Ultrasonic wave propagation in a graphene sheet, which is embedded in an elastic
medium, is studied using nonlocal elasticity theory incorporating small-scale effects.
The graphene sheet is modeled as an one-atom thick isotropic plate and the elastic
medium/substrate is modeled as distributed springs. For this model, the nonlocal
governing differential equations of motion are derived from the minimization of the
total potential energy of the entire system. After that, an ultrasonic type of wave
propagation model is also derived. The explicit expressions for the cut-off frequencies
are also obtained as functions of the nonlocal scaling parameter and the
-directional
wavenumber. Local elasticity shows that the wave will propagate even at higher
frequencies. But nonlocal elasticity predicts that the waves can propagate only up to
certain frequencies (called escape frequencies), after which the wave velocity
becomes zero. The results also show that the escape frequencies are purely a
function of the nonlocal scaling parameter. The effect of the elastic medium is
captured in the wave dispersion analysis and this analysis is explained with
respect to both local and nonlocal elasticity. The simulations show that the
elastic medium affects only the flexural wave mode in the graphene sheet.
The presence of the elastic matrix increases the band gap of the flexural
mode. The present results can provide useful guidance for the design of
next-generation nanodevices in which graphene-based composites act as a major
element.
