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An analysis of kinetic Monte Carlo simulation as a tool for modelling strain ageing
Scripta Materialia 53 (2005) 625–630 www.actamat-journals.com
An analysis of kinetic Monte Carlo simulation as a tool for modelling strain ageing A.J.P. Gater, S.G.R. Brown
*
EPSRC Engineering Doctorate Centre, Materials Research Centre, School of Engineering, University of Wales Swansea, Swansea, SA2 8PP, UK Received 17 February 2005; received in revised form 26 May 2005; accepted 30 May 2005 Available online 21 June 2005
Abstract The mechanism of strain ageing is briefly outlined, and details of a Monte Carlo algorithm for simulation of this process are given. The results of varying mesh shape, mesh size, free interstitial carbon and temperature on the performance of the algorithm are then critically assessed. 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Strain ageing; Dislocations; Kinetics
1. Introduction The threat of legislation regarding emissions and fuel consumption has led the automotive industry to think in terms of down gauging, and the use of novel high strength steels in the production of body panels, to reduce the weight of the Ôbody in whiteÕ [1]. There are a range of steels suitable for such drawing applications, with varied properties (Table 1) [2]. One currently experiencing a lot of interest is ultra low carbon bake hardenable sheet. The term Ôbake hardenableÕ steel refers to a range of highly formable strip steels with a yield stress that can be improved subsequent to forming via a heat treatment, producing a yield stress increase typically in the range 34–70 MPa [3]. The combination of an ultra low carbon (ULC) chemistry and the yield strength increase occurring after forming reduces the forces required in drawing operations, and thus the costs. Bake hardening has also been shown to produce an increase in surface dent resistance [4,5]. *
Corresponding author. Tel.: +44 1792 295284. E-mail address: [email protected] (S.G.R. Brown).
The yield stress increase is achieved by leaving controlled amounts of carbon in solution, typically 5– 15 wt. ppm [6] (5–15 wt. ppm = 0.0005–0.0015 wt.%) through vacuum degassing and careful control of micro alloying additions such as titanium and niobium; this level of carbon is not sufficient to be greatly detrimental to formability. During paint curing free interstitial carbon can diffuse to the cores of dislocations under a Ôdrift velocityÕ caused by an interaction of the strain fields around the dislocation and misfitting solute atoms. This carbon then forms solute atmospheres around the dislocations, locking them in place, and increasing the applied stress that is required to initiate dislocation glide. During atmosphere formation an initial yield strength increase of 20–30 MPa is observed, the maximum value corresponding with a locking ratio of around 0.9 atoms per atom plane threaded by dislocation [7]; further strengthening is then achieved through the formation of nano-scale precipitates around the dislocation. The higher the level of interstitial carbon present, the greater the yield stress increase that can be achieved, to a maximum value of 70–90 MPa [8,9]; however, at carbon levels in excess of 10 wt. ppm, the risk of
1359-6462/$ - see front matter 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2005.05.035
626
A.J.P. Gater, S.G.R. Brown / Scripta Materialia 53 (2005) 625–630
Table 1 Typical properties of steels suitable for deep drawing [2] Steel type
ry (MPa)
UTS (MPa)
Elongation (%)
n
rm
Interstitial free Al-Killed Rephosphorised Bake hardened HSLA
155 165–190 190–230 210 375
305 300 345–370 320 475
42 42 36 40 27
0.23 0.22 0.20 0.22 0.15
2.0 1.8 1.5 1.8 1.0
room temperature ageing increases and must be controlled through temper rolling [10]. For a thorough review of bake hardenable steels see Baird [11–14] and Baker et al. [1].
