arXiv subjects

Riccarda Rossi

Publications and source records attributed to Riccarda Rossi.

At least 37 records · Page 2Linked to original sources

arXiv2018

Rate-independent damage in thermo-viscoelastic materials with inertia

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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arXiv2018

From visco to perfect plasticity in thermoviscoelastic materials

We consider a thermodynamically consistent model for thermoviscoplasticity. For the related PDE system, coupling the heat equation for the absolute temperature, the momentum balance with viscosity and inertia for the displacement variable, and the flow rule for the plastic strain, we propose two weak solvability concepts, 'entropic' and 'weak energy' solutions, where the highly nonlinear heat equation is suitably formulated. Accordingly, we prove two existence results by passing to the limit in a carefully devised time discretization scheme. Furthermore, we study the asymptotic behavior of weak energy solutions as the rate of the external data becomes slower and slower, which amounts to taking the vanishing-viscosity and inertia limit of the system. We prove their convergence to a global energetic solution to the Prandtl-Reuss model for perfect plasticity, whose evolution is `energetically' coupled to that of the (spatially constant) limiting temperature.

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arXiv2018

Visco-Energetic solutions to some rate-independent systems in damage, delamination, and plasticity

This paper revolves around a newly introduced weak solvability concept for rate-independent systems, alternative to the notions of Energetic and Balanced Viscosity solutions. Visco-Energetic solutions have been recently obtained by passing to the time-continuous limit in a time-incremental scheme, akin to that for Energetic solutions, but perturbed by a `viscous' correction term, as in the case of Balanced Viscosity solutions. However, for Visco-Energetic solutions this viscous correction is tuned by a fixed parameter. The resulting solution notion turns out to describe a kind of evolution in between Energetic and Balanced Viscosity evolution. In this paper we aim to investigate the application of Visco-Energetic solutions to the paradigmatic example of perfect plasticity, and to nonsmooth rate-independent processes in solid mechanics such as damage and plasticity at finite strains. With the limit passage from adhesive contact to brittle delamination, we also provide a first result of Evolutionary Gamma-convergence for Visco-Energetic solutions. The analysis of these applications reveals the wide applicability of this solution concept and confirms its intermediate character.

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arXiv2018

Weighted Energy-Dissipation principle for gradient flows in metric spaces

This paper develops the so-called Weighted Energy-Dissipation (WED) variational approach for the analysis of gradient flows in metric spaces. This focuses on the minimization of the parameter-dependent global-in-time functional of trajectories \[ \mathcal{I}_\varepsilon[u] = \int_0^{\infty} e^{-t/\varepsilon}\left( \frac12 |u'|^2(t) + \frac1{\varepsilon}ϕ(u(t)) \right) \dd t, \] featuring the weighted sum of energetic and dissipative terms. As the parameter $\varepsilon$ is sent to~$0$, the minimizers $u_\varepsilon$ of such functionals converge, up to subsequences, to curves of maximal slope driven by the functional $ϕ$. This delivers a new and general variational approximation procedure, hence a new existence proof, for metric gradient flows. In addition, it provides a novel perspective towards relaxation.

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arXiv2017

Generation via variational convergence of Balanced Viscosity solutions to rate-independent systems

In this paper we investigate the origin of the Balanced Viscosity solution concept for rate-independent evolution in the setting of a finite-dimensional space. Namely, given a family of dissipation potentials $(Ψ_n)_n$ with superlinear growth at infinity and a smooth energy functional $\mathcal{E}$, we enucleate sufficient conditions on them ensuring that the associated gradient systems $(Ψ_n,\mathcal{E})$ Evolutionary Gamma-converge to a limiting rate-independent system, understood in the sense of Balanced Viscosity solutions. In particular, our analysis encompasses both the vanishing-viscosity approximation of rate-independent systems and their stochastic derivation.

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arXiv2017

Balanced Viscosity solutions to a rate-independent system for damage

This article is the third one in a series of papers by the authors on vanishing-viscosity solutions to rate-independent damage systems. While in the first two papers [KRZ13, KRZ15] the assumptions on the spatial domain $Ω$ were kept as general as possible (i.e. nonsmooth domain with mixed boundary conditions), we assume here that $\partialΩ$ is smooth and that the type of boundary conditions does not change. This smoother setting allows us to derive enhanced regularity spatial properties both for the displacement and damage fields. Thus, we are in a position to work with a stronger solution notion at the level of the viscous approximating system. The vanishing-viscosity analysis then leads us to obtain the existence of a stronger solution concept for the rate-independent limit system. Furthermore, in comparison to [KRZ13, KRZ15], in our vanishing-viscosity analysis we do not switch to an artificial arc-length parameterization of the trajectories but we stay with the true physical time. The resulting concept of Balanced Viscosity solution to the rate-independent damage system thus encodes a more explicit characterization of the system behavior at time discontinuities of the solution.

