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Auxetic deformations and elliptic curves.

Ciprian S Borcea1, Ileana Streinu2

  • 1Rider University, Lawrenceville, NJ 08648.

Computer Aided Geometric Design
|January 16, 2019
PubMed
Summary

Researchers developed a simpler algorithm to detect auxetic behavior in flexible periodic frameworks, crucial for materials science. This method identifies auxetic properties in 3D frameworks, including zeolites, using elliptic curves and Aronhold invariants.

Keywords:
Aronhold invariantsauxetic deformationelliptic curveperiodic frameworkzeolites

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Area of Science:

  • Materials Science and Engineering
  • Computational Materials Science
  • Solid Mechanics

Background:

  • Auxetic behavior describes materials that widen when stretched.
  • Detecting auxetic behavior in flexible periodic bar-and-joint frameworks is computationally challenging.
  • Current methods rely on complex semi-definite programming.

Purpose of the Study:

  • To develop a simpler algorithmic approach for detecting auxetic behavior in 3D periodic bar-and-joint frameworks.
  • To investigate the relationship between auxetic deformations and elliptic curves.
  • To provide a faster method for recognizing auxetic capabilities in specific material structures.

Main Methods:

  • Developed a new algorithmic approach for a specific class of 3D periodic bar-and-joint frameworks.
  • Established a connection between auxetic deformations and the properties of an associated elliptic curve.
  • Utilized classical Aronhold invariants of the cubic form defining the elliptic curve for algorithm development.

Main Results:

  • Presented a simpler and more efficient algorithm compared to existing semi-definite programming methods.
  • Demonstrated that auxetic behavior in these frameworks is linked to specific elliptic curve properties.
  • Achieved a fast algorithm for recognizing auxetic capabilities in relevant 3D frameworks.

Conclusions:

  • The new algorithm offers a more accessible method for identifying auxetic behavior in important material structures like zeolites.
  • The link to elliptic curves provides theoretical insight and computational advantages.
  • This work simplifies the analysis of auxetic properties in periodic frameworks.