Effect of Dislocation Density on the Initiationof Plastic Deformation on Fe­C Steels

Kaoru Sekido1,2,+1, Takahito Ohmura2,+2, Toru Hara2 and Kaneaki Tsuzaki1,2

1Doctoral Program in Materials Science and Engineering, University of Tsukuba, Tsukuba 305-0047, Japan2National Institute for Materials Science, Tsukuba 305-0047, Japan

The effect of pre-existing dislocations and interstitial carbon on the initiation of plastic deformation in interstitial free (IF) steel and ultralow carbon (ULC) steel were investigated by the nanoindentation technique. The critical load, Pc, of the pop-in phenomenon, which correspondsto plasticity initiation, appeared clearly on the loading curve. The Pc in high dislocation density materials was smaller than that in low dislocationdensity materials with no difference between IF and ULC while the Pc in low dislocation density materials was remarkably higher in ULC thanin IF. These results indicate that the interstitial carbon in the matrix does not affect the pop-in phenomenon when there are pre-existingdislocations or dislocation sources, and we discuss the reason for their occurrence occurs in high dislocation density materials.[doi:10.2320/matertrans.M2011356]

(Received November 21, 2011; Accepted February 6, 2012; Published March 28, 2012)

Keywords: plasticity initiation, dislocation density, nanoindentation, steel, carbon

1. Introduction

Nanoindentation is a valuable tool for studying thefundamental aspects of plasticity on a small scale such asdislocation nucleation.1­7) One of the characteristics ofnanoindentation is the detection of the pop-in behaviorwhich appears as a strain burst on the loading curve. Oneof the main mechanisms of this behavior is the initiationof plastic deformation such as dislocation nucleation andmultiplication.4,8) Plastic deformation of metal results in anincrease in dislocation density, which should affect the pop-inbehavior. Wang et al.9) reported that the pop-in behavior ofNi3Al crystals during plastic deformation was found tobe closely related to their crystal orientation, pre-existingdislocation density in the sample surface, the loading rate,and the holding time. The major pop-ins may be correlatedwith the special dislocation structure of the Kear­Wilsdorf(K-W) locks in the Ni3Al crystal that are formed during theloading process. Zbib et al.10) reported that the onset ofplasticity of a tungsten single crystal during nanoindentationdepends on the pre-existing dislocation density. They showedthat it is possible to generate a sudden transition from anelastic to a plastic deformation during nanoindentation evenwith substantial dislocation densities. Therefore, the dis-location activation models may be more appropriate for manymaterials rather than those that are based solely on thehomogeneous nucleation of a dislocation. Barnoush et al.11)

reported the effect of dislocation density on the pop-inbehavior using electron channeling contrast imaging (ECCI)in aluminum alloys. They demonstrated that only a very lowdislocation density is necessary for observing the pop-in, andat a high dislocation density the dislocation sources becomeactivated instead of dislocation nucleation. In the presentpaper, we discuss the effect of carbon on the relationshipbetween the dislocation density and the pop-in phenomenasince interstitial carbon is an important element of steel. In aprevious report, we discussed the effect of interstitial carbon

on the pop-in behavior in steel with low dislocation densityand showed that the average Pc where the pop-in occursin ultra low carbon (ULC) steel was higher than that ininterstitial free (IF) steel with no in-solution carbon.12) Thispaper describes the difference in the pop-in behavior of steelswith low and high dislocation densities and the effect ofcarbon on the movement of pre-existing dislocation.

