J Mech. Phrs. Solids, oi, 45, 99,:pp 26? Pergamon PII:S0022-5096(96000749 TWIN AND HABIT PLANE MICROSTRUCTURES DUE TO THE TETRAGONAL TO MONOCLINIC TRANSFORMATION OF ZIRCONIA N. K. SIMHA Division of Engineering and Applied Sciences, Mail Code: 104-44, California Institute of Technology Pasadena, CA 91125 U.S.A Receired 26 December 1995: in revised form 5 June 1996) ABSTRACT We ist construct Bain strains for the tetragonal to monoclinic (t-m) transformation of zirconia(ZrO, en examine the resulting twin and habit plane microstructures. The (t-m) transformation occurs via two paths transformation along path I has two Bain strains that involve shearing of a ectangular face of the tetragonal unit ccll, and shearing of the square base corresponds to path II. The monoclinic variants resulting from each of the three Bain strains can form 12 twins, and four of the twins corresponding to path Il are neither of type I nor of type Il Habit planes do not exist for the transformation along path I whereas transformation along path Il has: (+0.8139.+0.3898,-0.4309) ,(+0. 6489 0.6271.-0.4309),(+0.7804, +0.4530,-0.4309) We predict the exact twin planes observed by bailey [(1964)Phase transformation at high temperatures in hafnia and zirconia. Prac. Roy. Soc. 279A, 395-4121 Bansal and Heuer [(1972)On a martensitic phase transformation in Zirconia Zro tallographic 1281-1289]and Buljan et al. [(1976)Optical and X-ray the monoclinic++ in ZrO, J. Am. Ceram. Soc. 59, 351 habit planes that we predict have not yet been observed. C 1997 Elsevier Science Ltd ngle crysta ts reserved Keywords: A. microstructures, A. phase transformation. A, twinning, B ceramic material, B. strain 1. INTRODUCTION In the absence of stresses, single-crystal zirconia undergoes a martensitic tetragonal to monoclinic(t-+m)transformation around 950C(Ruff and Ebert 1929; Subbarac et aL., 1974). However, zirconia inclusions that are embedded in a ceramic matrix can be retained in the tetragonal phase even at room temperature. when these inclusions transform to the monoclinic phase in the stress field around a crack tip, they enha he fracture toughness of the ceramic matrix( Garvie et al., 1975). This phenomenon is called transformation toughening, and it has given rise to a new class of strong and tough ceramics called zirconia toughened ceramics(ztc)( Green et al., 1989) Transformation toughening increases the fracture toughness of these ceramics from 2 to 5 MPay/m to values as high as 18-20 MPaym(Tsukuma and Shimada, 1985 Evans, 1990) Martensitic transformations are diffusionless solid to solid phase transformations [for a review, see Wayman(1964)and Nishiyama(1978): the high symmetry phase
Pergamon J. Mwh. Phw Solids, Vol. 45, No. 2, pp. 261m 292, 1997 Copyright % I997 Elsevier Science Ltd Printed in Great Britain. All rights reserved PII : SOO22-5096(96)00074 0022-5096197 $17.00+0.00 TWIN AND HABIT PLANE MICROSTRUCTURES DUE TO THE TETRAGONAL TO MONOCLINIC TRANSFORMATION OF ZIRCONIA N. K. SIMHA Division of Engineering and Applied Sciences, Mail Code : 104-44, California Institute of Technology. Pasadena, CA 91125, U.S.A. (RcJceired 26 December 1995 : in revised form 5 June 1996) ABSTRACT We tirst construct Bain strains for the tetragonal to monoclinic (t -+ m) transformation of zircoma (ZrO?). and then examine the resulting twin and habit plane microstructures. The (t + m) transformation in zirconia occurs via two paths ; transformation along path I has two Bain strains that involve shearing of a rectangular face of the tetragonal unit cell, and shearing of the square base corresponds to path II. The monoclinic variants resulting from each of the three Bain strains can form 12 twins, and four of the twins corresponding to path II are neither of type I nor of type Il. Habit planes do not exist for the transformation along path I. whereas transformation along path II has: (+0.8139, kO.3898, -0.4309),, (kO.6489. k0.6271, -0.4309),, (+_0.7804, kO.4530, -0.4309),. We predict the exact twin planes observed by Bailey [(1964) Phase transformation at high temperatures in hafnia and zirconia. Proc. Rqv. SK 279A, 39554121, Bansal and Heuer [(1972) On a martensitic phase transformation in Zirconia ZrO,-I. Metallographic evidence. Actu Metull. 20, 1281-12891 and Buljan et al. [(1976) Optical and X-ray single crystal studies of the monoclinic- tetragonal transition in ZrO:. J. Am, Ceram. Sot. 59, 35 l-3541 ; additional twins and habit planes that we predict have not yet been observed. c: 1997 Elsevier Science Ltd. All rights reserved Keywords: A. microstructures, A. phase transformation. A. twinning, B. ceramic material. B. strain compatibility. 1. INTRODUCTION In the absence of stresses, single-crystal zirconia undergoes a martensitic tetragonal to monoclinic (t -+ m) transformation around 950°C (Ruff and Ebert, 1929 ; Subbarao rf al., 1974). However, zirconia inclusions that are embedded in a ceramic matrix can be retained in the tetragonal phase even at room temperature. When these inclusions transform to the monoclinic phase in the stress field around a crack tip, they enhance the fracture toughness of the ceramic matrix (Garvie et al., 1975). This phenomenon is called transformation toughening, and it has given rise to a new class of strong and tough ceramics called zirconia toughened ceramics (ZTC) (Green et al., 1989). Transformation toughening increases the fracture toughness of these ceramics from 2 to 5 MPa$ to values as high as 18-20 MPaJm (Tsukuma and Shimada, 1985 ; Evans, 1990). Martensitic transformations are diffusionless solid to solid phase transformations [for a review, see Wayman (1964) and Nishiyama (1978)] ; the high symmetry phase 261
