2. Similarity Numbers in Metal Cutting Testing and Modeling Viktor P. Astakhov CIRP 12

3. Viktor P. Astakhov CIRP12 20092 What seems to be a problem?

4. Viktor P. Astakhov CIRP12 20093 Introduction Studies of Metal Cutting 1.Analytical Studies 2.Numerical Studies 3.Experimental Studies Problems with Experimental Studies 1.High cost 2.Long time 3.Particularity of the obtained results

5. Viktor P. Astakhov CIRP12 20094 "In theory, there is no difference between theory and practice. But, in practice, there is." Jan L.A. van de Snepscheut

6. Viktor P. Astakhov CIRP12 20095 Introduction Example of the machining system (drilling)

7. Viktor P. Astakhov CIRP12 20096 Introduction Similarity Theory The similarity theory offers a better way to obtain a sound mathematical model of the complicated processes taking place in a complex technical system. Today it is largely used in the area of thermodynamic, fluid flow etc. This theory combines various information and knowledge about a complex process under study. Its basic principle is separation of a group of similar phenomena from a great class of phenomena by a general low. In such a context, similarity can be geometrical, physical etc.

8. Viktor P. Astakhov CIRP12 20097 Introduction Similarity Theory in Metal Cutting At the present stage, however, the similarity theory is not yet developed in metal cutting studies. Rather, a number of useful similarity criteria (numbers) are developed that can be used in modeling of the metal cutting process. The objective of this presentation is to discuss three most important similarity numbers as the chip compression ratio, the Péclet and the Poletica numbers.

9. Viktor P. Astakhov CIRP12 20098 Chip Compression Ratio (CCR) Tool-Chip Interface

10. Viktor P. Astakhov CIRP12 20099 CCR Significance of CCR The elementary work spent over plastic deformation of a unit volume of the work material calculates as K is the stress at ε=1, n is the strain-hardening coefficient. Knowing CCR, one not only assure the similarity of the deformation process but also calculate the power spent on the plastic deformation of the layer being removed and power spent due to friction at the tool-chip interface. These two are major contributors to the total power required by the cutting system

11. Viktor P. Astakhov CIRP12 200910 CCR Significance of CCR Aluminum 2024T6 Energy of plastic deformation, 67% Cohesive energy, 7% Flank energy, 6% Rake energy, 20% Steel 52100 Energy of plastic deformation, 63% Cohesive energy, 6%Flank energy, 9% Rake energy, 22% CCR is the simplest yet most important and most objective characteristic of the cutting process

12. Viktor P. Astakhov CIRP12 200911 Pe number Pèclet number Definition for metal cutting where v is the velocity of a moving heat source (the cutting speed) (m/s), w w is the thermal diffusivity of the work material (m 2 /s), k w is the thermoconductivity of the work material, (J/(m·s· o C)), (c p ·  ) w is the volume specific heat of work material, (J/(m 3 · o C)). The Péclet number is a similarity number, which characterizes the relative influence of the cutting regime (vt 1 ) with respect to the thermal properties of the workpiece material (w w ). If Pe>10 then the heat source (the cutting tool) moves over the workpiece faster than the velocity of thermal wave propagation in the work material so the thermal energy generated in cutting due to the plastic deformation of the work material and due to friction at the tool-chip interface does not affect the work material ahead of the tool. If Pe<10 then the thermal energy due to the plastic deformation and due to friction makes its strong contribution to the process of plastic deformation during cutting as its affect the mechanical properties of the work material.

13. Viktor P. Astakhov CIRP12 200912 Pe number Practical use in testing Influence of the cutting speed on CCR for different speeds. Work material – steel AISI 1030, tool material – carbide P20, rake angle γ n = 10 o, cutting edge angle κ r = 60o, depth of cut d w = 2mm Generalization of the experimental data using the Péclet number

14. Viktor P. Astakhov CIRP12 200913 Pe number Practical use in testing CCR vs. (a) the cutting speed for different feeds and (b) Pe criterion. Work material – tool steel H13, tool material – carbide M10, rake angle γ n =−10 o, cutting edge angle κ r = 60 o, depth of cut d w = 2mm CCR vs. Pe criterion for different rake angles. Work material – steel AISI 1045, tool material – carbide P20, cutting edge angle κ r = 60 o, depth of cut d w = 2mm

15. Viktor P. Astakhov CIRP12 200914 Po number Poletica number (Po-criterion) In metal cutting, the tool–chip contact length known as the length of the tool–chip interface determines major tribological conditions at this interface as temperatures, stresses, tool wear, etc. Moreover, all the energy required by the cutting system for chip removal passes through this interface. Therefore, it is of great interest to find out a way to asses this length. To deal with the problem, the Poletica criterion (Po-criterion) is introduced as the ratio of the contact length, l c to the uncut chip thickness, t 1

