Abstract
Analyzing the electrically-assisted stress relaxation behavior of Ti2AlNb alloy is beneficial for improving the forming accuracy of Ti2AlNb alloy thin-walled components. This study analyzes and models the deformation mechanism of Ti2AlNb alloy foils during the electrically-assisted stress relaxation process. The non-thermal effect of pulsed current on the stress relaxation of Ti2AlNb alloy foils was decoupled using forced air cooling. As the current density increases, it promotes dislocation recovery and O-phase precipitation, significantly accelerating the stress decrease in stress-driven stage I of the stress relaxation process. However, extensive precipitation of the O phase hinders dislocation movement, reducing the stress relaxation rate in stage II. A physically-based constitutive model was developed, considering the effects of pulsed current on activation energy, dislocation evolution, and phase transformation, with a relative average absolute error of 8.7%.
Keywords: Electrically-assisted stress relaxation, Ti2AlNb alloy foils, Physically-based constitutive model
Subject terms: Engineering, Materials science
Ti2AlNb alloy is a promising lightweight heat-resistant structural material for service at 650–750 ℃, due to its low density, high specific strength, good room-temperature plasticity, and high-temperature resistance. It has strong application potential in hot-end structures for aerospace and other equipment1,2. Due to its low elastic modulus and high strength, thin-walled Ti2AlNb alloy components are prone to springback defects during the forming process3. Stress relaxation is a common method for releasing residual stress in components and eliminating springback defects. However, stress relaxation methods based on thermomechanical loading are often time-consuming and energy-intensive4. Due to its high resistivity, the electrically-assisted plastic forming process of Ti2AlNb alloy components offers high heating efficiency and improved forming performance5. Moreover, since only the billet is heated, energy consumption is significantly reduced, and the service life of the mold is prolonged6.
Pulsed current has been used to eliminate residual stress and prevent springback defects in thin-walled components. For AZ31B magnesium alloy and QP980 high-strength steel, the athermal effect of pulsed current on stress relaxation has been confirmed by comparing electrically-assisted stress relaxation curves with isothermal test results7,8.
Increasing the current frequency and peak current can reduce the springback angle in V-bending of AZ31B magnesium alloy. When the frequency is 800 Hz and the peak current density is 100 A/mm2, the springback angle is only 0.1°9. For TC4 titanium alloy during U-bending, increasing the duty cycle from 10% to 30% results in an additional 10% reduction in the springback angle under the same current density (75 A/mm2) and deformation temperature (~ 750 ℃) 10. Applying low-frequency pulsed current reduced the residual stress in 316 L stainless steel welding joints by 40% without any significant temperature rise11. Therefore, the non-thermal effect of pulsed current further suppresses springback defects in metal thin-walled components12. Pulsed current can activate unconventional dislocation slip, such as prismatic slip in magnesium alloys and pyramidal slip in titanium alloys13,14. It can accelerate dislocation recovery and reduce dislocation density. Meanwhile, pulsed current promotes recrystallization nucleation and grain growth, and further accelerates phase transformations (e.g., α→β→α′ transformation in titanium alloys)10,12,15. These microstructural evolutions accelerate the stress relaxation process and inhibit the springback behavior of thin-walled components.
Ti2AlNb alloy typically consists of three phases: O phase, α2 phase, and β/B2 phase. During electrically-assisted compression of Ti2AlNb alloy, applying a pulsed current results in an additional 10% reduction in yield stress16. During deformation in the B2 + O phase region, pulsed current promotes the spheroidization of the lamellar O phase by activating twinning, and the local Joule heating effect at the O/B2 boundary accelerates the O→B2 phase transformation. During deformation in the B2 + α2 phase region, pulsed current increases the recrystallization nucleation rate by reducing the activation energy and aligns dislocation lines parallel to the current loading direction, thereby improving formability17. The phase transformations O→B2 + α2 and α2→B2 are also accelerated in the B2 + α2 phase region18. Pulsed current can also accelerate the dynamic precipitation of acicular O phase in the temperature range of 450–750 ℃ and enhance the precipitation strengthening effect19,20. Currently, there are no studies on the electrically-assisted stress relaxation of Ti2AlNb alloy foils.
