Research Group of Prof. Dr. M. Griebel
Institute for Numerical Simulation

Continuum Mechanical Simulation of Shape Memory Alloys


Marcel Arndt, Michael Griebel


In this project, we deal with the modeling and numerical simulation of martensitic phase transformations in shape memory alloys. We develop a model to describe the behavior of such alloys on the continuum mechanical level. This model involves the elastic stored energy, a rate-independent dissipation mechanism and higher order terms to describe capillarity and viscosity effects.

We develop and implement efficient techniques to solve our model numerically. These techniques include an incremental minimization scheme for the time evolution of the system, routines for the nonconvex mimimization and wavelet-based norm equivalences for a fast evalutation of the nonlocal higher-order contributions.


The pictures below show some snapshots of the time evolution of an Indium-Thallium shape memory alloy under tensile and compression hard device loading. The colors indicate the difference phases and variants. Yellow depicts to the austenitic phase, whereas red, blue and green correspond to the three martensitic variants.
Note the laminated microstructure in the second snapshot, which comes from an energy relaxation of the martensitic variants. The full time evolution can be seen in this movie.


Tomas Roubicek, University of Prague, Czech Republic
Vaclav Novak, Academy of Sciences, Prague, Czech Republic
Petr Sittner, Academy of Sciences, Prague, Czech Republic


[1] M. Arndt, M. Griebel, V. Novák, T. Roubícek, P. Šittner. Martensitic Transformation in NiMnGa Single Crystals: Numerical Simulation and Experiments. Technical Report, University of Bonn, 2005.
[2] M. Arndt. Upscaling from Atomistic Models to Higher Order Gradient Continuum Models for Crystalline Solids. Dissertation, Institute for Numerical Simulation, University of Bonn, 2004.
[3] M. Arndt, M. Griebel and T. Roubícek. Modelling and numerical simulation of martensitic transformation in shape memory alloys. Contin. Mech. Thermodyn. 15(5):463-485, 2003.
[4] M. Arndt. Modelling and numerical simulation of martensitic transformation. In: W. L. Wendland, M. Efendiev (eds.), Analysis and Simulation of Multifield Problems. Proceedings of the International Conference on Multifield Problems 2002, Stuttgart. Springer, 2003.
[5] M. Arndt. Upscaling Technique for the Atomistic-Continuum Simulation of Shape Memory Alloys with the EAM Potential. Proceedings of the Sixth World Congress on Computational Mechanics (WCCM 6), Beijing, 2004.

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