Springback control of Ti2AlNb alloy ultra-thin corrugated sheets via current-targeted loading during current-assisted roll forming process

Abstract

This paper proposes a novel current-assisted roll forming process for Ti₂AlNb alloy ultra-thin corrugated sheets, which enables local targeted loading of pulsed current, leveraging the electromigration effect to eliminate springback defects. First, the effect of pulsed current on the springback of Ti₂AlNb alloy foils during V-shaped bending was analyzed. The springback angle of the foils significantly decreases as the current density increases, particularly when it exceeds 20 A/mm². With a current density of 80 A/mm², the springback angle is only 0.3°. Subsequently, the deformation mechanism of Ti₂AlNb alloy ultra-thin corrugated sheets during the current-assisted roll forming process was elucidated. The strain field and current density field exhibit significant non-uniform distribution and dynamic evolution, with strain and current density concentrated at local bending corners. The springback angles of the corrugated sheet range from 0.2° to 0.6° under a loading current of 125 A. A prediction model for the forming angle of Ti₂AlNb alloy ultra-thin corrugated sheets was established based on a classical stress relaxation equation modified to include the effect of pulsed current and an approximate substitution of the current density history. The model achieved a relative average absolute error of 8.1%. The effective springback control of the Ti₂AlNb alloy ultra-thin corrugated sheet may be attributed to the electromigration effect of the pulsed current, since the Joule heating effect was suppressed.

Introduction

Metal honeycomb structures, composed of corrugated sheets stacked in parallel, are characterized by low relative density, high specific strength, and excellent thermal insulation performance, significantly promoting the lightweight development of aircraft¹. Ti₂AlNb alloy is a highly promising lightweight heat-resistant structural material for service in the 650–750 ℃ range^2,3. Therefore, the ultra-thin Ti₂AlNb alloy honeycomb structure, with its lightweight and high-temperature resistance properties, holds strong application potential for thermal protection components such as wing surfaces and skins in high-speed aircraft.

The preparation of ultra-thin metal honeycomb structures typically involves forming corrugated sheets from sheet metal through stamping or rolling, followed by fabricating honeycomb structures via spot welding or other connection methods⁴. However, forming corrugated sheets from hard-to-deform materials such as titanium alloys, Ti₂AlNb/TiAl alloys, and superalloys at room temperature is prone to springback defects^5,6. This often necessitates additional shaping processes or the application of hot forming to ensure dimensional accuracy. For instance, Satheesh K S et al. formed corrugated sheets from 0.1 mm thick nickel-based superalloy (Superni 263) using a combined process of roll bending and pressure shaping, subsequently fabricating honeycomb sandwich structures via vacuum brazing⁷. Tong G Q et al. achieved precise forming of 0.05 mm thick TC1 titanium alloy corrugated sheets through a similar combined process and fabricated ultra-thin honeycomb structures by spot welding⁸. Jiang S S et al. formed 2 mm thick TiAl alloy corrugated sheets via hot die pressing and fabricated multi-layer honeycomb structures through diffusion bonding⁹. Currently, there are no reported studies on ultra-thin (wall thickness ~ 0.1 mm) Ti₂AlNb alloy honeycomb structures.

Ti₂AlNb alloy has a low elastic modulus and high strength, making it susceptible to springback defects during sheet bending. Even after holding at 810 ℃ for 10 minutes, the springback angle during U-shaped bending of Ti₂AlNb alloy sheets still reaches 25°¹⁰. Moreover, hot bending or shaping processes can easily cause inevitable surface oxidation, which is unacceptable for Ti₂AlNb alloy ultra-thin corrugated sheets. The current-assisted forming process can rapidly heat the undeformed billet to the forming temperature, improve formability, and reduce component surface oxidation, offering significant advantages for efficient and green production¹¹. Furthermore, the suppressive effect of pulsed current on bending springback has been widely confirmed. Increasing the frequency and density of pulse current can reduce the springback angle in V-shaped bending of AZ31B sheets¹². When the current density reaches 120 A/mm², springback defects in Al-6111 alloy almost disappear¹³. With instantaneous high-density current loading (6 kV voltage), springback defects in TC4 titanium alloy are eliminated¹⁴. Low-frequency pulse current can reduce residual stress in 316 L stainless steel welded joints by 40% without significant temperature rise¹⁵. By controlling temperature rise¹⁶ or comparing with isothermal experiments¹⁴, the non-thermal effect of pulse current in suppressing springback has been verified. Pulsed current can activate unconventional dislocation slip (e.g., prismatic slip in magnesium alloys, pyramidal slip in titanium alloys), resulting in localized plastic deformation^17,18. It can also accelerate dislocation recovery and reduce dislocation density. Meanwhile, pulsed current promotes recrystallization nucleation and grain growth, and further accelerates phase transformation (e.g., α→β→α′ in titanium alloys)^19,20. All these microstructural evolution processes accelerate stress relaxation and inhibit springback defects in components.

