arXiv subjects

Hans-Dieter Alber

Publications and source records attributed to Hans-Dieter Alber.

3 recordsLinked to original sources

arXiv2017

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.

math-ph↗

arXiv2017

Asymptotics and numerical efficiency of the Allen-Cahn model for phase interfaces with low energy in solids

We study how the propagation speed of interfaces in the Allen-Cahn phase field model for phase transformations in solids consisting of the elasticity equations and the Allen-Cahn equation depends on two parameters of the model. The two parameters control the interface energy and the interface width but change also the interface speed. To this end we derive an asymptotic expansion of second order for the interface speed, called the kinetic relation, and prove that it is uniformly valid in both parameters. As a consequence we show that the model error is proportional to the interface width divided by the interface energy. We conclude that simulations of interfaces with low interface energy based on this model require a very small interface width, implying a large numerical effort. Effective simulations thus need adaptive mesh refinement or other advanced techniques. This version of the paper contains the proofs of Theorem 4.5 and Lemma 5.8, which are omitted in the version published in Continuum Mechanics and Thermodynamics.

math-ph↗

arXiv2013

The continuous theory of dislocations for a material containing dislocations to one Burgers vector only

We review the continuous theory of dislocations from a mathematical point of view using mathematical tools, which were only partly available when the theory was developed several decades ago. We define a space of dislocation measures, which includes Hausdorff measures representing the dislocation measures of single dislocation curves. The evolution equation for dislocation measures is defined on this space. It is derived from four basic conditions, which must be satisfied by the model.

math.AP↗