H.Y. Yu et al. Materials Science and Engineering B32(1995)153-158 where &i is the Kronecker d. Substituting(1)into (5 (1+v)(x-)B21 4(1 3入+2a)△ (1-2y)(4-)B1 10d2 rhm33 B 8 7(1-v- AP2d' 1+y)-2(3-v (2X1313-x)-5[A2mmk+223k 10d +区(2-7v)(4-4)B +(3A+21)(∑m3-2r (3+2)△ 02 2x(1-v) 5d210d4 22 “-+2AB-B +(32+2山)mm3] Σ13nmn=0 (11) where where r is the volume of the embryo Eqs.(8)and(11)show that the elastic interaction energy Eint is negative when the matrix is stiffer than m-∫smd,x-js,d (9) the inhomogeneity, i.eu>u', and vice versa. They also show that the magnitude of Eint decreases with increas ing d. Let us consider first the case when v=1/3: (10) and dQ=dx, dx2 dx3 becomes In this study the shape of the embryo in the earliest ge of nucleation is assumed to be spherical as pr124a△2+42+出△+4 (12) nucleus leads to a closed-form solution for the strain energy in the presence of inhomogeneities, since the and when d=a, 8)and (11) give mean value of the harmonic function of a sphere is equal to its value at the center. For a spherical embryo Ein 4 4u a=b=c).(7)reads 23(421_21 609 E。2u(1+v) 4(1 r9(1-v)15(1-v)9(1-v) 254 (13) 12815 (10) The volume integrals of the coupling tensors in 8 )are Eint 2 247 obtained by substituting(A2)into(9). They are 9605 1009 (1+v)(1-2v)(4-4)B21 x(1-yH. Y. Yu et al. / Materials Science and Engineering B32 (1995) 153-158 155 where 6,). is the Kronecker d. Substituting (1) into (5) and (6), one has Eoo- (q~ ,~..4- '9 .,__ ~a 2 (Z2mkk--3"~) 18 ~ 2 2 ----[~.Z~nrn33 nt- 2.Z3333 -(2 + 2.)r] 2 2 (2Z1313 -- Z')- [~l, Zmmkk 4- 2.Z33kk Ein t = + (3)]. + 2")(Zmm33 -- 2r)] (3;t+2.)a 2 Y* _~2 • 18 mmkk 2 (~'ZmmB3 + 2"Z*333) (7) 2 * ~zX , , --.S Z1313 6 ['~Zrnmkk"[-2/lgZ33kk +(32+ 2")~mm33] * .As ((32 + 2.) ) 3 ~- ~ Z*mml3"l-Zl*3mm ~Sr.~* q- 2. (Z3"313 -k y 1"333) ] 2 1~2"mm13 (8) ~'~1313-- Z.3kk_(l+v)(.--.')fl 2 1 ( 6a2/ 4Jr( 1 -- v) r ~5 4( 1 -- v) - ~5} (a--2v)(.--.')fl 1 ( 3a2/ Z3333 8er(1-v)("-"')fl (l+v)-2(3-v)Sd2 3a4 / + 10d4 } + [(2- 7v)(.-.')fl +(1- 1 1 ("-'"')fir2 ( 12a2+ 9a4/ 32at(l-v)d 3 .(1+v)-5dS- 10d 4] ("-"', + + 6-G d 3 * __ * i * m * Y~1333 -- Y~3313 -- "~rnml3 -- "~13rnm = 0 where 1 1 fl =. + (3 -4v). .'+(3 - 4v'). (11) where r is the volume of the embryo, f Zij~, = S ijkz dQ, Z'k, = S ij*kz dQ (9) f~ Q and dQ = dx 1 dx 2 dx 3. In this study the shape of the embryo in the earliest stage of nucleation is assumed to be spherical as pro￾posed in the Olson-Cohen model [9]. A spherical nucleus leads to a closed-form solution for the strain energy in the presence of inhomogeneities, since the mean value of the harmonic function of a sphere is equal to its value at the center. For a spherical embryo (a = b = c), (7) reads E=_2.(l+ v)&2q 8. 4.(1+ V)A~ r 9(1- v) 15(1- v) ~24 9(1- v) +.(7- 5v) 2 s (10) 30(1 - v) The volume integrals of the coupling tensors in (8) are obtained by substituting (A2) into (9). They are Eqs. (8) and (11) show that the elastic interaction energy Ein t is negative when the matrix is stiffer than the inhomogeneity, i.e.. >. ', and vice versa. They also show that the magnitude of Eint decreases with increas￾ing d. Let us consider first the case when v= 1/3: (10) becomes E= = 4/, A2 + 4. ~2 "l- 8._~ A~ "}- 4. $2 (12) r 9 5 9 15 and when d = a, (8) and (11 ) give 25(4 ) 128 ~ s2 (13) for. >.' and Eint 2 (~A2)+247 (~) 21 (~) r 15 960 ~2 +1~ A~ . (l+v)(1-2v)(.-.')fl 2 1 27 (4.) E"mk~-- zr(1-- V) r 4d 3 + 128 ~ s2 (14)