Wrought Ni-based superalloys play a dominant role in aero engines and industrial gas turbines based on several advantages [[1], [2], [3]]. First, the austenitic structure of nickel can withstand a wide range of temperatures and bond with numerous alloying elements, which is beneficial for material strength. Second, such alloys have excellent oxidation resistance. Third, desired mechanical properties can be obtained by optimizing the morphology, distribution, and volume fraction of γ′ precipitates. Generally, wrought Ni-based superalloys have service temperatures based on a high degree of alloying to intensify solution or precipitation strength [[4], [5], [6]]. For example, a Ni‒Co-based superalloy, which contains Co and Ti, can increase the service temperature by nearly 50 °C [7]. However, a high degree of alloying makes plastic deformational behavior (i.e. the Portevin-Le Châtelier (PLC) effect) more complex.

Plastic deformation behavior is the most fundamental and important mechanical behavior of alloys. The stress‒strain curves of uniform deformation are smooth before necking occurs in tensile tests. However, most alloys exhibit plastic instability with yield points [8] or serrated flow [9] phenomena. The yield point phenomenon generates an abrupt drop in the stress‒strain curve at the end of the elastic region. This drop in stress is followed by fluctuations in the curve, which correspond to local deformation bands (Lüders bands) in a tensile sample. The serrated flow phenomenon in the plastic region is also known as the PLC effect, which is common in most Ni-based superalloys [[10], [11], [12]]. The PLC effect often generates serrations in stress‒time curves and step changes in strain‒time curve. Additionally, tensile samples exhibit the propagation of strain localization, which is often referred to as PLC banding. Deformation bands, including Lüders bands and PLC bands, have a negative impact on the surface quality and further processing of alloys. Lüders bands only appear at the beginning of plastic deformation and can be eliminated via predeformation, while PLC bands can appear at any time during plastic deformation and cannot be removed effectively. Therefore, it is necessary to study the effects of deformation conditions on PLC and avoid the deformation region in which the PLC effect occurs.

Portevin and Châtelier first conducted detailed experimental studies on the plastic instability of aluminum alloys and proposed the concept of “serrated yielding” [13,14]. This plastic instability phenomenon was named as the PLC effect by Cottrell to honor their efforts [15]. Dynamic strain aging (DSA) has been widely accepted as the main mechanism of the PLC effect [[16], [17], [18]]. A serrated flow and negative strain rate sensitivity are the predominant characteristics in the DSA regime [19]. The critical strain of a serrated flow, which depends on the strain rates and temperatures, is a crucial factor in the PLC effect. Based on variations in critical strain, DSA can be divided into the normal DSA and inverse DSA regimes [20]. DSA is based on interactions between solute atoms and mobile dislocations. Therefore, all factors (including stacking fault (SF) energy and γ′ precipitates) that affect these interactions have an influence on the PLC effect [21,22].

Additionally, significant attention has been paid to the PLC effect in Ni-based superalloys based on its influence on mechanical properties. Valsan et al. [23] determined that DSA influences the deformation and fracture behaviors of the Nimonic PE 16 superalloy in a temperature range of 450-550 °C at low strain rates. Gopinath et al. [24] identified dislocations pinned by solutes because DSA can offset softening during the low-cycle fatigue testing of the U720Li alloy. Rezende et al. [25] reported that DSA and precipitates are responsible for the high strength of the Inconel 718 superalloy at elevated temperatures. Pu et al. [26] inferred that the intermediate-temperature embrittlement of the Ni-based UNS N10276 superalloy may be associated with a strong PLC effect. Therefore, the PLC effect can be utilized to enhance mechanical strength, but should be avoided during hot deformation processes. Understanding the PLC effect has great significance for designing and applying wrought Ni-based superalloys.

This paper presents a review of PLC effects in wrought Ni-based superalloys. Our main focuses can be divided into the experimental phenomena and micromechanisms of PLC effects in wrought Ni-based superalloys. Additionally, shortcomings and prospects are discussed in the final section.

