Grain size and temperature effect on the tensile behavior and deformation mechanisms of non-equiatomic Fe₄₁Mn₂₅Ni₂₄Co₈Cr₂ high entropy alloy

1. Introduction

High-entropy alloys (HEAs) are composed of multiple components added far beyond their binary solid solubility limits, yet they form a single-phase solid solution [1,2]. In recent years, they have attracted considerable interest due to the novelty of their promising mechanical properties. Among them, the HEAs composed of Co, Cr, Fe, Mn, and Ni have been widely studied because of their excellent mechanical properties, such as fracture toughness at room and cryogenic temperatures [[3], [4], [5], [6], [7], [8], [9], [10]]. Compared to the amount of studies on the mechanical properties of HEAs at low temperatures, their high-temperature properties have been relatively less studied [[11], [12], [13], [14], [15]].

He et al. [11] conducted strain-rate-change (SRC) tests in tension on an equiatomic CoCrFeNiMn HEA with a grain size of 12 μm, which was prepared by cold rolling followed by annealing at 1073 K for 1 h, in the temperature range between 1023 and 1123 K and in the strain rate range between 6.4 × 10^-7 and 8 × 10^-4 s^-1. They observed stress exponents (n) of 5 and 3 at high strain rates and low strain rates, respectively, and interpreted this result as due to dislocation-climb creep and solute-drag creep, respectively, as the dominant deformation mechanisms. Fu et al. [12] studied the tensile behavior of a CoCrFeNiMn alloy with a grain size of 25 μm in the temperature range between room temperature (RT) and 1073 K at a given strain rate of 10^-3 s^-1. They observed a decrease in the strength, strain hardening exponent and tensile fracture with increasing temperature. Jang et al. [13] studied the hot tensile deformation behavior of a CoCrFeNiMn alloy with a grain size of 167.6 μm in the temperature range between 973 and 1273 K at a given strain rate of 10^-3 s^-1. They observed that the flow stress drastically decreased, and the tensile ductility sharply increased above 1173 K due to the occurrence of dynamic recrystallization. Reddy et al. [14] reported obtaining a very fine grain size of 1.4 μm in a CoCrFeMnNi alloy using routes composed of cold rolling, annealing and warm rolling in sequence. They conducted SRC tests at 1023 K in the strain rate range between 10^-4 and 10^-2 s^-1. The alloy showed n = 2 at low strain rates and a tensile elongation of 300 %. Based on these observations, the authors suggested that grain-boundary-diffusion-controlled grain boundary sliding (GBS) is the rate controlling deformation mechanism at the corresponding temperature. Otto et al. [15] studied the temperature dependency of the tensile properties of a CoCrFeMnNi alloy with different grain sizes of 4.4, 50 and 155 μm in the temperature range between 77 and 1073 K at a given strain rate of 10^-3 s^-1. The strength increased as the grain size decreased at all temperatures and fracture strain tended to decrease as temperature increased.

The reviewed works above studied the effect of temperature, strain rate and grain size on the flow stress and tensile ductility, but the effect of grain size on the deformation mechanism at high temperatures (above 973 K), where a steady state is attained, has not been systematically studied. To understand the effect of grain size on the deformation behavior and mechanism at high temperatures, the data obtained from a series of temperatures and strain rates at different grain sizes are required. In this work, we studied the effect of grain size on the deformation behavior and mechanism of a nonequiatomic Fe₄₁Mn₂₅Ni₂₄Co₈Cr₂ HEA by preparing the samples with different grain sizes by annealing the Fe₄₁Mn₂₅Ni₂₄Co₈Cr₂ HEA processed by severe plastic deformation via high-ratio differential speed rolling (HRDSR). The Fe₄₁Mn₂₅Ni₂₄Co₈Cr₂ HEA consists of a single FCC solid solution phase and exhibits high thermodynamic stability as a single phase [16]. The thermodynamic stability of a single phase is important when developing a fundamental understanding of the deformation mechanism as a function of grain size because the strain rate and flow stress can be affected by the presence of secondary solid solution phases and/or intermetallic compounds, of which the volume fraction may change with temperature.

