Sergio Conti, Martin Lenz, Nora Lüthen, Martin Rumpf and Barbara Zwicknagl Geometry of martensite needles in shape memory alloys 2019 https://arxiv.org/abs/1912.02274
2018
M. Griebel, C. Rieger and B. Zwicknagl Regularized Kernel-Based Reconstruction in Generalized Besov Spaces Foundations of Computational Mathematics, 18(2): 459--508 2018 10.1007/s10208-017-9346-z
2017
Sergio Conti, Johannes Diermeier and Barbara Zwicknagl Deformation concentration for martensitic microstructures in the limit of low volume fraction Calc. Var. PDE, 56: 16 2017 10.1007/s00526-016-1097-1
Abstract: We consider a singularly-perturbed nonconvex energy functional which arises in the study of microstructures in shape memory alloys. The scaling law for the minimal energy predicts a transition from a parameter regime in which uniform structures are favored, to a regime in which the formation of fine patterns is expected. We focus on the transition regime and derive the reduced model in the sense of $Γ$-convergence. The limit functional turns out to be similar to the Mumford-Shah functional with additional constraints on the jump set of admissible functions. One key ingredient in the proof is an approximation result for $SBV^p$ functions whose jump sets have a prescribed orientation.
Abstract: We consider a partial differential inclusion problem which models stress-free martensitic inclusions in an austenitic matrix, based on the standard geometrically nonlinear elasticity theory. We show that for specific parameter choices there exist piecewise affine continuous solutions for the square-to-oblique and the hexagonal-to-oblique phase transitions. This suggests that for specific crystallographic parameters the hysteresis of the phase transformation will be particularly small.
Angkana Rüland, Christian Zillinger and Barbara Zwicknagl Higher Sobolev Regularity of Convex Integration Solutions in Elasticity: The Dirichlet Problem with Affine Data in int($K^lc$) 2017 https://arxiv.org/abs/1709.02880
2016
Sergio Conti and Barbara Zwicknagl Low volume-fraction microstructures in martensites and crystal plasticity Math. Models Methods App. Sci.: 1319-1355 2016 10.1142/S0218202516500317
Abstract: We study microstructure formation in two nonconvex singularly-perturbed variational problems from materials science, one modeling austenite-martensite interfaces in shape-memory alloys, the other one slip structures in the plastic deformation of crystals. For both functionals we determine the scaling of the optimal energy in terms of the parameters of the problem, leading to a characterization of the mesoscopic phase diagram. Our results identify the presence of a new phase, which is intermediate between the classical laminar microstructures and branching patterns. The new phase, characterized by partial branching, appears for both problems in the limit of small volume fraction, that is, if one of the variants (or of the slip systems) dominates the picture and the volume fraction of the other one is small.
Angkana Rüland, Christian Zillinger and Barbara Zwicknagl Higher Sobolev regularity of convex integration solutions in elasticity 2016 https://arxiv.org/abs/1610.02529
2015
Peter Bella, Michael Goldman and Barbara Zwicknagl Study of Island Formation in Epitaxially Strained Films on Unbounded Domains Arch. for Ration. Mech. and Anal., 218(1): 163-217 2015 http://dx.doi.org/10.1007/s00205-015-0858-x
Michael Griebel, Christian Rieger and Barbara Zwicknagl Multiscale approximation and reproducing kernel Hilbert space methods SIAM Journal on Numerical Analysis, 53(2): 852-873 2015 http://dx.doi.org/10.1137/130932144
2014
Irene Fonseca, Aldo Pratelli and Barbara Zwicknagl Shapes of Epitaxially Grown Quantum Dots Archive for Rational Mechanics and Analysis, 214(2): 359-401 2014 http://dx.doi.org/10.1007/s00205-014-0767-4