  We consider the tension-induced non-linearities of mechanical resonators, and
derive the Hamiltonian of the flexural modes up to the fourth order in the
position operators. This tension can be controlled by a nearby gate voltage. We
focus on systems which allow large deformations $u(x)\gg h$ compared to the
thickness $h$ of the resonator and show that in this case the third-order
coupling can become non-zero due to the induced dc deformation and offers the
possibility to realize equations of motion encountered in optomechanics. The
fourth-order coupling is relevant especially for relatively low voltages. It
can be detected by accessing the Duffing regime, and by measuring frequency
shifts due to mode-mode coupling.
