + R2 the displacement ﬁeld is determined in the ﬁxed coordinate system as follows 1 β1 y β2 y || 2 2 α + R b uz (x, y, t) = arctan arctan 2π(α2 + R2 ) x−Vt x−Vt 3 (10.3-8a) β1 y β2 y + arctan − arctan αRb⊥ 3 , x−Vt x−Vt 1 β1 y β2 y 2 2 arctan arctan R + α b⊥ wz (x, y, t) = 3 2π(α2 + R2 ) x−Vt x−Vt 3 β2 y β1 y || − arctan αRb3 . (10.3-8b) + arctan x−Vt x−Vt A1 =

The expressions for strains and stresses are omitted here due to limitation of space. We give the evaluation on the energy of the moving dislocation. Denote energy W per unit length on the moving dislocation which consists of the kinetic energy Wk and potential energy Wp deﬁned by the integrals 2 2 ∂uz ∂wz 1 dx1 dx2 , + Wk = ρ 2 ∂t ∂t Ω (10.3-9) 1 ∂uz ∂wz + Hij dx1 dx2 , σij Wp = 2 ∂t ∂t Ω

respectively, where the integration should be taken over a ring r0 < r < R0 , r0 denotes the size of the dislocation core, and R0 the size of so-called dislocation net

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similar to those in conventional crystals. which are introduced in Section 7.1. In general r0 ∼ 10−8 cm, and R0 ∼ 104 r0 . Substituting displacement formulas and corresponding stress formulas into (10.3-9), we obtain Wk = with

k k R0 ln , 4π r0

Wp =

k p R0 ln 4π r0

A21 A22 + , β1 β2 A21 1 2 2 2 kp = (μα + (K1 − K2 ))R + 2αR ) β1 + 2 β1 1 A22 2 2 2 + (μR + (K1 − K2 )α − 2αR ) β2 + 2 β2 ρV 2 (α2 + R2 ) kk = 2

(10.3-10)

(10.3-11)

and A1 , A2 given by (10.3-7). Therefore the total energy is W =

kk + kp R0 . ln 4π r0

(10.3-12)

It is concluded that when V → s2 , i.e., β2 → 0, this leads to the inﬁnity of the energy, and is invalid, and thus s2 is the limit of the velocity of a moving dislocation. Additionally, if V << s2 , the total energy can be written in the following simple form: 1 1 R0 || 2 1 = W0 + m0 V 2 , (10.3-13) W ≈ W0 + ρV 2 [(b3 )2 + (b⊥ ln 3) ] 2 4π r0 2 where W0 is the elastic energy per unit length of a rest screw dislocation, i.e., ||

||

2 ⊥ W0 = [μ(b3 )2 + R(b⊥ 3 ) + 2b3 b3 R]

1 R0 ln 4π r0

(10.3-14)

and m0 , the so-called “apparent” mass of the dislocation per unit length in the case considered 1 R0 || || ⊥ 2 m0 = [μ(b3 )2 + R(b⊥ . (10.3-15) ln 3 ) + 2b3 b3 R] 4π r0 It is evident that if V = 0, the solution reduces to that of static dislocation, which is given in Section 7.1. # # 1 − V 2 /c22 , s1 = c2 = μ/ρ Furthermore, if b⊥ 3 = 0, R = 0, then ε1 = μ, β1 = is the speed of transverse wave of the conventional crystal, the above solution reduces to b β1 y uz (x − V t, y) = arctan , 2π x−Vt μβ1 (x − V t) b , σyz = σzy = (10.3-16) 2π (x − V t)2 + β12 y 2 σxz = σzx = −

μβ1 y b , 2π (x − V t)2 + β12 y 2

10.4

Mode III moving Griﬃth crack in anti-plane elasticity

W ≈

1 μb + ρV 2 b2 2 2

m0 =

197

1 R0 , ln 4π r0

ρb2 R0 , ln 4π r0

this is exactly identical to the well known Eshellby solution for crystals[17] . The above discussion holds for anti-plane elasticity of three-dimensional cubic or one-dimensional hexagonal quasicrystals, only the material constants μ, K1 − K2 and R should be replaced by C44 , K44 and R44 , or by C44 , K2 and R3 , respectively.

10.4

Mode III moving Griﬃth crack in anti-plane elasticity

As another application of above elastodynamic theory, we study a moving Griﬃth crack of Mode III, which moves with constant speed V along x1 (see Fig. 10.4-1). Here we also take the ﬁxed coordinates (x1 , x2 , t) and the moving ones (x, y).

Fig.10.4-1

Moving Griﬃth crack of Mode III

In the moving coordinates the boundary conditions are # x2 + y 2 → ∞ : σij = 0, Hij = 0, y = 0, |x| < a : σyz = −τ, Hyz = 0.

(10.4-1)

Assume that the Laplace equations have the solution φ(x1 , y1 ) = ReF1 (z1 ),

ψ(x1 , y2 ) = ReF2 (z2 ),

(10.4-2)

where F1 (z1 ) and F2 (z2 ) are any analytic functions of z1 and z2 , and notation Re denotes the real part of a complex number. Because the boundary conditions (10.4-1) are more complicated than those given by (10.3-1), we must use conformal mapping a (10.4-3) z1 , z2 = ω(ζ) = (ζ + ζ −1 ) 2

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to solve the problem at ζ(= ξ + iη)-plane . After some calculations, we ﬁnd the solution F1 (z1 ) = F1 [ω(ζ)] = G1 (ζ) =

iΔ1 ζ, Δ

F2 (z2 ) = F2 [ω(ζ)] = G2 (ζ) =

iΔ2 ζ, Δ (10.4-4)

in which Δ = β1 β2 [(αμ + R2 )(α(K1 − K2 ) − R2 ) − R2 (α + (K1 − K2 ))(α − μ)], Δ1 = τ αβ2 (α(K1 − K2 ) − R2 ),

Δ2 = τ αβ1 R(α + (K1 − K2 )).

Because there is the inverse mapping as + + z1 2 z2 2 z1 z2 −1 −1 − − ζ = ω (z1 ) = − 1 = ω (z2 ) = − 1. a a a a

(10.4-5)

(10.4-6)

The manipulation afterward can also be done in z1 -plane/z2 -plane. The corresponding stresses are expressed by ⎧ ∂ ∂ ⎪ ⎪ ReF1 (z1 ) + R(α − μ)β2 ReF2 (z2 ), σyz = σzy = (αμ + R2 )β1 ⎪ ⎪ ∂y1 ∂y2 ⎪ ⎪ ⎪ ⎪ ⎨ σxz = σzx = (αμ + R2 ) ∂ ReF1 (z1 ) + R3 (α − μ)β2 ∂ ReF2 (z2 ), ∂x ∂x ∂ ∂ ⎪ ⎪ ReF1 (z1 ) + (α(K1 − K2 ) − R2 ) ReF2 (z2 ), Hzy = R3 (α + (K1 − K2 ))β1 ⎪ ⎪ ∂y ∂y ⎪ 1 2 ⎪ ⎪ ∂ ∂ ⎪ ⎩ Hzx = R3 (α + (K1 − K2 ))β1 ReF1 (z1 ) + (α(K1 − K2 ) − R2 ) ReF2 (z2 ). ∂x ∂x (10.4-7) Substituting (10.4-6) into (10.4-4) then into (10.4-7), the stresses can be evaluated in explicit form, e.g. σyz = σzy τ d 1 1 2 2 = − (αμ + R )β1 β2 (α(K1 − K2 ) − R ) 1 − cos θ − θ1 − θ2 Δ 2 2 (d1 d2 )1/2 τ D 1 1 + β1 β2 R2 (α + (K1 − K2 ))(α − μ) 1 − Θ Θ cos Θ − − 1 2 Δ 2 2 (D1 D2 )1/2 (10.4-8) with

# # # ⎧ d = x2 + y12 , d1 = (x − a)2 + y12 , d2 = (x + a)2 + y12 , ⎪ ⎪ ⎪ ⎪ # # # ⎪ ⎪ ⎪ D = x2 + y22 , D1 = (x − a)2 + y22 , D2 = (x + a)2 + y22 , ⎪ ⎪ ⎨ y y1 y1 1 ⎪ θ = arctan , θ1 = arctan , θ2 = arctan , ⎪ ⎪ x x−a x+a ⎪ ⎪ ⎪ y ⎪ ⎪ y2 y2 ⎪ ⎩ Θ = arctan 2 , Θ1 = arctan , Θ2 = arctan . x x−a x+a (10.4-9)

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199

It is easy to prove that (10.4-9) satisﬁes the relevant boundary conditions, and is the exact solution. Similarly, σxz = σzx , Hzx and Hzy can also be expressed explicitly. From (10.4-8), as y = 0, it yields xτ √ − τ, |x| > a, 2 (10.4-10) σyz (x, 0) = x − a2 −τ, |x| < a The stress presents singularity of order (x − a)−1/2 as x → a. The stress intensity factor for Mode III for phonon ﬁeld is # √ || KI = lim+ π(x − a)σyz (x, 0) = πaτ.

(10.4-11)

x→a

This is identical to the classical Yoﬀe solution[18] , there the stress intensity factor is also independent of crack moving speed V . Now we calculate energy of the moving crack, which is deﬁned by a [σzy (x, 0) ⊕ Hzy (x, 0)][uz (x, 0) ⊕ wz (x, 0)]dx W=2 0

1 1 = (Δ1 α−Δ2 R)τ πa = [αβ2 (α(K1 −K2 ) − R2 )−β1 R2 (α + (K1 − K2 ))]πa2 τ. Δ Δ (10.4-12) The crack energy release rate is G=

1 1 ∂W = [αβ2 (α(K1 −K2 )−R2 ) − β1 R2 (α + (K1 −K2 )](KI )2 . 2 ∂a 2Δ

(10.4-13)

The above discussion, results and conclusions hold too for anti-plane elasticity of three-dimensional cubic or one-dimensional hexagonal quasicrystals, only the material constants μ, K1 − K2 and R need to be replaced by C44 , K44 and R44 or by C44 , K2 and R3 , respectively. The results are given by the author in the Chinese edition of this book in 1999.

10.5

Elasto-/hydro-dynamics of quasicrystals and approximate analytic solution for moving screw dislocation in anti-plane elasticity

In the previous sections the formulation following Bak’s argument has been developed and certain results are obtained. The character of the work lies in the equations of motion being taken as ∂ 2 ui ∂σij ρ 2 = , ∂t ∂xj ∂Hij ∂ 2 wi , ρ 2 = ∂t ∂xj

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which are diﬀerent from those based on the argument of Lubensky et al. Meanwhile Rochal and Lorman[15] did not completely agree to the above equations even though they followed the Bak’s argument. They suggested the density ρ in the second equation listed above should be replaced by ρef , where ρef represents the generalized eﬀective phason density, but the meaning of the quantity is not very clear and its measurement is diﬃcult. Fan et al[13] , Rochal and Lorman[16] have respectively suggested a new version on elasto-/hydro-dynamics of quasicrystals, this compromises diﬀerent models, e.g. those proposed by Ref. [1] and Ref. [5]. At present the experimental data are not so suﬃcient to verify each model individually. It is the anthor’s opinion that it shall be highly desirable that all researchers collaborate the common features of diﬀerent models and ﬁnd a simplest version for elasto-/hydro-dynamics of quasicrystals, then carry out systematic, theoretical, numerical and experimental work; that would be beneﬁcial for promoting the study in the ﬁeld. Fan et al[13] presented results incorporating the arguments of both Bak and Lubensky et al. They suggested the equations of motion under linear and small deformation be written as ⎧ ∂ 2 ui ∂σij ⎪ ⎪ , ⎨ ρ 2 = ∂t ∂xj (10.5-1) ∂Hij ∂wi ⎪ ⎪ , = κ ⎩ ∂t ∂xj It is obvious that the ﬁrst equation of (10.5-1) is equation of conventional elastodynamics, in which ρ is the mass density as mentioned in the previous section, while the second one is diﬀusion equation, where κ = 1/Γw , in which Γw the kinematic coeﬃcient of phason ﬁeld of the material deﬁned by Lubensky et al[1] . It can be seen that the second equation of (10.5-1) is a linearized result of hydrodynamics of quasicrystals of Lubensky et al. The dynamic equations (10.5-1) should be named as the elasto-/hydro-dynamic equations for quasicrystals which are identical to equations (6) of Ref. [16]. This treatment is believed to be physically sound and reconciles the contradiction between the arguments of Bak and Lubensky et al; it also reconciles the contradiction between Refs. [5 ∼ 12] and Ref. [15], though they all follow the framework of Bak’s argument. As an example we discuss herein an approximate analytic solution of moving dislocation in anti-plane problem of three-dimensional icosahedral or cubic or onedimensional hexagonal quasicrystal, see Fan et al[13] . The stress-strain relations in this case are the same as (9.2-3), the diﬀerence lies in only the equations of motion which are given by (10.5.1), such that for anti-plane which become ⎧ 2 ⎪ ⎨ μ∇2 uz + R∇2 wz = ρ ∂ uz , ∂t2 (10.5-2) ⎪ 2 ⎩ R∇ uz + (K1 − K2 )∇2 wz = κ ∂wz . ∂t

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201

Equations (10.5-2) are wave-diﬀusion mixed type equations which are diﬀerent from (10.2-2), so the solution will be diﬃcult to obtain. The following boundary conditions for screw dislocation moving along direction Ox with speed V are considered as ⎧ # 2 2 ⎪ ⎨ x + y → ∞ : σij (x, y, t) → 0, ⎪ ⎩ duz = b3 , dwz = b⊥ 3. Γ

Hij (x, y, t) → 0, (10.5-3)

Γ

Solving boundary value problem (10.5-2), (10.5-3) is more complicated than solving either a pure wave propagation problem or a pure diﬀusion problem. We take the perturbation method along with variational method. Based on the physical consideration R/μ should be small and it can be taken as the perturbation parameter. Firstly, we calculate the zeroth-order perturbation solution. The governing equations (10.5-2) reduce to ⎧ ∂ 2 u0 ⎪ ⎪ ⎨ ρ 2 − μ∇2 u0 = 0, ∂t 0 ⎪ ∂w ⎪ ⎩ κ − (K1 − K2 )∇2 w 0 = 0, ∂t

(10.5-4)

where u0 , w0 denote the zeroth-order perturbation solution of the problem. A coordinate substitution x = x1 − V t

(10.5-5)

is used, here (x1 , y, t) denotes ﬁxed coordinate system and (x, y) the moving coordinate system, and ∂2 ∂2 = V2 2. 2 ∂t ∂x

(10.5-6)

⎧ 2 0 1 ∂ 2 u0 ∂ u ⎪ ⎪ + 2 = 0, ⎨ 2 ∂x β1 ∂y 2 0 ⎪ ∂ 2 w0 ∂ 2 w0 ⎪ 2 ∂w ⎩ + β = 0, + 2 ∂x2 ∂x ∂y 2

(10.5-7)

∂ ∂ = −V , ∂t ∂x Putting (10.5-6) into (10.5-4) yields

where β12 = 1 −

V2 , c2

β22 =

Vκ V = , K1 − K2 (K1 − K2 )Γw

(10.5-8)

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# in which c = μ/ρ for icosahedral quasicrystals is the velocity of transverse wave of phonon ﬁeld. Because the equations in (10.5-7) are decoupled, and considering dislocation conditions in (10.5-3) we have the following two separate dislocation problems such as ⎧ 2 0 1 ∂ 2 u0 ∂ u ⎪ ⎪ + 2 = 0, ⎨ 2 ∂x β1 ∂y2 (10.5-9a) ⎪ 0 ⎪ ⎩ du = b3 , Γ

⎧ 2 0 ∂ w ∂w0 ∂ 2 w0 ⎪ ⎪ + β22 = 0, + ⎨ 2 ∂x ∂x ∂y 2 ⎪ ⎪ ⎩ dw0 = b⊥ 3.

(10.5-9b)

Γ

The solution of problem (10.5-9a) is known, i.e.,

u0 =

b3 β1 y arctan . 2π x

(10.5-10)

To solve problem (10.5-9b) we put w0 = w1 + w2 , in which w1 satisﬁes

⎧ 2 ∂ w1 ∂ 2 w1 ⎪ ⎪ + = 0, ⎨ ∂x2 ∂y2 ⎪ ⎪ ⎩ dw1 = b⊥ 3,

(10.5-11)

(10.5-12)

Γ

so it has the solution w1 =

b⊥ y 3 arctan 2π x

(10.5-13)

and w2 satisﬁes ⎧ β2 b⊥ ∂ 2 w2 ∂ 2 w2 ⎪ 3y ⎪ 2 ∂w2 ⎨ + β = + , 2 2 2 2 + y2 ) ∂x ∂x ∂y 2π(x ⎪ ⎪ dw2 = 0. ⎩ Γ

(10.5-14)

10.5

Elasto-/hydro-dynamics of quasicrystals and approximate analytic solution...

203

The equation in problem (10.5-14) is more complex than that in (10.5-9a) and (10.5-12), so its analytic solution is very diﬃcult to obtain. We can do only approximate solution by variational principle (refer to Section 13.2 in Chapter 13). Omitting the detail the approximate solution is w2 = −

b⊥ β22 x2 y 3 . 2π (x2 + y 2 )(1 + β22 x)

(10.5-15)

From (10.5-11), (10.5-13) and (10.5-15), the zeroth-order approximate solution is

b⊥ w = 3 2π 0

y β22 x2 y arctan − 2 x (x + y 2 )(1 + β22 x)

.

(10.5-16)

Then we calculate the ﬁrst-order solution, the governing equations (10.5-2) are ⎧ ∂ 2 u1 ⎪ ⎪ ⎨ ρ 2 − μ∇2 u1 = R∇2 w 0 , ∂t 1 ⎪ ∂w ⎪ ⎩ κ − (K1 − K2 )∇2 w 1 = R∇2 u0 , ∂t

(10.5-17)

where u1 , w1 are denoted the ﬁrst-order perturbation solution of the problem. According to the boundary condition (10.5-3) and the governing equation (10.5-17), one ﬁnds the ﬁrst order pertubation solution by the variational method that u1 = −

b⊥ R 2β22 x2 y(3x2 − y 2 + 6β22 x3 + 2β24 x4 ) 3 , 2π (ρV 2 − μ) (x2 + y 2 )2 (1 + β22 x)3

b3 R 2(β13 − β1 )x3 y w =− . 2π(K1 − K2 ) (x2 + β12 y 2 )2 (1 + β22 x) 1

(10.5-18)

The approximate solution up to the ﬁrst pertubation of the moving dislocation problem is ⎧ uz (x1 , y, t) ≈ u0z (x1 , y, t)+u1z (x1 , y, t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ b3 β1 y b⊥ R 2β22 x2 y(3x2 −y 2 + 6β22 x3 +2β24 x4 ) ⎪ ⎪ , − 3 ⎨ = arctan 2 2π x 2π (ρV −μ) (x2 +y 2 )2 (1+β22 x)3 0 1 wz (x1 , y, t) ≈ wz (x1 , y, t)+wz (x1 , y, t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ b⊥ ⎪ β22 x2 y y b3 R 2(β13 −β1 )x3 y ⎪ ⎩ = 3 arctan − − , 2 2 2 2 2π x (1+β2 x)(x +y ) 2π(K1 − K2 ) (x +β12 y 2 )2 (1+β22 x) (10.5-19) where we recall x = x1 − V t. So the corresponding strains are as follows:

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⎧ ⎪ b⊥ β22 R β1 x 1 b3 ⎪ ⎪ ε = ε = + 3 ⎪ yz zy ⎪ 2 + β 2 y2 ⎪ 2 2π x π ρV 2 − μ ⎪ 1 ⎪ ⎪ ⎪ 6 − 12x4 y 2 + y 6 + 6(x7 − 3x5 y 2 )β 2 + 2(x8 − 3x6 y 2 )β 4 ⎪ 3x ⎪ 2 2 ⎪ , × ⎪ ⎪ ⎪ (x2 + y 2 )3 (1 + β22 x)3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ b b⊥ β22 R β1 y 1 ⎪ ⎪ εzx = εxz = − 3 2 + 3 ⎪ ⎪ 2 2 ⎪ 2 2π x + β1 y π ρV 2 − μ ⎪ ⎪ ⎪ ⎨ 3 3 5 6 14x y −2xy +(−3x y+38x4 y 3+x2 y 5 )β22 −8(x7 y−3x5 y 3 )β24−2(x8 y−3x6 y3 )β26 , × 2 2 2 3 4 ⎪ (x +y ) (1+β2 x) ⎪ ⎪ ⎪ ⎪ ⎪ 2 2 2 2 ⎪ b3 x3 (x2 − 3β12 y 2 )(β13 − β1 ) b⊥ R ⎪w 3 x(x + y + 2β2 xy ) ⎪ − = , ⎪ zy ⎪ 2 2 2 3 2 + y 2 )2 (1 + β 2 x) 2 ⎪ 2π (x π K − K 1 2 (1 + β2 x)(x + β1 y ) ⎪ 2 ⎪ ⎪ ⎪ ⎪ b⊥ y(2β22 x3 + y 2 + 4β22 xy 2 + x2 (1 + 2β24 y 2 )) ⎪ ⎪ wzx = − 3 ⎪ ⎪ ⎪ 2π (x2 + y 2 )2 (1 + β22 x)2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ b x2 y(x2 + 2β22 x3 − 2β12 β22 xy 2 − 3β12 y 2 )(β13 − β1 ) R ⎪ ⎪ + 3 ⎩ π K1 − K2 (1 + β22 x)2 (x2 + β12 y 2 )3

(10.5-20)

and the stresses can be expressed by

⎧ σyz = σzy ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2β22 Rμ 3x6 −12x4 y 2 +y 6 +6(x7 −3x5 y 2 )β22 +2(x8 −3x6 y 2 )β24 b⊥ ⎪ 3 ⎪ = ⎪ ⎪ ⎪ 2π ρV 2 −μ (x2 +y 2 )3 (1+β22 x)3 ⎪ ⎪ ⎪ ⎪ ⎪ 2 2 2 2 ⎪ b Rx(x +y +2β2 xy ) μβ1 x 2R2 x3 (x2 −3β12 y 2 )(β13 −β1 ) ⎪ ⎪ + 3 + − , ⎪ ⎪ 2 2 2 2 2 2 2 2 2 2 2 3 ⎪ (x +y ) (1+β2 x) 2π x +β1 y K1−K2 (1+β2 x)(x +β1 6y ) ⎪ ⎪ ⎪ ⎪ ⊥ 2 ⎪ b R 2β2 μ ⎪ ⎪ σzx = σxz = 3 ⎪ ⎪ ⎪ 2π ρV 2 −μ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 14x3 y 3−2xy 5+(−3x6 y+38x4 y 3 +x2 y 5 )β22−8(x7 y−3x5 y 3 )β24−2(x8 y−3x6 y 3 )β26 ) ⎪ ⎪ ⎪ × ⎪ ⎪ (x2 +y 2 )3 (1+β22 x)4 ⎪ ⎪ ⎪ ⎪ 2 3 2 2 2 2 ⎪ y(2β2 x +y +4β2 xy +x (1+2β24 y 2 )) ⎪ ⎪ ⎪ − ⎪ ⎪ (x2 +y 2 )2 (1+β22 x)2 ⎪ ⎪ ⎪ ⎪ ⎨ b 2R2 x2 y(x2 +2β22 x3 −2β12 β22 xy 2 −3β12 y 2 )(β13 −β1 ) μβ1 y − , + 3 2 2 2 ⎪ 2π K1 −K2 (1+β2 x)2 (x2 +β1 y 2 )3 x2 +β1 y 2 ⎪ ⎪ ⎪ ⊥ 2β 2 R2 3x6 −12x4 y 2 +y 6 +6(x7 −3x5 y 2 )β 2 +2(x8 −3x6 y 2 )β 4 ⎪ b ⎪ 3 2 2 2 ⎪ ⎪ ⎪Hzy = 2π ρV 2 −μ ⎪ (x2 +y 2 )3 (1+β22 x)3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ x(x2 +y 2 +2β22 xy2 ) b3 R 2x3 (x2−3β12 y 2 )(β13 −β1 ) β1 x ⎪ ⎪ + +(K1 −K2 ) 2 − . ⎪ ⎪ 2 2 2 2 2 2 2 2 2 2 3 ⎪ (x +y ) (1+β2 x) 2π x +β1 y (1+β2 x)(x +β1 y ) ⎪ ⎪ ⎪ ⎪ ⎪ b⊥ 2β22 R ⎪ ⎪ Hzx = 3 ⎪ ⎪ ⎪ 2π ρV 2 −μ ⎪ ⎪ ⎪ ⎪ ⎪ 14x3 y 3−2xy 5+(−3x6 y+38x4 y 3+x2 y 5 )β22−8(x7 y−3x5 y 3 )β24−2(x8 y−3x6 y 3 )β26 ⎪ ⎪ × ⎪ ⎪ ⎪ (x2+y 2 )3 (1+β22 x)4 ⎪ ⎪ ⎪ ⎪ 2 3 2 2 2 ⎪ y(2β2 x +y +4β2 xy +x2 (1+2β24 y 2 )) ⎪ ⎪ −(K −K ) ⎪ 1 2 ⎪ ⎪ (x2 +y 2 )2 (1+β22 x)2 ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ b R 2x y(x2 +2β22 x3 −2β12 β22 xy2 −3β12 y 2 )(β13 −β1 ) β1 y ⎪ ⎪ + 3 − . ⎩ 2 2 2 2π (1+β2 x)2 (x2 +β1 y 2 )3 x2 +β1 y2

(10.5-21)

For cubic or hexagonal quasicrystals, the solutions are exactly similar, with

10.5

Elasto-/hydro-dynamics of quasicrystals and approximate analytic solution...

205

μ, (K1 − K2 ) and R replaced by C44 , K44 and R44 or by C44 , K2 and R3 respectively in the above equations. It is easy to observe that the solution contains the contributions coming from three parts of wave propagation, diﬀusion and their interaction. The solution on phonon ﬁeld is dominated by wave propagation, and the solution on phason ﬁeld is dominated by diﬀusion, so the result is quite diﬀerent from the dynamic solution of the dislocation based on the Bak’s argument, which was discussed in Section 10.3 already. The present solution can reduce to that in static dislocation as V = 0, which can also reduce to the solution of crystals, this demonstrates the correctness of the work. The solution reveals that the phason ﬁeld resists motion of dislocation, this is the reason of diﬃculty of plastic deformation of quasicrystalline materials. The illustrations of the solution and comparison with other solutions for the case # V = 0.1c, c = μ/ρ are depicted in Figs.10.5-1∼10.5-4, in which the solution given by Ref. [7] has been introduced in Section 10.3. In the numerical computation we take the data of material constants as b⊥ 3 = 0.8b3

Fig.10.5-1

Variation of uz /b3 at x = 0.01 mm versus y

Fig.10.5-2

Variation of uz /b3 at y = 0.01 mm versus x

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Dynamics of elasticity and defects of quasicrystals

Fig.10.5-3

Variation wz /b3 at x = 0.01 mm versus y

Fig.10.5-4

Variation wz /b3 at y = 0.01 mm versus x

ρ = 4.1kg/cm3 ,

μ = 70GPa,

K1 = 74.2,

K2 = −37MPa,

1 = 4.8 × 10−19 m3 · s/kg = 4.8 × 10−10 cm3 · μs/g κ # given in Tables 9.1-1∼9.1-3. We recall x = x1 − V t, c = μ/ρ.

R = 0.004, μ

Γw =

10.6

Elasto-/hydro-dynamics and solutions of two-dimensional decagonal quasicrystals

In this section we would like to give a detailed description on the solution of twodimensional quasicrystals based on the elasto-/hydro-dynamics formulation[14] . The equations of deformation geometry and the generalized Hooke’s law are the same as before which will not be repeated here. By using the dynamic equations (10.5-1), i.e., the so-called elasto-/hydro-dynamic equations for quasicrystals, we can derive the mathematical formalism for the new

10.6

Elasto-/hydro-dynamics and solutions of two-dimensional decagonal...

207

dynamics of two-dimensional quasicrystals. As an application of the formulation, some dynamic crack solutions are given in this section. 10.6.1

The mathematical formulation of dynamic crack problems of decagonal quasicrystals

Among 200 quasicrystals observed to date, there are 70 two-dimensional decagonal quasicrystals; so this kind of solid phases plays an important role in the material. For simplicity, herein only point group 10mm two-dimensional decagonal quasicrystal will be considered. We denote the periodic direction as the z-axis and the quasiperiodic plane as the xy-plane. Assume that a crack in the solid along the periodic direction, i.e., the z-axis. It is obvious that elastic ﬁeld induced by a uniform tensile stress at upper and lower surfaces of the specimen is independent of z, so ∂/∂z = 0. In this case, the stress-strain relations are reduced to ⎧ σxx = L(εxx + εyy ) + 2M εxx + R(wxx + wyy ), ⎪ ⎪ ⎪ ⎪ ⎪ σ = L(εxx + εyy ) + 2M εyy − R(wxx + wyy ), ⎪ ⎪ ⎪ yy ⎪ ⎪ ⎪ ⎨ σxy = σyx = 2M εxy + R(wyx − wxy ), Hxx = K1 wxx + K2 wyy + R(εxx − εyy ), (10.6-1) ⎪ ⎪ ⎪ Hyy = K1 wyy + K2 wxx + R(εxx − εyy ), ⎪ ⎪ ⎪ ⎪ ⎪ Hxy = K1 wxy − K2 wyx − 2Rεxy , ⎪ ⎪ ⎪ ⎩ H = K w − K w + 2Rε , yx 1 yx 2 xy xy where L = C12 , M = (C11 − C12 )/2 are the phonon elastic constants, K1 and K2 the phason elastic constants, R phonon-phason coupling elastic constant. Substituting (10.6-1) into (10.5-1), we obtain the equations of motion of decagonal quasicrystals as follows: 2 ⎧ 2 2 2 2 ∂ wx ∂ 2 wx ∂ ux ∂ 2 wy ⎪ 2 ∂ ux 2 2 ∂ uy 2 ∂ ux 2 ⎪ + c − , = c1 + (c1 − c2 ) + c3 +2 ⎪ 2 ⎪ ∂t2 ∂x2 ∂x∂y ∂y 2 ∂x2 ∂x∂y ∂y 2 ⎪ ⎪ 2 ⎪ 2 2 2 ⎪ ∂ 2 uy ⎪ ∂ 2 wx ∂ wy ∂ 2 wy ⎪ 2 ∂ uy 2 2 ∂ ux 2 ∂ uy 2 ⎪ = c + (c − c ) + c − 2 + c − , ⎨ 2 1 2 1 3 ∂t2 ∂x2 ∂x∂y ∂y 2 ∂x2 ∂x∂y ∂y 2 2 ⎪ ∂ 2 uy ∂ wx ∂ 2 wx ∂ 2 ux ∂ 2 ux ∂wx ⎪ 2 2 ⎪ + − 2 = d − + d , ⎪ 1 2 ⎪ ∂t ∂x2 ∂y2 ∂x2 ∂x∂y ∂y 2 ⎪ ⎪ ⎪ ⎪ ⎪ ∂ 2 wy ∂ 2 ux ∂ 2 wy ∂ 2 uy ∂ 2 uy ∂wy ⎪ 2 ⎩ + + 2 = d21 − + d , 2 ∂t ∂x2 ∂y 2 ∂x2 ∂x∂y ∂y 2 (10.6-2) in which & & & + + + L + 2M M R K1 R K2 , c2 = , c3 = , d1 = , d2 = , d3 = , c1 = ρ ρ ρ κ κ κ (10.6-3)

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note that constants c1 , c2 and c3 have the meaning of elastic wave speeds, while d21 , d22 and d23 do not represent wave speeds, they are diﬀusive coeﬃcients. The decagonal quasicrystal with a crack is shown in Fig.10.6-1. It is a rectangular specimen with a central crack of length 2a(t) subjected to a dynamic or static tensile stress at its ends ED and F C, in which a(t)represents the crack length as a function of time, and for dynamic initiation of crack growth, the crack is stable, so a(t) = a0 = constant, for fast crack propagation, a(t) varies with time. At ﬁrst we consider dynamic initiation of crack growth, then study crack fast propagation.

Fig.10.6-1

The specimen with a central crack

Due to the symmetry of the specimen only the upper right quarter is considered. Referring to the upper right part and considering a ﬁx grips case, the following boundary conditions should be satisﬁed: ⎧ ux = 0, σyx = 0, wx = 0, Hyx = 0, on x = 0 for 0 y H, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ σxx = 0, σyx = 0, Hxx = 0, Hyx = 0, on x = L for 0 y H, σyy = p(t), σxy = 0, Hyy = 0, Hxy = 0, on y = H for 0 x L, (10.6-4) ⎪ ⎪ ⎪ σyy = 0, σxy = 0, Hyy = 0, Hxy = 0, on y = 0 for 0 x a(t), ⎪ ⎪ ⎪ ⎩ uy = 0, σxy = 0, wy = 0, Hxy = 0, on y = 0 for a(t) x L, where p(t) = p0 f (t) is a dynamic load if f (t) varies with time, otherwise it is a static load (i.e., if f (t) = const), and p0 = const with the stress dimension. The initial conditions are ⎧ ux (x, y, t)|t=0 = 0, uy (x, y, t)|t=0 = 0, ⎪ ⎪ ⎨ wx (x, y, t)|t=0 = 0, wy (x, y, t)|t=0 = 0, (10.6-5) ⎪ ∂uy (x, y, t) ⎪ ∂ux (x, y, t) ⎩ |t=0 = 0, |t=0 = 0. ∂t ∂t

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Elasto-/hydro-dynamics and solutions of two-dimensional decagonal...

209

For implementation of ﬁnite diﬀerence all ﬁeld variables in both governing equations (10.6-2) and boundary-initial conditions (10.6-4), (10.6-5) must be expressed by displacements and their derivatives. This can be done through the constitutive equations (10.6-1). The detail of the ﬁnite diﬀerence scheme is given in Appendix of this chapter. For the related parameters in this section, the experimentally determined mass density for decagonal Al-Ni-Co quasicrystal: ρ = 4.186 × 10−3 g · mm−3 is used and elastic phonon moduli are taken as C11 = 2.3433, C12 = 0.5741(1012 dyn/cm2 = 102 GPa) which are obtained by resonant ultrasound spectroscopy[19] , we have also chosen phason elastic constants K1 = 1.22 and K2 = 0.24(1012 dyn/cm2 = 102 GPa) estimated by Monto-Carlo simulation[20] and Γw = 1/κ = 4.8 × 10−19 m3 · s/kg = 4.8×10−10 cm3 ·μs/g[21] .The coupling constant R has been measured for some special cases recently, see Chapter 6 and Chapter 9 respectively. In computation we take R/M = 0.01 for coupling case corresponding to quasicrystals, and R/M = 0 for decoupled case which corresponds to crystals. 10.6.2

Examination on the physical model

In order to verify the correctness of the suggested model and the numerical simulation, we ﬁrst explore the specimen without a crack. We know that there are the fundamental solutions characterizing time variation natures based on wave propagation of phonon ﬁeld and on diﬀusion according to the mathematical physics ⎧ ⎨ u ∼ eiω(t−x/c) , 1 2 e−(x−x0 ) /Γw (t−t0 ) , ⎩ w∼ √ t − t0

(10.6-6)

where ω is a frequency and c is a speed of the wave, t is the time and t0 is a special value of t, x is a distance, x0 is a special value of x, and Γw is the kinetic coeﬃcient of phason deﬁned previously. Comparison results are shown in Fig.10.6-2 (a∼c), in which the solid line represents the numerical solution of quasicrystals and the dotted line represents fundamental solution of equation (10.6-6). From Figs. 10.6-2(a) and (b) we can see that both displacement components of phonon ﬁeld are in excellent agreement to the fundamental solutions. However, there are some diﬀerences because the phonon ﬁeld is inﬂuenced by phason ﬁeld and the phonon-phason coupling eﬀect. From Fig. 10.6-2(c), in the phason ﬁeld we ﬁnd that the phason mode presents diﬀusive nature in the overall tendency, but because of inﬂuence of the phonon and phonon-phason coupling, it also has some character of ﬂuctuation. So the model describes the dynamic behaviour of phonon ﬁeld and phason ﬁeld. This also shows the mathematical modelling of the present work is valid.

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(a) Displacement component of phonon ﬁeld ux versus time

(b) Displacement component of phonon ﬁeld uy versus time

(c) Displacement component of phason ﬁeld wx versus time Fig.10.6-2

Solutions without crack and comparison to fundamental solutions

10.6

Elasto-/hydro-dynamics and solutions of two-dimensional decagonal...

10.6.3

211

Testing the scheme and the computer program

10.6.3.1 Stability of the scheme The stability of the scheme is the core problem of ﬁnite diﬀerence method which depends upon the choice of parameter α = c1 τ /h, the ratio between time step and space step substantively. The choice is related to the ratio c1 /c2 , i.e., the ratio between speeds of elastic longitudinal and transverse waves of the phonon ﬁeld. To determine the upper bound for the ratio to guarantee the stability, according to our computational practice and experiences of computations for conventional materials, α = 0.8 is chosen in all cases. The computational results are stable. 10.6.3.2 Accuracy test The stability is only a necessary condition for successful computation. We must check the accuracy of the numerical solution. This can be realized through some comparison with some well known classical solutions (analytic as well as numerical solutions). For this purpose the material constants in the computation are chosen 3 as c1 = 7.34, c2 = 3.92 (mm/μs) and ρ = 5 × 103 kg/m , p0 = 1MPa which are the same as those in Refs. [22∼24] (but are diﬀerent from those listed in Subsection 10.6.1).At ﬁrst the comparison to the classical exact analytic solution is carried out, in this case we put wx = wy = 0 (i.e., K1 = K2 = R = 0) for the numerical solution. The comparison has been done with the key physical quantity—dynamic stress intensity factor, which is deﬁned by # KI (t) = lim π(x − a0 )σyy (x, 0, t). (10.6-7) x→a+ 0

The normalized dynamic stress intensity factor can be denoted as KI (t)/KIstatic , in which KIstatic is the corresponding static stress intensity factor, whose value here is √ taken as πa0 p0 (the reason for this refer to Subsection 10.6.3.3). For the dynamic initiation of crack growth in classical fracture dynamics there is the only exact analytic solution—the Maue’s solution[22] , but the conﬁguration of whose specimen is quite diﬀerent from that of our specimen. Maue studied a semi-inﬁnite crack in an inﬁnite body, and subjected to a Heaviside impact loading at the crack surface. While our specimen is a ﬁnite size rectangular plate with a central crack, and the applied stress is around the external boundary of the specimen. Generally, the Maue model cannot describe the interaction between wave and external boundary. However, consider a very short time interval, i.e., during the period between the stress wave from the external boundary arriving at the crack tip (this time is denoted by t1 ) and before the reﬂecting by external boundary stress wave emanating from the crack tip in the ﬁnite size specimen (the time is marked as t2 ). During this

212

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special short time interval our specimen can be seen as an “inﬁnite specimen”. The comparison given by Fig.10.6-3 shows the numerical results are in excellent agreement with those of Maue’s solution within the short interval in which the solution is valid. Our solution for the case of wx = wy = 0 is also compared with numerical solutions of conventional crystals, e.g. Murti’s solution[23] and Chen’s solutions[24] , which are also shown in Fig.10.6-3, it is evident, our solution is in good agreement with available solutions.

Fig.10.6-3

Comparison of our solution with analytic solution and other numerical solution for crystalline materials

10.6.3.3 Inﬂuence of mesh size (space step) The mesh size or the space step of the algorithm can inﬂuence the computational accuracy too. To check the accuracy of the algorithm we take diﬀerent space steps shown in Table 10.6-1, which indicates if h = a0 /40 the accuracy is good enough. The check is carried out through static solution, because the static crack problem in inﬁnite body of decagonal quasicrystals has exact solution, see Chapter 8, the normalized static intensity factor is equal to unit. In the static case, there is no wave propagation eﬀect, L/a0 3, H/a0 3 the eﬀect of boundary to solution is very weak, and for our present specimen L/a0 4, H/a0 8, which may be seen as an inﬁnite specimen, so the normalized static stress intensity factor is approximately but with highly precise equal to unit. The table shows that the algorithm is with a quite highly accuracy when h = a0 /40. Table 10.6-1

The normalized static S.I.F. of quasicrystals for diﬀerent space steps

h ¯ K Errors

a0 /10 0.9259 7.410%

a0 /15 0.94829 5.171%

a0 /20 0.96229 3.771%

a0 /30 0.97723 2.277%

a0 /40 0.99516 0.484%

10.6

Elasto-/hydro-dynamics and solutions of two-dimensional decagonal...

10.6.4

213

Results of dynamic initiation of crack growth

The dynamic crack problem indicates two “phases”in the process: the dynamic initiation of crack growth and fast crack propagation. In the phase of dynamic initiation of crack growth, the length of the crack is constant, assuming a(t) = a0 . The specimen with stationary crack subjected to a rapidly varying applied load p(t) = p0 f (t), where p0 is a constant with stress dimension and f (t) is taken as the Heaviside function. It is well known the coupling eﬀect between phonon and phason is very important, which reveals the distinctive physical properties including mechanical properties, and makes quasicrystals diﬀerent from the periodic crystals. So studying the coupling eﬀect is signiﬁcant. The dynamic stress intensity factor KI (t) for quasicrystals has the same deﬁnition given by (10.6-7), whose numerical results are plotted in Fig.10.6-4, where √ the normalized dynamics stress intensity factor K I (t)/ πa0 p0 is used. There are two curves in the ﬁgure, one represents quasicrystal, i.e.,R/M = 0.01, the other describes periodic crystals corresponding to R/M = 0, the two curves of the ﬁgure is apparently diﬀerent, though they are similar to some extent. Because of the phonon-phason coupling eﬀect, the mechanical properties of the quasicrystals are obviously diﬀerent from the crystals. Thus, the coupling eﬀect plays an important role. In Fig.10.6-4, t0 represents the time that the wave from the external boundary propagates to the crack surface, in which t0 = 2.6735μs. So the velocity of the wave propagation is ν0 = H/t0 = 7.4807km/s, which is just equal to the longitudinal # wave speed c1 = (L + 2M )/ρ. This indicates that for the complex system of coupled motion of wave propagation and diﬀusion the phonon wave propagation plays dominating role.