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arXiv2017

From Visco-Energetic to Energetic and Balanced Viscosity solutions of rate-independent systems

This paper focuses on weak solvability concepts for rate-independent systems in a metric setting. Visco-Energetic solutions have been recently obtained by passing to the time-continuous limit in a time-incremental scheme, akin to that for Energetic solutions, but perturbed by a `viscous' correction term, as in the case of Balanced Viscosity solutions. However, for Visco-Energetic solutions this viscous correction is tuned by a fixed parameter $μ$. The resulting solution notion is characterized by a stability condition and an energy balance analogous to those for Energetic solutions, but, in addition, it provides a fine description of the system behavior at jumps as Balanced Viscosity solutions do. Visco-Energetic evolution can be thus thought as `in-between' Energetic and Balanced Viscosity evolution. Here we aim to formalize this intermediate character of Visco-Energetic solutions by studying their singular limits as $μ\downarrow 0$ and $μ\uparrow \infty$. We shall prove convergence to Energetic solutions in the former case, and to Balanced Viscosity solutions in the latter situation.

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arXiv2017

Global existence for a nonlocal model for adhesive contact

In this paper we address the analytical investigation of a model for adhesive contact, which includes nonlocal sources of damage on the contact surface, such as the elongation. The resulting PDE system features various nonlinearities rendering the unilateral contact conditions, the physical constraints on the internal variables, as well as the integral contributions related to the nonlocal forces. For the associated initial-boundary value problem we obtain a global-in-time existence result by proving the existence of a local solution via a suitable approximation procedure and then by extending the local solution to a global one by a nonstandard prolongation argument.

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arXiv2016

Existence results for a coupled viscoplastic-damage model in thermoviscoelasticity

In this paper we address a model coupling viscoplasticity with damage in thermoviscoelasticity. The associated PDE system consists of the momentum balance with viscosity and inertia for the displacement variable, at small strains, of the plastic and damage flow rules, and of the heat equation. It has a strongly nonlinear character and in particular features quadratic terms on the right-hand side of the heat equation and of the damage flow rule, which have to be handled carefully. We propose two weak solution concepts for the related initial-boundary value problem, namely `entropic' and `weak energy' solutions. Accordingly, we prove two existence results by passing to the limit in a carefully devised time discretization scheme. Finally, in the case of a prescribed temperature profile, and under a strongly simplifying condition, we provide a continuous dependence result, yielding uniqueness of weak energy solutions.

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arXiv2016

Singular vanishing-viscosity limits of gradient flows: the finite-dimensional case

In this note we study the singular vanishing-viscosity limit of a gradient flow set in a finite-dimensional Hilbert space and driven by a smooth, but possibly non convex, time-dependent energy functional. We resort to ideas and techniques from the variational approach to gradient flows and rate-independent evolution to show that, under suitable assumptions, the solutions to the singularly perturbed problem converge to a curve of stationary points of the energy, whose behavior at jump points is characterized in terms of the notion of Dissipative Viscosity solution. We also provide sufficient conditions under which Dissipative Viscosity solutions enjoy better properties, which turn them into Balanced Viscosity solutions. Finally, we discuss the generic character of our assumptions.

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arXiv2016

Global existence results for viscoplasticity at finite strain

We study a model for rate-dependent gradient plasticity at finite strain based on the multiplicative decomposition of the strain tensor, and investigate the existence of global-in-time solutions to the related PDE system. We reveal its underlying structure as a generalized gradient system, where the driving energy functional is highly nonconvex and features the geometric nonlinearities related to finite-strain elasticity as well as the multiplicative decomposition of finite-strain plasticity. Moreover, the dissipation potential depends on the left-invariant plastic rate and thus, depends on the plastic state variable. The existence theory is developed for a class of abstract, nonsmooth, and nonconvex gradient systems, for which we introduce suitable notions of solutions, namely energy-dissipation-balance (EDB) and energy-dissipation-inequality (EDI) solutions. Hence, we resort to the toolbox of the direct method of the calculus of variations to check that the specific energy and dissipation functionals for our viscoplastic models comply with the conditions of the general theory.

