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FEM simulation of void coalescence in FCC crystals
Computational Materials Science 50 (2010) 411–418
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
FEM simulation of void coalescence in FCC crystals Wenhui Liu a,⇑, Hao Huang a, Jianguo Tang b a b
School of Electromechanical Engineering, Hunan University of Science and Technology, Xiangtan 411201, China School of Materials Science and Engineering, Central South University, Changsha 410083, China
a r t i c l e
i n f o
Article history: Received 28 June 2010 Received in revised form 12 August 2010 Accepted 30 August 2010 Available online 13 October 2010 Keywords: Void coalescence Crystal plasticity High-angle grain boundary Orientation factor
a b s t r a c t To investigate the coalescence behaviors of voids in FCC crystals, three bicrystal models were used to study the coalescence of voids in single crystals, voids at grain boundary and voids in two grains by using three-dimensional crystal plasticity finite element method, which was implemented with rate dependent crystal plasticity theory as user material subroutine. By comparison the width of inter-void ligament in bicrystals, significant effects of orientation factor and high-angle grain boundary on void coalescence were revealed: (1) Voids in soft orientation grains tend to coalesce much easier than that in hard orientation under the strain controlled boundary condition. (2) For void coalescence at grain boundary, with the orientation factor’s difference between the two grains increasing, larger deformation mismatch is induced between grains, and the void prefers to grow along grain boundary. (3) Large orientation factor’s difference accelerate void coalescence at grain boundary, but decelerate void coalescence between grains. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Many experimental work has been done to investigate the effect of microstructure about the recrystallized behavior on fracture toughness. Alarcon [1] found that there was less blunting of the crack tip in recrystallized microstructure than that of unrecrystallized microstructure by TEM, and the recrystallized microstructure occurred strain concentration preferentially at grain boundaries owing to the difficulty of transferring plasticity from favorably oriented grains to less favorably oriented adjacent grains. Dorward and Beerntsen [2] found that fracture in recrystallized Al–Zn– Mg–Cu alloys was predominantly intergranular in nature, and the presence of subgrains in unrecrystallized structures led to a combination of transgranular and intersubgranular fracture. Deshpande and Gokhale [3,4] showed that the area fraction of microvoid-induced transgranular fracture decreased with the increase of the volume fraction of recrystallized structures. Morere et al. [5] also showed that intergranular failure preferred to occur along recrystallized grains. Though some experimental results showed that recrystallized grains prefer to fail by intergranular failure, few micromechanical models have been built to demonstrate it. Crystal plasticity theory 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. The influence of plastic anisotropy on damage evolution was investigated by Qi and Bertram [6] by considering the process of creep damage in FCC single crystals. Shu [7] used an elasto⇑ Corresponding author. E-mail address: [email protected] (W.H. Liu). 0927-0256/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2010.08.033
viscoplastic strain gradient crystal plasticity formulation to study the deformation of a porous single crystal, and it was found that small voids had a tendency to grow slower than big voids. Orsini and Zikry [8] studied the void growth and interaction in FCC copper crystals using finite element analysis by employing a ratedependent formulation of crystal plasticity, and showed that the rotation of the crystalline lattice and plastic activity on slip systems were concentrated mainly in the ligament region between the voids. O’regan et al. [9] used a two-dimensional finite element analysis to simulate the growth and coalescence of voids by varying void volume fraction, loading state, lattice structure and orientation. Further investigations about the influence of different orientations on deformation behavior were made by Horstemeyer et al. [10]. The case of circular void embedded in a single crystal in plane strain conditions was analyzed by Kysar et al. [11] through slip-line theory. Potirniche and Hearndon [12] studied void growth and coalescence in FCC single crystals using crystal plasticity under uniaxial and biaxial loading conditions, in which a 2D plane strain unit cell with one and two cylindrical voids was created, and showed that void growth was strongly dependent on the crystallographic orientation with respect to the tensile axis for uniaxial tension conditions. Liu et al. [13] used a 3D unit cell including a spherical void and two spherical voids to study the effects of crystallographic orientation on void growth and coalescence in single crystals, and concluded that void growth direction and shape were significantly dependent on the crystallographic orientation. The effect of void growth in single crystals with uniformly distributed cylindrical voids was studied using a finite deformation strain gradient crystal plasticity theory with an intrinsic length parameter by Borg et al. [14]. Here, the conventional crystal plasticity theory
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W.H. Liu et al. / Computational Materials Science 50 (2010) 411–418
is adopted to demonstrate the effects of crystallographic orientation and misorientation on void coalescence.
C44 = 75 Gpa, reference slip rate c_ 0 = 0.001, strain rate sensitivity m = 0.02, the hardening parameters are h0 = 250 MPa, s0 = 16 MPa, ss = 190 MPa, a = 2.25, and latent hardening rate q = 1.4.
