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We consider a singularly-perturbed two-well problem in the context of planar geometrically linear elasticity to model a rectangular martensitic nucleus in an austenitic matrix. We derive the scaling regimes for the minimal energy in terms of the problem parameters, which represent the shape of the nucleus, the quotient of the elastic moduli of the two phases, the surface energy constant, and the volume fraction of the two martensitic variants. We identify several different scaling regimes, which are distinguished either by the exponents in the parameters, or by logarithmic corrections, for which we have matching upper and lower bounds.
@article{COCV_2020__26_1_A115_0,
author = {Conti, Sergio and Diermeier, Johannes and Melching, David and Zwicknagl, Barbara},
title = {Energy scaling laws for geometrically linear elasticity models for microstructures in shape memory alloys},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {26},
year = {2020},
doi = {10.1051/cocv/2020020},
mrnumber = {4185057},
zbl = {1459.49004},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2020020/}
}
TY - JOUR AU - Conti, Sergio AU - Diermeier, Johannes AU - Melching, David AU - Zwicknagl, Barbara TI - Energy scaling laws for geometrically linear elasticity models for microstructures in shape memory alloys JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2020 VL - 26 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/cocv/2020020/ DO - 10.1051/cocv/2020020 LA - en ID - COCV_2020__26_1_A115_0 ER -
%0 Journal Article %A Conti, Sergio %A Diermeier, Johannes %A Melching, David %A Zwicknagl, Barbara %T Energy scaling laws for geometrically linear elasticity models for microstructures in shape memory alloys %J ESAIM: Control, Optimisation and Calculus of Variations %D 2020 %V 26 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/cocv/2020020/ %R 10.1051/cocv/2020020 %G en %F COCV_2020__26_1_A115_0
Conti, Sergio; Diermeier, Johannes; Melching, David; Zwicknagl, Barbara. Energy scaling laws for geometrically linear elasticity models for microstructures in shape memory alloys. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 115. doi : 10.1051/cocv/2020020. http://geodesic.mathdoc.fr/articles/10.1051/cocv/2020020/
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This work was mainly done while all authors were at the University of Bonn and was partially supported by the Deutsche Forschungsgemeinschaft via project 211504053 - SFB 1060/A06.