This paper analyzes the effect of pulsed current on the springback behavior of Ti₂AlNb alloy foils during V-shaped bending. It then investigates the deformation mechanism of Ti₂AlNb alloy ultra-thin corrugated sheets during current-assisted roll forming and establishes a prediction model for the forming angle of the corrugated sheets.

Materials and methods

The Ti₂AlNb alloy foils, approximately 0.1 mm thick and 150 mm wide, were provided by the Institute of Metal Research, Chinese Academy of Sciences, as shown in Fig. 1a. The nominal composition is Ti–22Al–25Nb (in wt %). The foils underwent processing steps including hot rolling, cold rolling, and annealing heat treatment.

The microstructure of foils was characterized using electron backscattering diffraction technique. Sample surfaces were electrolytically polished (polishing solution: 6% perchloric acid + 34% butanol + 60% methanol, temperature − 20 ℃, time 45 s, current 0.86 A). The scanning step size was 0.2 μm, and the scanning area was a 25 μm × 25 μm rectangular region, as shown in Fig. 1c-e. The foils exhibit a distinct rolled microstructure, with fine grains elongated along the rolling direction. They consist of three phases: α₂, B2/β and O phase, with volume fractions of 61.2%, 34.3%, and 4.5%, respectively. The foil contains a large amount of deformed structure, and the average value of the Kernel average misorientation is 0.74°.

Uniaxial tensile tests of the Ti₂AlNb alloy foils at room temperature were conducted on a material testing machine (AG-X 50 kN, Shimadzu Corp.). The dimensions of the tensile specimens and the stress-strain curves are shown in Fig. 1b. The yield strength is approximately 708 MPa, the tensile strength is approximately 901 MPa, and the elongation is approximately 17.2%.

Fig. 1 [Images not available. See PDF.]

Initial Ti₂AlNb alloy foils, uniaxial tensile curve, and microstructure analysis: (a) initial Ti₂AlNb alloy foils; (b) uniaxial tensile curve at room temperature; (c) inverse pole figure; (d) Kernel average misorientation; (e) phase distribution.

The current-assisted V-shaped bending experiments on the foils were also conducted on the material testing machine (AG-X 50 kN, Shimadzu Corp.). A pulsed current was provided by a direct current power supply (MicroStar CRSLFP20-500, Dynatronix Inc.). The positive and negative poles of the power supply were connected to the upper and lower dies, respectively. The dies and platform were insulated using bakelite plates, as shown in Fig. 2(a). Initial bending samples were cut into 3 mm × 6 mm rectangles. The foil plane was perpendicular to the loading direction of both force and current. The rectangular foil sheet was first placed on the V-shaped lower die. The upper die moved down to apply a clamping force of 100 N, and then the pulse power supply was activated. A thermocouple inserted through the upper die measured the temperature of the bent sample in real time. Detailed experimental plans for the current-assisted V-shaped bending are provided in Table 1.

The schematic diagram of the current-assisted roll forming process for the Ti₂AlNb alloy ultra-thin corrugated sheet is shown in Fig. 2(b). Two assembly rollers rotate in opposite directions. The process begins by activating the pulse power supply. The initial foil is then placed between the rollers using wooden tweezers until engaged by the rollers. The positive and negative poles of the pulse power supply are connected to the upper and lower rollers, respectively, applying current perpendicular to the foil plane. This setup achieves localized high-density current loading at the bending positions of the corrugated sheet, aiding in the elimination of springback defects.

