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Phase-field Study for Texture Evolution in Polycrystalline Materials under Applied Stress
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ScienceDirect J. Mater. Sci. Technol., 2013, 29(10), 999e1004
Phase-field Study for Texture Evolution in Polycrystalline Materials under Applied Stress Yanli Lu*, Liuchao Zhang, Yingying Zhou, Zheng Chen, Jianguo Zhang State Key Lab of Solidification Processing, Northwestern Polytechnical University, Xi’an 710072, China [Manuscript received March 29, 2013, in revised form May 2, 2013, Available online 9 August 2013]
The formation and evolution of deformation texture in polycrystalline materials are studied by phase-field dynamic model. In addition, the driving force of texture evolution is also discussed. In this model, grains with different orientation are defined by a set of continuous non-conserved order parameter fields. Simulation results show that grains with preferred orientation grow at the expense of those with unfavorable orientations. It is more important that, elastic potential rather than elastic energy plays a crucial role in the evolution of texture whether the polycrystalline system is subjected to uniaxial stress or shear stress. KEY WORDS: Deformation texture; Polycrystalline; Applied stress; Phase-field model
1. Introduction Polycrystalline materials which contain numbers of grains separated by grain boundaries play a very important role in industry. These grains often have various microstructure characteristics such as grain size, morphology, topology and crystallographic orientation. When polycrystalline materials are under applied stress, there is always a pattern in the orientations that are present and a propensity for the occurrence of certain orientations. This tendency is known as texture, which greatly affects material properties. It has been found that the texture, in many cases, influences 20%e50% of the material properties[1]. Some properties that depend on the average texture of a material are as follows: Young’s modulus, Poisson’s ratio, strength, and toughness. Recently, advances in experimental characterization, such as the use of iterative thermomechanical processing (ITMP)[2e5], electron backscattered diffraction (EBSD)[6e8], and computer simulation[9e12], result in an increasing interest in fundamental understanding of the mechanisms underlying texture evolution during grain growth. David et al.[13] studied the formation of texture in a cubic close-packed metal by EBSD. Zhang et al.[14,15] investigated the texture development through statistical modeling
Corresponding author. Assoc. Prof., Ph.D.; Tel.: þ86 29 88460502; E-mail address: [email protected] (Y. Lu). 1005-0302/$ e see front matter Copyright Ó 2013, The editorial office of Journal of Materials Science & Technology. Published by Elsevier Limited. All rights reserved. http://dx.doi.org/10.1016/j.jmst.2013.08.011 *
and Monte Carlo (MC) simulations. Although the effect of texture on properties is exploited to produce materials with specific characteristics or behavior, the exact mechanisms about evolution of particular textures are incompletely understood. The phase-field dynamic model has recently emerged as a powerful computational approach to modeling and predicting microstructure evolution in materials[16e19]. Especially, Chen et al.[20] developed an effective phase-field approach. They uses an iterative-perturbation method to compute the displacement field in a polycrystalline system with arbitrary inhomogeneous distribution of elastic modules. In the present work, this approach is used to study the evolution of deformation texture in polycrystalline materials under applied stress and explore the mechanisms of particular textures evolving. 2. Theoretical Model Phase-field dynamic model[10,20] describes the microstructure of a polycrystalline system by using a set of continuous non-conserved order parameter fields hg ð! r Þ (g represents the ordinal number of corresponding grain and g ¼ 1.Q). The order parameter fields represent grains which have a given crystallographic orientation and the function P 2 ! 4ð! r ; tÞ ¼ Q g ¼ 1 hg ð r ; tÞ is used to distinguish the grain interior and the grain boundaries. In this model, grains are rotated with respect to a fixed coordinate system, so the elastic stiffness tensor of each grain is obtained by transforming the tensor with respect to a fixed coordinate system. Cpqrs denotes r Þ represents the stiffness tensor of the reference medium, Cijkl ð!
