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Numerical Solution of Coupled Dissipation Processes in the Mechanics of Shape Memory Alloys

Miroslav Frost 1, Alexej Moskovka 1, Petr Sedlák 1, Hanuš Seiner 1, Petr Pelech 2*, Barbora Benešová 2

1 Czech Academy of Sciences, Institute of Thermomechanics, Prague, Czechia; 2 Charles University, Faculty of Mathematics and Physics, Department of Mathematical Analysis, Prague, Czechia

Shape Memory Alloys & Transformation Mechanics · C223
Wednesday, 2 September 2026, 11:35–12:00 · Chair: Samuel Forest

Keywords: shape memory alloys, non-smooth potentials, alternate minimization

Shape memory alloys are often modeled via non-smooth potentials in the Generalized Standard Materials framework [1]. The non-differentiability of the minimized functional then makes the numerical analysis challenging. Alternate minimization scheme (AMS) is widely used as it reduces dimension of the problem and has been studied thoroughly when the split variables are not coupled by a non-smooth term, see e.g. [2]. Recent paper [3] extends the analysis to variables coupled by a non-smooth dissipation potential, which is a typical case for SMAs; the AMS converges to a limit which depends on the splitting and the solution differs from the one given by the full minimization scheme (FMS). As explained by M. Frost in the preceding contribution, the specific choice of splitting is dictated by physics and therefore the AMS (unlike FMS) reproduces the correct material response. While non-smooth dissipation potentials are iconic for SMAs, the free energy is usually considered to be continuously differentiable. Yet, a simple constraint on the state variables makes the indicator function appear in the energy function, and the resulting problem is then non-smooth in both rates and states. The limiting behavior of the AMS is then no longer covered by the results obtained in [3] and the strategy for describing the effective problem for AMS has to be modified. In the talk, we will reduce the constitutive model of SMAs given in the preceding contribution by M. Frost in order to make it mathematically feasible and highlight its important properties. We illustrate on simple examples why the FMS and AMS yield different solutions and describe the effective problem for AMS.

References

  1. B. Halphen and Q. S. Nguyen, Sur les matériaux standard généralisés, J. Mecanique, 14:39-–63, 1975.
  2. J. Bolte, S. Sabach, and M. Teboulle, Proximal alternating linearized minimization for nonconvex and nonsmooth problems, Math. Program., 146:459–-494, 2014, https://doi.org/10.1007/s10107-013-0701-9.
  3. A. Mielke, R. Rossi, and A. Stephan, On time-splitting methods for gradient flows with two dissipation mechanisms, Calc. Var., 64(63), 2025, https://doi.org/10.1007/s00526-024-02849-8.