准晶数学弹性理论及应用

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

This inter-disciplinary work covering the continuum mechanics of novel materials, condensed matter phvsics and partial differential equations discusses the mathematical theory of elasticity of quasicrystals (a new condensed matter) and its applications by setting up new partial differential equations of higher order and their solutions under complicated boundary value and initial value conditions. The new theories developed here dramatically simplify the solving of complicated elasticity equation systems. Large numbers of complicated equations involving elasticity are reduced to a single or a few partial differential equations of higher order. Systematical and direct methods of mathematical physics and complex variable functions are developed to solve the equations under appropriate boundary value and initial value conditions, and many exact analytical solutions are constructed.