2. Experimental Procedure

The chemical compositions of the IF and ULC used in thisstudy are shown in Table 1. The steel ingots were hot-rolledin a temperature range of 1423 and 1243K to 3.6mm sheets.The ULC sheets were cooled to room temperature at 50K/s,while the IF sheets were held at 973K to produce TiC carbideparticles and then cooled to room temperature. Note that theC content is 0.0158 at% in IF and 0.0734 at% in Ti, and the Ticontent is large enough to consume C by forming TiC. Thehot-rolled sheets were then cold-rolled to 1mm thickness andannealed at 1073K for 60 s to obtain a fully recrystallizedstructure. To obtain high dislocation density samples, therecrystallized samples were tensile deformed till fracture ataround 40% strain by a tensile load at room temperature,resulting in a dislocation density of 1014m¹2. To obtain lowdislocation density samples, the recrystallized samples werefurther annealed at 1123K for 7.2 ks and then held at 973Kfor 1.8 ks followed by cooling to room temperature in air. Theadditional annealing produced samples with a low dislocationdensity of 1011m¹2 which was lower than the dislocationdensity of 1012m¹2 before the tensile deformation anddetermined to more appropriate for the purpose of this study.Using the above procedure, we obtained four samples withdifferent dislocation densities for the two steels. The samples

Table 1 Chemical compositions (mass%) of IF and ULC.

C Si Mn P S Al Ti N Fe

IF 0.0034 0.009 0.1 0.002 0.0031 0.034 0.063 0.0016 Bal.

ULC 0.0038 0.009 0.1 0.003 0.0031 0.032 0.002 0.0015 Bal.+1Graduate Student, University of Tsukuba+2Corresponding author, E-mail: ohmura.takahito@nims.go.jp

Materials Transactions, Vol. 53, No. 5 (2012) pp. 907 to 912©2012 The Japan Institute of Metals

for nanoindentation were mechanically polished, and sub-sequently electropolished in a solution of 8% perchloric acid,10% butylcellosolve, 60% ethanol, and 22% water at 273Kunder a potential of 40V to remove the damaged layer.Nanoindentation experiments were carried out using aHysitron Triboindenter. The indentation tests were conductedunder the load controlled condition with a maximum load of1500 µN for the annealed samples and 1000 µN for thetensile deformed samples. A Berkovich indenter wasemployed, and the tip truncation was calibrated using areference specimen of fused silica. Analyses for the tipcalibration and the calculation of hardness were conductedusing the Oliver and Pharr method.13) The probed sitesand indent configurations on the specimen surfaces wereconfirmed before and after the indentation measurementswith the scanning probe microscope (SPM) capabilities of aTriboindenter. The indentation was made on several grainsand that were quite far from the grain boundaries and otherindent marks. The effect of crystallographic orientation wasexamined, and the results showed that the grain orientationdid not have much effect on the pop-in phenomenon allowingus to choose grains randomly for this study. The tensile testspecimens had a gauge length of 50mm and a thickness of1mm which conformed to the JIS-5 standard. The loadingaxis for the tensile tests was set parallel to the rollingdirection. The tensile tests were carried out with an AG-50kNG (Shimadzu Corporation) under a nominal strain rateof 2.8 © 10¹4 s¹1 at room temperature. The dislocationdensities were measured by the Scanning TransmissionElectron Microscopy (STEM). The samples for STEMobservations were mechanically thinned to 100 µm, andperforated by a twin-jet electropolisher, and the observationwas conducted on a JEOL-2010F operated at 200 kV.The dislocation density was determined by measuring thetotal length of dislocations on the STEM images under theassumption of a 100 nm foil thickness.

3. Results and Discussion

STEM images in Fig. 1 show typical dislocation structuresof fully-annealed and heavily deformed samples. The lowdislocation density steels show an inhomogeneous disloca-tion structure, and the density was measured in an area withrelatively high dislocation density. The dislocation density ofthe tensile deformed sample was estimated from the insideof the cell structure. The estimated dislocation densities ofthe annealed and the deformed samples were in the order of1011 and 1014m¹2, respectively, for both steels.

Figures 2(a) and 2(b) show the SPM images afterindentation for IF with low and high dislocation density,respectively. The indent mark is around 1µm. Someprecipitates of around 200 nm appear in the SPM imageswhich may be TiC since we found the TiC to be 100­200 nmwith STEM and EDS in the previous paper.12) The figureshows that the indentation was clearly made in an area with asufficiently smooth sample surface without any effects fromthe grain boundary or the precipitates.

