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search.filters.applied.f.access: openAccess × Author: Rossi, Riccarda × Subject: rate-independent systems ×

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    Balanced viscosity (BV) solutions to infinite-dimensional rate-independent systems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Mielke, Alexander; Rossi, Riccarda; Savaré, Giuseppe
    Balanced Viscosity solutions to rate-independent systems arise as limits of regularized rate-independent flows by adding a superlinear vanishing-viscosity dissipation. We address the main issue of proving the existence of such limits for infinite-dimensional systems and of characterizing them by a couple of variational properties that combine a local stability condition and a balanced energy-dissipation identity. A careful description of the jump behavior of the solutions, of their differentiability properties, and of their equivalent representation by time rescaling is also presented. Our techniques rely on a suitable chain-rule inequality for functions of bounded variation in Banach spaces, on refined lower semicontinuity-compactness arguments, and on new BV-estimates that are of independent interest.
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    Variational 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, Giuseppe
    We 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.
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    Some remarks on a model for rate-independent damage in thermo-visco-elastodynamics
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Lazzaroni, Giuliano; Rossi, Riccarda; Thomas, Marita; Toader, Rodica
    This note deals with the analysis of a model for partial damage, where the rate-independent, unidirectional ow rule for the damage variable is coupled with the rate-dependent heat equation, and with the momentum balance featuring inertia and viscosity according to Kelvin-Voigt rheology. The results presented here combine the approach from Roubí£ek [Rou09, Rou10] with the methods from Lazzaroni/Rossi/Thomas/Toader [LRTT14]. The present analysis encompasses, dierently from [Rou10], the monotonicity in time of damage and the dependence of the viscous tensor on damage and temperature, and, unlike [LRTT14], a nonconstant heat capacity and a time-dependent Dirichlet loading.
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    Rate-independent damage in thermo-viscoelastic materials with inertia
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Lazzaroni, Giuliano; Rossi, Riccarda; Thomas, Marita; Toader, Rodica
    We present a model for rate-independent, unidirectional, partial damage in visco-elastic materials with inertia and thermal effects. The damage process is modeled by means of an internal variable, governed by a rate-independent flow rule. The heat equation and the momentum balance for the displacements are coupled in a highly nonlinear way. Our assumptions on the corresponding energy functional also comprise the case of the Ambrosio-Tortorelli phase-field model (without passage to the brittle limit). We discuss a suitable weak formulation and prove an existence theorem obtained with the aid of a (partially) decoupled time-discrete scheme and variational convergence methods. We also carry out the asymptotic analysis for vanishing viscosity and inertia and obtain a fully rate-independent limit model for displacements and damage, which is independent of temperature.
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    Modeling solutions with jumps for rate-independent systems on metric spaces
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Mielke, Alexander; Rossi, Riccarda; Savaré, Giuseppe
    Rate-independent systems allow for solutions with jumps that need additional modeling. Here we suggest a formulation that arises as limit of viscous regularization of the solutions in the extended state space. Hence, our parametrized metric solutions of a rate-independent system are absolutely continuous mappings from a parameter interval into the extended state space. Jumps appear as generalized gradient flows during which the time is constant. The closely related notion of BV solutions is developed afterwards. Our approach is based on the abstract theory of generalized gradient flows in metric spaces, and comparison with other notions of solutions is given.