2. Modelling strain ageing A kinetic Monte Carlo simulation of strain ageing has been published by Soenen et al., accounting for both diffusion of carbon atoms to dislocations during ageing, and grain boundary segregation during the annealing process, and the results of this group have shown a good match to their experimental data [15]. As part of an ongoing research project the algorithm developed in that paper has been adopted for this study, in which the kinetics are controlled by a set of Arrhenius type equations controlling jump frequency, relating to jumps between positions of equivalent energy within the bulk material, and jumps to sites of higher or lower energy, within a critical radius around dislocations. The frequency of jumps between equivalent sites, fd, is controlled by an attempt frequency, m, and Ud the energy barrier to diffusion, which was taken as 82,000 J/mol, approximately equal to the activation energy for carbon diffusion, of 84,140 J/mol [15]. Ud D0 f ¼ fd ¼ m exp ð1Þ ; m¼ 2 RT a Here R is the gas constant, T the absolute temperature, D0 the coefficient of diffusivity for carbon in ferrite (2 · 106 m2/s) and a the lattice parameter of a body˚ ). Table 2 centred cubic (bcc) ferrite unit cell (2.87 A shows jump frequencies for interstitial carbon calculated using Eq. (1) with the parameters stated by Soenen et al. [15]; the chosen activation energy of 82,000 J/mol gives a Table 2 Modelled jump frequencies as a function of temperature Temperature (K)
273 298 323 373
Jump frequency (s1) (84.14 kJ/mol, 2E6 m2/s)
(82 kJ/mol, 2E6 m2/s)
0.0019 0.0432 0.5983 39.9106
0.0049 0.1024 1.3275 79.5801
jump frequency of around 0.1/s at 298 K, in agreement with the position of the carbon peak in internal friction studies. Where a diffusive jump moves an atom between sites two sites i and j with respective energies Ui and Uj the jump frequency fd is again observed if the energy of the target site is lower; if a jump takes an atom to a higher energy site the frequency is related exponentially to the energy difference between the two positions [15]: f ði ! jÞ ¼ fd ;
Ui > Uj Uj Ui f ði ! jÞ ¼ fd exp ; RT
ð2Þ Ui < Uj
ð3Þ
The program records the co-ordinates of all solute atoms and dislocations, and interaction energies are read from energy ÔtemplatesÕ if carbon atoms are within the calculated radius of interaction (rc). The critical radius of interaction allows for a minimum variation of 10% from the normal jump frequency and is calculated as [15]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Aa rc ¼ ð4Þ RT lnð1.1Þ The interaction energy of each site is calculated using [15]: U¼
A sin h r
ð5Þ
where (r, h) is the polar co-ordinate of the solute atom with respect to the dislocation line and A is the solute dislocation interaction parameter (taken as 7.5 · 106), T is the absolute temperature of the system and R is the gas constant. At the beginning of the program, carbon atoms are randomly placed throughout the mesh at the cell centres, and dislocations at the edges of unit cells. Each carbon atom has a local clock that keeps track of its location in time, this clock being updated every time the atom makes a diffusive jump. The algorithm proceeds as follows (Fig. 1): (1) Select that carbon atom for which the least amount of time has passed (the lagging atom). (2) Set the simulation time to the local time of the selected atom. (3) Using a random number generator determine in which of the four possible directions the carbon atom will jump, based on the relative jump frequencies. (4) Check whether the destination site is vacant; if so, perform the diffusive jump, if not, the selected atom remains stationary. (5) Calculate the jump frequencies for the atom in its new position.
A.J.P. Gater, S.G.R. Brown / Scripta Materialia 53 (2005) 625–630
627
Table 3 Simulations run in the evaluation of the Monte Carlo algorithm
Fig. 1. Diagrammatic representation of Monte Carlo algorithm.
(6) Update the atomsÕ local clock by the amount of time that will be required for it to perform its next diffusive jump. (7) Re-sort atoms by their local clocks. A periodic boundary condition was used in all cases. The algorithm should represent well the actual physics of dislocation locking, as it models the stepwise jumps of carbon atoms moving interstitially through a bodycentred cubic lattice, with frequency controlled by the diffusion equation, and the parameters that are shown in Table 2, which match well with observed phenomena.
