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Indentation as the basis for metal cutting
MECHANICS RESEARCH COMMUNICATIONS 0093-6413/80/060377-06502.00/0
Vol. 7(6),377-382,1980. Printed in the USA. Copyright (c) Pergamon Press Ltd
INDENTATION AS THE BASIS FOR METAL CUTTING
C. Sahay* and R.N. Dubey, Department of Mechanical Engineering, University of Waterloo, Waterloo, Ontario, Canada N2L 3GI
(Received 13 June 1980; accepted for print 25 July 1980)
Introduction
In a metal cutting operation, an external force acting through a mechanism induces relative velocity between the tool and workpiece. The force is applied on either the cutting tool or the workpiece while the other is generally held at rest. The amount of force required depends on the material under deformation (material properties of the metal, width and thickness of cut, etc.) and, to some extent, on the geometry of the tool and not on the mechanism of the force application. The process itself has, however, associated with it certain characteristics like cyclic variation in the cutting force and the associated frequency, the tendency of the chip to curl, the cutting ratio, etc. A real metal cutting operation is very complex. It involves the material constitutive property, high temperature, tool geometry, rigidity of tool and machine and the vibratory motion, etc. Any attempt for presenting a theory of metal cutting has to take into account all these factors and the resulting theory is likely to be a very complex one. However, any reasonable theory under any simplified situation, should be able to estimate the process characteristics in one unified theory and the present paper is a step in this direction. To facilitate the understanding of the cutting mechanism, we make some simplifying assumptions: We ignore the machine tool or cutting tool vibration and assume rigid-plastic behaviour for the workpiece. In our presentation we take advantage of the similarity in physics of indentation, shearing deformation and cutting.
Slip Line Field, Hodograph and Force Calculation Hill [8] considered the problem of indentation of wedge on specimen of finite width and thickness.
For small depths of indentation of the wedge into the
workpiece, the deformed material is displaced from under the wedge onto the
*On leave of absence from Indian School of Mines, Dhanbad (India) 377
378
C. SAHAY and R.N. DUBEY
the side and forms a coronet.
When the indentation increases the deformation
zone expands to touch either the bottom or the side of the workpiece depending on the width to depth ratio.
Whereas in the problem of shearing the critical
thickness determines the deformation zone, in metal cutting the width determines the deformation zone.
Sahay and Dubey [12] investigated
of a finite media having finite width by an unsymmetrical tool.
the indentation Fig. i shows
the slip line field associated with plastic deformation on account of indentation by an unsymmetrical wedge.
The unsymmetrical wedge may be assumed to
represent a metal cutting tool if half wedge angle of one side reduces to zero. Fig. 2 shows an indentation by an unsymmetrical wedge, modified to become a metal cutting tool, in a workpiece of finite width.
The tool works on the
material near one lateral face reducing indentation to a cutting situation. The equilibrium of the portion NCK'KIM, which is rigid, determines the length 'b' of segment JK, radius shear-plane.
'r' of sector KL and length 'a' of segment LM on the
Portion AF and FJ follow from the deformation in earlier stages.
The fact that line AFJKLM represents an s-line in the slip line field together with stress-free boundary at M is utilized in determining the orientation of AF with respect to AD.
C'
3'
,i
C
D'
FIG. i Slip line field in the deformation zone due to indentation by unsymmetrical wedge.
INDENTATION AS BASIS FOR METAL CUTTING
379
N
/ L
C
/
K
So
A
(i)
(ii) FIG. 2
(i) (ii)
Indentation by modified unsymmetrical wedge near the lateral surface of a workplece. Sllp line field for the above.
tc
b,eok' llj,k
"~,
P
a,foj,lk,l,m Vc
=, 0
FIG. 3 Hodograph for slip line field of Figure 2.
M
380
C. SAHAY and R.N. DUBEY
Fig. 3 shows the hodograph corresponding to the slip line field of Fig. 2. The lines om and ob represent the velocities of points m and b, respectively; 'b' is the point where the chip leaves contact with the tool.
Thus the radial
distance between these two velocity lines gives the chip thickness. city at inner radius would therefore be given by om. between 'm' and 'a' gives the uncut chip thickness.
Chip velo-
The horizontal distance It can be, therefore,
seen that other conditions remaining the same the chip-curl radius would increase with increase in the uncut chip thickness and also higher velocity of cutting would mean a higher chip-curl winding velocity as has been observed by Nakayama [i0], and Horne [9], for example. The cutting forces could be derived by determining the stress distribution along the face AB corresponding to the critical penetration and then integrating them over the tool-face.
It is possible to include the effect of
friction on the tool-face by including the appropriate tangential stress.
Frequency of Chip formation and Shear angle It is easily seen that, during the indentation,
the cutting force would in-
crease, reach a maximum value when the critical depth of indentation is reached and then suddenly drop as the material fails and the chip shears off.
