Permanent Strength of Interstitial-free Steel Processed by Severe Plastic Deformation and Subsequent Annealing

Abstract

The permanent strengths of interstitial-free (IF) steels with different grain sizes and dislocation densities processed by severe plastic deformation (SPD) and subsequent annealing are systematically investigated. Permanent strength, which is athermal and time-independent, corresponds to the fundamental capability to bear stresses caused by external forces. Sufficiently long-time (24 h) stress relaxation tests were carried out and experimental stress–relaxation time relationships were extrapolated to estimate the permanent strength that remained after an infinite time passed. The flow stresses observed in standard uniaxial tension tests increased with repeated SPD processes, and the fraction of permanent strength to the observed flow stress was mostly above 65%. The permanent strength also increased with repetition of SPD processes, and subsequent low-temperature annealing further augmented the permanent strength. During SPD processes, the dislocation-related strengthening was dominant, while the grain-size-related strengthening was minor, i.e., the Hall–Petch relation does not hold. On the other hand, after low-temperature annealing, the grain-size-related strengthening became dominant, quickly replacing the dislocation-related strengthening. In a coarse-grain region, the grain-size-related strength was consistent with the classical Hall–Petch relation. It was confirmed that the original Hall–Petch relation holds only in the coarse-grain region and it indicates “softening with grain coarsening due to annealing”, not “strengthening by grain refinement due to SPD”.

1. Introduction

Since the stress-bearing capability of a material governs the safety of structures, strengthening of materials has been one of the central purposes in the research and development of materials. Materials that are expected to bear stresses caused by external forces have been recognized to be structural materials. For the evaluation of material strength, uniaxial tensile tests at a strain rate ranging from 10⁻⁴ to 10⁻² /s are usually conducted. A flow stress temporarily observed during tensile loading consists of time-independent athermal and time-dependent thermal components.¹⁾ In any environment, e.g., under dead or live loads, varying temperatures, and long-time prestressing, the time-independent athermal strength is expected to be the basic stress bearing capability of materials and can be referred to as the permanent strength, which is identical to the athermal threshold stress. By contrast, thermal strength is temporal, which will relax and disappear eventually after some time at a constant strain state. Knowing the exact permanent strength is necessary in the design and construction of structures with a high level of safety. To the best of our knowledge, however, there are only a few studies on direct measurements of permanent strength in the literature. In previous studies (e.g., 1–6)), permanent strength was treated as one of the material parameters in material modeling and was not the primary target of those studies. Not many data on the permanent strength of structural metals have been accumulated so far.

Recently, Koizumi and coworkers^7,8) have systematically investigated the permanent strengths of industrial pure aluminum and copper that had various crystal grain sizes ranging from tens of micrometers to sub-micrometer produced by severe plastic deformation (SPD) and subsequent annealing. During the grain refining process by SPD, the permanent strengths of aluminum and copper reached maximum at an introduced plastic strain of 2 and decreased with further application of plastic strain. In particular, the permanent strength of aluminum became only one-seventh of the tensile flow stress observed in a standard strain rate tensile test (10⁻² /s). The permanent strength of aluminum was significantly increased (approximately five times) by subsequent low-temperature annealing, but that of copper did not change under any condition of annealing. Although SPD has been used to produce high-strength metals (e.g., 9–12)), the greater part of the flow stress observed in standard tensile tests is the time-dependent thermal strength that had emerged during SPD processes.^7,8) SPD produces ultrafine crystal grains, but does not necessarily yield a high permanent strength. The authors^7,8) have investigated the permanent strengths of only aluminum and copper so far. It is expected that the ratio of permanent strength to flow stress observed in a standard tensile test largely varies from material to material.

In this study, the permanent strengths of interstitial-free (IF) steel samples with different grain sizes and dislocation densities processed by SPD and subsequent annealing were systematically investigated. We emphasize the importance of the direct evaluation of the permanent strength of structural metals in the design and construction of a broad range of structures that must bear stresses for a long time.

2. Concept of the Present Study

The basic concept of the present study is the same as that in Refs. 7), 8), and we describe its essence briefly. The flow stress σ temporarily observed in a standard tensile test is assumed to be decomposed into the time-independent athermal component σ_i and the time-dependent thermal component σ^*.¹⁾   

$$\begin{array}{c}\sigma ={\sigma }_{\text{i}}+{\sigma }^{*}.\end{array}$$(1)

The component σ_i plays a role as the fundamental stress bearing capability independent of time and temperature, and thus we call it the permanent strength. Generally, in the design of structures, an allowable stress σ_a is predetermined by the yield stress at a plastic strain of 0.2% (referred to as σ_0.2, usually determined with a standard tensile test at a strain rate of 10⁻⁴–10⁻² /s) divided by a safety factor f that is properly chosen under various assumptions. Such a yield stress σ_0.2 consists of σ_i and σ^* as shown in Eq. (1). In general structural designs, it has been assumed that σ_i is not much smaller than σ_0.2, i.e., σ_0.2 ≈ σ_i. Thus, it has been implicitly considered that σ_i > σ_a = σ_0.2/f and no inelastic deformation occurs at an applied stress below σ_a. Therefore, the question regarding the real value of σ_i has been seldom raised as the main issue, and little attention has been paid to the direct measurements of σ_i. However, if σ_i < σ_a, an applied stress even below σ_a causes inelastic deformation and the premise that ordinary deformation is within the elastic limit does not hold. As shown in Refs. 7), 8), for aluminum and copper, σ_i varies significantly depending on the material and processing and heating histories. Thus, we should know the permanent strengths of all kinds of structural material to guarantee the fundamental safety of structures.

