Ultrasonic energy attenuation characteristics in plastic deformation of 2219-O aluminum alloy

In the ultrasonic-assisted metal forming process, the dislocations within the material are easier to move due to the absorption of ultrasonic energy, which can effectively promote material flow and improve the formability of components; this phenomenon is called the ultrasonic softening effect. The ultrasonic softening effect is generally treated as homogeneous at the whole materials for simplicity, while the attenuation of the ultrasonic energy along the propagation direction will bring inhomogeneous distribution of softening degree. In addition, the absorption of the ultrasonic energy by the material is also affected by the dislocation movements in the metal plastic processing procedure, resulting in the variation of the ultrasonic attenuation characteristics in the material with the plastic deformation, and the current research has little concerned about it. In this paper, the ultrasonic attenuation properties in 2219-O aluminum alloy with plastic strain were investigated. The influence of the dislocations and the dislocation movements caused by plastic deformation on the ultrasonic attenuation was characterized. The pre-strain specimen was designed to indicate the degree of plastic deformation of the material, and the specimen thickness direction was defined as the propagation direction of the ultrasonic energy. The experimental results and the microstructure observation showed that the absorption of ultrasonic energy by the material increases firstly and then decreases with the plastic strain increasing, which is related to the evolution of movable dislocations within the material. In order to accurately describe the ultrasonic energy attenuation characteristics in plastic deformation, the hardening equation of 2219-O aluminum alloy considering ultrasonic propagation distance and plastic strain was built, and the model accuracy was verified based on the experimental data.


Introduction
Ultrasound has the advantages of high frequency, concentrated sound energy, and strong propagation direction [1]. Some physical, chemical, and biological properties or states of the material can be altered by the action of ultrasonic energy on the material [2]. Studies have shown that applying ultrasonic vibration during metal processing can effectively reduce material yield stress and flow stress [3][4][5][6][7], reduce interfacial friction between mandrel and blanks [8,9], and improve the surface roughness of formed components [10].
At present, the research of the ultrasonic-assisted forming process mainly focuses on mechanism analysis, simulation modeling, and process exploration. Many research focused on revealing the acoustic softening mechanism, and the homogeneous softening effect in the material is supposed. Siddiq et al. [11] modified the evolution law of crystal plasticity by considering the effect of acoustic softening due to high-frequency vibration. This model exhibited good predictions for ultrasonic-assisted plastic deformation of polycrystalline aluminum. Yao et al. [12] combined the thermal activation model Arrhenius equation with the Gibbs free energy equation and proposed a unified acoustic plastic model to account for the acoustic softening phenomenon, which could accurately predict the stress-strain curves of aluminum specimens in the ultrasonic-assisted upsetting test. Wang et al. [13] proposed a mechanism that the athermal dislocation dynamics may change at the microstructure level during ultrasonic-assisted deformation, and the acoustic softening effect on the Hall-Petch behavior was modeled by incorporating a power function of acoustic energy density into the dislocation ejection work. The model could accurately predict the Hall-Petch slope in ultrasonic-assisted micro-tension at the lower strains.
Based on the acoustic theory [14], Shi et al. [15] investigated the attenuation of ultrasonic energy propagation in friction stir welding. The results showed that the energy attenuation caused by the absorption of the workpiece material exhibited an exponential attenuation law. The mechanical properties of the material at different positions are various, resulting in an inhomogeneous distribution of ultrasonic softening [2].
In the previous research on the ultrasonic-assisted spinning of ribbed components, our group compared the final rib heights with the simulation and the experiments; uniform material soften properties with the ultrasonic field were supposed, and the result showed the trend is correct, but somewhat different [16], as shown in Fig. 1. One of the important reasons is that the attenuation of ultrasonic energy when propagating in the material is not considered. Therefore, the real ultrasonic attenuation characteristics should be investigated to improve the accuracy of the simulation model.
In this paper, the attenuation characteristics of ultrasonic energy of 2219-O aluminum alloy in plastic deformation were theoretically analyzed. An experimental platform was built to measure the ultrasonic transfer attenuation degree. The relationship between ultrasonic transfer efficiency and plastic strain was investigated and explained based on microstructure observation. The hardening equation of 2219-O aluminum alloy considering ultrasonic propagation distance and plastic strain was established, and the experimental results validated its accuracy.

