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Elastic strain energy due to misfit strains in a polyhedral precipitate composed of low-index planes
Acta Materialia 55 (2007) 669–673 www.actamat-journals.com
Elastic strain energy due to misfit strains in a polyhedral precipitate composed of low-index planes Susumu Onaka
a,*
, Toshiyuki Fujii b, Masaharu Kato
a
a b
Department of Materials Science and Engineering, Tokyo Institute of Technology, 4259-J2-63 Nagatsuta, Yokohama 226-8502, Japan Department of Innovative and Engineered Materials, Tokyo Institute of Technology, 4259-J2-45 Nagatsuta, Yokohama 226-8502, Japan Received 16 June 2006; accepted 29 August 2006 Available online 15 November 2006
Abstract The dependence of precipitate shape on the elastic strain energy caused by anisotropy of the elastic moduli is discussed for polyhedral precipitates with purely dilatational misfit strains. By considering cubic structures for the precipitates and matrix phase, the elastic strain energy is calculated for the precipitate shapes of a {1 0 0} cube, a {1 1 0} rhombic-dodecahedron, a {1 1 1} octahedron, and the truncated shapes of all three polyhedra. The anisotropy of the elastic moduli of Cu is used for the numerical calculations. The polyhedral shapes of the {1 1 1} octahedron and {1 0 0} cube result in the highest and lowest values of the elastic strain energy per unit volume of precipitate. The calculated values of the elastic strain energy obtained for the truncated polyhedra are compared with values estimated using a geometrical method that considers the area changes of the {1 0 0}, {1 1 0} and {1 1 1} interfaces. 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Precipitates; Elastic strain energy; Misfit strains; Micromechanics
1. Introduction The shapes of precipitates in alloys are affected by various factors. Although some precipitate shapes obtained under certain conditions can be explained using kinetics, the stable shapes observed in alloys are best discussed with respect to energetics. These stable shapes sometimes show symmetry that reflects the symmetry of the alloy crystal structure. Polyhedral shapes composed of low-index planes are typical examples of such symmetrical shapes [1]. It is true that the anisotropy of the interface energy density between the precipitate and the matrix is one origin for the shapes of polyhedral precipitates. However, when precipitates have misfit strains, the resulting elastic strain energy is an important factor that affects the stability of the precipitate shape [2]. It is known that the anisotropy of the elastic moduli could cause a decrease in the elastic strain energy of precipitates following a change in shape *
Corresponding author. Tel. +81 45 924 5564; fax: +81 45 924 5566. E-mail address: [email protected] (S. Onaka).
from spherical to polyhedral [2]. In the present paper, we solve the inclusion problem of micromechanics for polyhedral precipitates with purely dilatational misfit strains and discuss the precipitate shape dependence of the elastic strain energy caused by the anisotropy of the elastic moduli. The solution to this problem is necessary when the equilibrium shape of precipitates is discussed in a quantitative manner. When a cube–cube orientation relationship exists between the cubic precipitates and cubic matrix phase, the precipitates often show shapes similar to polyhedra composed of the following low-index planes, {1 0 0}, {1 1 0} and {1 1 1} [1]. We first show a simple equation giving various polyhedra composed of the {1 0 0}, {1 1 0} and {1 1 1} planes [3]. Based on the geometrical equation and the anisotropy of the elastic moduli for Cu, we calculate the elastic strain energy due to the purely dilatational misfit strains in the polyhedral precipitates. Finally, we present the precipitate-shape dependence of the elastic strain energy for various polyhedral precipitates composed of the {1 0 0}, {1 1 0} and {1 1 1} planes.
