- Email: [email protected]
Elastic effects in metal cutting chip formation
Int. J. Mech. Sci. Vol. 22, pp. 457--466 Pergamon Press Ltd., 1980. Printedin Great Britain
ELASTIC EFFECTS IN METAL CUTTING CHIP FORMATION T. H. C. CHILDS School of Mechanical and Manufacturing Systems Engineering, University of Bradford, Bradford, BD7 IDP, England (Received 31 March 1980; in revised form 15 May 1980) Summary--A brief review of existing slip-line field theories and experimental studies of chip formation shows the theories to be inadequate because of their neglect of the elastic contact between the chip and tool which usually occurs beyond the plastically stressed contact region. It is shown how elastic contact effects can be incorporated into existing theories to obtain better agreement between theory and experiment. The main analytical method used is a simplification of Dewhurst's and Collins'ill matrix method of slip line field analysis whereby all generally curved slip-line field elements are replaced by circular arcs. The description of this simplification should be of more general ,interest than its particular application to metal cutting theory.
f l t k p r ~" A a th
NOTATION feed or undeformed chip thickness contact length between chip and tool chip thickness shear yield stress hydrostatic pressure mean friction stress between chip and tool 0.5 cos -~ (~/k) friction angle between chip and tool tool rake angle shear plane angle
~b [ slip line field angles, 0 [ defined in Fig. 1 8 I A, B, C, D, E labels on the slip-line fields a, b, c, r slip-line field element radii Fc cutting force Fr thrust force Other symbols are explained in the text.
1. INTRODUCTION
It is well-known and discussed in detail later that when a metal is machined orthogonally to form a continuous chip without a built-up edge, neither the chip thickness ratio (t/f), the chip/tool contact length ratio (l/y) nor the friction angle ;t between the chip and tool is uniquely determined by the tool rake angle a and the friction stress ratio (z/k). Slip-line field theories in which the metal is assumed to behave in a rigid-perfectly plastic manner have been developed by Lee and Shaffer[2], Kudo [3] and most recently by Dewhurst [4] to establish the ranges of allowable values of (l/f), (t/f) and A for particular values of (r/k) and a but in a significant number of experiments these quantities have been found to fall outside the ranges so determined. The inability of the rigid-perfectly plastic model to incorporate a shear yield stress varying with strain, strain-rate and temperature is believed by many, notably Oxley et al. [5], to be the main shortcoming of the model. The possible influences of elastic effects on chip formation have received little attention. Yet photoelastic[6], splittool [7], and visio-plasticity[8] experiments show that beyond the region of contact between a chip and a tool over which the chip is plastically stressed there usually exists an extensive region of elastic contact, the forces acting across which can account for up to 40% of the total force on the tool. This paper considers the influence MS Vol. 22. No. 8---A
457
458
T . H . C . CHILDS
of these elastic forces on chip formation, while maintaining the assumption of a plastic yield stress independent of strain, strain-rate or temperature. In the r i g i d - p e r f e c t l y p l a s t i c slip-line field t h e o r i e s r e f e r r e d to a b o v e , t h e r e s u l t a n t f o r c e a n d m o m e n t a c t i n g o n t h e field b o u n d a r y s e p a r a t i n g t h e rigid c h i p f r o m t h e p l a s t i c a l l y s t r e s s e d r e g i o n a r e t a k e n to be z e r o . (This is t h e f r e e - c h i p b o u n d a r y c o n d i t i o n . ) In t h e p r e s e n t w o r k a r a n g e of f o r c e s a n d m o m e n t s a r e a s s u m e d to act e x t e r n a l l y on t h e chip, a s s o c i a t e d w i t h a r a n g e o f a r b i t r a r y b u t r e a s o n a b l e e l a s t i c c o n t a c t s t r e s s d i s t r i b u t i o n s . Slip-line fields in e q u i l i b r i u m w i t h t h e s e f o r c e s a n d m o m e n t s h a v e b e e n o b t a i n e d a n d it is s h o w n t h a t t h e r a n g e s o f a l l o w a b l e v a l u e s of (I/f), (t[f) a n d A w h i c h r e s u l t f r o m t h e s e a r e e n l a r g e d to e n c o m p a s s e x p e r i m e n t a l r e s u l t s w h i c h c o u l d n o t b e e x p l a i n e d b y rigid m o d e l s . N o a t t e m p t has b e e n m a d e to s t u d y w h e t h e r t h e a s s u m e d e l a s t i c c o n t a c t c o n d i t i o n s a r e c o m p a t i b l e w i t h t h e flow o f t h e c h i p a n d in t h a t s e n s e t h e w o r k is t h e o r e t i c a l l y i n c o m p l e t e . T h e m a t h e m a t i c a l m e t h o d o f this p a p e r is a simplified d e v e l o p m e n t o f t h e m a t r i x a n a l y s i s u s e d b y D e w h u r s t to s t u d y f r e e - c h i p f o r m a t i o n [4]. A s e c o n d a r y p u r p o s e o f this p a p e r is to d e s c r i b e the simplification. T h i s is p a r t l y t h e s u b j e c t o f t h e n e x t s e c t i o n . A l s o in that s e c t i o n o t h e r r i g i d - p e r f e c t l y p l a s t i c slip-line field t h e o r i e s o f c h i p f o r m a t i o n a r e briefly r e v i e w e d a n d all are c o m p a r e d w i t h e x p e r i m e n t , so t h a t t h e y m a y later b e c o n t r a s t e d with the e l a s t i c - p e r f e c t l y p l a s t i c a n a l y s i s . 