Two generic theoretical frameworks: disordered elastic systems and elasto-plastic models. Threefold motivation: experimental/fundamental/methodological. Basic phenomenology and regimes of interest. Dynamical critical phenomena as a hallmark of disorder: analogy of depinning versus yielding transitions. Scope of the lecture (topics/tools).
Hamiltonian, Langevin dynamics, dynamical action. Observables: geometrical fluctuations and center-of-mass dynamics.
Static interface without disorder (thermal roughness). With disorder: roughness regimes and crossover scales. Larkin model.
Focus on the 1D interface: mapping to the 1+1 Directed Polymer. Connections to the 1D Kardar-Parisi-Zhang equation and universality class. Example of model reduction: effective 1D interface starting from a 2D Ginzburg-Laudau description.
Velocity-force characteristics. Fast-flow regime. Creep regime.
Depinning regime and thermal rounding. Additional degree of freedom, role of inertia. Oscillatory driving.
Elasto-plastic scenario in sheared amorphous materials. Stress redistribution: Eshelby propagator. Spatially-resolved vs mean-field versions: a glimpse in the zoology of models.
Tuning the mechanical noise: Hébraud-Lequeux model (Gaussian white noise); Lévy-flight model (power-law distributed noise). Role of structural disorder: Soft-Glassy-Rheology, disordered Hébraud-Lequeux, and KEP models. Fluidity models.
Britte-to-ductile transition: role of initial preparation. Quantitative calibration to atomistic simulations. Moving towards dense active matter: biological tissues mechanics. Oscillatory driving.
Analogy of depinning vs yielding: where do we stand? Exact benchmark in infinite dimension? And other open issues.