scholarly journals Finite auxetic deformations of plane tessellations

2013 ◽  
Vol 469 (2149) ◽  
pp. 20120465 ◽  
Author(s):  
Holger Mitschke ◽  
Vanessa Robins ◽  
Klaus Mecke ◽  
Gerd E. Schröder-Turk
Keyword(s):  
Deformation Behaviour ◽  
Deformation Mode ◽  
Energy Functional ◽  
Finite Deformations ◽  
Hexagonal Symmetry ◽  
Quadratic Equations ◽  
Crystallographic Symmetry ◽  
Auxetic Structures ◽  
Symmetry Constraints ◽  
Star Polygons

We systematically analyse the mechanical deformation behaviour, in particular Poisson's ratio, of floppy bar-and-joint frameworks based on periodic tessellations of the plane. For frameworks with more than one deformation mode, crystallographic symmetry constraints or minimization of an angular vertex energy functional are used to lift this ambiguity. Our analysis allows for systematic searches for auxetic mechanisms in archives of tessellations; applied to the class of one- or two-uniform tessellations by regular or star polygons, we find two auxetic structures of hexagonal symmetry and demonstrate that several other tessellations become auxetic when retaining symmetries during the deformation, in some cases with large negative Poisson ratios ν <−1 for a specific lattice direction. We often find a transition to negative Poisson ratios at finite deformations for several tessellations, even if the undeformed tessellation is infinitesimally non-auxetic. Our numerical scheme is based on a solution of the quadratic equations enforcing constant edge lengths by a Newton method, with periodicity enforced by boundary conditions.

Mathematical aspects of molecular replacement. V. Isolating feasible regions in motion spaces

2020 ◽  
Vol 76 (2) ◽  
pp. 145-162
Author(s):  
Bernard Shiffman ◽  
Shengnan Lyu ◽  
Gregory S. Chirikjian
Keyword(s):  
Configuration Space ◽  
A Priori ◽  
Rigid Bodies ◽  
Molecular Replacement ◽  
Space Group ◽  
Fundamental Domains ◽  
Crystallographic Symmetry ◽  
Symmetry Constraints ◽  
Translation Space ◽  
The Right

This paper mathematically characterizes the tiny feasible regions within the vast 6D rotation–translation space in a full molecular replacement (MR) search. The capability to a priori isolate such regions is potentially important for enhancing robustness and efficiency in computational phasing in macromolecular crystallography (MX). The previous four papers in this series have concentrated on the properties of the full configuration space of rigid bodies that move relative to each other with crystallographic symmetry constraints. In particular, it was shown that the configuration space of interest in this problem is the right-coset space Γ\G, where Γ is the space group of the chiral macromolecular crystal and G is the group of rigid-body motions, and that fundamental domains F Γ\G can be realized in many ways that have interesting algebraic and geometric properties. The cost function in MR methods can be viewed as a function on these fundamental domains. This, the fifth and final paper in this series, articulates the constraints that bodies packed with crystallographic symmetry must obey. It is shown that these constraints define a thin feasible set inside a motion space and that they fall into two categories: (i) the bodies must not interpenetrate, thereby excluding so-called `collision zones' from consideration in MR searches; (ii) the bodies must be in contact with a sufficient number of neighbors so as to form a rigid network leading to a physically realizable crystal. In this paper, these constraints are applied using ellipsoidal proxies for proteins to bound the feasible regions. It is shown that the volume of these feasible regions is small relative to the total volume of the motion space, which justifies the use of ellipsoids as proxies for complex proteins in MR searches, and this is demonstrated with P1 (the simplest space group) and with P212121 (the most common space group in MX).