2. RIGID-PERFECTLY PLASTIC SLIP-LINE FIELD THEORIES (2.1) The field variables The development of slip line field theories of free chip formation is summarised in Fig. I. Fig. l(a) in which various machining parameters are also defined shows the earliest field, proposed by Lee and Shaffer[2]. Its boundaries AC, BE and CE are straight lines. AB and BE are required to be equal for equilibrium reasons. The hydrostatic pressure is uniform throughout the field and is equal, for a free chip, to k. The only variable field angle is ~ which is determined by the rake face friction stress ratio (r/k), assumed to be constant over CE: ('c/k) = cos 2~'. Thus, for this field, each of the non-dimensional machining parameters such as (l/f), (t/f) and )t is uniquely determined by ('c/k) and a. An extension of Fig. l(a), introduced by Kudo[3], is shown in Fig. l(b). The slip line AB of Fig. l(a) is replaced by ABD, a centred-fan field of angular width 8. The boundary BC is constructed by the application of Hencky's theorems and the retained assumption of a uniform friction stress over CE. Equilibrium of the free chip still requires AD and DE to be of equal length and the hydrostatic pressure on them to equal k. However the extra field variable ~5removes the unique relationship between the non-dimensional machining parameters and ('c/k) and a. Kudo[3] also introduced the distortion, Fig. l(c), of Fig. l(a) to describe the formation of a curved chip caused by a variable friction stress on the rake face. Compatibility requires AB to be a circular arc, equilibrium that BE is identical to AB and compatibility again that the shape of ABE and its image in the hodograph plane differ only by a scale factor. Analysis in the physical and hodograph planes shows that the
• / A
/ ~
E
A
cA/0
Al
A
E
0
'~
FIG. 1. Metal cutting slip-line fields: a, after Ref. [2]; b-d, after Ref. [3]; c, after Ref. [4].
Elastic effects in metal cutting chip formation
459
friction stress distribution between C and E and the angular extent 0 of A B and B E is determined by the values of ~" at C and E and that the radius ratio (dr) of A B and B E depends in addition on ix. The hydrostatic pressure which varies throughout the field is also determined by ~" at C and E. At A, for example, PA/k varies from 1.0 to 1.013 depending on the value of 0. For this type of field a unique relation therefore exists between the machining parameters, tx and the friction stress distribution over CE. However there is no unique relationship involving the average friction stress on CE because the same average can be obtained from several different distributions of friction stress. Finally, Kudo combined his fields lb and lc to give that of Fig. l(d). Most recently, Dewhurst[4] has returned to the study of chip formation under conditions of constant friction stress along CE and has introduced a further extension and distortion of Lee and Shaffer's field. This is shown in Fig. l(e). A fan element BCD centered on C is seen. A B is required by compatibility to be a circular arc but the curvatures of the remaining elements BC, CD, BD and D E are allowed to vary along their lengths. The angular extent of BC is required by Hencky's first theorem to equal that of CD and that of CD, by the constant rake face friction stress boundary condition, to equal that of DE. However the angular extents 0, $ and )1 of AB, BD and DE are allowed to differ. Dewhurst, developing his and Collins'J1] earlier matrix technique for the numerical solution of slip-line field problems, shows that the stress and compatibility conditions of the machining problem are such that the field shape is completely determined by the four angles ~', 0, $, and 71. Further, for a given value of ~"there exists a range of solutions (0, ~k, ~) each of which together with a particular value of (PA/k) give a force and moment free chip. For these fields, therefore, no unique relation exists between the machining parameters, (*lk) and a. Dewhurst records only a limited number of results in his paper. In the next paragraphs an approximate form of his analysis is presented from which further results are derived and which is later developed to incorporate elastic contact effects. (2.2) An approximate form of Dewhurst's analysis Consider the approximate form of Dewhurst's field shown in Fig. 2 in which the curved elements DE, DC, B C and B D are replaced by circular arcs of radii a, b, c and d respectively and the radius of A B is written r. When 7) and $ are small c~-b
(1)
d ~- ~/b
(2)
D ' E = a)l + (b~2)/2
(3)
D' C = b~ - (a~2)12
(4)
D'CID'E = tan ~'.