The current-assisted roll forming device used in this study for the Ti₂AlNb alloy ultra-thin corrugated sheet is shown in Fig. 2(c). Two rollers are installed on a homemade rolling frame, insulated from the frame by ceramic sheets. The shape and size of the rolls are shown in Fig. 2(d). The rolls are made of 9Cr2Mo steel and undergo quenching and tempering treatment. The surface roughness of the rolls is required to be Ra0.8. The target corrugated sheet consists of semi-regular hexagons with a side length of 5.0 mm, thickness of 0.1 mm, width of 30 mm, and a corner radius of 0.1 mm. The position of the upper roller can be adjusted via the screw to regulate the gap between the upper and lower rollers. The gap is set at 0.15 mm.

A finite element model of the Ti₂AlNb alloy ultra-thin corrugated sheet formed by current-assisted roll forming was established using ABAQUS software. The model employs the Coupled thermal-electrical-structural analysis step, implicit integration algorithm, and the element type is Q3D8. Friction, heat conduction, and electrical conduction are set between the sheet and the roller. To reduce computational cost, the upper and lower rollers were hollowed out. The boundary conditions for the rollers and foil are shown in Fig. 2(e). An initial velocity was set for the foil to simulate the feeding process. The upper roller rotates actively while the lower roller is driven passively. Current was applied to the surface of the upper roller, and zero potential was set on the surface of the lower roller. The distribution of current density and temperature can be automatically calculated by ABAQUS software based on the applied current. The mechanical properties of the foil were defined based on the room temperature tensile stress-strain curve shown in Fig. 1(d). Other model parameters are listed in Table 2.

Fig. 2 [Images not available. See PDF.]

Schematic diagram and device for current-assisted V-shaped bending and roll forming of Ti₂AlNb alloy foils: (a) V-shaped bending device; (b) roll forming process; (c) roll forming device; (d) size of rolls; (e) finite element model.

Table 1. Detailed experimental plans for current-assisted V-shaped bending.

Current density (A/mm2)	Current frequency (Hz)	Duty cycle (%)	Time (s)	Bending angle (°)
0–80	100	30	10	120
30	40–140	30	10	120
30	100	10–70	10	120
30	100	30	0–12	120
30	100	30	10	60–150

Table 2. Finite element model parameters for current-assisted roll forming of Ti₂AlNb corrugated sheet.

Parameters	Values
Rotating speed of rollers n (r s− 1)	0.01
Initial velocity of foil v0 (mm s− 1)	1.0
Coefficient of friction	0.1
Density (t m− 3)	7.8 (roller) 4.5 (foil)
Electrical resistivity (µΩ m)	0.098 (roller) 0.9 (foil)
Thermal conductivity (W m− 1 ℃−1)	36.0 (roller) 15.0 (foil)
Thermal capacity (J kg− 1 ℃−1)	460.0 (roller) 684.0 (foil)
Emissivity	0.6 (foil)
Loading current (A)	0-125

Results

Current-assisted V-shaped bending

The temperature rise of Ti₂AlNb alloy foils during the current-assisted V-shaped bending process is shown in Fig. 3. Intense heat exchange between the thin foil (0.1 mm thickness) and the cold mold results in a relatively weak Joule heating effect from the pulsed current. As the current density, frequency, and duty cycle increase, the temperature rise of the foil gradually increases. The foil temperature peaks around 4 s after the power is turned on and then remains constant. Within the selected range of pulse current parameters, the maximum foil temperature is 133 ℃.

Fig. 3 [Images not available. See PDF.]

Temperature rise of Ti₂AlNb alloy foils during the current-assisted V-shaped bending process: (a) current density and current frequency; (b) duty cycle and time.

The effects of current parameters and bending angle on the forming angle of Ti₂AlNb alloy foils during current-assisted V-shaped bending are shown in Fig. 4. Overall, the forming angle gradually decreases with increasing current density, current frequency, duty cycle, and power-on time. Within the selected parameter range, current density has the greatest impact on the forming angle, while the duty cycle has a negligible effect. When the current density exceeds 20 A/mm², its effect on the forming angle increases significantly, indicating the existence of a current density threshold. As the power-on time increases to around 4 s, further increases have almost no effect on the forming angle. Under conditions of 80 A/mm² current density, 100 Hz frequency, 30% duty cycle, and 10 s power-on time, the forming angle is 120.3°. The springback angle is only 0.3°, indicating that springback is essentially eliminated. The springback angle of the Ti₂AlNb alloy foils increases with increasing bending angle.

