scholarly journals Fragmentary and incidental behaviour of columns, slabs and crystals

2014 ◽  
Vol 372 (2008) ◽  
pp. 20120032 ◽  
Author(s):  
Walter Whiteley
Keyword(s):  
Periodic Structure ◽  
Periodic Structures ◽  
Three Dimensional ◽  
The Real ◽  
Combinatorial Properties ◽  
Open Questions ◽  
Unit Cells ◽  
Periodic Frameworks

Between the study of small finite frameworks and infinite incidentally periodic frameworks, we find the real materials which are large, but finite, fragments that fit into the infinite periodic frameworks. To understand these materials, we seek insights from both (i) their analysis as large frameworks with associated geometric and combinatorial properties (including the geometric repetitions) and (ii) embedding them into appropriate infinite periodic structures with motions that may break the periodic structure. A review of real materials identifies a number of examples with a local appearance of ‘unit cells’ which repeat under isometries but perhaps in unusual forms. These examples also refocus attention on several new classes of infinite ‘periodic’ frameworks: (i) columns—three-dimensional structures generated with one repeating isometry and (ii) slabs—three-dimensional structures with two independent repeating translations. With this larger vision of structures to be studied, we find some patterns and partial results that suggest new conjectures as well as many additional open questions. These invite a search for new examples and additional theorems.

Design of Three-Dimensional, Triply Periodic Unit Cell Scaffold Structures for Additive Manufacturing

2018 ◽  
Vol 140 (7) ◽  
Author(s):  
Mazher Iqbal Mohammed ◽  
Ian Gibson
Keyword(s):  
Additive Manufacturing ◽  
Periodic Structures ◽  
Three Dimensional ◽  
Unit Cell ◽  
Final Size ◽  
Fused Filament Fabrication ◽  
Cell Structures ◽  
Triply Periodic ◽  
Unit Cells ◽  
Scaffold Structure

Highly organized, porous architectures leverage the true potential of additive manufacturing (AM) as they can simply not be manufactured by any other means. However, their mainstream usage is being hindered by the traditional methodologies of design which are heavily mathematically orientated and do not allow ease of controlling geometrical attributes. In this study, we aim to address these limitations through a more design-driven approach and demonstrate how complex mathematical surfaces, such as triply periodic structures, can be used to generate unit cells and be applied to design scaffold structures in both regular and irregular volumes in addition to hybrid formats. We examine the conversion of several triply periodic mathematical surfaces into unit cell structures and use these to design scaffolds, which are subsequently manufactured using fused filament fabrication (FFF) additive manufacturing. We present techniques to convert these functions from a two-dimensional surface to three-dimensional (3D) unit cell, fine tune the porosity and surface area, and examine the nuances behind conversion into a scaffold structure suitable for 3D printing. It was found that there are constraints in the final size of unit cell that can be suitably translated through a wider structure while still allowing for repeatable printing, which ultimately restricts the attainable porosities and smallest printed feature size. We found this limit to be approximately three times the stated precision of the 3D printer used this study. Ultimately, this work provides guidance to designers/engineers creating porous structures, and findings could be useful in applications such as tissue engineering and product light-weighting.

An Executable Notation, With Illustrations From Elementary Crystallography

1994 ◽  
Author(s):  
Donald B. Mclntyre
Keyword(s):  
Plate Tectonics ◽  
Periodic Structures ◽  
Dimensional Space ◽  
Reciprocal Lattice ◽  
Three Dimensional ◽  
Cubic Symmetry ◽  
Multiple Data ◽  
Special Cases ◽  
Unit Cells ◽  
Crystallographic Axes

