Micro-cutting force model of micro-jet induced by cavitation collapse in the ultrasonic field at micro-nano scale

The micro-jet is the main mechanism of cavitation on materials. More and more ultrasonic cavitation assisted machining uses the micro-jet effect in the industrial field. In this paper, the micro-jet impact is considered the micro-cutting process. To deeply analyze the micro-cutting mechanism of micro-jet, the size effect of materials was considered, and the micro-cutting force model of micro-jet was established and solved based on the spherical indentation test theory and the strain gradient plasticity theory. The results showed that the micro-cutting force and impact pressure of micro-jet with size effect are larger than those without size effect and the micro-cutting force of single micro-jet is within 2.35 N. The micro-cutting force of micro-jet increases exponentially with the increase of pit radius, but only slightly increases with the increase of pit depth. When the size effect is considered or not, the impact pressure of micro-jet is 1616–2922 MPa or 1615–1980 MPa, respectively. The increased ratio of micro-cutting force with size effect is 1+2m¯2a2G2bha2+h2σJC2-1×100%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left(\sqrt{1+\frac{2{\overline{m} }^{2}{a}^{2}{G}^{2}bh}{\left({a}^{2}+{h}^{2}\right){\sigma }_{\mathrm{JC}}^{2}}-1}\right)\times 100\%$$\end{document}, which can directly reflect the strength of size effect. The size effect is more obvious when the pit size is smaller, and the maximum increased ratio is 47.54%. The increased ratio increases nonlinearly with the decrease of pit radius and the increase of pit depth. It has a strong correlation with pit radius and pit depth. This paper can provide a theoretical viewpoint and support for the quantitative description of cavitation effect in ultrasonic-assisted machining.


Introduction
Cavitation is a common physical phenomenon because the local pressure in the liquid is less than its saturated vapor pressure. It was first proposed by Parsons and Barnaby in the nineteenth century, who pointed out that cavitation is the main reason for the corrosion of propeller blades and the decrease of service efficiency [1]. Due to the change of liquid ambient pressure, cavitation is characterized by the generation, growth, and collapse of a large number of cavitation bubbles. During the dynamic evolution and collapse of cavitation bubbles, micro-jet, shock wave, and high temperature are accompanied.
With the deepening understanding of the nature of cavitation and cavitation damage characteristics, it has been widely used in mechanical industry, chemical industry, biological medicine, and other fields, and a variety of cavitation assistive technologies have been derived, such as cavitation water jet [2], cavitation cleaning [3], cavitation peening [4], cavitation wastewater sludge treatment [5], and cavitation tumor treatment [6]. In the field of mechanical industry, the cavitation effect can be used in material surface modification, polishing, peening, and so on. To control cavitation easily, ultrasound is usually used to induce cavitation in industry, and the ultrasound of 20 kHz is the most common. Ultrasonic cavitation generates cavitation bubbles through ultrasonic negative pressure, which oscillates and collapses under the alternation of positive and negative ultrasonic pressure phases, and the strong cavitation effect can be produced [7].
The micro-jet, shock wave, and high temperature produced by the collapse of cavitation near the wall will cause special forms of material damage, which is called cavitation erosion [8]. Cavitation collapse is usually related to the sudden change of local pressure of the fluid, which is characterized by repeated random stress pulses of hundreds of megapascals. The result is a surface spreading damage characterized by micro pits, called cavitation pits, which are characterized by micrometer size, and their accumulation can lead to material failure. Naude and Ellis [9] found that the damage was mainly caused by the impact of high-speed micro-jet generated in the process of bubble collapse. In terms of the effect of cavitation on materials, the micro-jet plays a major role.
