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Void Shape Effects on Void Growth and Slip Systems Activity in Single Crystals
Rare Metal Materials and Engineering Volume 43, Issue 7, July 2014 Online English edition of the Chinese language journal Cite this article as: Rare Metal Materials and Engineering, 2014, 43(7): 1571-1576.
ARTICLE
Void Shape Effects on Void Growth and Slip Systems Activity in Single Crystals Sun Wanchao,
Lu Shan
Northwestern Polytechnical University, Xi’an 710072, China
Abstract: Cast void shape effects on void growth and slip system activity in single crystals were studied using crystal plasticity under various orientations of the crystalline lattice. A 3D unit cell with ellipsoidal void was set up using three-dimensional 12 potentially active slip systems; the spherical shape of void which is a special case of ellipsoid was also included. The numerical results show that the initial texture orientation, the ellipsoidal coordinate, the load coordinate system and the shape of void have a competitive effect on the evolution of voids. For triaxial tension conditions, the void fraction increase under the applied load is strongly dependent on the shape of void and the crystallographic orientation with respect to the load axis, as well as the activities on all the slip systems. When the symmetry of the unit cell is broken, the void experiences a rotation in spite of the load applied along <001> and <011> orientations with symmetry of the slip systems. An interesting feature is that, even in the case of anisotropic crystalline matrix materials, the overall effect of plastic anisotropy on damage evolution is diminished during non-spherical void growth. Key words: void growth; shape effect; single crystal; orthotropic; crystalline orientation; slip system
The ductile fracture occurring through the nucleation, growth and coalescence of voids, is a primary mode of material failure, so it is important to predict the condition of void growth and coalescence when a material is subjected to large plastic deformation. Analytical and computational investigations[1-12] were conducted to characterize and understand void growth and flow localization. Essentially all of these analytical and computational investigations were based on phenomenological inelastic constitutive relations such as J2 plasticity or Gurson’s[4] constitutive formulation, but there still exists much deficiency in this model [13-15]. Crystal plasticity model is a hot topic in the damage mechanics area, and most of the microstructures, such as inclusions, grain boundary and crystallographic slip can be taken into account. Most recently, based on this theory, the interrelated effects of porosity, initial void size, hydrostatic stress, geometrical softening, void distribution, and work-hardening rate on void growth, failure paths, ligament damage and coalescence behavior in face centre cubic (fcc) have been investigated [16-20], which were
implemented with the 2D circle-void unit cell or 3D model with spherical voids. Actually the voids which nucleate from second-phase particles have highly irregular shape, so it is necessary to address 3D model with non-spherical voids, and computational evaluations of this nature are valuable because they provide an approach to quantitatively assess micro-macro relations, and to illustrate the relation of void growth with void shape directly. In this paper, the rate-dependent crystal plasticity theory[12] is applied to investigate void growth in fcc single crystals. To illustrate the behavior of void growth for different shapes, the 3D unit cell including an ellipsoidal void was considered here. By changing the ratio of three axes for ellipsoid, three voids with different shapes were investigated.
1 Geometrical and Boundary Condition 1.1
Unit cell model
The ellipsoid was used to describe the shape of void (Fig.1). Assuming that the void nucleation has already occurred, a single ellipsoidal void was embedded in the unit
Received date: July 15, 2013 Corresponding author: Sun Wanchao, Ph. D., School of Power and Energy, Northwestern Polytechnical University, Xi’an 710072, P. R. China, E-mail: [email protected] Copyright © 2014, Northwest Institute for Nonferrous Metal Research. Published by Elsevier BV. All rights reserved.
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Sun Wanchao et al. / Rare Metal Materials and Engineering, 2014, 43(7): 1571-1576
cell which is a cubic with one side length L, as shown in Fig.2. Such a unit cell model which is a representative volume element of uniformly distributed voids in a medium can effectively represent the interaction effects between voids[16,17]. The finite element meshes used in the simulations are presented in Fig.2. In the case of the one-void unit cell, the initial void volume fraction fo is as follows:
f0 =
4 π r1 r2 r3 3 L3
R3
A R1: R2: R3=1:1:1
(1)
R1
1.3
Coordinate systems
The coordinate systems used in the simulations are presented in Fig.3. The x-y-z represents the load coordinate system, and it is the coordinate system where unit cell edge is located; <001>-<010>-<100> is the material coordinate system, where α0, β0, γ0 are the transform angles between material coordinate system and load coordinate system; R1-R2-R3 is the coordinate system where the three axis of ellipsoidal void are located, while α, β and γ are the transform angles between the ellipsoidal coordinate system and the load coordinate system.
