211 research outputs found
Generalized gradient flow structure of internal energy driven phase field systems
In this paper we introduce a general abstract formulation of a variational
thermomechanical model, by means of a unified derivation via a generalization
of the principle of virtual powers for all the variables of the system,
including the thermal one. In particular, choosing as thermal variable the
entropy of the system, and as driving functional the internal energy, we get a
gradient flow structure (in a suitable abstract setting) for the whole
nonlinear PDE system. We prove a global in time existence of (weak) solutions
result for the Cauchy problem associated to the abstract PDE system as well as
uniqueness in case of suitable smoothness assumptions on the functionals
Singular limit of an integrodifferential system related to the entropy balance
A thermodynamic model describing phase transitions with thermal memory, in
terms of an entropy equation and a momentum balance for the microforces, is
adressed. Convergence results and error estimates are proved for the related
integrodifferential system of PDE as the sequence of memory kernels converges
to a multiple of a Dirac delta, in a suitable sense.Comment: Key words: entropy equation, thermal memory, phase field model,
nonlinear partial differential equations, asymptotics on the memory ter
Global existence for a hydrogen storage model with full energy balance
A thermo-mechanical model describing hydrogen storage by use of metal
hydrides has been recently proposed in a paper by Bonetti, Fr\'emond and
Lexcellent. It describes the formation of hydrides using the phase transition
approach. By virtue of the laws of continuum thermo-mechanics, the model leads
to a phase transition problem in terms of three state variables: the
temperature, the phase parameter representing the fraction of one solid phase,
and the pressure, and is derived within a generalization of the principle of
virtual powers proposed by Fr\'emond accounting for micro-forces, responsible
for the phase transition, in the whole energy balance of the system. Three
coupled nonlinear partial differential equations combined with initial and
boundary conditions have to be solved. The main difficulty in investigating the
resulting system of partial differential equations relies on the presence of
the squared time derivative of the order parameter in the energy balance
equation. Here, the global existence of a solution to the full problem is
proved by exploiting known and sharp estimates on parabolic equations with
right hand side in L^1. Some complementary results on stability and steady
state solutions are also given.Comment: Key-words: phase transition model; hydrogen storage; nonlinear
parabolic system; existenc
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
On the strongly damped wave equation with constraint
A weak formulation for the so-called "semilinear strongly damped wave
equation with constraint" is introduced and a corresponding notion of solution
is defined. The main idea in this approach consists in the use of duality
techniques in Sobolev-Bochner spaces, aimed at providing a suitable
"relaxation" of the constraint term. A global in time existence result is
proved under the natural condition that the initial data have finite "physical"
energy.Comment: 21 page
Modeling and analysis of a phase field system for damage and phase separation processes in solids
In this work, we analytically investigate a multi-component system for
describing phase separation and damage processes in solids. The model consists
of a parabolic diffusion equation of fourth order for the concentration coupled
with an elliptic system with material dependent coefficients for the strain
tensor and a doubly nonlinear differential inclusion for the damage function.
The main aim of this paper is to show existence of weak solutions for the
introduced model, where, in contrast to existing damage models in the
literature, different elastic properties of damaged and undamaged material are
regarded. To prove existence of weak solutions for the introduced model, we
start with an approximation system. Then, by passing to the limit, existence
results of weak solutions for the proposed model are obtained via suitable
variational techniques.Comment: Keywords: Cahn-Hilliard system, phase separation, elliptic-parabolic
systems, doubly nonlinear differential inclusions, complete damage, existence
results, energetic solutions, weak solutions, linear elasticity,
rate-dependent system
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