Cavitation in ultrasonic-assisted machining technology, such as power ultrasonic honing, ultrasonic milling, and ultrasonic drilling, has gradually attracted the attention of researchers. Liew et al. [10] investigated the ultrasonic cavitation assisted EDM of ceramic materials and found that the upward flow and oscillation of cavitation bubbles can accelerate the discharge of debris and reduce the adhesion of debris on the workpiece surface. Ye et al. [11] observed the cavitation effect in ultrasonic honing and found that the violent vibration of cavitation bubble and the micro-jet released during its collapse will continuously impact the grinding fluid and the surrounding solid boundary, which plays an irreplaceable role in the purification of honing environment, the improvement of honing accuracy, and the suppression of honing noise. The cavitation dynamics in the ultrasonic honing environment was analyzed. The influence of ultrasonic honing factors such as honing pressure, rotation speed, and reciprocating speed of honing head on cavitation characteristics was obtained. The orthogonal test of ultrasonic honing cavitation was carried out. It was pointed out that the cavitation effect can improve the surface quality of workpieces in ultrasonic honing under certain conditions. Goto et al. [12] studied the effect of ultrasonic cavitation on micro vertical milling and obtained that the burr generation rate of ultrasonic cavitation assisted milling is less than 5%. Ohashi et al. [13] studied the application of cavitation in precision abrasive machining and concluded that a better machining effect can be obtained by using cavitation, and the fine surface can be easily processed with simple equipment. Liang et al. [14] applied ultrasonic cavitation to ultrasonic vibration drilling of stainless steel micropores and pointed out that this method can effectively improve chip breaking ability, greatly reduce thrust, prolong tool life, and obtain the best micro-hole machining quality. Toh [15] proposed the concept of ultrasonic cavitation peening and used ultrasonic cavitation to improve the surface properties of materials. In these ultrasonic cavitation assisted machining, the micro-jet produced by cavitation collapse is mainly utilized. Therefore, the micro-jet is helpful to material removal in ultrasonic-assisted machining. So, the detailed mechanism of micro-jet on materials is worthy of further study.
The micro-jet is caused by the asymmetric collapse of the bubble near the wall, which was first proposed by Kornfeld [16] in 1944. Hammitt [17] pointed out that the velocity of the micro-jet near the wall can reach 70-180 m/s when the bubble collapses. Most researchers believe that the velocity of micro-jet is several hundred meters per second and some believe that it can reach 950 m/s [18]. The high-speed micro-jet can produce the pressure of hundreds or even thousands of Mpa on the material surface, and the result is the formation of micro pits, namely, cavitation pits, which have been observed in a wide range of experiments. Therefore, the impact ability of micro-jet has aroused the interest of researchers. How to accurately obtain the impact force of micro-jet is a major difficulty. Soyama et al. [19] measured the impact force in the cavitation jet device by using piezoelectric polyvinylidene fluoride (PVDF) film pressure sensor. The results showed that when the jet velocity is 126-155 m/s, the maximum impact force can reach 200 N. However, under the same test conditions, the amplitude of impact force measured by the spherical nano indentation method is only 20 N [20]. The main reason is that although the size of the micro pressure sensor used to measure the impact force is very small, the radius of cavitation erosion is usually only submicron or micron, which is still very small compared with the micro pressure sensor. Therefore, the impact force measured by the pressure sensor at the same time may be the result of the collapse of several or even dozens of tiny bubbles at the same time. Besides, the collapse of the bubble only takes tens of microseconds or even a few microseconds [21]. The response frequency of different pressure sensors is different, so the measured pressure values are not the same.
The erosion point test method was first proposed by Knapp and Pasadena [22]. Its essence is to use the solid specimen itself as a substitute for the traditional pressure sensor and deduce the impact force generated by cavitation collapse through the data such as the geometric size and number of erosion points. Tzanakis et al. [23] used this method to analyze the geometric characteristics of cavitation pits. Reverse engineering was used to predict that the cavitation impact pressure was mostly 0.4-1 Gpa, and the corresponding micro-jet velocity was 200-700 m/s. However, the diameter of micro-jet is about several microns to tens of microns, the impact speed is fast, and the action time on materials is short. It is a typical micro scale short-term strong impact load, and the micro pits formed are also in submicron and micron levels. Therefore, the material will show a strong size effect in the process of micro-jet impacting materials. At this time, it is not accurate to explain this process with the traditional plastic theory, so it is necessary to develop a micro scale plasticity theory. Ma and Clarke [24], Nix and Gao [25] proposed the strain gradient plasticity (SGP) theory based on Taylor dislocation theory, which described the size effect in the deformation process of micro scale materials. It has been recognized and used for reference by the majority of researchers.
