Existence, numerical convergence, and evolutionary relaxation for a rate-independent phase-transformation model
- Heinz, Sebastian
- Mielke, Alexander
2010 Mathematics Subject Classification
- 49S05 35Q74 74C05 74N15
- Energetic solution, mutual recovery sequences, H-measures, laminates, two-phase material model, evolutionary Gamma-convergence
We revisit the two-well model for phase transformation in a linearly elastic body introduced and studied in A. Mielke, F. Theil, and V.I. Levita ``A variational formulation of rate--independent phase transformations using an extremum principle", Arch. Rational Mech. Anal., 162, 137-177, 2002 ([MTL02]). This energetic rate-independent model is posed in terms of the elastic displacement and an internal variable that gives the phase portion of the second phase. We use a new approach based on mutual recovery sequences, which are adjusted to a suitable energy increment plus the associated dissipated energy and, thus, enable us to pass to the limit in the construction of energetic solutions. We give three distinct constructions of mutual recovery sequences which allow us (i) to generalize the existence result in [MTL02], (ii) to establish the convergence of suitable numerical approximations via space-time discretization, and (iii) to perform the evolutionary relaxation from the pure-state model to the relaxed mixture model. All these results rely on weak converge and involve the H-measure as an essential tool.
- Phil. Trans. R. Soc. A, 374 (2016), pp. 20150171/1--20150171/23, DOI 10.1098/rsta.2015.0171 .