2.1 Sample preparation

In this study, UVSR experiments were conducted on the notched specimens of thin plates. The notched tensile specimen was designed based on modal analysis to obtain a uniform vibration shape of the specimen, as shown in Figure 1 (RD and TD are the rolling direction and transverse direction, respectively). The experimental specimens were made from a 6061-T6 aluminum alloy sheet with a thickness of 1 mm cut by wire electrical discharge machining. The chemical composition of the 6061-T6 aluminum alloy is listed in Table 1.

Figure 1全屏查看

Shape and size of 6061-T6 specimen (Unit: mm)

2.2 UVSR experiment

In this study, the modes of UVSR shown in Figure 2 were designed as: 1) the ultrasonic vibration was applied to the specimen while it was being stretched (TM1); 2) The specimen is unloaded after tensile deformation followed by the application of ultrasonic vibration (TM2); 3) The specimen after tensile deformation is treated with its natural ultrasonic frequency vibration pattern (designed as a bending vibration shape in this study, TM3); 4) The specimen is not unloaded after tensile deformation, and then ultrasonic vibration is applied (TM4). TM0 is the comparison operating condition without UVSR.

Figure 2全屏查看

Schematic diagram of UVSR model

The experimental platform of the UVSR was built based on a universal testing machine (UTM), as shown in Figure 3. The UTM controls the tensile movement of the specimen in the RD direction and stretches the specimen to 50% of the fracture displacement at a constant speed of 0.25 mm/min. Then, unloading is performed and residual stresses are generated in the specimen due to uneven plastic deformation. The ultrasonic vibrator is set perpendicular to the specimen surface by a stationary frame, which controls the linear excitation motion in the normal direction of the specimen, thus realizing the modes of TM1, TM2, and TM4. In addition, the elastic device is set symmetrically on both sides to improve the coupling between the vibrator and the specimen, which ensures the effective transmission of the ultrasonic energy field. The TM3 mode is realized by connecting the specimen clamping position to the horn.

Figure 3全屏查看

(a) UVSR experimental equipment, (b) UVSR schematic diagram, and (c) modes TM4 of UVSR

Ultrasound vibration with different energy densities can be realized by changing the output power of the ultrasound generator. The amplitudes of the vibrators at different powers were measured using a SOPTOPLV-S01 series laser vibrometer. The ultrasonic energy density can be expressed as [20]:

(1)

where represents the material transmission coefficient; represents the material density; and represent the amplitude and frequency of the ultrasonic vibration, respectively. The results are shown in Table 2. In addition, a FLIR-TG165 series infrared imaging thermometer gun was used to measure the specimen temperature during the UVSR to observe the temperature changes caused by ultrasonic vibration.

Residual stresses were measured using X-ray diffraction [21, 22]. The measurement was performed by the Rigaku AutoMATE II type X-ray stress gauge (Cr Target). The voltage and current are 40 kV and 40 mA, respectively. For the 6061-T6 aluminum alloy, (331) is used as the crystal face, and the diffraction angle varies from 128° to 149° at a scanning width of 0.08°.

3.1 Modal analysis modeling

The structure and dimensions of the ultrasonic vibrator and specimen were optimally designed based on the modal analysis of ABAQUS to achieve the mode TM3 described in section 2.2. The 3D model is shown in Figure 4, in which the ultrasonic transducer mainly consists of the front/rear metal cover and the piezoelectric ceramic. Corresponding to the setup of the UVSR experiment, the material of the front cover is a hard aluminum alloy, and that of the rear cover, horn, and tool is 45 steel. Lead zirconate titanate (PZT-4) was selected as the piezoelectric material of the ultrasonic transducer with the density of 7.5×10³ kg/m³, Poisson ratio of 0.3, and elastic modulus of 83 GPa. The 8-node hexahedral reduced integration element (C3D8R) is used to simplify the whole solid component in the numerical model.

Figure 4全屏查看

Finite element meshing of the whole model

3.2 Residual stress analysis modeling

The UVSR processes were simulated under the same conditions as those carried out in the previous experimentation. The simulation model is shown in Figure 5, in which the main relevant molds are established as rigid bodies and the specimens are established as deformable bodies based on the UVSR experimental setup. The specimen is simplified using the element types C3D8R.