2. Experimental methods

A HEA ingot with a nominal composition of Fe₄₁Mn₂₅Ni₂₄Co₈Cr₂ was produced by induction melting under vacuum. After homogenization at 1373 K for 13 h, the ingots were hot forged and then rolled to a 3 mm thickness at 1273 K. The rolled samples were homogenized at 1473 K for 2 h and then cooled by water quenching. The homogenized samples were subjected to HRDSR (with a speed ratio of 2) to a final thickness of 0.7 mm at room temperature. The rolling surfaces were maintained at 423 K during rolling.

The HRDSR processed samples were annealed at two temperatures, namely, 973 K for 1 h and 1473 K for 2 h, to produce different grain sizes. The samples annealed at 973 K and 1473 K are referred to here as ANN973 and ANN1473, respectively, and the as-received HRDSR-processed sample is referred to as as-HRDSRed.

Optical microscopy and electron backscattering diffraction (EBSD) analysis with a scanning step size of 0.1-2.5 μm were used to determine the grain size and characterize the microstructures. The observations were made on the longitudinal sections of the samples parallel to the rolling direction. The EBSD data were processed using TSL-OIM analysis software, and data points with a low confidence index (< 0.1) were eliminated from the EBSD data. Boundaries of annealing twins were excluded in determining the mean grain size (d). For the evaluation of the grain size, the tolerance angle was set to 5°.

The microstructure of the as-HRDSRed sample was examined using field emission transmission electron microscopy (JEM 2001 F, 200 keV) on an instrument equipped with energy dispersive spectrometry (EDS) due to the difficulty of observing the heavily deformed microstructures by optical microscopy and EBDS.

High-resolution X-ray diffraction (XRD, SmartLab) measurements were conducted with Cu K_α radiation to identify the crystallographic phases.

Dog-bone shaped tensile samples with a gauge length of 10 mm parallel to the rolling direction (RD) were prepared from the rolled samples. Tensile tests were conducted at 1 × 10^-3 s^-1 using a universal testing machine (UNITECH-M) in the temperature range of RT to 1173 K. At 973, 1073 and 1173 K, the tensile elongation tests were additionally performed at 10^-4, 10^-2 and 10^-1 s^-1.

The dynamic elastic moduli of homogenized Fe₄₁Mn₂₅Ni₂₄Co₈Cr₂ were measured as a function of temperature between 298 and 1123 K using the impulse excitation technique [17]. Sample geometries (plates) with sizes of 35 (L) × 15(w) ×1.5(t) mm were fabricated using electric discharge machining and subsequently polished with SiC papers. The tests were performed at a heating rate of 1 K min^-1 at a strain amplitude lower than 10^-6. The shear modulus G was calculated following the equation below [17]:

$ G=\frac{4Lmf^{2}_{t}}{bt}[B/(1+A)]$ (1)

where m is the weight of the sample, f_t is the torsional frequency and A and B are the correction factors.

3. Results

Fig. 1(a) shows the microstructure of the as-HRDSRed sample observed by TEM, and Fig. 1(b) and (c) show the microstructures of the ANN973 and ANN1473 samples observed by EBSD. The as-HRDSRed sample shows a heavily deformed microstructure with a high density of dislocations and ultrafine cell or subgrain boundaries (≤ 0.5 μm). The microstructures of the ANN973 and ANN1473 samples have equiaxed grains with a random texture, indicating that full recrystallization occurred during the annealing treatment of the as-HRDSRed sample. The grain sizes of the ANN973 and ANN1473 samples are 8.1 and 590.2 μm, respectively. Many annealing twins are found within the recrystallized grains.