Elementary crystallography is an ideal context for introducing students to mathematical geology. Students meet crystallography early because rocks are made of crystalline minerals. Moreover, morphological crystallography is largely the study of lines and planes in real three-dimensional space, and visualizing the relationships is excellent training for other aspects of geology; many algorithms learned in crystallography (e.g., rotation of arrays) apply also to structural geology and plate tectonics. Sets of lines and planes should be treated as entities, and crystallography is an ideal environment for introducing what Sylvester (1884) called "Universal Algebra or the Algebra of multiple quantity." In modern terminology, we need SIMD (Single Instruction, Multiple Data) or even MIMD. This approach, initiated by W.H. Bond in 1946, dispels the mysticism unnecessarily associated with Miller indices and the reciprocal lattice; edges and face-normals are vectors in the same space. The growth of mathematical notation has been haphazard, new symbols often being introduced before the full significance of the functions they represent had been understood (Cajori, 1951; Mclntyre, 1991b). Iverson introduced a consistent notation in 1960 (e.g., Iverson 1960, 1962, 1980). His language, greatly extended in the executable form called J (Iverson, 1993), is used here. For information on its availability as shareware, see the Appendix. Publications suitable as tutorials in , J are available (e.g., Iverson. 1991; Mclntyre, 1991 a, b; 1992a,b,c; 1993). Crystals are periodic structures consisting of unit cells (parallelepipeds) repeated by translation along axes parallel to the cell edges. These edges define the crystallographic axes. In a crystal of cubic symmetry they are orthogonal and equal in length (Cartesian). Those of a triclinic crystal, on the other hand, are unequal in length and not at right angles. The triclinic system is the general case; others are special cases. The formal description of a crystal gives prominent place to the lengths of the axes (a, b, and c) and the interaxial angles ( α, β, and γ). A canonical form groups these values into a 2 x 3 table (matrix), the first row being the lengths and the second the angles.

scholarly journals Number of Wavevectors for Each Frequency in a Periodic Structure

2017 ◽  
Vol 139 (5) ◽  
Author(s):  
Farhad Farzbod
Keyword(s):  
Periodic Structure ◽  
Degrees Of Freedom ◽  
Phonon Dispersion ◽  
Periodic Structures ◽  
Dispersion Curves ◽  
Neighbor Interaction ◽  
Phonon Dispersion Curves ◽  
Hexagonal Form ◽  
Vibration Properties ◽  
Unit Cells

Periodic structures have interesting acoustic and vibration properties making them suitable for a wide variety of applications. In a periodic structure, the number of frequencies for each wavevector depends on the degrees-of-freedom of the unit cell. In this paper, we study the number of wavevectors available at each frequency in a band diagram. This analysis defines the upper bound for the maximum number of wavevectors for each frequency in a general periodic structure which might include damping. Investigation presented in this paper can also provide an insight for designing materials in which the interaction between unit cells is not limited to the closest neighbor. As an example application of this work, we investigate phonon dispersion curves in hexagonal form of boron nitride to show that first neighbor interaction is not sufficient to model dispersion curves with force-constant model.

Homogenization of Vibrating Periodic Lattice Structures

2005 ◽  
Author(s):  
Stefano Gonella ◽  
Massimo Ruzzene
Keyword(s):  
Smart Materials ◽  
Periodic Structures ◽  
Three Dimensional ◽  
Periodic Pattern ◽  
Periodic Lattice ◽  
Lattice Structures ◽  
Equivalent Representation ◽  
Detailed Model ◽  
Wide Range ◽  
Unit Cells

The paper describes a homogenization technique for periodic lattice structures. The analysis is performed by considering the irreducible unit cell as a building block that defines the periodic pattern. In particular, the continuum equivalent representation for the discrete structure is sought with the objective of retaining information regarding the local properties of the lattice, while condensing its global behavior into a set of differential equations. These equations can then be solved either analytically or numerically, thus providing a model which involves a significantly lower number of variables than those required for the detailed model of the assembly. The methodology is first tested by comparing the dispersion relations obtained through homogenization with those corresponding to the detailed model of the unit cells and then extended to the comparison of exact and approximate harmonic responses. This comparison is carried out for both one-dimensional and two-dimensional assemblies. The application to three-dimensional structures is not attempted in this work and will be approached in the future without the need for substantial conceptual changes in the theoretical procedure. Hence the presented technique is expected to be applicable to a wide range of periodic structures, with applications ranging from the design of structural elements of mechanical and aerospace interest to the optimization of smart materials with attractive mechanical, thermal or electrical properties.