3. Model trials Trials have been aimed at assessing the affects of mesh size and shape on accuracy, and on the ability of the model to assess a range of temperatures and carbon contents (Table 3). To this end, experimental plots have been reproduced from papers by Elsen and Hougardy [6], De et al. [16] and Zhao et al. [17] to represent a range of experimental carbon contents and ageing temperatures. Additionally, a further run has been performed to include the impact of a strongly carbide forming element in solid solution. 225 wt. ppm of solute with an atomic
Mesh dimensions (Ô000 elements)
Mesh elements Temperature Carbon Carbide (Ô000,000 (K) (wt. ppm) former elements) (wt. ppm)
Size 10 · 10 15 · 15 22 · 22 35 · 35
100 225 484 1225
323 323 323 323
5 5 5 5
– – – –
Shape 15 · 15 19 · 12 28 · 8 56 · 4 112 · 2
225 228 224 224 224
323 323 323 323 323
5 5 5 5 5
– – – – –
Temperature/carbon 15 · 15 225 15 · 15 225 15 · 15 225 15 · 15 225 15 · 15 225 15 · 15 225
323 323 323 373 373 373
5 6 6.4 5 6 6.4
– – – – – –
Carbon stabiliser 15 · 15
323
5
225
225
weight of 51 was dispersed randomly over the area simulated on a 15,000 by 15,000 element grid, containing 5 wt. ppm of carbon and aged at 323 K. The substitutional atoms were placed in bcc ferrite lattice sites and an interaction was assumed with carbon atoms situated at the nearest neighbour octahedral sites, with a magnitude of 2 kJ/mol. Jumps to and from these sites were controlled by the jump equation as detailed previously for sites of non-equivalent energy. Where the placement of such a substitutional atom coincided with the interaction energy field resultant from a dislocation, the effects were considered to be cumulative.
4. Results Fig. 2 shows the effect of changing mesh size on the accuracy of the Monte Carlo simulation. Fig. 3 shows the effect of changing the mesh shape from a thin strip simulation (128,000 · 2000 elements) through to a square simulation of equivalent area (15,000 · 15,000 elements). In these simulations the effect of changing mesh shape appears negligible. Fig. 4 shows the ability of the technique to handle changing carbon content and temperature. The simulation runs are close to the reported data for experiments at 323 K and follow the same trend, producing sigmoid curves that are displaced to shorter times as the carbon concentration increases. Fig. 4 shows a poorer match to experimentally reported data for 373 K, and the curves no longer have
628
A.J.P. Gater, S.G.R. Brown / Scripta Materialia 53 (2005) 625–630
Fig. 2. Simulated strain ageing for steel containing 5 wt. ppm of carbon at 5% prestrain vs. experimental data taken from De et al. [16] on meshes of dimensions 10,0002, 15,0002, 22,0002 and 32,0002.
5. Discussion
Fig. 3. The effect of mesh shape on simulated strain ageing for a steel containing 5 wt. ppm of carbon at 5% prestrain vs. experimental data taken from De et al. [16] on a 15,0002 mesh.
a sigmoid shape. The general trend of shorter ageing times for higher carbon contents is repeated, but the overlap between simulated curves is greater than at 323 K. In Fig. 5 the addition of a substitutional element that forms strong dipoles with carbon can be seen to reduce the rate at which ageing occurs. The sigmoid curve is displaced to longer times, this effect being least pronounced at the start and end of the simulation.