1000
Y. Lu et al.: J. Mater. Sci. Technol., 2013, 29(10), 999e1004
the position-dependent stiffness tensor for a single grain in a fixed reference frame and it is given by[20] X Fg agip agjq agkr agls Cpqrs Cijkl ð! rÞ ¼ (1) g
agij
where is the transformation matrix representing the rotation of the coordinate system defined on a given grain g with respect to the fixed reference and is expressed in terms of the Euler angles q, j, and z as follows:
By rearranging and simplifying Eq. (10), we obtain: ) ( ! X v2 uk ð! rÞ h2g ð! C0ijkl ¼ Vj r Þagip agjq agkr agls Cpqrs 3 0kl 3 kl vrj vrl g " ! X v 2 ! g g g g 0 hg ð r Þaip ajq akr als Cpqrs Cijkl vrj g # vuk ð! rÞ vrl (11)
0
agij
1 cos qcos z sin qsin zcos j sin qcos z þ cos qsin zcos j sin zsin j ¼ @ cos qsin z sin qcos zcos j sin qsin z þ cos qcos zcos j cos zsin j A sin qsin j cos qsin j cos j
Here, 0q 2p, 0j 2p, 0z 2p. The spatially dependent elastic stiffness tensor Cijkl ð! r Þ can be written as a sum of a constant homogeneous part C0ijkl and a ’ position-dependent inhomogeneous perturbation Cijkl ð! r Þ. 0 0 Cijkl ð! r Þ ¼ Cijkl þ Cijkl
(3)
r Þ can be rewritten as[20] Thus Cijkl ð! 0 Cijkl ð! r Þ ¼ Cijkl þ
X
h
0 ! g g g g g ð r Þaip ajq akr als Cpqrs
!
0 Cijkl
(4)
g
Let 3 ij ð! r Þ denotes the total strain, when assuming linear elasticity, the local stress sij ð! r Þ can be given by[20] 0 0 sij ð! r Þ ¼ Cijkl þ Cijkl ð! r Þ 3 kl ð! r Þ 3 0kl ð! rÞ (5) By solving the mechanical equilibrium equation, the local elastic field can be obtained, vsij ¼ 0; vrj
i:e:;
Vj Cijkl ð! r Þ 3 kl ð! r Þ 3 0kl ð! rÞ ¼ 0
(6)
(2)
The polycrystalline microstructure at different time is obtained by modified CahneAllen equation[20]: vhg ð! r ; tÞ ¼ Lg ðuch þ uel Þ (12) vt where t represents simulation time step and is dimensionless, uch and uel represent the derivation of chemical and elastic parts of the free energy to hg ð! r Þ[20], vf hg 2kg V2 hg (13) uch ¼ vhg Here, kg is the gradient energy coefficient associated with the order parameter field hg ð! r Þ. When the position-dependent eigen-strain is set to be zero, uel can be written as: dFel rÞ ¼ ¼ hg ð! r Þagip agjq agkr agls Cpqrs d3 ij ð! r Þd3 kl ð! rÞ uel ð!
dhg
þ 3 ij 3 kl þ 2d3 ij ð! r Þ3 kl (14)
The total strain 3 ij ð! r Þ can be expressed as a sum of homogeneous and heterogeneous strains[20]: ! ! 3 ij ð r Þ ¼ 3 ij þ d3 ij ð r Þ (7)
Substituting Eqs. (13) and (14) to Eq. (12), the polycrystalline microstructure at different time is obtained by solving Eq. (12). 3. Results and Discussion
The homogeneous strain Z
3 ij
is defined as:
d3 ij ð! r Þd 3 r ¼ 0
(8)
r Þ is defined by The heterogeneous strain field d3 ij ð! displacement field as: rÞ 1 vui ð! r Þ vuj ð! d3 ij ð! (9) rÞ ¼ þ 2 vrj vri Substituting Eqs. (8) and (9) into Eq. (6), we get: rÞ Vj Cijkl ð!
