The recognition method for the fractal and the dynamic on the tool flank of a high-energy-efficiency milling cutter

The friction contact boundary between the tool flank of the milling cutter and the machining transition surface is important indicator to reveal the third deformation zone tool contact relationship and assessing the frictional wear performance of milling cutter. The existing models for friction boundary identification pay attention to the maximum width of accumulated friction and wear on the tool flank, ignoring the variability of the overall and local morphology of the friction boundary on the flank. Aimed at the influence of milling vibration on the instantaneous position of the cutter teeth and the machining transition surface, the solution and discrimination for the instantaneous position vector on the flank was proposed. Based on the mutagenicity of the instantaneous temperature and stress distribution, the influence of the instantaneous contact, extrusion, and deformation between the tool flank and the machined transition surface on the friction area was recognized. The calculation model of friction boundary of the flank was established. The irregularities of the distributions of the friction boundaries of the tool flank were revealed. The fractal recognition methods for instantaneous and cumulative friction boundary of the flank were proposed. And response was studied and verified with experiments. The results showed that it could effectively identify the irregular distribution of the friction boundary on the flank with the use of the above models. The formation and evolution of the friction boundary on the tool flank of the high-energy efficiency milling cutters were revealed.


Introduction
The high-energy efficiency milling cutter is a typical highspeed, interrupted cutting tool. The frictional contact boundary between the milling tool flank and the machined transition surface is not only premise of the tribology relations of the cutting tool in the third deformation zone, but directly affect the quality of machined surface and the service life of the milling cutter. It is one of the main control factors limiting the energy efficiency of milling cutter [1,2]. The dynamic distribution of the friction boundary on the milling tool flank was studied. The formation, evolution process, and key control variables of friction boundary on the milling tool flank were recognized. It has important guiding significance to reveal the tribology behavior between the milling cutter and the workpiece, and realize the high-energy efficiency milling process control.
The friction boundaries of the milling tool flank are closely related to the instantaneous boundaries of each cutter tooth [3,4]. Besides, recognizing the effects of cutting process parameters on cutting friction behavior is the key to achieve the instantaneous friction boundaries of each cutter tooth and the boundaries differences of all blades [5].
During cutting with high-energy efficiency milling cutter, the cutting load changes frequently, the instantaneous cutting behavior of cutter teeth is in an unstable state. This phenomenon lead to the instantaneous position and orientation of the tool flank and the formation process of the machined transition surface are variable. The instantaneous contact relationship between the tool flank and the machined transition surface is changing [6,7]. As the same time, the complexity and uncertainty of instantaneous contact stress, friction velocity distribution of cutting edge, and the tool flank are exacerbate, because of the diversity of the cutter 1 3 tooth error distribution and difference of influence of milling vibration on instantaneous cutting behavior [8,9].
Existing research on the milling tool flank had assumed that the instantaneous milling behavior of each cutter tooth of milling had the same variation. These researches ignored the difference between the influences of the vibration and on the instantaneous milling behavior of each cutter tooth. Therefore, the diversity of the instantaneous friction boundary of the flank of the cutter tooth could not be unveiled [10,11].
The experimental methods of the time-division cutting were used to research the change of the friction boundary on the tool flank. In order to obtain curve of the maximum value of the friction area of the tool flank with the increase of the cutting stroke, the maximum friction and wear width on the tool flank of the cutter tooth was measured by stopping the machine [12,13]. These studies ignored the information at different locations on the friction boundary of the milling cutter tooth [14,15]. As the same time, this method ignored the effect of dissipation of the thermal coupling field of the cutter teeth due to stopping measurements on the frictional behavior of the tool flank of the cutter teeth at different locations during long, uninterrupted cutting. When using the above method, it is impossible to correct calculation of dynamic distribution of instantaneous friction boundary of cutter tooth. The formation and evolution of the instantaneous friction boundary on the flank of the cutter teeth need to be revealed.