Fig. 2(a)-(c) shows the true stress-true strain curves for the ANN1473, ANN973 and as-HRDSRed samples obtained from the tensile test conducted in the temperature range of 298-1173 K at a given strain rate of 10^-3 s^-1. The yield stress (YS), ultimate tensile strength (UTS), UTS-YS and tensile elongation (%), which are measured or calculated from the engineering stress-engineering strain curves, are plotted as a function of temperature in Fig. 3(a)-(d). At RT, the strength of the as-HRDSRed sample is significantly higher than that of the ANN973 and ANN1473 samples: the UTS of the as-HRDSRed sample is 995 MPa, while the UTSs of the ANN973 and ANN1473 samples are 513 and 482 MPa, respectively. As the temperature increases, the strength decreases in all three alloys. The rapid stress decrease starts at approximately 873 K. The strength difference among the three alloys decreases as temperature increases. Beyond 973 K, the strength difference between the as-HRDSRed and ANN973 samples is quite small. The amount of UTS-YS, which represents the amount of strain hardening, is considerably larger in the ANN973 and ANN1473 samples than that of the as-HRDSRed sample at RT. ANN1473 has the largest UTS-YS among the three alloys. The UTS-YS value, however, gradually decreases with increasing temperature and almost disappears above 973 or 1073 K. In the temperature range between RT and 773 K, ANN1473 exhibits the largest tensile elongations (57 %-82 %) among the three alloys, while the as-HRDSRed sample exhibits the smallest tensile elongations (3 %-7 %), but this trend is reversed at temperatures above 873 K. The tensile elongation of ANN1473 gradually decreases with increasing temperature up to 773 K and then sharply decreases to 0.5 %-6 % above 873 K. For the as-HRDSRed sample, the tensile elongation is very small (6 %) at RT, and this low level of ductility is maintained at temperatures up to 773 K. The tensile elongation, however, increases rapidly at 873 K, reaching 61 % at 1073 K, and then decreases above 1173 K. For ANN973, the tensile elongation gradually decreases with increasing temperature and then suddenly drops to 37 % at 873 K. As temperature further increases, the tensile elongation increases up to 973 K and then decreases.

In the temperature range between RT and 773 K, where a large strain hardening prevails, the uniform strain may be determined based on a necking criterion of $\frac{d_{σ}}{d_{ε}}=σ$. The stresses associated with the onset of necking are marked (by yellow color symbols) on the curves in Fig. 2(a) and (b).

Fig. 4(a)-(c) shows the plots of true stress-true strain curves of ANN1473, ANN973 and the as-HRDSRed samples at various strain rates (at 10^-4, 10^-3, 10^-2 and 10^-1 s^-1) at different temperatures higher than 873 K (at 973, 1073 and 1173 K). As the temperature increases and the strain rate decreases, the steady state is rapidly reached after a decreased amount of strain hardening. The plot of log strain rate vs. log σ/G for the three alloys is shown in Fig. 5(a), where G is a function of the temperature and is empirically expressed by Eq. (2), which was obtained from the curve fitting to the experimental data shown in Fig. 5(d):

$ G=86.7-\frac{0.72}{exp(\frac{28.99}{T})-1}$ (2)

For plotting, the flow stresses were obtained at ε = 0.2 for the samples of as-HRDSRed and ANN973, while they were obtained at the peak stresses for ANN1473. The inverse of the slopes of the curves $(=\frac{dln(\frac{σ}{G})}{dln\dot{ε}})$ represent the strain rate sensitivity exponent (m). The values of m measured from the curves in Fig. 5(a)-(c) are plotted in Fig. 6(a). The tensile elongations of the three materials as a function of strain rate at different temperatures are plotted in Fig. 6(b). For all three materials, the m values tend to decrease as the strain rate increases and increase as the temperature increases. When compared at a given strain rate, the m value of the material with a small grain size is large. For example, at 1173 K/10^-4 s^-1, the m value of the as-HRDSRed sample is 0.51, the m value of ANN973 is 0.39 and the m value of ANN1473 is 0.18. The tensile elongations of the as-HRDSRed and ANN973 samples follow their m value trends as a function of strain rate and they are considerably larger than that of ANN1473 with smaller m values, indicating that the initiation and propagation of necking instability governs the tensile ductility behavior. For ANN1473, however, the elongation tends to increase as the strain rate increases. This will be discussed later.

