  We study a locally resonant granular material in the form of a precompressed
Hertzian chain with linear internal resonators. Using an asymptotic reduction,
we derive an effective nonlinear Schr\"odinger (NLS) modulation equation. This,
in turn, leads us to provide analytical evidence, subsequently corroborated
numerically, for the existence of two distinct types of discrete breathers
related to acoustic or optical modes: (a) traveling bright breathers with a
strain profile exponentially vanishing at infinity and (b) stationary and
traveling dark breathers, exponentially localized, time-periodic states mounted
on top of a non-vanishing background. The stability and bifurcation structure
of numerically computed exact stationary dark breathers is also examined.
Stationary bright breathers cannot be identified using the NLS equation, which
is defocusing at the upper edges of the phonon bands and becomes linear at the
lower edge of the optical band.