Serrated flow behavior is the main external manifestation of the PLC effect in wrought Ni-based superalloys. This behavior depends on the tensile temperature and strain rate [27]. Generally, the PLC effect can be divided into types A, B, and C. Fig. 1 presents the tensile stress‒strain curves of wrought Ni-based superalloys under different deformation conditions [28,29]. Serration shape changes with variations in tensile temperature are presented in Fig. 1(a). Serrations of types A, B, and C in the stress‒strain curves of the Inconel 718SPF alloy appear at temperatures of 227, 430, and 525 °C, respectively. Fig. 1(b) reveals that type-A, A + B, and B + C serrations in the stress‒strain curves of a wrought Ni-based superalloy occur at strain rates of 3 × 10^-3 s^-1, 3 × 10^-4 s^-1, and 8 × 10^-5 s^-1, respectively. In summary, type-A serrations appear at low tensile temperatures or large strain rates in the form of weak undulations in stress‒strain curves, which are characterized by a regular sequence of load drops alternating with smooth intervals. Type-C serrations typically occur at relatively low strain rates or high deformation temperatures and generate intense undulations and large stress drops in stress‒strain curves. Type-B serrations occur at intermediate strain rates and deformation temperatures, and the characteristics of type-B serrations lie between those of type-A and type-C serrations. In other words, the type of serrations changes from A to B, then to C with decreasing imposed strain rates and/or increasing deformation temperatures. This is because the diffusivity of solute atoms is a function of temperature and dislocation velocity is governed by the strain rate [30]. The higher diffusivity of solute atoms at higher temperatures and the slower dislocations at lower strain rates lead to more intense interactions between solute atoms and moving dislocations.

Stress drops also exhibit different characteristics corresponding to the three types of serrations [[31], [32], [33]]. Fig. 2 presents the distributions of stress drops for the Nimonic 263 superalloy at a strain rate of 4 × 10^-4 s^-1 at different deformation temperatures [34]. In the previous section, it was noted that the type of serrations changes from A to B, then to C with increasing deformation temperature. Based on the same rules, type-A, B, and C serrations appear at 200, 500, and 800 °C for the Nimonic 263 superalloy at a strain rate of 4 × 10^-4 s^-1. One can see that the distribution of stress drops corresponding to type-A serrations is exponential with a single peak at low stress amplitudes, whereas the stress drops corresponding to type-C serrations exhibits peak-shaped distributions with a maximum value at high stress drops magnitudes. The stress drops corresponding to type-B serrations exhibit intermediate distributions. Maj et al.[35] confirmed that the statistical analysis of stress drop magnitudes is a powerful method for describing the PLC effect based on a study of the Inconel 718 superalloy.

The occurrence of serrated yield phenomena always requires strain accumulation. Therefore, serrated flow appears on stress‒strain curves after reaching a certain strain value, which is referred to as the critical strain [30]. According to the relationships between critical strain and deformation conditions (strain rate or temperature), the PLC effect can be defined as a normal or inverse PLC effect. A normal PLC effect is often observed at high strain rates and/or low temperatures, where the critical strain increases with increasing strain rate or decreasing temperature. Inverse PLC effects exist at high temperatures and/or low strain rates, where the critical strain decreases with increasing strain rate or decreasing temperature [36]. Fig. 3 presents the variation in critical strain with strain rate and temperature for a wrought Ni-based superalloy [20]. One can see that a normal PLC effect occurs in the temperature range of 300-350 °C and an inverse PLC effect occurs in the temperature range of 400-500 °C.

Negative strain rate sensitivity is a distinguishing feature of the PLC effect. For a given temperature and strain, the strain rate sensitivity index can be calculated based on the following relationship [37]:

Where m is the strain rate sensitivity coefficient index, and σ1 and σ2 are the flow stress values corresponding to the strain rates of ε˙1 and ε˙2, respectively. For most materials, flow stress increases with strain rate (i.e., positive strain rate sensitivity). However, the strain rate sensitivity index is negative for materials that exhibit the PLC effect. Fig. 4 presents the variations in strain rate sensitivity with temperature for the Inconel 617 superalloy [38]. One can clearly see that the temperature range for the PLC effect is 300-700 °C, where serrations appear on the stress‒strain curves, based on the values of strain rate sensitivity.

Penning [39] proposed an N-shaped curve to describe the relationships between strain, strain rate, and stress when the PLC effect occurs, as shown in Fig. 5. In the plastic flow stage (ε˙∈[ε2,˙ε1˙]) , the material exhibits negative strain rate sensitivity and plastic instability. The strain rate jumps from ε˙2 to ε˙H when the strain rate reaches a value of ε˙2. The strain rate jumps to a small value of ε˙L when the strain rate is ε˙1 . Although the mode of the N-shaped curve was derived based on mathematical derivations, it can describe the spatiotemporal behavior of the PLC effect intuitively and effectively. Therefore, it has been accepted that the N-shaped curve is a precondition for serrated plastic instability.

