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Strain rate sensitivity of ultra-low carbon steels
Materials Science and Engineering A319– 321 (2001) 294– 298 www.elsevier.com/locate/msea
Strain rate sensitivity of ultra-low carbon steels S. Saimoto *, B.J. Diak Department of Materials and Metallurgical Engineering, Queen’s Uni6ersity, Nicol Hall, Kingston, Ont., Canada K7L3N6
Abstract Ultra-low carbon, interstitial free (IF) steels were prepared by an intermediate rolling reduction of 20% followed by a 3 or 9 day anneal at 400°C prior to further reduction and anneal. Previously described determination of the interstitial content using the Haasen plot intercept indicates that levels in the ppb range were attained. The strain rate sensitivity in the macro-plastic region defined by the slope of the Haasen plot increased after reducing solute levels below 1 ppm, and the work hardening mechanism was attributed to that of jogs and/or recombination of dissociated dislocations in bcc structures. A distinct strain rate sensitivity was found in the micro-strain region below 0.2% where the activation work was an order of magnitude larger than in the macro-plastic region indicating an activation distance of 4.5b. This measurement suggests that during micro-straining, glissile dislocations intersect, but as the jog density increases with straining, the dislocation cores become sessile and some recombination mechanism activates. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Activation volume; Activation work; Interstitial free steel; Strain rate sensitivity; Thermal activation; Work hardening
1. Introduction In the study of body-centered cubic (bcc) metals, the emphasis in past decades has been to understand the rapid temperature dependence of the flow stress below ambient temperature resulting in many theoretical studies [1] concerning the Peierls – Nabarro (P – N) force. However, above about 220 K for a Fe – 2%Mn steel, the P –N effect is replaced by another as yet indeterminate mechanism [2]. The variation of the carbon content from 40 ppm (wt.) to below an estimated 10 ppm (using the results [1,3] for purification by ZrH2) did not measurably affect the flow stress changes with temperature for these initially prestrained specimens. The early studies [4,5] showed that the change in stress due to a change in strain rate becomes insensitive to strain (stress) at 300 K, and hence the detailed studies using strain rate sensitivity measurements [6] did not seem fruitful. The greatest experimental difficulty was the appearance of the yield point and the Portevin –Le Chaˆtelier (PLC) effect due to the interstitial impurities, which could not be removed unless the iron specimens
underwent ZrH2 purification [3]. With zone refining followed by the ZrH2 treatment, the carbon content could be reduced to a few ppb for bcc metals [3,7]. More recently, Brunner and Diehl [8,9] measured strain rate sensitivity of iron crystals with interstitial content below 1 ppm. In our search for continuous recrystallization (CReX) of commercially available interstitialfree (IF) steel, we found that CReX would take place if carbon in titanium stabilized IF steel was reduced to below 1 ppm (wt.), resulting in a grain size of about 1 mm. The carbon content was measured using the Haasen plot [6]. The objective of this study is to precisely evaluate the strain rate sensitivity during tensile deformation of a fine grained ultra-low carbon steel obtained by CReX to quantitatively test the mechanisms of, (1) thermally activated recombination of dissociated dislocation cores as suggested by Escaig [10] and applied by Guyot and Dorn [1]; and (2) thermally activated double kink formation in screw dislocations as advocated by Brunner and Diehl [8,9].
2. Experimental procedures * Corresponding author. Tel.: + 1-613-5332754; fax: +1-6135336610. E-mail address: [email protected] (S. Saimoto).
A commercially supplied hot band (after hot rolling) of 4.09 mm in thickness was cold rolled by 21.8% then
0921-5093/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 5 0 9 3 ( 0 1 ) 0 1 0 8 1 - 4
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annealed at 400°C for 3 and 9 days, respectively. The IF steel hot band composition in ppm (wt.) was 19C–630Mn– 35Si –86P – 33S – 507Al – 505Ti – 35N– 20O and its recrystallization characteristics after subsequent rolling were reported earlier [11]. The 3.2 mm strip was further cold rolled to 0.76 mm. The rolling was carried out on a two-high Stanat mill with 100 mm diameter rolls. One piece from the 9 day intermediate anneal (9d-IA) strip was annealed for 1 h at 600°C to produce continuously recrystallized grains of about 1 mm, and another from the 3 day anneal (3d-IA) strip at 620°C for 1 h showed only partial CReX and resulted in predominantly 10 mm grains interspersed with 1 mm Fig. 3. Haasen plot of an Ti – IF steel as quenched 850°C previously reported in [6] as Fig. 76. In that figure a typographical error recorded |0 as 66.57 MPa rather than the correct 40.25 MPa cited here. Note that S1 does not appear for this heat treatment.
