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Advanced computer simulation of polycrystalline microstructure
Comput. Methods Appl. Mech. Engrg. 191 (2002) 3651–3667 www.elsevier.com/locate/cma
Advanced computer simulation of polycrystalline microstructure Sankaran Mahadevan *, Yaowu Zhao Department of Civil and Environmental Engineering, Vanderbilt University, Nashville, TN 37235, USA Received 12 January 2001; accepted 19 February 2002
Abstract This paper proposes an advanced two-dimensional method to simulate the microstructure of metallic dual-phase polycrystalline materials to enable accurate micro-mechanics computation. This is done by combining geometric and metallurgical principles to simulate microstructures that are closer to experimental observations than currently used methods. The method accounts for difference in grain growth velocities in different phases during solidification. If the ratio of grain growth velocities equals 1, the proposed method reduces to the currently used Voronoi tessellation method. If the ratio of grain growth velocities tends to 1, the method matches the physical situation of one grain type becoming embedded in the other grain type. The simulation results show that although the original grain core points are the same, the microstructure of the material will change with the ratio of grain growth velocities, including the lengths of the grain boundaries, the radii of curvature of the grain boundaries, and the triple point locations. Statistical analyses are conducted on the number of grain edges, grain diameter, grain area, and proportion of two phases of grains. Statistical analysis shows that the simulation results of the proposed method agree well with the experimental observation. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Microstructure; Grain; Grain growth; Velocity; Simulation; Alloys; Dual phase; Solidification; Tessellation
1. Introduction Simulation of the microstructure of modern alloys is an important step in the computational modeling of mechanical behavior and damage evolution. Alloys are often multiphase, or when they are single phase they are usually polycrystalline. The development of new materials and processes as well as the optimization of classical materials requires modeling more closely related to the microstructure [1]. Voronoi-based tessellation methods have been pursued for microstructure simulation [2–4], which are then used in micromechanics analysis [5–9]. The Voronoi-based methods yield microstructures where ‘‘all edges are straight lines, all cells are convex and each cell has at least three edges’’. Such microstructures are appropriate only
*
Corresponding author. Tel.: +1-615-322-3040. E-mail address: [email protected] (S. Mahadevan).
0045-7825/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 4 5 - 7 8 2 5 ( 0 2 ) 0 0 2 6 0 - 8
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Fig. 1. Optical photographs of different materials: (a) microstructure of a polished section of Al alloy at a magnification of 60 [10], (b) Ti alloy micro-structure [11], (c) microstructure of a polished section of Al alloy at a magnification of 60 [10], (d) distribution of spherical d0 in Al alloy [10].
for single-phase materials with all grains growing with the same velocity. Some studies [10–13] have presented two-dimensional optical photographs (Fig. 1) of different metal alloys. From these observations, it appears that Voronoi-based tessellation methods with straight boundaries and convex polygons do not really reflect the actual microstructure. Experimental evidence suggests that in order to accurately model phenomena such as recrystallization, shear localization and fracture, more detailed information concerning the effect of the metallurgical process on the geometry of the microstructure must be incorporated into the computational models [14–16]. In addition, it was found that a grain can have influence on the deformation field several grain diameters away [17]. This paper proposes a generalized method to simulate the two-dimensional microstructure of metallic materials that accounts for the aforementioned effects. The grains are allowed to grow with different velocities depending on phase difference, temperature variation, etc. If all the grain growth velocities are equal, then the method reduces to the Voronoi-based tessellation method. If the ratio of grain growth velocity is very large, then one phase gets embedded in the other phase. Between these two limits, this method will obtain realistic simulation of actual materials. This proposed method yields microstructures where the grains have both curved and straight edges, are not necessarily convex, and the minimum number of edges is only one. The simulation results show that although the original grain core points are the same, the microstructure of the material will change with the ratio of grain growth velocities, including the lengths of the grain boundaries, the radii of curvature of the grain boundaries, and the triple point locations of the grains. Statistical analyses are conducted on simulated microstructure results, such as the number of grain
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edges, grain diameter, grain area, and proportion of two phases of grains. Statistical analyses show that the results of the proposed method agree well with experimental observation.
2. Proposed method 2.1. Grain growth assumptions On the basis of a simplified three-dimensional model of the solidification process from liquid to solid phase of metallic materials, and some assumptions about grain growth, this section develops a method to simulate the microstructure and the grain growth process. Following are the basic assumptions: 1. The grain growth is isotropic; that is, if the grain core is the center of a circle, then the grain will grow at the same velocity in all directions. The grain growth will stop to form a grain boundary in one direction when it meets another growing grain boundary, but the grain will continue to grow along other directions. The grain growth process will end when its growth stops in all directions. 2. The velocity of grain growth is different in the different phases in the alloy. 3. After a boundary is formed between two grains, the grain boundary is unmovable and unchangeable. 4. There is no grain growth in the area of an already existing grain. That is, the grains are not allowed to overlap. 2.2. Grain boundary description The following observations can be made regarding the geometry of the microstructure and the grain boundaries, as a result of the above grain growth assumptions: (1) If two adjacent grains are in the same phase and they grow at the same velocity, then their grain boundary is a straight line, given by a mid-perpendicular (vertical line in Fig. 2), and the mid-perpendicular’s equation could be expressed as r1 ¼ r2 ;
ð1Þ
where r1 and r2 are the distances from a point s1 on the boundary to the grain core points A1 and A2 respectively. (2) If three adjacent grains are in the same phase and grow at equal velocity, then the grain boundary between every two grains is the mid-perpendicular, and the three grain boundaries will meet at one point
Fig. 2. Growth of two same-phase grains.
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Fig. 3. Growth of three same-phase grains.
(referred to as the triple point; Point T in Fig. 3). The location of the triple point could be obtained through the intersection of any two of the three mid-perpendiculars. (3) If two adjacent grains are in different phases, they will grow at different velocities (Fig. 4). Assume two grains with core points A1ðx1 ; y1 Þ and A2ðx2 ; y2 Þ grow at different velocities v1 , and v2 respectively, and v1 > v2 . Under the grain growth assumptions in Section 2.1, the grain boundary may be expressed as t1 ¼ t2 ;
ð2Þ
where t1 is the time of grain growth from core point A1ðx1 ; y1 Þ to the boundary, t2 is the time of grain growth from core point A2ðx2 ; y2 Þ to the boundary. Eq. (2) may be rewritten as l1 l2 ¼ ; v1 v2
ð3Þ
where l1 and l2 are the distances from the point s1 on the boundary to the grain core points A1 and A2 respectively. Let k ¼ ðv1 v2 Þ=v2 , then, Eq. (3) could be expressed as l1 ¼ ð1 þ kÞl2 ;
ð4Þ
Fig. 4. Growth of two grains in different phases with unequal velocities.
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where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 < l1 ¼ ðx x1 Þ2 þ ðy y1 Þ2 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : l ¼ ðx x Þ2 þ ðy y Þ2 : 2
2
2
Substitution of l1 and l2 into Eq. (4) leads to the following equation of a circle for the point s1ðx; yÞ, with center at Oða; bÞ and radius R: ðx aÞ2 þ ðy bÞ2 ¼ R2 ; where 8 ð1 þ kÞ2 x2 x1 > > > a¼ > 2 > > ð1 þ kÞ 1 > > > 2 > < ð1 þ kÞ y2 y1 b¼ ð1 þ kÞ2 1 > > > 2 > > ð1 þ kÞ > 2 2 > R ¼ > h i2 ½ðx2 x1 Þ þ ðy2 y1 Þ > > 2 : ð1 þ kÞ 1
ð5Þ
ðk 6¼ 0Þ
ð6Þ
However, considering the assumption in Section 2.1 that the grains cannot overlap, then the grain from A1 grows around the grain from A2. In Fig. 4, G1 and G2 are the tangent points from A1 to this circle. For a point s2ðx; yÞ on the boundary which is not between two grain core points A1 and A2, Eq. (2) will result in the following expression: l0 þ d1 d2 ¼ ; v1 v2
ð7Þ
where l0 ¼ jA1G1j which is a constant, d1 is the length of the arc G1Bs2, and d2 ¼ jA2s2j. Then Eq. (7) can be written as fd2 d1 ¼ l0 ;
ð8Þ
where f ¼ v1 =v2 . Eq. (8) has to be solved through numerical integration. For the reason of simplicity, we use part of an ellipse to approximate the curve of Eq. (8). The corresponding ellipse equation is ðx aÞ ð2RÞ2
2
2
þ
ðy bÞ ¼ 1; R2
ð9Þ
where a, b and R are the same as in Eq. (6), C is a constant with C > 1. Here, for simplicity, we arbitrarily choose C ¼ 2. Therefore, the grain boundary in Fig. 4 is the bold curve which is part of a circle on the left and part of an ellipse in the right. (4) In a two-phase material, some of the adjacent grains may be in different phases, and some other pairs may be in the same phase. Then the boundary between the same phase grains is the mid-perpendicular straight line, and the boundary between the grains in different phases is a curve. Consider three adjacent grains A1, A2 and A3, with core points ðx1 ; y1 Þ, ðx2 ; y2 Þ and ðx3 ; y3 Þ respectively. Assume that A1 and A2 are in the same phase, and A3 in another phase. Several cases may be considered as in Fig. 5, and the corresponding grain boundary equations are divided and listed in Table 1.