1 vuk ð! r Þ vul ð! rÞ 3 0kl ð! 3 kl þ þ r Þ ¼ 0 (10) 2 vrl vrk
In our simulation, the system size is chosen to be 512 512, which can ensure enough particles at later stage. The grid size along both Cartesian coordinate axes, Dx, is chosen to be 2.0 and the time step Dt ¼ 0.1. The elastic equilibrium equation (11) is solved by Fourier spectral iterative-perturbation method which is often used in a system with arbitrary elastic inhomogeneity and anisotropy[21]. In Eq. (12) the bulk free energy density is chosen to be X p p X p X 1 1 h2i h2j (15) f h1 ; h2 ; /hp ¼ h2i þ h4i þ 2 4 i¼1 i ¼ 1 j>i which is used in many phase-field simulations[22] and kg in Eq. (13) is chosen as 1.0. We assume the material to be elastically anisotropic with cubic elastic constants, C11 ¼ 450 GPa,
Y. Lu et al.: J. Mater. Sci. Technol., 2013, 29(10), 999e1004
1001
Fig. 1 Microstructure evolution of polycrystalline under uniaxial applied stress sxx ¼ 3.0 GPa: (a) t ¼ 100, (b) t ¼ 700, (c) t ¼ 1400, (d) t ¼ 2100.
C12 ¼ 150 GPa, C44 ¼ 300 GPa. In this case, buck modulus B ¼ 116.6 GPa, shear modulus G ¼ 100 GPa and Young’s modulus E ¼ 233.3 GPa. A system containing about six hundreds of grains with arbitrary asystallographic orientation and equiaxed grain size as initial microstructure is considered. In experimental method, this kind of material can be made by annealing at high temperature for a long time and then air cooling. In our simulation, different colors represent different crystallographic orientations varying from ½p radian to 3/2p radian, as shown in Fig. 1(a). Fig. 1 shows the microstructure evolution of polycrystalline under unaixal stress sxx ¼ 3.0 GPa. At the beginning, grains with different orientations scatter randomly in the system as shown in Fig. 1(a). With increasing time, grains begin to grow up, the number of grains with favor orientation increases and that of grains with unfavorable orientation decreases as shown in Fig. 1(b) and (c). At later stage, grains with favorable orientation become main part of the system as shown in Fig. 1(d). It can also be found that adjacent grains with the same or similar orientation (in this model, q is used to represent the orientation) have a tendency to grow toward each other. Salor et al.[13] employed EBSD mapping system to examine the grain boundary character distribution in a cubic close-packed metal, and they showed that there is a clear tendency for grain boundary to terminate some planes. Our simulation result is coincident with this experimental result, which proves that the simulation result is reasonable.
In order to quantitatively investigate the evolution of grain boundary orientation, the fraction of grain boundary orientation was counted at different time as shown in Fig. 2. It shows that at the initial stage, different grain boundary orientation has almost the same fraction which can be indicated by the almost straight line in blue color. The fraction of favored orientation increases as the time goes while fractions of grains with unfavor orientation decreases. At later stage, the curve has two peaks which shows
Fig. 2 Variation of grain boundary orientation fraction with orientation q at different time.
1002
Y. Lu et al.: J. Mater. Sci. Technol., 2013, 29(10), 999e1004
Fig. 3 Morphology evolution of grain boundary at different time: (a) t ¼ 1700, (b) t ¼ 1800, (c) t ¼ 1900, (d) t ¼ 2000.
that the grains with orientations of about 0.7854 (45 ) and 2.3562 (135 ) become main component in the system. Some grains in Fig. 1 whose positions are in the range of 112e162 in x axis and 323e373 in y axis are considered to investigate the grain boundary movement as shown in Fig. 3. In Fig. 3(a) grain I has an orientation of 0.4566 (26 ), grain II has an orientation about 0.7854 (45 ), grain III and grain IV have
orientations of about 2.3562 (135 ). From the grain boundary movement, it can be observed that grain I becomes smaller while other grains grow larger. Grain II, III, IV with favorable orientations grow at the expense of grain I. To investigate the driving force of grain boundary migration, we studied the elastic energy density (defined by fels ¼ Cijkl ð! r Þ3 elij ð! r Þ3 elkl ð! r Þ) and elastic potential profiles
Fig. 4 (a) Elastic energy density profile of each grain; (b) elastic potential density profile of each grain.
Y. Lu et al.: J. Mater. Sci. Technol., 2013, 29(10), 999e1004
1003
Fig. 5 Elastic potential density and elastic energy of polycrystalline subjected to shear stress sxy ¼ syx ¼ 2.0 GPa: (a) elastic potential density at t ¼ 1400; (b) elastic energy at t ¼ 1400; (c) elastic potential density at t ¼ 2100; (d) elastic energy at t ¼ 2100.