Fig.10.6-4

Normalized dynamics stress intensity factor (DSIF) versus time

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Dynamics of elasticity and defects of quasicrystals

There are many oscillations in the ﬁgure, especially the stress intensity factor. These oscillations characterize the reﬂection and diﬀraction between waves coming from the crack surface and the specimen boundary surfaces. The oscillations are inﬂuenced by the material constants and specimen geometry including the shape and size. 10.6.5

Results of the fast crack propagation

In this section, we focus on the discussion for the “phase” of fast crack propagation. To explore the inertia eﬀect caused by the fast crack propagation, the specimen is designed under the action of constant load P (t) = p rather than time-varying load, but the crack grows with high speed in this case. The problem for fast crack propagation is a nonlinear problem, because one part of the boundaries—crack is with unknown length beforehand. For this moving boundary problem, we must give additional condition for determining solution. That is, we must give a criterion checking crack propagation or crack arrest at the growing crack tip. This criterion can be imposed in diﬀerent ways, e.g., the critical stress criterion or critical energy criterion. The stress criterion is used in this paper:σyy < σc represents crack arrest, σyy = σc represents critical state and σyy > σc represents crack propagation. Here we take σc = 450M P a for decagonal Al-Ni-Co quasicrystals, which was obtained by referring measured value by Meng et al for decagonal Al-Cu-Co quasicrystals, refer to Ref.[2] in Chapter 8, the modiﬁcation by referring the hardness of alloys Al-Ni-Co and Al-Cu-Co, and the hardness on decagonal Al-Ni-Co can be found in paper given by Takeuchi et al[30] . The simulation of a fracturing process runs as follows: Given the specimen geometry and its material constants we ﬁrst solve the initial dynamic problem in the way previously described. When the stress σyy reaches a prescribed critical value σc the crack is extended by one grid interval. The crack now continues to grow, by one grid interval at a time, as long as the σyy stress level ahead of the propagating crack tip reaches the value of σc . During the propagation stage the time that elapses between two sequential extensions is recorded and the corresponding velocity is evaluated. The crack velocity for quasicrystals and periodic crystals is constructed in Fig.10.65, from the ﬁgure, we observe that the velocity in quasicrystals is lower than that of the periodic crystals; the phonon-phason coupling makes the quasicrystals being diﬀerent from periodic crystals. The reason for this is not so clear. Though Mikulla et al[25] gave an explanation for crack slowly propagation (i.e., the quasi-static crack growth) in decagonal quasi crystals, the case is quite diﬀerent from the present crack fast propagation. We ﬁnd that the fast crack propagating velocity is obviously diﬀerent in qua-

10.6

Elasto-/hydro-dynamics and solutions of two-dimensional decagonal...

215

sicrystals compared to the conventional engineering material. Next we will explore the velocity under diﬀerent loads in quasicrystal. The above described procedure was conducted, keeping the same geometry and material constants. With various loads, the relation between velocity and crack growth is constructed in Fig.10.6-6. The crack velocity increases with increasing applied load smoothly. It is understandable that as the load increases the time to reach the critical stress is less, so the velocity increases.

Fig.10.6-5

The crack velocity versus normalized crack size with diﬀerent coupling constants

Fig.10.6-6

Variation of crack velocity versus crack growth size for diﬀerent load levels

As shown in Fig.10.6-7, the calculated crack propagation results show some roughness as the load level increases. Currently there is no experimental observation for fast crack propagation, though Ebert et al[26] made some observation by

216

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Dynamics of elasticity and defects of quasicrystals

scanning tunneling microscopy for quasi-static crack growth. Because the fast propagation and quasi-static crack growth belong to two diﬀerent regimes, the comparison cannot be easily made.

Fig.10.6-7

Normalized crack growth size (a − a0 )/a0 of crack tip versus time for diﬀerent load levels

10.7 10.7.1

Elasto-/hydro-dynamics and applications to fracture dynamics of icosahedral quasicrystals Basic equations, boundary and initial conditions

The elasto-/hydro-dynamics of icosahedral Al-Pd-Mn quasicrystals is more interesting topic than that of decagonal Al-Ni-Co quasicrystals, especially a rich set of experimental data for elastic constants can be used for the computation described here. From the previous section we know there are lack of measured data for phason elastic constants, which are obtained by Monte Carlo simulation, and results in uncertainty for computation for decagonal quasicrystals. This shows the discussion on icosahedral quasicrystals is more necessary, and the formulation and numerical results are presented in this section. If considering only the plane problem, especially for the crack problems, there are much of similarities with those discussed in the previous section. We present herein only the distinguishing part that are diﬀerent. For the plane problem, i.e., ∂ = 0, (10.7-1) ∂z the linearized elasto-/hydro-dynamics of icosahedral quasicrystals has non-zero displacements uz , wz apart from ux , uy , wx , wy , so in the strain tensors,

10.7

Elasto-/hydro-dynamics and applications to fracture dynamics of...

1 εij = 2

∂uj ∂ui + ∂xj ∂xi

,

wij =

217

∂wi , ∂xj

it increases some non-zero components compared with those in two-dimensional quasicrystals. In connecting with this, in the stress tensors, the non-zero components increase too. With these reasons, the stress-strain relation presents diﬀerent nature with that of decagonal quasicrystals though the generalized Hooke’s law has the same form with that in one- and two-dimensional quasicrystals, i.e., σij = Cijkl εkl + Rijkl wkl ,

Hij = Rklij εkl + Kijkl wkl .

In particular the elastic constants are quite diﬀerent from those discussed in the previous sections, in which the phonon elastic constants can be expressed such as Cijkl = λδij δkl + μ(δik δjl + δil δjk )

(10.7-2)

and the phason elastic constant matrix [K] and phonon-phason coupling elastic one [R] are deﬁned by formula (9.1-6) in Chapter 9, which are not listed here again. Substituting these non-zero stress components into the equations of motion ρ

∂σij ∂ 2 ui = , 2 ∂t ∂xj

κ

∂wi ∂Hij = ∂t ∂xj

(10.7-3)

and through the generalized Hooke’s law and strain-displacement relation we obtain the ﬁnal dynamic equations as follows: 2 2 2 2 ∂ wx ∂ 2 ux ∂ux ∂ 2 wy ∂ 2 wx 2 ∂ ux 2 2 ∂ uy 2 ∂ ux 2 = c1 +c − , +θ +(c1 −c2 ) +c3 +2 ∂t2 ∂t ∂x2 ∂x∂y 2 ∂y2 ∂x2 ∂x∂y ∂y 2 ∂ 2 wy ∂ 2 uy ∂ 2 uy ∂ 2 uy ∂uy ∂ 2 ux ∂ 2 wx ∂ 2 wy = c22 +c21 − , +θ +(c21 −c22 ) +c23 −2 2 2 2 2 ∂t ∂t ∂x ∂x∂y ∂y ∂x ∂x∂y ∂y 2 2 2 ∂ ∂ wx ∂ 2 wx ∂ 2 uz ∂uz ∂2 ∂ 2 wy ∂ 2 wz ∂ 2 wz 2 2 = c + u , +θ + +c − −2 + z 2 3 ∂t2 ∂t ∂x2 ∂y 2 ∂x2 ∂y2 ∂x∂y ∂x2 ∂y2 2 ∂ ∂2 ∂wx ∂2 ∂2 +θwx = d1 w wz + +d − x 2 ∂t ∂x2 ∂y 2 ∂x2 ∂y2 2 ∂ ux ∂ 2 uy ∂ 2 ux ∂ 2 uz ∂ 2 uz − +d3 −2 + − ∂x2 ∂x∂y ∂y 2 ∂x2 ∂y2 2 2 2 2 ∂wy ∂ wz ∂ ∂ 2 ux ∂ 2 uy ∂ 2 uz ∂ ∂ uy + −d +2 −2 +θwy = d1 +d − , w y 2 3 ∂t ∂x2 ∂y 2 ∂x∂y ∂x2 ∂x∂y ∂y2 ∂x∂y 2 ∂ ∂2 ∂wz + +θwz = (d1 −d2 ) wz ∂t ∂x2 ∂y2 2 2 ∂ wx ∂ 2 wx ∂ ∂ 2 wy ∂2 +d uz , +d2 − −2 + 3 ∂x2 ∂y 2 ∂x∂y ∂x2 ∂y 2 (10.7-4)

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in which & c1 =

λ + 2μ , ρ

+ c2 =

& μ , ρ

c3 =

R , ρ

d1 =

K1 , κ

d2 =

K2 , κ

R , κ (10.7-5)

d3 =

note that constants c1 , c2 and c3 have the meaning of elastic wave speeds, while d1 , d2 and d3 do not represent wave speed, but are diﬀusive coeﬃcients and parameter θ may be understood as a manmade damping coeﬃcient as in the previous section. Consider an icosahedral quasicrystal specimen with a central crack shown in Fig.10.6-1, all parameters of geometry and loading are the same with those given in the previous section, but in the boundary conditions there are some diﬀerent points, which are given as below if only the upper right quarter of the specimen is considered: ⎧ u = 0, σyx = 0, σzx = 0, wx = 0, Hyx = 0, Hzx = 0, on x = 0 for 0 y H, ⎪ ⎪ ⎪ x ⎪ ⎪ ⎪ ⎪ ⎨ σxx = 0, σyx = 0, σzx = 0, Hxx = 0, Hyx = 0, Hzx = 0, on x = L for 0 y H, σyy = p(t), σxy = 0, σzy = 0, Hyy = 0, Hxy = 0, Hzy = 0, on y = H for 0 x L, ⎪ ⎪ ⎪ ⎪ σyy = 0, σxy = 0, σzy = 0, Hyy = 0, Hxy = 0, Hzy = 0, on y = 0 for 0 x a(t), ⎪ ⎪ ⎪ ⎩ u = 0, σ = 0, σ = 0, w = 0, H = 0, H = 0, on y = 0 for a(t) < x L. y

xy

zy

y

xy

zy

(10.7-6) The initial conditions are ux (x, y, t)|t=0 = 0,

uy (x, y, t)|t=0 = 0,

uz (x, y, t)|t=0 = 0,

wx (x, y, t)|t=0 = 0,

wy (x, y, t)|t=0 = 0,

wz (x, y, t)|t=0 = 0,

∂ux (x, y, t) |t=0 = 0, ∂t

∂uy (x, y, t) |t=0 = 0, ∂t

∂uz (x, y, t) |t=0 = 0. ∂t

10.7.2

(10.7-7)

Some results

We now concentrate on investigating the phonon and phason ﬁelds in the icosahedral 3 Al-Pd-Mn quasicrystal, in which we take ρ = 5.1g/cm and λ = 74.2, μ = 70.4GPa of the phonon elastic moduli, for phason ones K1 = 72, K2 = −37MPa (refer to Chapter 9) and the constant relevant to diﬀusion coeﬃcient of phason is Γw = [21] 1/κ = 4.8 × 10−10 cm3 · μs/g . On the phonon-phason coupling constant, there is no measured result for icosahedral quasicrystals so far, we take R/μ = 0.01 for quasicrystals, and R/μ = 0 for “decoupled quasicrystals” or crystals. The problem is solved by the ﬁnite diﬀerence method, the principle, scheme and algorithm are almost the same as shown in the previous section, and shall not be repeated here. The testing for the physical model, scheme, algorithm and computer program are similar to those given in Section 10.6.

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219

The numerical results for dynamic initiation of crack growth problem, the phonon and phason displacements are shown in Fig.10.7-1.

Fig.10.7-1

Displacement components of quasicrystals versus time

(a)displacement component ux ; (b)displacement component uy ; (c)displacement component wx ; (d)displacement component wy

The dynamic stress intensity factor KI (t) is deﬁned by KI (t) = lim

x→a+ 0

Fig.10.7-2

# π(x − a0 )σyy (x, 0, t)

Normalized dynamic stress intensity factor of central crack specimen under impact loading versus time

220

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˜ I (t) = KI (t)/√πa0 p0 and the normalized dynamics stress intensity factor (S.I.F.) K is used, the results are illustrated in Fig.10.7-2, in which the comparison with those of crystals are shown, one can see the eﬀects of phason and phonon-phason coupling are evident very much. For the fast crack propagation problem the primary results of the dynamic stress intensity factor versus time are given in Fig.10.7-3.

Fig. 10.7-3

Normalized stress intensity factor of propagating crack with constant crack speed versus time

Details of this work are reported in Ref. [27]. 10.7.3

Conclusion and discussion

In Sections 10.5∼10.7 a new model on dynamic response of quasicrystals is formulated based on both argument of Bak as well as argument of Lubensky et al. This model can be regarded as an elasto-/hydro-dynamics model for the material, or as a collaborating model of wave propagation and diﬀusion. This model is more complex than that of pure wave propagation model for conventional crystals, and analytic solution is very diﬃcult to obtain, except for a few simple examples given in Section 10.5. Numerical procedure based on ﬁnite diﬀerence algorithm are developed, the computed results conﬁrm the validity of wave propagation behaviour of phonon ﬁeld, and behaviour of diﬀusion of phason ﬁeld. The interaction between phonons and phasons are also recorded. The ﬁnite diﬀerence formalism is applied to analyze dynamic initiation of crack growth and crack fast propagation for two-dimensional decagonal Al-Ni-Co and three-dimensional icosahedral Al-Pd-Mn quasicrystals, the displacement and stress ﬁelds around the tip of stationary and propagating cracks are investigated. The

10.8

Appendix of Chapter 10: The detail of ﬁnite diﬀerence scheme

221

stress presents singularity with order r −1/2 , in which r denotes the distance measured from the crack tip. For the fast crack propagation, which is a nonlinear problem— moving boundary problem, one must provide additional condition for determining solution. For this purpose we give a criterion for checking crack propagation/crack arrest based on the critical stress criterion. Applications of this additional condition for determining solution enable us to conduct numerical simulation of the moving boundary value problem and examine the crack length-time evolution. However, much important and diﬃcult problems are left open for further study[31] .

10.8

Appendix of Chapter 10: The detail of ﬁnite diﬀerence scheme

Equations (10.6-2) subjected to conditions (10.6-4) and (10.6-5) are very complicated, analytic solution for the boundary-initial value problem is not available at present, which have to be solved by numerical method. Here we extend the method of ﬁnite diﬀerence of Shmuely and Alterman[28] scheme for analyzing crack problem for conventional engineering materials to quasicrystalline materials. A grid is imposed on the upper right of the specimen shown in Fig.10.8-1. For

Fig.10.8-1

Scheme of grid used in the text

convenience, the mesh size h is taken to be the same in both x and y directions. The grid is extended beyond the half step by adding four special grid lines x = −h/2, x = L + h/2, y = −h/2, y = H + h/2, which form the grid boundaries. Denoting the time step by τ and using central diﬀerence approximations, the ﬁnite diﬀerence formulation of equations (10.6-2), valid at the inner part of the grids is

222

Chapter 10

Dynamics of elasticity and defects of quasicrystals

⎧ ux (x, y, t + τ ) = 2ux (x, y, t) − ux (x, y, t − τ ) ⎪ ⎪ ⎪ τ 2 ⎪ ⎪ ⎪ ⎪ c1 [ux (x + h, y, t) − 2ux (x, y, t) + ux (x − h, y, t)] + ⎪ ⎪ ⎪ h 2 ⎪ ⎪ τ ⎪ ⎪ (c21 − c22 )[uy (x + h, y + h, t) + ⎪ ⎪ h ⎪ ⎪ ⎪ −uy (x + h, y − h, t) − uy (x − h, y + h, t) + uy (x − h, y − h, t)] ⎪ ⎪ ⎪ τ 2 ⎪ ⎪ ⎪ ⎪ c2 [ux (x, y + h, t) − 2ux (x, y, t) + ux (x, y − h, t)] + ⎪ ⎪ h 2 ⎪ ⎪ τ ⎪ ⎪ ⎪ + c3 [wx (x + h, y, t) − 2wx (x, y, t) + wx (x − h, y, t)] ⎪ ⎪ h ⎪ 2 ⎪ τ ⎪ ⎪ +2 c23 [wy (x + h, y + h, t) − wy (x + h, y − h, t) ⎪ ⎪ h ⎪ ⎪ ⎪ ⎪ −wy (x − h, y + h, t) + wy (x − h, y − h, t)] ⎪ ⎪ ⎪ τ 2 ⎪ ⎪ ⎪ − c3 [wx (x, y + h, t) − 2wx (x, y, t) + wx (x, y − h, t)], ⎪ ⎪ ⎪ h ⎪ ⎪ ⎪ uy (x, y, t + τ ) = 2uy (x, y, t) − uy (x, y, t − τ ) ⎪ ⎪ ⎪ τ 2 ⎪ ⎪ ⎪ ⎪ c2 [uy (x + h, y, t) − 2uy (x, y, t) + uy (x − h, y, t)] + ⎪ ⎪ ⎪ ⎪ hτ 2 ⎪ ⎪ ⎪ + (c21 − c22 )[ux (x + h, y + h, t) − ux (x + h, y − h, t) ⎪ ⎪ 2h ⎪ ⎪ ⎪ ⎪ −ux (x − h, y + h, t) + ux (x − h, y − h, t) ⎪ ⎪ ⎪ ⎪ τ 2 ⎪ ⎪ ⎪ + c1 [uy (x, y + h, t) − 2uy (x, y, t) + uy (x, y − h, t)] ⎪ ⎪ ⎪ ⎪ h 2 ⎪ ⎪ τ ⎪ ⎪ + c3 [wy (x + h, y, t) − 2wy (x, y, t) + wy (x − h, y, t)] ⎪ ⎪ ⎪ h τ 2 ⎪ ⎨ c23 [wx (x + h, y + h, t) − wx (x + h, y − h, t) −2 2h ⎪ ⎪ ⎪ −wx (x − h, y + h, t) + wx (x − h, y − h, t)] ⎪ ⎪ ⎪ τ 2 ⎪ ⎪ ⎪ ⎪ − c3 [wy (x, y + h, t) − 2wy (x, y, t) + wy (x, y − h, t)], ⎪ ⎪ h ⎪ ⎪ τ ⎪ ⎪ wx (x, y, t + τ ) = wx (x, y, t) + d22 2 [ux (x + h, y, t) − 2ux (x, y, t) + ux (x − h, y, t)] ⎪ ⎪ h ⎪ ⎪ ⎪ τ ⎪ ⎪ +d21 2 wx (x + h, y, t) + wx (x − h, y, t) − 4wx (x, y, t) ⎪ ⎪ h ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (x, y + h, t) + w (x, y − h, t) +w ⎪ x x ⎪ ⎪ ⎪ ⎪ ⎪ τ ⎪ 2 ⎪ uy (x + h, y + h, t) − uy (x + h, y − h, t) −2d ⎪ 2 ⎪ (2h)2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (x − h, y + h, t) + u (x − h, y − h, t) −u ⎪ y y ⎪ ⎪ ⎪ ⎪ τ ⎪ ⎪ −d22 2 [ux (x, y + h, t) − 2ux (x, y, t) + ux (x, y − h, t)] , ⎪ ⎪ h ⎪ ⎪ 2 τ ⎪ ⎪ ⎪ wy (x, y, t + τ ) = wy (x, y, t) + d2 h2 [uy (x + h, y, t) − 2uy (x, y, t) + uy (x − h, y, t)] ⎪ ⎪ ⎪ ⎪ τ ⎪ ⎪ +d21 2 wy (x + h, y, t) + wy (x − h, y, t) − 4wy (x, y, t) ⎪ ⎪ h ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ +w (x, y + h, t) + w (x, y − h, t) y y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ τ 2 ⎪ ⎪ ux (x + h, y + h, t) − ux (x + h, y − h, t) +2d 2 ⎪ ⎪ (2h)2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −u x (x − h, y + h, t) + ux (x − h, y − h, t) ⎪ ⎪ ⎪ ⎪ (10.8-1) τ ⎪ ⎩ −d22 2 [uy (x, y + h, t) − 2uy (x, y, t) + uy (x, y − h, t)] . h

10.8

Appendix of Chapter 10: The detail of ﬁnite diﬀerence scheme

223

The displacements at mesh points located at the special lines are determined by satisfying the boundary conditions, we obtain respectively for points on the grid lines x = −h/2 and x = L + h/2. h h − − ux L+2 h , y, t =ux L−2 h , y, t 2

2

1 2 1 ± 2 ±

wx

d21 (c21 − 2c22 ) + c23 d22 h2 uy L− h , y 2 c21 d21 − c23 d22 c23 (d21 − d23 ) h2 , y + h, t w y L− h 2 c21 d21 − c23 d22

h − =wx L−2 h , y, t

−h 2 , y, t L+ h 2

h 2 + h, t − uy L− h , y − h, t

− wy

2

h

, y − h, t L− h 2

,

2

(10.8-2a)

2

h d2 (c2 − 2c2 ) h2 2 ±2 22 21 2 22 uy L− , y + h, t − u , y − h, t y h L− h 2 2 c3 d2 − c1 d1 2 2 2 2 h h 1 c3 d2 − c1 d3 2 2 w , y + h, t − w , y − h, t , ± 2 2 y y h h L− 2 L− 2 2 c3 d2 − c21 d21 (10.8-2b) 1 h h h h −2 −2 2 2 ux L− h , y + h, t − ux L− h , y − h, t uy L+ h , y, t = uy L− h , y, t ± 2 2 2 2 2 h 1 c23 (d21 − d23 ) h2 2 , y + h, t − w , y − h, t , ± 2 2 w x x h h L− 2 L− 2 2 c2 d1 − c23 d22 (10.8-2c) wy

−h 2 , y, t L+ h 2

= wy

h 2

L− h 2

, y, t

h 1 c23 d22 − c22 d23 h2 2 wx L− h , y + h, t − wx L− , h , y − h, t 2 2 2 2 2 2 2 c2 d1 − c3 d2 (10.8-2d) where the equations (10.8-2a) and (10.8-2b) related to x = −h/2 is not valid. From the ﬁrst condition of (10.6-5), at x = 0, ux = 0 and wx = 0. To satisfy the condition the displacements ux and wx at x = −h/2 is approximated by ⎧ h h ⎪ ⎪ ⎨ ux x, − , t = −ux x, , t , 2 2 (10.8-3) h h ⎪ ⎪ ⎩ wx x, − , t = −wx x, , t . 2 2 ±

On the grid line y = −h/2 and y = H + h/2, we obtain 1 h h −h −h 2 2 2 2 uy x + h,H− ux x,H+ = ux x,H− ± − uy x − h,H− h ,t h ,t h ,t h ,t 2 2 2 2 2 2 2 2 h h 1 c (d − d ) 2 2 − wy x − h,H− , ± 23 2 1 2 3 2 wy x + h,H− h ,t h ,t 2 2 2 c2 d1 − c3 d2 (10.8-4a)

224

Chapter 10

Dynamics of elasticity and defects of quasicrystals

−h −h 2 2 wx x,H+ = wx x,H− h ,t h ,t 2

−h 2 uy x,H+ h 2

2

h h 1 c2 d2 − c22 d23 2 2 , t − w , t , ± 32 22 w x + h, x − h, y y h h H− 2 H− 2 2 c2 d1 − c23 d22 (10.8-4b) h 2 , t = uy x,H− h ,t 2

h h + d21 (c21 − 2c22 ) 1 2 2 ux x + h,H− − ux x − h,H− h ,t h ,t 2 2 2 2 2 2 2 c1 d1 − c3 d2 2 2 2 h h 1 c (d − d ) 2 2 − wx x − h,H− , ± 23 2 3 2 1 2 wx x + h,H− h ,t h ,t 2 2 2 c1 d1 − c3 d2 (10.8-4c) h 2 , t = wy x,H− h ,t ±

−h 2 wy x,H+ h 2

c23 d22

2

h h d2 (c2 − c2 ) 2 2 − ux x − h,H− ± 22 2 1 2 2 2 ux x + h,H− h ,t h ,t 2 2 c1 d1 − c3 d2 2 2 2 2 h h 1 c d − c3 d2 2 2 − wx x − h,H− , ± 12 32 wx x + h,H− h ,t h ,t 2 2 2 2 2 c1 d1 − c3 d2 (10.8-4d) in which, equations (10.8-4c) and (10.8-4d) related to y = −h/2 is valid only along the crack surface, namely, only for x a − h/2 at y = 0, in which the crack terminates. From the last condition of (10.6-5), at y = 0 and the ahead of the crack, uy = 0, wy = 0. To satisfy this condition the displacements uy and wy at y = −h/2 is approximated by ⎧ h h ⎪ ⎪ ⎨ uy x, − , t = −uy x, , t , 2 2 (10.8-5) ⎪ h h ⎪ ⎩ wy x, − , t = −wy x, , t . 2 2 In constructing the approximation (10.8-2∼10.8-5) we follow a method proposed by Shmuely and Peretz[29] which was also successfully employed in Ref. [28] for conventional engineering materials. According to this method, derivatives perpendicular to the boundary are proposed by uncentered diﬀerences and derivatives parallel to the boundary by centered diﬀerence. The real boundary can be considered as located at a distance of half the mesh size from the grid boundaries. The four grid corners require a special treatment. Diﬀerent methods of handling the discontinuities at such points have been proposed in the past. Here we found that satisfactory results are obtained when the displacements sought are extrapolated from those given along both sides of the corner in question. Accordingly, the components ux , uy , wx , wy at (−h/2, −h/2) are given by

References

ux uy

wx wy

225

h h h h h h ux ux ,− ,t + − ,− ,t = − , ,t uy uy 2 2 2 2 2 2 3h h h 3h ux u −0.5 ,− ,t + x − , ,t , uy uy 2 2 2 2

h h − ,− ,t 2 2

h h h h wx ,− ,t + − , ,t wy 2 2 2 2 3h h h 3h wx wx −0.5 ,− ,t + − , ,t . wy wy 2 2 2 2

w = x wy

(10.8-6)

Similar expressions are used for deriving the displacement components at (−h/2, H + h/2), (L + h/2, H + h/2)and(L + h/2, −h/2). By following relevant stability criterion of the scheme the computation is always stable and achieves high precision. Discussions on this aspect are omitted here due to space limitation.

References [1] Lubensky T C , Ramaswamy S and Joner J. Hydrodynamics of icosahedral quasicrystals. Phys Rev B, 1985, 32(11): 7444–7452 [2] Socolar J E S, Lubensky T C and Steinhardt P J. Phonons, phasons and dislocations in quasicrystals. Phys Rev B, 1986, 34(5): 3345–3360 [3] Bak P. Phenomenological theory of icosahedral in commensurate (quasiperiodic)order in Mn-Al alloys. Phys Rev Lett, 1985, 54(14): 1517–1519 [4] Bak P. Symmetry, stability and elastic properties of icosahedral in commensurate crystals. Phys Rev B, 1985, 32(9): 5764–5772 [5] Ding D H, Yang W G, Hu C Z et al. Generalized elasticity theory of quasicrystals. Phys Rev B, 1993, 48(10): 7003–7010 [6] Hu C Z, Wang R H, Ding D H. Symmetry groups, physical property tensors, elasticity and dislocations in quasicrystals. Reports on Progress in Physics, 2000, 63(1): 1–39 [7] Fan T Y, Li X F, Sun Y F. A moving screw dislocation in one-dimensional hexagonal quasicrystal. Acta Physica Sinica(Overseas Edition), 1999, 8(3): 288–295 [8] Fan T Y. A study on special heat of one-dimensional hexagonal quasicrystals. J Phys: Condens Matter, 1999, 11(45): L 513–L 517 [9] Fan T Y, Mai Y W. Partition function and state equation of point group 12mm two-dimensional quasicrystals. Euro Phys J B, 2003, 31(1): 25–27 [10] Fan T Y, Mai Y W. Elasticity theory, fracture mechanics and some relevant thermal properties of quasicrystalline materials. Appl Mech Rev, 2004, 57(5): 325–344 [11] Li C L, Liu Y Y. Phason-strain inﬂuences on low-temperature speciﬁc heat of the decagonal Al-Ni-Co quasicrystal. Chin Phys Lett, 2001, 18(4): 570–572

226

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Dynamics of elasticity and defects of quasicrystals

[12] Li C L, Liu Y Y. Low-temperature lattice excitation of icosahedral Al-Mn-Pd quasicrystals. Phys Rev B, 2001, 63(6): 064203 [13] Fan T Y , Wang X F, Li W et al. Elasto-hydrodynamics of quasicrystals. Phil Mag, 2009, 89(6): 501–512 [14] Zhu A Y, Fan T Y. Dynamic crack propagation in a decagonal Al-Ni-Co quasicrystal. J Phys: Condens Matter, 2008, 20(29): 295217 [15] Rochal S B, Lorman V L. Anisotropy of acoustic-phonon properties of an icosahedral quasicrystal at high temperature due to phonon-phason coupling Phys Rev B, 2000, 62(2): 874–879 [16] Rochal S B, Lorman V L. Minimal model of the phonon-phason dynamics on icosahedral quasicrystals and its application for the problem of internal friction in the i-AlPdMn alloys. Phys Rev B, 2002, 66(14): 144204 [17] Hirth J P, Lorthe J. Theory of Dislocations. 2nd ed On Eshelby’s solution one can refer to New York: John Wiely & Sons, 1982 [18] Yoﬀe E H. Moving Griﬃth crack. Phil Mag,1951, 43(10): 739–750 [19] Chernikov M A, Ott H R, Bianchi A et al. Elastic moduli of a single quasicrystal of decagonal Al-Ni-Co: evidence for transverse elastic isotropy. Phys Rev Lett, 1998, 80(2): 321–324 [20] Jeong H C, Steinhardt P J. Finite-temperature elasticity phase transition in decagonal quasicrystals. Phys Rev B, 1993, 48(13): 9394–9403 [21] Walz C. Zur Hydrodynamik in Quasikristallen. Diplomarbeit. Universitaet Stuttgart, 2003 [22] Maue A W. Die entspannungswelle bei ploetzlischem Einschnitt eines gespannten elastischen Koepores. Zeitschrift fuer angewandte Mathematik und Mechanik, 1954, 14(1): 1–12 [23] Murti V, Vlliappan S. The use of quarter point element in dynamic crack analysis. Engineering Fracture Mechanics, 1982, 23(3): 585–614 [24] Chen Y M. Numerical computation of dynamic stress intensity factor s by a Lagrangian ﬁnite-diﬀerence method (the HEMP code). Engineering Fracture Mechanics, 1975, 7(8): 653–660 [25] Mikulla R, Stadler J, Krul F et al. Crack propagation in quasicrystals. Phys Rev Lett, 1998, 81(15): 3163–3166 [26] Ebert Ph, Feuerbacher M, Tamura N et al. Evidence for a cluster-based on structure of Al-Pd-Mn single quasicrystals. Phys Rev Lett, 1996, 77(18): 3827–3830 [27] Wang X F, Fan T Y, Zhu A Y. Dynamic behaviour of the icosahedral Al-Pd-Mn quasicrystal with a Griﬃth Crack, Chin Phys B, 2009, 18(2), 709-714. (in Section 10.7 a part data are introduced from Zhu A Y and Fan T Y’s unpublished work: Fast crack propagation in three-dimensional icosahedral Al-Pd-Mn quasicrystals)

References

227

[28] Shmuely M, Alterman Z S. Crack propagation analysis by ﬁnite diﬀerences. Journal of Applied Mechanics, 1973, 40(4): 902–908 [29] Shmuely M, Peretz D. Static and dynamic analysis of the DCB problem in fracture mechanics. Int J of Solids and Structures, 1976, 12(1): 67–67 [30] Takeuchi S, Iwanaga H, Shibuya T. Hardness of quasicrystals. Japanese J Appl Phys, 1991, 30(3): 561–562 [31] Coddens G. On the problem of the relation between phason elasticity and phason dynamics in quasicrystals, Eur Phys. J B, 2006, 54(1): 37–65

Chapter 11 Complex variable function method for elasticity of quasicrystals In Chapters 7∼9 we frequently used the complex variable function method to solve problems of elasticity of quasicrystals, and many exact analytic solutions were obtained by the method. In those chapters we only provided the results, and the underlying principle and details of the method could not be discussed. Considering the relative new feature and particular eﬀect of the method, it is helpful to attempt a further discussion in depth. Of course this may lead to a slight repetition of relevant content of Chapters 7∼9. It is well-known that the so-called complex potential method in the classical elasticity is eﬀective, in general, mainly for solving harmonic and biharmonic partial diﬀerential equations, for these equations the solutions can be expressed by analytic √ functions of single complex variable z = x + iy, i = −1. In addition, the quasibiharmonic partial diﬀerential equations can be solved by analytic functions of two complex variables such as z1 = x + α1 y, z2 = x + α2 y, in which α1 , α2 are complex constants. The study of elasticity of quasicrystals has led to discovery of some multi-harmonic and multi-quasiharmonic equations, which cover a quite wide range of partial diﬀerential equations appearing in the ﬁeld to date and have been introduced in Chapters 5∼9. The discussion on the complex analysis for these equations is signiﬁcant. We know that the Muskhelishvili complex potentials method for the classical isotropic plane elasticity[1] , which solves mainly the biharmonic equation, made great contributions for a quite wide range of ﬁelds in science and engineering. And the complex potential method developed by Lekhnitzkii[2] for the classical anisotropic plane elasticity which solves mainly the quasi-biharmonic equation. The present formulation and solutions of the complex analysis for, e.g. quadruple and sextuple harmonic equations and quadruple-quasiharmonic equation, are at least a new development of the complex potential method of the classical elasticity. Though the new method is used to solve elasticity problems of quasicrystals at present, it may be extended into other disciplines of science and engineering in future. At ﬁrst we simply review the complex potential method for harmonic and biT. Fan, Mathematical Theory of Elasticity of Quasicrystals and Its Applications © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2011

230

Chapter 11 Complex variable function method for elasticity of quasicrystals

harmonic equations, which does not belong to a new innovation from point of view of methodology, then focus on those for quadruple and sextuple harmonic equations and quadruple quasiharmonic equation, and with discussions in detail presenting their new features from the angle of elasticity as well as complex potential method.

11.1

Harmonic and quasi-biharmonic equations in anti-plane elasticity of one-dimensional quasicrystals

The ﬁnal governing equations of elasticity of one-dimensional quasicrystals present the following two kinds discussed in Chapter 5: c44 ∇2 uz + R3 ∇2 wz = 0, (11.1-1) R3 ∇2 uz + K 2 ∇2 wz = 0, ∂4 ∂4 ∂4 ∂4 ∂4 + c c1 4 + c2 3 + c3 2 2 + c4 G = 0, 5 ∂x ∂x ∂y ∂x ∂y ∂x∂y 3 ∂y 4

(11.1-2)

in which equations (11.1-1) are actually two decoupled harmonic equations of uz and wz , whose complex variable function method was introduced in the Sections 8.1, 8.2 and 8.7, here we do not repeat any more. Equation (11.1-2) is a quasi-biharmonic equation which describes the phononphason coupling elasticity ﬁeld for some kinds of one-dimensional quasicrystal systems, Liu, Fan and Guo[3] and Liu[4] studied some solutions of them in terms of the complex variable function method, whose origin comes from the classical work of Lekhnitskii[2] , reader can ﬁnd some beneﬁcial hints in the monograph.

11.2

Biharmonic equations in plane elasticity of point group 12mm two-dimensional quasicrystals

From Chapter 6 we know that in elasticity of dodecagonal quasicrystals, the phonon and phason ﬁelds are decoupled each other. For such a plane elasticity we have the ﬁnal governing equations as follows: ∇2 ∇2 F = 0,

∇2 ∇2 G = 0.

The complex representation of solution of (11.2-1) is F (x, y) = Re[¯ z ϕ1 (z) + ψ1 (z)dz], G(x, y) = Re[¯ z π1 (z) + χ1 (z)dz],

(11.2-1)

(11.2-2)

where φ1 (z), ψ1 (z), π1 (z) and χ1 (z) are any analytic functions of complex variable √ z = x + iy, (i = −1). For these kinds of biharmonic equations Muskhelishvili[1]

11.3

The complex variable function method of quadruple harmonic equations and... 231

developed systematical complex variable function method, reader can ﬁnd some details in the well known monograph, we need not discuss them any more.

11.3

The complex variable function method of quadruple harmonic equations and applications in two-dimensional quasicrystals

As it was discussed in Chapters 6∼8, for point groups 5m and 10mm or point groups 5, 5 and 10, 10 quasicrystals, either by the displacement potential formulation or by the stress potential formulation we obtain the ﬁnal governing equation is quadruple harmonic equation, whose complex variable function method is newly created by Liu and Fan[5] based on the displacement potential formulation and by Li and Fan[6] based on the stress potential formulation. This complex potential method develops the methodology was used in the classical elasticity. It is necessary to give some further discussions in-depth. The representation in the following will be done based on the stress potential formulation, and discussion is given only for point groups 5, 5 and 10, 10 quasicrystals, because the point groups 5m and 10mm ones can be seen as a special case of the former. 11.3.1

Complex representation of solution of the governing equation

Because it is relatively simpler for the case of point groups 5m and 10mm, which belong to the special case of point group 5, 5 and point groups 10, 10, we here discuss only the ﬁnal governing equation of plane elasticity of pentagonal of point groups 5, 5 and decagonal quasicrystals of point groups 10, 10 ∇2 ∇2 ∇2 ∇2 G = 0,

(11.3-1)

where G(x, y) is the stress potential function. The solution of equation (11.3-1) is 1 2 1 3 (11.3-2) G = 2Re g1 (z) + z¯g2 (z) + z¯ g3 (z) + z¯ g4 (z) , 2 6 where gj (z)(j = 1, · · · , 4) are four analytic functions of a single complex variable z ≡ x + iy = reiθ . The bar denotes the complex conjugate hereinafter, i.e., z¯ = x − iy = re−iθ . We call these functions the complex stress potentials, or the complex potentials in brief. 11.3.2

Complex representation of the stresses and displacements

The Section 8.4 shows, from fundamental solution (11.3-2) one can ﬁnd the complex representation of the stresses as below:

232

Chapter 11 Complex variable function method for elasticity of quasicrystals ⎧ σxx = −32c1 Re(Ω (z) − 2g4 (z)), ⎪ ⎪ ⎪ ⎪ ⎪ σyy = 32c1 Re(Ω (z) + 2g4 (z)), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ σxy = σyx = 32c1 ImΩ (z), Hxx = 32R1 Re(Θ (z) − Ω (z)) − 32R2 Im(Θ (z) − Ω (z)), ⎪ ⎪ ⎪ H = −32R Im(Θ (z) + Ω (z)) − 32R Re(Θ (z) + Ω (z)), ⎪ xy 1 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Hyx = −32R1 Im(Θ (z) − Ω (z)) − 32R2 Re(Θ (z) − Ω (z)), ⎪ ⎩ Hyy = −32R1 Re(Θ (z) + Ω (z)) + 32R2 Im(Θ (z) + Ω (z),

where

⎧ ⎨

1 (IV ) (z) + z¯2 g4 (z), 2 ⎩ Ω (z) = g (IV ) (z) + z¯g (IV ) (z), 3 4 (IV )

Θ(z) = g2

(11.3-3)

(IV )

(z) + z¯g3

(11.3-4)

in which the prime, two prime, three prime and superscript (IV ) denote the ﬁrst to fourth order derivatives of gi (z) to variable z, in addition Θ (z) = dΘ(z)/dz, it is evident that Θ(z) and Ω (z) are not analytic functions. By some derivation from (11.3-3) we have the complex representation of the displacements such as ux + iuy = 32(4c1 c2 − c3 − c1 c4 )g4 (z) − 32(c1 c4 − c3 )(g3 (z) + zg4 (z)), (11.3-5) wx + iwy =

32(R1 − iR2 ) Θ(z) K1 − K 2

(11.3-6)

with constants c = M (K1 + K2 ) − 2(R12 + R22 ), c2 = 11.3.3

c + (L + M )(K1 + K2 ) , 4(L + M )c

c + M, K1 − K2 2 2 R + R2 K1 + K 2 c3 = 1 , c4 = . c c c1 =

(11.3-7)

The complex representation of boundary conditions

In the following we consider only the stress boundary value problem, i.e., at the boundary curve Lt the tractions (Tx , Ty ) and generalized tractions (hx , hy ) are given, and there are the stress boundary conditions such as σxx cos(n, x)+σxy cos(n, y) = Tx ,

σxy cos(n, x)+σyy cos(n, y) = Ty ,

(x, y) ∈ Lt , (11.3-8)

(x, y) ∈ Lt , (11.3-9) where T x , T y and hx , hy are tractions and generalized tractions at the boundary Lt where the stresses are prescribed. Hxx cos(n, x)+Hxy cos(n, y) = hx ,

Hxy cos(n, x)+Hyy cos(n, y) = hy ,

11.3

The complex variable function method of quadruple harmonic equations and... 233

From (11.3-8) and after some derivation the phonon stress boundary condition can be reduced to the equivalent form i (Tx + iTy )ds, z ∈ Lt . (11.3-10) g4 (z) + g3 (z) + zg4 (z) = 32c1 From equations. (11.3-9), (11.3-3) and (11.3-4), we have (R2 − iR1 )Θ(z) = i (hx + ihy )ds, z ∈ Lt . 11.3.4

(11.3-11)

Structure of complex potentials

11.3.4.1 Arbitrariness in the deﬁnition of the complex potentials For simplicity, we introduce the following new symbols (IV )

g2

(z) = h2 (z),

g3 (z) = h3 (z),

g4 (z) = h4 (z),

(11.3-12)

then equation (11.3-3) can be written as follows: σxx + σyy = 128c1 Re h4 (z),

(11.3-13)

σyy − σxx + 2iσxy = 64c1 Ω (z) = 64c1 [h3 (z) + z¯h4 (z)],

(11.3-14)

Hxy − Hyx − i(Hxx + Hyy ) = 64(iR1 − R2 )Ω (z),

(11.3.15)

(Hxx − Hyy ) − i(Hxy + Hyx ) = 64(R1 + R2 )Θ (z).

(11.3-16)

By inspection, the state of phonon and phason stresses is not altered if replacing h4 (z) by h4 (z) + Diz + γ,

(11.3-17)

h3 (z) by h3 (z) + γ ,

(11.3-18)

h2 (z) by h2 (z) + γ ,

(11.3-19)

in equations (11.3-13)–(11.3-16) where D is a real constant and γ, γ , γ are arbitrary complex constants. Now consider how these substitutions aﬀect the components of the displacements which were determined by the formulas (11.3-5) and (11.3-6).Direct substitution shows that ux + iuy = 32(4c1 c2 − c3 − c1 c4 )h4 (z) − 32(c1 c4 − c3 )(h3 (z) + zh4 (z)) +32(4c1 c2 − 2c3 )Diz + [32(4c1 c2 − c3 − c1 c4 )γ − 32(c1 c4 − c3 )γ ], (11.3-20)

234

Chapter 11 Complex variable function method for elasticity of quasicrystals

32(R1 − iR2 ) 1 2 32(R1 − iR2 ) wx + iwy = h2 (z) + zh3 (z) + z h4 (z) + γ . K1 − K 2 2 K1 − K 2 (11.3-21) Formulas (11.3-20) and (11.3-21) show that a substitution of the form (11.3-17), (11.3-19) will aﬀect the displacements, unless D = 0,

γ=

c1 c4 − c3 γ , γ = 0. 4c1 c2 − c3 − c1 c4

11.3.4.2 General formulas for ﬁnite multiply connected region Consider now the case when the region S, occupied by the quasicrystal, is multiply connected. In general the region is bounded by several simple closed contours s1 , s2 , · · · , sm , sm+1 , the last of these contours is to contain all the others, depicted in Fig. 11.3-1, i.e., a plate with holes. We assume that the contours do not intersect themselves and have no points in common. Sometimes we call s1 , s2 , · · · , sm as inner boundaries and sm+1 as outer boundary of the region. It is evident that the points z1 , z2 , · · · , zm are ﬁxed points in the holes, but located out of the material.

