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Related Concept Videos

Shearing Strain01:20

Shearing Strain

692
The shearing strain represents a cubic element's angular change when subjected to shearing stress. This type of stress can transform a cube into an oblique parallelepiped without influencing normal strains. The cubic element experiences a significant transformation when exposed solely to shearing stress. Its shape alters from a perfect cube into a rhomboid, clearly demonstrating the effect of shearing strain. The degree of this strain is considered positive if it reduces the angle between...
692
Lattice Centering and Coordination Number02:33

Lattice Centering and Coordination Number

10.2K
The structure of a crystalline solid, whether a metal or not, is best described by considering its simplest repeating unit, which is referred to as its unit cell. The unit cell consists of lattice points that represent the locations of atoms or ions. The entire structure then consists of this unit cell repeating in three dimensions. The three different types of unit cells present in the cubic lattice are illustrated in Figure 1.
Types of Unit Cells
Imagine taking a large number of identical...
10.2K
Singularity Functions for Shear01:26

Singularity Functions for Shear

234
In structural analysis, singularity functions are crucial in simplifying the representation of shear forces in beams under discontinuous loading. These functions describe discontinuous  variations in shear force across a beam with varying loads by using a single mathematical expression, regardless of the complexity of the loading conditions. The singularity functions are derived from creating a free-body diagram of the beam and then making conceptual cuts at specific points to examine the...
234
Stress: General Loading Conditions01:15

Stress: General Loading Conditions

395
To grasp the intricacy of real-world conditions where multiple loads are applied simultaneously to a structure, one might visualize a section passing through a specific point within a body, aligned parallel to the xy plane. This section is subjected to various forces, including original loads, normal forces, and shearing forces.
The shearing force, possessing potential directionality within the plane of the section, is simplified into two component forces running parallel to the x and y axes....
395
Three-Dimensional Analysis of Strain01:29

Three-Dimensional Analysis of Strain

329
Three-dimensional strain analysis is crucial for understanding how materials deform under stress, particularly in elastic, homogeneous materials. This method employs principal stress axes to simplify complex stress states into more understandable forms. Subjected to stress, a small cubic element within a material either expands or contracts along these axes, transforming into a rectangular parallelepiped. This transformation effectively illustrates the material's deformation. The principal...
329
Shear on the Horizontal Face of a Beam Element01:16

Shear on the Horizontal Face of a Beam Element

311
To understand shear on the flat side of a prismatic beam element, consider the vertical and horizontal shearing forces, and the normal forces, acting on the element. The element's upper (U) and lower (L) sections, which are divided by the beam's neutral axis, are examined. The equilibrium of these forces is determined by applying the equilibrium equation, which helps identify the horizontal shearing force. This force is directly related to the bending moments and the cross-section's...
311

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Indirect Fabrication of Lattice Metals with Thin Sections Using Centrifugal Casting
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A new method for lattice reduction using directional and hyperplanar shearing.

Cyril Cayron1

  • 1Laboratory of Thermo Mechanical Metallurgy (LMTM), PX Group Chair, EPFL, Rue de la Maladière 71b, Neuchâtel, 2000, Switzerland.

Acta Crystallographica. Section A, Foundations and Advances
|December 30, 2021
PubMed
Summary
This summary is machine-generated.

This study introduces a geometric lattice reduction method using shears, offering similar results to the Lenstra-Lenstra-Lovász algorithm for moderate dimensions. The new technique quantifies reduction via

Keywords:
algorithmhyperplanelattice reductionleft inverse

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Area of Science:

  • Crystallography and computational geometry.
  • Number theory and algorithm development.

Background:

  • Lattice reduction is crucial in various fields, including cryptography and computational geometry.
  • Existing methods like the Lenstra-Lenstra-Lovász (LLL) algorithm have limitations in certain applications.
  • A need exists for alternative lattice reduction techniques with potentially different properties.

Purpose of the Study:

  • To present a novel geometric method for lattice reduction.
  • To introduce a new metric, 'basis rhombicity', for evaluating the reduction process.
  • To compare the effectiveness of this method against established algorithms like LLL.

Main Methods:

  • A geometric approach utilizing cycles of directional and hyperplanar shears.
  • Quantification of deviation from cubicity using 'basis rhombicity' (sum of absolute metric tensor elements).
  • Testing and comparison with the Lenstra-Lenstra-Lovász (LLL) algorithm up to dimension 20.

Main Results:

  • The proposed geometric method achieves reduction levels comparable to the LLL algorithm for tested dimensions.
  • 'Basis rhombicity' effectively measures the deviation from a cubic lattice during reduction.
  • The method demonstrates applicability to reducing unit cells associated with hyperplanes.

Conclusions:

  • The presented geometric lattice reduction method is a viable alternative to existing algorithms.
  • The 'basis rhombicity' parameter provides a useful measure for assessing lattice quality.
  • This technique offers potential for applications involving unit cell reduction in specific geometric contexts.