1359-6454/$30.00 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2006.08.048
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S. Onaka et al. / Acta Materialia 55 (2007) 669–673
2. Simple equation giving polyhedra with various shapes
where
2.1. Cube composed of {1 0 0} planes
GI ðc; c; cÞ ¼ jgðc; c; cÞj þ jgðc; c; cÞj þ jgðc; c; cÞj pffiffiffi p þ jgðc; c; cÞj ; ðc ¼ 1= 3Þ
p
Using the shapes of a between the value of p in p
x1–x2–x3 orthogonal coordinate system, the sphere, a cube and an intermediate shape two, are given by choosing an appropriate the following equation [2–5]: p
p
jx1 =Rj þ jx2 =Rj þ jx3 =Rj ¼ 1
ðR > 0; p P 2Þ:
ð1Þ
A sphere with radius R is given by Eq. (1) when p = 2. By considering the xi axes of a cubic crystal as Æ1 0 0æ directions, a cube with edges 2R composed of {1 0 0} planes is given by Eq. (1) when p ! 1 [3,5]. Fig. 1 shows the precipitate shapes given by (1) for (a) p = 2; (b) p = 4; (c) p = 20, and (d) the x1–x2 cross-sections of these shapes. The shapes given by Eq. (1) are similar to those of a cube when p is larger than about 20. Using the spherical coordinates (r, h, u), (1) is rewritten as [3]: rc ðh; uÞ ¼
R ½G0 ð1; 0; 0Þ1=p
ð2Þ
;
where p
p
G0 ð1; 0; 0Þ ¼ jgð1; 0; 0Þj þ jgð0; 1; 0Þj þ jgð0; 0; 1Þj
p
and gða; b; cÞ ¼ aðsin h cos uÞ þ bðsin h sin uÞ þ cðcos hÞ: Onaka has shown that the shapes of various convex polyhedra are given by equations similar to Eq. (2) [3]. 2.2. Polyhedra composed of {1 0 0}, {1 1 0} and {1 1 1} planes The shape of a polyhedron composed of {1 0 0}, {1 1 0} and {1 1 1} planes is given by the following equation when p ! 1 [3]: rðh; uÞ ¼
R ; pffiffiffi p pffiffiffi p 1=p ½G0 ð1; 0; 0Þ þ ð 3aÞ GI ðc; c; cÞ þ ð 2bÞ GII ðj; j; 0Þ ð3Þ
p
p
and GII ðj; j; 0Þ ¼ jgðj; j; 0Þjp þ jgðj; j; 0Þjp þ jgð0; j; jÞjp p
p
þ jgð0; j; jÞj þ jgðj; 0; jÞj pffiffiffi p þ jgðj; 0; jÞj ðj ¼ 1= 2Þ: The parameters a(>0) and b(>0) in the right-hand side of Eq. (3) determine the shape and area fractions of the {1 0 0}, {1 1 0} and {1 1 1} polyhedron interfaces, respectively. Fig. 2 reveals the changes in polyhedral shape given by Eq. (3) when p ! 1 [3], where the shapes of these polyhedra in each region are shown by insets. The region for a {1 0 0} cube with edges 2R is given when a 6 1/3 and b 6 1/2, while the region for a {1 1 1} octahedron occurs when 1 6 a and b 6 a. The (x1, x2, x3) coordinates of one of the vertices of the octahedron are given by (0, 0, R/a). The region for a {1 1 0} rhombic-dodecahedron is given when 1 6 b and 3a/2 6 b. The (x1, x2, x3) coordinates of one of the rhombic-dodecahedron vertices are given by (0, 0, R/b). Other regions represent the so-called truncated polyhedra composed of two or three kinds of low-index planes. For example, when b = 1/2, the shape change resulting from truncation of a cube to give an octahedron occurs with an increase in apfrom by ffiffiffi ffiffiffi 1/3 to 1. The shapepgiven Eq. (3) for a ¼ 1= 2 2 1 0:55 and b ¼ 1= 2 0:71 has six square {1 0 0} planes, 12 square {1 1 0} planes and eight equilateral-triangular {1 1 1} planes, corresponding to the shape of a (small) rhombicuboctahedron [6]. 3. Elastic strain energy due to misfit strains 3.1. Outline of calculation method The elastic strain energy due to misfit precipitates has been evaluated by solving micromechanical inclusion problems [7]. When a cube–cube orientation relationship exists between a misfit precipitate and the matrix phase, the misfit precipitate can be treated as an inclusion with uniform and purely dilatational eigenstrains eij ¼ e dij , where dij is Kronecker’s delta. When the inclusion is embedded in an
Fig. 1. The shapes given by Eq. (1) when: (a) p = 2; (b) p = 4; (c) p = 20 and (d) the x1 x2 cross-sections of these shapes.
S. Onaka et al. / Acta Materialia 55 (2007) 669–673 {110}
1.0
B
2.6
{110}-{111}
{100}-{110}
Φ
{100}{111}{110}
671
2.5
{111}
2.4
β 0.5
2
{100}-{111}
0
1/3 0.5 2/3
α
10 p
20
30 40
Fig. 3. The p dependence of U = E/{C44 (e*)2V} for changes in polyhedral shape from a sphere to a {1 0 0} cube, as given by Eq. (2).
