《复合材料 Composites》课程教学资源（学习资料）第五章陶瓷基复合材料_On the heterogeneous nucleation of martensite

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MATERIAL B ELSEVIER Materials Science and Engineering B32( 1995)153-154 On the heterogeneous nucleation of martensite H Y. YuS.C Sanday, B B Rath Naval Research Laboratory, Washington, DC 20375-5000, USA Abstract Martensitic nucleation near inhor ear elasticity. the coherent strain energy due to the formation of a martensite embryo decreases wher is elastically stiffer than the matrix and vice versa. A maximum reduction of 20% in strain energy is calculated ase when the embryo is formed near a free surface. The results are consistent with the experimental observations of preferential nucleation of martensite at a free surface. a possible explanation for the nature of the "pre- existing martensite embryo"in the Kaufman and Cohen model of a nucleation site is also proposed: the dislocation loop in the parent phase is itself the site for the embryo such that it will transform into martensite during transformation. The calculated critical characteristics of this embryo are in good agreement with the model of Chen and chiao and their experimental results Keywords: Martensitic transformation; Inclusion near interfaces; Nucleation 1. Introduction tion of the energy involved in martensitic nucleation leads to the conclusion that heterogeneous nucleation Studies of martensitic transformation have been must be the typical case and therefore special nucleat rekindled in recent years. This is due to applications ing sites must be considered. This led Kaufman and involving the shape memory effect and alloys with high Cohen [2] to introduce the idea that frozen-in marten- damping characteristics. Martensitic transformations in site embryos exist. This model remained popular for ceramics are also being discovered and this field is some years, but a detailed search for such embryos led becoming increasingly popular because of the realiza- to a negative conclusion[3]. It has been proposed by tion of transformation toughening of ceramics. Marten- Cech and Turnbull [4] that lattice defects are respon sitic transformations are also being found in solid gases sible for the nucleation of martensite. It has also been such as oxygen and helium. The classical A15 super- shown experimentally [5-7 that the nucleation of conducting compounds such as V3Si and Nb3 Sn are martensite does take place at the location of stress reported to undergo martensitic transformations just concentrations such as dislocations, stacking faults and above the superconducting transition temperature. grain boundaries. Several theoretical models have been Finally, the importance of hydrogen in metals and the developed for heterogeneous nucleation of martensite. geometrical similarity of the formation of certain Cohen and Olson [8 have proposed that dislocation metallic hydrides with classical martensitic transforma- arrays with specific structures suitable for the nuclea tions have prompted a more detailed focus on the tion of martensite are the nucleation sites. Olson and characteristics and crystallography of martensitic Cohen [9, 10] also suggested that an embryo could be transformations formed on a close-packed plane by interacting with a Except when the transformation driving force is short wall of evenly spaced pre-existing lattice disloc sufficiently large, as in the case of Fe-Co precipitates tions. The strain energy of a defect interacting with the formed in a Cu matrix reported recently [1 where the strain field of the nucleus that is attached to the defect nucleation is homogeneous, a straightforward calcula- has been computed for dislocation loops [ 6, 11] and for a straight dislocation line[12]. Some recent develop ments concerning martensitic transformation and On-site scientist at the Naval Research Laboratory from Geo- heterogeneous martensitic nucleation have been Centers Inc, Fort Washington, MD, USA. reviewed in detail by Wayman [13]and Cohen[14 0921-5107/95/$.500 1995-Elsevier Science S.A. All rights reserved SSD/0921-5107(95)03005-0

