A. Uniaxial creep

Creep of the Al_0.15CoCrFeNi and Al_0.6CoCrFeNi HEAs was explored by employing the uniaxial stress-relaxation tests.^{Reference Youssef, Zaddach, Niu, Irving and Koch11} The HEA samples, with a diameter of 5 mm and a gauge length of 25 mm, were loaded up to a strain of 0.2% at a loading rate of 8 × 10⁻⁵ s⁻¹ and temperature of 580–700 °C (0.3–0.5 T _m, where T _m is the melting temperature), followed by a holding period of 4 h for stress relaxation.^{Reference Youssef, Zaddach, Niu, Irving and Koch11} The recorded stress-relaxation traces at different temperatures for the Al_0.15 and Al_0.6 HEAs are given in Figs. 1(a) and 1(b), respectively.^{Reference Youssef, Zaddach, Niu, Irving and Koch11} From the slopes of the curves in these two figures, one can tell that the Al_0.60 HEA exhibits a faster relaxation rate than the Al_0.60 one. The difference stems from their distinct microstructures—a single fcc structure for the Al_0.15 HEA and a fcc + bcc duplex structure for the Al_0.60 HEA.

FIG. 1.

Uniaxial-stress relaxation curves of the (a) fcc Al_0.15CoCrFeNi and (b) fcc + bcc Al_0.6CoCrFeNi HEAs at varied temperatures. (c) Measured data in the form of $\ln \left( {{{\dot{\varepsilon }}_{\rm{c}}}\exp \left( {{Q / {RT}}} \right)} \right)$ versus ln σ (symbols) for the two HEAs collected from different testing temperatures along with the best linear fitting lines from which unique values of Q and n were determined for each alloy. (d) The measured data in the form of $RT\ln {\dot{\varepsilon }_{\rm{c}}}$ versus σ (symbols) for the Al_0.6CoCrFeNi HEAs collected from different testing temperatures along with the best linear-fitting lines from which the activation volume, V*, was determined.^{Reference Cao, Shang, Zhao, Cheng, Wang and Wang29}

Consider that creep of the Al_0.15 and Al_0.60 HEAs follows the classic Norton’s law^{Reference Dieter and Bacon39}

(1)$${\dot{\varepsilon }_{\rm{c}}} = A{\sigma ^n}\exp \left( { - {Q \over {RT}}} \right)\quad ,$$

where ${\dot{\varepsilon }_{\rm{c}}}$ is the creep-strain rate. σ is the applied stress. T is the temperature. R is the gas constant. n is the stress exponent. Q is the activation energy. A is a material constant. By assuming that the deformation of the alloys at high temperatures is composed of the elastic component and viscous (or creep) component, the total applied strain of ε = 0.2% can be written as^{Reference Dieter and Bacon39}

(2)$$\varepsilon = {\varepsilon _{\rm{e}}} + {\varepsilon _{\rm{c}}}\quad .$$

Taking the time derivative to both sides of Eq. (2) and applying the Hooke’s law to the elastic part, the creep rate can be formulated in the form of the stress-relaxation rate^{Reference Dieter and Bacon39}

(3)$${\dot{\varepsilon }_{\rm{c}}} = - {\dot{\varepsilon }_{\rm{e}}} = - {1 \over E}{{{\rm{d}}\sigma } \over {{\rm{d}}t}}\quad ,$$

where E is the Young’s modulus. With Eq. (3), one can calculate the creep rate, ${\dot{\varepsilon }_{\rm{c}}}$, of both the Al_0.15 and Al_0.60 HEAs at different temperatures from the stress-relaxation data in Figs. 1(a) and 1(b). By plotting the $\ln {\dot{\varepsilon }_{\rm{c}}} - \ln \sigma$ data and performing a linear fit with Eq. (1), one can derive the values of the stress exponent, n, at different temperatures, as shown in Fig. 2(c) in Ref. Reference Cao, Shang, Zhao, Cheng, Wang and Wang29. In a similar fashion, but plotting and fitting the $\ln {\dot{\varepsilon }_{\rm{c}}} - {1 / T}$ data points with Eq. (1), one can extract the activation energy, Q, at different stress levels, as given in Fig. 2(d) in Ref. Reference Cao, Shang, Zhao, Cheng, Wang and Wang29.

In a consolidated creep law for an individual alloy, both n and Q should not depend on the applied stress. Therefore, a single value of n or Q can be obtained by first plotting $\ln \left( {{{\dot{\varepsilon }}_{\rm{c}}}\exp \left( {{Q / {RT}}} \right)} \right)\;{\rm{versus}}\;\ln \sigma$ to cluster the data at different temperatures together, and then fitting with Eq. (1). Such an operation for the Al_0.15 and Al_0.60 HEAs are given in Fig. 1(c),^{Reference Cao, Shang, Zhao, Cheng, Wang and Wang29} in which the value of Q needs to be adjusted iteratively to reach the best fit. After obtaining a single Q value for each alloy, the values of n and A turn out to be the slope and intercept of the best fitting line, respectively. The deduced n and Q for the Al_0.15 and Al_0.60 HEAs are 5.56 and 8.82, and 385 kJ/mol and 334 kJ/mol,^{Reference Cao, Shang, Zhao, Cheng, Wang and Wang29} respectively. From these values, it is evident that the Al_0.15 HEA has greater creep resistance than the Al_0.60 HEA. Since n = 3–8 normally signifies dislocation creep^{Reference Dieter and Bacon39} in crystalline metals, the creep mechanisms in both the Al_0.15 and Al_0.60 HEAs, therefore, were believed to be managed by dislocation motion.^{Reference Cao, Shang, Zhao, Cheng, Wang and Wang29}

It is known that the activation volume, V*, in a plastic deformation can be calculated according to

(4)$${V^ * } = RT{{\partial \ln {{\dot{\varepsilon }}_{\rm{c}}}} \over {\partial \sigma }}\quad ,$$

where R is the gas constant and T is the temperature.^{Reference Cao, Shang, Zhao, Cheng, Wang and Wang29,Reference Caillard and Martin40} By plotting $RT\ln {\dot{\varepsilon }_{\rm{c}}}$ as a function of σ for the HEAs at different testing temperatures and performing linear fittings, the values of V* were derived from the slopes of the fitting lines. In Fig. 1(d), an example of this operation is provided for the Al_0.60 HEA.^{Reference Cao, Shang, Zhao, Cheng, Wang and Wang29} The calculated V* values for the Al_0.15 and Al_0.60 alloys at varied temperatures range from 3.15 × 10⁻⁴ to 5.22 × 10⁻⁴ m³/mol and 4.3 × 10⁻⁴ to 6.74 × 10⁻⁴ m³/mol, respectively.^{Reference Cao, Shang, Zhao, Cheng, Wang and Wang29} For the fcc-single-phase Al_0.15 HEA, taking ${a \over 2}\left\langle {110} \right\rangle$ as the Burger’s vector and a = 3.606 Å as the lattice constant, V* was calculated to be in the range of 32–54b.^{Reference Miracle and Senkov3,Reference Cao, Shang, Zhao, Cheng, Wang and Wang29} With the established knowledge that a V* value ranging approximately from b³ to 1000b³ is related to dislocation cross-slip,^{Reference Cao, Shang, Zhao, Cheng, Wang and Wang29,Reference Somekawa and Schuh41} it was concluded that creep in the Al_0.15 HEA was dominated by the dislocation slide. This finding essentially agrees with what has arrived through the stress exponent.

Moreover, Tsao et al. investigated the creep behavior of the Ni_47.9Al_10.2Co_16.9Cr_7.4Fe_8.9Ti_5.8Mo_0.9Nb_1.2W_0.4C_0.4 HEA with fcc + L1₂ phases at 750–982 °C using uniaxial creep tests^{Reference Tsao, Yeh, Kuo, Kakehi, Murakami, Yeh and Jian30} and found the excellent creep resistance for this alloy. Specifically, the creep-rupture life of the HEA loaded at 159 MPa and at 982 °C was measured to be 114 h, comparable to some first-generation Ni-based superalloys under similar conditions, for example, NX-188 DS and Rene’ 80 superalloys.^{Reference Tsao, Yeh, Kuo, Kakehi, Murakami, Yeh and Jian30,Reference Howes42,Reference Ohtomo and Saiga43} Its creep activation energy was determined to be 290 kJ/mol,^{Reference Tsao, Yeh, Kuo, Kakehi, Murakami, Yeh and Jian30} higher than 230 kJ/mol of CMSX-2 superalloys.^{Reference Rouault-Rogez, Dupeux and Ignat44} The exceptional creep resistance of this alloy was thought of originating from the stable γ′ phases and lower stacking fault energy.^{Reference Tsao, Yeh, Kuo, Kakehi, Murakami, Yeh and Jian30}

B. Spherical nanoindentation creep

Lee et al. had applied spherical nanoindentation to probe the creep behavior of the CoCrFeMnNi HEAs with an fcc structure.^{Reference Lee, Seok, Zhao, Choi, He, Lu, Suh, Ramamurty, Kawasaki, Langdon and Jang33} In their tests, a spherical indenter with a tip radius of R = 38.6 μm was used to load the samples to different maximum load levels, P _max, at room temperature and a constant loading rate of dP/dt = 0.5 mN⁻¹. The samples were, then, held at P _max for 1000 s for creeping, and unloaded thereafter. The creep tests were conducted on the coarse-grained samples with a grain size of ∼46 μm and the nanocrystalline samples with a grain size of ∼49 and ∼33 nm, in both elastic and elastic–plastic regimes. Figure 2(a) gives the load–displacement (P–h) curves of the three HEA samples in the elastic regime.^{Reference Lee, Seok, Zhao, Choi, He, Lu, Suh, Ramamurty, Kawasaki, Langdon and Jang33} The loading curves were ascertained to be elastic because P _max applied on each sample was below their respective yielding load, as determined from a set of separate elastic–plastic transition tests (Fig. 3 in Ref. Reference Lee, Seok, Zhao, Choi, He, Lu, Suh, Ramamurty, Kawasaki, Langdon and Jang33). This assertion can also be easily verified by noticing that the loading and unloading curves run nearly in parallel in Fig. 2(a). A conspicuous feature in Fig. 2(a) is that the penetration depth increases for all three samples at the load-holding stage, implying that they have experienced creep deformation even in the elastic regime. This time-dependent deformation is plastic in nature rather than an elasticity as it remains upon unloading. Holding the samples in the elastic–plastic regime (higher P _max) undoubtedly incurs a greater magnitude of creep deformation.^{Reference Lee, Seok, Zhao, Choi, He, Lu, Suh, Ramamurty, Kawasaki, Langdon and Jang33}

FIG. 2.

