Prediction method for grain growth

Abstract

To provide a method for predicting grain growth in carburization processing.SOLUTION: A method includes: a step of dividing a model space, obtained by dispersing deposits into a space formed of austenite grains, into a plurality of cells, and imparting a deposit object volume fraction f and a crystal azimuth to each of the cells; a step of calculating mobility M of a grain boundary calculated from the crystal azimuth and a temperature T; a step of calculating a radius RL of curvature of the grain boundary with a circumferential cell, and calculating driving force ΔG of the grain growth given by multiplying grain boundary energy σ, calculated from the crystal azimuth, by a function of the radius RL of curvature, a deposit diameter r and the deposits object volume fraction; a substitution step of extracting a substitution candidate cell with the smallest driving force ΔG among adjacent cells, and substituting the substitution candidate cell when the grain growth speed v obtained by multiplying the driving force ΔG of the substitution candidate cell by the mobility M is larger than a generated uniform random number z; and a resetting step of imparting a deposit diameter r after a time step, wherein the steps from mobility calculation to resetting are repeated.SELECTED DRAWING: Figure 3

Description

[Technical field]

[0001] The present invention relates to a method for predicting grain growth of austenite grains during carburization. [Background technology]

[0002] Carburizing is widely used as a surface hardening treatment for parts used in power transmission, such as gears, pulleys, and shafts. However, if the carburizing treatment is carried out at high temperatures to rapidly progress the carburizing treatment, or if cold forging is performed before carburizing, abnormal grain growth is likely to occur during the carburizing treatment. Such abnormal grain growth during carburizing increases the amount of heat treatment strain and causes a decrease in the fatigue strength of the parts.

[0003] Here, in order to suppress abnormal grain growth during carburizing, it is effective to obtain the so-called "pinning effect" by finely dispersing precipitates such as AlN and NbC in the steel. Therefore, if it is possible to predict the occurrence of abnormal grain growth from the pinning effect of these precipitates, it will be possible to determine the amount of particles required to obtain the necessary pinning, that is, the amount of particle-forming elements such as Nb and Al to be added to the steel, and process conditions such as the appropriate carburizing temperature.

[0004] For example, Patent Document 1 discloses a method for predicting in advance structural abnormalities such as abnormal grain growth in steel bar or section products that have two-dimensional distributions of strain, strain rate, and temperature in the cross section during rolling. In this method, three-dimensional rolling analysis calculations are performed for each rolling pass, and temperature analysis calculations are performed for the temperature distribution in the cross section between each pass and during the final cooling process, and the final structure is predicted taking into account the strain, strain rate, and temperature distribution in the cross section. [Prior art documents] [Patent documents]

[0005] [Patent Document 1] Japanese Patent Application Publication No. 6-182412 Summary of the Invention [Problem to be solved by the invention]

[0006] As described above, obtaining the pinning effect of precipitates is effective in suppressing abnormal grain growth during carburizing. However, precipitates are not necessarily uniformly dispersed in steel due to the effects of solidification segregation, and it has been difficult to predict the relationship between pinning by precipitates and abnormal grain growth.

[0007] On the other hand, several methods for simulating the growth of crystal grains have been proposed, including the Cellular Automation method, which divides space into cells and calculates the interactions between adjacent cells. By using this method, it was considered to predict the occurrence of abnormal grain growth by introducing the effect of the dispersion state of precipitates into the simulation of the grain growth of austenite grains during carburizing treatment.

[0008] The present invention has been made in view of the above circumstances, and an object of the present invention is to provide a method for predicting grain growth during carburizing treatment, including prediction of the occurrence of abnormal grain growth of austenite grains during carburizing treatment, by using a cellular automaton method to introduce the effect of the dispersed state of precipitates into a simulation of the grain growth of austenite grains. [Means for solving the problem]

