Y Wang, A G Khachaturyan/Materials Science and Engineering A 438-440(2006)55-63 microcrack propagation at mesoscales [12]and with phase field with a homogeneous strain that is responsible for the martensitic model of dislocation core structures at microscales [13]. These features of the transformation advances have paved the road for realistic nano-and meso-scale <3 A key step in developing a phase field model is to formulate phase field modeling of MTs involving simultaneous plastic the total free energy of a system as a function of the chosen deformation and fracture. The latter could have important appli- SOP fields. In the discussion of MTs, the total free energy is cations to transformation toughening in ceramics. usually separated into two competing terms, i. e, the chemical This article provides an overview of the applications of the free energy that provides the driving force for the transformation phase field methods to computer simulation of MTs at two dif- and the elastic strain energy that suppresses the transformation ferent length scales. Thus, at the mesoscopic level, the examples The chemical free energy of a homogeneous system is usually include fundamental characteristics of microstructural evolution approximated by a Landau polynomial expansion with respect to (stain-accommodating packing of martensitic variants, auto- the SOPs [31-33]. For example, the simplest fourth-order poly catalytic effect on nucleation and growth, and transformation nomial approximation of the Landau free energy used [34-38 hysteresis)during a generic cubic ->tetragonal MT in a sin- for a proper fcc- bcc MT is presented as a sum of the symmetry gle crystal [14] and a more complex cubic->trigonal MT in invariants a polycrystalline system [15]. At the microscopic leve scopic phase field model of dislocation core structures [13, 161 f(n1, n2, n3= Ar discussions focus on the most recent developments of micro- 1(m2+n+n)-32(m+n+n nd its application to the study of micromechanics of nucle ation and growth of martensite[17]. It should be pointed out +A3(m+n+32 (1) that extensive work has been done at the mesoscopic level in pursuing ferroelastic and phase field approaches to martensitic where np(p=1, 2, 3)are the SOPs that describe the tetragonal transformations [18-27] and the applications discussed in this distortion of the three orientation variants along the Bain path article are just a few examples from our own work, by no means andAi(i=1, 2, 3)are the expansion coefficients. For an improper complete MT the sixth-order polynomial forming independent symmet invariants of the appropriate order has been introduced [14] 2. Phase field method at meso-scale and its applications to martensitic transformations f(m1,n2,…,m)=(m+n+…+听 Using gradient thermodynamics of non-uniform systems [28-30], the theory of microelasticity [1,6,7), and continuum A fields of conserved and non-conserved parameters, the phase m+n+…+m field method describes spatial-temporal evolution of arbitrary microstructures consisting of various types of extended defects A3(+n+… such as homo- and hetero-phase interfaces, antiphase domain boundaries, ferromagnetic and ferroelectric domain walls, and, most recently, dislocations and cracks (for the most recent 6+听吗 reviews, see [12]). A typical example of the conserved fields is +…+n2-272-1m2 concentration that characterizes chemical non-uniformity and a typical example of the non-conserved fields is the long-range (⑦m+n2+…+m) order parameter that characterizes crystal symmetry change during an order-disorder transformation. The microstructures developed in a MT can be characterized fully by a set of non- +6+吃 onserved fields, the structural order parameter(SOP)fields Different from phase field models of solidification where the where np (p=l, 2,..., n) are the SOPs that characterize the non-conserved field is introduced as a numerical technique to amplitudes of the optimal displacement modes and Ai(i=1 avoid boundary tracking, the non-conserved fields employed in 2,..., 6) are the expansion coefficients. The free energies pre- phase field models of solid-state phase transformations have sented in Eqs. (1)and(2)are specific free energies of structurall well defined physical meanings. In a proper MT such as the homogeneous finite elements with uniform values(equilibrium face centered cubic to body centered cubic ( cc- bcc) lattice or non-equilibrium) of the SOP, np. The minima of the free rearrangement in iron, for example, the SOP distinguishing the energy correspond to the equilibrium free energies of the stress- martensite and austenite phases is the tetragonal Bain distor- free parent and product phases. The minima for the product tion(a homogeneous strain that transforms an fcc lattice to a(martensite) phase are degenerate, with each of them describing bcc lattice). In an improper MT such as the cubic>tetragonal a crystallographically equivalent orientation domain transformation in Zro2 and many MTs in ferroelastic materi- In a structurally non-uniform system such as a mixture of als, the SOP is the amplitude of a"soft "optical displacement martensite and austenite phases, the finite elements characte mode that results in a mutual displacement of atoms within a unit ized by the Landau free energy have different values of np cell of the parent phase(shuffle). This primary SOP is coupled According to gradient thermodynamics [28-30], the chemical56 Y. Wang, A.G. Khachaturyan / Materials Science and Engineering A 438–440 (2006) 55–63 microcrack propagation at mesoscalses[12] and