(defined by fpot ¼ ðCijkl ð! r Þ Cpqrs Þ3 elij ð! r Þ3 elkl ð! r Þ) of different grains as shown in Fig. 4. From color contrast, it can be observed that the elastic energy density in grain II, III, IV with preferred orientation is larger than that in grain I with unfavorable orientation as shown in Fig. 4(a). From Fig. 4(b), it can be observed that grain I with unfavorable orientation has higher elastic potential profile, grain I gradually decreases in order to reduce the total energy. Therefore, it can be concluded that the elastic potential plays a more critical role in deciding the movement of grain boundary and grains with low elastic potential density grows up first. Texture evolution of polycrystalline subjected to shear stress sxy ¼ syx ¼ 2.0 GPa was also investigated as shown in Fig. 5. In Fig. 5(a), grain A has high elastic potential density, while Fig. 5(b) shows that it has low elastic energy, when time goes, it decreases to reduce the total energy of the polycrystalline system, which indicates that elastic potential plays a more critical role in deciding the movement of grain boundary when the polycrystalline system is subjected to shear stress. Fig. 5 also shows that three grains marked as B1, B2 and B3 respectively evolved into one grain marked as B at the final stage, which is due to their similar orientation. 4. Conclusion A phase-field dynamic model is used to study the formation and evolution of texture in polycrystalline under applied stress and investigate the mechanisms underlying it. It is observed that the grains with preferred crystallographic orientation have an
advancing in growth and grow at the expense of grains with unfavorable crystallographic orientation. Grains with similar orientations have a tendency to grow toward each other and merge into one grain. Elastic potential is a key parameter in deciding the evolution of texture whether the polycrystalline system is subjected to uniaxial stress or shear stress. Acknowledgments The authors appreciate the financial support from the National Natural Science Foundation of China (Grant Nos. 51174168, 51274167) and NPU Foundation for Fundamental Research (Grant No. JC20120222). The authors sincerely thank Prof. Long-Qing Chen in Department of Materials Science and Engineering, Penn State University of USA for his helpful discussion. REFERENCES [1] Olaf Engler, Valerie Randle, Introduction to Texture Analysis: Macrotexture, Microtexture, and Orientation Mapping, second ed., Taylor & Francis, New York, 2010. [2] J. Gruber, A.D. Rollett, G.S. Rohrer, Acta Mater. 58 (2010) 14e19. [3] S.Y. Li, L. Yang, L.M. Lei, C.Y. Wei, H.R. Zhang, J. Mater. Sci. Technol. 28 (2012) 1015e1024. [4] C.S. Kim, Y. Hu, G.S. Rohrer, V. Randle, Scripta Mater. 52 (2005) 633e637. [5] Z.P. Zeng, Y.S. Zhang, S. Jonsson, Mater. Sci. Eng. A 513e514 (2009) 83e90. [6] J.N. Deng, Y.L. Yang, Y.D. Wang, J.G. Chen, R.L. Peng, J. Mater. Sci. Technol. 25 (2009) 219e224.
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[7] G.S. Rohrer, V. Randle, C.S. Kim, Y. Hu, Acta Mater. 54 (2006) 4489e4502. [8] J. Gruber, H.M. Miller, T.D. Miller, G.S. Rohrer, A.D. Rollett, Acta Mater. 57 (2009) 6102e6112. [9] N. Moelans, B. Blanpain, P. Wollants, Comput. Coupling Phase Diagrams Thermochemistry 32 (2008) 268e294. [10] L.Q. Chen, Annu. Rev. Mater. Res. 32 (2002) 113e140. [11] M. Greenwood, N. Provatas, J. Rottler, Phys. Rev. Lett. 105 (2010) 045702. [12] E. Biyikli, D. Canadinc, H.J. Maier, T. Niendorf, S. Top, Mater. Sci. Eng. A 527 (2010) 5604e5612. [13] D.M. Saylor, B.S. El Dasher, A.D. Rollett, G.S. Rohrer, Acta Mater. 52 (2004) 3649e3655. [14] L.Z. Zhang, A.D. Rollett, T. Bartel, D. Wu, M.T. Lusk, Acta Mater. 60 (2012) 1201e1210.