N, K SIMHA austenite) transforms to a low symmetry phase(martensite), when the temperature is decreased. In experiments it is typically observed that twin-free austenite is separated by an interface, called the habit plane, from twinned martensite(see Fig. I)(Wayman 1964: Nishiyama, 1978). The martensite appears in parallel bands; the band consisting of one variant of martensite is separated from a neighbouring band consisting of a second variant of martensite by a twin plane, and such a pair of bands, consisting of elated variants, is periodically repeated. If the temperature is further decreased, the twinned martensite consumes the austenite. The transformation and companying microstructure are reversible, but exhibit hysteresis The(t-m) transformation in zirconia is accompanied by a dilatation of about 4%and a shear strain of about 15%(see Scction 3). Since zirconia inclusions in ZTC are invariably twinned following the transformation, the average shear of the inclusion is, in general, less than 15%. Toughening depends on the average strain of zirconia inclusions due to the(t+m) transformation(Budiansky et al., 1983; Evans et al 1981: Simha and Truskinovsky, 1994). Consequenly the amount of toughening depends on twinning. In addition, the critical stresses that trigger the(t- m)trans- formation in retained inclusions(Budiansky and Truskinovsky, 1993)and the critical size of inclusions that can be retained(Evans et al, 1981)are influenced by twinning Hence, in order to understand toughening and further enhance the material properties of ZTC, it is essential to determine the microstructure due to the(t-m)trans- formation of zirconia Experimental investigation of the(t-mtransformation of zirconia is complicated by the high transformation temperature and by the transformation dilatation which causes cracking in a specimen; consequently there are only a few studies that use single-crystal zirconia(Bansal and Heuer, 1972, 1974; Buljan et al., 1976), and although they find some of the possible twin planes, it is debatable whether they observe the habit plane microstructure(see Section 4). Lam and Zhang(1992) find twins by using the solution elucidated by Ericksen(1985), but they do not discuss habit planes. Bansal and Heuer(1974)and Kriven et al. (1981) have predicted some but not all possible, habit planes by using the crystallographic theory. Thus, until neither theoretical nor experimental studies have found all the twins and habit In this paper we apply the nonlinear theory of martensite(Ball and James, 1987, 1992)to find all the twins and habit planes for the(t+ m) transformation of zirconia In various materials, twins predicted by using the nonlinear theory have agreed very well with experiments(Ball and James, 1992). The nonlinear theory was used by Ball and James(1987)to analyse the habit plane microstructures for the cubic to tetragonal transformation of InTl; since then the nonlinear approach has been successfully used to study wedge-like microstructures(Bhattacharya, 1991; Hane, 1995), to explain accommodation in martensite(Bhattacharya, 1992), and to examine various micro- structures in CuAINi single crystals( Chu, 1993; Shield, 1995) In Section 2 we first characterise the Bain(transformation) strains for tetragona LO monoclinic transformations by viewing the Bainl strain as a mapping from the tetragonal point group to the monoclinic point group. We then discuss aspects of the nonlinear theory that are essential for finding twin and habit plane microstructures for a given Bain strain, a finite set of distinct martensite phases called variants are
262 N. K. SIMHA (austenite) transforms to a low symmetry phase (martensite), when the temperature is decreased. In experiments it is typically observed that twin-free austenite is separated by an interface, called the habit plane, from twinned martensite (see Fig. 1) (Wayman, 1964 ; Nishiyama, 1978). The martensite appears in parallel bands ; the band consisting of one variant of martensite is separated from a neighbouring band consisting of a second variant of martensite by a twin plane, and such a pair of bands, consisting of symmetry related variants, is periodically repeated. If the temperature is further decreased, the twinned martensite consumes the austenite. The transformation and accompanying microstructure are reversible, but exhibit hysteresis. The (t + m) transformation in zirconia is accompanied by a dilatation of about 4% and a shear strain of about 15% (see Section 3). Since zirconia inclusions in ZTC are invariably twinned following the transformation, the