In the present study, as in Refs. 7), 8), the stress relaxation test for a long-time duration is adopted for the evaluation of σ_i. The remaining stress after stress relaxation for an infinite time is defined as σ_i. In practice, sufficiently long-time (24 h) stress relaxation tests are conducted and the remaining stress–relaxation time relationship recorded is extrapolated to estimate σ_i at infinite time passed.⁷⁾ Since stress relaxation tests are carried out at room temperature, a change in the material microstructure due to variations in temperature is expected to be excluded. In the experimental investigation in this paper, using the stress relaxation tests, flow stresses σ observed in tensile tests at standard and ultralow strain rates are decomposed into σ_i and σ^*.

3. Experimental Procedure

3.1. Material

Samples were made of IF steel rods¹ with a diameter of 10 mm and a length of 60 mm. The chemical composition is shown in Table 1.

Table 1. Sample composition (wt%).

C	Si	Mn	P	S	Al	Ti	Fe
< 0.001	< 0.003	< 0.003	< 0.002	< 0.003	0.028	0.051	Bal.

¹  Laboratory samples provided by Nippon Steel Corporation.

3.2. Equal Channel Angular-pressing (ECAP)

Samples were prepared using the ECAP process,^13,14) which is a type of SPD process. The equivalent strain ε^ECAP applied to a sample is approximately expressed by the following relation:¹⁴⁾   

$$\begin{array}{c}{\epsilon }^{\text{ECAP}}=\frac{n}{\sqrt{3}}\left\{2\text{cot}\left(\frac{\varphi }{2}+\frac{\psi }{2}\right)+\psi \text{cosec}\left(\frac{\varphi }{2}+\frac{\psi }{2}\right)\right\},\end{array}$$(2)

where n is the number of ECAP passes, ϕ (= 90 deg for the die used in this study) is the inner angle of the bend, and ψ (= 36.87 deg) is the outer angle of the bend. The nominal diameter of the channel was 10 mm. The starting sample before ECAP processing is designated as sample “0 passes”. According to Eq. (2) with the present values of ϕ and ψ, ε^ECAP per one ECAP pass is about 1.0.

ECAP processing was performed using a hydraulic universal testing machine (Shimadzu UH-500kN) with a crosshead speed of 30 mm/min at room temperature. Molybdenum disulfide paste (SUMICO Moly Paste 500) was used as the lubricant. The ECAP procedure called “Route Bc”¹⁵⁾ was adopted, in which the sample was rotated 90° around its longitudinal axis at each pass, to obtain equiaxed crystal grains. Schematic diagrams for the ECAP die and the material coordinate system are respectively shown in Figs. 1(a) and 1(b).

Fig. 1.

Schematic diagrams for experimental procedure. (a) Die used for ECAP processing; (b) Sample coordinates for ECAP processing; (c) Methods for controlling the strain rate during the tensile and stress relaxation tests; (d) Shape and dimensions of the specimen for the material tests.

3.3. Test Samples

Two groups of samples were prepared following Refs. 7), 8), 16). The first group consisted of samples with grains successively refined by repeated ECAP processes. The second group consisted of samples first processed by ECAP with 16 passes and subsequently annealed step by step at various temperatures for different durations to produce coarser grains. Henceforth, the first group is referred to as samples in the “refining approach” and the second group is referred to as samples in the “coarsening approach”. In the coarsening approach, two-step annealing^11,17) aiming at suppressing the local growth of crystal grains was performed. Table 2 shows the designation of the samples. Details of measurements of grain size and dislocation density will be given in subsections 3.4 and 3.5, respectively.

Table 2. Designations and details of samples.

