Application of anisotropic inclusion theory to the energy evaluation for the matrix channel deformation and rafting geometry of γ−γ′ Ni superalloys

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Abstract

Uniaxial plastic strain along [0 0 1] is introduced in three types of matrix domains in γ−γ′ nickel superalloys: uniformly in the γ matrix, the horizontal matrix channels normal to [0 0 1] and the vertical channels normal to [1 0 0] and [0 1 0]. Using a mean field method, an elastic energy change due to the introduction of plastic strain (elongation or compression) in the three types of domains is calculated. It is also shown that the particle shape change from a cuboid to a disk, parallel to {0 0 1}, due to the elastic energy decrease, occurs only when the precipitate misfit is present. If there is no plastic strain in the matrix, these disks can be formed with random orientations, parallel to three types of {0 0 1} planes. It is only when there is a plastic strain in the matrix in addition to a precipitate misfit that the alignment of disks occurs on particular types of {0 0 1}. The choice of the alignment plane depends on the sign of the ratio of the plastic strain to the precipitate misfit strain.

Introduction

γ′ precipitates in nickel base superalloys change their shape from cuboidal to disk-like during the early stages of a creep test. One research group has shown that this shape change, called rafting, is primarily initiated by plastic deformation in the γ matrix [1], [2]. The matrix plastic deformation, and a subsequent change in the precipitate shape can change the elastic energy, indicating that rafting is an energy decreasing process. Thus, we have analysed the elastic energy change caused by a shape change of the γ′ particles after plastic deformation is introduced in the γ matrix [3]. It was found that the elastic energy became smaller when the γ′ particles become disk-like parallel to {0 0 1}, after tensile or compressive plastic strain is introduced in the matrix along [0 0 1]. However, this investigation assumed that the plastic deformation took place uniformly in the matrix.

Prior to the above study, Ichitsubo and Tanaka also analysed the subject of rafting [4], [5], [6]. Recently they pointed out a deficiency in our study, which ignored the discriminating role of horizontal and vertical matrix channel deformation modes in the γ matrix [7]. When the largest component in magnitude of plastic strain is along [0 0 1], horizontal channels are defined as matrix subdomains normal to [0 0 1] and vertical channels are defined as those normal to [1 0 0] and [0 1 0]. We further note that there are several crucial experimental observations, obtained by transmission electron microscopy [8], [9] or X-ray diffraction [10], which have reported a localized plastic deformation in the horizontal matrix channels during the early stages of a tensile creep test in alloys having a negative lattice parameter mismatch. These results were explained by noting that the superposition of the internal stress due to misfit strain and the applied stress resulted in different forces acting on gliding matrix dislocations. The matrix channels oriented normal to a tensile external stress, which experience a higher total stress field, yield before those oriented parallel to the applied stress, which experience a lower total stress. The roles of channels, normal and parallel to the applied stress, are reversed when misfit strain is positive.

There are important theoretical studies with respect to channel deformation. FEM analysis was performed by Pollock and Argon [8] and Socrate and Parks [11] in order to evaluate the total stress field in the matrix channels. The results of the calculations conducted in [8] indicated that horizontal (vertical) matrix channels experienced a larger Von Mises equivalent stress under an external tensile stress, when γ′ particles had a negative (positive) lattice parameter misfit with respect to the matrix. Accordingly, they showed that the channels which experienced more stress yielded before those experiencing less stress. Socrate and Parks [11] used an energy–momentum tensor, in the context of isotropic elasticity, in order to determine the force acting on a γ–γ′ interface and correctly predicted the occurrence of rafting when the amount of plastic strain in the matrix was more than two times the initial misfit value. Under these conditions, the initial misfit and elastic properties cease to be important [12]. These studies and comments motivated us to reexamine the rafting phenomenon, by paying attention to the different roles of the horizontal and vertical channels in plastic deformation. We will take a simpler approach based on energetics, using anisotropic elasticity. The method employed is essentially the same as used before [3]. However, in the present study, the elastic energy is decomposed into two contributing factors. It is hoped that, due to this decomposition, the present paper provides clearer analytical insight into the physics behind the selection of plastic domains and the occurrence of rafting.

Section snippets

Analysis

In order to conduct the analysis in a simple and intuitive way, we assume that the γ matrix and γ′ phases have the same elastic constants. This is firstly because our previous paper has shown that this assumption gives nearly the same result as the analysis when γ′ is 15% elastically harder than γ. Secondly, this assumption still predicts the occurrence of rafting in a successful manner without the complication of elastic mismatch. To begin with, the elastic state due to the presence of

Numerical results and discussion

Using the numerical values of the elastic constants given in Section 2.1, the α parameter is plotted against ɛP(V)/ɛ0 in Fig. 2. We also used f = 0.7 and F1 = 0.11 As ɛP(V)/ɛ0 increases from l1, α increases from −0.5 to the asymptotic value,α¯=(1F3)(C11C12)2(C11+C12)

It should be noted that this value never reaches 0. This is due to the large difference in the dimensions of a vertical channel along the [1 0 0] and [0 1 0] directions, as mentioned before. The factor 1  F3 in the numerator indicates the

Conclusion

In order to primarily discuss the rafting occurring in γ–γ′ Ni alloys, the plastic deformation in horizontal channels and vertical channels in the matrix is examined quantitatively. The deformation of these types of channels is discriminated by the precipitate misfit and the reason for the discrimination is discussed by examining the interaction between the precipitate misfit and plastic strain. Rafting geometry is also discussed for each type of channel operation. It is also confirmed that

Acknowledgements

N. Ratel is grateful to the ILL for financial support and T. Mori acknowledges visiting fellowship funding from an EPSRC platform grant.

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