Stability of microstructure for tetragonal to monoclinic martensitic transformations

Pavel Belik; Mitchell Luskin

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 34, Issue: 3, page 663-685
  • ISSN: 0764-583X

Abstract

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We give an analysis of the stability and uniqueness of the simply laminated microstructure for all three tetragonal to monoclinic martensitic transformations. The energy density for tetragonal to monoclinic transformations has four rotationally invariant wells since the transformation has four variants. One of these tetragonal to monoclinic martensitic transformations corresponds to the shearing of the rectangular side, one corresponds to the shearing of the square base, and one corresponds to the shearing of the plane orthogonal to a diagonal in the square base. We show that the simply laminated microstructure is stable except for a class of special material parameters. In each case that the microstructure is stable, we derive error estimates for the finite element approximation.

How to cite

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Belik, Pavel, and Luskin, Mitchell. "Stability of microstructure for tetragonal to monoclinic martensitic transformations." ESAIM: Mathematical Modelling and Numerical Analysis 34.3 (2010): 663-685. <http://eudml.org/doc/197438>.

@article{Belik2010,
abstract = { We give an analysis of the stability and uniqueness of the simply laminated microstructure for all three tetragonal to monoclinic martensitic transformations. The energy density for tetragonal to monoclinic transformations has four rotationally invariant wells since the transformation has four variants. One of these tetragonal to monoclinic martensitic transformations corresponds to the shearing of the rectangular side, one corresponds to the shearing of the square base, and one corresponds to the shearing of the plane orthogonal to a diagonal in the square base. We show that the simply laminated microstructure is stable except for a class of special material parameters. In each case that the microstructure is stable, we derive error estimates for the finite element approximation. },
author = {Belik, Pavel, Luskin, Mitchell},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Martensitic transformation; microstructure; nonconvex variational problem; simple laminate; tetragonal; monoclinic; volume fraction; Young measure; finite element; error estimate; stability; uniqueness; simply laminated microstructure; tetragonal to monoclinic martensitic transformations; energy density; rotationally invariant wells; error estimates; finite element approximation},
language = {eng},
month = {3},
number = {3},
pages = {663-685},
publisher = {EDP Sciences},
title = {Stability of microstructure for tetragonal to monoclinic martensitic transformations},
url = {http://eudml.org/doc/197438},
volume = {34},
year = {2010},
}

TY - JOUR
AU - Belik, Pavel
AU - Luskin, Mitchell
TI - Stability of microstructure for tetragonal to monoclinic martensitic transformations
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 34
IS - 3
SP - 663
EP - 685
AB - We give an analysis of the stability and uniqueness of the simply laminated microstructure for all three tetragonal to monoclinic martensitic transformations. The energy density for tetragonal to monoclinic transformations has four rotationally invariant wells since the transformation has four variants. One of these tetragonal to monoclinic martensitic transformations corresponds to the shearing of the rectangular side, one corresponds to the shearing of the square base, and one corresponds to the shearing of the plane orthogonal to a diagonal in the square base. We show that the simply laminated microstructure is stable except for a class of special material parameters. In each case that the microstructure is stable, we derive error estimates for the finite element approximation.
LA - eng
KW - Martensitic transformation; microstructure; nonconvex variational problem; simple laminate; tetragonal; monoclinic; volume fraction; Young measure; finite element; error estimate; stability; uniqueness; simply laminated microstructure; tetragonal to monoclinic martensitic transformations; energy density; rotationally invariant wells; error estimates; finite element approximation
UR - http://eudml.org/doc/197438
ER -

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