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arXiv2016

From Adhesive to Brittle Delamination in Visco-Elastodynamics

In this paper we analyze a system for brittle delamination between two visco-elastic bodies, also subject to inertia, which can be interpreted as a model for dynamic fracture. The rate-independent flow rule for the delamination parameter is coupled with the momentum balance for the displacement, including inertia. This model features a nonsmooth constraint ensuring the continuity of the displacements outside the crack set, which is marked by the support of the delamination parameter. A weak solvability concept, generalizing the notion of energetic solution for rate-independent systems to the present mixed rate-dependent/rate-independent frame, is proposed. Via refined variational convergence techniques, existence of solutions is proved by passing to the limit in approximating systems which regularize the nonsmooth constraint by conditions for adhesive contact. The presence of the inertial term requires the design of suitable recovery spaces small enough to provide compactness but large enough to recover the information on the crack set in the limit.

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arXiv2015

A rate-independent gradient system in damage coupled with plasticity via structured strains

This contribution deals with a class of models combining isotropic damage with plasticity. We are inspired by It has been inspired by a work by Freddi and Royer-Carfagni, 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.

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arXiv2014

"Entropic" solutions to a thermodynamically consistent PDE system for phase transitions and damage

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 {\em 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 $L^1$. 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 \emph{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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arXiv2014

Balanced-Viscosity solutions for multi-rate systems

Several mechanical systems are modeled by the static momentum balance for the displacement $u$ coupled with a rate-independent flow rule for some internal variable $z$. We consider a class of abstract systems of ODEs which have the same structure, albeit in a finite-dimensional setting, and regularize both the static equation and the rate-independent flow rule by adding viscous dissipation terms with coefficients $\varepsilon^α$ and $\varepsilon$, where $0<\varepsilon \ll 1$ and $α>0$ is a fixed parameter. Therefore for $α\neq 1$ $u$ and $z$ have different relaxation rates. We address the vanishing-viscosity analysis as $\varepsilon \downarrow 0$ of the viscous system. We prove that, up to a subsequence, (reparameterized) viscous solutions converge to a parameterized curve yielding a Balanced Viscosity solution to the original rate-independent system, and providing an accurate description of the system behavior at jumps. We also give a reformulation of the notion of Balanced Viscosity solution in terms of a system of subdifferential inclusions, showing that the viscosity in $u$ and the one in $z$ are involved in the jump dynamics in different ways, according to whether $α>1$, $α=1$, and $α\in (0,1)$.

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arXiv2014

Analysis of a model coupling volume and surface processes in thermoviscoelasticity

We focus on a highly nonlinear evolutionary abstract PDE system describing volume processes coupled with surfaces processes in thermoviscoelasticity, featuring the quasi-static momentum balance, the equation for the unidirectional evolution of an internal variable on the surface, and the equations for the temperature in the bulk domain and the temperature on the surface. A significant example of our system occurs in the modeling for the unidirectional evolution of the adhesion between a body and a rigid support, subject to thermal fluctuations and in contact with friction. We investigate the related initial-boundary value problem, and in particular the issue of existence of global-in-time solutions, on an abstract level. This allows us to highlight the analytical features of the problem and, at the same time, to exploit the tight coupling between the various equations in order to deduce suitable estimates on (an approximation) of the problem. Our existence result is proved by passing to the limit in a carefully tailored approximate problem, and by extending the obtained local-in-time solution by means of a refined prolongation argument.

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arXiv2014

A quasilinear differential inclusion for viscous and rate-independent damage systems in non-smooth domains

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 influence of viscosity in the description of the jump regime.

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arXiv2013

On the transversality conditions and their genericity

In this note we review some results on the transversality conditions for a smooth Fredholm map $f: X \times (0,T) \to Y$ between two Banach spaces $X,Y$. These conditions are well-known in the realm of bifurcation theory and commonly accepted as "generic". Here we show that under the transversality assumptions the sections $C(t)=\{x:f(x,t)=0\}$ of the zero set of $f$ are discrete for every $t\in (0,T)$ and we discuss a somehow explicit family of perturbations of $f$ along which transversality holds up to a residual set. The application of these results to the case when $f$ is the $X$-differential of a time-dependent energy functional $E:X\times (0,T)\to R$ and $C(t)$ is the set of the critical points of $E$ provides the motivation and the main example of this paper.

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