(1) There is good agreement between simulated and reported ageing curves, and Fig. 2 shows that mesh size appears to have a small observable effect on the performance of the simulation technique. However, in these graphs the comparison is between three individual runs against the reported result. Because the carbon atoms are placed randomly at the start of the simulation run it may be that averaging the predicted ageing behaviour over several runs at one condition may reduce or eliminate this effect. Using the current algorithm runs containing approximately one billion elements had a total simulation time in excess of 50 h; conversely, those runs containing 225 million elements had a run time of below 6 h. This result is interesting in that it suggests that the use of relatively small grids in the KMC simulation of strain aging may be capable of producing useful results. (2) Fig. 3 shows mesh shape to have no discernible effect on the technique in the range over which simulations were run. The minimum mesh width used was 2000 elements, with an average dislocation spacing of around 1000 mesh elements in all trials; as a periodic boundary condition was used it is unsurprising that the mesh shape had little effect on the simulation result, and it seems likely that the effect will be negligible provided the mesh width is larger than the average disloca-
A.J.P. Gater, S.G.R. Brown / Scripta Materialia 53 (2005) 625–630
629
Fig. 4. Simulated strain ageing for steels containing 5 wt. ppm, 6 wt. ppm and 6.4 wt. ppm of carbon at 5% prestrain at 323 K and 373 K vs. experimental data taken from De et al. [16], Zhao et al. [17] and Elsen and Hougardy [6] on a 15,0002 element mesh.
Fig. 5. Simulated strain ageing for a steels containing 5 wt. ppm of carbon, and 5 wt. ppm of carbon with 225 wt. ppm of a strong carbide former with atomic weight 51 at 5% prestrain and 323 K on a 15,0002 element mesh.
range from 1100 to 500 unit cells. It is therefore suggested that thin strip simulations should be performed with a mesh width in excess of this upper value (2200 mesh elements in the current model). (3) From Fig. 4, the technique seems suited for the simulation of changing carbon content, though the range covered in this investigation was limited. The accuracy of the model seems sensitive to temperature, performing significantly less well at 373 K than in trials at 323 K. This effect, and model alterations with which it can be prevented, are the subject of ongoing simulations. (4) The demonstration shown in Fig. 5, while highly simplified, illustrates that this technique may have potential for modelling the effects of carbide formers such as vanadium in solution, which have been suggested as having a delaying effect on low temperature strain ageing when added in sufficiently large quantities [20]. The authors are unaware of this extension to the technique having been used before. This is the subject of ongoing research that will also include accelerated ageing experiments for different carbide forming elements. Acknowledgements This work was carried out under the Engineering Doctorate Scheme; the help and support of Corus Group and the funding of EPSRC is gratefully acknowledged.
Fig. 6. Evolution of dislocation density as a function of prestrain, adapted from Ref. [18]. Original paper by Amiot and Despujols [19].
tion spacing; if this condition is not met, the dislocation density would be artificially increased as the distance between each dislocation and itself would be shorter than the average dislocation spacing. In the region of 1–5% prestrain it has been shown that dislocation density, q, varies between roughly 1 · 1013 and 6 · 1013 m2 (Fig. 6) [18]; accepting that average dislocation spacing can be taken as q0.5 [15] this corresponds to a
References [1] Baker LJ, Daniel SR, Parker JD. Mater Sci Technol 2002;18:355. [2] Asensio J, Romano G, Martinez VJ, Verdeja JI, Pero-Sanz JA. Mater Charact 2001;47:119. [3] Fekete JR, Strugala DC, Yao Z. JOM 1992(January):17. [4] Kurasawa M, Satoh S, Obara T, Tsunoyama K. Kawasaki Steel Giho 1987;19:119. [5] Hirose Y, Nakata M, Kojima N, Mizui N, The comparison in dent-resistance between high strength steel with excellent formability and bake-hardenable steel. In: Proc 20th biennial IDDGR Õ98 congress, sheet metal forming beyond 2000, Genval, Brussels, Belgium, June 1998. p. 407.
