A constitutive equation for graphene based on density functional theory

Mei Xu, Jeffrey T. Paci, Jay Oswald, Ted Belytschko

Research output: Contribution to journalArticlepeer-review

57 Scopus citations


An anisotropic strain energy function is proposed for tensile loading in graphene that provides a nonlinear, hyperelastic constitutive equation. In the proposed function, the energy depends on the principal invariants of the right Cauchy-Green tensor and the strains in the zigzag and armchair directions. The use of the zigzag and armchair strains gives the model the ability to account for anisotropic behavior at moderate deformations. The constitutive law parameters are determined by a least squares fit to the energies predicted by density functional theory (DFT) calculations, and a good match is obtained to the DFT results for zigzag and armchair graphene sheets with various loading combinations. The law is applied in a continuum calculation of nanoindentation of a graphene membrane. The force-deflection predicted with this model show excellent agreement with analogous experimental results, thus providing a strong link between DFT calculations and nanoexperiments.

Original languageEnglish (US)
Pages (from-to)2582-2589
Number of pages8
JournalInternational Journal of Solids and Structures
Issue number18
StatePublished - Sep 15 2012


  • Anisotropic
  • Elastic constants
  • Graphene
  • Nanoindentation

ASJC Scopus subject areas

  • Modeling and Simulation
  • Materials Science(all)
  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering
  • Applied Mathematics


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