Modelling of the Strain Hardening Behaviour of Die-Cast Magnesium-Aluminium-Rare Earth Alloy

2020 ◽  
Vol 35 ◽  
pp. 1-8
Author(s):  
Hua Qian Ang
Keyword(s):  
Strain Hardening ◽  
Tensile Deformation ◽  
Deformation Behaviour ◽  
Strain Curve ◽  
Deformation Mode ◽  
Prismatic Slip ◽  
Stress Strain Curve ◽  
Stress Strain ◽  
Die Cast ◽  
Cast Magnesium

The tensile deformation behaviour of magnesium alloy AE44 (Mg-4Al-4RE) under strain rates ranging from 10-6 to 10-1 s-1 has been investigated. Present study shows that the deformation mode begins with the activation of elastic (Stage 1), followed by <a> basal slip and twinning (Stage 2), <a> prismatic slip (Stage 3) and finally to <c+a> pyramidal slip (Stage 4). The commencement of these deformation mechanisms results in four distinct stages of strain hardening in the stress-strain curve. In this work, the four stages of deformation behaviour are modelled, and an empirical equation is proposed to predict the entire stress-strain curve. Overall, the model predictions are in good agreement with the experimental data. This study on the decomposition of stress-strain curve into four stages provides insights into the contribution of individual deformation mechanism to the overall deformation behaviour and opens a new way to assess mechanical properties of die-cast magnesium alloys.

The Role of Texture, Temperature and Strain Rate in the Activity of Deformation Twinning

2005 ◽  
Vol 495-497 ◽  
pp. 1037-1042 ◽  
Author(s):  
Donald W. Brown ◽  
Sean R. Agnew ◽  
S.P. Abeln ◽  
W.R. Blumenthal ◽  
Mark A.M. Bourke ◽  
...  
Keyword(s):  
Mechanical Properties ◽  
Plastic Deformation ◽  
Deformation Mode ◽  
Relative Activity ◽  
Deformation Twinning ◽  
Underlying Structure ◽  
Cubic Metals ◽  
Pyramidal Slip ◽  
Crystallographic Symmetry

Plastic deformation in cubic metals is relatively simple due to the high crystallographic symmetry of the underlying structure. Typically, one unique slip mode can provide arbitrary deformation. This is not true in lower symmetry hexagonal metals, where prismatic and basal slip (the usual favored modes) are insufficient to provide arbitrary deformation. Often, either pyramidal slip and/or deformation twinning must be activated to accommodate imposed plastic deformation. The varied difficulty of activating each of these deformation mechanisms results in a highly anisotropic yield surface and subsequent mechanical properties. Further, the relative activity of each deformation mode may be manipulated through control of the initial crystallographic texture, opening new opportunities for the optimization of mechanical properties for a given application.

Mechanical Properties of As-Grown and Ion-Beam-Modified GaN Films

2000 ◽  
Vol 649 ◽  
Author(s):  
S.O. Kucheyev ◽  
J.E. Bradby ◽  
J.S. Williams ◽  
M.V. Swain ◽  
M. Toth ◽  
...  
Keyword(s):  
Mechanical Properties ◽  
Deformation Behavior ◽  
Ion Beam ◽  
Deformation Behaviour ◽  
Deformation Mode ◽  
Spherical Indenter ◽  
Force Microscopy ◽  
Atomic Force ◽  
Slip Bands ◽  
Doping Type

ABSTRACTThe deformation behavior of as-grown and ion-beam-modified wurtzite GaN films is studied by a nanoindentation with a spherical indenter. Atomic force microscopy (AFM) and cathodoluminescence are used to characterize the deformation mode. No systematic dependence of the mechanical properties on the film thickness ( at least for thicknesses from 1.8 to 4 μm) as well as on doping type is observed. Results strongly suggest that (i) slip is the major contributor to the plastic deformatio of crystalline GaN and (ii) slip nucleation (rater than a phae transformation) is responsible for “pop-in” events observed during loading. Indentation with an ∼ 4.2 μm radius spherical indenter at maximum loads up to 900 mN does not produce any cracking visible by AFM in crystalline GaN. Instead, under such loads, indentation results in a pronounced elevation of the material around the impression. Implantation disorder dramatically changes the deformation behaviour of GaN. In particular, implanation-produced defects in crystalline GaN syppress (i) “pop-in” events during loading, (ii) slip bands observed by AFM, and (iii) the plastic component of deformation. GaN amorphized by in bombardment exhibits plastic flow even for very low loads. The values of hardness and elastic modulus of amorphous GaN are dramatically reduced compared to those of as-grown GaN.