Figure 3 shows the typical load-displacement curvesobtained by nanoindentation. Figures 3(a) and 3(b) representthe curves for the steels with low dislocation density and high

dislocation density, respectively. A clear pop-in behaviorappears in the low dislocation density steels in Fig. 3(a), andthe critical load for the pop-in in ULC is higher than that inthe IF. On the other hand, the pop-in behavior is not clear insteels with high dislocation density shown in Fig. 3(b). Tomake sure a transition in deformation mode from elastic toelastoplastic and obtain the critical load for the initiation ofplastic deformation, we used the Hertz contact theory.14)

It was applied to the first point where a deviation appearedon the P­h curve in Fig. 3(b). The load P and penetrationdepth h are given by the following equations:

P ¼ 4

3E�Ri

12h

32 ; ð1Þ

1

E� ¼ ð1� vs2Þ

Es

þ ð1� vi2Þ

Ei

; ð2Þ

where Ri is the curvature of an indenter tip, E* is the reducedmodulus deduced from eq. (2), vs and vi are the Poisson’sratios of the specimen and the indenter, and Es and Ei are theYoung’s moduli of the specimen and the indenter. The P­hcurves that were obtained by substituting 200GPa for E* and230 nm for Ri into eq. (1) are shown as a broken line inFig. 3. The E* is estimated from the unloading curve inFig. 3, and Ri is determined by measuring a standard

Fig. 1 STEM images for IF and ULC with (a),(b) low dislocation densityand (c),(d) high dislocation density.

Fig. 2 SPM images for IF with (a) low dislocation density, and (b) highdislocation density after indentation. The maximum load is 1500µN forlow dislocation density and 1000µN for high dislocation density.

K. Sekido, T. Ohmura, T. Hara and K. Tsuzaki908

sample.15) The calculated curves roughly fit in the data thatwas obtained experimentally before the pop-in phenomenonin all the materials, which confirms that the pop-inphenomenon in the present study corresponds to the plasticityinitiation. Even though the pop-in phenomenon is not clear inFig. 3(b), the critical load for the pop-in can be found usingthe Hertz contact curve by considering it as a deviation fromthe broken line. The critical load Pc at which the pop-inoccurs and the corresponding excursion depth "h are definedin Fig. 3. According to the figure, the first pop-in appears at alower Pc with a high dislocation density than with a lowdislocation density. This tendency agrees with the report byBarnoush et al. in which Al with high dislocation density didnot exhibit a pop-in while Al with low dislocation densityshowed a clear pop-in.11)

Figures 4(a) to 4(d) are the plots of Pc vs. "h for IF andULC with low and high dislocation density. The followingthings can be determined from these figures. First, the averagePc in the ULC is higher than that in the IF with lowdislocation density. Second, the average Pc in high dislocationdensity materials is lower than that in low dislocation densitymaterials. Third, the average Pc in the ULC is almost thesame as that in the IF with high dislocation density.

Leipner et al.16) described the critical stress ¸n for thedislocation nucleation in GaAs using nanoindentation. Theyproposed the equation:

¸n ¼Gb

³ e3r0

2� ¯

1� ¯; ð3Þ

where G is the shear modulus, e is the Euler number, b is themagnitude of the burgers vector, r0 is the cutoff radius atthe dislocation core, and ¯ is Poisson’s ratio. We obtained¸n µ 9.7GPa using the typical values of r0 = b/3, b =0.29 nm, G = 83GPa, and ¯ = 0.3 for ferrite. On the otherhand, the maximum shear stress ¸max underneath the indentercan also be determined from the Hertz contact theory14) as:

¸max ¼ 0:18E�

Ri

� �23

P13 : ð4Þ

Table 2 shows the calculated ¸max values for each material,which was obtained by substituting the value of E* =200GPa, and Ri = 230 nm, and P with the maximum Pc

value from Fig. 4 into eq. (4). ¸max in low dislocation densitymaterials is larger than ¸n, suggesting that dislocationnucleation can occur. The fact that these ¸max values areclose to the ideal strength also supports the occurrence ofdislocation nucleation. In high dislocation density materials,¸max is lower than ¸n, indicating that the plasticity initiationdoes not depend on dislocation nucleation but on anothermechanism with a lower critical stress.