[0009] The method according to the present invention is a method for predicting the grain growth of austenite grains in a carburizing process, in which a model space in which precipitates are dispersed in a space consisting of austenite grains is divided into a plurality of cells, and the initial precipitate diameter r0, the temperature T of the carburizing process, and the maximum value z of uniform random numbers z are calculated. max and an initial setting step of giving each of the cells a precipitate volume fraction f and a crystal orientation; a mobility calculation step of calculating, for each of the cells, a mobility M of the grain boundary calculated from the crystal orientation and the temperature T; and a radius of curvature R of the grain boundary between each of the cells and the surrounding cells. LAfter calculating, the grain boundary energy σ calculated from the crystal orientation is multiplied by the radius of curvature R L a driving force calculation step of calculating a driving force ΔG of grain growth given by multiplying a function of a precipitate diameter r and a precipitate volume fraction f, and a uniform random number z (0≦z≦z) generated for a grain growth velocity v obtained by extracting a replacement candidate cell having the smallest driving force ΔG from among the adjacent cells for each of the cells and multiplying the driving force ΔG of the replacement candidate cell by the mobility M. max ), a replacement step of replacing the cell with the replacement candidate cell when the initial precipitate diameter r0 is larger than the initial precipitate diameter r0, and a resetting step of giving the precipitate diameter r after a time step from the initial precipitate diameter r0, wherein the resetting step is repeated from the mobility calculation step.

[0010] According to this feature, the simulation is based on a model in which grain growth occurs when the grain growth rate v according to the grain growth kinetics is greater than a uniform random number z that is randomly given, and the space is divided into a plurality of cells, and the time evolution of grain growth is calculated step by step using a cellular automaton method, making it possible to simulate the grain growth of austenite grains by introducing the influence of the dispersion state of precipitates that are non-uniformly dispersed in the space.

[0011] In the above-described invention, the initial setting step may include selecting a designated cell whose mobility is to be emphasized and providing an emphasis coefficient for the mobility in the designated cell, and the mobility calculation step may include multiplying the calculated value for the designated cell by the emphasis coefficient to obtain the mobility M. According to this feature, it is possible to simulate a case where the mobility is emphasized by allocating a high mobility only to a specific cell due to non-uniformity in mobility, and only a specific crystal grain has an orientation or corresponding grain boundary different from those of the surrounding crystal grains, resulting in a high mobility.

[0012] In the above-mentioned invention, in the cell at the position x, the driving force ΔG is ΔG[x], the grain boundary energy σ is σ[x], and the radius of curvature R L R L[x], the precipitate diameter r is r[x], the precipitate volume fraction f is f[x], and p and q are coefficients,

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[0013] In the above-mentioned invention, in the initial setting step, the grain boundary energy E0 per unit length and the crystal orientation of each cell are given, and the grain boundary energy σ[x] is the sum of the grain boundary energies of the cells surrounding the cell x and the cells having different crystal orientations.

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[0014] In the above-mentioned invention, the initial setting step may be characterized in that the crystal orientation is given by a quaternion. With this feature, it is possible to simplify the calculation of the orientation difference.

[0015] In the above-mentioned invention, in the initial setting step, the radius of curvature R of the grain boundary is L A numerical value b is given to specify a cell range for which the driving force is to be calculated, and in the driving force calculation step, the radius of curvature R of the grain boundary is calculated. L of,

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[0016] In the above-mentioned invention, the precipitate diameter r for each time step is, at time t,

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[0017] [Figure 1] FIG. 13 shows the range of other cells around cell x that should be considered when calculating curvature. [Diagram 2] 1 is a graph showing the distribution of crystal misorientation in (a) the method by Moelans et al. and (b) the method according to the present embodiment. [Diagram 3] FIG. 1 is a flow diagram illustrating a method in one embodiment according to the present invention. [Figure 4] FIG. 13 is a structural diagram showing the results of a simulation without mobility enhancement and without precipitates. [Diagram 5] 13 is a graph showing the transition of the average particle size during the simulation. [Figure 6] FIG. 13 is a tissue diagram showing a simulation result when mobility is emphasized. [Figure 7] 13 is a graph showing the ratio of maximum grain size to average grain size as a simulation result when only the magnification for emphasizing mobility is changed. [Figure 8] 13 is a graph showing the ratio of maximum grain size to average grain size as a simulation result when only the number of cells emphasizing the mobility is changed. [Figure 9] FIG. 13 is a structural diagram showing a simulation result in the case where precipitates are present. [Figure 10] 1 is a graph showing the ratio of the maximum grain size to the average grain size as a simulation result when the volume fraction f0 of precipitates is changed. [Figure 11] 1 is a map showing changes in the volume fraction of precipitates used in a simulation. [Figure 12] FIG. 13 is a structural diagram showing a simulation result based on a change in the volume fraction of precipitates using the map. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0018] First, an outline of the method for predicting grain growth in this embodiment will be described.