2. Simulation model 3. Results and discussion A user-defined crystal plasticity material subroutine is used to implement the rate dependent constitutive relationship in ABAQUS. In FCC crystals, plastic deformation occurs along the {1 1 1}h1 1 0i family of slip systems, which means 12 potentially active slip systems as shown in Table 1. Bunge angles are used in the simulation analysis. In Bunge’s system, the crystallographic axes [1 0 0], [0 1 0], and [0 0 1], are designated x, y, z, and the external sample coordinate system are designated X, Y, Z, and the final orientation of the crystal coordinate system with respective to the sample coordinate system is defined by rotations expressed in three angles u1, U, u2. To analyze the influence of crystallographic orientation and misorientation on void coalescence in FCC crystals, various initial orientations are considered. The crystallographic orientations analyzed in this study are the combinations of Cu (90°, 35°, 45°), Bs (35°, 45°, 90°), Cube (0°, 0°, 0°) and Goss (0°, 45°, 90°). The combinations of Goss and Cube orientation (A1), Cu and Cube orientation (A2), Bs and Goss orientation (A3), Cu and Goss orientation (A4) are studied, and the misorientations for A1, A2, A3 and A4 are 45°, 56°, 35° and 55°, individually. To illustrate the coalescence behaviors in FCC crystals, three different kinds of 3D unit cells including two spherical voids are considered here, as shown in Fig. 1. The inter-void ligament between the two voids is d/2, and the initial void volume fraction of the unit cell is 0.01, so the ratio of void diameter (d) to side length (l) is 0.212. Fig. 1a–c are the unit cells for modeling the coalescence of voids in single crystals, voids at grain boundary, and voids in two grains, respectively. To illustrate the coalescence behaviors of voids in FCC crystals, the finite element calculations are performed using a constant strain control. The constant strain control boundary conditions are similar to that considered by Liu et al. [13,15], as shown in Fig. 1. All unit cells are subjected to the same triaxial strain field by applying a proportional displacement along axis X, Y and Z. The axis X is parallel to sample rolling direction, and the axis Z is parallel to plate normal direction. The applied displacement in X direction is d, while in Y and Z direction, they are ad and bd respectively. In this study, the strain controlled boundary condition with the following value a = b = 0.235 is employed. The average macroscopic strain can be obtained by:
ei ¼ ln
li þ ui li
ð1Þ
where ui is the prescribed displacement of the ith direction, and li is the original length of the unit cell. The volume average macroscopic stresses are obtained by the volume average of the microstructural stress in the unit cell.
r ij ¼
1 V
Z
rij dV
ð2Þ
V
Here, the parameter definition and the rate dependent crystal plasticity constitutive theory are the same with those in Ref. [13], and the crystal model is considered as having anisotropic elasticity with elasticity tensor C11 = 170 GPa, C12 = 124 GPa,
In this section, the effects of crystallographic orientation and misorientation on void coalescence are analyzed by using crystal plasticity finite element method. Fig. 2 demonstrates the curves of the computed von Mises equivalent stress vs. equivalent strain eX under the prescribed displacement conditions a = b = 0.235 for single crystals with various crystallographic orientations. It is shown that the equivalent stress is increasing according to the orders of Cube, Goss, Brass and Copper orientation. 3.1. Void coalescence in single crystals Schacht et al. [16] and Liu et al. [13] investigated void growth behaviors in single crystals by using constant strain control, and found that void growth and deformation behaviors significantly depend on the crystallographic orientation. Here, the same method is used to show the coalescence of voids in FCC crystals. Figs. 3–5 show plastic deformation that characterizes the coalescence behavior of two voids in single crystals in cross section XZ. The simulation results show the plastic flow localization and heterogeneous distribution of plastic deformation, and indicate that the deformation and the shortest width of inter-void ligament positioned between the two voids are dependent on the initial orientation of the crystalline lattice. The plastic deformation of Cube orientation is shown in Fig. 3, which indicates that the peak of the local plastic strain in the inter-void ligament region reaches 2.917 at the corresponding eX = 0.05. In Fig. 3, an intensive shear band which is formed directly between the two voids, represents a large coalescence effect in Cube orientation, and the ratio of the minimal width of the intervoid ligament (dmin) with original inter-void ligament (0.5d) is minimal for the four crystallographic orientation. In Fig. 4, Bs orientation indicates that the peak local plastic strain in the inter-void ligament region reaches 2.548 at the corresponding eX = 0.05. This orientation shows a small coalescence effect, and a shear band is formed between the two voids. Fig. 5a and b are the contour of plastic deformation for Cu and Goss orientation at the corresponding eX = 0.05. The inter-void ligament region in Cu orientation reaches 2.092, which is minimal in the four typical crystallographic orientations, and the value dmin/ 0.5d is 0.0562 at the corresponding eX = 0.05, which is the maximal value in the four crystallographic orientations. The peak local plastic strain of Goss orientation reaches 2.331 at the corresponding eX = 0.05, and voids tend to coalesce obviously at eX = 0.05. Horstemeyer [10,17] analyzed void growth and coalescence at the micromechanical length scale by using axisymmetric and planar unit cells, and concluded that void coalescence was determined by the inter-void ligament distance. Fig. 6 shows the evolution of the value dmin/0.5d with applied strain eX. In Fig. 6, the ligament width for Cube orientation decreases rapidly at small strain. For Copper, Goss and Brass orientation, the ligament widthes change relative slowly compared with Cube orientation at small strain, and it could be inferred that voids in soft orientation tend to coa-
Table 1 Slip systems in FCC crystals. m
111
s Slip systems
011 a1
1 1 1 01 1 a2
0 11 a3
1 01 b1
11 1 101 b2
10 1 b3
011 c1
1 11 101 c2
1 0 ] 1 c3
1 01 d1
01 1 d2
110 d3
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Fig. 1. Unit cells for void coalescence.