average shear of the inclusion is, in general, less than 15%. Toughening depends on the average strain of zirconia inclusions due to the (t + m) transformation (Budiansky et al., 1983 ; Evans et al., 1981; Simha and Truskinovsky, 1994). Consequently the amount of toughening depends on twinning. In addition, the critical stresses that trigger the (t + m) transformation in retained inclusions (Budiansky and Truskinovsky, 1993) and the critical size of inclusions that can be retained (Evans et al., 198 1) are influenced by twinning. Hence, in order to understand toughening and further enhance the material properties of ZTC, it is essential to determine the microstructure due to the (t + m) transformation of zirconia. Experimental investigation of the (t -+ m) transformation of zirconia is complicated by the high transformation temperature and by the transformation dilatation which causes cracking in a specimen; consequently there are only a few studies that use single-crystal zirconia (Bansal and Heuer, 1972, 1974 ; Buljan et al., 1976), and although they find some of the possible twin planes, it is debatable whether they observe the habit plane microstructure (see Section 4). Lam and Zhang (1992) find twins by using the solution elucidated by Ericksen (1985), but they do not discuss habit planes. Bansal and Heuer (1974) and Kriven et al. (1981) have predicted some, but not all possible, habit planes by using the crystallographic theory. Thus, until now, neither theoretical nor experimental studies have found all the twins and habit planes. In this paper we apply the nonlinear theory of martensite (Ball and James, 1987, 1992) to find all the twins and habit planes for the (t -+ m) transformation of zirconia. In various materials, twins predicted by using the nonlinear theory have agreed very well with experiments (Ball and James, 1992). The nonlinear theory was used by Ball and James (1987) to analyse the habit plane microstructures for the cubic to tetragonal transformation of InTl; since then the nonlinear approach has been successfully used to study wedge-like microstructures (Bhattacharya, 1991 ; Hane, 1995), to explain accommodation in martensite (Bhattacharya, 1992), and to examine various microstructures in CuAlNi single crystals (Chu, 1993 ; Shield, 1995). In Section 2 we first characterise the Bain (transformation) strains for tetragonal to monoclinic transformations by viewing the Bain strain as a mapping from the tetragonal point group to the monoclinic point group. We then discuss aspects of the nonlinear theory that are essential for finding twin and habit plane microstructures : for a given Bain strain, a finite set of distinct martensite phases called variants are
Transformation of zirconia 26 Fig 1. Habit plane cubic austenite from(twinned)orthorhombic martensite in a CuAINi alloy picture courtesy of Dr C. Chu). The length of the picture corresponds to 1.5 mm on the specimen. We have been unable to find similar observations in zirconia
Transformation of zirconia 263 Fig (pi< 1, Habit plane separating cubic austenite from (twinned) orthorhombic martensite in a CuAlNi i :ture courtesy of Dr C. Chu). The length of the picture corresponds to 1.5 mm on the specimen. have been unable to find similar observations in zirconia. dloy We
Transformation of zirconia dentified Then the Hadamard condition provides a relation that twin- related variants must satisfy: the general solution of the twinning equation( Ball and James, 1987) and the procedure for classifying twins as type l or type Il(Zanzotto, 1988 )are given Next, the habit plane microstructure is analysed this involves finding the variants that form the martensite laminate and their volume fractions. the normal to the habit plane, and the average strain of the martensite In Section 3 we apply the nonlinear heory to the(t++ m) transformation of zirconia: we identify the Bain strains and then analyse twin and habit plane microstructures. We compare our predictions with experimental observations and other theoretical predictions in Section 4, and in Section 5 we discuss our results Notation: We use greek letters to denote scalars(e g a E ) bold faced small roman letters to denote vectors(e.g beo), and bold faced capital roman letters to denote second order tensors and 3 x 3 matrices (e.g. C). The transpose of C is C, its inverse isC. its trace is Tr C, and its determinant is Det C. The scalar product of vectors b and c is b'c; their vector product is b A c. The tensor product a@ b maps vectors to vectors, i.e(a@ b)e=a(bc)for any vector c. The gradient operator is denoted by v, the identity matrix by l, and unit vectors have a hat on top(e. g n 2. THE NONLINEAR THEORY OF MARTENSITE In thermoelasticity a free energy is associated with the material undergoing the martensitic transformation. Let Q be a regular crystalline body; from the lattice viewpoint, the crystal is a Bravais or a I-lattice(Pitter, 1985). Let y: 92+sbe its deformation and F= Vy the deformation gradient (Det F>0). The free energy per unit reference volume is assumed to depend on the deformation gradient Fa.er temperature 6 =中(F,) Free energy is frame indifferent, i.e. rigid body rotations do not effect φ(F,的)=(RF,0), for all rotations r (2.l a rotation r is a 3 x 3 matrix that satisfies RR=L DetR=+1 The polar decomposition theorem states that a matrix F with positive determinant ha F= RU, where R is a rotation, U=U and positive definite (2.3) Using the polar decomposition theorem, we see that the free energy depends on the deformation gradient F only through the right stretch tensor U φ=d(F,)=φ(U,),U=√FF (2.4) The point group of austenite g is the group of all rotations that restore the