Fig. 1. The Haasen plot of D|/(TD ln m; ), which is inversely proportional to the activation volume, vs. corrected stress (|− |0) for 9-day and 3-day intermediate anneal Ti –IF steel specimens, tensile tested at 180°C with the measured parameters recorded in Table 1. Note that 9d-IA data is offset vertically.
grains, which cover less than 20% of the area. Although increasing the anneal temperature resulted in a slightly larger but uniform grain size, the carbon content increased due to some resolution of the TiC and these complications will be discussed elsewhere. Tensile specimens were cut and prepared from these sheets with final gauge dimensions of about 35× 1.9×0.76 mm. The tensile testing was performed in a specially designed apparatus [6] to permit strain rate change tests without strain transients. In order to avoid the effect of the P –N force and to clearly assess the dislocation– dislocation interactions in an environment more or less free of carbon solutes, these specimens were tested in the temperature range of 60–220°C (9 0.1°C) by immersion in a silicon oil bath. The nominal strain rate was 2.83×10 − 5 s − 1 with rate changes of 1/4 and 1/10. Due to the low interstitial content, no inverse PLC, solute drag and/or sub-microscopic precipitate effects (discussed in [6]) were evident.
3. Results
Fig. 2. The Haasen plot of specimens tensile tested at 220°C with the measured values recorded in Table 1. Note the distinct and parallel separation of the 1/10 and 1/4 rate change data points. Note that 9d-IA data is offset vertically.
Figs. 1 and 2 show the Haasen plots for the 9d-IA and 3d-IA specimens at 180 and 220°C which are characterized by two distinct regions; first, in the microstrain, a region of very low strain rate sensitivity (S1) and second, in the macro-strain past the 0.2% offset strain, an order of magnitude higher (S2). The ordinate in these figures are measured values which are inversely proportional to the activation volume, and the slope of the curve is defined as the strain rate sensitivity. Note that unlike our previous result for titanium stabilized IF steel quenched from 850°C (Fig. 3), the 9d-IA plots
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Fig. 4. Tensile stress vs. strain curves tested at indicated temperatures and offset along the strain axes by units of 0.005. The horizontal arrows mark the positions of |0 and the vertical ones the location of 0.002 strain. The ultra-fine grain size of 1 mm results in the high flow stresses with reduced ductility.
show a distinctive two-stage effect whereby the transition from the low S1 to the larger S2 occurs at the inception of the macro-strain (Fig. 4). This effect is also clearly measurable at 60 and 120°C. Thus if this steeper slope was due to thermal activation over fine precipitates [6], it should show a temperature dependence which is not apparent in Table 1. Such observation, which signals an abrupt change in the activation volume has been previously reported for the case of Mo [12]. The much smaller micro-strain effect in 3d-IA specimens is attributed to the fact that with larger grain sizes this regime is very short prior to dislocation multiplication and intersections and the present apparatus can not clearly resolve this region. The amount of interstitial atoms in solution was determined by back-extrapolation of the Haasen plot to the 0.02% offset stress (|0) at test temperatures of 180 or 220°C [6,13,14] since it is determinable prior to the onset of PLC (this choice of |0 approximately coincides with the so-called anelastic stress [12], and the corrected flow stress and our definition of effective stress is defined as |− |0).
Fig. 5. The macro-region strain rate sensitivity (S2) vs. the amount of carbon in solution as determined from the Haasen intercept for various Ti – IF steels tested in this laboratory.