Fig. 4 [Images not available. See PDF.]

Effect of current parameters and bending angle on the forming angle of Ti₂AlNb alloy foils during current-assisted V-shaped bending: (a) current density and current frequency (bending angle of 120°); (b) duty cycle and time (bending angle of 120°); (c) bending angle.

Calculation of springback angle of V-shaped bending

Springback essentially represents the release of residual stress in the component. During the pressure holding and current application stages, the V-shaped bending samples undergo stress relaxation. The residual stress in the sample during the uniaxial tensile stress relaxation process can be described by the classical stress relaxation equation²¹:

1

where is the stress relaxation limit, t is the stress relaxation time, is a fitting constant, is the reference time.

Pulsed current accelerates the stress relaxation process, with current density and frequency having particularly significant influences²². Therefore, Eq. (1) can be rewritten as:

2

where , , , are fitting constants, is the current density, is the current frequency and is the reference current density.

The axial strain and radial stress during the bending process of thin sheets can be disregarded, that is, and . Based on Mises yielding criterion and the incremental theory, the relationships between axial stress and circumferential stress , as well as between equivalent stress and circumferential stress can be derived:

3

4

The residual stress measured in the uniaxial tensile stress relaxation experiment is the equivalent stress . According to the equivalent stress in Mises yielding criterion, the circumferential stress during sheet metal bending deformation is related to the residual stress as follows¹²:

5

In the rigid-ideal plasticity assumption, the value of circumferential stresses on the inner and outer sides of the sheet are uniform. The outer side experiences tensile stress ( ) while the inner side experiences compressive stress ( ). This assumption neglects the elastic strain, the uneven stress distribution in the thickness direction, and the offset of the neutral layer of the strain. However, in engineering models, the rigid-ideal plasticity assumption is an effective simplification tool for calculating bending moments. At this point, the circumferential stress serves as the equivalent stress that represents the actual stress distribution. The bending moment (M) and the springback angle ( ) can be calculated as¹³:

6

7

where b is the width and h is the thickness of the bending sample, is the radius of the neutral layer, is the equivalent Young’s modulus under plain strain condition, is the inertia moment.

Based on Eq. (2) to (7), the springback angle for the current-assisted V-shaped bending of Ti₂AlNb alloy foil can be calculated. Equation (2) to (7) were implemented in a Python program. The parameters were determined through an iterative process of comparing the calculated springback angles with the experimental results and gradually narrowing down the range of parameter values. The relative average absolute error (RAAE ) was used to indicate the fitting accuracy. The RAAE value of the optimal fitting result is 5.3% as shown in Fig. 5. Parameters for the springback calculation model are listed in Table 3.

Fig. 5 [Images not available. See PDF.]

Calculation of forming angle for Ti₂AlNb alloy foils during current-assisted V-shaped bending: (a) current density and current frequency; (b) duty cycle and time.

Table 3

Parameters of the springback calculation model for current-assisted V-shaped bending of Ti₂AlNb foils.

Parameters	Values
Stress relaxation limit (MPa)	0
Reference time (s)	2.14
Reference current density (A/mm2)	36.94
Width of bending sample b (mm)	3
Thickness of bending sample h (mm)	0.1
Radius of neutral layer (mm)	0.1
Equivalent Young’s modulus (MPa)
Young’s modulus (MPa)	31,157
Poisson’s ratio
Inertia moment (mm4)
Fitting constant	117.28
Fitting constant	338.27
Fitting constant	2.26E5
Fitting constant	4.01
Fitting constant	-2.06

Current-assisted roll forming of Ti₂AlNb alloy ultra-thin corrugated sheet

Simulation results for the stress and strain of the Ti₂AlNb alloy ultra-thin corrugated sheet during current-assisted roll forming are shown in Fig. 6(a)-(c). The corrugated sheet exhibits obvious non-uniform deformation during the process, with plastic deformation primarily occurring at the local bending corners, while the straight wall areas undergo negligible plastic deformation. The maximum equivalent plastic strain is 0.139, which is less than the ultimate tensile strain of the Ti₂AlNb alloy foils. The distribution of equivalent stress is basically consistent with the equivalent plastic strain distribution, but it is more dispersed. The maximum value is 876.6 MPa. Each corrugation is divided into a front corner and a back corner, with the strain and stress in the front corner slightly greater than those in the back corner. The forming time for each corner is approximately 10 s. During forming, stress and strain increase rapidly; after forming is complete, the strain remains constant while the stress drops rapidly.

