Studying the Hall-Petch effect regarding sub-micrometer steel (0.6% C)

Rodolfo Rodríguez Baracaldo¹, José Maria Cabrera Marrero², Jose Antonio Benito Páramo³

¹ PhD in Materials Engineering, Universidad Politécnica de Cataluña. Professor, Universidad Nacional de Colombia. rrodriguezba@unal.edu.co

² PhD in Industrial Engineering, Universidad Politécnica de Cataluña. Professor, Universidad Politécnica de Cataluña. Jose.maria.cabrera@upc.edu

³ Ph.D. in Chemistry, Universidad de Barcelona. Professor, Universidad Politécnica de Cataluña.Josep.a.benito@upc.edu

ABSTRACT

This paper study the synthesis and mechanical characterisation of steel (0.6% C) having lower than 1 micron grain size. There was severe plastic deformation in high pressure planetary ball milling and consolidation for obtaining bulk samples at temperatures between 350°C and 500°C. Studying grain size evolution showed that samples without subsequent heat treatment retained their nanocrystalline structure. Grain growth was controlled in heat-treated samples due to many nucleation points and the presence of cementite precipitates. The results obtained regarding hardness and grain size satisfactory agreed with the Hall-Petch ratio. The influence of the synthesis and mechanical characterisation techniques used in this work were compared to results mentioned in several references.

Keywords: alloy steel, heat treatment, Hall-Petch relationship

Received: March 5th 2010 Accepted: November 24th 2010

Introduction

Reduced grain size has a strong effect on crystalline materials' mechanical properties. Small grain size increases the strength of a material, thereby promoting performance regarding high mechanical requirements. Hall (1951) and Petch (1953) studied mechanical properties in materials having less than one micrometer grain size, called ultra fine grain (UFG), and materials having less than 100 nm grain size, called nanocrystalline (NC) during the middle of the last century; they separately established a relationship between yield strength sy and grain size D:

This expression is known as the Hall-Petch relationship where σ₀ refers to the frictional stress required to move unlocked dislocations along a glide plane; σ₀depends on temperature, strain and impurity level in material. Such frictional stress becomes increased as a function of constant K which depends on the material but is independent of temperature and average grain size D (Dieter 1988), taking into account the relationship between hardness and yield strength H = (2.5 to 3) o₀(Courtney 2000). The Hall-Petch relationship can be expressed in terms of hardness as:

Although equation 2 is widely accepted in crystalline materials having grain sizes greater than one micron, the important work of Chokshi et al ., (1989) has shown that this relationship is not so evident regarding UFG and NC materials. The idea of pile-up dislocation has been the physical explanation of the Hall-Petch relationship. However, the number of dislocations piled-up decreases as grain size decreases with a fixed level of stress, such number is thus a function of applied stress and the distance to source (Dieter 1988). Regarding critical grain size, it is not feasible to refer to piled-up dislocations to explain plastic strain; therefore, the Hall-Petch relationship does not explain the mechanical behaviour of ultrafine and nanocrystalline material (Meyers et al ., 2006; Pande and Cooper, 2009).

This paper analyses the synthesis and mechanical characterisation of an iron alloy having 0.6% C NC/UFG grain size structure. Mechanical properties were studied through hardness testing and their relationship to grain size from the perspective of the Hall-Petch ratio. This paper compares the results of this work with work reported in the pertinent literature, analysing possible reasons for disagreement between authors when the mechanical behaviour of iron and steel having grain sizes in the UFG and NC range was analysed.

Materials and methods

Irregular shaped 75 to 160 micron particle size steel powder was used for this work. The initial powder, having 10 micron grain size and 1.2 ± 0.1 GPa hardness, was treated by severe plastic deformation during planetary ball milling for 60 hours. Stainless steel containers and balls having a 27:1 ball-powder ratio were used. A stationary argon atmosphere was used to control oxidation during milling. The powder's final composition (% wt.) was 0.58% C, 0.33% O, 0.27% Cr, 0.03% Si, 0.2% Mn and Fe base, with 9.3 GPa final hardness and 12 ± 4 nm grain size. The milled NC structure powder was consolidated by warm consolidation to obtain bulk material; warm compression took place at 350°C to 500°C and 850 MPa pressure for one hour. For further details regarding obtaining the NC powder and consolidation please refer to Rodriguez-Baracaldo (2006). The specimen's wide grain size range was obtained by 650°C to 900°C heat treatment for 30 minutes.

The grain size was identified by transmission electron microscopy (TEM). The specimens were mechanically buffed to 80 microns and then polished (Gatan Duo Ion Mill model 600). Grain size was determined by bright and dark field imagines to better identify grain boundaries, avoiding overlapping grains. Milled powder and bulk samples' hardness was evaluated by Vickers indentations using MVA-HO Akashi equipment with a 200g load (1.96 N) on the specimens' upper and lower surfaces. The values were obtained by averaging at least 15 measurements on each specimen. Nanometer sized cementite precipitates were studied by combining TEM and X-ray diffraction techniques.

