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Description

Understanding deformations of macroscopic thin plates and shells has a long and rich history, culminating with the Foeppl-von Karman equations in 1904, 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 fluctu- ations in thin elastic membranes fundamentally alter the long wavelength physics, as exemplified by experiments from the McEuen group at Cornell that twist and bend individual atomically-thin free-standing graphene sheets (with vK = 10^13!) We review here the remarkable properties of thermalized sheets, where enhancements of the bending rigidity by factors of ∼ 5000 have now been observed. We then move on to discuss thin amorphous spherical shells with a uniform nonzero curvature, accessible for example with soft matter experiments on diblock copolymers. This curvature couples the in-plane stretching modes with the out-of-plane undulation modes, giving rise to qualitative differences in the fluctuations of thermal spherical shells compared to flat membranes. Inter- esting effects arise because a shell can support a pressure difference between its interior and exterior. Thermal corrections to the predictions of classical shell theory for microscale shells diverge as the shell radius tends to infinity.