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Fig. 5. Grain boundaries between three adjacent grains (A1 and A2 with velocity v1 , A3 with velocity v2 ): (a) v1 > v2 , (b) v2 > v1 , (c) long growth time between A1 and A2, and v1 > v2 , (d) long growth time between A1 and A2, and v2 > v1 , and (e) the location of A3 is between A1 and A2, and v1 > v2 .
2.3. Proposed microstructure simulation algorithm In two-dimensional analysis, an arbitrary grain shape is an irregular polygon, whose boundary is formed during the solidification phase from liquid to solid stage. If there are other grains surrounding a grain, the grain is referred to as an interior grain in the following discussion, and a grain along the cell boundary is referred to as a border grain. The steps of the proposed method for the simulation of the microstructure are as follows: 1. Use Monte Carlo sampling to generate the core points of the grains in a given area. (Here, it is assumed that the area is a square and the edge length equals 1 unit.) 2. Randomly choose one point (for example, point ðx1 ; y1 Þ) as the starting point O (Fig. 6), then calculate the time to grow to the other points and get the shortest time core point A, say point ðx2 ; y2 Þ. Note that different grains have different growth velocities. 3. Compute the angles of the other grain core points with the line OA. Then form an array B of size (n 2), arranged according to increasing angle values in the counter-clockwise direction. 4. Assuming B, the first point in this array, is an adjacent core point, find the location of the triple point T of the triangle OAB. 5. Calculate the growth times of the other core points from the triple point T. 6. If any of these growth times is less than the time of OT, then B is not an adjacent core point. Replace B with the next core point in the array B and repeat steps 3–5. 7. If all the growth times computed in step 5 are greater than the time of OT, then B is an adjacent core point to O. Now replace core point A with B and repeat the above process with OB as the reference line.
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Table 1 Equations to determine boundaries of three adjacent dual-phase grains Case
Equation
Case 1: (Fig. 5(a)) A1 and A2 with v1 , A3 with v2 , and v1 > v2
Case 2: (Fig. 5(b)) A1 and A2 with v1 , A3 with v2 , and v2 > v1
Explanation
ðx aÞ2 þ ðy bÞ2 ¼ R2 r1 ¼ r2
ðx aÞ2 þ ðy bÞ2 ¼ R2 r1 ¼ r2
8 > ð1 þ kÞ2 x3 x1 > > a¼ > > > ð1 þ kÞ2 1 > > >
> > < ð1 þ kÞ2 y3 y1 v1 v2 b¼ 6¼ 0 k¼ 2 v2 ð1 þ kÞ 1 > > > > 2 > > ð1 þ kÞ > > ½ðx3 x1 Þ2 þ ðy3 y1 Þ2 R¼ > > > ½ð1 þ kÞ2 1 2 : 8 ð1 þ kÞ2 x1 x3 > > a¼ > > > ð1 þ kÞ2 1 > > >
> > < ð1 þ kÞ2 y1 y3 v1 v2 b¼ 6¼ 0 k¼ 2 v2 ð1 þ kÞ 1 > > > > 2 > > ð1 þ kÞ > > ½ðx1 x3 Þ2 þ ðy1 y3 Þ2 R¼ > > : ½ð1 þ kÞ2 1 2
Case 3: (Fig. 5(c)) A1 and A2 with v1 , A3 with v2 , and v1 > v2 (long growth time between A1 and A2)
8 2 2 > < ðx a1 Þ þ ðy b1 Þ ¼ 1 R21 ð2R1 Þ2 > : 2 ðx a2 Þ þ ðx b2 Þ2 ¼ R22
8 > ð1 þ kÞ2 x3 x1 > > ¼ a 1 > > > ð1 þ kÞ2 1 > > >
> > < ð1 þ kÞ2 y3 y1 v1 v2 k ¼ 6 ¼ 0 b1 ¼ v2 ð1 þ kÞ2 1 > > > > 2 > > ð1 þ kÞ > > ½ðx3 x1 Þ2 þ ðy3 y1 Þ2 R1 ¼ > > > ½ð1 þ kÞ2 1 2 : 8 > ð1 þ kÞ2 x3 x2 > > a2 ¼ > > > ð1 þ kÞ2 1 > > >
> > < ð1 þ kÞ2 y3 y2 v1 v2 k¼ b2 ¼ 6 0 ¼ 2 v2 ð1 þ kÞ 1 > > > > 2 > > ð1 þ kÞ >R ¼ > ½ðx3 x2 Þ2 þ ðy3 y2 Þ2 > 2 > > ½ð1 þ kÞ2 1 2 :
Case 4: (Fig. 5(d)) A1 and A2 with v1 , A3 with v2 , and v2 > v1 , (long growth time between A1 and A2)
8 2 2 > < ðx a1 Þ þ ðy b1 Þ ðx a1 Þ2 ðy b1 Þ2 > þ : 2 R21 8 ð2R1 Þ 2 > ðx a2 Þ þ ðy b2 Þ2 < ðx a2 Þ2 ðy b2 Þ2 > þ : R22 ð2R2 Þ2
8 > ð1 þ kÞ2 x1 x3 > > a ¼ 1 > > > ð1 þ kÞ2 1 > > >
> > < ð1 þ kÞ2 y1 y3 v1 v2 6¼ 0 k¼ b1 ¼ 2 v2 ð1 þ kÞ 1 > > > > 2 > > ð1 þ kÞ > > ½ðx1 x3 Þ2 þ ðy1 y3 Þ2 R1 ¼ > > > ½ð1 þ kÞ2 1 2 :
¼ R21 ¼1 ¼ R22 ¼1
8 ð1 þ kÞ2 x2 x3 > > > a2 ¼ > > > ð1 þ kÞ2 1 > > > >
> > ð1 þ kÞ2 y2 y3 v1 v2 < k ¼ b2 ¼ ¼ 6 0 v2 ð1 þ kÞ2 1 > > > > > > ð1 þ kÞ2 > > > ½ðx2 x3 Þ2 þ ðy2 y3 Þ2 R2 ¼ > > > ½ð1 þ kÞ2 1 2 : (continued on next page)
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Table 1 (continued) Case
Equation
Case 5: (Fig. 5(e)) A1 and A2 with v1 , A3 with v2 , and v1 > v2
Explanation 2
2
ðx a1 Þ þ ðy b1 Þ ¼ R21 ðx a2 Þ2 þ ðy b2 Þ2 ¼ R22
8 ð1 þ kÞ2 x3 x1 > > ¼ a > 1 > > ð1 þ kÞ2 1 > > >
> > < ð1 þ kÞ2 y3 y1 v1 v2 k ¼ b1 ¼ ¼ 6 0 v2 ð1 þ kÞ2 1 > > > > 2 > > ð1 þ kÞ > > ½ðx3 x1 Þ2 þ ðy3 y1 Þ2 R1 ¼ > > : ½ð1 þ kÞ2 1 2 8 ð1 þ kÞ2 x3 x2 > > > a2 ¼ > > ð1 þ kÞ2 1 > > >
> > < ð1 þ kÞ2 y3 y2 v1 v2 6¼ 0 k¼ b2 ¼ 2 v2 ð1 þ kÞ 1 > > > > 2 > > ð1 þ kÞ > > R2 ¼ ½ðx3 x2 Þ2 þ ðy3 y2 Þ2 > > : ½ð1 þ kÞ2 1 2
Fig. 6. Process of forming the grain boundary.