The micro pits formed by micro-jet impingement have high similarities with spherical indentations. Therefore, the formation of micro pits can be regarded as the formation of spherical indentations, which provides a method for analyzing the mechanism of micro-jet impingement. The measurement of stress-strain response of materials by spherical indentation test has been studied in detail [26,27]. The size effect of materials in nano-micro scale spherical indentation has also aroused great interest of researchers. With the decrease of indentation size or indenter size, the measured hardness increases significantly, which is commonly referred to as the indentation size effect (ISE). Al-Rub and Faruk [28] predicted the size effect of nanoindentation under the spherical indenter. Sun et al. [29] studied the strain gradient in the spherical indentation and proposed a method to determine the intrinsic characteristic length of the material. It was found that the strain gradient in the spherical indentation was related to both the indentation radius and the indentation depth.
To sum up, it is necessary to consider the size effect of materials to explain the effect of micro-jet on materials from the micro scale. This is also the key to understanding the effect of cavitation in ultrasonic-assisted machining, especially in the environment where micro-jet impingement is used to improve the material removal rate. Therefore, the micro-jet impact is considered a micro-cutting process in this paper. Considering the size effect of materials, based on the spherical indentation test theory and strain gradient plasticity theory, the micro-cutting force model of micro-jet is established from the micro scale, which can provide a viewpoint and support for quantitative description of cavitation effect in ultrasonic-assisted machining.

Theory
The micro-jet is regarded as a spherical indenter, and the micro pit is regarded as a spherical indentation. Considering the size effect in the formation of the micro pit, the stress-strain of the material is obtained by using the spherical indentation test theory and the strain gradient plasticity theory, and finally, the micro-cutting force model of microjet is established. To express the size effect in the formation of micro pits on the surface of materials under the micro-jet impingement, the mechanism-based strain gradient plasticity (MSG) theory was applied, and the shear strength τ of the material can be expressed by dislocation density [30]: where α is an empirical constant and taken as 0.5 [31]; G is the shear modulus; b is the magnitude of Burgers vector; and ρ t , ρ s , and ρ g are the densities of total dislocation, statistically stored dislocations, and geometrically necessary dislocations, respectively. Equation (1) describes the relationship between shear strength and dislocation density. The flow stress σ can be described as follows: where m is the Taylor factor, which is taken as 3 for FCC polycrystals. The essence of metal plastic deformation is the dislocation movement. At the micro scale, there are not only statistically stored dislocations, but also additional geometrically necessary dislocations due to the compatibility of crystal grids. Geers et al. [32] believed that the existence of size effect is related to the grain orientation, grain size, the number of grains in the direction of material thickness, and boundary constraints, among which the grain orientation is most critical. Due to the random distribution of statistically stored dislocations, dislocation slip will produce uniform plastic deformation. The geometrically necessary dislocations are usually arranged regularly and periodically with the same sign, so they have a strong blocking effect on the slip. This will result in local non-uniform plastic deformation and large strain gradient, which makes the material hardening and strengthening. In the process of plastic deformation, when the grain orientation is the same, the geometrically necessary dislocations are obvious. This will produce additional non-uniform deformation, which will enhance the hardening effect and material anisotropy, resulting in the size effect phenomenon of "the smaller, the stronger." Therefore, the size effect is caused by geometrically necessary dislocations. When the size effect of material is not taken into account, i.e., the macro scale, the flow stress is only related to the density of statistically stored dislocations density, and Eq. (2) becomes: where σ JC is the flow stress of material described by J-C model at the traditional macro scale, so The study of plastic theory showed that the hardness is directly proportional to the flow stress, and the coefficient is 3 [30]. Chen and Tsai [33] confirmed that the theoretical proportional relationship between hardness and flow stress is still valid in the samples with reduced size through microhardness test and finite element analysis. Therefore, the relationship between material hardness and flow stress is as follows: where k is the proportional coefficient and k ≈ 3. When the size effect is not considered, the hardness is H 0 , which is only related to the statistically stored dislocations.