Beukel [40] determined that negative strain rate sensitivity stems from the obstruction of dislocation motion by solute atoms. The solute composition at arrested dislocations cannot instantaneously respond to changes in the strain rate, meaning that it must transform gradually to accommodate a new quasi-steady state. During this transient period, the variation in flow stress also reaches a steady state, as shown in Fig. 6 [41]. According to changes in flow stress associated with the variation of strain rate, the strain rate sensitivity coefficient index can be written as [41,42]

Where m_i is the instantaneous strain rate sensitivity and m_f is the quasi-steady-state strain rate sensitivity. Numerous experiments have revealed that m_i is always positive at all strains, whereas m_f decreases with increasing of stain and eventually becomes negative [[43], [44], [45]]. It is clear that the strain rate sensitivity (m₀) is negative and the PLC effect occurs when mf>mi. Thus far, experiments and simulations on the PLC effect have largely been conducted based on negative strain rate sensitivity.

PLC banding is a macro phenomenon of strain localization, which is also associated with serrated flows on stress‒strain curves. PLC bands can be observed via shadowgraph [46], laser scanning extensometer [47], infrared thermal images [48], digital speckled pattern interferometry [49], and digital image correlation [[50], [51], [52]]. Similar to serrations, PLC bands are also divided into three types (A, B, and C) [53]. In general, type-A PLC bands have a characteristic of continuous propagation. Type-B bands are discontinuous or hopping based on the propagation of localized bands. Type-C bands occur at random intervals along the gauge length [54]. To date, little work has been conducted on the PLC bands of superalloys because all of the aforementioned observation methods have limitations at high temperatures.

Han [55] analyzed the nucleation, propagation, and temporal and spatial dynamic features of PLC bands for simple Ni-1at.%C binary alloys based on digital image correlation because Ni and C are the basic elements of wrought Ni-based superalloys. Fig. 7 presents the strain fringe patterns at a strain rate of 1 × 10^-3 s^-1 according to a stress‒strain curve with type-A serrations. The first nucleation of a PLC band occurs at position I, which is located on the end of the fixture, as shown in frame 1634. Following nucleation, the PLC band broadens slightly, then propagates to the other side. During propagation, the PLC band exhibits apparent expansion to position II (frame 1638). The interval time for this propagation is less than 0.2 s. Propagation stops when the PLC band expands to the middle of the gauge length. Next, a new PLC band nucleates in position III (frame 1642). This new PLC band propagates toward one side first, then propagates to the other side. It is worth noting that position III falls within the gauge length and is close to midpoint.

Fig. 8 presents the spatial distributions of strain along the centerline of a tested specimen at a strain rate of 1 × 10^-3 s^-1. In the cumulative strain contours, one can see that the PLC band propagates along the direction of the dotted arrows. Furthermore, the strain fringe pattern is composed of the propagation processes of multiple PLC bands (solid arrowheads). Successive PLC bands exhibit spatiotemporal correlation. Specifically, new PLC bands nucleate at the edges of previous bands. In particular, there is clear spatiotemporal correlation between the PLC bands in positions I and II.

Fig. 9 presents the strain fringe patterns at a strain rate of 1 × 10^-4 s^-1 for stress‒strain curves with type-B serrations. A PLC band with a “\” shape nucleates first and stops expanding once it reaches a certain width as the tensile testing progresses. Next, a second PLC band with a “/” shape nucleates to the side of the \-shaped PLC band. The shape of the third PLC band is the same as that of the second band. The fourth PLC band nucleates from the same side of the tensile sample. It should be noted that type-B PLC bands do not exhibit continuous propagation and do not tend to follow other types of bands (Fig. 10). The average propagation time for neighboring PLC bands is approximately 1.2 s.

When the strain rate is 1 × 10^-5 s^-1, the average propagation time for neighboring PLC bands is approximately 14.4 s. Fig. 11 presents strain fringe patterns corresponding to stress‒strain curves with type-C serrations. PLC bands nucleate in the middle of the tensile sample. A total of 17 PLC bands formed during the process of tensile testing, as shown in Fig. 12. The PLC bands propagate along one side from frame 10,808 to frame 11,010, then propagate along the other side from frame 11,010 to frame 11,053. Neighboring PLC bands exhibit spatiotemporal correlation because they are of type A. The nucleation sites of type-C PLC bands are randomly distributed within the gauge length, whereas the nucleation sites of the type-A PLC bands are located at the end of the fixture.

In general, type-A PLC bands exhibit the characteristic of continuous propagation, but initial nucleation sites are not always located at the ends of fixtures. This phenomenon has been explored by investigating PLC bands in steels using the digital image correlation method [56]. Zavattieri et al. [57] reported that the inhomogeneity of microstructures can generate nucleation sites for type-A PLC bands within the gauge length, meaning that the stress concentrations caused by microstructures are more significant than those inherent to testing fixtures. Although type-B PLC bands typically exhibit the characteristic of jump propagation, they are sometimes regular. Type-B PLC bands without jumping propagation were observed by Halim et al. [50] and Renard et al. [58]. Type-C PLC bands exhibit obvious spatial coupling, which may stem from stress concentrations.