The reason that the ULC has a higher Pc than the IF in lowdislocation density materials was already discussed in theprevious paper.12) Since there are no dislocations beneath theindenter, a dislocation loop nucleates once an indent is made.We used the model that is based on a Frank­Read sourcegenerated by a double cross slip. Three load levels indicatedas P1, P2, and Pc are shown in Fig. 5(a). We assume that ashear loop nucleates at a defect free area when the load is P1,and the first cross slip and a subsequent double cross slipoccur at P2 to form a Frank­Read source with the Frank­Read length lc. When the applied shear stress reaches thecritical shear stress ¸c given as:

¸c ¼Gb

lc; ð5Þ

the Frank­Read source is activated at Pc. Hereafter, the effectof interstitial carbon will be discussed using this model. Thecritical loop size is estimated to be around 2 nm from eqs. (4)and (5). The distance between carbons is estimated to bearound 4 nm from the 0.0177 at%C contents when thesegregation of carbon is not considered. Since the order ofthe distance between carbons is comparable to the criticaldislocation loop size, we believe that there is a highprobability that the carbon interacts with the nucleateddislocation loop. The effect of carbon on edge dislocations is

Fig. 3 Typical load-displacement curves for IF and ULC with (a) lowdislocation density and (b) high dislocation density.

Table 2 The ¸max for IF and ULC with low and high dislocation density atthe maximum Pc.

Pc

(µN)¸max

(GPa)

IF (low dislocation density) 500 13.02

ULC (low dislocation density) 1433 18.49

IF (high dislocation density) 140 8.51

ULC (high dislocation density) 100 7.61

Effect of Dislocation Density on the Initiation of Plastic Deformation on Fe­C Steels 909

higher than that on screw dislocations since the bindingenergy between the carbon and the edge dislocation ishigher.17) Thus, the mobility of the edge component shouldbe lower in ULC than in IF, resulting in different dislocationloop sizes as shown in Fig. 5(b) with thicker and thinnerdislocation lines, respectively. Therefore lc in ULC becomesshorter than that in IF, even though the load has beenmaintained for a certain period of time in both steels. From

eqs. (4) and (5), the relationship between lc and Pc can bedefined as:

1

lc/ Pc

13 : ð6Þ

Accordingly, ULC with a shorter lc needs a higher Pc thanthat of IF for the initiation of plastic deformation.

High dislocation density materials have a lower Pc thanlow dislocation density materials. In high dislocation densitymaterials, the microstructure could contain numerous dis-location sources generated by dislocation interactions duringthe tensile deformation, and some dislocation sources may beactivated at a lower shear stress than the shear stress ¸c forthe indentation induced dislocation source. Accordingly, theonset of the plastic deformation under an indentation-inducedstress is presumably dominated by the multiplication of a pre-existing dislocation underneath the indenter. Consequently,the plastic deformation is initiated at a lower load.

When we considered the effect of interstitial carbon on thepop-in event in high dislocation density materials, there wasno significant difference in Pc between the IF in Fig. 4(c) andthe ULC in Fig. 4(d). Next the thermal activation process ofthe dislocation motion is considered to determine the effect ofinterstitial carbon on the pop-in event experimentally. Sinceinterstitial carbon is a short-range obstacle, the passingmechanism of dislocation on the interstitial carbon shouldbe a thermal activation process. Therefore, if the pop-inbehavior is affected by the interstitial carbon in ULC, the Pc

would show an indentation rate dependence mentioned in theprevious paper.12) Figures 6(a) and 6(b) show the indentationrate dependence with low and high dislocation density

Fig. 4 The relationship between Pc and "h for IF and ULC with (a),(b) low dislocation density and (c),(d) high dislocation density.