2
R1
Fig.1 Y
2.1 Void growth 2.1.1 Void growth in <001> crystal orientation Fig.4 indicates that in the case of A-1 the peak normalized void volume fraction reaches 1.059 for the corresponding εx=0.1, while those of for B-1, B-2, B-3 and B-4 are 1.047 1.040 1.359 and 1.309, respectively. As shown in Fig.4, the growth rate of A-1 is slow, which is consistent with the previous result [17]. A more rapid increase occurs in the case of B-3 and B-4. It's worth noting that in the case of B-1 and B-2,
Geometrical of ellipsoidal void a
L aδ bδ δ
δ bδ Z
b
L L
X
aδ
Fig.2
Boundary conditions (a) and finite element mesh (b) a
z
z
010
R2
β0 α0
β γ0
001
x
y α
γ
R1
y
x N
Fig.3
b
R3
100
Results and Discussion
In this section, the results from the finite element simulations relating to the ellipsoidal void effects on the void growth are analyzed. The strain controlled boundary conditions were employed in this study and the load triaxiality factor a=b=–0.235, as shown in Fig.3. The initial void volume fraction is f0=0.01 and L=50 µm. The interrelated combination of crystal orientations, ellipsoidal shape and ellipsoid orientations for the analysis and corresponding symbols in this study are listed in Table 1.
R3 R2 C R1: R2: R3=5:3:3
Boundary condition
For all orientation, all unit cells were subjected to the same triaxial strain field by applying a proportional displacement along the three axes x, y and z. The applied displacement in the x direction is δ, while in the y and z direction, it is aδ and bδ, respectively, as shown in Fig.3. Similar boundary conditions were also considered by Shu[21].
R3 R2 B R1: R2: R3=5:5:3
Where r1, r2, r3 are the half-axis length of three axes elliposoidal sphere respectively. f is the void volume fraction calculated during solution and R is the normalized void volume fraction (2) R = f f0
1.2
R2
R1
N
Material coordinate (a) and ellipsoidal coordinate (b) vs. load coordinate system
a lower increase occurs than A-1, but the difference is minor. Fig.5 indicates that corresponding to C-1, C-2, C-3 and C-4, the peak normalized void volume fractions reach 1.121 1.113 1.345 and 1.233, respectively, for the corresponding εx=0.1. Compared with A-1, the void tends to grow easily in the cases of C-1 and C-2, especially in the cases of C-3 and C-4.
2.1.2
Void growth in <011> crystal orientation
Ellipsoidal shape B shows a relatively similar distribution between <011> and <001> crystal orientation, as shown in Fig.4 and Fig.6. In Fig.6, the peak normalized void volume fractions of shapes A-2, B-5, B-6, B-7, and B-8 reach 1.336 1.056 1.036 1.438 and 1.379, respectively,
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Sun Wanchao et al. / Rare Metal Materials and Engineering, 2014, 43(7): 1571-1576
Interrelated combination of crystal orientations, ellipsoidal shape and ellipsoid orientations for the analysis and cor
Normalized Void Volume Fraction
responding symbols 001 (α0/β0/γ0=0/0/0) α/β/γ Shape A Shape B Shape C 0/0/0 A-1 B-1 C-1 0/90/0 / B-2 C-2 0/30.96/0 / B-3 C-3 0/45/22.99 / B-4 C-4
Fig.4
1.4
011 (α0/β0/γ0=45/0/0) Shape A Shape B Shape C A-2 B-5 C-5 / B-6 C-6 / B-7 C-7 / B-8 C-8 Normalized Void Volume Fraction
Table 1
(α0 /β0 /γ0 =0/0/0)
1.3
A-1 B-1 B-2 B-3 B-4
1.2 1.1 1.0
0
0.02
0.04 0.06 0.08 Applied Strain
0.10
Normalized void volume fraction evolution with the ap-
Fig.5
Fig.7 shows the evolution of the normalized void volume fractions for five states proposed for study. The largest variation in void volume fraction is experienced by the C-7, with an increase of 1.384 for a total applied strain of 0.1. C-5 and C-6 indicate approximately the same increase in void fraction of approximately 1.15 times as much as the initial void volume, while C-8 reaches 1.279.
Void growth in <111> crystal orientation
Fig.6
Fig.8 indicates that in the case of A-3 the peak normalized void volume fraction reaches 1.326 for the corresponding εx=0.1 while those of B-9 B-10 B-11 and B-12 are 1.067 1.050 1.482 and 1.401 respectively. Fig.9 indicates that corresponding to C-9 C-10 C-11 and C-12, the peak normalized void volume fractions reach 1.171 1.172 1.415 and 1.301, respectively, for the corresponding εx=0.1.
