MATERIALS 兴 HIENGE& ENGIEERING ELSEVIER Materials Science and Engineering A 438-440(2006)55-63 www.elsevier.com/locate/msea Multi-scale phase field approach to martensitic transformations Yunzhi Wang,, Armen G Khachatur a Department of Materials Science and Engineering. The Ohio State University, Columbus, OH 43210, USA b Department of Ceramic and Materials Engineering. Rutgers University, 607 Taylor Road, Piscataway, NJ08854, USA Received 18 August 2005: received in revised form 17 March 2006: accepted 17 April 2006 Because of its flexibility in treating complex geometries and topological changes, the phase field method has been used widely in modeling microstructural evolution during various phase transformations, grain growth and, most recently, plastic deformations. We review formulations and applications of the method in the context of martensitic transformations(MTs). Examples are chosen to illustrate the capabilities of the method at both mesoscopic and microscopic length scales. At the mesoscopic level we present simulation predictions on structural configurations of critical nuclei generated by homogeneous nucleation through Langevin thermal fluctuations under large undercooling, formation of herringbone structures by autocatalytic growth, and microstructural evolution and transformation hysteresis in a polycrystalline alloy under uniaxial stresses. At the microscopic level we discuss new developments in phase field model of dislocation dissociation and core structure and phase field model of mislocation-a new elementary defect introduced to describe the initiation and growth of martensite. o 2006 Elsevier B. v. All rights reserved Keywords: Martensitic transformation; Phase field; Nucleation; Dislocation; Multi-scale modelin 1. Introduction formation of a critical nucleus configuration in an arbitrary pr existing microstructure and its subsequent growth without any The essential features of martensitic transformations(MTs) a priori assumptions about its shape, orientation, spatial loca- are characterized by(a) thermoelastic two-phase equilibrium tion, and arrangement of different orientation variants within that violates the Gibbs phase rule;(b) non-ergodicity that it. The conventional treatment that considers an isolated parti leads to path-dependent equilibrium microstructure and trans- cle of a given shape without capturing the effect of pre-existing formation hysteresis;(c)unique morphological patterns; and microstructures may not be sufficient to quantitatively describe (d) stress-induced transformation and shape memory effect. the nucleation and growth processes leading to self-organization These features are associated with elastic strain accommoda- of multivariant and multiphase martensitic microstructures. tion of different orientation variants of the martensitic phase in The work by Cahn [2-5] on spinodal a parent phase matrix, which dominates the transformation ther- coherent fluctuations is, in fact, the earliest application of modynamics, kinetics and crystallography. The strain-induced the phase field method to coherent transformations in solids long-range elastic interactions among the orientation variants where the transformation-induced coherency strain was consid and between them and pre-existing strain-carrying defects such ered. The extension of the method to arbitrary microstructures as dislocations, precipitates, microcracks and grain boundaries produced by diffusional and displacive transformations with determine the activation pathways for nucleation and growt arbitrary transformation strains is made possible by using the and control the overall transformation kinetics. Since the elastic microelasticity theory of Khachaturyan and Shatalov(Ks the strain energy is in general a function of size, shape, orien- ory)[1, 6, 7] who developed a reciprocal-space formulation of the tation and spatial arrangement of precipitates [1], a rigorous strainenergy as an explicit functional of arbitrary continuous dis- treatment of microstructural evolution during MTs requires a tributions of structural and/or compositional non-uniformities kinetic approach that is able to describe self-consistently the The method has been applied to various coherent phase trans- formations including MTs and many complicated strain-induced morphological patterns have been predicte Corresponding author. Tel. +1 614 292 0682: fax: +1 614 292 1537. see[8-11)). Most recently, the KS theap. ed (for recent reviews vas successfully inte- E-mail address: wang.363@osu.edu(Y. Wang). grated with the phase field models of dislocation dynamics and 0921-5093/S-see front matterMaterials Science and Engineering A 438–440 (2006) 55–63 Multi-scale phase field approach to martensitic transformations Yunzhi Wang a,∗, Armen G. Khachaturyan b a Department of Materials Science and Engineering, The Ohio State University, Columbus, OH 43210, USA b Department of Ceramic and Materials Engineering, Rutgers University, 607 Taylor Road, Piscataway, NJ 08854, USA Received 18 August 2005; received in revised form 17 March 2006; accepted 17 April 2006 Abstract Because of its flexibility in treating complex geometries and topological changes, the phase field method has been used widely in modeling microstructural evolution during various phase transformations, grain growth and, most recently, plastic deformations. We review formulations and applications of the method in the context of martensitic transformations (MTs). Examples are chosen to illustrate the capabilities of the method at both mesoscopic and microscopic length scales. At the mesoscopic level we present simulation predictions on structural configurations of critical nuclei generated by homogeneous nucleation through Langevin thermal fluctuations under large undercooling, formation of herringbone structures by autocatalytic growth, and microstructural evolution and transformation hysteresis in a polycrystalline alloy under uniaxial stresses. At the microscopic level we discuss new developments in phase field model of dislocation dissociation and core structure and phase field model of mislocation—a new elementary defect introduced to describe the initiation and growth of martensite. © 2006 Elsevier B.V. All rights reserved. Keywords: Martensitic transformation; Phase field; Nucleation; Dislocation; Multi-scale modeling 1. Introduction The essential features of martensitic transformations (MTs) are characterized by (a) thermoelastic two-phase equilibrium that violates the Gibbs phase rule; (b) non-ergodicity that leads to path-dependent equilibrium microstructure and trans￾formation hysteresis; (c) unique morphological patterns; and (d) stress-induced transformation and shape memory effect. These features are associated with elastic strain accommoda￾tion of different orientation variants of the martensitic phase in a parent phase matrix, which dominates the transformation ther￾modynamics, kinetics and crystallography. The strain-induced long-range elastic interactions among the orientation variants and between them and pre-existing strain-carrying defects such as dislocations, precipitates, microcracks and grain boundaries determine the activation pathways for nucleation and growth and control the overall transformation kinetics. Since the elastic strain energy is in general a function of size, shape, orien￾tation and spatial arrangement of precipitates [1], a rigorous treatment of microstructural evolution during MTs requires a kinetic approach that is able to describe self-consistently the ∗ Corresponding author. Tel.: +1 614 292 0682; fax: +1 614 292 1537. E-mail address: wang.363@osu.edu (Y. Wang). formation of a critical nucleus configuration in an arbitrary pre￾existing microstructure and its subsequent growth without any a priori assumptions about its shape, orientation, spatial loca￾tion, and arrangement of different orientation variants within it. The conventional treatment that considers an isolated parti￾cle of a given shape without capturing the effect of pre-existing microstructures may not be sufficient to quantitatively describe the nucleation and growth processes leading to self-organization of multivariant and multiphase martensitic microstructures. The work by Cahn [2–5] on spinodal decomposition and coherent fluctuations is, in fact, the earliest application of the phase field method to coherent transformations in solids where the transformation-induced coherency strain was consid￾ered. The extension of the method to arbitrary microstructures produced by diffusional and displacive transformations with arbitrary transformation strains is made possible by using the microelasticity theory of Khachaturyan and Shatalov (KS the￾ory)[1,6,7] who developed a reciprocal-space formulation of the strain energy as an explicit functional of arbitrary continuous dis￾tributions of structural and/or compositional non-uniformities. The method has been applied to various coherent phase trans￾formations including MTs and many complicated strain-induced morphological patterns have been predicted (for recent reviews see [8–11]). Most recently, the KS theory was successfully inte￾grated with the phase field models of dislocation dynamics and 0921-5093/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2006.04.123