We prove nonexistence of breathers (spatially localized and time-periodic
oscillations) for a class of Fermi-Pasta-Ulam lattices representing an
uncompressed chain of beads interacting via Hertz's contact forces. We then
consider the setting in which an additional on-site potential is present,
motivated by the Newton's cradle under the effect of gravity. Using both direct
numerical computations and a simplified asymptotic model of the oscillator
chain, the so-called discrete p-Schr\"odinger (DpS) equation, we show the
existence of discrete breathers and study their spectral properties and
mobility. Due to the fully nonlinear character of Hertzian interactions,
breathers are found to be much more localized than in classical nonlinear
lattices and their motion occurs with less dispersion. In addition, we study
numerically the excitation of a traveling breather after an impact at one end
of a semi-infinite chain. This case is well described by the DpS equation when
local oscillations are faster than binary collisions, a situation occuring e.g.
in chains of stiff cantilevers decorated by spherical beads. When a hard
anharmonic part is added to the local potential, a new type of traveling
breather emerges, showing spontaneous direction-reversing in a spatially
homogeneous system. Finally, the interaction of a moving breather with a point
defect is also considered in the cradle system. Almost total breather
reflections are observed at sufficiently high defect sizes, suggesting
potential applications of such systems as shock wave reflectors