Simulation results for the current density and temperature of the Ti₂AlNb alloy ultra-thin corrugated sheet during current-assisted roll forming are shown in Fig. 6(d) and (e). During the rolling process, the contact position and area between the roller and the foil constantly change, resulting in a dynamic variation of the current density distribution. Taking the front corner of a corrugation as an example, when the roller first contacts the foil, the electrified area is very small and the current density increases rapidly. As the electrified area gradually increases, the current density gradually decreases until it reaches zero.

With an applied current of 100 A, the transient maximum current density achieved by the corrugated sheet is 117.6 A/mm². Similar to the current-assisted V-shaped bending described earlier, intense heat exchange between the foil and the rollers limits the temperature of the corrugated sheet to less than 50 ℃. Thus, the Joule heating effect caused by the current can be disregarded.

Fig. 6 [Images not available. See PDF.]

Simulation results for Ti₂AlNb alloy ultra-thin corrugated sheet: (a) distributions of equivalent plastic strain and equivalent stress; (b) stress evolution; (c) strain evolution; (d) evolution of the current density and temperature in the bending front corner; (e) distribution of the current density.

Forming angles, wall thicknesses and microstructures of the experimentally formed Ti₂AlNb alloy ultra-thin corrugated sheet components are shown in Fig. 7. The roller rotational speed was 0.01 r/s, current frequency 100 Hz, and duty cycle 30%. The experimental results of the thickness distribution of corrugated sheets are basically consistent with the simulation results. The straight wall areas show almost no thinning, while at the bending corner position, there is a significant thinning, with the maximum thinning rate being approximately 8%.

Without electrical power, the corrugated sheet shows obvious springback, with the springback angle at the back corner greater than that at the front corner; the forming angles were 128.2° and 126.1°, respectively. As the loading current increased from 0 A to 125 A (transient maximum current density 147 A/mm²), the forming angles of the corrugated sheet gradually decreased, reaching 120.6° and 120.2° at the back and front corners, respectively. The difference in forming angles between the front and back corners decreased from 2.1° to 0.4°. Based on the finite element simulation results and the component surface morphology, the Joule heating effect is weak, suggesting that the non-thermal effect of the current primarily suppresses the springback defect.

The microstructure of the formed corrugated sheet with a loading current of 125 A is shown in Fig. 7(d) and (e). Compared with the initial foil, the average value of the Kernel average misorientation decreased from 0.74° to 0.33°, indicating a significant reduction in the dislocation density. The contents of each phase did not change significantly, among which the content of the O phase was 5.1%.

Fig. 7 [Images not available. See PDF.]

Experimentally formed Ti₂AlNb alloy ultra-thin corrugated sheet components: (a) forming angle; (b) wall thickness; (c) corrugated sheet components; (d) Kernel average misorientation; (e) phase distribution.

Calculation of forming angle of Ti₂AlNb alloy ultra-thin corrugated sheet

Based on the simulation results, the evolution history of the current density at the forming corner position of the Ti₂AlNb alloy ultra-thin corrugated sheet can be obtained. For simplicity, the current density loading history is simplified into n segments of constant current density loading, with the current loading time for each segment being , as shown in Fig. 8(a). The springback suppression effect of each constant current density segment can be predicted using the springback prediction model for Ti₂AlNb alloy foils during current-assisted V-shaped bending, allowing calculation of the forming angle for the corrugated sheet during current-assisted roll forming.

Let be the forming angle of the corrugated sheet without current application, and be the reduction value of the springback angle caused by the current loading in the i-th segment. Then, the forming angle of the current-assisted formed corrugated sheet can be expressed as:

8

The reduction value of the springback angle caused by the current loading in the i-th segment has a linearly positive correlation with the reduction in residual stress caused by the current loading in the i-th segment , as expressed by the following equation:

9

where represents the forming angle after the current loading in the (i-1)-th segment.