Crystal Structure Analysis of Cu-Zr Alloys by Convergent Beam Diffraction

1981 ◽  
Vol 39 ◽  
pp. 354-355
Author(s):  
A. F. Marshall ◽  
J. W. Steeds ◽  
D. Bouchet ◽  
S. L. Shinde ◽  
R. G. Walmsley
Keyword(s):  
Crystal Structure ◽  
Point Group ◽  
Intermetallic Phase ◽  
Three Dimensional ◽  
Crystal Structure Analysis ◽  
Ion Milling ◽  
Composition And Structure ◽  
Unit Cells ◽  
Sputter Deposited ◽  
Convergent Beam

Convergent beam electron diffraction is a powerful technique for determining the crystal structure of a material in TEM. In this paper we have applied it to the study of the intermetallic phases in the Cu-rich end of the Cu-Zr system. These phases are highly ordered. Their composition and structure has been previously studied by microprobe and x-ray diffraction with sometimes conflicting results.The crystalline phases were obtained by annealing amorphous sputter-deposited Cu-Zr. Specimens were thinned for TEM by ion milling and observed in a Philips EM 400. Due to the large unit cells involved, a small convergence angle of diffraction was used; however, the three-dimensional lattice and symmetry information of convergent beam microdiffraction patterns is still present. The results are as follows:1) 21 at% Zr in Cu: annealed at 500°C for 5 hours. An intermetallic phase, Cu3.6Zr (21.7% Zr), space group P6/m has been proposed near this composition (2). The major phase of our annealed material was hexagonal with a point group determined as 6/m.

scholarly journals Imaging of bi-anisotropic periodic structures from electromagnetic near-field data

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Dinh-Liem Nguyen ◽  
Trung Truong
Keyword(s):  
Inverse Scattering ◽  
Maxwell Equations ◽  
Field Data ◽  
Periodic Structures ◽  
Near Field ◽  
Scattering Problem ◽  
Three Dimensional ◽  
Inverse Scattering Problem ◽  
Factorization Method ◽  
Unique Determination

AbstractThis paper is concerned with the inverse scattering problem for the three-dimensional Maxwell equations in bi-anisotropic periodic structures. The inverse scattering problem aims to determine the shape of bi-anisotropic periodic scatterers from electromagnetic near-field data at a fixed frequency. The factorization method is studied as an analytical and numerical tool for solving the inverse problem. We provide a rigorous justification of the factorization method which results in the unique determination and a fast imaging algorithm for the periodic scatterer. Numerical examples for imaging three-dimensional periodic structures are presented to examine the efficiency of the method.

Prediction of internal geometry and tensile behavior of 3D woven solid structures by mathematical coding

2021 ◽  
pp. 152808372110013
Author(s):  
Vivek R Jayan ◽  
Lekhani Tripathi ◽  
Promoda Kumar Behera ◽  
Michal Petru ◽  
BK Behera
Keyword(s):  
Volume Fraction ◽  
Three Dimensional ◽  
Areal Density ◽  
Woven Fabrics ◽  
Tensile Behavior ◽  
Geometrical Parameters ◽  
Fiber Volume ◽  
Acceptable Error ◽  
Internal Geometry ◽  
Unit Cells

The internal geometry of composite material is one of the most important factors that influence its performance and service life. A new approach is proposed for the prediction of internal geometry and tensile behavior of the 3 D (three dimensional) woven fabrics by creating the unit cell using mathematical coding. In many technical applications, textile materials are subjected to rates of loading or straining that may be much greater in magnitude than the regular household applications of these materials. The main aim of this study is to provide a generalized method for all the structures. By mathematical coding, unit cells of 3 D woven orthogonal, warp interlock and angle interlock structures have been created. The study then focuses on developing code to analyze the geometrical parameters of the fabric like fabric thickness, areal density, and fiber volume fraction. Then, the tensile behavior of the coded 3 D structures is studied in Ansys platform and the results are compared with experimental values for authentication of geometrical parameters as well as for tensile behavior. The results show that the mathematical coding approach is a more efficient modeling technique with an acceptable error percentage.

