Asymptotic modeling of thin linearly quasicrystalline plates (original) (raw)

An exact solution for a multilayered two-dimensional decagonal quasicrystal plate

International Journal of Solids and Structures, 2014

By extending the pseudo-Stroh formalism to two-dimensional decagonal quasicrystals, an exact closedform solution for a simply supported and multilayered two-dimensional decagonal quasicrystal plate is derived in this paper. Based on the different relations between the periodic direction and the coordinate system of the plate, three internal structure cases for the two-dimensional quasicrystal layer are considered. The propagator matrix method is also introduced in order to treat efficiently and accurately the multilayered cases. The obtained exact closed-form solution has a concise and elegant expression. Two homogeneous quasicrystal plates and a sandwich plate made of a two-dimensional quasicrystal and a crystal with two stacking sequences are investigated using the derived solution. Numerical results show that the differences of the periodic direction have strong influences on the stress and displacement components in the phonon and phason fields; different coupling constants between the phonon and phason fields will also cause differences in physical quantities; the stacking sequences of the multilayer plates can substantially influence all physical quantities. The exact closed-form solution should be of interest to the design of the two-dimensional quasicrystal homogeneous and laminated plates. The numerical results can also be employed to verify the accuracy of the solution by numerical methods, such as the finite element and difference methods, when analyzing laminated composites made of quasicrystals.

Small-angle grain boundaries in quasicrystals

Physical Review Letters, 1989

The Read-Shockley treatment of small-angle grain boundaries in crystals is generalized to the case of quasicrystals. The dependence of the grain-boundary energy on the angle of mismatch between abutting quasicrystalline grains is calculated. It is found that, even for a symmetric tilt boundary in a quasicrystal, dislocations with at least two types of Burgers vectors are required; these dislocations have to be arranged quasiperiodically along the boundary. The possible clumping of these dislocations to form composites is discussed. Explicit calculations are presented for a pentagonal quasicrystal.

Non-linear generalized elasticity of icosahedral quasicrystals

Journal of Physics A: Mathematical and General, 2002

Quasicrystals can carry, in addition to the classical phonon displacement field, a phason displacement field, which requires a generalized theory of elasticity. In this paper, the third-order strain invariants (including phason strain) of icosahedral quasicrystals are determined. They are connected with 20 independent third-order elastic constants. By means of non-linear elasticity, phason strains with icosahedral irreducible Γ 4 -symmetry can be obtained by phonon stress, which is impossible in linear elasticity.

Elastic and anelastic behaviour of icosahedral quasicrystals

1987

A theory is developed, based on theoretical-group analysis, which describes the linear, reversible, time-dependent response of an icosahedral quasicrystal, containing point defects, to stress the field known as anelastic relaxation. We obtain also anelastic relaxation relationships for the practical Young, shear and Poisson moduli.

Elastic Green's function of icosahedral quasicrystals

The European Physical Journal B, 1998

The elastic theory of quasicrystals considers, in addition to the "normal" displacement field, three "phason" degrees of freedom. We present an approximative solution for the elastic Green's function of icosahedral quasicrystals, assuming that the coupling between the phonons and phasons is small.

Geometric models for continuous transitions from quasicrystals to crystals

Starting from variable p-veetors half-stars whíeh verify Hadwiger's theorem, the cut-projeetion method is used here. The strip ofprojeetion is projeeted on a rotatory subspaee and a variable tiling is obtained. Two out standing examples are developed. The first, a eontinuous evolution from a two-dímensional octagonal quasilattiee to two square lattiees 45° rotated in between. The seeond is a eontinuous evolution from a three-dimensional Penrose tiling to an f.e.e. vertex lattiee. Physieal applieations to quasierystal-<:rystal transitions are poínted out. After quasicrystalline phases were discovered (Shechtman, Blech, Gratias and Cahn 1984), some theoretical (El ser and Henley 1985, Kramer 1987) and experimental works (Guyot and Audier 1985, Urban, Moser and Kronmüller 1985, Audier and Guyot 1986a, b, Guyot, Audier and Lequette 1986) began to pay attention to the close and systematic relationship between quasicrystals and crystals. RecentIy, many works have pointed in the same direction (Poon, Dmowski, Egami, Shen and Shiflet 1987, Zhou, Li, Ye and Kuo 1987, Yamamoto and Hiraga 1988, Zhang, Wang and Kuo 1988, Sadananda, Singh and Imam 1988, Yu-Zhang, Bigot, Chevalier, Gratias, Martin and Portier 1988, Fitz Gerald, Withers, Stewart and Calka 1988, Yang 1988, Henley 1988, Chandra and Suryanarayana 1988, Cahn, Gratias and Mozer 1988). Some authors even state that the transition from quasicrystalline to crystalline phases is continuous over a range ofintermediate phases (Reyes-Gasga, Avalos-Borja and José-Yacamán 1988, Zhou, Ye, Li and Kuo 1988). We present here a geometric model to describe simple and plausible continuous evolutions from quasilattices to lattices. Our method is a version ofthe well known cutprojection method (Kramer and Neri 1984, Duneau and Katz 1985, EIser 1986). In the above mentioned work, EIser and Henley (1985) modified the cut-projection method to allow study of the connection between crystal and quasicrystal structures. These authors tilted the strip of projection with respect to the hypercubic lattice (defined in the hyperspace EP) but they fixed the projection hyperplane (or projection subspace P, p> n). So, different hypercubic roofs were projected in such a way that the quasicrystal structure was the limit of a discontinuous sequence of periodic structures. In this work, we develop the contrary strategy and we describe a lattice as an atrophical quasilattice. We fix the particular strip (in the p-dimensional hypercubic lattice of EP) which generates the standard quasiperiodic tiling but we rotate the projection hyperplane (or 0950--0839/

Phason-elastic energy in a model quasicrystal

Journal of Non-Crystalline Solids, 2004

The standard two-dimensional decagonal binary tiling quasicrystal with Lennard-Jones potentials is metastable at zero temperature with respect to one phason strain mode. By calculating the frequencies of local environments as a function of phason strain, a correction for the potentials is predicted, which stabilizes the quasicrystal. : 61.44.Br, 62.20.Dc

Elastic theory of icosahedral quasicrystals - application to straight dislocations

The European Physical Journal B, 2001

In quasicrystals, there are not only conventional, but also phason displacement fields and associated Burgers vectors. We have calculated approximate solutions for the elastic fields induced by two-, three-and fivefold straight screw-and edge-dislocations in infinite icosahedral quasicrystals by means of a generalized perturbation method. Starting from the solution for elastic isotropy in phonon and phason spaces, corrections of higher order reflect the two-, three-and fivefold symmetry of the elastic fields surrounding screw dislocations. The fields of special edge dislocations display characteristic symmetries also, which can be seen from the contributions of all orders.