Preface
Chapter 1 Crystals
1.1 Periodicity of crystal structure, crystal cell
1.2 Three-dimensional lattice types
1.3 Symmetry and point groups
1.4 Reciprocal lattice
1.5 Appendix of Chapter 1: Some basic concepts
References
Chapter 2 Framework of the classical theory of elasticity
2.1 Review on some basic concepts
2.2 Basic assumptions of theory of elasticity
2.3 Displacement and deformation
2.4 Stress analysis and equations of motion
2.5 Generalized Hooke's law
2.6 Elastodynamics, wave motion
2.7 Summary
References
Chapter 3 Quasicrystal and its properties
3.1 Discovery of quasicrystal
3.2 Structure and symmetry of quasicrystals
3.3 A brief introduction on physical properties of quasicrystals
3.4 One-, two- and three-dimensional quasicrystals
3.5 Two-dimensional quasicrystals and planar quasicrystals
References
Chapter 4 The physical basis of elasticity of quasicrystals
4.1 Physical basis of elasticity of quasicrystals
4.2 Deformation tensors
4.3 Stress tensors and the equations of motion
4.4 Free energy and elastic constants
4.5 Generalized Hooke's law
4.6 Boundary conditions and initial conditions
4.7 A brief introduction on relevant material constants of quasicrystals
4.8 Summary and mathematical solvability of boundary value or initial- boundary value problem
4.9 Appendix of Chapter 4: Description on physical basis of elasticity of
quasicrystals based on the Landau density wave theory
References
Chapter 5 Elasticity theory of one-dimensional quasicrystals and simplification
5.1 Elasticity of hexagonal quasicrystals
5.2 Decomposition of the problem into plane and anti-plane problems
5.3 Elasticity of monoclinic quasicrystals
5.4 Elasticity of orthorhombic quasicrystals
5.5 Tetragonal quasicrystals
5.6 The space elasticity of hexagonal quasicrystals
5.7 Other results of elasticity of one-dimensional quasicrystals
References
Chapter 6 Elasticity of two-dimensional quasicrystals and simplification
6.1 Basic equations of plane elasticity of two-dimensional quasicrystals:
point groups 5m and 10mm in five- and ten-fold symmetries
6.2 Simplification of the basic equation set: displacement potential function method
6.3 Simplification of the basic equations set: stress potential function method
6.4 Plane elasticity of point group 5, ■ pentagonal and point group 10, ■ decagonal quasicrystals
6.5 Plane elasticity of point group 12mm of dodecagonal quasicrystals
6.6 Plane elasticity of point group 8mm of octagonal quasicrystals, displacement potential
6.7 Stress potential of point group 5, ■ pentagonal and point group 10, ■ decagonal quasicrystals
6.8 Stress potential of point group 8mm octagonal quasicrystals
6.9 Engineering and mathematical elasticity of quasicrystals
References
Chapter 7 Application I: Some dislocation and interface problems
and solutions in one- and two,dimensional quasicrystals
7.1 Dislocations in one-dimensional hexagonal quasicrystals
7.2 Dislocations in quasicrystals with point groups 5m and 10mm symmetries
7.3 Dislocations in quasicrystals with point groups 5, ■ five-fold and 10, ■ ten-fold symmetries
7.4 Dislocations in quasicrystals with eight-fold symmetry
7.5 Dislocations in dodecagonal quasicrystals
7.6 Interface between quasicrystal and crystal
7.7 Conclusion and discussion
References
Chapter 8 Application II: Solutions of notch and crack problems of one-and two-dimensional quasicrystals
8.1 Crack problem and solution of one-dimensional quasicrystals
8.2 Crack problem in finite-sized one-dimensional quasicrystals
8.3 Griffith crack problems in point groups 5m and 10mm quasicrystals
based on displacement potential function method
8.4 Stress potential function formulation and complex variable function
method for solving notch and crack problems of quasicrystals of point groups 5, ■ and 10, ■
8.5 Solutions of crack/notch problems of two-dimensional octagonal quasicrystals
8.6 Other solutions of crack problems in one-and two-dimensional quasicrystals
8.7 Appendix of Chapter 8: Derivation of solution of Section 8.1
References
Chapter 9 Theory of elasticity of three-dimensional quasicrystals and its applications
9.1 Basic equations of elasticity of icosahedral quasicrystals
9.2 Anti-plane elasticity of icosahedral quasicrystals and problem of
interface between quasicrystal and crystal