It has close relationship between the contact boundary of the instantaneous tool-working friction pair on the tool flank and the machining transition surface formed by instantaneous position and unsteady cutter tooth cutting edge under vibration [16][17][18]. The instantaneous frictional boundary pattern on the tool flank showed a complex and variable irregular distribution because of the multiple scales of relative motion and mutual contact, extrusion and deformation with non-smooth, discontinuous and multi-tooth asynchronous.
In the research on the measurement and characterization of the friction and wear on the flank of the cutter tooth, NA Krishnan et al. researched the effect of size effect on the wear behavior of TiAlN coated WC micro end mills [19]. Li et al. proposed a new method for predicting the incremental wear rate of milling cutter flanks and optimization the milling parameters of the CNC [20]. Chen et al. raised a back-propagation neural network (BPNN) to predicting tool wear [21]. Saha et al. solved the theoretical model depend on the state of wear of the milling tool [22].
Existing research included between the friction, the wear of milling cutter, the geometric relationship of cutting positions, the dimensional effect of frictional wear on the tool flank, the cutting force, the incremental relationship for the wear rate, and the modeling approach to the frictional wear problem. The above method had laid the foundations for the research of the distribution of instantaneous and cumulative frictional boundary formation on the tool flank. However, it is not yet possible to fully quantify the overall and local irregularities of the instantaneous friction boundary and the dynamics of its scale on the tool flank. It was necessary to study this in depth.
Therefore, in this research, the variability of the instantaneous geometric contact between the tool flank of the cutter tooth and the machined transition surface were studied. And the abrupt variability of the instantaneous contact temperature and instantaneous contact stress distribution on the tool flank of the cutter tooth were studied. A method for calculating the friction boundary on the tool flank was proposed. The dynamic of the fractal dimension and scale coefficient of the instantaneous frictional boundary on the tool flank of the cutter tooth was studied. The differences in the evolution of the instantaneous friction boundary and the distribution of the cumulative friction boundary on tool flank were revealed. The identification method of the fractal and its dynamic properties of the frictional boundary of the tool flank of the milling cutter were put forward. And analyzing the influence of the process variables and the experimental verification was conducted.

The solution and discrimination of the instantaneous position vector on the flank of the cutter tooth under vibration
High efficiency milling cutter structure, tooth error, milling method, cutting parameters, and milling vibration directly affect the instantaneous contact relationship between the tool flank and the machined transition surface. Among them, the instantaneous cutting position of the indexable high feed milling cutter and its teeth under vibration is shown in Fig. 1. In Fig. 1, D is the diameter of the cutter shank. l is the overhang of the cutter. L is the total length of the cutter. e zmin is the axial lowest tip point of the cutter tooth. e 1 is the tip point of the cutter with the maximum radius of gyration. r max is the maximum turning radius of the cutter tooth. o-xyz is the workpiece coordinate system, o 0 -x 0 y 0 z 0 is the coordinate of the cutter structure, where o 0 is the center of rotation of the axial lowest tip point of the milling cutter, which is coplanar with the axial lowest tip point of the milling cutter. x 0 axis is parallel to the cutting speed direction of the maximum tip point of the radius of rotation. y 0 axis is parallel to the radial direction of the maximum tip point of the radius of rotation. z d axis is the rotary axis of the milling cutter and points to the tool shank direction. o i -x i y i z i is the coordinate of cutter tooth i, where o i is the gyration center on the cutter tooth. y i axis is the direction perpendicular to the bottom surface of the cutter tooth through the origin o i . z i axis passes through o i and points to the tip point. x i passes through o i and is perpendicular to both y i and z i . η is the mounting angle between z i and z 0 axis. e is any point on the tool flank of the cutter tooth i. r i is the radius of rotation of the cutter tooth i. Δr i is the radial error of the cutter tooth i. Δz i is the axial error of the cutter tooth i. φ i is the angle between y 0 axis and o 0 e 1 in the x 0 o 0 y 0 plane. v f is the feed rate, a p is the milling depth, a e is the milling width, i is the number of the cutter tooth.