The sine hyperbolic Garofalo equation, which is a well-known phenomenological equation, has been widely utilize to express the relationship between the strain rate, flow stress and temperature for metallic alloys over wide strain rates and temperature ranges where power law creep and power-law breakdown (PLB) occur [19]. When the effect of the elastic modulus on creep is taken into consideration, which is important when the temperature dependency of the elastic modulus is considerable in the investigated temperature range, the Garofalo equation can be presented using the modulus compensated flow stress as:

$\dot{ε}= A\ sinh (\frac{σ}{G})^{n} exp (-\frac{Qc}{RT})$ (3)

where $\dot{ε}$ is the strain rate, σ is the flow stress, Q_c is the activation energy for plastic flow, R is the gas constant, n is the stress exponent, A is the material constant, T is the absolute temperature, and α is the fitting parameter. Eq. (3) can be reduced to Eqs. (4) and (5) at low stresses where power-law creep governs plastic flow and at high stresses where PLB (instead of power-law creep) occurs, respectively:

$\dot{ε}=A(\frac{σ}{G})^{n_{1}} exp (-\frac{Qc}{RT})$ (4)

$\dot{ε}=Aexp(β\frac{σ}{G})exp (-\frac{Qc}{RT})$ (5)

To calculate the Qc value using Eq. (3), determination of the α value is necessary in advance. The α value can be obtained by measuring the β and n1 values from the linear fit to the data points in the plots of log $\dot{ε}$ - log $(\frac{σ}{G})$ and log $\dot{ε}$ - $\frac{σ}{G}$ (not shown here), respectively, and then averaged for different temperatures. From the average n1 and β values, the average α value was calculated. Using the average α value, the Qc value was calculated by measuring the average slopes of the fitting lines in the plot of log ε˙ - log sinh(α$\frac{σ}{G}$) at different temperatures and the plot of log sinh(α$\frac{σ}{G}$) - 1/T at different strain rates using Eq. (6):

$Q_{c}=R[\frac{\partial l\dot{n}ε}{\partial ln[sinh(α\frac{σ}{G})]}]_{T}[\frac{\partial ln[sinh(α\frac{σ}{G})]}{\partial (1/T)}]_{\dot{ε}}=RnS$ (6)

The plots for measuring N and S values of the ANN1473, ANN973 and as-HRDSRed samples are shown in Fig. 7(a)-(f). The α and Qc values for the as-HRDSRed, ANN973 and ANN1473 samples calculated following the above procedures are 508.7 and 189.2 kJ/mol and 515.8 and 224.5 kJ/mole, and 957.9 and 185.5 kJ/mol, respectively. The Qc values of the three materials are reasonably similar to one another and their average is 199.7 kJ/mol.