9.3 Phonon-phason decoupled plane elasticity of icosahedral
quasicrystals
9.4 Phonon-phason coupled plane elasticity of icosahedral quasicrystals--
displacement potential formulation
9.5 Phonon-phason coupled plane elasticity of icosahedral quasicrystals--
stress potential formulation
9.6 A straight dislocation in an icosahedral quasicrystal
9.7 An elliptic notch/Griffith crack in an icosahedral quasicrystal
9.8 Elasticity of cubic quasicrystals--the anti-plane and axisymmetric deformation
References
Chapter 10 Dynamics of elasticity and defects of quasicrystals
10.1 Elastodynamics of quasicrystals followed the Bak's argument
10.2 Elastodynamics of anti-plane elasticity for some quasicrystals
10.3 Moving screw dislocation in anti-plane elasticity
10.4 Mode III moving Griftith crack in anti-plane elasticity
10.5 Elast0-/hydro-dynamics of quasicrystals and approximate analytic
solution for moving screw dislocation in anti-plane elasticity
10.6 Elasto-/hydro-dynamics and solutions of two-dimensional decagonal quasicrystals
10.7 Elasto-/hydro-dynamics and applications to fracture dynamics of icosahedral quasicrystals
10.8 Appendix of Chapter 10: The detail of finite difference scheme
References
Chapter 11 Complex variable function method for elasticity of quasicrystals
11.1 Harmonic and quasi-biharmonic equations in anti-plane elasticity of one-dimensional quasicrystals
11.2 Biharmonic equations in plane elasticity of point group 12mm two-dimensional quasicrystals
11.3 The complex variable function method of quadruple harmonic
equations and applications in two-dimensional quasicrystals
11.4 Complex variable function method for sextuple harmonic equation
and applications to icosahedral quasicrystals
11.5 Complex analysis and solution of quadruple quasiharmonic equation
11.6 Conclusion and discussion
References
Chapter 12 Variational principle of elasticity of quasicrystals
numerical analysis and applications
12.1 Basic relations of plane elasticity of two-dimensional quasicrystals
12.2 Generalized variational principle for static elasticity ofquasicrystals
12.3 Finite element method
12.4 Numerical examples
References
Chapter 13 Some mathematical principles on solutions of elasticity of quasicrystals
13.1 Uniqueness of solution of elasticity of quasicrystals
13.2 Generalized Lax-Milgram theorem
13.3 Matrix expression of elasticity of three-dimensional qnasicrystals
13.4 The weak solution of boundary value problem of elasticity of quasicrystals
13.5 The uniqueness of weak solution
13.6 Conclusion and discussion
References
Chapter 14 Nonlinear behaviour of quasicrystals
14.1 Macroscopic behaviour of plastic deformation of quasicrystals
14.2 Possible scheme of plastic constitutive equations
14.3 Nonlinear elasticity and its formulation
14.4 Nonlinear solutions based on simple models
14.5 Nonlinear analysis based on the generalized Eshelby theory
14.6 Nonlinear analysis based on the dislocation model
14.7 Conclusion and discussion
14.8 Appendix of Chapter 14: Some mathematical details
References
Chapter 15 Fracture theory of quasicrystals
15.1 Linear fracture theory of quasicrystals
15.2 Measurement of GIC
15.3 Nonlinear fracture mechanics
15.4 Dynamic fracture
15.5 Measurement of fracture toughness and relevant mechanical
parameters of quasicrystalline material
References
Chapter 16 Remarkable conclusion
References
Major Appendix: On some mathematical materials
Appendix I Outline of complex variable functions and some additional calculations
A.I.1 Complex functions, analytic functions
A.I.2 Cauchy's formula
A.I.3 Poles
A.I.4 Residue theorem
A.I.5 Analytic extension
A.I.6 Conformal mapping
A.I.7 Additional derivation of solution (8.2-19)
A.I.8 Additional derivation of solution (11.3-53)
A.I.9 Detail of complex analysis of generalized cohesive force model for plane
elasticity of two-dimensional point groups 5m, 10mm and 10, 10 quasicrystals
A.I.10 On the calculation of integral (9.2-14)
Appendix II Dual integral equations and some additional calculations.
A.II.1 Dual integral equations
A.II.2 Additional derivation on the solution of dual integral equations(8.3-8)
A.II.3 Additional derivation on the solution of dual integral equations(9.8-8)
References
Index
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