(5)
and
b may be related to a by noting that
and that
y, Fy ~~
-:,,.
~
Y/
C FIG. 2. A circular arc approximation to Fig. l(e).
460
T . H . C . CHILDS It follows from (3), (4) and (5) that
b/a = (t + "0/2)/(! - "0t/2)
(6)
where t = tan ~. a may be related to r by noting, after Dewhurst, that both the shape of A B D E and of CDE is the same as its image in the hodograph plane. It may hence be shown that
a/r = g/( l
-
"0~bg~)
(7)
where g has been written for the r.h.s, of equation (6). The shape of the field thus depends only on the friction parameter r and the field angle variables "0, 4, and 0. Approximate force and moment relations in the approximate slip-line field are readily written down. Consider, for example, the forces and moment Fx, F, and M acting on unit width of the chip through the element A B of the field where the cartesian coordinates x and y are chosen, as shown in Fig. 2, respectively parallel and perpendicular to the tangent to A B at A (the so-called Mikhlin coordinate system) and M is the moment about B. Series expansions given by Dewhurst and Collins[I] may be truncated to give
Fx/kr ~ (x/r) + (pB/k )(y/r) + 03/3
(8)
F d k r = ( - y / r ) + (pB/k )(x/r) + 02
(9)
M/kr 2 ~ (p,/k)(y/r) + (5/3)(x/r)(y/r)
(10)
where x and y are the projected lengths of A B in the x- and y-directions:
xlr~ 0
(11)
y/r ~ 02/2
(12)
and p , is the hydrostatic pressure at B which, because A B is a /3 slip-line, may alternatively be written (PA - 2 k O ) where PA is the hydrostatic pressure at A. Similar expressions for the force and moment components acting across B D and D E may also be written down and all may then be suitably combined to obtain expressions for the resultant non-dimensional force and moment on the chip in terms of (pA/k), ~, "0, t# and 0. A computer programme has been written to find the combinations of these parameters which yield a free chip. The programme is identical in principle to Dewhurst's[4], using the Powell optimisation routine in the NAG library. However the approximations which have been described give rise to a drastically shortened programme. The results of the computations are shown graphically as the solid lines in Fig. 3 in which the variations of "0, ~ and 0 with (pa/k) a r e shown separately for values of (r/k) of 0.98, 0.70, 0.36 and 0.01. For each value of ('dk) the values of "0, ~ and O plotted against (pa/k) have a common turning point which defines a minimum possible value of (pa/k). This minimum value varies with (,r/k) from 0-57 when (r/k) = 0.98-0.77 when (z[k)=O.Ol. These results are in close agreement with Dewhurst's who records that minimum possible values of (pA/k) vary from 0.56 to 0.77 when (z[k) varies from 1.0 to 0.
~[ /IT/kl,0.98 ,1,
//
~
4,
).98
0-7 a,O**
/__o, ~ro
0
/iI/0"010"6
0.6
0.70 O'36 0"01
./,;//
/ 0-4 |
0.70
0.2 ~
2
0.1
~
o
o:7
0-4
O,4
0.3 O'2 0.1
.1
o:6
o'-7 IpA/kl
° 0"6
0.7
0.8
0"9
FIG. 3. The dependence of '0, ~band 0 on pA/k for sr = 0.1 (r/k = 0.98) to sr = 0.78, (r/k = 0.01 t, also showing by hatching the solution limits for ct = 0 and 30°.
