N.K. SIMHA F=I+@h, for some vector bE 9 Using (2. 18)and(2.26), we obtain the condition for compatibility of the habit plane as (U+(1-p)a⑧i)=I+b②h In Fig. I it can be seen that close to the habit plane the martensite twin bands cease to be parallel and undergo splitting, and we have depicted this region as the transition zone in Fig. 4. Although we neglected this transition zone in obtaining(2.27), Bal and Jamcs(1987) includc cncrgy contributions from the austenite, martensite, and transition zone in their analysis of energy minimization. They then show that as the bands become finer and finer at fixed volume fraction of the martensite variants, the total energy decreases, and in the limit the average deformation gradient martensite tends to f while the volume of the transition zone shrinks to zero Given martensite variants U U, and the twin (R a, n) that relates them we now investigate the conditions for the existence of a rotation Q, vectors b, h, and a scalar that satisfy the habit plane equation(2.27); according to the existence theorem due to Ball and James(1987), a solution(Q, H, b@ h) of(2.27)exists, if and only if 1+8*/2≤0 and TrL2-DetU2-2+|a|2/(286*)≥0. (2.28) where U(U2-D)- (2.29) To find solutions, one first evaluates the scalar u* from *=(1+√1+2/6)/2 (2.30) thuS A*E[/2, 1). Ball and James(1987)show that there are four distinct solutions (Q1,k*,b⑧1),(Q2,*,b2⑧五2),(Q3,1-1*,b⑧五1),and(Q,1-p*,b4⑧h)fon u*E(1/2, 1), and when u*=1/2 the four solutions collapse to two; vectors b and h are found as follows: find eigenvalues K< K2<K3 of the matrix Ch:=[U+(1-)ia[U4+(1-)a⑧旬 (2.31) where H-H*or u-1-A*, then /-kk1+x√k3-1k3 where=+1,p#0 is a constant chosen such that 6=1, and k, k, are normalised eigenvectors of Ch corresponding to KI, K3, respectively. Rotation Q is given by Q=(I+bh)[U,+(1-p)a②旬 (2.34) For a given martensite laminate, specified by twin related variants U, U, and a corresponding twinning shear a and normal f, there are up to four solutions for the habit plane equation. In Fig. 2 wells are depicted as circles and twins by light dashed
212 N. K. SIMHA F, = I+ b @ fi, for some vector b E g3. Using (2.18) and (2.26), we obtain the condition for compatibility of the habit plane as Q(U,+(l-p)a@ii) = I+b@h. (2.27) In Fig. 1 it can be seen that close to the habit plane the martensite twin bands cease to be parallel and undergo splitting, and we have depicted this region as the transition zone in Fig. 4. Although we neglected this transition zone in obtaining (2.27) Ball and James (1987) include energy contributions from the austenite, martensite, and transition zone in their analysis of energy minimization. They then show that as the bands become finer and finer at fixed volume fraction of the martensite variants, the total energy decreases, and in the limit the average deformation gradient of the martensite tends to F,, while the volume of the transition zone shrinks to zero. Given martensite variants Ui, Uj and the twin (R, a, fi) that relates them, we now investigate the conditions for the existence of a rotation Q, vectors b, i;, and a scalar p that satisfy the habit plane equation (2.27) ; according to the existence theorem due to Ball and James (1987), a solution (Q, ,LL, b 0 h) of (2.27) exists, if and only if where 1+6*/2<0 and TrU~-DetU~-2+fa]2/(26*)30, (2.28) 6* = a.U,(U,‘--I))‘% (2.29) To find solutions, one first evaluates the scalar ,M* from /L* =(l +,/i+2:a*)/2; (2.30) thus p* E [l/2,1). Ball and James (1987) show that there are four distinct solutions (Q,, p*, h 0 h), (Qzr p*, b 0 &I, (Qj, 1 - P*, b3 0 f&h and (Q4, 1 -P*, h 0 id for ~*~(1/2, l), and when p* = l/2 the four solutions collapse to two; vectors b and ii are found as follows : find eigenvalues ti, < ICY < ICY of the matrix C,:=[U,+(l-~)iiOa][U,+(l-j~)a@fi], (2.31) where p = p* or p = 1 -,u*, then (2.32) i;=p-’ (2.33) where x = + 1, p # 0 is a constant chosen such that [ii/ = 1, and t,. i;, are normalised eigenvectors of C, corresponding to K,, x3, respectively. Rotation Q is given by Q=(I+b@h)[Ui+(l-p)a@ii]-‘. (2.34) For a given martensite laminate, specified by twin related variants Ui, U, and a corresponding twinning shear a and normal fi, there are up to four solutions for the habit plane equation. In Fig. 2 wells are depicted as circles and twins by light dashed
