A physically-based model considering dislocation–solute atom dynamic interactions for a nickel-based superalloy at intermediate temperatures

Materials and Design 183 (2019) 108122

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A physically-based model considering dislocation–solute atom dynamic interactions for a nickel-based superalloy at intermediate temperatures Y.C. Lin a,b,⁎, Hui Yang a, Dao-Guang He a,⁎⁎, Jian Chen c a b c

School of Mechanical and Electrical Engineering, Central South University, Changsha 410083, China State Key Laboratory of High Performance Complex Manufacturing, Changsha 410083, China School of Energy and Power Engineering, Changsha University of Science and Technology, Changsha 410114, China

H I G H L I G H T S

G R A P H I C A L

A B S T R A C T

• Serrated flow features of a nickel-based superalloy with different initial microstructures were study. • Serrated flow features are prominently influenced by deformation parameters and initial microstructures. • A physically-based constitutive model is developed considering dislocationsolute atoms dynamic interactions. • The evolution of moving dislocation, solute atoms concentration, and the strain rate dependent stress are analyzed. • The established model can well described the intermediate-temperature the serrated flow features.

a r t i c l e

i n f o

Article history: Received 6 July 2019 Received in revised form 1 August 2019 Accepted 12 August 2019 Available online 14 August 2019 Keywords: Superalloy Flow behavior Serrated flow Dislocation-solute atom interaction Numerical analysis

a b s t r a c t The uniaxial tensile tests are performed to study flow behavior of a nickel-based superalloy with different initial microstructures at an intermediate temperature range (473–973 K). The experimental results show that the obvious serrated flow characteristics can be observed from the flow stress curves. Meanwhile, it is concluded that the occurrence of serrated flow features not only associate with deformation parameters, but also are prominently influenced by different initial microstructures. Furthermore, based on the plastic deformation mechanism of the investigated superalloy and dislocation-solute atoms dynamic interactions, a physically-based constitutive model is developed. The dynamic characteristics of the developed model are discussed, i.e., the evolution of moving dislocation, solute atoms concentration, and the strain rate dependent stress are analyzed. Additionally, various types of serrations are numerically simulated by appropriate parameters. Also, the predicted results show a good coincident with the observed data, suggesting that the established model can accurately describe the intermediate-temperature flow behaviors involving the serrated flow features of the investigated nickel-based superalloy. © 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction ⁎ Correspondence to: Y.C. Lin, State Key Laboratory of High Performance Complex Manufacturing, Changsha 410083, China. ⁎⁎ Corresponding author. E-mail addresses: [email protected] (Y.C. Lin), [email protected] (D.-G. He).

During the intermediate-temperature deformation, Portevin-Le Chatelier (PLC) effect is an unstable and heterogeneous plastic flow phenomenon and it is commonly observed in some Ni-based

https://doi.org/10.1016/j.matdes.2019.108122 0264-1275/© 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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Fig. 1. The detailed testing samples and its geometry dimension (unit: mm).

superalloys [1–5]. Especially, this phenomenon manifests itself by propagating strain bands of localization plastic deformation [6], and associated with serrated flow features on the stress-strain curves [7,8]. Recently, abundant investigations have been denoted to studying the serrated flow characteristics of metallic alloys. For instance, the refined composite multiscale entropy method was used to model the serrated flow behavior of carburized steel alloys [9]. Luo et al. [10] found that the PLC effects have prominent effects on thermo-mechanical properties in Ti\\Mo alloys. Krishna et al. [11] concluded that the PLC effects or serrated flow prominently affected by the dislocation density and grain size of an AA5083 alloy. Min et al. [12] concluded that the influences of deformation parameter on spatiotemporal behavior of PLC bands in a TWIP steel. Mazière et al. [13] researched that the critical plastic strain for onset of serrated flow behavior of an aluminum alloy. Prasad et al. [14] demonstrated the effects of PLC effects on the lowcycle fatigue performances in an IMI 834 alloy. Meanwhile, several scholars summarized that the occurrence of serrated flow can change plasticity characteristics of engineering materials, such as ultimate tensile strength [15,16], tensile ductility [17,18], surface roughening [19,20] and type of rupture [21].