Elastic effects in metal cutting chip formation
461
Dewhurst gives detailed results for the variation of field angles with (pA/k) when (r/k) = 0.5. These are included as the dotted lines in Fig. 3. It may be inferred by interpolation that the approximate and exact analyses are in good agreement over the lower branches of the curves but as rl, 4, and 0 increase with (pa/k) along the upper branches the approximate analysis, while remaining good in its calculation of ~O, increasingly overestimates 77 and underestimates 0, by up to 10% as the bounds of the values included in Fig. 3 are approached. It is not surprising that the approximate analysis deteriorates as rl, 4, and 0 increase as the basis of the approximations is that these angles are small. (2.3) The ranges of validity of the 1ields For all the fields of Fig. I the regions between the slip-lines A B and the free surface of the workpiece and between the slip-lines A B or, for fields lb and ld, the slip-lines A D and the free surface of the chip are required to be rigid. This imposes limits on the allowable values of the field angles. For the fields of Figs. l(a), (c) and (e), following the work of Hill[9], it can be shown that
(pA/k)<~2y + 1 -- 3 r / 2
(13)
2 cos (/3 - ~'/4) - 1 ~<(pa/k) <~2/3 + 1 - ~r/2
(14)
and
for the fields to be valid where y is the angle between the free surface of the workpiece and the tangent to A B at A and /3 is that between the chip free-surface and the tangent to A B at A. If (13) is violated the workpiece becomes overstressed and if (14) is violated, the chip becomes overstressed. For the fields of Figs. l(b) and (d), equation (14) remains valid provided PA is regarded as the hydrostatic stress on A D at A and /3 as the angle between the chip free-surface and the tangent to A D at A. In that case equation (13) must be modified to
(pA/k) <- 2y -- 28 + 1 - 3~r]2.
(15)
Calculations in the next section indicate that for a given rake face mean friction stress extreme values of the machining parameters are associated with Kudo's straight chip fields, Fig. l(b), and Dewhurst's fields. Kudo's fields are always limited by overstressing of the workpiece. Remembering that pA/k = 1.0 for Kudo's field equation (15) gives the limit to 8 as 8 ~<0.393 - (r + a)/2.
(16)
Dewhurst's fields may be limited by either equation (13) or (14). In Fig. 3 the hatched regions show the limits to the field angles calculated from the approximate analysis for rake angles of 0 and 30*. For each value of ('dk), the lower limit to the field angles is zero (i.e. Lee and Shaffer's solution), unless a > (¢r/4 - r) in which case (see Dewhurst) the lower limit is determined by equation (13). The upper limit is usually determined by the left-hand inequality of equation (14) but for large values of ('dk) and small values of a volume conservation can impose a more severe constraint. Volume conservation requires that point A of the slip-line field lies above the tool tip C: in Fig. 3, the lower branch of the limiting boundary for a = 0 is determined by volume conservation. A comparison of the limiting values shown in Fig. 3 with those published in Ref. [4] again shows good agreement between the approximate theory and Dewhurst's analysis. (2.4) Non-dimensional machining parameters For all the fields of Fig. !, non-dimensional machining parameters can be calculated from (z/k), a, the field angles and (pa/k). In the most simple case, Fig. l(a), they depend only on ('dk) and a ; in addition it can readily be shown that three of them, (t[l), (da - a) where ~b is the shear plane angle, and ;t depend only on ('dk) while the other parameters such as (t/f), (Ill) Fc/(fk) and F74(fk) can be expressed in terms of them and a :
(t/l) = cos ~r(cos r + sin ~') / (~-a)=¢ ,~ = ~-/4- ~r j
(t/f)
=
(17)
cos (~ - a ) / s i n (d' - a + a )
(I/f) = (t/f)/(t/l) Fc/(fk) = cos (A - a)/[sin (4' - a + a ) cos (d' - a + ~t)] Fr/(fk) = [Fc/(fk)] tan (A - a )
(18)
Equations (18) depend on the field variables only through equations (17) and are valid for all the fields of Fig. I hut equations (17) vary with the type of field: for Dewhurst's field for example ((t[l), ((a- a) and h. depend on rl, 4,, 0 and (PA/k) as well as on (r/k) and are computed from the composition of equations (8), (9), (I 1) and (12) and the equivalent equations for the elements BC, BD and DE of the field. However for all the fields (t/I), (~b- t~) and A remain independent or almost independent of a ; the calculation of the values of these for the various slip-line fields most compactly enables the predictions of the field theories to be considered. Figs. 4(a), 5(a) and 6(a) are a partial record of such calculations in which the range of values of (t/I), (d~ - a) and A allowed by equations ( 1 3 ) , ( 1 4 ) o r ( 1 5 ) are shown by cross-hatching for two values of
462
T . H . C . CHILDS
II1
2"0 k
2"0
1.6
1.6
1.2
1.2
0.8
~
0-4
•
-0-5
£-0"1
0"8
db qb~Sreee~ e
0"4
i,I. -i n=5 2
O
l
i
0"2
0"4
i
i
i
0-6
0"8
1"0
5
1 i
o:2
0"6
' 0"4
{T/k]
0"8
1.'0
[rVk]
FIG. 4. The allowable ranges of (t/l) and (r/k): a, for a rigid-plastic material; b, allowing for elastic contact between the chip and tool: • and O, my own data; [] from Ref. [10]; [] from Ref. [ll]. a
b
!