(a) Creep tests conducted in the elastic regime of the nanocrystalline (nc) and coarse-grained (cg) CoCrFeMnNi HEAs with a fcc phase at room temperature by spherical nanoindentation. (b) Uniaxial creep-strain rate (converted from the indentation creep-strain rate), $\dot{\varepsilon }_{\rm{u}}^{{\rm{creep}}}$, as a function of the holding time for the nanocrystalline HEAs with a grain size of ∼33 nm at different load levels, with the inset showing the indentation creep strain, $\varepsilon _{\rm{i}}^{{\rm{creep}}}$, as a function of the holding time. (c) Measured creep data in the form of $\ln {\dot{\varepsilon }_{{\rm{ssc}}}}$ versus ln σ (symbols) along with the best linear-fitting lines from which the stress exponent, n, was determined. (d) Measured creep data in the form of $\ln {\dot{\varepsilon }_{{\rm{ssc}}}}$ versus σ (symbols) along with the best linear-fitting lines from which the activation volume, V*, were determined.^{Reference Lee, Seok, Zhao, Choi, He, Lu, Suh, Ramamurty, Kawasaki, Langdon and Jang33}

To quantify the stress exponent and activation volume in a spherical nanoindentation test, a very first step is to obtain a history of the creep-strain rate during the load-holding stage. To this end, the nanoindentation-creep strain was computed as $\varepsilon _{\rm{i}}^{{\rm{creep}}} = {{0.2\left( {a - {a_0}} \right)} / R}$, where a ₀ and a are the contact radii at the onset of creep and an arbitrary instant of creep, respectively.^{Reference Lee, Seok, Zhao, Choi, He, Lu, Suh, Ramamurty, Kawasaki, Langdon and Jang33} The instantaneous contact radius was estimated to be $a = \sqrt {2hR - {h^2}}$,^{Reference Lee, Seok, Zhao, Choi, He, Lu, Suh, Ramamurty, Kawasaki, Langdon and Jang33,Reference Choi, Yoo, Kim, Seok, Wang and Jang45} in which h is taken to be the actually measured depth for simplicity instead of the contact depth. An example of the calculated nanoindentation creep strain, $\varepsilon _{\rm{i}}^{{\rm{creep}}}$, as a function of the holding time, t _hold, for the ∼33 nm nanocrystalline HEA at four different P _max values is displayed as the inset in Fig. 2(b).^{Reference Lee, Seok, Zhao, Choi, He, Lu, Suh, Ramamurty, Kawasaki, Langdon and Jang33} In general, the nanoindentation $\varepsilon _{\rm{i}}^{{\rm{creep}}} - {t_{{\rm{hold}}}}$ curves consist of transient and steady-state creep regimes, resembling the uniaxial creep–time relation but with the tertiary stage missing. The nanoindentation $\varepsilon _{\rm{i}}^{{\rm{creep}}} - {t_{{\rm{hold}}}}$ curves can then be fitted with^{Reference Lee, Seok, Zhao, Choi, He, Lu, Suh, Ramamurty, Kawasaki, Langdon and Jang33}

(5)$$\varepsilon _{\rm{i}}^{{\rm{creep}}} = \alpha \left( {1 - {{\rm{e}}^{ - r{t_{{\rm{hold}}}}}}} \right) + \omega {t_{{\rm{hold}}}}\quad ,$$

where α, r, and ω are fitting constants. Differentiating the final form of Eq. (5) with respect to t _hold then leads to the history of the creep-strain rate, i.e., $\dot{\varepsilon }_{\rm{i}}^{{\rm{creep}}} = f\left( {{t_{{\rm{hold}}}}} \right)$. With the empirical relation of $\dot{\varepsilon }_{\rm{u}}^{{\rm{creep}}} \approx 0.01\dot{\varepsilon }_{\rm{i}}^{{\rm{creep}}}$,^{Reference Lee, Seok, Zhao, Choi, He, Lu, Suh, Ramamurty, Kawasaki, Langdon and Jang33,Reference Wang, Lai, Huang and Nieh46} one can derive the equivalent uniaxial creep strain rate, $\dot{\varepsilon }_{\rm{u}}^{{\rm{creep}}}$, with the already computed $\dot{\varepsilon }_{\rm{i}}^{{\rm{creep}}}$. The variations of the resulting $\dot{\varepsilon }_{\rm{u}}^{{\rm{creep}}}$ with t _hold for the ∼33 nm nanocrystalline HEA at the varied P _max are plotted in Fig. 2(b), from which it is seen that for all load levels, a steady-state creep strain rate is attained after a loading period of ∼400 s.

Recall that the creep-stress exponent of $n = {{\partial \ln {{\dot{\varepsilon }}_{{\rm{ssc}}}}} \over {\partial \ln \sigma }}$, where ${\dot{\varepsilon }_{{\rm{ssc}}}}$ is the steady-state creep rate.^{Reference Lee, Seok, Zhao, Choi, He, Lu, Suh, Ramamurty, Kawasaki, Langdon and Jang33} Determining ${\dot{\varepsilon }_{{\rm{ssc}}}}$ at t _hold = 1000 s and estimating σ from the measured hardness, H, at the same t _hold by σ = H/3 with C being ∼3,^{Reference Lee, Seok, Zhao, Choi, He, Lu, Suh, Ramamurty, Kawasaki, Langdon and Jang33,Reference Johnson47} the $\ln {\dot{\varepsilon }_{{\rm{ssc}}}} - \ln \sigma$ data points can be constructed [Fig. 2(c)], from which n was determined to be the slope of the linear fits. With the averaged data points, n was determined to be ∼1 for the nanocrystalline HEAs and ∼3 for the coarse-grained HEA. The activation volume was defined as ${V^ * } = \sqrt 3 kT{{\partial \ln {{\dot{\varepsilon }}_{{\rm{ssc}}}}} \over {\partial \sigma }}$,^{Reference Lee, Seok, Zhao, Choi, He, Lu, Suh, Ramamurty, Kawasaki, Langdon and Jang33} where k is the Boltzmann’s constant. In a similar fashion, the values of V* for three different HEA samples were determined from the slopes of the linear fits of the $\ln {\dot{\varepsilon }_{{\rm{ssc}}}} - \sigma$ data, as given in Fig. 2(d). The V* values were ultimately converted to be ∼4.6b³ for the coarse-grained HEA and ∼0.8b³ for the nanocrystalline HEAs using a Burger’s vector of ${\bi{b}} = l \times {{\sqrt 2 } \over 2} \approx 2.25 \times {10^{ - 10}}\;{\rm{m}}$, where l is the lattice constant and is equal to ∼3.6 Å for the CoCrFeMnNi HEAs.^{Reference Lee, Seok, Zhao, Choi, He, Lu, Suh, Ramamurty, Kawasaki, Langdon and Jang33}

The values of n and V* are important as they are usually implicative of steady-state creep mechanisms in a material. Specifically, n ≈ 1 hints a diffusion-mediated creep (through lattice or grain boundaries) in the nanocrystalline HEAs, whereas n ≈ 3 in the coarse-grained HEA is indicative of dislocation creep.^{Reference Lee, Seok, Zhao, Choi, He, Lu, Suh, Ramamurty, Kawasaki, Langdon and Jang33} From the activation-volume perspective, in fcc metals, the dislocation glide mechanism corresponds to V* values in the range of ∼100b³–1000b³, and the diffusion-mediated creep takes over when V* goes smaller down to ∼b³.^{Reference Lee, Seok, Zhao, Choi, He, Lu, Suh, Ramamurty, Kawasaki, Langdon and Jang33,Reference Conrad48–Reference Frost and Ashby50} As a result, V* = 0.8b³ and 4.6b³ in the nanocrystalline HEAs and coarse-grained HEAs indicate a diffusion creep mechanism for the both alloy groups.^{Reference Lee, Seok, Zhao, Choi, He, Lu, Suh, Ramamurty, Kawasaki, Langdon and Jang33} This conclusion complies with that deduced from the stress exponent for the nanocrystalline HEAs but opposes that for the coarse-grained HEA. However, it was argued that creep in the coarse-grained HEA should be still governed by the dislocation slide, and the reduced value of V* was artificially caused by the interfacial diffusion between the indenter tip and sample surface, which is particularly pronounced for shallow indentations.^{Reference Lee, Seok, Zhao, Choi, He, Lu, Suh, Ramamurty, Kawasaki, Langdon and Jang33}

Spherical nanoindentation tests were also used to study the room-temperature creep behavior of the CoCrFeCuNi HEA film (∼1.45 μm thick), at the as-deposited state with an fcc structure and annealed state with an fcc + bcc structure.^{Reference Ma, Peng, Wen and Zhang34} For both states, the creep deformation increases with increasing nanoindentation-holding load and loading rate. By defining the strain-rate sensitivity as $m = {{\partial \ln H} \over {\partial \ln {{\dot{\varepsilon }}_{\rm{c}}}}}$ along with H = P/A, A = 2πR(h _i − 0.75 × P/S), and ${\dot{\varepsilon }_{\rm{c}}} = {1 \over {\sqrt A }}{{{\rm{d}}\sqrt A } \over {{\rm{d}}t}}$ (H is the hardness, P is the load, A is the contact area, h _i is the indentation displacement, and S is the stiffness), m was determined from the slope of the linear fit to the $\ln H - \ln {\dot{\varepsilon }_{\rm{c}}}$ data points.^{Reference Ma, Peng, Wen and Zhang34} When the creep load increases from 15 to 50 mN, the value of m increases from 0.1 to 0.25 for the as-deposited alloy and decreases from 0.27 to 0.21 for the annealed alloy.^{Reference Ma, Peng, Wen and Zhang34} Since the strain-rate sensitivity is the reciprocal of the stress exponent, i.e., =1/n, the corresponding n values for the as-deposited and annealed alloys are calculated to be 4–10 and 3.7–4.8, respectively. With the loading rate increasing from 0.1 to 5 mN/s, the value of m increases from 0.14 to 0.4 for the as-deposited alloy and remains at ∼0.27 for the annealed alloy.^{Reference Ma, Peng, Wen and Zhang34} Converting m into n, the former and latter are 2.5–7.14 and 3.7, respectively. The calculated n values indicate that creep in both the as-deposited and annealed CoCrFeCuNi HEA films is dominated by dislocation glide or climb.