Generally, the suitable constitutive descriptions can describe the flow behaviors involving PLC effects of materials. In past years, some models have been developed to characterize the flow behaviors considering serrated flow characteristics in many alloys. For instance, Benallal et al. [22] and Sheikh et al. [23] proposed suitable constitutive models to depict the serrated flow features of hot deformed aluminum alloys. According to the negative strain rate sensitivity of PLC effect [24], Dierke et al. [25] established a suitable statistical model to characterize the serrated flow features in an Al\\Mg alloy. Rizzi et al. [26] developed a physically-motivated model associated with dynamic response of PLC effect. Garg et al. [27] and Gupta et al. [28] established the accurate constitutive equations to describe the dynamic strain aging characteristics of an austenitic stainless steel. Ni-based superalloys with excellent properties are extensively applied in critical components of modern nuclear and aerospace industries [29–32]. The complicated precipitates (γ', γ″ and δ) and multiple chemical compositions greatly affect the flow behaviors and mechanical properties of these alloys [33–36]. Lin et al. [37] proposed the physically-based constitutive description for hot flow characteristics of a nickel-based superalloy in the compressive process. Tang et al. [38] researched the microstructural evolution of a Ni-based superalloy in the hot forming process, and proposed a unified model to describe the flow behavior. Considering the evolution of dislocation substructures, Chen et al. [39] established the artificial intelligent model to characterize the high-temperature deformation behavior of an IN718 alloy. Wen et al. [40] studied that the influences of different aging heat treatments on microstructures and processing maps of a GH4169 superalloy. He et al. [41] analyzed the dissolution mechanisms and kinetics of δ phase in a GH4169 superalloy during the hot deformation.

Fig. 2. Three kinds of initial microstructures with different heat treatment status of: (a) ST; (b) HS; (c) SAT; (d) SAD patterns of γ' and γ″ phases.

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Obviously, many investigators have studied the microstructural evolution and high-temperature flow behaviors in nickel-based superalloys. But, few investigations focused on developing accurate constitutive models to describe the intermediate-temperature deformation behaviors at low strain rates, especially considering the serrated flow characteristics. In present research, the intermediatetemperature (473–973 K) deformation tests were conducted on a typical Ni-based superalloy under three initial microstructural conditions. Based on experimental data and major plastic deformation mechanism, a physically-based model is established to characterize the relationships between flow stresses and deformation conditions, especially considering the serrated flow characteristics. Meanwhile, the numerical analyses for different serrations types and the predictive capabilities of developed model were discussed in details. 2. Material and experimental procedures In this research, the experimental material employed is a commercial forged GH4169 superalloy, and its alloy elements (wt%) is 52.82Ni ‐ 18.96Cr ‐ 5.23Nb ‐ 3.01Mo ‐ 1.00Ti ‐ 0.59Al ‐ 0.03C ‐ 0.01Co ‐ (bal.) Fe. According to ISO 6982-2 [42], the dumbbell-like testing samples were machined. The detailed testing sample and its geometry dimension are presented in Fig. 1. Before preforming uniaxial tensile tests, three kinds of heat treatment procedures including ST (solution treated), HS (solution plus γ′/γ″ phases aging precipitation), and SAT (solution plus δ phase aging precipitation) were conducted. The above heat treatment and its effects on initial microstructures of this alloy are reported by authors' previous investigations [2,3]. Fig. 2 exhibits three kinds of initial microstructures of this alloy with different heat treatment procedures. Uniaxial tensile experiments were performed on a MTS–GWT2015 testing machine at 473–973 K and 0.0001–0.001 s−1. The detailed schedules for uniaxial tensile test are illustrated in Fig. 3. For the tensile experiments, all the testing samples were kept at the target temperatures for 10 min, and the temperature gradient in heating furnace was controlled within 1 K. Besides, all measured results were recorded using the industrial computer. 3. Intermediate-temperature flow behaviors and serrated flow characteristics Fig. 4 depicts the flow stress curves and transition schematics of serration types for the researched superalloy with three kinds of initial microstructures under testing conditions. From Fig. 4a, c and e, the obvious

