Effective limit for a viscous model with periodic friction coefficients

In a microscopic scale, the evolution of internal variables during a rate-dependent process can be described by a family of differential inclusions with oscillating terms having a tiny period 0 < ϵ << 1. These differential inclusions could model the balance between the energy gain and dissipation. Our goal is to relate such a family of models in Rn, n = 1, as ϵ approaches zero, with a single model that retains a certain amount of data from the original system while eliminating the oscillations, which means that the limit model has fewer parameters. A typical example is dry friction with a periodic friction coefficient, which is the main focus of this thesis. Our means is to supply a homogenization formula that fits periodic functions with specific growth features such as the sum of the dissipation potential and its dual. The latter sum’s integral happens to be essential in finding the homogenized limit system. Our model’s periodicity is induced by a dissipation potential that is thought to be convex, Lipschitz-continuous and non C1. This is new compared to the literature, where in the context of homogenization the dissipation potential is assumed to be C1-smooth. Our more general assumptions require refined arguments in the proofs. First, we show that our models given as subdifferential inclusions have equivalent formulations under certain assumptions, including the dual and the energetic formulation. Second, we prove the uniqueness of a regular solution for each ϵ > 0 and n = 1 using the dual counterpart of our model which happens to be an ordinary differential equation. In the situation n ≥ 2, the right-hand side of the ordinary differential equation results from the inf of another function, so it is difficult to calculate its subdifferential explicitly. We solve this problem using the Moreau-Yosida regularization. In general, this leads only to a H¨older-continuous (but not Lipschitz-continuous) nonlinearity. Hence, only local existence of solutions can be guaranteed. Finally, in one dimension, a homogenization formula is provided for certain functionals that have a p-growth property (1 < p < ∞). This formula is crucial in the sense that it allows to characterize the limiting system in our model by passing to the Γ-limit when ϵ approaches 0 in the energetic formulation. This notion of convergence is appropriate for our study since we show at the end that the limiting model is still expressed in terms of an energy-dissipation model with a dissipation potential (effective) without an oscillating term.