scholarly journals The Curled Up Dimension in Quasicrystals

Crystals ◽  
2021 ◽  
Vol 11 (10) ◽  
pp. 1238
Author(s):  
Fang Fang ◽  
Richard Clawson ◽  
Klee Irwin
Keyword(s):  
Shear Strain ◽  
Line Defect ◽  
Geometrical Frustration ◽  
Project Method ◽  
Topological Change ◽  
Parallel Lines ◽  
Higher Dimensional ◽  
Aperiodic Order

Most quasicrystals can be generated by the cut-and-project method from higher dimensional parent lattices. In doing so they lose the periodic order their parent lattice possess, replaced with aperiodic order, due to the irrationality of the projection. However, perfect periodic order is discovered in the perpendicular space when gluing the cut window boundaries together to form a curved loop. In the case of a 1D quasicrystal projected from a 2D lattice, the irrationally sloped cut region is bounded by two parallel lines. When it is extrinsically curved into a cylinder, a line defect is found on the cylinder. Resolving this geometrical frustration removes the line defect to preserve helical paths on the cylinder. The degree of frustration is determined by the thickness of the cut window or the selected pitch of the helical paths. The frustration can be resolved by applying a shear strain to the cut-region before curving into a cylinder. This demonstrates that resolving the geometrical frustration of a topological change to a cut window can lead to preserved periodic order.

scholarly journals Structural transformations in quasicrystals induced by higher dimensional lattice transitions

2012 ◽  
Vol 468 (2141) ◽  
pp. 1452-1471 ◽  
Author(s):  
Giuliana Indelicato ◽  
Tom Keef ◽  
Paolo Cermelli ◽  
David G. Salthouse ◽  
Reidun Twarock ◽  
...  
Keyword(s):  
Three Dimensional ◽  
Higher Dimensions ◽  
Structural Transformations ◽  
Icosahedral Symmetry ◽  
Local Transformation ◽  
Project Method ◽  
Higher Dimensional ◽  
Transformation Mechanisms ◽  
Transition Paths ◽  
Penrose Tilings

We study the structural transformations induced, via the cut-and-project method, in quasicrystals and tilings by lattice transitions in higher dimensions, with a focus on transition paths preserving at least some symmetry in intermediate lattices. We discuss the effect of such transformations on planar aperiodic Penrose tilings, and on three-dimensional aperiodic Ammann tilings with icosahedral symmetry. We find that locally the transformations in the aperiodic structures occur through the mechanisms of tile splitting, tile flipping and tile merger, and we investigate the origin of these local transformation mechanisms within the projection framework.

Higher-Dimensional Space of Nanoworld

2021 ◽  
pp. 1-30
Keyword(s):  
Fundamental Domain ◽  
Dimensional Space ◽  
Golden Section ◽  
Convex Bodies ◽  
Three Dimensional ◽  
Geometrical Model ◽  
Parallel Lines ◽  
Higher Dimensional ◽  
Diffraction Patterns ◽  
Fifth Order

In this chapter, a geometrical model to accurately describe the distribution of light points in diffraction patterns of quasicrystals is proposed. It is shown that the proposed system of parallel lines has axes of the fifth order and periodically repeating the fundamental domain of the quasicrystals. This fundamental domain is 4D-polytope, called the golden hyper-rhombohedron. It consists of eight rhombohedrons densely filling the 4D space. Faces of the hyper-rhombohedron are connected by the golden section; they can be scaled as needed. On this universal lattice of the vertices of the golden hyper-rhombohedrons, famous crystallographic lattices—Bravais, Delone, Voronoi, etc.—can be embedded. On the lattice of the vertices of the golden hyper-rhombohedrons, projections of all regular three-dimensional convex bodies—Plato's bodies—can be constructed.