[0019] This is a method for predicting the grain growth of austenite grains in a structure during heat treatment of steel in the austenite single-phase region, such as carburizing, annealing, and normalizing. Simulation is performed based on a model in which grain growth occurs when the grain growth rate v according to grain growth kinetics is greater than a randomly given value. The method is based on a prediction method in which the space is divided into multiple cells corresponding to crystal grains, and the time evolution of grain growth is calculated step by step using a cellular automaton method.

[0020] Here, the driving force ΔG of grain growth is calculated by taking into account the influence of precipitates that become pinning particles that hinder grain growth. At this time, it is assumed that the pinning particles are non-uniformly distributed in space, and the pinning effect is also non-uniformly distributed in space. Then, a unique crystal orientation is given to each cell, which is a crystal grain, and the driving force ΔG and mobility M according to the crystal orientation difference at the crystal grain boundary (hereinafter referred to as grain boundary) are calculated, and the results are substituted into the formula (ΔG × M) of the grain growth rate v obtained from the kinetics of grain growth and compared with the randomly given value described above. Then, as described above, when the grain growth rate v is greater than this randomly given value, it is replaced with an adjacent cell to express the grain growth. By repeating such steps, grain growth can be predicted.

[0021] The crystal orientation of the cells was given by a quaternion to simplify the calculation of the orientation difference. Also, taking into account the non-uniformity of mobility, the mobility of a specific cell was made greater than that of other cells. Furthermore, in order to introduce the effect of the curvature of the grain boundary on the driving force of grain growth in the cellular automaton method, the radius of curvature was calculated using the box counting method. Also, the process in which the pinning effect decreases due to Ostwald ripening of the pinning particles was calculated, and this was gradually reflected in the driving force of grain growth.

[0022] The method for predicting grain growth will be described in detail below.

[0023] Several methods have been proposed for simulating metal structures, such as the phase-field method, the cellular automaton method, and the Monte Carlo method. In this embodiment, the cellular automaton method, which has a relatively low calculation cost and is not as probabilistic as the Monte Carlo method, is used. In the cellular automaton method, it is necessary to define the rules for moving to the next step. Here, a grain growth model using the cellular automaton method for the grain growth behavior of austenite grains during heat treatment was constructed as follows.

[0024] Here, we apply the general theory of grain growth to the cellular automaton method as follows. In the general theory of grain growth, the driving force ΔG of grain growth per unit volume is expressed as formula (1) using the radius of curvature of the grain boundary. Here, R L is the radius of curvature, and σ is the grain boundary energy. Each cell is given a code to identify it. For example, the position of the cell (vector indicating the coordinates) is x, and it can be expressed as [x]. In other words, the grain boundary energy of cell x is σ[x], and the radius of curvature of cell x is R L Alternatively, in a two-dimensional space, the coordinates of the cell's position, x and y, may be used to express the position as [x, y].

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[0025] In addition, the mobility of grain boundaries is said to be temperature dependent and is expressed by equation (2).

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[0026] Although there is already a lot of knowledge about the constants required for formulas (1) and (2), the method of handling curvature using the cellular automaton method is not common. This is because the cellular automaton method handles space by dividing it discretely. To solve this problem, in this embodiment, the box counting method proposed by Pierre et al. is used to introduce the effect of the radius of curvature on the driving force of grain growth. In the box counting method, the radius of curvature R is calculated as shown in formula (3). L Represents.

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[0027] Here, b represents the cell range to be considered around cell x, as shown in Figure 1. For example, if b=5, the curvature is calculated taking into account the second nearest cell. δ x,x’ is 0 when cell x and cell x' are different grains, and is 1 when they are the same grain.

[0028] We will now explain how to handle crystal grains. A method that is often used when using the cellular automaton method or the phase field method is to assign a unique grain number to each cell to identify the crystal grain. In this case, for example, as shown in equation (4), the orientation difference is calculated based on the absolute value of the difference between grain numbers divided by the maximum grain number.

