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- ItemRate-independent evolution of sets(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Rossi, Riccarda; Stefanelli, Ulisse; Thomas, MaritaThe goal of this work is to analyze a model for the rate-independent evolution of sets with finite perimeter. The evolution of the admissible sets is driven by that of a given time-dependent set, which has to include the admissible sets and hence is to be understood as an external loading. The process is driven by the competition between perimeter minimization and minimization of volume changes. In the mathematical modeling of this process, we distinguish the adhesive case, in which the constraint that the (complement of) the `external load' contains the evolving sets is penalized by a term contributing to the driving energy functional, from the brittle case, enforcing this constraint. The existence of Energetic solutions for the adhesive system is proved by passing to the limit in the associated time-incremental minimization scheme. In the brittle case, this time-discretization procedure gives rise to evolving sets satisfying the stability condition, but it remains an open problem to additionally deduce energy-dissipation balance in the time-continuous limit. This can be obtained under some suitable quantification of data. The properties of the brittle evolution law are illustrated by numerical examples in two space dimensions.
- ItemA vanishing viscosity approach to a rate-independent damage model(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Knees, Dorothee; Rossi, Riccarda; Zanini, ChiaraWe analyze a rate-independent model for damage evolution in elastic bodies. The central quantities are a stored energy functional and a dissipation functional, which is assumed to be positively homogeneous of degree one. Since the energy is not simultaneously (strictly) convex in the damage variable and the displacements, solutions may have jumps as a function of time. The latter circumstance makes it necessary to recur to suitable notions of weak solution. However, the by-now classical concept of global energetic solution fails to describe accurately the behavior of the system at jumps. Hence, we consider rate-independent damage models as limits of systems driven by viscous, rate-dependent dissipation. We use a technique for taking the vanishing viscosity limit, which is based on arc-length reparameterization. In this way, in the limit we obtain a novel formulation for the rate-independent damage model, which highlights the interplay of viscous and rate-independent effects in the jump regime, and provides a better description of the energetic behavior of the system at jumps.
- ItemBalanced-Viscosity solutions to infinite-dimensional multi-rate systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Mielke, Alexander; Rossi, RiccardaWe consider generalized gradient systems with rate-independent and rate-dependent dissipation potentials. We provide a general framework for performing a vanishing-viscosity limit leading to the notion of parametrized and true Balanced-Viscosity solutions that include a precise description of the jump behavior developing in this limit. Distinguishing an elastic variable $u$ having a viscous damping with relaxation time $eps^alpha$ and an internal variable $z$ with relaxation time $eps$ we obtain different limits for the three cases $alpha in (0,1)$, $alpha=1$ and $alpha>1$. An application to a delamination problem shows that the theory is general enough to treat nontrivial models in continuum mechanics.
- ItemA metric approach to a class fo doubly nonlinear evolution euations and applications(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Rossi, Riccarda; Mielke, Alexander; Savaré, GiuseppeThis paper deals with the analysis of a class of doubly nonlinear evolution equations in the framework of a general metric space. We propose for such equations a suitable metric formulation (which in fact extends the notion of Curve of Maximal Slope for gradient flows in metric spaces, see [5]), and prove the existence of solutions for the related Cauchy problem by means of an approximation scheme by time discretization. Then, we apply our results to obtain the existence of solutions to abstract doubly nonlinear equations in reflexive Banach spaces. The metric approach is also exploited to analyze a class of evolution equations in $L^1$ spaces.
- ItemVariational convergence of gradient flows and rate-independent evolutions in metric spaces(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Mielke, Alexander; Rossi, Riccarda; Savaré, GiuseppeWe study the asymptotic behaviour of families of gradient flows in a general metric setting, when the metric-dissipation potentials degenerate in the limit to a dissipation with linear growth. We present a general variational definition of BV solutions to metric evolutions, showing the different characterization of the solution in the absolutely continuous regime, on the singular Cantor part, and along the jump transitions. By using tools of metric analysis, BV functions and blow-up by time rescaling, we show that this variational notion is stable with respect to a wide class of perturbations involving energies, distances, and dissipation potentials. As a particular application, we show that BV solutions to rate-independent problems arise naturally as a limit of p-gradient flows, p>1, when the exponents p converge to 1.