C. Sharp nanoindentation creep

Berkovich nanoindentation has been used for investigating the creep behavior of a coarse-grained Al_0.3CoCrFeNi HEA with a single fcc structure at room temperature.^{Reference Zhang, Yu, Cheng, Zhang, Diao, Shi, Chen, Chen, Feng, Bai, Jing, Ma, Liaw, Li and Liu31} The creep deformation of this alloy is sensitive to both applied loads and loading rates, as given in Figs. 3(a) and 3(c),^{Reference Zhang, Yu, Cheng, Zhang, Diao, Shi, Chen, Chen, Feng, Bai, Jing, Ma, Liaw, Li and Liu31} that is, the nanoindentation creep depth increases with increasing both the creep load and loading rate. With a power-law creep equation like Eq. (1) and some relations in nanoindentation tests, Zhang et al.^{Reference Zhang, Yu, Cheng, Zhang, Diao, Shi, Chen, Chen, Feng, Bai, Jing, Ma, Liaw, Li and Liu31} ultimately derived

(6)$$\ln {H^{ - n}} = A - {Q \over {RT}} + \ln t\quad ,$$

where A is a constant. Obviously, lnH is linearly scaled with lnt. Therefore, conducting a linear fit to the ln H − lnt data enables the determination of the stress exponent, n. The values of n determined at different peak loads are plotted in Fig. 3(b).^{Reference Zhang, Yu, Cheng, Zhang, Diao, Shi, Chen, Chen, Feng, Bai, Jing, Ma, Liaw, Li and Liu31} It is seen that as the load increases from 500 to 900 μN, the value of n ramps from ∼10 up to ∼90. The dependence of the stress exponent on the loading time and/or loading rate is graphed in Fig. 3(d).^{Reference Zhang, Yu, Cheng, Zhang, Diao, Shi, Chen, Chen, Feng, Bai, Jing, Ma, Liaw, Li and Liu31} Now, as the loading rate extends from 0.06 to 5 mN/s, n gradually increases from ∼40 to ∼90.^{Reference Zhang, Yu, Cheng, Zhang, Diao, Shi, Chen, Chen, Feng, Bai, Jing, Ma, Liaw, Li and Liu31} A striking feature of this work is that the value of n is extremely higher than those typically found for crystalline metals (<10). Based on the large n value, the authors argued that the there was a change in the creep mechanism from the diffusion flow to dislocation climb and slip with increasing indentation loads. This trend might be true for some nanoindentation tests as the self-diffusion along the free surfaces of the specimens and the indenter/specimen interface may play a dominant role in shallow indentation depths. However, claiming that a deformation mechanism transition based on n varying from ∼10 to ∼90 is somewhat misleading. This is because a diffusion-dominant creep mechanism in traditional crystalline metals generally occurs at n = 1. The extremely large n value was speculated by the authors to come from three resources: (i) Large grains of the alloy make the grain boundaries barely contribute to creep; (ii) Multiple components and distorted lattices retard the dislocation movement; (iii) The nanoscaled phases with highly concentrated solute atoms can effectively resist creep due to the sluggish diffusion in the multi-element matrix. Nevertheless, justifications of these speculated causes are lacking.

FIG. 3.

Creep tests conducted on the fcc Al_0.3CoCrFeNi HEAs by Berkovich nanoindentation at (a) different maximum loads and (c) loading rates. Variation of the stress exponent and hardness with (b) the applied loads and (d) loading time.^{Reference Zhang, Yu, Cheng, Zhang, Diao, Shi, Chen, Chen, Feng, Bai, Jing, Ma, Liaw, Li and Liu31}

Ma et al. used the Berkovich nanoindentation to study the creep characteristics of the fcc CoCrFeNiCu and bcc CoCrFeNiCuAl_2.5 HEA films, which have the thicknesses of 1.45 μm and 1.75 μm,^{Reference Ma, Feng, Debela, Peng and Zhang35} respectively. Since the strain-rate sensitivity is defined as $m = {{\partial \ln \sigma } \over {\partial \ln {{\dot{\varepsilon }}_{\rm{c}}}}}$ with $\sigma \approx {H / {3\sqrt 3 }}$ and ${\dot{\varepsilon }_{\rm{c}}} = {1 \over {{h_{\rm{c}}}}}{{{\rm{d}}{h_{\rm{c}}}} \over {{\rm{d}}t}}$ (h _c = h _i − 0.72 × P/S is the contact depth), m was determined from the slope of the linear fit to the $\ln H - \ln {\dot{\varepsilon }_{\rm{c}}}$ data. m was determined to be 0.035 and 0.0048 for the CoCrFeNiCu and CoCrFeNiCuAl_2.5 HEAs, respectively.^{Reference Ma, Feng, Debela, Peng and Zhang35} Again using n = 1/m, the stress exponents of the CoCrFeNiCu and CoCrFeNiCuAl_2.5 HEAs are calculated to be 28.57 and 208.33, respectively. Hence, this work suffers from the same issue as the one by Zhang et al.,^{Reference Zhang, Yu, Cheng, Zhang, Diao, Shi, Chen, Chen, Feng, Bai, Jing, Ma, Liaw, Li and Liu31} that is, the determined values of the stress exponent, n, are overly large.

D. Comparison of different creep tests

The techniques used for investigating the creep behavior of HEAs and probing their stress exponent, n, and activation volume, V*, include the uniaxial stress-relaxation, spherical-nanoindentation, and Berkovich-nanoindentation experiments, as summarized in Table I. Each testing method has its own advantages and disadvantages. A uniaxial stress-relaxation test is a closest means to mimic the standard creep tests and can directly give out the uniaxial creep strain rate. If the creep-strain rate is calculated correctly, the derived n, V*, and hence the inferred creep mechanism should be reliable. Nevertheless, the correct calculation of these parameters relies on the validity of Eq. (2). ε = ε_e + ε_c is only valid when the samples are loaded below the yield point, i.e., no plastic deformation is involved. Moreover, due to the uniform deformation, creep tests by this method only become appreciable at high temperatures. Nanoindentation experiments, either a spherical- or sharp-type, overcome the temperature restriction due to the highly localized large strain and can even detect significant creep at room temperature. This convenience, plus the less demanding sample size, essentially facilitates the creep characterization of HEAs. But one of the issues of using nanoindentation for creep experiments is that the direct use of the nanoindentation creep-strain rate for calculating n and V* is in question.^{Reference Dean, Bradbury, Aldrich-Smith and Clyne51–Reference Su, Herbert, Sohn, LaManna, Oliver and Pharr54} In the research of HEAs, most authors chose to directly use the nanoindentation-creep-strain rate^{Reference Zhang, Yu, Cheng, Zhang, Diao, Shi, Chen, Chen, Feng, Bai, Jing, Ma, Liaw, Li and Liu31,Reference Ma, Peng, Wen and Zhang34,Reference Ma, Feng, Debela, Peng and Zhang35} except that Lee et al. converted it into an equivalent uniaxial creep-strain rate using a relation of $\dot{\varepsilon }_{\rm{u}}^{{\rm{creep}}} \approx 0.01\dot{\varepsilon }_{\rm{i}}^{{\rm{creep}}}$ prior to fitting for n and V* values.^{Reference Lee, Seok, Zhao, Choi, He, Lu, Suh, Ramamurty, Kawasaki, Langdon and Jang33} Unreasonably large n values for HEAs were calculated from the creep data by the Berkovich nanoindentation tests, as given in Table I. The improper usage of the nanoindentation creep-strain rate might be part of causes. This factor may also contribute to the very different n values measured for the as-cast CoCrFeCuNi HEA films by the spherical nanoindentation and Berkovich nanoindentation tests, i.e., 2.5–10 for the former and 28.57 for the latter. The n values from the uniaxial stress relaxation tests^{Reference Cao, Shang, Zhao, Cheng, Wang and Wang29} and corrected spherical nanoindentation tests^{Reference Lee, Seok, Zhao, Choi, He, Lu, Suh, Ramamurty, Kawasaki, Langdon and Jang33} are in the reasonable range in general.

TABLE I.

The derived stress exponent, n, and activation volume, V*, of HEAs from different creep tests.

Note: b is the Burger’s vector.

A. Fatigue strength and life

High-cycle four-point bending fatigue tests have been carried out on the Al_0.5CoCrCuFeNi HEAs to investigate their fatigue strength and life at room temperature. In the early work by Hemphill et al.,^{Reference Hemphill, Yuan, Wang, Yeh, Tsai, Chuang and Liaw20} the testing samples were made from commercial-purity (> 99%) raw materials, followed by annealing at 100 °C for 6 h, water quenching, and cold rolling with a 84% reduction in thickness. The fatigue tests ran in a load-controlled mode, and different load levels in a sinusoidal waveform were swept through the samples at a frequency of 10 Hz. In all tests, the samples were so oriented that the direction of the stress on the tensile or compressive surface is parallel to the rolling direction. The maximum tensile stress takes place on the convex side of the bent samples, specifically confined to the span of the two outer pins. Assuming stress uniformity in this domain, the peak tensile stress was estimated to be $\sigma = {{3P\left( {{S_0} - {S_i}} \right)} \over {2B{W^2}}}$, where P, S ₀, S _i, B, and W are the applied load, outer span length, inner span length, sample thickness, and sample height, respectively. The samples were tested at a stress ratio of R = 0.1 until fatigue failure or reaching 10⁷ cycles.