A 0
5
1.0
Fig. 2. A map showing the variation in the shape of the {1 0 0}–{1 1 0}– {1 1 1} polyhedra given by Eq. (3) whenp ! 1.
elastically anisotropic material, the elastic strain energy depends on the inclusion shape even if the eigenstrains are purely dilatational [2,7]. We evaluate this shape dependence of the elastic strain energy caused by the anisotropy of the elastic moduli, where the shape of the inclusion changes as shown in Fig. 2. Numerical calculations are needed to evaluate the shape dependence of the elastic strain energy. When numerical calculations are performed, the single equation, Eq. (3), which gives various inclusion shapes, is a very useful tool [2]. Here, we assume an infinitely extended material containing an inclusion. The elastic moduli of the inclusion are assumed to be the same as those for the matrix. For materials with cubic structures, the elastic moduli are written with three independent values in the Voight notation: C11, C12 and C44. Using C11, C12 and C44, the well-known anisotropy ratio Ar is given by: Ar ¼ 2C 44 =ðC 11 C 12 Þ
ð4Þ
and the Poisson ratio m1 0 0 for the Æ1 0 0æ extension is given by: m100 ¼ C 44 =ðC 11 þ C 12 Þ: ð5Þ Both Ar and m1 0 0 are dimensionless. In the present paper, the elastic moduli of Cu with Ar = 3.2 and m1 0 0 = 0.42 [8] have been used to calculate the elastic strain energy. When Ar and m1 0 0 are fixed, the inclusion-shape dependence of the elastic strain energy E due to eij ¼ e dij is discussed with respect to variations in the dimensionless value U = E/{C44(e*)2V}, where V is the volume of an inclusion. Using previous results [9] to give derivatives of the elastic Green’s functions for cubic materials, it is possible to calculate U for inclusions of various shapes. Details of the calculation method used are shown in our previous paper [2]. 3.2. The p dependence of elastic strain energy We first consider the p dependence of U = E/ {C44(e*)2V}, where the change in U is caused by changes
in the inclusion shape from a sphere to a polyhedron. Fig. 3 shows the p dependence of U, corresponding to the change in shape from sphere to cube, as given by Eq. (2). As shown in Fig. 3, U decreases with the shape change to give a {1 0 0} cube [3]. Although U decreases rapidly at the initial increase in p from 2 upwards, the p dependence of U becomes smaller for larger values of p, as shown in Fig. 3. When p is larger than about 20, U approaches a certain limiting value for p ! 1. This dependence of p on U is common for various polyhedra shown in Fig. 2. We hence adopt the values of U for the polyhedral shapes given by Eq. (3) when p = 40. 3.3. The a and b dependences of elastic strain energy We have calculated U = E/{C44(e*)2V} for the various polyhedra shown in Fig. 2. Fig. 4 shows a three-dimensional (3D) representation revealing the a and b dependences of U. Three regions for the {1 0 0} cube, the {1 1 0} rhombic-dodecahedron and the {1 1 1} octahedron are indicated in Fig. 4 by broken lines and insets. The values of U for the representative polyhedra are:
2.8 2.6 2.4 1.0
β
Φ
{100}
2.2 0.5 0
1/3
0
0.5
2/3
1.0
α
Fig. 4. A 3D representation showing the a and b dependences of U.
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S. Onaka et al. / Acta Materialia 55 (2007) 669–673
U1 1 1 ¼ 2:67 for the f1 1 1g octahedron U1 1 0 ¼ 2:63 for the f1 1 0g rhombic-dodecahedron U1 0 0 ¼ 2:39 for the f1 0 0g cube: The values U111 = 2.67 and U1 0 0 = 2.39 correspond to the highest and lowest values of U for the various polyhedra shown in Fig. 2.
