P.M. Kelly, L.R. Francis Rose/ Progress in Materials Science 47(2002)463-557 473 ORIGINAL ORIGINAL LATTICE SHAPE Apply the L,LS Then the Bain strain BL Finally, RBL Fig. 3. Schematic illustration of the steps involved in the phenomenological theory of martensitic trans- formation. The changes to the crystal structure are shown on the left and to the macroscopic transforming The inputs required by the phenomenological theory are 1. The crystal structures and lattice parameters of the parent and product phases 2. The lattice correspondence between these two structures (i.e. which atom in the parent structure becomes which atom in the product structure) 3. The elements(plane and direction) of the lattice invariant shear L The lattice parameters, the crystal structures and the correspondence allow the Bain strain B to be calculated. An essential requirement for the Bain strain is that the principal strains of B must differ in sign (i.e. two tensile strains and one compressive
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P.M. Kelly, L.R. Francis Rose/ Progress in Materials Science 47(2002)463-557 strain or two compressive strains and one tensile strain). This ensures that there is a set of lines (vectors) that are unextended as a result of the Bain strain. This set of unextended lines is known as the bain cone, and a line somewhere in this cone will end up lying in the habit plane. The elements of the lattice invariant shear L are also input assumptions. They normally chosen to be the elements of a known slip or twinning system in the parent or product structures. For the phenomenological theory to lead to a real solution for S, the plane of the lattice invariant shear must intersect the Bain cone [ 19]. Provided this requirement is satisfied, the value of the magnitude 'g of the lattice invariant shear L that results in an undistorted plane can be calculated. All that then remains is to calculate the rigid body rotation R required to ensure that this undistorted lane is also unrotated The outputs of the phenomenological theory are 1. The habit plane of the martensite plates(the invariant plane) 2. The magnitude g of the lattice invariant shear. 3. The magnitude and direction of the shape strain 4. The orientation relationship between the parent and product phases. This is a combination of the original correspondence and the rigid body rotation R Before going on in Section 3 to compare the predictions of the phenomenological theory with experiment for the tetragonal to monoclinic (t-m) transformation in zirconia, some important features of the phenomenological theory and its predic tions need to be considered. In particular these relate to the nucleation of the mar tensitic transformation, the importance of surface and strain energies in determining the shape of the transformed product, the shape strain and its interaction with applied stress(stress-induced transformation) and the possibility of forming self- accommodating martensite plates 2. 2. Correspondences, variants and twins In a martensitic transformation the correspondence between the two crystal structures- parent and product is particularly important, since, together with he lattice parameters and crystal structures of the two phases, the correspondence determines the Bain strain B. In general, even with simple crystal structures, there are a number of possible correspondences that could be followed. If the parent and product crystal structures have some ordered arrangement of atoms- such as the carbon atoms in ferrous martensite it may be possible to identify a unique cor respondence between the two phases. However, in many cases this is not possible and decisions between different possible correspondences have to be based on othe criteria The usual practice is to select corresponding unit cells such that changes in lengt and mutual inclination of the edges of these unit cells are minimised For the tetragonal to monoclinic transformation in zirconia there are three possi- ble, simple correspondences that depend on which monoclinic axis is derived from
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P. M. Kelly, L.R. Francis Rose/Progress in Materials Science 47(2002 )463-557 475 the unique 'c axis of the parent tetragonal phase. The tetragonal'c' axis (ctcan become the 'a','b' orc axis of the monoclinic product phase(am, bm or cm). Hence, the three correspondences denoted by A, B or C [27]. As shown in Table 1, either correspondence A, where ct becomes am, or correspondence B, where ct becomes b has the worst match between the tetragonal and the monoclinic lattice parameters and consequently has the largest Bain strain [27]. However, with correspondence B one of the principle strains- that links c, and bm-is almost zero. This means that in strain for this correspondence is very close to an invariant plane strain and transformation with this correspondence would be expected to require a particularly small lattice invariant shear as will be demonstrated later in Section 3. 3. Note hat the small value of g associated with this correspondence compensates for the relatively large Bain strain and makes this correspondence more likely than corre- spondence A. Not surprisingly therefore, correspondence A is rare. The only reported occurrence of correspondence A is in the transformation in yttria stabilised zirconia [28-30. In this case, which will be discussed later, the presence of twinned tetragonal domains forces correspondence A to be adopted. The third correspondence--C invariably has the smallest Bain strain(see Table 1)and so, despite the fact that it is not as close to an invariant plane strain as correspondence B, it is the favoured correspondence. This is particularly so with Mg-PSZ where the lattice parameters are such that, even with correspondence B, the magnitude of the lattice invariant shear(g)remains relatively high. As will be shown later, the combined experimental data available indicates that with the exception referred to above(correspondence a in certain circumstances in Y-TZP)the tetragonal to monoclinic transformation in zirconia invariably follows either correspondence B or correspondence C. The simple A, B or C notation proposed by Riven et al. [27] suffers from the inability to distinguish different variants of a particular correspondence. A variant of the correspondence can arise if crystallographically equivalent, yet distinct versions The Bain strain and values of 'g for the three basic correspondences in the t-m transformation in zirconia Reference YTZP YTZP Ce.TZP Correspondence A 1.04398 04870 1.05946 √(n B 04310 104926 101561 01690 m 0.0287 Typical values of 'g 0.0017 0.0287 0.0352 Experimentally observed 4. B and c