For the constitutive model of electrically-assisted stress relaxation, an electric parameter 
(where 
represents the peak current density and 
represents current frequency), was incorporated into the classical stress relaxation model, successfully predicting the electrically-assisted stress relaxation behavior of AZ31B magnesium alloys and QP980 high-strength steels7,8. A constitutive model for 2219 aluminum alloy has been established, considering the introduced electron wind force and the reduced activation energy for dislocation movement due to pulsed current21. However, physical mechanisms such as accelerated phase transformation and enhanced dislocation recovery during the electrically-assisted forming oprocess of Ti2AlNb alloy should also be incorporated into its constitutive model for electrically-assisted stress relaxation.
In this paper, the non-thermal effect of pulsed current on the stress relaxation of Ti2AlNb alloy foils was decoupled using forced air cooling. The deformation mechanism was analyzed through various microstructural observations. A physically-based constitutive model was developed, considering the effects of pulsed current on activation energy, dislocation evolution, and phase transformation.
Materials and methods
Ti2AlNb alloy foil with a thickness of 0.1 mm and a width of ~ 100 mm was used, as shown in Fig. 1(a). The nominal composition was Ti–22Al–25Nb (in wt%). The foils were provided by the Institute of Metal Research, Chinese Academy of Sciences, and underwent successive hot rolling, cold rolling, and incomplete annealing treatments. The microstructure of the initial foils was analyzed using a scanning electron microscope. The sample surfaces were electrolytically polished (polishing solution: 6% perchloric acid + 34% butanol + 60% methanol, temperature − 20 °C, time 45 s, current 0.86 A). The step size for the electron backscatter diffraction test was 0.3 μm.
Fig. 1.
Initial Ti2AlNb alloy foils and the microstructure analysis: (a) initial Ti2AlNb alloy foils; (b) backscattered electron image; (c) inverse pole figure; (d) phase distribution.
The foil exhibits a distinct rolled microstructure, with fine grains elongated along the rolling direction, as shown in Fig. 1. Based on energy-dispersive X-ray spectroscopy element distribution (Table 1) and electron backscatter diffraction phase composition analysis, the foil is composed of three phases: α2, B2/β and O phase. The volume fractions of α2, B2/β and O phase are 53%, 36.9%, and 10.1%, respectively. The α2 and B2/β phases are distributed alternately in a lath-like microstructure, and the O phase is distributed in an acicular structure within the B2/β phase. According to the inverse pole figure shown in Fig. 1(c), each phase exhibits similar grain orientations, resulting in a distinct rolling texture. After incomplete annealing, the foil retains a significant amount of deformed structure, with a low-angle grain boundary content of 39.1%.
Table 1.
Elements distribution at typical locations in the initial Ti2AlNb alloy foils.
| Element At% | Point A | Point B | Point C |
|---|---|---|---|
| Ti | 56.48 | 53.27 | 55.02 |
| Al | 26.89 | 20.36 | 22.25 |
| Nb | 16.63 | 25.36 | 22.73 |
| Phase | α2 | β/B2 | O |
An electrically-assisted stress relaxation test platform was established based on a material testing machine (AG-X 50 kN, Shimadzu Corp.), as shown in Fig. 2(a). A pulsed current was provided by a DC power supply (MicroStar CRSLFP20-500, Dynatronix Inc.). The positive and negative poles of the power supply were connected to the upper and lower tensile clamps, respectively. The clamps and the platform were insulated using bakelite plates. The temperature distribution of the sample was monitored in real time using an infrared thermography camera (FLIR T660).
This study aims to verify the effect of pulsed current on the springback defects of Ti2AlNb alloy thin-walled components formed at room temperature. Therefore, during the electrically-assisted stress relaxation experiment, the foil was first loaded to 880 MPa at room temperature, corresponding to an engineering strain of approximately 10%. Then, the pulse power supply was activated to conduct the stress relaxation test. Figure 2(b) shows the room-temperature tensile curve of the Ti2AlNb alloy foil and the initial stress value for stress relaxation. The current loading in this paper is achieved using a DC pulse power supply. This paper primarily analyzes the influence of peak current density on stress relaxation. The peak current density is defined as the ratio of the peak current to the cross-sectional area of the sample, with values ranging from 10 to 60 A/mm2. The frequency and duty cycle of the pulsed current were fixed at 100 Hz and 30%, respectively. This means the pulse period was 10 ms, the pulse width was 3 ms, and the pulse interval was 7 ms. The total stress relaxation time, i.e., the duration of the pulsed current loading, was 200 s. The electrically-assisted stress relaxation with air cooling was performed using an air compressor.