Transformation of zirconia 265 identified. Then the Hadamard condition provides a relation that twin-related variants must satisfy; the genera1 solution of the twinning equation (Ball and James, 1987) and the procedure for classifying twins as type I or type II (Zanzotto, 1988) are given. Next, the habit plane microstructure is analysed; this involves finding the variants that form the martensite laminate and their volume fractions, the normal to the habit plane, and the average strain of the martensite. In Section 3 we apply the nonlinear theory to the (t + m) transformation of zirconia : we identify the Bain strains and then analyse twin and habit plane microstructures. We compare our predictions with experimental observations and other theoretical predictions in Section 4, and in Section 5 we discuss our results. Notation : We use greek letters to denote scalars (e.g. 01 E &!), bold faced small roman letters to denote vectors (e.g. be%!‘), and bold faced capita1 roman letters to denote second order tensors and 3 x 3 matrices (e.g. C). The transpose of C is CT, its inverse is C’. its trace is Tr C, and its determinant is Det C. The scalar product of vectors b and c is b - c ; their vector product is b A c. The tensor product a @ b maps vectors to vectors, i.e. (a 0 b)c = a(b - c) for any vector c. The gradient operator is denoted by V, the identity matrix by 1, and unit vectors have a hat on top (e.g. ii). 2. THE NONLINEAR THEORY OF MARTENSITE In thermoelasticity a free energy is associated with the material undergoing the martensitic transformation. Let fi be a regular crystalline body ; from the lattice viewpoint, the crystal is a Bravais or a l-lattice (Pitteri, 1985). Let y : R -+ W3 be its deformation and F = Vy the deformation gradient (Det F > 0). The free energy per unit reference volume 4 is assumed to depend on the deformation gradient F and temperature 0 4 = &F, 0). Free energy 4 is frame indifferent, i.e. rigid body rotations do not effect 4 $(F, 0) = 4(RF, O), for all rotations R ; (2.1) a rotation R is a 3 x 3 matrix that satisfies RTR=I, DetR= +I. (2.2) The polar decomposition theorem states that a matrix F with positive determinant has a unique representation F = RU, where R is a rotation, U = UT and positive definite. (2.3) Using the polar decomposition theorem, we see that the free energy 4 depends on the deformation gradient F only through the right stretch tensor U 4 = ~(F,B) = 4(u,8), u = JFTF. (2.4) The point group of austenite 8, is the group of all rotations that restore the
NK SIMHA austenite unit cell. Free energy should reflect the symmetry of austenite(Ball and James, 1992), thus d(U,O)=φ(QUQ,)forl!Q∈ (2.5) Let a be the austenite point group and m the martensite point group. When austenite undergoes a martensitic transformation, its symmetry group changes from a to m Given ga and ym, a positive definite symmetric 3 x 3 matrix Uo that satisfies {R∈RUR=Uo}= is called a Bain or transformation strain. For fixed a and m the eigenvectors of the Bain strain are unique; however, the eigenvalues depend on the lattice parameters of le austenite and martensite phases. This definition is due to Ball and James(1992) nd they characterise the Bain strains for cubic to tetragonal, cubic to orthorhombic, and orthorhom bic to monoclinic transformations et(e1,e2, e3 be orthonormal, then the eight rotations that comprise the tetragonal point group乡 ={1Q(,Q(2,eQ(π,e,Q(,Q(兀,,Q(xe土e2)((26) where by Q(a, e)we mean a rotation of a radians about axis e. If a rotation Q+ I atisfies Q= I for some integer k, then rotation Q is a k-fold rotation and its axis is a k-fold axis: if a k-fold and m-fold rotation with k m have a common axis. then the axis is said to be k-fold. Thus the tetragonal point group t has one four-fold axis s)and four two-fold axes(e,e2, e,+e2). Any of the four 2-fold axes or the 4-fold axis of the tetragonal point group t can become the 2-fold axis of the monoclinic point group gm, and we now characterise the Bain strains for tetragonal to monoclinic Theorem 1(Bain strain). Let er, e, e, be the orthonormal tetragonal basis with as the 4-fold axis of gr. Suppose that a-m is the set of all positive definite symmetric 3 x 3 matrices that satisfy Q∈92:QUQ"=U}= (2.7) ()If gm=[I, Q0) where e is a two-fold axis of g, then hm={ne⑧e+n2旬2⑧2+n33②自3η>0.,η2>0,n3>0,明2≠3 (e,自2,自3) orthonormal,2≠土e3,2≠±(e,A的},(28) (II)If m= L, QE, then thim={:山十n22②自2+ne3⑧e:n1>0,n2>0,n3> n≠n2、,,e) orthonormal..≠±e,自≠土e,山≠士e1±e)√2 (2.9)