Figure 5全屏查看

Illustration of UVSR process (Unit: mm)

The pre-stretch deformation is first performed to obtain the stress-strain field in the UVSR simulation analysis, which is then used as a predefined field to assign initial residual stresses to the specimen. A periodic amplitude profile () is applied to the tool for simulating the dynamic behavior of its excited specimen [25], as shown in Figure 5. In addition, backpressure was applied along the normal direction of the specimen by setting spring feature. Remarkably, the ultrasonic vibration is a high-speed instantaneous dynamic response process, and therefore the ABAQUS explicit time integration algorithm is used for the simulation of the ultrasonic vibration process. Subsequently, the self-equilibration process of the specimen was then simulated using the ABAQUS standard implicit algorithm.

3.3 Ultrasound plasticity constitutive model

The Johnson-Cook (J-C) model is the classical constitutive model used to describe the dynamic mechanical response of metals. In the J-C model, the equivalent stress is expressed as [26]:

(2)

where , , , and are material constants, respectively; , and are equivalent plastic strain, strain rate and reference strain rate, respectively; , and are the material current temperature, the reference temperature and the melting temperature of material, respectively.

Researches [1] have found that a significant softening phenomenon is presented during ultrasonic vibration-assisted metal forming. The J-C model has been modified by scholars [27, 28] to describe the dynamic mechanical response of metal under ultrasonic vibration by studying the influence law of amplitude on each parameter in the model, yet its expression is relatively complex and the solution process is cumbersome. Previous studies have found that the ultrasonic softening effect is similar to the softening behavior in the temperature field [29], and its mechanical response is closely related to the acoustic energy density. Given this, the ultrasonic vibration J-C (UVJ-C) model is proposed by introducing a new ultrasonic softening function . In addition, the rate-dependent constitutive model embedded in ABAQUS does not include the ultrasonic plasticity model. Therefore, the application of UVJ-C model needs to develop a material subroutine. Among them, the user subroutines VUHARD and UHARD are available for explicit and standard modules, respectively.

3.4 Parameter determination

The determination of material parameters is generally obtained by fitting the experimental data. It is indicated that the ultrasonic softening effect function can be expressed in the form of allometric function based on the experimental data under different ultrasonic parameters by nonlinear fitting (Figures 6(a)-(c)), which is:

Figure 6全屏查看

(a) Stress-strain curves at different strain rates; (b) Stress-strain curves at different ultrasonic parameters; (c) Ultrasonic softening effect curve; (d) Temperature field evolution curves

(3)

where and represent the acoustic softening constant and exponent, respectively. Therefore, the UVJ-C model is specifically expressed as:

(4)

and the acoustic softening rate σ_r is expressd as:

where σ_uv and σ_nuv represent the equivalent stress with and without ultrasonic vibration assistance, respectively.

The determination of the material constants in the J-C model is crucial for the reliability of the simulation results [30], and further parameter identification is necessary. Therefore, the calibration of model parameters is implemented in MATLAB using the advantages of genetic algorithms [31]. In addition, the experiment was performed at room temperature and there was no significant sharp increase in specimen temperature during the different ultrasonic vibration modes (Figure 6(d)). Therefore, the thermal softening effect was neglected in the UVSR simulation. The final optimization results for the model material constants are demonstrated in Table 3.

4.1 Modal prediction validation

The solid granules are uniformly sprinkled on the surface of the specimen, which were driven to the region of smaller amplitude after applying ultrasonic vibration to visualize the specimen vibration shape. The specimen vibration shape after TM3 treatment is shown in Figure 7.

Figure 7全屏查看

Vibration shape distribution: (a) By simulation; (b) By experiment (t₂>t₁)

It can be observed that the specimen is in the shape of bending vibration, which is the required shape for the experiment. The nodal lines in the simulation cloud diagram of the specimen match well with the morphology of the solid granules after redistribution in the experiment, which indicates the effectiveness of the modal prediction of the ultrasonic vibration system. ZHANG et al [19] have successfully observed the vibration shape of welded aluminum alloy plates using a similar approach.

4.2 Tensile experiment results

The specimens were stretched after UVSR to evaluate the effect of different ultrasonic vibrational modes on the mechanical properties of the specimens. The forming force-displacement curves of the 6061-T6 aluminum alloy are illustrated in Figure 8.

