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Research Papers

An Approach to Calculate the Elastic Interaction Energy of Inhomogeneous Precipitates: Application to γ′-Ni3Ti in A-286 Steel

[+] Author and Article Information
Mati Shmulevitsh

Department of Material Engineering,
Ben Gurion University of the Negev,
Beer-Sheva 84105, Israel;
NRCN,
Beer-Sheva 9001, Israel

Roni Z. Shneck

Department of Material Engineering,
Ben Gurion University of the Negev,
Beer-Sheva 84105, Israel

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received March 8, 2018; final manuscript received April 28, 2018; published online May 21, 2018. Assoc. Editor: Shaoxing Qu.

J. Appl. Mech 85(8), 081003 (May 21, 2018) (9 pages) Paper No: JAM-18-1131; doi: 10.1115/1.4040117 History: Received March 08, 2018; Revised April 28, 2018

The elastic interaction energy between several precipitates is of interest since it may induce ordering of precipitates in many metallurgical systems. Most of the works on this subject assumed homogeneous systems, namely, the elastic constants of the matrix and the precipitates are identical. In this study, the elastic fields, and self and interaction energies of inhomogeneous anisotropic precipitates have been solved and assessed, based on a new iterative approach using the quasi-analytic Fourier transform method. This approach allows good approximation for problems of several inhomogeneous precipitates in solid matrix. We illustrate the calculation approach on γ-Ni3Ti precipitates in A-286 steel and demonstrate that the influence of elastic inhomogeneity is in some incidences only quantitative, while in others it has essential effect. Assuming homogeneous system, disk shape precipitate is associated with minimum elastic energy. Only by taking into account different elastic constants of the precipitate, the minimum self-energy is found to be associated with spherical shape, and indeed, this is the observed shape of the precipitates in A-286 steel. The elastic interaction energy between two precipitates was calculated for several configurations. Significant differences between the interactions in homogeneous and inhomogeneous were found for disk shape morphologies. Only quantitative differences (9% higher interaction between inhomogeneous precipitates) were found between two spherical precipitates, which are the actual shape.

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References

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Figures

Grahic Jump Location
Fig. 1

Comparison of the energy density of homogeneous spherical precipitate to Eshelby equivalent inclusion method and to the results of three iterations made by the approach of Chu et al.

Grahic Jump Location
Fig. 2

The total elastic energy of the system including precipitate and matrix per unit volume of the precipitate as a function of its shape for homogeneous system and for two calculation methods of inhomogeneous systems (Chu third-order approximation and Eshelby equivalent precipitate, designated inhomogeneous)

Grahic Jump Location
Fig. 3

Profiles of the εxx (a) and εyy (b) strains across spherical γ′-Ni3Ti precipitate with the austenite FCC matrix along the y axis for homogeneous spherical precipitate, Eshelby “equivalent precipitate” and the three iterations made by the approach of Chu et al.

Grahic Jump Location
Fig. 4

Profiles of the εxx strain of two spherical γ′-Ni3Ti precipitates located at two different distances: long distance, d = 6R (the precipitates are centered at x1 = −3 and x2 = 3)—weak overlap between the strain fields; and short distance, d = 3R (the precipitates are centered at x1 = −1.5 and x2 = 1.5)—strong overlap of the strain fields

Grahic Jump Location
Fig. 5

The total elastic energy of two inhomogeneous spherical γ′-Ni3Ti precipitates as a function of the interparticle distance along three crystallographic axes

Grahic Jump Location
Fig. 6

The total elastic energy of two spherical homogeneous and inhomogeneous precipitates located at 2.4R along ⟨100⟩, as functions of the transformation strain εT

Grahic Jump Location
Fig. 7

(a) The total elastic energy of pairs of two homogeneous precipitates parallel to each other with different aspect ratios (from a thin disk to a sphere) as a function of the interparticle distance. The difference between the minimum energy at short interparticle distance and the energy at long distance is the interaction energy, shown in figure (b). The interaction in homogenous system decreases with decreasing aspect ratio.

Grahic Jump Location
Fig. 8

The total elastic energy of two homogeneous and inhomogeneous sphere γ′-Ni3Ti precipitates along ⟨100⟩ as a function of the interparticle distance

Grahic Jump Location
Fig. 9

(a) Comparison of the total elastic energy between two disk-shaped precipitates (with aspect ratio 0.3) at different configurations in homogenous and inhomogeneous systems shown in figures (b)–(d). Each configuration is described by the maps of the εxx strain in the XY plane, generated in a homogeneous system by the disk shaped precipitates. (b) Configuration A—two disks are e parallel to each other, (c) configuration B—two disks located side by side, and (d) configuration C—two disks have vertical arrangements.

Grahic Jump Location
Fig. 10

Displacement maps of homogeneous (a) and inhomogeneous (b) disk shape precipitates that are parallel to each other in equilibrium with the FCC matrix at the XY plane

Grahic Jump Location
Fig. 11

(a) The inhomogeneous actual problem containing a precipitate with transformation strains εijT and elastic constants CijklP that are different from the matrix, (b) the reference system containing the matrix with a precipitate identical in its initial form and transformation strains to those of the first system, and elastic constants identical to the matrix, and (c) the displacements in the matrix εC of the inhomogeneous system A and in the reference system εco. Δu are the additional displacements that should be added to the displacements in the reference system in order to obtain the displacements in the actual system A. Δu are due to the excess elastic stresses in the reference precipitate relative to the actual one.

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