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.

References

References
1.
Eshelby
,
J. D.
,
1961
, “
Elastic Inclusion and Inhomogeneities
,”
Progress in Solid Mechanics
, Vol.
2
,
N. I.
Sneddon
, and
R.
Hill
, eds.,
North Holland
,
Amsterdam, The Netherlands
, pp.
89
140
.
2.
Kinoshita
,
N.
, and
Mura
,
T.
,
1971
, “
Elastic Fields of Inclusions in Anisotropic Media
,”
Phys. Status Solidi A
,
5
(
3
), pp.
759
768
.
3.
Mura
,
T.
,
1987
,
Micromechanics of Defects in Solids
,
Martinus Nijhoff Publishers
,
The Hague, The Netherlands
.
4.
Khachaturyan
,
A. G.
,
1983
,
Theory of Structural Transformations in Solids
,
Wiley
,
New York
.
5.
Kang
,
S. J.
,
Kim
,
Y. W.
,
Kim
,
M.
, and
Zuo
,
J.
,
2014
, “
Determination of Interfacial Atomic Structure, Misfits and Energetics of Ω Phase in Al–Cu–Mg–Ag Alloy
,”
Acta Mater.
,
81
, pp.
501
511
.
6.
Fallah
,
V.
,
Korinek
,
A.
,
Opoku
,
N. O.
,
Raeisinia
,
B.
,
Gallerneault
,
M.
,
Provatas
,
N.
, and
Esmaeili
,
S.
,
2015
, “
Atomic-Scale Pathway of Early Stage Precipitation in Al–Mg–Si Alloys
,”
Acta Mater.
,
82
, pp.
457
467
.
7.
Ji
,
Y. Z.
,
Issa
,
A.
,
Heo
,
T. W.
,
Saal
,
J. E.
,
Wolverton
,
C.
, and
Chen
,
L.
,
2014
, “
Predicting β′ Precipitate Morphology and Evolution in Mg–RE Alloys Using a Combination of First-Principles Calculations and Phase-Field Modeling
,”
Acta Mater.
,
76
, pp.
259
271
.
8.
Shmulevitsh
,
M.
,
Meshi
,
L.
,
Pinkas
,
M.
, and
Shneck
,
R. Z.
,
2015
, “
Elastic Consideration of the Precipitation in Model Alloys of Maraging Steels: Theory and Experimental Validation
,”
J. Mater. Sci.
,
50
(
14
), pp.
4970
4979
.
9.
Lee
,
J. K.
,
Barnett
,
D. M.
, and
Aaronson
,
H. I.
,
1977
, “
The Elastic Strain Energy of Coherent Ellipsoidal Precipitates in Anisotropic Crystalline Solids
,”
Metall. Trans. A
,
8
(
6
), pp.
963
970
.
10.
Li
,
Z.
,
Li
,
Y.
,
Sunand
,
J.
, and
Feng
,
X. Q.
,
2011
, “
An Approximate Continuum Theory for Interaction Between Dislocation and Inhomogeneity of Any Shape and Properties
,”
J. Appl. Phys.
,
109
(
11
), p.
113529
.
11.
Li
,
Y.
,
Li
,
Z.
,
Wang
,
X.
, and
Sun
,
J.
,
2010
, “
Analytical Solution for Motion of an Elliptical Inclusion in Gradient Stress Field
,”
J. Mech. Phys. Solids
,
58
(
7
), pp.
1001
1010
.
12.
Shneck
,
R. Z.
,
2001
, “
Microstructural Shape Evolution of γ′ in Nickel-Based Superalloys by Stress-Assisted Diffusion
,”
Philos. Mag., A
,
81
(
2
), pp.
383
389
.
13.
Seitz
,
E.
, and
Fountaine
,
D.
,
1978
, “
Elastic Interaction Energy Calculations for Guinier-Preston Zones in Al-Cu and Cu-Be
,”
Acta Mater.
,
26
(
11
), pp.
1671
1679
.
14.
Lee
,
J. K.
, and
Johnson
,
W. C.
,
1977
, “
Elastic Strain Energy and Interactions of Thin Square Plates Which Have Undergone a Simple Shear
,”
Scr. Metall.
,
11
(
6
), pp.
477
484
.
15.
Johnson
,
W. C.
, and
Lee
,
J. K.
,
1982
, “
An Integral Equation Approach to the Elastic Interaction of Two Precipitates
,”
Phys. Status Solidi A
,
71
(
2
), pp.
589
602
.
16.
Johnson
,
W. C.
,
Earmme
,
Y. Y.
, and
Lee
,
J. K.
,
1980
, “
Approximation of the Strain Field Associated With an Inhomogeneous Precipitate—Part 2: The Cuboidal Inhomogeneity
,”
ASME J. Appl. Mech.
,
47
(
4
), pp.
781
788
.
17.
Chu
,
H. J.
,
Pan
,
E.
,
Ramsey
,
J. J.
,
Wang
,
J.