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PLC effects and serrated flow characteristics are noticed from the flow stress curves. Clearly, the flow stresses are sensitive to the thermomechanical parameters and initial microstructures. Meanwhile, different serrations types (type A/A + B/B/C) can be obtained in the partial enlarged views. According to authors' previous reports [2,3], the detailed descriptions on morphological features for different types of serrations in this alloy can be demonstrated, i.e., the morphological characteristics for type A serrations are upper the true stress-true strain curves, and they predominantly occurs at the 473–573 K. Moreover, the serrations of type A + B/B, which are always obtained at the intermediate temperature range (573–773 K), lead to severe fluctuations on the upper or lower true stress-true strain curves. Also, during the relatively high deformation temperature range (773–973 K), type C serrations always appear, and they are below the flow stress curves. In addition, due to the appearance of dynamic recovery [2,3,43] and the generation of some strengthening precipitates particularly at 973 K and lower strain rates [44–46], the flow characteristics at 973 K are obviously distinct from those at other intermediate temperatures. As exhibited in Fig. 4b, d and f, it can be easily concluded that the transition of serrations types with the applied deformation conditions, i.e., with the increased deformation temperature, the transition type of serration flow characteristics can be arranged as follow: Type A → Type B → Type A + B → Type C. Also, the raised strain rate can significantly limit the range of serrated flow features. At present, there are some explanations on the micro-mechanism of PLC effects in Ni-based superalloys, mainly including interactions of micro-twinning and dislocation slip [47,48], dislocations shearing second precipitations [49,50], formation of stacking faults [51–53], etc. However, according to authors' previous reports [2,3], the dominant plastic mechanism of serrated flow features or PLC effects in the studied superalloy is the dislocation across slip, and the fluctuations of different serrated types in flow stress curves mainly depend on the dislocationssolute atoms combination interactions. 4. Constitutive model and discussions 4.1. A physically-based model involving dislocation-solute atom dynamic interactions 4.1.1. Evolution of plastic deformation during the thermal activation process Generally, the plastic deformation, which is significantly affected by deformation temperatures and strain rates, is governed by the thermal activation behavior [54]. Due to the externally applied stress, the plastic deformation results in the movement of the dislocations in matrix. According to the research of Orowan [55], the applied strain rate can be written as: ε_ ¼ bρm v

ð1Þ

where ε_ is the applied strain rate, b represents the Burgers vector, ρm represents the density of moving dislocations and v is the mean velocity of the moving dislocations. In the thermal activation process, Kocks et al. [56] formulated the following equation to describe the mean velocity of the moving dislocations,
 ΔG v ¼ lη exp − kT

Fig. 3. The detailed schedules for uniaxial tensile test.

ð2Þ

where l represents a slip distance of the moving dislocations in the thermal activation process, η is the attempt frequency for the mobile dislocations, ΔG denotes the free energy for thermal activation process, k represents the Boltzmann constant and T denotes the absolute temperature.

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Fig. 4. Flow stress curves and transition schematics of serration types for the researched superalloy with different initial microstructures: (a-b) ST; (c-d) HS; (e-f) SAT. (Note: (a), (c) and (e) curves are horizontally separated at a strain interval of 0.05.)

Moreover, McCormick [57] suggested that the free energy for thermal activation process, ΔG, is calculated by following relation: ΔG ¼ ΔG0 −σ e V e

ð3Þ

where ΔG0 denotes the reference free energy, σe represents the effective stress and Ve is the effective volume in the thermal activation process. Hence, the applied strain rate can be expressed as:
 ΔG0 σe ε_ ¼ ηεΩ exp − þ kT kT=V e

ð4Þ

4.1.2. Evolution of mobile dislocations movement Because the researched superalloy has the face-centered-cubic crystal structure, the overcome of peierls nabarro (P\\N) barriers is relatively unimportant. During the plastic deformation, the obstacles of moving dislocations are consisted of short-range barriers and longrange barriers. Usually, the interactions of dislocations and crystal defects may bring out the short-range barriers, which are overcome by thermal activation movement. While the long-range barriers, which are the athermal part of the externally applied stress, are independent of the deformation conditions. Therefore, the externally applied stress can be described by the resistance of the dislocations movement [54], i.e., σ ext ¼ σ long þ σ short

where εΩ = ρmbl represents the elementary strain, and its physical implication is an increase strain once arrested moving dislocation accomplished a successful thermal activation movement.