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[0029] Therefore, in this embodiment, instead of giving each crystal grain a grain number, a crystal orientation is given to each grain, as in the case of a real structure. In this case, a quaternion is used to represent the orientation in order to simplify the calculation. The use of a quaternion to represent the crystal orientation has been proposed, for example, by Takahashi et al.

[0030] If the quaternions representing the orientations of cubic crystals a and b are qa and qb, respectively, then the orientation difference θ a,b can be simply expressed as (Equation 5).

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[0031] Here, e i are 24 types of quaternions that represent the symmetry operations of a cubic crystal. Also, (p, q) represents the inner product of the quaternions p and q. For example, when Euler angles or lattice vectors are used, more complicated operations are required, but by using quaternions as in this embodiment, the orientation difference can be obtained at a lower calculation cost.

[0032] In addition, since the four real parts of the quaternion can be treated equally, there is an advantage that when random orientation is given, it is easy to give random orientation by simply giving uniform random numbers to each of the four real parts and normalizing them. For example, as shown in FIG. 2(a), when grain numbers are given to crystal grains and grain boundary misorientation is defined by the method by Moelans et al., the distribution of crystal misorientation is significantly different from the theoretical random misorientation distribution shown by the dashed line. On the other hand, as shown in FIG. 2(b), when the crystal orientation is directly given randomly according to this embodiment, the distribution of crystal misorientation is close to the theoretical distribution shown by the dashed line.

[0033] Next, we will explain how to handle grain boundary energy. We applied the following formula (6) by Read and Shockrey to the grain boundary energy, and incorporated the crystal orientation dependence of the grain boundary energy. However, the grain boundary energy E0 per unit length is given, and θ x x' is the inclination angle of the crystal orientation between cell x and cell x'.

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[0034] The grain boundary energy σ[x] of each cell is expressed as the sum of the grain boundary energies with the surrounding cells, as shown in equation (7).

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[0035] Regarding the mobility of grain boundaries, we applied the formula (8) proposed by Humphry et al. to introduce the crystal orientation dependence of the mobility.

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[0036] In addition, considering the non-uniformity of mobility, we devised a model that emphasizes mobility by assigning high mobility only to certain crystal grains and performed simulations. For example, this assumes that only certain crystal grains have a different crystal orientation or coincidence boundaries from the surrounding crystal grains, which increases their mobility.

[0037] Finally, the effect of precipitates will be explained. It is said that precipitates have a pinning effect on grain boundaries during heat treatment and suppress grain coarsening. According to Zener, the driving force ΔG of grain growth per unit volume is expressed as formula (9) using coefficients p and q when the pinning effect is introduced. In this example, Zener's formula (9'), in which these coefficients are determined, is used.

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[0038] Furthermore, the precipitate diameter (radius of the precipitate) may be increased over time by the growth of the precipitate. For example, the precipitate diameter r at time t is set to undergo Ostwald ripening according to the following formula (10) with respect to the initial precipitate diameter r0.

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[0039] Furthermore, assuming that there is a bias in the distribution of precipitates due to the effects of segregation in the metal structure, the grain growth behavior was confirmed using equation (11), which gives a distribution to the volume fraction f so that the volume fraction f has a gradient with respect to position. The distribution of f according to equation (11) can be expressed, for example, as a map as shown in Figure 11, which will be described later.

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[0040] The grain growth rate v is expressed by the above-mentioned driving force ΔG and mobility M as shown in equation (12).

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[0041] In this example, when ΔG decreases, if the grain growth rate v is equal to or greater than a certain value z given by a random number, the grains are replaced by adjacent crystal grains to become the same phase so that ΔG is minimized. This is the rule shown in the following formula (13).

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[0042] As shown in FIG. 3, an example of a combination of the process steps of the above-mentioned method is shown in a flow chart.

[0043] First, the initial structure and constants are set (S1: Initial setting step). The model space, in which precipitates are dispersed in a space consisting of austenite grains, is divided into multiple cells, and the initial precipitate diameter r0, temperature T, and maximum value z of uniform random number z are set. max The precipitate volume fraction f and crystal orientation are determined for each cell.

[0044] Next, the counting of time steps is started (S2). The process described below is carried out for each time step count.

