College
College of Engineering and Polymer Science
Date of Last Revision
2026-04-28 12:34:15
Major
Applied Mathematics
Honors Course
498
Number of Credits
2
Degree Name
Bachelor of Science
Date of Expected Graduation
Spring 2026
Abstract
Graphene, a single-atom-thick layer of carbon arranged in a hexagonal lattice, exhibits exceptional mechanical, electrical, and thermal properties that make it a promising material for a wide range of engineering applications. This paper presents a mathematical framework for modeling the mechanical behavior of graphene, with a focus on atomistic-to-continuum approaches. We begin with a onedimensional Frenkel-Kontorova model that represents graphene as a discrete chain of particles interacting with both their nearest neighbors through harmonic spring potentials and an underlying substrate through van der Waals forces. Numerical simulations of this discrete model demonstrate the commensurate-toincommensurate phase transition, revealing how geometric mismatch between the graphene layer and substrate drives the formation of periodic structural defects known as domain walls. We then derive a continuum energy functional from the discrete model by applying a Taylor series expansion and identifying the discrete sums as Riemann integrals. This upscaling procedure yields a continuous total energy functional that captures the same physics as the atomistic model while being far more tractable analytically. These results establish a mathematical connection between atomic-scale interactions and a continuum description of graphene's mechanical behavior.
Research Sponsor
J. Patrick Wilber
First Reader
Stefan Forcey
Second Reader
Andreas Aristotelous
Honors Faculty Advisor
James Cossey
Proprietary and/or Confidential Information
No
Community Engaged Scholarship
No
Recommended Citation
Sanor, Douglas M., "Mathematical Model of Graphene" (2026). Williams Honors College, Honors Research Projects. 2137.
https://ideaexchange.uakron.edu/honors_research_projects/2137
Included in
Ordinary Differential Equations and Applied Dynamics Commons, Other Applied Mathematics Commons, Partial Differential Equations Commons