4. Discussion The horizontal line A in Fig. 2 shows the change in polyhedron shape by truncation between the {1 0 0} cube and the {1 1 1} octahedron. The vertical line B in Fig. 2, on the other hand, corresponds to the line between the {1 0 0} cube and the {1 1 0} rhombic-dodecahedron. The variations of U along lines A and B are shown by solid curves in Fig. 5a and b, respectively. The broken horizontal
line in Fig. 5a and b shows Usphere = 2.56 for a sphere. The shape changes along lines A and B are also shown graphically by insets in Fig. 5a and b. The variation of U = E/{C44(e*)2V} caused by the shape change along line A has been calculated by Sato and Johnson [10]. Converting their result to the present a dependence of U, the result by Sato and Johnson is indicated by the dotted curve in Fig. 5a. Overall, the present result is in good agreement with the result of Sato and Johnson [10]. However, the a dependence of U for the slightly truncated cube at around a 1/3 is different, as shown in Fig. 5a. The truncation of the polyhedron along line A in Fig. 2 causes the changes in the area S1 0 0 of the {1 0 0} interfaces, and the area S111 of the {1 1 1} interfaces of the polyhedron. Using the total area Stotal = S1 0 0 + S111, the shape change along line A is considered as a decrease in the ratio S1 0 0/Stotal from unity to zero, and the increase of the ratio S111/Stotal from zero to unity. Here, we consider Ug A given by: UgA ¼ U100 ðS 100 =S total Þ þ U111 ðS 111 =S total Þ
2.7
for the geometrically estimated U value of the truncated polyhedron between the {1 0 0} cube and the {1 1 1} octahedron. Using U111 = 2.67, U1 0 0 = 2.39, and recognising the purely geometrical factors (S1 0 0/Stotal) and (S111/Stotal) as functions of a, the variation of Ug A has been calculated as shown by the broken curve in Fig. 5a. From Fig. 5a, we find that Ug A reproduces well the actual variation of U = E/{C44(e*)2V}. For line B in Fig. 2, the truncation between the {1 0 0} cube and the {1 1 0} rhombic-dodecahedron, Ug B is given by:
a eq. (6)
2.6
S-J
Φ 2.5 2.4 0.4
0.6
0.8
1.0
1/3
2.7
b eq. (7)
2.6 Φ 2.5
2.4 0.5
0.6
0.7
0.8
0.9
ð6Þ
1.0
β Fig. 5. The variations of U = E/{C44 (e*)2V} caused by the truncation of polyhedra along (a) line A and (b) line B, which are shown by solid curves. The shape changes of these polyhedra are graphically shown by insets. The broken horizontal line corresponds to Usphere = 2.56 for a sphere. The dotted curve in (a) is the result reported by Sato and Johnson [10]. The broken curves in (a) and (b) show the values given by Eqs. (6) and (7).
UgB ¼ U100 ðS 100 =S total Þ þ U110 ðS 110 =S total Þ;
ð7Þ
where S110 is the area of the {1 1 0} interfaces, and Stotal in (7) is Stotal = S1 0 0 + S110. The variation of Ug B has been calculated as shown by the broken curve in Fig. 5b. We find that Ug B also approximately reproduces U on line B in Fig. 2. pffiffiffi For the rhombicuboctahedron given by a ¼ 1= 2 2 1 pffiffiffi 0:55 and b ¼ 1= 2 0:71, U = E/{C44(e*)2V} is obtained as U = 2.56 as a result of the micromechanical calculations. On the other hand, the geometrical method considering S1 0 0, S110 and S111 gives Ug = 2.57, which is very close to U = 2.56. We find that U for the truncated polyhedra of the{1 0 0} cube, the {1 1 0} rhombic-dodecahedron and the {1 1 1} octahedron are reproduced by U1 0 0, U110 and U111 with the geometrical method considering S1 0 0, S110 and S111. 5. Summary We provide a solution for the inclusion problem of micromechanics for polyhedral precipitates with purely dilatational misfit strains. By considering cubic structures
S. Onaka et al. / Acta Materialia 55 (2007) 669–673
for the precipitates and matrix phase, the elastic strain energies have been calculated for the precipitate shapes of a {1 0 0} cube, a {1 1 0} rhombic-dodecahedron, a {1 1 1} octahedron and the truncated shapes of all three polyhedra. Using the elastic moduli of Cu for the numerical calculations, the dependence of precipitate shape on the elastic strain energy caused by the anisotropy of the elastic moduli has been discussed. The polyhedral shapes of the {1 1 1} octahedron and the {1 0 0} cube result in the highest and lowest elastic strain energies per unit volume of precipitate. We have found that the calculated values of the elastic strain energy for truncated polyhedra are reproduced well by the values estimated using a geometrical method that considers the area changes of the {1 0 0}, {1 1 0} and {1 1 1} interfaces.