266 N. K. SIMHA austenite unit cell. Free energy 4 should reflect the symmetry of austenite (Ball and James, 1992) thus 4(U, 13) = &QUQ’,O) for all Q EY’,. (2.5) 2.1. Bain strain Let P’, be the austenite point group and .Ym the martensite point group. When austenite undergoes a martensitic transformation, its symmetry group changes from 9, to Pm. Given 9, and P,,,, a positive definite symmetric 3 x 3 matrix U, that satisfies {RE$,:RU,,R~ = U,} = P,,, is called a Bain or transformation strain. For fixed P’, and Pm the eigenvectors of the Bain strain are unique ; however, the eigenvalues depend on the lattice parameters of the austenite and martensite phases. This definition is due to Ball and James (1992), and they characterise the Bain strains for cubic to tetragonal, cubic to orthorhombic, and orthorhombic to monoclinic transformations. Let {C,, i$, C,} be orthonormal, then the eight rotations that comprise the tetragonal point group P’t are PPt = jl,Q(~,Ej),Q(~,(,).Q(n,0,),Q(~,~,,,Q(,,C,),Q(n.B, +b,}> (2.6) where by Q(m,iZ) we mean a rotation of a radians about axis 6. If a rotation Q # I satisfies Qk = I for some integer k, then rotation Q is a k-fold rotation and its axis is a k-fold axis ; if a k-fold and m-fold rotation with k > m have a common axis, then the axis is said to be k-fold. Thus the tetragonal point group YPt has one four-fold axis (6,) and four two-fold axes (C,, &, 6, +&). Any of the four 2-fold axes or the 4-fold axis of the tetragonal point group P’t can become the 2-fold axis of the monoclinic point group P’,, and we now characterise the Bain strains for tetragonal to monoclinic transformations. Theorem 1 (Bain strain). Let {C,, &, e,} be the orthonormal tetragonal basis with & as the 4-fold axis of Pt. Suppose that @‘+m 1s the set of all positive definite symmetric 3 x 3 matrices that satisfy {QEP~:QUQ= = U} = 9,. (2.7) (I) If P,,, = {I,Q:} w h ere & is a two-fold axis of P,, then %! f’” = {V,~o~+yIzEIz O%+%Q3 8% :Vl > 0, Vz > 0, g, > 0, ylz # Y/3, (C, B2, ii,) orthonormal, 8, # fi&, Ei, # f (i& A 6)}, (2.8) (II) If g,,, = {I, Q&j, then @ I?” = (yllfi, OQ, +r/ 252 @ Q2 +@3 @ 63 : VI > 0, q2 > 0, ‘I3 > 0, yl #r12,(fi,,i12,C3) orthonormal,ii, # +6,, 8, # +g2, 8, # k(i4,_+6,>/,,h). (2.9)
on of zirconia Proof. If a rotation Q satisfies(2.7)for a given positive definite symmetric 33 matrix U. then it leaves the eigenspace of matrix U invariant. i.e Uu= nueU(Qu)=n(Qu) Part(n). Let (n, e),(nz, G2),(n3.G3)be eigenvalue eigenvector pairs of matrix U Noting that Qe=-1+2e@e, we obtain QFU(Q2)=U Let e,=e, Ae.Since Qae=,Q=一e1,Q}t=±e, and Qe e=一, we require that the eigenvectors of U not lie along &, e Suppose Qi,=ou, and Qu,=wu, with ∈{1、-1;andQ∈{Q,Qa}, then the necessary and sufficient conditions for QE. Qa to satisfy (2.7)is n,= n so we set n2* n3 Part(In. Let (n, G,),(n2, u2),(3, e,) be the three eigenvalue-eigenvector pairs of matrix the们=mS0 we set nr≠ Let e be a20 Id axis of p,then QE U(Q)'=U Necessary and sufficient conditions for the 4-fold otations to sati ee,=0, and suppose that e,=e, Ae. Since Qie=e. Qe=-e, and Qi e3 e3, we require that the eigenvectors of U not lie along a 2-fold axis Theorem I is another case of Theorem 2. 10 of Ball and James(1992). Hertog(1987) has obtained the Bain strain when the tetragonal symmetry group is 1, Q(z/2.e3) Q(3T/2, e3), Q(z, e3) and the monoclinic symmetry group is 1, Q(. e3)). Sets Wi -m (2. 8)and whm(2.9)characterise all possible Bain strains that accompany a tetragonal to monoclinic transformation. Examples of Bain strains belonging to set wi mare cases(i)and(ii)in Fig. 5 and to set wii m are cases(iii)and(iv). Lattice parameters of the tetragonal and monoclinic phases determine the values of (n 2. 3)(see Section 3). Since vector e can lie along any of the four 2-fold axes of j, five Bain strains are possible for a tetragonal to monoclinic transformation, and we will use experimental observations to decide upon the bain strains that operate in zirconia 2.2. Variants and energy wells I he free energy is constructed such that at higher temperatures austenite has lower energy, whereas at lower temperatures martensite has lower energy; the energy of both phases is assumed to be equal at a critical transformation temperature e It is natural to identify the austenite at the critical temperature as the reference ,日)=φ(Un,B), where Ua is a bain strain accompanying an austenite to martensite transformation at temperature 0. Due to material symmetry (2.5)we see that the energy o has the same value at matrice U,=Q..Qi, Q Distinct matrices u.i=1.2.,. v are called variants of martensite and the number of variants is given by