Ultrasonic attenuation parameter definition
In order to clarify the attenuation characteristics of ultrasonic energy along the propagation direction in the plastic deformation, the propagation of ultrasonic energy in the workpiece needs to be explained first.
According to the acoustic theory [14], the attenuation of the ultrasonic wave along the propagation distance is exponential attenuation in a solid medium. The schematic diagram of the propagation is shown in Fig. 2.
Based on Fig. 2, the ultrasonic attenuation expression can be presented as where U(x) represents ultrasonic energy at propagation distance x, U 0 is the initial input of ultrasonic energy, α is the attenuation coefficient, and x is propagation distance.
The theoretical analysis in this research is based on the following assumptions: (1) Ultrasonic attenuation occurs in the direction of propagation and is one dimensional; (2) The input of ultrasonic energy is constant; (3) The proportion of environmental consumption of ultrasonic energy is ignored.
In this study, the ultrasonic transfer efficiency is defined to characterize the attenuation characteristics of ultrasonic energy. According to Eq. (1), it is assumed that under the influence of plastic deformation, the output ultrasonic energy U out can be expressed as where U out is the output ultrasonic energy, U 0 is the initial input ultrasonic energy, and t is workpiece thickness. Due to the ultrasonic attenuation is composed of various attenuation forms, it is difficult to measure the ultrasonic attenuation coefficient α. When the workpiece thickness is determined, e − t is constant, that is, U out ∕U 0 is a fixed value. Therefore, we can directly measure the energy ratio of input and output to obtain the ultrasonic transfer efficiency. Ultrasonic energy U is related to frequency, amplitude, and medium and can be expressed as where m is the workpiece density, and and f are the ultrasonic vibration amplitude and frequency, respectively.
As shown in Eq. (3), since the ultrasonic is transmitted in the fixed material, the only parameter that can be changed is the amplitude. Thus, the parameter of ultrasonic transfer efficiency is introduced, which is defined as the ratio of output amplitude to input amplitude 0 . Based on Eq. (2), the specific expression of ultrasonic transfer efficiency can be obtained: In Eq. (4), parameter can be labeled according to experiments.

Experimental principle
Penetration method is generally used to measure ultrasonic attenuation property. And the scheme of measuring the ultrasonic transfer coefficient is shown in Fig. 3. The ultrasonic tool was placed on the material surface, and a piezoelectric acceleration sensor was installed on the corresponding position of the other side to collect ultrasonic vibration signals.
In the metal forming process, the strain continues to increase from elastic stage to plastic stage, under the action of the external load. It is difficult to use in situ methods to measure the ultrasonic attenuation effect, and a method of fixed strain value was used in this research. First, the attenuation efficiency under ultrasonic loading conditions was measured, then the strain value was sequentially increased to obtain the relationship between strain and attenuation characteristics. If the interval of strain values is sufficiently small, it can be regarded that the attenuation characteristic in the continuous deformation process is obtained. The following experimental schemes are designed: (1) Pre-strain test: the uniaxial tensile test was carried out to obtain samples with different pre-strain, which can be used to characterize the plastic deformation; (2) Measurement experiment of ultrasonic transfer efficiency: The different ultrasonic amplitudes were applied to the pre-strain samples, and the ultrasonic energy transfer coefficient was measured.

Experimental material
2219-O aluminum alloy with 3 mm thickness was used in this study, and the chemical composition is shown in Table 1. The uniaxial tensile specimen was cut along the 0° direction from the rolled sheet according to the ASTM B211/B211M-2019 standard, and the specimen gauge length is 50 mm, as shown in Fig. 4. The tensile stress-strain curve of 2219-O aluminum alloy was obtained from SUNS electronic universal testing machine, as shown in Fig. 5. And the corresponding parameters are shown in Table 2.

Pre-tensile specimen preparation
Six groups of samples were selected for research. The uniaxial tensile test was carried out at a tensile rate of 1.5 mm/ min. Since the ultimate tensile displacement of the sample was about 9 mm, as shown in Table 2. The samples were stretched from 2 to 7 mm, respectively, and six samples of numbers 0-5 with different pre-strains were obtained, as shown in Table 3.