ELSEVIER Materials Science and Engineering B32 ( 1995 ) 153-158 MATERIALS SCIENCE & ENGINEERING B On the heterogeneous nucleation of martensite H.Y. YuS.C. Sanday, B.B. Rath Naval Research Laboratory, Washington, DC 20375-5000, USA Abstract Martensitic nucleation near inhomogeneities is modeled using linear elasticity. The coherent strain energy due to the formation of a martensite embryo decreases when the inhomogeneity is elastically stiffer than the matrix and vice versa. A maximum reduction of 20% in strain energy is calculated for the case when the embryo is formed near a free surface. The results are consistent with the experimental observations of preferential nucleation of martensite at a free surface. A possible explanation for the nature of the "preexisting martensite embryo" in the Kaufman and Cohen model of a nucleation site is also proposed: the dislocation loop in the parent phase is itself the site for the embryo such that it will transform into martensite during transformation. The calculated critical characteristics of this embryo are in good agreement with the model of Chen and Chiao and their experimental results. Keywords: Martensitic transformation; Inclusion near interfaces; Nucleation 1. Introduction Studies of martensitic transformation have been rekindled in recent years. This is due to applications involving the shape memory effect and alloys with high damping characteristics. Martensitic transformations in ceramics are also being discovered and this field is becoming increasingly popular because of the realization of transformation toughening of ceramics. Martensitic transformations are also being found in solid gases such as oxygen and helium. The classical A15 superconducting compounds such as V3Si and Nb3Sn are reported to undergo martensitic transformations just above the superconducting transition temperature. Finally, the importance of hydrogen in metals and the geometrical similarity of the formation of certain metallic hydrides with classical martensitic transformations have prompted a more detailed focus on the characteristics and crystallography of martensitic transformations. Except when the transformation driving force is sufficiently large, as in the case of Fe-Co precipitates formed in a Cu matrix reported recently [1] where the nucleation is homogeneous, a straightforward calcula- 1On-site scientist at the Naval Research Laboratory from GeoCenters Inc., Fort Washington, MD, USA. tion of the energy involved in martensitic nucleation leads to the conclusion that heterogeneous nucleation must be the typical case and therefore special nucleating sites must be considered. This led Kaufman and Cohen [2] to introduce the idea that frozen-in martensite embryos exist. This model remained popular for some years, but a detailed search for such embryos led to a negative conclusion [3]. It has been proposed by Cech and Turnbull [4] that lattice defects are responsible for the nucleation of martensite. It has also been shown experimentally [5-7] that the nucleation of martensite does take place at the location of stress concentrations such as dislocations, stacking faults and grain boundaries. Several theoretical models have been developed for heterogeneous nucleation of martensite. Cohen and Olson [8] have proposed that dislocation arrays with specific structures suitable for the nucleation of martensite are the nucleation sites. Olson and Cohen [9,10] also suggested that an embryo could be formed on a close-packed plane by interacting with a short wall of evenly spaced pre-existing lattice dislocations. The strain energy of a defect interacting with the strain field of the nucleus that is attached to the defect has been computed for dislocation loops [6,11] and for a straight dislocation line [12]. Some recent developments concerning martensitic transformation and heterogeneous martensitic nucleation have been reviewed in detail by Wayman [13] and Cohen [14]. 0921-5107/95/$9.50 © 1995 - Elsevier Science S.A. All rights reserved SSD1 0921-5107(95)03005-0

154 H.Y. Yuet al Materials Science and Engineering B32(1995)153-158 The dependence of transformed particle fraction on (0, 0, d )near the interface of a bimaterial as shown in particle size found by means of small particle experi- Fig. 1(b). The Lame constants of the parent phase are a ments indicates that martensite nucleates at a free and u and the Lame constants of the inhomogeneity surface or near a free surface for Fe-22Ni-049C pow- are A' and u. The two solids are perfectly bonded der [ 15]. Magee [15 also proposed that the absence of together. The elastic constants of the martensite and surface nucleation in the case of Fe-24 2Ni-36Mn the parent phase(matrix) are assumed to be the same powder may be due to a slight oxide skin. It has also The martensite embryo is an ellipsoidal inclusion with been shown [16] that for Fe-24Ni-3Mn alloy with a semiaxes a, b and c. The habit plane of the martensite clean surface the sample has a greater concentration of embryo is parallel to the interface x3=0. Let the stress martensite near the surface than in the interior and free transformation strain of the martensite embryo none of the samples plated with an adherent layer of consist of a uniform dilatation A, an expansion 5 nickel 0.001 inch thick has any preferential concentra- normal to the habit plane and a simple shear s on the tion of martensite near the surface. The preferential habit plane. Thus the stress-free transformation strain surface nucleation is thought to be associated with the components are free surface itself, e.g. the lack of three-dimensional surface. The increased martensitic transformation tem- i=,s4 constraint on the transformation shape change at the perature in iron-base alloyed thin films is also attri- buted to this free-surface phenomenon [17. However, and all other components are zero. The total energy no theoretical model has been provided to explain change associated with the transformation is hese phenomena In this study the effect of inhomogeneities on the Etot"Eur+ Echem estrain strain energy of martensitic transformation will be where Esurf corresponds to the coherent interfacial presented. A possible mechanism of heterogeneous energy between the matrix and the martensite, Echet martensitic nucleation is also proposed refers to the chemical driving force of martensitic transformation and Estrain is the coherent strain energy due to the misfit between the two lattices the elastic 2. Heterogeneous nucleation near inhomogeneities strain energy change is made up of two terms When the size of the inhomogeneity is much larger Estrain+Ei than the martensite embryo as shown in Fig. 1(a), the in which Ea represents the self-strain energy of the problem can be modeled as an embryo formed at point embryo when formed within the constraints of the extent and Eint represents the elastic interaction energy matrixλμ) embryo(λ.μ) between the inhomogeneity and the Bain strain(trans formation strain)of the embryo Following the treatment given by the authors for inclusion problem in bimaterials [18, 19, the strained strain inside the embryo @2 is e where Suk is the Eshelby tensor [20] for a homo- geneous inclusion in an isotropic infinite solid and Suka is the coupling tensor due to the presence of the inhomogeneity(details are given in the Appendix). The matrix(λ.) strain energy change due to the formation of the embryo is given by (3)with inhomogeneityλ,g) E。=-∫smuo+25m-12n-2d Model for the calculation of the effect of an inhomogeneity on En --4(sm, 0, +2ustwleHe,ds martensitic nucleation