The Haasen intercept is lower for the 9d-IA specimen than that for 3d-IA in Figs. 1 and 2 indicating that the carbon/interstitial content is lower due to the effect of the longer intermediate anneal of 9 days at 400°C. The deduced compositions of these and previous specimens are given in Table 1 from which S2 can be compared with the solute composition (Fig. 5). Note that the minimum S2 value in Fig. 5 refers to the as-quenched case shown in Fig. 3 in which a small upper deflection in the Haasen plot denoted as S3 occurs, and is characterized by a divergence between 1/4 and 1/10 strain rate change responses. On the other hand, the inverse activation volumes due to 1/4 and 1/10 rate changes are distinctly separated in the IA specimens, which have a lower interstitial content. The strain rate sensitivity measured from the 1/4 and 1/10 rate changes are nearly but not identical depending on the test conditions. Hence, the higher strain slope (S3 in Fig. 3) appears to be comparable to S2 for the 3d-IA specimen suggesting that the divergence may be due to solute carbon removal by dynamic carbon segregation after the generation of dislocations during the initial
Table 1 Calculated thermodynamic response values from the Haasen plots (k is the Boltzmann constant) Specimen
9d-05
9d-08
9d-07
9d-06
3d-02
3d-03
As-quenched [6]
Test temperature (°C) S1 (1/10)×10−6 (1 K−1) S2 (1/10)×10−5 (1 K−1) S2 (1/4)×10−5 (1 K−1) Haasen intercept ×10−4 (MPa K−1) ppm (wt.) DW k/S1 (1/10) (eV) DW k/S2 (1/10) (eV) Activation volume in b 3 at (|−|0)= 160 Mpa
60 6.09 5.27 6.84 2.14 0.215 14.2 1.64 646
120 4.47 6.19 7.06 2.18 0.223 19.3 1.39 657
180 5.28 5.69 6.60 2.08 (1.66)a 0.203 (0.129)a 16.3 1.51 474 (506)a
220 5.30 6.23 6.42 1.16 0.063 16.3 1.38 614
180 – 3.86 3.16 4.68 1.03 – 2.73 –
220 – 3.12 3.19 3.07 0.442 – 2.70 –
220 – 1.01 1.01 4.61 1.00 – 8.62 –
a
A second test indicated that |0 may be lowered to 221 (bracketed values) from 228 MPa.
S. Saimoto, B.J. Diak / Materials Science and Engineering A319–321 (2001) 294–298
straining. A calculation of the amount of atom planes available to capture at least one carbon atom in the dislocation core [15] using the dislocation density, z, which corresponds to the shear stress at the point of divergence according to the relation, ~ = hvbz 0.5, where v =80.65 GPa [16], b = 0.24824 nm and h =0.4 [17], results in about 1 ppm (temperature corrections for these parameters were not used in these preliminary calculations). This means that the divergence of the 1/4 and 1/10 curves with straining in Fig. 3 could be a consequence of a change in dislocation– dislocation interaction with dynamic reduction of solute carbon to below 1 ppm (wt.). From Fig. 5 it appears that the interstitial solutes start to measurably affect the deformation mechanisms at about 1 ppm (wt.). An alternative interpretation derived from the study of face-centered cubic (fcc) aluminum is to attribute the divergence between 1/4 and 1/10 rate changes to the deformation generated vacancies which recover during the time required to mechanically impose a strain rate change [18]. However, one would expect to observe the divergence to increase with strain whereas the results in Figs. 1 and 2 suggest a constant or a slightly decreasing effect. Nevertheless, the activation work deduced from DW =k/S, where k is the Boltzmann constant, shows that S2 correlates to the energy of vacancy formation of about 1.5 eV (Table 1). In contrast, S1 exceeds that for interstitial formation (vb 3 =7.7 eV) by a factor of 2. In the case of aluminum, the large DW becomes independent of temperature above 0.25Tm (180°C for iron), and is attributed to the strain field near the dislocation core, whereas at 4.2 K the smaller value is attributed to vacancy formation [19]. In comparison, for iron the energy to form a double kink in the screw dislocation is about 0.92 eV [8,9], which is much lower than the above data-deduced values. Also, the thermally activated double kink formation [8,9] predicts a large temperature dependence of the activation volume, V, and this is not observed between 60 and 220°C (Table 1), where V is normalized at the corrected flow stress of 160 MPa. At 180°C, a second test indicates that |0 may be lower resulting in the bracketed values and the anomalous deviation becomes less apparent.
