Statistical Mechanics of Mutilated Sheets and Shells
Understanding deformations of macroscopic thin plates and shells has a long and rich history, culminating with the Foeppl-von Karman equations in 1904, a precursor of general relativity characterized by a dimensionless coupling constant (the "Foeppl-von Karman number") that can easily reach vK = 10^7 in an ordinary sheet of writing paper. However, thermal fluctuations in thin elastic membranes fundamentally alter the long wavelength physics, as exemplified by experiments that twist and bend individual atomically-thin free-standing graphene sheets (with vK = 10^13!) We first discuss thermalized graphene sheets. With this system, it may be possible to study the quantum mechanics of two dimensional Dirac massless fermions in a fluctuating curved space whose dynamics resembles a simplified form of general relativity. We then move on to analyze the physics of sheets mutilated with holes, puckers and stitches. Holes dramatically reduce the crumpling temperature, while puckers and stitches lead to Ising-like phase transitions that strongly affect the physics of the fluctuating sheet. We also discuss thin shells with a background curvature that couples in-plane stretching modes with the out-of-plane undulation modes, giving rise to qualitative differences between thermalized spherical shells compared to flat membranes.
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