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[6] Elsen P, Hougardy P. Steel Res 1993;64:431. [7] Wilson DV, Russel B. Acta Metall 1960;8:468. [8] Van Snick A, Lips K, Vandeputte S, De Cooman BC, Dilewijns J. In: Bleck W, editor. Proc conf on modern LC and ULC sheet steels for cold forming—processing and properties, vol. 2. Aachen: Verlag Mainz; 1998. p. 413. [9] Massardier V, Lavaire N, Soler M, Merlin J. Scripta Mater 2004;50:1435. [10] Prahdan R. International conference on steel in motor vehicle manufacture. Dusseldorf, Germany: Verlag Stahleisen; 1990. p. 60. [11] Baird JD. Iron Steel 1963;36:186.
[12] [13] [14] [15] [16] [17] [18] [19] [20]
Baird JD. Iron Steel 1963;36:326. Baird JD. Iron Steel 1963;36:368. Baird JD. Iron Steel 1963;36:400. Soenen B, De AK, De Cooman BC. Acta Mater 2004;52:3483. De AK, De Blauwe K, Vandeputte S, De Coomen BC. J Alloys Compd 2000;310:405. Zhao JZ, De AK, De Cooman BC. Mater Lett 2000;44:374. Berbenni S, Favier V, Lemoine X, Berveiller M. Scripta Mater 2004;51:303. Amiot B, Despujols JJ. Met Technol 1983;10:161. Taylor KA, Speer JG. 39th MWSP conf proc, vol. 35, 1998. p. 49.
An analysis of kinetic Monte Carlo simulation as a tool for modelling strain ageing A.J.P. Gater, S.G.R. Brown
*
EPSRC Engineering Doctorate Centre, Materials Research Centre, School of Engineering, University of Wales Swansea, Swansea, SA2 8PP, UK Received 17 February 2005; received in revised form 26 May 2005; accepted 30 May 2005 Available online 21 June 2005
Abstract The mechanism of strain ageing is briefly outlined, and details of a Monte Carlo algorithm for simulation of this process are given. The results of varying mesh shape, mesh size, free interstitial carbon and temperature on the performance of the algorithm are then critically assessed. 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Strain ageing; Dislocations; Kinetics
1. Introduction The threat of legislation regarding emissions and fuel consumption has led the automotive industry to think in terms of down gauging, and the use of novel high strength steels in the production of body panels, to reduce the weight of the Ôbody in whiteÕ [1]. There are a range of steels suitable for such drawing applications, with varied properties (Table 1) [2]. One currently experiencing a lot of interest is ultra low carbon bake hardenable sheet. The term Ôbake hardenableÕ steel refers to a range of highly formable strip steels with a yield stress that can be improved subsequent to forming via a heat treatment, producing a yield stress increase typically in the range 34–70 MPa [3]. The combination of an ultra low carbon (ULC) chemistry and the yield strength increase occurring after forming reduces the forces required in drawing operations, and thus the costs. Bake hardening has also been shown to produce an increase in surface dent resistance [4,5]. *
Corresponding author. Tel.: +44 1792 295284. E-mail address: [email protected] (S.G.R. Brown).
The yield stress increase is achieved by leaving controlled amounts of carbon in solution, typically 5– 15 wt. ppm [6] (5–15 wt. ppm = 0.0005–0.0015 wt.%) through vacuum degassing and careful control of micro alloying additions such as titanium and niobium; this level of carbon is not sufficient to be greatly detrimental to formability. During paint curing free interstitial carbon can diffuse to the cores of dislocations under a Ôdrift velocityÕ caused by an interaction of the strain fields around the dislocation and misfitting solute atoms. This carbon then forms solute atmospheres around the dislocations, locking them in place, and increasing the applied stress that is required to initiate dislocation glide. During atmosphere formation an initial yield strength increase of 20–30 MPa is observed, the maximum value corresponding with a locking ratio of around 0.9 atoms per atom plane threaded by dislocation [7]; further strengthening is then achieved through the formation of nano-scale precipitates around the dislocation. The higher the level of interstitial carbon present, the greater the yield stress increase that can be achieved, to a maximum value of 70–90 MPa [8,9]; however, at carbon levels in excess of 10 wt. ppm, the risk of
1359-6462/$ - see front matter 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2005.05.035
626
A.J.P. Gater, S.G.R. Brown / Scripta Materialia 53 (2005) 625–630
Table 1 Typical properties of steels suitable for deep drawing [2] Steel type
ry (MPa)
UTS (MPa)
Elongation (%)
n
rm
Interstitial free Al-Killed Rephosphorised Bake hardened HSLA
155 165–190 190–230 210 375
305 300 345–370 320 475
42 42 36 40 27
0.23 0.22 0.20 0.22 0.15
2.0 1.8 1.5 1.8 1.0
room temperature ageing increases and must be controlled through temper rolling [10]. For a thorough review of bake hardenable steels see Baird [11–14] and Baker et al. [1].
