Y Wang, A.G. Khachaturyan/ Materials Science and Engineering A 438-440 (2006)55-63 Fig. 5. Time-evolution of dislocation substructures on a(111) plane of an fcc crystal during annealing simulated by the phase field method [13] nucleation of martensite on pre-existing dislocations and grain term that is proportional only to the dislocation line length and boundaries within a single framework. describes the effective dislocation core energy y treating a dislocation loop as a thin platelet misfitting inclusion(e.g, a thin martensitic platelet of a single variant) sas eing formulated in the phase field framework, the model able to describe collective motion of dislocations of arbi a phase field description of the spatial-temporal evolution of trary configurations and dislocation-precipitate interactions at gliding dislocations was developed [60]. In this approach a new meso-scales. Topological changes such as nucleation, multipli set of sops is introduced to characterize the structural non- cation annihilation and reaction of dislocations are accounted uniformities associated with dislocations. A region enclosed by for naturally without explicitly tracking the moving dislocation a dislocation loop on a slip plane can be regarded as a"trans- lines(see, e.g., Fig. 5(13). To apply the model to study the formed region"that has been plastically deformed by dislocation micromechanisms of MTs, however, one needs to resolve the glide and the boundaries between the transformed regions of dif- details of dislocation core structures because dissociation of lat ferent type and number of slips are lines of dislocations. These tice dislocations is essential in dislocation models of martensite slipped regions on each individual slip plane are characterized embryos [54-59. Recently, a microscopic phase field model by the SOPs, n(a, ma, r), where, a and ma index the slip plane of dislocation dissociation and core structures [13, 16] has been and slip direction, respectively. The number of the structural developed, which employs ab initio GSF energy as the model order parameters required equals to the number of operative slip input. It has been shown quantitatively systems in a crystal. field dislocation model is a 3D generalization of the Peierls- The time-evolution of the dislocation loops is treated in a sim- Nabarro(P-N) model [62, 63]. When applied to straight dislo- ilar way as how the evolution of martensitic particles is described cations, a complete agreement of the phase field model with in the phase field model introduced above, e.g., by Eq (6). Simi- the P-N model for core structures [64] has been achieved using lar to the energetics of phase field model of MT discussed earlier, the same set of ab initio data of GSF energies [65](Fig. 6) the total energy of a dislocation system consists of three terms the crystalline energy that replaces the local specific chemical free energy, the gradient energy, and the elastic strain energy. The crystalline energy is a periodic energy landscape associ- ated with a general plastic deformation in a crystal by arbitrary o linear combinations of localized simple shears(slips)associ- ated with all operative slip systems [13]. when projected onto a particular slip plane, the potential energy landscape reduces to 0. 6- n(r) the so-called generalized stacking fault(GSF)energy [61]. The elastic energy equation is identical to the one formulated for 0. 4 MT(e.g, Eq(4)), with the SFTS of martensite replaced by the 0.3 eigenstrain of a dislocation loop. The gradient energy accounts dn(x) for contributions from spatial non-uniformity of n(a, ma, r)in the slip plane(inhomogeneity in the slip displacement). To take 0.1) into account the fact that the crystal lattice is the same on both sides of the slip plane and thus the slip plane does not introduce interfacial energy, the gradient terms have been formulated [60] in such a form that their values are finite only at the disloc- Fig. 6. Quantitative comparison of the core structures of an edge dislocation in Al calculated by the phase field model (lines)[ 16] and the Peierls model (poin tion line and vanish at the slip plane. This results in an energy [64] with the same GSF energy [65]Y. Wang, A.G. Khachaturyan / Materials Science and Engineering A 438–440 (2006) 55–63 61 Fig. 5. Time-evolution of dislocation substructures on a (1 1 1) plane of an fcc crystal during annealing simulated by the phase field method [13]. nucleation of martensite on pre-existing dislocations and grain boundaries within a single framework. By treating a dislocation loop as a thin platelet misfitting inclusion (e.g., a thin martensitic platelet of a single variant), a phase field description of the spatial-temporal evolution of gliding dislocations was developed [60]. In this approach a new set of SOPs is introduced to characterize the structural non￾uniformities associated with dislocations. A region enclosed by a dislocation loop on a slip plane can be regarded as a “trans￾formed region” that has been plastically deformed by dislocation glide and the boundaries between the transformed regions of dif￾ferent type and number of slips are lines of dislocations. These slipped regions on each individual slip plane are characterized by the SOPs, η(α, m, r), where, α and m index the slip plane and slip direction, respectively. The number of the structural order parameters required equals to the number of operative slip systems in a crystal. The time-evolution of the dislocation loops is treated in a sim￾ilar way as how the evolution of martensitic particles is described in the phase field model introduced above, e.g., by Eq. (6). Simi￾lar to the energetics of phase field model of MT discussed earlier, the total energy of a dislocation system consists of three terms: the crystalline energy that replaces the local specific chemical free energy, the gradient energy, and the elastic strain energy. The crystalline energy is a periodic energy landscape associ￾ated with a general plastic deformation in a crystal by arbitrary linear combinations of localized simple shears (slips) associ￾ated with all operative slip systems [13]. When projected onto a particular slip plane, the potential energy landscape reduces to the so-called generalized stacking fault (GSF) energy [61]. The elastic energy equation is identical to the one formulated for MT (e.g, Eq. (4)), with the SFTS of martensite replaced by the eigenstrain of a dislocation loop. The gradient energy accounts for contributions from spatial non-uniformity of η(α, m, r) in the slip plane (inhomogeneity in the slip displacement). To take into account the fact that the crystal lattice is the same on both sides of the slip plane and thus the slip plane does not introduce interfacial energy, the gradient terms have been formulated [60] in such a form that their values are finite only at the disloca￾tion line and vanish at the slip plane. This results in an energy term that is proportional only to the dislocation line length and describes the effective dislocation core energy. Being formulated in the phase field framework, the model is able to describe collective motion of dislocations of arbi￾trary configurations and dislocation-precipitate interactions at meso-scales. Topological changes such as nucleation, multipli￾cation, annihilation and reaction of dislocations are accounted for naturally without explicitly tracking the moving dislocation lines (see, e.g., Fig. 5 [13]). To apply the model to study the micromechanisms of MTs, however, one needs to resolve the details of dislocation core structures because dissociation of lat￾tice dislocations is essential in dislocation models of martensite embryos [54–59]. Recently, a microscopic phase field model of dislocation dissociation and core structures [13,16] has been developed, which employs ab initio GSF energy as the model input. It has been shown quantitatively [13,16] that the phase field dislocation model is a 3D generalization of the Peierls￾Nabarro (P-N) model [62,63]. When applied to straight dislo￾cations, a complete agreement of the phase field model with the P-N model for core structures [64] has been achieved using the same set of ab initio data of GSF energies [65] (Fig. 6). Fig. 6. Quantitative comparison of the core structures of an edge dislocation in Al calculated by the phase field model (lines) [16] and the Peierls model (points) [64] with the same GSF energy [65]