scholarly journals On the study of the dynamic response in 3D additively manufactured auxetic structures

2021 ◽  
Vol 250 ◽  
pp. 06004
Author(s):  
Niklas Fjeldberg ◽  
Jesús Pernas ◽  
David Varas ◽  
Jordi Martín
Keyword(s):  
Dynamic Response ◽  
Split Hopkinson Pressure Bar ◽  
Deformation Behaviour ◽  
Experimental Tests ◽  
Hopkinson Pressure Bar ◽  
Deformation Response ◽  
Unit Cells ◽  
Auxetic Structures ◽  
Split Hopkinson ◽  
Pressure Bar

The main aim of this work is to analyse the quasi-static and the dynamic response and deformation behaviour of polymeric auxetic structures manufactured with an SLA (stereolitography) additive manufacturing technique. To this end, different experimental tests were performed using a universal servo-hydraulic instron machine and a Split Hopkinson Pressure Bar (SHPB). The study focuses on the understanding of the mechanical deformation response of the metamaterial depending on the type of load, the amount and distribution of the unit cells and the strain rate applied.

A Comparison of the microstructure and tensile behaviour of irradiated fcc and bcc metals

1998 ◽  
Vol 540 ◽  
Author(s):  
N. Baluc ◽  
C. Bailat ◽  
Y. Dai ◽  
M.I. Luppo ◽  
R. Schaublin ◽  
...  
Keyword(s):  
Proton Irradiation ◽  
Deformation Behaviour ◽  
Deformation Mode ◽  
High Energy ◽  
Tensile Behaviour ◽  
High Energy Proton ◽  
Bcc Metals ◽  
Energy Proton ◽  
Defect Accumulation ◽  
Fission Neutrons

AbstractAs part of an on-going research program, findings are presented from a comparison of the microstructures and associated tensile properties of fcc and bcc materials after high energy proton irradiation, to fluences between 10−4 and 1 dpa, at 300-320 K. Results for this comparison between Cu, Pd, 304 and 316 stainless steel on one side and Fe and the F82H ferritic-martensitic low activation steel on the other are discussed, showing a strong difference in defect accumulation behaviour between the differing crystal structures. The overall deformation behaviour is similar, with an initial localised deformation taking place in all cases, even though the actual deformation mode itself might be different. Furthermore, a comparison is made with some of the materials that have also been irradiated with fission neutrons, showing no influence of the PKA spectra for these irradiation conditions.

Using crystallographic symmetry in diffraction and contrast analysis

1989 ◽  
Vol 47 ◽  
pp. 504-505
Author(s):  
U. Dahmen ◽  
K.H. Westmacott
Keyword(s):  
Electron Microscopy ◽  
High Symmetry ◽  
Two Phase ◽  
Tem Analysis ◽  
Crystallographic Symmetry ◽  
The Matrix ◽  
Contrast Analysis ◽  
Practical Tool ◽  
Convergent Beam ◽  
Beam Diffraction

Despite the increased use of convergent beam diffraction, symmetry concepts in their more general form are not commonly applied as a practical tool in electron microscopy. Crystal symmetry provides an abundance of information that can be used to facilitate and improve the TEM analysis of crystalline solids. This paper draws attention to some aspects of symmetry that can be put to practical use in the analysis of structures and morphologies of two-phase materials.It has been shown that the symmetry of the matrix that relates different variants of a precipitate can be used to determine the axis of needle- or lath-shaped precipitates or the habit plane of plate-shaped precipitates. By tilting to a special high symmetry orientation of the matrix and by measuring angles between symmetry-related variants of the precipitate it is possible to find their habit from a single micrograph.

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