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Comparison of a Rapidely Converging Phase Field Model for Interfaces in Solids with the Allen-Cahn Model

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Abstract

We compare two phase field models for interfaces in elastic solids carrying low surface energy. One model has hybrid properties of a Hamilton-Jacobi and a parabolic equation, the other is the Allen-Cahn model. For vanishing width of the interface we construct asymptotic solutions of second order for the hybrid model and of first order for the Allen-Cahn model. These constructions show that the width of the diffused interface necessary for tracking accurately the sharp interface can be chosen much larger for the hybrid model than for the Allen-Cahn model, and that moreover the hybrid model can describe interfaces with nonlinear kinetic relation. This explains why numerical simulations based on the hybrid model are considerably more effective. These simulations are discussed in the last section.

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Acknowledgement

Helpful remarks of the referees are gratefully acknowledged.

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Authors and Affiliations

  1. Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr. 7, 64289, Darmstadt, Germany

    Hans-Dieter Alber

  2. Departamento de Matemáticas, Universidad del País Vasco, Apartado 644, 48080, Bilbao, Spain

    Peicheng Zhu

  3. IKERBASQUE, Basque Foundation for Science, 48011, Bilbao, Spain

    Peicheng Zhu

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  1. Hans-Dieter Alber
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  2. Peicheng Zhu
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Correspondence to Hans-Dieter Alber.

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Alber, HD., Zhu, P. Comparison of a Rapidely Converging Phase Field Model for Interfaces in Solids with the Allen-Cahn Model. J Elast 111, 153–221 (2013). https://doi.org/10.1007/s10659-012-9398-x

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  • DOI: https://doi.org/10.1007/s10659-012-9398-x

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