154 H.Y. Yu et al. / Materials Science and Engineering B32 (1995) 153-158 The dependence of transformed particle fraction on particle size found by means of small particle experiments indicates that martensite nucleates at a free surface or near a free surface for Fe-22Ni-0.49C powder [15]. Magee [15] also proposed that the absence of surface nucleation in the case of Fe-24.2Ni-3.6Mn powder may be due to a slight oxide skin. It has also been shown [16] that for Fe-24Ni-3Mn alloy with a clean surface the sample has a greater concentration of martensite near the surface than in the interior and none of the samples plated with an adherent layer of nickel 0.001 inch thick has any preferential concentration of martensite near the surface. The preferential surface nucleation is thought to be associated with the free surface itself, e.g. the lack of three-dimensional constraint on the transformation shape change at the surface. The increased martensitic transformation temperature in iron-base alloyed thin films is also attributed to this free-surface phenomenon [17]. However, no theoretical model has been provided to explain these phenomena. In this study the effect of inhomogeneities on the strain energy of martensitic transformation will be presented. A possible mechanism of heterogeneous martensitic nucleation is also proposed. 2. Heterogeneous nucleation near inhomogeneities When the size of the inhomogeneity is much larger than the martensite embryo as shown in Fig. l(a), the problem can be modeled as an embryo formed at point embryo (~., Ix) matrix (k, It) N~ (a) X3 embryo (~,, It) matrix (~., It xl I inhomogeneity (~', It') I1 X 2 (b) Fig. 1. (a) A martensite embryo near an inhomogeneity. (b) Model for the calculation of the effect of an inhomogeneity on martensitic nucleation. (0, 0, d) near the interface of a bimaterial as shown in Fig. l(b). The Lam6 constants of the parent phase are 2 and At and the Lam6 constants of the inhomogeneity are 2' and At'. The two solids are perfectly bonded together. The elastic constants of the martensite and the parent phase (matrix) are assumed to be the same. The martensite embryo is an ellipsoidal inclusion with semiaxes a, b and c. The habit plane of the martensite embryo is parallel to the interface x3 = 0. Let the stressfree transformation strain of the martensite embryo consist of a uniform dilatation A, an expansion normal to the habit plane and a simple shear s on the habit plane. Thus the stress-free transformation strain components are T T A T A T S e11= e22 e33=~+ ~, e13 (1) 3' .5 2 and all other components are zero. The total energy change associated with the transformation is Eto t = Esurf + Echem + Estrain (2) where Esurf corresponds to the coherent interfacial energy between the matrix and the martensite, Ech .... refers to the chemical driving force of martensitic transformation and Estrain is the coherent strain energy due to the misfit between the two lattices. The elastic strain energy change is made up of two terms Estrain = E oo + Ein t (3) in which E~ represents the self-strain energy of the embryo when formed within the constraints of the surrounding homogeneous parent phase of infinite extent and Ein t represents the elastic interaction energy between the inhomogeneity and the Bain strain (transformation strain) of the embryo. Following the treatment given by the authors for the inclusion problem in bimaterials [18,19], the constrained strain inside the embryo f2 is C o0 • T e~j = (S~jkt + S~jkt) ekt (4) where S0~I is the Eshelby tensor [20] for a homogeneous inclusion in an isotropic infinite solid and S~k t is the coupling tensor due to the presence of the inhomogeneity (details are given in the Appendix). The strain energy change due to the formation of the embryo is given by (3) with f o0 ao T T T T Eoo = _1 [( )~Sm,~tbq + 2AtSqkt)ekt --J, emm6ij-- 2Atei/]eij dff2 Q (5) f * ~ ,,~* \ T T Eint = __1 .J (~,Srnmkl6ij + •At3i/kdek/e O. dff~ (6)

H. 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 proposed 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 increasing 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)

H.Y. Yu et al./Materials Science and Engineering B32(1995)153-158 for uc. 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 transformation 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 lenticular, 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)