Fig. 2.
Test platform of electrically-assisted stress relaxation and selection of initial loading stress: (a) test platform; (b) initial value of stress relaxation.
Results and discussion
Electrically-assisted stress relaxation behavior
Figure 3 shows the evolution of temperature and stress during the electrically-assisted stress relaxation process, where the start time of stress relaxation coincides with the start of pulsed current loading. The sample heats up rapidly after power is applied and reaches a steady-state temperature within 5 s. As the peak current density increases, the steady-state temperature of the sample rises significantly. Meanwhile, due to the higher heat dissipation rate, air cooling significantly affects the sample temperature. Under the same current parameters, the steady-state temperature of the sample with air cooling is significantly lower than that without air cooling. The steady-state temperatures are nearly identical (~ 440 °C) for a peak current density of 20 A/mm2 without air cooling and 50 A/mm2 with air cooling.
Fig. 3.
Evolution of temperature and stress during the electrically-assisted stress relaxation process: (a) sample temperature over time; (b) steady-state sample temperature; (c) stress relaxation curves; (d) ultimate residual stress; (e) stage Ⅰ and Ⅱ; (f) steady stress relaxation rate.
The stress relaxation curve can be divided into two stages: the short-term stage of rapid stress decrease (stage Ⅰ) and the long-term stage of uniform stress decrease (stage Ⅱ). Stage Ⅰ is a variable-rate relaxation stage, and stage Ⅱ is an approximately steady-rate relaxation stage22. The critical point for the transition from stage I to stage II is when the second derivative of stress equals zero.
As the peak current density increases, along with rising temperature and prolonged relaxation time, the ultimate residual stress gradually decreases. When the steady-state temperature was ~ 440 °C, the ultimate residual stress of the sample with a current density of 50 A/mm2 was dramatically lower (~ 484 MPa) than that of the sample with 20 A/mm2. This indicates that during electrically-assisted stress relaxation, the non-thermal effect of pulsed current contributes to additional stress reduction.
In stage I, the stress level is relatively high, and the process is primarily stress-driven. In stage II, the stress is relatively low, and the process is thermally activated23. The steady stress relaxation rates under different test conditions are shown in Fig. 4(d). Without forced air cooling, the steady-state stress relaxation rate increases significantly with current density, mainly due to the pronounced Joule heating effect of the pulsed current. However, under forced cooling, the steady-state stress relaxation rate first increases and then decreases with increasing current density. This indicates that although a high current density reduces the ultimate residual stress, it decreases the steady-state stress relaxation rate.
Fig. 4.
Backscattered electron images of samples without air cooling after stress relaxation: (a) 10 A/mm2; (b) 20 A/mm2; (c) 30 A/mm2; (d) 40 A/mm2.
Microstructure evolution during electrically-assisted stress relaxation
Figure 4 shows the backscattered electron images of samples without air cooling after stress relaxation. At peak current densities of 10 and 20 A/mm2, the sample microstructure changes slightly and is almost identical to the initial foil due to the lower stress relaxation temperature. When the peak current density reaches 30 A/mm2, relatively obvious O-phase precipitation occurs in the sample. As the peak current increases to 40 A/mm2, the O phase gradually thickens and fragments under stress loading. Meanwhile, several equiaxed α2 phase edges transform into rim-O phase, attributed to the peritectoid reaction between α2 and B2β phases.
Figure 5 shows the backscattered electron images of samples with air cooling after stress relaxation. When the peak current density is 40 A/mm2 or less, the sample microstructure is almost identical to the initial foil. As the peak current density increases to 50 A/mm2, numerous fine acicular O phases precipitate in the sample. When the peak current density is further increased to 60 A/mm2, the sample exhibits coarsening of the O phase and transformation of the α2 phase into rim-O phase. The combined effect of these two processes promotes the spheroidization of the β/B2 phase.