/
"~. ". .
• o~ /
.4
""
/ •
o
"'y
• ~
20 t
%. :o
.
b
40
"H
20
, , ~ ~°o~.~' .. . "
, /'b~.,:"
SIIIII sis~ / 0 ,, "
•
~6
0~8
1:0
0
~ ,,0-5
0.2'
0.4 '
[~/k]
0.1
0.2
0---6
0-8 '
1-0 '
l~k]
FIG. 5. The allowable ranges of )t and "r/k: a and b, hatching and symbols as for Fig. 4.
II I,p-al
•o ~
~.o.
3O
30
0
20
0.1
,, 0-5 0-2
40
I
•
10 rig2
I QO
•
°o°q6 1 rim2
-%
1'o
~o 3'0 ,o .t °
' 50
-10
0
~o
0
~6
36
) 40
50
20
FIG. 6. The allowable range of (d~ - a ) and A: a and b, hatching and symbols as for Fig. 4.
Elastic effects in metal cutting chip formation
463
rake angle, 0 and 30°. Figs. 4(b), 5(b) and 6(b) are the subject of Section 3 of this paper, while the experimental results are considered in Section (2.5). Fig. 4(a) shows the variation of the allowable range of (t/l) with (z/k). The dashed line is the unique relation predicted by Lee and Shaffer. For both rake angles, the upper limit to (tll) is provided by Dewhurst's fields, constrained by equation (14). For a = 0 the lower limit to (t]l) is provided by Kudo's field Fig. l(b) constrained by equation (15) while for c~ = 30O the lower limit is also provided by Kudo's field while that limit is less than the Lee and Shaffer prediction (i.e. for (z/k) > 0.87) but for (r/k) < 0.87 the lower limit is obtained from Dewhurst's field constrained by equation (13). Fig. 5(a) shows the allowable variation of ;t with (r/k) subject to the same constraints as those which led to Fig. 4(a): a relation much closer to unique is predicted between A and (r/k) than between (t[I) and
(r/k).
Rather than illustrating the allowable relations between (~b- a) and (z/k) Fig. 6(a) shows the solution range for ( ~ b - a ) and A because this approaches the familiar manner of recording machining data of plotting ~b against (3. - a). The non-uniqueness between (• - a) and A is marked, the solution ranges being limited by the same constraints that gave Fig. 4(a). (2.5) Comparisons with experiment The shortcomings of the rigid-perfectly plastic theories are most readily illustrated by referring to experiments on the machining of steels at speeds above those which give built-up-edge formation, for which much data exists. The experimental results in Figs. 4, 5 and 6 have been collected from my own work, Zorev's[10] and that of Eggleston, Herzog and Thomsen [l l], (other worker's results have also been reviewed and are in agreement with those presented here). The solid symbols refer to experiments on non-free cutting carbon and alloy steels and the open symbols to tests on re-sulpherised, so called free-cutting, steels. In my own tests (circular symbols) the contact length I has been assessed from series of experiments with restricted contact tools, noting the longest restricted length for which small reductions in length influenced chip formation. Zorev and Eggleston and his co-workers observed the length over which chip material adhered to the rake race of the tool. The data encompasses tests for which 0 < a < 30°. It is clear that only with respect to the relationship between (th - c~) and ,~ do the experimental results lie mainly within the rigid-perfectly plastic solution ranges. When machining parameters which involve the contact length are considered (z as well as (t/l) is such a parameter as its calculation requires an estimate of I) the rigid-perfectly plastic theory is inadequate. 3. ELASTIC-CONTACT EFFECTS Fig. 7(a) shows the main model used to introduce elastic effects into the analysis of chip formation. The contact length CF of total length (1 + n)l is considered to be composed of a plastic portion CE of length l and an elastic part EF of length nl. (z/k) is assumed constant over CE and the normal pressure (pn/k) is governed by the Dewhurst-type slip-line field ABCDE. Over EF, (z/k) and (pn/k) are arbitrarily assumed to decrease parabolically from their values at E to zero at F, giving rise to a resultant force R. (The parabolic variation accords roughly with observation but the analysis to be described could readily be repeated with other assumed distributions.) The programme written to find PA/k, r/, ~ and 0 for free chip formation (Section 2.2) was adapted to find, for a range of r and n, the values of pA]k, 77, $ and 0 which would give rise to a resultant force over ABDE equal and opposite to the elastic-generated R. As an example, to be compared with Fig. 3(b), the solid lines of Fig. 8 show the calculated variation of ~b with (pa]k) for r = 0.1 and for values of n from 0 to 20 while the hatched areas define the allowable solution ranges for a = 0 and 30°. The incorporation of elastic effects
Pn
FIG. 7. a, the main model incorporating elastic contact effects; b, a further model when ~b= 0.