Experiment platform
The energy transfer coefficient measurement test platform is shown in Fig. 6. The selected ultrasonic vibration device is HJ20-3500, as shown in Fig. 6a. The ultrasonic frequency of the ultrasonic generator is 20 kHz, and the ultrasonic generator can change the ultrasonic amplitude by adjusting the power; the minimum ultrasonic amplitude can be set to 3 μm.
The data acquisition (DAQ) system includes a piezoelectric acceleration sensor, data acquisition card, and acquisition software, as shown in Fig. 6b. The selected piezoelectric sensor is SA-IE50G, which has a measurement frequency range of 0.2-25 kHz and a sensitivity of 99.5 mV/g. Data acquisition hardware and software includes a data acquisition card, constant current adapter, and DAQ data acquisition software. The piezoelectric sensor was glued to the upper surface of the pre-strain sample to measure the vibration signal transmitted by ultrasonic vibrations along the thickness of the sample, and then the sample was fixed on the upper platform by the adhesive method. The ultrasonic tool was in contact with the lower surface of the pre-strain sample to apply ultrasonic amplitude. During the test, the ultrasonic amplitude can be adjusted to 3, 6, and 9 μm by changing the power of the ultrasonic generator. Then, the data acquisition system was used to collect the vibration signal, and the frequency spectrum of the vibration signal was further analyzed to obtain the ultrasonic vibration amplitude data.
TEM observation was used to obtain the dislocation configuration at different strains under the ultrasonic amplitudes; the samples were prepared from the tested specimen, and the selected observation position was the cross section of the position marked by the box line in Fig. 7. The cross section was thinned to 40 µm with a precision ion thinning instrument (PIPS 695).

Ultrasonic energy attenuation characteristics
The pre-strain results obtained in the uniaxial tensile test are shown in Table 3. Sample #0 is the initial stage of plastic deformation. In order to explore the relationship between ultrasonic attenuation and plastic deformation, it is assumed that ultrasonic energy is not lose at all in sample #0, which means that U out = U in .
The results of partial data of the ultrasonic energy signal measured are shown in Fig. 8a. The MATLAB spectrum analysis program was written to conduct spectrum analysis on the data within a fixed time. The obtained amplitude is shown in Fig. 8b, and its corresponding frequency is around 20 kHz, which proves that the frequency of ultrasonic keeps constant in the propagation process.
The input ultrasonic amplitudes were 3 μm, 6 μm, and 9 μm, and the corresponding output amplitudes were measured respectively. The results of ultrasonic transfer efficiency changing with pre-strain are shown in Table 4. Figure 9 shows the variation law of ultrasonic transfer efficiency with   pre-strain and input ultrasonic amplitude. It can be seen that the values of ultrasonic transfer efficiency do not change significantly with the change of the input ultrasonic amplitude. But the ultrasonic transfer efficiency decreases with the increase of pre-strain, but reaches the minimum value at a certain pre-strain, and then continues to increase with the increase of pre-strain.
In the plastic deformation stage, the dislocation atoms are easily activated, due to the input of ultrasonic energy, which makes the deformation of the material easier. Figure 10 shows the tensile stress-strain curve of 2219-O aluminum alloy with and without ultrasonic vibration. The shaded envelope area represents the total reduction in stress under different ultrasonic amplitude conditions.
(a) Transfer coefficient measurement system (b) Data acquisition system  It can be seen from Fig. 10 that stress level reduction varies with strain increases, which means that there is a variation in the absorption of ultrasonic energy by the material as the plastic deformation progresses. And the variation trend of stress reduction with the increase of strain due to ultrasonic energy can be obtained, as shown in Fig. 11. The variation trend of stress reduction remains the same under different ultrasonic amplitude conditions, but with the increase of ultrasonic amplitude, the degree of stress reduction increases. Combining the stress reduction trend in Fig. 11 with the ultrasonic transfer efficiency in Fig. 9, we can find that when the ultrasonic energy transfer efficiency decreases, the corresponding ultrasonic energy absorption degree increases, which further proves that in the plastic deformation process, the absorption of ultrasonic energy by the material is not monotonically increasing with the increase of strain.