MODIFICATION OF CASIMIR FORCES DUE TO BAND GAPS IN PERIODIC STRUCTURES

2002 ◽  
Vol 17 (06n07) ◽  
pp. 798-803 ◽  
Author(s):  
C. VILLARREAL ◽  
R. ESQUIVEL-SIRVENT ◽  
G. H. COCOLETZI
Keyword(s):  
Density Of States ◽  
Casimir Force ◽  
Periodic Structures ◽  
Local Density ◽  
Band Gaps ◽  
Basic Unit ◽  
Local Density Of States ◽  
Casimir Forces ◽  
Unit Cells ◽  
The Difference

The Casimir force between inhomogeneous slabs that exhibit a band-like structure is calculated. The slabs are made of basic unit cells each made of two layers of different materials. As the number of unit cells increases the Casimir force between the slabs changes, since the reflectivity develops a band-like structure characterized by frequency regions of high reflectivity. This is also evident in the difference of the local density of states between free and boundary distorted vacuum, that becomes maximum at frequencies corresponding to the band gaps. The calculations are restricted to vacuum modes with wave vectors perpendicular to the slabs.

The crystal structure of myoglobin. VI. Seal myoglobin

1960 ◽  
Vol 258 (1293) ◽  
pp. 181-201 ◽  
Keyword(s):  
Tertiary Structure ◽  
Heavy Atom ◽  
Three Dimensional ◽  
Sperm Whale ◽  
Fourier Synthesis ◽  
Heavy Atoms ◽  
Isomorphous Replacement ◽  
Unit Cells ◽  
The Common ◽  
Sperm Whale Myoglobin

Myoglobin from the common seal ( Phoca vitulina ) when crystallized from ammonium sulphate forms monoclinic crystals with space group the unit cell, a = 57·9Å, b = 29·6Å, c = 106·4Å, β = 102°15', contains four molecules. The method of isomorphous replacement has been used in an investigation of the centrosymmetric b -axis projection in which it has been possible to determine signs for nearly all the h0l reflexions having spacings greater than 4Å. Three independent heavy-atom derivatives were employed and the signs so determined have been used to compute a map of the electron density projected on the (010) plane. This projection has been interpreted in terms of the molecule of sperm-whale myoglobin, as deduced by Bodo, Dintzis, Kendrew & Wyckoff (1959) from a three-dimensional Fourier synthesis to 6Å resolution. The results of the interpretation show that the two myoglobin molecules are very similar in form (tertiary structure) in spite of the differences in their amino-acid composition. The relative orientation of the two unit cells with respect to the myoglobin molecule is given and a comparison is made of the positions of the heavy atoms in each molecule.

Robot Assisted Modal Analysis on a Stationary Bladed Wheel

2014 ◽  
Author(s):  
L. Bertini ◽  
B. Monelli ◽  
P. Neri ◽  
C. Santus ◽  
A. Guglielmo
Keyword(s):  
Three Dimensional ◽  
Laser Doppler Vibrometer ◽  
Testing Time ◽  
Real Representation ◽  
The Real ◽  
Complex Formulation ◽  
Frequency Matching ◽  
Input Multiple Output ◽  
Multiple Measurements ◽  
Single Input

This paper shows an automated procedure to experimentally find the eigenmodes of a bladed wheel with highly three-dimensional geometry. The stationary wheel is supported in free-free conditions, neglecting stress-stiffening effects. The single input / multiple output approach was followed. The vibration speed was measured by means of a laser-Doppler vibrometer, and an anthropomorphic robot was used for accurate orientation and positioning of the measuring laser beam, allowing multiple measurements during a limited testing time. The vibration at corresponding points on each blade was measured and the data elaborated in order to find the initial (lower frequency) modes. These modal shapes were then compared to finite element simulations and accurate frequency matching and exact number of nodal diameters obtained. Being the modes cyclically harmonic, the complex formulation could be attractive, being not affected by the angular phase of the mode representation. Nevertheless, stationary modes were experimentally detected, rather than rotating, and then the real representation was necessary. The discrete Fourier transform of the blade displacements easily allowed to find both the angular phase and the correct number of nodal diameters. Successful MAC experimental to analytical comparison was finally obtained with the real representation after introducing the proper angular phase for each mode.

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