The reduction in residual stress caused by the current loading in the i-th segment can be calculate based on Eq. (2), changing the time to the time of the i-th segment and the current density to the loading current density in the i-th segment , as follows:

10

The front and back corners of the corrugated sheet require separate calculations, differing in the initial forming angle without current and the current density loading history. The calculation results for the forming angle are shown in Fig. 8(b), with a calculated RAAE value of 8.1%.

Fig. 8 [Images not available. See PDF.]

Calculation of forming angle of Ti₂AlNb alloy ultra-thin corrugated sheet: (a) schematic diagram of calculation strategy; (b) calculation results.

Discussion

During the current-assisted forming of Ti₂AlNb alloy, pulsed current promotes dislocation movement and annihilation, reducing dislocation density²³. It also promotes the spheroidization of the lamellar O phase by activating twinning and increases the recrystallization nucleation rate by reducing the activation energy^24,25. Meanwhile, pulsed current accelerates phase transformation due to enhanced element diffusion, such as the precipitation of acicular O phase in the 450–750 ℃ range, O→B2 transformation in the B2 + O phase region, and O→B2 + α₂ and α₂→B2 in the B2 + α₂ phase region^26,27. All these microstructural evolution processes accelerate the release of residual stress in Ti₂AlNb alloy²⁸, which is beneficial for controling springback defects.

Unlike existing studies on current-assisted forming of Ti₂AlNb alloy, the Joule heating effect in this study is significantly suppressed due to intense heat exchange between the foil and the cold mold. Therefore, the springback control of the Ti₂AlNb alloy ultra-thin corrugated sheet should be primarily attributed to the electromigration effect of the pulsed current. Electromigration refers to the phenomenon where atoms or vacancies migrate under the action of drifting electrons due to the influence of pulsed current²⁹. According to electron wind theory, kinetic energy transfer between drifting electrons and atoms can promote the movement of vacancies, dislocations, and other defects even at room temperature. Enhanced dislocation movement promotes dislocation annihilation, resulting in a decrease in lattice defects such as dislocations and small-angle grain boundaries, thereby releasing residual stress in components^30,31. Pulsed current can also alter dislocation movement behavior, significantly affecting dislocation morphology and forming a regular distribution related to the direction of the pulsed current³². The influence of pulsed current on dislocation movement and morphology is likely the main reason for controlling springback defects in Ti₂AlNb alloy ultra-thin corrugated sheets.

Conclusions

This paper analyzed the deformation behavior of Ti₂AlNb alloy foils during current-assisted V-shaped bending and current-assisted roll forming, and established a prediction model for the springback angle of the corrugated sheets. The following conclusions were reached:

1. As the current density increases, particularly beyond 20 A/mm², the springback angle of V-shaped Ti₂AlNb alloy foils decreases significantly. Under conditions of 80 A/mm² current density, 100 Hz frequency, 30% duty cycle, and 10 s power-on time, the springback angle is only 0.3°.

2. The strain field and current density field of the corrugated sheet exhibit significant non-uniform distribution and dynamic evolution, with strain and current density concentrated at the local bending corners. As the loading current increases to 125 A, the springback angles at the back and front corners are 0.6° and 0.2°, respectively.

3. A prediction model for the forming angle of Ti₂AlNb alloy ultra-thin corrugated sheets was established based on a classical stress relaxation equation modified to include the effect of pulsed current and an approximate substitution of the current density history. The model achieved a relative average absolute error of 8.1%.

4. Due to intense heat exchange between the foil and the cold mold, the Joule heating effect is significantly suppressed. The springback control of the Ti₂AlNb alloy ultra-thin corrugated sheet may be attributed to the electromigration effect of the pulsed current, which enhances dislocation movement and regular arrangement, thereby reducing residual stress.

Author contributions

**Jie Zhao** : Writing – original draft, Visualization, Investigation, Formal analysis. **Chengqian Wang** : Writing – review & editing, Visualization, Investigation. **Yingming Tu** : Visualization, Investigation, Formal analysis, Conceptualization. **Min Cui** : Funding, Validation, Supervision. **Cainian Jing** : 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 ([email protected]) at that time.

Declarations

Competing interests

The authors declare no competing interests.

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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