From Fig. 1, the relationship matrix between the instantaneous coordinate of the milling cutter and its cutter teeth and the workpiece coordinate under the influence of vibration and cutter tooth error was shown in Eq. (1)-Eq. (9). (1) where, Q 1 , Q 2 , Q 3 , Q 4 , Q 5 are rotation matrices. M 1 , M 2 , M 3 are translation matrices. where the conversion relation of the Φ i between the cutter tooth coordinate and the workpiece coordinate was shown in Eq. (11).
where, θ y e and θ z e are the included angle between the normal vector �� ⃗ N of any point e on the flank and the negative of the direction unit vector � ⃗ y and ⃗ z on the workpiece coordinate system.
where the terminal point coordinate e 1 (x 1 , y 1 , z 1 ) is the normal vector �� ⃗ N.
According to Fig. 1 and Eq. (4) and Eq. (5), the instantaneous position vectors between the cutter tooth tool flank and the machined transition surface could be obtained, as Fig. 2.where the instantaneous position vector of the selected point e on the cutter tooth tool flank could be expressed as follows: The instantaneous position vector of the mapping point e′ on the machined transition surface along the normal vector direction of this point could be expressed as follows: The instantaneous velocity of motion v vn (t) of the selected point on the tool flank relative to the mapped point on its machined transition surface was shown in Eq. (18).
where, v x (t), v y (t), and v z (t) are the components of the motion velocity v vn (t) at any point on the tool flank in the x, y, and z coordinate directions, respectively, and their directions are the opposite directions of the friction velocity v n (t).
According to Eq. (15)-Eq. (18), the instantaneous position vector relationship between the tool flank of the cutter tooth and the machined transition surface could be expressed as follows: From Eq. (19), the discriminant conditions of the instantaneous contact between the tool flank of the cutter tooth and the machining transition surface could be expressed as follows: When ���� ⃗ ee ′ does not satisfy Eq. (20), the state between the cutter tooth tool flank and the machined transition surface is not contact. When ���� ⃗ ee ′ satisfy Eq. (20), between the tool flank and the machined transition surface and mutual extrusion is mutual contact. When ���� ⃗ ee ′ is 0, the state of the instantaneous geometric contact between the tool flank and the machined transition surface is the critical. Based on the above models and the indexable high feed milling cutter provided by the company and its cutting titanium alloy process conditions, the cutting vibration experiments were conducted on the milling machining center. The instantaneous normal vector and instantaneous geometric contact boundary of the milling tool flank were solved. In order to solve the instantaneous geometric contact relationship of milling cutters under vibration, the cutting experiment of the high feed cutter was carried out in VDL-1000E three-axis milling center using the high feed milling cutter and its cutting conditions for titanium alloy provided by the enterprise.
Before milling experiment, Zoller 1610A tool setting instrument was using to measure the axial length and turning radius of each cutter tooth of the milling cutter, obtaining the axial error and radial error.
The milling cutter diameter is 32 mm. The number of teeth is 3. The milling method is dry and smooth milling. The total cutting stroke L m of the milling cutter is 5 m. The milling is carried out 20 times in layers, respectively. The milling parameters and tooth error are shown in Table 1. The workpiece material component was shown in Table 2, where f z is feed per tooth.
The vibration acceleration signal during the milling process was detected using a PCB triaxial acceleration sensor and a DH5922 instantaneous signal test and analysis system, as shown in Fig. 3. In Fig. 3, m is the milling cutter layered. cutting, Δt is the cutting cycle of the milling cutter, a x , a y , and a z are the milling vibration acceleration signals along the feed rate direction, the cutting width direction, and the cutting depth direction, respectively.
Based on the above experimental results, the Eqs. (13) and (14) were used to solve for the instantaneous normal vector angle of the tool flank. The results are shown in Fig. 4.