Fig. 11.3-1

Finite multi-connected region

Similar to the discussion of the classical elasticity theory (refer to [1]), we can obtain h4 (z) =

m

Ak ln(z − zk ) + h4∗ (z),

(11.3-22)

k=1

h4 (z) =

m

Ak z ln(z − zk ) +

k=1

h3 (z) =

m

m

γk ln(z − zk ) + h4∗ (z),

(11.3-23)

k=1

γk ln(z − zk ) + h3∗ (z),

(11.3-24)

k=1

recalling zk denote ﬁxed points outside the region S, h3∗ (z), h4∗ (z) are holomorphic (analytic and single-valued, refer to the Major Appendix of this book) in S, Ak real constants, γk , γk complex constants.

11.3

The complex variable function method of quadruple harmonic equations and... 235

By substituting (11.3-22)∼(11.3-24) into (11.3-16) one can ﬁnd that m

h2 (z) =

γk ln(z − zk ) + h2∗ (z),

(11.3-25)

k=1

h2∗ (z)is holomorphic in S and γk are complex constants. Consideration will be given to the condition of single-valuedness of phonon displacements. By equation (11.3-5), one has ux + iuy = 32(4c1 c2 − c3 − c1 c4 )h4 (z) − 32(c1 c4 − c3 )(h3 (z) + zh4 (z)). (11.3-26) Substituting (11.3-23)∼(11.3-25) into (11.3-26), it is immediately seen that [ux+iuy ]sk = 2πi{[32(4c1 c2−c3−c1 c4 )+32(c1 c4−c3 )]Ak z+32(4c1 c2−c3−c1 c4 )γk+γk (z)}, (11.3-27) in which [ ]Sk denotes the increase undergone by the expression in brackets for one anti-clockwise circuit of the contour sk . Hence it is necessary and suﬃcient for the single-valuedness of phonon displacements that in the formulas (11.3-22)∼(11.3-25) Ak = 0,

32(4c1 c2 − c3 − c1 c4 )γk + γk = 0.

(11.3-28)

Similar to the above mentioned discussion, by equation (11.3-6), one has [wx + iwy ]sk =

32(R1 − iR2 ) (−2πi)γk . K1 − K 2

(11.3-29)

Hence it is necessary and suﬃcient for the single-valuedness of phason displacements if γk = 0. (11.3-30) It will now be shown shat the quantities γk , γk may be very simply expressed in terms of Xk , Yk , where (Xk , Yk ) denotes the resultant vector of the external stresses, exerted on the contour sk . By (11.3-10), applying it to the contour sk , one has −32c1 i[h4 (z) + h3 (z) + zh4 (z)]sk = Xk + iYk

with

(11.3-31)

Xk =

Tx ds, sk

Yk =

Ty ds. sk

In the present case the normal vector n must be directed outwards with respect to the region Sk . Consequently, the contour sk must be traversed in the clockwise direction. Taking this fact into consideration, one obtains −2πi(γk − γk ) =

i (Xk + iYk ). 32c1

(11.3-32)

236

Chapter 11 Complex variable function method for elasticity of quasicrystals

By equations (11.3-28), (11.3-31) and (11.3-32), one has Ak = 0, γk = d1 (Xk + iYk ),

γk = d2 (Xk − iYk ),

(11.3-33)

where 4c1 c2 − c3 − c1 c4 2c1 π[32(4c1 c2 − c3 − c1 c4 ) + 1] (11.3-34) and which are independent of the suﬃx k. So that ⎧ m ⎪ ⎪ ⎪ (z) = d (Xk + iYk ) ln(z − zk ) + h4∗ (z), h ⎪ 4 1 ⎪ ⎪ ⎨ k=1 m (11.3-35) ⎪ h (z) = d (Xk − iYk ) ln(z − zk ) + h3∗ (z), 3 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ h (z) = h k=1 (z). d1 =

1 , 64c1 π[32(4c1 c2 − c3 − c1 c4 ) + 1]

2

d2 = −

2∗

We can conclude that the complex functions h2 (z), h3 (z), h4 (z) must be expressed by formulas (11.3-35) to assure the single-valuedness of stresses and displacements, where h2∗ (z), h3∗ (z), h4∗ (z) are holomorphic in S. 11.3.4.3 Case of inﬁnite regions From the point of view of application in Chapter 9, this chapter and Chapters 14, 15 the consideration of inﬁnite regions is likewise of major interest. We assume that the contour sm+1 goes to inﬁnity. Since equations (11.3-13) and (11.3-14) are similar to the classical elasticity theory, we have

h4 (z) = d1 (X + iY ) ln z + (B + iC)z + h04 (z), (11.3-36) h3 (z) = d2 (X − iY ) ln z + (B + iC )z + h03 (z), where B, C, B , C are unkonwn real constants to be determined and X=

m k=1

Xk , Y =

m

Yk ,

k=1

h03 (z), h04 (z) are functions, holomorphic in region S, including the point at inﬁnity,i.e, for suﬃciently large |z| they may be expanded into series of the form h04 (z) = a0 +

a1 a2 + 2 + ··· , z z

h03 (z) = a0 +

a1 a2 + 2 + ··· . z z

(11.3-37)

11.3

The complex variable function method of quadruple harmonic equations and... 237

On the basis of (11.3-2) the state of stresses will not be altered by assuming a0 = a0 = 0. By the theorem of Laurent expansion, the function h2∗ (z) may be represented in region S including point at inﬁnity by the series h2∗ (z) =

+∞

cn z n .

(11.3-38)

−∞

Substituting equations (11.3-36) and (11.3-38) into equation (11.3-16), one has (Hxx − Hyy ) − i(H xy + Hyx ) +∞ d 1 2d 2 1 cn nz n−1 + z¯ − 2 + h03 (z) + z¯2 + h04 (z) , = 2 × 32(R1 + R2 z 2 z3 −∞ (11.3-39) hence it follows that for the stresses to remain ﬁnite as |z| → ∞, one must have cn = 0,

n 2.

It is obvious that the phonon and phason stresses will be bounded, if these conditions are satisﬁed. Hence one has ﬁnally ⎧ ⎨ h4 (z) = d1 (X + iY ) ln z + (B + iC)z + h04 (z), h3 (z) = d2 (X − iY ) ln z + (B + iC )z + h03 (z), (11.3-40) ⎩ h2 (z) = (B + iC )z + h02 (z), where B , C are unknown real constants to be determined, h02 (z) is function, holomorphic in region S, including the point at inﬁnity, thus it has the form similar to that of (11.3-37), a a (11.3-41) h02 (z) = a0 + 1 + 22 + · · · . z z We have assumed that a0 = a0 = 0 already, now further assume a0 = 0, i.e., h04 (∞) = h03 (∞) = h02 (∞) = 0, then from (11.3-40) and (11.3-13)∼(11.3-16) one can determine ⎧ (∞) (∞) (∞) (∞) (∞) ⎪ σxx − σyy σxy σxx + σyy ⎪ ⎪ , B = , C = , B = ⎪ ⎪ 128c1 64c1 32c1 ⎪ ⎪ ⎪ ⎨ (∞) (∞) (∞) (∞) R2 (Hxy − Hyx ) − R1 (Hxx + Hyy ) B = , 2 2 ⎪ 64(R1 − R2 ) ⎪ ⎪ ⎪ ⎪ (∞) (∞) (∞) (∞) ⎪ R1 (Hxy − Hyx ) − R2 (Hxx + Hyy ) ⎪ ⎪ ⎩ C = 64(R12 − R22 ) (∞)

in which C has no usage, we put it to be zero, and σij applied stresses at point of inﬁnity.

(11.3-42)

(∞)

and Hij represent the

238

Chapter 11 Complex variable function method for elasticity of quasicrystals

11.3.5

Conformal mapping

If we constrain our discussion only for case of stress boundary value problems, then the problems will be solved under boundary conditions (11.3-10) and (11.3-11). For some complicated regions solutions of the problems cannot be directly obtained in the physical plane (i.e., the z-plane). We must use a conformal mapping z = ω(ζ)

(11.3-43)

to transform the region studied in the plane onto interior of the unit circle γ in the mapping plane (say e.g. ζ-plane). Substituting (11.3-43) into (11.3-40) we have ⎧ ⎨ h4 (z) = Φ4 (ζ) = d1 (X + iY ) ln ω(ζ) + Bω(ζ) + Φ40 (ζ), h3 (z) = Φ3 (ζ) = d2 (X − iY ) ln ω(ζ) + (B + iC )ω(ζ) + Φ30 (ζ), (11.3-44) ⎩ h2 (z) = Φ2 (ζ) = (B + iC )ω(ζ) + Φ20 (ζ), where Φj (ζ) = hj [ω(ζ)],

Φj0 (ζ) = h0j [ω(ζ)],

In addition hj (z) =

j = 1, · · · , 4.

Φi (ζ) . ω (ζ)

At the mapping plane the boundary conditions (11.3-10) and (11.3-11) stand for i Φ (σ) = (Tx + iTy )ds, Φ4 (σ) + Φ3 (σ) + ω(σ) 4 (11.3-10 ) 32c1 ω (σ) (R2 − iR1 )Θ(σ) = i

(hx + ihy )ds,

(11.3-11 )

where σ = eiϕ represents the value of ζ at the unit circle (i.e., ρ = 1). From these boundary value equations we can determine the unknown functions Φi (ζ)(i = 2, 3, 4). 11.3.6

Reduction of the boundary value problem to function equations

Due to Φ1 (ζ) = 0, we now have three unknown functions Φj (ζ)(j = 2, 3, 4). Taking conjugate of (11.3-10 ) yields i Φ (σ) =− Φ4 (σ) + Φ3 (σ) + ω(σ) 4 ω (σ) 32c1

(Tx − iTy )ds.

(11.3-10 )

Substituting the ﬁrst one of equations (11.3-4) into (11.3-11 ), then multiplying dσ/2πi(σ − ζ) to both sides of (11.3-10 ) and (11.3-11 ) and integrating along the unit circle γ leads to

11.3

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

The complex variable function method of quadruple harmonic equations and... 239

1 2πi 1 2πi

γ

1 Φ4 (σ)dσ + σ−ζ 2πi

γ

Φ4 (σ)dσ 1 + σ−ζ 2πi

γ

1 Φ3 (σ)dσ + σ−ζ 2πi

γ

Φ3 (σ)dσ 1 + σ−ζ 2πi

γ

1 1 ω(σ) Φ 4 (σ)dσ = 32c1 2πi ω (σ) σ − ζ

γ

ω(σ) Φ4 (σ)dσ 1 1 = ω (σ) σ − ζ 32c1 2πi

γ

γ

tdσ σ−ζ tdσ σ−ζ

1 Φ2 (σ)dσ ω(σ) Φ 3 (σ)dσ 1 + ⎪ 2πi γ σ − ζ 2πi γ ω (σ) σ − ζ ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ ω(σ) Φ4 (σ)dσ ω(σ) ω (σ) Φ4 (σ)dσ 1 ⎪ ⎪ ⎪ + − ⎪ ⎪ 2πi γ [ω (σ)]2 σ − ζ [ω (σ)]3 σ−ζ ⎪ γ ⎪ ⎪ ⎪ ⎪ 1 1 hdσ ⎪ ⎪ , ⎩ = R1 − iR2 2πi γ σ − ζ (11.3-45) where t = i (Tx + iTy )ds, t = −i (Tx − iTy )ds, h = i (h1 + ih2 )ds in equations (11.3-45), which are the function equations to determine complex potentials Φj (ζ) which are analytic in the interior of the unit circle γ, and satisfy boundary value conditions (11.3-45) at the unit circle. 11.3.7

Solution of the function equations

According to the Cauchy’s integral formula (refer to Appendix I) 1 2πi

γ

Φi (σ) dσ = Φi (ζ), σ−ζ

1 2πi

γ

Φi (σ) dσ = Φi (0), σ−ζ

|ζ| < 1.

So that (11.3-45) are reduced to ⎧ 1 i 1 ω(σ) Φ4 (σ)dσ tdσ ⎪ ⎪ (ζ) + Φ (0) + = , Φ ⎪ 4 3 ⎪ 2πi σ − ζ 32c 2πi σ −ζ ⎪ 1 γ ω(σ) γ ⎪ ⎪ ⎪ ⎪ ⎪ ω(σ) Φ4 (σ)dσ tdσ 1 i 1 ⎪ ⎪ ⎪ Φ (0) + Φ (ζ) + = − , 3 ⎪ ⎨ 4 2πi γ ω(σ) σ − ζ 32c1 2πi γ σ − ζ 2 ⎪ ω(σ) Φ3 (σ)dσ 1 ω(σ) Φ4 (σ)dσ 1 ⎪ ⎪ ⎪ + Φ (ζ) + 2 ⎪ ⎪ 2πi γ ω (σ) σ − ζ 2πi γ [ω (σ)]2 σ − ζ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ i ⎪ ω(σ) ω (σ) Φ4 (σ)dσ 1 hdσ ⎪ − ⎩ = . (σ)]3 [ω σ − ζ R − iR 2πi σ −ζ 1 2 γ γ

(11.3-46)

The calculation of integrals in (11.3-46) depends upon conﬁguration of the sample, so the mapping function ω(ζ), and the applied stresses t and h. In the following we will give some concrete solutions for given conﬁguration and applied traction. 11.3.8

Example 1 Elliptic notch/crack problem and solution

We here calculate the stress and displacement ﬁelds induced by an elliptic notch

Chapter 11 Complex variable function method for elasticity of quasicrystals

240

x2 y 2 + = 1 in an inﬁnite plane of icosahedral quasicrystal, see Fig.11.3-2, a2 b2 the edge of which is subjected to a uniform pressure p.

L :

Fig. 11.3-2

An elliptic notch in a decagonal quasicrystal

The boundary conditions can be expressed by equations (11.3-10) and (11.3-11), for simplicity, we assume hx = hy = 0. Thus ⎧ ⎪ ⎨ i (Tx + iTy )ds = i (−p cos(n, x) − ip cos(n, y))ds = −pz = −pω(σ), ⎪ ⎩ i (hx + ihy )ds = 0. (11.3-47) In addition in this case in formulas (11.3-44) X = Y = 0, B = 0,

B = C = 0,

B = C = 0,

(11.3-48)

so Φi (ζ) = Φi0 (ζ), but in the following we omit the superscript of the functions Φi0 (ζ) for simplicity. The conformal mapping is 1 + mζ , (11.3-49) z = ω(ζ) = R0 ζ to transform the region containing ellipse at the z-plane onto the interior of the unit circle at the ζ-plane, refer to Fig.11.3-3, where ζ = ξ + iη = ρeiϕ and R0 = a+b a−b ,m= . 2 a+b Substituting (11.3-48) and (11.3-49) into the ﬁrst two equations of function equation (11.3-46), one obtains

11.3

The complex variable function method of quadruple harmonic equations and... 241

Fig.11.3-3

Conformal mapping from the exterior of the elliptic hole at z-plane onto the interior of the unit circle at ζ-plane

⎧ pR0 (1 + m2 )ζ ⎪ ⎪ , ⎨ Φ3 (ζ) = 32c1 mζ 2 − 1 ⎪ pR0 ⎪ ⎩ Φ4 (ζ) = − mζ. 32c1

(11.3-50a) (11.3-50b)

Substitution of ω(σ) σ2 + m = σ , ω (σ) mσ 2 − 1

2

ω(σ) ω (σ) 2σ(σ 2 + m)2 = − ω (σ)3 (mσ 2 − 1)3

and (11.3-50a)(11.3-50b) into the third equation of (11.3-46) yields 1 Φ2 (σ)dσ σ(σ 2 + m)2 Φ4 (σ)dσ 1 σ 2 + m Φ3 (σ)dσ 1 + + = 0. σ 2πi γ σ − ζ 2πi γ mσ 2 − 1 σ − ζ 2πi γ (mσ 2 − 1)3 σ − ζ (11.3-50c) By Cauchy’s integral formula, we have 1 Φ2 (σ)dσ = Φ2 (ζ), 2πi γ σ − ζ σ2 + m Φ3 (σ)dσ ζ2 + m 1 σ = ζ Φ (ζ), 2πi γ mσ 2 − 1 σ − ζ mζ 2 − 1 3 1 σ(σ 2 + m)2 Φ4 (σ)dσ ζ(ζ 2 + m)2 Φ (ζ). = 2 3 2πi γ (mσ − 1) σ − ζ (mζ 2 − 1)3 4 Substituting these equations and (11.3-50a), (11.3-50b) into (11.3-50c), one obtains Φ2 (ζ) =

pR0 ζ(ζ 2 + m)[(1 + m2 )(1 + mζ 2 ) − (ζ 2 + m)] . 32c1 (mζ 2 − 1)3

(11.3-50d)

Utilizing the above mentioned results, the phonon and phason stresses can be determined at the ζ-plane, so the displacements.

242

Chapter 11 Complex variable function method for elasticity of quasicrystals

If take m = 1, from (11.3-50) we can obtain solution of the Griﬃth crack, in particular the explicit solution at z-plane can be explored by taking inversion ζ = ω −1 (z) = # z/a − z 2 /a2 − 1 (as m = 1) into the relevant formulas. The concrete results are given in Section 8.4, which are omitted here. 11.3.9

Example 2 Inﬁnite plane with an elliptic hole subjected to a tension at inﬁnity

In this case X = Y = 0, B = C = 0,

Tx = Ty = 0,

B=

t = t = h = 0,

p , 64c1

B = C = 0,

(11.3-51)

so that from (11.3-44), h4 (z) = Φ4 (ζ) = Bω(ζ) + Φ40 (ζ), h3 (z) = Φ3 (ζ) = Φ30 (ζ), h2 (z) = Φ2 (ζ) =

(11.3-52)

Φ20 (ζ),

Substituting (11.3-52) into (11.3-45), we obtain the similar equations on functions Φj0 (ζ)(j = 2, 3, 4), so the solution is similar to (11.3-50). 11.3.10

Example 3 Inﬁnite plane with an elliptic hole subjected to a distributed pressure at a part of surface of the hole

The problem is shown in Fig.11.3-4, but the conformal mapping is diﬀerent to that of example 1 and example 2.

Fig. 11.3-4

Inﬁnite plane with an elliptic hole subjected to a distributed pressure at a

part of surface of the hole and its conformal mapping at ζ-plane, in which the region at z-plane is mapped onto the exterior of the unit circle at ζ-plane.

11.4

Complex variable function method for sextuple harmonic equation and...

243

In terms of the similar procedure we ﬁnd the solution[9] as follows ⎧ mR0 σ2 1 p σ2 − ζ ⎪ ⎪ · − ln + z (ζ) = · + z ln ln(σ − ζ) − z ln(σ − ζ) Φ ⎪ 4 1 1 2 2 ⎪ 32c1 2πi ζ σ1 σ1 − ζ ⎪ ⎪ ⎪ ⎪ ⎪ + ip(d1 − d2 )(z1 − z2 ) ln ζ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 p (1 + m2 )R0 ζ σ2 R0 (σ1 − σ2 )(1 + mζ 2 ) ⎪ ⎪ (ζ) = · + Φ · − ln 3 ⎪ ⎪ 32c1 2πi (ζ 2 − m) σ1 (ζ 2 − m) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − z2 ln(σ2 − ζ) + z1 ln(σ1 − ζ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (1 + m2 ) ⎪ ⎪ + d ) · (z − z ) ln ζ + (z − z ) − ip(d , 1 2 1 2 1 2 ⎪ ⎨ ζ2 − m σ2 1 pR0 (mζ 2 + 1)(ζ 2 + m) σ2 − σ1 ⎪ ⎪ · ln Φ (ζ) = · + 2 ⎪ ⎪ 32c1 2πi (ζ 2 − m)3 σ1 (σ2 − ζ)(σ1 − ζ) ⎪ ⎪

⎪ 2 ⎪ ⎪ (mζ 1 + 1) p σ2 − σ1 ⎪ ⎪ · 2 + · × 2Re z2 · ⎪ 2 ⎪ 32c 2πi (ζ − m) (σ − ζ)(σ1 − ζ) ⎪ 2 ⎪ 1 ⎪ ⎪ (σ2 − ζ)(σ1 − ζ) + (σ2 + σ1 − 2ζ)(σ2 − σ1 ) m ⎪ ⎪ ⎪ + z · ζ − − R 2 0 ⎪ ⎪ ζ (σ2 − ζ)(σ1 − ζ) ⎪ ⎪

⎪ 2 2 ⎪ (mζ + 1)(ζ + m) 1 ⎪ ⎪ ⎪ ip d1 (z1 − z2 − z1 + z2 ) ⎪ 2 3 ⎪ (ζ − m) ζ − σ1 ⎪ ⎪ ⎪ ⎪ 1 1 1 ⎪ ⎪ ⎩ + (d2 − d1 )(z1 − z2 ) 2 + + ζ ζ (ζ − σ1 )2 (11.3-53) where m m z1 = R σ1 + , z2 = R σ2 + . σ1 σ2 The detail is given in Major appendix of this book.

11.4

Complex variable function method for sextuple harmonic equation and applications to icosahedral quasicrystals

Plane elasticity of icosahedral quasicrystals has been reduced to a sextuple harmonic equation to solve in Chapter 9, where we have shown the solution procedure of the equation for a notch/crack problem by complex variable function method, we here provide further discussion in-depth from view-point of complex function theory. The aim is to develop the complex potential method for higher order multi-harmonic equations. Though there are some similar natures in the following description with that introduced in the previous section, the discussion here is necessary, because the governing equation and boundary conditions for icosahedral quasicrystals are quite diﬀerent from those for decagonal quasicrystals.

244

Chapter 11 Complex variable function method for elasticity of quasicrystals

11.4.1

The complex representation of stresses and displacements

In Section 9.5 by the stress potential we obtain the ﬁnal governing equation under the approximation R2 /μK1 1 ∇2 ∇2 ∇2 ∇2 ∇2 ∇2 G = 0.

(11.4-1)

Fundamental solution of equation (11.4-1) can be expressed in six analytic functions of complex variable z, i.e., G(x, y) = Re[g1 (z) + z¯g2 (z) + z¯2 g3 (z) + z¯3 g4 (z) + z¯4 g5 (z) + z¯5 g6 (z)],

(11.4-2)

where gi (z) are arbitrary analytic functions of z = x + iy, the bar denotes the complex conjugate. By equations (11.4.1),(11.4-2) and (9.5-2), (9.5-3), the stresses can be expressed as follows: ⎧ ⎪ σxx + σyy = 48c2 c3 RImΓ (z), σyy − σxx + 2iσxy = 8ic2 c3 R(12Ψ (z) − Ω (z)), ⎪ ⎪ ⎪ ⎪ 24λR ⎪ ⎪ σzy − iσzx = −960c3 c4 f6 (z), σzz = c2 c3 ImΓ (z), ⎪ ⎪ ⎪ (μ + λ) ⎪ ⎪ ⎪ H − H − i(H + H ) = −96c c Ψ (z) − 8c c RΩ (z), ⎪ yx xx yy 2 5 1 2 ⎨ xy Hyx + Hxy + i(Hxx − Hyy ) = −480c2 c5 f6 (z) − 4c1 c2 RΘ (z), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Hyz + iHxz = 48c2 c6 Γ (z) − 4c2 R2 (2K2 − K1 )Ω (z), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 24R2 ⎪ ⎪ c2 c3 ImΓ (z), ⎩ Hzz = (μ + λ) (11.4-3) where ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

z f6 (z), Ψ (z) = f5 (z) + 5¯ Γ (z) = f4 (z) + 4¯ z f5 (z) + 10¯ z 2 f6 (z), Ω (z) = f3 (z) + 3¯ z f4 (z) + 6¯ z 2 f5 (z) + 10¯ z 3 f6 (z), (IV)

z f3 (z) + 3¯ z 2 f4 (z) + 4¯ z 3 f5 (z) + 5¯ z 4 f6 Θ(z) = f2 (z) + 2¯

R(2K2 − K1 )(μK1 + μK2 − 3R2 ) , 2(μK1 − 2R2 ) ⎪ 1 ⎪ ⎪ c3 = K2 (μK2 − R2 ) − R(2K2 − K1 ), ⎪ ⎪ ⎪ R ⎪ ⎪ ⎪ ⎪ (μK2 − R2 )2 ⎪ ⎪ , c2 = μ(K1 − K2 ) − R2 − ⎪ ⎪ ⎪ μK1 − 2R2 ⎪ ⎪ ⎪ ⎪ 1 μK1 − 2R2 ⎪ ⎪ c4 = c1 R + c3 K1 + ⎪ ⎪ ⎪ 2 λ+μ ⎪ ⎪ ⎪ ⎪ μK2 − R2 ⎪ ⎩ c5 = 2c4 − c1 R, c6 = (2K2 − K1 )R2 − 4c4 . μK1 − 2R2 c1 =

(z),

(11.4-4)

11.4

Complex variable function method for sextuple harmonic equation and...

245

In the above expressions,the function g1 (z)is not used and to be assumed g1 (z) = 0 so f1 (z) = 0 for simplicity, we have introduced the following new symbols: (9) (8) (7) g2 (z) = f2 (z), g3 (z) = f3 (z), g4 (z) = f4 (z), (11.4-5) (6) (5) g5 (z) = f5 (z), g6 (z) = f6 (z), (n)

where gi denotes n-th derivative with the argument z. Similar to manipulation in the previous section, the complex representations of displacement components can be written as follows (here we have omitted the rigid body displacements): ⎧ 2c2 ⎪ ⎪ + iu = −6c R Γ (z) − 2c3 c7 RΩ (z), + c u ⎪ y x 3 7 ⎪ ⎪ μ+λ ⎪ ⎪ ⎪ ⎪ 4 ⎪ ⎪ (240c10 Imf6 (z) uz = ⎪ 2 ⎪ μ(K + K ⎪ 1 2 ) − 3R ⎪ ⎪ ⎪ ⎨ +c1 c2 R2 Im(Θ(z) − 2Ω (z) + 6Γ (z) − 24Ψ (z))), (11.4-6) ⎪ R ⎪ ⎪ ⎪ ⎪ wy + iwx = − c1 (μK1 − 2R2 ) (24c9 Ψ (z) − c8 Θ(z)), ⎪ ⎪ ⎪ ⎪ ⎪ 4(μK2 − R2 ) ⎪ ⎪ wz = (240c10 Imf6 (z) ⎪ ⎪ ⎪ (K1 − 2K2 )R(μ(K1 + K2 ) − 3R2 ) ⎪ ⎪ ⎩ +c1 c2 R2 Im(Θ(z) − 2Ω (z) + 6Γ (z) − 24Ψ (z))), in which c7 =

c2 K1 + 2c1 R , μK1 − 2R2

c9 = c8 + 2c3 c4

11.4.2

c8 = c1 c3 R(μ(K1 − K2 ) − R2 ), (μK2 − R2 )2 c2 − , c10 = c1 c3 R2 − c4 (c3 R − c2 K1 ). μK1 − 2R2 (11.4-7)

The complex representation of boundary conditions

The boundary conditions of plane elasticity of icosahedral quasicrystals can be expressed as follows: σxx l + σxy m = Tx , Hxx l + Hxy m = hx ,

σyx l + σyy m = Ty , Hyx l + Hyy m = hy ,

σzx l + σzy m = Tz , Hzx l + Hzy m = hz

(11.4-8) (11.4-9)

for (x, y) ∈ L, which represents the boundary of a multi-connected quasicrystalline material, and dy dx l = cos(n, x) = , m = cos(n, y) = − , ds ds T = (Tx , Ty , Tz ) and h = (hx , hy , hz ) denote the surface traction vector and generalized surface traction vector, n represents the outward unit normal vector of any point of the boundary, respectively.

246

Chapter 11 Complex variable function method for elasticity of quasicrystals

Utilizing equation (11.4-3) and the ﬁrst two formulas of equations (11.4-8) ,one has −4c2 c3 R[3(f4 (z)+4¯ z f5 (z)+10¯ z 2 f6 (z))−(f3 (z)+3zf4 (z)+6z 2 f5 (z)+10z 3 f6 (z))] = i (Tx + iTy )ds, z ∈ L. (11.4-10) Taking conjugate on both sides of equation (11.4-10) yields −4c2 c3 R[3(f4 (z)+4zf5 (z)+10z 2 f6 (z))−(f3 (z)+3¯ z f4 (z)+6z 2 f5 (z)+10¯ z 3 f6 (z) = −i (Tx − iTy )ds, z ∈ L. (11.4-11) Similarly, from equations (11.4-3) and the ﬁrst two formulas of (11.4-9) ,one obtains (11.4-12) 48c2 (2c4 − c1 R)Ψ (z) + 2c1 c2 RΘ(z) = i (hx + ihy )ds, z ∈ L. Furthermore we assume Tz = hz = 0,

(11.4-13)

for simplicity and by the third equations in (11.4-8) and (11.4-9) and the formulas of (11.4-3) and (11.4-13),one has f6 (z)+f6 (z) = 0, 4c11 Re[f5 (z)+5¯ z f6 (z)]+(2K2 −K1 )RRe[f4 (z)+4¯ z f5 (z)+10¯ z 2 f6 (z)+20f6 (z)] = 0,

in which c11 = (2K2 − K1 )R −

z∈L

(11.4-14)

4c4 (μK2 − R2 ) . (μK1 − 2R2 )R

(11.4-15)

As we have shown in the previous section, complex analytic functions (i.e., the complex potentials) must be determined by boundary value equations, and the discussion is as below. 11.4.3

Structure of complex potentials

11.4.3.1 The arbitrariness of the complex potentials For explicit description, equation (11.4-3) can be written as follows: ⎧ σzy − iσzx = −960c3 c4 f6 (z), ⎪ ⎪ ⎪ ⎪ ⎪ c1 (σyy − σxx − 2iσxy ) + ic2 [Hxy − Hyx + i(Hxx + Hyy )] = −192ic2 c3 c4 Ψ (z), ⎪ ⎪ ⎪ ⎨ 2c1 (Hzy + iHzx ) − R(2K2 − K1 )[Hxy − Hyx + i(Hxx + Hyy )] ⎪ = 96c3 cR(2K2 − K1 )Ψ (z) + 96c1 c3 c6 Γ (z), ⎪ ⎪ ⎪ ⎪ ⎪ c5 (σyy − σxx + 2iσxy ) + ic2 R[Hxy − Hyx − i(Hxx + Hyy )] = −16ic2 c3 c4 Ω (z), ⎪ ⎪ ⎩ Hyx + Hxy + i(Hxx − Hyy ) = −480c2 c5 f6 (z) − 4c1 c2 RΘ (z). (11.4-16)

11.4

Complex variable function method for sextuple harmonic equation and...

247

Similar to the discussion of two-dimensional quasicrystal, from the equations,it is obvious that a state of phonon and phason stresses is not altered, if one replaces fi (z) by fi (z) + γi ,

i = 2, · · · , 6,

(11.4-17)

where γi are arbitrary complex constants. Now consider how these substitutions aﬀect the components of the displacement vectors which were determined by the formula (11.4-6). Substituting (11.4-13) into (11.4-8)∼(11.4-12) shows that if the complex constants γi (i = 2, · · ·, 6) satisfy ⎧ 2c2 ⎪ ⎪ + c7 γ4 + c7 γ3 = 0, ⎪ 3 ⎪ ⎨ μ+λ 24c9 γ5 − c8 γ2 = 0, (11.4-18) ⎪ ⎪ 2c c ⎪ 9 2 ⎪ γ5 − γ4 = 0, ⎩ 40c10 γ6 − c1 c3 R2 4 1 − c8 (μ + λ)c7 then the substitution (11.4-17) will not aﬀect the displacements. 11.4.3.2 General formulas for ﬁnite multiply connected region Consider now the case when the region S, occupied by the body, is multiply connected,see Fig 11.3-1. Since the stress must be single-valued and equation (11.4-16) σzy − iσzx = −960c3 c4 f6 (z).

(11.4-19)

we know that f6 (z)is holomorphic and hence single-valued in the region with contour sm+1 , so the complex function can be expressed as z f6 (z)dz + constant, (11.4-20) f6 (z) = z0

where z0 denotes ﬁxed point. By equation(11.4-20), we have f6 (z) = bk ln(z − zk ) + f6∗ (z),

(11.4-21)

f6∗ (z) is holomorphic in the region with contour sm+1 . Substituting (11.4-21) into the second formula of equations (11.4-16), i.e., c1 (σyy − σxx − 2iσxy ) + ic2 [Hxy − Hyx + i(Hxx + Hyy )] = −192ic2 c3 c4 Ψ (z) shows that f5 (z) is holomorphic in the region enclosed by contour sm+1 , so one has f5 (z) = ck ln(z − zk ) + f5∗ (z), f5∗ (z) is holomorphic in the region of interior of contour sm+1 .

(11.4-22)

248

Chapter 11 Complex variable function method for elasticity of quasicrystals

Similar to the above mentioned discussion, by equations (11.4-16)∼(11.4-18), the complex functions fi (i = 2, 3, 4) can be written as ⎧ ⎨ f4 (z) = dk ln(z − zk ) + f4∗ (z), f3 (z) = ek ln(z − zk ) + f3∗ (z), (11.4-23) ⎩ f2 (z) = tk ln(z − zk ) + f2∗ (z) where dk , ek and tk are complex constants, fi∗ (z)(i = 2, 3, 4) is holomorphic in the region with contour sm+1 . By substituting (11.4-21)∼(11.4-23) into the complex expressions of displacements, the condition of single-valuedness of displacements will be given as follows: ⎧ 2c2 ⎪ ⎪ −3 + c7 dk + c7 ek = 0, ⎪ ⎨ μ+λ (11.4-24) ⎪ 24c9 ck + c8 tk = 0, ⎪ ⎪ ⎩ 240c10 bk + c1 c3 R2 (tk − 2ek + 6dk − 24ck ) = 0.s Applying the boundary conditions given above to the contour sk and by equation (11.4-24), we know that the above complex contants may be very simply expressed in terms of surface traction and generalized surface traction as ⎧ c1 c3 R2 c9 12c2 ⎪ ⎪ dk + 24 1 + ck , bk = ⎪ ⎪ 240c10 (μ + λ)c7 c8 ⎪ ⎪ ⎪ c8 ⎪ ⎪ ⎪ (hx − ihy ), c = ⎪ ⎪ k −96π[c c (2c − c1 R) − c1 c3 R] 3 8 4 ⎪ ⎪ ⎨ c8 tk = (hx + ihy ), (11.4-25) 4π[c3 c8 (2c4 − c1 R) − c1 c3 R] ⎪ ⎪ ⎪ ⎪ (μ + λ)c7 ⎪ ⎪ (Tx + iTy ), dk = ⎪ ⎪ 24πc c R(2c ⎪ 2 3 2 + (μ + λ)c7 ) ⎪ ⎪ ⎪ ⎪ 2c + (μ + λ)c7 ⎪ ⎩ ek = − 2 (Tx − iTy ). 16πc22 c3 R We can easily extend the above results to the case there are m inner boundaries. 11.4.4

Case of inﬁnite regions

From the point of view of application the consideration of inﬁnite regions is likewise of major interest. We assume that the contour sm+1 has entirely moved to inﬁnity. Similar to the discussion of two-dimensional quasicrystal, we have ⎧ m m ⎪ ⎪ f (z) = b ln z + f (z), f (z) = ck ln z + f5∗∗ (z), ⎪ 6 k 6∗∗ 5 ⎪ ⎪ ⎪ k=1 k=1 ⎪ ⎪ m m ⎨ f4 (z) = dk ln z + f4∗∗ (z), f3 (z) = ek ln z + f3∗∗ (z), (11.4-26) ⎪ ⎪ k=1 k=1 ⎪ ⎪ m ⎪ ⎪ ⎪ ⎪ f2 (z) = tk ln z + f2∗∗ (z), ⎩ k=1

11.4

Complex variable function method for sextuple harmonic equation and...

249

fi∗∗ (z)(i = 2, · · · , 6) are functions, holomorphic outside sm+1 , not including the point at inﬁnity. By the theorem of Laurent expansion, the function h2∗ (z) may be represented outside sm+1 by the series +∞

fi∗∗ (z) =

ain z n

(i = 2, · · · , 6.)

(11.4-27)

n=−∞

Substituting the ﬁrst equation of (11.4-26) and (11.4-27) into equation (11.4-16),one $m % has ∞ 1 (11.4-28) bk + na6n z n−1 , σzy − iσzx = −960c3 c4 z −∞ k=1

hence it follows that for the stress to remain ﬁnite as |z| → ∞, one must have a6n = 0,

n 2.

(11.4-29)

Similarly, from equations (11.4-15)∼(11.4-18), to make the stress be bounded, the following conditions are also be satisﬁed ain = 0,

n 2, (i = 2, · · · , 6.)

(11.4-30)

So we can obtain the expressions of the complex function fi (z)(i = 2, · · · , 6) for the stress to remain ﬁnite as |z| → ∞,for example f6 (z) =

m

bk ln z + (B + iC)z + f60 (z),

(11.4-31)

k=1

where B, C are unknown real constants to be determined, f60 (z) is function, holomorphic outside sm+1 , including the point at inﬁnity. The determination of unknown contants B, C etc is similar to that given in subsection 11.3.4, but the details are omitted here due to limitation of the space. 11.4.5

Conformal mapping and function equations at ζ- plane

We now have ﬁve equations of boundary value (11.4-10)∼(11.4-12) and (11.4-14), from which the unknown functions fi (z)(i = 2, · · · , 6)will be determined, in addition we have assumed that f1 (z) = 0, because it has no usage. For some complicated regions the function equations cannot be directly solved at the physical plane (i.e., the z - plane), and the conformal mapping is particular meaningful in the case. Assume that a conformal mapping z = ω(ζ)

(11.4-32)

is used to transform the region at z- plane onto the interior of the unit circle γ at ζ- plane. Under the mapping the unknown functions fi (z) become fi (z) = fi [ω(ζ)] = Φi (ζ) i = 2, 3, · · · , 6.