8. After identifying all the triple points of the starting point O, choose another point as a starting point and repeat steps 2–7 until all the core points are explored. 9. Plot the polygons connecting the identified triple points corresponding to each grain core point, to generate the microstructure. 3. Simulation results The microstructure simulation results are shown in Figs. 7 and 8 with different numbers of grains and with different ratios of grain growth velocities. Fig. 7 considers 16 grains, and Fig. 8 considers 32 grains. The ratio of grain growth velocities (f ¼ v1 =v2 ) ranges from 1 to 15 in both figures. Figs. 7(h) and 8(h) are the ideal honeycombs used in some studies [8]. 3.1. Statistical evaluation of the simulated microstructure In order to check the effectiveness of the above simulation of the metal alloy microstructure, it is necessary to perform statistical analysis of the grain edge, grain size, etc. This is discussed in this section.
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Fig. 7. Simulated micro-structure for different grain growth velocity ratios (for 16 grains): (a) v1 ¼ v2 (single phase), (b) v1 ¼ 1:1v2 , (c) v1 ¼ 1:2v2 , (d) v1 ¼ 1:3v2 , (e) v1 ¼ 1:4v2 , (f) v1 ¼ 1:5v2 , (g) v1 ¼ 15v2 , (h) ideal honeycomb structure.
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Fig. 8. Simulated microstructure for different grain growth velocity ratios (for 32 grains): (a) v1 ¼ v2 (single phase), (b) v1 ¼ 1:1v2 , (c) v1 ¼ 1:2v2 , (d) v1 ¼ 1:3v2 , (e) v1 ¼ 1:4v2 , (f) v1 ¼ 1:5v2 , (g) v1 ¼ 15v2 , (h) ideal honeycomb structure.
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3.1.1. Grain edge statistics Metallography theory [18] indicates that the following two factors determine the grain shape during the solidification of a polycrystalline metallic material: (1) the grains must fill the space and not have gaps between them; and (2) the potential energy of the grain must be minimized. Smith [19] studied the grain shapes of several materials, and concluded that the average number of edges for a grain approaches 517 ( ¼ 5.143). This indicated that the most likely grain shape is either a pentagon or a hexagon under equilibrium. It can actually be seen in experiments that most of the grain shapes in the two-dimensional crosssection of a metallograph are pentagonal or hexagonal. A statistical analysis of the number of grain edges in the microstructures simulated in Figs. 7 and 8 is conducted, and the results are shown in Tables 2 and 3 respectively. It is seen that these statistics agree well with experimentally observed average value observed by Smith [19], and by Kurt and Carpay [12], and are in between their average values. The mean number of edges in a two-dimensional Poisson–Voronoi tessellation method [2] is 6.0011 which is outside the range of experimental observation. 3.1.2. Grain size statistics The grain average diameter and its area directly reflect the effectiveness of microstructure simulation results. According to the equiaxial grain calculation method, the grain average diameter may be deduced from the grain area [10]. Thus, the grain average diameter may be expressed as rffiffiffiffi Si di ¼ 2 ði ¼ 1; 2; . . . ; mÞ; ð10Þ p
Table 2 Statistical analysis of grain edges in Fig. 7 Edges
Fig. 7(a)
Fig. 7(b)
Fig. 7(c)
Fig. 7(d)
Fig. 7(e)
Fig. 7(f)
Fig. 7(g)
1 2 3 4 5 6 7 8
0 0 0 6 7 1 1 1
0 0 1 5 5 3 1 1
0 0 1 5 5 3 1 1
0 0 1 5 5 3 1 1
0 1 0 5 5 1 4 0
1 0 0 5 3 6 1 0
5 0 1 2 3 2 2 1
Mean
5
5.0625
5.0625
5.0625
5.0625
4.9357
4.0625
Fig. 8(g)
Table 3 Statistical analysis of grain edges in Fig. 8 Edges
Fig. 8(a)
Fig. 8(b)
Fig. 8(c)
Fig. 8(d)
Fig. 8(e)
Fig. 8(f)
1 2 3 4 5 6 7 8
0 0 0 6 13 10 2 1
0 0 0 6 14 8 3 1
0 0 1 5 15 7 3 1
0 0 1 6 13 8 2 2
0 0 1 7 14 6 3 1
0 0 1 7 13 7 2 2
Mean
5.3438
5.3438
5.2813
5.3125
5.1875
5.25
7 3 0 3 7 6 5 1 4.3438
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where di is the ith grain average diameter, Si is the ith grain area, and m is the number of grains in the microstructure. A chi-square statistical test at a significant level a ¼ 0:1 is performed to compare the normal and lognormal distributions to describe the statistics of the grain average diameter. The normal distribution is observed to fit the simulated data more accurately, and is shown in Figs. 9 and 10 for different values of grain numbers. However, the lognormal distribution, which does not allow negative values, may be more reasonable based on physical reasoning. The estimates of mean and standard deviation of the grain diameter are shown in Tables 5 and 6. The distribution of the grain area for each value of grain number is observed to follow a two-parameter gamma distribution at a significance level a ¼ 0:1 (Figs. 11 and 12) when the data of grain areas in Figs. 7 and 8 are subjected to a chi-square test. The estimates of mean and standard deviation of grain area and the corresponding parameters a and b of the gamma distribution are shown in Tables 7 and 8.
Fig. 9. Grain diameter distribution.
Fig. 10. Grain diameter distribution.
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Table 4 Statistical analysis of grain edges in other studies
Average grain edge number
Simulated results [10]
Experimental observation [19]
Experimental observation [12]
Tessellation [2]
5:36–5:55
5.143
5:56–5:78
6.0011
Table 5 Statistical analysis of the grain diameter l^d (mm) r^d (mm) COV
Fig. 7(a)
Fig. 7(b)
Fig. 7(c)
Fig. 7(d)
Fig. 7(e)
Fig. 7(f)
0.2742 0.0684 0.2494
0.2726 0.0748 0.2743
0.2711 0.0805 0.2967
0.2694 0.0866 0.3213
0.2685 0.0893 0.3326
0.2672 0.0944 0.3531
Table 6 Statistical analysis of the grain diameter l^d (mm) r^d (mm) COV
Fig. 8(a)
Fig. 8(b)
Fig. 8(c)
Fig. 8(d)
Fig. 8(e)
Fig. 8(f)
0.1903 0.0520 0.2734
0.1899 0.0545 0.2869
0.1894 0.0586 0.3096
0.1890 0.0605 0.3202
0.1879 0.0645 0.3433
0.1845 0.0682 0.3696
Fig. 11. Grain area distribution (area unit: mm2 ).
The grain average diameter is found to follow a normal distribution, with COV values for different grain sizes ranging from 0.2492 to 0.3531. Other experimental studies [12] observed that the grain average diameter follows a lognormal distribution, with COV values ranging from 0.328 to 0.569 for different annealing times. The method to calculate the grain average diameter in this paper directly uses Eq. (6), which is different from the planar cross section method used in Ref. [12] that tends to increase the median planar diameter because smaller grains are less likely to be encountered. In this paper, the grain area is found to follow a two-parameter gamma distribution, with statistical parameters as listed in Table 4. The COV values for different grain sizes range from 0.4830 to 0.6016. In comparison, the two-dimensional tessellation method [2] obtains a gamma distribution for the grain area with COV values of 0.518.
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Fig. 12. Grain area gamma distribution (area unit: mm2 ).
