  We study a model of two layers, each consisting of a d-dimensional elastic
object driven over a random substrate, and mutually interacting through a
viscous coupling. For this model, the mean-field theory (i.e. a fully connected
model) predicts a transition from elastic depinning to hysteretic plastic
depinning as disorder or viscous coupling is increased. A functional RG
analysis shows that any small inter-layer viscous coupling destablizes the
standard (decoupled) elastic depinning FRG fixed point for d <= 4, while for d
> 4 most aspects of the mean-field theory are recovered. A one-loop study at
non-zero velocity indicates, for d<4, coexistence of a moving state and a
pinned state below the elastic depinning threshold, with hysteretic plastic
depinning for periodic and non-periodic driven layers. A 2-loop analysis of
quasi-statics unveils the possibility of more subtle effects, including a new
universality class for non-periodic objects. We also study the model in d=0,
i.e. two coupled particles, and show that hysteresis does not always exist as
the periodic steady state with coupled layers can be dynamically unstable. It
is also proved that stable pinned configurations remain dynamically stable in
presence of a viscous coupling in any dimension d. Moreover, the layer model
for periodic objects is stable to an infinitesimal commensurate density
coupling. Our work shows that a careful study of attractors in phase space and
their basin of attraction is necessary to obtain a firm conclusion for
dimensions d=1,2,3.
