  We propose a mathematical model to describe the athermal fluctuations of thin
sheets driven by the type of random driving that might be experienced prior to
weak crumpling. The model is obtained by merging the F\"oppl-von K\'arm\'an
equations from elasticity theory with techniques from out-of-equilibrium
statistical physics to obtain a nonlinear strongly coupled $\phi^{4}$-Langevin
field equation with spatially varying kernel. With the aid of the
self-consistent expansion (SCE), this equation is analytically solved for the
structure factor of a fluctuating sheet. In contrast to previous research which
has suggested that the structure factor follows an anomalous power-law, we find
that the structure factor in fact obeys a logarithmically corrected rational
function. Numerical simulations of our model confirm the accuracy of our
analytical solution.