Thus, if
c is the critical indentation and V one chip T = c/V c.
is cutting velocity, time for formation of c The frequency of chip formation therefore equals Vc/C, c
being directly proportional to uncut chip thickness.
The above statement is in
agreement with the general observations in metal cutting reported by Eugene [7] and Bizeaul and LeMaitre [4]. As a further observation one can easily see that the direction of flow of material during the early stages of deformation is ADEC and during the final formation of chip is AFJKLM, a cycle of chip formation. [1,2], Benerjee and Palmer
i.e., the orientation of slip-plane changes during This is in accord with observations of Albrecht
[3], and Pekalharing
[ii].
The formation of Chevron
pattern during inhomogeneous shearing of directionally solidified silver-copper eutectic by Cooke and Rice [6] can also be explained in the light of the change of shear angle during cutting.
Friction at Interface The effect of friction at the interface of the tool and the workpiece can be -~n,".n~.n',".~,,:::,r'l
~'n
th~ ~n],,~on tbroumh a chan~e in inclination of the slid line
INDENTATION AS BASIS FOR METAL CUTTING
with the tool face. to limiting value
381
For increasing frictional values, the angle '~' decreases of 0 °.
The over stressing of material around the wedge
imposes another constraint on the limiting value of '~'; this also depends on tool angle, 0.
In the case this constraint governs the angle '~', tool and a
portion of material adjacent to it do not move relative to each other.
This
portion of material can be looked upon as appendage to the tool and in the context of metal cutting referred to as built up edge. Other effects of friction on tool face could be studied from the slip line field and the hodograph.
A higher coefficient of friction gives higher value
of '~' and hence higher stresses at tool-work interface.
This also reduces
the angle between direction of cutting and shear line at the tool point.
The
average shear angle is therefore seen to reduce with increasing friction. Consequently the chip thickness would increase.
Such observations have been
made by Shaw [13]. Conclusion The present proposal, therefore, takes due account of the various observations in metal cutting.
The theory is also able to provide estimates of the cutting
forces, cutting ratio, frequency of chip formation and force fluctuation, chipcurl radius as well as incorporate the effects of friction to a limited extent. Of course the results obtained herein are limited in their accuracy because of assumed simplification in material properties and also the cutting condition. References i.
P. Albrecht, New Development in the Theory of Metal Cutting Process Part II, Vol. 83, pp. 557-571 (1961).
2.
P. Albrecht, Self Induced Vibration in Metal Cutting, Trans. ASME, JEI, Vol. 84, pp. 405-417 (1962).
3.
H. Banerjee, and W.B. Palmer, Metal Cutting with a Discontinuous Chip, Proc. of the Int. Mach. Tool Des., pp. 405-415 (1965).
.
5.
.
D. Bizeaul, and F. LeMaitre, A Study of the Periodic Deformation Produced on Machined Surfaces, Annal of CIRP, Vol. 24/1, pp. 39-41 (1975). N.H. Cook, P. Jhaveri, and N. Nayak, The Mechanism of Chip Curl and its Importance in Metal Cutting, Trans. ASME, Ser. B, Vol. 85, pp. 374-380 (1963). W.B.H. Cooke and W.B. Rice, Inhomogeneous Shearing during Continuous Chip Formation, Trans. ASME, Ser. B., JEI, Vol. 95, pp. 844-848 (1973).
382
C. SAHAY and R.N. DUBEY
7.
E. Eugene, Experimental Study on the Combined Influence of the Sharpening Angle, Microtecnic, Vol. XI, No. 5, pp. 223-230 (1957).
8.
R. Hill, A Theoretical Investigation of the Effect of Specimen Size in the Measurement of Hardness, Phil. Mag., Vol. 41, pp. 745-753 (1950).
9.
J.G. Horne, A New Model for Initial Chip Curl in Continuous Cutting, Int. J. Mech. Sci., Vol. 20, pp. 739-745 (1978).
i0.
K. Nakayama, Chip Curl in Metal Cutting Process, Bull. of the Faculty of Eng'g., Yokohama National Univ., Vol. ii, pp. 1-13 (1962).
ii.
A.J. Pekalharing, Discussion on Contribution a l'etude des Phenomenes Periodiques dans les Deformation Dynamics by LeMaitre, Bizeaul and Nell, Annals of CIRP, Vol. 23/2, pp. 317-318 (1974).
12.
C. Sahay and R.N. Dubey, A Theoretical Model for Incipient Plastic Deformation in Orthogonal Cutting submitted for publication.
13.
M.C. Shaw, Metal Cutting Principles, M.I.T. Press, pp. 3-16, 3-17 (1965).
14.
B. Worthington, and A.H. Redford, Chip Curl and the Action of the Groove Type Chip Former, Int. J. Mach. Tool Des. Res., Vol. 13, pp. 257-270 (1973).
