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

2. 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” 2x3x5x

3. Buckling Transition (Lidmar, Mirny and Nelson 2003)  = 120  = 15000  = 230 Color indicates energy purple = zero Color indicates Gaussian curvature purple = negative

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

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

6. Effective stiffness to applied forces 12 

7. Large  limits: Vertex: Longitudinal deformation of triangular face K~aY/log(b  ) Edge: Ridge scaling (Witten et al.) in presence of force K~Y/  1/2 Face: Axial force on residual curvature K~Y/  2/3 Small  limit: Response of sphere to isotropic pressure: NK~12  Y Medium  : Soft mode concentrated on vertices

8. Conclusions Degeneracies governed by symmetry Nondegenerate mode softens at transition Vertex elasticity softens first at transition Vertices stiffen beyond the transition (Transition smeared by Gaussian curvature)