Transformation of zirconia 267 Proqf: If a rotation Q satisfies (2.7) for a given positive definite symmetric 3 x 3 matrix U. then it leaves the eigenspace of matrix U invariant, i.e. Uu = Y/U- U(Qu) = Y](Qu). Parr (I). Let (r/, C), (q2, a,), (vi, a,) be eigenvalueeeigenvector pairs of matrix U. Noting that Qz = -I+26 @ @, we obtain QzU(Qz)’ = U. Let &I = Gj A 6. Since Q;;‘$ = &, Q:r”G = --gI, Q$,ei,i? = +6,~, and Q;,i$ = -Cl. we require that the eigenvectors of U not lie along Cl, &. Suppose QO, = rut% and Qti, = ~6, with CL) E (1, - 1) and Q E {Q&, Qa,}, then the necessary and sufficient conditions for QZ,, QZ, to satisfy (2.7) is q2 = Y/~, so we set q2 # yli. Part (II). Let (vi, ai), (yap, a,), (q3, e,) be the three eigenvalueeeigenvector pairs of matrix U. then Qi,U(Qi,)T = U. Necessary and sufficient conditions for the 4-fold rotations to satisfy (2.7) is PI, = q2. so we set YI, # ylz. Let 6 be a 2-fold axis of Y,, then 6.6, = 0, and suppose that &I = 8, A 6. Since QzC = 6, Qt GI = -Cl, and Qz 8, = -&, we require that the eigenvectors of U not lie along a 2-fold axis. Theorem 1 is another case of Theorem 2. IO of Ball and James (1992). Hertog ( 1987) has obtained the Bain strain when the tetragonal symmetry group is {I, Q(rc/2,6,), Q(3rc/2, e,), Q(rc, iZ,)] and the monoclinic symmetry group is [I, Q(x, 6,)). Sets 9&i *“’ (2.8) and ~$,ul;“’ (2.9) characterise all possible Bain strains that accompany a tetragonal to monoclinic transformation. Examples of Bain strains belonging to set +Y)+“’ are cases (i) and (ii) in Fig. 5 and to set &I;+” are cases (iii) and (iv). Lattice parameters of the tetragonal and monoclinic phases determine the values of (q,, q2, qj) (see Section 3). Since vector i? can lie along any of the four 2-fold axes of .P,, five Bain strains are possible for a tetragonal to monoclinic transformation, and we will use experimental observations to decide upon the Bain strains that operate in zirconia. 2.2. Variants and energy wells The free energy 4 is constructed such that at higher temperatures austenite has lower energy, whereas at lower temperatures martensite has lower energy ; the energy of both phases is assumed to be equal at a critical transformation temperature 0,. It is natural to identify the austenite at the critical temperature as the reference configuration, then $(I,@,) = &U”, O,), (2.10) where U,) is a Bain strain accompanying an austenite to martensite transformation at temperature 8,. Due to material symmetry (2.5) we see that the energy 4 has the same value at matrices U, = Q,U,Q’, Q;E:Y,,. (2.1 I) Distinct matrices U,, i = 1,2, . , v are called variants of martensite, and the number of variants is given by
N. K SIMHA 1- QRU ig. 2. The tetragonal (bold)and four monoclinic wells two twin-related variants. the bold dashed line denotes habit microstructures with tetragonal on one side and a monoclinic laminate on the other Order of austenite point group gpa Order of martensite point group m see Van Tandeloo and Amelinckx (1974). For a tetragonal to monoclinic trans- formation(2. 12)gives v= 4 Frame indifference (2. 1 )implies that the free energy has the same value at matrices u and ru where r is a rotation the sct =RI: Ris a rotation, is called the austenite well. and the union of martensite wells M=Ur-1;: R is a rotation and U, a martensite variant). Following Ball and James(1992), we schematically depict wells as circles in the space of deformation gradients. Figure 2 shows the austenite and four martensite wells for a tetragonal to monoclinic transformation. The free energy o is lower on the austenite well at temperatures 0>0, whereas at temperatures 0 B it is lower on the mar tensite wells, and at the critical temperature 0 it is the same on both austenite and martensite wells(2.10) 2.3 Twins a deformation that results in a twin is continuous(coherent) and has different constant deformation gradients on either side of the twin plane Suppose deformation y: Q2+9 has constant gradients F+ and F- on either side of a plane with normal
268 N. K. SIMHA Fig. 2. The tetragonal (bold) and four monoclinic wells are depicted schematically by circles. Each light dashed line indicates a twin, and points on it correspond to average deformation gradients of laminates of two twin-related variants. The bold dashed line denotes habit plane microstructures with tetragonal phase on one side and a monoclinic laminate on the other. Order of austenite point group .P’, ’ = Order of martensite point group P, ’ (2.12) see Van Tandeloo and Amelinckx (1974). For a tetragonal to monoclinic transformation (2.12) gives v = 4. Frame indifference (2.1) implies that the free energy 4 has the same value at matrices U and RU where R is a rotation. The set zz! = {RI : R is a rotation) is called the austenite well, and the union of martensite wells is (2.13) A= (J:'=,{Ru,:R IS a rotation and U, a martensite variant}. (2.14) Following Ball and James (1992), we schematically depict wells as circles in the space of deformation gradients. Figure 2 shows the austenite and four martensite wells for a tetragonal to monoclinic transformation. The free energy 4 is lower on the austenite well at temperatures 8 > O,, whereas at temperatures fl < 19, it is lower on the martensite wells, and at the critical temperature 8, it is the same on both austenite and martensite wells (2. IO). 2.3. Twins A deformation that results in a twin is continuous (coherent) and has different constant deformation gradients on either side of the twin plane. Suppose deformation y : R + @ has constant gradients F, and F_ on either side of a plane with normal ij