Table 7 Statistical analysis of the grain area l^s (mm) r^s (mm) COV a b
Fig. 7(a)
Fig. 7(b)
Fig. 7(c)
Fig. 7(d)
Fig. 7(e)
Fig. 7(f)
0.0625 0.0302 0.4830 4.2808 0.0146
0.0625 0.0322 0.5150 3.7651 0.0166
0.0625 0.0336 0.5383 3.4530 0.0181
0.0625 0.0352 0.5632 3.1566 0.0198
0.0625 0.0360 0.5766 3.0193 0.0207
0.0625 0.0376 0.6016 2.7655 0.0226
Table 8 Statistical analysis of the grain area l^s (mm) r^s (mm) COV a b
Fig. 8(a)
Fig. 8(b)
Fig. 8(c)
Fig. 8(d)
Fig. 8(e)
Fig. 8(f)
0.0313 0.0136 0.4345 5.3051 0.0059
0.0313 0.0143 0.4569 4.8906 0.0064
0.0313 0.0155 0.4952 4.0755 0.00768
0.0313 0.0163 0.5208 3.6867 0.00849
0.0313 0.0173 0.5527 3.2741 0.00956
0.0313 0.0186 0.5942 2.8626 0.01105
In all three cases (grain shape, grain diameter, and grain area), the statistics of the simulated microstructure according to the proposed method agree well with those of the observed microstructure [2,12,19], for single phase materials. 3.1.3. Two-phase alloy material statistics The properties of a multi-phase alloy material are determined not only by the properties of the individual phases, but also by the phase proportion. The different phase grains in Figs. 7(b)–(g) and 8(b)–(g) grow at different velocities. Assume the dimensions of the cell in Figs. 7 and 8 to be 100 lm 100 lm, and the grain with velocity v1 , as a-phase, and the grain with velocity v2 as b-phase. Statistical analysis is conducted for each phase in Figs. 7 and 8, including the diameter of each phase, and the proportion of the two phases.
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3.1.4. Area proportion of a-phase and b-phase The total grain surface area in each phase may be expressed as
PN a Sa ¼ i¼1 S PN b i Sb ¼ i¼1 Si
for a-phase grains; for b-phase grains;
ð11Þ
where Na and Nb are the number of grains in a-phase and b-phase respectively, and Sa and Sb are the grain areas of a-phase and b-phase respectively. The proportions of a-phase and b-phase grains in the microstructure are therefore Sa =S and Sb =S, respectively, where S is the total area of the cell (100 lm 100 lm). These results are shown in Tables 9 and 10 for the simulated microstructures in Figs. 7 and 8. 3.1.5. Average diameter of a-phase and b-phase Referring to Eq. (6), the average diameter of a-phase and b-phase grains may be computed as Da ¼
2
PNa pffiffiffiffiffiffiffiffiffi Si =p i¼1 ; Na
Db ¼
2
PNb pffiffiffiffiffiffiffiffiffi Si =p i¼1 ; Nb
ð12Þ
where Da and Db are the average diameter of a-phase and b-phase, shown in Tables 9 and 10. The minimum and maximum diameters of the a-phase and b-phase grains are also shown in Tables 9 and 10. The data in Tables 9 and 10 show that the grain diameters and area proportion of the a and b phases in the simulated results will change with the ratio of velocities. The diameter and proportion of the a-phase will increase, while the diameter and proportion of the b-phase will decrease as the ratio of grain growth velocities (f ¼ v1 =v2 ) increases. Statistical analysis shows that the ratio of grain growth velocities (f ¼ v1 =v2 ) is a major factor that affects the microstructure of multi-phase metallic alloys even if the grain core points are kept in the same location.
Table 9 Two-phase alloy simulation results Specimen Fig. Fig. Fig. Fig. Fig.
7(b) 7(c) 7(d) 7(e) 7(f)
Phase proportion (%)
Diameter of a (lm)
a
b
l^da
rda
dmax
Diameter of b (lm) l^db
rdb
dmax
70.17 73.72 76.62 79.50 81.94
29.83 26.28 23.38 20.50 18.06
27.63 28.32 28.86 29.38 29.83
7.31 7.53 7.71 7.90 8.03
38.86 39.48 40.02 40.36 40.64
26.50 24.85 23.31 21.79 20.28
8.57 8.78 9.00 9.16 9.18
37.73 36.01 34.34 32.53 30.21
Table 10 Two-phase alloy simulation results Specimen Fig. Fig. Fig. Fig. Fig.
8(b) 8(c) 8(d) 8(e) 8(f)
Phase proportion (%)
Diameter of a (lm)
a
b
l^da
rda
dmax
Diameter of b (lm) l^db
rdb
dmax
74.65 78.65 80.99 83.30 85.98
25.35 21.35 19.01 16.70 14.02
20.13 20.67 20.95 21.25 21.44
5.28 5.43 5.54 5.64 5.75
25.79 26.92 27.78 28.27 28.91
16.93 15.58 14.41 13.37 12.24
4.53 4.60 4.57 4.63 4.84
23.24 22.09 21.02 20.12 19.24
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Fig. 13. One type of grains embedded in another type of grains.
3.1.6. Grain shape change with different velocity The lengths and radii of the grain boundaries, and the locations of triple points, are all affected by grain growth velocity during the solidification process. As shown in Figs. 7(a) and 8(a) when all grains grow with equal velocity, all the grain boundaries are straight lines in the proposed simulation method which reduces to the Voronoi tessellation method. When the adjacent grain growth velocities are not equal, the grain boundaries between them become curved, as showed in Figs. 7(b)–(f) and 8(b)–(f), where the ratio of adjacent grain velocities (f ¼ v1 =v2 ) ranges from 1.1 to 1.5. The length and radius of the curved boundary become shorter as the ratio of grain growth velocity increases, because the grain with higher velocity will grow around the grain with smaller velocity. The corresponding triple point locations also change as the ratio of grain growth velocity changes. As the ratio of grain growth velocity tends to a large value, one kind of grain will be embedded in another grain, as showed in Figs. 7(g) and 8(g) when the ratio of adjacent grain velocities (f ¼ v1 =v2 ) is equal to 15, and the proposed method is able to simulate this case. Therefore, the proposed method provides a better simulation model for stress analysis and material constants estimation even for complicated material structure as shown in Fig. 13 (simulated with f ¼ v1 =v2 ¼ 25). When the core points are uniformly distributed in the given area, and all the grain growth velocities are equal, the simulation results would be honeycombs as shown in Figs. 7(h) and 8(h).
4. Conclusion A generalized 2-D method has been developed in this paper to simulate the microstructure of metallic polycrystalline materials. This method assumes that adjacent grains in different phases will grow at different velocities during solidification. The grain growth velocity affects the shape of the grain, including number of edges, grain diameter, grain area, the length and radii of grain boundary, and the location of triple points. If the ratio of growth velocity is equal to 1, the proposed method reduces to the Voronoi tessellation method. If the ratio of growth velocity tends to 1, it simulates the case where one type of grain embedded in the other type of grain. The grain growth velocity is affected by different types of grains during solidification process, and also by uneven temperature. The simulation results show that the edge number and triple point location of some grains will change with the increasing ratio of grain growth velocity even if the
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original core points are the same. The proposed method is able to model these various situations effectively. Statistical analysis shows that the results of the proposed method agree well with experimental observation.