Fig. 2. Equivalent stress vs. equivalent strain.
lesce much easier than that in hard orientation under the load condition from Fig. 6. 3.2. Void coalescence at grain boundary Figs. 7–9 show the plastic deformation of unit cells with two voids located at the grain boundary for A1, A2, A3 and A4. Fig. 7a and b are the case for A1 at eX = 0.025 and eX = 0.05 individually. From Fig. 7, it can be found that the inter-void ligament in Z direction is still large, and there is small void interaction at eX = 0.025. The plastic deformation of A2 is shown in Fig. 8. In Fig. 8, an intensive shear band which is formed directly between the two
voids, represents a large coalescence effect, and the width of inter-void ligament is smaller than that in A1. From Fig. 2, it can be found that the equivalent stress at the same equivalent strain is increasing according to the orders of Cube, Goss, Brass and Copper orientation, that is to say, the orientation factor’s difference in A2 is larger than that in A1. It can be noticed that A2 shows more intensive void interaction effects than that in A1 from Figs. 7 and 8. Firstly, the local strain peak in A2 is larger than the maximal local strain in A1. Secondly, the width of inter-void ligament in A2 is smaller than that in A1, and A2 shows an intensive coalescence effect at small strain. All this evidences reveal that A2 shows more coalescence effects than A1, and A2 may prefer to fail by intergranular fracture than A1. It can be noticed that the void growth behaviors in the two grains are quite different, and the void growth velocity in soft grain is larger than that in hard one, for example, the void growth velocity in Cube orientation is larger than that in Cu orientation in Fig. 8, and corners are induced around the void. Nemat-Nasser et al. [18] experimentally observed non-circular shapes of the outer and the inner boundaries. Nemat-Nasser et al. [18] and Solanki et al. [19] using both finite element and atomistic methods, 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 is spherical initially, becomes an irregular shape, and some corners are induced, which are more obvious at large equivalent plastic deformation. Table 2 is the active slip systems and its corresponding shear strain increments (dc) for eX = 0.004 in regions A and B in Fig. 8. Because the rate dependent crystal plasticity constitutive theory used here assumes all the 12 slip systems may be activated, in this work, we think the slip system whose shear strain increments are less than 3% maximal shear strain increment is not active. Table 2 is the
Fig. 3. Contour plots of equivalent plastic strain for Cube at strain eX (a) 0.02 and (b) 0.05.
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Fig. 4. Contour plots of equivalent plastic strain for Bs at strain eX (a) 0.02 and (b) 0.05.
Fig. 5. Contour plots of equivalent plastic strain for Cu and Goss.
different active slip systems in regions A and B, and shows that there are eight active slip systems in region A, and three active slip systems in region B meaning relative smaller deformation occurred in hard orientation, which may cause deformation discontinuity around the void, and form the corner-like region. Kysar et al. [11] explained the discontinuity in lattice rotation through slip-line theory. Here, due to the crystallographic misorientation and different active slip systems of the two grains in the cell, the plastic deformation is heterogeneous, especially at the grain boundary near the void, which leads to the formation of corners at the regions, then crack will initiate and grow along the grain boundary. Fig. 9 shows the plastic deformation of the unit cells with two voids located at grain boundary for A3 and A4 at the corresponding eX = 0.05. Compared with A3, the plastic deformation distribution between the two grains in A4 is more heterogeneous, and the width of inter-void ligament is shorter, which means that A4 shows a larger coalescence effect than A3. Fig. 6. The width of inter-void ligament in single crystals.
Fig. 7. Contour plots of equivalent plastic strain of A1 with voids located at grain boundary (a) eX = 0.025 and (b) eX = 0.05.
W.H. Liu et al. / Computational Materials Science 50 (2010) 411–418
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Fig. 8. Contour plots of equivalent plastic strain of A2 with voids located at grain boundary (a) eX = 0.025 and (b) eX = 0.05.
Fig. 9. Contour plots of equivalent plastic strain of A3 and A4 with voids located at grain boundary.
Table 2 The active slip systems and shear strain increment. Region A Region B
Active slip systems Strain increment (106) Active slip systems Strain increment (106)
a1 930
a2 629
b2 125
a1 2551
a3 193
b3 1109
c2 811
d1 2282
d2 122
Fig. 10 shows the evolution of the value dmin/0.5d for void coalescence at grain boundary in A1, A2, A3 and A4. From Fig. 10a and b,
it can be found that the ligament width of A1 is larger than A2, and A3 is larger than A4. Figs. 7–9 show plastic deformation that characterizes the growth and coalescence behaviors of two voids in cross section XZ. From Figs. 7–9 it can be found that the plastic deformation distribution between the two grains in A1 and A3 is relatively homogeneous, and the orientation factor’s difference between two grains in A1 and A3 is small, but for A2 and A4, in which the orientation factor’s difference between two grains is larger, the widthes of inter-void ligaments in cells are smaller, and cells show more intensive coalescence effects in A2 and A4 due to deformation mismatch, and cells may prefer to fail by intergranular fracture. It is
Fig. 10. The width of inter-void ligament for voids located at grain boundary.
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W.H. Liu et al. / Computational Materials Science 50 (2010) 411–418
well known that the orientation factor’s difference between adjacent grains in the recrystallized structure is large in general, then deformation mismatch is induced among grains during deformation, and the material prefers to fail by intergranular fracture. It is just because of the orientation factor’s difference in recrystallized grain that voids and cracks will initiate and grow easily due to deformation mismatch.