Fig. 8(a) shows the plots of log Z vs. log sinh(α$\frac{σ}{G}$), where Z is the Zener-Holloman parameter (=$\dot{ε}$exp($\frac{Qc}{RT}$)). For plotting, the α and Qc values evaluated from the data of the as-HRDSRed, ANN973 and ANN1473 samples are used. A very high degree of correlation is obtained between the parameters for each material (R²≥0.99). The n value decreases from 4.95 to 3.01 as the grain size decreases. The almost same quality of correlation is obtained when averages (α =660.8 and Qc =199.7 kJ/mol) of the α and Qc values of the three materials are used (Fig. 8(b)). Fig. 8(c) shows the plots of log Z vs. log σ/G for the as-HRDSRed, ANN973 and ANN1473 samples, where the average Qc ( = 199.7 kJ/mol) is used. The curve for ANN1473 is on the far left side of the curves for the as-HRDSRed and ANN973 samples because the flow stresses for ANN1473 were taken at the very small strains in the stress-strain curves due to early fracture. A comparison of the curves for the as-HRDSRed and ANN973 samples indicates that the former has a slightly lower flow stress than the latter at small Z values (i.e., at low strain rates and high temperatures), while an opposite result occurs at high strain rates. Fig. 8(d) shows the plot of the m values measured from the slopes of the fitting curves shown in Fig. 8(c), which are presented as a function of Z. The m values for the as-HRDSRed sample are largest, and the m values for ANN1473 are the smallest. The difference in m value among the three materials, however, tends to decrease as the Z value increases and nearly vanishes at large Z values.

A processing map, which is the superimposition of an instability map onto a power dissipation map, has been used to evaluate the hot workability of materials [20,21]. The power dissipation refers to the amount of power consumed for the microstructural change with respect to the total power during deformation and it is expressed by the efficiency of power dissipation, η, which is defined as [22]:

$η=2(1-\frac{1}{σ\dot{ε}}\int_{0}^{\dot{ε}}σd\dot{ε})$ (7)

An instability map characterizes the instability zones of the sample during hot deformation. The instability criterion is described as follows [22]:

ξ=2m-η (8)

where ξ($\dot{ε}$) is the instability parameter. When ξ($\dot{ε}$) is negative, adiabatic shear bands, cracks or flow localizations occurs in the microstructure.

The processing maps for the ANN1473, ANN973 and as-HRDSRed samples are constructed using the variation of η values with temperature and strain rate at ε = 0.2 and are presented in Fig. 9(a)-(c). The contour values in the processing maps represent the η values as a percentage and the red boundaries represent the flow-instability boundaries. In all three alloys, η tends to increase with decreasing strain rate and increasing temperature. As grain size decreases, η value increases and the domains associated with high η (≥ 30) extends to an increased strain rate and a decreased temperature. For the as-HRDSRed and ANN973 samples, the flow instability domain occurs in the temperature range of 973-1123 K and at high strain rates (above ∼10^-2 s^-1), but it disappears above 1123 K. These results indicate that the alloy with fine grains is more suitable for hot working and an optimum temperature condition for hot working at high strain rates is obtained above 1123 K.

Grain growth in HEAs with a single phase is expected to occur rapidly at high temperatures due to the lack of secondary phases that could pin grain boundaries. Therefore, there is a high chance of appreciable grain growth during the sample heating and holding stage prior to tensile loading at high temperatures. The microstructures after the sample heating and holding (20 min in total) at 973, 1073 and 1173 K were examined to measure the grain sizes just before the tensile loading. The microstructures of the as-HRDSRed and ANN973 samples examined by EBSD at selective conditions are shown in Fig. 10(a)-(d). For the as-HRDSRed sample, full recrystallization occurred during the sample heating and holding stage at all the three temperatures. The grain sizes at 973, 1073 and 1173 K are 6.8, 8.6 and 13.5 μm, respectively, indicating that grain growth after recrystallization occurred more rapidly as the testing temperature increased. The grain size of ANN973 measured at 1173 K also shows the occurrence of considerable grain growth during the sample heating and holding stage (from 8.1to 14.6 μm). The grain sizes measured after the sample heating and holding stage at all the testing conditions are listed in Table 1. The results of grain-size measurement just befor the application of tensile loading suggest that grain growth during sample heating and holding should be considered in evaluating the effect of the grain size on the deformation behavior and mechanism at high temperatures.