Initial material (If steel)
Samples
Refining approach (grain size: decrease)			Coarsening approach (grain size: increase)
Name	Average grain size d [μm]	Average dislocation density ρ [×1014 m−2]	Name	Average grain size d [μm]	Average dislocation density ρ [×1014 m−2]
0 passes	345	–	16 passes-O1 (400°C-0.5 h)	1.07	0.686
1 pass	128	2.771	16 passes-O2 (400°C-6 h)	1.14	0.188
2 passes	8.25	2.640	16 passes-O3 (400°C-6 h + 500°C-0.5 h)	2.75	0.005
4 passes	13.8	3.232	16 passes-O4 (400°C-6 h + 550°C-0.5 h)	6.21	–
8 passes	0.86	3.259	16 passes-O5 (400°C-6 h + 600°C-0.5 h)	45.6	–
12 passes	0.81	3.239	16 passes-O6 (400°C-6 h + 750°C-0.5 h)	325	–
16 passes	0.80	2.739	–	–	–

3.4. Electron Backscattered Diffraction Pattern (EBSD) Measurements

Electron backscattered diffraction pattern (EBSD) measurements were carried out to determine the average grain size d of each sample (hereafter simply referred to as the ‘‘grain size’’). Plate-shaped specimens (5 mm × 5 mm) for EBSD measurements were fabricated by wire-cut electrical discharge machining along the extrusion direction–transverse direction (ED–TD) plane at the center of the specimen diameter. The observation surfaces were mirror-finished by mechanical polishing with abrasive-coated papers of #600, #800, #1000, #1500, and #2000 followed by buff polishing with an alumina suspension (particles with diameters of 1 μm and 0.3 μm) and electropolishing with an electrolyte (methanol:perchloric acid = 10:1) at −40°C and 15 V for 120 s. For the EBSD measurements, a field emission scanning electron microscope (FE-SEM) (JEOL JSM-7200F) with orientation imaging microscopy (OIM) (EDAX DigiView 3) was used, and EDAX TEAM and OIM analysis 8 were respectively used for data acquisition and analysis. To evaluate the grain size by OIM analysis, a threshold for the misorientation angle between orientations of neighboring grains must be specified. While an angle of 5 deg is generally used, an angle of 2 deg has often been used for as-SPD-processed metals in previous studies.^18,19) In this study, an angle of 2 deg was used for samples in the refining approach, while an angle of 5 deg was used for samples in the coarsening approach. EBSD data with a confidence index (CI) of less than 0.1 were excluded before evaluating the grain size.

3.5. X-ray Diffraction (XRD) Measurements

The dislocation density of the specimens was evaluated on the basis of XRD line profiles. Plate-shaped specimens with dimensions of 20 mm in ED and 10 mm in TD were cut from the rod samples, and the observation surfaces were mirror-finished similarly to the samples for the EBSD measurements. The reflection peaks for (110), (200), (211), (220), and (310) were measured using an XRD instrument (Bruker D8 ADVANCE) with a scintillation counter at a sampling interval of 0.01 deg, a scan rate of 0.025 deg/min, and a sample rotation speed of 15 /min. Using the full width at half-maximum (FWHM) of the reflection peaks, we calculated the dislocation density by the Williamson–Hall plot method²⁰⁾ with correction for the effects of elastic anisotropy in Fe crystals²¹⁾ (calculation method is described in Refs. 16), 22)).

3.6. Uniaxial Tensile Test

Uniaxial tensile tests were conducted at an ultralow strain rate ($\dot{\epsilon }$ = 1×10⁻⁶ /s) and a standard strain rate ($\dot{\epsilon }$ = 1×10⁻² /s) using a universal testing machine (Shimadzu Autograph AG-X plus 300 kN). All the tests were carried out at room temperature. Strains were measured using a pair of wire-strain gauges (TML FLA-2-11) placed on opposite positions on the specimen surfaces. The two gauges were connected in the form of an opposite arm half-bridge with two 3-wire active gauges. In the low-strain-rate tests ($\dot{\epsilon }$ = 1×10⁻⁶ /s), the strain rate was imposed by feedback control using the strain gauges up to the first 7200 s. After then, the strain rate was controlled by a crosshead speed of 0.00156 mm/min. In the standard-strain-rate tests ($\dot{\epsilon }$ = 1×10⁻² /s), the strain rate was imposed by feedback control using the strain gages throughout the test. Schematic diagrams of the strain rate control method and specimen dimensions are respectively shown in Figs. 1(c) and 1(d).

3.7. Stress Relaxation Tests and Evaluation of Permanent Strength

Stress relaxation tests were conducted at logarithmic tensile strains ε of 0.005, 0.01 and 0.015 for 24 h each at room temperature (except for samples “16 passes-O4, -O5 and -O6” for which the stress relaxation test was carried out only at ε of 0.01). The strain rate during tensile loading towards each strain level was set to be $\dot{\epsilon }$ = 5×10⁻⁴ /s. (except for samples “16 passes-O4, -O5 and -O6” for which the strain rate was $\dot{\epsilon }$ = 1×10⁻⁶ /s). In order to keep the strain constant during the stress relaxation tests, the feedback control using the strain gages was used. The equipment and conditions were the same as those used in the uniaxial tensile tests. To determine σ_i, which is the remaining stress after an infinite time duration during the stress relaxation test, experimental curves of stress versus relaxation time were fitted to the following function:   

$$\begin{array}{c}\sigma ={\sigma }_{\text{i}}-A_1{e}^{-\frac{t}{x_1}}-A_2{e}^{-\frac{t}{x_2}}-A_3{e}^{-\frac{t}{x_3}},\end{array}$$(3)

where σ is the flow stress during the relaxation test, t is the relaxation time, and A_j and x_j are fitting parameters. It was confirmed in Ref. 8) that Eq. (3) provides good fitting results for σ–t relations for SPD-processed FCC (face-centered cubic) metals.