673
References [1] [2] [3] [4] [5]
[6] [7] [8] [9] [10]
Yazawa Y, Furuhara T, Maki T. Acta Mater 2004;52:3727. Onaka S, Kobayashi N, Fujii T, Kato M. Mat Sci Eng A 2003;A347:42. Onaka S. Phil Mag Let 2006;86:175. Onaka S. Phil Mag Let 2001;81:265. Jaklic A, Leonardis A, Solina F. Segmentation and recovery of superquadrics. Computational imaging and vision, 20. Dordrecht: Kluwer Academic; 2000. Cromwell PR. Polyhedra. Cambridge: Cambridge University Press; 1997. Mura T. Micromechanics of defects in solids. Dordrecht: Martinus Nijhoff; 1987. Hirth JP, Lothe J. Theory of dislocations. 2nd ed. Malabar: Krieger; 1992. p. 837. Barnett DM. Phys Stat Sol (b) 1972;49:741. Sato S, Johnson WC. Metall Trans A 1992;23A:2761.
Elastic strain energy due to misfit strains in a polyhedral precipitate composed of low-index planes Susumu Onaka
a,*
, Toshiyuki Fujii b, Masaharu Kato
a
a b
Department of Materials Science and Engineering, Tokyo Institute of Technology, 4259-J2-63 Nagatsuta, Yokohama 226-8502, Japan Department of Innovative and Engineered Materials, Tokyo Institute of Technology, 4259-J2-45 Nagatsuta, Yokohama 226-8502, Japan Received 16 June 2006; accepted 29 August 2006 Available online 15 November 2006
Abstract The dependence of precipitate shape on the elastic strain energy caused by anisotropy of the elastic moduli is discussed for polyhedral precipitates with purely dilatational misfit strains. By considering cubic structures for the precipitates and matrix phase, the elastic strain energy is calculated for the precipitate shapes of a {1 0 0} cube, a {1 1 0} rhombic-dodecahedron, a {1 1 1} octahedron, and the truncated shapes of all three polyhedra. The anisotropy of the elastic moduli of Cu is used for the numerical calculations. The polyhedral shapes of the {1 1 1} octahedron and {1 0 0} cube result in the highest and lowest values of the elastic strain energy per unit volume of precipitate. The calculated values of the elastic strain energy obtained for the truncated polyhedra are compared with values estimated using a geometrical method that considers the area changes of the {1 0 0}, {1 1 0} and {1 1 1} interfaces. 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Precipitates; Elastic strain energy; Misfit strains; Micromechanics
1. Introduction The shapes of precipitates in alloys are affected by various factors. Although some precipitate shapes obtained under certain conditions can be explained using kinetics, the stable shapes observed in alloys are best discussed with respect to energetics. These stable shapes sometimes show symmetry that reflects the symmetry of the alloy crystal structure. Polyhedral shapes composed of low-index planes are typical examples of such symmetrical shapes [1]. It is true that the anisotropy of the interface energy density between the precipitate and the matrix is one origin for the shapes of polyhedral precipitates. However, when precipitates have misfit strains, the resulting elastic strain energy is an important factor that affects the stability of the precipitate shape [2]. It is known that the anisotropy of the elastic moduli could cause a decrease in the elastic strain energy of precipitates following a change in shape *
Corresponding author. Tel. +81 45 924 5564; fax: +81 45 924 5566. E-mail address: [email protected] (S. Onaka).
from spherical to polyhedral [2]. In the present paper, we solve the inclusion problem of micromechanics for polyhedral precipitates with purely dilatational misfit strains and discuss the precipitate shape dependence of the elastic strain energy caused by the anisotropy of the elastic moduli. The solution to this problem is necessary when the equilibrium shape of precipitates is discussed in a quantitative manner. When a cube–cube orientation relationship exists between the cubic precipitates and cubic matrix phase, the precipitates often show shapes similar to polyhedra composed of the following low-index planes, {1 0 0}, {1 1 0} and {1 1 1} [1]. We first show a simple equation giving various polyhedra composed of the {1 0 0}, {1 1 0} and {1 1 1} planes [3]. Based on the geometrical equation and the anisotropy of the elastic moduli for Cu, we calculate the elastic strain energy due to the purely dilatational misfit strains in the polyhedral precipitates. Finally, we present the precipitate-shape dependence of the elastic strain energy for various polyhedral precipitates composed of the {1 0 0}, {1 1 0} and {1 1 1} planes.