Figure 8全屏查看

Forming force-displacement curves: (a) Different treatment modes; (b) Treatment of short duration (TM2) and low amplitude (TM3); (c) TM1 treatment at different amplitudes; (d) TM1 treatment at different durations

The different ultrasonic vibration models all lead to a decrease in forming force compared to mode TM0 (specimen S0) (Figure 8(a)). Among them, the largest decrease in forming force was observed in mode TM3 (specimen S3), which could reach 4.05%. Moreover, mode TM1 improves the specimen ductility compared to mode TM0, whereas modes TM2 and TM3 act as inhibitors of specimen ductility.

The force-displacement curves for specimens exposed to low-amplitude vibration for a short time in modes TM2 and TM3 are illustrated in Figure 8(b). It was found that the ductility of specimens S4 and S5 improved when compared to specimens S2 and S3 under the same ultrasound vibration mode, suggesting that the suppression of ductility in specimens S4 and S5 was caused by the vast ultrasound energy field and the long vibration time. The forming force-displacement curves of the specimens treated with different amplitudes and times in mode TM1 are demonstrated in Figures 8(c) and (d); the ductility was found to show a tendency to increase and then decrease, which further supports the existence of an inflection point in the effect of ultrasound vibration on the mechanical properties of the material. In addition, specimens S7 and S9 exhibited higher strength and ductility compared to specimen S0, similar to the mechanical phenomena observed by ZOHREVAND et al [18] in 2205 dual-phase pre-strained stainless steel after ultrasonic vibration, which indicates that the strength and ductility of the material can be increased simultaneously after ultrasonic vibration treatment.

4.3 Fractography

Figure 9 shows the scanning electron microscopy (SEM) images of the specimens at different locations of tensile fracture after treatment with different ultrasonic vibration modes. The approximate positions of the electron microscope scans are marked with red dots in the figure.

Figure 9全屏查看

Specimens fracture morphology: (a) Notched root (TM0); (b) Specimen center (TM0); (c) Notched root (TM1); (d) Specimen center (TM1); (e) Notched root (TM2); (f) Specimen center (TM2); (g) Notched root (TM3); (h) Specimen center (TM3)

It is observed that the fracture morphology of the specimens consists of dimples, tearing ridges, and cleavage steps, and therefore the fracture mechanism belongs to tough-brittle mixed fracture. The fracture morphology is shown in Figures 9(a)-(d) after the vibration of modes TM0 and TM1, respectively, which shows a large number of dimples, and the difference in fracture morphology at different locations is not obvious. Compared with TM0, the fracture surface of the specimens treated with mode TM1 is flat, the distribution of dimples is denser and uniform, and the dimples are relatively smaller and more uniformly distributed in size to the extent that it is not easy to form individual weak points, which improves the fracture toughness of the specimen.

The fracture morphology after the vibration of modes TM2 and TM3 is shown in Figures 9(e)-(h), respectively. The number of dimples is relatively small and shallow. There is a clear difference in the morphology of the fracture at different positions of the specimen, where the number of dimples at the notch position is significantly greater than that at the central position. Therefore, it can be determined that the fracture occurred from the notch position. Moreover, after the mode TM2 vibration, it can be observed that the fracture has obvious cleavage steps characteristics, which leads to poor plasticity, consistent with the tensile test results. It is concluded that mode TM1 is more effective in inhibiting the fracture for the same ultrasonic energy field and vibration time.

4.4 Residual stress under ultrasonic vibration

4.4.1 UVSR model validation

The FEM model was validated using the digital image correlation (DIC) online measurement system (Figure 10(a)). It is found that the strain cloud diagram generated by the numerical simulation (Figure 10(b)) is consistent with the experimental one (Figure 10(c)), which proves that the established model can effectively predict the plastic flow behavior of the notched sample. However, the DIC is unable to measure the strain field of the specimen under ultrasonic vibration conditions due to the masking of the ultrasonic vibrator. Therefore, the residual stress data of the specimen obtained by X-ray diffraction were employed as a reference for its simulation results in order to verify the reliability of the finite element simulation results under ultrasonic vibration conditions. The relationship between the diffraction angle and the azimuth angle of diffraction crystal plane is illustrated in Figure 11.