Results and Discussion

Microstructural analysis

Grain size evolution analysis can be divided into three groups: samples without subsequent heat treatment, samples with heat treatment at lower than austenitic transformation temperature and samples with heat treatment higher than the austenitic transformation temperature. Figure 1 shows TEM images for the three scenarios. Ferrite grain size increased as compaction temperature increased in consolidated samples without subsequent heat treatment; however, average grain size was conserved in the nanocrystalline or low ultrafine range (see Figure 1a). Grain growth in samples having heat treatment at temperatures lower than austenitic transformation temperature (c 760°C) was controlled by two factors: nanocrystalline structure has many nucleation points, meaning a lot of points for recrystallisation to start. This scenario created competition between them, resulting in limited grain growth. Moreover, the presence of cementite precipitates operated as obstacles controlling ferrite grain growth, as shown in Figure 1b and 1c. Heat treatment higher than total ferrite-austenite transformation temperature lost all previous nanocrystalline structure, which increased exponentially with increasing temperature (Fig. 1d).

Mechanical analysis

Table 1 summarises the samples' average ferrite grain size obtained in different conditions, along with microhardness values. Decreased micro-hardness value due to increasing treatment temperature to 760°C was attributed to recrystallisation and normal grain growth. Although temperatures above 760°C (total transformation ferrite-austenite) were applied, hardness values did not become dramatically reduced, probably due to precipitates effect on the steel. Kimura et al., (1996) have shown similar behaviour in iron powder deformed by mechanical milling; heat treatment above 500°C produced continuous softening of the powder. It should be noted that Kimura et al., pointed out the presence of contamination elements obtained during milling which restricted ferrite grain growth; however, this was not quantified in their research. The response of this NC steel (0.6% C) produced by warm compaction of powder had a ferrite grain size-dependent hardness values. The range of grain sizes studied initially presented high hardness which decreased as grain size increased.

The following will analyse this pattern and its relation to grain size in NC and UFG steel samples (i.e. regarding the Hall-Petch ratio). Figure 2 shows decreasing hardness when grain size increased, having 7.4 ± 0.3 GPa values in 15 nm to 2.3 ± 0.2 GPa grain size samples where average grain size was 2,800 nm. Considering that the Hall-Petch effect is usually expressed in terms of hardness and the inverse of the square root of grain size, Figure 3 shows this ratio accompanied by an additional upper axis for grain size.

Figure 3 shows the samples' hardness (greater than 30 nm average grain size, obtained by consolidation at 425°C) maintained acceptable linearity according to the Hall-Petch ratio setting, whereas consolidated samples having 15 nm grain size (obtained by consolidation at 350°C) showed a clear deviation from the linearity obtained in the other results. It is likely that the unusual hardness values obtained in samples having less than 30 nm grain size were affected by a greater than 6% porosity level as a result of the low consolidation temperature used. Sanders et al., (1997) have pointed out that FCC and BCC metals having nanocrystalline structure do not satisfactorily agree with the Hall-Petch effect due to the strong influence of porosity. Therefore, the results for samples having 15 nm grain size were not considered when analysing the Hall-Petch ratio.

The fit equation obtained (see Figure 3) intercepted with the vertical axis giving H₀= 1.413 GPa and K=1.257 GPa·µm ^0.5 slope. The frictional stress value for the steel being studied was very high, H₀ = 1.413 GPa, considering the H₀ = 0.096 GPa determined by Petch (1953) on interstitial free pure iron studding having grain sizes from 20 to 100 microns. The high value obtained meant that high stress was required to move a free dislocation on the slip plane of the steel being studied. This behaviour may have been mainly due to two factors: the presence of cementite precipitates working as additional barriers to dislocation movement and large internal strain resulting in high dislocation density. The K coefficient varied according to the structural characteristics of the steels being studied. Pure iron analysed by Petch (1953) had 0.663 GPa µm^0.5, such value becoming increased according to grain boundary nature and quantity. K = 1.257 GPa µm^0.5 for the steel being studied suggested that this steel's grain boundary was a more effective barrier to dislocation movement than grain boundaries in pure iron and low carbon steel.