The type of indenter affects micro indentation hardness. For the conical indenter, Nix and Gao developed a dislocation model to estimate the density of geometrically necessary dislocations and derived the relationship between hardness H and indentation depth h in the strain gradient plasticity theory [25]: where h * is a parameter of length dimension, and its calculation formula is as follows: where θ is the angle between the surface of the conical indenter and the surface of the pressed material. However, the micro pit formed by micro-jet impinging on the material surface is more consistent with the micro indentation under the spherical indenter. Swadener et al. [34] extended the model to spherical indentation and established the following relationship: and R is the radius of the indenter.
According to the geometric conditions of the spherical indentation, it can be concluded that where a and h are the radius and depth of indentation, respectively. The hardness H can be obtained as follows: According to the relationship between hardness and load in the indentation theory, H = F / (πa 2 ), the JC micro-cutting force F of micro-jet at micro scale can be obtained as follows: Equation (12) reflects the size effect of material at micro scale. As long as the macro scale stress σ JC is obtained, the micro-cutting force F can be obtained. J-C model [35] is the most widely used semi-empirical constitutive model, which can describe the influence of high strain, high strain rate, and high temperature on the material, and also conforms to the dynamic response characteristics of materials impacted by micro-jet. However, the micro-jet impinges on the material surface in the liquid environment, so the temperature influence term in the J-C model is ignored, which is expressed by the following formula: where ε is the strain; ̇ is the strain rate; ̇0 is the reference strain rate; and A, B, n, and c are the material constants. The strain rate of the material can reach the order of 10 4 -10 6 s −1 in the process of micro-jet impingement and micro pit formation, and strain can be expressed as So far, the micro-cutting force model of micro-jet has been established completely, which can be obtained by combining Eqs. (12)- (14). Aluminum alloy is widely used to study cavitation pitting and cavitation damage characteristics. Al7075 is the most selected material, so this paper selects Al7075 for analysis. Its parameters [36] are shown in Table 1.

Results and discussed
In reference [37][38][39][40], Al7075 was used to study cavitation damage, and the size characteristics of cavitation pit were obtained. Based on that, Al7075 is used as the research material to calculate the micro-cutting force of microjet, and the pit radius is 1-20 μm, and the pit depth was 0.05-3 μm. The micro-cutting force of micro-jet is shown in Fig. 1. It can be seen that the variation of micro-cutting force with and without size effect is similar and the former is larger than the latter. The maximum micro-cutting forces with and without size effect are 2.331 N and 2.308 N, respectively. Although the micro-cutting force produced by a single micro-jet is not very large, in the process of intense ultrasonic cavitation, there will be a large number of bubbles that collapse to produce micro-jets, which have an obvious effect on the material. There may be dozens or even hundreds of micro-jets impinging on the wall at the same moment, which is also one of the main reasons for the large difference in the results of micro-jet impact pressure measured by the pressure sensor. Overall, the micro-cutting force of micro-jet increases with the increase of pit radius and pit depth, but the influence of pit radius is more obvious.
To further analyze the relationship between micro-cutting force of micro-jet and pit radius and pit depth, the variation curves of micro-cutting force of micro-jet with pit radius and pit depth are obtained, as shown in Figs. 2 and 3, respectively. It can be seen that the micro-cutting force of micro-jet increases exponentially with the increase of the pit radius in Fig. 2, while it increases only slightly with the increase of the pit depth. Therefore, the micro-cutting force of micro-jet is mainly affected by the pit radius, and the influence of pit depth is not significant. The pit radius is closely related to the micro-jet radius, which can be regarded as equal, so it can be considered that the micro-cutting force of micro-jet is mainly related to the micro-jet radius.