ð5Þ

where σext denotes the externally applied stress, σlong represents the athermal part of the externally applied stress denoting long-range

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barriers of the moving dislocation and σshort represents thermal part of the externally applied stress denoting short-range barriers of the moving dislocation. As reported by Chen et al. [58], the contribution of the long-range stress between the moving dislocations to the externally applied stress can be written as: σ long ¼ hεm

ð6Þ

where h is the hardening coefficient, ε is the total plastic strain and m is the hardening power exponent. Notably, h and m represent as material constants (m b 1). The dynamic interactions between mobile dislocations and solute atoms lead to the short-range barriers of dislocation movement. Due to the limited regime of the short-range barriers, the corresponding stress field only partially superimposes on the long-range stress field. According to assumption of Jiang et al. [59], the relationship between the mobile dislocations and solute atoms can be written as: σ short ¼ ξC 

ð7Þ

where ξ represents the pinning strength of the solute atoms towards moving dislocations, C ∗ is the concentration of the solute atoms towards the moving dislocation lines. As mentioned previously, the serrated flow characteristics depend on the moving dislocations and solute atoms combination interactions. Thus, the appearance of serrated stress on flow stress curves can be denoted as: σ serrated ¼ σ short

ð8Þ

Furthermore, the above initial internal stress can be determined as: σ0 = ΔG0/Ve. Meanwhile, introducing a material parameter: S0 = kT/Ve [26,60], the effective stress σe can be written as:
ε_ Þ þ σ0 σ e ¼ S0 In ηεΩ

ð9Þ

where S0 denotes the instantaneous strain rate sensitivity. Hence, the total flow stress can be described using the following equations: 8 > σ ¼ σ e þ σ long þ σ short ; > > >
 > > ε_ > > > ; σ e ¼ σ 0 þ S0 In > > ηεΩ < σ long ¼ hεm ; > > σ short ¼ ξC > > > σ serrated ¼ σ short ; > > > > > ΔG0 > :σ 0 ¼ Va

ð10Þ

diffusion ability, ta is the effective pinning time for solute atoms towards moving dislocations and n is a material parameters related to the exponent of pipe diffusion characteristics [3,4]. Because of the serrated flow phenomenon reflecting an unstable plastic deformation condition, the effective pinning time cannot immediately respond to the change in applied strain rate, and it obeys the following equation [63]: ta ta ¼ 1− dt tw

   C  ¼ C s 1− exp −βt na

ð11Þ

where Cs∗ represents the saturation value of the solute atom concentration, β represents a material parameters denoting the solute atoms

ð12Þ

where tw represents the waiting time for moving dislocations on barriers, and it can be expressed by [64]: tw ¼

εΩ ε_

ð13Þ

Binding with the Eqs. (12)–(13) and neglecting the high order terms, the differential forms of Eq. (11) can be denoted as:   ðn−1Þ  ε_ C_ ¼ λ1 C s −C  C  =C s =n−λ2 C  εΩ

ð14Þ

where λ1 and λ2 are materials constants. As reported by Rizzi et al. [26], the two terms on the right side of Eq. (14) involve the dynamic interactions for moving dislocation and solute atoms, i.e., corresponding to the pinning effect and unpinning effect, respectively.  When the applied strain rate ε_ is zero, Eq. (14) can be written as C_ ¼ λ1 ðC s −C  ÞðC  =C s Þðn−1 Þ=n . The parameter λ1 ‐1 greatly affects the characteristic time scale of the pinning process [59]. As the pinning time is too short (t≪λ1 ‐1 ), the diffusion rates of solute atoms towards the moving dislocations linearly increase. Thus, the diffusion rate of the solute atom is determined as C ∗ ≈ Cs∗(λ1t/n)n. As the pinning time is too long (t≫λ1 ‐1 ), the solute atom concentration around the moving dislocations nearly approaches the saturation value (C ∗ ≈ Cs∗). Then, PLC effects are induced. However, the second term on the right side of Eq. (14) denotes unpinning process due to the thermal activation process, and it may lead to the reduction of the solute concentration. So, the waiting time tw of moving dislocations on localized barriers is the feature time scale of this process. In summary, the two time features scales λ1 ‐1 and εΩ =ε_ dominate the occurrence of serrated flow characteristics. Finally, the physically-based model involving dislocation-solute atoms dynamic interactions for the intermediate temperature flow behavior of the investigated superalloy can be summarized as: 8 σ ¼ σ e þ σ long þ σ short > >
 > > ε_ > > > σ e ¼ σ 0 þ S0 In > > ηεΩ > > >σ m > > long ¼ hε > <σ ¼ ξC  short