- ItemEntropic solutions to a thermodynamically consistent PDE system for phase transitions and damage(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Rocca, Elisabetta; Rossi, RiccardaIn this paper we analyze a PDE system modelling (non-isothermal) phase transitions and damage phenomena in thermoviscoelastic materials. The model is thermodynamically consistent: in particular, no small perturbation assumption is adopted, which results in the presence of quadratic terms on the right-hand side of the temperature equation, only estimated in L1. The whole system has a highly nonlinear character. We address the existence of a weak notion of solution, referred to as entropic, where the temperature equation is formulated with the aid of an entropy inequality, and of a total energy inequality. This solvability concept reflects the basic principles of thermomechanics as well as the thermodynamical consistency of the model. It allows us to obtain global-in-time existence theorems without imposing any restriction on the size of the initial data. We prove our results by passing to the limit in a time discretization scheme, carefully tailored to the nonlinear features of the PDE system (with its entropic formulation), and of the a priori estimates performed on it. Our time-discrete analysis could be useful towards the numerical study of this model.
- ItemCoupling rate-independent and rate-dependent processes: Existence results(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Rossi, Riccarda; Thomas, MaritaWe address the analysis of an abstract system coupling a rate-independent process with a second order (in time) nonlinear evolution equation. We propose suitable weak solution concepts and obtain existence results by passing to the limit in carefully devised time-discretization schemes. Our arguments combine techniques from the theory of gradient systems with the toolbox for rate-independent evolution, thus reflecting the mixed character of the problem. Finally, we discuss applications to a class of rate-independent processes in visco-elastic solids with inertia, and to a recently proposed model for damage with plasticity.
- ItemA rate-independent gradient system in damage coupled with plasticity via structured strains(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Bonetti, Elena; Rocca, Elisabetta; Rossi, Riccarda; Thomas, MaritaThis contribution deals with a class of models combining isotropic damage with plasticity. It has been inspired by a work by Freddi and Royer-Carfagni [FRC10], including the case where the inelastic part of the strain only evolves in regions where the material is damaged. The evolution both of the damage and of the plastic variable is assumed to be rate-independent. Existence of solutions is established in the abstract energetic framework elaborated by Mielke and coworkers (cf., e.g., [Mie05, Mie11b]).
- ItemA quasilinear differential inclusion for viscous and rate-independent damage systems in non-smooth domains(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Knees, Dorothee; Rossi, Riccarda; Zanini, ChiaraThis paper focuses on rate-independent damage in elastic bodies. Since the driving energy is nonconvex, solutions may have jumps as a function of time, and in this situation it is known that the classical concept of energetic solutions for rate-independent systems may fail to accurately describe the behavior of the system at jumps. Therefore, we resort to the (by now well-established) vanishing viscosity approach to rate-independent modeling and approximate the model by its viscous regularization. In fact, the analysis of the latter PDE system presents remarkable difficulties, due to its highly nonlinear character. We tackle it by combining a variational approach to a class of abstract doubly nonlinear evolution equations, with careful regularity estimates tailored to this specific system relying on a q-Laplacian type gradient regularization of the damage variable. Hence, for the viscous problem we conclude the existence of weak solutions satisfying a suitable energy-dissipation inequality that is the starting point for the vanishing viscosity analysis. The latter leads to the notion of (weak) parameterized solution to our rate-independent system, which encompasses the in uence of viscosity in the description of the jump regime.
- ItemBV solutions and viscosity approximations of rate-independent systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Mielke, Alexander; Rossi, Riccarda; Savaré, GiuseppeIn the nonconvex case solutions of rate-independent systems may develop jumps as a function of time. To model such jumps, we adopt the philosophy that rate independence should be considered as limit of systems with smaller and smaller viscosity. For the finite-dimensional case we study the vanishing-viscosity limit of doubly nonlinear equations given in terms of a differentiable energy functional and a dissipation potential which is a viscous regularization of a given rate-independent dissipation potential. The resulting definition of `BV solutions' involves, in a nontrivial way, both the rate-independent and the viscous dissipation potential, which play a crucial role in the description of the associated jump trajectories. We shall prove a general convergence result for the time-continuous and for the time-discretized viscous approximations and establish various properties of the limiting $BV$ solutions. In particular, we shall provide a careful description of the jumps and compare the new notion of solutions with the related concepts of energetic and local solutions to rate-independent systems.
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