References [1] P. Gilormini, Y. Brechet, Syntheses: mechanical properties of heterogeneous media: Which material for which model? Which model for which material?, Model. Simul. Mater. Sci. Engrg. 7 (1999) 805–816. [2] S. Kumar, S.K. Kurtz, Properties of a two-dimensional Poisson–Voronoi tesselation: a Monte-Carlo study, Mater. Charact. 31 (1993) 55–68. [3] S. Kumar, S.K. Kurtz, J. Banavar, M.G. Sharma, Properties of three-dimensional, Poisson–Voronoi tesselation: a Monte-Carlo Study, J. Statist. Phys. 67 (3/4) (1992) 523–551. [4] J.B. Parse, J.A. Wert, A geometrical description of particle distributions in materials, Modeling Simul. Mater. Sci. Engrg. 1 (1993) 275–296. [5] R.L. Mullen, R. Ballarini, Y. Lin, A.H. Heuer, Monte Carlo simulation of effective elastic constants of polycrystalline thin films, Acta Mater. 45 (6) (1997) 2247–2255. [6] S. Kumar, S.K. Kurtz, V.K. Agarwala, Micro-stress distribution within polycrystalline aggregate, Acta Mech. 114 (1996) 203–216. [7] J.S. Matthew, C.H. Wilson, J.G. Lorna, The effects of non-periodic microstructure on the elastic properties of two-dimensional cellular solids, Int. J. Mech. Sci. 37 (11) (1995) 1161–1177. [8] S. Grant, D.E.G. Xiang, J.S. Matthew, J.G. Lorna, Modeling fatigue damage accumulation in two-dimensional Voronoi honeycombs, Int. J. Mech. Sci. 42 (2000) 645–656. [9] Y. Liu, Y. Kageyama, S. Murakami, Creep fracture modeling by use of continuum damage variable based on Voronoi simulation of grain boundary cavity, Int. J. Mech. Sci. 40 (2–3) (1998) 147–158. [10] Y. Zhao, The research of computational micro-mechanics on Al alloy fatigue microcracks, Ph.D. Dissertation, Northwestern Polytechnology University, Xi’an, PR China, 1997. [11] Y. Shao, The research of computational micro-mechanics, Ph.D. Dissertation, Northwestern Polytechnology University, Xi’an, PR China, 1990. [12] S.K. Kurt, F.M.A. Carpay, Microstructure and normal grain growth in metals and ceramics. Part II. Experiment, J. Appl. Phys. 51 (11) (1980) 5745–5754. [13] J.M.J. den Toonder, J.A.W. van Dommelen, F.P.T. Baaijens, The relation between single crystal elasticity and the effective elastic behavior of polycrystalline materials: theory, measurement and computation, Modeling Simul. Mater. Sci. 7 (1999) 909–928. [14] W.H. Hunt Jr., O. Richmond, R.D. Yong, Fracture initiation in particle hardened materials with high volume fraction, in: Proceedings of the Sixth International Conference on Composite Materials, London, UK, vol. 2, 1987, p. 2209. [15] P.E. Magnusen, E.M. Dubensky, D.A. Koss, The effect of void arrays on the void linking during ductile fracture, Acta Metall. 36 (1988) 1503–1509. [16] M. Blicharski, R. Becker, H. Hu, Deformation texture of channel-die deformed aluminum bicrytals with S orientation, Acta. Metall. Mater. 41 (1993) 2007–2016. [17] R. Becker, O. Richmond, Incorporation of microstructural geometry in material modeling, Modeling Simul. Mater. Sci. Engrg. 2 (1994) 439–454. [18] Y. Song, The Boundary and Strength of Metal Material, Xi’an Jiaotong University Press, Xi’an, 1987. [19] C.S. Smith, A Search for Structure, MIT Press, Cambridge, 1981.
Advanced computer simulation of polycrystalline microstructure Sankaran Mahadevan *, Yaowu Zhao Department of Civil and Environmental Engineering, Vanderbilt University, Nashville, TN 37235, USA Received 12 January 2001; accepted 19 February 2002
Abstract This paper proposes an advanced two-dimensional method to simulate the microstructure of metallic dual-phase polycrystalline materials to enable accurate micro-mechanics computation. This is done by combining geometric and metallurgical principles to simulate microstructures that are closer to experimental observations than currently used methods. The method accounts for difference in grain growth velocities in different phases during solidification. If the ratio of grain growth velocities equals 1, the proposed method reduces to the currently used Voronoi tessellation method. If the ratio of grain growth velocities tends to 1, the method matches the physical situation of one grain type becoming embedded in the other grain type. The simulation results show that although the original grain core points are the same, the microstructure of the material will change with the ratio of grain growth velocities, including the lengths of the grain boundaries, the radii of curvature of the grain boundaries, and the triple point locations. Statistical analyses are conducted on the number of grain edges, grain diameter, grain area, and proportion of two phases of grains. Statistical analysis shows that the simulation results of the proposed method agree well with the experimental observation. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Microstructure; Grain; Grain growth; Velocity; Simulation; Alloys; Dual phase; Solidification; Tessellation
1. Introduction Simulation of the microstructure of modern alloys is an important step in the computational modeling of mechanical behavior and damage evolution. Alloys are often multiphase, or when they are single phase they are usually polycrystalline. The development of new materials and processes as well as the optimization of classical materials requires modeling more closely related to the microstructure [1]. Voronoi-based tessellation methods have been pursued for microstructure simulation [2–4], which are then used in micromechanics analysis [5–9]. The Voronoi-based methods yield microstructures where ‘‘all edges are straight lines, all cells are convex and each cell has at least three edges’’. Such microstructures are appropriate only
*
Corresponding author. Tel.: +1-615-322-3040. E-mail address: [email protected] (S. Mahadevan).
0045-7825/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 4 5 - 7 8 2 5 ( 0 2 ) 0 0 2 6 0 - 8
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Fig. 1. Optical photographs of different materials: (a) microstructure of a polished section of Al alloy at a magnification of 60 [10], (b) Ti alloy micro-structure [11], (c) microstructure of a polished section of Al alloy at a magnification of 60 [10], (d) distribution of spherical d0 in Al alloy [10].
for single-phase materials with all grains growing with the same velocity. Some studies [10–13] have presented two-dimensional optical photographs (Fig. 1) of different metal alloys. From these observations, it appears that Voronoi-based tessellation methods with straight boundaries and convex polygons do not really reflect the actual microstructure. Experimental evidence suggests that in order to accurately model phenomena such as recrystallization, shear localization and fracture, more detailed information concerning the effect of the metallurgical process on the geometry of the microstructure must be incorporated into the computational models [14–16]. In addition, it was found that a grain can have influence on the deformation field several grain diameters away [17]. This paper proposes a generalized method to simulate the two-dimensional microstructure of metallic materials that accounts for the aforementioned effects. The grains are allowed to grow with different velocities depending on phase difference, temperature variation, etc. If all the grain growth velocities are equal, then the method reduces to the Voronoi-based tessellation method. If the ratio of grain growth velocity is very large, then one phase gets embedded in the other phase. Between these two limits, this method will obtain realistic simulation of actual materials. This proposed method yields microstructures where the grains have both curved and straight edges, are not necessarily convex, and the minimum number of edges is only one. The simulation results show that although the original grain core points are the same, the microstructure of the material will change with the ratio of grain growth velocities, including the lengths of the grain boundaries, the radii of curvature of the grain boundaries, and the triple point locations of the grains. Statistical analyses are conducted on simulated microstructure results, such as the number of grain
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edges, grain diameter, grain area, and proportion of two phases of grains. Statistical analyses show that the results of the proposed method agree well with experimental observation.
2. Proposed method 2.1. Grain growth assumptions On the basis of a simplified three-dimensional model of the solidification process from liquid to solid phase of metallic materials, and some assumptions about grain growth, this section develops a method to simulate the microstructure and the grain growth process. Following are the basic assumptions: 1. The grain growth is isotropic; that is, if the grain core is the center of a circle, then the grain will grow at the same velocity in all directions. The grain growth will stop to form a grain boundary in one direction when it meets another growing grain boundary, but the grain will continue to grow along other directions. The grain growth process will end when its growth stops in all directions. 2. The velocity of grain growth is different in the different phases in the alloy. 3. After a boundary is formed between two grains, the grain boundary is unmovable and unchangeable. 4. There is no grain growth in the area of an already existing grain. That is, the grains are not allowed to overlap. 2.2. Grain boundary description The following observations can be made regarding the geometry of the microstructure and the grain boundaries, as a result of the above grain growth assumptions: (1) If two adjacent grains are in the same phase and they grow at the same velocity, then their grain boundary is a straight line, given by a mid-perpendicular (vertical line in Fig. 2), and the mid-perpendicular’s equation could be expressed as r1 ¼ r2 ;
ð1Þ
where r1 and r2 are the distances from a point s1 on the boundary to the grain core points A1 and A2 respectively. (2) If three adjacent grains are in the same phase and grow at equal velocity, then the grain boundary between every two grains is the mid-perpendicular, and the three grain boundaries will meet at one point
Fig. 2. Growth of two same-phase grains.
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Fig. 3. Growth of three same-phase grains.
(referred to as the triple point; Point T in Fig. 3). The location of the triple point could be obtained through the intersection of any two of the three mid-perpendiculars. (3) If two adjacent grains are in different phases, they will grow at different velocities (Fig. 4). Assume two grains with core points A1ðx1 ; y1 Þ and A2ðx2 ; y2 Þ grow at different velocities v1 , and v2 respectively, and v1 > v2 . Under the grain growth assumptions in Section 2.1, the grain boundary may be expressed as t1 ¼ t2 ;
ð2Þ
where t1 is the time of grain growth from core point A1ðx1 ; y1 Þ to the boundary, t2 is the time of grain growth from core point A2ðx2 ; y2 Þ to the boundary. Eq. (2) may be rewritten as l1 l2 ¼ ; v1 v2
ð3Þ
where l1 and l2 are the distances from the point s1 on the boundary to the grain core points A1 and A2 respectively. Let k ¼ ðv1 v2 Þ=v2 , then, Eq. (3) could be expressed as l1 ¼ ð1 þ kÞl2 ;
ð4Þ
Fig. 4. Growth of two grains in different phases with unequal velocities.