Fig. 5 (a) A schematic P­h curve. (b) The model for the relationshipbetween dislocation movement and a pop-in.

K. Sekido, T. Ohmura, T. Hara and K. Tsuzaki910

materials, respectively. In low dislocation density materials,the Pc in ULC shows a clear indentation rate dependence,while in high dislocation density materials, neither IF norULC shows any indentation rate dependence. These resultsindicate that there is no effect of interstitial carbon on thepop-in event for the high dislocation density materials.

The effect of carbon on high dislocation density materialsis considered as follows. As mentioned above, we proposedthat the interstitial carbon provides a higher friction stressagainst an edge dislocation movement while the nucleatedshear loop develops into a Frank­Read source. On the otherhand, the high dislocation density materials already containmany dislocation sources which have a longer Frank­Readlength with a lower critical load for multiplication once anindent is made. Therefore the dislocation nucleation isunnecessary and the process of shear loop growth, which isaffected by carbon, does not occur.

Shim et al. reported that a difference in the indenter sizeproduced different highly stressed zones and explained therelationship between the size of a highly stressed zone andthe dislocation density through a schematic image.18) We alsodrew a schematic image of our results in Fig. 7. Figures 7(a)and 7(b) show the image of IF and ULC with low dislocationdensity, and Figs. 7(c) and 7(d) show the image of IF andULC with high dislocation density. The highly stressed zoneof the ULC with low dislocation density in Fig. 7(b) consists

of carbon with few pre-existing dislocations. On the otherhand, the highly stressed zone of the ULC with highdislocation density in Fig. 7(d) contains not only carbonbut also many dislocations. Next we discuss the two reasonswhy the carbon does not have any effect on the ULC withhigh dislocation density when the dislocation multiplicationoccurs.

The first reason is that the dislocation without any carbonpinning could be the dominating mechanism. Britton et al.19)

discussed the relationship between the amount of carbonand the dislocation density for the pop-in using the Fe­0.01mass%C polycrystal (bcc) based on the model byCottrell and Bilby.20) The model describes the effect ofcarbon contents ½ (mass%) on the stress­strain curve ofsteels with different dislocation densities μ (m¹2). The yielddrop is shown to appears on the stress­strain curve when

½=μ � 10�18 ðmass%m2Þ; ð7Þin contrast, the yield drop does not appear when

½=μ � 10�19 ðmass%m2Þ: ð8ÞThus, they concluded that pop-ins do not occur when thesample contains many dislocations since there is not enoughcarbon contents to pin all the dislocations, which correspondsto the case of the ULC with high dislocation density in ourstudy. The carbon content in the ULC is 0.0038mass% with alow dislocation density of 1011m¹2 and a high dislocationdensity of 1014m¹2. ½/μ is estimated to be 10¹14 (mass%m2)for low dislocation density, which satisfies eq. (7) resulting ina pop-in and 10¹17 (mass%m2) for high dislocation densitywhich is two orders higher with the critical value in eq. (8).However, the carbon content could be overestimated in thegrain interior because we ignored the carbon segregation tothe grain boundaries. Additionally, the dislocation densitycould be underestimated since we measured it within thedislocation cells and did not count the dislocations on the cellwalls. Therefore, the actual value of ½/μ could be muchlower than the estimated value and correspond to the case ofeq. (8). Consequently, we could assume the possibility thatsome of the pre-existing dislocations might not be pinned by

Fig. 6 Indentation rate dependence for IF and ULC with (a) lowdislocation density and (b) high dislocation density.

Fig. 7 The schematic images showing IF and ULC with (a),(b) lowdislocation density and (c),(d) high dislocation density during nano-indentation.