1359-6454/$30.00 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2006.08.048
670
S. Onaka et al. / Acta Materialia 55 (2007) 669–673
2. Simple equation giving polyhedra with various shapes
where
2.1. Cube composed of {1 0 0} planes
GI ðc; c; cÞ ¼ jgðc; c; cÞj þ jgðc; c; cÞj þ jgðc; c; cÞj pffiffiffi p þ jgðc; c; cÞj ; ðc ¼ 1= 3Þ
p
Using the shapes of a between the value of p in p
x1–x2–x3 orthogonal coordinate system, the sphere, a cube and an intermediate shape two, are given by choosing an appropriate the following equation [2–5]: p
p
jx1 =Rj þ jx2 =Rj þ jx3 =Rj ¼ 1
ðR > 0; p P 2Þ:
ð1Þ
A sphere with radius R is given by Eq. (1) when p = 2. By considering the xi axes of a cubic crystal as Æ1 0 0æ directions, a cube with edges 2R composed of {1 0 0} planes is given by Eq. (1) when p ! 1 [3,5]. Fig. 1 shows the precipitate shapes given by (1) for (a) p = 2; (b) p = 4; (c) p = 20, and (d) the x1–x2 cross-sections of these shapes. The shapes given by Eq. (1) are similar to those of a cube when p is larger than about 20. Using the spherical coordinates (r, h, u), (1) is rewritten as [3]: rc ðh; uÞ ¼
R ½G0 ð1; 0; 0Þ1=p
ð2Þ
;
where p
p
G0 ð1; 0; 0Þ ¼ jgð1; 0; 0Þj þ jgð0; 1; 0Þj þ jgð0; 0; 1Þj
p
and gða; b; cÞ ¼ aðsin h cos uÞ þ bðsin h sin uÞ þ cðcos hÞ: Onaka has shown that the shapes of various convex polyhedra are given by equations similar to Eq. (2) [3]. 2.2. Polyhedra composed of {1 0 0}, {1 1 0} and {1 1 1} planes The shape of a polyhedron composed of {1 0 0}, {1 1 0} and {1 1 1} planes is given by the following equation when p ! 1 [3]: rðh; uÞ ¼
R ; pffiffiffi p pffiffiffi p 1=p ½G0 ð1; 0; 0Þ þ ð 3aÞ GI ðc; c; cÞ þ ð 2bÞ GII ðj; j; 0Þ ð3Þ
p
p
and GII ðj; j; 0Þ ¼ jgðj; j; 0Þjp þ jgðj; j; 0Þjp þ jgð0; j; jÞjp p
p
þ jgð0; j; jÞj þ jgðj; 0; jÞj pffiffiffi p þ jgðj; 0; jÞj ðj ¼ 1= 2Þ: The parameters a(>0) and b(>0) in the right-hand side of Eq. (3) determine the shape and area fractions of the {1 0 0}, {1 1 0} and {1 1 1} polyhedron interfaces, respectively. Fig. 2 reveals the changes in polyhedral shape given by Eq. (3) when p ! 1 [3], where the shapes of these polyhedra in each region are shown by insets. The region for a {1 0 0} cube with edges 2R is given when a 6 1/3 and b 6 1/2, while the region for a {1 1 1} octahedron occurs when 1 6 a and b 6 a. The (x1, x2, x3) coordinates of one of the vertices of the octahedron are given by (0, 0, R/a). The region for a {1 1 0} rhombic-dodecahedron is given when 1 6 b and 3a/2 6 b. The (x1, x2, x3) coordinates of one of the rhombic-dodecahedron vertices are given by (0, 0, R/b). Other regions represent the so-called truncated polyhedra composed of two or three kinds of low-index planes. For example, when b = 1/2, the shape change resulting from truncation of a cube to give an octahedron occurs with an increase in apfrom by ffiffiffi ffiffiffi 1/3 to 1. The shapepgiven Eq. (3) for a ¼ 1= 2 2 1 0:55 and b ¼ 1= 2 0:71 has six square {1 0 0} planes, 12 square {1 1 0} planes and eight equilateral-triangular {1 1 1} planes, corresponding to the shape of a (small) rhombicuboctahedron [6]. 3. Elastic strain energy due to misfit strains 3.1. Outline of calculation method The elastic strain energy due to misfit precipitates has been evaluated by solving micromechanical inclusion problems [7]. When a cube–cube orientation relationship exists between a misfit precipitate and the matrix phase, the misfit precipitate can be treated as an inclusion with uniform and purely dilatational eigenstrains eij ¼ e dij , where dij is Kronecker’s delta. When the inclusion is embedded in an
Fig. 1. The shapes given by Eq. (1) when: (a) p = 2; (b) p = 4; (c) p = 20 and (d) the x1 x2 cross-sections of these shapes.
