  The structural flexibility of low dimensional nanomaterials offers unique
opportunities for studying the impact of strain on their physical properties
and for developing innovative devices utilizing strain engineering. A key
towards such goals is a device platform which allows the independent tuning and
reliable calibration of the strain. Here we report the fabrication and
characterization of graphene nanoelectromechanical resonators(GNEMRs) on
flexible substrates. Combining substrate bending and electrostatic gating, we
achieve the independent tuning of the strain and sagging in graphene and
explore the nonlinear dynamics over a wide parameter space. Analytical and
numerical studies of a continuum mechanics model, including the competing
higher order nonlinear terms, reveal a comprehensive nonlinear dynamics phase
diagram, which quantitatively explains the complex behaviors of GNEMRs.