2. Modelling strain ageing A kinetic Monte Carlo simulation of strain ageing has been published by Soenen et al., accounting for both diffusion of carbon atoms to dislocations during ageing, and grain boundary segregation during the annealing process, and the results of this group have shown a good match to their experimental data [15]. As part of an ongoing research project the algorithm developed in that paper has been adopted for this study, in which the kinetics are controlled by a set of Arrhenius type equations controlling jump frequency, relating to jumps between positions of equivalent energy within the bulk material, and jumps to sites of higher or lower energy, within a critical radius around dislocations. The frequency of jumps between equivalent sites, fd, is controlled by an attempt frequency, m, and Ud the energy barrier to diffusion, which was taken as 82,000 J/mol, approximately equal to the activation energy for carbon diffusion, of 84,140 J/mol [15]. Ud D0 f ¼ fd ¼ m exp ð1Þ ; m¼ 2 RT a Here R is the gas constant, T the absolute temperature, D0 the coefficient of diffusivity for carbon in ferrite (2 · 106 m2/s) and a the lattice parameter of a body˚ ). Table 2 centred cubic (bcc) ferrite unit cell (2.87 A shows jump frequencies for interstitial carbon calculated using Eq. (1) with the parameters stated by Soenen et al. [15]; the chosen activation energy of 82,000 J/mol gives a Table 2 Modelled jump frequencies as a function of temperature Temperature (K)
273 298 323 373
Jump frequency (s1) (84.14 kJ/mol, 2E6 m2/s)
(82 kJ/mol, 2E6 m2/s)
0.0019 0.0432 0.5983 39.9106
0.0049 0.1024 1.3275 79.5801
jump frequency of around 0.1/s at 298 K, in agreement with the position of the carbon peak in internal friction studies. Where a diffusive jump moves an atom between sites two sites i and j with respective energies Ui and Uj the jump frequency fd is again observed if the energy of the target site is lower; if a jump takes an atom to a higher energy site the frequency is related exponentially to the energy difference between the two positions [15]: f ði ! jÞ ¼ fd ;
Ui > Uj Uj Ui f ði ! jÞ ¼ fd exp ; RT
ð2Þ Ui < Uj
ð3Þ
The program records the co-ordinates of all solute atoms and dislocations, and interaction energies are read from energy ÔtemplatesÕ if carbon atoms are within the calculated radius of interaction (rc). The critical radius of interaction allows for a minimum variation of 10% from the normal jump frequency and is calculated as [15]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Aa rc ¼ ð4Þ RT lnð1.1Þ The interaction energy of each site is calculated using [15]: U¼
A sin h r
ð5Þ
where (r, h) is the polar co-ordinate of the solute atom with respect to the dislocation line and A is the solute dislocation interaction parameter (taken as 7.5 · 106), T is the absolute temperature of the system and R is the gas constant. At the beginning of the program, carbon atoms are randomly placed throughout the mesh at the cell centres, and dislocations at the edges of unit cells. Each carbon atom has a local clock that keeps track of its location in time, this clock being updated every time the atom makes a diffusive jump. The algorithm proceeds as follows (Fig. 1): (1) Select that carbon atom for which the least amount of time has passed (the lagging atom). (2) Set the simulation time to the local time of the selected atom. (3) Using a random number generator determine in which of the four possible directions the carbon atom will jump, based on the relative jump frequencies. (4) Check whether the destination site is vacant; if so, perform the diffusive jump, if not, the selected atom remains stationary. (5) Calculate the jump frequencies for the atom in its new position.
