# Mechanik der Quasikristalle - Zhibin Wang. M.Sc.

**Analytical concepts to investigate the mechanical behavior of quasicrystals including defects**

**Summary :**

Quasicrystals (QCs) have become the focus of theoretical and experimental studies in the physics of condensed matter since the first discovery of the icosahedral QC in Al-Mn alloys. The discovery of QCs reveals a new symmetry in solids, i.e. quasiperiodic symmetry. This changes the traditional concept of classifying solids into two classes: crystals and non-crystals. The electronic, optic, magnetic, thermal and mechanical properties of the material have been extensively investigated in experimental and theoretical analyses, showing their complex structure and unusual properties. For large single-grain QCs, over one hundred different alloys with thermodynamic stability have been produced. This suggests that QCs may become a new class of functional and structural materials, which have many prospective engineering applications.

To analyze the mechanical behavior of QCs, different analytical approaches have been derived within this project generalizing the classical theory of elasticity towards this new interesting class of materials. For this purpose, some direct and systematic methods of mathematical physics and function theory are extended for QCs. One feature making mathematical solutions different from those known for most of the crystalline materials is the coupling of two different fields, i.e. phonon and phason fields. This coupling has an essential effect on all kinds of boundary value problems.

Without any ad hoc assumptions, a refined theory of QC thick plates has been deduced systematically and directly from linear elastic theory of QCs by using the general solution and the Lur’e symbolic method. In the case of homogeneous boundary conditions, the refined theory is exact in the sense that a solution satisfies all the equations in elastic theory and consists of three parts: the biharmonic part, the shear part and the transcendental part. In the case of non-homogeneous boundary conditions, two special cases of boundary conditions are considered, i.e. pure normal surface loadings and pure shear surface loadings.

Many engineering structures are composed of two or more materials with different physical properties. The problem of an elastic body with two joined dissimilar materials with forces applied at an arbitrary point is fundamental to the development of an elastic theory and has a vital significance in the structural design. The Green's function solutions in quasicrystalline materials can be applied to calculate phonon and phason fields caused by an externally or internally applied force. The present study deals with the problem of a combination of line phonon forces and line phason forces applied to infinite planes and bi-material planes consisting of two half-planes of dissimilar QC materials bonded together accounting for imperfect boundary conditions at the interface.

The performance of materials is influenced by the presence of defects such as dislocations, inclusions, cracks etc. Studies of defects in QCs have attracted extensive attention not only because of their importance towards structural analyses but also to gain a deeper understanding of the mechanical and physical properties of QCs. In order to understand quantitatively the influence of a dislocation and the interaction with other defects on the mechanical behavior of QCs, it is necessary to determine the elastic field induced by a dislocation and the interaction of dislocations in QCs, including interactions between a dislocation and a crack.

For Mixed-Mode crack problems in QCs the exact expressions for the phonon and phason fields near the crack tip as well as the remote fields have been derived using a complex potential method and a generalized Lekhnitskii's formalism. Field intensity factors of phonon and phason fields as well as energy release rates and Irwin’s relationship have also been determined. In addition, fracture mechanical weight functions, i.e. Green’s functions for intensity factors, are derived from the theorem of Maxwell and Betti. Thus, a framework of fracture mechanics of quasicrystalline materials is put forward, aiming for the derivation of a suitable fracture criterion.

Based on the framework of irreversible thermodynamics, configurational forces in QCs are derived, quantifying the driving forces on dislocations, cracks and other discontinuities in the material space.