Fig. 11(a)-(d) show the IPF and GB maps of the gauge regions of the as-HRDSRed, ANN1473 and ANN973 samples after the tensile elongation tests at 1073 and 1173 K - 10^-3 s^-1. The grain sizes of the ANN1473, ANN973 and as-HRDSRed samples measured after the tensile elongation tests at 1173 K - 10^-3 s^-1 are 450 ±11.2, 16.2 ±0.97 and 15.6 ±1.06 μm, respectively. The grain size of the as-HRDSRed sample after the tensile elongation test at 1073 K - 10^-3 s^-1 is 8.9 ±0.53 μm. These results indicate that the amount of grain growth during tensile deformation is small at all testing conditions compared to that occurs during the sample heating and holding stage. Fig. 12 shows the XRD test results for ANN973 measured on the gauge and grip regions after the tensile elongation tests at different temperatures at the lowest strain rate of 10^-3 s^-1. The XRD curves contain no peaks for secondary phase even at the highest temperature of 1173 K, confirming the high thermal stability of a single-phase in Fe₄₁Mn₂₅Ni₂₄Co₈Cr₂.

Fig. 13(a)-(c) show the fractured surfaces of the tensile samples of the three materials tested at RT, 673, 973 and 1173 K at a given strain rate of 10^-3 s^-1. For ANN1473 at RT, numerous dimples with diameters of 10-30 μm are present on the flat surfaces, indicating the occurrence of ductile fracture as a result of the nucleation, growth, and coalescence of microvoids. At 673 K, the fracture morphology is similar to that observed at RT. At 1173 K, however, the fracture mode completely changed from ductile fracture to brittle intergranular fracture. Only a limited number of shallow dimples are found on the facets of grains. ANN973 shows ductile fracture with many dimples at RT. At 973 and 1173 K, fine globules are observed on the fractured surfaces, indicating the occurrence of ductile intergranular fracture. For the as-HRDSRed sample at RT and 673 K, the number of dimples is smaller and they are shallower than those observed in ANN1473, indicating that less ductile fracture took place in the as-HRDSRed sample. At 1173 K, fine globules are observed as in ANN973. The typical size of the globules is 10-20 μm, suggesting that dynamic recrystallization occurred near the fracture region of the tensile sample (indicated by arrows on the GB map in Fig. 11(c)) where plastic deformation was locally concentrated and intergranular cavitation resulted in intergranular separation.

4. Discussion

Using the true stress-true strain curves, work hardening rates as a function of strain were calculated for ANN973 and ANN1473 at temperatures below 773 K, where distinct strain hardening prevails, and the plots are shown in Fig. 14(a) and (b). The curves can be subdivided into two stages depending on the slopes. Both materials show the same trend in the first stage, where the work hardening rate rapidly decreases due to the elastic-plastic transition.TIn the second stage, the two materials show different behavior. A monotonic decrease in the work hardening rate occurs in ANN973 with a fine grain size, indicating that plastic deformation is governed by dislocation slip. This behavior is typical for metallic alloys. On the other hand, the work hardening rate first decreases and then increases with strain in ANN1473 with a coarse grain size. Asghari-Rad et al. [23] observed a plateau in V₁₀Cr₁₅Mn₅Fe₃₅Co₁₀Ni₂₅ with a coarse grain size (82.7 μm) at RT, and Sun et al. [24] observed a plateau followed by a slight increase in the work hardening rate - for CoCrFeMnNi with a grain size of 88.9 μm at RT. Both authors claimed that this strain hardening behavior is induced by mechanical twinning during deformation.

Laplanche et al. [25] showed the effect of temperature on mechanical twinning and proposed that the critical stress for twining is independent of temperature. In addition to the effect of temperature on mechanical twinning, there is another factor (i.e., grain size) that controls the transition of deformation mechanisms from dislocation slip to mechanical twinning [23,24]. Sun et al. [24] investigated the influence of grain size on the deformation behavior of CrMnFeCoNi at RT and found that mechanical twinning is suppressed with grain refinement due to an increased critical stress for twinning. They proposed that the critical stress for mechanical twinning (σT), which is expressed by Eq. (9):