The stress range (Δσ = σ_max − σ_min = 0.9σ_max) versus cycles to failure (i.e., S–N data) obtained at different fatigue-load levels are plotted as symbols in Fig. 4(a).^{Reference Hemphill, Yuan, Wang, Yeh, Tsai, Chuang and Liaw20} One of the most striking features of the measured data is that it manifests a great extent of scatter. For instance, at a high stress level approaching the yield strength (1284 MPa) of the alloy, i.e., Δσ = 1125 MPa, the failure of the samples spreads in a wide range of cycles from 35,000 to 450,000. The data scattering in the lifetime is further exaggerated as the applied stress range ramps down. In spite of a wide spread in the testing data, the general trend that the number of cycles to failure increases with the decreasing stress level, found in most of the structural materials,^{Reference Suresh55} still holds. The fatigue strength (or endurance limit) of the alloy, gauged as the Δσ value when the specimens reach 10⁷ fatigue cycles without failure, reads as a lower bound of 540 MPa and an upper bound of 945 MPa in Fig. 4(a).^{Reference Hemphill, Yuan, Wang, Yeh, Tsai, Chuang and Liaw20} The fatigue ratio, defined as ${{\Delta \sigma } \over {{\sigma _{{\rm{UTS}}}}}}$, where σ_UTS is the ultimate tensile strength of the materials, is thus calculated to be in the range of 0.402–0.703.

FIG. 4.

Fatigue S–N data of the (a) as-cast^{Reference Hemphill, Yuan, Wang, Yeh, Tsai, Chuang and Liaw20} and (b) wrought^{Reference Tang, Yuan, Tsai, Yeh, Lundin and Liaw21} Al_0.5CoCrCuFeNi HEAs with a dual fcc structure. The curves in (a) are the predicted quantile lives with the Weibull mixture predictive model^{Reference Hemphill, Yuan, Wang, Yeh, Tsai, Chuang and Liaw20} while those in (b) are the predicted stress amplitude versus median fatigue life by the random endurance-limit model.^{Reference Tang, Yuan, Tsai, Yeh, Lundin and Liaw21} (c) Effect of defects counts on the fatigue life of the as-cast Al_0.5CoCrCuFeNi HEA at various stress levels.^{Reference Hemphill, Yuan, Wang, Yeh, Tsai, Chuang and Liaw20} (d) Two microstructural morphologies observed by SEM in the as-cast Al_0.5CoCrCuFeNi HEA. Left: a lamellar flow pattern with alternating α (dark) and β (light) phases. Right: a random orientation of α and β phases.^{Reference Hemphill, Yuan, Wang, Yeh, Tsai, Chuang and Liaw20}

It was suspected that the large scatter in the fatigue testing data in Fig. 4(a) was attributed to defects inheriting from the raw materials, or introduced during the casting and rolling processes. To testify this hypothesis, Tang et al.^{Reference Tang, Yuan, Tsai, Yeh, Lundin and Liaw21} fabricated three groups of Al_0.5CoCrCuFeNi HEAs and processed them under three different conditions to acquire distinct amounts of macro- or microdefects, and then comparatively investigated their fatigue properties. Condition-1 samples were made by following the exactly same procedure as by Hemphill et al.^{Reference Hemphill, Yuan, Wang, Yeh, Tsai, Chuang and Liaw20} The Condition-2 samples were prepared in the same fashion, but an extra machining step was added to remove shrinkage pores and macrosegregation before cold rolling.^{Reference Tang, Yuan, Tsai, Yeh, Lundin and Liaw21} The Condition-3 samples were formulated by utilizing the high-purity raw materials (> 99.9 wt%), while the remaining steps were identical to Condition-1. Evidently, the Condition-1 samples retain a greatest amount of defects, followed by the Condition-2 samples, and finally Condition-3 samples. The high-cycle four-point bending tests of the three groups of HEA samples were conducted at the same conditions as in Hemphill et al.,^{Reference Hemphill, Yuan, Wang, Yeh, Tsai, Chuang and Liaw20} and the fatigue data were presented as symbols in Fig. 4(b) as the stress amplitude versus cycles to failure (S–N data), where the stress amplitude was defined as ${\sigma _{\rm{a}}} = {{\Delta \sigma } \over 2}$. Comparing the fatigue data from the three different conditions, it is seen that a significant degree of scatter in the fatigue life at a given σ_a is still very noticeable for the Condition-1 and Condition-2 samples. By contrast, the Condition-3 samples exhibit a much tighter distribution in the fatigue life. As an example, at σ_a = 495 MPa, the fatigue failure of the two Condition-2 samples varies from 2.7 × 10⁴ to 1.3 × 10⁶ cycles, whereas the failure of the two Condition-3 ones are in the range of 6.0 × 10⁴–9.8 × 10⁴ cycles. Considering the samples from all three conditions, the fatigue strength based on the stress amplitude, σ_a, is evaluated to have a lower bound of 270 MPa and an upper bound of 473 MPa. Converting these values into the stress range, Δσ, the resulting fatigue strength is from 540 to 946 MPa, which essentially agrees with the result found by Hemphill et al.^{Reference Hemphill, Yuan, Wang, Yeh, Tsai, Chuang and Liaw20} The fatigue ratio, ${{\Delta \sigma } \over {{\sigma _{{\rm{UTS}}}}}}$, is calculated to be 0.402 and 0.704.

3. Predictive modeling

The fatigue S–N relation of HEAs can be predicted by virtue of various statistical models. One of such models is the random endurance-limit model,^{Reference Tang, Yuan, Tsai, Yeh, Lundin and Liaw21} in which the fatigue life, n, is thought to be dependent on the applied stress, s, and the fatigue-endurance limit, γ, through

(7)$$\log \left( n \right) = {\beta _0} + {\beta _1}\log \left( {s - \gamma } \right) + \varepsilon ,\;\left( {s > \gamma } \right)\quad ,$$

where β₀ and β₁ are material constants. ε is a random error term.

Next, let V = log(γ), and it follows a probability density function (pdf) given by

(8)$${f_{\rm{V}}}\left( {v;{\mu _{\rm{r}}},{\sigma _{\rm{r}}}} \right) = {{{\varphi _{\rm{V}}}} \over {{\sigma _{\rm{r}}}}}\left( {{{v - {\mu _{\rm{r}}}} \over {{\sigma _{\rm{r}}}}}} \right)\quad ,$$

where μ_r and σ_r are the location and scale parameters, respectively.

Then, with x = log(s) and W = log(n), the marginal pdf and cumulative distribution function (cdf) of W are given, respectively, as

(9)$${f_W}\left( {w;x,\theta } \right) = \int_{ - \infty }^x {{1 \over {\sigma {\sigma _{\rm{r}}}}}{\varphi _{W|V}}} \left( {{{w - \left[ {{\beta _0} + {\beta _1}\log \left( {\exp \left( x \right) - \exp \left( v \right)} \right)} \right]} \over \sigma }} \right){\varphi _V}\left( {{{v - {\mu _{\rm{r}}}} \over {{\sigma _{\rm{r}}}}}} \right){\rm{d}}v\quad ,$$

and

(10)$${F_W}\left( {w;x,\theta } \right) = \int_{ - \infty }^x {{1 \over {{\sigma _{\rm{r}}}}}{\Phi _{W|V}}} \left( {{{w - \left[ {{\beta _0} + {\beta _1}\log \left( {\exp \left( x \right) - \exp \left( v \right)} \right)} \right]} \over \sigma }} \right){\varphi _V}\left( {{{v - {\mu _{\rm{r}}}} \over {{\sigma _{\rm{r}}}}}} \right){\rm{d}}v\quad ,$$

where both φ_W|V and φ_V are assumed to be the standardized normal pdf. Φ_W|V is the conditional cdf of W at a given V. θ = (β₀, β₁, σ, μ_r, σ_r). σ is a scale parameter.

To conveniently apply the above statistical model on the fatigue data of the Al_0.5CoCrCuFeNi HEAs, the fatigue data of the samples in each testing condition were denoted by (n _i, s _i, δ_i), i = 1, 2, …, m, where m is the total number of observations in the condition, s _i is the applied stress magnitude for the ith sample, n _i is the number of cycles to failure or 10⁷ run-out cycles, as indicated by the binary indicator of δ_i = 1 or δ_i = 0. The unknown parameters, θ, were estimated by the maximum likelihood method, that is, maximizing the likelihood function of the data given by

(11)$$L\left( \theta \right) = \prod\limits_{i = 1}^m {{{\left[ {{f_{\rm{W}}}\left( {{w_i};{x_i},\theta } \right)} \right]}^{{\delta _i}}}{{\left[ {1 - {F_{\rm{W}}}\left( {{w_i};{x_i},\theta } \right)} \right]}^{\left( {1 - {\delta _i}} \right)}}} \quad ,$$

where w _i = log(n) and x _i = log(s) for i = 1, 2, …, m. Numerical methods were used to compute the model parameters for the different conditions of Al_0.5CoCrCuFeNi HEA samples.

After having the model parameters determined from Eq. (11), the p-quantile fatigue life, n _p(s), at the stress level, s, can be calculated from

(12)$$\Pr \left( {n \le {n_{\rm{p}}}\left( s \right);s,\theta } \right) = p = {F_{\rm{W}}}\left( {\log \left( {{n_{\rm{p}}}\left( s \right)} \right);\log \left( s \right),\theta } \right)\quad .$$

The median fatigue life at p = 0.5 was computed to describe the average S–N behavior of the three conditions of HEA samples, and the predictions are shown in Fig. 4(b) as curves. In general, the mathematical predictions can reasonably capture the S–N relation of the HEAs in all three conditions.