A-1 C-1 C-2 C-3 C-4
1.2 1.1 1.0
0
0.02
0.04 0.06 0.08 Applied Strain
0.10
1.5 (α0 /β0 /γ0 =45/0/0)
1.4 A-2 B-5 B-6 B-7 B-8
1.3 1.2 1.1 1.0
0
0.02
0.04 0.06 0.08 Applied Strain
0.10
plied strain: A vs. B
Discussion
From Fig.5 to Fig.9, the difference in the normalized void volume fraction in the case of the A is significantly increased from <001> to <011> and <111> crystal orientation. The relative influences of lattice orientation on void growth is similar to the results found by Sangyul Ha[22] and W H Liu[23]. In the present work, much larger normalized void growth has been observed for B and C corresponding α/β/γ=0/30.96/0 and α/β/γ=0/45/22.9 than α/β/γ=0/0/0 and α/β/γ=0/90/0. However, an interesting feature is that, al
1.3
Normalized void volume fraction evolution with the ap-
Normalized Void Volume Fraction
2.1.4
(α0 /β0 /γ0 =0/0/0)
plied strain: A vs. C
for the corresponding εx=0.1. When the strain is applied along <011> crystal orientation, A-2, B-5, B-7, and B-8 tend to grow more easily compared to the strain applied along <001> crystal orientation.
2.1.3
1.4
Normalized void volume fraction evolution with the ap-
Normalized Void Volume Fraction
plied strain: A vs. B
111(α0/β0/γ0=45/35.26/0) Shape A Shape B Shape C A-3 B-9 C-9 / B-10 C-10 / B-11 C-11 / B-12 C-12
Fig.7
1.4
(α0 /β0 /γ0 =45/0/0)
1.3
A-2 C-5 C-6 C-7 C-8
1.2 1.1 1.0
0
0.02
0.04 0.06 0.08 Applied Strain
0.10
Normalized void volume fraction evolution with the applied strain: A vs. C
though it increases from <001> to <011> and <111> crystal orientation in the case of the B and C apart from B-6, the
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Sun Wanchao et al. / Rare Metal Materials and Engineering, 2014, 43(7): 1571-1576
Fig.8
Table 2
1.5 (α0 /β0 /γ0 =45/35.26/0)
1.4 A-3 B-9 B-10 B-11 B-12
1.3 1.2
A
B
1.1 1.0
0
0.02
0.04 0.06 0.08 Applied Strain
0.10
Normalized void volume fraction evolution with the ap-
Normalized Void Volume Fraction
plied strain: A vs. B
Fig.9
1.5
C (α0 /β0 /γ0 =45/35.26/0)
1.4
1.2 1.1
z x y
0
0.02
0.04 0.06 0.08 Applied Strain
z y x
0.10
Normalized void volume fraction evolution with the applied strain: A vs. C
normalized void volume fractions do not increase so rapidly as in the case of A. This implies that the overall effect of plastic anisotropy on damage evolution is diminished during non-spherical void growth. To better interpret the effects of the texture orientation on different shapes of void growth, a comparison between A, B and C for each orientation is tabulated in Table 2.
2.2
between A, B and C for each orientation <001> <011> <111> A-1 A-2 Increase A-3 Increase 1.059 1.336 26.16% 1.326 25.21% B-1 B-5 B-9 1.047 1.056 0.86% 1.067 1.9% B-2 B-6 B-10 1.040 1.036 -0.48% 1.050 0.96% B-3 B-7 B-11 1.359 1.438 5.81% 1.482 9.05% B-4 B-8 B-12 1.309 1.379 5.35% 1.401 7.03% C-1 C-5 C-9 1.121 1.166 4.01% 1.171 4.46% C-2 C-6 C-10 1.113 1.157 3.95% 1.172 5.30% C-3 C-7 C-11 1.345 1.384 2.90% 1.415 5.20% C-4 C-8 C-12 1.233 1.279 2.73% 1.301 5.52%
A-3 C-9 C-10 C-11 C-12
1.3
1.0
Comparisons of normalized void volume fraction
z x
Fig.10
Contour plots of equivalent strain for A-1 in orientation <001> with strain εx=0.1
z x y
Lattice rotation
Fig.10 (Fig.11) indicates that in the case of A-1 (A-2) in orientation <001> (<011>) the peak local equivalent strain reaches a value of 0.584 (0.613) for the corresponding remote strains of 0.1. The distribution of the equivalent strain is relatively symmetrical, and consequently, no void or unit cell distortion and rotation are recorded. The absent deformation of B-1, B-2, B-5, B-6, C-1, C-2, C-5 and C-6 are similar to A-1 and A-2, due to the fact that the shape of void and its distribution do not change the symmetry of the unit cell. Fig.12 indicates the void experience rotation in spite of the load applied along <001> and <011> orientations with symmetry of the slip systems in the cases of B-3, B-4, B-7, B-8, C-3, C-4, C-7 and C-8. The transform angles (α, β and γ) between ellipsoidal coordinate and the load coordinate break the symmetry of the unit cell, as well as the symmetry of the slip systems. It is noticed that the peak local equivalent strain