4. Results and Discussion

4.1. Grain Size and Dislocation Density

Examples of inverse pole figure (IPF) maps obtained with the EBSD measurements are shown in Fig. 2. The grain size after the first ECAP pass was still large (d = 128 μm), and variations in crystal orientations in each grain (i.e., a typical deformation structure) are observed. After 16 ECAP passes, the grain size was reduced to 0.80 μm. In the annealed specimens, the grains became coarser and there was no crystal orientation variation in each grain. Samples “16 passes-O2” (Fig. 2(d)) and “16 passes-O4” (Fig. 2(e)) with slightly different heat treatment histories had very different textures as seen in EBSD maps. For sample “16 passes-O4”, IPF maps of four randomly chosen locations were confirmed to be very similar. It has been suggested that certain orientations may be preferentially recrystallized under the influence of the deformation structure during the formation of recrystallized aggregate structure in pure iron that has been subjected to severe cold rolling.²³⁾

Fig. 2.

Examples of EBSD crystal orientation maps for samples of (a) 0 passes, (b) 1 pass, (c) 16 passes, (d) 16 passes-O2, (e) 16 passes-O4 and (f) 16 passes-O6. The average grain sizes for all the samples are shown in Table 2. (Online version in color.)

The calculated dislocation densities are plotted with respect to the grain size in Fig. 3. In the refining approach, the dislocation density was nearly constant after 1 ECAP pass. In the coarsening approach, the dislocation density in sample “16 passes-O1” decreased to about a quarter of that in sample “16 passes”. For samples “16 passes-O4”, “16 passes-O5”, and “16 passes-O6”, the dislocation densities could not be determined from the measured XRD line profiles. This is likely because the dislocation density is too low to be captured by the Williamson–Hall plot method.

Fig. 3.

Relationship between dislocation density and grain size. Dislocation density, ρ, are shown in Table 2. (Online version in color.)

4.2. Effect of Loading Strain Rate on Flow Stress in Uniaxial Tensile Tests

Experimental curves of true stress (σ) versus logarithmic strain (ε) at the two different imposed strain rates in the refining approach are shown in Fig. 4 (resulting strain rates calculated from the recorded data are shown in Fig. S1). In the ultralow-strain-rate tests ($\dot{\epsilon }$ = 1×10⁻⁶ /s; Fig. 4(a)), σ_0.2 of sample “1 pass” increased 5.9 times compared with that of sample “0 passes”, where σ_0.2 is the yield stress at a plastic strain of 0.2%, which corresponds to an intersection between a σ–ε curve and a line from the origin with a slope of the Young’s modulus (206 GPa, a general value for steel with neglect of the crystallographic texture effects). Subsequently, σ_0.2, as well as σ, gradually increased with repeated ECAP processes and saturated at 12 ECAP passes. The grain size, which was 128 μm after 1 ECAP pass, decreased to 0.80 μm after 16 ECAP passes, while σ_0.2 increased only by a factor of 1.6 at most. In the standard-strain-rate tests ($\dot{\epsilon }$ = 1×10⁻² /s; Fig. 4(b)), σ increased stepwise with repeated ECAP processes. In comparison with the results for $\dot{\epsilon }$ = 1×10⁻⁶ /s, the difference in σ_0.2 widens after 2 ECAP passes, indicating that the repeated ECAP processes augmented the strain rate sensitivity. For aluminum⁷⁾ and copper,⁸⁾ in the ultralow-strain-rate tests ($\dot{\epsilon }$ = ~10⁻⁷ /s), an inversion phenomenon in which σ of a sample with more ECAP passes was smaller than that of a sample with fewer ECAP passes was observed, but this was not the case for the IF steel used in this study.

Fig. 4.

Experimental curves of true stress versus logarithmic strain at (a) $\dot{\epsilon }$ = 1×10⁻⁶ /s and at (b) $\dot{\epsilon }$ = 1×10⁻² /s in the refining approach in which samples underwent only ECAP processes. (Online version in color.)

Experimental curves of true stress (σ) versus logarithmic strain (ε) for different strain rates in the coarsening approach are shown in Fig. 5 (resulting strain rates calculated from the recorded data are shown in Fig. S2). In the ultralow-strain-rate tests ($\dot{\epsilon }$ = 1×10⁻⁶ /s; Fig. 5(a)), σ decreased stepwise from 16 ECAP passes with the progress of annealing treatments. The same trend is observed in the case of the standard-strain-rate tests (at $\dot{\epsilon }$ = 1×10⁻² /s; Fig. 5(b)).

Fig. 5.

