# Approximation of compressible viscoplastic models with general nonlinearity for granular flows

Abstract : This thesis is motivated by a research program between the LAMA (Mathematics, Univ. Gustave Eiffel) and the Institut de Physique du Globe of Paris (Earth Sciences) on granular media and their mathematical description.We consider here a continuous description: the material is described as a fluid with viscoplastic rheology, that allows us to model the transition between static (solid) states and mobile (liquid) states. Incompressible models have been used since the introduction of the so called $mu(I)$ rheology (Jop et al. 2006). However such models do not represent accurately real flows, even in laboratory experiments. Recent studies indicate that volume variations, even if not significantly large, play a key role in the dynamics. Therefore compressible models have been recently considered (Barker et al. 2017). Although particular rheologies such as Bingham or Herschel-Bulkley models have been often considered in mathematical studies such as Malek et al. 2010, not much can be found on general nonlinearities in terms of the trace and the norm of the strain rate tensor. We consider here compressible models with general nonlinearities $sigmainpartial F(D)$ where $sigma$ is the stress, $D$ is the strain rate and $F$ is a convex viscoplastic potential.Under technical assumptions on $F$ such as subquadratic growth and superlinearity we prove the existence of solutions to the associated variational problem. This is obtained in the viscous as well as in the inviscid cases. We establish Euler-Lagrange characterizations of these solutions. No regularity is assumed on $F$, thus yield stress rheologies are included. Numerical methods for viscoplastic laws have been classically used: augmented Lagrangian or regularization methods. However these methods were designed merely for Bingham or Herschel-Bulkley fluids, and moreover their cost is still too high for applications to real configurations.Here we consider an iterative but explicit method in the sense that there is no linear system to solve, inherited from the minimizing of total variation functionals used in imaging (Chambolle, Pock 2011). It is applicable to any kind of nonlinearity, and includes a kind of projection on some convex sets. We prove the convergence of the method discretized in space with finite elements. Numerical tests confim the theoretical results
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Hoai-Duc Nguyen. Approximation of compressible viscoplastic models with general nonlinearity for granular flows. Mechanics of materials [physics.class-ph]. Université Paris-Est, 2020. English. ⟨NNT : 2020PESC2044⟩. ⟨tel-03337186⟩

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