According to Fig. 4, the instantaneous normal vector position of the characteristic point on the tool flank changed continuously by the influence of the milling vibration during the process from cutting in to cutting out the workpiece. This phenomenon led to the variability and instability of the instantaneous normal vector position on the tool flank. It could directly affect the instantaneous contact relationship between the tool flank and the machined transition surface.
According to Eq. (1)-Eq. (20), the instantaneous boundaries of the geometric contact between the tool flank and the machined transition surface could be obtained. Take the 14 th layered cutting of the cutting motion cycles 331 and the instantaneous contact angle 45° as an example, where the instantaneous boundaries of the geometry contact on the tool flank are shown in Fig. 5.
According to the above analysis results, the influence of the cutter tooth error, the milling vibration and other factors, milling cutter cutting attitude were unstable state. The instantaneous normal vector on the tool flank was frequent changes. This phenomenon led to the instantaneous position vector relationship between the tool flank and the machined transition surface changes constantly. And the instantaneous geometric contact boundaries were variability.
The instantaneous geometric contact of the tool flank and the machined transition surface under vibration was revealed by using above methods. On this basis, the friction boundary of the tool flank could be further identified by using the finite element thermodynamic coupling field distribution between the tool flank and the machined transition surface.

Calculation method of the friction boundary on the tool flank of the cutter tooth under vibration
Using the Fig. 1 and Eq. (1)-Eq. (14), Table 1 and Fig. 3, the thermal coupling field boundary condition of the high feed milling cutter cutting titanium could be constructed. The instantaneous equivalent stress and instantaneous temperature distribution of the tool flank of the cutter tooth were obtained. Take the 14 th layered cutting of the cutting  According to the Eq. (15) and Eq. (18), along the direction of the normal vector and the instantaneous velocity of the machined transition surface, the selected point on the  tool flank of instantaneous normal stress and tangential stress were obtained as shown in Fig. 6. According to Fig. 6, normal stress σ and tangential stress τ could be expressed as follows: where σ q j is the component of the normal stress in the direction of the normal stress. The τ q j is the component of the tangential stress in the direction of the normal stress. The j is the three directional components of the tetrahedron. And the vector angle between the three components of the equivalent stress and the normal stress could be expressed as follows: The vector angle between the three components of the equivalent stress and the tangential stress could be expressed as Eq. (23).
According to Figs. 2 and 6, cutting edge structure retention under the thermodynamic coupling not only influence the formation of machined transition surface, but also the upper and lower boundary of instantaneous contact of the  tool flank of the cutter tooth and the machined transition surface could be determined. From Fig. 6, the state of tool flank and machined transition surface changed from contact to non-contact along the direction away from the cutting edge of the cutter teeth. The transfer mode and distribution of temperature, normal stress, and tangential stress change under the changed due to the variation of contact medium and the change rate showed obvious mutation. The number of meshes divided by the cutter teeth in the simulation process was 620,000, and minimum size of the grid was 0.3 mm. In order to make the simulation results more accurate when the step distance did not exceed 1/3 of the minimum element size of the workpiece grid element.
The temperature and stress distribution of the tool flank under vibration could be obtained by the above method. According to Fig. 6(a) and (b), the equivalent stress at different positions, the temperature change rate, the normal stress change rate and the tangential stress change rate along the V direction are shown in Figs. 7, 8, 9, and 10. According to Fig. 7, structural damage due to instantaneous contact generation between the cutting edge and workpiece was identified by using the instantaneous equivalent stress on the tool flank of the cutter tooth and yield strength criterion. The upper boundary of instantaneous friction was obtained, as shown in Fig. 11.where V 0 (U) is the original   According to Figs. 8, 9, and 10, the abrupt boundary curve of the temperature, the normal stress and tangential stress of the tool face in the direction away from the cutting edge were obtained. Combined with Fig. 6, the influence of the relative motion, mutual contact, extrusion, and deformation between the tool flank and the machined transition surface were comprehensively considered. Using Eq. (24), instantaneous friction lower boundary was obtained, as shown in Fig. 12.