When the fatigue-life data can seemly be classified into two groups [e.g., a strong group with long fatigue lives and a weak group with short fatigue lives in Fig. 4(a)], a Weibull mixture predictive model is more suitable. The pdf and cdf of the model are given, respectively, by

(13)$$f\left( {{{\left. {N\left( S \right)} \right|}_{\rm{p}}},{\alpha _{\rm{w}}}\left( S \right),{\beta _{\rm{w}}},{\alpha _{\rm{s}}}\left( S \right),{\beta _{\rm{s}}}} \right) = p{{{\beta _{\rm{w}}}} \over {{\alpha _{\rm{w}}}\left( S \right)}}{\left( {{{N\left( S \right)} \over {{\alpha _{\rm{w}}}\left( S \right)}}} \right)^{{\beta _{\rm{w}}} - 1}}\exp \left( { - {{\left( {{{N\left( S \right)} \over {{\alpha _{\rm{w}}}\left( S \right)}}} \right)}^{{\beta _{\rm{w}}}}}} \right) + \left( {1 - p} \right){{{\beta _{\rm{s}}}} \over {{\alpha _{\rm{s}}}\left( S \right)}}{\left( {{{N\left( S \right)} \over {{\alpha _{\rm{s}}}\left( S \right)}}} \right)^{{\beta _{\rm{s}}} - 1}}\exp \left( { - {{\left( {{{N\left( S \right)} \over {{\alpha _{\rm{s}}}\left( S \right)}}} \right)}^{{\beta _{\rm{s}}}}}} \right)\quad ,$$

and

(14)$$F\left( {{{\left. {N\left( S \right)} \right|}_{\rm{p}}},{\alpha _{\rm{w}}}\left( S \right),{\beta _{\rm{w}}},{\alpha _{\rm{s}}}\left( S \right),{\beta _{\rm{s}}}} \right) = p\left[ {1 - \exp \left( { - {{\left( {{{N\left( S \right)} \over {{\alpha _{\rm{w}}}\left( S \right)}}} \right)}^{{\beta _{\rm{w}}}}}} \right)} \right] + \left( {1 - p} \right)\left[ {1 - \exp \left( { - {{\left( {{{N\left( S \right)} \over {{\alpha _{\rm{s}}}\left( S \right)}}} \right)}^{{\beta _{\rm{s}}}}}} \right)} \right]\quad ,$$

where the subscripts, w and s, denote the weak and strong groups, respectively. P is the fraction of the samples that belong to the weak group. N is the cycles to failure. α_w(S) and α_s(S) are the Weibull scale parameters of the weak and strong groups, which scale with the stress, S, according to

(15)$$\log \left( {{\alpha _{\rm{w}}}\left( S \right)} \right) = {\gamma _{{\rm{w}},0}} + {\gamma _{{\rm{w}},1}}\log \left( S \right)\quad ,$$

and

(16)$$\log \left( {{\alpha _{\rm{s}}}\left( S \right)} \right) = {\gamma _{{\rm{s}},0}} + {\gamma _{{\rm{s}},1}}\log \left( S \right)\quad .$$

From Eqs. (13)–(16), the model contains 7 unknown parameters, that is, p, γ_w,0, γ_w,1, β_w, γ_s,0, γ_s,1, and β_s. Once again, these unknown model parameters were evaluated using the maximum likelihood method, with the likelihood function being

(17)$$L\left( {p,{\gamma _{{\rm{w}},0}},{\gamma _{{\rm{w}},1}},{\beta _{\rm{w}}},{\gamma _{{\rm{s}},0}},{\gamma _{{\rm{s}},1}},{\beta _{\rm{s}}}} \right) = \prod\limits_{i = 1}^m {f{{\left( {{N_i}} \right)}^{{\delta _i}}}{{\left[ {1 - F\left( {{N_i}} \right)} \right]}^{\left( {1 - {\delta _i}} \right)}}} \quad .$$

Once the values of the model parameters are determined, if a fatigue-life data at the ith sample, N _i, belongs to the strong or weak group can be decided by considering which group has a higher likelihood value. Also, the p-quantile fatigue lives of the samples in the strong and weak groups are, respectively, calculated as

(18)$${N_{{\rm{p}},{\rm{w}}}}\left( S \right) = \exp \left( {{\gamma _{{\rm{w}},0}} + {\gamma _{{\rm{w}},1}}\log \left( S \right)} \right){\left( { - \log \left( {1 - p} \right)} \right)^{{1 / {{\beta _{\rm{w}}}}}}}\quad ,$$

and

(19)$${N_{{\rm{p}},{\rm{s}}}}\left( S \right) = \exp \left( {{\gamma _{{\rm{s}},0}} + {\gamma _{{\rm{s}},1}}\log \left( S \right)} \right){\left( { - \log \left( {1 - p} \right)} \right)^{{1 / {{\beta _{\rm{s}}}}}}}\quad .$$

With the determined model parameters of p = 0.369, γ_w,0 = 15.238, γ_w,1 = −0.555, β_w = 3.773, γ_s,0 = 126.454, γ_s,1 = −16.245, and β_s = 0.612, the predicted S–N curves at p = 0.025, 0.5, and 0.975 for both the strong and weak groups are plotted in Fig. 4(a).^{Reference Hemphill, Yuan, Wang, Yeh, Tsai, Chuang and Liaw20} The median S–N curves may be used for predicting the fatigue life or the endurance limit of the HEAs, and the 2.5 and 97.5 quantiles are useful in the construction of a 95% predictive interval and quantify the scatter in the fatigue life.

4. Comparison with other alloys

The fatigue strength of the Al_0.5CoCrCuFeNi HEAs tested under three different conditions^{Reference Tang, Yuan, Tsai, Yeh, Lundin and Liaw21} was compared with some other commonly encountered structural alloys, including steels, aluminum alloys, nickel alloys, titanium alloys, bulk metallic glasses (BMGs), etc. The comparison is shown in Fig. 5(a) as plots of stress amplitude versus cycles to failure and compiled in Table II. For a clean presentation, the endurance limits for the crystalline alloys other than HEAs were plotted as dashed lines, and the data for BMGs are shown as a rectangle. Despite that the endurance limit of the HEAs covers a wide range from 270 to 473 MPa in the stress amplitude, σ_a (540–946 MPa in the stress range of Δσ), its lower bound still exceeds that of the majority of other alloys. Its upper bound is surpassed by some BMGs, mostly because these BMGs have higher strengths. To offset the strength effect, the stress amplitude was normalized by the ultimate tensile strength of the alloys, and the obtained fatigue ratio is replotted in Fig. 5(b).^{Reference Tang, Yuan, Tsai, Yeh, Lundin and Liaw21} As a consequence, the BMG fatigue rectangle is shifted downward relative to the HEAs, and the upper bound of the fatigue ratios of the HEAs becomes outperforming all alloys in comparison. Certainly, the lower half of the endurance limit range of the HEAs still overlaps with that of BMGs and some other alloys.

FIG. 5.

Fatigue data of the wrought Al_0.5CoCrCuFeNi HEAs in comparison with conventional alloys and BMGs. (a) Stress amplitude versus cycles to failure. (b) Fatigue ratio versus cycles to failure. (c) Fatigue-endurance limit versus ultimate tensile strength. (d) Fatigue ratio versus ultimate tensile strength.^{Reference Tang, Yuan, Tsai, Yeh, Lundin and Liaw21}

TABLE II.

Fatigue strength and fatigue ratio of the HEAs in comparison with some conventional alloys.

As the fatigue strengths of structural alloys are usually linearly proportional to their ultimate tensile strength (σ_UTS), it is easy to understand that the exceptional fatigue strength of the HEAs is partly attributed to their high strength (σ_UTS = 1344 MPa). This assertion is justified by viewing the fatigue-endurance limit versus σ_UTS in Fig. 5(c) as well as the fatigue ratio versus σ_UTS in Fig. 5(d). Overall, in comparison with many conventional alloys, the Al_0.5CoCrCuFeNi HEAs manifest favorable fatigue resistance and may have great potentials for structural applications that require prolonged fatigue lives.

B. Fatigue-crack growth rate

The material’s resistance to crack growth has been studied for the AlCrFeNi₂Cu, Al_0.2CrFeNiTi_0.2, and CrMnFeCoNi HEAs.^{Reference Seifi, Li, Yong, Liaw and Lewandowski36,Reference Thurston, Gludovatz, Hohenwarter, Laplanche, George and Ritchie37} The first two consist of fcc + bcc phases, whereas the last possesses a single fcc structure. For the AlCrFeNi₂Cu and Al_0.2CrFeNiTi_0.2 HEAs, three-point-bending tests were conducted to monitor the fatigue-crack growth of the as-cast, single-edge notch (SEN) samples at a cyclic frequency of 20 Hz in room-temperature air with a relative humidity of 40%, in accordance with the ASTM E647 standard.⁵⁶ The samples, notches, and precracks were sized in accordance with the ASTM E399 standard,⁵⁷ and the crack growth during the tests was monitored and controlled using a direct current potential drop technique.^{Reference Seifi, Li, Yong, Liaw and Lewandowski36} The fatigue tests on the CrMnFeCoNi HEAs were performed on disc-shaped compact tension [DC(T)] samples recrystallized at 1073 K for 1 h,^{Reference Thurston, Gludovatz, Hohenwarter, Laplanche, George and Ritchie37} also following the ASTM E647 standard.⁵⁶ After machining the samples and notches according to the ASTM E1820 standard⁵⁸ and performing suitable fatigue precracking, the cyclic loading was applied at a frequency of 25 Hz with a constant load ratio of R = 1. The crack length was monitored dynamically using both the optical-microscopy techniques and strain-gauge measurement on the back-face of the samples.^{Reference Thurston, Gludovatz, Hohenwarter, Laplanche, George and Ritchie37}

In consideration of the factors that affect the fatigue-crack-propagation behavior, the fatigue-crack-growth rates of the AlCrFeNi₂Cu and Al_0.2CrFeNiTi_0.2 HEAs had been surveyed at the load ratio of R = 0.1, 0.3, and 0.7,^{Reference Seifi, Li, Yong, Liaw and Lewandowski36} and the CrMnFeCoNi HEA had been tested at the temperatures of 293 and 198 K.^{Reference Thurston, Gludovatz, Hohenwarter, Laplanche, George and Ritchie37} The key results acquired from these investigations are presented as follows.