[15] L.Z. Zhang, T. Bartel, M.T. Lusk, Comp. Mater. Sci. 48 (2010) 419e425. [16] M.A. Zaeem, H.B. Yin, S.D. Felicelli, J. Mater. Sci. Technol. 28 (2012) 137e146. [17] M.T. Wang, B.Y. Zong, G. Wang, J. Mater. Sci. Technol. 24 (2008) 829e834. [18] G. Choudhuri, S. Chakraborty, D. Srivastava, G.K. Dey, Results Phys. 3 (2013) 7e13. [19] H.K. Yeddu, A. Borgenstam, J. Agren, Acta Mater. 61 (2013) 2595e2606. [20] S. Bhattacharyya, T.W. Heo, K. Chang, L.Q. Chen, Model. Simul. Mater. Sci. Eng. 19 (2011) 035002. [21] P. Yu, S.Y. Hu, L.Q. Chen, Q. Du, J. Comp. Phys. 208 (2005) 34e50. [22] L.Q. Chen, W. Yang, Phys. Rev. B 50 (1994) 752e756.
ScienceDirect J. Mater. Sci. Technol., 2013, 29(10), 999e1004
Phase-field Study for Texture Evolution in Polycrystalline Materials under Applied Stress Yanli Lu*, Liuchao Zhang, Yingying Zhou, Zheng Chen, Jianguo Zhang State Key Lab of Solidification Processing, Northwestern Polytechnical University, Xi’an 710072, China [Manuscript received March 29, 2013, in revised form May 2, 2013, Available online 9 August 2013]
The formation and evolution of deformation texture in polycrystalline materials are studied by phase-field dynamic model. In addition, the driving force of texture evolution is also discussed. In this model, grains with different orientation are defined by a set of continuous non-conserved order parameter fields. Simulation results show that grains with preferred orientation grow at the expense of those with unfavorable orientations. It is more important that, elastic potential rather than elastic energy plays a crucial role in the evolution of texture whether the polycrystalline system is subjected to uniaxial stress or shear stress. KEY WORDS: Deformation texture; Polycrystalline; Applied stress; Phase-field model
1. Introduction Polycrystalline materials which contain numbers of grains separated by grain boundaries play a very important role in industry. These grains often have various microstructure characteristics such as grain size, morphology, topology and crystallographic orientation. When polycrystalline materials are under applied stress, there is always a pattern in the orientations that are present and a propensity for the occurrence of certain orientations. This tendency is known as texture, which greatly affects material properties. It has been found that the texture, in many cases, influences 20%e50% of the material properties[1]. Some properties that depend on the average texture of a material are as follows: Young’s modulus, Poisson’s ratio, strength, and toughness. Recently, advances in experimental characterization, such as the use of iterative thermomechanical processing (ITMP)[2e5], electron backscattered diffraction (EBSD)[6e8], and computer simulation[9e12], result in an increasing interest in fundamental understanding of the mechanisms underlying texture evolution during grain growth. David et al.[13] studied the formation of texture in a cubic close-packed metal by EBSD. Zhang et al.[14,15] investigated the texture development through statistical modeling
Corresponding author. Assoc. Prof., Ph.D.; Tel.: þ86 29 88460502; E-mail address: [email protected] (Y. Lu). 1005-0302/$ e see front matter Copyright Ó 2013, The editorial office of Journal of Materials Science & Technology. Published by Elsevier Limited. All rights reserved. http://dx.doi.org/10.1016/j.jmst.2013.08.011 *
and Monte Carlo (MC) simulations. Although the effect of texture on properties is exploited to produce materials with specific characteristics or behavior, the exact mechanisms about evolution of particular textures are incompletely understood. The phase-field dynamic model has recently emerged as a powerful computational approach to modeling and predicting microstructure evolution in materials[16e19]. Especially, Chen et al.[20] developed an effective phase-field approach. They uses an iterative-perturbation method to compute the displacement field in a polycrystalline system with arbitrary inhomogeneous distribution of elastic modules. In the present work, this approach is used to study the evolution of deformation texture in polycrystalline materials under applied stress and explore the mechanisms of particular textures evolving. 2. Theoretical Model Phase-field dynamic model[10,20] describes the microstructure of a polycrystalline system by using a set of continuous non-conserved order parameter fields hg ð! r Þ (g represents the ordinal number of corresponding grain and g ¼ 1.Q). The order parameter fields represent grains which have a given crystallographic orientation and the function P 2 ! 4ð! r ; tÞ ¼ Q g ¼ 1 hg ð r ; tÞ is used to distinguish the grain interior and the grain boundaries. In this model, grains are rotated with respect to a fixed coordinate system, so the elastic stiffness tensor of each grain is obtained by transforming the tensor with respect to a fixed coordinate system. Cpqrs denotes r Þ represents the stiffness tensor of the reference medium, Cijkl ð!