(11.4-33)

250

Chapter 11 Complex variable function method for elasticity of quasicrystals

Substituting (11.4-32) and (11.4-33) into the ﬁrst relation of boundary condition (11.4-14) and multiplying dσ/2π(σ−ζ) to both sides of the equation, then integrating along the unit circle γ and |ζ| < 1 yields 1 2πi

γ

1 Φ6 (σ) dσ + σ−ζ 2πi

γ

Φ6 (σ) dσ = 0. σ−ζ

This shows Φ6 (ζ) = 0

(11.4-34)

according to the Cauchy integral formula. Substitution of (11.4-32)∼(11.4-34) into boundary conditions (11.4-10)∼ (11.4-12) and the second one of conditions (11.4-14) and taking manipulation similarly to above leads to boundary value equations for determining the unknown functions Φi (ζ)(i = 2, · · · , 5) at ζ - plane, i.e., Φ4 (σ) ω(σ) Φ5 (σ) Φ3 (σ) 4 1 dσ + dσ − dσ (σ) σ − ζ σ − ζ 2πi ω 2πi γ γ γ σ−ζ 1 ω(σ) Φ4 (σ) [ω(σ)]2 Φ5 (σ) [ω(σ)]2 ω (σ) 1 dσ dσ − 6 −3 − Φ5 (σ) 2 3 2πi γ ω (σ) σ − ζ 2πi γ σ −ζ ω (σ) ω (σ) 1 1 t = dσ, 4c2 c3 2πi γ σ − ζ (11.4-35) 4 1 3 Φ4 (σ) ω(σ) Φ4 (σ) Φ3 (σ) dσ + dσ − dσ 2πi γ σ − ζ 2πi γ ω (σ) σ − ζ 2πi γ σ − ζ 2 2 ω(σ) Φ3 (σ) ω(σ) Φ5 (σ) ω(σ) ω (σ)Φ5 (σ) dσ 1 1 −3 − dσ − 6 i 2πi γ ω (σ) σ − ζ 2π γ [ω (σ)]2 [ω (σ)]3 σ−ζ 1 t 1 dσ, = 4c2 c3 R 2πi γ σ − ζ (11.4-36) 1 Φ2 (σ) ω(σ) Φ3 (σ) 1 dσ + 2 dσ+ 2πi γ σ − ζ 2πi γ ω (σ) σ − ζ 2 2 2 ω(σ) Φ4 (σ) ω(σ) ω (σ)Φ4 (σ) ω(σ) Φ 5 (σ) dσ 1 1 3 − + 4 2πi γ [ω (σ)]2 [ω (σ)]3 σ−ζ 2πi γ [ω (σ)]3 3 3 3 ω(σ) ω (σ)Φ5 (σ) ω(σ) ω (σ)Φ5 (σ) ω(σ) ω (σ)Φ5 (σ) dσ −3 + 3 − 4 [ω (σ)]5 [ω (σ)]4 σ−ζ [ω (σ)] h 1 dσ, = 2πi σ−ζ (11.4-37) 4c11 (2K2 − K1 )R Φ5 (σ) ω(σ) Φ5 (σ) Φ4 (σ) dσ + +4 dσ = 0, (11.4-38) 2πi γ σ − ζ 2πi σ−ζ ω (σ) σ − ζ γ 3 2πi

11.4

Complex variable function method for sextuple harmonic equation and...

in which t = i

(Tx + iTy )ds, t = −i

251

(Tx − iTy )ds, h = i

(h1 + ih2 )ds. For

given conﬁguration and applied stresses we can obtain the solution by solving these function equations. 11.4.6

Example: Elliptic notch problem and solution

We consider an icosahedral quasicrystal solid with an elliptic notch, which penetrates through the medium along the z-axis direction,the edge of the elliptic notch subjected to the uniform pressure p, similar to Fig11.3-2. Since the measurement of generalized traction has not been reported so far, for simplicity, we assume that hx = 0, hy = 0. However the calculation can not be completed at the z-plane owing to the complicity, we have to employ the conformal mapping 1 + mζ (11.4-39) z = ω(ζ) = R0 ζ to transform the exterior of the ellipse at the z-plane onto the interior of the unit circle γ at the ζ− plane, in which R0 = (a + b)/2,

m = (a − b)/(a + b)

(11.4-39)

Substituting (11.4-39) and( 11.4-34) into (11.4-35)-(11.4-38), considering ω(σ) σ2 + m = σ ω (σ) mσ 2 − 1 and ζ

ζ2 + m ζ2 + m (ζ) = ζ Φ (α1 + 2α2 ζ + 3α3 ζ 2 + · · ·) 5 mζ 2 − 1 mζ 2 − 1 1 Φ4 (σ) dσ = Φ4 (ζ) 2πi σ−ς γ

are analytic in |ζ| < 1 and continuous in the unit circle γ, by means of Cauchy integral formula, from equation(11.4-34), we have σ 2 + m Φ5 (σ) ζ2 + m 1 σ dσ = ζ Φ (ζ) 2πi mσ 2 − 1 σ − ς mζ 2 − 1 5 γ

Substituting ω(σ) ω (σ)

=−

1 mσ 2 + 1 , σ σ2 − m

ω(σ)2 ω (σ) 3

ω (σ)

=

2σ(mσ 2 + 1)2 (σ 2 − m)3

252

Chapter 11 Complex variable function method for elasticity of quasicrystals

into equation(11.4-34), and note that 1 mζ 2 + 1 1 mζ 2 + 1 − Φ (ζ) = − ζ ζ2 − m 4 ζ ζ2 − m

2ζ(mζ 2 + 1)2 2ζ(mζ 2 + 1)2 Φ (ζ) = 5 (ζ 2 − m)3 (ζ 2 − m)3

β2 β3 β1 + 2 + 3 2 + · · · ζ ζ

α1 + 2

α2 α3 +3 2 +··· ζ ζ

are analytic in |ζ| > 1 and continuous in the unit circle γ, by means of Cauchy integral formula and analytic extension of the complex variable function theory, from equation(11.4-34), we obtain 1 Φ3 (σ) ω(σ) Φ4 (σ) dσ = 0, dσ = 0 σ−ς 2πi ω (σ) σ − ς γ γ 2 dσ 1 ω(σ) Φ5 (σ) ω(σ)2 ω (σ) − Φ5 (σ) =0 2 3 2πi σ −ς ω (σ) ω (σ) γ 1 2πi

Substituting the above results into equation(11.4-34), with the help of equation (11.4-38),one has Φ4 (ζ) =

(2K2 − K1 )R0 pmζ(ζ 2 + m) R0 pmζ − 12c2 c3 R 2c2 c3 C 11 (mζ 2 − 1)

Φ5 (ζ) =

2 −K1 )R0 − (2K 48c2 c3 C 11 pmζ

(11.4-36)

Similar to the above discussion, by equations (11.4-36) and (11.4-37), one has Φ2 (ζ) = −

R0 pζ(ζ 2 + m)(m3 ζ 2 + 1) 2c2 c3 R (mζ 2 − 1)3

(2K2 −K1 )R0 pmζ 3 (ζ 2 +m)[m2 ζ 6 −(m3 +4m)ζ 4 +(2m4 +4m2 +5)ζ 2 +m] + 2c2 c3 C 11 (mζ 2 −1)5 Φ3 (ζ)

=−

R0 pζ(m2 + 1) (2K2 − K1 )R0 pmζ 3 (ζ 2 + m)(mζ 2 − m2 − 2) − 4c2 c3 R (mζ 2 − 1) 12c2 c3 C 11 (mζ 2 − 1)3 (11.4-41)

The elliptic notch problem is solved. The solution of the Griﬃth crack subjected to a uniform pressure can be obtained corresponding to the case m = 1, R0 = a/2 of the above solution. The solution of crack can be expressed explicitly in the z−plane, the concrete results refer to Section 9.7 in Chapter 9.

11.5

11.5

Complex analysis and solution of quadruple quasiharmonic equation

253

Complex analysis and solution of quadruple quasiharmonic equation

In Chapters 6∼8 we have known that the plane elasticity of octagonal quasicrystals is governed by the ﬁnal equation (∇2 ∇2 ∇2 ∇2 − 4ε∇2 ∇2 Λ2 Λ2 + 4εΛ2 Λ2 Λ2 Λ2 )F = 0 either by displacement potential or by stress potentail, in which ⎧ ∂2 ∂2 ∂2 ∂2 ⎪ 2 2 ⎪ + , Λ = − , ⎨ ∇ = ∂x2 ∂y2 ∂x2 ∂y2 ⎪ R2 (L + M )(K2 + K3 ) ⎪ ⎩ ε= [M (K1 + K2 + K3 ) − R2 ][(L + 2M )K1 − R2 ]

(11.5-1)

(11.5-2)

Due to the appearance of operator Λ2 it seems there is no any connection with complex variable functions in solving equation (11.5-1). But if we rewrite it as 8 ∂ ∂8 ∂8 ∂8 ∂8 + 4(1 − 4ε) 6 2 + 2(3 + 16ε) 4 4 + 4(1 − 4ε) 2 6 + 8 F = 0, ∂x8 ∂x ∂y ∂x ∂y ∂x ∂y ∂y (11.5-3) then ﬁnd that this is one of typical multi-quasiharmonic partial diﬀerential equation with 8th order, and there is complex representation of solution such as F (x, y) = 2Re

4

Fk (zk ),

zk = x + μk y,

(11.5-4)

k=1

in which functions Fk (zk ) are analytic functions of complex variable zk (k = 1, 2, 3, 4), and μk = αk + iβk (k = 1, 2, 3, 4) are complex parameters and determined by the roots of the following eigenvalue equation μ8 + 4(1 − 4ε)μ6 + 2(3 + 16ε)μ4 + 4(1 − 4ε)μ2 + 1 = 0.

(11.5-5)

We have shown in Chapters 7 and 8 some solutions of dislocations (based on displacement potentail formulation) and notches/cracks (based on stress potential formulation) can be found in terms of this complex analysis. In the procedure it must carry out some calculations on determinants of 4th order, so the solution expressions are quite lengthy, but which are analytic substantively.

11.6

Conclusion and discussion

The discovery of quadruple and sextuple harmonic equations is interesting for the elasticity. This chapter gives a comprehensive discussion on the complex analysis for solving the equations, we think the study is preliminary.

254

Chapter 11

Complex variable function method for elasticity of quasicrystals

In some sense, the complex potential approach above mentioned is a new development of Muskhelishvili approach of the classical plane elasticity. Apart from the development to the complex potential theory and method, the conformal mapping technique has also been extended. According to the monograph [1], application of the conformal mapping is limited within the rational function class. But it can be extended into the transcendental function class, and some exact analytic solutions for more complicated cracked conﬁgurations are achieved, see e.g. Chapter 8. The method is eﬀective not only for solving elasticity problems but also for solving some simple plasticity problems, see e.g. Fan and Fan[10] , Li and Fan[11,12] and Fan and Tang[13] .

References [1] Muskhelishvili N I. Some Basic Problems of the Mathematical Theory of Elasticity. Groningen: P Noordhoﬀ., 1956 [2] Lekhnitskii S G. Theory of Elasticity of an Anisotropic Body. Holden-Day. San Francisco, 1963 [3] Liu G T, Fan T Y, Guo R P. Governing equations and general solutions of plane elasticity of one-dimensional quasicrystals. Int J Solid and Structures, 2004, 41(14): 3949−3959 [4] Liu G T. The complex variable function method of the elastic theory of quasicrystals and defects and auxiliary equation method for solving some nonlinear evolution equations. Dissertation. Beijing Institute of Technology, 2004 (in Chinese) [5] Liu G T, Fan T Y. The complex method of the plane elasticity in 2D quasicrystals point group 10mm ten-fold rotation symmetry notch problems. Science in China E, 2003, 46(3): 326−336 [6] Li L H, Fan T Y. Final governing equation of plane elasticity of icosahedral quasicrystals– stress potential method. Chin Phys Lett, 2006, 24(9): 2519−2521 [7] Li L H, Fan T Y. Complex function method for solving notch problem of point group 10 and 10 two-dimensional quasicrystal based on the stress potential function, J. Phys.: Condens. Matter, 2006, 18(47): 10631−10641 [8] Li L H, Fan T Y. Complex function method for notch problem of plane elasticity of icosahedral quasicrystals, Science in China, G, 2008, 51(6): 773−780 [9] Li W, Fan T Y. Study on elastic analysis of crack problem of two-dimensional decagonal quasicrystals of point group 10,10. Mod Phys Lett, B, 2009, 23(16), 1989–1999 [10] Fan T Y, Fan L. Plastic fracture of quasicrystals. Phil Mag, 2008, 88(4): 321−335 [11] Li W, Fan T Y. Study on plastic analysis of crack in decagonal Al-Ni-Co quasicrystals. Mod. Phys. Lett, B, in press, 2009

References

255

[12] Li W, Fan T Y. Plastic solution of crack in three-dimensional Al-Pd-Mn quasicrystals, Phil.Mag., 2009, 89(31), 2823–2831 [13] Fan T Y, Tang Z Y, 2009, The strict theory of complex variable function method of sextuple harmonic equation and applications, J. Math. Phys., accepted, 2010

Chapter 12 Variational principle of elasticity of quasicrystals, numerical analysis and applications From Chapter 5 to Chapter 11, we developed mainly analytic theories and methods. The elasticity problems of quasicrystals were reduced to boundary value or initial-boundary value problems of some partial diﬀerential equations to solve, in which some analytic methods were used. For some boundary value problems, these methods are extremely powerful, even capable of obtaining exact analytic solutions. In Chapter 14 and Major Appendix we will further develop the analytic method in studying some problems such as nonlinear deformation etc. However there are limitations themselves for these analytic methods. In general, they can only treat some problems with simple conﬁgurations and simple boundary conditions; while for more complicated problems, the methods cannot display their power, one must use numerical methods. In Chapter 10 we have applied the ﬁnite diﬀerence method to determining the numerical solutions for some elasto-/hydro-dynamics problems of quasicrystals. In this chapter we derive a variational principle of elasticity of quasicrystals, which is the foundation of the subsequent development of the ﬁnite element method for problems of quasicrystals. Discretization is the main feature of ﬁnite diﬀerence method and ﬁnite element method. It has been shown that solutions obtained by these two schemes can approach to the analytic solutions as the discreted mesh (or element) tends to inﬁnitesimal. In the other hand, in contrast to the analytic solutions (or classical solutions), ﬁnite element solutions belong to one kind of socalled weak solutions (or generalized solutions) according to the modern theory of partial diﬀerential equations. The further mathematical principle on the weak solutions of elasticity of quasicrystals will be developed in Chapter 13, which will help us to understand that the ﬁnite element method is one of important tools to implement weak solutions from other angles.

T. Fan, Mathematical Theory of Elasticity of Quasicrystals and Its Applications © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2011

258 Chapter 12

12.1

Variational principle of elasticity of quasicrystals, numerical analysis...

Basic relations of plane elasticity of two-dimensional quasicrystals

Here we only consider the two-dimensional quasicrystals, and further assume that along the periodic direction, i.e., along the x3 -axis all ﬁeld variables are uniform, so that ∂ = 0. (12.1-1) ∂x3 This assumption and the fact w3 = 0

(12.1-2)

reduce the elasticity to a plane phonon-phason elasticity and an anti-plane pure phonon elasticity, respectively, and the solution is greatly simpliﬁed. In the following only the plane elasticity is considered. In this case, the generalized Hooke’s law stands for

σij = Cijkl εkl + Rijkl wkl ,

(12.1-3)

Hij = Kijkl wkl + Rklij εkl , where Cijkl = Lδij δkl + M (δik δjl + δil δjk ), L = C12 ,

M=

C11 − C12 , 2

Kijkl = (K1 − K2 − K3 )δik δjl + K2 δij δkl + K3 δil δ

(12.1-4) (12.1-5) (12.1-6)

+2(K2 + K3 )(δi1 δj2 δk1 δl2 + δi2 δj1 δk2 δl1 ), Rijkl = R1 (δi1 − δ)(δij δkl − δik δjl + δil δjk ) +R2 [(1 − δij )δkl + δij (δi1 − δi2 )(δk1 δl2 − δk2 δl1 )] i, j, k, l = 1, 2. (12.1-7) and the deﬁnitions about C12 , C12 (orL, M ), K1 , K2 , K3 , R1 and R2 have been discussed in detail in Chapters 6∼8. For the points group 10mm, R1 > 0, R2 = 0, K1 > 0, K2 > 0, K2 + K3 = 0; for the point group 8mm, R1 > 0, R2 = 0, K1 > 0, K2 > 0, K3 > 0; for the point group 10,10 R1 > 0, R2 > 0, K1 > 0, K2 > 0, K2 + K3 = 0; for the point group 12mm, R1 = 0, R2 = 0,K1 > 0,K2 > 0,K3 > 0 (but some measured results show K2 < 0, this does not inﬂuence the extreme value of energy functional, only inﬂuences the minimum value of the functional in the proof for theorem of variational principle of elasticity of quasicrystals, in Section 12.2). The strains in above relevant formulas are denoted by

12.2

Generalized variational principle for static elasticity of quasicrystals

1 εij = 2

∂uj ∂ui + ∂xj ∂xi

,

wij =

∂wi ∂xj

259

(12.1-8)

and stresses satisfy the equilibrium equations ∂Hij ∂σij + fi = 0, + gi = 0. ∂xj ∂xj

(12.1-9)

The above formulas hold in any interior point of region Ω , and at boundary Γt , the stresses satisfy the boundary conditions

σij nj = Ti , (x1 , x2 ) ∈ Γt , Hij nj = hi

(12.1-10)

and at boundary Γu the displacements satisfy the boundary conditions

ui = u ¯i , ¯i , wi = w

(x1 , x2 ) ∈ Γu ,

(12.1-11)

where Ti is the traction vector, hi the generalized traction vector at boundary Γt , u ¯i and w ¯i the given displacements at boundary Γu , ni the unit outward normal vector at any point of the boundary and Γ = Γu + Γt . In the following only the static problems are studied, and the initial value conditions will not be concerned. For the dynamic problems, the initial value conditions must be used, which have been discussed in Chapter 10.

12.2

Generalized variational principle for static elasticity of quasicrystals

The variational principle in mathematical physics is one of basic principles, which reveals that the extreme value (or stationary value) of energy functional of a system is equivalent to the governing equations and the corresponding boundary value (or initial-boundary value) conditions of the system. Accordingly solutions of initialboundary value problem of the partial diﬀerential equations can be converted to determine the extreme value of the corresponding energy functional. And the latter will be implemented by a discretization procedure, one among them is the ﬁnite element method. We here extend the minimum potential energy principle of classical elasticity to describe elasticity of quasicrystals. Theorem(Variational principle of elasticity of quasicrystals) For suﬃcient smooth boundary, if all ui and wi satisfy equations of deformation geometry (12.1-8)

260 Chapter 12

Variational principle of elasticity of quasicrystals, numerical analysis...

and displacement boundary conditions (12.1-11), let the energy functional of quasicrystals

Π =

F dΩ + Ω

Ω

(fi ui + gi wi )dΩ +

Γt

(Ti ui + hi wi )dΓ

(12.2-1)

to take a minimum value, then they will be the solution satisfying the equilibrium equations (12.1-9) and the stress boundary conditions (12.1-10), in which F is deﬁned by wij εij ⎧ ⎪ ⎪ F = σ dε + Hij dwij = Fu + Fw + Fuw , ij ij ⎪ ⎪ ⎪ 0 0 ⎪ ⎪ ⎪ ⎪ 1 ⎨ Fu = Cijkl εkl , (12.2-2) 2 ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ Fw = 2 Kijkl wkl , ⎪ ⎪ ⎪ ⎩ Fuw = Rijkl εij wkl or by (4.4-1), Ω is the region occupied by the quasicrystal and Γ is the boundary of Ω. In addition, the conditions in the theorem are suﬃcient and necessary. Proof (1) Necessity Assume that the functional Π takes its extreme value, i.e.,δ Π = 0. From (12.21), ∂F ∂F δΠ = δεij + δwij − fi δui − gi δwi dΩ ∂εij ∂wij Ω + (Ti δui + hi δwi )dΓ = 0 (12.2-3) Γt

where σij =

∂F , ∂εij

Hij =

∂F , ∂wij

(12.2-4)

which have been introduced in Chapter 4. ∂F are symmetric, therefore we obtain Noting that the suﬃxes of quantities ∂εij Ω

∂F δεij dΩ = ∂εij

Ω

∂F δ ∂εij

∂ui ∂xj

Making use of the Green formula to above formula yields

dΩ .

12.2

Generalized variational principle for static elasticity of quasicrystals

Ω

∂F δεij dΩ = ∂εij

Ω

∂ ∂xj

= Γu +Γt

261

∂ ∂F ∂F δui dΩ − δui dΩ ∂εij ∂xj ∂εij Ω ∂F ∂ ∂F nj δui dΓ − δui dΩ . ∂εij ∂x ∂ε j ij Ω

Because the displacements are given at the boundary Γu , at which δui = δ u ¯i = 0, the above formula has been reduced to ∂F ∂F ∂F ∂ δεij dΩ = nj δui dΓ − (12.2-5) δui dΩ . ∂εij Ω ∂εij Γt ∂εij Ω ∂xj Due to wij = ∂wi /∂xj , a similar analysis to what adopted just above gives rise to Ω

∂F δwij dΩ = ∂wij

Γt

∂F nj δwi dΓ − ∂wij

Ω

∂ ∂xj

∂F ∂wij

δui dΩ .

(12.2-6)

Substituting (12.2-5) and (12.2-6) into (12.2-3) leads to ∂ ∂ ∂F ∂F + fi δui + + gi δwi dΩ δ Π =− ∂x ∂εij ∂x ∂wij j Ω j ∂F ∂F nj − Ti δui + nj − hi δwi dΓ = 0. + ∂ε ∂w ij ij Γt

(12.2-7)

Since δui and δwi are of arbitrary and independent variation at region Ω and boundary Γ there follows ∂ ∂ ∂F ∂F + fi = 0, + gi = 0, (x1 , x2 ) ∈ Ω , ∂xj ∂εij ∂xj ∂wij ∂F ∂F nj − hi = 0, nj − Ti = 0, (x1 , x2 ) ∈ Γt . ∂wij ∂εij Substituting (12.2-4) into the above formulas yields the equilibrium equations and stress boundary conditions. This shows that ui and wi satisfying equations of deformation geometry, stress-strain relations and displacement boundary conditions and making energy functional to have minimum value, should be the solution satisfying the equilibrium equations and stress boundary conditions. (2) Suﬃciency The suﬃciency of the conditions given by the theorem means that, if among all sets of the displacements ui and wi satisfy relations of deformation geometry, displacement boundary, the one that satisﬁes the equilibrium equations and stress boundary conditions makes in (12.2-1) an extremum. In what follows, we show that Π is minimum.

262 Chapter 12

Variational principle of elasticity of quasicrystals, numerical analysis...

Suppose that quantities ui , εij , wi and wij obey the stress-strain relations (12.1-3) and satisfy the displacement boundary conditions (12.1-13). Let

∗ εij = εij + δεij , u∗i = ui + δui , (12.2-8) ∗ = wij + δwij , wi∗ = wi + δwi wij through the displacement- strain relations, we have ⎧ ⎨ δε = 1 (δu + δu ), ij i,j j,i 2 ⎩ δw = δw , ij

(12.2-9)

i,j

where ui,j = ∂ui /∂xj etc. ∗ ) can be expanded into the Taylor series as follows: The free energy F (ε∗ij , wij ∗ F (ε∗ij , wij ) = F (εij + δεij , wij + δwij )

∂F ∂F δεij + δwij ∂εij ∂wij 1 ∂2F 1 ∂2F ∂2F + δεij δεkl + δwij δwkl + δεij δwkl +· · · . 2 ∂εij ∂εkl 2 ∂wij ∂wkl ∂εij ∂wkl (12.2-10) If the free energy is a homogeneous quantity of strain components of second order, then the expansion of energy functional corresponding to (12.2-10) does not contain terms higher than second order, i.e., = F (εij , wij ) +

Π ∗ = Π + δ Π + δ 2 Π + O(δ 3 ), in which

(12.2-11)

∂F ∂F δεij + δwij − fi δui − gi δwi dΩ − (Ti δui + hi δwi )dΓ , ∂εij ∂wij Ω Γt (12.2-12)

1 ∂2F ∂2F 1 ∂2F 2 δεij δεkl + δwij δwkl + δεij δwkl dΩ . δ Π = 2 ∂εij ∂εkl 2 ∂wij ∂wkl ∂εij ∂wkl Ω (12.2-13) Applying the Green formula to (12.2-12) leads to δΠ =

δ Π = 0. Because ui and wi (through the corresponding σij and Hij ) satisfy the equilibrium equations and stress boundary conditions, this means that the energy functional takes extreme value. According to the discussion in Ref.[2], we do some extension, i.e., L + M > 0, M > 0, K1 > 0, K2 > 0, K3 > 0, M K1 > R2 ,then elasticity of two-dimensional quasicrystals presents stability, in the case the stress-strain elastic matrix should be positive deﬁnite, so

12.3

Finite element method

263

δ 2 Π > 0. This ensures that δ Π = 0 takes not only an extreme value, but also the minimum value of the energy functional. But recent experimental results and some simulation show that there may be K2 < 0, this does not inﬂuence the energy taking extreme value, but the extreme value may not be minimum value. Collaborating variational principle with theory of functional analysis, we can prove that existence, uniqueness and stability of solution of the boundary value problem (12.1-3),(12.1-8)∼(12.1-11).This is concerned not only with the numerical implementation, but also with other topics, the detailed discussion of which will be given in Chapter 13 or see Guo and Fan[4] . The variational principle can be extended to dynamic case, in which it is needed only to extend the energy functional (12.2-1) to be as follows: .. F dΩ + [(fi − ρ ui )ui + (gi − κw˙ i )wi )dΩ + (Ti ui + hi wi )dΓ , (12.2-14) Π = Ω

Ω

Γt

where the meaning of ρ and κ can be found in Chapter 10. From (12.2-14) we can obtain the corresponding variational equation similar to (12.2-3), but which is equivalent to the elasto-/hydro-dynamic equations and related boundary and initial conditions of quasicrystals. The further discussion is omitted here.

12.3

Finite element method

A discretization of the variational equation and domain Ω yields the ﬁnite element algorithm. Dividing the quasicrystalline body studied into N elements, and we here use the harmonic element of tetragon, i.e., we require that (1) Within every element, displacements are continuous and single-valued; (2) Along the inter-element boundaries, the displacements are continuous, i.e.,

(m)

(m )

ui

= ui

(m) wi

=

(m ) ui ,

(x1 , x2 ) ∈ Γ (mm ) ,

(12.3-1)

where Γ (mm ) represents the interface of element m and element m ; (3) At the displacement boundary Γu ,the displacements should satisfy the conditions such as

¯i , ui = u (12.3-2) (x1 , x2 ) ∈ Γu . ¯i , wi = w

264 Chapter 12

Variational principle of elasticity of quasicrystals, numerical analysis...

By satisfying these conditions, the discrete form of the energy functional Π is given by Π∗ =

N m=1

Ω (m)

(m) (m) (m) (m) ui −gi wi )dΩ −

(F (m) −fi

(m)

Γt

(Ti

(m) (m) (m) (m) ui +hi wi )dΓ .

(12.3-3) Taking variation to (12.3-3) and letting it to be zero, by using the method similar to that adopted in the previous section, we can show that within an element the equilibrium equations are satisﬁed by ⎧ (m) ⎪ ∂F ∂ (m) ⎪ ⎪ + fi = 0, ⎨ ∂xj ∂εij (12.3-4) (x1 , x2 ) ∈ Ωm , (m) ⎪ ∂ ∂F (m) ⎪ ⎪ + gi = 0, ⎩ ∂xj ∂wij the stress boundary conditions are satisﬁed by ⎧ ⎪ ∂F (m) (m) (m) ⎪ ⎪ nj = T i , ⎨ ∂εij (m) ⎪ ∂F (m) (m) ⎪ ⎪ nj = h i , ⎩ ∂wij

(m)

(x1 , x2 ) ∈ Γt

(12.3-5)

and at the interface between elements the displacement continuous conditions (12.3-1) are satisﬁed. The discrete energy functional (12.3-3) can also be written in the following matrix form N

1 (m) T (m) (m) (m) T (m) Π = {u } {K }{u } − {u } {R } , 2 m=1 ∗

(12.3-6)

where column vector {u(m) } represents the element displacement (including phonons and phasons) vector, {u(m) }T is its transpose, the row vector, and {K (m) } the element stiﬀness matrix , i.e., T [K (m) ] = [B] [D][B]Δdx1 dx2 , (12.3-7) Ωm

in which, Δ denotes the thickness of the element and Δ=1.0 for plane strain case (i.e., the thick body, and we do not consider plane stress case, i.e., the thin body), [B] = [B1 , B2 , · · · Bn ] = [L][I]

(12.3-8)

is the element strain matrix , n is the number of nodes with diﬀerential operator

12.3

Finite element method

matrix,

265

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ [L] = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

∂ ∂x 0

⎤

∂ ∂y

0 ∂ ∂y ∂ ∂x

0

0

0

0

0

0

0

0

∂ ∂y

0

0

0

0 0 ∂ ∂x 0

⎥ ⎥ 0 ⎥ ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ∂ ⎥ ⎥ ∂y ⎥ ⎥ ⎥ 0 ⎥ ⎥ ∂ ⎦ ∂x

[Bi ] = [L][Ii ]

(12.3-9)

(12.3-9 )

and [I] = [I1 , I2 , · · · , In ] is the element interpolation matrix, n is the number of nodes as pointed out just before, here n = 4, i.e., ⎤ ⎡ Ii 0 0 0 ⎢ 0 Ii 0 0 ⎥ ⎥ (12.3-10) [Ii ] = ⎢ ⎣ 0 0 Ii 0 ⎦ , i = 1, 2, 3, 4. 0 0 0 Ii where Ii is the i-th node interpolation function. In (12.3-7) matrix [D] denotes the plane elastic constants matrix of two-dimensional quasicrystals, namely ⎡ ⎤ L + 2M L 0 R1 R1 R2 −R2 ⎢ L ⎥ L + 2M 0 −R1 −R1 −R2 R2 ⎢ ⎥ ⎢ 0 ⎥ 0 2M 2R2 2R2 −2R1 2R1 ⎢ ⎥ ⎢ ⎥. R R 2R K K 0 0 [D] = ⎢ 1 1 2 1 2 ⎥ ⎢ R1 ⎥ −R 2R K K 0 0 1 2 2 1 ⎢ ⎥ ⎣ R2 ⎦ −R2 2R1 0 0 K1 + K2 + K3 K3 −R2 R2 2R1 0 0 K3 K1 + K 2 + K 3 (12.3-11) The strain vector and stress vector corresponding to (12.3-11) are arranged by [εij , wij ] = [ε11 , ε22 , ε12 , w11 , w22 , w12 , w21 ], [σij , Hij ] = [σ11 , σ22 , σ12 , H11 , H22 , H12 , H21 ]. The external force vector {R(m) } in (12.3-6) is

f T (m) T T [I] {R } = [I] Δdx1 dx2 + ΔdΓ , (m) g h Ωm Γ t

in which

(12.3-12)

266 Chapter 12

Variational principle of elasticity of quasicrystals, numerical analysis...

T f = [f1x , f1y , g2x , g2y , · · · , f4x , f4y , g4x , g4y ], g

T T = [T1x , T1y , h1x , h1y , · · · , T4x , T4y , h4x , h4y ] h

f T and , are the transposes of the row vectors, x = x1 , y = x2 . g h After assemblage of the element matrix, (12.3-6) may be rewritten in terms of matrix form 1 Π ∗ = {u}T [K]{u} − {u}T {R}. (12.3-13) 2 Carrying out the operation δΠ ∗ = 0,

(12.3-14)

we obtain the ﬁnite element equation [K]{u} = {R},

(12.3-15)

where [K], {u} and {R} are the global stiﬀness matrix, displacement and external force vectors generated by assemblage of element stiﬀness matrixes, element displacement and external force vectors, and {u} is solution vector unknown to be determined. By solving algebraic equation set (12.3-15) the displacement vector can be determined, and so do the strains and stresses. The stresses of quasicrystals at the Gauss integration points can be evaluated through the element node displacements in accordance with the following version

σ H

(m)

= [S]

u m

(m)

= [D][B]

u m

(m)

= [D][L][I]

u m

(m) , (12.3-16)

where [S] = [D][B] = [D][L][I] is the element stress matrix (m)T σ = [σxx , σyy , σxy , Hxx , Hyy , Hxy , Hyx ] H is the phonon and phason stress vector in element m and

u w

(m)T = [u1x , u1y , w1x , w1y , · · · u4x , u4y , w4x , w4y ]

is the node displacement (including phonones and phasons) vector at element m.

12.4

Numerical examples

12.4

267

Numerical examples

To illustrate the application of the ﬁnite element method to stress analysis of quasicrystals, some numerical results are displayed in this section Example 1 A hollow cylinder of octagonal quasicrystal material subjected to an inner pressure is shown in Fig. 12.4-1.

Fig. 12.4-1

Hollow cylinder of octagonal quasicrystal subjected to a pressure

There are boundary conditions r = r0 : σrr = −p, σrθ = 0, Hrr = 0, Hrθ = 0 r = r1 : σrr = 0, σrθ = 0, Hrr = 0, Hrθ = 0.

(12.4-1)

The phonon and phason stresses, displacements and free energy (strain energy density) are ploted versus radial distance in Figs.12.4-2∼12.4-6 for diﬀerent material constants, respectively. In the Fig.12.4-3, it seems that the phason stress is one order of magnitude less than the phonon stress, but this by no means shows that the phason stress are less than phonon stress in general. One of the reasons resulting in the lower phason stresses compared with the phonon stresses lies in the boundary conditions (12.4-1). Instead of (12.4-1), if we take r = r0 : σrr = 0, σrθ = 0, Hrr = q, Hrθ = 0, r = r1 : σrr = 0, σrθ = 0, Hrr = 0, Hrθ = 0,

(12.4-2)

then the computation shows that the phason stresses are greater than the phonon stresses. This suggests that the boundary conditions play an important role in the analysis. Due to the lack of the knowledge especially lack of the measuring data

268 Chapter 12

Variational principle of elasticity of quasicrystals, numerical analysis...

on the generalized tractions hi , one often supposes that the tractions to be zero, this inﬂuences the correctness of the computational results, especially to those of the phason ﬁeld. Even so, the computaion reveals the eﬀects of the phason and phonon-phason coupling.

Fig.12.4-2 1. 2. 3. 4.

Fig.12.4-3 1. 2. 3. 4.

Phonon stress versus radial distance

K1 /M = 0.8, K2 /M = 0.6, K3 /M = 0.4, R1 /M = 0.1; K1 /M = 0.7, K2 /M = 0.6, K3 /M = 0.4, R1 /M = 0.1; K1 /M = 0.8, K2 /M = 0.7, K3 /M = 0.4, R1 /M = 0.1; K1 /M = 0.8, K2 /M = 0.6, K3 /M = 0.4, R1 /M = 0.2

Phason stress versus radial distance

K1 /M = 0.8, K2 /M = 0.6, K3 /M = 0.4, R1 /M = 0.1; K1 /M = 0.7, K2 /M = 0.6, K3 /M = 0.4, R1 /M = 0.1; K1 /M = 0.8, K2 /M = 0.7, K3 /M = 0.4, R1 /M = 0.1; K1 /M = 0.8, K2 /M = 0.6, K3 /M = 0.4, R1 /M = 0.2

12.4

Numerical examples

Fig.12.4-4 1. 2. 3. 4.

Phonon displacement versus radial distance

K1 /M = 0.8, K2 /M = 0.6, K3 /M = 0.4, R1 /M = 0.1; K1 /M = 0.7, K2 /M = 0.6, K3 /M = 0.4, R1 /M = 0.1; K1 /M = 0.8, K2 /M = 0.7, K3 /M = 0.4, R1 /M = 0.1; K1 /M = 0.8, K2 /M = 0.6, K3 /M = 0.4, R1 /M = 0.2

Fig.12.4-5 1. 2. 3. 4.

269

Phason displacement versus radial distance

K1 /M = 0.8, K2 /M = 0.6, K3 /M = 0.4, R1 /M = 0.1; K1 /M = 0.7, K2 /M = 0.6, K3 /M = 0.4, R1 /M = 0.1; K1 /M = 0.8, K2 /M = 0.7, K3 /M = 0.4, R1 /M = 0.1; K1 /M = 0.8, K2 /M = 0.6, K3 /M = 0.4, R1 /M = 0.2

Fig.12.4-6 shows that the variation of the free energy (or the strain energy density) in the cylinder as function of coordinates, versus radial distance for diﬀerent material constants. The mesh conﬁguration of the phason ﬁeld presents some changes before and after deformation, which are shown in Fig. 12.4-7, this explores the anisotropic

270 Chapter 12

Variational principle of elasticity of quasicrystals, numerical analysis...

behaviour of the phason ﬁeld of octagonal quasicrystal. The axisymmetry of the mesh conﬁguration of phonon ﬁeld remains after deformation, because the phonon ﬁeld in xy-plane is the isotropic plane of elasticity. The results are identical to those discussed by Refs. [2,3].

Fig.12.4-6

Strain energy density of hollow cylinder versus radial distance

1 1.75E+0.5, 2 2.50E+05, 3 3.61E+0.5, 4 4.55E+05, 5 5.40E+05, 6 6.41E+05, 7 7.34E+05

12.4

Numerical examples

Fig. 12.4-7

271

The mesh of phason ﬁeld before and after deformation:

K1 /M = 0.8, K2 /M = 0.6, K3 /M = 0.4, R1 /M = 0.1[9]

Example 2 Numerical solution of Griﬃth crack in an octagonal quasicrystal with point group 8mm[9] The problem has been solved analytically and given by Zhou and Fan[8] but it is more diﬃcult than that for decagonal[4] . Here the numerical solutions for two diﬀerent materials R/M =0.0 and R/M =0.1, and other elastic constants are the same as given in example 1. The results are depicted by Fig. 12.4-8. The solution for the former material, is the solution for dodecagonal quasicrystal in fact. All results are in agreement with those of analytic solutions, see Chapter 8 and the Refs. [5]∼[7] of this chapter for more detailed discussions.

Fig. 12.4-8

FEM numerical results of normalized phonon stress near the crack tip versus normalized distance 1. K1 /M = 0.8, K2 /M = 0.6, K3 /M = 0.4, R/M = 0.0 2. K1 /M = 0.7, K2 /M = 0.6, K3 /M = 0.4, R/M = 0.1

272

Chapter 12

Variational principle of elasticity of quasicrystals, numerical analysis...

References [1] Hu H C. Variational Principles of Elasticity, Beijing. Science Press, 1981 (in Chinese) [2] De P, Pelcorits R A. Linear elasticity theory of pentagonal quasicrystals. Phys Rev B, 1987 35(28): 8609−8620 [3] Ding D H, Yang W G, Hu C Z et al. Generalized elasticity theory of quasicrystals. Phy Rev B, 1993 48(16). 7003−7010 [4] Li X F, Fan T Y, Sun Y F. A decagonal quasicrystal with a Griﬃth crack. Phil Mag A, 1999, 79(8): 1943−1952 [5] Guo L H, Fan T Y. Solvability of boundary value problems of elasticity of threedimensional quasicrystal. Appl Math Mech, English Edition 2007, 28(8): 1061−1070 [6] Guo Y C, Fan T Y. Mode II Griﬃth crack in decagonal quasicrystals. Appl Math Mech, English Edition, 2001, 22(10), 1311−1317 [7] Fan T Y and Mai Y W. Elasticity theory, fracture mechanics and some relevant thermal properties of quasicrystalline materials, Appl. Mech. Rev, 2004, 57(5), 325−344. [8] Zhou W M, Fan T Y, Plane elasticity and crack problem of octagonal quasicrystals. Chin. Phys. 2001, 10 (6), 277−284 [9] Wu X F, Numerical analysis of power-law hardening material and quasicrystalline material with defects, Dissertation, Beijing Institute of Technology, 1998 (in Chinese)

Chapter 13 Some mathematical principles on solutions of elasticity of quasicrystals Starting from Chapter 4 we studied the elasticity of quasicrystals and gave many solutions. A further discussion on some common features of the solutions will be oﬀered in this chapter.

13.1

Uniqueness of solution of elasticity of quasicrystals

Theorem Assume quasicrystal occupied by region Ω is in equilibrium under the boundary conditions σij nj = Ti , Hij nj = hi , xi ∈ Γt , (13.1-1) ui = u ¯i , wi = w ¯i , xi ∈ Γu , i.e., if the equations ∂Hij ∂σij + fi = 0, + gi = 0, ∂xj ∂xj coupled by

and

xi ∈ Ω

σij = Cijkl εkl + Rijkl wkl , Hij = Rklij εkl + Kijkl wkl

1 εij = 2

∂uj ∂ui + ∂xj ∂xi

,

wij =

(13.1-2)

(13.1-3)

∂wi ∂xj

(13.1-4)

satisfying boundary conditions (13.1-1), then the boundary value problem (13.1-1)∼(13.1-4) has a unique solution. Proof If the conclusion is not true, suppose there are two solutions of equations (13.1-2)∼(13.1-4) under boundary conditions (13.1-1): (1)

(1)

ui ⊕ w i ,

(1)

(1)

εij ⊕ wij ,

ui ⊕ w i , εij ⊕ wij , (1)

(1)

σij ⊕ Hij ,

(2)

(2)

(2)

(2)

(2)

(2)

σij ⊕ Hij ,

T. Fan, Mathematical Theory of Elasticity of Quasicrystals and Its Applications © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2011

274 Chapter 13

Some mathematical principles on solutions of elasticity of quasicrystals

both solutions satisfy equations of deformation geometry, stress-strain relationship, equilibrium equations and boundary conditions. According to the linear superposition principle, the diﬀerence between these two solutions (1) (2) (1) (2) (ui − ui ) ⊕ (wi − wi ) = Δui ⊕ Δwi , (1)

(2)

(1)

(2)

(σij − σij ) ⊕ (Hij − Hij ) = Δσij ⊕ ΔHij should be the solution of the problem too. The “diﬀerences” should satisfy zero boundary conditions. The work done by the external forces is 0= (Δf ⊕ Δg) · (Δu ⊕ Δw)dΩ Ω

+ Γ

(ΔT ⊕ Δh) · (Δu ⊕ Δw)dΓ = 2

ΔU dΩ ,

(13.1-5)

Ω

where ΔU represents the strain energy density corresponding to the diﬀerences of two sets of solutions, and the derivation of the last step of (13.1-5) used the Gauss theorem. Because ΔU is the positive deﬁnite quadratic form of Δεij and Δwij , there is ΔU 0,

(13.1-6)

considering the left-handside of (13.1-5) be zero, it follows that ΔU = 0.

(13.1-7)

Based on the positive deﬁnite property of ΔU , Δεij and Δwij must be zero, (1) (2) (1) (2) apart from only rigid displacements, so that εij = εij , wij = wij , etc. At the same time, at boundary Γu , one has Δu = 0,

Δw = 0.

This means the rigid-body displacements cannot exist, it must be (1)

ui

(2)

= ui ,

(1)

wi

(2)

= wi .

The theorem is a powerful tool in the study of elasticity of quasicrystals, e.g. uniqueness of the solution ensures that diﬀerent approaches for a problem of elasticity of quasicrystals will not result in diﬀerent solutions so long as the equations of elasticity of quasicrystals and the prescribed boundary conditions are satisﬁed.

13.2

13.2

Generalized Lax-Milgram theorem

275

Generalized Lax-Milgram theorem

Consider a quasicrystal described by equations (13.1-2)∼(13.1-4) and boundary conditions (13.1-1). In the case the displacements satisfy the following partial diﬀerential equations: ∂ 2 uk ∂ 2 wk + Rijkl = fi (x, y, z)), − Cijkl ∂xj ∂xl ∂xj ∂xl (x, y, z) ∈ Ω . (13.2-1) ∂ 2 uk ∂ 2 wk = gi (x, y, z), − Rklij + Kijkl ∂xj ∂xl ∂xj ∂xl At boundary Γu , they satisfy homogenous conditions ui |Γu = 0,

wi |Γu = 0.

(13.2-2)

And at boundary Γt , the corresponding stresses satisfy non-homogenous conditions ⎧ ∂uk ∂wk ⎪ ⎪ ⎪ ⎨ Cijkl ∂x + Rijkl ∂x nj |Γt = Ti (x, y, z), l l (13.2-3) ⎪ ∂u ∂w ⎪ k k ⎪ + Kijkl + nj |Γt = hi (x, y, z), ⎩ Rklij ∂xl ∂xl in which x1 = x, x2 = y, x3 = z, Γ = Γu + Γt . If ui , wi ∈ C 2 (Ω )∩C 1 (∂Ω ) and satisfy equation (13.2-1) and boundary conditions (13.2-2) and (13.2-3), then they will be called a classical solution of boundary value problem (13.2-1)∼(13.2-3), where ∂Ω = Γ = Γu + Γt . The boundary value problem (13.2-1)∼(13.2-3) can be transformed into a variational problem. To this end, let us introduce the norm ⎧ 2 2 2 ⎪ ∂u ∂u ∂u ⎪ i i i 2 ⎪ dΩ , + + ⎪ ui 1,Ω = ⎪ ⎨ ∂x ∂y ∂z Ω (13.2-4) 2 2 2 ⎪ ⎪ ∂w ∂w ∂w ⎪ i i i 2 ⎪ dΩ . + + ⎪ wi 1,Ω = ⎩ ∂x ∂y ∂z Ω This norm is suitable only for case of homogeneous boundary conditions (13.2-2), otherwise the norm will not be in this form. With the norm in (13.2-4), denote the space deﬁning ui (x, y, z) and wi (x, y, z) is by H (Ω ). If introduce inner product ⎧ % $ (1) (2) (1) (2) (1) (2) ⎪ ∂ui ∂ui ∂ui ∂ui ∂ui ∂ui ⎪ (1) (2) ⎪ + + dΩ , (u , ui ) = ⎪ ⎪ ⎨ i ∂x ∂x ∂y ∂y ∂z ∂z Ω (13.2-5) % $ (1) (2) (1) (2) (1) (2) ⎪ ⎪ ∂wi ∂wi ∂wi ∂wi ∂wi ∂wi ⎪ (1) (2) ⎪ + + dΩ , ⎪ ⎩ (wi , wi ) = ∂x ∂x ∂y ∂y ∂z ∂z Ω

276 Chapter 13

Some mathematical principles on solutions of elasticity of quasicrystals

then H (Ω ) is a Hilbert space, it is also named Sobolev space. We can deﬁne space V such as V = {(ui , wi ) ∈ H (Ω ), (ui )Γu = 0, (wi )Γu = 0}.