1. Temperature effect

The blue and green circles in Fig. 6 depict the crack-growth rate, ${{{\rm{d}}a} \over {{\rm{d}}N}}$, of the CrMnFeCoNi HEA at 198 and 293 K, respectively, as a function of the applied stress-intensity-factor range, ΔK.^{Reference Thurston, Gludovatz, Hohenwarter, Laplanche, George and Ritchie37} It is seen that at high ΔK, i.e., >10 MPa m^1/2, the crack growth rates of the alloy at the two different temperatures are nearly identical. The temperature effect on the crack-growth rate kicks in when ΔK moves toward small values. Specifically, at the same level of ΔK, the crack-growth rate at 198 K is approximately one order of magnitude smaller than that at 293 K. Treating the threshold fatigue stress-intensity-factor range, ΔK _th, as the maximum value of ΔK that gives a crack-growth rate approaching 10⁻¹¹ m/cycle, ΔK _th values for the alloys tested at 293 and 198 K were determined to be ∼4.8 and ∼6.3 MPa m^1/2.^{Reference Thurston, Gludovatz, Hohenwarter, Laplanche, George and Ritchie37} Clearly, the ΔK _th increases by ∼30% as the testing temperature decreases from 293 to 198 K. This is a clear implication that the CrMnFeCoNi HEA is more resistant to crack propagation at cryogenic temperatures, essentially consistent with the previous findings that reduced temperatures can lead to the improved yield strength, ductility, and fracture toughness.^{Reference Gludovatz, Hohenwarter, Catoor, Chang, George and Ritchie13,Reference Gludovatz, Hohenwarter, Thurston, Bei, Wu, George and Ritchie59} Besides, a modest change in the Paris exponent from 3.5 at 293 K to 4.5 at 198 K was noted.^{Reference Thurston, Gludovatz, Hohenwarter, Laplanche, George and Ritchie37}

FIG. 6.

The fatigue-crack-growth behavior of the CrMnFeCoNi, AlCrFeNi₂Cu, and Al_0.2CrFeNiTi HEAs in comparison with some other advanced materials.

Examination of the crack-propagation path at the sectioned mid-thickness plane by BSE and EBSD scans revealed that the transgranular fracture was the dominant crack propagation mode in the sample tested at 293 K.^{Reference Thurston, Gludovatz, Hohenwarter, Laplanche, George and Ritchie37} By contrast, the intergranular fracture prevailed in the sample at 198 K.^{Reference Thurston, Gludovatz, Hohenwarter, Laplanche, George and Ritchie37} As the intergranular fracture tends to cause a more deviated crack-propagation path than the transgranular fracture, more pronounced physical contact will occur between neighboring crack flanks in the sample tested at 198 K. Combining with the alloy’s higher strength at cryogenic temperatures, more pronounced roughness-induced crack closure would take place, and the alloy, thus, exhibited greater crack growth resistance at 198 K.

3. Comparison with other alloys

The full trajectories of fatigue-crack-growth rate of the CrMnFeCoNi,^{Reference Thurston, Gludovatz, Hohenwarter, Laplanche, George and Ritchie37} AlCrFeNi₂Cu,^{Reference Seifi, Li, Yong, Liaw and Lewandowski36} and Al_0.2CrFeNiTi^{Reference Seifi, Li, Yong, Liaw and Lewandowski36} HEAs were compared with some selected advanced alloys in Fig. 6, including the austenitic 304L steel,^{Reference Thurston, Gludovatz, Hohenwarter, Laplanche, George and Ritchie37} a twinning-induced plasticity (TWIP) steel,^{Reference Thurston, Gludovatz, Hohenwarter, Laplanche, George and Ritchie37} Inconel 718 superalloy,^{Reference Xiao, Chen and Chaturvedi62} and Zr_41.2Ti_13.8Cu_12.5Ni₁₀Be_22.5 BMG.^{Reference Gilbert, Ritchie and Johnson63} It can be seen that among these materials, the two types of Al-containing HEAs bear the highest ΔK _th and moderate Paris slope, m. They are, therefore, in the possession of favorable fatigue-crack-growth resistance in comparison with the Mn-containing HEA and nearly all other materials in comparison. The fatigue resistance of the Mn-containing HEA exceeds the Inconel 718 superalloy and is comparable with that of the 304L steel, TWIP-XIP1000 steel, and Zr_41.2Ti_13.8Cu_12.5Ni₁₀Be_22.5 BMG. In addition to these alloys, Table III compiles the fatigue threshold and Paris slope of more structural materials tested in comparable conditions. Out of the majority of these alloys, both the Mn-containing and Al-containing HEAs manifest favorable fatigue-crack-growth resistance. Particularly for the CrMnFeCoNi HEA, a remarkable combination of high strength, good ductility, excellent fracture toughness, and outstanding fatigue-crack growth resistance makes them a very promising structural material for applications under harsh environments.

TABLE III.

Fatigue-crack-growth properties and testing conditions of the HEAs in comparison with selected conventional alloys.

^a RT = room temperature.

C. Fractography

As conventional alloys, the fatigue failure of HEAs also experiences crack initiation, slow and stable crack propagation, and fast fracture. This process is illustrated in Fig. 7 by the SEM micrographs of the fracture surfaces of the Condition-3 Al_0.5CoCrCuFeNi HEA.^{Reference Tang, Yuan, Tsai, Yeh, Lundin and Liaw21} The alloy was made from high-purity raw materials (> 99.9 wt%) and homogenized at 1000 °C for 6 h, followed by water quenching and cold rolling, and fatigued under the three-point bending test at a frequency of 10 Hz, room temperature, and R = 1. Failure occurred at a stress amplitude of 495 MPa with a fatigue life of 97,715 cycles.^{Reference Tang, Yuan, Tsai, Yeh, Lundin and Liaw21} Figure 7(a) first gives an overview of the fatigue-fracture surface of the sample, from which the regions (b), (c), and (d) were, respectively, identified as the crack-initiation region, stable crack-propagation region, and fast fracture region.^{Reference Tang, Yuan, Tsai, Yeh, Lundin and Liaw21} It is seen that the fatigue crack is initiated from the corner of the tensile-stress surface. A more detailed examination of the crack-initiation region in Fig. 7(b) reveals many pores.^{Reference Tang, Yuan, Tsai, Yeh, Lundin and Liaw21} It is, therefore, reasoned that these pores raise up the stress locally and initiate the cracks. Figure 7(c),^{Reference Tang, Yuan, Tsai, Yeh, Lundin and Liaw21} a magnified view of the stable crack-propagation region, reveals a feature of combined striations and cleavage facets, which indicates that the cracks propagate in both ductile and brittle modes. The orientation of the striations is somewhat varied, rather than having a well-defined direction relative to the crack-propagation direction. This kind of striation morphology might stem from cracks initiating from multiple pores, and their propagation is also deviated by interior pores from time to time.^{Reference Tang, Yuan, Tsai, Yeh, Lundin and Liaw21} The fast-fracture region, shown closely in Fig. 7(d),^{Reference Tang, Yuan, Tsai, Yeh, Lundin and Liaw21} consists of numerous dimples and tear ledges, which are typical ductile fracture features. The ductile fracture of the Al_0.5CoCrCuFeNi HEA at the final stage of fatigue is found by SEM observations to be accommodated by a multitude of slip bands.^{Reference Tang, Yuan, Tsai, Yeh, Lundin and Liaw21} The Al_0.5CoCrCuFeNi HEAs obtained in other conditions exhibit similar fractographic characteristics.^{Reference Tang, Yuan, Tsai, Yeh, Lundin and Liaw21}

FIG. 7.

SEM graphs of the fracture surfaces of a Condition-3 Al_0.5CoCrCuFeNi HEA, failed at 97,715 fatigue cycles at a stress amplitude of 495 MPa, room temperature, and a load ratio of R = 0.1, showing the (a) overview, (b) crack-initiation region, (c) crack-propagation region, and (d) fast-fracture region.^{Reference Tang, Yuan, Tsai, Yeh, Lundin and Liaw21}

On the other hand, the fatigue-fracture surface of the AlCrFeNi₂Cu HEA in the fast-fracture phase features only a small amount of isolated dimple-like regions whereas the majority is the faceted brittle fracture regions.^{Reference Seifi, Li, Yong, Liaw and Lewandowski36} And the proportion and size of the dimple-like region becomes even reduced on the fracture surface of the Al_0.2CrFeNiTi HEAs.^{Reference Seifi, Li, Yong, Liaw and Lewandowski36} Therefore, the fatigue fracture in these two alloys is primarily dominated by the brittle failure, with the small-scale ductile failure.