1000
Y. Lu et al.: J. Mater. Sci. Technol., 2013, 29(10), 999e1004
the position-dependent stiffness tensor for a single grain in a fixed reference frame and it is given by[20] X Fg agip agjq agkr agls Cpqrs Cijkl ð! rÞ ¼ (1) g
agij
where is the transformation matrix representing the rotation of the coordinate system defined on a given grain g with respect to the fixed reference and is expressed in terms of the Euler angles q, j, and z as follows:
By rearranging and simplifying Eq. (10), we obtain: ) ( ! X v2 uk ð! rÞ h2g ð! C0ijkl ¼ Vj r Þagip agjq agkr agls Cpqrs 3 0kl 3 kl vrj vrl g " ! X v 2 ! g g g g 0 hg ð r Þaip ajq akr als Cpqrs Cijkl vrj g # vuk ð! rÞ vrl (11)
0
agij
1 cos qcos z sin qsin zcos j sin qcos z þ cos qsin zcos j sin zsin j ¼ @ cos qsin z sin qcos zcos j sin qsin z þ cos qcos zcos j cos zsin j A sin qsin j cos qsin j cos j
Here, 0q 2p, 0j 2p, 0z 2p. The spatially dependent elastic stiffness tensor Cijkl ð! r Þ can be written as a sum of a constant homogeneous part C0ijkl and a ’ position-dependent inhomogeneous perturbation Cijkl ð! r Þ. 0 0 Cijkl ð! r Þ ¼ Cijkl þ Cijkl
(3)
r Þ can be rewritten as[20] Thus Cijkl ð! 0 Cijkl ð! r Þ ¼ Cijkl þ
X
h
0 ! g g g g g ð r Þaip ajq akr als Cpqrs
!
0 Cijkl
(4)
g
Let 3 ij ð! r Þ denotes the total strain, when assuming linear elasticity, the local stress sij ð! r Þ can be given by[20] 0 0 sij ð! r Þ ¼ Cijkl þ Cijkl ð! r Þ 3 kl ð! r Þ 3 0kl ð! rÞ (5) By solving the mechanical equilibrium equation, the local elastic field can be obtained, vsij ¼ 0; vrj
i:e:;
Vj Cijkl ð! r Þ 3 kl ð! r Þ 3 0kl ð! rÞ ¼ 0
(6)
(2)
The polycrystalline microstructure at different time is obtained by modified CahneAllen equation[20]: vhg ð! r ; tÞ ¼ Lg ðuch þ uel Þ (12) vt where t represents simulation time step and is dimensionless, uch and uel represent the derivation of chemical and elastic parts of the free energy to hg ð! r Þ[20], vf hg 2kg V2 hg (13) uch ¼ vhg Here, kg is the gradient energy coefficient associated with the order parameter field hg ð! r Þ. When the position-dependent eigen-strain is set to be zero, uel can be written as: dFel rÞ ¼ ¼ hg ð! r Þagip agjq agkr agls Cpqrs d3 ij ð! r Þd3 kl ð! rÞ uel ð!
dhg
þ 3 ij 3 kl þ 2d3 ij ð! r Þ3 kl (14)
The total strain 3 ij ð! r Þ can be expressed as a sum of homogeneous and heterogeneous strains[20]: ! ! 3 ij ð r Þ ¼ 3 ij þ d3 ij ð r Þ (7)
Substituting Eqs. (13) and (14) to Eq. (12), the polycrystalline microstructure at different time is obtained by solving Eq. (12). 3. Results and Discussion
The homogeneous strain Z
3 ij
is defined as:
d3 ij ð! r Þd 3 r ¼ 0
(8)
r Þ is defined by The heterogeneous strain field d3 ij ð! displacement field as: rÞ 1 vui ð! r Þ vuj ð! d3 ij ð! (9) rÞ ¼ þ 2 vrj vri Substituting Eqs. (8) and (9) into Eq. (6), we get: rÞ Vj Cijkl ð!