(13.2-6)

and the inner product at space V (13.2-5), then V is also a Hilbert space and V ⊂ H (Ω ).

(13.2-7)

Deﬁne the space H = {X = (u1 , u2 , u3 , w1 , w2 , w3 ) ∈ (H (Ω ))61 , (ui )Γu = 0, (wi )Γu = 0} and the norm X1,Ω

3 1/2 2 2 = [ui 1,Ω + wi 1,Ω ) .

(13.2-8)

(13.2-9)

i=1

For any X = (u1 , u2 , u3 , w1 , w2 , w3 ) ∈ (H (Ω ))61 , by using the strain-displacement relations 1 ∂ui ∂uj , εij (X) = εji (X) = + 2 ∂xj ∂xi wij (X) =

∂wi ∂xj

and the stress-strain relations of quasicrystals σij (X) = σji (X) = Cijkl εkl (X) + Rijkl wkl (X), Hij = Rklij εkl (X) + Kijkl wkl (X), Deﬁne the bilinear functional [σij (X (1) )εij (X (2) ) + Hij (X (1) )wij (X (2) )]dΩ , B(X (1) , X (2) ) =

(13.2-10)

which possesses symmetry, continuity and positive deﬁnite properties. Take a linear functional l(X) = [fi ui + gi wi ]dΩ + [Ti ui + hi wi ]dΓ ,

(13.2-11)

Ω

Ω

Γt

where (f1 , f2 , f3 , g1 , g2 , g3 ) ∈ (L2 (Ω ))6 , (T1 , T2 , T3 , h1 , h2 , h3 ) ∈ (L2 (Ω ))6 are given functions at Ω and Γ respectively. One can prove that l(X) is a continuous functional of X at region Ω , and L2 represents the Lebesgue square integrable meaning. In what follows, we state a number of theorems on the solution of the problem of elasticity of quasicrystals.

13.2

Generalized Lax-Milgram theorem

277

Theorem 1 The variational problem associated with the boundary value problem (13.2-1)∼(13.2-3) is to obtain X ∈ H so that X B(X, Y ) = l(Y ),

∀Y ∈ H,

(13.2-12)

in which l(Y ) is a linear functional deﬁned by (13.2-11). (Proof is omitted.) Theorem 2 If X = (u1 , u2 , u3 , w1 , w2 , w3 ) ∈ (C 2 (Ω ))6 is a classical solution of boundary value problem (13.2-1)∼(13.2-3), then X must be solution of the variational problem (13.2-12), i.e., X is also a generalized solution (or weak solution). Alternatively, if X is a solution of the variational problem (13.2-12), and X ∈ (C 2 (Ω ))6 , then must be the classical solution of the boundary value problem (13.2-1)∼(13.2-3). (Proof is omitted.) Theorem 3(Generalized Lax-Milgram theorem) Assume that H is the Hilbert space deﬁned above, for elasticity of quasicrystals, B(X, Y ) is a bilinear functional at H × H and satisﬁes B(X, Y ) = f (X), ∀Y ∈ H, (13.2-13) admits a unique solution X ∈ H and X

1 f (H) , α

(13.2-14)

where (H) is the dual space of H consisting of all bounded linear functionals and equipped with the norm f (Y ) f (H) = sup . Y ∈H,Y =0 Y This theorem gives an extension of Lax-Milgram theorem[3] . (Proof is omitted.) Theorem 4 Assume J(X) =

1 B(X, X) − l(X) 2

(13.2-15)

and H is the same as mentioned previously, B(X, Y ) is a bilinear functional at H ×H deﬁned by (13.2-10), then the variational problem for having X ∈ H so that J(X) being minimum J(X0 ) = min J(X) (13.2-16) X∈H

in which (1) Exists solution, and the number of solutions does not exceed one; (2) If there is a solution of problem (13.2-16), which must be the solution of problem (13.2-12), and vise versa;

278 Chapter 13

Some mathematical principles on solutions of elasticity of quasicrystals

(3) If X ∗ is their solution, then J(X) − J(X ∗ ) =

1 B(X − X ∗ , X − X ∗ ), 2

∀X ∈ H.

(13.2-17)

(Proof is omitted.) The discussion of this section may provide preparedness for the following text.

13.3

Matrix expression of elasticity of three-dimensional quasicrystals

In the previous section the variational problem on elasticity of quasicrystals was discussed in which the concept of weak solution (generalized solution) is mentioned. In the following sections we will deal with the weak solution of elasticity of threedimensional icosahedral quasicrystals, for simplicity a matrix expression for the elasticity will be used. From the discussion of previous chapters we know that the basic equations for elasticity of quasicrystals except cubic quasicrystals ⎧ ∂ 2 ui ∂ 2 wi ∂Hij ∂wi ∂σij ⎪ (13.3-1) ⎪ + fi = ρ 2 , + gi = ρ 2 or κ , ⎪ ⎪ ∂ t ∂xj ∂t ∂t ⎪ ⎨ ∂xj 1 ∂ui ∂uj ∂wi (13.3-2) , wij = εij = + , ⎪ ⎪ ⎪ 2 ∂x ∂x ∂xj j i ⎪ ⎪ ⎩ σij = Cijkl εkl + Rijkl wkl , Hij = Kijkl wkl + Rklij εkl , (13.3-3) where xi ∈ Ω , i, j = 1, 2, 3, t > 0, u = (ux , uy , uz ) represents the phonon displacement vector, w = (wx , wy , wz ) the phason one, εij , wij the phonon and phason strains, σij , Hij the corresponding phonon and phason stresses, Cijkl , Kijkl , Rijkl the elastic constants, fi , gi the body force and generalized body force densities, ρ the mass density and κ = 1/Γw , respectively. Denoting ∂Ω = (∂Ω )u + (∂Ω )σ to express the boundary of region occupied by the quasicrystal there are the boundary conditions x ∈ (∂Ω )u : ui = u0i , x ∈ (∂Ω )σ : σij nj = Ti ,

wi = wi0 ,

(13.3-4)

Hij nj = hi ,

(13.3-5)

where u0i and wi0 represent the known functions at boundary (∂Ω )u , Ti and hi the prescribed traction and generalized traction density at boundary (∂Ω )σ , nj the unit outward normal vector at any point of the boundary. There are initial conditions for dynamic problem ui |t=0 = ai (x), (or ui |t=0 = ai (x),

wi |t=0 = bi (x), wi |t=0 = bi (x),

u˙ i |t=0 = ci (x), u˙ i |t=0 = ci (x))

w˙ |t=0 = di (x),

(13.3-6)

13.3

Matrix expression of elasticity of three-dimensional quasicrystals

279

where ai , bi , ci , di are known functions, x = (x1 , x2 , x3 ) ∈ Ω . Putting ˜ T = (u1 , u2 , u3 , w1 , w2 , w3 , )1×6 , U

˜ T = (fi gi )1×6 = (f1 , f2 , f3 , g1 , g2 , g3 )1×6 , F

˜ T = (σ11 , σ22 , σ33 , σ12 , σ23 , σ31 , H11 , H22 , H33 , H12 , H23 , H31 , H13 , H21 , σ H32 )1×15 , T

˜ = (ε11 , ε22 , ε33 , 2ε12 , 2ε23 , 2ε31 , w11 , w22 , w33 , w12 , w23 , w31 , w13 , w21 , ε w32 )1×15 , ⎡ ⎡ ⎡ ˜ =⎣ ∂

⎤

˜ (1) ∂

0

0

˜ (2) ∂

˜ (1) ∂

⎦,

⎢ ⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎣

∂1 0 0

0 ∂2 0

0 0 ∂3

∂2 0 ∂3

∂1 ∂3 0

0 ∂2 ∂1

∂i =

∂ , ∂xi

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎦

˜ (2) ∂

∂1 0 0 ∂2 0 0 ∂3 0 0

0 ∂2 0 0 ∂3 0 0 ∂1 0

0 0 ∂3 0 0 ∂1 0 0 ∂2

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

˜ T, ∂T, σ ˜,F ˜ , ∂, σ, ˜ T, F ˜ T, ε ˜T denote the transpose of U ˜ ε ˜, σ i = (σi1 , σi2 , σi3 ) the U i-th row of matrix (σij )3×3 , H i = (Hi1 , Hi2 , Hi3 ) the i-th row of matrix (Hij )3×3 . So that, (13.3-2) can be rewritten as the matrix form (13.3-2 )

˜. ˜ = ∂U ε Note that

∂σ1j ∂xj

∂σ2j ∂xj

∂σ3j ∂xj

∂H1j ∂xj

∂H2j ∂xj

∂H3j ∂xj

T

= ∂Tσ ˜ , then (13.3.1)

can be rewritten by ¨ . = ρU . ˜ +F ∂Tσ . Putting ⎡ ⎢ ⎢ ⎢ ⎢ C=⎢ ⎢ ⎢ ⎢ ⎣

··· ··· ··· ··· ··· ···

C11ij C22ij C33ij C12ij C23ij C31ij

··· ··· ··· ··· ··· ···

⎤

⎡

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎣

6×6

C1111 C2211 C3311 C1211 C2311 C3111

C1122 C2222 C3322 C1222 C2322 C3122

C1133 C2233 C3333 C1233 C2333 C3133

(13.3-1 )

C1112 C2212 C3312 C1212 C2312 C3112

C1123 C2223 C3323 C1223 C2323 C3123

C1131 C2231 C3331 C1231 C2331 C3131

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

,

6×6

280 Chapter 13

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ K =⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡

··· ··· ··· ··· ··· ··· ··· ··· ···

⎢ ⎢ ⎢ R =⎢ ⎢ ⎢ ⎣ ⎡ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎣

··· ··· ··· ··· ··· ···

··· ··· ··· ··· ··· ··· ··· ··· ···

K11ij K22ij K33ij K12ij K23ij K31ij K13ij K21ij K32ij

K1111 ⎢ K2211 ⎢ ⎢ K3311 ⎢ ⎢ K1211 ⎢ =⎢ ⎢ K2311 ⎢ K3111 ⎢ ⎢ K1311 ⎢ ⎣ K2111 K3211 ⎡

Some mathematical principles on solutions of elasticity of quasicrystals

K1122 K2222 K3322 K1222 K2322 K3122 K1322 K2122 K3222 R11ij R22ij R33ij R12ij R23ij R31ij

R1111 R2211 R3311 R1211 R2311 R3111

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

K1133 K2233 K3333 K1233 K2333 K3133 K1333 K2133 K3233

··· ··· ··· ··· ··· ···

R1122 R2222 R3322 R1222 R2322 R3122

⎤

K1112 K2212 K3312 K1212 K2312 K3112 K1312 K2112 K3212

K1123 K2223 K3323 K1223 K2323 K3123 K1323 K2123 K3223

K1131 K2231 K3331 K1231 K2331 K3131 K1331 K2131 K3231

K1113 K2213 K3313 K1213 K2313 K3113 K1313 K2113 K3213

K1121 K2221 K3321 K1221 K2321 K3121 K1321 K2121 K3221

⎤ K1132 K2232 ⎥ ⎥ K3332 ⎥ ⎥ K1232 ⎥ ⎥ K2332 ⎥ , ⎥ K3132 ⎥ ⎥ K1332 ⎥ ⎥ K2132 ⎦ K3232 9×9

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ 6×9

R1133 R2233 R3333 R1233 R2333 R3133

R1112 R2212 R3312 R1212 R2312 R3112

R1123 R2223 R3323 R1223 R2323 R3123

then

D = (dij )15×15 =

R1131 R2231 R3331 R1231 R2331 R3131 C

R

RT

K

R1113 R2213 R3313 R1213 R2313 R3113

R1121 R2221 R3321 R1221 R2321 R3121

R1132 R2232 R2232 R1232 R2332 R3132

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

,

6×9

.

Here the order of index i, j of C is the same with those of the phonon strain tensor, the order of index i, j of K, R is the same with those of phason strain tensor, and RT is the transpose of R. From the above expressions one can ﬁnd that due to the symmetry of C and K (see e.g. (4.4-3) and (4.4-5)), the matrix D is symmetric. The generalized Hooke’s law (13.3-3) can be rewritten as ˜ = D˜ σ ε,

(13.3-3 )

13.3

Matrix expression of elasticity of three-dimensional quasicrystals

281

(13.3-1 ) and (13.3-2 ) can be collected as below: ¨ ˜ +F ˜ = ρU . . ∂ T D∂ U Put

⎡ ⎢ ⎢ ⎢ A(x) = ⎢ ⎢ ⎢ ⎣

a1 (x) a2 (x) a3 (x) b1 (x) b2 (x) b3 (x)

⎤

⎡

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎢ ⎢ ⎢ B(x) = ⎢ ⎢ ⎢ ⎣

,

6×1

⎡

⎢ ⎢ ⎢ 0 ˜ =⎢ σ ⎢ ⎢ ⎣ ⎡ ˜ (1) ∂ n

⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎣ ⎡

˜ (2) ∂ n

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

T1 T2 T3 h1 h2 h3

c1 (x) c2 (x) c3 (x) d1 (x) d2 (x) d3 (x)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎡ ,

˜n = ⎣ ∂

(13.3-7)

⎤

⎡

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎢ ⎢ ⎢ ˜ U =⎢ ⎢ ⎢ ⎣ 0

,

6×1

u01 u02 u03 w10 w20 w30

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

,

6×1

⎤

˜ (1) ∂ n

0

0

˜ (2) ∂ n

⎦,

6×1

cos(n, x1 ) 0 0 0 cos(n, x2 ) 0 0 0 cos(n, x3 ) cos(n, x2 ) cos(n, x1 ) 0 0 cos(n, x3 ) cos(n, x2 ) cos(n, x3 ) 0 cos(n, x1 ) cos(n, x1 ) 0 0 0 0 cos(n, x2 ) 0 0 cos(n, x3 ) 0 0 cos(n, x2 ) 0 cos(n, x3 ) 0 0 0 cos(n, x1 ) cos(n, x3 ) 0 0 0 0 cos(n, x1 ) 0 0 cos(n, x2 )

⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

˜ (1) , ∂ ˜ (2) are obtained from the diﬀerential operator matrices ∂ ˜ ,∂ ˜ (1) , ∂ ˜ (2) ˜n , ∂ where ∂ n n through a replacement ∂ i by cos(n, xi ). Equation (13.3-4) can be rewritten as ˜ (x, t) = U ˜ 0, U

x ∈ (∂Ω )u .

(13.3-4 )

Considering the similarity of the left-handside of (13.3-5) with the ﬁrst term of (13.3˜ Tσ ˜ 0 , x ∈ (∂Ω )σ , In addition by using 1), then (13.3-5) can be rewritten as ∂ n˜ = σ (13.3-2 ) and (13.3-3 ) there is ˜U ˜ T D∂ ˜ =σ ˜ 0, ∂ n

x ∈ (∂Ω )σ .

(13.3-5 )

282 Chapter 13

Some mathematical principles on solutions of elasticity of quasicrystals

¨ . If the quasicrystal is in static equilibrium: ρU = 0 (i.e., the inertia forces vanish), in the case it needs not the initial conditions, and the boundary value problem of quasicrystals is interpreted as follows:

˜U ˜ T D∂ ˜ =F ˜ , x ∈ Ω , t > 0, −∂ " " ˜ T D∂ ˜U ˜ (x, t)" ˜ 0, ∂ ˜ (x, t)" U =U =σ ˜ 0, n (∂Ω)u (∂Ω)σ

(13.3-8) (13.3-9)

where ∂Ω = (∂Ω )u + (∂Ω )σ .

13.4

The weak solution of boundary value problem of elasticity of quasicrystals

For simplicity only the displacement boundary problem (or say the Dirichlet prob˜ (x) ∈ (C 2 (Ω ˜ ∈ (L2 (Ω ))6 , if U ˜ ))6 lem) is dealt with in the following. Suppose that F being the solution of problem (13.3-8), (13.3-9), then for any vector function η = 1 T η3×1 = η11 η21 η31 η12 η22 η32 1×6 ∈ (C0∞ (Ω ))6 , multiplying the both 2 η3×1 sides of (13.3-8) by η (by making scalar product), and then integrating along Ω , we have T ˜ ˜ ˜ ˜ · ηdx. (−∂ D ∂ U ) · ηdx = F (13.4-1) Ω

Ω

From (13.3-3) and Chapter 4 known σij = σji , in terms of Gauss formula, (13.3-2 ) and (13.3-3 ), there exists ∂σij 1 ∂Hij 2 ˜ T D∂ ˜U ˜ · ηdx = − −∂ ηi + ηi dx ∂xj ∂xj Ω Ω ∂ηi1 ∂ηi2 ∂ 1 2 (σij ηi + Hij ηi ) − σij + hij = − dx ∂xj ∂xj ∂xj Ω ∂η 1 ∂η 2 (σij ηi1 +Hij ηi2 )nj dS + = − σij i +Hij i dx ∂xj ∂xj ∂Ω Ω $ % 1 ∂ηj1 ∂η 2 1 ∂ηi1 + σij + Hij i dx σij = 2 ∂xj 2 ∂xi ∂xj Ω ˜ )εij (η 1 ) + Hij (U ˜ )wij (η 2 )]dx [(σij (U = Ω

˜ ) · ε˜(η)dx = σ ˜ (U

= Ω

Ω

˜ T D∂ ˜U ˜ dx, (∂η)

(13.4-2)

13.5

The uniqueness of weak solution

283

(13.4-1) and (13.4-2) yield ˜ T D∂ ˜U ˜ dx = ˜ · ηdx. ˜)·ε ˜ U ˜(η)dx = (∂η) F σ( Ω

Ω

C0∞

(13.4-3)

Ω

is dense in (13.4-3) holds for ∀η(x) ∈ (H01 (Ω ))6 . Because ˜ (x) ∈ (C (Ω ))6 , and (13.4-3) is valid ∀η(x) ∈ (H 1 (Ω ))6 , we can In contrast, if U 0 do derivation in counter order of the above procedure, and ﬁnd (13.4-1) through the fundamental lemma of variational method[2] . So that we have ˜ (x) ∈ (H 1 (Ω ))6 , and (13.4-3) holds ˜ ∈ (L2 (Ω ))6 , if U Deﬁnition Assume F 0 ˜ (x) being the weak solution (or generalized solution) ∀η(x) ∈ (H01 (Ω ))6 , then say U of the boundary value problem ˜U ˜ T D∂ ˜ (x) = F ˜ (x), x ∈ Ω , t > 0 (13.3-8) −∂ ˜ (x) |∂Ω = 0. (13.3-9 ) U

13.5

H01 (Ω ), 2 ¯

The uniqueness of weak solution

Making use of (·, ·) to express inner product in L2 (Ω ), and the corresponding norm 1/2 2 v dx , for the scalar function v ∈ L2 (Ω ). And in terms of (·, ·)1 is · : v = Ω

to denote the inner product in H01 (Ω ), the corresponding norm is ·1 : v1 = 3 2 1/2 2 1/2 3 ∂v ∂v 2 v dx+ dx , and semi-norm is |·|1 : |v|1= dx , Ω Ω ∂xk Ω ∂xk k=1

k=1

for scalar function v ∈ L2 (Ω ). Note 1 Norm ·1 is equivalent to semi-norm |·|1 . Note 2 The norm and semi-norm of vector function v = (v1 , v2 , · · · , vn ) ∈ (H01 (Ω ))n (it is marked by H01 (Ω ) some times)are denoted as ·1 and |·|1 , such as 2 n n n 3 ∂vi v21 = vi 21 = vi2 dx + dx = |v|2 dx + |vx |2 dx, ∂x k Ω Ω Ω Ω i=1 i=1 i=1 k=1

|v|21 =

n 3 i=1 k=1

Ω

∂vi ∂xk

2

dx,

∂v1 ∂v2 ∂v ∂vn = , ,··· , . Obvi∂xk ∂xk ∂xk ∂xk ∂xk i=1 i=1 k=1 ously, the Note 1 holds for vector function v, too. Lemma(Korn inequality[4,5] ) Assume Ω is a bounded region with boundary ∂Ω of suﬃcient smooth in Rn , and ∀v = (v1 , v2 , · · · , vn ) ∈ H01 (Ω ), there is 2 n ∂vi ∂vk 2 + dx c1 v1 , ∂x ∂x k i Ω 2

where |v| =

n

2

vi2 , |vx | =

i,k=1

2 n 3 ∂vi

,

284 Chapter 13

Some mathematical principles on solutions of elasticity of quasicrystals

in which the positive constant c1 is only dependent on Ω . Theorem Suppose Ω is a bounded region in R3 and with suﬃcient smooth boundary ∂Ω , if real symmetric matrix D = (dij ) satisﬁes the inequality λ1

15

15

ξi2

i=1

ξi dij ξj λ2

i,j=1

15

ξi2 ,

i=1

where λ1 , λ2 are positive constants, then for any F˜ ∈ (L2 (Ω ))6 , displacement boundary value problem (13.3-8), (13.3-9 ) exists unique weak solution (or generalized solution). T ˜ dx, then (13.4-3) can be rewritten as ˜ (∂η) D∂ U Proof Put U , η = Ω

˜ , η = (F ˜ , η), U

∀η ∈ (H01 (Ω ))6 .

(13.5-1)

At ﬁrst we prove ·, · is a new inner product at (H01 (Ω ))6 . For this purpose it ˜,U ˜ 0, and U ˜,U ˜ = 0 ⇔ U ˜ = 0, ∀U ˜ ∈ (H 1 (Ω ))6 . needs to prove: U 0 In the following we give only an outline of the proof, the detail is omitted. In ˜ , η in (13.5-1) is positive deﬁnite bilinear functional at H01 (Ω ), the proof addition U can be done from the Lax-Milgram theorem (see Section 13.2). Due to the assumption, matrix D = (dij )15×15 being positive deﬁnite, matrix D = (dij )15×15 and unit matrix I are in contract, i.e., there exists a reversible matrix C such that D = CTC (note that here C is not the phonon elastic constant matrix). Then T ˜ ˜ ˜U ˜U ˜ ˜ ˜ ˜ ˜ )T (C T C)∂ ˜ dx U , U = (∂ U ) D ∂ U dx = (∂ Ω

Ω

T

˜U ˜U ˜ ) (C ∂ ˜ )dx 0, (C ∂

= Ω

.,U . = 0 ⇔ U

.U .U .U . )T (C ∂ . )dx = 0 ⇔ C ∂ . = 0. (C ∂

Ω

.U . = 0, i.e., Because C is reversible, ∂ ∂ui ∂uj ∂wi ∂ui = 0, + = 0 (i = j), = 0, ∂xi ∂xj ∂xi ∂xj

i, j = 1, 2, 3.

∂wi ˜ |∂Ω = 0, and wi = 0 at = 0 that wi should be constant, besides U ∂xj ˜ = 0 at the boundary. In similar analysis we ﬁnd that ui = 0 at boundary. Thus U boundary. It follows

13.5

The uniqueness of weak solution

285

In this way we have proved ·, · is a new inner product at (H01 (Ω ))6 , the corre.,U . 12 . . (1) = U sponding norm is U Secondly, at (H01 (Ω ))6 the new norm ·(1) is equivalent to the initial norm ·1 . We are going to give the proof about this. From the Cauchy’s inequality, the assumption of the theorem and Note 1, there is 15 / /2 /˜/ ˜U ˜U ˜U ˜U ˜ )i dij (∂ ˜ )j dx λ2 ˜ )T (∂ ˜ )dx (∂ (∂ /U / = (1)

i,j=1

Ω

= λ2 Ω

λ2

Ω

⎡ ⎤ 2 2 2 3 3 3 ∂uj ∂ui ∂ui ∂wi ⎥ ⎢ + + + ⎣ ⎦ dx ∂xi ∂xj ∂xi ∂xj i,j=1 i=1 i,j=1

⎧ 2 ⎪ 3 ⎨ ∂u ⎪ ⎩ i=1

Ω

2λ2

3 i,j=1

Ω

i

∂xi

∂ui ∂xj

i<j

+2

3

i,j=1 i<j

2

+

∂wi ∂xj

⎫ 2 2 2 ⎪ 3 ∂uj ∂wi ⎬ ∂ui + dx + ∂xj ∂xi ∂xj ⎪ ⎭ i,j=1

2

dx

" "2 / /2 "˜" /˜/ = 2λ2 "U " 2c /U / . 1

1

In other hand, from the assumption of the theorem, the Korn inequality and Note 1 there is 15 / /2 /˜/ ˜U ˜U ˜U ˜ )i dij (∂ ˜ )j dx λ1 ˜ )T (∂ ˜ )dx (∂˜U (∂ /U / = (1)

i,j=1

Ω

= λ1 Ω

= λ1 Ω

Ω

⎡ ⎤ 2 2 2 3 3 3 ∂ui ∂ui ∂wi ⎥ ∂uj ⎢ + + + ⎣ ⎦ dx ∂x ∂x ∂x ∂xj i j i i,j=1 i=1 i,j=1 ⎡

i<j

3

⎣1 4 i,j=1

∂uj ∂ui + ∂xj ∂xi

2

⎤ 2 3 ∂wi ⎦ + dx ∂xj i,j=1

2 2 3 1 ∂ui ∂wi dx + λ1 dx λ1 c2 4 ∂xj ∂xj i,j=1 Ω Ω

" " / /2 1 /˜/ " ˜ "2 min λ1 c2 , λ1 "U / . " c /U 4 1 1 Consequently, we have proved the equivalency between the new norm ·(1) and the initial norm ·1 .

286 Chapter 13

Some mathematical principles on solutions of elasticity of quasicrystals

˜ ∈ (L2 (Ω ))6 , by using Schwarz inequality and the fact that the Finally, for F 1 embedding H0 (Ω ) ⊂ → L2 (Ω ) is a compact embedding, we have " " " / / " / / / / " " /˜/ /˜/ ˜ · ηdx" / ˜/ " M ∀η ∈ (H01 (Ω ))6 , F F · η M F · η / / / / 1 /F / · η1 , 1 " " " " Ω ˜ · ηdx (∀η ∈ (H 1 (Ω ))6 ) is a unique continuous linear functional i.e., η → F 0 Ω

at (H01 (Ω ))6 . Therefore, from Riesz theorem, there must be a unique y F˜ ∈ (H01 (Ω ))6 , such that ˜ · ηdx = y ˜ , η , ∀η ∈ (H01 (Ω ))6 . F F Ω

Thus (13.5-1) is rewritten as 3 4 ˜ , η = y ˜ , η , U F

∀η ∈ (H01 (Ω ))6 .

˜ = y ˜ is the unique weak solution (generalized solution) of displaceThis shows U F ment value problem (13.3-8), (13.3-9 ). In the above proof, using Korn inequality is crucial (the second Korn inequality will be used for the stress boundary value problem, but this discussion is not included here).

13.6

Conclusion and discussion

In some extent the discussion Sections 13.3∼13.5 is a development of Section 13.2. These discussions provide a basis for the numerical methods developed in Chapters 10 and 12 from point of view of weak solution. Of course for the stress boundary value problems and dynamic problems the discussion needs to do some extensions of the previous discussions. It is evident that the discussion in Sections 13.3∼13.5 is valid for any systems of one-, two- and three-dimensional quasicrystals, except cubic quasicrystals, for which wij = ∂wi /∂xj should be replaced by wij = (∂wi /∂xj +∂wj /∂xi )/2 in the equations (13.3-2) only. The diﬀerence of the formulation for diﬀerent quasicrystal systems lies in the elastic constant matrix D, one can directly use the above formulation for any quasicrystal systems by substituting the concrete matrix D into the relevant formulas.

References [1] Fan T Y, Mai Y W. Elasticity theory, fracture mechanics and some relevant thermal properties of quasicrystalline materials. Appl Mech Rev, 2004, 57(5): 325–344

References

287

[2] Courant R, Hilbert D. Methods of Mathematical Physics. New York: Interscience Publisher Inc. 1955 [3] Oden J J, Reddy J H. An Introduction to the Mathematical Theory of Finite Element. New York: John Wiley & Sons. 1976 [4] Guo L H, Fan T Y. Solvability on boundary-value problems of elasticity of threedimensional quasicrystals. Applied Mathematics and Mechanics, 2007, 28(8): 1061– 1070 [5] Fikera G. Existence Theorems of Elasticity Theory. Moscow: World Press, 1974 (in Russian) [6] Kondratjev W A, Oleinik O A. Korn Inequality of Boundary Value Problems for Systems of Theory of Elasticity in Boundless Region. Moscow: UMN Press, 1988 (in Russian)

Chapter 14 Nonlinear behaviour of quasicrystals From Chapter 4 to Chapter 13 we mainly discussed the elasticity and relevant properties of quasicrystals, which belong to linear regime both physically and mathematically. Their mathematical treatment is relatively easy though the calculations are quite complex. The current chapter is to give a simple description on deformation and fracture of quasicrystals with nonlinear behaviour, considering the great diﬃculty in this topic. For the conventional engineering materials including crystalline material, the nonlinear behaviour means mainly plasticity. In the study on the classical plasticity there are two diﬀerent theories, one is the macroscopic plasticity theory, which is based on some assumptions concluded from certain experimental data, and the other is so-called crystal plasticity theory, which is based on the mechanism of motion of dislocation, and in some extent can be seen as a “microscopic” theory. The diﬃculty for quasicrystal plasticity lies in lack of both enough macro- and micro-data. At present the macroscopic experiments have not, as yet, been properly undertaken. Though there is some work on the mechanism in microscopy of the plasticity, the data are very limited. This leads to the constitutive law of quasicrystals being essentially unknown. Due to the reason the systematic mathematical analysis on deformation and fracture for the material is not available so far. In spite of these diﬃculties, study on plasticity of quasicrystals has aroused a great deal of attention of researchers[1∼8] . But the analytic quantitative work may be at an infant stage. Considering the readers’ interest and the development level, it is beneﬁcial to give a brief discussion on some simple problems of nonlinear behaviour of the material with some simple models and by extending results in the study of linear regime. Of course, these discussions are not complete, which may provide some hints for further development of the area. This chapter is arranged as follows. First, we discuss some experimental results on the nonlinear deformation behaviour of quasicrystals, then describe a possible plastic constitutive equation of the material. In view of the diﬃculty for setting up the equations, we turn to introduce nonlinear elastic constitutive equations of quasicrystals which are available at present though not equivalent to the plastic T. Fan, Mathematical Theory of Elasticity of Quasicrystals and Its Applications © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2011

290

Chapter 14

Nonlinear behaviour of quasicrystals

constitutive equations. The Sections 14.4 and 14.5 may be seen as applications of the macro-constitutive law in which some nonlinear solutions on quasicrystals are presented. In Section 14.6 another version of the study based on the dislocation model or “microscopic model” is exhibited, which achieves the same results given in Sections 14.4 and 14.5.

14.1

Macroscopic behaviour of plastic deformation of quasicrystals

At medium and low temperature, quasicrystals exhibit brittleness, but they present plasticity-ductility at high temperature. In addition, near the high stress concentration zone, e.g. around dislocation core or crack tip, plastic ﬂow appears. It was observed in experiments that plastic deformation of quasicrystals is induced by motion of dislocations in the material. This reveals the important connection between plasticity and the structural defects in quasicrystals. In some extent there are similarities between the phenomena of crystals and quasicrystals. But the latter presents salient structural features, fundamentally diﬀering from those of conventional crystals. The plasticity of quasicrystals must be studied in a quite diﬀerent way from that for studying the classical plasticity. The fundamental step in studying plasticity of quasicrystals is, of course, experimental observation. As in previous discussion, this chapter mainly deals with icosahedral and decagonal quasicrystals. Recently there are many experimental data for Al-Mn-Pd icosahedral quasicrystals, which are the most intensively studied and possess a very high brittle-ductile transition temperature. At strain rate of 10−5 /s the ductile range sets in at about 690 ◦ C corresponding to a homologous temperature (i.e., the absolute temperature scaled by the absolute melting temperature) of about 0.82. At lower strain rates of 10−6 /s and below limited ductility can be observed down to temperature about 480 ◦ C. Fig. 14.1-1 shows the stress-strain curves of the quasicrystal in the hightemperature range 730 − 800 ◦ C at strain rate 10−5 /s. As another class of icosahedral quasicrystals, icosahedral Zn-Ma-Dy, the stressstrain curves measured by experiments are similar to those given by Fig. 14.1-1. Feuerbacher and Urban[9] , Guyot and Canova[10] , Feuerbacher et al[11] , among others, discussed the constitutive equation based on the experimental data of dislocation density and dislocation velocity, e.g. the plastic strain rate and the applied stress in a power-law form σ )m , (14.1-1) ε˙p = B(σ/ˆ in which B and m are temperature dependent parameters and σ ˆ is the internal

14.1

Macroscopic behaviour of plastic deformation of quasicrystals

291

variable that can be regarded as a reference stress representing the current microstructural state of the material and is used to accommodate the model to the description of diﬀerent materials or hardening mechanism[12] . Combining relevant information, formula (14.1-1) can be used to well predict experimental curves, e.g. recorded by Fig. 14.1-1. It should be pointed out, though there are some similar forms to (14.1-1) in the classical plasticity, they are quite diﬀerent substantively. For example, the current parameters B, m and σ ˆ are diﬀerent from those appearing in relevant formulas in the classical plasticity. Those parameters in the classical plasticity were measured from pure macroscopic approach rather than dislocation model, some detail about the latter can be found in Refs. [9-12], which, from the angle of methodology, are diﬀerent from those adopted in the classical plasticity. Unfortunately there is lack of the comprehensive macroscopic experimental data (e.g. the data arising from multiaxial loading condition) for quasicrystal plasticity so far.

Fig. 14.1-1

Stress-strain curves of icosahedral Al-Mn-Pd single quasicrystals at strain rate of 10−5 /s[9]

Decagonal quasicrystals are an interesting topic for plastic deformation studies since the phase possesses quasiperiodic as well as periodic lattice directions. Therefore the inﬂuences of periodicity and quasiperiodicity on the plastic deformation can be directly revealed on one material by investigating the plastic properties in diﬀerent deformation geometries—an anisotropic behaviour is expected. In Ref. [9] the experimental data for stress-strain curves for Al-Ni-Co decagonal quasicrystals are depicted in Fig. 14.1-2 (note that: the results obtained for the basic cobalt (b-Co) modiﬁcation of decagonal Al-Ni-Co), it presents plastic properties diﬀerent from those of icosahedral quasicrystals. The reason for this is the distinguishing nature of quasicrystalline lattice between the two phases. It is diﬀerent

292

Chapter 14

Nonlinear behaviour of quasicrystals

from icosahedral phase, decagonal phase possesses quasiperiodic as well as periodic lattice directions. Therefore, the inﬂuences of periodicity and quasiperiodicity on the plastic deformation lead to an anisotropic behaviour. To reveal this feature the Fig. 14.1-2 gives the data to those orientations as A , A⊥ and A45 respectively. It is obvious that the eﬀect of anisotropy is important.

Fig. 14.1-2

(a) Stress-strain curves of decagonal b-Co decagonal Al-Ni-Co single

quasicrystals for diﬀerent specimen orientations at strain rate of 10−5 /s, (b) Deﬁnition of the specimen orientations A⊥ , A and A45 [9]

The experimental data are worthy of a theoretical study.

14.2

Possible scheme of plastic constitutive equations

The formula (14.1-1) results from the ﬁtting of experimental data and assists in studying plastic constitutive equations of quasicrystals. From point of view macroscopically, the result is obtained for uniaxial loading condition as, at present there is lack of data on multiaxial loading condition. We assume that, if suﬃcient experimental data are availabe on yield surface/loading surface for some quasicrystals, then we can obtain an equation for the yield surface such as (14.2-1) Φ = σeﬀ − Y = 0, where σeﬀ = σe + f (Hij ) denotes a generalized eﬀective stress in which σe represents the eﬀective stress of the phonon stresses σij and f (Hij ) the part coming from phason stresses Hij . If (14.2-2) Y = σY = const,

14.2

Possible scheme of plastic constitutive equations

293

in which σY is the uniaxial yield limit of the material, then (14.2-1) represents the initial yield surface. On the other hand if Y = Y (h),

(14.2-3)

where h is a parameter related to deformation history, then (14.2-1) describes the evolution law of material deformation. When the yield surface/loading surface like (14.2-1) is available, one can construct the following plastic constitutive equations as ⎧ ∂Φ 1 ⎪ , σ˙ eﬀ ⎪ ε˙ij = ⎨ H(σeﬀ ) ∂σij (14.2-4) ⎪ ∂Φ 1 ⎪ ⎩ w˙ ij = σ˙ eﬀ H(σeﬀ ) ∂Hij provided that the ﬂow rule of the isotropic hardening is taken, where Φ is the above mentioned yield surface/loading surface function. The dot over physical quantities denotes the variation rate of the quantities, and H(σeﬀ ) the hardening modulus of the material, which can be calibrated by a test of a simple stress-strain state, e.g. given by (14.1-1). Then adding the elastic constitutive relationship to (14.2-4) the elastic-plastic constitutive equation will be set up. The elastic constitutive law was fully discussed in previous chapters. The assumed constitutive law shown in equations (14.2-4) which are incremental plastic equations, can describe eﬀect of deformation history including loading/unloading state within the deformation process. This may be a complete constitutive equation. There is a possible and relative simpler constitutive law belonging to total plasticity theory or deformation plasticity theory. That is if deﬁning the effective stress σeﬀ and eﬀective strain εeﬀ , in which the former was introduced above, and the latter has a similar deﬁnition and consists of phonon strains εij as well as phason strains wij , then between the strains and stresses there are relations ⎧ 1 3εeﬀ 1 ⎪ ⎪ − δ = − δ ε σ σ , ε ⎪ ij kk ij ij kk ij ⎨ 3 2σeﬀ 3 (14.2-5) ⎪ ⎪ ⎪ wij − 1 wkk δij = 3εeﬀ Hij − 1 Hkk δij ⎩ 3 2σeﬀ 3 in which εkk = εxx + εyy + εzz ,

wkk = wxx + wyy + wzz ,

σkk = σxx + σyy + σzz ,

Hkk = Hxx + Hyy + Hzz and we assume that

εeﬀ =

(e)

εeﬀ , σeﬀ < σ0 , A(σeﬀ )n , σeﬀ > σ0 ,

(14.2-6)

294

Chapter 14

Nonlinear behaviour of quasicrystals

where σ0 , A and n are the material constants of the quasicrystals, which can be (e) measured through a uniaxial test, εeﬀ represents the quantity at elastic deformation stage and σ0 the uniaxial tensile yield stress. In contrast to (14.2-4), the equations (14.2-5) can not describe deformation history, as they are substantively nonlinear elastic constitutive equations rather than plastic constitutive equations related to quasicrystalline materials. However they can describe plastic deformation in the case of proportional loading and no unloading. It is evident either (14.2-4) or (14.2-5) belong to a supposed incremental plastic constitutive law or total plastic constitutive law for quasicrystals. One cannot say whether they are correct or not due to lack of enough experimental data. If one has the constitutive equations (14.2-4) or (14.2-5) then by coupling the equations of deformation geometry 1 ∂ui ∂uj ∂wi εij = + (14.2-7) , wij = 2 ∂xj ∂xi ∂xj and the equilibrium equations ∂σij = 0, ∂xj

∂Hij = 0, ∂xj

(14.2-8)

the basic framework of the theory of macro-plasticity of quasicrystals, in the sense of incremental or total deformation, of quasicrystals can be set up. At present there is lack of such data, so the equations (14.2-1), (14.2-2) and equations (14.2-4) have not been established yet. With the same reason the equations (14.2-5) have not been set up either. This is the major diﬃculty of macro-plasticity theory currently. It is evident the possible theory is nonlinear, because the material parameters are not constants any more, and the mechanical behaviour is dependent with the history of deformation process in general. The solution is of course more diﬃcult than that for elastic deformation. Due to relative simplicity of the equations (14.2-5), for some simple conﬁgurations, e.g. anti-plane elasticity of one-dimensional hexagonal, three-dimensional cubic and three-dimensional icosahedral quasicrystals, one can probe into plastic analysis by using the proposed constitutive equations.

14.3

Nonlinear elasticity and its formulation

As pointed out in the previous section due to lack of constitutive equations of plasticity of quasicrystals up to now, analysis of plastic deformation for the material is not available. By this reason we can perform some simpler nonlinear elastic analysis ﬁrst, though the mechanisms of nonlinear elastic deformation and plastic deformation are quite diﬀerent. Furthermore, there are some diﬀerences between

14.4

Nonlinear solutions based on simple models

295

the following nonlinear elasticity and the total plasticity introduced by equations (14.2-5). We here do not constrain the concrete form of relationship between stresses and strains. The results obtained in the following may provide some useful hints for further plastic analysis. Consider the following nonlinear elastic constitutive relations i.e., deﬁne the free energy (or strain energy density) F (εij , wij ) =

εij 0

σij dεij +

wij

0

Hij dwij ,

(14.3-1)

then there is σij =

∂F , ∂εij

Hij =

∂F . ∂wij

(14.3-2)

So the formulas (14.3-1) and (14.3-2) can be seen as constitutive law for linear as well as nonlinear elasticity of quasicrystals, which, in general, cannot describe the plastic deformation. If there is proportional loading and without unloading the relationship can give appropriate description of the plastic deformation. In above formulas εij and wij are phonon and phason strain tensors given by εij =

1 2

∂uj ∂ui + ∂xj ∂xi

,

wij =

∂wi ∂xj

(14.3-3)

where ui and wi denote the phonon and phason displacement vectors, σij and Hij the phonon and phason stress tensors, respectively, which satisfy the equilibrium equations (if the body forces and generalized body forces are omitted) ∂σij = 0, ∂xj

∂Hij = 0. ∂xj

(14.3-4)

The equations (14.3-1)∼(14.3-4) are the basic equations describing nonlinear elastic deformation of a quasicrystal. In Section 14.5 we will give some applications of formulation (14.3-1)∼(14.3-4), and show that it constitutes the basis of those nonlinear analysis of Section 14.4.

14.4

Nonlinear solutions based on simple models

In this section we give some nonlinear solutions based on some simple models which substantively are extended from the framework of Eshelby s work[13] , the classical Eshelby hypothesis was set up for linear elasticity, which can be extended into nonlinear elasticity even total plasticity (or deformation plasticity). Fan and Fan[14] carried out the investigation.