, and
Xue
,
C. X.
,
2011
, “
A General Perturbation Method for Inhomogeneities in Anisotropic and Piezoelectric Solids With Applications to Quantum-Dot Nanostructures
,”
Int. J. Solids Struct.
,
48
(
5
), pp.
673
679
.
18.
Seifollahi
,
M.
,
Razavi
,
S. H.
,
Kheirandish
,
S.
, and
Abbasi
,
S. M.
,
2013
, “
The Mechanism of η Phase Precipitation in A286 Superalloy During Heat Treatment
,”
J. Mater. Eng. Perform.
,
22
(
10
), pp.
3063
3069
.
19.
Savoie
,
M.
,
Esnouf
,
C.
,
Fournier
,
L.
, and
Delafosse
,
D.
,
2007
, “
Influence of Ageing Heat Treatment on Alloy A-286 Microstructure and Stress Corrosion Cracking Behavior in PWR Primary Water
,”
J. Nucl. Mater.
,
360
(
3
), pp.
222
230
.
20.
De Cicco
,
H.
,
Luppo
,
M. I.
,
Raffaeli
,
H.
,
Gaetano
,
J. D.
,
Gribaudo
,
L. M.
, and
Ovejero-García
,
J.
,
2005
, “
Creep Behavior of an A286 Type Stainless Steel
,”
Mater. Charact.
,
55
(
2
), pp.
97
105
.
21.
Nembach
,
E.
, and
Neite
,
G.
,
1985
, “
Precipitation Hardening of Superalloys by Ordered γ′-Particles
,”
Prog. Mater. Sci.
,
29
(
3
), pp.
177
319
.
22.
Thompson
,
A. W.
, and
Brooks
,
J. A.
,
1982
, “
The Mechanism of Precipitation Strengthening in an Iron-Base Superalloy
,”
Acta Mater.
,
30
(
12
), pp.
2197
2203
.
23.
Zarestky
,
J.
, and
Stassis
,
C.
,
1987
, “
Lattice Dynamics γ-Fe
,”
Phys. Rev. B
,
35
(
9
), pp.
4500
4502
.
24.
Zhang
,
H. L.
,
Al-Zoubi
,
N.
,
Johansson
,
B.
, and
Vitos
,
L.
,
2011
, “
Alloying Effects on the Elastic Parameters of Ferromagnetic and Paramagnetic Fe From First-Principles Theory
,”
J. Appl. Phys.
,
110
(
7
), p.
073707
.
25.
Cao
,
Y.
,
Zhu
,
J.
,
Liu
,
Y.
,
Lai
,
Z.
, and
Nong
,
Z.
,
2013
, “
First-Principles Studies of the Structural, Elastic, Electronic and Thermal Properties of γ-Ni3Ti
,”
Physica B
,
412
, pp.
45
49
.
26.
Ardell
,
A. J.
,
Nicholson
,
R. B.
, and
Eshelby
,
J. D.
,
1966
, “
On the Modulated Structure of Aged Ni-Al Alloys: With an Appendix on the Elastic Interaction Between Inclusions
,”
Acta Mater.
,
14
(
10
), pp.
1295
1309
.
27.
Lee
,
J. K.
, and
Johnson
,
W. C.
,
1978
, “
Calculation of Elastic Strain Field of Cuboidal Precipitate in Anisotropic Matrix
,”
Phys. Status Solidi A
,
46
(
1
), pp.
267
272
.
28.
Baldan
,
A.
,
2002
, “
Review Progress in Ostwald Ripening Theories and Their Applications to the γ′-Precipitates in Nickel-Base Superalloys
,”
J. Mater. Sci.
,
37
(12), pp.
2379
2405
.
29.
Beerends
,
R. J.
,
Ter Morsche
,
H. G.
,
van den Berg
,
J. C.
, and
van de Vrie
,
E. M.
,
2003
,
Fourier and Laplace Transforms
,
Cambridge University Press
,
New York
, Chap. 4.
30.
Völkla
,
R.
,
Glatzelb
,
U.
, and
Feller-Kniepmeierc
,
M.
,
1988
, “
Measurement of the Lattice Misfit in the Single Crystal Nickel Based Superalloys CMSX-4, SRR99 and SC16 by Convergent Beam Electron Diffraction
,”
Acta Mater.
,
46
(
12
), pp.
4395
4404
.
31.
Ecob
,
R. C.
,
Ricks
,
R. A.
, and
Porter
,
A. J.
,
1982
, “
The Measurement of Precipitate/Matrix Lattice Mismatch in Nickel-Base Superalloys
,”
Scr. Metall.
,
16
(
9
), pp.
1085
1090
.
32.
Nathal
,
M. V.
,
MacKay
,
R. A.
, and
Garlick
,
R. G.
,
1988
, “
Lattice Parameter Variations During Aging in Nickel-Base Superalloys
,”
Scr. Metall.
,
22
(
9
), pp.
1421
1424
.
33.
Balluffi
,
R. W.
,
2012
,
Introduction to Elastic Theory for Crystal Defects
,
Cambridge University Press
,
New York
, Chap. 3.6.
You do not currently have access to this content.