4. Discussion The observation in Figs. 1 and 2 of the clear separation of the Haasen plot depending on the amount of rate change in the macro-strain region has not been reported previously to our knowledge. Unfortunately the resolution of the testing parameters was insufficient to show if this effect also held in the micro-strain region. A higher strain rate sensitivity with ultra high purity was indicated in the 1/10 rate change plot of Stein and Low [7], and this trend is consistent with the
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current finding shown in Fig. 5. The strain rate sensitivity, S, is related to the activation volume, V=bld (where l is the interobstacle segment length of the dislocation, b is the Burgers vector, and d is the activation distance) by (|− |o)V=k/S. Previously, we argued that the region of the Haasen plot where the Cottrell–Stokes relation is obeyed must be independent of the amount of strain rate due to the fact
d #l l #d #V + =b # ln m; ( ln m; # ln m;
(1)
is zero [18]. This implies that if the obstacle is the forest dislocation the changes in l and in d are equal and opposite. The effect may be the same for the case of a large strain centre such as an interstitial site. However, if the obstacle is a jog or constriction caused by the dislocation intersections, than a change in d must still occur, while changes in l may be negligible or not correlated. The fact that the Haasen plot can be linearized and the slopes due to 1/4 and 1/10 rate changes are nearly equal suggests that the latter condition can apply. Since V appears to change inversely with the corrected flow stress, it suggests that the intersected dislocation retains a memory of the number of intersections. In Escaig’s theory [10] of recombination of sessile dissociated dislocation to produce glissile screw dislocations by thermal activation, Guyot and Dorn [1] clearly show that the activation volume should obey an inverse quadratic relationship to the effective stress instead of a linear one as suggested by the Cottrell–Stokes relation. A quadratic relation may hold near the ultimate stress (not shown), but not near the macro-yield point, and hence it is difficult to clearly separate the two mechanisms. The resolution of this apparent discrepancy may be that the dislocation–forest intersection converts the glissile to a sessile configuration, which is estimated to be energetically small of the order of 1/15 eV [10]. Thus the recombination distance is controlled by the inter-jog distance, which must correlate to the forest dislocation density and mean slip distance. However, unlike the forest theory of hardening, a memory effect is retained and the work hardening formalism should parallel that of jog dragging. In the micro-strain region, on the other hand, V is large and l is comparable to the grain size. Meakin [12] attributed this observation in Mo single crystals to the motion of pre-existing kinks, widely separated on dislocations threading the crystallites. However, Meakin’s estimate of the activation work, DW$10 − 2vb 3, is orders of magnitudes smaller than the present measured values (Table 1). Thus DW values, larger than vb 3, must be due to the strain field interactions between the glissile dislocations, which result in a large d. The intersecting dislocations will result in a Cottrell–Stokes forest hardening relationship prior to the chopping up
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of the segments with many jogs. This jog creation process consequently converts a glissile core into a sessile one with a reduction in the long range strain field. A mechanism is needed to explain the reduction of DW upon the inception of macro-deformation. Since DW = k/S= hvb 2d [6] and using S2, h(d/b) $ 0.18 and using an h value of 0.4 [17], d must be about 0.5b suggesting a partial dislocation in conformity to the recombination mechanism. On the other hand, using S1, d must be 4.5b. These results suggests that the dislocation density measurement should correlate with S1 rather than with S2. This means that the inter-jog spacing would become smaller than the inter-forest one as the dislocation traverses its mean slip distance. If this mechanism is truly operational, the mean slip distances can not rapidly increase in terms of the multiples of cell size due to the jog hardening effect. The consequence of such hardening is that diffuse necking is promoted by the inherent resistance to micro-banding. Thus this may be the essential difference between bcc and fcc metal deformation.
5. Conclusion Precision strain rate sensitivity measurements of ultra-low carbon, titanium IF steel result in a Haasen plot with two linear regions, the first in the micro-strain region, and the second after the inception of macroplastic flow. The measured activation work suggests that the former is associated with the intersection of glissile dislocations, whereas the latter is due to jog hardening and/or the recombination of sessile partial dislocation arrays. These quantitative measurements on bcc iron of high purity in the ppb range can be used to affirm the various theories of thermally activated flow.
Acknowledgements The authors wish to thank the Natural Sciences and Engineering Research Council of Canada and Materials Manufacturing Ontario for their continued support. We thank Dr H. Kobayashi of Dofasco Inc. for suggesting such studies, and C. Lutchman for technical assistance.