y
z
z y x
y
x
Fig.11
Contour plots of equivalent strain for A-2 in orientation <011> with strain εx=0.1
at the same load strain increases according to the order of B-4, C-3, B-8, C-4, C-7, C-8, B-3 and B-7. Nemat-Nasser et al.[24] and Solanki et al.[25], used both finite element and atomistic methods to revealed that the plastic flow localization induced corner formation in the void shape along with crack formation and growth. In the present work, the same phenomenon is observed. The void which has an ellipsoidal
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z
y
y
x
z
y
x
y
z
x
z
z
y x
z
x
y
z
x
x y
z
y
Fig.13
Octahedral {111} planes and <110> family of slip directions for a single crystal
x
Table 3 Fig.12
Contour plots of equivalent strain for B and C in orientation <001> and <011> with strain ε x = 0 . 1 A
shape initially, develops an irregular shape, and induces some corners where the dislocations will be initiated, which is more obvious at larger deformation.
2.3
Slip systems activity
The four octahedral {111} planes and the twelve corresponding <110> slip directions (three on each plane) are shown in Fig.13. The direction of load application with respect to the crystal orientation (<001>-<010>-<100> axes in Fig.13) determines the resolved shear stress for each of the {111}<110> slip plane/slip direction sets (or slip systems). To better interpret the combinational effects of the initial texture orientation, the ellipsoidal coordinate, the load coordinate system and the shape of void on the activated slip systems, a comparison for each orientation is tabulated in Table 3. It is noticed that the activated slip systems is mainly determined by the initial texture orientation, the ellipsoidal coordinate, the load coordinate system and the shape of void. In spite of the fact that the activated slip systems of A-1, A-2, B-1, B-2, B-5, B-6, C-1, C-2, C-5 and C-6 are very different from each other, the symmetry of the activated slip systems are not changed. This is the reason why there is no void or unit cell distortion and rotation to be recorded, as pointed out in section 2.2. However, in the
B
C
Activated slip system for different shapes and
orientations <001> A-1 3,-4,9,-10 B-1 3,-4,9,-10 B-2 -1,3,-4,6,-7,9,-10,12 B-3 -8,9,-10,11 B-4 5,-8 C-1 3,-4,9,-10 C-2 1,6,7,12 C-3 -2,5 C-4 -8
<011> A-2 -4,9 B-5 -1,3,-7,9 B-6 -1,3,-4,6,-7, 9,-10,12 B-7 -1,3 B-8 -6 C-5 -1,3,-7,9 C-6 4,-6,10,-12 C-7 -7,9 C-8 -1,3
<111> A-3 2 B-9 1,2 B-10 5,8 B-11 1,2 B-12 4,7 C-9 1,2 C-10 6,9 C-11 1,2 C-12 3
case of all remains in Table 3, there is no symmetry in the activated slip systems.
3 Conclusions 1) For triaxial tension conditions, the void fraction increase under the applied load is strongly dependent on the shape of void and the crystallographic orientation with respect to the load axis. 2) Even in the case of anisotropic crystalline matrix materials, the overall effect of plastic anisotropy on damage
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evolution is noteworthy diminished during non-spherical void growth, but differences are still present. 3) The transform angles (α, β and γ) between ellipsoidal coordinate system and the load coordinate system break the symmetry of the unit cell, as well as the symmetry of the slip systems. When the symmetry of the unit cell is broken, the void experiences rotation in spite of the load applied along <001> and <011> orientations with symmetry of the slip systems. 4) The activities on all the slip systems are influenced by the combination of the tensile axis, crystal orientation and different shapes of void.