According to Fig. 12, V N (U,t), V σ (U,t), V τ (U,t), V T (U,t) are the geometric contact, the normal stress, the tangential stress, and the temperature criterion identification curves on the tool flank of the cutter tooth, respectively. V ξ (U,t) is the instantaneous friction boundary on the tool flank at moment t.
During the full time cutting of the milling cutter, the accumulation result of the instantaneous friction boundary of the tool flank were used to analyze and evaluate the degree of friction and wear of milling tool flank, as shown in Eq. (25).
According to Eq. (25), V s (U) and V p (U) are the accumulate friction upper and lower boundaries, respectively. V s (U,t) is instantaneous frictional upper boundary curve on the tool flank at time t. V min (U, t) is the minimum value of the instantaneous frictional boundary on the tool flank at time t.
Using the solution result and Eq. (25) of the instantaneous friction boundary of the tool flank, cumulative friction boundary on the tool flank of three cutter teeth were obtained after the milling cutter completes 20 times of layered milling under the Table 1, as shown in Fig. 13.where V i = V s (U) and V i = V p (U) are the upper and lower boundary of cumulative friction on the flank of three cutter teeth of milling cutter under the scheme 1, respectively.
According to Figs. 7,8,9,10,11,12, and 13, the upper and lower friction boundaries of instantaneous and cumulative on the each cutter tooth showed irregularity, diversity and complex geometry. With the above method, the difference between the instantaneous and cumulative friction boundary of the tool flank with different cutter teeth could be identified. But it is impossible to quantitatively distinguish the evolution process of the friction boundary of each tooth flank from the shape and structure. So it is necessary to further explore.

Fractal and its time-varying of the friction boundary on the milling tool flank
In order to quantitatively describe the dynamics variation of the instantaneous friction boundary on the tool flank, irregularity, diversity, and complexity of distribution of the friction boundary patterns on the tool flank were characterized based on fractal theory.  The calculation methods of the friction boundary on the tool flank and solution method of the fractal feature parameter were adopted. The power spectral density function of the instantaneous friction contact boundary of the three cutter's flank in Table 1 could be calculated by using above methods respectively. Take the results of the 14 th layered milling, 331 st cycle, and contact angle of 45° as an example.
The power spectral density function for the results of the instantaneous friction contact boundary on the tool flank of the three cutter teeth are shown in Figs. 14 and 15.
The fractal dimension and scale coefficient of the instantaneous and cumulative friction boundary on the tool flank at different cycles during the cut-in to cut-out process of the milling cutter could be obtained by the above method. In comparatively close. The cutter teeth with the different layered cutting periods in the same cutting cycle were compared. The fractal characteristic parameters of instantaneous friction boundary on the tool flank were significantly different.
The curves of the fractal dimension of the instantaneous frictional upper and lower boundaries on the tool flank showed an overall irregular as serrate shape. Where, the fractal dimension of the upper boundary appeared high variability in the middle of the cutting period, low variability in early and later cutting period. And the fractal dimension of the lower boundary was a trend of high variability in early cutting period and decreasing thereafter.
The overall trend of the scale coefficient of instantaneous friction upper boundary on the tool flank were overall gradually increasing and clearly oscillating in the middle and end of the cutting period. The scale coefficient of the lower boundary on the tool flank overall increased first and decreased gradually.
The fractal dimension of the cumulative friction upper and lower boundaries versus the instantaneous friction boundary at the end of the cutting period was slightly different. And the scale coefficient of the cumulative friction lower boundary and instantaneous friction boundary at the end of the cutting period had significant differences.