Fig. 5.
Backscattered electron images of samples with air cooling after stress relaxation: (a) 10 A/mm2; (b) 20 A/mm2; (c) 30 A/mm2; (d) 40 A/mm2; (e) 50 A/mm2; (f) 60 A/mm2.
Figure 6 shows transmission electron microscopy images and electron backscatter diffraction analysis of samples with different peak current densities but similar deformation temperatures (~ 440 °C), specifically, a peak current density of 20 A/mm2 without air cooling and 50 A/mm2 with air cooling. The sample with the lower peak current density shows significant dislocation pile-up in the α2 phase, and the O phase content is relatively low, with a size of approximately 40 nm. In contrast, the sample with the higher peak current density shows almost no dislocations, but contains numerous fine acicular O phases within the β/B2 phase, with a size of approximately 15 nm.
Fig. 6.
Transmission electron microscope figures and electron backscatter diffraction analysis of samples with different peak current densities and a similar deformation temperature: (a)/(c) 20 A/mm2 and without air cooling; (b)/(d) 50 A/mm2 and with air cooling.
Phase distribution maps based on electron backscatter diffraction confirm the above results. Under high peak current density, more O phase precipitates within the β/B2 phase. The volume fraction of the O phase under high current density (50 A/mm2) was 18.6%, whereas under low current density (20 A/mm2) it was only 12.3%. According to the distribution of geometrically necessary dislocation density, significant dislocation accumulation often occurs near the precipitated O phase. The local geometrically necessary dislocation density near the precipitated O phase can reach 60 × 1014/mm2, as calculated from the kernel average misorientation. This indicates that the precipitated O phase hinders dislocation movement and affects the release of residual stress.
Mechanism and modeling of electrically-assisted stress relaxation
Many studies have found that pulsed current promotes dislocation movement and annihilation, reducing dislocation density24. The most common explanation is the electron wind force theory, where momentum transfer from drifting electrons to crystal defects facilitates dislocation movement25. However, quantitative calculations of the electron wind force suggest it is insufficient to drive dislocation movement26. In-situ transmission electron microscopy observations revealed non-directional migration of incoherent twin boundaries under pulsed current, challenging the electron wind theory27. Local Joule heating appears to be the main reason for promoting dislocation movement28. Increased current density leads to decreased dislocation density, which undoubtedly reduces material residual stress and accelerates the stress relaxation process.
During the electrically-assisted forming of titanium alloys, the local Joule heating effect caused by pulsed current at crystal defects increases the diffusion coefficient. The high local diffusion coefficient may accelerate the diffusion of stable β-phase elements such as Mo and V, promoting the nucleation and growth of the β phase at relatively lower macroscopic temperatures25,28. Within the aging temperature range of Ti2AlNb alloy, the local Joule heating effect of pulsed current can also induce rapid precipitation and growth of acicular O phases, enhancing precipitation strengthening19,20. In this study, when the stress relaxation temperature is approximately 440 °C, increasing the current density from 20 A/mm2 to 50 A/mm2 leads to significant precipitation of acicular O phases. Second-phase precipitation significantly reduces creep activation energy23, resulting in rapid stress decrease during the stress-driven stage I. The rapid growth of acicular O phase under pulsed current promotes the spheroidization of the lamellar βB2 phase. However, extensive O-phase precipitation hinders dislocation movement, leading to dislocation accumulation. This, to some extent, reduces the stress relaxation rate in stage II.
Figure 7 summarizes the effect of increased current density on the stress relaxation mechanism of Ti2AlNb alloy foils.
Fig. 7.
Effect of increased current density on the stress relaxation mechanism of Ti2AlNb alloy foils.
Generally, stress relaxation is regarded as a specific form of creep behavior. During stress relaxation process, the total strain of the sample remains constant, while elastic strain gradually transforms into creep strain29.
 |
1 |
 |
2 |
 |
3 |
where 
, 
, 
and 
are the total strain, elastic strain, plastic strain, and creep strain, respectively. E is the elastic modulus of the material. 
refers to the plastic strain generated during initial loading and remains unchanged during subsequent stress relaxation relaxation process.