A.J.P. Gater, S.G.R. Brown / Scripta Materialia 53 (2005) 625–630
627
Table 3 Simulations run in the evaluation of the Monte Carlo algorithm
Fig. 1. Diagrammatic representation of Monte Carlo algorithm.
(6) Update the atomsÕ local clock by the amount of time that will be required for it to perform its next diffusive jump. (7) Re-sort atoms by their local clocks. A periodic boundary condition was used in all cases. The algorithm should represent well the actual physics of dislocation locking, as it models the stepwise jumps of carbon atoms moving interstitially through a bodycentred cubic lattice, with frequency controlled by the diffusion equation, and the parameters that are shown in Table 2, which match well with observed phenomena.
3. Model trials Trials have been aimed at assessing the affects of mesh size and shape on accuracy, and on the ability of the model to assess a range of temperatures and carbon contents (Table 3). To this end, experimental plots have been reproduced from papers by Elsen and Hougardy [6], De et al. [16] and Zhao et al. [17] to represent a range of experimental carbon contents and ageing temperatures. Additionally, a further run has been performed to include the impact of a strongly carbide forming element in solid solution. 225 wt. ppm of solute with an atomic
Mesh dimensions (Ô000 elements)
Mesh elements Temperature Carbon Carbide (Ô000,000 (K) (wt. ppm) former elements) (wt. ppm)
Size 10 · 10 15 · 15 22 · 22 35 · 35
100 225 484 1225
323 323 323 323
5 5 5 5
– – – –
Shape 15 · 15 19 · 12 28 · 8 56 · 4 112 · 2
225 228 224 224 224
323 323 323 323 323
5 5 5 5 5
– – – – –
Temperature/carbon 15 · 15 225 15 · 15 225 15 · 15 225 15 · 15 225 15 · 15 225 15 · 15 225
323 323 323 373 373 373
5 6 6.4 5 6 6.4
– – – – – –
Carbon stabiliser 15 · 15
323
5
225
225
weight of 51 was dispersed randomly over the area simulated on a 15,000 by 15,000 element grid, containing 5 wt. ppm of carbon and aged at 323 K. The substitutional atoms were placed in bcc ferrite lattice sites and an interaction was assumed with carbon atoms situated at the nearest neighbour octahedral sites, with a magnitude of 2 kJ/mol. Jumps to and from these sites were controlled by the jump equation as detailed previously for sites of non-equivalent energy. Where the placement of such a substitutional atom coincided with the interaction energy field resultant from a dislocation, the effects were considered to be cumulative.
4. Results Fig. 2 shows the effect of changing mesh size on the accuracy of the Monte Carlo simulation. Fig. 3 shows the effect of changing the mesh shape from a thin strip simulation (128,000 · 2000 elements) through to a square simulation of equivalent area (15,000 · 15,000 elements). In these simulations the effect of changing mesh shape appears negligible. Fig. 4 shows the ability of the technique to handle changing carbon content and temperature. The simulation runs are close to the reported data for experiments at 323 K and follow the same trend, producing sigmoid curves that are displaced to shorter times as the carbon concentration increases. Fig. 4 shows a poorer match to experimentally reported data for 373 K, and the curves no longer have
628
A.J.P. Gater, S.G.R. Brown / Scripta Materialia 53 (2005) 625–630
Fig. 2. Simulated strain ageing for steel containing 5 wt. ppm of carbon at 5% prestrain vs. experimental data taken from De et al. [16] on meshes of dimensions 10,0002, 15,0002, 22,0002 and 32,0002.