T. H. C. CHILDS
464
0.;t. |n 0
2 3.255
lilt/
8
12
20
::O:oo
0.1
0 0-5
0"7
0-9
a 1.1 •
1.3
1.5
1.7
1.9
2-1
IpA/kl FIG. 8. The dependence of ~b on pA/k for r = 0' ! and values of the elastic parameter n from 0 to 20, also showing by hatching the solution limits for a = 0 and 30 °.
is seen to displace the relation between qJ and (pA/k) to both higher th and higher (pA/k) values. Qualitatively similar shifts occur in the values of rl and 0. The boundary of the solution range, for example abcde for a =30 °, is determined by various constraints: over ab, n = 0; over bc the left-hand inequality and over cd the right-hand inequality of equation (14) limits the solutions; the position of de is determined by equation (13); the requirement that ~b/>0 gives rise to the part ea of the boundary. Further slip-line fields can be developed from the solutions for which q~= 0 by allowing the slip-line A B C (Fig. 7a) to develop into a fan, as shown in Fig. 7(b). These solutions are then limited by equation (15). The ranges of the machining parameters calculated from these new solutions are shown in Figs. 4(b), 5(b) and 6(b). In these the contact length I' is the sum of the plastic and elastic parts and the friction stress • ' is the average value over I'. In Fig. 4(b) the solution ranges for a = 0 and 30° are shown for r = 0-1 and 0.5 (i.e. for ('rlk) over the plastic contact length equal to 0.98 and 0-54). The allowable variations of A with (r'/k) depend only slightly on a: in Fig. 5(b) the dashed lines bound the entire solution range and in particular the hatched regions define the solution ranges for ~"= 0.1,0.2 and 0-5. In Fig. 6(b) the solution ranges for a = 0 and 30° are again shown for ~"= 0.1 and 0.5. In addition the upper boundary of the range for ~"= 0 and the lower boundary for r = 0-2 are also shown. In all three figures an indication is given by the lightly dashed lines of the relationships between the machining parameters and the elastic contact parameter n. The incorporation of elastic effects is seen in Fig. 4(b) to extend the solution range to lower values of (t/l') and in Fig. 5(b) to higher values of )t for each value of ('r'/k). The experimental results then fall within the allowable ranges. The only extension to the solution range in Fig. 6(b) occurs at high values of A where little experimental data lies. It is interesting to note however that a maximum value of A of 55° is predicted.
4. DISCUSSION AND CONCLUSIONS T h e e x i s t e n c e of an elastic c o n t a c t r e g i o n b e t w e e n the chip a n d tool has long b e e n r e c o g n i s e d e x p e r i m e n t a l l y b u t i g n o r e d in m e c h a n i c a l t h e o r i e s of chip f o r m a t i o n . This p a p e r s h o w s h o w elastic c o n t a c t f o r c e s c a n be i n c o r p o r a t e d into p e r f e c t l y plastic slipline field t h e o r i e s of chip f o r m a t i o n b y r e g a r d i n g t h e m as e x t e r n a l l y a p p l i e d f o r c e s r e l a t e d to the a s s u m e d length of elastic c o n t a c t . T h e m a i n a s s u m p t i o n is that the friction stress m a y be a s s u m e d to be c o n s t a n t o v e r the plastic c o n t a c t b e t w e e n the chip a n d tool a n d to d e c r e a s e p a r a b o l i c i a l l y o v e r the elastic c o n t a c t part a n d that the ratio b e t w e e n the n o r m a l p r e s s u r e o n the rake face a n d the f r i c t i o n stress is c o n s t a n t o v e r the elastic c o n t a c t region. G i v e n this, it is s h o w n that for a n y g i v e n c o n s t a n t f r i c t i o n stress o v e r the plastic c o n t a c t length the a d d i t i o n of a v a r i a b l e elastic c o n t a c t length e x t e n d s the m a c h i n i n g p a r a m e t e r s o l u t i o n r a n g e in a m a n n e r in a g r e e m e n t with e x p e r i m e n t (see Figs. 4-6). T h e a n a l y s i s is a p p r o x i m a t e . C i r c u l a r arc a p p r o x i m a t i o n s to the field c h a r a c teristics are m a d e in b o t h the p h y s i c a l p l a n e a n d the h o d o g r a p h b u t t h e s e h a v e b e e n s h o w n in S e c t i o n (2.2) to i n t r o d u c e little e r r o r into the c a l c u l a t i o n s . T h e e x i s t e n c e of a n elastic c o n t a c t l e n g t h implies a small c o m p o n e n t of plastic flow of the chip into the
Elastic effectsin metal cuttingchip formation
465
