  Many complex systems respond to a continuous input of energy by an
accumulation of stress over time, interrupted by sudden energy releases called
avalanches. Recently, it has been pointed out that several basic features of
avalanche dynamics are induced at the microscopic level by relaxation
processes, which are neglected by most models. During my thesis, I studied two
well-known models of avalanche dynamics, modified minimally by the inclusion of
some forms of relaxation.
  The first system is that of a viscoelastic interface driven in a disordered
medium. In mean-field, we prove that the interface has a periodic behaviour
(with a new, emerging time scale), with avalanche events that span the whole
system. We compute semi-analytically the friction force acting on this surface,
and find that it is compatible with classical friction experiments. In finite
dimensions (2D), the mean-field system-sized events become local, and numerical
simulations give qualitative and quantitative results in good agreement with
several important features of real earthquakes.
  The second system including a minimal form of relaxation consists in a toy
model of avalanches: the Directed Percolation process. In our study of a
non-Markovian variant of Directed Percolation, we observed that the
universality class was modified but not completely. In particular, in the
non-Markov case an exponent changes of value while several scaling relations
still hold. This picture of an extended universality class obtained by the
addition of a non-Markovian perturbation to the dynamics provides promising
prospects for our first system.