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where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 < l1 ¼ ðx x1 Þ2 þ ðy y1 Þ2 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : l ¼ ðx x Þ2 þ ðy y Þ2 : 2
2
2
Substitution of l1 and l2 into Eq. (4) leads to the following equation of a circle for the point s1ðx; yÞ, with center at Oða; bÞ and radius R: ðx aÞ2 þ ðy bÞ2 ¼ R2 ; where 8 ð1 þ kÞ2 x2 x1 > > > a¼ > 2 > > ð1 þ kÞ 1 > > > 2 > < ð1 þ kÞ y2 y1 b¼ ð1 þ kÞ2 1 > > > 2 > > ð1 þ kÞ > 2 2 > R ¼ > h i2 ½ðx2 x1 Þ þ ðy2 y1 Þ > > 2 : ð1 þ kÞ 1
ð5Þ
ðk 6¼ 0Þ
ð6Þ
However, considering the assumption in Section 2.1 that the grains cannot overlap, then the grain from A1 grows around the grain from A2. In Fig. 4, G1 and G2 are the tangent points from A1 to this circle. For a point s2ðx; yÞ on the boundary which is not between two grain core points A1 and A2, Eq. (2) will result in the following expression: l0 þ d1 d2 ¼ ; v1 v2
ð7Þ
where l0 ¼ jA1G1j which is a constant, d1 is the length of the arc G1Bs2, and d2 ¼ jA2s2j. Then Eq. (7) can be written as fd2 d1 ¼ l0 ;
ð8Þ
where f ¼ v1 =v2 . Eq. (8) has to be solved through numerical integration. For the reason of simplicity, we use part of an ellipse to approximate the curve of Eq. (8). The corresponding ellipse equation is ðx aÞ ð2RÞ2
2
2
þ
ðy bÞ ¼ 1; R2
ð9Þ
where a, b and R are the same as in Eq. (6), C is a constant with C > 1. Here, for simplicity, we arbitrarily choose C ¼ 2. Therefore, the grain boundary in Fig. 4 is the bold curve which is part of a circle on the left and part of an ellipse in the right. (4) In a two-phase material, some of the adjacent grains may be in different phases, and some other pairs may be in the same phase. Then the boundary between the same phase grains is the mid-perpendicular straight line, and the boundary between the grains in different phases is a curve. Consider three adjacent grains A1, A2 and A3, with core points ðx1 ; y1 Þ, ðx2 ; y2 Þ and ðx3 ; y3 Þ respectively. Assume that A1 and A2 are in the same phase, and A3 in another phase. Several cases may be considered as in Fig. 5, and the corresponding grain boundary equations are divided and listed in Table 1.
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Fig. 5. Grain boundaries between three adjacent grains (A1 and A2 with velocity v1 , A3 with velocity v2 ): (a) v1 > v2 , (b) v2 > v1 , (c) long growth time between A1 and A2, and v1 > v2 , (d) long growth time between A1 and A2, and v2 > v1 , and (e) the location of A3 is between A1 and A2, and v1 > v2 .
2.3. Proposed microstructure simulation algorithm In two-dimensional analysis, an arbitrary grain shape is an irregular polygon, whose boundary is formed during the solidification phase from liquid to solid stage. If there are other grains surrounding a grain, the grain is referred to as an interior grain in the following discussion, and a grain along the cell boundary is referred to as a border grain. The steps of the proposed method for the simulation of the microstructure are as follows: 1. Use Monte Carlo sampling to generate the core points of the grains in a given area. (Here, it is assumed that the area is a square and the edge length equals 1 unit.) 2. Randomly choose one point (for example, point ðx1 ; y1 Þ) as the starting point O (Fig. 6), then calculate the time to grow to the other points and get the shortest time core point A, say point ðx2 ; y2 Þ. Note that different grains have different growth velocities. 3. Compute the angles of the other grain core points with the line OA. Then form an array B of size (n 2), arranged according to increasing angle values in the counter-clockwise direction. 4. Assuming B, the first point in this array, is an adjacent core point, find the location of the triple point T of the triangle OAB. 5. Calculate the growth times of the other core points from the triple point T. 6. If any of these growth times is less than the time of OT, then B is not an adjacent core point. Replace B with the next core point in the array B and repeat steps 3–5. 7. If all the growth times computed in step 5 are greater than the time of OT, then B is an adjacent core point to O. Now replace core point A with B and repeat the above process with OB as the reference line.
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Table 1 Equations to determine boundaries of three adjacent dual-phase grains Case
Equation
Case 1: (Fig. 5(a)) A1 and A2 with v1 , A3 with v2 , and v1 > v2
Case 2: (Fig. 5(b)) A1 and A2 with v1 , A3 with v2 , and v2 > v1
Explanation
ðx aÞ2 þ ðy bÞ2 ¼ R2 r1 ¼ r2
ðx aÞ2 þ ðy bÞ2 ¼ R2 r1 ¼ r2
8 > ð1 þ kÞ2 x3 x1 > > a¼ > > > ð1 þ kÞ2 1 > > >
> > < ð1 þ kÞ2 y3 y1 v1 v2 b¼ 6¼ 0 k¼ 2 v2 ð1 þ kÞ 1 > > > > 2 > > ð1 þ kÞ > > ½ðx3 x1 Þ2 þ ðy3 y1 Þ2 R¼ > > > ½ð1 þ kÞ2 1 2 : 8 ð1 þ kÞ2 x1 x3 > > a¼ > > > ð1 þ kÞ2 1 > > >
> > < ð1 þ kÞ2 y1 y3 v1 v2 b¼ 6¼ 0 k¼ 2 v2 ð1 þ kÞ 1 > > > > 2 > > ð1 þ kÞ > > ½ðx1 x3 Þ2 þ ðy1 y3 Þ2 R¼ > > : ½ð1 þ kÞ2 1 2
Case 3: (Fig. 5(c)) A1 and A2 with v1 , A3 with v2 , and v1 > v2 (long growth time between A1 and A2)
8 2 2 > < ðx a1 Þ þ ðy b1 Þ ¼ 1 R21 ð2R1 Þ2 > : 2 ðx a2 Þ þ ðx b2 Þ2 ¼ R22
8 > ð1 þ kÞ2 x3 x1 > > ¼ a 1 > > > ð1 þ kÞ2 1 > > >
> > < ð1 þ kÞ2 y3 y1 v1 v2 k ¼ 6 ¼ 0 b1 ¼ v2 ð1 þ kÞ2 1 > > > > 2 > > ð1 þ kÞ > > ½ðx3 x1 Þ2 þ ðy3 y1 Þ2 R1 ¼ > > > ½ð1 þ kÞ2 1 2 : 8 > ð1 þ kÞ2 x3 x2 > > a2 ¼ > > > ð1 þ kÞ2 1 > > >
> > < ð1 þ kÞ2 y3 y2 v1 v2 k¼ b2 ¼ 6 0 ¼ 2 v2 ð1 þ kÞ 1 > > > > 2 > > ð1 þ kÞ >R ¼ > ½ðx3 x2 Þ2 þ ðy3 y2 Þ2 > 2 > > ½ð1 þ kÞ2 1 2 :
Case 4: (Fig. 5(d)) A1 and A2 with v1 , A3 with v2 , and v2 > v1 , (long growth time between A1 and A2)
8 2 2 > < ðx a1 Þ þ ðy b1 Þ ðx a1 Þ2 ðy b1 Þ2 > þ : 2 R21 8 ð2R1 Þ 2 > ðx a2 Þ þ ðy b2 Þ2 < ðx a2 Þ2 ðy b2 Þ2 > þ : R22 ð2R2 Þ2
8 > ð1 þ kÞ2 x1 x3 > > a ¼ 1 > > > ð1 þ kÞ2 1 > > >
> > < ð1 þ kÞ2 y1 y3 v1 v2 6¼ 0 k¼ b1 ¼ 2 v2 ð1 þ kÞ 1 > > > > 2 > > ð1 þ kÞ > > ½ðx1 x3 Þ2 þ ðy1 y3 Þ2 R1 ¼ > > > ½ð1 þ kÞ2 1 2 :
¼ R21 ¼1 ¼ R22 ¼1
8 ð1 þ kÞ2 x2 x3 > > > a2 ¼ > > > ð1 þ kÞ2 1 > > > >
> > ð1 þ kÞ2 y2 y3 v1 v2 < k ¼ b2 ¼ ¼ 6 0 v2 ð1 þ kÞ2 1 > > > > > > ð1 þ kÞ2 > > > ½ðx2 x3 Þ2 þ ðy2 y3 Þ2 R2 ¼ > > > ½ð1 þ kÞ2 1 2 : (continued on next page)
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Table 1 (continued) Case
Equation
Case 5: (Fig. 5(e)) A1 and A2 with v1 , A3 with v2 , and v1 > v2
Explanation 2
2
ðx a1 Þ þ ðy b1 Þ ¼ R21 ðx a2 Þ2 þ ðy b2 Þ2 ¼ R22
8 ð1 þ kÞ2 x3 x1 > > ¼ a > 1 > > ð1 þ kÞ2 1 > > >
> > < ð1 þ kÞ2 y3 y1 v1 v2 k ¼ b1 ¼ ¼ 6 0 v2 ð1 þ kÞ2 1 > > > > 2 > > ð1 þ kÞ > > ½ðx3 x1 Þ2 þ ðy3 y1 Þ2 R1 ¼ > > : ½ð1 þ kÞ2 1 2 8 ð1 þ kÞ2 x3 x2 > > > a2 ¼ > > ð1 þ kÞ2 1 > > >
> > < ð1 þ kÞ2 y3 y2 v1 v2 6¼ 0 k¼ b2 ¼ 2 v2 ð1 þ kÞ 1 > > > > 2 > > ð1 þ kÞ > > R2 ¼ ½ðx3 x2 Þ2 þ ðy3 y2 Þ2 > > : ½ð1 þ kÞ2 1 2
Fig. 6. Process of forming the grain boundary.
