R.C. Pond et al. /Materials Science and Engineering A 438-440(2006)109-112 5139 11364 Fig. 1. Schematic illustration of parent and product lattices viewed along eg: (a)orthorhombic and(double)monoclinic cells for ZrO2, and(b) bcc and hcp cells for Ti. Unit sphere and deformation ellipsoids for(c)ZrO2 and(d) Ti; the principal axes and habit plane inclinations are shown 2. The phenomenological and topological models transferred from one crystal to the other, some being sheared into their new locations and the remainder shuffling [5]. How The central proposition of the PTMc is that the ever, because the densities of the two crystals are not generally parent-martensite interface is an invariant plane(IP)of the total equal, this process does not correspond to b being parallel to the shape transformation [5-7]. The classical theories are algorithms TP analogously to the case of dislocations in single crystals 3] for finding this plane given the lattice deformation, correspon- In the cases of Zro2 and Ti, the horizontal planes in Fig. I(a and dence and mode of LID. In other words, no strain arises in either b), respectively, are candidate TPs when coherently strained, as of the crystals because all vectors parallel to the HP remain depicted in Fig. 2(a and b). Moreover, disconnections with b invariant in the transformation. The IP can be found by iden- as indicated in Fig. 2, with magnitudes defined in the reference tifying any two non-parallel invariant vectors. If one princi coherent interface, would traverse these terraces conservatively strain of the lattice deformation is zero, the associated Eigenvec- a more detailed impression of the formation of the disconnec tore constitutes one invariant vector, so the solution for the IP tion for Ti, indicating the step height h, is shown in Fig. 3. The becomes the 2-D problem of finding an invariant vector vi per- coordinate frame used in Figs. 2 and 3 is referred to as the terrace endicular to ei after the requisite rigid-body rotation Ris carried frame out The IP solutions for ZrO2 and Ti are depicted in Fig. 1; the Now the spacing of disconnections necessary to accom- unit cells of the parent(orthorhombic)and product(monoclinic) modate the misfit strains can be calculated. Following Olson lattices are shown in Fig. 1(a) for ZrO2 projected along ei, andand Cohen [10], the coherency strain can be modelled as similarly for Ti(B and a). The corresponding deformation ellip- an array of dislocations, as indicated in Fig. 4(a); the total soids are depicted in Fig. I(c and d); for ZrO2 the inclination Burgers vector of these defects per unit length of ter- of the IP to the horizontal is 13.529 and r(with respect to the race, bey, is equal numerically to the total strain which is onfiguration shown in Fig. I(a))is a clockwise rotation(of the given by (e(a)yy -e()yy)(l+e(a)yy +e(B)yy+e(a)yye(B)yy) monoclinic crystal) about ei by 0.035, while for Ti these are where e(B)yy represents the 11.364 and 0.528(of a)anticlockwise, respectively. orthorhombic/B crystal parallel to y, and e(a)yy signifies the Using the TM, one first has to identify a candidate TP and compression of the monoclinic/o crystal in this direction: the disconnection that satisfy the criteria for diffusionless trans- total misfit strain on the terrace parallel to y is hereafter written formation [4]. In fact, the criteria are rarely satisfied for dis- as Eyy. When both crystals are equally strained, the magnitude of onnections in general inter-phase interfaces, but solutions can Eyy is 3. 84 and 3.80% for ZrO2 and Ti, respectively. As shown in be found, for example, in coherent interfaces. When a discon- Figs. 2 and 3, disconnections have Burgers vector components nection moves conservatively across a terrace plane, atoms are b, and bx, in addition to their step character. Thus, introducing110 R.C. Pond et al. / Materials Science and Engineering A 438–440 (2006) 109–112 Fig. 1. Schematic illustration of parent and product lattices viewed along ei; (a) orthorhombic and (double) monoclinic cells for ZrO2, and (b) bcc and hcp cells for Ti. Unit sphere and deformation ellipsoids for (c) ZrO2 and (d) Ti; the principal axes and habit plane inclinations are shown. 2. The phenomenological and topological models The central proposition of the PTMC is that the parent–martensite interface is an invariant plane (IP) of the total shape transformation [5–7]. The classical theories are algorithms for finding this plane given the lattice deformation, correspon￾dence and mode of LID. In other words, no strain arises in either of the crystals because all vectors parallel to the HP remain invariant in the transformation. The IP can be found by iden￾tifying any two non-parallel invariant vectors. If one principal strain of the lattice deformation is zero, the associated Eigenvec￾tor ei constitutes one invariant vector, so the solution for the IP becomes the 2-D problem of finding an invariant vector vi per￾pendicular to ei after the requisite rigid-body rotation R is carried out. The IP solutions for ZrO2 and Ti are depicted in Fig. 1; the unit cells of the parent (orthorhombic) and product (monoclinic) lattices are shown in Fig. 1(a) for ZrO2 projected along ei, and similarly for Ti ( and ). The corresponding deformation ellip￾soids are depicted in Fig. 1(c and d); for ZrO2 the inclination of the IP to the horizontal is 13.529◦ and R (with respect to the configuration shown in Fig. 1(a)) is a clockwise rotation (of the monoclinic crystal) about ei by 0.035◦, while for Ti these are 11.364◦ and 0.528◦ (of ) anticlockwise, respectively. Using the TM, one first has to identify a candidate TP and disconnection that satisfy the criteria for diffusionless trans￾formation [4]. In fact, the criteria are rarely satisfied for dis￾connections in general inter-phase interfaces, but solutions can be found, for example, in coherent interfaces. When a discon￾nection moves conservatively across a terrace plane, atoms are transferred from one crystal to the other, some being sheared into their new locations and the remainder shuffling [5]. How￾ever, because the densities of the two crystals are not generally equal, this process does not correspond to b being parallel to the TP analogously to the case of dislocations in single crystals [3]. In the cases of ZrO2 and Ti, the horizontal planes in Fig. 1(a and b), respectively, are candidate TPs when coherently strained, as depicted in Fig. 2(a and b). Moreover, disconnections with b as indicated in Fig. 2, with magnitudes defined in the reference coherent interface, would traverse these terraces conservatively; a more detailed impression of the formation of the disconnec￾tion for Ti, indicating the step height h, is shown in Fig. 3. The coordinate frame used in Figs. 2 and 3 is referred to as the terrace frame. Now the spacing of disconnections necessary to accom￾modate the misfit strains can be calculated. Following Olson and Cohen [10], the coherency strain can be modelled as an array of dislocations, as indicated in Fig. 4(a); the total Burgers vector of these defects per unit length of ter￾race, bcy, is equal numerically to the total strain which is given by (e()yy − e()yy)(1 + e()yy + e()yy + e()yye()yy) −1, where e()yy represents the uniaxial expansion of the orthorhombic/ crystal parallel to y, and e()yy signifies the compression of the monoclinic/ crystal in this direction: the total misfit strain on the terrace parallel to y is hereafter written as εyy. When both crystals are equally strained, the magnitude of εyy is 3.84 and 3.80% for ZrO2 and Ti, respectively. As shown in Figs. 2 and 3, disconnections have Burgers vector components by and bz, in addition to their step character. Thus, introducing