The classic power-law creep relationship is widely used to describe the relationships among creep strain rate, stress, temperature, and activation energy30.
 |
4 |
where A is the material constant, n is the stress exponent, Q is the activation energy, R is the gas constant, and T is the stress relaxation temperature.
During the creep process, a certain creep resistance (
) must be overcome to generate creep strain31,32. For creep behavior controlled by dislocation movement, factors such as lattice friction, solute atoms, precipitated phases, grain boundaries, and dislocation density hinder dislocation movement, contributing to creep resistance33.
 |
5 |
 |
6 |
where 
represents the lattice friction resistance, while 
, 
, 
, and 
respectively represent solid solution strengthening, precipitation strengthening, fine grain strengthening and deformation strengthening.
Based on the aforementioned microstructural evolution analysis, the main changes during the electrically-assisted stress relaxation of Ti2AlNb foils occur in dislocation density and precipitated O phases. Therefore, in this paper, 
, 
and 
are combined into one term 
, which is related to the deformation temperature34:
 |
7 |
 |
8 |
where 
is a fitting constant and 
is the reference temperature.
During the stress relaxation process, the precipitation of the O phase hinders dislocation movement and increases creep resistance. The strengthening effect is related to the content and size of the O phase. The strengthening effect due to the size of the O phase follows the Hall-Petch relationship35.
 |
9 |
where 
is a fitting constant, 
is the volume fraction of the O phase, and 
is the thickness of the precipitated O phase.
Under the stress relaxation conditions selected in this paper (temperature below 900 °C), the O phase content gradually increases with temperature, which can be described by the Johnson-Mehl-Avrami model36. Since the O phase precipitates during stress relaxation, the effect of time on its content must be considered.
 |
10 |
where 
and 
are fitting constants, 
is the initial volume fraction of β/B2, 
is the phase transition temperature for O→β/B2. Based on the phase diagram of Ti–22Al–xNb alloy, 
is set to 1000℃.
The strengthening effect due to dislocation interactions can be described by the Taylor model37.
 |
11 |
 |
12 |
where 
is the dislocation density normalized by the maximum dislocation density 
, 
is a fitting constant.
During the hot deformation process, both dislocation multiplication due to deformation and dislocation annihilation due to dynamic recovery occur, typically described by the Kocks-Mecking Eq38.:
 |
13 |
where 
and 
are fitting constants. 
is related to dynamic recovery and is influenced by deformation temperature, strain rate, and activation energy.
 |
14 |
where 
and 
are fitting constants.
When pulsed current passes through a metal, it inevitably causes a temperature rise due to the Joule heating effect, which can be expressed as39:
 |
15 |
where, 
represents the fraction of electrical energy converted into thermal energy, I represents the applied current, and t represents the time interval, 
, 
and 
represent the resistance, density, and heat fusion of the material, respectively, and 
represents the specimen volume.
When the frequency and duty cycle of the pulsed current remain constant, the steady-state temperature is proportional to the square of the current density. The expression can be simplified as:
 |
16 |
where 
is a fitting constant, 
is the applied current density. The coefficient 
can be obtained by fitting the relationship between current density and the steady-state temperature shown in Fig. 3(b).
Dislocation movement is an energy-activated process. In electrically-assisted forming, high-speed moving electrons interact with lattice defects such as dislocations, enhancing atomic vibration at dislocation cores and thereby reducing the activation energy required for dislocation movement27. The activation energy is also related to the deformation temperature and can be expressed as 34,37:
 |
17 |
where 
and 
are fitting constants, 
is the reference current density.
The local Joule heating effect of pulsed current promotes dislocation movement and annihilation, increasing the dislocation recovery rate33.
 |
18 |
Pulsed current accelerates element diffusion and promotes O-phase precipitation. Therefore, the O-phase content is rewritten as40:
 |
19 |
The parameters of the constitutive model were determined by referencing literature and fitting the stress relaxation curves, as listed in Table 2. Figure 8(a) and (b) shows the fitting results for the stress relaxation curves of Ti2AlNb alloy foils during electrically-assisted stress relaxation. The relative average absolute error (RAAE
) is used to indicate the accuracy of the calculated stress relaxation curves, and the calculated RAAE value is 8.7%.
Fig. 8.
