Mechanical properties of warped membranes

We explore how a frozen background metric affects the mechanical properties of solid planar membranes at zero temperature. Our focus is a special class of ``warped membranes'' with a preferred random height profile characterized by random Gaussian variables $h(q)$ in Fourier space with zero mean and variance $< |h(q)|^2 > \sim q^{-m}$. Using statistical physics tools to treat this quenched random disorder, we find that in the linear response regime, similar to thermally fluctuating polymerized membranes, an increasing scale-dependent effective bending rigidity, while the Young and the shear moduli are reduced. Compared to flat plates of the same thickness $t$, the bending rigidity of warped membranes is increased by a factor $\sim h_v/t$ while the in-plane elastic moduli are reduced by $\sim t/h_v$, where $h_v =\sqrt{ < |h(x)|^2 > }$ describes the frozen height fluctuations. Interestingly, $h_v$ is system size dependent for warped membranes characterized with $m>2$. Numerical results show good agreement with theoretical predictions, which are now being tested experimentally, where warped membranes are prepared with 3D printers.