Different types of PLC bands are indicative of the processes of nucleation and propagation for shear bands, which can reduce stress concentrations and homogeneous deformation. There are no evident corresponding relationships between the spatial coupling features of PLC bands and the mechanisms of DSA. On the micro level, PLC bands with different types result from the collective motion of mass mobile dislocations that are unpinned by solutes. Therefore, different types of PLC bands exhibit no distinct differences based on micro-mechanisms. This explains the close relationship between the spatiotemporal behaviors of PLC bands and the shape, size, and thickness of a sample.

Cui et al. [59] determined that both normal and inverse PLC effects occur at temperatures ranging from 300 to 500 °C for a Ni‒Co-based superalloy, as shown in Fig. 13(a). The defining tensile characteristic for the normal PLC effect is dislocation movement (Fig. 13(b)), while a high density of SFs is a characteristic of the inverse PLC effect (Fig. 13(c)). Therefore, it can be concluded that the inverse PLC effect has a relationship with SFs. The influence of SFs on the PLC effect has also been investigated in TWIP steels. Lee et al. [60] hypothesized that the PLC effect in a high-manganese TWIP steel stemmed from point defects or SFs breaking free from point defects. Lee et al. [61] claimed that C atom transitions and regular arrangements in SFs cause the PLC effect in TWIP steels.

The SF energy (SFE) of wrought Ni-based superalloys decreases with increasing Co contents within a certain range [62]. Wrought Ni-based superalloys with low SFE tend to exhibit many SFs or micro-twins during tensile testing. To confirm the influence of SFs or micro-twins on the PLC effect, wrought Ni-based superalloys with five different levels of Co content were subjected to tensile testing under different deformation conditions. All of the others elements in the alloys had the same mass fractions. The tested alloys had the same initial microstructures, but the temperature regimes for the PLC effect exhibited noticeable differences, as shown in Fig. 14 [29]. The temperature regime for the PLC effect shifts toward higher temperatures with increasing Co content.

Fig. 15 presents the true stress‒strain curves of the five superalloys with different Co contents under the same tensile conditions [29]. It is evident that the type of serrations transitions from C to A with decreasing SFE (increasing Co content). Generally, the PLC effect can be explained by DSA, which considers serrations formed by the interaction between solutes and dislocations. Therefore, the diffusion of solutes is a precondition for the PLC effect. The activation energy for DSA determines which atoms interact with dislocations. This energy lies in the range of 85-145 kJ/mol according to Ref, [63]. The alloys discussed above have similar activation energy values, implying that Co is not responsible for the mechanism of DSA [10]. In the results of microstructure observations, high densities of SFs or micro-twins were identified in the DSA regime.

To analyze the relationships between SFs (micro-twins) and the PLC effect in wrought Ni-based superalloys, tensile samples were pretreated with creep to generate high densities of SFs or micro-twins (Fig. 16(a)) [64]. The average stress drop is only 10 MPa for the sample free of SFs (micro-twins), but it can reach 150 MPa for a sample subjected to creep deformation, as shown in Fig. 16(b) [64]. The main deformation mechanism is dislocation slip for the tensile sample without creep pretreatment. The microstructures resulting from creep pretreatment include large amounts of SFs (micro-twins). Therefore, there is a strong relationship between SFs (micro-twins) and the PLC effect in wrought Ni-based superalloys. The influence of SFE on the PLC effect is involved in the deformation mechanisms of SFs (micro-twins).

For precipitation-strengthened wrought Ni-based superalloys, γ′ precipitates make a significant contribution to strength improvement by hindering dislocation movement during plastic deformation [65]. Pink et al. [66] hypothesized that the serrations on the stress‒strain curves of Al‒Li alloys are related to the shearing of precipitates. Sun et al. [67] stated that precipitates should not be ignored when considering the PLC effect in the A2024 aluminum alloy and solute clouds. However, the influence of γ′ precipitates on the PLC effect in wrought Ni-based superalloys are still unclear. To reveal the influence mechanisms of γ′ precipitates on the PLC effect, tensile tests of wrought Ni-based superalloys with different γ′ contents and morphologies were conducted systematically.

Four wrought Ni-based superalloys with various volume fractions of γ′ precipitates were analyzed to investigate their PLC effect [22]. Fig. 17 presents the stress‒strain curves obtained with different γ′ contents. One can see that the serrations change continuously from type-B to type-C serrations with increasing γ′ contents. The serration amplitude is large when the content of γ′ precipitates reaches 30 vol.%. The cause of these serration changes is that more γ′ precipitates prevent dislocation movement and a large number of atoms pin mobile dislocations, which increases the pinning force around dislocations.

