Stability of microstructure for tetragonal to monoclinic martensitic transformations

Free access article

Issue		ESAIM: M2AN Volume 34, Number 3, May-June 2000


Page(s)		663 - 685
DOI		10.1051/m2an:2000161

DOI: 10.1051/m2an:2000161
M2AN, Vol. 34, N $^{\rm o}$ 3, 2000, pp. 663-685

Stability of microstructure for tetragonal to monoclinic martensitic transformations

Pavel Belik
School of Mathematics, University of Minnesota, 206 Church Street SE, Minneapolis, MN 55455, USA; (belik@math.umn.edu)

Mitchell Luskin
School of Mathematics, University of Minnesota, 206 Church Street SE, Minneapolis, MN 55455, USA; (luskin@math.umn.edu)

Received: March 29, 1999. Revised: November 25, 1999

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.

Keywords and phrases: Martensitic transformation, microstructure, nonconvex variational problem, simple laminate, tetragonal, monoclinic, volume fraction, Young measure, finite element, error estimate

AMS Subject Classification: 49J45, 65N15, 65N30, 73C50, 73G05, 73K20, 73V05

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