1 vuk ð! r Þ vul ð! rÞ 3 0kl ð! 3 kl þ þ r Þ ¼ 0 (10) 2 vrl vrk
In our simulation, the system size is chosen to be 512 512, which can ensure enough particles at later stage. The grid size along both Cartesian coordinate axes, Dx, is chosen to be 2.0 and the time step Dt ¼ 0.1. The elastic equilibrium equation (11) is solved by Fourier spectral iterative-perturbation method which is often used in a system with arbitrary elastic inhomogeneity and anisotropy[21]. In Eq. (12) the bulk free energy density is chosen to be X p p X p X 1 1 h2i h2j (15) f h1 ; h2 ; /hp ¼ h2i þ h4i þ 2 4 i¼1 i ¼ 1 j>i which is used in many phase-field simulations[22] and kg in Eq. (13) is chosen as 1.0. We assume the material to be elastically anisotropic with cubic elastic constants, C11 ¼ 450 GPa,
Y. Lu et al.: J. Mater. Sci. Technol., 2013, 29(10), 999e1004
1001
Fig. 1 Microstructure evolution of polycrystalline under uniaxial applied stress sxx ¼ 3.0 GPa: (a) t ¼ 100, (b) t ¼ 700, (c) t ¼ 1400, (d) t ¼ 2100.
C12 ¼ 150 GPa, C44 ¼ 300 GPa. In this case, buck modulus B ¼ 116.6 GPa, shear modulus G ¼ 100 GPa and Young’s modulus E ¼ 233.3 GPa. A system containing about six hundreds of grains with arbitrary asystallographic orientation and equiaxed grain size as initial microstructure is considered. In experimental method, this kind of material can be made by annealing at high temperature for a long time and then air cooling. In our simulation, different colors represent different crystallographic orientations varying from ½p radian to 3/2p radian, as shown in Fig. 1(a). Fig. 1 shows the microstructure evolution of polycrystalline under unaixal stress sxx ¼ 3.0 GPa. At the beginning, grains with different orientations scatter randomly in the system as shown in Fig. 1(a). With increasing time, grains begin to grow up, the number of grains with favor orientation increases and that of grains with unfavorable orientation decreases as shown in Fig. 1(b) and (c). At later stage, grains with favorable orientation become main part of the system as shown in Fig. 1(d). It can also be found that adjacent grains with the same or similar orientation (in this model, q is used to represent the orientation) have a tendency to grow toward each other. Salor et al.[13] employed EBSD mapping system to examine the grain boundary character distribution in a cubic close-packed metal, and they showed that there is a clear tendency for grain boundary to terminate some planes. Our simulation result is coincident with this experimental result, which proves that the simulation result is reasonable.
In order to quantitatively investigate the evolution of grain boundary orientation, the fraction of grain boundary orientation was counted at different time as shown in Fig. 2. It shows that at the initial stage, different grain boundary orientation has almost the same fraction which can be indicated by the almost straight line in blue color. The fraction of favored orientation increases as the time goes while fractions of grains with unfavor orientation decreases. At later stage, the curve has two peaks which shows
Fig. 2 Variation of grain boundary orientation fraction with orientation q at different time.
