Soft modes near the buckling transition of icosahedral shells Outline: Mass and spring model...

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

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

Mass and Spring Model

a

ks

n1 n2

Es=(1/2)ksΣ(|r1-r2|-a)2

Eb=(1/2)kbΣ|n1-n2|2

kb

Dimensionless combination: ~ ks R2/kb

“Foppl-von Karman Number”2x 3x 5x

Buckling Transition(Lidmar, Mirny and Nelson 2003)

= 120 = 15000 = 230

Color indicates energypurple = zero

Color indicates Gaussian curvaturepurple = negative

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

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

Effective stiffness to applied forces

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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~12Y

Medium :Soft mode concentrated on vertices

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)