5. Discussion
Fig. 3. The effect of mesh shape on simulated strain ageing for a steel containing 5 wt. ppm of carbon at 5% prestrain vs. experimental data taken from De et al. [16] on a 15,0002 mesh.
a sigmoid shape. The general trend of shorter ageing times for higher carbon contents is repeated, but the overlap between simulated curves is greater than at 323 K. In Fig. 5 the addition of a substitutional element that forms strong dipoles with carbon can be seen to reduce the rate at which ageing occurs. The sigmoid curve is displaced to longer times, this effect being least pronounced at the start and end of the simulation.
(1) There is good agreement between simulated and reported ageing curves, and Fig. 2 shows that mesh size appears to have a small observable effect on the performance of the simulation technique. However, in these graphs the comparison is between three individual runs against the reported result. Because the carbon atoms are placed randomly at the start of the simulation run it may be that averaging the predicted ageing behaviour over several runs at one condition may reduce or eliminate this effect. Using the current algorithm runs containing approximately one billion elements had a total simulation time in excess of 50 h; conversely, those runs containing 225 million elements had a run time of below 6 h. This result is interesting in that it suggests that the use of relatively small grids in the KMC simulation of strain aging may be capable of producing useful results. (2) Fig. 3 shows mesh shape to have no discernible effect on the technique in the range over which simulations were run. The minimum mesh width used was 2000 elements, with an average dislocation spacing of around 1000 mesh elements in all trials; as a periodic boundary condition was used it is unsurprising that the mesh shape had little effect on the simulation result, and it seems likely that the effect will be negligible provided the mesh width is larger than the average disloca-
A.J.P. Gater, S.G.R. Brown / Scripta Materialia 53 (2005) 625–630
629
Fig. 4. Simulated strain ageing for steels containing 5 wt. ppm, 6 wt. ppm and 6.4 wt. ppm of carbon at 5% prestrain at 323 K and 373 K vs. experimental data taken from De et al. [16], Zhao et al. [17] and Elsen and Hougardy [6] on a 15,0002 element mesh.
Fig. 5. Simulated strain ageing for a steels containing 5 wt. ppm of carbon, and 5 wt. ppm of carbon with 225 wt. ppm of a strong carbide former with atomic weight 51 at 5% prestrain and 323 K on a 15,0002 element mesh.
range from 1100 to 500 unit cells. It is therefore suggested that thin strip simulations should be performed with a mesh width in excess of this upper value (2200 mesh elements in the current model). (3) From Fig. 4, the technique seems suited for the simulation of changing carbon content, though the range covered in this investigation was limited. The accuracy of the model seems sensitive to temperature, performing significantly less well at 373 K than in trials at 323 K. This effect, and model alterations with which it can be prevented, are the subject of ongoing simulations. (4) The demonstration shown in Fig. 5, while highly simplified, illustrates that this technique may have potential for modelling the effects of carbide formers such as vanadium in solution, which have been suggested as having a delaying effect on low temperature strain ageing when added in sufficiently large quantities [20]. The authors are unaware of this extension to the technique having been used before. This is the subject of ongoing research that will also include accelerated ageing experiments for different carbide forming elements. Acknowledgements This work was carried out under the Engineering Doctorate Scheme; the help and support of Corus Group and the funding of EPSRC is gratefully acknowledged.
Fig. 6. Evolution of dislocation density as a function of prestrain, adapted from Ref. [18]. Original paper by Amiot and Despujols [19].
tion spacing; if this condition is not met, the dislocation density would be artificially increased as the distance between each dislocation and itself would be shorter than the average dislocation spacing. In the region of 1–5% prestrain it has been shown that dislocation density, q, varies between roughly 1 · 1013 and 6 · 1013 m2 (Fig. 6) [18]; accepting that average dislocation spacing can be taken as q0.5 [15] this corresponds to a
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