The results showed that the distribution pattern of the instantaneous friction upper and lower boundaries on the tool flank underwent different evolutionary processes. The cumulative friction boundary was not the same as the instantaneous friction boundary at the end of the cutting period in terms of scale and complexity of the distribution pattern. The reasons were the constantly stacked of the different evolvement of the instantaneous frictional boundary on the tool flank of each cutter tooth. The improved grey correlation analysis method was adopted. The fractal dimension and scale coefficient of the instantaneous friction lower boundary  Table 3. Fig. 20 Fractal characteristic parameters of abrupt change boundary of the instantaneous normal stress on the tool flank In Table 3, γ N is the correlation degree of the fractal characteristic parameters between the instantaneous geometric contact boundary and the instantaneous friction lower boundary on the tool flank, γ T is the correlation degree of the fractal characteristic parameters between the instantaneous normal stress field boundary and the instantaneous friction lower boundary on the tool flank, γ σ is the correlation degree of the fractal characteristic parameters between the instantaneous normal stress field boundary and the instantaneous friction lower boundary on the tool flank, γ τ is the correlation degree of the fractal characteristic parameters between the instantaneous tangential stress field boundary and the instantaneous friction lower boundary on the tool flank.
According to Figs. 18, 19, 20, and 21, compared to the abrupt change boundaries of variation rate of instantaneous normal stress and instantaneous tangential stress, there were significantly different change between the instantaneous geometric contact boundary and the instantaneous normal stress field boundary. According to the Table 3, all the correlation degrees were calculated to be above 0.63 and greater than 0.60, which was a strong and positive correlation. The fractal dimension of instantaneous friction lower boundary was mainly  determined by the abrupt change boundary of variation rate in instantaneous normal stress and the instantaneous tangential stress. The influence of instantaneous geometric contact boundary was minimal. Each factor had a close influence on the instantaneous friction lower boundary. The results showed that the distribution patterns and the evolution of the instantaneous friction lower boundary on the tool flank were directly influence by the multi-temporal variability of the instantaneous relative motion, mutual contact, extrusion and deformation of the tool flank and the machined transition surface.  The complexity and diversity of instantaneous friction boundaries on the milling tool flank were quantitatively characterized by fractal dimension and scale coefficient. The evolution of the instantaneous friction boundary over time and the characteristics of the cumulative friction boundary distribution on the tool flank were revealed. The influence of the friction boundary for the instantaneous relative movement, the mutual contact, the extrusion, and deformation between the tool flank and the machined transition surface were identified by the improved grey correlation analysis method. And the milling process solutions were judged by its response to process variables.
Take Table 1 as scheme 1. Using the same milling cutter, workpiece, mounting method, cutting method, and inspection method as in scheme 1, keeping cutting efficiency, cutting depth and cutting width unchanged, increasing cutting  . 22 Identification method of fractal characteristics of friction boundary on the tool flank speed, reducing the feed per tooth accordingly and adjusting the tooth error distribution, then the instantaneous contact relationship between teeth and transition surface could be changed. Based on the above, the milling experiments could be implemented for solving the friction boundaries of the tool flank as shown as Table 4. Vibration acceleration signals could be obtained using the above experimental scheme, as shown in Fig. 23.
According to Fig. 22, the friction boundaries of the tool flank of each tooth and their fractal dimensions and scale coefficients were obtained. And it was compared with scheme 1, as shown in Fig. 24.
According to Figs. 23 and 24, the milling vibration varied following the varying milling tool speed, feed per tooth and tooth error. This caused the variation of the fractal characteristic parameters of instantaneous friction boundary of tool flank. Although the fractal dimension and scale coefficients of both schemes generally showed similar time-varying properties, differences in the rate of evolution of the process over the different cutting cycles still existed.
Compared with scheme 1, the fractal dimensions of the upper boundary of the instantaneous friction of scheme 2 decreased in the middle and later cutting periods. Its change of fractal dimension of the instantaneous friction lower boundary was similar to scheme 1 and slightly higher in the early cutting period. In the middle and later cutting period, the variation range and level of the fractal dimension of the lower boundary of instantaneous friction were significantly reduced. The fractal dimension of the upper and lower boundary of cumulative friction in scheme 2 was significantly reduced.