Experimental curves of true stress versus logarithmic strain at (a) $\dot{\epsilon }$ = 1×10⁻⁶ /s and (b) $\dot{\epsilon }$ = 1×10⁻² /s in the refining approach in which samples first underwent 16 ECAP passes followed by different annealing treatments as specified in Table 2. (Online version in color.)

4.3. Separation of Permanent and Temporal Strengths

Results of stress relaxation tests for the refining and coarsening approaches are shown in Fig. 6 (resulting strain rates calculated from the recorded data are shown in Supplementary Fig. S3). Relationships between remaining stress ratio and relaxation time with their curve-fitting results are shown in Fig. 7. The remaining stress ratio is defined as σ during relaxation divided by that at the beginning of relaxation. In Fig. 7, only the results of stress relaxation tests at ε = 0.01 are shown. Results for the other conditions and the corresponding curve fitting results are shown in Fig. S4, Fig. S5, and Table S2. In the refining approach (Fig. 6(a)), the remaining stress ratio lowered as the number of ECAP passes increased. The lowest remaining stress ratio was observed for samples “16 passes” and “12 passes”, which was 0.72, and thus about 30% of flow stress disappeared owing to stress relaxation. In the coarsening approach (Fig. 6(b)), the remaining stress ratio was generally maintained above 90% for all the annealing conditions. This indicates that stress relaxation can be suppressed by annealing.

Fig. 6.

Experimental curves of true stress versus logarithmic strain with stress relaxation tests at logarithmic strains of 0.005, 0.01, and 0.015 for 24 h each: (a) refining approach; (b) coarsening approach. (For details, see Table S3). Dashed lines indicate curve fitting results for σ_i predicted as asymptotic values after infinite time duration at the three different strain values (indicated by solid circles). (Online version in color.)

Fig. 7.

Relationships between remaining stress ratio and relaxation time during stress relaxation test at a logarithmic strain of 0.01: (a) refining approach; (b) coarsening approach. Dashed lines indicate curve fitting results obtained using Eq. (3). (Online version in color.)

The dashed lines in Fig. 7 show the curve fitting results obtained using Eq. (3) for the stress relaxation test results. All convergence calculations for the curve fitting were performed using the Levenberg–Marquardt method.^24,25) The evaluation of σ_i was based on the assumption that the material microstructure did not change significantly during stress relaxation.^7,8) We compared the stress value observed in the ultralow-strain-rate tensile tests and that observed in the stress relaxation tests at the moments when the same plastic strain rate occurred. The relative difference (RD) between the stress values in the tensile and relaxation tests at the same plastic strain rate was mostly within ±10% (Supplementary Fig. S6), where RD = (σ_relax − σ_tens)/σ_tens with σ_relax and σ_tens being the stresses observed in the relaxation and tensile tests, respectively. That is, if the observed flow stresses σ are the same, the same plastic strain rate occurs in both uniaxial tensile and stress relaxation tests. Furthermore, σ in each reloading process after each stress relaxation test returned to almost the same level as that before relaxation (see Fig. 6). Therefore, we find no serious inconsistency with the assumption that no significant change in the material microstructure occurred during stress relaxation.

The decomposition of the flow stress at 0.2% plastic strain, σ_0.2 (0.2% proof stress), measured in the uniaxial tensile tests into σ_i and σ^* is shown in Fig. 8. The difference between σ^* at $\dot{\epsilon }$ = 1×10⁻⁶ /s (blue bars) and that at $\dot{\epsilon }$ = 1×10⁻² /s (red bars) represents the extent of the strain rate sensitivity. To quantify the strain rate sensitivity, we adopt a simple power law ${\sigma }^{*}=K{\dot{\epsilon }}^m$. Calculated values of m are shown in Fig. 8. For several samples, the variation of σ stopped immediately after the start of the relaxation test. In those cases, σ_i was taken as σ at 24 h (for samples “0 passes” and “16 passes-O3”) or 20 h (for samples “16 passes-O5” and “16 passes-O6”). In the refining approach, σ_i at 0 ECAP passes increased five times after the first ECAP process. Then, σ_i increased gently with subsequent ECAP processes and σ_i was nearly constant after 8 ECAP passes. Thus, σ_i increased with repeated SPD processes in the case of IF steel, unlike the behavior of aluminum and copper that exhibited a significant decrease in σ_i with repeated SPD processes.^7,8)

Fig. 8.

Breakdown of tensile flow stress σ at a plastic strain of 0.002 (0.2% proof stress, σ_0.2) into athermal component (permanent strength) σ_i and thermal component (temporal strength) σ^* for uniaxial tension tests at $\dot{\epsilon }$ = 1×10⁻⁶ /s and $\dot{\epsilon }$ = 1×10⁻² /s for If steel. “P” stands for “pass” or “passes”. Values of strain rate sensitivity, m, have been evaluated assuming the simple relation ${\sigma }^{*}=K{\dot{\epsilon }}^m$. (Online version in color.)

