Continuum Mechanics and Thermodynamics

, Volume 29, Issue 3, pp 757–803 | Cite as

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

  • Hans-Dieter AlberEmail author
Original Article


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.


Allen–Cahn phase field model for solids Asymptotic expansion Propagation speed of phase interfaces Kinetic relation Model error Numerical efficiency 


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© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Fachbereich MathematikTechnische Universität DarmstadtDarmstadtGermany

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