Similarity Numbers in Metal Cutting Testing and Modeling

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Similarity Numbers in Metal Cutting Testing and
Modeling
Viktor P. Astakhov
CIRP 12
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What seems to be a problem?
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Studies of Metal Cutting
Introduction

• Analytical Studies
• Numerical Studies
• Experimental Studies

Problems with Experimental Studies

• High cost
• Long time
• Particularity of the obtained results

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"In theory, there is no difference between theory
and practice. But, in practice, there is." Jan
L.A. van de Snepscheut
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Example of the machining system (drilling)
Introduction
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Similarity Theory
Introduction

• 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.

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Similarity Theory in Metal Cutting
Introduction

• 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.

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Chip Compression Ratio (CCR)
Tool-Chip Interface
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Significance of CCR
CCR
The elementary work spent over plastic
deformation of a unit volume of the work material
calculates as
K is the stress at e1, 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
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Significance of CCR
CCR
Aluminum 2024T6
Steel 52100
Energy of plastic deformation, 67
Energy of plastic deformation, 63
Rake energy, 20
Rake energy, 22
Cohesive energy, 7
Flank energy, 6
Cohesive energy, 6
Flank energy, 9
CCR is the simplest yet most important and most
objective characteristic of the cutting process
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Pèclet number
Pe number
Definition for metal cutting
where v is the velocity of a moving heat source
(the cutting speed) (m/s), ww is the thermal
diffusivity of the work material (m2/s), kw is
the thermoconductivity of the work material,
(J/(msoC)), (cp ?)w is the volume specific
heat of work material, (J/(m3oC)).
The Péclet number is a similarity number, which
characterizes the relative influence of the
cutting regime (vt1) with respect to the thermal
properties of the workpiece material (ww). If
Pegt10 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 Pelt10 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.
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Practical use in testing
Pe number
Influence of the cutting speed on CCR for
different speeds. Work material steel AISI
1030, tool material carbide P20, rake angle ?n
10o, cutting edge angle ?r 60o, depth of cut
dw 2mm
Generalization of the experimental data using the
Péclet number
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Practical use in testing
Pe number
CCR vs. Pe criterion for different rake angles.
Work material steel AISI 1045, tool material
carbide P20, cutting edge angle ?r 60o, depth
of cut dw 2mm
CCR vs. (a) the cutting speed for different feeds
and (b) Pe criterion. Work material tool steel
H13, tool material carbide M10, rake angle ?n
-10o, cutting edge angle ?r 60o, depth of cut
dw 2mm
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Poletica number (Po-criterion)
In metal cutting, the toolchip contact length
known as the length of the toolchip 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, lc to the uncut chip
thickness, t1
Po number
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Practical use in testing
Po number
Influence of chip compression ratio on
Po-criterion in machining steel AISI E9310, tool
material P20 (79WC, 15TiC, 6Co), cutting feed
f 0.07 - 0.43mm/rev and cutting edge angle ?r
70o
Influence of chip compression ratio on
Po-criterion in machining beryllium copper
UNSC17000 of different hardnesses. Tool material
M30 (92WC, 8Co)
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Practical use in testing
Introduction
Influence of chip compression ratio on
Po-criterion in machining various work materials
using different tool materials and tool rake
angles
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Other important numbers
A number
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 t1, b1T are the uncut chip
thickness and the true chip width, respectively,
m, c? is the volumetric heat capacity of the work
material, J/(m3 oK) ?c is the cutting
temperature, oC Fp is the power components of
the force, N.
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The D-criterion which calculates as
D, E, F numbers
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 ?1 (m) with respect to the uncut chip
thickness t1 (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, kt and kw are
thermal conductivities of tool and work
materials, J/(m s oC), respectively, ßn is the
normal tool wedge angle etn is the acute angle
in the reference plane between the major (side)
and minor cutting edges.
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Machinability test
Machinability
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
or
where n1 and m1 are constants to be determines
experimentally.
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Machinability test
The latter equation can be re-written for the
optimum cutting speed vo (the speed that
corresponds to the optimal cutting temperature
?o) as
Machinability
To determine constants n1 and m1, the power
components of the force, Fp 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 n1 Pe when A
1 and m1 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
(66WC30TiC4Co), tool geometry ?n 12o, an
10o, ?r 45o, ?r1 25o, rn 1 mm, similarity
numbers F 1.48, D 0.0126-0.1500, E 0.06 -
0.76.
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Machinability test
Machinability
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
(0.08C1Cr56Ni1Co1Al), tool material carbide
M30 (92WC8Co), tool geometry ?n 12o, an
12o, ?r 45o, ?r1 45o, rn 1 mm, cutting
regime dw 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
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Machinability test
on
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 t1 - v
curves as shown. This t1 - 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 n20.034 and m2 0.81.
SIMPLE, physically-grounded, straightforward test
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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 authors 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.
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THANK YOU
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