3.3. Void coalescence between two grains Figs. 11–13 show plastic deformation that characterizes coalescence behavior of voids in two grains for A1, A2, A3 and A4. Fig. 11 is the coalescence behavior in A1, and the peak local plastic strain in
the inter-void ligament region reaches 3.543 at the corresponding
eX = 0.05. The void coalescence between grains in A2 is shown in Fig. 12. From Fig. 12, it could be found that the peak of the local plastic strain in the inter-void ligament region reaches 3.252 which is smaller than that in A1, and the width of inter-void ligament is larger than that in A1, which means that A2 shows smaller coalescence effect than A1. Fig. 13 shows the plastic deformation of unit cells with voids in two grains for A3 and A4 at the corresponding eX = 0.05. Compared with A4, the width of inter-void ligament in A3 is shorter, which means that A3 shows a larger coalescence effect than A4. From Figs. 11–13, it could be found that the plastic deformation distribution between the two grains in A1–A4 is relatively
Fig. 11. Contour plots of equivalent plastic strain of A1 with voids in two grains (a) eX = 0.025 and (b) eX = 0.05.
Fig. 12. Contour plots of equivalent plastic strain of A2 with voids in two grains (a) eX = 0.025 and (b) eX = 0.05.
Fig. 13. Contour plots of equivalent plastic strain of A3 and A4 with voids in two grains.
W.H. Liu et al. / Computational Materials Science 50 (2010) 411–418
homogeneous, and cells show smaller coalescence effects compared with Figs. 7–9. Fig. 14 shows the evolution of the value dmin/0.5d for void coalescence between two grains in A1–A4. Converse to Fig. 10, the lig-
417
ament width of A1 is smaller than A2, and A3 is smaller than A4, which means that void coalescence behaviors between two grains are different from void coalescence behaviors at grain boundary, and the effects of orientation factor’s difference on void
Fig. 14. The width of inter-void ligament for voids in two grains.
Fig. 15. Comparison of the width of inter-void ligament for different combinations.
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W.H. Liu et al. / Computational Materials Science 50 (2010) 411–418
coalescence at grain boundary and void coalescence between grains are different. Simulation results show that large orientation factor’s difference accelerate void coalescence at grain boundary, but decelerate void coalescence between grains. Figs. 11–13 show plastic deformation that characterizes the growth and coalescence behaviors of voids in two grains. Integrated with Fig. 14, it can be found that the value dmin/0.5d in A1 and A3 is relatively small, and the orientation factor’s difference between two grains in A1 and A3 is small. But for A2 and A4, in which the orientation factor’s difference between two grains is larger, the width of inter-void ligaments in cells are larger, and cells show relatively smaller coalescence effects. So it could be concluded that large orientation factor’s difference is unfavorable for void coalescence between grains. Fig. 15 shows the comparison of the value dmin/0.5d for void coalescence at grain boundary and void coalescence between grains. From Fig. 15, it could be found that void coalescence at grain boundary is faster than that between grains. Simulation results show that the plastic deformation for void coalescence between grains is relatively homogeneous, and grain boundaries show a tendency to reduce void coalescence effect, but for void coalescence at grain boundary, the plastic deformation between grains is inhomogeneous, and the void tends to grow along grain boundary. 4. Conclusion Three bicrystal models were created to simulate the coalescence behavior and the plastic deformation distribution around voids under the strain controlled boundary condition, and the rate dependent crystal plasticity theory was applied to investigate void coalescence behaviors. The computed results show: 1. Voids in soft orientation grains tend to coalesce much easier than that in hard orientation under the strain controlled boundary condition. 2. For void coalescence at grain boundary, with the orientation factor’s difference between the two grains increasing, larger deformation mismatch is induced between grains, and the void prefers to coalesce along grain boundary.
3. Large orientation factor’s difference is favorable for void coalescence at grain boundary, unfavorable for void coalescence between grains. It is just because of large orientation factor’s difference in recrystallized grain that voids will grow and coalesce easily along grain boundary due to deformation mismatch, and recrystallized grains prefer to fail by intergranular failure.
Acknowledgements The present work is supported by the National Science Foundation of China (Project No. 50905188), and Hunan Provincial Natural Science Foundation of China (Project Nos. 10JJ3062 and 10JJ8002), which are greatly acknowledged by the authors.