Fig. 18 presents the variation in serration type as a function of deformation conditions and γ′ content [68]. It should be noted that increasing γ′ content causes the PLC effect region to shift toward lower temperatures and higher strain rates. Additionally, the type of serrations changes significantly with variation in γ′ content. Generally speaking, large serrations indicate a strong PLC effect in alloys, where there are strong interactions between mobile dislocations and a solute atmosphere. According to Ref. [53], the relationship between serration amplitude and the concentration of solute atoms along dislocation lines can be described as

Where γ represents the pinning strength of the solutes and C_s denotes the solute concentration along the dislocation lines. The delay time for changes in solute concentrations along dislocation lines increases with increasing γ′ content and temperature, or decreasing strain rate, which leads to the increase in C_s. Therefore, increasing serration amplitude (Δσ) causes the type of serrations to change from A to C.

The specific influence of γ′ content on the PLC effect was further analyzed based on deformation mechanisms. Fig. 19 presents the shapes of serrations for alloys with different γ′ contents and their corresponding deformation microstructures [22,68]. These two alloys exhibit similar microstructures at the same temperature. The alloys begin deforming at 300 °C and exhibit dislocation bands, while SFs can be observed at 500 °C. However, the SF density for the 30 vol.% alloy is much greater than that for the 5 vol.% alloy.

Fig. 20(a) reveals that γ matrix compositions do not change with γ′ content. Furthermore, similar activation energies in the range of 65-100 kJ/mol indicate that the diffusing atomic species associated with the PLC effect are not altered by γ′ precipitates, as shown in Fig. 20(b) [22]. However, the serration characteristics are significantly influenced by the fraction of γ′ precipitates. Increasing γ′ precipitate content leads to high-density SFs, which also influence DSA via the Suzuki effect and the SF intersections of different slip planes. Therefore, γ′ precipitates can promote the PLC effect in wrought Ni-based superalloys.

Cui et al. [69] inferred that the tertiary γ′ precipitate (particle diameter below 50 nm) fraction and size had significant effects on the serrated flow of a wrought Ni-based superalloy, whose microstructures were controlled via solution and aging treatments. This is because variations in precipitates can alter the substitutional solute atom concentration and average spacing between precipitates. Therefore, the mobile dislocation mode and interactions between substitutional solute atoms and dislocations, which are believed to be responsible for DSA, and also change following different heat treatments [70]. To analyze the influence of γ′ precipitates on the PLC effect in wrought Ni-based superalloys further, Nimonic 263 alloys subjected to different heat treatments were examined in Refs. [71,72]. Fig. 21 presents the microstructures of the tested alloys. With increasing aging time, the mean radius of the γ′ precipitates increases linearly and the mean edge-to-edge inter-precipitate distance increases significantly, while the area number density of γ′ precipitates decreases significantly. The volume fraction of γ′ precipitates increases to approximately 14% after aging for 2 h and reaches a thermodynamic equilibrium state (approximately 17%) after aging for 50 h. The morphology of the γ′ precipitates remains spherical with prolonged aging time. Although the concentrations of Al, Ti, and Mo change significantly, the leading element (Cr) concentration related to the PLC effect in Nimonic 263 alloys only changes slightly in the matrix and precipitates.

Fig. 22 plots the stress‒strain curves of Nimonic 263 alloys with different aging times tested at 500 °C with a constant strain rate of 4 × 10^-4 s^-1 [71,72]. It is apparent that the mean stress drop increases initially with increasing aging time, then decreases when the aging time increases over 50 h. The critical strain exhibits a similar variation trend to the stress drop amplitude and reaches a maximum value after 50 h of aging. The type of serrations changes from A + B to C with increasing aging time. The mean delay time of mobile dislocations at obstacles and the mean free-flying time of mobile dislocations after overcoming obstacles were also calculated in Ref. [71]. The change in the mean delay time is similar to that in the mean stress drop and critical strain with increasing aging time, while the variation in mean free-flying time is less obvious.

The deformation mechanisms of wrought Ni-based superalloys must be influenced by the size of γ′ precipitates. For example, the deformation mechanism of a wrought Ni-based superalloy [73] changes from anti-phase boundary (APB) shearing to SF shearing to Orowan bypassing with increasing precipitate size. Fig. 23 illustrates deformation mechanisms based on the mean radius of γ′ precipitates [71]. The deformation mechanisms are controlled by dislocation slipping in the matrix, APB-coupled a/2 < 101 > -type dislocations shearing γ′ precipitates, and slip bands continuously cutting γ and γ′ phases when the γ′ precipitate mean radius is less than 28 nm. The deformation mechanisms shift to APB-coupled pairs of a/2 < 101 > -type dislocations shearing γ′ precipitates, Shockley partial dislocation continuously shearing γ and γ′ phases, and matrix dislocations bypassing γ′ precipitates when the size of γ′ precipitates is between 28 and 45 mm. The deformation mechanisms are dominated by the Orowan bypassing process when the γ′ precipitate size is greater than 45 mm. According to the variation in PLC characteristics, there are close relationships between the PLC effect and deformation mechanisms.