Figure 4 compares this work's results with the most representative results for UFG/NC iron and steel exclusively obtained by severe plastic deformation, especially results obtained by mechanical milling. Some results regarding strength published in the references referred to metal material hardness using σ_y H = 3 (Dieter, 1988). The Figure illustrates two kinds of hardness values; the first group maintained linearity and agreement with the HP relationship. This group includes work by Jang and Koch (1990), Malow and Koch (1998), Kimura et al.. (1995), Yin et al.. (2001), Sakai et al., 2000, Belyakov et al., (2001-03), Takaki et al., (2001), Jia et al., (2003) and Khan et al., (2000). A second group of results had a pattern which was completely independent of grain size; this was particularly noticeable in NC steels obtained by dynamic deformation methods, such as dynamic impact (Korznikov et al., 1995), mechanical grinding (Xu et al., 2002) and ball impact (Todoke et al., 2002; Umemoto et al., 2003).

It should be made clear that the high dispersion of results could have been due to marked differences in processing and mechanical characterisation, which must be noted here because they affect studying Hall-Petch ratio parameters. One of the main limitations of NC iron and steel obtained by mechanical milling is the presence of second phases due to contamination during milling. The work of Jang and Koch (1990) and Malow and Koch (1998) studied microhardness value variation regarding iron powder. They pointed out the presence of oxygen and nitrogen but did not quantify them and did not analyse their influence on material properties. It should be remembered that the results obtained in Jang and Koch's work clearly showed that high hardness was related to small grain size, which was not obtained in the steel used in this research. Other characterisation studies on unbounded powder have been made by Kimura et al., (1995) and Yin et al., (2001); they even identified the presence of foreign elements like oxygen in their work (but did not quantify them).

The remaining work shown in Figure 4 carried out mechanical tests on consolidated samples obtained from NC iron powder. Sakai et al., (2000), Belyakov et al., (2001-03) and Takaki et al., (2001) obtained samples via warm roll bounding (600°C to 800°C) of powder encapsulated in steel tubes whilst Jia et al., (2003) and Khan et al., (2000) obtained the samples by uniaxial compression. Despite the wide variation in results from a Hall-Petch ratio perspective, it can be concluded that NC and UFG materials produced by mechanical milling had H₀ friction stress values higher than the 0.096 MPa reference value determined by Armstrong et al., (1962) for pure iron. The H₀ value for iron produced by mechanical milling was analogous to that found in the steel used in this research, showing the strong influence of elements becoming incorporated during milling. A different pattern appeared concerning iron with oxides; the work of Sakai et al., (2000) and Belyakov et al., (2001-03) studied samples with oxygen (0.2% to 0.6% wt.) deliberately incorporated into NC powder during mechanical milling. These samples having a significant amount of oxide precipitates in the ferrite matrix had high hardness values for samples having grain sizes in the low UFG range. It is possible that oxides located in grain boundaries increased the dislocation movement blocking effect, as indicated by Srinivasarao et al., (2008) and Oh-shi et al., (2007).

NC steel samples obtained by high-energy deformation methods, such as dynamic impact, high-energy mechanical milling and ball impact, have revealed an independent grain size pattern. These methods led to considerable reduction in grain size, increasing the difficulty in determining their size and increased the error level when studying hardness values. The above difficulties partly explained grain size independence. An alternative explanation for variations in Hall-Petch ratio for steel has been proposed by Takaki et al., (2001) and Hidaka et al., (2001) who have proposed that there is some stress relaxation mechanism in the grain boundaries which becomes effective when ferrite grain size is less than 100 nm. This mechanism would allow a relaxation of the crystal structure, markedly changing the grain size effect on the material's hardness. Tejedor et al., (2008) and Rodríguez et al., (2010) have used nanoindentation techniques for studying strain rate sensitivity to find the possible presence of deformation mechanisms different to the classical dislocation pile-up model for explaining the Hall-Petch ratio. The authors of this work have pointed out that the classical dislocation movement mechanisms (which are not sensitive to strain rate) become blocked by carbon atoms in the nanometer range for the steel being studied. This situation generates alternative processes not explained by the Hall-Petch ratio. The research is expected to provide experimental results for determining deformation mechanisms for steel having a nanocrystalline structure.

Conclusions

According to the experimental results obtained in this work, it can be concluded that analysis of grain size evolution has shown that the samples without subsequent heat treatment did retain their structure in the nanometer or low ultrafine range. In heat-treated samples where lower than total austenitic-transformation temperature was analysed, controlled grain growth was due to the numerous nucleation points and cemen-tite precipitation. Heat treatment above total austenitic transformation temperature produced significant grain growth with loss of all previous nanocrystalline structure.

The results obtained regarding hardness and grain size agreed with the Hall-Petch ratio according to the lineal expression H (GPa)= 1.413 + 1.257 D ^-0.5. Deviation from linearity was clear in samples having less than 30 nm grain size. This pattern was not adjusted to the classic dislocation pile-up model explaining the Hall-Petch effect; such singularity could be explained by the effect of porosity or the presence of a different deformation mechanism to that of the classic dislocation pile-up model explaining the Hall-Petch ratio.

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