References [1] O.E. Alarcon, A.M.M. Nazar, W.A. Monteiro, Mater. Sci. Eng. 138 (1991) 275– 285. [2] R.C. Dorward, D.J. Beerntsen, Metall. Mater. Trans. A 26 (1995) 2481–2484. [3] N.U. Deshpande, A.M. Gokhale, D.K. Denzer, et al., Metall. Mater. Trans. A 29 (1998) 1191–1201. [4] A.M. Gokhale, N.U. Deshpande, D.K. Denzer, et al., Metall. Mater. Trans. A 29 (1998) 1203–1210. [5] B. Morere, J.C. Ehrstrom, P.J. Gregson, et al., Metall. Mater. Trans. A 31 (2000) 2503–2515. [6] W. Qi, A. Bertram, Int. J. Plasticity 15 (1999) 1197–1215. [7] J.Y. Shu, Int. J. Plasticity 14 (10–11) (1998) 1085–1107. [8] V.C. Orsini, M.A. Zikry, Int. J. Plasticity 17 (2001) 1393–1417. [9] T.L. O’regan, D.F. Quinn, M.A. Howe, et al., Comput. Mech. 20 (1997) 115–121. [10] M.F. Horstemeyer, S. Ramaswamy, Int. J. Plasticity 9 (2000) 5–28. [11] J.W. Kysar, Y.X. Gan, M. Arauza, Int. J. plasticity 21 (2005) 1481–1520. [12] G.P. Potirniche, J.L. Hearndon, M.F. Horstemeyer, et al., Int. J. Plasticity 22 (2006) 921–942. [13] W.H. Liu, X.M. Zhang, J.G. Tang, et al., Comput. Mater. Sci. 40 (2007) 130–139. [14] U. Borg, C.F. Niordson, J.W. Kysar, Size effects on void growth in single crystals with distributed voids, Int. J. Plasticity 24 (2008) 688–701. [15] W.H. Liu, X.M. Zhang, J.G. Tang, Mech. Mater. 41 (2009) 799–809. [16] T. Schacht, N. Untermann, E. Steck, Int. J. Plasticity 19 (10) (2003) 1605–1626. [17] M.F. Horstemeyer, M.M. Matalanis, A.M. Sieber, Int. J. Plasticity 16 (2000) 979– 1015. [18] S. Nemat-nasser, T. Okinaka, V. Nesterenko, et al., Philos. Mag. A 78 (5) (1998) 1151–1174. [19] K. Solanki, M.F. Horstemeyer, M.I. Baskes, et al., Mech. Mater. 37 (2005) 317– 330.
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
FEM simulation of void coalescence in FCC crystals Wenhui Liu a,⇑, Hao Huang a, Jianguo Tang b a b
School of Electromechanical Engineering, Hunan University of Science and Technology, Xiangtan 411201, China School of Materials Science and Engineering, Central South University, Changsha 410083, China
a r t i c l e
i n f o
Article history: Received 28 June 2010 Received in revised form 12 August 2010 Accepted 30 August 2010 Available online 13 October 2010 Keywords: Void coalescence Crystal plasticity High-angle grain boundary Orientation factor
a b s t r a c t To investigate the coalescence behaviors of voids in FCC crystals, three bicrystal models were used to study the coalescence of voids in single crystals, voids at grain boundary and voids in two grains by using three-dimensional crystal plasticity finite element method, which was implemented with rate dependent crystal plasticity theory as user material subroutine. By comparison the width of inter-void ligament in bicrystals, significant effects of orientation factor and high-angle grain boundary on void coalescence were revealed: (1) Voids in soft orientation grains tend to coalesce much easier than that in hard orientation under the strain controlled boundary condition. (2) For void coalescence at grain boundary, with the orientation factor’s difference between the two grains increasing, larger deformation mismatch is induced between grains, and the void prefers to grow along grain boundary. (3) Large orientation factor’s difference accelerate void coalescence at grain boundary, but decelerate void coalescence between grains. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Many experimental work has been done to investigate the effect of microstructure about the recrystallized behavior on fracture toughness. Alarcon [1] found that there was less blunting of the crack tip in recrystallized microstructure than that of unrecrystallized microstructure by TEM, and the recrystallized microstructure occurred strain concentration preferentially at grain boundaries owing to the difficulty of transferring plasticity from favorably oriented grains to less favorably oriented adjacent grains. Dorward and Beerntsen [2] found that fracture in recrystallized Al–Zn– Mg–Cu alloys was predominantly intergranular in nature, and the presence of subgrains in unrecrystallized structures led to a combination of transgranular and intersubgranular fracture. Deshpande and Gokhale [3,4] showed that the area fraction of microvoid-induced transgranular fracture decreased with the increase of the volume fraction of recrystallized structures. Morere et al. [5] also showed that intergranular failure preferred to occur along recrystallized grains. Though some experimental results showed that recrystallized grains prefer to fail by intergranular failure, few micromechanical models have been built to demonstrate it. Crystal plasticity theory 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. The influence of plastic anisotropy on damage evolution was investigated by Qi and Bertram [6] by considering the process of creep damage in FCC single crystals. Shu [7] used an elasto⇑ Corresponding author. E-mail address: [email protected] (W.H. Liu). 0927-0256/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2010.08.033
viscoplastic strain gradient crystal plasticity formulation to study the deformation of a porous single crystal, and it was found that small voids had a tendency to grow slower than big voids. Orsini and Zikry [8] studied the void growth and interaction in FCC copper crystals using finite element analysis by employing a ratedependent formulation of crystal plasticity, and showed that the rotation of the crystalline lattice and plastic activity on slip systems were concentrated mainly in the ligament region between the voids. O’regan et al. [9] used a two-dimensional finite element analysis to simulate the growth and coalescence of voids by varying void volume fraction, loading state, lattice structure and orientation. Further investigations about the influence of different orientations on deformation behavior were made by Horstemeyer et al. [10]. The case of circular void embedded in a single crystal in plane strain conditions was analyzed by Kysar et al. [11] through slip-line theory. Potirniche and Hearndon [12] studied void growth and coalescence in FCC single crystals using crystal plasticity under uniaxial and biaxial loading conditions, in which a 2D plane strain unit cell with one and two cylindrical voids was created, and showed that void growth was strongly dependent on the crystallographic orientation with respect to the tensile axis for uniaxial tension conditions. Liu et al. [13] used a 3D unit cell including a spherical void and two spherical voids to study the effects of crystallographic orientation on void growth and coalescence in single crystals, and concluded that void growth direction and shape were significantly dependent on the crystallographic orientation. The effect of void growth in single crystals with uniformly distributed cylindrical voids was studied using a finite deformation strain gradient crystal plasticity theory with an intrinsic length parameter by Borg et al. [14]. Here, the conventional crystal plasticity theory
412
W.H. Liu et al. / Computational Materials Science 50 (2010) 411–418
is adopted to demonstrate the effects of crystallographic orientation and misorientation on void coalescence.