with phase field model of dislocation core structures at microscales [13]. These advances have paved the road for realistic nano- and meso-scale phase field modeling of MTs involving simultaneous plastic deformation and fracture. The latter could have important appli￾cations to transformation toughening in ceramics. This article provides an overview of the applications of the phase field methods to computer simulation of MTs at two dif￾ferent length scales. Thus, at the mesoscopic level, the examples include fundamental characteristics of microstructural evolution (stain-accommodating packing of martensitic variants, auto￾catalytic effect on nucleation and growth, and transformation hysteresis) during a generic cubic→tetragonal MT in a sin￾gle crystal [14] and a more complex cubic→trigonal MT in a polycrystalline system [15]. At the microscopic level, the discussions focus on the most recent developments of micro￾scopic phase field model of dislocation core structures [13,16] and its application to the study of micromechanims of nucle￾ation and growth of martensite [17]. It should be pointed out that extensive work has been done at the messoscopic level in pursuing ferroelastic and phase field approaches to martensitic transformations [18–27] and the applications discussed in this article are just a few examples from our own work, by no means complete. 2. Phase field method at meso-scale and its applications to martensitic transformations Using gradient thermodynamics of non-uniform systems [28–30], the theory of microelasticity [1,6,7], and continuum fields of conserved and non-conserved parameters, the phase field method describes spatial-temporal evolution of arbitrary microstructures consisting of various types of extended defects such as homo- and hetero-phase interfaces, antiphase domain boundaries, ferromagnetic and ferroelectric domain walls, and, most recently, dislocations and cracks (for the most recent reviews, see [12]). A typical example of the conserved fields is concentration that characterizes chemical non-uniformity and a typical example of the non-conserved fields is the long-range order parameter that characterizes crystal symmetry change during an order-disorder transformation. The microstructures developed in a MT can be characterized fully by a set of non￾conserved fields, the structural order parameter (SOP) fields. Different from phase field models of solidification where the non-conserved field is introduced as a numerical technique to avoid boundary tracking, the non-conserved fields employed in phase field models of solid-state phase transformations have well defined physical meanings. In a proper MT such as the face centered cubic to body centered cubic (fcc→bcc) lattice rearrangement in iron, for example, the SOP distinguishing the martensite and austenite phases is the tetragonal Bain distor￾tion (a homogeneous strain that transforms an fcc lattice to a bcc lattice). In an improper MT such as the cubic→tetragonal transformation in ZrO2 and many MTs in ferroelastic materi￾als, the SOP is the amplitude of a “soft” optical displacement mode that results in a mutual displacement of atoms within a unit cell of the parent phase (shuffle). This primary SOP is coupled with a homogeneous strain that is responsible for the martensitic features of the transformation. A key step in developing a phase field model is to formulate the total free energy of a system as a function of the chosen SOP fields. In the discussion of MTs, the total free energy is usually separated into two competing terms, i.e., the chemical free energy that provides the driving force for the transformation and the elastic strain energy that suppresses the transformation. The chemical free energy of a homogeneous system is usually approximated by a Landau polynomial expansion with respect to the SOPs [31–33]. For example, the simplest fourth-order poly￾nomial approximation of the Landau free energy used [34–38] for a properfcc→bcc MT is presented as a sum of the symmetry invariants f (η1, η2, η3) = A1 2 (η2 1 + η2 2 + η2 3) − A2 3 (η3 1 + η3 2 + η3 3) + A3 4 (η2 1 + η2 2 + η2 3) 2 (1) where ηp (p = 1, 2, 3) are the SOPs that describe the tetragonal distortion of the three orientation variants along the Bain path and Ai (i = 1, 2, 3) are the expansion coefficients. For an improper MT the sixth-order polynomial forming independent symmetry invariants of the appropriate order has been introduced [14] f (η1, η2,...,ηn) = A1 2 (η2 1 + η2 2 +···+ η2 n) + A2 4 (η4 1 + η4 2 +···+ η4 n) + A3 4 (η2 1 + η2 2 +···η2 n) 2 + A4 6 (η2 1η2 2η2 3 + η2 1η2 2η2 4 +· · · + η2 n−2η2 n−1η2 n) + A5 6 (η6 1 + η6 2 +···+ η6 n) + A6 6 (η2 1 + η2 2 +···+ η2 n) 3 (2) where ηp (p = 1, 2, ..., n) are the SOPs that characterize the amplitudes of the optimal displacement modes and Ai (i = 1, 2, ..., 6) are the expansion coefficients. The free energies pre￾sented in Eqs.(1) and (2) are specific free energies of structurally homogeneous finite elements with uniform values (equilibrium or non-equilibrium) of the SOP, ηp. The minima of the free energy correspond to the equilibrium free energies of the stress￾free parent and product phases. The minima for the product (martensite) phase are degenerate, with each of them describing a crystallographically equivalent orientation domain. In a structurally non-uniform system such as a mixture of martensite and austenite phases, the finite elements character￾ized by the Landau free energy have different values of ηp. According to gradient thermodynamics [28–30], the chemical