[0045] Next, the driving force ΔG is calculated for all cells (S3: driving force calculation step). That is, the radius of curvature R of the grain boundary between the surrounding cells is calculated by the formula (3). L The grain boundary energy σ calculated from the crystal orientation is multiplied by the radius of curvature R of the grain boundary. L The driving force for grain growth, ΔG, given by multiplying the precipitate diameter r and the precipitate volume fraction f is calculated using equation (9).

[0046] Furthermore, the driving force when a cell is replaced by an adjacent cell is calculated for all cell combinations (S4). Here, the driving force when a cell is replaced by an adjacent cell is calculated in advance in order to detect a cell that will be replaced with a cell that minimizes the driving force, which will be described later.

[0047] Then, the mobility M is calculated for all the cells (S5: mobility calculation step). Here, the mobility M is calculated from the crystal orientation and the temperature T by the formula (8).

[0048] Here, a designated cell for which mobility is to be emphasized is selected in advance, and it is determined whether the cell for which the mobility has been calculated is a designated cell (S6). If it is a designated cell for which mobility is to be emphasized (S6: Yes), the mobility is multiplied by an emphasis coefficient that is a predetermined magnification to obtain an emphasized mobility (S7). If it is not a designated cell for which mobility is to be emphasized (S6: No), the mobility is left as it is.

[0049] Then, the grain growth rate v when the cell is replaced with the cell that minimizes the driving force is calculated (S8). First, the driving force ΔG after The adjacent cell with the smallest driving force ΔG is extracted as a replacement candidate cell. Then, the driving force ΔG of the replacement candidate cell is smaller than that of the original cell. after If is smaller, a substitution decision is made.

[0050] In the replacement judgment, the grain growth rate v of the replacement candidate cell is compared with the uniform random number z (S9). after The grain growth rate v is obtained by multiplying the mobility M by the uniform random number z(0≦z≦z max) and compared. If the grain growth rate v is greater than the uniform random number z (S9: Yes), it is determined that the cell should be replaced with the replacement candidate cell. In other words, it is determined that the crystal grains corresponding to the original cell and the crystal grains corresponding to the replacement candidate cell will become the same phase and grain growth will occur. In this case, the original cell and the replacement candidate cell are replaced as one cell (S10: replacement step). On the other hand, if the grain growth rate is not greater than the uniform random number z (S9: No), it is determined that grain growth will not occur and the cell is not replaced.

[0051] Then, the diameter of the precipitates that will become pinning particles is reset (S11: resetting step). Here, taking into consideration the growth of the precipitates due to the Ostwald ripening described above, the precipitate diameter r when the time corresponding to the progress of one time step is advanced based on Equation 10 is reset.

[0052] The above process is carried out for all cells.

[0053] Finally, the process returns to the start of the time step (S2) and repeats the process until a specified time step is reached (S12). That is, a time corresponding to one count of the time step is set, and counting is repeated until the time when grain growth is to be observed by simulation is reached. Note that the order of the calculation of the driving force (S3 and S4) and the calculation of the mobility (S5) can be reversed, so that the calculation of the mobility is performed first.

[0054] This makes it possible to simulate the grain growth process and even predict the occurrence of abnormal grain growth.

[0055] Next, the results of an actual simulation performed on a two-dimensional cell using the above-mentioned method will be described.

[0056] The initial structure given to the cellular automaton method for grain growth was based on the cellular automaton method for phase transformation, and calculations were performed using the cellular automaton method starting from several initial states with different average grain sizes. In this case, calculations were performed with 1px set to 1μm, and the initial average grain size was set to 3.19μm. Periodic boundary conditions were used as boundary conditions.

[0057] In the simulation, calculations were performed using Python code and code written in C. The parameters used in the calculations are shown in Table 1.

[0058] [Table 1]

[0059] First, a simulation was performed under conditions where mobility was not emphasized and precipitates were not present. The initial structure and the structures after 100, 500, and 1000 steps are shown in Fig. 4. The crystal orientation is shown in the IPF map.