In the coarsening approach, σ_i of sample “16 passes-O1” increased by 13% relative to that of as-ECAP-processed sample “16 passes”, indicating hardening by annealing. Significant hardening by annealing with a fivefold increase in σ_i was observed in aluminum,^7,8) while no such hardening was observed in copper.⁸⁾ In SPD-processed metals with fine grains and high dislocation density, hardening by annealing could occur since the heat treatment would reduce the number of mobile dislocations in the grains and the remaining dislocations would form clusters near grain boundaries.²⁶⁾ In fact, in this study, the dislocation density was significantly reduced after low-temperature annealing (Fig. 3). This reorganization of the internal microstructure may suppress the thermal activation of dislocation motion and contribute to the increase in σ_i. Hardening by annealing was maintained up to “16 passes-O2”. With further annealing, recovery of the internal microstructure and/or recrystallization occur. Consequently, both σ_0.2 and σ_i decreased significantly in the samples further annealed. The sample with the highest σ_i was “16 passes-O2”, accompanied by the significant reduction of the strain rate sensitivity. In the case of IF steel, the combination of SPD processing and low-temperature annealing is effective for developing high strength.

The relationship between m and the stress relaxation ratio is shown in Fig. 9. The stress relaxation ratio is defined as (1– remaining stress ratio) at infinite time and is obtained as the average value of the stress relaxation test results for the different values of ε. No significant correlation was found between m and stress relaxation ratio with the correlation coefficient r of −0.424. The maximum stress relaxation ratio was 29% for sample “16 passes”. In the cases of copper and aluminum, positive correlations between m and stress relaxation ratio were observed: i.e., r = 0.637 for copper and r = 0.786 for aluminum,⁸⁾ and the maximum stress relaxation ratios reached 50% for copper and 80% for aluminum, which were much greater than that for IF steel. There is no significant variation in m values from the samples in the refining approach to sample “16 passes-O4” in the coarsening approach. Although the m values for samples “1 pass” and “16 passes” were nearly identical, their stress relaxation ratios were very different. Values of σ_0.2 and σ_i for $\dot{\epsilon }$ = 1×10⁻⁶ /s are similar to each other at 1 ECAP pass, but they are markedly different at 16 ECAP passes. The condition σ_0.2 ≈ σ_i does not hold at 16 ECAP passes, and the observed flow stress includes a non-negligible magnitude of σ^*. Previous studies suggested that stress relaxation behavior in SPD material with a large strain rate dependence might be strongly related to grain boundary sliding (e.g., 27,28)) and it is possible that grain boundary sliding is enhanced at a low strain rate (e.g., 29–33)). Samples with grain refinement promoted by repeated SPD processes are considered to contain both an increase in mobile dislocation density^{34,35,36,37,38,39)} and the effect of grain boundary sliding. This may be a reason for the increase in σ^* at $\dot{\epsilon }$ = 1×10⁻⁶ /s in the samples with 12 and 16 ECAP passes.

Fig. 9.

Relationship between strain rate sensitivity (m) and stress relaxation ratio for IF steel. The “stress relaxation ratio” is defined as (1– remaining stress ratio) at infinite time duration (using extrapolations with Eq. (3)) and was calculated as an average of values obtained in the three stress relaxation tests at logarithmic strains of 0.005, 0.01, and 0.015 for most samples (exceptions are sample “1 pass” with only two relaxation test results forε = 0.005 and 0.01, and “16 passes-O4–O6” with only one stress relaxation test result forε = 0.01). (Online version in color.)

Samples “16 passes-O5” and “16 passes-O6” exhibited m values greater than 0.3, which are significantly larger than those of the other samples. In both the samples, annealing at temperatures above the recrystallization point (600°C and 750°C) coarsened the grain (d > 45 μm), and no lattice distortion inside the grains is seen (Fig. 2(f)). Michalak⁴⁰⁾ produced pure iron (99.96 wt% purity) with an average grain size of 100 μm by recrystallization at 700°C after cold working, decomposed the tensile flow stress σ into σ_i and σ^* using Eq. (1), and observed m of about 0.2. Thus, the reproducibility of large m values for well recrystallized iron with high purity (including IF steel) is confirmed. It is well known that m at room temperature decreases as the purity of Fe decreases owing to alloying.^{40,41,42,43,44)}

In general, accumulation of lattice defects such as dislocations in a plastically deformed metal causes the increase in internal energy and leads to the unstableness of microstructures. Annealing releases the internal energy and induces recrystallization, which stabilizes the microstructures. The internal energy of metals processed by SPD is generally higher than that of general metals produced by conventional processing methods. Thus, recrystallization would be promoted faster in metals processed by SPD than in conventionally processed metals. In fact, the grain sizes of samples “0 passes” and “16 passes-O6” are almost identical (~300 μm), but σ_i of the latter is clearly smaller than σ_i of the former. The σ_i of sample “16 passes-O6” is almost identical to the frictional stress σ₀ (52.3 MPa: the average value of σ₀ in Refs. 45,46,47,48,49,50)) of ultralow-carbon steel, which can be considered to have fewer dislocations than sample “0 passes”. These views suggest that the increase in the m-values of samples “16 passes-O5” and “16 passes-O6” is a direct consequence of the viscous flow characteristics of mobile dislocations in iron.