Effect of Dislocation Density on the Initiation of Plastic Deformation on Fe­C Steels 911

carbon and are able to move in the same way as dislocationsin the IF, since many dislocation sources exist underneath theindenter. The schematic image of this case is shown inFig. 7(d1).

The other reason is that the critical stress for dislocationsource activation is more dominant than the occurrence of theunpinning from carbon. We estimated the practical balancebetween the carbon contents and the dislocation density forthe ULC with high dislocation density. In this estimation, allthe carbon exists in the grain interior with no segregation orprecipitation. The number of carbon atoms per unit volumeNc

v in the ULC estimated from the composition of carbon isas follows. The carbon contents in the ULC is 0.0177 at%,thus the mean spacing of carbon is estimated to be around4 nm, and Nc

v is calculated to be 1.6 © 1025m¹3. On the otherhand, the number of carbon atoms existing on a dislocationNd

v is estimated from the dislocation density. We assume thespacing of the carbon to be 0.29 nm, which is the nearestneighbor of the octahedral site. Nd

v is obtained by dividingthe dislocation density (1014m¹2) by the spacing of carbonatoms (0.29 nm) and calculated to be 3.4 © 1023m¹3. Fromthese estimations, Nc

v is much larger than Ndv, indicating that

the carbon content seems to be high enough to pin all thedislocations as schematically shown in Fig. 7(d2). Thus,another possibility should be considered. When all thedislocations or dislocation sources beneath the indenter arepinned by carbon, the critical stress ¸p for the dislocationmultiplication of pre-existing dislocation based on a linetension model for a tensile deformation induced Frank­Readtype source is expressed as:

¸p ¼Gb

lp; ð9Þ

where lp is the tensile deformation induced Frank­Readlength. ¸p is dominant if the stress for unpinning from thecarbon is lower than ¸p. It is not easy to estimate the stress forunpinning from the carbon; however, the yield stress givenby the tensile test is around 300MPa,12) and the dislocationcan move at this stress level for the initiation of thedeformation. On the other hand, the ¸max calculated from Pc

in eq. (4) is around 8GPa as shown in Table 2. Since there isa stress distribution underneath the indenter and the positionof the activated dislocation source could be far from that ofthe ¸max, the actual ¸p might be lower than 8GPa. However,the ¸p could still be much higher than the yield stress level.Thus, the critical stress ¸p for the activation of the tensiledeformation induced pre-existing dislocation source is domi-nant for the pop-in and the unpinning from the carbon has noeffect on the ULC with high dislocation density. In this case,the Pc is associated with the ¸p in eq. (9); hence ¸c and lc ineq. (5) have no relation to Pc. Therefore, the effect of in-solution carbon on the dislocation mobility leading to thedislocation source formation does not have to be considered.

4. Summary

The effect of pre-existing dislocation and interstitial carbonon the pop-in behavior was investigated using the IF and

ULC steels. The steels with high dislocation density showeda smaller Pc than those with low dislocation density sincethey contained many pre-existing dislocations and dislocationsources that can multiply at a lower load. There was nodifference in the average Pc value between the IF and ULCwith high dislocation density. The ULC also did not showany indentation rate dependence, suggesting that theinterstitial carbon does not have any effect on the pop-inbehavior. One presumable reason for this behavior is thatsome of the pre-existing dislocation sources might not bepinned by carbon. Another factor might be that the stress forunpinning is lower than the critical stress against the linetension for activating the tensile deformation induceddislocation source. Therefore the stress to initiate the plasticdeformation does not depend on the carbon content whenthe materials contain sufficient dislocations or dislocationsources.

Acknowledgements

The steel sheets in this study were supplied by SteelResearch Laboratories, Nippon Steel Corporation. This studywas supported by JST, CREST. K.S. acknowledges theNational Institute for Materials Science (NIMS) for theprovision of a NIMS Graduate Research Assistantship.

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