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Related Concept Videos

Transformation of Plane Strain01:12

Transformation of Plane Strain

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When analyzing elongated structures like bars subjected to uniformly distributed loads, it is essential to understand the transformation of plane strain when coordinate axes are rotated. This transformation helps to assess how material deformation characteristics vary with orientation, which is crucial in materials science and structural engineering.
Under plane strain conditions, typical for members where one dimension significantly exceeds the others, deformations and resultant strains are...
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Three-Dimensional Analysis of Strain01:29

Three-Dimensional Analysis of Strain

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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...
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Shearing Strain01:20

Shearing Strain

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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 the...
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Cyclohexane does not exist in a planar form due to the high angle and torsional strain it would experience in the planar structure. Instead, it adopts non-planar chair and boat conformations.
The chair form is the most stable and derives its name from its resemblance to the “easy chair.” In the chair conformation, two carbon atoms are arranged out-of-plane — one above and one below, minimizing the torsional strain. In the chair form, the bond angle is very close to the ideal...
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Sigmatropic rearrangements are a class of pericyclic reactions in which a σ bond migrates from one part of a π system to another. These are intramolecular rearrangements where the total number of σ and π bonds remain unchanged.
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Consider a conductor in electrostatic equilibrium. The net electric field inside a conductor vanishes, and extra charges on the conductor reside on its outer surface, regardless of where they originate.
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Related Experiment Video

Updated: Nov 17, 2025

Fabrication of Gate-tunable Graphene Devices for Scanning Tunneling Microscopy Studies with Coulomb Impurities
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Numerical quasiconformal transformations for electron dynamics on strained graphene surfaces.

François Fillion-Gourdeau1,2, Emmanuel Lorin3,4, Steve MacLean1,2,5

  • 1Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1.

Physical Review. E
|February 19, 2021
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Summary

We mathematically modeled low-energy electrons on strained graphene surfaces using the Dirac equation. Our methods simplify calculations and show strained graphene can focus electron wave packets.

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Area of Science:

  • Condensed Matter Physics
  • Materials Science
  • Theoretical Physics

Background:

  • Modeling electron dynamics on graphene surfaces is crucial for understanding its electronic properties.
  • Deformations in graphene introduce curvature, complicating the mathematical description via the Dirac equation.

Purpose of the Study:

  • To develop simplified mathematical models for low-energy electron dynamics on strained graphene surfaces.
  • To investigate the focusing effects of localized surface deformations on electron wave packets.

Main Methods:

  • Mathematical modeling using the Dirac equation in curved spacetime.
  • Employing two simplification strategies: diagonal metric approximation and isothermal coordinates via quasiconformal transformations (Beltrami equation).
  • Utilizing a least-squares finite-element method for the Beltrami equation and a pseudospectral method for the Dirac equation.

Main Results:

  • The Dirac equation in curved space-time was effectively simplified for strained graphene.
  • A numerical scheme was developed to solve the Beltrami and Dirac equations accurately.
  • Scattering of electrons on Gaussian-shaped deformations demonstrated focusing of electron wave packets.

Conclusions:

  • The developed methods provide a tractable approach to studying electron dynamics on deformed graphene.
  • Localized strained regions on graphene surfaces can act as electron focusing lenses.