Vol. 7(6),377-382,1980. Printed in the USA. Copyright (c) Pergamon Press Ltd
INDENTATION AS THE BASIS FOR METAL CUTTING
C. Sahay* and R.N. Dubey, Department of Mechanical Engineering, University of Waterloo, Waterloo, Ontario, Canada N2L 3GI
(Received 13 June 1980; accepted for print 25 July 1980)
Introduction
In a metal cutting operation, an external force acting through a mechanism induces relative velocity between the tool and workpiece. The force is applied on either the cutting tool or the workpiece while the other is generally held at rest. The amount of force required depends on the material under deformation (material properties of the metal, width and thickness of cut, etc.) and, to some extent, on the geometry of the tool and not on the mechanism of the force application. The process itself has, however, associated with it certain characteristics like cyclic variation in the cutting force and the associated frequency, the tendency of the chip to curl, the cutting ratio, etc. A real metal cutting operation is very complex. It involves the material constitutive property, high temperature, tool geometry, rigidity of tool and machine and the vibratory motion, etc. Any attempt for presenting a theory of metal cutting has to take into account all these factors and the resulting theory is likely to be a very complex one. However, any reasonable theory under any simplified situation, should be able to estimate the process characteristics in one unified theory and the present paper is a step in this direction. To facilitate the understanding of the cutting mechanism, we make some simplifying assumptions: We ignore the machine tool or cutting tool vibration and assume rigid-plastic behaviour for the workpiece. In our presentation we take advantage of the similarity in physics of indentation, shearing deformation and cutting.
Slip Line Field, Hodograph and Force Calculation Hill [8] considered the problem of indentation of wedge on specimen of finite width and thickness.
For small depths of indentation of the wedge into the
workpiece, the deformed material is displaced from under the wedge onto the
*On leave of absence from Indian School of Mines, Dhanbad (India) 377
378
C. SAHAY and R.N. DUBEY
the side and forms a coronet.
When the indentation increases the deformation
zone expands to touch either the bottom or the side of the workpiece depending on the width to depth ratio.
Whereas in the problem of shearing the critical
thickness determines the deformation zone, in metal cutting the width determines the deformation zone.
Sahay and Dubey [12] investigated
of a finite media having finite width by an unsymmetrical tool.
the indentation Fig. i shows
the slip line field associated with plastic deformation on account of indentation by an unsymmetrical wedge.
The unsymmetrical wedge may be assumed to
represent a metal cutting tool if half wedge angle of one side reduces to zero. Fig. 2 shows an indentation by an unsymmetrical wedge, modified to become a metal cutting tool, in a workpiece of finite width.
The tool works on the
material near one lateral face reducing indentation to a cutting situation. The equilibrium of the portion NCK'KIM, which is rigid, determines the length 'b' of segment JK, radius shear-plane.
'r' of sector KL and length 'a' of segment LM on the
Portion AF and FJ follow from the deformation in earlier stages.
The fact that line AFJKLM represents an s-line in the slip line field together with stress-free boundary at M is utilized in determining the orientation of AF with respect to AD.
C'
3'
,i
C
D'
FIG. i Slip line field in the deformation zone due to indentation by unsymmetrical wedge.
INDENTATION AS BASIS FOR METAL CUTTING
379
N
/ L
C
/
K
So
A
(i)
(ii) FIG. 2
(i) (ii)
Indentation by modified unsymmetrical wedge near the lateral surface of a workplece. Sllp line field for the above.
tc
b,eok' llj,k
"~,
P
a,foj,lk,l,m Vc
=, 0
FIG. 3 Hodograph for slip line field of Figure 2.
M
380
C. SAHAY and R.N. DUBEY
Fig. 3 shows the hodograph corresponding to the slip line field of Fig. 2. The lines om and ob represent the velocities of points m and b, respectively; 'b' is the point where the chip leaves contact with the tool.
Thus the radial
distance between these two velocity lines gives the chip thickness. city at inner radius would therefore be given by om. between 'm' and 'a' gives the uncut chip thickness.
Chip velo-
The horizontal distance It can be, therefore,
seen that other conditions remaining the same the chip-curl radius would increase with increase in the uncut chip thickness and also higher velocity of cutting would mean a higher chip-curl winding velocity as has been observed by Nakayama [i0], and Horne [9], for example. The cutting forces could be derived by determining the stress distribution along the face AB corresponding to the critical penetration and then integrating them over the tool-face.
It is possible to include the effect of
friction on the tool-face by including the appropriate tangential stress.
Frequency of Chip formation and Shear angle It is easily seen that, during the indentation,
the cutting force would in-
crease, reach a maximum value when the critical depth of indentation is reached and then suddenly drop as the material fails and the chip shears off.