8. After identifying all the triple points of the starting point O, choose another point as a starting point and repeat steps 2–7 until all the core points are explored. 9. Plot the polygons connecting the identified triple points corresponding to each grain core point, to generate the microstructure. 3. Simulation results The microstructure simulation results are shown in Figs. 7 and 8 with different numbers of grains and with different ratios of grain growth velocities. Fig. 7 considers 16 grains, and Fig. 8 considers 32 grains. The ratio of grain growth velocities (f ¼ v1 =v2 ) ranges from 1 to 15 in both figures. Figs. 7(h) and 8(h) are the ideal honeycombs used in some studies [8]. 3.1. Statistical evaluation of the simulated microstructure In order to check the effectiveness of the above simulation of the metal alloy microstructure, it is necessary to perform statistical analysis of the grain edge, grain size, etc. This is discussed in this section.
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Fig. 7. Simulated micro-structure for different grain growth velocity ratios (for 16 grains): (a) v1 ¼ v2 (single phase), (b) v1 ¼ 1:1v2 , (c) v1 ¼ 1:2v2 , (d) v1 ¼ 1:3v2 , (e) v1 ¼ 1:4v2 , (f) v1 ¼ 1:5v2 , (g) v1 ¼ 15v2 , (h) ideal honeycomb structure.
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Fig. 8. Simulated microstructure for different grain growth velocity ratios (for 32 grains): (a) v1 ¼ v2 (single phase), (b) v1 ¼ 1:1v2 , (c) v1 ¼ 1:2v2 , (d) v1 ¼ 1:3v2 , (e) v1 ¼ 1:4v2 , (f) v1 ¼ 1:5v2 , (g) v1 ¼ 15v2 , (h) ideal honeycomb structure.
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3.1.1. Grain edge statistics Metallography theory [18] indicates that the following two factors determine the grain shape during the solidification of a polycrystalline metallic material: (1) the grains must fill the space and not have gaps between them; and (2) the potential energy of the grain must be minimized. Smith [19] studied the grain shapes of several materials, and concluded that the average number of edges for a grain approaches 517 ( ¼ 5.143). This indicated that the most likely grain shape is either a pentagon or a hexagon under equilibrium. It can actually be seen in experiments that most of the grain shapes in the two-dimensional crosssection of a metallograph are pentagonal or hexagonal. A statistical analysis of the number of grain edges in the microstructures simulated in Figs. 7 and 8 is conducted, and the results are shown in Tables 2 and 3 respectively. It is seen that these statistics agree well with experimentally observed average value observed by Smith [19], and by Kurt and Carpay [12], and are in between their average values. The mean number of edges in a two-dimensional Poisson–Voronoi tessellation method [2] is 6.0011 which is outside the range of experimental observation. 3.1.2. Grain size statistics The grain average diameter and its area directly reflect the effectiveness of microstructure simulation results. According to the equiaxial grain calculation method, the grain average diameter may be deduced from the grain area [10]. Thus, the grain average diameter may be expressed as rffiffiffiffi Si di ¼ 2 ði ¼ 1; 2; . . . ; mÞ; ð10Þ p
Table 2 Statistical analysis of grain edges in Fig. 7 Edges
Fig. 7(a)
Fig. 7(b)
Fig. 7(c)
Fig. 7(d)
Fig. 7(e)
Fig. 7(f)
Fig. 7(g)
1 2 3 4 5 6 7 8
0 0 0 6 7 1 1 1
0 0 1 5 5 3 1 1
0 0 1 5 5 3 1 1
0 0 1 5 5 3 1 1
0 1 0 5 5 1 4 0
1 0 0 5 3 6 1 0
5 0 1 2 3 2 2 1
Mean
5
5.0625
5.0625
5.0625
5.0625
4.9357
4.0625
Fig. 8(g)
Table 3 Statistical analysis of grain edges in Fig. 8 Edges
Fig. 8(a)
Fig. 8(b)
Fig. 8(c)
Fig. 8(d)
Fig. 8(e)
Fig. 8(f)
1 2 3 4 5 6 7 8
0 0 0 6 13 10 2 1
0 0 0 6 14 8 3 1
0 0 1 5 15 7 3 1
0 0 1 6 13 8 2 2
0 0 1 7 14 6 3 1
0 0 1 7 13 7 2 2
Mean
5.3438
5.3438
5.2813
5.3125
5.1875
5.25
7 3 0 3 7 6 5 1 4.3438
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where di is the ith grain average diameter, Si is the ith grain area, and m is the number of grains in the microstructure. A chi-square statistical test at a significant level a ¼ 0:1 is performed to compare the normal and lognormal distributions to describe the statistics of the grain average diameter. The normal distribution is observed to fit the simulated data more accurately, and is shown in Figs. 9 and 10 for different values of grain numbers. However, the lognormal distribution, which does not allow negative values, may be more reasonable based on physical reasoning. The estimates of mean and standard deviation of the grain diameter are shown in Tables 5 and 6. The distribution of the grain area for each value of grain number is observed to follow a two-parameter gamma distribution at a significance level a ¼ 0:1 (Figs. 11 and 12) when the data of grain areas in Figs. 7 and 8 are subjected to a chi-square test. The estimates of mean and standard deviation of grain area and the corresponding parameters a and b of the gamma distribution are shown in Tables 7 and 8.
Fig. 9. Grain diameter distribution.
Fig. 10. Grain diameter distribution.
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Table 4 Statistical analysis of grain edges in other studies
Average grain edge number
Simulated results [10]
Experimental observation [19]
Experimental observation [12]
Tessellation [2]
5:36–5:55
5.143
5:56–5:78
6.0011
Table 5 Statistical analysis of the grain diameter l^d (mm) r^d (mm) COV
Fig. 7(a)
Fig. 7(b)
Fig. 7(c)
Fig. 7(d)
Fig. 7(e)
Fig. 7(f)
0.2742 0.0684 0.2494
0.2726 0.0748 0.2743
0.2711 0.0805 0.2967
0.2694 0.0866 0.3213
0.2685 0.0893 0.3326
0.2672 0.0944 0.3531
Table 6 Statistical analysis of the grain diameter l^d (mm) r^d (mm) COV
Fig. 8(a)
Fig. 8(b)
Fig. 8(c)
Fig. 8(d)
Fig. 8(e)
Fig. 8(f)
0.1903 0.0520 0.2734
0.1899 0.0545 0.2869
0.1894 0.0586 0.3096
0.1890 0.0605 0.3202
0.1879 0.0645 0.3433
0.1845 0.0682 0.3696
Fig. 11. Grain area distribution (area unit: mm2 ).
