134 research outputs found
Thermodynamics and analysis of rate-independent adhesive contact at small strains
We address a model for adhesive unilateral frictionless Signorini-type
contact between bodies of heat-conductive viscoelastic material, in the linear
Kelvin-Voigt rheology, undergoing thermal expansion. The flow-rule for
debonding the adhesion is considered rate-independent and unidirectional, and a
thermodynamically consistent model is derived and analysed as far as the
existence of a weak solution is concerned
A degenerating PDE system for phase transitions and damage
In this paper, we analyze a PDE system arising in the modeling of phase
transition and damage phenomena in thermoviscoelastic materials. The resulting
evolution equations in the unknowns \theta (absolute temperature), u
(displacement), and \chi (phase/damage parameter) are strongly nonlinearly
coupled. Moreover, the momentum equation for u contains \chi-dependent elliptic
operators, which degenerate at the pure phases (corresponding to the values
\chi=0 and \chi=1), making the whole system degenerate. That is why, we have to
resort to a suitable weak solvability notion for the analysis of the problem:
it consists of the weak formulations of the heat and momentum equation, and,
for the phase/damage parameter \chi, of a generalization of the principle of
virtual powers, partially mutuated from the theory of rate-independent damage
processes. To prove an existence result for this weak formulation, an
approximating problem is introduced, where the elliptic degeneracy of the
displacement equation is ruled out: in the framework of damage models, this
corresponds to allowing for partial damage only. For such an approximate
system, global-in-time existence and well-posedness results are established in
various cases. Then, the passage to the limit to the degenerate system is
performed via suitable variational techniques
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.Comment: 27 page
BV solutions and viscosity approximations of rate-independent systems
In 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
Rate-independent evolution of sets
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. \par 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.Comment: Dedicated to Alexander Mielke on the occasion of his 60th birthda
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
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
Global attractors for gradient flows in metric spaces
We develop the long-time analysis for gradient flow equations in metric
spaces. In particular, we consider two notions of solutions for metric gradient
flows, namely energy and generalized solutions. While the former concept
coincides with the notion of curves of maximal slope, we introduce the latter
to include limits of time-incremental approximations constructed via the
Minimizing Movements approach.
For both notions of solutions we prove the existence of the global attractor.
Since the evolutionary problems we consider may lack uniqueness, we rely on the
theory of generalized semiflows introduced by J.M. Ball. The notions of
generalized and energy solutions are quite flexible and can be used to address
gradient flows in a variety of contexts, ranging from Banach spaces to
Wasserstein spaces of probability measures.
We present applications of our abstract results by proving the existence of
the global attractor for the energy solutions both of abstract doubly nonlinear
evolution equations in reflexive Banach spaces, and of a class of evolution
equations in Wasserstein spaces, as well as for the generalized solutions of
some phase-change evolutions driven by mean curvature
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