The micro-cutting force of a single micro-jet is very small. Numerically, the micro-cutting force with size effect seems to increase little compared with that without size effect. Therefore, to further study the effect of size effect, the increased ratio of micro-cutting force of micro-jet is analyzed. It can be seen from Eqs. (9) and (12) that there is a difference √ 1 + 2m 2 a 2 G 2 bh (a 2 +h 2 ) 2 JC between considering and not considering the size effect. Therefore, the increased ratio of micro-cutting force is shown in Fig. 4, which can directly reflect the strength of size effect. It can be found that the size effect is more obvious when the pit size is smaller, and the maximum increased ratio is 47.54%, which occurs at a = 1 μm and h = 1 μm. In this paper, the increased ratio is not particularly high, which is due to the high strength of Al7075. It can be seen from the increased ratio formula that the size effect is negatively correlated with σ JC . The smaller the σ JC is, the stronger the size effect is. That is to say, the lower the strength of the metal materials, the stronger the size effect, such as aluminum and copper [41]. Correspondingly, the metal materials with higher strength may show a weaker size effect, which is easy to understand and obvious.  Figure 5 shows the variation of the increased ratio with the pit radius under different pit depths. The increased ratio increases nonlinearly with the decrease of the pit radius. With the continuous increase of the pit radius, the increased ratio remains at a stable low value. Figure 6 shows the variation of the increased ratio with the pit depth under different pit radiuses. Although the increased ratio increases first and then decreases slightly with the pit depth when the pit radius is small, the increased ratio increases with the pit depth from the overall trend. The increased ratio has a strong correlation with the pit radius and pit depth.
According to the micro-cutting force of micro-jet, the impact pressure of micro-jet can be obtained: The impact pressure of micro-jet under different pit sizes is shown in Fig. 7. It can be seen that the impact pressure of micro-jet with size effect is significantly greater than that without size effect, which also indicates the necessity of considering the size effect in the process of micro-jet impingement. The impact pressures of the micro-jet with and without size effect are 1616-2922 MPa and 1615-1980 MPa, respectively. The results of this study are compared with those of reference [39], where Al7075 is also selected as the research object. The results show that the impact pressure of micro-jet considering the size effect is consistent with the result of 1.5-3 GPa in reference [39], which verifies the accuracy of this study, and also explains the necessity and rationality of considering the size effect. At present, in the absence of research on the microcutting force of micro-jet induced by cavitation collapse, with size effect, a=1 μm with size effect, a=5 μm with size effect, a=10 μm with size effect, a=20 μm without size effect, a=1 μm without size effect, a=5 μm without size effect, a=10 μm without size effect, a=20 μm we propose a quantitative method, which makes up for the vacancy, but is not deep enough and needs further exploration. In the follow-up, we hope to make breakthroughs in the forward construction and solution of the micro-cutting force model of micro-jet, the direct measurement platform, and the accurate measurement method of size effect.

Conclusions
1. The micro-jet impingement induced by cavitation collapse is considered a micro-cutting process. Considering the size effect of material, based on the spherical indentation test theory and strain gradient plasticity theory, the micro-cutting force model of micro-jet is established from the micro scale. The micro-cutting force of a single micro-jet is within 2.35 N, and the micro-cutting force with size effect is larger than that without size effect. The micro-cutting force of micro-jet increases exponentially with the increase of pit radius, but only slightly increases with the increase of pit depth. 2. The increased ratio of micro-cutting force of micro-jet is √ 1 + 2m 2 a 2 G 2 bh (a 2 +h 2 ) 2 JC − 1 × 100% , which can directly reflect the strength of size effect. The size effect is more obvious when the pit size is smaller, and the maximum increased ratio is 47.54%, which occurs at a = 1 μm and h = 1 μm. From the overall trend, the increased ratio increases nonlinearly with the decrease of pit radius and increases with the increase of pit depth. The increased ratio has a strong correlation with the pit radius and pit depth. When the size effect is considered or not, the impact pressure of micro-jet is 1616-2922 MPa or 1615-1980 MPa, respectively. The former is more accurate and reasonable.

Declarations
Ethics approval Not applicable.

Consent to participate Not applicable.
Consent for publication Not applicable.

Competing interests
The authors declare no competing interests.