C44 = 75 Gpa, reference slip rate c_ 0 = 0.001, strain rate sensitivity m = 0.02, the hardening parameters are h0 = 250 MPa, s0 = 16 MPa, ss = 190 MPa, a = 2.25, and latent hardening rate q = 1.4.
2. Simulation model 3. Results and discussion A user-defined crystal plasticity material subroutine is used to implement the rate dependent constitutive relationship in ABAQUS. In FCC crystals, plastic deformation occurs along the {1 1 1}h1 1 0i family of slip systems, which means 12 potentially active slip systems as shown in Table 1. Bunge angles are used in the simulation analysis. In Bunge’s system, the crystallographic axes [1 0 0], [0 1 0], and [0 0 1], are designated x, y, z, and the external sample coordinate system are designated X, Y, Z, and the final orientation of the crystal coordinate system with respective to the sample coordinate system is defined by rotations expressed in three angles u1, U, u2. To analyze the influence of crystallographic orientation and misorientation on void coalescence in FCC crystals, various initial orientations are considered. The crystallographic orientations analyzed in this study are the combinations of Cu (90°, 35°, 45°), Bs (35°, 45°, 90°), Cube (0°, 0°, 0°) and Goss (0°, 45°, 90°). The combinations of Goss and Cube orientation (A1), Cu and Cube orientation (A2), Bs and Goss orientation (A3), Cu and Goss orientation (A4) are studied, and the misorientations for A1, A2, A3 and A4 are 45°, 56°, 35° and 55°, individually. To illustrate the coalescence behaviors in FCC crystals, three different kinds of 3D unit cells including two spherical voids are considered here, as shown in Fig. 1. The inter-void ligament between the two voids is d/2, and the initial void volume fraction of the unit cell is 0.01, so the ratio of void diameter (d) to side length (l) is 0.212. Fig. 1a–c are the unit cells for modeling the coalescence of voids in single crystals, voids at grain boundary, and voids in two grains, respectively. To illustrate the coalescence behaviors of voids in FCC crystals, the finite element calculations are performed using a constant strain control. The constant strain control boundary conditions are similar to that considered by Liu et al. [13,15], as shown in Fig. 1. All unit cells are subjected to the same triaxial strain field by applying a proportional displacement along axis X, Y and Z. The axis X is parallel to sample rolling direction, and the axis Z is parallel to plate normal direction. The applied displacement in X direction is d, while in Y and Z direction, they are ad and bd respectively. In this study, the strain controlled boundary condition with the following value a = b = 0.235 is employed. The average macroscopic strain can be obtained by:
ei ¼ ln
li þ ui li
ð1Þ
where ui is the prescribed displacement of the ith direction, and li is the original length of the unit cell. The volume average macroscopic stresses are obtained by the volume average of the microstructural stress in the unit cell.
r ij ¼
1 V
Z
rij dV
ð2Þ
V
Here, the parameter definition and the rate dependent crystal plasticity constitutive theory are the same with those in Ref. [13], and the crystal model is considered as having anisotropic elasticity with elasticity tensor C11 = 170 GPa, C12 = 124 GPa,
In this section, the effects of crystallographic orientation and misorientation on void coalescence are analyzed by using crystal plasticity finite element method. Fig. 2 demonstrates the curves of the computed von Mises equivalent stress vs. equivalent strain eX under the prescribed displacement conditions a = b = 0.235 for single crystals with various crystallographic orientations. It is shown that the equivalent stress is increasing according to the orders of Cube, Goss, Brass and Copper orientation. 3.1. Void coalescence in single crystals Schacht et al. [16] and Liu et al. [13] investigated void growth behaviors in single crystals by using constant strain control, and found that void growth and deformation behaviors significantly depend on the crystallographic orientation. Here, the same method is used to show the coalescence of voids in FCC crystals. Figs. 3–5 show plastic deformation that characterizes the coalescence behavior of two voids in single crystals in cross section XZ. The simulation results show the plastic flow localization and heterogeneous distribution of plastic deformation, and indicate that the deformation and the shortest width of inter-void ligament positioned between the two voids are dependent on the initial orientation of the crystalline lattice. The plastic deformation of Cube orientation is shown in Fig. 3, which indicates that the peak of the local plastic strain in the inter-void ligament region reaches 2.917 at the corresponding eX = 0.05. In Fig. 3, an intensive shear band which is formed directly between the two voids, represents a large coalescence effect in Cube orientation, and the ratio of the minimal width of the intervoid ligament (dmin) with original inter-void ligament (0.5d) is minimal for the four crystallographic orientation. In Fig. 4, Bs orientation indicates that the peak local plastic strain in the inter-void ligament region reaches 2.548 at the corresponding eX = 0.05. This orientation shows a small coalescence effect, and a shear band is formed between the two voids. Fig. 5a and b are the contour of plastic deformation for Cu and Goss orientation at the corresponding eX = 0.05. The inter-void ligament region in Cu orientation reaches 2.092, which is minimal in the four typical crystallographic orientations, and the value dmin/ 0.5d is 0.0562 at the corresponding eX = 0.05, which is the maximal value in the four crystallographic orientations. The peak local plastic strain of Goss orientation reaches 2.331 at the corresponding eX = 0.05, and voids tend to coalesce obviously at eX = 0.05. Horstemeyer [10,17] analyzed void growth and coalescence at the micromechanical length scale by using axisymmetric and planar unit cells, and concluded that void coalescence was determined by the inter-void ligament distance. Fig. 6 shows the evolution of the value dmin/0.5d with applied strain eX. In Fig. 6, the ligament width for Cube orientation decreases rapidly at small strain. For Copper, Goss and Brass orientation, the ligament widthes change relative slowly compared with Cube orientation at small strain, and it could be inferred that voids in soft orientation tend to coa-
Table 1 Slip systems in FCC crystals. m
111
s Slip systems
011 a1
1 1 1 01 1 a2
0 11 a3
1 01 b1
11 1 101 b2
10 1 b3
011 c1
1 11 101 c2
1 0 ] 1 c3
1 01 d1
01 1 d2
110 d3
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Fig. 1. Unit cells for void coalescence.