Transformation of zirconia y=y(x) y=RU X RU reference deformed Fig. 3. A twinning deformation in the reference configuration, then the Hadamard compatibility condition says that deformation y is continuous on Q2 if and only if F+=F+p⑧q for some pes. Let F, and F, belong to different martensite wells and let R be a rotation. Suppose F; and R, are constant deformation gradients on either side of a win plane with normal iEg' in the reference configuration, then by using the Hadamard jump condition (2. 15)we get F,=F,+ag where ae is the twinning shear. Let F=R U; and F;=RU; where U; and U,are two martensite variants(2. 11), then(2.16)becomes RRRU=U+Ra⑧i R=RRR a=Ra andn=n (2.17) to obtain the twinning equation RU =U+an Equation (2. 18)can be interpreted as the requirement for a coherent planc interfacc, having normal f in the reference configuration, between regions with constant defor- mation gradients U; and RU, (see Fig 3)and with a as the shear vector. The twin plane normal in the deformed configuration is given by i=U7 f/UF Al, and by taking determinants on either side of (2. 18), it follows that a i=0(Ball and James 1987) thus the twin normal ii and shear a correspond to crystallographic twinning elements K and n, respectively
Transformation of zirconia reference deformed Fig. 3. A twinning deformation. in the reference configuration, then the Hadamard compatibility condition says that deformation y is continuous on CI if and only if F, =F_+p@tj (2.15) for some PE @. Let Fi and F, belong to different martensite wells and let R be a rotation. Suppose F, and RF, are constant deformation gradients on either side of a twin plane with normal HEW’ in the reference configuration, then by using the Hadamard jump condition (2.15) we get RF, = F, + I @ ;. (2.16) where 5 E g3 is the twinning shear. Let F, = R,U, and F, = R,U, where U, and U, are two martensite variants (2.1 l), then (2.16) becomes R;RR;U, = U, + Rf 10 r? Set R=RTRR,, a=RT1, andfi=h to obtain the twinning equation (2.17) RU,= U,+a@ii. (2.18) Equation (2.18) can be interpreted as the requirement for a coherent plane interface, having normal ii in the reference configuration, between regions with constant deformation gradients U, and RU, (see Fig. 3) and with a as the shear vector. The twin plane normal in the deformed configuration is given by ii = U,-’ fi/lU,’ Al, and by taking determinants on either side of (2.18), it follows that a * ii = 0 (Ball and James. 1987) ; thus the twin normal ii and shear a correspond to crystallographic twinning elements K, and y,, respectively
N. K. SIMHA Given a pair of martensite variants (U, U) we now find a rotation R, shear vector and twin plane normal f that satisfy the twinning equation(2. 18). Define C:=UFIUZUI then C=(I+U; 8a)(+aUF n), and the six parameters that determine shear a and normal n can be obtained from proposition 4 of Ball and James(1987)[also see Ericksen(1980)and Gurtin(1983). Suppose A1< A2<h, are the eigenvalues of the matrix CL, solutions for twinning shear a and normal f exist, if and only if λ1≥0andλ2=1 (2.20) When Ct+ l, shear a and normal i are given by -4,e+yA2 (2.21) where x=+l, p*0 is a constant chosen such that a=I and e, e3 are normalised eigenvectors of C, corresponding to ii, a,, respectively. Rotation R is found using R=(U+a⑧n)U (2.23) For given U. U (Ui+U, there are either two solutions(R, a, n) of the twinning equation(2. 18)(corresponding to x=+I)or none. Twin related variants are atically depicted on the well diagram(Fig. 2)by means of a light dashed line connecting variant U, with deformation gradicnt RU Thus there are cither nonc or ' o dashed lines between any pair of martensite wells. Suppose there are v martensite variants, then there are v(v-1)/2 pairs of martensite variants, and if all pairs are twin elated, then the martensite can form v(v-I)twins. Suppose we take(2. 16)to be the ic twinning equation with a solution(R, a, h); then it is not possible to say how many such twins the martensite can form since each variant U: corresponds to infinitely many deformation gradients F,=R U, where R, is a rotation. Moreover, (2. 18)involves only variants U, and U Thus it is advantageous to consider(2. 18)as he basic twinning cquation To classify twins, we rewrite the twinning equation(2. 18)as RU;=[+a nju where i=U-i is the twin plane normal in the deformed configuration; variant RU can be obtained by shearing variant U:. One class of solutions(R, a, n) has the property that the two twin related variants are related also by a 2-fold rotation(Zanzotto 1988),i RU,=[+a⑧司U;=Ql (2.24) where He sa, the austenite point group, and the rolation Q=-+2n g ninl, for type I twins, (2.25 for type II tw
