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.

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 Methods for tuning plasmonic and photonic optical resonances in high surface area porous electrodes

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Lauren M. Otto ◽  
E. Ashley Gaulding ◽  
Christopher T. Chen ◽  
Tevye R. Kuykendall ◽  
Aeron T. Hammack ◽  
...  
Keyword(s):  
Photonic Crystal ◽  
Photonic Crystals ◽  
Titanium Nitride ◽  
Surface Area ◽  
Periodic Structures ◽  
High Surface ◽  
High Surface Area ◽  
Three Dimensional ◽  
Porous Electrodes ◽  
Wide Range

AbstractSurface plasmons have found a wide range of applications in plasmonic and nanophotonic devices. The combination of plasmonics with three-dimensional photonic crystals has enormous potential for the efficient localization of light in high surface area photoelectrodes. However, the metals traditionally used for plasmonics are difficult to form into three-dimensional periodic structures and have limited optical penetration depth at operational frequencies, which limits their use in nanofabricated photonic crystal devices. The recent decade has seen an expansion of the plasmonic material portfolio into conducting ceramics, driven by their potential for improved stability, and their conformal growth via atomic layer deposition has been established. In this work, we have created three-dimensional photonic crystals with an ultrathin plasmonic titanium nitride coating that preserves photonic activity. Plasmonic titanium nitride enhances optical fields within the photonic electrode while maintaining sufficient light penetration. Additionally, we show that post-growth annealing can tune the plasmonic resonance of titanium nitride to overlap with the photonic resonance, potentially enabling coupled-phenomena applications for these three-dimensional nanophotonic systems. Through characterization of the tuning knobs of bead size, deposition temperature and cycle count, and annealing conditions, we can create an electrically- and plasmonically-active photonic crystal as-desired for a particular application of choice.

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.

scholarly journals Homogenization of fully nonlinear rod lattice structures: on the size of the RVE and micro structural instabilities

2021 ◽  
Author(s):  
Ludwig Herrnböck ◽  
Paul Steinmann
Keyword(s):  
Large Deformations ◽  
Mechanical Response ◽  
Three Dimensional ◽  
Homogenization Method ◽  
Volume Element ◽  
Lattice Structures ◽  
Micro Scale ◽  
Macro Scale ◽  
Unit Cells ◽  
Structural Instabilities

AbstractThis work investigates the possibility of applying two-scale computational homogenization to rod lattice structures emerging, for instance, from additive manufacturing. The influence of the number of unit cells within the representative volume element (RVE), thus, the RVE’s size on the homogenized mechanical response is studied for occurring microscopic structural instabilities. Therein, the macro-scale, described in terms of three-dimensional continuum mechanics, is coupled to the micro-scale described by geometrically exact rods, enabling arbitrary large deformations and rotations. A special feature of the presented framework is that the rods building the lattice structures are not restricted to deform purely elastically but may deform inelastically. The mechanical response of lattice structures is investigated by applying the developed homogenization method to an exemplary lattice. Under special loads the structure reaches an instable state and may buckle. The appearance of instabilities depends on the geometric properties of the lattice’s underlying rods and the RVE’s size.

scholarly journals Application of Photopolymer Materials in Holographic Technologies

Polymers ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 2020 ◽  
Author(s):  
Nadezhda Vorzobova ◽  
Pavel Sokolov
Keyword(s):  
3D Printing ◽  
Periodic Structures ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Polymer Materials ◽  
Solar Concentrators ◽  
Three Dimensional Objects ◽  
Wide Range ◽  
Material Fabrication ◽  
Dimensional Object

The possibility of the application of acrylate compositions and Bayfol HX photopolymers in holographic technologies is considered. The holographic characteristics of materials, their advantages, and limitations in relation to the tasks of obtaining holographic elements based on periodic structures are given. The conditions for obtaining controlled two and multichannel diffraction beam splitters are determined with advantages in terms of the simplicity of the fabrication process. The diffraction and selective properties of volume and hybrid periodic structures by radiation incidence in a wide range of angles in three-dimensional space are investigated, and new properties are identified that are of interest for the development of elements of holographic solar concentrators with advantages in the material used and the range of incidence angles. A new application of polymer materials in a new method of holographic 3D printing for polymer objects with arbitrary shape fabrication based on the projection of a holographic image of the object into the volume of photopolymerizable material is proposed, the advantage of which, relative to additive 3D printing technologies, is the elimination of the sequential synthesis of a three-dimensional object. The factors determining the requirements for the material, fabrication conditions, and properties of three-dimensional objects are identified and investigated.

