Soft modes near the buckling transition of icosahedral shells Outline: Mass and spring model Buckling transition Eigenvalue spectra Soft mode Effective stiffness (vertex/edge/face) with Jack Lidmar and David Nelson

Mass and Spring Model a ksks n1n1 n2n2 E s =(1/2)k s Σ(|r 1 -r 2 |-a) 2 E b =(1/2)k b Σ|n 1 -n 2 | 2 kbkb Dimensionless combination:  ~ k s R 2 /k b “Foppl-von Karman Number”

Buckling Transition Lidmar, Mirny and Nelson (2003)  = 120  =  = 230 Color indicates energy purple indicates zero Color indicates Gaussian curvature purple = negative

Eigenvalue spectrum of model (h=4,k=0)

Nondegenerate eigenvalues of model (h=8,k=0)

Effective stiffness to applied forces 12 

Conclusions Degeneracies governed by symmetry Nondegenerate mode softens at transition Vertex elasticity softens first at transition Vertices are stiffest beyond the transition