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 nucleation154 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 experi￾ments indicates that martensite nucleates at a free surface or near a free surface for Fe-22Ni-0.49C pow￾der [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 concentra￾tion 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 tem￾perature in iron-base alloyed thin films is also attri￾buted 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 stress￾free 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 (trans￾formation strain) of the embryo. Following the treatment given by the authors for the inclusion problem in bimaterials [18,19], the con￾strained 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 homo￾geneous 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)