Microstructure analysis
In order to explore the non-monotonic phenomenon, dislocation density and its configuration of samples with different pre-strains were observed. The internal dislocation configuration of 2219-O aluminum alloy under different prestrain conditions is shown in Fig. 12, in which the short rod structure with orthogonal sparse distribution is the second phase CuAl 2 [17]. It can be seen from Fig. 12 that most dislocations present the irregular linear distribution. Relevant studies show that [18] the proportion of ultrasonic scattering attenuation caused by the precipitated phase and grain boundary is very small, which is mainly caused by internal friction of dislocation damping. Therefore, our study mainly analyzes the reason of internal friction caused by dislocation configuration. Generally, as the pre-strain increases, the dislocation density and its configuration also change. Sample #0 in Fig. 12 is the original dislocation configuration with linear distribution, and it can be seen that there are a small number of dislocations in the material.
In samples #2 and #3, the dislocation density increases gradually because of metal work hardening. It is worth noting that the dislocation distribution of sample #1 is similar to that of sample #3, and the ultrasonic transfer efficiency of sample #1 is close to that of sample #3. While the dislocation density of sample #2 is lower than that of sample #3, the ultrasonic transfer efficiency of sample #2 is lower than that of sample #3, which further indicates that the ultrasonic transfer coefficient is correlated with dislocation density.
As the deformation continues to increase, dislocation walls at the bottom right of the TEM image of sample #4 will form, when the dislocation multiplication and entanglement persist to a certain extent. During the formation process of the dislocation walls, the dislocation distribution is obviously different, which is manifested that the dislocation density at the dislocation walls is larger, while the dislocation density nearby is relatively small. In addition, with the further increase of pre-strain, there will be obvious dislocation cell formation at the top right of the TEM image of sample #5. The dislocation density distribution is lower inside the dislocation cell and higher in the wall of the dislocation cell.
Related studies have shown that mobile dislocations slip under strain while interacting with dislocation walls and eventually transform into immobile dislocations and dislocation walls [19]. Besides, the vibration of movable dislocation at its position is the main factor causing energy consumption [20]. Therefore, combined with the analysis of experimental results and microscopic observation results, it can be seen that with the occurrence of deformation, the increase of the above three kinds of dislocations leads to the significant increase of the overall dislocation density of the material. The increase of movable dislocation density leads to the increase of ultrasonic energy loss.
When the structures such as dislocation walls and dislocation cells are formed due to the multiplication and entanglement of dislocations, the movable dislocation densities decrease gradually, and the ultrasonic energy loss decreases accordingly, which means that the degree of ultrasonic attenuation decreases. Therefore, taking the small range of the corresponding strain of sample #4 as the lower limit of ultrasonic transfer efficiency, the transfer efficiency gradually increases after the formation of dislocation walls, dislocation cells, and other structures.
The experimental results show that the ultrasonic energy transfer efficiency varies with the plastic deformation. And the analysis of the microscopic observation results shows the internal cause of the variation law of the ultrasonic transfer efficiency. In order to accurately describe the ultrasonic energy attenuation characteristics in plastic deformation, the corresponding model needs to be established.

Establishment of hardening equation
According to the analysis of the material microstructure, after the generation of dislocation cell, it can be believed that with the continuous increase of deformation, the ultrasonic energy transfer efficiency will continue to improve until it approaches 1. Based on the simulation deformation strain results in existing studies, the maximum plastic strain value at the position of the inner ribs is about 0.1 [16]. Therefore, it is assumed that the transfer coefficient reached 1 when the true strain was 0.1 and remained constant if the strain continues to increase. Based on the above analysis, in order to carry out complete parameter fitting, the range of prestrain was expanded. Not only the pre-strain data obtained in the experiment was used, but also a data point was set, that is, when the pre-strain is 0.15, the ultrasonic transfer efficiency is 1.
To further accurately describe the law of ultrasonic energy transfer coefficient changing with strain, the Sigmoid function was used to describe the change of ultrasonic energy transfer efficiency in a small strain range of 0-0.2. The definition of the Sigmoid function can be expressed as This function has the characteristics of monotonic and continuous, and it is easy to divide the predicted value into two parts using this function. In other words, the center point can be defined as the limit value, and the corresponding output y can be divided into two parts on the left and right sides of the limit value. Therefore, on the basis of the experimental results, the Sigmoid function is suitable for describing the variation law of ultrasonic transfer efficiency that first decreases and then increases with the increase of pre-strain.
According to the Sigmoid function, the ultrasonic energy transfer efficiency can be defined as where A and B are fitting coefficients, and 0 is the pre-strain value at the minimum transfer efficiency.
The data point with ultrasonic energy transfer efficiency of 1 when the strain was set at 0.15 was added to the original experimental data, and the transfer efficiency results of different input ultrasonic amplitudes changing with strain were fitted according to Eq. (6), and the fitting results were shown in Fig. 13. The R-square goodness of fit in the first half can reach 0.962, 0.730, and 0.928, respectively, and the overall fitting result is good. The results of fitting parameters are shown in Table 5. It is considered that ultrasonic transfer efficiency is only related to strain, since the results of ultrasonic transfer efficiency do not change significantly with the input amplitude. The values of fitting parameters A and B can be obtained by averaging the fitting results. In the first half of the monotonically decreasing part, the fitting parameters A and B are defined as A 1 and B 1 , while in the second half of the monotonically increasing part, the fitting parameters A and B are defined as A 2 and B 2 .
In the previous work, based on the study of the ultrasonicassisted uniaxial tensile process of 2219-O aluminum alloy, we have established the hardening equation of 2219-O aluminum alloy with ultrasonic softening effect, which can be expressed as [16] where is the stress without ultrasonic vibration, which can be expressed as g( ) 1−m represents the stress reduction caused by ultrasonic vibration: where m is 0.77, and is ultrasonic amplitude.
Due to the occurrence of ultrasonic energy attenuation, the ultrasonic amplitude is varied at different positions in the propagation process, and the change of ultrasonic amplitude is related to the propagation distance, strain, and input amplitude. Based on Eqs. (4) and (6), the expression of ultrasonic amplitude changing with the propagation distance and strain can be obtained: where f x 0 ( ) is the expression of the transfer efficiency changing with strain obtained by the experiment when the material thickness is x 0 , 0 is the input ultrasonic amplitude, and x is the propagation distance.
Based on the above deduction and analysis, the hardening equation of 2219-O aluminum alloy considering ultrasonic attenuation characteristics is shown in Eq. (11). x 0 is 3 mm in this study, and the specific expression of f x 0 ( ) is shown in Eq. (6). The specific parameters were fitted by the experimental results of attenuation, and the established hardening equation could provide a theoretical model for the simulation of ultrasonic-assisted plastic forming considering ultrasonic attenuation.

