H Y, Yu et al. Marerials Science and Engineering B32(1995)153-158 for Estrain =0, where Agchem is the chemical free-energy Appendix change per unit volume of martensite formed and is negative at temperatures T< To (To being the equilib The eshelby tensor Soak and the tensor S in rium temperature at which the martensite and its Eq (4)are parent phase coexist)and y is the interfacial energy per unit area. The critical thickness of the embryo is Sa obtained by letting Etot-0, which gives 8(1-甲w+(1=y(O+p0 (22) (A1) chem Using the proposed model, the values of a*and c* nay be estimated for ZrO, particles in order to com pare them with the values estimated from the model of Smmkk (1+v)(1-2v)(-)B Chen and Chiao [6]. Adopting the material parameters (1-v) E=0,s=0.154,b=0.52nm,△ gch=2.04×103J m3and y=0. 2 Jm 2used by Chen and Chiao, one S3kk (1+v)(-u)B has from(20)and(22 4x(1-v) These are comparable with the values of a*=18.1 nm S*k=-(1+v)u-u,B a*≈16.7nm c*≈196nm (23) (@1+2x33) and cs 3.3 nm given by Chen and Chiao which were qualitatively supported by their experimental(trans (1-2v)(-u)B mission electron micre )results. No definite (2r3-73-2x3313) 4 argument may be given as to which model is more valid. The experimental evido embryo nucleating from a single stacking fault[] could $3333 1(4-)B(-(1-4v)r3+2xr33 support both models, since the stacking fault could be 8x(1-v) considered as a loop of Shockley-Frank partial dis- locations [22 The advantage of the proposed model 4(2+v)x3-2x33 rests mostly on its simplicity +2(2-7v)(-)B+(1-v)k(B-B)@3} onclusions S 4x(1-v) A-')B(xr31-@3-4x3④3 The fo demonstrate that martensite favors nucleation near x313)+(1-v(B-B)3 inhomogeneities with stiffness less than the parent ids and free surfaces. This (1-2v)(-)B 33-2x3Φ13) can also be applied to other solid state transformations 4x1-)"( that involve strain energy changes. A model of the pre- (A2 existing martensite embryo in Kaufman and Cohens theory has also been proposed In this new model the (4-u)B dislocation loops in the parent phase are themselves S33-8x(1-v) (1-4vr1313-2x3r the so-called"pre-existing embryos". The strain energy loop)provides the strain energy needed for the nuclea- 1+v)x3313+2x3④ uB-u B tion of the martensite embryo during transfermation The advantage of this model is its simplicit (-)B IT1313+2 Acknowledgment 1-4x13-2x3cH31 This work has been partially sponsored by the Office of Naval Research through the Naval Research Laborator 1(g-2+2(1B-1BAH. Y. Yu et al. / Materials Science and Engineering B32 (1995) 153-158 157 for Estrain = 0, where Agchem is the chemical free-energy change per unit volume of martensite formed and is negative at temperatures T< T O (T O being the equilib￾rium temperature at which the martensite and its parent phase coexist) and ~'11 is the interfacial energy per unit area. The critical thickness of the embryo is obtained by letting Eto t = O, which gives S i/kl c* - 2~1 (22) A&h~m and Using the proposed model, the values of a* and c* may be estimated for ZrO 2 particles in order to com￾pare them with the values estimated from the model of Chen and Chiao [6]. Adopting the material parameters ~=0, s=0.154, bs=0.52 nm, Agchem=2.04Xl08 J m-3 and VII = 0.2 J m 2 used by Chen and Chiao, one has from (20) and (22) a* = 16.7 nm, c* = 1.96 nm (23) These are comparable with the values of a* = 18.1 nm and c* = 3.3 nm given by Chen and Chiao which were qualitatively supported by their experimental (trans￾mission electron microscopy) results. No definite argument may be given as to which model is more valid. The experimental evidence of the e-martensite embryo nucleating from a single stacking fault [5] could 53333 support both models, since the stacking fault could be considered as a loop of Shockley-Frank partial dis￾locations [22]. The advantage of the proposed model rests mostly on its simplicity. Appendix The Eshelby tensor Si/~kt and the tensor S*kt in Eq. (4) are 1 1 - v) [tp' ij ' +(1 - 6j, + ,t,I.,aj + (~ljk(~i, "1- (~],jl(~ik)-- 2 v¢~j6k,] (11) . (l+v)(1-2v)(tt-,u')fl II S mmkk ~" (ID,33 ~(1 - v) _1_ t . (1 t-v)(tt-_/j )fl[( _4v)~i,~33 ,, 533kk = 4~(1 - v) 1 - 2x30,333 ] , (1+ II + 2X3@ 333) Sl3k~= 4~(1- v) ~P,31 ,.. * (1--2V)(/AZ-fl')flf"~ll -70~33-2x30,3,3)n atom33 "~- 4er(1 -- 1 "P) \z~l '3333 1 {(tt- tt')fl[-(1- 4V)F,3333 n + 2x3F,33333 ,, 8zr(1- v) 4(2 + II ,~ 2~11 1 -- V)X3(I),333 -- ZX31¥,3333] + 2[(2 - 7 v)(kt-/t')fl + (1 - 'l,')[At(fl-31)](~l,133} 4. Conclusions The foregoing arguments have been presented to demonstrate that martensite favors nucleation near inhomogeneities with stiffness less than the parent phase, such as voids and free surfaces. This conclusion can also be applied to other solid state transformations that involve strain energy changes. A model of the pre￾existing martensite embryo in Kaufman and Cohen's theory has also been proposed. In this new model the dislocation loops in the parent phase are themselves the so-called "pre-existing embryos". The strain energy stored in this distorted parent phase (the dislocation loop) provides the strain energy needed for the nuclea￾tion of the martensite embryo during transformation. The advantage of this model is its simplicity. Acknowledgment This work has been partially sponsored by the Office of Naval Research through the Naval Research Laboratory. S 1~333 = 1 t 11 _ 11 4=(1- v)[(/~-/~ )fl(x3F,33313- O,~3 4x30 313 2~II x . - x3 ' 3313 ) * (1 - '( fl - 3' • (1 -- mY) (/A--/A')3 11 II lI S mml3 "~- -4--~1= -~) (2F3313- 3 (I~, 13 -- 2X31I),133) (A2) • (/'g --/'g ; )fl [( II 1I S3313 = ~(~7~ ~,1 -- 41,,)F,3313 - 2x3F 33133 + 41)OI113 +4(1+ i, ~ 2._n 1 /zfl-/~'fl' 1-')23(I),313 -t- (1)1,113 Z~x3qJ'3313] 4~r S1313 ~7~( i 7~ [FI"313 + 2x3Fl, I13,33 _2(1_v)~I,111_4x3dp,iu3_., 2~1I , , Z;X3qJ,1313] 8,7l: /V-l- .' *{12 + 2(tiff--/2'fl')(I){Ill