Fabrication defects and limitations of AlSi10Mg lattice structures manufactured by selective laser melting

2021 ◽  
pp. 146442072110127
Author(s):  
Enrico Ossola ◽  
Andrew A Shapiro ◽  
Andre Pate ◽  
Samad Firdosy ◽  
Eugenio Brusa ◽  
...  
Keyword(s):  
Process Parameters ◽  
Cross Sections ◽  
High Surface ◽  
Three Dimensional ◽  
Lattice Structures ◽  
Unit Cells ◽  
Octet Truss ◽  
Overhang Length ◽  
Fabrication Defects

Additive manufacturing has enabled the production of lattice structures with tailored mechanical properties. However, process limitations still exist, affecting the quality of the struts, practically limiting sizes and types of printable unit cells. Typically, long, thin, unsupported horizontal struts exhibit large deviations from ideal geometries, due to high surface roughness and internal porosity. AlSi10Mg specimens were designed and fabricated by laser powder bed fusion to investigate the role of strut orientation, size, and overhang length using different sets of process parameters. Visual inspection, three-dimensional scanning, and metallographic inspection of the cross-sections were performed. A quality control methodology based on dimensional and geometric tolerances has been defined in order to quantitatively characterize the quality of the struts. Optimized process parameters were selected and used to fabricate octet-truss specimens which were then characterized by compression testing.

The Design Process of Additively Manufactured Mesoscale Lattice Structures: A Review

2018 ◽  
Vol 18 (4) ◽  
Author(s):  
Francesco Tamburrino ◽  
Serena Graziosi ◽  
Monica Bordegoni
Keyword(s):  
Design Process ◽  
Three Dimensional ◽  
Lattice Structures ◽  
Functional Requirements ◽  
Software Developers ◽  
Holistic View ◽  
Whole Process ◽  
Unit Cells ◽  
3D Printed ◽  
The Impact

This review focuses on the design process of additively manufactured mesoscale lattice structures (MSLSs). They are arrays of three-dimensional (3D) printed trussed unit cells, whose dimensions span from 0.1 to 10.0 mm. This study intends to detail the phases of the MSLSs design process (with a particular focus on MSLSs whose unit cells are made up of a network of struts and nodes), proposing an integrated and holistic view of it, which is currently lacking in the literature. It aims at guiding designers' decisions with respect to the settled functional requirements and the manufacturing constraints. It also aims to provide an overview for software developers and researchers concerning the design approaches and strategies currently available. A further objective of this review is to stimulate researchers in exploring new MSLSs functionalities, consciously considering the impact of each design phase on the whole process, and on the manufactured product.

scholarly journals Design and fabrication considerations for three dimensional scaffold structures

2017 ◽  
Vol 2 (2) ◽  
pp. 120 ◽  
Author(s):  
Mazher Iqbal Mohammed
Keyword(s):  
3D Printing ◽  
Three Dimensional ◽  
Porous Structures ◽  
Dimensional Changes ◽  
Design Engineers ◽  
Wide Range ◽  
Unit Cells ◽  
Traditional Manufacturing ◽  
Optimal Material ◽  
Accuracy And Repeatability

<p>Porous three dimensional structures have seen extensive investigation among design engineers for a wide range of novel applications. The fabrication of such designs would not be possible using traditional manufacturing approaches owing to the dimensional intricacy of such structures, but have now become a distinct possibility owing to the maturity of 3D printing technologies. In this study, we have examined the creation of novel unit cells from mathematic surface renderings as a basis for creating tailored porous structures, before realising the final designs through Fuse Deposition Modelling (FDM) 3D printing. We examined the use of Gyroid and Schwarz primitive (P) surfaces to create novel unit cells not typically found in design software libraries. We then transpose these structures into several test geometries comprising a cylinder, cuboid and tetrahedron, which will adequately test limits of design and fabrication in regular and irregularly shaped structures. It was found that the porosities of the resulting models could be adjusted through discrete dimensional changes in the unit cell and digital wrapping procedures. It was also found that models could be fabricated using FDM printing to a minimum pore diameter of approximately 1mm with a high degree of accuracy and repeatability. Ultimately this work will provide guidance to engineering's when creating porous structures and could find usefulness in applications where optimal material usage versus porosity are required, such as in high throughput 3D fluidic applications, such as heat exchangers and tissue engineered structures.</p>

Reconfigurable Modular Chain: A Reversible Material for Folding Three-Dimensional Lattice Structures

2017 ◽  
Vol 9 (2) ◽  
Author(s):  
Zhe Xu ◽  
Connor McCann ◽  
Aaron M. Dollar
Keyword(s):  
Structural Design ◽  
Three Dimensional ◽  
Space Structures ◽  
Modular Robots ◽  
Lattice Structures ◽  
Rhombic Dodecahedron ◽  
Wide Range ◽  
Different Shapes ◽  
High Level ◽  
Functional Components

A wide range of engineering applications, ranging from civil to space structures, could benefit from the ability to construct material-efficient lattices that are easily reconfigurable. The challenge preventing modular robots from being applied at large scales is mainly the high level of complexity involved in duplicating a large number of highly integrated module units. We believe that reconfigurability can be more effectively achieved at larger scales by separating the structural design from the rest of the functional components. To this end, we propose a modular chainlike structure of links and connector nodes that can be used to fold a wide range of two-dimensional (2D) or three-dimensional (3D) structural lattices that can be easily disassembled and reconfigured when desired. The node geometry consists of a diamondlike shape that is one-twelfth of a rhombic dodecahedron, with magnets embedded on the faces to allow a forceful and self-aligning connection with neighboring links. After describing the concept and design, we demonstrate a prototype consisting of 350 links and experimentally show that objects with different shapes can be successfully approximated by our proposed chain design.

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