H.Y. Yu et al./Materials Science and Engineering B32(1995)153-158 for u<u. For pure shear transformation strain, i.e of about 19.5% in the strain energy is obtained when the nucleation takes place near a free surface. Taking typical values of A+5=0.05 and s=0. 18 for steels 21, the reduction in energy is 22.6%when A=0 and 22. 4% when 5=0 These numbers show that regardless of the transforma reduced about 20% when the embryo is formed near a free surface or void, while the strain energy will increase by about 20% when the embryo is nucleated near a hard inclusion. This is consistent with the experimental observations on the lack of preferential concentration of martensite near the surface when the surface is coated with a harder layer For more detailed calculations the embryo can be assumed to be in the shape of an oblate spheroi a=b>c. By substituting(Al)into(9)and(7), ond Fig. 2. A martensite embryo formed by the transformation of a dislocation loop E。2(1+v) (1+v r9(1-y) 4(1-v)a°3(1-y)a where 8(1 (15) (b :+b where v is the poisson ratio of the matrix and the embryo and (15)is the same as the expression obtained is the cut-off distance such that two dislocation by Christian [21]. The interaction energy Eint can then elements do not interact when they are closer than this be obtained numerically by using(8), (9)and(A2 distance. After the transformation the strain energy of the martensite disc, using an ellipsoidal thin disc as an approximation, is given by(10)for zero dilatation as 3. Nature of the pre-existing embryo E The idealized shape of a plate of martensite is len- 252+(2-v)s ticular, i. e. a minimum strain energy configuration, a situation much like that involved in the formation of a Let us assume that the strain energy of the embryo is mechanical twin. The model proposed here is shown in provided entirely by the strain energy of the dislocation Fig 2, where a circular dislocation loop with radius a loop, i.e. Eloop-Eembryo, and use the approximations component b, is assumed to exist in a parent phase of I5-na-be b infinite extent. Unlike the model given by Chen and which may be deduced from the relationship given by Chiao 6] where the martensite is nucleated around the Eshelby [23] for the equivalence of inclusions and loop, it is assumed that the loop itself, i. e the distorted dislocation loops. The critical dimension a" of the parent phase, will transform into a martensite embryo embryo or the dislocation loop can be estimated from with Bain transformation strain e33=s and(16)-(19)as ei3=e31=$/2 in the shape of a thin disc with radius a and thickness c Before the transformation the strain energy of the dislocation loop is[22] 2b2In 12|-1+(2-wblm2+n2-2 Loop 2b.In [2b2+(2-v)b 丌a4(x(1-vla Eq (2)becomes +(2-v)b (16) Etot=*cAgchem 2a*271 (21)156 H.Y. Yu et al. / Materials Science and Engineering B32 (1995) 153-158 for j~ ~/~'. For pure shear transformation strain, i.e. A=~=0, a reduction of about 19.5% in the strain energy is obtained when the nucleation takes place near a free surface. Taking typical values of A + ~ = 0.05 and s = 0.18 for steels [21], the reduction in energy is 22.6% when A = 0 and 22.4% when ~ = 0. These numbers show that regardless of the transforma￾tion strain composition the strain energy will be reduced about 20% when the embryo is formed near a free surface or void, while the strain energy will increase by about 20% when the embryo is nucleated near a hard inclusion. This is consistent with the experimental observations on the lack of preferential concentration of martensite near the surface when the surface is coated with a harder layer. For more detailed calculations the embryo can be assumed to be in the shape of an oblate spheroid with a = b ">c. By substituting (A1) into (9) and (7), one has _ ~(1+ v) CA~ E~o 2/2(l+V) A2_~ 7r/~ c~2+3(i--v) a r 9( 1- v) 4( 1- v) a 7rkt(2- v) c 2 + s (15) 8(1- v) a where v is the Poisson ratio of the matrix and the embryo and (15) is the same as the expression obtained by Christian [21]. The interaction energy Ein t can then be obtained numerically by using (8), (9) and (A2). 3. Nature of the pre-existing embryo The idealized shape of a plate of martensite is len￾ticular, i.e. a minimum strain energy configuration, a situation much like that involved in the formation of a mechanical twin. The model proposed here is shown in Fig. 2, where a circular dislocation loop with radius a and Burgers vector with edge component be and shear component b~ is assumed to exist in a parent phase of infinite extent. Unlike the model given by Chen and Chiao [6] where the martensite is nucleated around the loop, it is assumed that the loop itself, i.e. the distorted parent phase, will transform into a martensite embryo with Bain transformation strain e~3=~ and elY3 = e~l = s/2 in the shape of a thin disc with radius a and thickness c. Before the transformation the strain energy of the dislocation loop is [22] Eloop 2 tea 4(7r(ff-v)a[2b~[ln(~)-I 1 +(2- v)b~ [ln (~) - 2]1 (16) X3 Xl dislocation loop ~X 2 embryo Fig. 2. A martensite embryo formed by the transformation of a dislocation loop. where (b~ + --s,h2]l/2 p = (17) 8 is the cut-off distance such that two dislocation elements do not interact when they are closer than this distance. After the transformation the strain energy of the martensite disc, using an ellipsoidal thin disc as an approximation, is given by (10) for zero dilatation as Eembryo 7rkt C [2~2 + (2 -- V)S z] r 8( 1- V) a (18) Let us assume that the strain energy of the embryo is provided entirely by the strain energy of the dislocation loop, i.e. Eloop = Eembryo, and use the approximations r~ = :Tra2 be, rs = 7raZ bs (19) which may be deduced from the relationship given by Eshelby [23] for the equivalence of inclusions and dislocation loops. The critical dimension a* of the embryo or the dislocation loop can be estimated from (16)-(19) as [ lln / -2c-7~1/2 32a*/ - 1 +(2-v)b~[ln( 32a* /-2] 2b:[ /(be_l._bs) ) (be2~b~ffs)X/2j 2 _7r [2b~ +(2-v)b~] (20) 2 Eq. (2) becomes Etot = ~a*2 CAgchem + 27ra*2 ~11 (21)