10
1987, 35: 577 11
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Saje M, Pan J, Needleman A. International Journal of Fracture [J], 1982, 19: 163
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Aoki S, Kishimoto K, Takeya A et al. International Journal of Fracture[J], 1984, 24: 267
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Needleman A, Kushner A S. European Journal of Mechanics [J], 1990, 9: 193
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Potirniche G P, Hearndon J L, Horstemeyer M F et al. International Journal of Plasticity [J], 2006, 22: 921
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Biswasa P, Narasimhanb R, Tewaria A. Materials Science and Engineering A [J], 2011, 528: 823
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Borg Ulrik, Niordson Christian F et al. International Journal of Plasticity[J], 2008, 24: 688
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Javier Segurado, Javier Llorca. Acta Materialia[J], 2009, 57:
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Liu W H, Zhang X M, Tang J G et al. Computational Materials Science[J], 2007, 40:130
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25 Solanki K, Horstemeyer M F, Baskes M I et al. Mechanics of Materials[J], 2005, 37: 317
1576
ARTICLE
Void Shape Effects on Void Growth and Slip Systems Activity in Single Crystals Sun Wanchao,
Lu Shan
Northwestern Polytechnical University, Xi’an 710072, China
Abstract: Cast void shape effects on void growth and slip system activity in single crystals were studied using crystal plasticity under various orientations of the crystalline lattice. A 3D unit cell with ellipsoidal void was set up using three-dimensional 12 potentially active slip systems; the spherical shape of void which is a special case of ellipsoid was also included. The numerical results show that the initial texture orientation, the ellipsoidal coordinate, the load coordinate system and the shape of void have a competitive effect on the evolution of voids. For triaxial tension conditions, the void fraction increase under the applied load is strongly dependent on the shape of void and the crystallographic orientation with respect to the load axis, as well as the activities on all the slip systems. When the symmetry of the unit cell is broken, the void experiences a rotation in spite of the load applied along <001> and <011> orientations with symmetry of the slip systems. An interesting feature is that, even in the case of anisotropic crystalline matrix materials, the overall effect of plastic anisotropy on damage evolution is diminished during non-spherical void growth. Key words: void growth; shape effect; single crystal; orthotropic; crystalline orientation; slip system
The ductile fracture occurring through the nucleation, growth and coalescence of voids, is a primary mode of material failure, so it is important to predict the condition of void growth and coalescence when a material is subjected to large plastic deformation. Analytical and computational investigations[1-12] were conducted to characterize and understand void growth and flow localization. Essentially all of these analytical and computational investigations were based on phenomenological inelastic constitutive relations such as J2 plasticity or Gurson’s[4] constitutive formulation, but there still exists much deficiency in this model [13-15]. Crystal plasticity model is a hot topic in the damage mechanics area, and most of the microstructures, such as inclusions, grain boundary and crystallographic slip can be taken into account. Most recently, based on this theory, the interrelated effects of porosity, initial void size, hydrostatic stress, geometrical softening, void distribution, and work-hardening rate on void growth, failure paths, ligament damage and coalescence behavior in face centre cubic (fcc) have been investigated [16-20], which were
implemented with the 2D circle-void unit cell or 3D model with spherical voids. Actually the voids which nucleate from second-phase particles have highly irregular shape, so it is necessary to address 3D model with non-spherical voids, and computational evaluations of this nature are valuable because they provide an approach to quantitatively assess micro-macro relations, and to illustrate the relation of void growth with void shape directly. In this paper, the rate-dependent crystal plasticity theory[12] is applied to investigate void growth in fcc single crystals. To illustrate the behavior of void growth for different shapes, the 3D unit cell including an ellipsoidal void was considered here. By changing the ratio of three axes for ellipsoid, three voids with different shapes were investigated.
1 Geometrical and Boundary Condition 1.1
Unit cell model
The ellipsoid was used to describe the shape of void (Fig.1). Assuming that the void nucleation has already occurred, a single ellipsoidal void was embedded in the unit
Received date: July 15, 2013 Corresponding author: Sun Wanchao, Ph. D., School of Power and Energy, Northwestern Polytechnical University, Xi’an 710072, P. R. China, E-mail: [email protected] Copyright © 2014, Northwest Institute for Nonferrous Metal Research. Published by Elsevier BV. All rights reserved.
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Sun Wanchao et al. / Rare Metal Materials and Engineering, 2014, 43(7): 1571-1576
cell which is a cubic with one side length L, as shown in Fig.2. Such a unit cell model which is a representative volume element of uniformly distributed voids in a medium can effectively represent the interaction effects between voids[16,17]. The finite element meshes used in the simulations are presented in Fig.2. In the case of the one-void unit cell, the initial void volume fraction fo is as follows:
f0 =
4 π r1 r2 r3 3 L3
R3
A R1: R2: R3=1:1:1
(1)
R1
1.3
Coordinate systems
The coordinate systems used in the simulations are presented in Fig.3. The x-y-z represents the load coordinate system, and it is the coordinate system where unit cell edge is located; <001>-<010>-<100> is the material coordinate system, where α0, β0, γ0 are the transform angles between material coordinate system and load coordinate system; R1-R2-R3 is the coordinate system where the three axis of ellipsoidal void are located, while α, β and γ are the transform angles between the ellipsoidal coordinate system and the load coordinate system.