S. Onaka et al. / Acta Materialia 55 (2007) 669–673 {110}
1.0
B
2.6
{110}-{111}
{100}-{110}
Φ
{100}{111}{110}
671
2.5
{111}
2.4
β 0.5
2
{100}-{111}
0
1/3 0.5 2/3
α
10 p
20
30 40
Fig. 3. The p dependence of U = E/{C44 (e*)2V} for changes in polyhedral shape from a sphere to a {1 0 0} cube, as given by Eq. (2).
A 0
5
1.0
Fig. 2. A map showing the variation in the shape of the {1 0 0}–{1 1 0}– {1 1 1} polyhedra given by Eq. (3) whenp ! 1.
elastically anisotropic material, the elastic strain energy depends on the inclusion shape even if the eigenstrains are purely dilatational [2,7]. We evaluate this shape dependence of the elastic strain energy caused by the anisotropy of the elastic moduli, where the shape of the inclusion changes as shown in Fig. 2. Numerical calculations are needed to evaluate the shape dependence of the elastic strain energy. When numerical calculations are performed, the single equation, Eq. (3), which gives various inclusion shapes, is a very useful tool [2]. Here, we assume an infinitely extended material containing an inclusion. The elastic moduli of the inclusion are assumed to be the same as those for the matrix. For materials with cubic structures, the elastic moduli are written with three independent values in the Voight notation: C11, C12 and C44. Using C11, C12 and C44, the well-known anisotropy ratio Ar is given by: Ar ¼ 2C 44 =ðC 11 C 12 Þ
ð4Þ
and the Poisson ratio m1 0 0 for the Æ1 0 0æ extension is given by: m100 ¼ C 44 =ðC 11 þ C 12 Þ: ð5Þ Both Ar and m1 0 0 are dimensionless. In the present paper, the elastic moduli of Cu with Ar = 3.2 and m1 0 0 = 0.42 [8] have been used to calculate the elastic strain energy. When Ar and m1 0 0 are fixed, the inclusion-shape dependence of the elastic strain energy E due to eij ¼ e dij is discussed with respect to variations in the dimensionless value U = E/{C44(e*)2V}, where V is the volume of an inclusion. Using previous results [9] to give derivatives of the elastic Green’s functions for cubic materials, it is possible to calculate U for inclusions of various shapes. Details of the calculation method used are shown in our previous paper [2]. 3.2. The p dependence of elastic strain energy We first consider the p dependence of U = E/ {C44(e*)2V}, where the change in U is caused by changes
in the inclusion shape from a sphere to a polyhedron. Fig. 3 shows the p dependence of U, corresponding to the change in shape from sphere to cube, as given by Eq. (2). As shown in Fig. 3, U decreases with the shape change to give a {1 0 0} cube [3]. Although U decreases rapidly at the initial increase in p from 2 upwards, the p dependence of U becomes smaller for larger values of p, as shown in Fig. 3. When p is larger than about 20, U approaches a certain limiting value for p ! 1. This dependence of p on U is common for various polyhedra shown in Fig. 2. We hence adopt the values of U for the polyhedral shapes given by Eq. (3) when p = 40. 3.3. The a and b dependences of elastic strain energy We have calculated U = E/{C44(e*)2V} for the various polyhedra shown in Fig. 2. Fig. 4 shows a three-dimensional (3D) representation revealing the a and b dependences of U. Three regions for the {1 0 0} cube, the {1 1 0} rhombic-dodecahedron and the {1 1 1} octahedron are indicated in Fig. 4 by broken lines and insets. The values of U for the representative polyhedra are:
2.8 2.6 2.4 1.0
β
Φ
{100}
2.2 0.5 0
1/3
0
0.5
2/3
1.0
α
Fig. 4. A 3D representation showing the a and b dependences of U.
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U1 1 1 ¼ 2:67 for the f1 1 1g octahedron U1 1 0 ¼ 2:63 for the f1 1 0g rhombic-dodecahedron U1 0 0 ¼ 2:39 for the f1 0 0g cube: The values U111 = 2.67 and U1 0 0 = 2.39 correspond to the highest and lowest values of U for the various polyhedra shown in Fig. 2.
