  We study circular nanomechanical graphene resonators by means of continuum
elasticity theory, treating them as membranes. We derive dynamic equations for
the flexural mode amplitudes. Due to geometrical nonlinearity these can be
modeled by coupled Duffing equations. By solving the Airy stress problem we
obtain analytic expressions for eigenfrequencies and nonlinear coefficients as
functions of radius, suspension height, initial tension, back-gate voltage and
elastic constants, which we compare with finite element simulations. Using
perturbation theory, we show that it is necessary to include the effects of the
non-uniform stress distribution for finite deflections. This correctly
reproduces the spectrum and frequency tuning of the resonator, including
frequency crossings.