Figure 10全屏查看

(a) DIC experimental setup; (b) Strain obtained from simulation; (c) Strain obtained from experiments

Figure 11全屏查看

Relationship between and of 6061-T6 aluminum alloy: (a) Position P1 (TM0); (b) Position P2 (TM0); (c) Position P1 (TM3); (d) Position P2 (TM3)

The measurements were carried out at the position away from the notch position P1 and at the notch position P2, since the arc notch is the main deformation region during stretching and in consideration of the economy of residual stress measurement. It is found from Figures 11(a) and (c) that appears to oscillate significantly for , which is related to the presence of the texture of the material under testing. The presence of texture disrupts the original regularity in the variation of the average interplanar spacing of the diffracting crystal planes. The work of van HOUTTE et al [32] provides a more detailed explanation for this. The method for the measurement of residual stress at position P1 will have an inestimable error; therefore, the experimental data at position P2 are adopted as the reference for the simulation results. The degree of residual stress release can be described by employing the stress relief ratio (, where and represent the value of the initial residual stress and the residual stress after UVSR, respectively). The residual stress parallel to the RD direction under different ultrasonic modes is shown in Table 4.

Table 4全屏查看

Measurement results of residual stress for 6061-T6 aluminum alloy

Vibration mode	Experiment		Simulation
	Residual stress/MPa	Relieving ratio/%	Residual stress/MPa	Relieving ratio/%
TM0	-132.5		-154.8
TM1	-109.0	17.7	-130.7	15.6
TM2	-96.8	26.9	-103.7	33.0
TM3	-45.8	65.4	-51.3	66.9
TM4	-110.2	16.8	-128.1	17.2

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The experimental results show that the residual stress can be effectively released by superimposed ultrasonic vibration during the parts manufacturing process (TM1) and the parts under load holding state (TM4). Compared with other vibration modes, mode TM3 demonstrates the highest residual stress release efficiency, reaching 65.4%. The experimental results demonstrate good agreement with the finite element simulation results, which further proves the correctness of the UVSR model.

4.4.2 Residual stress distribution

The cloud diagram of in mode TM0 obtained by finite element modeling is illustrated in Figure 12(a). The residual stresses are distributed symmetrically along the center of the specimen, and there is a stress concentration at the notched position. The reason is that the notched region produces non-uniform plastic deformation after the load is applied to the specimen, which is elongated.

Figure 12全屏查看

Residual stress cloud diagram: (a) TM0; (b) TM1; (c) TM2; (d) TM3; (e) TM4 (Conditions: amplitude of 10.16 μm, time of 120 s)

The deformation is large relative to the other areas. The elongated region is compressed under unloading conditions, which introduces residual compressive stresses in the notched region, and balanced by residual tensile stresses away from the notch. The values in different vibration modes are shown in Figures 12(b)-(e). It can be found that the distribution of is more uniform for modes TM2 and TM3 compared to modes TM1 and TM4.

The on paths PD1 and PD2 after different modes is output in Figure 13 to further analyze the distribution of residual stress. In the absence of ultrasonic vibration excitation (TM0), the first decreases slowly and then rises to the maximum tensile stress (73.93 MPa) at the center of the specimen along the path PD1. The at the root of the notch is presented as the maximum compressive stress (-158.59 MPa), and then values along the PD2 path gradually evolves into tensile stress. After the ultrasonic vibration, the overall trend of the residual stresses under modes TM1 and TM4 is almost the same as that of mode TM0. The values under modes TM2 and TM3 are significantly lower, and the overall fluctuations are smaller than those of modes TM1 and TM4. However, unlike mode TM3, residual compressive stresses were introduced on the specimen surface after mode TM2 treatment, which could reach up to -96.86 MPa.

Figure 13全屏查看

Residual stress in different treatment modes: (a) Path PD1; (b) Path PD2

The residual stress parallel to the TD directions is illustrated in Figure 14. In the mode TM0, has the same trend as in the PD2 path. However, the difference is that the maximum compressive stress (-51.99 MPa) appeared at 6.29 mm from the specimen center in the PD1 path, then rises to the maximum tensile stress (54.92 MPa), and finally it decreases to 29.22 MPa at the specimen center, which presents a wavy distribution.