2
R1
Fig.1 Y
2.1 Void growth 2.1.1 Void growth in <001> crystal orientation Fig.4 indicates that in the case of A-1 the peak normalized void volume fraction reaches 1.059 for the corresponding εx=0.1, while those of for B-1, B-2, B-3 and B-4 are 1.047 1.040 1.359 and 1.309, respectively. As shown in Fig.4, the growth rate of A-1 is slow, which is consistent with the previous result [17]. A more rapid increase occurs in the case of B-3 and B-4. It's worth noting that in the case of B-1 and B-2,
Geometrical of ellipsoidal void a
L aδ bδ δ
δ bδ Z
b
L L
X
aδ
Fig.2
Boundary conditions (a) and finite element mesh (b) a
z
z
010
R2
β0 α0
β γ0
001
x
y α
γ
R1
y
x N
Fig.3
b
R3
100
Results and Discussion
In this section, the results from the finite element simulations relating to the ellipsoidal void effects on the void growth are analyzed. The strain controlled boundary conditions were employed in this study and the load triaxiality factor a=b=–0.235, as shown in Fig.3. The initial void volume fraction is f0=0.01 and L=50 µm. The interrelated combination of crystal orientations, ellipsoidal shape and ellipsoid orientations for the analysis and corresponding symbols in this study are listed in Table 1.
R3 R2 C R1: R2: R3=5:3:3
Boundary condition
For all orientation, all unit cells were subjected to the same triaxial strain field by applying a proportional displacement along the three axes x, y and z. The applied displacement in the x direction is δ, while in the y and z direction, it is aδ and bδ, respectively, as shown in Fig.3. Similar boundary conditions were also considered by Shu[21].
R3 R2 B R1: R2: R3=5:5:3
Where r1, r2, r3 are the half-axis length of three axes elliposoidal sphere respectively. f is the void volume fraction calculated during solution and R is the normalized void volume fraction (2) R = f f0
1.2
R2
R1
N
Material coordinate (a) and ellipsoidal coordinate (b) vs. load coordinate system
a lower increase occurs than A-1, but the difference is minor. Fig.5 indicates that corresponding to C-1, C-2, C-3 and C-4, the peak normalized void volume fractions reach 1.121 1.113 1.345 and 1.233, respectively, for the corresponding εx=0.1. Compared with A-1, the void tends to grow easily in the cases of C-1 and C-2, especially in the cases of C-3 and C-4.
2.1.2
Void growth in <011> crystal orientation
Ellipsoidal shape B shows a relatively similar distribution between <011> and <001> crystal orientation, as shown in Fig.4 and Fig.6. In Fig.6, the peak normalized void volume fractions of shapes A-2, B-5, B-6, B-7, and B-8 reach 1.336 1.056 1.036 1.438 and 1.379, respectively,
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Sun Wanchao et al. / Rare Metal Materials and Engineering, 2014, 43(7): 1571-1576
Interrelated combination of crystal orientations, ellipsoidal shape and ellipsoid orientations for the analysis and cor
Normalized Void Volume Fraction
responding symbols 001 (α0/β0/γ0=0/0/0) α/β/γ Shape A Shape B Shape C 0/0/0 A-1 B-1 C-1 0/90/0 / B-2 C-2 0/30.96/0 / B-3 C-3 0/45/22.99 / B-4 C-4
Fig.4
1.4
011 (α0/β0/γ0=45/0/0) Shape A Shape B Shape C A-2 B-5 C-5 / B-6 C-6 / B-7 C-7 / B-8 C-8 Normalized Void Volume Fraction
Table 1
(α0 /β0 /γ0 =0/0/0)
1.3
A-1 B-1 B-2 B-3 B-4
1.2 1.1 1.0
0
0.02
0.04 0.06 0.08 Applied Strain
0.10
Normalized void volume fraction evolution with the ap-
Fig.5
Fig.7 shows the evolution of the normalized void volume fractions for five states proposed for study. The largest variation in void volume fraction is experienced by the C-7, with an increase of 1.384 for a total applied strain of 0.1. C-5 and C-6 indicate approximately the same increase in void fraction of approximately 1.15 times as much as the initial void volume, while C-8 reaches 1.279.