270 N. K. SIMHA Given a pair of martensite variants (U, Uj) we now find a rotation R, shear vector a, and twin plane normal ti that satisfy the twinning equation (2.18). Define c, : = U,:’ u; u,-‘, (2.19) then C, = (I + U; ’ ii 0 a)(1 + a @ U,: ’ a), and the six parameters that determine shear a and normal ti can be obtained from proposition 4 of Ball and James (1987) [also see Ericksen (1980) and Gurtin (1983)]. Suppose i, ,Oand;l, = I. (2.20) When C, # I, shear a and normal fi are given by (2.21) where x = f 1, p # 0 is a constant chosen such that ]Ei] = 1 and i?,, & are normalised eigenvectors of C, corresponding to &, &, respectively. Rotation R is found using R =(Ui+a@n)U,-‘. (2.23) For given Uj, U, (Ui # U,) there are either two solutions (R, a, n) of the twinning equation (2.18) (corresponding to x = + 1) or none. Twin related variants are schematically depicted on the well diagram (Fig. 2) by means of a light dashed line connecting variant U, with deformation gradient RU,. Thus there are either none or two dashed lines between any pair of martensite wells. Suppose there are v martensite variants, then there are v(v - 1)/2 pairs of martensite variants, and if all pairs are twin related, then the martensite can form v(v - 1) twins, Suppose we take (2.16) to be the basic twinning equation with a solution (8, H, ?I) ; then it is not possible to say how many such twins the martensite can form, since each variant Ui corresponds to infinitely many deformation gradients F, = RiUi, where R, is a rotation. Moreover, (2.18) involves only variants Uj and U,. Thus it is advantageous to consider (2.18) as the basic twinning equation. To classify twins, we rewrite the twinning equation (2.18) as RLJ, = [I+ a 0 ii]U, where ii = U; ’ ii is the twin plane normal in the deformed configuration ; variant RU, can be obtained by shearing variant Ui. One class of solutions (R, a, fi) has the property that the two twin related variants are related also by a 2-fold rotation (Zanzotto, 1988), i.e. RU, = [I+ a @ ii]U, = QU,H, where HE P’,, the austenite point group, and the rotation (2.24) Q= i -1+2ii @ ii/liil*, for type I twins, -1+2a@ ailal*, for type II twins ; (2.25)
Transformation of zirconia martensite austenite 围园昌喜 Fig, 4 Schematic representation of Fig. I. The transition zone(shaded )corresponds to the region where he martensite bands cease to he parallel and split in Fig. I rotation Q is of r radians about the twin plane normal in the deformed configuration for type I twins and about the shear vector for type II twins. If a twin is both type I and type Il, then it is called a compound twin In summary, then, the point groups and lattice parameters of the austenite and martensite determine the bain strains for a martensitic transformation For each bai strain, the symmetry of the free energy delivers a finite set of martensite variants The twinning equation(2.18)and its solution (2.21),(2.22)show that twins are ompletely determined by these variants. Consequently one can find all the possible wins between the martensite variants 2. 4. The habit plane microstructure In Fig. 4. we schematically represent the habit plane microstructure shown in Fig 1. Austenite, with the deformation gradient L, is separated frill Martensite by tlie habit plane with the normal he, and the martensite comprises of a periodic arrangement of parallel bands with deformation gradients QU, and QRU: Q and R are rotations(2. 2), and the martensite variants UU, satisfy the twinning equation (2.18), i.e. Q(RU -U,=a@n). Let the volume fraction of the variant with the deformation gradient QU, be !E(0, 1), then the average deformation gradient in the martensite region Is F=nQU, +(1-HQRU (2.26) Austenite, with the deformation gradient identity l, is separated by the habit plane viewed tant deformation gradient equal the average deformation gradient Fu. Then the Hadamard condition(2. 15)gives
Transformation of zirconia Fig. 4. Schematic representation of Fig. 1. The transition zone (shaded) corresponds to the region where the martensite bands cease to be parallel and split in Fig. I. rotation Q is of 71 radians about the twin plane normal in the deformed configuration for type I twins and about the shear vector for type II twins. If a twin is both type I and type II, then it is called a compound twin. In summary, then, the point groups and lattice parameters of the austenite and martensite determine the Bain strains for a martensitic transformation. For each Bain strain, the symmetry of the free energy 4 delivers a finite set of martensite variants. The twinning equation (2.18) and its solution (2.21), (2.22) show that twins are completely determined by these variants. Consequently one can find all the possible twins between the martensite variants. 2.4. The habit plane microstructure In Fig. 4, we schematically represent the habit plane microstructure shown in Fig. 1. Austenite, with the deformation gradient I, is separated from martensite by the habit plane with the normal ii~.G%~, and the martensite comprises of a periodic arrangement of parallel bands with deformation gradients QU, and QRU, : Q and R are rotations (2.2), and the martensite variants U,. U, satisfy the twinning equation (2.18), i.e. Q(RU,-U, = a @ ii). Let the volume fraction of the variant with the deformation gradient QU, be ~E(O, l), then the average deformation gradient in the martensite region is F,, = PQU,+(~-P)QRU,. (2.26) Austenite, with the deformation gradient identity I, is separated by the habit plane from martensite, viewed as a region with a constant deformation gradient equal to the average deformation gradient F,,. Then the Hadamard condition (2.15) gives