16. Viktor P. Astakhov CIRP12 200915 Po number Practical use in testing Influence of chip compression ratio on Po-criterion in machining steel AISI E9310, tool material P20 (79%WC, 15%TiC, 6%Co), cutting feed f = 0.07 − 0.43mm/rev and cutting edge angle κ r = 70 o Influence of chip compression ratio on Po- criterion in machining beryllium copper UNSC17000 of different hardnesses. Tool material – M30 (92%WC, 8%Co)

17. Viktor P. Astakhov CIRP12 200916 Introduction Practical use in testing Influence of chip compression ratio on Po-criterion in machining various work materials using different tool materials and tool rake angles

18. Viktor P. Astakhov CIRP12 200917

19. Viktor P. Astakhov CIRP12 200918 A number Other important numbers One of the most important is the A-criterion. Since it first derived and studied by Silin, it may be referred as Silin criterion. It calculates as and characterizes the part of the thermal energy (heat) absorbed by the chip relative to the whole amount of heat generated in the deformation zone. In this equation t 1, b 1T are the uncut chip thickness and the true chip width, respectively, m, cρ is the volumetric heat capacity of the work material, J/(m 3 o K); θ c is the cutting temperature, o C; F p is the power components of the force, N.

20. Viktor P. Astakhov CIRP12 200919 D, E, F numbers The D-criterion which calculates as and characterizes the uncut chip cross-section. The E-criterion or relative sharpness of the cutting edge which calculates and characterizes the influence of the cutting edge radius ρ 1 (m) with respect to the uncut chip thickness t 1 (m). The F-criterion which calculates as characterizes influence of the tool geometry with respect to the thermal conductivities of tool and work materials. In tis equation, k t and k w are thermal conductivities of tool and work materials, J/(m s o C), respectively, β n is the normal tool wedge angle; ε tn is the acute angle in the reference plane between the major (side) and minor cutting edges.

21. Viktor P. Astakhov CIRP12 200920 Machinability Machinability test where constants n3, m4 – m7 are to be determined experimentally using a suitable design of experiment techniques In experimental studies of machinability when a specific tool (tool material, tool holder etc) and workpiece (dimensions and work material) were selected for test, it often sufficient at the first stage of the study to consider the following relationship where n 1 and m 1 are constants to be determines experimentally. or

22. Viktor P. Astakhov CIRP12 200921 Machinability Machinability test The latter equation can be re-written for the optimum cutting speed v o (the speed that corresponds to the optimal cutting temperature  o ) as To determine constants n 1 and m 1, the power components of the force, F p and the cutting temperature  c are measured simultaneously. If the test results are plotted on a double logarithmic A versus Pe diagram (the same module along both axes) as shown, then n 1 = Pe when A = 1 and m 1 = tan  1. For data shown in the figure, the machinability equations becomes Experimental determination of the constants of Eq. (15): (a) work material - stainless steel AISI 303, tool material: carbide P01 (66%WC30%TiC4%Co), tool geometry: γ n = 12 o, α n = 10 o, κ r = 45 o, κ r1 = 25 o, r n = 1 mm, similarity numbers: F = 1.48, D = 0.0126- 0.1500, E = 0.06 - 0.76.

23. Viktor P. Astakhov CIRP12 200922 Machinability Machinability test The foregoing analysis leads to a new approach to machinability determination using the following procedure. Five - seven different cutting feeds should be selected for the study. The depth of cut should be kept the same for all tests. The number of tests corresponds to that of the selected cutting feeds. In each test, the cutting speed is varied and the cutting force and cutting temperature are measured. As shown in the figure. Workpiece material: nickel-based high alloy (0.08%C1%Cr56%Ni1%Co1%Al), tool material: carbide M30 (92%WC8%Co), tool geometry: γ n = 12 o, α n = 12 o, κ r = 45 o, κ r1 = 45 o, r n = 1 mm, cutting regime: d w = 1 mm, f, mm/rev, =1- 0.074, 2- 0.11, 3- 0.15, 4- 0.25, 5- 0.30, 6- 0.34, 7- 0.39

24. Viktor P. Astakhov CIRP12 200923 on Machinability test The optimum cutting speed is defined for each feed as that corresponding to the minimum stabilized value of the cutting force. Plotting the results on a double logarithmic the true uncut chip thickness versus cutting speed (same module along both axes), one can obtain a t 1 - v curves as shown. This t 1 - v curve may be considered linear within a certain range of the uncut chip thickness. The equation for this linear proportion of the curve is written In which constants n 2 =0.034 and m 2 = 0.81. SIMPLE, physically-grounded, straightforward test

25. Viktor P. Astakhov CIRP12 200924 Conclusions To narrow the gap between the metal cutting theory and practice, a sound similarity approach should be developed to utilize the full power of the similarity theory. In the author’s opinion, the basic set of the relevant similarity numbers should be developed in metal cutting and the three basic theorem of similarity should be used to determine the necessary and sufficient conditions of similarity of cutting process. The three first similarity numbers discussed here, namely, CCR, the Péclet and Poletica criteria are of a great help in metal cutting studies.

26. Viktor P. Astakhov CIRP12 200925 Direction of spending THANK YOU The happy end