$σ_{T}=M\frac{γ}{b_{p}}+\frac{k_{T}}{\sqrt{d}}$ (9)

where M is the Taylor factor (3.06 [26]), γ is the stacking fault energy (21 mJ/m² [27]), b_p is the Burgers vector of a partial dislocation (1.46 × 10^-10 m [28]), k_T is the Hall-Petch constant for twinning ( = 980 MPa μm^1/2 [24]) and d is the grain size. The calculated σT values and the maximum flow stress upon necking (marked on the curves in Fig. 2(d)-(f)) are plotted as a function of the grain size in Fig. 15. When the maximum flow stress is higher than the critical twinning stress, mechanical twins can be activated from the beginning of plastic deformation. For ANN1473, at all temperatures below 773 K, the maximum flow stresses are close to or higher than σT. On the other hand, for ANN973, the maximum flow stresses are lower than σT, indicating that activation of mechanical twinning does not occur during deformation. Therefore, the higher tensile ductility of ANN1473 compared to that of ANN973 in the temperature range between RT and 773 K can be attributed to the continuous activation of mechanical twins with strain in ANN1473, leading to the increased strain hardening.

As the temperature increases, the strain hardening effect decreases as the level of dynamic recovery increases. Thus, tensile elongation decreases with increasing temperature. When dynamic recovery prevails with further increase in temperature, however, the tensile ductility increases due to increased strain-rate sensitivity. The deformation behavior of metals in a steady state is often well depicted by simultaneously considering the contributions of two deformation mechanisms to plastic flow: grain boundary sliding (GBS) associated with n₁=2 and dislocation-climb creep associated with n₁=5[29,30]. In this case, the strain rate is expressed by Eq. (10) [29,30]:

$\dot{ε}=A\frac{D^{*}_{eff}}{d^{2}}(\frac{σ}{G})^{2}+B\frac{D^{*}_{eff}}{b^{2}}(\frac{σ}{G})^{5}$ (10)

where $D^{*}_{eff}$ is the effective diffusion coefficient determined by considering the lattice (D_L) and grain boundary diffusion (D_gb) coefficients [31,32], and D_eff is the effective diffusion coefficient determined by considering the lattice and pipe diffusion (D_p) coefficients [31,32]. As D_L, D_gb and D_p for Fe₄₁Mn₂₅Ni₂₄Co₈Cr₂ are unknown, we modify Eq. (10) using the Q_c value measured in this study:

$\dot{ε}=A'\frac{exp(-\frac{199700}{RT})}{d^{2}}(\frac{σ}{G})^{2}+B'\frac{exp(-\frac{199700}{RT})}{b^{2}}(\frac{σ}{G})^{5}$ (11)

From the fitting to the data for the as-HRDSRed sample at 1173 K where n₁ = 2 at low strain rates and n₁= 5 at high strain rates are well observed (Fig. 5(c)), the values of A' and B' were determined using the grain size (d = 13.5 μm) measured just prior to tensile loading: A' = 60 m²/s and B' = 15 m²/s. Using the A' and B' values, curve fitting was attempted at other temperatures for the as-HRDSRed and at all temperatures for the ANN1473 and ANN973 samples. The fitting curves by Eq. (11) are shown as solid curves in Fig. 5(a)-(c). Good fitting was obtained at all the conditions, supporting the usefulness of Eq. (11) in describing the deformation behavior at high temperatures above 973 K. There are regions (typically above 10^-2 s^-1) where deviation occurs from the prediction by Eq. (11). These regions represent PLB that occurs at high strain rates. For the as-HRDSRed and ANN973 samples, GBS and dislocation climb creep are predicted to govern at low strain rates and high strain rates, respectively. For ANN1473, on the other hand, dislocation climb creep is predicted to dominate plastic flow over the entire investigated strain rate range at all the temperatures. The curve fitting result with Eq. (11) on the data of Reddy et al. [14] and He et al. [11] for CoCrFeNiMn are shown in Fig. 16. For the data of Reddy et al. [14], a good curve fitting is obtained at d = 4.4 μm, which is the grain size measured at the gauge region after the tensile elongation test. For the data of He et al. [11], the initial grain size was used for curve fitting because the grain sizes just before tensile loading and after the tensile elongation tests are unknown. The curves predicted by Eq. (11) are spaced apart on the left side of the experimental data, but the curve shapes and slopes are similar to those obtained from the experimental data. This result suggests that the observation of n = 3 at low strain rates below 3×10^-4 s^-1 in a CoCrFeNiMn HEA may represent the transition from GBS with n₁ = 2 to dislocation-climb creep with n₁ = 5 rather than solute-drag creep associated with n ₁= 3, which was claimed by He et al. [11].