4. Discussion The horizontal line A in Fig. 2 shows the change in polyhedron shape by truncation between the {1 0 0} cube and the {1 1 1} octahedron. The vertical line B in Fig. 2, on the other hand, corresponds to the line between the {1 0 0} cube and the {1 1 0} rhombic-dodecahedron. The variations of U along lines A and B are shown by solid curves in Fig. 5a and b, respectively. The broken horizontal
line in Fig. 5a and b shows Usphere = 2.56 for a sphere. The shape changes along lines A and B are also shown graphically by insets in Fig. 5a and b. The variation of U = E/{C44(e*)2V} caused by the shape change along line A has been calculated by Sato and Johnson [10]. Converting their result to the present a dependence of U, the result by Sato and Johnson is indicated by the dotted curve in Fig. 5a. Overall, the present result is in good agreement with the result of Sato and Johnson [10]. However, the a dependence of U for the slightly truncated cube at around a 1/3 is different, as shown in Fig. 5a. The truncation of the polyhedron along line A in Fig. 2 causes the changes in the area S1 0 0 of the {1 0 0} interfaces, and the area S111 of the {1 1 1} interfaces of the polyhedron. Using the total area Stotal = S1 0 0 + S111, the shape change along line A is considered as a decrease in the ratio S1 0 0/Stotal from unity to zero, and the increase of the ratio S111/Stotal from zero to unity. Here, we consider Ug A given by: UgA ¼ U100 ðS 100 =S total Þ þ U111 ðS 111 =S total Þ
2.7
for the geometrically estimated U value of the truncated polyhedron between the {1 0 0} cube and the {1 1 1} octahedron. Using U111 = 2.67, U1 0 0 = 2.39, and recognising the purely geometrical factors (S1 0 0/Stotal) and (S111/Stotal) as functions of a, the variation of Ug A has been calculated as shown by the broken curve in Fig. 5a. From Fig. 5a, we find that Ug A reproduces well the actual variation of U = E/{C44(e*)2V}. For line B in Fig. 2, the truncation between the {1 0 0} cube and the {1 1 0} rhombic-dodecahedron, Ug B is given by:
a eq. (6)
2.6
S-J
Φ 2.5 2.4 0.4
0.6
0.8
1.0
1/3
2.7
b eq. (7)
2.6 Φ 2.5
2.4 0.5
0.6
0.7
0.8
0.9
ð6Þ
1.0
β Fig. 5. The variations of U = E/{C44 (e*)2V} caused by the truncation of polyhedra along (a) line A and (b) line B, which are shown by solid curves. The shape changes of these polyhedra are graphically shown by insets. The broken horizontal line corresponds to Usphere = 2.56 for a sphere. The dotted curve in (a) is the result reported by Sato and Johnson [10]. The broken curves in (a) and (b) show the values given by Eqs. (6) and (7).
UgB ¼ U100 ðS 100 =S total Þ þ U110 ðS 110 =S total Þ;
ð7Þ
where S110 is the area of the {1 1 0} interfaces, and Stotal in (7) is Stotal = S1 0 0 + S110. The variation of Ug B has been calculated as shown by the broken curve in Fig. 5b. We find that Ug B also approximately reproduces U on line B in Fig. 2. pffiffiffi For the rhombicuboctahedron given by a ¼ 1= 2 2 1 pffiffiffi 0:55 and b ¼ 1= 2 0:71, U = E/{C44(e*)2V} is obtained as U = 2.56 as a result of the micromechanical calculations. On the other hand, the geometrical method considering S1 0 0, S110 and S111 gives Ug = 2.57, which is very close to U = 2.56. We find that U for the truncated polyhedra of the{1 0 0} cube, the {1 1 0} rhombic-dodecahedron and the {1 1 1} octahedron are reproduced by U1 0 0, U110 and U111 with the geometrical method considering S1 0 0, S110 and S111. 5. Summary We provide a solution for the inclusion problem of micromechanics for polyhedral precipitates with purely dilatational misfit strains. By considering cubic structures
S. Onaka et al. / Acta Materialia 55 (2007) 669–673
for the precipitates and matrix phase, the elastic strain energies have been calculated for the precipitate shapes of a {1 0 0} cube, a {1 1 0} rhombic-dodecahedron, a {1 1 1} octahedron and the truncated shapes of all three polyhedra. Using the elastic moduli of Cu for the numerical calculations, the dependence of precipitate shape on the elastic strain energy caused by the anisotropy of the elastic moduli has been discussed. The polyhedral shapes of the {1 1 1} octahedron and the {1 0 0} cube result in the highest and lowest elastic strain energies per unit volume of precipitate. We have found that the calculated values of the elastic strain energy for truncated polyhedra are reproduced well by the values estimated using a geometrical method that considers the area changes of the {1 0 0}, {1 1 0} and {1 1 1} interfaces.
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