References [1] P. Guyot, J.E. Dorn, Can. J. Phys. 45 (1967) 983 –1016. [2] P. Wynblatt, A. Rosen, J.E. Dorn, Trans. AIME 233 (1965) 651 – 655. [3] A.S. Keh, Y. Nakada, Can. J. Phys. 45 (1967) 1101 –1120. [4] Z.S. Basinski, J.W. Christian, Aust. J. Phys. 13 (1960) 299 –308. [5] B.L. Mordike, P. Haasen, Phil. Mag. 7 (1962) 459 –474. [6] B.J. Diak, K.R. Upadhyaya, S. Saimoto, Prog. Mater. Sci. 43 (1998) 223 – 363. [7] D.F. Stein, J.R. Low, Acta Metall. 14 (1966) 1183 –1194. [8] D. Brunner, J. Diehl, Phys. Stat. Sol. A124 (1991) 155 –170. [9] D. Brunner, J. Diehl, Phys. Stat. Sol. A125 (1991) 203 –216. [10] B. Escaig, J. Phys. 29 (1968) 225 – 239. [11] S. Saimoto, H. Jin, B.J. Diak, in: T. Sakai, H.G. Suzuki (Eds.), Recrystallization and Related Phenomena, Japan Institute of Metals, 1999, pp. 781 – 786. [12] J.D. Meakin, Can. J. Phys. 45 (1967) 1121 – 1134. [13] S. Saimoto, J.N.A. Starling, B.J. Diak, J.D. Boyd, in: M. Koiwa, K. Otsuka, T. Miyazaki (Eds.), Solid – Solid Phase Transformations’99, Japan Institute of Metals, 1999, pp. 1557 – 1560. [14] K.R. Upadhyaya, S. Saimoto, in: E.B. Hawbolt, S. Yue (Eds.), Phase Transform. During Thermal/Mechanical Processing of Steel, TMS/CIMM, 1995, pp. 331 – 341. [15] G.R. Speich, Trans. AIME 245 (1969) 2553 – 2564. [16] W.C. Leslie, The Physical Metallurgy of Steels, McGraw-Hill, New York, 1981, p. 113. [17] D. McLean, Can. J. Phys. 45 (1967) 973 – 982. [18] S. Saimoto, M.S. Duesbery, Acta Metall. 32 (1984) 147 –155. [19] S. Saimoto, H. Sang, Acta Metall. 31 (1983) 1873 – 1881.
Strain rate sensitivity of ultra-low carbon steels S. Saimoto *, B.J. Diak Department of Materials and Metallurgical Engineering, Queen’s Uni6ersity, Nicol Hall, Kingston, Ont., Canada K7L3N6
Abstract Ultra-low carbon, interstitial free (IF) steels were prepared by an intermediate rolling reduction of 20% followed by a 3 or 9 day anneal at 400°C prior to further reduction and anneal. Previously described determination of the interstitial content using the Haasen plot intercept indicates that levels in the ppb range were attained. The strain rate sensitivity in the macro-plastic region defined by the slope of the Haasen plot increased after reducing solute levels below 1 ppm, and the work hardening mechanism was attributed to that of jogs and/or recombination of dissociated dislocations in bcc structures. A distinct strain rate sensitivity was found in the micro-strain region below 0.2% where the activation work was an order of magnitude larger than in the macro-plastic region indicating an activation distance of 4.5b. This measurement suggests that during micro-straining, glissile dislocations intersect, but as the jog density increases with straining, the dislocation cores become sessile and some recombination mechanism activates. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Activation volume; Activation work; Interstitial free steel; Strain rate sensitivity; Thermal activation; Work hardening
1. Introduction In the study of body-centered cubic (bcc) metals, the emphasis in past decades has been to understand the rapid temperature dependence of the flow stress below ambient temperature resulting in many theoretical studies [1] concerning the Peierls – Nabarro (P – N) force. However, above about 220 K for a Fe – 2%Mn steel, the P –N effect is replaced by another as yet indeterminate mechanism [2]. The variation of the carbon content from 40 ppm (wt.) to below an estimated 10 ppm (using the results [1,3] for purification by ZrH2) did not measurably affect the flow stress changes with temperature for these initially prestrained specimens. The early studies [4,5] showed that the change in stress due to a change in strain rate becomes insensitive to strain (stress) at 300 K, and hence the detailed studies using strain rate sensitivity measurements [6] did not seem fruitful. The greatest experimental difficulty was the appearance of the yield point and the Portevin –Le Chaˆtelier (PLC) effect due to the interstitial impurities, which could not be removed unless the iron specimens
underwent ZrH2 purification [3]. With zone refining followed by the ZrH2 treatment, the carbon content could be reduced to a few ppb for bcc metals [3,7]. More recently, Brunner and Diehl [8,9] measured strain rate sensitivity of iron crystals with interstitial content below 1 ppm. In our search for continuous recrystallization (CReX) of commercially available interstitialfree (IF) steel, we found that CReX would take place if carbon in titanium stabilized IF steel was reduced to below 1 ppm (wt.), resulting in a grain size of about 1 mm. The carbon content was measured using the Haasen plot [6]. The objective of this study is to precisely evaluate the strain rate sensitivity during tensile deformation of a fine grained ultra-low carbon steel obtained by CReX to quantitatively test the mechanisms of, (1) thermally activated recombination of dissociated dislocation cores as suggested by Escaig [10] and applied by Guyot and Dorn [1]; and (2) thermally activated double kink formation in screw dislocations as advocated by Brunner and Diehl [8,9].