As the grain size decreases, the contribution of GBS to the total deformation increases, and thus, the m value increases, resulting in an increased tensile elongation. As the temperature increases, the contribution of GBS to the total deformation increases, but when the increase in the grain size due to grain growth is significant, the contribution of GBS to total deformation can decrease despite an increase in the temperature. This explains why the tensile elongation increases with increasing temperature and then decreases in the ANN973 and as-HRDSRed samples with further increase in temperature (Fig. 3(d)). The catastrophic and sudden loss of tensile ductility above 873 K in ANN1473 is suspected to be due to the nucleation of voids or cracks that are nucleated at grain boundaries owing to grain-boundary weakening and the initiation of grain boundary sliding. Li et al. [33] observed a significant hot embrittlement at 1973-1173 K in 10Cr12Ni3Mo2VN steel with coarse grains, which was attributed to intergranular fracture induced by grain boundary sliding. In ANN1473, tensile elongation tends to increase with increasing strain rate at temperatures above 973 K (Fig. 6(b)), which may be attributed to suppression of GBS at a high strain rate. In the ANN973 and as-HRDSRed samples, a brittle intergranular fracture phenomenon is not observed at high temperatures. This may be because stress concentrations by GBS is easily released due to grain boundary diffusion that is enhanced by the presence of a large number of grain boundaries in fine-grained microstructures.

5. Conclusions

The grain size impact on the tensile properties of nonequiatomic Fe₄₁Mn₂₅Ni₂₄Co₈Cr₂ in the temperature range between RT and 1173 K was studied by preparing samples with three different grain sizes: ≤0.5, 8.1 and 590.2 μm, and the following results were obtained.

1 In the temperature range between RT and 773 K, where strain hardening prevailed, the material with a grain size of 590.2 μm exhibited the largest tensile elongations of 57 %-82 % among the samples studied due to its high strain hardening capability induced by mechanical twinning. The material with ultrafine grains exhibited the smallest tensile elongations of 3 %-7 % due to its greatly reduced strain hardening ability after severe plastic deformation but exhibited the highest strength due to its very fine grain size.

2 At temperatures above 873 K, where strain hardening considerably decreased and steady-state flow prevailed, the material with the finest grain size exhibited the largest tensile elongations of 30 %-85 % due to its high strain rate sensitivity (m) (0.1-0.5). The material with the large grain size of 590.2 μm exhibited very small elongations of 0.2 %-8 % due to the small m values (0.1-0.2) and occurrence of brittle intergranular fracture.

3 The tensile elongation behavior and deformation mechanism of the materials with different grain sizes in the temperature range between 973 and 1173 K, where steady state flow prevailed, could be explained in terms of the simultaneous contribution of two independent deformation mechanisms: grain boundary sliding and dislocation-climb creep. The analysis results indicate that in evaluating the deformation behavior and identifying the deformation mechanism, it is important to consider rapid grain growth during the sample heating and holding stage. This is because the contribution of grain boundary sliding to the total deformation is significant when grain size is small.

Data availability

The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.

Acknowledgements

This research was financially supported by the Basic Research Laboratory Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (Project No. NRF 2015-041523).

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