[0060] FIG. 5 shows the change in average particle size. Here, the parabolic law is expressed as follows: <R g > 2 - <r0> 2 ∝t-t0 Where: <R g > is the average particle size, <r0>is the initial average grain size, t is time, and t0 is the reference time. In the figure, the dashed line shows a straight line fitted using the least squares method, starting from the timing when the calculation settles from the initial grain size and grain growth begins. As shown in the figure, the average grain size of the crystal grains shows a progression close to a parabolic law, which is considered to be steady grain growth.

[0061] Next, Figure 6 shows the grain growth behavior when the mobility is emphasized at a magnification of 10 times and the number of emphasized cells is 10. Abnormal grain growth is observed when the mobility is emphasized. Emphasizing the mobility can be considered equivalent to introducing the effects of coincidence boundaries and segregation. For example, Olmsted et al. have reported that the mobility of coincidence boundaries is much higher than that of other grain boundaries. There are also reports that segregation reduces mobility. The criteria for judging abnormal grain growth are the maximum grain size R in the structure, max The average particle size <R g When the value divided by > exceeds 3, it is determined that abnormal grain growth has occurred. In other words, R max / <R g > is used as a discriminant for abnormal grain growth, and when the value of this discriminant exceeds 3, it is determined that abnormal grain growth has occurred.

[0062] Furthermore, as shown in Figures 7 and 8, simulations were performed for two cases: when only the magnification for emphasizing mobility was changed, and when only the number of cells for emphasizing mobility was changed. Both figures show the ratio of the maximum grain size to the average grain size after 1000 steps. Here, 10 random number seeds were prepared before the calculation started, and the calculation was performed 10 times under the same conditions.

[0063] In Figure 7, abnormal grain growth began to be observed when the mobility coefficient, which is the multiplication factor of mobility, reached 3 or more, and it can be seen that abnormal grain growth occurred in all cases when the mobility coefficient reached 7 or more.

[0064] In Fig. 8, if there is even one cell that emphasizes mobility, abnormal grain growth is observed. On the other hand, if the ratio of the number of cells that emphasize mobility exceeds 20% of the initial number of cells, abnormal grain growth is no longer observed. This is because if the number of cells that emphasize mobility becomes too large, only the crystal grains corresponding to the emphasized cells undergo normal grain growth.

[0065] Figure 9 shows an example of the structure resulting from calculations after 1000 steps when a simulation was performed under conditions in which precipitates are present. Figure 10 shows the effect on the discriminant equation for abnormal grain growth when the volume fraction of precipitates, f0, is changed. These results show that the occurrence of abnormal grain growth can be suppressed by increasing the volume fraction of precipitates. In this study, the precipitate diameter, r, was fixed at 5 nm regardless of position. Here, the magnification of the mobility to be emphasized was set to 10 times, and the number of cells for emphasizing the mobility was set to 10. As a result, when there is no bias in the volume fraction of precipitates due to position, the precipitate volume fraction, f0, was set to approximately 1.3 × 10 -4 If this is the case, abnormal grain growth can be suppressed.

[0066] Here, we discuss the correspondence with Gladman’s critical condition equation for abnormal grain growth. Gladman’s equation is as follows.

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[0067] As shown in FIG. 11, a simulation was performed assuming that the precipitates are biased and modulated in the vertical direction so that the volume fraction of the precipitates is represented by shading, forming stripes extending in the horizontal direction of the paper. FIG. 12 shows the structure after 1000 steps. When the number m (modulation) of biased stripes in the simulated range is 4 or less (excluding the case of m=0 where there is no bias), abnormal grain growth occurs. In particular, abnormal grain growth occurs in the part where the volume fraction f0 of the precipitates is low. In this way, if there is a bias in the volume fraction of the precipitates, abnormal grain growth is more likely to occur. In particular, the greater the bias, the more likely abnormal grain growth tends to occur. This shows that the greater the bias in the precipitate distribution, the more disadvantageous it is for suppressing abnormal grain growth.

[0068] As described above, in this example, a grain growth simulation model was proposed that takes into account factors that have not been considered before, such as mobility and biased precipitates. By performing a simulation taking these factors into account, it became possible to predict abnormal grain growth during carburizing. This makes it easier to control the structure, which can be said to lead to faster material development.

