Supercritical and Subcritical Pitchfork Bifurcations in a Buckling Problem for a Graphene Sheet Between two Rigid Substrates
Buchtel College of Arts and Sciences
Date of Last Revision
Number of Credits
Bachelor of Science
Date of Expected Graduation
In this paper we study a model of the buckling of a sheet of graphene between two rigid substrates. We seek to understand the buckling of the sheet as the substrate separation is varied with a fixed load on each end of the sheet. We write down the expression for total energy of the system and from it derive a 2-point nonlinear boundary-value problem whose solutions are equilibrium configurations of the sheet. We cannot get an explicit solution. Instead, we perform a bifurcation analysis by using asymptotics to approximate solutions on the bifurcating branches near the bifurcation points. The bifurcating parameter is the separation between the rigid substrates. We find that the bifurcations are supercritical or subcritical pitchfork bifurcations. We perform a parametric study to understand how the nature of the bifurcations is influenced by other physical parameters in the problem. This leads to insight on the orientation of the buckled graphene sheet depending on substrate separation.
J. Patrick Wilber
Honors Faculty Advisor
Curtis B. Clemons
Grdadolnik, Jake, "Supercritical and Subcritical Pitchfork Bifurcations in a Buckling Problem for a Graphene Sheet Between two Rigid Substrates" (2021). Williams Honors College, Honors Research Projects. 1269.
Ordinary Differential Equations and Applied Dynamics Commons, Other Applied Mathematics Commons