Research Article

Stability of microstructure for tetragonal to monoclinic martensitic transformations

Belik, Pavel^a1 and Luskin, Mitchell^a2

^a1 School of Mathematics, University of Minnesota, 206 Church Street SE, Minneapolis, MN 55455, USA; (belik@math.umn.edu)

^a2 School of Mathematics, University of Minnesota, 206 Church Street SE, Minneapolis, MN 55455, USA; (luskin@math.umn.edu)

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.

(Received March 29 1999)

(Revised November 25 1999)

(Online publication April 15 2002)

Key Words:

• Martensitic transformation;
• microstructure;
• nonconvex variational problem;
• simple laminate;
• tetragonal;
• monoclinic;
• volume fraction;
• Young measure;
• finite element;
• error estimate

Mathematics Subject Classification:

• 49J45;
• 65N15;
• 65N30;
• 73C50;
• 73G05;
• 73K20;
• 73V05