lines. We now schematically depict the habit plane equation(2.27)by the bold dashed line in the well diagram. Points on the light dashed line that connects variants QI and QRU correspond to the average deformation gradient of martensite laminates built from the two variants(U, U) and are specified by the volume fraction of one of the variants; the dark dashed line connects such a martensite laminate to the austenite well. The habit plane equation can have up to four solutions, however to prevent cluttering, we draw only one bold dashed line Suppose there are v martensite variants; let each pair of the martensite variants be twin related, then the martensite can form v(v-1)twins. In addition, if each of the martensite twins can form a habit plane microstructure, there are up to 4v (v-1)habit planes. It is possible that some of the twin and habit planes are crystallographically equivalent due to the symmetry of austenite, nevertheless these simple estimates are useful for applications 2.5 Experimentalists usually use the crystallographic theory of martensite( Wechsler et al., 1953: Bowles and MacKenzie, 1954), hence we rewrite the habit plane equation (2.27)in the notation of the crystallographic theory Consider a deformation that takes an austenite unit cell to a martensite unit cell the corresponding deformation gradient is, in general, not symmetric, We identify terms QU, in(2.27)as the polar decomposition of the gradient of a deformation that brings austenite to the martensite variant U, and, then, rewrite(2.27)in a form presented by Wayman (1964)and Nishiyama (1978)as QBP=P (2.35) B=U,P=(1+(1-pUa⑧的).P1=I+b⑧h in the language of the crystallographic theory Q is called the rigid body rotation, B is the Bain strain(or lattice deformation ), P is the lattice invariant shear, and P, is the shape strain. a lattice correspondence identifies the structural unit in the austenite that transforms to martensite and specifies the accompanying deformation In applying the crystallographic theory one postulates a lattice correspondence, e. g. the austenite c axis cu is parallel to the martensite b axis bm(ca ll bm). The lattice correspondence determines a deformation gradient F, the right stretch tensor of which is the Bain strain B F. The dyad the lattice invariant shear P is assumed; Wechsler et al.(1953)and Bowles and acKenzie(1954)provide procedures to find the rigid body rotation Q, amount of twinning(1-p)IU; al, magnitude Ib] and direction b/bl of the shadd is usually habit plane normal h. An orientation relationship is calculated as QR at specified through its action on a vector and a normal in the autenite
Transformation of zirconia 273 lines. We now schematically depict the habit plane equation (2.27) by the bold dashed line in the well diagram. Points on the light dashed line that connects variants QU, and QRU, correspond to the average deformation gradient of martensite laminates built from the two variants (U,, U,) and are specified by the volume fraction of one of the variants ; the dark dashed line connects such a martensite laminate to the austenite well. The habit plane equation can have up to four solutions, however, to prevent cluttering, we draw only one bold dashed line. Suppose there are v martensite variants ; let each pair of the martensite variants be twin related, then the martensite can form v(v- 1) twins. In addition, if each of the martensite twins can form a habit plane microstructure, there are up to 4v(v - 1) habit planes. It is possible that some of the twin and habit planes are crystallographically equivalent due to the symmetry of austenite, nevertheless these simple estimates are useful for applications. 2.5. Crystallo,yraphic theory Experimentalists usually use the crystallographic theory of martensite (Wechsler et al., 1953 ; Bowles and MacKenzie, 1954), hence we rewrite the habit plane equation (2.27) in the notation of the crystallographic theory. Consider a deformation that takes an austenite unit cell to a martensite unit cell ; the corresponding deformation gradient is, in general, not symmetric. We identify terms QU, in (2.27) as the polar decomposition of the gradient of a deformation that brings austenite to the martensite variant U, and, then, rewrite (2.27) in a form presented by Wayman (1964) and Nishiyama (1978) as where QBP = P,, (2.35) B = U,, P=(I+(l-p)U,-‘a@;), P, = I+b@G; (2.36) in the language of the crystallographic theory Q is called the rigid body rotation, B is the Bain strain (or lattice deformation), P is the lattice invariant shear, and P, is the shape strain. A lattice correspondence identifies the structural unit in the austenite that transforms to martensite and specifies the accompanying deformation. In applying the crystallographic theory one postulates a lattice correspondence, e.g. the austenite c axis ca is parallel to the martensite b axis b,(c, 11 b,). The lattice correspondence determines a deformation gradient F, the right stretch tensor of which is the Bain strain B = dm. The dyad in the lattice invariant shear P is assumed ; Wechsler et a/. (1953) and Bowles and MacKenzie (1954) provide procedures to find the rigid body rotation Q, amount of twinning (1 -p)(U;’ al, magnitude lb1 and direction b/lb1 of the shape strain, and habit plane normal b. An orientation relationship is calculated as QB and is usually specified through its action on a vector and a normal in the autenite