Fig. 2. Equivalent stress vs. equivalent strain.
lesce much easier than that in hard orientation under the load condition from Fig. 6. 3.2. Void coalescence at grain boundary Figs. 7–9 show the plastic deformation of unit cells with two voids located at the grain boundary for A1, A2, A3 and A4. Fig. 7a and b are the case for A1 at eX = 0.025 and eX = 0.05 individually. From Fig. 7, it can be found that the inter-void ligament in Z direction is still large, and there is small void interaction at eX = 0.025. The plastic deformation of A2 is shown in Fig. 8. In Fig. 8, an intensive shear band which is formed directly between the two
voids, represents a large coalescence effect, and the width of inter-void ligament is smaller than that in A1. From Fig. 2, it can be found that the equivalent stress at the same equivalent strain is increasing according to the orders of Cube, Goss, Brass and Copper orientation, that is to say, the orientation factor’s difference in A2 is larger than that in A1. It can be noticed that A2 shows more intensive void interaction effects than that in A1 from Figs. 7 and 8. Firstly, the local strain peak in A2 is larger than the maximal local strain in A1. Secondly, the width of inter-void ligament in A2 is smaller than that in A1, and A2 shows an intensive coalescence effect at small strain. All this evidences reveal that A2 shows more coalescence effects than A1, and A2 may prefer to fail by intergranular fracture than A1. It can be noticed that the void growth behaviors in the two grains are quite different, and the void growth velocity in soft grain is larger than that in hard one, for example, the void growth velocity in Cube orientation is larger than that in Cu orientation in Fig. 8, and corners are induced around the void. Nemat-Nasser et al. [18] experimentally observed non-circular shapes of the outer and the inner boundaries. Nemat-Nasser et al. [18] and Solanki et al. [19] using both finite element and atomistic methods, 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 is spherical initially, becomes an irregular shape, and some corners are induced, which are more obvious at large equivalent plastic deformation. Table 2 is the active slip systems and its corresponding shear strain increments (dc) for eX = 0.004 in regions A and B in Fig. 8. Because the rate dependent crystal plasticity constitutive theory used here assumes all the 12 slip systems may be activated, in this work, we think the slip system whose shear strain increments are less than 3% maximal shear strain increment is not active. Table 2 is the
Fig. 3. Contour plots of equivalent plastic strain for Cube at strain eX (a) 0.02 and (b) 0.05.
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Fig. 4. Contour plots of equivalent plastic strain for Bs at strain eX (a) 0.02 and (b) 0.05.
Fig. 5. Contour plots of equivalent plastic strain for Cu and Goss.
different active slip systems in regions A and B, and shows that there are eight active slip systems in region A, and three active slip systems in region B meaning relative smaller deformation occurred in hard orientation, which may cause deformation discontinuity around the void, and form the corner-like region. Kysar et al. [11] explained the discontinuity in lattice rotation through slip-line theory. Here, due to the crystallographic misorientation and different active slip systems of the two grains in the cell, the plastic deformation is heterogeneous, especially at the grain boundary near the void, which leads to the formation of corners at the regions, then crack will initiate and grow along the grain boundary. Fig. 9 shows the plastic deformation of the unit cells with two voids located at grain boundary for A3 and A4 at the corresponding eX = 0.05. Compared with A3, the plastic deformation distribution between the two grains in A4 is more heterogeneous, and the width of inter-void ligament is shorter, which means that A4 shows a larger coalescence effect than A3. Fig. 6. The width of inter-void ligament in single crystals.
Fig. 7. Contour plots of equivalent plastic strain of A1 with voids located at grain boundary (a) eX = 0.025 and (b) eX = 0.05.
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Fig. 8. Contour plots of equivalent plastic strain of A2 with voids located at grain boundary (a) eX = 0.025 and (b) eX = 0.05.
Fig. 9. Contour plots of equivalent plastic strain of A3 and A4 with voids located at grain boundary.