[0069] In this embodiment, the prediction of grain growth was performed using the cellular automaton method. Then, an attempt was made to reproduce the behavior of abnormal grain growth and inhibition by precipitates, and the following conclusions were obtained. First, in the cellular automaton method, abnormal grain growth was generated by emphasizing the mobility of grain boundaries, and abnormal grain growth occurred when the magnification of emphasis was set to 3 times or more. This suggests that the mobility of grain boundaries may play an important role in abnormal grain growth. Such abnormal grain growth can be suppressed by the pinning effect of precipitates. However, when there is a bias in the concentration of precipitates in the structure and there are places where the pinning effect is weak, it becomes difficult to suppress abnormal grain growth. From the above, it was possible to reproduce abnormal grain growth at a relatively low calculation cost by using the cellular automaton method. In other words, it is considered that the above-mentioned embodiment is an effective means for reproducing abnormal grain growth behavior during heat treatment such as carburization.

[0070] Although the representative embodiments of the present invention and modified examples based thereon have been described above, the present invention is not necessarily limited to these, and a person skilled in the art will be able to find various alternative embodiments and modified examples without departing from the spirit of the present invention or the scope of the appended claims. [Explanation of symbols]

[0071] S1 Initial Setup Steps S3 Driving force calculation step S5 Mobility calculation step S10 Substitution step S11 Resetting step

Claims

1. 1. A method for predicting grain growth of austenite grains in a carburizing process, comprising the steps of: The model space in which precipitates are dispersed in the space consisting of austenite grains is divided into a plurality of cells, and the initial precipitate diameter r ０ , the temperature T of the carburizing treatment, the maximum value z of the uniform random number z max and an initial setting step of giving each of the cells a precipitate volume fraction f and a crystal orientation; a mobility calculation step of calculating a mobility M of a grain boundary calculated from the crystal orientation and the temperature T for each of the cells; For each of the cells, the radius of curvature R of the grain boundary between the surrounding cells Ｌ After calculating, the grain boundary energy σ calculated from the crystal orientation is multiplied by the radius of curvature R Ｌ a driving force calculation step of calculating a driving force ΔG for grain growth given by multiplying the precipitate diameter r by a function of the precipitate volume fraction f; For each of the cells, a replacement candidate cell having the smallest driving force ΔG among the adjacent cells is extracted, and a uniform random number z (0≦z≦z) generated for a grain growth rate v obtained by multiplying the driving force ΔG of the replacement candidate cell by the mobility M is calculated. max ), replacing the cell with the replacement candidate cell; The initial precipitate diameter r ０ a resetting step of giving the precipitate diameter r after a time step from and repeating the mobility calculation step through the resetting step.

2. In the initial setting step, a selection of a designated cell for emphasizing the mobility and an emphasis coefficient of the mobility in the designated cell are given; 2. The method for predicting grain growth according to claim 1, wherein said mobility calculation step multiplies the calculated value by an emphasis coefficient to obtain said mobility M for said specified cell.

3. In the cell at the position x, the driving force ΔG is ΔG[x], the grain boundary energy σ is σ[x], and the radius of curvature R Ｌ R Ｌ [x], the precipitate diameter r is r[x], the precipitate volume fraction f is f[x], and p and q are coefficients, [0010] 3. The method for predicting grain growth according to claim 1, wherein the prediction is given by:

4. In the initial setting step, the grain boundary energy E ０ And, I gave it to him. The grain boundary energy σ[x] is the sum of the grain boundary energies of the cells surrounding the cell x and having different crystal orientations. [0025] (However, δ x,x' is 0 when cell x and cell x' are different, and 1 when they are the same. x x' is the inclination angle of the crystal orientation between cell x and cell x'.

4. The method for predicting grain growth according to claim 3, wherein the prediction is given by:

5. 5. The method for predicting grain growth according to claim 4, wherein in said initial setting step, said crystal orientation is given as a quaternion.

6. In the initial setting step, the radius of curvature R of the grain boundary Ｌ A value b is given to specify the cell range in which to calculate In the driving force calculation step, the radius of curvature R of the grain boundary Ｌ of, [0030] (However, δ x,x' is 0 if cell x and cell x' are different, and 1 if they are the same.) 4. The method for predicting grain growth according to claim 3, wherein the prediction is given by:

7. The precipitate diameter r for each time step is given by: [0045] 2. The method for predicting grain growth according to claim 1, wherein the prediction is given by:

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