Figure 14全屏查看

Residual stress in different treatment modes: (a) Path PD1; (b) Path PD2

Ultrasonic vibration modes result in different releasing effects on the residual stresses in different directions by comparing the residual stresses after ultrasonic vibration. The is significantly reduced after the treatment of modes TM1 and TM3 compared to modes TM2 and TM4, and the overall changes tended to be flat, especially since the residual stress is almost zero after mode TM1 excitation. However, mode TM1 is not significant for the release of , which indicates that the internal stresses released by the ultrasonic vibration are correlated with the direction of residual stresses.

Comprehensive comparison, the study concludes that TM3 has the best effect on residual stress relief, followed by TM2, TM1 and TM4. However, TM3 is the treatment mode of ultrasonic vibration based on the natural frequency of the sample itself, which is limited by the geometric size, shape, and material of the sample, so it is not easy to achieve in industrial applications.

4.4.3 Effects of ultrasonic vibration factors

The effects of variations in ultrasonic vibration factors on the effectiveness of UVSR are analyzed based on the FEM. The orthogonal experimental analysis method was used to carry out simulation experiments to improve efficiency. The simulation experiment scheme is designed according to the level table with four factors and four levels (as shown in Table 5) to analyze the effect of the amplitude (), the excitation time (), the contact area () between vibrator and specimen and back force () on UVSR.

Table 5全屏查看

Four-factor four-level table of the simulation experiment

No.	/μm	/s	/mm2	/N
1	8.29	60	13	15
2	10.16	120	64	35
3	11.96	180	177	55
4	15.46	240	660	75

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The average values of the stress relief ratio on the PD2 path for different ultrasonic vibration factors were obtained by extracting the equivalent stresses based on the numerical model. The priorities of the influence of the various factors on the UVSR are calculated using a range analysis method. The results show that the contact area is the most important factor affecting UVSR, followed by the back force, the amplitude, and the excitation time. As far as the study is concerned, the best technological parameters of UVSR are obtained through range analysis: ---, that is, contact area of 660 mm², back pressure of 75 N, amplitude of 11.96 μm, excitation time of 60 s.

4.5 Discussion of residual stress relaxation mechanisms

Residual stress exists in an object to maintain internal equilibrium when there is no external force acting on it. The effect of ultrasonic vibration on the specimen under modes TM2 and TM3 consists of two aspects by analyzing the force on the specimen during the UVSR process. 1) The alternating excitation force (shown in Figure 15(a)) provided by ultrasonic vibration causes the specimen to generate pulsating stress, relying on the sum of the pulsating stress and initial residual stress exceeding the yield strength of the material, resulting in a localized micro elastic-plastic deformation; 2) Ultrasonic softening effect increases the possibility of its plasticization, that is, the application of ultrasonic vibrations during deformation results in a significant decrease in the yield strength of the material, which is different from that of conventional low-frequency VSR.

Figure 15全屏查看

(a) Force analysis of specimens in UVSR; (b) Longitudinal wave particle motion (R: Reflected wave; T: Transmitted wave; I: Incident wave)

The mode TM4 is in the loaded state and the residual stress release mechanism is similar to that of TM2 and TM3. It is different from mode TM4 reling on the sum of the pulsating stress and the stress provided by the load to produce localized microelastic-plastic deformation for stress release. However, mode TM1 is different from them. The residual stress release is due to ultrasonic vibrations playing the role of coordinating the deformation.

The vibrator will generate ultrasonic waves in the vibration direction during the UVSR process. It spreads in the medium as a stress longitudinal wave, and the particle displacements for the longitudinal wave are shown in Figure 15(b). After contact with the specimen surface, a part of the ultrasonic waves will be reflected. Another part will be transmitted to the internal material, which will have a certain effect on the internal structure. Based on dislocation dynamics, part of the ultrasonic energy is absorbed by the crystal defects when ultrasound is transmitted into the material, decreasing the potential barriers that impede the dislocation motion and inducing dislocation slip, leading to an additional evolution of the internal structure of the material.

According to the principle of the minimum energy, the ultrasonic vibrations in this process will harmonize the internal energy distribution of the material to a minimum and stable equilibrium state, which is therefore manifested as a reduction of residual stresses when the external loads are stopped. It has been found that superimposed ultrasonic vibration during plastic deformation of materials leads to a decrease in dislocation density [33], an increase in grain size and the proportion of twin boundaries [34], and it has been found that ultrasonic vibration facilitates grain rotation [35]. All the above phenomena are traces left behind by ultrasonic vibration in the process of coordinated plastic deformation.