JOURNAL OF THE Journal of the Mechanics and Physics of Solids PHYSICS OF SOLID ELSEVIER 52(2004)2057-2077 Modeling and simulation of martensitic phase transitions with a triple point Patrick W. donda. Johannes zimmer a Dinision of Engineering and Applied Science, California Institute of Technology, Mail Stop 104-4 Pasadena ca 9/125. USa b Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig. Germany Received 24 November 2003: received in revised form 17 February 2004; accepted 2 March 2004 A framework for modeling complex global energy landscapes in a piecewise manner is pre- sented. Specifically, a class of strain-dependent energy functions is derived for the triple point of Zirconia(ZrO2), where tetragonal, orthorhombic(ortho) and monoclinic phases are stable. A simple two-dimensional framework is presented to deal with this symmetry breaking. An explicit energy is then fitted to the available elastic moduli of Zirconia in this two-dimensional setting. First, we use the orbit space method to deal with symmetry constraints in an easy way. Sec- ond, we introduce a modular(piecewise) approach to reproduce or model elastic moduli, energy barriers and other characteristics independently of each other in a sequence of local steps. This allows for more general results than the classical Landau theory (understood in the sense that the energy is a polynomial of invariant polynomials ) The class of functions considered here is strictly larger. Finite-element simulations for the energy constructed here demonstrate the pattern formation in Zirconia at the triple point. C 2004 Elsevier Ltd. All rights reserved PACS61.50Ks;,62.20.-x;81.30.Kf Keywords: A. Microstructures; A. Phase transformations; B. Elastic material 1. Introduction This paper provides a framework for modeling complex energetic landscapes, such as atomistic potentials or energies describing materials that undergo phase transitions. Corresponding author.Tel:+49-341-99-59-545;fax:+49-341-99-59-633 E-mail addresses: pwd(@caltech.edu(P w. DondI), zimmer(@mis. mpg. de (J. Zimmer ) URL //www.mis.mpg.de/zimme 0022-5096/S-see front matter e 2004 Elsevier Ltd. All rights reserved doi:10.1016/jmps200403.001Journal of the Mechanics and Physics of Solids 52 (2004) 2057 – 2077 www.elsevier.com/locate/jmps Modeling and simulation of martensitic phase transitions with a triple point Patrick W. Dondla, Johannes Zimmerb;∗ aDivision of Engineering and Applied Science, California Institute of Technology, Mail Stop 104-44, Pasadena, CA 91125, USA bMax-Planck-Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig, Germany Received 24 November 2003; received in revised form 17 February 2004; accepted 2 March 2004 Abstract A framework for modeling complex global energy landscapes in a piecewise manner is pre￾sented. Speci3cally, a class of strain-dependent energy functions is derived for the triple point of Zirconia (ZrO2), where tetragonal, orthorhombic (orthoI) and monoclinic phases are stable. A simple two-dimensional framework is presented to deal with this symmetry breaking. An explicit energy is then 3tted to the available elastic moduli of Zirconia in this two-dimensional setting. First, we use the orbit space method to deal with symmetry constraints in an easy way. Sec￾ond, we introduce a modular (piecewise) approach to reproduce or model elastic moduli, energy barriers and other characteristics independently of each other in a sequence of local steps. This allows for more general results than the classical Landau theory (understood in the sense that the energy is a polynomial of invariant polynomials). The class of functions considered here is strictly larger. Finite-element simulations for the energy constructed here demonstrate the pattern formation in Zirconia at the triple point. ? 2004 Elsevier Ltd. All rights reserved. PACS: 61.50.Ks; 62.20.−x; 81.30.Kf Keywords: A. Microstructures; A. Phase transformations; B. Elastic material 1. Introduction This paper provides a framework for modeling complex energetic landscapes, such as atomistic potentials or energies describing materials that undergo phase transitions. ∗ Corresponding author. Tel.: +49-341-99-59-545; fax: +49-341-99-59-633. E-mail addresses: pwd@caltech.edu (P.W. Dondl), zimmer@mis.mpg.de (J. Zimmer). URL: http://www.mis.mpg.de/zimmer/ 0022-5096/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jmps.2004.03.001