Verification of the hardening equation
In order to verify the correctness of the corrected equation, the stress-strain data obtained in the uniaxial tensile test were respectively substituted into the existing hardening equation considering the ultrasonic softening effect shown     11), the ultrasonic propagation distance was set to 0.1 mm to ensure that the transfer distance was basically the same as that in Eq. (2). On this basis, using the tensile test data when the ultrasonic amplitude is 3 μm, the stress-strain curve under the action of ultrasonic was obtained by calculation, as shown in Fig. 14. Figure 14 shows the comparison between the experimental results and the calculation results of the hardening equations. It can be seen that the calculation results of Eqs.
(2) and (11) are in good agreement with the experimental results, which proves the accuracy of the established modified hardening equation. The variation trend of stress in Fig. 14 corresponds to the variation law of ultrasonic transfer efficiency in Fig. 9. When the true strain is about 0.05, the ultrasonic transfer efficiency is the lowest, which means that the ultrasonic energy absorption effect is more obvious at this point, corresponding to further ultrasonic softening, so the stress will first decrease and then increase. And the stress-strain curve after the strain is 0.05 is more consistent with the experimental data, which indicates that the hardening equation considering ultrasonic attenuation is more in line with the actual ultrasonic-assisted plastic forming process.
By changing the x in Eq. (10), the stress-strain curves under different ultrasonic propagation distances can be calculated, as shown in Fig. 15. With the increase of ultrasonic propagation distance, the tensile stress gradually increases, which indicates that the ultrasonic softening effect is gradually weakening in this process. And this change is consistent with the attenuation form of ultrasonic energy in plastic deformation described in Eq. (6). Besides, under the condition of different ultrasonic propagation distances, the ultrasonic energy efficiency also shows a trend of first decreasing and then increasing with the increase of strain, and with the increase of ultrasonic propagation distance, this variation trend is more obvious. Therefore, the hardening equation established in this paper can also characterize the acoustoplastic properties of materials under different ultrasonic propagation distances.

Conclusion
In this paper, the influence of the dislocation within the materials and the dislocation movements caused by plastic deformation on the ultrasonic attenuation characteristics was studied. The experimental platform for measuring ultrasonic transfer efficiency was established, and the variation law of ultrasonic transfer efficiency related to plastic strain was obtained, which was further explained based on microscopic analysis. Finally, the hardening equation of 2219-O aluminum alloy considering ultrasonic propagation distance and plastic strain was established, and the model accuracy was verified based on the experimental data. The specific conclusions are as follows: 1. The degree of ultrasonic attenuation is affected by the thickness of the material and also related to the plastic strain of the material. For 2219-O aluminum alloy, the ultrasonic transfer efficiency decreases first and then increases with the increase of pre-strain. When the prestrain reaches 0.05, the ultrasonic transfer efficiency is the lowest. 2. In the plastic deformation stage, the absorption of ultrasonic energy by movable dislocation density is the reason for the variation of ultrasonic transfer efficiency. The mobile dislocation density will gradually increase with increasing strain, but due to dislocation entanglement, movable dislocations will gradually transform into dislocation cells and dislocation walls, resulting in nonmonotonic changes in ultrasonic transfer efficiency. 3. The hardening equation of 2219-O aluminum alloy considering ultrasonic propagation distance and plastic strain can provide more accurately express to the acoustoelastic properties of materials under different ultrasonic propagation distances. This hardening equation can be applied to the ultrasonic-assisted spinning process, and the influence of the ultrasonic attenuation characteristics on the forming height of the rib can be explored, so as to obtain a more accurate simulation result. Data availability Not applicable.
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Competing interests
The authors declare no competing interests.