The grain average diameter is found to follow a normal distribution, with COV values for different grain sizes ranging from 0.2492 to 0.3531. Other experimental studies [12] observed that the grain average diameter follows a lognormal distribution, with COV values ranging from 0.328 to 0.569 for different annealing times. The method to calculate the grain average diameter in this paper directly uses Eq. (6), which is different from the planar cross section method used in Ref. [12] that tends to increase the median planar diameter because smaller grains are less likely to be encountered. In this paper, the grain area is found to follow a two-parameter gamma distribution, with statistical parameters as listed in Table 4. The COV values for different grain sizes range from 0.4830 to 0.6016. In comparison, the two-dimensional tessellation method [2] obtains a gamma distribution for the grain area with COV values of 0.518.
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Fig. 12. Grain area gamma distribution (area unit: mm2 ).
Table 7 Statistical analysis of the grain area l^s (mm) r^s (mm) COV a b
Fig. 7(a)
Fig. 7(b)
Fig. 7(c)
Fig. 7(d)
Fig. 7(e)
Fig. 7(f)
0.0625 0.0302 0.4830 4.2808 0.0146
0.0625 0.0322 0.5150 3.7651 0.0166
0.0625 0.0336 0.5383 3.4530 0.0181
0.0625 0.0352 0.5632 3.1566 0.0198
0.0625 0.0360 0.5766 3.0193 0.0207
0.0625 0.0376 0.6016 2.7655 0.0226
Table 8 Statistical analysis of the grain area l^s (mm) r^s (mm) COV a b
Fig. 8(a)
Fig. 8(b)
Fig. 8(c)
Fig. 8(d)
Fig. 8(e)
Fig. 8(f)
0.0313 0.0136 0.4345 5.3051 0.0059
0.0313 0.0143 0.4569 4.8906 0.0064
0.0313 0.0155 0.4952 4.0755 0.00768
0.0313 0.0163 0.5208 3.6867 0.00849
0.0313 0.0173 0.5527 3.2741 0.00956
0.0313 0.0186 0.5942 2.8626 0.01105
In all three cases (grain shape, grain diameter, and grain area), the statistics of the simulated microstructure according to the proposed method agree well with those of the observed microstructure [2,12,19], for single phase materials. 3.1.3. Two-phase alloy material statistics The properties of a multi-phase alloy material are determined not only by the properties of the individual phases, but also by the phase proportion. The different phase grains in Figs. 7(b)–(g) and 8(b)–(g) grow at different velocities. Assume the dimensions of the cell in Figs. 7 and 8 to be 100 lm 100 lm, and the grain with velocity v1 , as a-phase, and the grain with velocity v2 as b-phase. Statistical analysis is conducted for each phase in Figs. 7 and 8, including the diameter of each phase, and the proportion of the two phases.
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3.1.4. Area proportion of a-phase and b-phase The total grain surface area in each phase may be expressed as
PN a Sa ¼ i¼1 S PN b i Sb ¼ i¼1 Si
for a-phase grains; for b-phase grains;
ð11Þ
where Na and Nb are the number of grains in a-phase and b-phase respectively, and Sa and Sb are the grain areas of a-phase and b-phase respectively. The proportions of a-phase and b-phase grains in the microstructure are therefore Sa =S and Sb =S, respectively, where S is the total area of the cell (100 lm 100 lm). These results are shown in Tables 9 and 10 for the simulated microstructures in Figs. 7 and 8. 3.1.5. Average diameter of a-phase and b-phase Referring to Eq. (6), the average diameter of a-phase and b-phase grains may be computed as Da ¼
2
PNa pffiffiffiffiffiffiffiffiffi Si =p i¼1 ; Na
Db ¼
2
PNb pffiffiffiffiffiffiffiffiffi Si =p i¼1 ; Nb
ð12Þ
where Da and Db are the average diameter of a-phase and b-phase, shown in Tables 9 and 10. The minimum and maximum diameters of the a-phase and b-phase grains are also shown in Tables 9 and 10. The data in Tables 9 and 10 show that the grain diameters and area proportion of the a and b phases in the simulated results will change with the ratio of velocities. The diameter and proportion of the a-phase will increase, while the diameter and proportion of the b-phase will decrease as the ratio of grain growth velocities (f ¼ v1 =v2 ) increases. Statistical analysis shows that the ratio of grain growth velocities (f ¼ v1 =v2 ) is a major factor that affects the microstructure of multi-phase metallic alloys even if the grain core points are kept in the same location.
Table 9 Two-phase alloy simulation results Specimen Fig. Fig. Fig. Fig. Fig.
7(b) 7(c) 7(d) 7(e) 7(f)
Phase proportion (%)
Diameter of a (lm)
a
b
l^da
rda
dmax
Diameter of b (lm) l^db
rdb
dmax
70.17 73.72 76.62 79.50 81.94
29.83 26.28 23.38 20.50 18.06
27.63 28.32 28.86 29.38 29.83
7.31 7.53 7.71 7.90 8.03
38.86 39.48 40.02 40.36 40.64
26.50 24.85 23.31 21.79 20.28
8.57 8.78 9.00 9.16 9.18
37.73 36.01 34.34 32.53 30.21
Table 10 Two-phase alloy simulation results Specimen Fig. Fig. Fig. Fig. Fig.
8(b) 8(c) 8(d) 8(e) 8(f)
Phase proportion (%)
Diameter of a (lm)
a
b
l^da
rda
dmax
Diameter of b (lm) l^db
rdb
dmax
74.65 78.65 80.99 83.30 85.98
25.35 21.35 19.01 16.70 14.02
20.13 20.67 20.95 21.25 21.44
5.28 5.43 5.54 5.64 5.75
25.79 26.92 27.78 28.27 28.91
16.93 15.58 14.41 13.37 12.24
4.53 4.60 4.57 4.63 4.84
23.24 22.09 21.02 20.12 19.24
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Fig. 13. One type of grains embedded in another type of grains.
3.1.6. Grain shape change with different velocity The lengths and radii of the grain boundaries, and the locations of triple points, are all affected by grain growth velocity during the solidification process. As shown in Figs. 7(a) and 8(a) when all grains grow with equal velocity, all the grain boundaries are straight lines in the proposed simulation method which reduces to the Voronoi tessellation method. When the adjacent grain growth velocities are not equal, the grain boundaries between them become curved, as showed in Figs. 7(b)–(f) and 8(b)–(f), where the ratio of adjacent grain velocities (f ¼ v1 =v2 ) ranges from 1.1 to 1.5. The length and radius of the curved boundary become shorter as the ratio of grain growth velocity increases, because the grain with higher velocity will grow around the grain with smaller velocity. The corresponding triple point locations also change as the ratio of grain growth velocity changes. As the ratio of grain growth velocity tends to a large value, one kind of grain will be embedded in another grain, as showed in Figs. 7(g) and 8(g) when the ratio of adjacent grain velocities (f ¼ v1 =v2 ) is equal to 15, and the proposed method is able to simulate this case. Therefore, the proposed method provides a better simulation model for stress analysis and material constants estimation even for complicated material structure as shown in Fig. 13 (simulated with f ¼ v1 =v2 ¼ 25). When the core points are uniformly distributed in the given area, and all the grain growth velocities are equal, the simulation results would be honeycombs as shown in Figs. 7(h) and 8(h).
4. Conclusion A generalized 2-D method has been developed in this paper to simulate the microstructure of metallic polycrystalline materials. This method assumes that adjacent grains in different phases will grow at different velocities during solidification. The grain growth velocity affects the shape of the grain, including number of edges, grain diameter, grain area, the length and radii of grain boundary, and the location of triple points. If the ratio of growth velocity is equal to 1, the proposed method reduces to the Voronoi tessellation method. If the ratio of growth velocity tends to 1, it simulates the case where one type of grain embedded in the other type of grain. The grain growth velocity is affected by different types of grains during solidification process, and also by uneven temperature. The simulation results show that the edge number and triple point location of some grains will change with the increasing ratio of grain growth velocity even if the
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original core points are the same. The proposed method is able to model these various situations effectively. Statistical analysis shows that the results of the proposed method agree well with experimental observation.
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