Thus, if
c is the critical indentation and V one chip T = c/V c.
is cutting velocity, time for formation of c The frequency of chip formation therefore equals Vc/C, c
being directly proportional to uncut chip thickness.
The above statement is in
agreement with the general observations in metal cutting reported by Eugene [7] and Bizeaul and LeMaitre [4]. As a further observation one can easily see that the direction of flow of material during the early stages of deformation is ADEC and during the final formation of chip is AFJKLM, a cycle of chip formation. [1,2], Benerjee and Palmer
i.e., the orientation of slip-plane changes during This is in accord with observations of Albrecht
[3], and Pekalharing
[ii].
The formation of Chevron
pattern during inhomogeneous shearing of directionally solidified silver-copper eutectic by Cooke and Rice [6] can also be explained in the light of the change of shear angle during cutting.
Friction at Interface The effect of friction at the interface of the tool and the workpiece can be -~n,".n~.n',".~,,:::,r'l
~'n
th~ ~n],,~on tbroumh a chan~e in inclination of the slid line
INDENTATION AS BASIS FOR METAL CUTTING
with the tool face. to limiting value
381
For increasing frictional values, the angle '~' decreases of 0 °.
The over stressing of material around the wedge
imposes another constraint on the limiting value of '~'; this also depends on tool angle, 0.
In the case this constraint governs the angle '~', tool and a
portion of material adjacent to it do not move relative to each other.
This
portion of material can be looked upon as appendage to the tool and in the context of metal cutting referred to as built up edge. Other effects of friction on tool face could be studied from the slip line field and the hodograph.
A higher coefficient of friction gives higher value
of '~' and hence higher stresses at tool-work interface.
This also reduces
the angle between direction of cutting and shear line at the tool point.
The
average shear angle is therefore seen to reduce with increasing friction. Consequently the chip thickness would increase.
Such observations have been
made by Shaw [13]. Conclusion The present proposal, therefore, takes due account of the various observations in metal cutting.
The theory is also able to provide estimates of the cutting
forces, cutting ratio, frequency of chip formation and force fluctuation, chipcurl radius as well as incorporate the effects of friction to a limited extent. Of course the results obtained herein are limited in their accuracy because of assumed simplification in material properties and also the cutting condition. References i.
P. Albrecht, New Development in the Theory of Metal Cutting Process Part II, Vol. 83, pp. 557-571 (1961).
2.
P. Albrecht, Self Induced Vibration in Metal Cutting, Trans. ASME, JEI, Vol. 84, pp. 405-417 (1962).
3.
H. Banerjee, and W.B. Palmer, Metal Cutting with a Discontinuous Chip, Proc. of the Int. Mach. Tool Des., pp. 405-415 (1965).
.
5.
.
D. Bizeaul, and F. LeMaitre, A Study of the Periodic Deformation Produced on Machined Surfaces, Annal of CIRP, Vol. 24/1, pp. 39-41 (1975). N.H. Cook, P. Jhaveri, and N. Nayak, The Mechanism of Chip Curl and its Importance in Metal Cutting, Trans. ASME, Ser. B, Vol. 85, pp. 374-380 (1963). W.B.H. Cooke and W.B. Rice, Inhomogeneous Shearing during Continuous Chip Formation, Trans. ASME, Ser. B., JEI, Vol. 95, pp. 844-848 (1973).
382
C. SAHAY and R.N. DUBEY
7.
E. Eugene, Experimental Study on the Combined Influence of the Sharpening Angle, Microtecnic, Vol. XI, No. 5, pp. 223-230 (1957).
8.
R. Hill, A Theoretical Investigation of the Effect of Specimen Size in the Measurement of Hardness, Phil. Mag., Vol. 41, pp. 745-753 (1950).
9.
J.G. Horne, A New Model for Initial Chip Curl in Continuous Cutting, Int. J. Mech. Sci., Vol. 20, pp. 739-745 (1978).
i0.
K. Nakayama, Chip Curl in Metal Cutting Process, Bull. of the Faculty of Eng'g., Yokohama National Univ., Vol. ii, pp. 1-13 (1962).
ii.
A.J. Pekalharing, Discussion on Contribution a l'etude des Phenomenes Periodiques dans les Deformation Dynamics by LeMaitre, Bizeaul and Nell, Annals of CIRP, Vol. 23/2, pp. 317-318 (1974).
12.
C. Sahay and R.N. Dubey, A Theoretical Model for Incipient Plastic Deformation in Orthogonal Cutting submitted for publication.
13.
M.C. Shaw, Metal Cutting Principles, M.I.T. Press, pp. 3-16, 3-17 (1965).
14.
B. Worthington, and A.H. Redford, Chip Curl and the Action of the Groove Type Chip Former, Int. J. Mach. Tool Des. Res., Vol. 13, pp. 257-270 (1973).