D. Deformation and failure mechanisms

The TEM analyses had been applied to unveil the deformation mechanism underlying the remarkable fatigue resistance of the Al_0.5CoCrCuFeNi HEA.^{Reference Tang, Yuan, Tsai, Yeh, Lundin and Liaw21} Although it was found that nanotwins and tangled dislocations in the alloy had been triggered in the cold-rolling process prior to fatigue tests, cyclic loading indeed induced a higher density of nanotwins and tangled dislocations particularly at the crack-initiation site. Specifically, two sets of differently oriented nanotwins, $\left[ {2\bar{1}1} \right]$ and $\left[ {21\bar{1}} \right]$ ones, were created at the crack-initiation site. The increase in the amount of nanotwins takes place prior to crack initiation and induces extra work hardening, thereby offering more resistance to crack initiation. As a result, the Al_0.5CoCrCuFeNi HEA yields favorable fatigue strength and fatigue ratio when compared with conventional alloys.^{Reference Hemphill, Yuan, Wang, Yeh, Tsai, Chuang and Liaw20,Reference Seifi, Li, Yong, Liaw and Lewandowski36}

A. Fracture toughness

Fracture toughness is a measure of crack resistance or damage tolerance of materials, which plays a critical role in understanding the fracture mechanism of materials as well as designing practically used structural components.^{Reference Anderson38} Acquiring the damage-tolerance values of HEAs is indispensable for a complete assessment of their suitability in structural applications. As of now, fracture toughness of several single-phase HEAs, multiphase HEAs, and medium-entropy alloys (MEAs) has been measured.

1. Single-phase HEAs and MEAs

Gludovatz et al.^{Reference Gludovatz, Hohenwarter, Catoor, Chang, George and Ritchie13} conducted detailed fracture-toughness studies on a five-component equiatomic CoCrFeMnNi HEA, and a three-component equiatomic CoCrNi MEA^{Reference Gludovatz, Hohenwarter, Thurston, Bei, Wu, George and Ritchie59} using fatigue-precracked compact-tension specimens, both of which possess a recrystallized, single-phase fcc microstructure. The five-component alloy, with a uniform grain size of ∼6 μm, shows a crack-initiation fracture toughness, K _JIc, of 217 MPa m^1/2 at room temperature (RT) and a crack-growth fracture toughness, K, of more than 300 MPa m^1/2 after ∼2 mm crack propagation. As given in Fig. 8(a), the K _JIc value remains almost unchanged with lowering the temperature to 198 K (low temperature, LT) and cryogenic nitrogen temperature of 77 K (LNT), despite the strength remarkably increases (the yield strength, σ_y, and ultimate tensile strength, σ_UTS, increase by ∼85% and ∼70%, respectively). These astonishing toughness-fracture values are comparable to those of austenitic stainless steels, such as 304L and 316L at RT, and best cryogenic steels (e.g., 5Ni and 9Ni) at LNT. Such high toughness is attributed to a full ductile fracture by microvoid coalescence, in which microvoids nucleate at the crack tip due to stress concentration at Cr-rich and/or Mn-rich oxides [Fig. 8(b)], then grow together, and form a macroscopic flaw to fracture. These particles are easily introduced from processing, e.g., melting, and casting for the reactive element of Mn. The initial-crack-tip opening displacements at crack initiation (CTOD_i) are 57 ± 19 μm at RT and 49 ± 13 μm at LNT, which contribute to the exceptional crack initiation toughness of this alloy. The other contribution is derived from the steady strain hardening (the strain-hardening exponent, n, of ∼0.4 at RT, LT, and LNT), which can effectively suppress plastic instability. The steady strain hardening permits the increase of toughness with crack extension, ultimately resulting in values over ∼300 MPa m^1/2 for all testing temperatures. Such high steady strain hardening intrinsically comes from the plastic deformation by the predominated mode of the dislocation planar slip at RT, and deformation-induced nanotwinning at LNT, as illustrated in Fig. 8(c). Twinning enhances continuous strain hardening by introducing new twin boundaries when deformed at the liquid nitrogen temperature. Moreover, the samples tested at 77 K show more deflected crack paths on the fracture surface than those tested at RT, evidencing the enhanced toughness at LNT.

FIG. 8.

(a) J-integral crack–resistance curves of the fcc CrMnFeCoNi HEAs at 77 K, 200 K, and 293 K, from which the fracture toughness, K _JIC, can be determined.^{Reference Gludovatz, Hohenwarter, Catoor, Chang, George and Ritchie13} (b) Evidence of the formation of microvoids from Cr- or Mn-rich particles in the alloy at room temperature, suggesting the ductile failure.^{Reference Gludovatz, Hohenwarter, Catoor, Chang, George and Ritchie13} (c) Left: the EBSD images in the vicinity of the crack tip in the deformed CrMnFeCoNi HEA at 293 K, showing the formation of annealing twins and dislocation-induced grain misorientations. Right: the BSE and EBSD images in the wake of the propagated crack in the deformed alloy at 77 K, showing pronounced cell structures resulting from dislocation activities as well as deformation-induced nanotwins.^{Reference Gludovatz, Hohenwarter, Catoor, Chang, George and Ritchie13} (d) Comparison of the HEAs and other materials in the Ashby map of the fracture toughness versus yield strength.^{Reference Gludovatz, Hohenwarter, Thurston, Bei, Wu, George and Ritchie59}

The ternary CoCrNi MEA, following recrystallization, features a nonuniform microstructure with grain sizes of ∼5–50 μm. The fracture toughness, K _JIc, at crack initiation is 208 MPa m^1/2 and increases to ∼290 MPa m^1/2 as the crack extends by ∼2 mm.^{Reference Gludovatz, Hohenwarter, Thurston, Bei, Wu, George and Ritchie59} The crack initiation and growth toughness increase by 31 and 48%, respectively, when decreasing the temperature to LNT. The fracture toughness of the three-component alloy is comparable to that of the five-component alloy at RT, but higher by ∼25% at LNT. The high toughness values of the ternary alloy are associated with the fully ductile fracture by microvoid coalescence, with a more pronounced stretch zone at crack initiation. Microvoids of the ternary alloy are initiated at the Cr-rich particles, whereas in the five-component HEA, the initiation of microvoids is found at Cr-rich and Mn-rich particles of a higher volume fraction. The exceptional damage-tolerance of the ternary alloy at RT also results from the continuous steady strain hardening induced by a combined deformation mode of the pronounced dislocation slip and nanotwinning. Deformation-induced nanotwinning becomes the major mode at LNT to carry the plastic strain with increasing twinning activity, providing a steady source of strain hardening. Thus, the strength, ductility, and toughness of the CoCrNi MEA are simultaneously enhanced, exceeding those of the CoCrFeMnNi HEA [Fig. 8(d)].

2. Multiphase HEAs

With three-point bending of SEN samples, Seifi et al.^{Reference Seifi, Li, Yong, Liaw and Lewandowski36} reported that the conditional fracture toughness values, K _q, of two as-vacuum arc-cast Al_0.2CrFeNiTi_0.2 and AlCrFeNi₂Cu HEAs are in the range of 32–35 and 40–45 MPa m^1/2, respectively. Both the Al_0.2CrFeNiTi_0.2 and AlCrFeNi₂Cu HEAs have a mixed microstructure consisting of bcc and fcc phases, and their toughness values are far below that of the recrystallized, single-phase fcc HEAs and MEAs.^{Reference Gludovatz, Hohenwarter, Catoor, Chang, George and Ritchie13,Reference Gludovatz, Hohenwarter, Thurston, Bei, Wu, George and Ritchie59} The toughness of these alloys may be enhanced by introducing fatigue precrack with a notch-root radius of ∼100 μm.^{Reference Seifi, Li, Yong, Liaw and Lewandowski36} Increasing the testing temperature to 473 K increases the toughness of these two HEAs as well, and the load versus displacement curves start showing a nonlinear regime, opposed to the linear relation at RT. A combination of brittle features and some isolated ductile regions are observed on all fracture surfaces of the two HEAs tested at RT and 473 K [Fig. 9(a)]. The brittle and ductile features are roughly corresponding to the microstructural characteristics, e.g., the size and volume fraction of the bcc and fcc constituents, respectively. The Al_0.2CrFeNiTi_0.2 alloy is relatively less tough with fewer ductile regions and larger-scale brittle features, resulting from a less volume fraction of the fcc phase, as compared to that of the AlCrFeNi₂Cu alloy.

FIG. 9.

(a) SEM image of the fatigue-fractured surface of the AlCrFeNi₂Cu HEA, showing the cleavage failure along with some isolated dimple features.^{Reference Seifi, Li, Yong, Liaw and Lewandowski36} (b) Fractograph of the AlCoCrCuFeNi HEA exhibiting cleavage facets and river-like lines.^{Reference Roy, Roy, Daoud, Glatzel and Ray64} (c) Fracture surface of the AlCoCrFeNi HEA sintered by the powder metallurgy at 1273 K.^{Reference Mohanty, Maity, Mukhopadhyay, Sarkar, Gurao, Bhowmick and Biswas65} (d) Secondary electron image of the fracture surface of the AlCoCrFeNi HEA.^{Reference Zhang, Han, Meng, Su and Li66}

Roy et al.^{Reference Roy, Roy, Daoud, Glatzel and Ray64} determined the fracture toughness of the as-cast Al₂₃Co₁₅Cr₂₃Cu₈Fe₁₅Ni₁₅ HEA, by conducting three-point-bending tests on single-edge-notched bend (SENB) and chevron-notched rectangular bars (CVNRBs). The microstructure is dendritic with a large primary grain size, within which Fe–Co-rich bcc precipitates with sizes of 200–500 nm disperse in an ordered bcc Al–Ni-rich matrix, and a small volume fraction of fcc Cu-rich particles precipitate at grain boundaries. The fracture-toughness values from the SENB and CVNRB samples, K _q and K _Icv, are 5.8 ± 0.2 MPa m^1/2 and 5.4 ± 0.2 MPa m^1/2, respectively. Again, these fracture values are far lower than those of the fully recrystallized CoCrFeMnNi HEAs and CoCrNi MEAs. The K _q value is slightly higher than the K _Icv value due to the fact that the SENB samples are not fatigue precracked. The fracture surface exhibits four distinguishing characteristics, i.e., river-line-type features, featureless facets, regions mixed river lines and featureless facets, and secondary cracks [Fig. 9(b)]. Thus, the cracking of the material is of a cleavage type, similar to conventional crystalline materials.

