117 research outputs found
A sharp interface model for the propagation of martensitic phase boundaries
A model for the quasistatic evolution of martensitic phase boundaries is presented. The model is essentially the gradient flow of an energy that can contains elastic energy due to the underlying change in crystal structure in the course of the phase transformation and surface energy penalizing the area of the phase boundary. This leads to a free boundary problem with a nonlocal velocity that arises from the coupling to the elasticity equation. We show existence of solutions under a technical convergence condition using an implicit time-discretization
Optimization of the branching pattern in coherent phase transitions
Branching can be observed at the austenite-martensite interface of
martensitic phase transformations. For a model problem, Kohn and M\"uller
studied a branching pattern with optimal scaling of the energy with respect to
its parameters. Here, we present finite element simulations that suggest a
topologically different class of branching patterns and derive a novel, low
dimensional family of patterns. After a geometric optimization within this
family, the resulting pattern bears a striking resemblance to our simulation.
The novel microstructure admits the same scaling exponents but results in a
significantly lower upper energy bound.Comment: 6 pages, 4 figures, 2 tables. correction of minor typesetting error
A gradient system with a wiggly energy and relaxed EDP-convergence
If gradient systems depend on a microstructure, we want to derive a
macroscopic gradient structure describing the effective behavior of the
microscopic effects. We introduce a notion of evolutionary Gamma-convergence
that relates the microscopic energy and the microscopic dissipation potential
with their macroscopic limits via Gamma-convergence. This new notion
generalizes the concept of EDP-convergence, which was introduced in
arXiv:1507.06322, and is called "relaxed EDP-convergence". Both notions are
based on De Giorgi's energy-dissipation principle, however the special
structure of the dissipation functional in terms of the primal and dual
dissipation potential is, in general, not preserved under Gamma-convergence. By
investigating the kinetic relation directly and using general forcings we still
derive a unique macroscopic dissipation potential.
The wiggly-energy model of James et al serves as a prototypical example where
this nontrivial limit passage can be fully analyzed.Comment: 43 pages, 8 figure
Pinning of interfaces in random media
For a model for the propagation of a curvature sensitive interface in a time
independent random medium, as well as for a linearized version which is
commonly referred to as Quenched Edwards-Wilkinson equation, we prove existence
of a stationary positive supersolution at non-vanishing applied load. This
leads to the emergence of a hysteresis that does not vanish for slow loading,
even though the local evolution law is viscous (in particular, the velocity of
the interface in the model is linear in the driving force).Comment: 15 Page
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