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Many complex systems respond to a continuous input of energy by anaccumulation of stress over time, interrupted by sudden energy releases calledavalanches. Recently, it has been pointed out that several basic features ofavalanche dynamics are induced at the microscopic level by relaxationprocesses, which are neglected by most models. During my thesis, I studied twowell-known models of avalanche dynamics, modified minimally by the inclusion ofsome forms of relaxation.  The first system is that of a viscoelastic interface driven in a disorderedmedium. In mean-field, we prove that the interface has a periodic behaviour(with a new, emerging time scale), with avalanche events that span the wholesystem. We compute semi-analytically the friction force acting on this surface,and find that it is compatible with classical friction experiments. In finitedimensions (2D), the mean-field system-sized events become local, and numericalsimulations give qualitative and quantitative results in good agreement withseveral important features of real earthquakes.  The second system including a minimal form of relaxation consists in a toymodel of avalanches: the Directed Percolation process. In our study of anon-Markovian variant of Directed Percolation, we observed that theuniversality class was modified but not completely. In particular, in thenon-Markov case an exponent changes of value while several scaling relationsstill hold. This picture of an extended universality class obtained by theaddition of a non-Markovian perturbation to the dynamics provides promisingprospects for our first system.

Viscoelastic Interfaces Driven in Disordered Media and Applications to Friction