4.1.3. Evolution of solute atom concentration Generally, the metallic materials always distort and mismatch owing to the existence of solute atoms, resulting in forming the stress field around solute atoms [26]. This stress field and moving dislocations interactions lead to the formation of the Cottrell atmosphere near the dislocation lines. Because the solute atoms diffusion in matrix can be shortly hindered moving dislocations, the evolution of solute concentration in the matrix is governed by the effective pinning time. Meanwhile, Cottrell et al. [61,62] concluded that the evolution of the solute concentration C ∗ in the matrix, i.e.,

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> σ serrated ¼ σ short > > > > > ΔG0 > > σ0 ¼ > > > Va > >    >  ε_ > : C_ ¼ λ1 C s −C  C  =C s ðn−1Þ =n−λ2 C  εΩ

ð15Þ

4.2. Dynamic response analysis of the proposed constitutive model Usually, the dynamic interactions between the moving dislocations and solute atoms gradually achieve an equilibrium when the applied the strain rate is constant [26]. With the abrupt changed in strain rate, the above steady state of pinning effects and unpinning effects break. Also, this may bring out that the saturation concentration of solute atoms towards the moving dislocations suddenly change. But, the solute atoms concentration cannot immediately respond to this variation due

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to relating to the pinning time of solute atoms towards mobile dislocations. So, this evolution only reaches a new steady state with gradual change. In fact, the final strain rate dependent flow stress may be above or below the initial value due to responding this transient process. In order to comprehensively understand the gradual change in the flow characteristics, introducing the asymptotic strain rate sensitivity St→∞, i.e., St→∞

∂σ ∂C  ¼ j ¼ S0 þ ξ ∂Inε_ ∂Inε_ ;t ε

ð16Þ

Then, the strain rate dependent flow stress Δσ ε_ affected by sudden

strain rates can be represented as:   ε_ 2 Δσ ε_ ¼ S0 In þ ξC_  Δt ε_ 2 ε_ 1

ð17Þ

where ε_ 1 and ε_ 2 are the strain rate before and after the abrupt change, respectively. Δt is the asymptotic time. Fig. 5 presents the strain rate dependent stress reflecting sudden strain rates. As reported in literatures [26,59,60], some model parameters can be determined as λ1 = 0.06, λ2 = 0.08, η = 1 × 1011 Hz, εΩ = 3 × 10−4, η = 65, Cs∗ = 0.15, S0 = 1.08MPa. Obviously, with the sudden change of strain rate, the strain rate dependent flow stresses first instantaneously increase the maximum value, and then gradually decrease to the new equilibrium value at the strain rate after the abrupt change. Meanwhile, the increments of the strain rate dependent flow stresses have different values. These results imply that the negative strain rate sensitivity occurs at the applied strain rate range. 4.3. Numerical analysis and validation of the proposed constitutive model 4.3.1. Coupling conditions of the numerical simulation To adequately considerate different strains at different elements of the sample during the uniaxial tensile testing, the samples separated into N sections perpendicular to the tensile direction, and coupled through the force loading F. Based on the research of Lebyodkin et al. [65], each section, which apart from the end sections i = 1 and i = N, obeys the following equation, which can be represented as: 8 > < σ ðiÞ ¼ σ e ðiÞ þ σ long ðiÞ þ σ short ðiÞ−κ f½ε ði−1Þ−εðiÞ þ ½εði þ 1Þ−ε ðiÞg ðn−1Þ=n    ε_ ðiÞ  > :C_ ðiÞ ¼ λ1 C s −C  ðiÞ C  ðiÞ=C s −λ2 C ðiÞ ð18Þ εΩ where κ represents the coupling coefficient of each element and κ{[ε(i − 1) − ε(i)] + [ε(i + 1) − ε(i)]} relates to the incompatibilities of macroscopic strain between the adjacent sections [65].