296

14.4.1

Chapter 14

Nonlinear behaviour of quasicrystals

Generalized Dugdale-Barenblatt model for anti-plane elasticity for some quasicrystals

The crack problem of nonlinear deformation of quasicrystals is very interesting. For solving this problem, one of the available models is the generalized DugdaleBarenblatt model (or DB model for simplicity)[15,16] originated from the classical nonlinear fracture theory which was used for conventional structural materials including crystalline materials. The linear solutions for a crack for anti-plane for some quasicrystals have been discussed in Chapters 8 and 9. We now discuss only the nonlinear solution. Assume that there is an atomic cohesive force zone at the crack tip with length d shown in Fig. 14.4-1, the value of which is unknown at moment to be determined. In continuum theory of quasicrystals the atomic cohesive force zone is the plastic zone macroscopically, and the distribution of atomic cohesive force must be evaluated by experimental observation. Due to lack of the data of the new material, if we assume that the atomic cohesive is a constant τc , the shear yield limit (or shear yield strength) of the material, within the zone, then the problem is simpliﬁed. In this version the nonlinear boundary value problem is linearized already. So the previous formulation exhibited e.g. in Chapter 8 can be used to solve the present problem. The most eﬀective method for solving the problem is the complex variable function method, but the formulation and calculation are quite complicated and lengthy, which are much more complex than those given in previous chapters. The details are omitted, we here list only the ﬁnal results: 2θ1 ζ2 1 2i 2i e2iθ1 − ζ 2 + F1 (ζ) = τ − τ τc ln −2iθ1 , F2 (ζ) = 0, c 1 2 2 C44 πζ ζ − 1 2πi C44 e − ζ2 (14.4-1)

Fig. 14.4-1

Atomic cohesive force zone near the crack tip for anti-plane elasticity for some quasicrystals

14.4

Nonlinear solutions based on simple models

in which F1 (ζ) = φ (ζ) +

297

R3 ψ (ζ), C44

F2 (ζ) = ψ (ζ) +

R3 φ (ζ) K2

and φ(ζ) = φ1 (t) = φ1 (ω(ζ)),

ψ(ζ) = ψ1 (t) = ψ1 (ω(ζ)),

where

a+d (ζ + ζ −1 ) 2 represents the conformal mapping from t-plane (xy-plane) onto ζ-plane, under the mapping the region of xy-plane is transformed onto the interior of the unit circle at ζ-plane, and φ (ζ) and ψ (ζ) are derivatives of the functions to the new complex variable ζ, and θ1 is the angle of point at the unit circle corresponding to the crack tip (i.e., y = 0, x = a, and there is relation cos θ1 = a/(a + d) between the corresponding points at t-plane and ζ-plane). From the solution, the size of plastic zone d is determined as πτ1 d = a sec −1 , (14.4-2) 2τc t = ω(ζ) =

i.e., θ1 =

πτ1 and the crack tip tearing displacement is 2τc 4K2 τc a πτ1 ln sec . δIII = π(C44 K2 − R32 ) 2τc

(14.4-3)

It is obvious that the results are very simple and explicit, and the eﬀects of phason to plastic deformation and fracture are explored. If taking the tearing displacement as a nonlinear fracture parameter, then a fracture criterion for one-dimensional hexagonal quasicrystal is δIII = δIIIC (14.4-4) is suggested in which δIIIC is the critical value the crack tip tearing displacement, measured by experiment, a material constant. In the low stress level case, i.e., τ1 /τc 1, the deformation is linear elastic, in which 2 1 πτ1 πτ1 + ··· =1+ sec 2τc 2 2τc if we remain the ﬁrst two terms then 2 4 πτ1 1 πτ1 1 πτ1 ln sec + + ··· = 2τc 2 2τc 12 2τc

298

Chapter 14

so that in the linear case

Nonlinear behaviour of quasicrystals

GIII . (14.4-5) τc is equivalent to the energy release rate GIII in the δIII =

This shows that parameter δIII linear case. The above results are obtained for one-dimensional hexagonal quasicrystals, which hold for three-dimensional icosahedral quasicrystals, if the material constants C44 , K2 and R3 are replaced by μ, K1 − K2 and R; which hold for three-dimensional cubic quasicrystals too, if the constants are replaced by C 44 , K44 and R44 , respectively. 14.4.2

Generalized Dugdale-Barenblatt model for plane elasticity of twodimensional point groups 5m, 10mm and 5, 5, 10, 10 quasicrystals

The linear analysis for a crack in plane elasticity in point groups 5m and 10mm as well as in groups 5, 5 and 10, 10 of two-dimensional quasicrystals are given in Sections 8.3 and 8.4 in terms of the Fourier method and complex variable function method respectively, where we pointed out that if taking R1 = R, R2 = 0 then the solution on point groups 5, 5 and 10, 10 will exactly reduces to that on point groups 5m and 10mm. In the following we discuss the nonlinear solutions only based on the complex variable function method. For two-dimensional quasicrystals a similar generalized Dugdale-Barenblatt model to that given in the Subsection 14.4.1 is suggested as below, refer to Fig. 14.4-2. That is, there is the so-called atomic cohesive force zone, or the plastic zone macroscopically, with length d, temporarily unknown and to be determined. If we suppose that the distribution of the atomic cohesive force distribution and magnitude of the atomic cohesive force σc = σc (x) are known, further more σc =constant within the zone, then the nonlinear problem is linearized already, in which σc represents the

Fig. 14.4-2

The generalized DB model for two-dimensional decagonal quasicrystal

14.4

Nonlinear solutions based on simple models

299

macro-tensile yield strength of the quasicrystal. Therefore the formulation introduced above can be used to solve the present problem. The powerful method for the problem is the complex variable function-conformal mapping method, which was displayed brieﬂy in the Subsection 14.4.1. But the calculation is much more complicated and lengthy than those listed in the previous subsection. The detail of complex variable function-conformal mapping method is given in Subsection 11.3.10 in Chapter 11 and Subsection A.I.9 in Major Appendix, here we list the ﬁnal plastic solution only as follows. The size of the plastic zone is determined as πp −1 , (14.4-6) d = a sec 2σc which is similar to that given by (14.4-2), but the meaning is diﬀerent. And the crack tip opening displacement is for point groups 5m and 10mm quasicrystals 2σc a 1 πp K1 + ln sec (14.4-71 ) δI = π L+M M K1 − R 2 2σc and for point groups 5, 5, 10, 10 ones 2σc a 1 πp K1 δI = + ln sec , π L+M M K1 − (R12 + R22 ) 2σc

(14.4-72 )

which is also similar to that obtained by (14.4-3) in which L = C12 , M = (C11 − C12 )/2 are the phonon elastic constants, K1 the phason elastic constant and R, R1 , R2 the phonon-phason coupling constants respectively. It is very interesting that the solution (14.4-7) exactly covers the solutions of crystals and conventional structural materials, as K1 = R = 0, e.g. for classical conventional materials ⎧ π p π p 4(1 − ν)σc a (1 + κ)σc a ⎪ ⎪ ln sec ln sec = , ⎪ ⎪ πμ 2 σc πμ 2 σc ⎪ ⎪ ⎪ ⎨ plane strain state, δt = CT OD = ⎪ (1 + κ )σc a π p π p 4σs a ⎪ ⎪ ln sec ln sec = , ⎪ ⎪ πμ 2 σc (1 + ν)πμ 2 σc ⎪ ⎪ ⎩ plane stress state, 3−ν for plane stress, ν is the Poisson’s ratio 1+ν of the materials. So the solutions of crystals and conventional structural materials are the speciﬁc cases of our solution of quasicrystals. Fig. 14.4-3 shows the normalized crack tip opening displacement δI /a versus normalized stress p/σc and gives the comparison between quasicrystals and crystals as well as the comparison between point groups 10mm and 10, 10 quasicrystals. The detail of solution including point group 10, 10 is given in Subsection A.I.9 of Major Appendix of this book. where κ = 3−4ν for plane strain, κ =

300

Chapter 14

Fig. 14.4-3

Nonlinear behaviour of quasicrystals

Normalized crack tip opening displacement versus normalized stress and comparison[27]

Based on the parameter δI given by (14.4-7) we can suggest the nonlinear fracture criterion for pentalgonal and/or decagonal quasicrystals for the Mode I loading as δI = δIC ,

(14.4-8)

in which δIC denotes critical value of the crack opening displacement, measured by experiment, a material constant. In linear elastic case, i.e., p/σc 1, through a similar analysis like in the previous subsection the crack tip opening displacement reduces to δI =

GI , σc

(14.4-9)

where GI is deﬁned by (8.3-18), this gives the simple connection between these two parameters, so the criterion (14.4-8) reduces to energy release rate criterion for linear elastic case discussed in Chapter 8. 14.4.3

Generalized Dugdale-Barenblatt model for plane elasticity of threedimensional icosahedral quasicrystals

The schematic ﬁgure of the model for icosahedral quasicrystals is similar to Fig. 14.4.2, the plastic zone is the same given by (14.4-6), and the crack tip opening displacement is as follows π p 1 c4 σc a + · ln sec · δI = lim 2uy (x, 0) = lim 2uy (x, 0) = 2 ϕ→ϕ2 x→l λ + μ c2 π 2 σc (14.4-10)

14.5

Nonlinear analysis based on the generalized Eshelby theory

in which (μK2 − R2 )2 , c2 = μ(K1 − K2 ) − R − μK1 − 2R2 2

301

1 μK1 − 2R2 c4 = c1 R + c2 K1 + 2 λ+μ (14.4-11)

R(2K2 − K1 )(μK1 + μK2 − 3R2 ) . 2(μK1 − 2R2 ) The variation of the normalized crack tip opening displacement versus the normalized applied stress is shown in Fig. 14.4-4.

with c1 =

Icosahedral quasicrystal Al-Mn-Pd Icosahedral quasicrystal Al-Cu-Li Conventional crystal aluminum

δI/a

p/σc

Fig. 14.4-4

Normalized crack tip opening displacement of icosahedral quasicrystals versus normalized applied stress [27]

14.5

Nonlinear analysis based on the generalized Eshelby theory

In the previous section we obtained some nonlinear solutions for one-, two-and threedimensional quasicrystals by using some simple physical models. We will show that those solutions have some inherent relations to the generalized Eshelby theory originated from the classical reference [13] for crystals. Fan and Fan[14] did some work to extend the classical Eshelby model for crystals to that for quasicrystals. 14.5.1

Generalized Eshelby energy-momentum tensor and generalized Eshelby integral

Fan and Fan[14] (also refer to Ref. [18]) deﬁned the generalized Eshelby energymomentum tensor as G = F n1 − σij nj

∂ui ∂wj − Hij nj , ∂x1 ∂x1

(14.5-1)

302

Chapter 14

Nonlinear behaviour of quasicrystals

where F is deﬁned by (14.3-1) and ni the unit vector of outward normal at any point of an arc in a quasicrystal, and at which there are σij nj = Ti ,

Hij nj = hi ,

where Ti denotes the traction vector and hi the generalized traction vector, shown in Fig. 14.5-1, respectively.

Fig. 14.5-1

The path of the generalized Eshelby integral

Furthermore we deﬁne an integral such as E= GdΓ ,

(14.5-2)

Γ

in which Γ is an integration path enclosing a crack tip in the material. To memory Eshelby the integral is named generalized Eshelby integral. The integral exhibits path independency. In Section 14.8, i.e., Appendix of Chapter 14, the proof on the path-independency of the integral is given. Because the integral presents the above important character, it has some applications in fracture analysis of quasicrystals. Another important character is the value of the integral which is equal to the energy release rate GIII (under Mode III loading), or GII (under Mode II loading), or GI (under Mode I loading) of one-, two- and three-dimensional quasicrystals when the material is in linear elastic deformation state. The mathematical proof about this will be provided in Section 14.8, i.e., Appendix of Chapter 14. The features of the generalized Eshelby integral (14.5-2) exhibited above imply that it may be signiﬁcant for the analysis of nonlinear fracture of quasicrystals. In the following an application for this purpose is discussed.

14.5

Nonlinear analysis based on the generalized Eshelby theory

14.5.2

303

Relation between crack tip opening displavement and the generalized Eshelby integral

By using the path-independency of E-integral (14.5-2), we take the integration path Γ shown in Fig. 14.5-2 and let the path close to the surface of the plastic zone as , and along segment AB and

close as possible. Now the integration path is ACB

BC, i.e., dy = 0, so that GdΓ = E=

T1 = 0,

T 2 = σc ,

h2 = Hc

∂u ∂wy dx + Hc dx ∂x ∂x A Γ ACB = σc [(uy )B − (uy )A ] + Hc [(wy )B − (wy )A ] ≈ σc [(uy )B − (uy )A ] = σc δI . (14.5-3)

Fig. 14.5-2

GdΓ =

B

σc

The integration path for evaluating crack tip opening displacement

This proves the relation between the generalized Eshelby integral and crack tip opening displacement, it shows the E-integral presents an equivalency to the crack tip opening displacement, where σc is the atomic cohesive force (or the plastic yield strength in macroscopic sense) and the “generalized atomic cohesive force” of the quasicrystal material. In microscopy the quantity is meaningful, but it has not been measured macroscopically at present, the eﬀect of which is omitted in formula (14.5-3) by this reason. For Mode II and Mode III crack the proof is similar. Since by the generalized BCS model and generalized DB model one can obtain the same result on crack tip opening displacement, which is also as a direct result of the generalized Eshelby integral for quasicrystals, we realize that the generalized Eshelby energy-momentum tensor theory is the uniform physical basis of the two models.

304

14.5.3

Chapter 14

Nonlinear behaviour of quasicrystals

Some further interpretation on application of E-integral to the nonlinear fracture analysis of quasicrystals

The above subsections and the appendices exhibit the generalized Eshelby integral can be as a uniform basis of the generalized BCS model as well as the generalized DB model for nonlinear fracture analysis of some one-, two- and three-dimensional quasicrystals. We suggest taking the E-integral as a fracture parameter, and E = Ec

(14.5-4)

as fracture criterion, where Ec is the critical value of the integral, a material constant, which can be measured through some conventional specimens, the discussion will be given in Section 14.8 (the Appendix of Chapter 14). Though the measurement of the critical value of the generalized Eshelby integral may be easier, but the evaluation of values of E-integral for plastic deformation of quasicrystals is very diﬃcult. Therefore the implementation of the criterion (14.54) is not so convenient in practice. Instead, people can use fracture criterion of crack tip opening displacement considering the equivalency between the crack tip opening displacement δ(δI , δII , δIII ) and E-integral, we suggest taking the crack tip opening displacement as a fracture parameter. Elasto-plastic crack solutions for some one- and two-and three-dimensional quasicrystals have been found based on the generalized Dugdale-Barenblatt model, refer to Refs. [14, 18, 27], in which the size of the plastic zone and crack tip opening displacement are determined, with these data an equivalent plastic fracture criterion is suggested as δ = δc ,

(14.5-5)

in which δc represents the critical value of the crack tip opening displacement, a material constant. The evaluation of crack tip opening displacement has been exactly completed for large plates with central crack and narrow plastic zone and introduced in Section 14.4 or refer to Refs. [14, 18, 27], and some approximate solution for other conﬁguration can be obtained by complex variable function method (an outline of the method may be interpreted in Refs. [14, 18, 27]) and other methods (mainly the approximate methods and numerical methods). The shortcoming of this criterion lies in some diﬃculties of the determination of δc , but which can be obtained through relation Ec δc = . (14.5-6) σc and the Section 14.8 shows the measurement of Ec is easier. Thus, the collaboration of criterions (14.5-4) and (14.5-5) makes the nonlinear fracture analysis for quasicrystals possible. Some details will be given in the Appendix of this chapter (i.e., Section 14.8).

14.6

Nonlinear analysis based on the dislocation model

14.6

305

Nonlinear analysis based on the dislocation model

We pointed out at the begining of this chapter that in the study on the classical plasticity there are two diﬀerent theories, one is the macroscopic plasticity theory, and the other is so-called crystal plasticity theory, in some extent the latter can be seen as a “microscopic” theory. which is based on the mechanism of motion of dislocation. We discussed quite more dislocation solutions in Chapter 7 for one- and two-dimensional quasicrystals and in Chapter 9 for three-dimensional quasicrystals, those results are helpful to explore the plastic deformation and fracture of the material. Because this analysis needs not to rely on nonlinear macroscopic constitutive law and it reduces a great diﬃculty in mathematical treatment. Here we focus on the discussion concerning plastic ﬂow around crack tip for some one-, two- and three-dimensional quasicrystals, respectively. 14.6.1

Screw dislocation pile-up for hexagonal or icosahedral or cubic quasicrystals

For the topic the basis is the generalized BCS model developed by Fan and coworkers[8] , which is stated as below. Assume there is a Griﬃth crack with length 2l along the direction of z axis in a one- or three-dimensional quasicrystal, subjected to a longitudinal shear stress τ (∞) at inﬁnity see Fig. 14.6-1. Around the crack tip there is a dislocation pile-up with size d which is unknown so far to be determined. Within the zone of dislocation pile-up the material presents plastic ﬂow, the stress σyz is equal to the critic shear stress τc , the atomic cohesive force, or so-called the ﬂow limit of the material macroscopically. For simplicity the external applied stress at the inﬁnity can be removed, instead it is applied at the crack surface. The latter is equivalent with the former from point of view of fracture behaviour for any studied systems. This is the generalized BCS model

Fig. 14.6-1

Schematic picture of a crew dislocation pile-up coupled with a crack in anti-plane elasticity of quasicrystals

306

Chapter 14

Nonlinear behaviour of quasicrystals

for icosahedral quasicrystals. The origin of the model was given by Bilby, Cottrell and Swinden[20] and Bilby, Cottrell, Smith et al[21] for crystals, we developed model to the study for some one- and two-dimensional quasicrystals[8] . The model can be formulated by the following boundary conditions for one-dimensional hexagonal quasicrystals: (x2 + y 2 )1/2 → ∞ : σij = 0, Hij = 0, y = 0, |x| < l : σyz = −τ (∞) , Hyz = 0, y = 0, l < |x| < l + d : σyz = −τ (∞) + τc = 0, Hyz = 0.

(14.6-1)

Some detailed statement for the formulation can refer to Ref. [18]. The boundary conditions (14.6-1) has been shown that the nonlinear (plastic deformation) problem is linearized mathematically. So it has been transformed into a linear problem (or an “equivalent” elasticity problem) described by the ﬁnal governing equations (14.6-2) ∇2 uz = 0, ∇2 wz = 0. and boundary conditions (14.6-1), detail about this can be found in the text of Chapters 5, 7 and 8. The nonlinear fracture problem can be solved if we can obtain the solution of equation (14.6-2) under boundary conditions (14.6-1). It is obvious the problem is complicated. Though we can solve the problem in the methodology developed in Chapter 8, but there is simpler procedure to solve it if we are interested in only some plastic deformation parameters around crack tip and if we have a dislocation solution, the latter is given in Section 7.1. For this purpose we ﬁrst introduce a dislocation density function f (ξ), then problem (14.6-1) and (14.6-2) can transformed into the following singular integral equation problem such as f (ξ)dξ τ (x) = , (14.6-3) A L ξ−x in which ξ denotes the dislocation source point coordinate, x denotes the ﬁeld point coordinate at the real axis and L represents interval (l, l + d). From the dislocation solution given in Section 7.1 the constantA is deﬁned by

|x| < l, −τ (∞) , (14.6-4) τ (x) = (∞) −τ + τc , l < |x| < l + d,

A=−

(C44 K2 − R32 )b3 . 2πK2

In terms of the Muskhelishvili[26] theory, Fan et al[8] solved the integral equation (14.6-3) under condition (14.6-4), the solution is & & 1 x + (l + d) ξ − (l + d) dξ f (x) = − 2 τ (ξ) π A x − (l + d) L ξ + (l + d) ξ−x

14.6

Nonlinear analysis based on the dislocation model

307

&

x + (l + d) l (∞) i 2τc arccos −τ π x − (l + d) l+d " " " " " (l + d)2 − lx " " (l + d)2 + lx " τc " " " " . arc cosh " − arc cosh " + 2 π A (l + d)(l − x) " (l + d)(l + x) "

1 =− 2 π A

(14.6-5)

Because the function f (x) should be a real number, the factor multiplying the imaginary number i must be zero in the ﬁrst term of right-handside of (14.6-5), this leads to l 2τc arccos − τ (∞) π = 0, l+d i.e.,

(∞) πτ d = l sec −1 . 2τ c

(14.6-6)

This is the same as (14.4-2) if l is replaced by a. From solution (14.6-5) we evaluate amount of dislocations N (x) such as x f (ξ)dξ. (14.6-7) N (x) = 0

Substituting (16.6-5) (coupled with (14.6-6)) into (14.6-7), we can get N (l + d) and N (l), so the amount of dislocation motion is δIII =

b3 [N (l

2b lτc + d) − N (l)] = 32 π A

l+d ln l

4K2 τc l = ln sec π(C44 K2 − R32 )

πτ (∞) . 2τc (14.6-8)

This is the same as (14.4-3) apart from diﬀerence of notations. 14.6.2

Edge dislocation pile-up for pentagonal or decagonal twodimensional quasicrystals

For plane elasticity the corresponding problem is “edge” dislocation shown in Fig. 14.6-2, here we consider the pile-up in the pentagonal or decagonal two-dimensional quasicrystals. There are the boundary conditions ⎧ 2 ⎨ (x + y 2 )1/2 → ∞ : σij = 0, Hij = 0, y = 0, |x| < l : σyx = −τ (∞) , Hyx = 0, σyz = 0, Hyz = 0, ⎩ y = 0, l < |x| < l + d : σyx = −τ (∞) + τc , Hyx = 0, σyz = 0, Hyz = 0, (14.6-9) and the governing equation (14.6-10) ∇2 ∇2 ∇2 ∇2 F = 0. Instead solving the boundary value problem (14.6-9) and (14.6-10), one can determine the dislocation density which satisﬁes the integral equation (14.6-3) if the relevant quantity A is known beforehand.

308

Chapter 14

Fig. 14.6-2

Nonlinear behaviour of quasicrystals

Schematic picture of an “edge” dislocation pile-up coupled with a crack in a plane elasticity of quasicrystals

In the singular integral equation the shear stress distribution function, τ (x) is the same as before, but here A is replaced by:

A=

(L + M )(M K1 − R2 ) b1 . π (L + M )K1 + (M K1 − R2 )

(14.6-11)

which has been found in Section 7.2 for point groups 5m and 10mm quasicrystals. According to the condition concerning factor multiplying the imaginary i be zero in (14.6-5), the plastic zone size is determined by (14.6-6), this is similar to (14.4-6), but the physical meaning is somewhat diﬀerent. By a similar calculation on amount of dislocation motion we can obtain the slip of the crack tip such as (∞) (∞) 2τ 2b l τc (L+M ) K1 1 2τ 2τc l δII = 21 ln sec + = ln sec . 2 π A M +K1 πτc π M +K1 M K1 −R πτc (14.6-12) It is evident that the result of (14.6-12) is similar to that of (14.4-7), but the physical meaning is diﬀerent. According to the macroscopic fracture mechanics, the crack tip slip corresponding to the Mode II crack tip displacement (i.e., the crack tip sliding displacement) δII . But for the Mode I crack tip opening displacement (14.4-7) there is no physical basis of dislocation model, though the similar mathematical treatment can be easily given, which is omitted here. Even if this, the dislocation model provides a powerful support to the nonlinear analysis of previous sections. 14.6.3

Edge dislocation pile-up for three-dimensional icosahedral quasicrystals (x2 + y 2 )1/2 → ∞ :

σij = 0,

Hij = 0,

14.7

Conclusion and discussion

|x| < l :

y = 0,

309

σyx = −τ (∞) , Hyx = 0, σyz = 0, Hyz = 0, σyy = 0, Hyy = 0,

y = 0, l < |x| < l + d : σyx = −τ (∞) + τc , Hyx = 0, σyz = 0, Hyz = 0, σyy = 0, Hyy = 0, ∇2 ∇2 ∇2 ∇2 ∇2 ∇2 F (x, y) = 0. The derivation can be oﬀered similar to above. Because the formulas are too lengthy (please refer to Section 9.6), the discussion is omitted. The dislocation model provides a powerful support to the generalized cohesive force model and generalized Eshelby model presented in previous sections, even if there is diﬃculty physically, we are unable to derive δI directly from the model, and results in a slight faulty aspect of the model.

14.7

Conclusion and discussion

This chapter discussed the deformation and fracture exhibiting nonlinear behaviour of quasicrystals, this is a topic which has not been well studied so far. The discussion in the chapter shows that the nonlinear elastic constitutive equation possesses some meaning for the investigation due to lack of plastic constitutive equation at present. The generalized Eshelby principle based on the nonlinear elastic constitutive law plays an important role in revealing the fracture behaviour of quasicrystals under nonlinear deformation which may be a basis of generalized DB model in some extent. We realize that the generalized Eshelby principle and generalized DB model are based on the macro-nonlinear continuous mechanics of quasicrystals. The generalized BCS model developed in Section 14.6 emphasizes dislocation mechanism for the plastic deformation, it is diﬀerent from those of the generalized DB hypothesis as well as the generalized Eshelby principle at physical basis and methodology. The attractiveness is that results derived from these quite diﬀerent hypothesizes achieve the exact agreement in the study.

14.8

Appendix of Chapter 14—Some mathematical details

14.8.1

Proof on path-independency of E-integral

Consider a single-connected region D with a closed boundary C. From the Green formula the ﬁrst, second and third terms of the left-handside of the integral (14.5-2) will be ∂F F n1 dΓ = F dx2 = dx1 dx2 , ∂x1 C C σij nj C

∂ui ∂wi + Hij nj ∂x1 ∂x1

dΓ = D

D

∂ ∂xj

σij

∂ui ∂wi + Hij ∂x1 ∂x1

dx1 dx2 .

310

So that

Chapter 14

GdΓ = C

D

∂F ∂ − ∂x1 ∂xj

Nonlinear behaviour of quasicrystals

∂ui ∂wi σij + Hij ∂x1 ∂x1

dx1 dx2 .

(14.8-1)

Meantime, (14.3-2) provides that ∂F ∂εij ∂F ∂wij ∂F ∂εij ∂wij = + = σij + Hij ∂x1 ∂εij ∂x1 ∂wij ∂x1 ∂x1 ∂x1 ∂ 1 ∂ui ∂ ∂uj ∂wi = σij + + Hij ∂x1 2 ∂xj ∂xi ∂x1 ∂xj ∂ ∂ ∂ui ∂wi + Hij , = σij ∂x1 ∂xj ∂x1 ∂xj where the symmetry of tensor εij (given by the ﬁrst equation of (14.3-3)) is used. The above result can be rewritten as ∂F ∂ui ∂wi ∂ ∂ ∂σij ∂ui ∂Hij ∂wi = + σij − Hij − ∂x1 ∂xj ∂x1 ∂xj ∂x1 ∂xj ∂x1 ∂xj ∂x1 ∂ui ∂wi ∂ σij , (14.8-2) = + Hij ∂xj ∂x1 ∂x1 in the last step in driving (14.8-2) the equilibrium equations (14.3-4) are used. Substituting (14.8-2) into (14.8-1) we have GdΓ = 0. C

If taking C = Γ + BB − Γ + A A, in which Γ is a path similar to Γ , these two curves intersect the upper face of the crack at points B and B , and the lower face at points A and A , respectively. Due to dx2 = dy = 0, at the segments BB and A A, there are GdΓ = 0, BB +A A

Ti = 0,

hi = 0

GdΓ =

Γ

GdΓ . Γ

Because Γ and Γ are arbitrary paths satisfying the requirements of E-integral deﬁnition, the last equality proves its path-independency. 14.8.2

Proof on the equivalency of E-integral to the energy release rate under the linear elastic case for quasicrystals

In the text of this chapter we argue that the E-integral is equivalent to the energy release rate when the quasicrystals are in linear elastic deformation state, the

14.8

Appendix of Chapter 14—Some mathematical details

311

mathematical proof about this is given here. The proof can be done for one-, twoand three-dimensional quasicrystals and for Mode I, Mode II and Mode III cracks. For simplicity here we only discuss Mode III crack for one-dimensional hexagonal quasicrystals. At the linear elastic case, the generalized Hooke’s law σij = Cijkl εkl + Rijkl wkl , Hij = Kijkl wkl + Rklij εkl is reduced to σyz = σyz = 2C44 εyz + R3 wzy , σzx = σxz = 2C44 εzx + R3 wzx , Hzy = K2 wzy + 2R3 εzy , Hzx = K2 wzx + 2R3 εzx for the phonon-phason coupling anti-plane shearing (or longitudinal shearing) state in elasticity of one-dimensional hexagonal quasicrystals, one ﬁnds the free energy (or strain energy density) is 1 2 2 + wzy ) + R3 (εzx wzx + εyz wzy ). F = C44 (ε2xz + ε2yz ) + K2 (wzx 2

(14.8-3)

Substituting the crack solution given in Chapter 8 into (14.8-3) we have F =

1 ⊥ 2 ⊥ {C44 [K2 KIII − R3 KIII ] + 2K2 [C44 KIII − R3 KIII ]2 4(C44 K2 − R32 )

⊥ ⊥ + 2R3 [K2 KIII − R3 KIII ][C44 KIII − R3 KIII ]},

(14.8-4)

√ √ ⊥ where KIII = πaτ1 and KIII = πaτ2 are the stress intensity factors associated with phonon and phason ﬁelds respectively. Integrating the quantity given by (14.8-4) around a path enclosing the crack tip, then one can obtain the ﬁrst term of the E-integral. Because of the pathindependency of the integral, we can take a half-circle with the crack tip as its origin and with radius r, such that

−π

F dy = π

−π

F r cos θdθ = 0, π

where (r, θ) denote the crack tip coordinates. Now we calculate the second and third terms of the integral.

(14.8-5)

312

Chapter 14

Nonlinear behaviour of quasicrystals

According to the deﬁnition of Section 14.3, it is known that between the traction, generalized traction and the phonon, phason stresses there are Tz = σzx nx + σzy ny = σzx cos θ + σzy sin θ, hz = Hzx nx + Hzy ny = Hzx cos θ + Hzy sin θ, by substituting the crack solution given in Section 8.1 into the above formulas, we ﬁnd 1 1 1 1 ⊥ KIII sin θ, hz = √ KIII Tz = √ sin θ. (14.8-6) 2 2 2πr 2πr Based on the crack solution given in Section 8.1, there are ⎧ K 2 τ 1 − R3 τ 2 √ 1 ⎪ 2ar sin θ, ⎨ uz = 2 C44 K2 − R3 2 C τ − R3 τ1 √ 1 ⎪ ⎩ wz = 44 2 2ar sin θ. C44 K2 − R32 2

(14.8-7)

In addition, ∂ ∂r ∂ ∂θ ∂ ∂ ∂ = + = cos θ − sin θ . ∂x ∂r ∂x ∂θ ∂x ∂r ∂θ Substituting these results and relations into the second and third terms of the integral we obtain −

−π π

σij nj

∂ui ∂wi + Hij nj ∂x1 ∂x1

dΓ =

⊥ 2 ⊥ K2 (KIII )2 + C44 (KIII ) − 2R3 KIII KIII . 2 C44 K2 − R3 (14.8-8)

From (14.8-5) and (14.8-8), we ﬁnd that

EIII =

⊥ 2 ⊥ K2 (KIII )2 + C44 (KIII ) − 2R3 KIII KIII = GIII . 2 C44 K2 − R3

which is just (8.1-25) given in Chapter 8. Similarly one can obtain EI = GI

(14.8-9)

(14.8-10)

and EII = GII ,

(14.8-11)

where the suﬃxes mean the Mode I, Mode II and Mode III of the cracks. The above demonstration indicates that the generalized Eshelby integral is equivalent to the energy release rate of crack in linear elasticity of quasicrystals. Under plastic deformation of quasicrystals, in general, the generalized Eshelby integral does not represent energy release rate, the reason of this is the unloading can appear due to crack extension, and stress-strain relations described by (14.3-1)

14.8

Appendix of Chapter 14—Some mathematical details

313

and (14.3-2) do not remain one to one correspondence, the physical background of the E-integral does not hold. But if we deﬁne the potential energy per unit thickness for a plane (i.e., a two-dimensional) region Ω occupied by a quasicryslal, and denote the boundary of the region by Γ F dxdy − (Ti ui + hi wi )dΓ , (14.8-12) V = Ω

Γ

then the total potential energy is Π = BV,

(14.8-13)

where B denotes the thickness of the specimen, and there is relation E=−

δΠ 1 δΠ =− , δA B δa

(14.8-14)

where E-integral can be expressed only by the “diﬀerence quotient” rather than the “diﬀerential quotient”. The diﬀerence quotient is not the energy release rate, only represents the ratio between individual diﬀerence value of the energy over diﬀerence value of crack length for the same conﬁguration specimen but with diﬀerent initial crack lengths. Considering the situation the physical meaning of (14.8-14) is left for further discussion. Even if there is the faulty aspect, (14.8-14) provides useful application for calibrating the measurement data of the integral in experiments. The relevant discussion will be done in the next subsection. 14.8.3

On the evaluation of the critic value of E-integral

The E-integral provides not only the theoretical basis for generalized BCS model as well as generalized DB model but also an eﬀective tool for measuring the material constants Ec and δc . We mentioned that the measurement of δc is diﬃcult, but the measurement of Ec is easier. Due to the connection between these two quantities the value of the former can be obtained from that of the latter. At low and conventional temperature quasicrystals present brittle behaviour, the fracture belongs to linear elastic fracture, and the measurement of fracture toughness can be carried out by indentation technique, see e.g. Meng et al[19] . At high temperature, the materials present large plastic deformation, the measurement can be done by the bending experiment. The three point bending specimen, see Fig. 14.8-1, is one of common specimens. There are two procedures either by multi-specimens testing or by single specimen testing. For simplicity we here discuss the single specimen testing only. For the three point bending specimen shown in Fig. 14.8-1, based on formula (14.8-14) the value of E-integral is approximately by

314

Chapter 14

Fig. 14.8-1

Nonlinear behaviour of quasicrystals

Three point bending specimen

2U , (14.8-15) B(W − a) where W represents the width, B the thickness of the specimen and U the area under P -Δ curve (shown by Fig. 14.8-2), i.e., U = P dΔ, (14.8-16) E=

in which P is the load (force per unit thickness) at loading point, and Δ the displacement of the same point. When the initiation of crack growth is observed, then the value of E-integral is marked as the fracture toughness of the quasicrystal. If the phason ﬁeld is absent, the material is degenerated to conventional structural material. In this case, the E-integral is reduced to conventional Eshelby integral or J-integral, the latter was introduced by Rice[22] and Cherepanov[23] . Begley and Landes[24,25] put forward the experimental study on J-integral, further promoting the development of nonlinear fracture theory and its applications for conventional engineering materials. These experiences would be helpful for Fig. 14.8-2 The depiction of the the experimental study of nonlinear fracture of specimen deformation energy quasicrystalline materials.

References [1] Calliard D. Dislocation mechanism and plasticity of quasicrystals: TEM observations in icosahedral Al-Pd-Mn. Materials Sci Forum, 2006, 509(1): 49–56 [2] Geyer B, Bartsch M, Wollgarten M et al. Plastic deformation of icosahedral Al-PdMn single quasicrystals. Experimental results. Phil Mag A, 2000, 80(7): 1151–1164

References

315

[3] Messerschmidt U, Bartsch M, Wollgarten M et al. Plastic deformation of icosahedral Al-Pd-Mn single quasicrystals. II. Interpretation of experimental results. Phil Mag A, 2000, 80(7): 1165–1181 [4] Urban K, Wollgarten M. Dislocation and plasticity of quasicrstals. Materials Sci Forum, 1994, 150–151(2): 315–322 [5] Wollgarten M, Bartsch M, Messerschmidt U et al. In-situ observation of dislocation motion in icosahedral Al-Pd-Mn single quasicrystals. Phil Mag Lett, 1995, 71(1): 99–105 [6] Feuerbacher M, Metzmacher C, Wollgarten M et al. 1997, Plastic deformation of decagonal Al-Ni-Co quasicrystals. Phil Mag Lett, 76(4): 396–375 [7] Messerschmidt U, Bartsch M, Feuerbacher M et al. Friction mechanism of dislocation motion in icosadedral Al-Pd-Mn single quasicrystals. Phil Mag A, 1999, 79(11): 2123–2135 [8] Fan T Y, Trebin H-R, Messerschmidt U et al. Plastic ﬂow coupled with a Griﬃth crack in some one- and two-dimensional quasicrystals. J Phys: Condens Matter, 2004, 16(47): 5229–5240 [9] Feuerbacher M, Urban K. Platic behaviour of quasicrystalline materials. in: Quasicrystals, Trebin H R, Wiely Press, Berlin, 2003 [10] Guyot P, Canova G. The plasticity of icosahedral quasicrystals. Phil Mag A, 1999, 79(11): 2815–2822 [11] Feuerbacher M, Schall P, Estrin Y et al. Aconstitntive model for quasicystal plasticity. Phil Mag Lett, 2001, 81(7): 473–482 [12] Estrin Y. in: Uniﬁed Constitutive Laws of Plastic Deformation, Krausz A, Krausz K, Academic Press. New York, 1996 [13] Eshelby J D. The continuum theory of dislocations in crystals, Solid State Physics (ed by Seits F et al). Vol. 3. New York: Academic Press, 1956 [14] Fan T Y, Fan L. Plastic fracture of quasicrystals. Phil Mag, 2008, 88(4): 523–535 [15] Dugdale D S. Yielding of steel sheets containing slits. J Mech Phys Solids, 1960, 32(2): 105–108 [16] Barenblatt G I. The mathematical theory of equilibrium of crack in brittle fracture. Advances in Applied Mechanics. 1962, 7: 55–129 [17] Fan T Y, Mai Y W. Elasticity theory, fracture mechanics and some relevant thermal properties of quasicrystalline materials. Appl Mech Rev, 2004, 57(5): 325–344 [18] Fan T Y, Fan L. Relation between generalized Eshelby integral and generalized BCS and generalized D B models for some one- and two-dimensional quasicrystals. Chin. Phys. B in press, 2010 [19] Meng X M, Tong B Y, Wu Y K. Mechanical properties of quasicrystals Al65 Cu20 Co15 . Acta Metallurgica Sinica A, 1994, 30(3): 61–64 (in Chinese)

316

Chapter 14

Nonlinear behaviour of quasicrystals

[20] Bilby B A, Cottrell A H, Swinden K H. The spread of plastic yield from a notch. Proc R Soc A, 1963, 272(2): 304–314 [21] Bilby B A, Cottrell A H, Smith E et al. Plastic yealding from sharp notches, ibid, 1964, 279(1): 1–9 [22] Rice J R. A path independent integral and approximate analysis of strain concentration by notches and cracks. J Appl Mech, 1968, 35(4): 379–386 [23] Cherepanov G P. On crack propagation in solids. Int J Solids and Structures, 1969, 5(8): 863–871 [24] Begley G T, Landes J D. 1972, The J-integral as a fracture criterion, Fracture Toughness, ASTM STP 514, American Society for Testing and Materials, Philadelphia, 1–20 [25] Landes J D, Begley G T. 1972, The eﬀect of specimen geometry on JIC , Fracture Toughness, ASTM STP 514, American Society for Testing and Materials, Philadelphia, 24–29 [26] Muskhelishvili N I. Singular Integral Equations. Groningen, Noordhoﬀ, 1956 [27] Li W, Fan T Y. Study on plastic analysis of crack problem in icosahedral quasicrystals. Phil. Mag., 2009, 89(31): 2823–2831

Chapter 15 Fracture theory of quasicrystals Quasicrystals are brittle, study on fracture behaviour of the material is signiﬁcant. In the previous chapters many crack problems are investigated, and the exact analytic, approximate and numerical solutions are constructed. This provides a basis for discussing the fracture theory. The exact solutions of crack problem of quasicrystals present particular importance, which reveal the essential nature describing fracture behaviour of this kind material. The exact analytic solution on a Griﬃth crack in decagonal quasicrystals was given by Li and Fan[1] in 1999. Afterward Fan[2] , Fan and Mai[3] developed the idea of linear elastic fracture mechanics of quasicrystals based on the common feature of crack solutions obtained in that date. Rudhart et al[4] discussed the relevant topic from other point of view. Recently Fan and Guo[5] , Zhu and Fan[6] , Li and Fan[7] observed the analytic methods and solutions of cracks in three-dimensional icosahedral quasicrystals, Fan and Fan[8,9] , Li and Fan[15] found the plastic analytic solutions of cracks, Zhu and Fan[10,11] carried out the numerical analysis on dynamic crack of quasicrystals, respectively. And the measurement of fracture toughness of the material is also reported[12] . The quite fruitful information mentioned above suggests the need to give a summary on the fracture study of quasicrystals. This chapter undertakes such a task and may put forward some pointers for the fracture mechanics of quasicrystals.