Table 2 The active slip systems and shear strain increment. Region A Region B
Active slip systems Strain increment (106) Active slip systems Strain increment (106)
a1 930
a2 629
b2 125
a1 2551
a3 193
b3 1109
c2 811
d1 2282
d2 122
Fig. 10 shows the evolution of the value dmin/0.5d for void coalescence at grain boundary in A1, A2, A3 and A4. From Fig. 10a and b,
it can be found that the ligament width of A1 is larger than A2, and A3 is larger than A4. Figs. 7–9 show plastic deformation that characterizes the growth and coalescence behaviors of two voids in cross section XZ. From Figs. 7–9 it can be found that the plastic deformation distribution between the two grains in A1 and A3 is relatively homogeneous, and the orientation factor’s difference between two grains in A1 and A3 is small, but for A2 and A4, in which the orientation factor’s difference between two grains is larger, the widthes of inter-void ligaments in cells are smaller, and cells show more intensive coalescence effects in A2 and A4 due to deformation mismatch, and cells may prefer to fail by intergranular fracture. It is
Fig. 10. The width of inter-void ligament for voids located at grain boundary.
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well known that the orientation factor’s difference between adjacent grains in the recrystallized structure is large in general, then deformation mismatch is induced among grains during deformation, and the material prefers to fail by intergranular fracture. It is just because of the orientation factor’s difference in recrystallized grain that voids and cracks will initiate and grow easily due to deformation mismatch.
3.3. Void coalescence between two grains Figs. 11–13 show plastic deformation that characterizes coalescence behavior of voids in two grains for A1, A2, A3 and A4. Fig. 11 is the coalescence behavior in A1, and the peak local plastic strain in
the inter-void ligament region reaches 3.543 at the corresponding
eX = 0.05. The void coalescence between grains in A2 is shown in Fig. 12. From Fig. 12, it could be found that the peak of the local plastic strain in the inter-void ligament region reaches 3.252 which is smaller than that in A1, and the width of inter-void ligament is larger than that in A1, which means that A2 shows smaller coalescence effect than A1. Fig. 13 shows the plastic deformation of unit cells with voids in two grains for A3 and A4 at the corresponding eX = 0.05. Compared with A4, the width of inter-void ligament in A3 is shorter, which means that A3 shows a larger coalescence effect than A4. From Figs. 11–13, it could be found that the plastic deformation distribution between the two grains in A1–A4 is relatively
Fig. 11. Contour plots of equivalent plastic strain of A1 with voids in two grains (a) eX = 0.025 and (b) eX = 0.05.
Fig. 12. Contour plots of equivalent plastic strain of A2 with voids in two grains (a) eX = 0.025 and (b) eX = 0.05.
Fig. 13. Contour plots of equivalent plastic strain of A3 and A4 with voids in two grains.
W.H. Liu et al. / Computational Materials Science 50 (2010) 411–418
homogeneous, and cells show smaller coalescence effects compared with Figs. 7–9. Fig. 14 shows the evolution of the value dmin/0.5d for void coalescence between two grains in A1–A4. Converse to Fig. 10, the lig-
417
ament width of A1 is smaller than A2, and A3 is smaller than A4, which means that void coalescence behaviors between two grains are different from void coalescence behaviors at grain boundary, and the effects of orientation factor’s difference on void
Fig. 14. The width of inter-void ligament for voids in two grains.
Fig. 15. Comparison of the width of inter-void ligament for different combinations.
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coalescence at grain boundary and void coalescence between grains are different. Simulation results show that large orientation factor’s difference accelerate void coalescence at grain boundary, but decelerate void coalescence between grains. Figs. 11–13 show plastic deformation that characterizes the growth and coalescence behaviors of voids in two grains. Integrated with Fig. 14, it can be found that the value dmin/0.5d in A1 and A3 is relatively small, and the orientation factor’s difference between two grains in A1 and A3 is small. But for A2 and A4, in which the orientation factor’s difference between two grains is larger, the width of inter-void ligaments in cells are larger, and cells show relatively smaller coalescence effects. So it could be concluded that large orientation factor’s difference is unfavorable for void coalescence between grains. Fig. 15 shows the comparison of the value dmin/0.5d for void coalescence at grain boundary and void coalescence between grains. From Fig. 15, it could be found that void coalescence at grain boundary is faster than that between grains. Simulation results show that the plastic deformation for void coalescence between grains is relatively homogeneous, and grain boundaries show a tendency to reduce void coalescence effect, but for void coalescence at grain boundary, the plastic deformation between grains is inhomogeneous, and the void tends to grow along grain boundary. 4. Conclusion Three bicrystal models were created to simulate the coalescence behavior and the plastic deformation distribution around voids under the strain controlled boundary condition, and the rate dependent crystal plasticity theory was applied to investigate void coalescence behaviors. The computed results show: 1. Voids in soft orientation grains tend to coalesce much easier than that in hard orientation under the strain controlled boundary condition. 2. For void coalescence at grain boundary, with the orientation factor’s difference between the two grains increasing, larger deformation mismatch is induced between grains, and the void prefers to coalesce along grain boundary.
3. Large orientation factor’s difference is favorable for void coalescence at grain boundary, unfavorable for void coalescence between grains. It is just because of large orientation factor’s difference in recrystallized grain that voids will grow and coalesce easily along grain boundary due to deformation mismatch, and recrystallized grains prefer to fail by intergranular failure.
Acknowledgements The present work is supported by the National Science Foundation of China (Project No. 50905188), and Hunan Provincial Natural Science Foundation of China (Project Nos. 10JJ3062 and 10JJ8002), which are greatly acknowledged by the authors.
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