8 research outputs found
Viscosity solutions to a new phase-field model for martensitic phase transformations
summary:We investigate a new phase-field model which describes martensitic phase transitions, driven by material forces, in solid materials, e.g., shape memory alloys. This model is a nonlinear degenerate parabolic equation of second order, its principal part is not in divergence form in multi-dimensional case. We prove the existence of viscosity solutions to an initial-boundary value problem for this model
An alternative to the Allen-Cahn phase field model for interfaces in solids - numerical efficiency
The derivation of the Allen-Cahn and Cahn-Hilliard equations is based on the
Clausius-Duhem inequality. This is not a derivation in the strict sense of the
word, since other phase field equations can be fomulated satisfying this
inequality. Motivated by the form of sharp interface problems, we formulate
such an alternative equation and compare the properties of the models for the
evolution of phase interfaces in solids, which consist of the elasticity
equations and the Allen-Cahn equation or the alternative equation. We find that
numerical simulations of phase interfaces with small interface energy based on
the alternative model are more effective then simulations based on the
Allen-Cahn model.Comment: arXiv admin note: text overlap with arXiv:1505.0544
Solvability via viscosity solutions for a model of phase transitions driven by configurational forces
In the present article, we are interested in an initial boundary value
problem for a coupled system of partial differential equations arising in
martensitic phase transition theory of elastically deformable solid materials,
e.g., steel. This model was proposed and investigated in previous work by Alber
and Zhu in which the weak solutions are defined in a standard way, however the
key technique is not applicable to multi-dimensional problem. Intending to
solve this multi-dimensional problem and to investigate the sharp interface
limits of our models, we thus define weak solutions in a different way by using
the notion of viscosity solution, then prove the existence of weak solutions to
this problem in one space dimension, yet the multi-dimensional problem is still
open.Comment: 21 page
Spherically symmetric solutions to a model for phase transitions driven by configurational forces
We prove the global in time existence of spherically symmetric solutions to
an initial-boundary value problem for a system of partial differential
equations, which consists of the equations of linear elasticity and a
nonlinear, non-uniformly parabolic equation of second order. The problem models
the behavior in time of materials in which martensitic phase transitions,
driven by configurational forces, take place, and can be considered to be a
regularization of the corresponding sharp interface model. By assuming that the
solutions are spherically symmetric, we reduce the original multidimensional
problem to the one in one space dimension, then prove the existence of
spherically symmetric solutions. Our proof is valid due to the essential
feature that the reduced problem is one space dimensional.Comment: 25 page