15.1

Linear fracture theory of quasicrystals

The descriptions on crack solutions in diﬀerent chapters reveal some common nature of stress ﬁeld and displacement ﬁeld around crack tip. For all exact solutions of cracks of one-, two- and three-dimensional quasicrystals in linear elasticity regime the resulting expressions of stress and displacement components hold with respect to arbitrary variables x and y. In addition we found quite a lot of solutions of cracks of dynamic state and nonlinear behaviour of quasicrystals, some common feature are also exhibited. At ﬁrst we analyze the linear elastic solutions. As indicted in Section 8.1, within the framework of linear elasticity and inﬁnite-sharp crack model, near crack tip, i.e., (15.1-1) r1 /a 1, T. Fan, Mathematical Theory of Elasticity of Quasicrystals and Its Applications © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2011

318

Chapter 15

Fracture theory of quasicrystals −1/2

stresses for phonons as well as for phasons appear singular in order of r1 (r1 →0), and other terms can be ignored when compared to this term. Although stress singularity is implausible, it is the result of idealized mathematical model. Quite a few researchers indicated its severe weakness in theory and the paradox of its methodology[1,2] . However, prior to the actual establishment of more reasonable fracture theory, we still continue using this theory in expectable near future. If temporarily accepting this theory, we focus on the ﬁeld variables near crack tip. If in (8.3-14) only keeping the term in order of (r1 /a)−1/2 , we have ⎧ ⎪ KI 1 1 3 ⎪ ⎪ √ θ θ θ 1 − sin = cos sin σ ⎪ xx 1 1 1 , ⎪ ⎪ 2 2 2 2πr1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ KI 1 3 1 ⎪ ⎪ √ σ = cos sin θ θ θ 1 + sin ⎪ yy 1 1 1 , ⎪ ⎪ 2 2 2 2πr1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ KI 1 3 ⎪ ⎪ √ σ = σ = cos θ1 cos θ1 , ⎪ xy yx ⎪ 2 2 2πr ⎪ 1 ⎪ ⎪ ⎪ ⎨ d21 K 3 3 5 (15.1-2) Hxx = − √ I sin θ1 2 sin θ1 + sin θ1 cos θ1 , ⎪ 2 2 2 2πr1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ d21 KI 3 5 ⎪ ⎪ Hyy = √ ⎪ sin2 θ1 cos θ1 , ⎪ ⎪ 2 2 2πr ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ d21 KI 3 5 ⎪ ⎪ √ H = − sin2 θ1 sin θ1 , ⎪ xy ⎪ ⎪ 2 2 2πr 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ d21 KI 3 5 3 ⎪ ⎪ √ H = sin θ − sin θ sin θ θ 2 cos ⎩ yx 1 1 1 1 , 2 2 2 2πr1 where d21 = R(K1 − K2 )/4(M K1 − R2 ) given in Section 8.3 and # √ || 2π(x − a)σyy (x, 0) = π ap, KI = lim x→a+

in which

⎧ ⎨

(15.1-3)

x − 1 , |x| > a, σyy (x, 0) = x2 − a2 ⎩ −p, |x| < a, p

√

(15.1-4)

It is one of results of Sections 8.3 and 8.4, and (15.1-3) represents a physical parameter describing fracture behaviour of quasicrystals under Mode I loading condition. The physical meaning of the generalized surface tractions hi = Hij nj is clear, but have not been measured so far, we do not considered hi at the physical boundary || (simply assume zero). Therefore, we only obtain the KI , but the stress intensity factor for phason ﬁeld still exits if we do not assume the generalized tractions hi to be || zero. Can we use the stress intensity factor KI in the parallel space (physical space)

15.1

Linear fracture theory of quasicrystals

319

as the physical parameter to control the crack stability (instability) in quasicrystals? This only depends upon tests. || It can be found that KI given by (15.1-3) is not directly related to the material constants of quasicrystals in the case for any self-equilibrium applied stress state (this is similar to that for conventional fracture mechanics for structural materials). Nevertheless, it does not mean that it cannot be used as the physical parameter to govern the crack stability/instability in quasicrystals. Further study is needed in this topic. But the displacement ﬁeld concerning crack is strongly related with material constants (this is also similar to that of conventional fracture mechanics for structural materials), we must distinguish the results for diﬀerent quasicrystal systems. For point groups 5m and 10mm it can be found from (8.3-17) that ⎧ ⎧ |x| > a, ⎪ ⎪ ⎪ ⎨ 0, ⎪ ⎪ ⎨ uy (x, 0) = √ K1 p 1 ⎪ (15.1-5) + a2 − x2 , |x| < a, ⎩ 2 ⎪ 2 M K − R L + M 1 ⎪ ⎪ ⎪ ⎩ wy (x, 0) = 0, |x| < ∞. The strain energy of the system due to the existence of crack is a WI = 2 (σyy (x, 0) ⊕ Hyy (x, 0)(uy (x, 0) ⊕ wy (x, 0))dx 0

=2

0

a

σyy (x, 0)uy (x, 0)dx

πa2 p2 = 4

1 K1 + L+M M K1 − R2

,

(15.1-6)

which is called crack strain energy with the suﬃx “I” to indicate the Mode I crack. It can be found from solution in Sections 8.3 and 8.4 that under the assumption of generalized surface traction hi = Hij nj being free, the crack strain energy is still relevant to both the elastic constant K1 of the phason ﬁeld and the phonon-phason coupling elasticity constant R apart from relevant to the phonon elastic constants L = C12 , M = (C11 − C12 )/2. In Section 8.3, we deﬁne the strain energy release rate (crack growth force) for point groups 5m and 10mm, 1 πap2 K1 1 ∂W1 GI = = + 2 ∂a 4 L+M M K1 − R 2 1 1 K1 = + (KI )2 , (15.1-7) 4 L+M M K1 − R

320

Chapter 15

Fracture theory of quasicrystals

for point groups 5, 5 and 10, 10 quasicrystals GI =

L(K1 + K2 ) + 2(R12 + R22 ) 2 (KI ) , 8(L + M )c

(15.1-8)

c = M (K1 + K2 ) − 2(R12 + R22 ) and for icosahedral quasicrystals GI

c1 =

1 = 2

1 c7 + μ + λ c3

(KI )2 ,

(15.1-9)

R(2K2 − K1 )(μK1 + μK2 − 3R2 ) 2(μK1 − 2R2 )

c3 = μ(K1 − K2 ) − R2 − c7 =

(μK2 − R2 )2 , μK1 − 2R2

c3 K1 + 2c1 R , μK1 − 2R2

which also indicate that the strain energy release rate depends on not only the phonon elastic constants, but also the phason elastic constants and phonon-phason coupling constants. In the above relations, for point groups 5m and 10mm, due to L + M > 0, M K1 − R2 > 0, M + L > 0 and crack energy WI and crack energy release rate GI are all positive, meaningful in physical sense. For point groups 5, 5 and 10, 10, icosahedral and other quasicrystals too. Considering the obvious physical meaning of GI , we recommend GI = GIC

(15.1-10)

as the crack initiation criterion, where GIC is the critical value, a material constant determined experimentally. With the availability of explicit expression GI , the measurement of GIC is convenient, to be discussed in the next section. The above results have been documented in Chapters 8, 9 and the relevant references. With these common features of cracks in quasicrystalline materials, the fundamental of fracture theory for the material can be set up.

15.2

Measurement of GIC

Meng et al[12] measured the fracture toughness for Al65 Cu20 Co15 decagonal quasicrystal by using nonstandard specimen, because the expressions on the stress intensity factors and the energy release rate were not available then. During the characterization of mechanical properties of quasicrystals, similar to conventional structural materials, standard samples are expected to use, such as

15.2

Measurement of GIC

321

cracked samples. Here we recommend three-point bending specimen shown by Fig. 14.8-1 of Appendix of Chapter 14 and compact tension specimen shown by Fig. 15.21 for determining GIC . The corresponding GI expressions are obtained by extending the formula (15.1-7) and others.

Fig. 15.2-1

Compact tension specimen for measuring fracture toughness of quasicrystalline material

15.2.1

Characterization of GI and GIC of three-point bending quasicrystal samples ||

Due to KI independent of material constants, according to fracture mechanics, the stress intensity factor corresponding to the three-point bending specimen as shown in Fig.14.8-1 of Appendix of Chapter 14 is a 3/2 a 5/2 a 1/2 PS 29 KI = − 4.6 + 21.8 W W W BW 3/2 (15.2-1) a 7/2 a 9/2 −37.6 + 38.7 . W W Therefore extension of (15.1-17) leads to

a 3/2 1 K1 1 PS a 1/2 + − 4.6 29 GI = 4 L+M M K1 − R2 W W BW 3/2 a 5/2 a 7/2 a 9/2 2 +21.8 − 37.6 + 38.7 W W W

(15.2-2)

322

Chapter 15

Fracture theory of quasicrystals

for point groups 5m and 10mm, where S is the sample span, B the sample thickness, W the sample width, a the crack length plus the size of the machined notch, and P is the external force (per unit length). Finally, the GIC value can be determined by measuring the critical external force PC . For other quasicrystal systems there are similar results. Characterization of GI and GIC of compact tension quasicrystal sample

15.2.2

It can be found in fracture mechanics that the stress intensity factor of the compact tensile sample as shown in Fig. 15.2-1 is a 3/2 a 1/2 PS − 185.5 29.6 KI = 3/2 W W BW a 5/2 a 7/2 a 9/2 . (15.2-3) + 655.7 − 1017.0 + 638.9 W W W Therefore for point groups 5m and 10mm quasicrystals.

a 1/2 1 P K1 + 29.6 L+M M K1 − R 2 W BW 3/2 a 3/2 a 5/2 a 7/2 a 9/2 2 −185.5 + 655.7 − 1017.0 + 638.9 , W W W W (15.2-4)

1 GI = 4

where B, W , a, and P have the same meanings above. The GIC value can be determined by measuring the critical external force PC . For other quasicrystal systems there are similar results.

15.3

Nonlinear fracture mechanics

In the regime for nonlinear deformation of quasicrystals, the stress intensity factor and energy release rate cannot be used as a fracture parameter, and we must carry out elasto-plastic analysis and the exact solutions of the crack problems have not been obtained except a few of special cases. Fortunately this diﬃcult topic has been discussed in Chapter 14 in detail, now it is needed listing some key points only. Instead stress intensity factor and energy release rate the crack tip opening displacement or Eshelby integral may be a parameter characterizing the mechanical behaviour of crack tip under nonlinear deformation of quasicrystals. These quantities are strongly related to material constants, so the discussion must be done for distinguishing quasicrystal systems.

15.4

Dynamic fracture

323

For one-dimensional hexagonal quasicrystals we have obtained the crack tip sliding displacement for Mode III crack as πτ1 4K2 τc a ln sec (15.3-1) δIII = π(C44 K2 − R32 ) 2τc and for two-dimensional quasicrystals with point groups 5m and 10mm the crack tip opening displacement for Mode I crack is 2σc a 1 πp K1 δI = + ln sec . (15.3-2) π L+M M K1 − R2 2σc For other quasicrystal systems the results are similar. The plastic zone size around the crack tip is πτ1 −1 d = a sec 2τc for one-dimensional hexagonal quasicrystals and πp d = a sec −1 2σc for two-dimensional quasicrystals with point groups 5m and 10mm. And we have fracture criterion for mode I crack δI = δIC .

(15.3-3)

For Mode II and Mode III crack there are similar criterion, which have been discussed in Chapter 14. As pointed out in Chapter 14 the Eshelby integral can also be a fracture parameter, and based on which one can set up a fracture criterion, the full discussion can be found there. The experimental measurement of nonlinear fracture toughness of quasicrystals has been introduced in Section 14.8 of Chapter 14, it does not mention any more.

15.4

Dynamic fracture

As we have known from Chapter 10, the study on dynamics for quasicrystals presents a diﬃcult situation, so the study for dynamic fracture. Nevertheless the Chapter 10 provides some beneﬁcial data for us. By taking the so-called elasto-/hydro-dynamic equation system for quasicrystals, the dynamic crack initiation problem can be solved by ﬁnite diﬀerence method. In linear case the crack dynamic initiation can be described by dynamic stress intensity factor, for some samples the results are listed in Chapter 10, which are sensitive functions of loading type, loading rate and sample geometry including crack geometry. One of results can be seen from the Fig. 15.4-1.

324

Chapter 15

Fig. 15.4-1

Fracture theory of quasicrystals

Normalized dynamic stress intensity factor of central stationary crack

specimen under Heaviside impact loading (for icosahedral Al-Pd-Mn quasicrystals)

With the results we can propose the fracture criterion for dynamic crack initiation ˙ KI (t) = KId (σ),

(15.4-1)

in which KI (t) is the dynamic stress intensity factor evaluated by diﬀerent ap˙ represents the dynamic fracture toughness for initiation of crack proaches, KId (σ) growth of the material and is measured by test, a material constant, but is function of loading rate σ. ˙ While for fast crack propagation/crack arrest problems we have results, e.g. shown in Fig. 15.4-2 for the central crack specimen there is fracture criterion such as

Fig. 15.4-2

Normalized dynamic stress intensity factor of propagating crack with

constant crack speed of central crack specimen (for icosahedral Al-Pd-Mn quasicrystals)

15.5

Measurement of fracture toughness and relevant mechanical parameters of ...

KI (t) KID (V ),

325

(15.4-2)

where KI (t) is also the dynamic stress intensity factor, a computational quantity, while KID (V ) denotes the fracture toughness for fast propagating crack, which must be measured by test, a material constant, but is the function of crack speed V = da/dt. In (15.4-2), the equality sign represents crack propagation condition, and the inequality sign marks the crack arrest condition.

15.5

Measurement of fracture toughness and relevant mechanical parameters of quasicrystalline material

Ref. [12] reported the measurement of fracture toughness of two-dimensional decagonal quasicrystal Al65 Cu20 Co15 as well as three-dimensional icosahedral Al-Li-Cu quasicrystal, the authors used the indentation approach. 15.5.1

Fracture toughness

The material is locally pressed, the crack around the indentation will appear as the compressible stress reaches a certain value, this describes the ability of fracture of the material along the direction of the compressible stress. When the crack length 2a is 2.5 times greater than the length 2c of diagonal of the indentation, then the fracture toughness can be evaluated by + √ c , (15.5-1) KIC = 0.203HV c 3 a in which HV denotes hardness of the material. Their results of measurement for decagonal Al-Ni-Co quasicrystal is √ KIC = 1.0 ∼ 1.2MPa m

(15.5-2)

with HV = 11.0 ∼ 11.5GPa, and for icosahedral Al-Li-Cu quasicrystal is √ KIC = 0.94MPa m,

(15.5-3)

in which HV = 4.10GPa. The values of fracture toughness for general alloys for black metals measured by Ma et al[13] are much greater than the above data, those for aluminum alloys and √ other colour metals are also, e.g., for aluminum 33MPa m refer to Fan[14] . So one ﬁnds that quasicrystals are very brittle. The author of the book minds the indirect measurement for fracture toughness of quasicrystals through indentation perhaps is not so exact, because formula (15.5-1) is empirical, the exact measurement should use the stress intensity factor formula. Due to the high brittleness of the material, maybe it is easy by taking the indentation approach.

326

15.5.2

Chapter 15

Fracture theory of quasicrystals

Tension strength

The tensile strength σc is measured through the formula 2 σc = 0.187Pa/a ,

(15.5-4)

σc = 450MPa

(15.5-5)

σc = 550MPa

(15.5-6)

they obtained before annealing and after annealing for decagonal Al-Ni-Co quasicrystal. The Fig. 15.5-1 shows the SEM morphology of grain interior containing large hole, Fig. 15.5-2 shows diagram of indentation crack under applied load 100g before annealing and after annealing, Fig. 15.5-3 shows the SEM fractograph and fracture feature for decagonal Al65 Cu20 Co15 quasicrystal.

Fig. 15.5-1

Fig. 15.5-2

SEM morphology of grain interior hole

Morphology of cracks near impression under 100g load

References

327

Fig. 15.5-3

SEM fractograph

References [1] Li X F, The defect problems and their analytic solutions in the theory of elasticity of quasicrystals, Dissertation, Beijing Institute of Technology, Beijing, 1999 (in Chinese); Li X F, Fan T Y, Sun Y F. A decagonal quasicrystal with a Griﬃth crack. Phil Mag A, 1999, 79(8): 1943–1952 [2] Fan T Y. Mathematical theory of elasticity and defects of quasicrystals. Advances of Mechanics, 2000, 30(2): 161–174 (in Chinese) [3] Fan T Y, Mai Y W. Theory of elasticity, fracture mechanics and some relevant thermal properties of quasicrystalline materials. Appl Mech Rev, 2004, 57(5): 235– 244 [4] Rudhart C, Gumbsch P, Trebin H R. Crack propagation in quasicrystals. Quasicrystals, Trebin H R. Berlin: Wiely Press, 2003 [5] Fan T Y, Guo L H. The ﬁnal governing equation of plane elasticity of icosahedral quasicrystals. Phys Lett A, 2005, 341(5): 235–239 [6] Zhu A Y, Fan T Y. Elastic analysis of Mode II Griﬃth crack in an icosahedral quasicrystal. Chinese Physics, 2007, 16(4): 1111–1118 [7] Li L H, Fan T Y. Complex variable method for plane elasticity of icosahedral quasicrystals and elliptic notch problem. Science in China, G, 2008, 51(6): 1–8 [8] Fan T Y, Fan L. Plastic fracture of quasicrystals. Phil Mag, 2008, 88(4): 323–335 [9] Fan T Y, Fan L. Relation between generalized Eshelby integral and generalized BCS model and generalized DB model. Chin. Phys. B, in press, 2010 [10] Zhu A Y, Fan T Y. Dynamic crack propagation of decagonal Al-Ni-Co decagonal quasicrystals. J Phys: Condens Matter, 2008, 20(29): 295217 [11] Wang X F, Fan T Y and Zhu A Y. Dynamic behaviour or the icosahedral Al-Pd-Mn quasicrystal with a Griﬃth crack, Chin. Phys. B, 2009, 18(2): 709–714. or Zhu A

328

Chapter 15

Fracture theory of quasicrystals

Y and Fan T Y, Fast crack propagctim in three-dimensional icosahedral Al-Pd-Mn quasicrystals, J. Phys. A, 2008, submitted [12] Meng X M, Tong B Y, Wu Y K. Mechanical behaviour of Al65 Cu20 Co15 quasicrystal. Acta Metallurgy Sinica, 1994, 30(1): 61–64 (in Chinese) [13] Ma D L. Handbook for Parameters of Fracture Mechanics Behaviour of General Black Metals. Beijing: Industrial Press, 1994 (in Chinese) [14] Fan T Y. Foundation of Fracture Theory, Beijing: Science Press, 2003 (in Chinese) [15] Li W, Fan T Y. Study on elastic analysis of crack problem decagonal Al-Ni-Co quasicrystals of point group 10, 10. J Phys: Int. Mod. Phys. Lett. B, 2009, 23(16), 1989–1999.

Chapter 16 Remarkable Conclusion The text of discussion on elasticity of quasicrystals and some applications is ended till the Chapter 15. This work in some extent is a development of crystal (or classical) elasticity. The crystal (or classical) elasticity is a branch of theoretical physics, see e.g. Sommerfeld[1] and Landau and Lifshitz[2] . Since the time of Hooke, Bernoulli, Euler, Navie, Cauchy, Saint-Venant, among others, the classical theory of elasticity has been mature, representing one of the most elegant branches of continuum mechanics, which challenges the applied mathematics and becomes one of sources of many powerful methods of mathematical physics. The study of elasticity of quasicrystals not only enlarges the scope of the classical elasticity, but also strengthens the link between elasticity and other branches of physics, e.g. the elementary excitation theory of condensed matter[3] , symmetry and symmetry-breaking[4] , hydrodynamics[5] etc. It strengthens the link between elasticity and applied mathematics as well, for instance, the partial diﬀerential equations, complex variable function method, group theory, discrete geometry, numerical analysis etc, those have been partly concerned in the text. In particular, the mathematical solutions of elasticity of quasicrystals promote the development of partial diﬀerential equations of higher order, applied complex analysis, the Fourier method, weak solution theory and some numerical methods in continuum mechanics and applied mathematics, which have been touched upon in the previous chapters. The study of the elasticity also promotes the development of defect theories (dislocation theory[6] , fracture mechanics[7] etc), plasticity[8−10] and elasto-/hydro-dynamics[11,12] . It is believed that extensive research on the new area will bring forth useful results of theoretical interest and practical importance. Due to the space limitation, some relevant subjects could not be included in the volume, but they present important meaning, for examples, the eﬀect of phason strain ﬁeld to phase transition of crystal-quasicrystal[13−22] , speciﬁc heat of quasicrystals[23−27] based on the classical Debye[28] theory, evaluating indirectly elastic modulus of phason and phonon-phason coupling[29,30] . The readers may refer to the following references on these interesting subjects.

T. Fan, Mathematical Theory of Elasticity of Quasicrystals and Its Applications © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2011

330

Please number this as Chapter 16

References [1] Sommerfeld A. Vorlesungen ueber theoretische Physik. Vol. V: Elastische Theorie. Wiesbaden: Diederich Verlag, 1952 [2] Landau L D, Lifshitz E M. Theoretical Physics. Vol. VII: Theory of Elasticity. Oxford: Pergamon Press, 1986 [3] Bak P. Symmetry, stability and elastic properties of icosahedral incommensurate crystals. Phys Rev B, 1985, 32(9): 5764–5772 [4] Horn P M, Malzfeldt W, DiVincenzo D P et al. Systematics of disorder in quasiperiodic Material. Phys Rev Lett, 1986, 57(12): 1444–1447 [5] Lubensky T C. Introduction to Quasicrystals. Jaric M V. Boston: Academic Press, 1988. [6] Hu C Z, Wang R H, Ding D H. Symmetry groups, physical property tensors, elasticity and dislocations in quasicrystals. Rep Prog Phys, 2000, 63(1): 1–39 [7] Fan T Y, Mai Y W. Elasticity theory, fracture mechanic and some thermal properties of quasicrystalline materials. Appl Mech Rev, 2004, 57(5), 325–344 [8] Feuerbacher M, Urban K. Platic behaviour of quasicrystalline materials. in: Quasicrystals. Trebin H R, Wiely Press, Berlin, 2003 [9] Fan T Y, Trebin H R, Messerschmidt U et al. Plastic ﬂow coupled with a crack in some one- and two-dimensional quasicrystals. J Phys: Condens Matter, 2004, 16(47): 5229–5240 [10] Fan T Y, Fan L. Plastic fracture of quasicrystals. Phil Mag, 2008, 88(4): 523–535 [11] Fan T Y, Wang X F, Li W et al. Elasto-hydrodynamics of quasicrystals. Phil Mag, 2009, 89(6), 501–512 [12] Wang X F, Fan T Y. Dynamic behaviour of the icosahedral AL-Pd-Mn quasicrystals with a Griﬃth crack. Chin Phys, 2009, 18(2), 709–714 [13] Audier M, Guyot P. Al4 Mn quasicrystal atomic structure, diﬀraction data and Penrose tiling. Phil Mag B, 1986, 53(1): L43–L51 [14] Jaric M V, Gratias D. Extended Icosahedral Structures. Boston: Academic, 1989 [15] Jaric M V, Lundquist S. Proc Adiatico Research Conf Qusicrystals, Singapore: World Scientiﬁc, 1990 [16] Elser V, Henley C L. Crystal and quasicrystal structures in Al-Mn-Si alloys. Phys Rev Lett, 1985, 55(26): 2883–2886 [17] Henley C L, Elser V. Quasicrystal structure of (Al, Zn)49 Mg32 . Phil Mag B, 1986, 53(3): L59–L61 [18] Lubensky T C, Ramaswamy S, Toner J. Hydrodynamics of icosahedral quasicrystals. Phys Rev B, 1985, 32(11): 7444–7452 [19] Lubensky T C, Socolar J E S, Steinhardt P J et al. Distortion and peak broadening in quasicrystal diﬀraction patterns. Phys Rev Lett, 1986, 57(12): 1440–1443

References

331

[20] Li F H. in Crystal-Quasicrystal Transitions. Jacaman M J, Torres M, Amsterdam: Elsevier Sci Publ. 1993 [21] Li F H, Teng C M, Huang Z R et al. In between crystalline and quasicrystalline states. Phil Mag Lett, 1988, 57(1): 113–118 [22] Xie T Y, Fan L, et al. Study on interface of quasicrystal-crystal. Chin. Phys. B, submitted 2009 [23] Fan T Y. A study of speciﬁc heat of one-dimensional hexagonal quasicrystals. J Phys: Condens, Matter, 1999, 11(45): L513–L517 [24] Li C L, Liu Y Y. Phason-strain inﬂuence on low-temperature speciﬁc heat of the decagonal Al-Ni-Co quasicrystal. Chin Phys Lett, 2001, 18(4): 570–573 [25] Li C L, Liu Y Y. Lower-temperature lattice excitation of icosahedral AL-MN-pd quasicrystals. Phys Rev B, 2001, 63(6): 064203 [26] Fan T Y, Mai Y W. Partition function and state equation of point group 12mm dodecagonal quasicrystals. Euro Phys J B, 2003, 31(2): 25–27 [27] Wang J, Fan T Y. An analytic study on speciﬁc heat of icosahedral Al-Pd-Mn quasicrystals. Modern Phys Lett B, 2008, 22(17): 1651–1659 [28] Debye P. Eigentuemlichkeit der speziﬁschen Waermen bei tiefen Temperaturen. Arch de Gen´eve, 1912, 33(7): 256–258 [29] Edagawa K, Giso Y. Experimental evaluation of phonon-phason coupling in icosahedral quasicrystals. Phil. Mag, 2007, 87(1): 77–95 [30] Edagawa K. Phonon-phason coupling in decagonal quasicrystals. Phil Mag, 2007, 87(7): 2789–2798

Major Appendix: On Some mathematical materials Mathematical solutions for individual boundary value problems were given in the previous chapters, in which crack problems are more diﬃcult than dislocation problems. Due to the complexity of boundary conditions for the formers, we must use some special techniques, in which the complex function method collaborating conformal mapping and Fourier analysis collaborating dual integral equations are the most eﬀective procedures. The use of the procedures were exhibited in the previous chapters. Though some appendices in relevant chapters were given in the text for discussing some special problems, we list some necessary additional materials on complex variable functions and dual integral equations in separate two parts in the present appendix. It is interesting in particular, between these two parts there are some close inherent connections. Perhaps these materials are not necessary for readers majoring applied mathematics, but which may be helpful for young scholars and post graduate students who are not majoring applied mathematics.

Appendix I

Outline of complex variable functions and some additional calculations

It is enlightened that Muskhelishvili[1] gave extensive description in detail on complex variable functions in due presentation of elasticity in his classical monograph, this is very beneﬁcial to readers. However there is no the possibility for the present book. We list here a few of points only of the function theory which were frequently cited in the text. These can be referred for readers who are advised to read books of I. I. Privalov[2] and M. A. Lavrentjev and B. V. Schabat[3] for the further details. Other knowledge has been provided in due succession of the text of Chapters 7∼9 and 11. The present contents can also be seen as a supplement in reading the material given in Chapters 7∼9 and 11 if it is needed. The importance of complex variable functions is not only in deriving the solutions by the complex potential formulation but also in dealing with the solutions by integral transforms and dual integral equations to be discussed in the next section of the appendix.

334

A.I.1

Major Appendix: On Some mathematical materials

Complex functions, analytic functions

√ Usually z = x + iy is denoted as a complex variable in which −1 = i, or z = reiθ , # y , the and r = x2 + y 2 , called the modulus of the complex number, θ = arctan x argument of z. Assume f (z) to be a function of one complex variable, or complex function in abbreviation, denote f (z) = P (x, y) + iQ(x, y),

(A.I-1)

in which both P (x, y) and Q(x, y) are functions with real variables, and called the real and imaginary parts, respectively, and marked by P (x, y) = Ref (z),

Q(x, y) = Imf (z).

There is a sort of complex functions called analytic functions (or regular functions, and single-valued analytic functions are called holomorphic functions) which have important applications in many branches of mathematics, physics and engineering. The concepts related with this are discussed as follows. The complex function f (z) is analytic in a given region, this means that it can be expanded in the neighbourhood of any point z0 of the region into a non-negative integer power series (i.e., the Taylor series) of the form f (z) =

∞

an (z − z0 )n ,

(A.I-2)

n=0

in which an is a constant (in general, a complex number). The concept will be used frequently in later calculation. Another deﬁnition of an analytic function is that if the complex function f (z) given in the region, whose real part P (x, y) and imaginary part Q(x, y) are single valued, have continuous partial derivatives of the ﬁrst order, and satisfy CauchyRiemann conditions such as ∂Q ∂P ∂Q ∂P = , =− (A.I-3) ∂x ∂y ∂y ∂x in the region. This kind of function, P and Q are named mutually conjugate harmonic ones. From (A.I-3) it follows that 2 2 ∂ ∂ ∂2 ∂2 2 ∇2 P = P = 0, ∇ Q = 0. + Q = + ∂x2 ∂y2 ∂x2 ∂y 2 This concept will also be often used in the following. An analytic function can also be deﬁned in integral form. Assuming f (z) is an analytic function f (z) in a certain region D at z-plane, and Γ is any simple smooth

Appendix I

Outline of complex variable functions and some additional calculations 335

closed curve (sometime called simple curve for simplicity), we can obtain that f (z) is analytic in the region if f (z)dz = 0. (A.I-4) Γ

The result is known as the Cauchy’s integral theorem (or simply called the Cauchy’s theorem) which is frequently used in some chapters in the text and in subsequent description here. The theory of complex functions proves that the above deﬁnitions are mutually equivalent. A.I.2

Cauchy’s formula

An important result of the Cauchy’s theorem is so-called the Cauchy’s formula, i.e., if f (z) analytic in a single-connected region D + bounded by a closed curve Γ and continuous in D+ + Γ (Fig. A.I-1), then 1 2πi

Γ

f (t) dt = f (z), t−z

(A.I-5)

in which z is an arbitrary point in D+ .

Fig. A.I-1

A ﬁnite region D+

Proof Taking z as the centre, ρ as the radius, make a small circle in D+ . According to Cauchy’s theorem (A.I-4), Γ

f (t) dt = t−z

γ

f (t) dt. t−z

(A.I-6)

As f (z) is analytic in D + and continuous in D+ + Γ , there is a small number ε > 0, for any point t and γ, if ρ is suﬃcient small, such as |f (t) − f (z)| < ε

336

Major Appendix: On Some mathematical materials

and note that |t − z| = ρ , hence lim ε→0

Γ

f (t) dt = t−z

γ

f (z) dt. t−z

(A.I-7)

Just as mentioned in the beginning that f (z) is analytic in D + , the value of the integral f (z) dt t γ −z will not be changed when ρ is reducing. Therefore the limit mark in the left-handside of (A.I-7) can be removed. In addition 2π iθ f (z) dt ρe dθ = 2πif (z). dt = f (z) = f (z) t − z t − z ρeiθ 0 γ γ Based on (A.I-6) and this result, formula (A.I-5) is proved. In formula (A.I-5), if z is taken its values in region D− consisting of the points lying outside Γ (see Fig. A.I-1), then 1 f (t) dt = 0. (A.I-8) 2πi Γ t − z In fact, this is a direct consequence of the Cauchy’s theorem, because in the case the integrand f (ζ)/(ζ − z) as function of ζ is analytic in region D + where ζ denotes the point in the region D+ . Suppose all conditions are the same as those for (A.I-5), then 1 2πi

Γ

f (t) dt = f (0). t−z

(A.I-9)

Proof For simplicity here the proof is given for the case Γ as a circle. Being analytic in the region D + , f (z) may be expanded as non-negative integer power series, in which, taking z0 = 0, such that 1 f (0)z 2 + · · · . 2! 1 at the circle Γ , here The function f (z) in formula (A.I-9) is the value of f¯ z 1 1 1 1 f¯ = f (0) + f (0) + f (0) 2 + · · · z z 2! z f (z) = a0 + a1 z + a2 z 2 + · · · = f (0) + f (0)z +

is an analytic function in D− . From the Cauchy’s formula,

1 dt 1, k = 0, = 0, k > 0, 2πi Γ tk (t − z)

Appendix I

Outline of complex variable functions and some additional calculations 337

such that (A.I-9) is proved. In contrast to above, current function f (z) is analytic in D − (including z = ∞), then

1 f (z) −f (z) + f (∞), z ∈ D− , (A.I-10) dt = f (∞), z ∈ D+ . 2πi Γ t − z The proof of this formula can be oﬀered in the similar manner to that for (A.I-5), but the following points must be noted: (i) The analytic function f (z) in D − (includingz = ∞) may be expanded as the following series: 1 1 f (z) = c0 + c1 + c2 2 + · · · ; z z

− c0 1 0, z ∈ D , (ii) dt = c0 , z ∈ D + . 2πi Γ t − z where c0 = f (∞) = 0. If all conditions are same as that for formula (A.I-10), there exists 1 2πi A.I.3

Γ

f (t) dt = 0. t−z

(A.I-11)

Poles

Suppose a ﬁnite point in z-plane (i.e., z is not a point at inﬁnity), and in the neighbourhood of the point, the function presents the form as follows: f (z) = G(z) + f0 (z),

(A.I-12)

in which f0 (z) is an analytic function in the neighbourhood of point a and G(z) =

A1 Am A0 + + ··· + , z − a (z − a)2 (z − a)m

(A.I-13)

where A1 , A2 , · · · , Am are constants, such that f (z) is called having a pole with order m, z = a is the pole. If a is a point at inﬁnity, f0 (z) in (A.I-12) is regular at point at inﬁnity (i.e., f (t) = c0 + c1 z −1 + c2 z −2 + · · · ), while at z = ∞, G(z) = A0 + A1 z + · · · + Am z m ,

(A.I-14)

then we say that f (z) has a pole of order m at z = ∞. A.I.4

Residue theorem

If the function f (z) has pole a with order m, its integral may be evaluated simply by computing residue.

338

Major Appendix: On Some mathematical materials

What is the meaning of the residue? Suppose f (z) analytic in the neighbourhood of point z = a, but except z = a, and inﬁnite at z = a. In this case the point z = a is named isolated singular point. The residue of the function f (z) at point z = a is the value of the integral 1 f (z)dz, 2πi Γ in which Γ represents any closed contour enclosing point z = a. For a residue we will use the resignation as Res f (a). If z = a is a m-order pole of f (z), its residue may be evaluated from the following formula and dm−1 1 lim m−1 {(z − a)m f (z)} , Resf (a) = (A.I-15) (m − 1)! z→a dz obviously the integral is

f (z)dz = 2πiResf (a). Γ

So evaluation of integrals may be reduced to the calculation of derivatives, it is greatly simpliﬁed. Especially if z = a is a ﬁrst order pole, then Resf (a) = lim (z − a)f (z), z→a

(A.I-16)

the calculating is much simpler. What follows the residue theorem is introduced as: let the function f (z) be analytic in region D and continuous in D+Γ except at ﬁnite isolated poles a1 , a2 , · · · , an , then n f (z)dz = 2πi Resf (ak ), (A.I-17) Γ

k=1

where Γ represents the boundary of region D. Example Calculate the integral ∞ 1 1 e−iωt dω = I 2π −∞ −mω 2 + k

(A.I-18)

in terms of the residue theorem, where m and k are positive constants. Though the integral is a real integral, it is diﬃcult to evaluate because the integration limit is inﬁnite and there are two singular points at the integration path, but is easily completed by using the residue theorem. At ﬁrst we extend the real variable ω to a complex one, i.e., put ω = ω1 + iω2 , and ω1 , ω2 are real variables. At the complex plane ω, a half-circle with origin (0, 0) and radius R → ∞ is taken as an additional integral path referring to Fig. A.I-2. Along the real axis, the integrand

Appendix I

Outline of complex variable functions and some additional calculations 339

# # of the integral has two poles (− k/m, 0) and ( k/m,0), the value of the integral is equal to ∞ 1 1 −iωt dω = I1 = lim + + + + + e , R→∞,r→0 2π −∞ −mω 2 + k CR 1 2 3 C1 C2 (A.I-19)

Fig. A.I-2

Integration path at ω-plane

where the ﬁrst integral in the right-handside of (A.I-19) is carried out on path of the grand half-circle, the second and fourth ones are on the path along the real axis # # # # except intervals (−r − k/m, − k/m + r) and (−r + k/m, k/m + r), the ﬁfth # and sixth ones are on two small half-circle arcs C1 and C2 with origins (− k/m, 0) # and ( k/m, 0) and radius r respectively. Because the integrand in the interior enclosing by the integration path in (A.I-19) is analytic, according to the Cauchy’s theorem (referring to formula (A.I-3)) I1 = 0.

(A.I-20)

Based on the bebaviour of the integrand and the Jordan lemma, the ﬁrst one in the right-handside of (A.I-19) must be zero. So that

lim

+

R→∞,r→0

and

lim

R→∞,r→0

1

+ 2

+ 3

+ 2

3

=0

+ C1

+

1

C2

= I = − lim

r→0

+ C1

. C2

# # At arc C1 : ω + k/m = reiθ1 , dω = ireiθ1 dθ1 , and at C2 : ω − k/m = reiθ2 , dω = ireiθ2 dθ2 . Substituting these into the above integrals and after some simple calculations, we obtain + π k I= # sin t (A.I-21) m m k/m

340

Major Appendix: On Some mathematical materials

This result will be used in the evaluation of some integrals in Chapter 9, which can be seen the Subsection AI.10 of this appendix. A.I.5

Analytic extension

A function f1 (z) analytic at region D1 , if one can construct another function f2 (z) analytic at region D2 , D1 and D2 have common bounding Γ , furthermore, f1 (z) = f2 (z),

z ∈ Γ,

we can say that f1 (z) and f2 (z) are analytic extension to each other, we can also f1 (z) as z ∈ D1 say that function F (z) = f2 (z) as z ∈ D2 analytic at D = D1 + D2 is an analytic extension of f1 (z) as well as f2 (z). A.I.6

Conformal mapping

In the text of Chapters 7∼9, and 11, by using one or several analyic functions which are also named complex potentials, we have expressed the solutions of harmonic, biharmonic, quadruple harmonic, sextuple harmonic equations, and quasi-biharmonic and quasi-quadruple harmonic equations, which is called the complex representation. We can see that only the complex representation is the ﬁrst step for solving some boundary value problems. For some problems with complicated boundaries, one must utilize the conformal mapping to transform the problem into the mapping plane, in this plane the function equations concluded from the boundary conditions are formulated at a unit circle, and the calculation can be put forward, in some cases, exact analytic solutions are available. The so-called conformal mapping is that the complex variable z = x + iy and another one ζ = ξ + iη can be connected by z = ω(ζ),

(A.I-22)

in which ω(ζ) is a single-valued analytic function of ζ = ξ + iη in some region. Except certain points the inversion of mapping (A.I-22) exists. If for a certain region the mapping is single-valued, we say it is a single-valued conformal mapping. In general, the mapping is single-valued, but the inversion ζ = ω −1 (z) is impossible single-valued. It has the following properties: (1) A angle at point z = z0 after the mapping becomes a angle at point ζ = ζ0 , but the both angles have the same value of the argument, and the rotation is either in the same direction, this is the ﬁrst kind of conformal mapping, or in counter direction, which is the second kind of conformal mapping.

Appendix I

Outline of complex variable functions and some additional calculations 341

(2) If ω(ζ) is analytic and single-valued in region Ω and transforms the region into region D, then the inversion ζ = ω −1 (z) is analytic and single-valued in region D and maps D onto Ω . (3) If D is a region and c is a simple closed curve in it, and its interior belongs to D, and if ω −1 (z) is analytic, and maps c onto a closed curve γ at Ω region bilaterally single-valued, then ω(ζ) is analytic and single-valued in the region and maps D onto the interior of Ω . In the text we mainly used the following two kinds of conformal mapping, i.e., (1) Rational function conformal mapping, e.g. ω(ζ) =

c + a0 + a1 ζ + · · · + an ζ n ζ

or ω(ζ) = Rζ + b0 + b1

(A.I-23)

1 1 + · · · + bn n , ζ ζ

(A.I-24)

in which, c, a0 , a1 , · · · , an , R, b0 , b1 , · · · , bn are constants. These mappings can be used in studying inﬁnite region with a crack at physical plane onto the interior of unit circle at mapping plane. In the monograph of Muskhelishvili[3] , he postulated that his method is only suitable for this kind of mapping functions. Fan[4] extended it to transcendental mapping functions and achieved exact analytic solutions for crack problems for complicated conﬁguration. (2) Transcendental functions, e.g. H (1 + ζ)2 ω(ζ) = ln 1 + π (1 − ζ)2

(A.I-25)

and

πa 6 5# 2W 1 − ζ 2 tan arctan − a, (A.I-26) π 2W which can be used to transform a ﬁnite specimen with a crack onto the interior of unit circle at mapping plane, where H, W and a represent sample sizes and crack size. ω(ζ) =

A.I.7

Additional derivation of solution (8.2-19)

Formula (8.2-19) is obtained from the integral of (8.2-6) in which the conformal mapping ω(ζ) is given by (8.2-16). The diﬃculty of the calculation lies in the integral path is a part of circle rather than whole circle. Substitution (8.2-16) into the right-handside of (8.2-6) yields p F (ζ) = − 2πi

1 −1

ω(σ) p dσ = − (σ − ζ)2 2πi

1

−1

ω (σ) dσ, σ−ζ

342

Major Appendix: On Some mathematical materials

where ω (σ) = −

4Hα(1 − β) 1 − σ (1 + σ)2 . 2 π 1 + σ [(1 + σ) + α(1 − σ 2 )2 ][(1 + σ)2 + βα(1 − σ)2 ]

Substituting eiϕ by (1 + ix)/(1 − ix) yields that 1−σ = −2x, 1+σ

dσ =

2i dx, (1 − ix)2

(1 + σ)2 =

4 , (1 − ix)2

1 1 − ix (1 − ix)[1 − ζ − ix(1 + ζ)] 1 − ζ − x2 (1 + ζ) − 2ix − . = = σ−ζ 1 − ζ + ix(1 + ζ) (1 − ζ)2 + x2 (1 + ζ)2 (1 − ζ)2 + x2 (1 + ζ)2 So that F (ζ) = − =− =

pHα(1 − β) π2

4pHα(1 − β) π2

1 −1

1 0

(1 −

ix[1 − ζ − x2 (1 + ζ) − 2ix] dx − βαx2 )[(1 − ζ)2 + x2 (1 + ζ)2 ]

αx2 )(1

x2 dx (1 − αx2 )(1 − βαx2 )[(1 − ζ)2 + x2 (1 + ζ)2 ]

α(1 − β)(1 − ζ 2 ) 2pH 2 2 π [α(1 − ζ) + (1 + ζ)2 ][βα(1 − ζ)2 + (1 + ζ)2 ] 1+ζ 1+ζ × arctan − arctan 1−ζ −1 − ζ √ √ √ √ 4pH α arctanh α βα arctanh α − . − 2 π α(1 − ζ)2 + (1 + ζ)2 γα(1 − ζ)2 + (1 + ζ)2 (A.I-27)

In the last step the evaluation is used of the Mathematica 3.0[19] . By considering √ √ 1+ α √ = 2 arctanh α, A = ln 1− α √ √ 1 + βα √ M = ln = 2 arctanh βα, 1 − βα 1+ζ 1+ζ i−ζ i arctan = − arctan = ln , −1 + ζ 1−ζ 2 1 − iζ then (A.I-27) is just the formula (8.2-19). A.I.8

Additional derivation of solution (11.3-53)

In example 3 of Section 11.3, the calculation is quite lengthy, here we provide some details on the evaluation. As an example, we can show the derivation on function Φ4 (ζ), which is Φ4 (ζ) = d1 (X + iY ) ln ζ + Bω(ζ) + Φ4∗ (ζ),

Appendix I

Outline of complex variable functions and some additional calculations 343

in which the ﬁrst two terms are known referring to the text, and the single-valued analytic function Φ4∗ (ζ) satisﬁes the following boundary condition: Φ4∗ (σ) + Φ3∗ (σ) +

ω(σ) ω (σ)

· Φ4∗ (σ) = f0 ,

where f0 =

i 32c1

(Tx + iTy )ds − (d1 − d2 )(X + iY ) ln σ −

ω(σ) ω (σ)

· d1 (X − iY ) · σ

−2Bω(σ) − (B − iC )ω(σ) (A.I-28) and (referring to Fig. 11.3-4) Tx = −p cos(n, x),

Ty = −p cos(n, y) at z1 M z2 , ⎧ ⎪ ⎪ ⎪ ⎨ ipdz, z1 M z2 , (Tx + iTy )ds = ⎪ ⎪ ⎪ ⎩ 0, z Nz ,

2

(A.I-29)

1

(Tx + iTy )ds = ip(z1 − z2 ).