Calculated results for electrically-assisted stress relaxation of Ti2AlNb alloys foils: (a) stress relaxation curves without air cooling; (b) stress relaxation curves with air cooling; (c) relaxed stress caused by thermal and non-thermal effects; (d) percentage of relaxed stress caused by thermal and non-thermal effects.
Figure 8(c) shows the calculated relaxed stress caused by the thermal and non-thermal effects of pulsed current. The thermal effect of pulsed current has a dominant influence on stress relaxation, especially under deformation conditions without air cooling. As the peak current density increases, the relaxed stress caused by the non-thermal effect gradually increases. However, when the current density is less than 20 A/mm2 or greater than 50 A/mm2, the change in relaxed stress is relatively small.
Figure 8(d) shows the calculated percentage of relaxed stress caused by the thermal and non-thermal effects of pulsed current. Under deformation conditions without air cooling, as the peak current density increases, the percentage of relaxed stress caused by the non-thermal effect gradually increases, reaching a peak of 20.5% at 30 A/mm2. Under deformation conditions with air cooling, the proportion first increases and then decreases with increasing current density, reaching a peak of 47.1% at 50 A/mm2.
Thus, the contribution of the non-thermal effect of pulsed current to stress relaxation is closely related to current density and cooling conditions.
Table 2.
Parameters in the constitutive model for Ti2AlNb alloys foils during electrically-assisted stress relaxation process.
| Parameters | Meaning | Value or expression |
|---|---|---|
|
Fitting constants | 172.3 |
|
129.4 | |
|
0.0021 | |
|
0.033 | |
|
0.08 | |
|
191.5 | |
|
4.8E8 | |
|
|
|
|
0.17 (air cooling) 0.55 (no air cooling) | |
|
262.6 | |
|
−0.06 | |
|
0.04 | |
|
0.11 | |
|
0.05 | |
|
Elasticity modulus (MPa) | 22.5 
|
| A | Material constant | 0.3 
|
| n | Stress exponent | 0.0068 2.6 |
|
Reference temperature (K) | 300 |
|
Maximum dislocation density (mm−2) | 1E16 |
|
Reference current density (A/mm2) | 15 |
| R | Gas constant (J/mol/K) | 8.314 |
Conclusions
This study analyzed the deformation behavior and microstructural evolution of Ti2AlNb alloy foils during electrically-assisted stress relaxation and developed a physically-based constitutive model. The following conclusions were reached.
The non-thermal effect of pulsed current on the stress relaxation of Ti2AlNb alloy foils was decoupled using forced air cooling. When the deformation temperature was approximately 440 °C, increasing the current density from 20 A/mm2 to 50 A/mm2 reduced the ultimate residual stress from 596 MPa to 484 MPa.
Increasing current density promotes dislocation recovery and O-phase precipitation, significantly accelerating the stress decrease in the stress-driven stage I. However, extensive O-phase precipitation hinders dislocation movement, reducing the stress relaxation rate in stage II.
A physically-based constitutive model was developed, considering the effects of pulsed current on activation energy, dislocation evolution, and phase transformation. The model demonstrates good agreement with experimental data, with a relative average absolute error of 8.7%.
Author contributions
Jie Zhao: Writing – original draft, Visualization, Investigation, Formal analysis. Tianyi Gao: Writing – review & editing, Visualization, Investigation. Bao Qu: Visualization, Investigation, Formal analysis, Conceptualization. Min Cui: Funding, Validation, Supervision. Rongfu Xu: Validation, Supervision.
Funding
This work was financially supported by the National Natural Science Foundation of China (No. 52205345) and the Special Funding for Taishan Scholars Project (No. tsqnz20240828).
Data availability
Data will be made available on request, please contact Jie Zhao (zhaojiemse@163.com) at that time.
Declarations
Competing interests
The authors declare no competing interests.
Footnotes
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Contributor Information
Min Cui, Email: cuimin23@sdjzu.edu.cn.
Rongfu Xu, Email: rongfu@sdjzu.edu.cn.
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Associated Data
This section collects any data citations, data availability statements, or supplementary materials included in this article.
Data Availability Statement
Data will be made available on request, please contact Jie Zhao (zhaojiemse@163.com) at that time.