2. Experimental procedures * Corresponding author. Tel.: + 1-613-5332754; fax: +1-6135336610. E-mail address: [email protected] (S. Saimoto).
A commercially supplied hot band (after hot rolling) of 4.09 mm in thickness was cold rolled by 21.8% then
0921-5093/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 5 0 9 3 ( 0 1 ) 0 1 0 8 1 - 4
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annealed at 400°C for 3 and 9 days, respectively. The IF steel hot band composition in ppm (wt.) was 19C–630Mn– 35Si –86P – 33S – 507Al – 505Ti – 35N– 20O and its recrystallization characteristics after subsequent rolling were reported earlier [11]. The 3.2 mm strip was further cold rolled to 0.76 mm. The rolling was carried out on a two-high Stanat mill with 100 mm diameter rolls. One piece from the 9 day intermediate anneal (9d-IA) strip was annealed for 1 h at 600°C to produce continuously recrystallized grains of about 1 mm, and another from the 3 day anneal (3d-IA) strip at 620°C for 1 h showed only partial CReX and resulted in predominantly 10 mm grains interspersed with 1 mm Fig. 3. Haasen plot of an Ti – IF steel as quenched 850°C previously reported in [6] as Fig. 76. In that figure a typographical error recorded |0 as 66.57 MPa rather than the correct 40.25 MPa cited here. Note that S1 does not appear for this heat treatment.
Fig. 1. The Haasen plot of D|/(TD ln m; ), which is inversely proportional to the activation volume, vs. corrected stress (|− |0) for 9-day and 3-day intermediate anneal Ti –IF steel specimens, tensile tested at 180°C with the measured parameters recorded in Table 1. Note that 9d-IA data is offset vertically.
grains, which cover less than 20% of the area. Although increasing the anneal temperature resulted in a slightly larger but uniform grain size, the carbon content increased due to some resolution of the TiC and these complications will be discussed elsewhere. Tensile specimens were cut and prepared from these sheets with final gauge dimensions of about 35× 1.9×0.76 mm. The tensile testing was performed in a specially designed apparatus [6] to permit strain rate change tests without strain transients. In order to avoid the effect of the P –N force and to clearly assess the dislocation– dislocation interactions in an environment more or less free of carbon solutes, these specimens were tested in the temperature range of 60–220°C (9 0.1°C) by immersion in a silicon oil bath. The nominal strain rate was 2.83×10 − 5 s − 1 with rate changes of 1/4 and 1/10. Due to the low interstitial content, no inverse PLC, solute drag and/or sub-microscopic precipitate effects (discussed in [6]) were evident.
3. Results
Fig. 2. The Haasen plot of specimens tensile tested at 220°C with the measured values recorded in Table 1. Note the distinct and parallel separation of the 1/10 and 1/4 rate change data points. Note that 9d-IA data is offset vertically.
Figs. 1 and 2 show the Haasen plots for the 9d-IA and 3d-IA specimens at 180 and 220°C which are characterized by two distinct regions; first, in the microstrain, a region of very low strain rate sensitivity (S1) and second, in the macro-strain past the 0.2% offset strain, an order of magnitude higher (S2). The ordinate in these figures are measured values which are inversely proportional to the activation volume, and the slope of the curve is defined as the strain rate sensitivity. Note that unlike our previous result for titanium stabilized IF steel quenched from 850°C (Fig. 3), the 9d-IA plots
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Fig. 4. Tensile stress vs. strain curves tested at indicated temperatures and offset along the strain axes by units of 0.005. The horizontal arrows mark the positions of |0 and the vertical ones the location of 0.002 strain. The ultra-fine grain size of 1 mm results in the high flow stresses with reduced ductility.
show a distinctive two-stage effect whereby the transition from the low S1 to the larger S2 occurs at the inception of the macro-strain (Fig. 4). This effect is also clearly measurable at 60 and 120°C. Thus if this steeper slope was due to thermal activation over fine precipitates [6], it should show a temperature dependence which is not apparent in Table 1. Such observation, which signals an abrupt change in the activation volume has been previously reported for the case of Mo [12]. The much smaller micro-strain effect in 3d-IA specimens is attributed to the fact that with larger grain sizes this regime is very short prior to dislocation multiplication and intersections and the present apparatus can not clearly resolve this region. The amount of interstitial atoms in solution was determined by back-extrapolation of the Haasen plot to the 0.02% offset stress (|0) at test temperatures of 180 or 220°C [6,13,14] since it is determinable prior to the onset of PLC (this choice of |0 approximately coincides with the so-called anelastic stress [12], and the corrected flow stress and our definition of effective stress is defined as |− |0).