4.4. Effects of Grain Size on Permanent Strength

It has been generally believed that grain refinement promoted by SPD increases permanent strength, uncritically supposing Hall–Petch-like effect. This view is not always true, as shown in Fig. 8. Samples “1 pass” and “16 passes” had the extremely different grain sizes of 128 μm and 0.60 μm, respectively, but, the difference in the permanent strength σ_i between these two samples was small (Fig. 8). As clearly demonstrated for aluminum and copper in Refs. 7), 8), Hall–Petch relation^51,52) does not indicate “increase in strength due to grain refinement”, but only represents “reduction in strength due to grain coarsening by annealing in a coarse-grain region (generally above 5–10 μm) ”. Furthermore, the original Hall–Petch relation was intended to be applied to the rate-independent strength σ_i; however, in a tremendous amount of existing research (e.g., 11,12,16,53–64)), Hall–Petch-like relations have been erroneously applied to the total stress (i.e., flow stress) σ, causing much confusion.

In this section, we further decompose σ_i into the dislocation-related strength component σ_ρ and the grain-size-related strength component σ_gs to examine the genuine grain size effects. Strictly, σ_gs may involve or should be separated into the grain size and grain boundary effects. We cannot quantitatively separate these two at present. The permanent strength at a plastic strain of 0.2% (here written as σ_i(0.2)), which has been quantified in Fig. 8, is further decomposed into σ_ρ and σ_gs⁶⁵⁾ as   

$$\begin{array}{c}{\sigma }_{\text{i}\left(0.2\right)}={\sigma }_{\text{gs}}+{\sigma }_{\rho }.\end{array}$$(4)

Although the friction stress σ₀ (the initial yield stress of an ultracoarse grain material) frequently appears in Eq. (4) separately, here it (52.3 MPa for IF steel, which is the average σ₀ taken from Refs. 45,46,47,48,49,50), Table S4) is included in σ_gs for a simple expression. The classical Taylor’s law^66,67) is used to evaluate σ_ρ:   

$$\begin{array}{c}{\sigma }_{\rho }=M\alpha \mu b\sqrt{\rho },\end{array}$$(5)

where M is the Taylor factor, which is taken to be 2.75⁶⁸⁾ neglecting crystallographic texture effects, μ is the shear modulus (79 GPa for iron), b is the magnitude of the Burgers vector (0.249 nm for iron⁶⁹⁾), ρ is the measured dislocation density shown in Table 2, and α is a nondimensional coefficient often taken to be from 0.2 to 0.7. The procedure for determining α in Eq. (5) is as follows. At the first ECAP pass, σ_i(0.2) was sufficiently large despite the coarse grain size of 128 μm, so it is assumed that the strength was attributable almost entirely to dislocation-related strength. We assume that σ_gs in the coarse grain size range does not deviate greatly from the classical Hall–Petch relation. Using the Hall–Petch coefficient $\overline{k}$ = 143.1 MPa μm^1/2, which is the average k taken from Refs. 45,46,47,48,49,50) (Table S4), we assume that the relation σ_i(0.2) = σ_gs + σ_ρ = 52.3 + 143.1d^−1/2 + Mαμb$\sqrt{\rho }$ [MPa] holds for sample “1 pass”. From this relation, α was determined as 0.38 using the grain size d (= 128 μm), σ_i(0.2), and ρ calculated with the data from XRD measurements. The same value of α was used in data analysis for both the refining approach and the coarsening approach.

Figure 10 shows σ_i(0.2) and its decomposition into σ_gs and σ_ρ with the grain size d. Substituting the dislocation density of each sample and the already determined α (= 0.38) into Eq. (5) gives σ_ρ. Subsequently, σ_gs was calculated by substituting σ_ρ and σ_i(0.2) into Eq. (4). For the specimens whose ρ could not be determined owing to the very low dislocation density (i.e., “0 passes”, “16 passes-O4” to “-O6”), we assume σ_gs = σ_i(0.2) and σ_ρ = 0.

Fig. 10.

Results of breakdown of σ_i at a plastic strain of 0.2% (i.e., σ_i(0.2)) into grain-size-related strength σ_gs and dislocation-related strength σ_ρ plotted versus the reciprocal of the square root of d for IF steel: (a) refining approach; (b) coarsening approach. (Online version in color.)