tool and hence a slight modification of the velocity boundary conditions in the plastic-rigid hodograph. This has been ignored as being likely to have only a small influence on forces and flow in the slip-line field compared to that of the circular arc approximations first mentioned. Variable friction stress over the plastic contact length has not been analysed mainly because it is not known how to incorporate such variation into Dewhurst's analysis but also because it is thought that it would not lead to any extension of the solution range: in the absence of elastic contact the solution range associated with the variable friction stress field of Fig. l(c) is much less than that associated with Dewhurst's field, Fig. l(e). The allowable elastic contact lengths have been found to be limited by overstressing criteria: for example in Fig. 8 no solution exists for n 1>20 if a = 30°, whereas for ~"= 0-5 and o~ = 0, n ~<5 (see Figs. 4b or 6b). In a complete theory there would be additional compatability limits on n: a tightly curled chip is likely to be associated with smaller lengths of elastic contact than straight chips. It is not known at present whether values of n as large as 20 ever exist in practice. For values of n < 2, Fig. 6(b) shows that the relationship between (~b- a) and A depends but little on n whereas Figs. 4(b) and 5(b) show n strongly influences the relationship between A and (r'/k) and (t[l') and (r'/k). This is important to the experimentalist. Good estimates of (r/k) over the plastic contact length can, if the theory is correct and n < 2 , be made from measurements of (d~-c~) and A but estimates made by dividing the rake face friction force by the measured total contact area are likely to give very different answers. Some of the experimental points in Fig. 5(b) are associated with n values between 2 and 5 and a number, mostly Zorev's, with values still greater. To some extent this may be the result of overestimating the contact length; this point is returned to later. However, it is clearly of the greatest interest to look more closely at what determines elastic contact length, to determine for example whether the values of ~ and n associated with the data of Figs. 4-6 vary understandably with speed, feed, material properties or with the machine tool structure. Only then will it be established if the present model is adequate or whether variable plastic friction stresses, non-parabolic elastic contact stress distributions or work-hardening effects as discussed by Oxley [5] or Childs and Rowe[12] still are significant. Nevertheless some preliminary comments on the data in Figs. 4-6 may be made. The interpretation of machining data may be made more or less easy by the manner of its display. Historical reasons have made it most usual to present the results of tests as the variation of ~b with (A - a ) . In Section (2.4) it is argued that the variations of (~b- a) and A as well as of (t/l) with (r/k) are more fundamental. The experimental results of Figs. 4-6, when interpreted in the light of the theory developed here, give weight to this. The data for non-free-cutting steels is seen to be clustered in regions of higher ~" values than that for free-cutting steels; generally the non-free-cutting steels are associated with plastic contact friction stresses > 0.9 k while equivalent values for the free-cutting steels are between 0.7 and 0-9k. This is qualitatively as might be expected. More detailed comments must await further work. It will only be noted here that the experimental interpretation of what is the total contact length is of central importance to the comparison of the theory and experiment. In Figs. 4--6 the data points fall into clusters not only depending on the type of steel cut but also on whether contact length was determined by restricted contact cutting tests or by visual inspection of the tool face. Restricted contact tests give shorter estimates of contact length than do visual inspections. Finally a comment may be made on the use of the numerical optimisation technique with circular arc slip-line field approximations for analysing other slip-line field problems. The method will be useful for those mixed problems like the present in which the equilibrium and velocity boundary conditions are coupled. Where they can be uncoupled simpler methods of solution retaining the circular arc approximation can be and have been used[13].
466 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
T . H . C . CHILDS REFERENCES P. Dewhurst and I. F. Collins, Int. J. Num. Meth. Engng 7, 357-78 (1973). E. H. Lee and B. W. Shaffer, J. Appl. Mech. 18, 405-13 (1951). H. Kudo, Int. J. Mech. Sci. 7, 43-55 (1965). P. Dewhurst, Proc. R. Soc. Lond. A360, 587-610 (1978). P. L. B. Oxley and W. F. Hastings, Phil. Trans. R. Soc. Lond. A252, 565-84 (1976). H. Chandrasekaran and D. V. Kapoor, Trans. A S M E Bg'/, 495-502, (1965). S. Kato, K. Yamaguchi and M. Yamada, Trans. A S M E B94, 683-9, (1972). T. H. C. Childs, Proc. I. Mech. E. Lond. 186 Ptl, 717-27 (1973). R. Hill, J. Mech. Phys. Solids 2, 278--85 (1954). N. N. Zorev, Metal Cutting Mechanics. Pergamon Press, Oxford (1966). D. M. Eggleston, R. Herzog and E. G. Thomsen, Trans. A S M E B81,263-79 (1959). T. H. C. Childs and G. W. Rowe, Rep. Prog. Phys. 36, 223-88 (1973). F. A. A. Crane and J. M. Alexander, J. Inst. Met. 96, 289-300 (1968).