Fig. 5. The macro-region strain rate sensitivity (S2) vs. the amount of carbon in solution as determined from the Haasen intercept for various Ti – IF steels tested in this laboratory.
The Haasen intercept is lower for the 9d-IA specimen than that for 3d-IA in Figs. 1 and 2 indicating that the carbon/interstitial content is lower due to the effect of the longer intermediate anneal of 9 days at 400°C. The deduced compositions of these and previous specimens are given in Table 1 from which S2 can be compared with the solute composition (Fig. 5). Note that the minimum S2 value in Fig. 5 refers to the as-quenched case shown in Fig. 3 in which a small upper deflection in the Haasen plot denoted as S3 occurs, and is characterized by a divergence between 1/4 and 1/10 strain rate change responses. On the other hand, the inverse activation volumes due to 1/4 and 1/10 rate changes are distinctly separated in the IA specimens, which have a lower interstitial content. The strain rate sensitivity measured from the 1/4 and 1/10 rate changes are nearly but not identical depending on the test conditions. Hence, the higher strain slope (S3 in Fig. 3) appears to be comparable to S2 for the 3d-IA specimen suggesting that the divergence may be due to solute carbon removal by dynamic carbon segregation after the generation of dislocations during the initial
Table 1 Calculated thermodynamic response values from the Haasen plots (k is the Boltzmann constant) Specimen
9d-05
9d-08
9d-07
9d-06
3d-02
3d-03
As-quenched [6]
Test temperature (°C) S1 (1/10)×10−6 (1 K−1) S2 (1/10)×10−5 (1 K−1) S2 (1/4)×10−5 (1 K−1) Haasen intercept ×10−4 (MPa K−1) ppm (wt.) DW k/S1 (1/10) (eV) DW k/S2 (1/10) (eV) Activation volume in b 3 at (|−|0)= 160 Mpa
60 6.09 5.27 6.84 2.14 0.215 14.2 1.64 646
120 4.47 6.19 7.06 2.18 0.223 19.3 1.39 657
180 5.28 5.69 6.60 2.08 (1.66)a 0.203 (0.129)a 16.3 1.51 474 (506)a
220 5.30 6.23 6.42 1.16 0.063 16.3 1.38 614
180 – 3.86 3.16 4.68 1.03 – 2.73 –
220 – 3.12 3.19 3.07 0.442 – 2.70 –
220 – 1.01 1.01 4.61 1.00 – 8.62 –
a
A second test indicated that |0 may be lowered to 221 (bracketed values) from 228 MPa.
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straining. A calculation of the amount of atom planes available to capture at least one carbon atom in the dislocation core [15] using the dislocation density, z, which corresponds to the shear stress at the point of divergence according to the relation, ~ = hvbz 0.5, where v =80.65 GPa [16], b = 0.24824 nm and h =0.4 [17], results in about 1 ppm (temperature corrections for these parameters were not used in these preliminary calculations). This means that the divergence of the 1/4 and 1/10 curves with straining in Fig. 3 could be a consequence of a change in dislocation– dislocation interaction with dynamic reduction of solute carbon to below 1 ppm (wt.). From Fig. 5 it appears that the interstitial solutes start to measurably affect the deformation mechanisms at about 1 ppm (wt.). An alternative interpretation derived from the study of face-centered cubic (fcc) aluminum is to attribute the divergence between 1/4 and 1/10 rate changes to the deformation generated vacancies which recover during the time required to mechanically impose a strain rate change [18]. However, one would expect to observe the divergence to increase with strain whereas the results in Figs. 1 and 2 suggest a constant or a slightly decreasing effect. Nevertheless, the activation work deduced from DW =k/S, where k is the Boltzmann constant, shows that S2 correlates to the energy of vacancy formation of about 1.5 eV (Table 1). In contrast, S1 exceeds that for interstitial formation (vb 3 =7.7 eV) by a factor of 2. In the case of aluminum, the large DW becomes independent of temperature above 0.25Tm (180°C for iron), and is attributed to the strain field near the dislocation core, whereas at 4.2 K the smaller value is attributed to vacancy formation [19]. In comparison, for iron the energy to form a double kink in the screw dislocation is about 0.92 eV [8,9], which is much lower than the above data-deduced values. Also, the thermally activated double kink formation [8,9] predicts a large temperature dependence of the activation volume, V, and this is not observed between 60 and 220°C (Table 1), where V is normalized at the corrected flow stress of 160 MPa. At 180°C, a second test indicates that |0 may be lower resulting in the bracketed values and the anomalous deviation becomes less apparent.
