  We present a model of a mechanical system with a vibrational mode spectrum
identical to the spectrum of electronic excitations in a tight-binding model of
graphene. The model consists of point masses connected by elastic couplings,
called "tri-bonds," that implement certain three-body interactions, which can
be tuned by varying parameters that correspond to the relative hopping
amplitudes on the different bond directions in graphene. In the mechanical
model, this is accomplished by varying the location of a pivot point that
determines the allowed rigid rotations of a single tri-bond. The infinite
system constitutes a Maxwell lattice, with the number of degrees of freedom
equal to the number of constraints imposed by the tri-bonds. We construct the
equilibrium and compatibility matrices and analyze the model's phase diagram,
which includes spectra with Weyl points for some placements of the pivot and
topologically polarized phases for others. We then discuss the edge modes and
associated states of self stress for strips cut from the periodic lattice.
Finally, we suggest a physical realization of the tri-bond, which allows access
to parameter regimes not available to experiments on (strained) graphene and
may be used to create other two-dimensional mechanical metamaterials with
different spectral features.