Void growth in <111> crystal orientation
Fig.6
Fig.8 indicates that in the case of A-3 the peak normalized void volume fraction reaches 1.326 for the corresponding εx=0.1 while those of B-9 B-10 B-11 and B-12 are 1.067 1.050 1.482 and 1.401 respectively. Fig.9 indicates that corresponding to C-9 C-10 C-11 and C-12, the peak normalized void volume fractions reach 1.171 1.172 1.415 and 1.301, respectively, for the corresponding εx=0.1.
A-1 C-1 C-2 C-3 C-4
1.2 1.1 1.0
0
0.02
0.04 0.06 0.08 Applied Strain
0.10
1.5 (α0 /β0 /γ0 =45/0/0)
1.4 A-2 B-5 B-6 B-7 B-8
1.3 1.2 1.1 1.0
0
0.02
0.04 0.06 0.08 Applied Strain
0.10
plied strain: A vs. B
Discussion
From Fig.5 to Fig.9, the difference in the normalized void volume fraction in the case of the A is significantly increased from <001> to <011> and <111> crystal orientation. The relative influences of lattice orientation on void growth is similar to the results found by Sangyul Ha[22] and W H Liu[23]. In the present work, much larger normalized void growth has been observed for B and C corresponding α/β/γ=0/30.96/0 and α/β/γ=0/45/22.9 than α/β/γ=0/0/0 and α/β/γ=0/90/0. However, an interesting feature is that, al
1.3
Normalized void volume fraction evolution with the ap-
Normalized Void Volume Fraction
2.1.4
(α0 /β0 /γ0 =0/0/0)
plied strain: A vs. C
for the corresponding εx=0.1. When the strain is applied along <011> crystal orientation, A-2, B-5, B-7, and B-8 tend to grow more easily compared to the strain applied along <001> crystal orientation.
2.1.3
1.4
Normalized void volume fraction evolution with the ap-
Normalized Void Volume Fraction
plied strain: A vs. B
111(α0/β0/γ0=45/35.26/0) Shape A Shape B Shape C A-3 B-9 C-9 / B-10 C-10 / B-11 C-11 / B-12 C-12
Fig.7
1.4
(α0 /β0 /γ0 =45/0/0)
1.3
A-2 C-5 C-6 C-7 C-8
1.2 1.1 1.0
0
0.02
0.04 0.06 0.08 Applied Strain
0.10
Normalized void volume fraction evolution with the applied strain: A vs. C
though it increases from <001> to <011> and <111> crystal orientation in the case of the B and C apart from B-6, the
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Sun Wanchao et al. / Rare Metal Materials and Engineering, 2014, 43(7): 1571-1576
Fig.8
Table 2
1.5 (α0 /β0 /γ0 =45/35.26/0)
1.4 A-3 B-9 B-10 B-11 B-12
1.3 1.2
A
B
1.1 1.0
0
0.02
0.04 0.06 0.08 Applied Strain
0.10
Normalized void volume fraction evolution with the ap-
Normalized Void Volume Fraction
plied strain: A vs. B
Fig.9
1.5
C (α0 /β0 /γ0 =45/35.26/0)
1.4
1.2 1.1
z x y
0
0.02
0.04 0.06 0.08 Applied Strain
z y x
0.10
Normalized void volume fraction evolution with the applied strain: A vs. C
normalized void volume fractions do not increase so rapidly as in the case of A. This implies that the overall effect of plastic anisotropy on damage evolution is diminished during non-spherical void growth. To better interpret the effects of the texture orientation on different shapes of void growth, a comparison between A, B and C for each orientation is tabulated in Table 2.