For uniaxial tensile testing with invariant applied strain rates, the proposed model needs to obey the following elastic constraint equation [26], σ_ T ¼ E

! j V 1X − ε_ ðiÞlðiÞ L L i¼1

ð19Þ

where σT is the total applied stress of the tensile sample, E is the modulus of elasticity between the sample and testing machine, V is the tensile _ is speed of the testing machine, L is the gauge length of sample, εðiÞ strain rate of the i ‐ th elements of sample, and l(i) is the length of the i ‐ th elements of sample. When t = tn, the variables of i ‐ th elements εn(i), ε_ n ðiÞ, Cs(n)(i), σn(i) and σT (n) can be determined. Then, the value of εn+1(i) can be estimated by εn(i). Cs(n+1)(i) and σT (n+1) can be determined from Eqs. (18)–(19). Therefore, the total applied loading FT at this moment can be described as: F T ¼ A

σ T ðnþ1Þ  1 þ εT ðnþ1Þ

ð20Þ

where εT (n+1) = Vt/L represent the average strain of tensile sample. Generally, due to the third law of Newton, the applied loading along the tensile sample is equal everywhere during deformation process. Thus, the applied stress of the i ‐ th elements of tensile sample at the moment of t = tn is represented as: σ nþ1 ðiÞ ¼


 FT σ nþ1 ðiÞ 1 þ εnþ1 ðiÞ þ E A

ð21Þ

Therefore, Eqs. (18)–(21) form a group of closed solution space. Once the initial value of this model is given, the equations can be solved smoothly. 4.3.2. Serrated flow characteristics of the numerical simulation The numerical simulation of developed model is conducted under the different deformation parameters and different microstructural conditions. To introduce the unstable plastic flow, a group of random amplitude stress from 0 to 20 MPa was introduced the initial internal stress σ0(i). Based on some parameters of literatures [26,59,60] and many trial-error tests, the model parameters are obtained, i.e., λ1 = 0.06, λ2 = 0.08, κ = 3, η = 1 × 1011 Hz, εΩ = 3 × 10−4, ξ = 65, n = 1/3, Cm = 0.15, S0 = 1.08MPa, E = 1.2 × 105 MPa, L = 64 mm, N = 64, A = 9π mm2, σ0 = 35 MPa, ε0(i) = 0.002, h ∈ [1200, 1800], m ∈ [0, 1]. Fig. 6 shows typical serrated flow characteristics between the measured and simulated results at different deformation parameters. Here, the strain rate is 0.0003 s−1 and the deformation temperatures are 473 K, 673 K and 873 K, respectively. It is clearly found that various

Fig. 5. Strain rate dependent stress reflecting the sudden strain rates: (a) ε_ 1 ¼ 0:0001 s‐1 →ε_ 2 ¼ 0:001 s‐1 ; (b) ε_ 1 ¼ 0:0003 s‐1 →ε_ 2 ¼ 0:001 s‐1 .

Y.C. Lin et al. / Materials and Design 183 (2019) 108122

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Fig. 6. Typical serrated flow characteristics between measured and simulated results at different deformation parameters (strain rate is 0.0003 s−1): (a-b) SAT superalloy tested at 473K; (c-d) HS superalloy tested at 673K; (e-f) ST superalloy tested at 873K.

types of serrations shown on the flow curves, and the various serration amplitudes in accorded with the experimental findings. It is worth noting that the numerical analysis of each element in the sample is different due to the introduction of initial random amplitude stress and the coupling conditions. Meanwhile, this inconsistent development of each element in the sample is controlled by the proposed constitutive model. In the calculation process, the variations of parameters h and m mainly result in the different serration flow characteristics. 4.3.3. Validation of the developed constitutive model To obtain the prediction ability of the developed constitutive model, the correlation index (R) and the average absolute relative error (AARE) between the experimental and predicted curves are evaluated.   N  ∑i¼1 Ei −E P i −P qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð22Þ 2 2 N  N  ∑i¼1 P i −P ∑i¼1 Ei −E

AAREð%Þ ¼

N  1X Ei −P i j  100 N i¼1  Ei

ð23Þ

where N is the number of the experimental data, Ei and Pi correspond to the experimental and predicted data, respectively. E and P are the average values for Ei and Pi, respectively.