These variations in deformation mechanisms cause yield strength to increase and then decrease, which indicates that the hindering effects of obstacles to mobile dislocations correspondingly grow stronger initially and then grow weaker. The mean delay time for mobile dislocations at obstacles and the mean stress drop are directly reflected by the hindering effects. Dislocations impeded by γ′ precipitates can overcome obstacles via shearing or bypassing mechanisms when the loading stress is enough. Therefore, the mean delay time for mobile dislocations at obstacles is closely linked to the content and size of γ′ precipitates. Additionally, the delay time reaches a maximum value when the aging time is 50 h. Solutes rapidly gather at arrested dislocations by diffusion during the delay time. A higher concentration of solutes clustered at dislocations leads to greater pinning force, resulting in a maximum mean stress drop after aging for 50 h. Consequently, γ′ precipitates influence the PLC effect in Nimonic 263 alloys by hindering dislocation motion.

For the sake of conveniently analyzing the relationship between γ′ precipitates and critical strain, elementary incremental strain was introduced in Ref [71]. as follows:

Where Ω is the elementary incremental strain, ε˙ is the rate of dislocation movement, t_w is the delay time for mobile dislocations, b is the Burgers vector, and ρm and ρf are the densities of mobile and forest dislocations, respectively. The PLC effect can occur when the elementary incremental strain lies in the interval of τ0ε˙X13/2,τ0ε˙X23/2 as shown in Fig. 24. Here, τ0 is the relaxation time associated with diffusion, and X₁ and X₂ are two solutions that cause negative strain rate sensitivity to appear.

When the aging time is less than 50 h, the inhibition of dislocations via shearing mechanisms increases with increasing content and size of γ′ precipitates, which reduces the density of mobile dislocations. The curve of elementary incremental strain vs. strain moves toward to the bottom right, meaning that the critical strain increases with increasing aging time, as shown in Fig. 24. When the aging time is greater than 50 h, the edge-to-edge inter-precipitate distance increases with increasing aging time, allowing dislocations to bypass γ′ precipitates more easily, meaning that the density of mobile dislocations increases again. Therefore, the curve of elementary incremental strain vs. strain moves toward the top left, meaning that the critical strain decreases with increasing aging time. In conclusion, γ′ precipitates can directly influence the PLC effect in wrought Ni-based superalloys by altering deformation mechanisms.

Ni-based superalloys with different shapes of γ′ precipitates were prepared by different heat treatment processes, as shown in Fig. 25 [74]. The shapes of the γ′ precipitates in the three alloys flowers, spheres, and cubes, respectively. These alloys have similar sizes (0.7-1.2 μm) and volume fractions (approximately 46%) of γ′ precipitates.

Fig. 26(a) presents enlarged stress‒strain curves at 400 °C. The serrations are of type A and the PLC effect exhibits normal characteristics when the γ′ precipitates are flowers or cubes. Although the PLC effect is still normal, the type of serrations is A + B when the γ′ precipitates are spheres. Fig. 26(b) presents enlarged stress‒strain curves at 500 °C. The serrations transform to type C and the PLC effect exhibits inverse characteristics for the alloys with flower-shaped or cubic γ′ precipitates. The serrations change to type B and the PLC effect is normal when the γ′ precipitates are spheres.

The microstructures after tensile testing were observed to analyze the influence of γ′ precipitate shape on the PLC effect in Ni-based superalloys. Dislocation pileup occurs around γ′ precipitates and some SFs run through matrices and precipitates in the alloys with flower-shaped or cubic γ′ precipitates, which exhibit tensile deformation at 400 °C. The main deformation mechanisms are dislocation slipping and SFs. Masses of dislocations shear γ′ precipitates and form dislocation arrays that accumulate in front of obstacles when alloys with spherical γ′ precipitates exhibit tensile deformation at 400 °C. Additionally, the degree of dislocation pileup around γ′ precipitates decreases significantly compared to that in the alloys with flower-shaped or cubic γ′ precipitates. When tensile testing is conducted at 500 °C, the main deformation mechanism is dislocation slipping. Additionally, some APB-coupled dislocations shear γ′ precipitates in the alloy with flower-shaped γ′ precipitates. High micro-twin density appears in some regions of the alloy with cubic γ′ precipitates. The micro-twins run through matrices and precipitates in the alloy with spherical γ′ precipitates. In summary, it is easier to inhibit dislocation motion for γ′ precipitates with complex shapes based on larger mismatches. In other words, the influence of flower-shaped and cubic γ′ precipitates on the PLC effect in wrought Ni-based superalloys is stronger than that of spherical γ′ precipitates.