2.2
between A, B and C for each orientation <001> <011> <111> A-1 A-2 Increase A-3 Increase 1.059 1.336 26.16% 1.326 25.21% B-1 B-5 B-9 1.047 1.056 0.86% 1.067 1.9% B-2 B-6 B-10 1.040 1.036 -0.48% 1.050 0.96% B-3 B-7 B-11 1.359 1.438 5.81% 1.482 9.05% B-4 B-8 B-12 1.309 1.379 5.35% 1.401 7.03% C-1 C-5 C-9 1.121 1.166 4.01% 1.171 4.46% C-2 C-6 C-10 1.113 1.157 3.95% 1.172 5.30% C-3 C-7 C-11 1.345 1.384 2.90% 1.415 5.20% C-4 C-8 C-12 1.233 1.279 2.73% 1.301 5.52%
A-3 C-9 C-10 C-11 C-12
1.3
1.0
Comparisons of normalized void volume fraction
z x
Fig.10
Contour plots of equivalent strain for A-1 in orientation <001> with strain εx=0.1
z x y
Lattice rotation
Fig.10 (Fig.11) indicates that in the case of A-1 (A-2) in orientation <001> (<011>) the peak local equivalent strain reaches a value of 0.584 (0.613) for the corresponding remote strains of 0.1. The distribution of the equivalent strain is relatively symmetrical, and consequently, no void or unit cell distortion and rotation are recorded. The absent deformation of B-1, B-2, B-5, B-6, C-1, C-2, C-5 and C-6 are similar to A-1 and A-2, due to the fact that the shape of void and its distribution do not change the symmetry of the unit cell. Fig.12 indicates the void experience rotation in spite of the load applied along <001> and <011> orientations with symmetry of the slip systems in the cases of B-3, B-4, B-7, B-8, C-3, C-4, C-7 and C-8. The transform angles (α, β and γ) between ellipsoidal coordinate and the load coordinate break the symmetry of the unit cell, as well as the symmetry of the slip systems. It is noticed that the peak local equivalent strain
y
z
z y x
y
x
Fig.11
Contour plots of equivalent strain for A-2 in orientation <011> with strain εx=0.1
at the same load strain increases according to the order of B-4, C-3, B-8, C-4, C-7, C-8, B-3 and B-7. Nemat-Nasser et al.[24] and Solanki et al.[25], used both finite element and atomistic methods to revealed that the plastic flow localization induced corner formation in the void shape along with crack formation and growth. In the present work, the same phenomenon is observed. The void which has an ellipsoidal
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z
y
y
x
z
y
x
y
z
x
z
z
y x
z
x
y
z
x
x y
z
y
Fig.13
Octahedral {111} planes and <110> family of slip directions for a single crystal
x
Table 3 Fig.12
Contour plots of equivalent strain for B and C in orientation <001> and <011> with strain ε x = 0 . 1 A
shape initially, develops an irregular shape, and induces some corners where the dislocations will be initiated, which is more obvious at larger deformation.
2.3
Slip systems activity
The four octahedral {111} planes and the twelve corresponding <110> slip directions (three on each plane) are shown in Fig.13. The direction of load application with respect to the crystal orientation (<001>-<010>-<100> axes in Fig.13) determines the resolved shear stress for each of the {111}<110> slip plane/slip direction sets (or slip systems). To better interpret the combinational effects of the initial texture orientation, the ellipsoidal coordinate, the load coordinate system and the shape of void on the activated slip systems, a comparison for each orientation is tabulated in Table 3. It is noticed that the activated slip systems is mainly determined by the initial texture orientation, the ellipsoidal coordinate, the load coordinate system and the shape of void. In spite of the fact that the activated slip systems of A-1, A-2, B-1, B-2, B-5, B-6, C-1, C-2, C-5 and C-6 are very different from each other, the symmetry of the activated slip systems are not changed. This is the reason why there is no void or unit cell distortion and rotation to be recorded, as pointed out in section 2.2. However, in the
B
C
Activated slip system for different shapes and
orientations <001> A-1 3,-4,9,-10 B-1 3,-4,9,-10 B-2 -1,3,-4,6,-7,9,-10,12 B-3 -8,9,-10,11 B-4 5,-8 C-1 3,-4,9,-10 C-2 1,6,7,12 C-3 -2,5 C-4 -8
<011> A-2 -4,9 B-5 -1,3,-7,9 B-6 -1,3,-4,6,-7, 9,-10,12 B-7 -1,3 B-8 -6 C-5 -1,3,-7,9 C-6 4,-6,10,-12 C-7 -7,9 C-8 -1,3
<111> A-3 2 B-9 1,2 B-10 5,8 B-11 1,2 B-12 4,7 C-9 1,2 C-10 6,9 C-11 1,2 C-12 3
case of all remains in Table 3, there is no symmetry in the activated slip systems.
3 Conclusions 1) For triaxial tension conditions, the void fraction increase under the applied load is strongly dependent on the shape of void and the crystallographic orientation with respect to the load axis. 2) Even in the case of anisotropic crystalline matrix materials, the overall effect of plastic anisotropy on damage
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evolution is noteworthy diminished during non-spherical void growth, but differences are still present. 3) The transform angles (α, β and γ) between ellipsoidal coordinate system and the load coordinate system break the symmetry of the unit cell, as well as the symmetry of the slip systems. When the symmetry of the unit cell is broken, the void experiences rotation in spite of the load applied along <001> and <011> orientations with symmetry of the slip systems. 4) The activities on all the slip systems are influenced by the combination of the tensile axis, crystal orientation and different shapes of void.
10
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