X + iY =

Multiplying both sides of the above equation by

1 dσ , and integrating around 2πi σ − ζ

the unit circle, we have Φ4∗ (σ) ω(σ) Φ4∗ (σ) Φ3∗ (σ) f0 1 1 1 1 dσ + dσ + dσ = dσ, 2πi γ σ − ζ 2πi γ ω (σ) σ − ζ 2πi γ σ − ζ 2πi γ σ − ζ (A.I-30) in which according to the Cauchy’s integral formula (referring to formula (A.I-5)) there is Φ4∗ (σ) 1 dσ =Φ4∗ (ζ) 2πi γ σ − ζ and in terms of analytic extension principle and the Cauchy’s theorem (referring to Sections A.I-5 and A.I-1) 1 2πi

γ

ω(σ) Φ4∗ (σ) =0 ω (σ) σ − ζ

γ

Φ3∗ (σ) dσ = const. σ−ζ

and according to formula (A.I-9) 1 2πi

344

Major Appendix: On Some mathematical materials

So that (A.I-30) reduces to 1 2πi

Φ4∗ (ζ) =

γ

f0 dσ + const σ−ζ

And substituting (A.I-28) and (A.I-29) into the right-handside yields 1 2πi

γ

pz2 σ2 dσ m dσ + σ σ−ζ 2πi σ1 σ − ζ σ1 ln σ p(z1 − z2 ) 1 dσ + 2πi 2πi γ σ − ζ p(z1 − z2 ) 1 σ 2 + m dσ − , 2πi 2πi γ 1 − mσ 2 σ − ζ

f0 pR0 dσ = σ−ζ 2πic1

σ2

σ+

in which

σ2

σ1 σ2 σ1

m σ2 m σ2 − ζ m dσ + ζ+ = σ2 − σ1 − ln ln , σ+ σ σ−ζ ζ σ1 ζ σ1 − ζ

dσ σ1 − ζ = ln . σ−ζ σ2 − ζ

In accordance with the Cauchy’s theorem (referring to formula (A.I-4)) 1 2πi

γ

σ 2 + m dσ = 0, 1 − mσ 2 σ − ζ

because the integrand is single-valued analytic function in the region outside the unit circle γ. The remaining term is 1 ln σ dσ. I(ζ) = 2πi γ σ − ζ For calculating it we consider dI 1 ln σ 1 1 = dσ = − ln σd dζ 2πi γ (σ − ζ)2 2πi γ σ−ζ ln σ σ=σ1 1 1 1 exp i(ϕ1 + 2π) 1 dσ + =− =− ln − σ − ζ σ=σ 2πi γ σ(σ − ζ) 2πi σ1 − ζ exp(iϕ1 ) ζ 1

2πi 1 = − . σ1 − ζ ζ So that I(ζ) = ln(σ1 − ζ) − ln ζ + const.

Appendix I

Outline of complex variable functions and some additional calculations 345

Hence function Φ4∗ (ζ) is determined so the function Φ4 (ζ) in which the constant term is omitted: 1 p σ2 − ζ mR0 σ2 · + z ln · − ln + z1 ln(σ1 − ζ) − z2 ln(σ2 − ζ) Φ4 (ζ) = 32c1 2πi ζ σ1 σ1 − ζ +ip(d1 − d2 )(z1 − z2 ) ln ζ, which is just the ﬁrst formula of (11.3-53), others can be similarly derived. In the derivation the classical work of Muskhelishvili[1] is referred. 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

The elasticity solution (11.3-53) based on complex analysis can be used to solve the present problem. The generalized Dugdale-Barenblatt model or generalized cohesive force model for plane elasticity of the quasicrystals can be reduced to solve the equation ∇2 ∇2 ∇2 ∇2 G = 0

(A.I-31)

under boundary conditions ⎧ # ⎪ x2 + y 2 → ∞, σyy = p, σxx = σxy = 0, Hxx = Hyy = Hxy = Hyx = 0 ⎪ ⎨ σyy = σxy = 0, Hyy = Hyx = 0 y = 0, |x| < a, ⎪ ⎪ ⎩ σyy = σc , σxy = 0, Hyy = Hyx = 0 y = 0, a < |x| < a + d, (A.I-32) which can be decomposed into two cases, among them one is ⎧ # x2 + y 2 → ∞, ⎨ σxx = σxy = σyy = 0, Hxx = Hyy = Hxy = Hyx = 0 σ = σxy = 0, Hyy = Hyx = 0 y = 0, |x| < a, ⎩ yy a < |x| < a + d σyy = σc , σxy = 0, Hyy = Hyx = 0 (A.I-33) and another #

σyy = p, σxx = σxy = 0, Hxx = Hyy = Hxy = Hyx = 0 x2 + y 2 → ∞, y = 0, |x| < a + d. σyy = σxy = 0, Hyy = Hyx = 0 (A.I-34) The solution of problem (A.I-31), (A.I-33) can be obtained from (11.3-53), i.e., if put m = 1, R0 = (a + d)/2 then the elliptic hole reduced to a Griﬃth crack with half-length (a + d). In Fig. 11.3-3, let z1 = (a + d, +0), z2 = (a, +0), from (11.3-53), we can obtain a solution, similarly put z1 = (a + d, −0), z2 = (a, −0), z1 = (−a − d, +0), z2 = (−a, +0), z1 = (−a − d, −0), z2 = (−a, −0)

346

Major Appendix: On Some mathematical materials

respectively from (11.3-53), one can ﬁnd other corresponding three solutions, by superposing which one can obtain solution ⎧ 1 1 σc (a + d)ϕ2 1 σc (1) ⎪ ⎪ · − Φ4 (ζ) = · · ⎪ ⎪ 32c π ζ 32c 2πi 1 1 ⎪ ⎪ ⎪ ⎨ σ −ζ σ2 + ζ (ζ − σ2 )(ζ + σ2 ) 2 z ln + ln − a ln ⎪ σ2 − ζ σ2 + ζ (ζ + σ2 )(ζ − σ2 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Φ (1) (ζ) = 1 · σc (a + d)ϕ2 · 2ζ − 1 · σc a ln (ζ − σ2 )(ζ + σ2 ) , ⎩ 3 32c1 π ζ 2 − 1 32c1 2πi (ζ + σ2 )(ζ − σ2 ) (A.I-35) iϕ where σ = e represents the value of ζ at the unit circle in the mapping plane and σ2 = eiϕ2 , a = (a + d) cos ϕ2 . And the solution of problem (A.I-31), (A.I-34) is known, i.e., ⎧ 1 p 1 (2) ⎪ ⎪ ⎨ Φ4 (ζ) = − 32c1 2 (a + d) ζ , ⎪ ζ p ⎪ ⎩ Φ3(2) (ζ) = − . (a + d) 32c1 (ζ 2 − 1)

(A.I-36)

(1)

The superposition of (A.I-35) and (A.I-36) gives the total solution Φ4 (ζ) = Φ4 (ζ)+ (2) (1) (2) Φ4 (ζ), Φ3 (ζ) = Φ3 (ζ) + Φ3 (ζ), for example the ﬁrst term of Φ4 (ζ) is −

1 p σc (a + d)ϕ2 1 1 1 · (a + d) + · 32c1 2 ζ 32c1 π ζ

(A.I-37)

and Φ2 (ζ) has not been listed here because it is too lengthy. So the stresses and displacements are determined already. In addition, we know that σij , Hij ∼

Φ (ζ) ω (ζ)

(A.I-38)

here Φ(ζ) means Φ4 (ζ) or Φ3 (ζ), and ω (ζ) ∼ 1/(1 − ζ 2 ). From Section 14.4 in the text we know that there is no stress singularity at the Dugdale-Barenblatt crack tip, this fact and conjunct with (A.I-37) and (A.I-38) require that the value of formula (A.I-37) must be zero, which leads to (14.4-6). Considering this the ﬁnal version of Φ4 (ζ) is σ2 − ζ σ2 + ζ (ζ − σ2 )(ζ + σ2 ) 1 σc z ln + ln − a ln . (A.I-39) Φ4 (ζ) = − · 32c1 2πi σ2 − ζ σ2 + ζ (ζ + σ2 )(ζ − σ2 ) And the displacement at the crack surface presents the form uy (x, 0) = (128c1 c2 − 64c3 )Im(Φ4 (ζ))ζ=σ .

(A.I-40)

Appendix I

Outline of complex variable functions and some additional calculations 347

After some calculation we ﬁnd that sin(ϕ2 −ϕ) (4c1 c2 −2c3 ) σc (a + d) (sin ϕ2 −sin ϕ) · cos ϕ ln −cos ϕ2 ln , · uy (x, 0) = c1 2π sin(ϕ2 +ϕ) (sin ϕ2 +sin ϕ) (A.I-41) so the crack tip opening displacement δI = CT OD = lim 2uy (x, 0) = lim 2uy (x, 0) x→l

=

(8c1 c2 − 4c3 )σc a ln sec c1 π

ϕ→ϕ2

π σ (∞) 2 σc

(A.I-42) ,

in which the constants c1 , c2 , c3 are deﬁned in Section 11.3, so the solution holds for point group 10, 10 decagonal quasicrystals as well. When we assume R1 = R, R2 = 0 in the equation (A.I-42), δI will be the corresponding solution of point group 10mm decagonal quasicrystals, i.e., 2σc a 1 π σ (∞) K1 + ln sec , δI = CT OD = π L+M M K1 − R2 2 σc which is just the (14.4-7). If let K1 = R = 0, L = λ, M = μ in above formula then it exactly reduces to the classical Dugdale solution holding for engineering material (or structural material) including crystalline material (referring to Section 14.4). The more details can be found in article given by [25]. A.I.10

On the calculation of integral (9.2-14)

In formulas (9.2-14) y > 0 result in the integrals being convergent. We let ξ to extend to complex number ξ1 + iξ2 , take integration path at complex ξ-plane similar to Fig. A.I-2, and by physical consideration k(K1 − K2 ) > 0, μ(K1 − K2 ) − R2 > 0, and k = μ(c) /h, then one can ﬁnd the integrand of (9.2-14) in the interior of the region bounded by the integration path is analytic except poles (1)

ξ1 =

R2

k(K1 − K2 ) k(K1 − K2 ) (2) < 0, ξ1 = − 2 >0 − μ(K1 − K2 ) R − μ(K1 − K2 )

(A.I-43)

at real axis ξ1 , and by a generalized Jordan lemma the integral along the big halfcircle is zero. # # (1) (2) If in Fig. A.I-2 we put ω = ξ, ω1 = ξ1 , ω2 = ξ2 , k/m = ξ1 , − k/m = ξ1 , according to the additional integration path at complex ξ-plane, through the similar manner for evaluating integral (A.I-18) then we can obtain results (9.2-15) and (9.2(1) (2) 16) respectively where ξ1 , ξ1 are deﬁned by (A.I-43).

348

Major Appendix: On Some mathematical materials

Appendix II

Dual integral equations and some additional calculations

Application of integral transforms in elasticity of quasicrystals may be eﬀective and more widely used than that of complex variable function method, and has got much analytic solutions, see e.g. Li[20] , Zhou and Fan[21] , Zhou[22] , Zhu and Fan[23,24] etc, due to the space limitation, the results have not been introduced. The application of the method to crack problems often leads to some dual integral equations, so it is beneﬁcial for discussion about this. A.II.1

Dual integral equations

It is well-known the Fourier transform or Hankel transform are very useful tool in solving partial diﬀerential equations which has been shown in Chapters 7∼9 though the introduction is very limited. For non-harmonic or non-multiharmonic equations the complex potential method is not eﬀective, we have to use the Fourier transform, Hankel transform, Mellin transform or others. After the transform, the boundary value problems of the dislocations are reduced to some algebraic equations to solve (this is relatively simpler), while those of the cracks are concluded for solving the following dual integral equations ⎧ ∞ ⎪ ⎪ y α f (y)Jν (xy)dy = g(x), 0 < x < 1, ⎨ 0 (A.II-1) ∞ ⎪ ⎪ ⎩ f (y)Jν (xy)dy = 0, x>1 0

or ⎧ ∞ n ⎪ αj ⎪ y ajk fj (y)Jνj (xy)dy = gj (x), 0 < x < 1, ⎪ ⎪ ⎨ 0 k=1 ∞ n ⎪ ⎪ ⎪ ⎪ ajk fj (y)Jνj (xy)dy = 0, x > 1, ⎩ 0

j = 1, 2, · · · , n (A.II-2)

k=1

or ⎧ ∞ ∞ ⎪ ⎪ g1 (ξ1 , ξ2 , s, x1 , x2 )f (ξ1 , ξ2 , s)Jα (ξ1 x1 )Jβ (ξ2 x2 )dξ1 dξ2 = h(x1 , x2 , s), ⎪ ⎪ ⎪ 0 0 ⎪ ⎪ ⎪ ⎨ (x1 , x2 ) ∈ Ω1 , ∞ ∞ ⎪ ⎪ ⎪ g2 (ξ1 , ξ2 , s, x1 , x2 )f (ξ1 , ξ2 , s)Jα (ξ1 x1 )Jβ (ξ2 x2 )dξ1 dξ2 = 0, ⎪ ⎪ ⎪ 0 0 ⎪ ⎪ ⎩ (x1 , x2 ) ∈ Ω2 . (A.II-3)

Appendix II

Dual integral equations and some additional calculations

349

Among them the equations (A.II-1) are the simplest ones, which will be discussed in the following only. The equations (A.II-2) are dealt with multi-unknown functions, and (A.II-3) are the two-dimensional dual integral equations, these two kinds of dual integral equations are more complicated. In equations (A.II-1), f (x) is unknown function to be determined, g(x) is known one, α and ν are constants, Jν (xy) the ﬁrst kind Bessel function of ν order. Titchmarsh[6] and Busbridge[7] gave the analytic solution of the equations. Various authors[8∼16] discussed the solutions with diﬀerent methods. Here only the procedure of Refs. [6,7] is introduced. Titchmarsh[6] gave formal solution for the case α > 0. Busbridge extended the discussion to the case α > −2 and gave proof for the existence of the solution. The solution is given through a complex integral as follows k+i∞ Γ(1/2 + ν/2 + s/2) 1 ψ(s)x−s ds, 2s−α (A.II-4) f (x) = 2πi k−i∞ Γ(1/2 + ν/2 + α/2 − s/2) in which s = σ + iτ and C+i∞ 1 Γ(1/2 + ν/2 − α/2 + w/2) g(α + 1 − w) · dw, ψ(s) = 2πi C−i∞ Γ(1/2 + ν/2 + w/2) w−s

(A.II-5)

where w = u + iv (in which v represents the imaginary part of complex variable w, do not confuse with ν—the suﬃx of the Bessel function, which represents the order of the Bessel function), σ < u and g(α + 1 − w) =

1

g(x)xα−w dx

0

in above formulas Γ(x)represent the Euler gamma function. The solution (A.II-4) holds for both α > 0 and α > −2. For α > 0, the solution can be expressed by real integral as 1 (2x)1−α/2 1 1+α/2 f (x) = μ Jν+α/2 (μx)dμ g(ρμ)ρν+1 (1 − ρ2 )α/2−1 dρ Γ(α/2) 0 0 (A.II-4 ) and for α > −2, which is in form 1 2−α/2 x−α y ν+1 (1 − y2 )α/2 g(y)dy μ1+α/2 Jν+α/2 (x) f (x) = Γ(1 + α/2) 0 (A.II-4 ) 1 1 + 0

y α+1 (1 − y 2 )α/2 dy

0

(xu)2+α/2 g(yu)Jν+1+α/2 (xu)du .

1 < ν + 1, the Mellin transforms of g(x) 2 and f (x) exist, the latter is analytic in the strip region −ν < Re s = σ < α and Theorem

If α > −2, − ν − 1 < α −

350

Major Appendix: On Some mathematical materials σ−α+ε

has the order O(|t| ) (ε > 0, t → ∞), where s = σ + iτ is the Mellin transform parameter, then equations (A.II-1) have one and only one solution (A.II-4). Proof Because the strict proof given by Busbridge [7] is very lengthy, we cannot quote its all details here, instead, only a rough outline of the proof is ﬁgured out in the following. One can ﬁnd that in the proof a quite lot of complex variable function knowledge are used, this seems that the theory on dual integral equations presents inherent connection with complex analysis. So the Section I of this appendix is helpful for the present discussion too. 1 At ﬁrst assume that 0 < α < 2, −ν − 1 < α − < ν + 1 and the Mellin transform 2 of f (x) ∞ f (s) = f (x)xs−1 dx, s = σ + iτ 0

is analytic in region −ν < σ < α, and assume as ε > 0, as t → ∞ has order −α+ε ). O(|t| According to the deﬁnition, the Mellin transform of y α Jν (xy) is ∞ 2α+s−1 Γ(α/2 + ν/2 + s/2) J α (s) ≡ [yα Jν (xy)]y s−1 dy = α+s . (A.II-6) x Γ(1 − α/2 + ν/2 − s/2) 0 Recall that s = σ + iτ . By using the notations of relevant Mellin transform the left-handside of the ﬁrst and second equations (A.II-1) become ∞ C+i∞ 1 α y f (y)Jν (xy)dy = f (s)J α (1 − s)ds, 2πi C−i∞ 0 ∞ C+i∞ 1 f (y)Jν (xy)dy = f (s)J 0 (1 − s)ds 2πi C−i∞ 0 and substituting (A.II-6) into the above formulas yields C+i∞ α−s 1 2 Γ(1/2 + α/2 + ν/2 − s/2) f (s)ds = g(x), 0 < x < 1, 2πi C−i∞ x1−s Γ(1/2 − α/2 + ν/2 + s/2) C+i∞ α−s 2 Γ(1/2 + ν/2 − s/2) 1 f (s)ds = 0, x > 1. 2πi C−i∞ Γ(1/2 + ν/2 + s/2) Put 2α−s Γ(1/2 + ν/2 + s/2) f (s) = ψ(s). (A.II-7) Γ(1/2 + α/2 + ν/2 − s/2) Then the above equations reduce to ⎧ C+i∞ Γ(1/2 + ν/2 + s/2) 1 ⎪ ⎪ ψ(s)xs−1−α ds = g(x), 0 < x < 1, ⎪ ⎨ 2πi C−i∞ Γ(1/2 + ν/2 − α/2 + s/2) C+i∞ ⎪ ⎪ 1 Γ(1/2 + ν/2 − s/2) ⎪ ⎩ ψ(s)xs−1 ds = 0, x > 1. 2πi C−i∞ Γ(1/2 + ν/2 + α/2 − s/2) (A.II-8)

Appendix II

Dual integral equations and some additional calculations

351

Multiplying xα−w to the ﬁrst one of (A.II-8), where w = u + iv and σ − u > 0, then integrating over (0, 1) to x and C+i∞ ds Γ(1/2 + ν/2 + s/2) 1 ψ(s) = g(α − w + 1), (A.II-9) 2πi C−i∞ Γ(1/2 + ν/2 − α/2 + s/2) s−w where u < C and

g(α − w + 1) =

1

g(x)xα−w dx.

0

The left-handside of equation (A.II-9) is analytic everywhere in the strip zone −ν < σ < α −α+ε

). If we move the inteexcept the simple pole s = w, and behaves order O(|t| gration path from σ = C to σ = C < u, see Fig. A.II-1, based on the Cauchy’s integral formula (referring to formula (A.I-5)), C +i∞ Γ(1/2 + ν/2 + s/2) 1 ds ψ(s) 2πi C −i∞ Γ(1/2 + ν/2 − α/2 + s/2) s−w = g(α − w + 1) −

Γ(1/2 + ν/2 + w/2) ψ(w). Γ(1/2 + ν/2 − α/2 + w/2)

This translation of the integration line corresponds to form a closed region, the value of the integral around the closed region is just equal to the second term of the above formula including the sign of the term. Because the left-handside is analytic as u > C , so the right-handside. In addition, ψ(w) −

Γ(1/2 + ν/2 − α/2 + w/2) g(α − w + 1) Γ(1/2 + ν/2 + α/2)

is analytic for the case 1 1 1 1 + ν − α + w = 0, −1, −2, · · · . 2 2 2 2

Fig. A.II-1

The integration path in s = σ + iτ -plane

(A.II-10)

352

Major Appendix: On Some mathematical materials

Integrating function (A.II-10) along a big rectangle whose corners are C − iT,

C + iT,

−T + iT,

and one can ﬁnd the absolute values " " " −T +iT " " " ", " " C+iT "

−T − iT,

T > |v|

of the integrals " " " " " −T −iT " " C−iT " " " " " " ", " " " −T +iT " " −T −iT "

−α/2+ε

have order O(|t| ), and the value of ε can be always taken less than α/2, then they can tend to zero as T → ∞. According to the Cauchy’s integral theorem (referring to formula (A.I-4)) C+i∞

ds Γ(1/2 + ν/2 − α/2 + w/2) 1 g(α − s + 1) = 0, u < C. ψ(s) − 2πi C−i∞ Γ(1/2 + ν/2 + s/2) s−w (A.II-11) −w to the second one of equations (A.II-8), where σ − w < 0, Similarly, multiplying x then integrating to x over (1, ∞) C +i∞ 1 ds Γ(1/2 + ν/2 − s/2) ψ(s) = 0. (u > C ) 2πi C −i∞ Γ(1/2 + ν/2 + α/2 − s/2) s−w Moving the integration path and ﬁnd that C+i∞ 1 ds ψ(w) = ψ(s) . (u < C) 2πi C−i∞ s−w

(A.II-12)

Comparing (A.II-11) and (A.II-12), we ﬁnd (A.II-5), substituing it into (A.II-7), then taking inversion of the Mellin transform yields (A.II-4), the theorem is proved. In above g(α − s + 1) =

1

g(ξ)ξ α−s dξ.

0

In addition, one has

1 1 η s−w−1 dη. = s−w 0 If exchange integral order, then (A.II-5) may be rewritten as −s C+i∞ 1 1 Γ(1/2 + ν/2 − α/2 + s/2) ξ 1 ψ(ω) = g(ξ)ξ α dξ η −w−1 dη× ds, 2πi C−i∞ Γ(1/2 + ν/2 + s/2) η 0 0 in which the integration[26,27] −s C+i∞ 1 Γ(1/2 + ν/2 − α/2 + s/2) ξ ds 2πi C−i∞ Γ(1/2 + ν/2 + s/2) η ⎧ 2 ⎨ ξ 1+ν−α (η 2 − ξ 2 )α/2−1 η 1−ν , η ξ, = Γ(α/2) ⎩ 0, 0 < η < ξ.

Appendix II

Dual integral equations and some additional calculations

So that 2 ψ(w) = Γ(α/2)

1

g(ξ)ξ 0

1+ν

dξ

1

353

η −w−ν (η 2 − ξ 2 )α/2−1 dη.

ξ

By exchanging the integral order (this means the integration zone at ξη-plane is changed), we may ﬁnd that 1 1 2 η −w−ν dη g(ξ)ξ 1+ν (η 2 − ξ 2 )α/2−1 dξ ψ(w) = Γ(α/2) 0 0 1 1 2 α−ω η dη g(ξ)ξ 1+ν (1 − ξ 2 )α/2−1 dξ. = Γ(α/2) 0 0 Substituting it into (A.II-4) yields 1 1 2 η α dη g(ηζ)ζ 1+ν (1 − ζ 2 )α/2−1 dζ f (x) = Γ(α/2) 0 0 C+i∞ Γ(1/2 + ν/2 + s/2) 1 × 2s−α (xη)−s ds. 2πi C−i∞ Γ(1/2 + ν/2 + α/2 − s/2) By utilizing inversion of the Mellin transform[26,27] , 1 2πi

C+i∞ C−i∞

2s−α

Γ(1/2 + ν/2 + s/2) (at)−s ds = 2−α/2 (at)1−α/2 Jν+α/2 (at), Γ(1/2 + ν/2 + α/2 − s/2)

then one ﬁnds (AII.4 ). For the case α > −2, the derivation is similar. For some details reader can refer to Busbridge[7] . In the following some examples are discussed in detail, which are the dual integral equations appearing in Chapters 8 and 9 respectively, where only the solutions were listed without derivation detail. A.II.2

Additional derivation on the solution of dual integral equations (8.3-8) ∞ ⎧ 2 ⎪ ⎪ [C(ξ) |ξ| − 6D(ξ)] cos(ξx)dξ = −p, 0 < x < a, ⎪ ⎪ d11 0 ⎪ ⎪ ⎪ ⎪ ∞ ⎪ ⎪ ⎪ ⎪ ξ −1 [C(ξ) |ξ| − 6D(ξ)] cos(ξx)dξ = 0, x > a, ⎨ 0 (A.II-13) ∞ ⎪ 2 ⎪ ⎪ D(ξ) cos(ξx)dξ = 0, 0 < x < a, ⎪ ⎪ d12 0 ⎪ ⎪ ⎪ ⎪ ∞ ⎪ ⎪ ⎪ ⎩ ξ −1 D(ξ) cos(ξx)dξ = 0, x > a. 0

354

Major Appendix: On Some mathematical materials

It is evident that the second pair of dual integral equations (A.II-6) has the zero solution, i.e., D(ξ) = 0, and we only consider the ﬁrst pair in (A.II-13), which is ∞ ⎧ 2 ⎪ ⎪ C(ξ) |ξ| cos(ξx)dξ = −p, 0 < x < a, ⎨ d11 0 (A.II-14) ∞ ⎪ ⎪ −1 ⎩ ξ C(ξ) |ξ| cos(ξx)dξ = 0, x > a. 0

Because

cos(ξx) =

πξx 2

1/2 J−1/2 (ξx)

and denoting ξ

1/2

C(ξ) = f (ξ),

η = aξ,

x ρ= , a

g(ρ) = a

πad11 2ρ

then (A.II-14) is reduced to ∞ ⎧ 2 ⎪ ⎪ ηf (η)J−1/2 (ηρ)dη = g(ρ), 0 < ρ < 1, ⎨ d 11 0 ∞ ⎪ ⎪ ⎩ f (η)J−1/2 (ηρ)dη = 0, ρ > 1,

1/2 p,

(A.II-14 )

0

which becomes one of the standard dual integral equations shown by (A.II-1) with α = 1,

1 ν=− , 2

g(ρ) = g0 ρ−1/2 ,

g0 = const = a(πad11 )1/2 p.

In this case it is very easy to calculate the solution of dual integral equations (A.II-14) (or (A.II-14 )) by formulas (A.II-4) and (A.II-5), but the key step is choosing the integration path. In previous introduction on Titchmarsh-Busbridge solution, we mentioned that it must require that −ν > k > α, −ν < C < α and k < C. At present case ν = −1/2, α = 1 such that 1/2 < k < 1 and 1/2 < C < 1. The concrete calculation is as below. Due to 1 1 g0 g(α + 1 − t) = g(ρ)ρα−t dρ = g0 ρ−1/2 ρ1−t dρ = , 3/2 − t 0 0 where t = t1 + it2 represents a complex variable, and requires that t1 < 3/2. Substituting the relevant data and the above result into (A.II-5), we have C+i∞ 1 Γ(−1/4 + t/2) 1 1 ψ(s) = g0 dt, 2πi C−i∞ Γ(1/4 + t/2) t − s 3/2 − t the integration path is shown in Fig. A.II-2. In this case the integrand has only one pole at point t = 3/2 of ﬁrst order. According to the formula for evaluating residue (A.I-15) the above integral is very easy obtained as

Appendix II

Dual integral equations and some additional calculations

ψ(s) = g0

√ Γ(1/2) π 1 = g0 , Γ(1) 3/2 − s 3/2 − s

355

(A.II-15)

√ 1 whereas Γ(1) = 1, Γ = π. Substituting the result into formula (AII.4) leads 2 to k+i∞ s−α √ 1 2 Γ(1/4 + s/2) −s η ds. f (η) = g0 π (A.II-16) 2πi k−i∞ Γ(1/4 − s/2) In terms of the inversion of the Mellin transform[26,27] , 1 2πi

2s−λ

k−i∞

−λ/2

=2

k+i∞

1−λ/2

(βη)

Γ(1/2 + μ/2 + s/2) (βη)−s ds Γ(1/2 + μ/2 + λ/2 − s/2)

(A.II-17)

Jμ+λ/2 (βη).

In formula (A.II-16), μ = −1/2, λ = 3, β = 1, so that √ f (η) = g0 π(2η)−1/2 J1 (η) and C(ξ) = ξ −1/2 f (ξ) =

πad11 −1 ξ J1 (aξ). 2

This is just the result given by (8.3-10).

Fig. A.II-2

The integration path in t = t1 + it2 -plane

The calculation through (A.II-4 ) and (A.II-4 ) yields the same result, so the correctness of the result is demonstrated. A.II.3

Additional derivation on the solution of dual integral equations (9.8-8)

In Section 9.8, the dual integral equations

356

Major Appendix: On Some mathematical materials

∞ 0

ξAi (ξ)J0 (ξr)dξ = Mi p0 , 0 < r < a, (A.II-18)

∞

Ai (ξ)J0 (ξr)dξ = 0,

0

r>a

is solved and obtained the solution (9.8-8). We here give the detail for the derivation. According to the standard type of the equations here α = 1, ν = 0, g(ρ) = g0 = const, so 1 g0 , Ret = t1 < 2, g(α + 1 − t) = g(ρ)ρα−t dρ = 2−t 0 ψ(s) = g0

1 2πi

C+i∞

C−i∞

1 1 1 Γ(1/2) 2 dt = g0 √ , Γ(1/2 + t/2) t − s 2 − t π2−s

in which s = σ + iτ , and the integral is evaluated through the residue of pole t = 2 and the integration path is shown in Fig. A.II-3. Substituting the result into (A.II-4) yields 2 1 Ai (ξ) = f (ξ) = g0 √ π 2πi

Fig. A.II-3

k+i∞ k−i∞

2g0 2s Γ(1/2 + s/2) −s ξ ds = √ ξ −1/2 J3/2 (ξ), Γ(2 − s/2) 2π

Integration path at the t = t1 + it2 -plane

which is just the solution (9.8-8), a little bit diﬀerence with that lies in we here used the normalized coordinate ρ = r/a. In the last step of the calculation the inversion of the Mellin transform (A.II-17) was used. The above two subsections demonstrate the eﬀect and simplicity of complex variable function method in evaluating solutions of Titchmarsh-Busbridge dual integral equations.

References

357

The system of dual integral equations (AII-2) and its applications are discussed by Fan[17] , the two-dimensional dual integral equations (AII-3) are solved approximately by Fan and Sun[18] , in which some applications are also given.

References [1] Muskhelishvili N I. Some Basic Problems of the Mathematical Theory of Elasticity, Groningen: Noordhoﬀ Ltd, 1953 [2] Privalov I I. Introduction to Complex Variable Functions Theory. Moscow: Science, 1984 (in Russian) [3] Lavrentjev M A, Schabat B V. Methods of Complex Functions Theory. 6th ed. Moscow, Science, 1986 (in Russian) [4] Fan T Y. Semi-inﬁnite crack in a strip. Chin Phys Lett, 1990, 8(9): 401–404 [5] Shen T W, Fan T Y. Two collinear semi-inﬁnite cracks in a strip. Eng Fract Mech, 2003, 70(8): 813–822 [6] Titchmarsh B C. Introduction to the Fourier Integrals. Oxford: Clarenden Press, 1937 [7] Busbridge I W. Dual integral equations. Math Soc Proc, 1938, 44(2): 115–129 [8] Weber H. Ueber die Besselschen Functionen und ihre Anwendung auf die Theorie der electrischen Stroeme. J fuer reihe und angewandte Mathematik, 1873, 75(1): 5–105 [9] McDonald H M. The electrical distribution induced on a disk plased in any ﬂuid of force. Phil Mag, 1895, 26(1): 257–260 [10] King L V. On the acoustic radiation pressure on circular disk. Roy Soc London Proc Ser A, 1935, 153(1): 1–16 [11] Copson E T. On the problem of the electric disk. Edinburg Math Soc Proc, 1947, 8(1): 5–14 [12] Tranter C J. On some dual integral equations. Quar J Math Ser 2, 1951, 2(1): 60–66 [13] Sneddon I N. Fourier Transforms. New York: McGraw-Hill, 1951 [14] Gordon A N. Dual integral equations. London Math Soc J, 1954, 29(5): 360–369 [15] Noble B. On some dual integral equations. Quar J Math Ser 2, 1955, 6(2): 61–67 [16] Noble B. The solutions of Bessel function dual integral equations by a multiplying factor method. Cambridge Phil Soc Proceeding, 1963, 59(4): 351–362 [17] Fan T Y. Dual integral equations and system of dual integral equations and their applications in solid mechanics and ﬂuid mechanics. Mathematica Applicata Sinica, 1979, 2(3): 212–230 (in Chinese) [18] Fan T Y, Sun Z F. A class of two-dimensional dual integral equations and applications. Appl Math Mech, 2007, 28(2): 247–252 [19] Wolfran St. The Mathematica Book. 3 rd . Wolfran St. Cambridge: Cambridge University Press, 1996

358

Major Appendix: On Some mathematical materials

[20] Li X F. The defect problems and their analytic solutions in the theory of elasticity of quasicrystals. Dissertation. Beijing Institute of Technology, 1999 (in Chinese) [21] Zhou W M, Fan T Y. Plane elasticity problem of two-dimensional octagonal quasicrystal and crack problem. Chin Phys, 2001, 10(8): 743–747 [22] Zhou W M. Mathematical analysis of elasticity and defects of two- and three-dimensional quasicrystals. Dissertation. Beijing: Beijing Institute of Technology, 2000 (in Chinese) [23] Zhu A Y, Fan T Y, 2007, Elastic analysis of a mode II crack in an icosahedral quasicrystal, Chin. Phys., 16(5), 1111–1119 [24] Zhu A Y, Fan T Y, 2009, Elastic analysis of a Griﬃth crack in an icosahedral AlPd-Mn quasicrystal, Int. J. Modern Phys. B, 23(10), 1–16 [25] Li W, Fan T Y, Study on plastic analysis of crack problem in icosahedral quasicrystals. Phil. Mag., 2009, 89(31): 2823–2831. [26] Watson G N. A Treatise on the Theory of Bessel Functions. New York: Amazon Com, 1955 [27] Erdelyi A. Higher Transcendental Functions. Vol.2. New York: McGraw-Hill, 1953

Index A

Conjugate vector of reciprocal vector, 45

Absolute temperature, 6

Complex variable function, 127

Acoustic wave, 6

Conformal mapping, 132

Al-Mn alloy, 24

Compact embedding, 280

Amount of dislocation, 300

Compact tension specimen, 314

Analytic function, 56

Complete crystal, 3

Analytic extension, 332

Constitutive law, 282

Angular momentum conservation, 36

Contact problem, 99

Anti-plane elasticity, 54

Coordinate frame, 52

Aphomus (glassy structure), 10

Coordinate transformation, 52

“Apparent” mass of dislocation, 192

Crack, 99

Atomic cohesive force, 289

Crack speed, 193

Axisymmetric case, 182

Crack growth force, 131

B

Crack strain energy release rate, 131

Bessel function, 185

Crack tip opening displacement, 292

Biharmonic equation, 86

Crack tip tearing displacement, 290

Bilinear functional, 271

Crystal-quasicrystal coexisting, 124

Bloch theorem, 28

Crystal-quasicrystal phase transition, 128

Body force, 6

Cubic system, 22

Boundary conditions, 40

Cubic quasicrystal, 37 D

Bragg reﬂection, 24 Brillouin zone, 28

Decagonal quasicrystal, 28

Brittle fracture, 130

Decagonal Al-Ni-Co quasicrystal, 285

Burgers vector, 102

Decomposition, 54 C

Deformation, 16

Cauchy’s formula, 327

Deformation energy, 35

Cauchy’s integral, 132

Deformation tensor, 34

Cauchy-Riemann relations, 127

“diﬀerenc quotient”, 313

Classical elasticity, 12

Diﬀraction spot, 24

Classical solution, 271

Diﬀraction pattern, 70

Column vector, 258

Diﬀusion, 48

360

Index

Direct sum, 27

Fermi energy, 29

Discrete form of the energy functional, 258

Fibonacci sequence, 46

Discrete spectrum, 7

Finite diﬀerence scheme, 205

Dislocation, 99

Finite multi-connected region p229

Dislocation core, 103

Finite element method, 257

Dislocation density function, 299

Five-fold symmetry, 25

Dislocation network, 103

Flow rule of isotropic hardening, 256

Dislocation pile-up, 298

Fourier transform, 105

Dislocation velocity, 190

Fracture criterion, 131

Displacement, 16

Fracture toughness, 318

Displacement boundary condition, 40

Free energy, 8

Displacement potential function, 61 Dodecagonal quasicrystal, 28

Function equation, 233 Fundamental solution, 205

Dual integral equations, 137 Dynamic fracture, 316 Dynamic initiation of crack growth, 204 Dynamic stress intensity factor, 207 E Eight-fold symmetry, 25 Elastic constant matrix, 19 Elastic wave, 6

G Galilean transformation, 190 Gauss theorem, 36 Generalized BCS model, 295 Generalized body force, 36 Generalized Dugdal-Barenblatt model, 289 Generalized Eshelby theory, 294 Generalized Eshelby energy-momentum ten-

Elasticity, 15

sor, 294

Elastodynamics, 22 Elasto-/hydro-dynamics, 38 Electronic microscopy, 24 Element interpolation matrix, 259

Generalized Hooke’s law, 18 Generalized Eshelby integral, 295 Generalized Lax-Milgram theorem, 269

Element stiﬀness matrix, 258

Generalized solution (weak solution), 271

Element strain matrix, 258

Generalized traction, 40

Elementary excitation, 7

Generalized variational principle, 253

Elliptic notch, 144

Global stiﬀness matrix, 260

Embedding, 280

Griﬃth crack, 125 H

Energy functional, 254 Engineering elasticity, 96

Hall eﬀect, 28

Entropy, 7

Hankel transform, 184

Equilibrium state, 7

Hardening modulus, 286

Euler gamma function, 341 F Fast crack propagation, 204

Hardness, 318 Hexagonal system, 21 Higher dimensional space, 27

Index

361

M

Hilbert space, 270

Manmade damping coeﬃcient, 214

Holomorphic, 230 Horizontal mapping, 3 I

Mathematical elasticity, 96 Matrix representation of elasticity, 272

Icosahedral Al-Pd-Mn quasicrystal, 214

Mesh, 217

Incommensurate phase, 10

Miller index, 27

Incremental plasticity, 286

Mellin transform, 340

Initial condition, 40

Metric defect, 99

Inner product, 270

Minimum value, 257

Inner variable, 284

Mixed boundary condition, 40

Interface, 101

Moessbauer spectroscopy, 48

Inversion, 4

Momentum conservation, 36

Inversion of Fourier transform, 106

Monoclinic system, 19

Inversion of Mellin transform, 345

Monte-Carlo simulation, 68

Isotropic body, 22

Moving boundary problem, 210 J

Jordan lemma, 331

Moving Griﬃth crack, 193 Mutually conjugate harmonic functions, 326

K

N

Kinetic coeﬃcient, 37

Neutron scattering, 48

Korn inequality, 277

NMR (nuclear magnetic resonance), 48 L

Node, 285

Lattice dynamics, 6

Non-crystallographic point groups, 27

Lattice wave, 6

Nonlinear elasticity, 287

Lattice vibration, 6

Nonlinear behaviour of quasicrystals, 282

Laue class, 27

Nonlinear fracture mechanics, 315

Law of crystal symmetry, 3

Norm, 277

Lebesgue square integrable, 270

Number of nodes, 285

Light conductivity rate, 28

O

Linear fracture theory, 310

One-dimensional quasicrystals, 28

Linear functional, 271

Order parameter, 9

Linear transformation, 14

Orientational symmetry, 3

Loading surface, 285

Orthorhombic system, 20

Local discontinuity, 99

P

Local isomorphism, 26

Paradigm, 7

Longitudinal shear, 125

Parallel space, 27

Long-range order, 11

Path independency, 295

Long-wavelength, 6

Penrose tiling, 26

362

Index

Penny-shaped crack, 184

Row vector, 258 S

Pentagonal quasicrystal, 28 Periodicity, 1

SEM morphology, 319

Perturbation, 197

SEM fractolograph, 320

Phonetic frequency mode, 7

Semi-norm, 277

Phonon, 6

Sextuple harmonic equation, 167

Phonon strain tensor, 35

Singular integral equation, 299

Phason, 10

Sobolev space, 270

Phason strain tensor, 35

Solvability, 281

Photon, 6

Space elasticity, 61

Photonic frequency mode, 6

Space step, 207

Photonic crystal, 28

Speciﬁc heat, 6

Planar quasicrystal, 29 Planck quantum theory, 6 Plane elasticity, 54

Stability (of ﬁnite diﬀerence scheme), 207 Strain tensor, 16 Strain rate, 283

Plastic constitutive equation, 285 Plasticity, 282

Stress potential function, 77 Stress singularity, 99

Plastic zone, 280

Stress tensor, 17

Point operation, 3

St. Venant boundary condition, 98

Point group, 4

Symmetry-breaking, 7

Poles, 329

T

Positive deﬁnite, 256 Taylor expansion, 16

Power law form, 283

Ten-fold symmetry, 25

Projection, 46

Tetragonal system, 3

Pseudogap p29 Q

Theorem of Laurent expansion, 232

Quadruple harmonic equation, 76

Three-dimensional quasicrystal, 28

Quanta, 6

Three point bending specimen, 307

Quantum, 6

Time step, 207

Quasicrystal, 12

Topological defect, 99

Quasi-particle, 6

Total plasticity, 286

Quasiperiodic symmetry, 26

Traction, 18

R

Transcendental mapping function, 333

Reciprocal lattice, 5

Translational symmetry, 1

Residue theorem, 329

Triclinic system, 2

Rhombohedral system, 3

Twelve-fold symmetry, 25

Rotation-inversion, 4

Two-dimensional quasicrystal, 28

Index

363

U

Vertical mapping, 3

Uniqueness of solution, 267 V

X X-ray diﬀraction patterns, 1

Vertical space, 27

Y

Variational principle, 252

Yield surface, 285

Variational method

Yield strength, 285