Effects of local shear strain on the zigzag graphene nanoribbon with a topological line defect

2020 ◽  
Vol 123 ◽  
pp. 114195
Author(s):  
Li-Hua Qu ◽  
Xiao-Long Fu ◽  
Chong-Gui Zhong ◽  
Jian-Min Zhang
Keyword(s):  
Shear Strain ◽  
Graphene Nanoribbon ◽  
Line Defect ◽  
Local Shear

scholarly journals Aperiodic crystals and beyond

2014 ◽  
Vol 70 (a1) ◽  
pp. C521-C521
Author(s):  
Uwe Grimm ◽  
Michael Baake
Keyword(s):  
Bragg Diffraction ◽  
Penrose Tiling ◽  
General Introduction ◽  
Ordered Structures ◽  
Model Sets ◽  
Higher Dimensional ◽  
Aperiodic Order ◽  
Discrete Structures ◽  
Aperiodic Crystals ◽  
Recent Monograph

Following Dan Shechtman's discovery of quasicrystals in 1982, the realm of crystallography has been extended to include structures that lack translational periodicity. While periodic crystals can be modelled as decorations of lattices, aperiodic crystals require more general discrete structures such as point sets or tilings in space for the description of their structure. For a mathematical introduction to the field of aperiodic order, we refer to the recent monograph [1]. Interesting examples are obtained by projections from higher-dimensional lattices, leading to model sets which have pure Bragg diffraction, though the Bragg peaks are, in general, dense in space. All symmetries that have been experimentally observed in quasicrystals can be reproduced in this way, and some of the resulting structures are standard examples of tilings that are frequently used in quasicrystal modelling, such as the famous Penrose tiling. But there exists a plethora of ordered structures beyond cut and project sets, some of which have even weirder properties. After a general introduction to aperiodically ordered structures, a couple of examples of such systems are briefly described, offering a glimpse at the largely unexplored world of order beyond (aperiodic) crystals.

Parallel Lines

2004 ◽  
Author(s):  
Nina Davenport
Keyword(s):  
Parallel Lines

Analysis of magneto rheological elastomers for energy harvesting systems

2020 ◽  
Vol 64 (1-4) ◽  
pp. 439-446
Author(s):  
Gildas Diguet ◽  
Gael Sebald ◽  
Masami Nakano ◽  
Mickaël Lallart ◽  
Jean-Yves Cavaillé
Keyword(s):  
Magnetic Field ◽  
Energy Harvesting ◽  
Shear Strain ◽  
Magnetic Induction ◽  
Magnetic Particles ◽  
Homogeneous Magnetic Field ◽  
Electrical Signal ◽  
Harvesting Systems ◽  
Dc Bias ◽  
Magneto Rheological

Magneto Rheological Elastomers (MREs) are composite materials based on an elastomer filled by magnetic particles. Anisotropic MRE can be easily manufactured by curing the material under homogeneous magnetic field which creates column of particles. The magnetic and elastic properties are actually coupled making these MREs suitable for energy conversion. From these remarkable properties, an energy harvesting device is considered through the application of a DC bias magnetic induction on two MREs as a metal piece is applying an AC shear strain on them. Such strain therefore changes the permeabilities of the elastomers, hence generating an AC magnetic induction which can be converted into AC electrical signal with the help of a coil. The device is simulated with a Finite Element Method software to examine the effect of the MRE parameters, the DC bias magnetic induction and applied shear strain (amplitude and frequency) on the resulting electrical signal.

On the nominal transverse shear strain to characterize the severity of creasing

TAPPI Journal ◽  
2018 ◽  
Vol 17 (04) ◽  
pp. 231-240
Author(s):  
Douglas Coffin ◽  
Joel Panek
Keyword(s):  
Shear Strain ◽  
Transverse Shear ◽  
Elastic Limit ◽  
Experimental Results ◽  
Transverse Shear Strain ◽  
Maximum Strain ◽  
Machine Direction ◽  
Engineering Approach ◽  
Wide Range ◽  
Analytic Expression

A transverse shear strain was utilized to characterize the severity of creasing for a wide range of tooling configurations. An analytic expression of transverse shear strain, which accounts for tooling geometry, correlated well with relative crease strength and springback as determined from 90° fold tests. The experimental results show a minimum strain (elastic limit) that needs to be exceeded for the relative crease strength to be reduced. The theory predicts a maximum achievable transverse shear strain, which is further limited if the tooling clearance is negative. The elastic limit and maximum strain thus describe the range of interest for effective creasing. In this range, cross direction (CD)-creased samples were more sensitive to creasing than machine direction (MD)-creased samples, but the differences were reduced as the shear strain approached the maximum. The presented development provides the foundation for a quantitative engineering approach to creasing and folding operations.

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