DSA is a generally accepted micromechanism of the PLC effect [[75], [76], [77]]. However, specific microprocess descriptions (e.g., the dispersal mode of solutes and interactions between solutes and dislocations) are still under discussion based on a lack of direct observational evidence.

Cottrell and Jaswon [78] first presented DSA to explain the PLC effect based on the relationship between the velocity of solute diffusion and the velocity of dislocation movement. If the velocity of solute diffusion is similar to or slightly less than that of dislocation movement, the PLC effect can occur. Dislocations move with solute atoms, which increases overall resistance to dislocation motion. When an applied load can overcome this resistance, dislocations can move freely because they are released from solute constraints. The solute atoms then catch up dislocation again based on diffusion. Such repetitive processes lead to the instability of flow stress, which manifests as macroscopic serrations in stress‒strain curves. However, Cottrell [15] determined that solute atoms diffusion is too slow to pin mobile dislocations. To resolve this conflict, a vacancy theory was proposed, where the concentration of vacancies is increased by plastic deformation or quenching, which improves the diffusion rate of solute atoms. Therefore, the solute atoms can generate pinning and cause the PLC effect.

A phenomenological model based on Cottrell’s theory can successfully predict the function of critical strain, temperature, and strain rate, which is described as follows [79,80]:

Where εc is the critical strain, l is the effective radius of the atomic atmosphere, ε˙ is the strain rate, T is the temperature, k is the Boltzmann constant, b is the Burgers vector, D₀ is the diffusion frequency factor, Q_m is the effective activation energy for solute migration, m is the correlation index between vacancy concentration and critical strain, β is the correlation index between dislocation density and critical strain, and K and N are coefficients related to the vacancy concentration and mobile dislocation density, respectively. However, the critical strain calculated using Eq. (5) differs significantly from experimental values and Eq. (5) does not consider the effects of solute atomic concentration on critical strain [81].

McCormick [80] hypothesized that dislocation motion is discontinuous [82,83] and that solute atoms diffuse into dislocations only if dislocations are arrested by obstacles. The arrested dislocations then break free and move to subsequent obstacles. The conditions for the PLC effect are t_a<t_w, where t_a is the aging time required to pin an arrested dislocation and t_w is the delay time at an obstacle. According to this hypothesis, the relationship between critical strain, solute concentration, strain rate, and temperature can be defined as follows:

Where C₁ is the solute concentration on the arrested dislocation line, C₀ is the solute concentration of the alloy, and A is a constant. Eq. (6) not only successfully forecasts the occurrence of serrated flows, but also considers the solute concentration on the arrested dislocation line. Furthermore, Beukel [40] and Reed-Hill et al. [84,85] extended and optimized the DSA theory.

Both Cottrell’s and McCormick’s theories are based on the hypothesis that vacancies contribute to the bulk diffusion of solute atoms. However, this hypothesis falls apart for Ni‒C alloys [86,87]. Mulford and Kocks [88] proposed the Mulford‒Kocks model based on dislocation arresting theory [89], which introduced the pipe diffusion mechanism of solute atoms, during their study of DSA in the Inconel 600 alloy. The dislocation arresting theory suggests that solute atmospheres form around forest dislocations and are then transmitted from forest dislocations to arrested dislocations. Additionally, a solute atmosphere can simply pin a portion of the mobile dislocation line at the forest dislocation junction. The solute atoms can then drain back into the forest dislocations after unpinning. This theory resolves the contradiction between the rate of bulk diffusion and temperature (strain rate), and is widely accepted as the primary DSA mechanism [[90], [91], [92]].

Recently, Picu and Zhang [93] determined that the activation energy required for pipe diffusion in the absence of vacancies is similar to that required for bulk diffusion. Additionally, vacancy-assisted pipe diffusion is limited by a lack of driving force. Curtin et al. [94,95] found that calculated critical strain rates for the PLC effect are many orders of magnitude smaller than experimental values. The strength of a fully formed solute atmosphere can reach as high as 500 to 5000 MPa, causing the calculated binding energy to diverge from the actual value. Subsequently, the “cross-core” diffusion mechanism was proposed, where solutes directly cross the slip plane from the compression to the tensile side in the core of a dislocation. Legros et al. [96] first observed the pipe diffusion phenomenon via in situ transmission electron microscopy and confirmed that pipe diffusion velocity is almost three orders of magnitude greater than bulk diffusion velocity. This discovery provided strong support for the pipe diffusion theory.