Fig. 7 depicts the comparisons between the predicted and measured flow stresses of the investigated superalloy with different initial microstructures. Clearly, the deviations between the predicted and measured data are minor. From Fig. 7d, the values of R and AARE for the developed model are 0.9950 and 4.51%, respectively, which adequately implies that the developed model can accurately forecast the intermediate temperature deformation behaviors considering the serrated flow features of the investigated alloy. As mentioned previously, due to the appearance of dynamic recovery and the generation of strengthening phases (γ' and γ″), the lager errors between the predicted and measured flow stress at 973 K through the developed model. Hence, these predicted results are not presented in Fig. 7. Up to now, although there are few literatures on the models to predict the serrated flow characteristics of nickel-based superalloys, some scholars still developed suitable constitutive models to predict the serrated flow characteristics of other alloys. For example, Chen and Nemat-Nasser [66] established model for predicting dynamic strain aging behaviors of a titanium alloy, Sheikh and Serajzadeh [23] used the tradition neural networks to forecast the serrated flow behavior of an AA5083 alloy. Also, Dierke et al. [25] proposed the model for predicting the Portevin– LeChatelier effect in an Al\\Mg alloy. With comparison of results by these works, it can be noticed that the proposed physically-based constitutive model has the better accuracy, especially for forecasting the serrated types.

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Fig. 7. Comparisons between the predicted and measured flow stresses under different initial microstructural conditions: (a) ST; (b) HS; (c) SAT; (d) R and AARE.

5. Conclusions

Originality and plagiarism

The intermediate temperature flow behaviors of the investigated superalloy with different initial microstructures are researched by uniaxial tensile tests. A physically-based model involving dislocation-solute atoms dynamic interactions is developed. Several significant findings are listed as:

It is an original research work which has not been published previously, that it is not under consideration for publication elsewhere, and that if accepted it will not be published elsewhere in the same form, in English or in any other language, without the written consent of the Publisher. Should you need to contact me, please use the below address or call me at +86-013469071208 or via email at [email protected], [email protected].

(1) The intermediate temperature deformation characteristics not only associate with deformation parameter, but also are prominently influenced by different initial microstructures. Meanwhile, various serrated flow characteristics can be obviously observed on the flow stress curves. (2) Based on the plastic deformation mechanism of the studied alloy, a dislocation-solute atom dynamic constitutive model is developed. The evolution of moving dislocations and solute atoms are analyzed, i.e., the moving dislocations are hindered by solute atoms at lower strain rates, while the unpinning effects of moving dislocations occur at higher strain rates. (3) The prediction ability of the developed model is verified. Different serrations types are obtained by appropriate parameters. Also, the high predictability suggests that the developed model can greatly predicted the intermediate-temperature flow behaviors of the investigated alloy considering the serrated flow features.

Authorship of the paper All the authors have made a significant contribution to the conception, design, execution, or interpretation of the reported study, i.e., the first author carried out the research concept and design. The second and third authors carried out the collection and assembly of data, and wrote the article. The fourth author jointly discussed and revised this work according to review comments.

Declaration of competing interest There are no any conflicts of interest in this submission. Acknowledgments This work was supported by the National Natural Science Foundation Council of China (Grant No. 51775564), and Key Laboratory of Efficient & Clean Energy Utilization, College of Hunan Province (No. 2018NGQ001), China. References [1] C.G. Tian, G.M. Han, C.Y. Cui, X.F. Sun, Effects of Co content on tensile properties and deformation behaviors of Ni-based disk superalloys at different temperatures, Mater. Des. 88 (2015) 123–131. [2] Y.C. Lin, H. Yang, Y. Xin, C.Z. Li, Effects of initial microstructures on serrated flow features and fracture mechanisms of a nickel-based superalloy, Mater. Charact. 144 (2018) 9–21. [3] Y.C. Lin, H. Yang, X.M. Chen, D.D. Chen, Influences of initial microstructures on Portevin-Le Chatelier effect and mechanical properties of a Ni-Fe-Cr-base superalloy, Adv. Eng. Mater. 20 (2018), 1800234. [4] W. Chen, M.C. Chaturvedi, On the mechanism of serrated deformation in aged Inconel 718, Mater. Sci. Eng. A 229 (1997) 163–168. [5] P. Maj, J. Zdunek, J. Mizera, K.J. Kurzydlowski, B. Sakowicz, M. Kaminski, Microstructure and strain-stress analysis of the dynamic strain aging in Inconel 625 at high temperature, Met. Mater. Int. 23 (2017) 54–67. [6] M. Abbadi, P. Hähner, A. Zeghloul, On the characteristics of Portevin–Le Chatelier bands in aluminum alloy 5182 under stress-controlled and strain-controlled tensile testing, Mater. Sci. Eng. A 337 (2002) 194–201.

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