In the refining approach (Fig. 10(a)), σ_ρ is dominant for all the conditions except for sample “0 passes”. The thick blue line in Fig. 10(a) shows the Hall-Petch line computed for σ_gs. The “Hall–Petch zone” indicates a region determined from the maximum and minimum of σ₀ and k values taken from Refs. 45,46,47,48,49,50) (Table S4). The calculated Hall–Petch coefficient k of 80.0 MPa μm^1/2 for σ_gs is much smaller than the $\overline{k}$ reported in the literature^{45,46,47,48,49,50)} (Table S4). Although the grain refinement was significantly promoted by repeated ECAP processes, the increase in σ_gs was gentle. The grain refinement itself does not largely contribute to the increase in σ_gs. Note that, in the cases of copper and aluminum, the Hall-Petch coefficients for σ_gs were negative.⁸⁾

In the coarsening approach (Fig. 10(b)), the σ_gs values of samples “16 passes-O1” and “16 passes-O2” increased more than two times from that of sample “16 passes”. The grain-size-related strength σ_gs occupies a majority of σ_i(0.2) for samples “16 passes-O1” and “16 passes-O2”. The σ_gs values of samples “16 passes-O3” to “-O6” gradually approach the Hall–Petch zone, accompanied by grain coarsening. It is confirmed that the Hall–Petch relation is consistent with the reduction in strength associated with grain coarsening during annealing process. The same is true for copper and aluminum as shown in Ref. 8).

Tanaka et al.⁷⁰⁾ proposed a new strengthening law in which the strength of steel is not based on the conventional additive law of Eq. (4), but, the dislocation-related strengthening (i.e., Taylor’s law) or the grain size-related strengthening (conventional Hall-Petch relation) functions alternatively, depending on the conditions of work hardening and accumulation of dislocations near the grain boundary. In our data, as seen in Fig. 10(a), σ_i(0.2) in the refining approach mainly consisted of σ_ρ, but contained non-negligible σ_gs. On the other hand, in the coarsening approach (Fig. 10(b)), σ_gs dramatically increased after low-temperature annealing, but non-negligible σ_ρ still remained. The proposed switching law⁷⁰⁾ that accounts for the strengthening either by the Taylor relation or alternatively by the Hall–Petch relation would be satisfactory as the first approximation, which is rather a rough quantitative approximation.

5. Conclusions

In this study, the permanent strength of IF steel processed by SPD and subsequent annealing was systematically investigated. Repetition of SPD processes accompanied by the increase in the dislocation density and the decrease in grain size contributes to the increase in the permanent strength σ_i, unlike in the cases of aluminum and copper exhibiting a significant decrease in σ_i for repeated SPD processes.⁸⁾ The main conclusions are as follows.

(1) The flow stresses σ of IF steel observed in tensile tests at ultralow and standard strain rates both monotonically increased with repeated SPD processes, unlike the σ of aluminum and copper at an ultralow strain rate that decreased with SPD.⁸⁾ From 1 to 16 ECAP passes, the fraction of the permanent strength σ_i to the total flow stress σ was mostly above 65% (Fig. 8).

(2) The permanent strength σ_i increased with repetition of SPD processes and its maximum was reached at the 16th ECAP pass in the refining approach. The σ_i values of samples “16 passes-O1” and “16 passes-O2” were higher than that of sample “16 passes”. In the case of IF steel, a relatively high σ_i could be acquired only with SPD processing and a further increase in σ_i can be gained by subsequent annealing. The former behavior is very different from that in aluminum and copper, which exhibited a significant reduction in σ_i during repeated SPD processes.⁸⁾

(3) In the refining approach, the dislocation-related strength σ_ρ was dominant, while the grain-size-related strength σ_gs was minor. A straight-line approximation of σ_gs versus d^−1/2 showed the Hall-Petch coefficient k of 80 MPa μm^1/2, which was 56% of $\overline{k}$ of 143 MPa μm^1/2 for a coarse grain region (d > 18 μm: this threshold d value is the average of the minimum grain sizes shown in Refs. 45,46,47,48,49,50)), as shown in Fig. 10(a). On the other hand, in the coarsening approach, in samples “16 passes-O1” and “16-passes-O2”, σ_gs became quickly dominant as if σ_gs replaced σ_ρ (Fig. 10(b)). In the conventional coarse grain region (d > 10 μm), σ_gs was consistent with the classical Hall–Petch relation (Fig. 10(b)). Thus, it was also confirmed for IF steel, as well as for aluminum and copper,⁸⁾ that the original Hall–Petch relation holds only in the coarse grain region and indicates softening with grain coarsening due to annealing, not strengthening by grain refinement due to SPD.

Acknowledgement

The authors are grateful for the support from Nippon Steel Corporation including the provision of IF steel samples. T. K. is grateful for the support provided by JSPS KAKENHI Grant Number JP21K14051. The authors acknowledge Polytechnic University of Japan, where T. K. previously worked, for the use of FE-SEM with OIM and universal testing machines.

Supporting Information

This material, which contains Figs. S1–S6 and Tables S1–S4, is available on the website at https://doi.org/10.2355/isijinternational.ISIJINT-2022-328.

Data Availability

The data that support the findings of this study are available from the corresponding author on reasonable request.

Authors’ Contributions

T. K. and M. K. conceived the project and wrote the paper together. T. K. and T. T. performed the experiments. T. K. and M. K. performed the detailed analysis of the experimental data. All authors discussed the results and contributed to the manuscript.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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