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P.M. Kelly, L.R. Francis Rose/ Progress in Materials Science 47(2002)463-557 of the correspondence could be followed in a particular transformation. For exam- ple, in the case of the tetragonal to monoclinic transformation in zirconia following correspondence C(where c, becomes cm), either of the two tetragonala'axes could become the 'a monoclinic axis. These two variants of the correspondence are crys- tallographically equivalent, but in the case of a single crystal of the parent tetragonal phase would result in monoclinic crystals oriented at right angles to each other Hayakawa and his co-workers [28-30] introduced a more comprehensive system of describing the different correspondences between tetragonal and monoclinic zirconia that allowed the different variants of a single correspondence to be identified and labelled. To distinguish these and other variants of the correspondence in the t-m transformation in zirconia, the Hayakawa et al notation assumes that the two crys- tallographically equivalent tetragonal'a' axis are distinguishable from one another ence, a tetragonal zirconia crystal is arbitrarily denoted by the axes at, bt and ct. even though at and bt are crystallographically equivalent. The example discussed above consists of the two variants ABC(where a, becomes am, b becomes bm and ct becomes cm) and BaC(where a, becomes minus bm, b becomes am, and c becomes cm). These two variants are related by a 90 rotation about the cr axis, which is parallel to cm. Note that the change of sign of the axis symbol from B to B (i.e. making a, become minus bm) was done to preserve a right-handed set of axes in both parent and product lattices. The Hayakawa et al correspondence notation system [28-30] will be used throughout the present paper, whenever a specific variant of the three basic correspondences A, B and C needs to be distinguished Variants of the three correspondences are illustrated in Fig. 4. In practice, selecting a correspondence will often suggest the orientation relationship between the two phases(and vice versa). This occurs because the magnitude of the rotation R elatively small (usually 100 or less)so that corresponding vectors in the two pha ses will be inclined to one another at fairly small angles. In the case of the t-m transformation in zirconia, the three basic correspondences A, B and C relate the nit cell vectors in the two phases. In the tetragonal parent phase these three vectors form an orthogonal set, while in the monoclinic product phase the angle B between the two vectors am and cm is M9g. Hence, in the orientation relationship it is impossible for all three corresponding pairs of vectors in the two phases to be approximately parallel (i.e. within a degree or so of each other). The monoclinic b, axis is perpendicular to both am and cm so that, if the orientation relationship was such that bm was parallel to of the tetragonal axes, then one or other(but not both) of the monoclinic axes am or cm could be parallel to a tetragonal axis. This suggests two possible orientation relationships, both of which would have bm approximately parallel to a tetragonal axis(deviation of <1-20). One of the two possible orientation relationships would then involve an approximate parallelism between am and the corresponding tetragonal axis and between the plane(001)m and a plane of the type(100) or(001). The other would have cm approximately paralle to a tetragonal axis and (100)m approximately parallel to a plane of the type (100) or(001)t As shown in Fig 4. these two orientation relationships combine with each of the three basic correspondences to give a total of six possible orientation relationships- A-l, A-2, B-1, B-2, C-1 and C-2
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P.M. Kelly, L.R. Francis Rose/ Progress in Materials Science 47(2002)463-557 The possible number of variants in any particular transformation depends on the point group symmetry of the parent and product crystal structures and the corre- spondence between them. If centres of symmetry are ignored, then for each corre- spondence type(A, B, or C) there are eight distinguishable variants(actually four ets of crystallographically equivalent pairs) for the t-m transformation in zirconia These variants can be combined in different ways. For example, variants ABC and A BC are related by a 180 rotation about [100l 180 rotation about a. If these variants obeyed orientation relationship C-2, then, because c, is parallel to cm nd b. to bm in both variants, this pair can potentially lead to two monoclinic This tetragonal crystal can transform to monoclinic so that the ct axis am(far left-correspondence BCA), or bm(middle- correspondence CAB), or cm(far right-correspondence ABC) Correspondence Correspondence CAB R A-1 OR C. (001)m~l(100 (001m~l(o01n [100Jm-001J [100]m~l010 00lm~100l 0101m-l100 010】m~oo1 010lm-l[0101 OR B-2 OR C2 (100m~l1(001 100)m~l1(010) 100)m~(100 [001]m-ll [010 [00Im~100t [0101~o0l Fig 4. Schematic diagram showing the three general correspondences A, B and C for the t-m transfor mation in zirconia, together with the two expected orientation relationships for each correspondence
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