  In this article, we have considered a nonlinear nonlocal time dependent
fourth order equation demonstrating the deformation of a thin and narrow
rectangular plate. We propose $C^1$ conforming virtual element method (VEM) of
arbitrary order, $k\ge2$, to approximate the model problem numerically. We
employ VEM to discretize the space variable and fully implicit scheme for
temporal variable. Well-posedness of the fully discrete scheme is proved under
certain conditions on the physical parameters, and we derive optimal order of
convergence in both space and time variable. Finally, numerical experiments are
presented to illustrate the behaviour of the proposed numerical scheme.