1002
Y. Lu et al.: J. Mater. Sci. Technol., 2013, 29(10), 999e1004
Fig. 3 Morphology evolution of grain boundary at different time: (a) t ¼ 1700, (b) t ¼ 1800, (c) t ¼ 1900, (d) t ¼ 2000.
that the grains with orientations of about 0.7854 (45 ) and 2.3562 (135 ) become main component in the system. Some grains in Fig. 1 whose positions are in the range of 112e162 in x axis and 323e373 in y axis are considered to investigate the grain boundary movement as shown in Fig. 3. In Fig. 3(a) grain I has an orientation of 0.4566 (26 ), grain II has an orientation about 0.7854 (45 ), grain III and grain IV have
orientations of about 2.3562 (135 ). From the grain boundary movement, it can be observed that grain I becomes smaller while other grains grow larger. Grain II, III, IV with favorable orientations grow at the expense of grain I. To investigate the driving force of grain boundary migration, we studied the elastic energy density (defined by fels ¼ Cijkl ð! r Þ3 elij ð! r Þ3 elkl ð! r Þ) and elastic potential profiles
Fig. 4 (a) Elastic energy density profile of each grain; (b) elastic potential density profile of each grain.
Y. Lu et al.: J. Mater. Sci. Technol., 2013, 29(10), 999e1004
1003
Fig. 5 Elastic potential density and elastic energy of polycrystalline subjected to shear stress sxy ¼ syx ¼ 2.0 GPa: (a) elastic potential density at t ¼ 1400; (b) elastic energy at t ¼ 1400; (c) elastic potential density at t ¼ 2100; (d) elastic energy at t ¼ 2100.
(defined by fpot ¼ ðCijkl ð! r Þ Cpqrs Þ3 elij ð! r Þ3 elkl ð! r Þ) of different grains as shown in Fig. 4. From color contrast, it can be observed that the elastic energy density in grain II, III, IV with preferred orientation is larger than that in grain I with unfavorable orientation as shown in Fig. 4(a). From Fig. 4(b), it can be observed that grain I with unfavorable orientation has higher elastic potential profile, grain I gradually decreases in order to reduce the total energy. Therefore, it can be concluded that the elastic potential plays a more critical role in deciding the movement of grain boundary and grains with low elastic potential density grows up first. Texture evolution of polycrystalline subjected to shear stress sxy ¼ syx ¼ 2.0 GPa was also investigated as shown in Fig. 5. In Fig. 5(a), grain A has high elastic potential density, while Fig. 5(b) shows that it has low elastic energy, when time goes, it decreases to reduce the total energy of the polycrystalline system, which indicates that elastic potential plays a more critical role in deciding the movement of grain boundary when the polycrystalline system is subjected to shear stress. Fig. 5 also shows that three grains marked as B1, B2 and B3 respectively evolved into one grain marked as B at the final stage, which is due to their similar orientation. 4. Conclusion A phase-field dynamic model is used to study the formation and evolution of texture in polycrystalline under applied stress and investigate the mechanisms underlying it. It is observed that the grains with preferred crystallographic orientation have an
advancing in growth and grow at the expense of grains with unfavorable crystallographic orientation. Grains with similar orientations have a tendency to grow toward each other and merge into one grain. Elastic potential is a key parameter in deciding the evolution of texture whether the polycrystalline system is subjected to uniaxial stress or shear stress. Acknowledgments The authors appreciate the financial support from the National Natural Science Foundation of China (Grant Nos. 51174168, 51274167) and NPU Foundation for Fundamental Research (Grant No. JC20120222). The authors sincerely thank Prof. Long-Qing Chen in Department of Materials Science and Engineering, Penn State University of USA for his helpful discussion. REFERENCES [1] Olaf Engler, Valerie Randle, Introduction to Texture Analysis: Macrotexture, Microtexture, and Orientation Mapping, second ed., Taylor & Francis, New York, 2010. [2] J. Gruber, A.D. Rollett, G.S. Rohrer, Acta Mater. 58 (2010) 14e19. [3] S.Y. Li, L. Yang, L.M. Lei, C.Y. Wei, H.R. Zhang, J. Mater. Sci. Technol. 28 (2012) 1015e1024. [4] C.S. Kim, Y. Hu, G.S. Rohrer, V. Randle, Scripta Mater. 52 (2005) 633e637. [5] Z.P. Zeng, Y.S. Zhang, S. Jonsson, Mater. Sci. Eng. A 513e514 (2009) 83e90. [6] J.N. Deng, Y.L. Yang, Y.D. Wang, J.G. Chen, R.L. Peng, J. Mater. Sci. Technol. 25 (2009) 219e224.
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