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Author: Rossi, Riccarda × End date: 2019 × Start date: 2010 × Has files: Yes ×

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Now showing 1 - 10 of 18
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    Some remarks on a model for rate-independent damage in thermo-visco-elastodynamics
    (Bristol : IOP Publ., 2016) 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 flow 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 Roubicek [1, 2] with the methods from Lazzaroni/Rossi/Thomas/Toader [3]. The present analysis encompasses, differently from [2], the monotonicity in time of damage and the dependence of the viscous tensor on damage and temperature, and, unlike [3], a nonconstant heat capacity and a time-dependent Dirichlet loading.
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    From nonlinear to linear elasticity in a coupled rate-dependent/independent system for brittle delamination
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Rossi, Riccarda; Thomas, Marita
    We revisit the weak, energetic-type existence results obtained in [RT15] for a system for rateindependent, brittle delamination between two visco-elastic, physically nonlinear bulk materials and explain how to rigorously extend such results to the case of visco-elastic, linearly elastic bulk materials. Our approximation result is essentially based on deducing the MOSCO-convergence of the functionals involved in the energetic formulation of the system. We apply this approximation result in two different situations at small strains: Firstly, to pass from a nonlinearly elastic to a linearly elastic, brittle model on the time-continuous level, and secondly, to pass from a time-discrete to a time-continuous model using an adhesive contact approximation of the brittle model, in combination with a vanishing, super-quadratic regularization of the bulk energy. The latter approach is beneficial if the model also accounts for the evolution of temperature.
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    Nonsmooth analysis of doubly nonlinear evolution equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Mielke, Alexander; Rossi, Riccarda; Savaré, Giuseppe
    In this paper we analyze a broad class of abstract doubly nonlinear evolution equations in Banach spaces, driven by nonsmooth and nonconvex energies. We provide some general sufficient conditions, on the dissipation potential and the energy functional, for existence of solutions to the related Cauchy problem. We prove our main existence result by passing to the limit in a time-discretization scheme with variational techniques. Finally, we discuss an application to a material model in finite-strain elasticity.
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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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    Entropic solutions to a thermodynamically consistent PDE system for phase transitions and damage
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Rocca, Elisabetta; Rossi, Riccarda
    In 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.
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    Rate-independent evolution of sets
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Rossi, Riccarda; Stefanelli, Ulisse; Thomas, Marita
    The 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.
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    A vanishing viscosity approach to a rate-independent damage model
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Knees, Dorothee; Rossi, Riccarda; Zanini, Chiara
    We 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.
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    A 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, Chiara
    This 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.
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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.