For wrought Ni-based superalloys, the interactions between solute atoms and dislocations are complex based on large additions of different elements. Additionally, there are interactions between different elements, meaning that multiple elements should be considered when the influence of a particular element is investigated. The acting elements in the DSA mechanism should be divided into two categories: substitutional elements (Cr, W, and Mo), and interstitial elements (C and B). Some researchers have concluded that C atoms dominate the normal PLC effect, while substitutional elements influence the inverse PLC effect in age-hardened Ni alloys [97]. Li et al. [98] determined that the dragging effect of hydrogen atoms on dislocations contributed to the PLC effect in the Inconel 718 superalloy when hydrogen embrittlement behavior was investigated at room temperature. Beese et al. [99] confirmed that interstitial C is the primary control element for the PLC effect in Inconel 6255. Kuang and Was [100] hypothesized that the serrations on the stress‒strain curves of alloy 690 at 360 °C result from C atmospheres formed at dislocations. Gopinath et al. [36] believed that substitutional atoms are responsible for the normal PLC effect in alloy 720 Li because the pinning of dislocations by interstitials in face-centered cubic alloys is weak. Han et al. [101] also inferred that the diffusion of substitutional elements (Cr, Mo) is responsible for the PLC effect in the Nimonic 263 superalloy based on calculations of activation energy. Furthermore, their conclusions were verified through comparisons to PLC effects in simple binary alloys and the Nimonic 263 alloy [34]. Max et al. [102] concluded that Mo solute atom diffusion plays a significant role in the PLC effect in the Inconel 718 alloy. Rao et al. [38] reported that the diffusion of C through dislocation cores is the controlling mechanism of the PLC effect in Inconel 617 at low temperatures, while substitutional elements (Cr and No) operate at higher temperature ranges. Based on this conflicting information, we believe that the influence of solute atoms on the PLC effect in wrought Ni-based superalloys should be studied in combination with specific components.

According to the discussion above, the pinning and unpinning of solute atoms with dislocations are generally considered to be the main causes of the normal PLC effect. Regardless, the mechanisms of the inverse PLC effect are still unclear. Xu et al. [103] studied the influence of SFs on the PLC effect in a wrought Ni-based superalloy using transmission electron microscopy. The density of SFs increases with increasing temperature and reaches a maximum value at 450 °C, as shown in Fig. 27. These results indicate that SFs play a crucial role in the inverse PLC effect. In the microstructure observations in Fig. 28, SFs largely occur along one slip plane, but expand to other planes with increasing strain. The corresponding stress‒strain curve contains both upward and downward serrations. The solute atoms are concentrated around the SFs and work to pin the partial dislocations on both sides of SFs at low strain rates. When the applied load reaches a certain threshold, the occurrence of unpinning results in upward serrations in the stress‒strain curve. With increasing extent of deformation, numerous non-parallel SFs occur and act as obstacles to each other, which leads to downward serrations in the stress‒strain curve.

In conclusion, studies on DSA mechanisms have largely focused on the interactions of solute clouds, mobile dislocations, and the obstacles of forest dislocations or precipitates. Experimental methods mainly rely on the static observation and analysis of deformed specimens. The detailed processes of modes and paths of solute diffusion, as well as the unpinning of dislocations, are difficult to observe. More direct evidence regarding the interaction between solute atoms and dislocations on the atomic scale can be obtained by applying electron microscopy in future studies.

This paper reviewed the PLC effect in wrought Ni-based superalloys, which have been featured in numerous publications, including work from our group, in recent years. The main conclusions of our research are as follows:

(1)The PLC effect in Co-rich wrought Ni-based superalloys is caused by unpinning, which can be explained by interactions of mobile dislocations with substitutional solute (Cr) atmospheres. Increasing Co content (or decreasing SFE) causes the PLC region to move toward higher temperatures. Furthermore, the large amplitudes of stress drops in low-SFE superalloys are related to SFs (or micro-twins).

(2)γ′ precipitates, temperature, and strain rate are the controlling factors of the PLC effect in wrought Ni-based superalloys. The PLC effects in wrought Ni-based superalloys with different γ′ precipitates are all essentially attributed to DSA. γ′ precipitates only promote the occurrence of the PLC effect by obstructing dislocation movement.

(3)All of the PLC bands corresponding to type-A, B, and C serrations for wrought Ni-based superalloys result from strain localization. Their propagation characteristics are influenced by external conditions and internal microstructures. The angle between PLC bands and the tensile direction is approximately 60° during propagation.

This work was financially supported by the National Natural Science Foundation of China (Nos. 51671189 and 51271174) and the Ministry of Science and Technology of China (Nos. 2017YFA0700703 and 2019YFA0705304).