Elasticity Of Quasicrystals Research Articles

Elasticity theory, fracture mechanics, and some relevant thermal properties of quasi-crystalline materials

A review is given on the basic concepts and fundamental framework of the theory of elasticity for quasi-crystalline materials, including some 1D, 2D, and 3D quasi-crystals. The elasticity of quasi-crystals embodies some new concepts, field variables, and equations. It is much more complicated and beyond the scope of classical elasticity which holds only for conventional structural materials, including crystalline materials. Hence, some well-developed methods in classical elasticity cannot be directly applied to solve the problems of elasticity of quasi-crystalline materials. But the ideas of the classical theory of elasticity provide beneficial insight to treat this new subject. A decomposition and superposition procedure is suggested to simplify the elasticity problems of 1D and 2D quasi-crystals. Application of displacement and stress potentials further simplifies the problems. The large number of complicated equations involving elasticity is reduced to a single or a few partial differential equations of higher order by this technique. Also, efforts have been made to simplify the equations for 3D cubic quasi-crystals to a single partial differential equation of higher order. Simplification of the basic equations provides the possibility to solve boundary value or initial-boundary value problems of elasticity. For this purpose, some direct and systematic methods of mathematical physics and function theory are developed, and a series of analytic (classical) solutions, mainly for dislocations and cracks in materials, are derived. In addition, attention is drawn to those variational problems and generalized solutions (weak solutions) of boundary value problems and numerical implementation by the finite element method. The above may be seen as a development of the theory and methodology akin to those of classical elasticity. Based on the exact solutions of crack problems with different configurations under different motion states for different quasi-crystal systems, we put forward a framework of fracture mechanics of quasi-crystalline materials. This may be seen as an extension of the development of fracture mechanics for conventional structural materials. Also, some elastodynamic problems for some 1D and 2D quasi-crystals are studied, related results for dislocation and crack dynamics are found, and possible connections with certain thermal properties of quasi-crystalline materials, eg, specific heat and other thermo-dynamic functions, are discussed. There are 75 references cited in this review article.

Elasticity of a one-dimensional tiling model and its implication to the phason elasticity of quasicrystals

A one-dimensional tiling model with matching rule energy (antiferromagnetic Ising Hamiltonian) is studied. We present an analytic study of a transition from the unlocked phase, where free energy is proportional to the square gradient of the perp-space field $[f\ensuremath{\sim}(\ensuremath{\partial}{w)}^{2}],$ to the locked phase $(f\ensuremath{\sim}|\ensuremath{\partial}w|)$ in perp-space elasticity. The phase diagram and the temperature dependence of the elastic constant in the unlocked phase show similarity with the two-dimensional Penrose tiling. The results imply that the unlocking transition of a two-dimensional Penrose tiling model is related to the disordering transition in a one-dimensional Ising model.

Finite-temperature elasticity phase transition in decagonal quasicrystals.

We present evidence for a finite-temperature phase transition in the phason elasticity of quasicrystals. A tiling model for energetically stabilized decagonal quasicrystals has been studied in an extensive series of Monte Carlo simulations. The Hamiltonian (energetics) of the model is given by nearest-neighbor Penrose-like matching rules between three-dimensional unit cells. A new order parameter and diagnostics have been introduced. We show that a transition from locked-phason to unlocked-phason dynamics occurs at finite temperature. In the unlocked phase, phasons can be thermodynamically excited even though the quasicrystal is energetically stabilized at low temperatures.

Direct tests of the entropic model of quasicrystals.

Assuming an entropic origin for phason elasticity in quasicrystals, we derive predictions for the temperature dependence of grain-boundary structure and free energy, the nature of the elastic instability in these systems, and the behavior of sound damping near the instability. We believe that these will provide decisive tests of the entropic model for quasicrystals.

Measurement of glasslike and crystalline elasticity in quasicrystals.

We have measured the Young's modulus and internal friction of several families of aluminum-based quasicrystals from 0.01 to 300 K and from 200 Hz to 8 kHz with a vibrating-reed technique. In the metastable aluminum--transition-metal quasicrystals we see a unique large linear temperature dependence of the internal friction in the 1--100-K range; this is one of the very few measured properties of quasicrystals that differ from metallic glasses or alloys. We do not see the large linear dependence in quasicrystals from the stable families, providing an observable difference in the physical properties of the stable and metastable families. We see glassy behavior in some of the quasicrystals that is as large as that observed in metallic glasses. Finally, we will present low-temperature speed-of-sound data that cannot be explained by standard models of glasses and alloys.