The fracture toughness, K _I ≈ 3.9 MPa m^1/2, measured using the single-edge-V-notched beam method, is obtained for an annealed, spark-plasma-sintered (SPS) AlCoCrFeNi HEA.^{Reference Mohanty, Maity, Mukhopadhyay, Sarkar, Gurao, Bhowmick and Biswas65} Such a low toughness value is closely related to the full brittle fracture initiated from the nanosized σ-phase precipitates [Fig. 9(c)]. A much higher toughness of K _q ≈ 25.2 MPa m^1/2 for a SPS AlCoCrFeNi HEA is achieved in three-point-bending tests of the SENB specimens.^{Reference Zhang, Han, Meng, Su and Li66} A combination of the cleavage fracture in the bcc phase and microvoid coalescence in the fcc phase appears on the fracture surface [Fig. 9(d)]. The difference in the fracture toughness of the two sintered HEAs is associated with their microstructures, in which the tough HEA shows no appearance of the brittle nanocrystalline σ-phase.

B. Fracture under quasi-static uniaxial loading

The most studied fracture type is induced by quasi-static uniaxial loading, especially tension and compression. The fracture behaviors upon being subjected to uniaxial tension or compression stresses, depending on whether the individual HEA is ductile or brittle, is closely related to the measured properties, such as strength and ductility.

1. Quasi-static uniaxial tensile loading

The single-phase fcc CoCrFeMnNi alloy, the most studied HEA, shows dimples with Cr- and/or Mn-oxides on the fracture surface under uniaxial tension loading, which are formed by microvoid nucleation at the particles and coalescence.^{Reference Gludovatz, Hohenwarter, Catoor, Chang, George and Ritchie13,Reference Zaddach, Scattergood and Koch67,Reference Gali and George68} On the other hand, no particles occur inside the dimples on the fracture surface for the CoCrFeNi HEA due to the absence of the reactive element of Mn.^{Reference Zaddach, Scattergood and Koch67} Much less necking is found when decreasing the tested temperature down to 77 K.^{Reference Gludovatz, Hohenwarter, Catoor, Chang, George and Ritchie13,Reference Gali and George68} This is because the high strain-hardening shows up at low temperatures and tends to suppress necking instability. Such a ductile fracture is even unchanged in a hydrogen-charged CoCrFeMnNi HEA. Specifically, only ∼5% reduction in the tensile ductility is noted relative to the uncharged counterpart.^{Reference Zhao, Lee, Seok, Lee, Phaniraj, Suh, Ha, Kim, Ramamurty and Jang69} However, the intergranular fracture appears at the edges when the hydrogen concentration is high. Even so, a ductile mode of microvoid coalescence is still observed in the inner regions where the hydrogen concentration is low.^{Reference Luo, Li and Raabe70} When increasing the temperature to 1073 K and testing at a strain rate of 8 × 10⁻⁴ s⁻¹, intergranular fracture prevails with cracks perpendicular to the tensile stress axis.^{Reference He, Zhu, Zhou, Liu, Nieh and Lu71} Furthermore, processing routines, not only temperature, significantly influence the fracture behaviors of single-phase HEAs. The additively manufactured Al_0.3CoCrFeNi HEA is fractured by an intergranular mode.^{Reference Joseph, Stanford, Hodgson and Fabijanic72} A Bridgman-solidification-synthesized CoCrFeNiAl_0.3 HEA single crystal is tension-fractured by a shear mode with a relatively smooth surface, and a much less necking, compared to that of the polycrystalline specimen.^{Reference Ma, Zhang, Qiao, Wang, Gao, Jiao, Yang and Zhang73} The nanocrystalline CoCrFeMnNi HEA, prepared by severe plastic deformation using high-pressure torsion, shows a ductile fracture with dimples in sizes of several hundred nanometers^{Reference Schuh, Mendez-Martin, Völker, George, Clemens, Pippan and Hohenwarter7} [Fig. 10(a)]. The fracture mechanism transforms into an intergranular mode in a 723 K-annealed specimen due to the formation of multiphase particles at grain boundaries [Figs. 10(b) and 10(c)]. Further increasing the annealing temperature to 873 and 973 K, very dense dimple-containing particles are observed [Figs. 10(d) and 10(e)]. Coarser-sized dimples without particles are found when annealing at 1073 K^{Reference Schuh, Mendez-Martin, Völker, George, Clemens, Pippan and Hohenwarter7} [Fig. 10(f)]. A full ductile fracture covered by dimples is also shown in a cold-rolled, refractory HfNbTaTiZr HEA with a bcc structure,^{Reference Senkov and Semiatin4} and some other bcc-type alloys, including the Hf₂₅Nb₂₅Ti₂₅Zr₂₅,^{Reference Wu, Cai, Wang, Si, Zhu, Wang and Hui74} TaHfZrTi,^{Reference Huang, Wu, He, Wang, Liu, An, Wu and Lu75} and Hf_0.5Nb_0.5Ta_0.5Ti_1.5Zr HEAs.^{Reference Sheikh, Shafeie, Hu, Ahlström, Persson, Veselý, Zýka, Klement and Guo76}

FIG. 10.

Fractographs of the nanocrystalline fcc CoCrFeMnNi HEAs (a) subjected to severe plastic deformation, and continual annealing at (b) 450 °C for 1 h, (c) 450 °C for 15 h, (d) 600 °C for 1 h, (e) 700 °C for 1 h, and (f) 800 °C for 1 h.^{Reference Schuh, Mendez-Martin, Völker, George, Clemens, Pippan and Hohenwarter7}

A ductile fracture mode also exhibits in a tension-fractured, precipitation-strengthened CoCrFeNiMo_0.3 HEA, in which microvoids initiate along the grain boundary–precipitate interfaces and large intermetallic particles.^{Reference Liu, Lu, He, Luan, Wang, Liu, Liu, Chen and Liu77} Oxides of Cr and Mn inside dimples, again, are observed in the as-cast, multiphase Fe_0.4Cr_0.4NiMn_xCu (0 ≤ x ≤ 1.4) HEAs consisting of fcc and bcc phases, at which microvoids are nucleated.^{Reference Rao, Wang, Zhu, Chen, Wang, Si, Wu and Hui78} A mixed fracture feature with regions of dimpled and faceted appearance is present in a fractured specimen for a direct-laser-fabricated Al_0.6CoCrFeNi HEA, in which voids and cracks are formed at the fcc/bcc interface.^{Reference Joseph, Stanford, Hodgson and Fabijanic79} With the addition of Al, a homogenized AlCoCrFeNi HEA exhibits a transgranular fracture.^{Reference Ghassemali, Sonkusare, Biswas and Gurao80} Cracks are initiated at the triple junction of the bcc grains, and the fcc precipitates at the bcc grain boundaries block crack propagation. Thereby, the cracks are branched into new cracks, redirected, or even terminated at grain boundaries.^{Reference Ghassemali, Sonkusare, Biswas and Gurao80} Intergranular fracture is also reported in both as-cast and homogenized AlCoCrFeNi HEAs undergoing uniaxial tension at 973 K.^{Reference Tang, Senkov, Parish, Zhang, Zhang, Santodonato, Wang, Zhao, Yang and Liaw81} The intergranular cracks and/or voids form at the intersections of several grain boundaries or grain boundaries perpendicular to the principal tensile stress axis.^{Reference Tang, Senkov, Parish, Zhang, Zhang, Santodonato, Wang, Zhao, Yang and Liaw81} An eutectic AlCoCrFeNi_2.1 HEA, consisting of B2 and A1 phases, shows fracture features mainly of trench-like appearance when tested at ambient and cryogenic temperatures.^{Reference Lu, Gao, Jiang, Chen, Wang, Jie, Kang, Zhang, Guo, Ruan, Zhao, Cao and Li82}

C. Fracture under high strain-rate loading

1. Impact loading

A series of the as-cast Al_xCoCrFeNi (x = 0, 0.1, 0.75, and 1.5)^{Reference Xia, Gao and Zhang86} and Al_0.1, Al_0.3 HEAs^{Reference Li and Zhang87} with a fully recrystallized microstructure are Charpy-impact-tested at temperatures from 298 to 77 K using V-notch specimens. The recrystallized Al_0.1 and Al_0.3 alloys are predominately ductile-fractured at 298 K by microvoid nucleation and coalescence. The fracture surface shows brittle fracture features in a small amount of regions initiated by a high degree of shock hardening.^{Reference Li and Zhang87} With the addition of Al, the area of dimple-like regions decreases, i.e., the Al_0.3 samples contain more dimples than the Al_0.1 ones. When lowering the temperature to 77 K, the dimple-like regions gradually reduce and the brittle-fracture regions increase, with the alloys exhibiting a high density of tear edges and low impact energies.^{Reference Xia, Gao and Zhang86,Reference Li and Zhang87} However, an improvement in Charpy-impact energies with decreasing temperature is observed for the as-cast Al₀ and Al_0.1 specimens.^{Reference Xia, Gao and Zhang86} This trend is associated with the sizes of dimples on fracture surfaces (Fig. 11). The 77 K-tested specimens show much smaller and deeper dimples, compared to the specimens tested at 200 and 298 K [Figs. 11(a)–11(f)]. As for the Al_0.75 and Al_1.5 HEAs, brittle fracture features, e.g., river-like patterns and multifaceted surfaces, are present on the fracture surface, indicating a primary cleavage fracture mode^{Reference Xia, Gao and Zhang86} [Figs. 11(g)–11(i)]. Therefore, very low levels of impact energies (0.64–3.58 J) are obtained at 298, 200, and 77 K.^{Reference Xia, Gao and Zhang86}

FIG. 11.

The fractographs of the fcc CoCrFeNi (a–c), fcc Al_0.1CoCrFeNi (d–f), and bcc Al_1.5CoCrFeNi (g–i) HEAs failed in Charpy-impact tests at 77 K (a, d, and g), 200 K (b, e, and h), and 298 K (c, f, and i).^{Reference Xia, Gao and Zhang86}