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-1BA

H. 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 equilibrium 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 compare 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 (transmission 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 dislocations [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 preexisting 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 nucleation 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

158 H.Y. Yu et al. / Materials Science and Engineering B32 (1995) 153-158 S1212- 8oz(1- v) [(3- n n ~ 2~11 1 4v)l-" 1212 + 2X31-' 12123 -- AX3t'Pj212] /~ - p' p + 8 + +/,') x [(1 - 2v)(2p' - p)fl - (1 - " L~' " ..... n )/.l p J~J,1212 S 1213 =" 8~r(1 - v) [(3 - 4v)Fl1213 + 2x3rl~2,33 - 4(1 - v)x3(I)lll21 ~ 2~I1 1 , --ZX3q) 1213J 1 [/u-p' +-- 8Jr Ip+ /u' I t~t\lTr]l I + 2V11112)- 2(Pfl- P p :w 112] , (/.2 -- ~U;)~ U ll 2~11 \ S 1312 - 8,7['( 1 - v) [F'1312 + 2(x3F'13123 - x31"P'1312) -- 4 ])X3 (I)I, I112] tU -- ~ut ll 1 {/u-/u' 4zr/,u + ,u' ,ufl + ,u'fl' IJ~II 112 where ~l=! 1 Ri dff2', 1 O" = ! aft', ~n = f ~ll dx3, Fl=f R1 dr2' f2 F II= f R 2 dr' n (~II = f u~jH dx 3 RI = [(Xl _ xrl )2 + (x 2 _ x~)2 + (x 3 - ,~A -- -~3/"' ~2]1/2j R2 =[(x, -X'l) 2 + + (x3 + d-x )q '/2 and dr2' -- dx'~ dx~ dx~. References [1] M. Lin, G.B. Olson and M. Cohen, Acta Metall., 41 (1992) 253. [2] L. Kaufman and M. Cohen, Prog. Met. Phys., 7(1958) 165. [3] M.H. Korenko and M. Cohen, Scr. Metall., 8 (1974) 751. [4] R.E. Cech and D. Turnbull, Trans. A1ME, 206 (1956) 124. [5] J.W. Brooks, M.H. Loretto and R.E. Smallman, Acta Metall., 27(1979) 1839. [6] I.-W. Chen and Y.-H. Chiao, Acta Metall., 33 (1985) 1827. [7] T. Saburi and S. Nenno, Proc. Int. Conf. on Martensitic Transformations, 1986, p. 176. [8] M. Cohen and G.B. Olson, Trans. Jpn. Inst. Met., Suppl., 17 (1976)93. [9] G.B. Olson and M. Cohen, Metall. Trans. A, 7(1976) 1897. [ 10] G.B. Olson and M. Cohen, Metall. Trans. A, 7{ 1976) 1905. [11] K.E. Eastefling and A.R. Th61dn, Acta Metall., 24 (1976) 333. I12] M. Suesawa and H.E. Cook, Acta Metall., 28(1980) 423. I13] C.M. Wayman, in H.I. Aaronson, D.E. Laughlin, R.E Sekerka and C.M. Wayman (eds.), Proc. Int. Conf. on Solid Phase Transformations, TMS-AIME, Warrendale, PA, 1981,p. 1119. [14] M. Cohen, Mater. Trans., JIM, 33 (1992) 178. [15] C.L. Magee, Metall. Trans., 2(1971 ) 2419. [ 16] V. Raghavan and M. Cohen, Metall. Trans., 2 ( 1971 ) 2409. [17] H. Warlimont, Trans. A1ME, 221 (1961) 1270. [ 18] H.Y. Yu and S.C. Sanday, Proc. R. Soc. Lond. A, 434 ( 1991 ) 520. [19] H.Y. Yu and S.C. Sanday, Proc. R. Soc. Lond. A, 439(1992) 659. [20] J.D. Eshelby, Proc. R. Soc. Lond. A, 241 (1957) 376. [21] J.W. Christian, Acta Metall., 6 (1958) 377. [22] J.P. Hirth and J. Lothe, Theory of Dislocations, McGrawHill, New York, 1968, p. 144. [23] J.D. Eshelby, in I.N. Sneddon and R. Hill (eds.), Progress in Solid Mechanics, Vol. 2, North-Holland~ Amsterdam, 1961, p. 89

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《复合材料 Composites》课程教学资源（学习资料）第五章陶瓷基复合材料_On the heterogeneous nucleation of martensite

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《复合材料 Composites》课程教学资源（学习资料）第五章陶瓷基复合材料_On the heterogeneous nucleation of martensite