4. Discussion The observation in Figs. 1 and 2 of the clear separation of the Haasen plot depending on the amount of rate change in the macro-strain region has not been reported previously to our knowledge. Unfortunately the resolution of the testing parameters was insufficient to show if this effect also held in the micro-strain region. A higher strain rate sensitivity with ultra high purity was indicated in the 1/10 rate change plot of Stein and Low [7], and this trend is consistent with the
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current finding shown in Fig. 5. The strain rate sensitivity, S, is related to the activation volume, V=bld (where l is the interobstacle segment length of the dislocation, b is the Burgers vector, and d is the activation distance) by (|− |o)V=k/S. Previously, we argued that the region of the Haasen plot where the Cottrell–Stokes relation is obeyed must be independent of the amount of strain rate due to the fact
d #l l #d #V + =b # ln m; ( ln m; # ln m;
(1)
is zero [18]. This implies that if the obstacle is the forest dislocation the changes in l and in d are equal and opposite. The effect may be the same for the case of a large strain centre such as an interstitial site. However, if the obstacle is a jog or constriction caused by the dislocation intersections, than a change in d must still occur, while changes in l may be negligible or not correlated. The fact that the Haasen plot can be linearized and the slopes due to 1/4 and 1/10 rate changes are nearly equal suggests that the latter condition can apply. Since V appears to change inversely with the corrected flow stress, it suggests that the intersected dislocation retains a memory of the number of intersections. In Escaig’s theory [10] of recombination of sessile dissociated dislocation to produce glissile screw dislocations by thermal activation, Guyot and Dorn [1] clearly show that the activation volume should obey an inverse quadratic relationship to the effective stress instead of a linear one as suggested by the Cottrell–Stokes relation. A quadratic relation may hold near the ultimate stress (not shown), but not near the macro-yield point, and hence it is difficult to clearly separate the two mechanisms. The resolution of this apparent discrepancy may be that the dislocation–forest intersection converts the glissile to a sessile configuration, which is estimated to be energetically small of the order of 1/15 eV [10]. Thus the recombination distance is controlled by the inter-jog distance, which must correlate to the forest dislocation density and mean slip distance. However, unlike the forest theory of hardening, a memory effect is retained and the work hardening formalism should parallel that of jog dragging. In the micro-strain region, on the other hand, V is large and l is comparable to the grain size. Meakin [12] attributed this observation in Mo single crystals to the motion of pre-existing kinks, widely separated on dislocations threading the crystallites. However, Meakin’s estimate of the activation work, DW$10 − 2vb 3, is orders of magnitudes smaller than the present measured values (Table 1). Thus DW values, larger than vb 3, must be due to the strain field interactions between the glissile dislocations, which result in a large d. The intersecting dislocations will result in a Cottrell–Stokes forest hardening relationship prior to the chopping up
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of the segments with many jogs. This jog creation process consequently converts a glissile core into a sessile one with a reduction in the long range strain field. A mechanism is needed to explain the reduction of DW upon the inception of macro-deformation. Since DW = k/S= hvb 2d [6] and using S2, h(d/b) $ 0.18 and using an h value of 0.4 [17], d must be about 0.5b suggesting a partial dislocation in conformity to the recombination mechanism. On the other hand, using S1, d must be 4.5b. These results suggests that the dislocation density measurement should correlate with S1 rather than with S2. This means that the inter-jog spacing would become smaller than the inter-forest one as the dislocation traverses its mean slip distance. If this mechanism is truly operational, the mean slip distances can not rapidly increase in terms of the multiples of cell size due to the jog hardening effect. The consequence of such hardening is that diffuse necking is promoted by the inherent resistance to micro-banding. Thus this may be the essential difference between bcc and fcc metal deformation.
5. Conclusion Precision strain rate sensitivity measurements of ultra-low carbon, titanium IF steel result in a Haasen plot with two linear regions, the first in the micro-strain region, and the second after the inception of macroplastic flow. The measured activation work suggests that the former is associated with the intersection of glissile dislocations, whereas the latter is due to jog hardening and/or the recombination of sessile partial dislocation arrays. These quantitative measurements on bcc iron of high purity in the ppb range can be used to affirm the various theories of thermally activated flow.
Acknowledgements The authors wish to thank the Natural Sciences and Engineering Research Council of Canada and Materials Manufacturing Ontario for their continued support. We thank Dr H. Kobayashi of Dofasco Inc. for suggesting such studies, and C. Lutchman for technical assistance.
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