At the same time, the scale coefficients of the upper and lower boundaries of the instantaneous and cumulative friction of the cutter tooth flank in scheme 2 were lower than those in scheme 1. Correlation degrees on the instantaneous friction lower boundary of the two schemes are shown in Fig. 25.
As shown as Fig. 25, γ ND , γ TD , γ σD , γ τD are the correlation degree of the fractal dimension of the instantaneous geometric contact boundary, instantaneous temperature field boundary, instantaneous normal stress field boundary, instantaneous tangential stress field boundary, and instantaneous friction lower boundary on the tool flank of the cutter tooth, respectively. γ NG , γ TG , γ σG , γ τG are the correlation degree of the scale coefficient of the instantaneous geometric contact boundary, instantaneous temperature field boundary, instantaneous normal stress field boundary, instantaneous tangential stress field boundary, and instantaneous friction lower boundary on the tool flank of the cutter tooth, respectively. According to Fig. 25, all correlation degrees were above 0.62 and greater than 0.60. It was shown that the formation and evolution of the lower boundary of the instantaneous friction of the milling cutter flank were strongly related to the factors such as  instantaneous geometric contact boundary of tool flank and fractal characteristic of the abrupt change boundaries of variation rate of the instantaneous temperature, the instantaneous normal stress, and instantaneous tangential stress. As the same time, the responses of the fractal characteristic parameter of the different cutter teeth were different when the changes in process design variables such as milling cutter speed, feed per tooth, and cutter tooth error occurred. The response of the correlation degrees of the instantaneous temperature, instantaneous normal stress, and instantaneous tangential stress of the tool flank were more obvious than the instantaneous geometric contact boundary of the tool flank. The results showed that the process design variables mainly changed the distribution of the instantaneous contact deformation between the tool flank and the machining transition surface, and then affected the irregularity and scale range of the instantaneous friction lower boundary distribution of the tool flank.
The results showed that the dynamic behavior of the instantaneous friction boundary on the tool flank and its correlation with the instantaneous geometric contact boundary on the tool flank and the instantaneous thermal coupling field characteristic of the abrupt change boundary of the variation rate could be identified with the use of the above models. This method provided a basis for further identifying the friction evolution process and process control variables of the milling tool flank.

Identification and experimental verification of the cumulative friction boundary on the milling tool flank
The solution results of the upper and lower boundary of cumulative friction on the tool flank were obtained by using the method of the scheme 2, as shown in Fig. 26. According to Figs. 12 and 26, the comparison results of the fractal dimension and scale coefficient of the cumulative friction boundaries on the tool flank of the two schemes are shown in Table 5.
In Table 5, D s , G s are the fractal dimension and scale coefficient of the cumulative friction upper boundary on the tool flank of the milling cutter. D p , G p are the fractal dimension and scale coefficient of the cumulative friction lower boundary on the milling tool flank.
According to Table 5, under the condition of the same cutting efficiency, the fractal dimension and scale coefficient of the cumulative friction boundary on the tool flank of the each cutter tooth were less than scheme 1. The above analysis results showed that it could improve the irregularity and scale range of the cumulative friction boundary distribution on the milling tool flank when rotation speed, feed per tooth, cutter tooth error, and other process design variables were adjusted. The milling cutter and their cutting process program could be judged by the Fig. 22.
The cumulative friction boundaries on the tool flank of each cutter tooth for the two schemes were detected by the VHX-100 Ultra Deep Field Microscope. And the  Table 6. According to Figs. 27, 28, and Table 6, the cumulative friction boundary of the milling cutter flank under the two schemes were obviously different. The responses of the upper and lower boundary fractal characteristic parameters were consistent. As the same time, it could be seen that there was a certain difference between the calculated and the measured results of.the cumulative friction boundary on the flank of each cutter tooth. During the experiment, there was deviation between the measured value and the actual value of milling vibration due to the influence of the random factors. To further verify the correctness of the solution model of the friction boundary of the milling cutter flank and the identification method of the dynamic distribution, the relative error analysis between the solution of the cumulative friction boundary on the tool flank of the milling cutter and the experimental results was carried out according to the following formula. The results were shown in Table 7.
where λ D is the relative error of the fractal dimension of the cumulative friction boundary, λ G is the relative error of the scale coefficient of the cumulative friction boundary. D, G are the solution results of the fractal dimension and scale coefficient, respectively. D 0 , G 0 are the experimental results of the fractal dimension and scale coefficient, respectively.
In Table 7, λ Ds , λ Gs are the relative error of the fractal characteristic parameters of the cumulative friction upper boundary, respectively. λ Dp , λ Gp are the relative error of the fractal characteristic parameters of the cumulative friction lower boundary, respectively.   According to Table 7, the average relative error value of the fractal characteristic parameters of the cumulative friction upper and lower boundary curves were no greater than 15.1% for both the experimental of the solved tool flank of different cutter teeth of the milling cutter. The results showed that friction boundary solution was in good agreement with the experimental results.
In summary, the evolution of the instantaneous friction boundary and cumulative friction boundary distribution on the tool flank of a milling cutter tooth could be identified with the use of the above models effectively. And the influence characteristic of the friction boundary affected by cutting parameters, cutter tooth error, milling vibration and other factors could be revealed. The above model and method provided the basis for the accurate assessment of the tool life.

Conclusion
(1) A method for solution and discrimination of the instantaneous position vector of the tool flank under vibration was proposed. The instantaneous mapping relationship between the tool flank normal vector and machined transition surface was revealed. And the instantaneous geometric contact boundary between the tool flank and the machining transition surface could be obtained under the multi factors comprehensive effect. The results showed that the variability of instantaneous geometric contact boundary and the instantaneous geometric contact relation could be revealed using of the above methods.
(2) The instantaneous equivalent stress of cutter tooth cutting edge and the distribution mutagenicity of the instantaneous temperature, normal stress, and tangential stress of the tool flank under vibration were adopted. The solution model of the upper and lower boundary of instantaneous friction of the tool flank was constructed, and the calculation method of the accumulated friction boundary of the tool flank was proposed. The results showed that the instantaneous contact and friction between the milling tool flank and the machining transition surface were instable. The geometry of the upper and lower boundary of instantaneous and cumulative friction of cutter teeth was complex and irregular, and there were some differences of the friction boundary of the tool flank with different cutter teeth.
(3) The results of fractal characteristic and the timevarying properties of the milling tool flank showed that the evolution processes of the distribution pattern of upper and lower boundary of instantaneous friction on the tool flank were different. As the same time, the instantaneous relative motion, the multi-temporal variability of the mutual contact, extrusion, and deformation of the tool flank under vibration directly influenced the distribution patterns and their evolution of the instantaneous friction boundary on the tool flank. The correlation between instantaneous frictional boundary morphology and thermal-stress coupling field parameters are bigger than 0.62, which means a strong positive correlation. The abovementioned factors are the main reason for the scale ranges and the distribution irregularity of accumulated friction boundary on tool flank.
(4) An identification method for the fractal characteristic and the dynamic properties of friction boundary of the tool flank was proposed. The method revealed the dynamic formation process of the instantaneous friction boundary of multiple cutter teeth using the changing of the fractal dimension and scale coefficient on the instantaneous friction boundary. The irregularity and diversity of the distribution of the cumulative friction boundary on the tool flank were quantitative characterized by using of the instantaneous friction boundary. The smallest relative error between the calculated and experimental results of the upper and lower boundaries of the cumulative friction on tool flank is 1.3%, and the average relative error is 6.65%. The analysis results showed that the above method could reveal the influence of cutting parameters, tooth error, milling vibration, and other process variables on the friction boundary and its formation process on the tool flank of the milling cutter.

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Data availability The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Declarations
Ethics approval and consent to participate This chapter does not contain any studies with human participants or animals performed by any of the authors.

Consent for publication
The authors declare no competing interests.

Competing interests
The authors declare no competing interests.