ELASTIC EFFECTS IN METAL CUTTING CHIP FORMATION T. H. C. CHILDS School of Mechanical and Manufacturing Systems Engineering, University of Bradford, Bradford, BD7 IDP, England (Received 31 March 1980; in revised form 15 May 1980) Summary--A brief review of existing slip-line field theories and experimental studies of chip formation shows the theories to be inadequate because of their neglect of the elastic contact between the chip and tool which usually occurs beyond the plastically stressed contact region. It is shown how elastic contact effects can be incorporated into existing theories to obtain better agreement between theory and experiment. The main analytical method used is a simplification of Dewhurst's and Collins'ill matrix method of slip line field analysis whereby all generally curved slip-line field elements are replaced by circular arcs. The description of this simplification should be of more general ,interest than its particular application to metal cutting theory.
f l t k p r ~" A a th
NOTATION feed or undeformed chip thickness contact length between chip and tool chip thickness shear yield stress hydrostatic pressure mean friction stress between chip and tool 0.5 cos -~ (~/k) friction angle between chip and tool tool rake angle shear plane angle
~b [ slip line field angles, 0 [ defined in Fig. 1 8 I A, B, C, D, E labels on the slip-line fields a, b, c, r slip-line field element radii Fc cutting force Fr thrust force Other symbols are explained in the text.
1. INTRODUCTION
It is well-known and discussed in detail later that when a metal is machined orthogonally to form a continuous chip without a built-up edge, neither the chip thickness ratio (t/f), the chip/tool contact length ratio (l/y) nor the friction angle ;t between the chip and tool is uniquely determined by the tool rake angle a and the friction stress ratio (z/k). Slip-line field theories in which the metal is assumed to behave in a rigid-perfectly plastic manner have been developed by Lee and Shaffer[2], Kudo [3] and most recently by Dewhurst [4] to establish the ranges of allowable values of (l/f), (t/f) and A for particular values of (r/k) and a but in a significant number of experiments these quantities have been found to fall outside the ranges so determined. The inability of the rigid-perfectly plastic model to incorporate a shear yield stress varying with strain, strain-rate and temperature is believed by many, notably Oxley et al. [5], to be the main shortcoming of the model. The possible influences of elastic effects on chip formation have received little attention. Yet photoelastic[6], splittool [7], and visio-plasticity[8] experiments show that beyond the region of contact between a chip and a tool over which the chip is plastically stressed there usually exists an extensive region of elastic contact, the forces acting across which can account for up to 40% of the total force on the tool. This paper considers the influence MS Vol. 22. No. 8---A
457
458
T . H . C . CHILDS
of these elastic forces on chip formation, while maintaining the assumption of a plastic yield stress independent of strain, strain-rate or temperature. In the r i g i d - p e r f e c t l y p l a s t i c slip-line field t h e o r i e s r e f e r r e d to a b o v e , t h e r e s u l t a n t f o r c e a n d m o m e n t a c t i n g o n t h e field b o u n d a r y s e p a r a t i n g t h e rigid c h i p f r o m t h e p l a s t i c a l l y s t r e s s e d r e g i o n a r e t a k e n to be z e r o . (This is t h e f r e e - c h i p b o u n d a r y c o n d i t i o n . ) In t h e p r e s e n t w o r k a r a n g e of f o r c e s a n d m o m e n t s a r e a s s u m e d to act e x t e r n a l l y on t h e chip, a s s o c i a t e d w i t h a r a n g e o f a r b i t r a r y b u t r e a s o n a b l e e l a s t i c c o n t a c t s t r e s s d i s t r i b u t i o n s . Slip-line fields in e q u i l i b r i u m w i t h t h e s e f o r c e s a n d m o m e n t s h a v e b e e n o b t a i n e d a n d it is s h o w n t h a t t h e r a n g e s o f a l l o w a b l e v a l u e s of (I/f), (t[f) a n d A w h i c h r e s u l t f r o m t h e s e a r e e n l a r g e d to e n c o m p a s s e x p e r i m e n t a l r e s u l t s w h i c h c o u l d n o t b e e x p l a i n e d b y rigid m o d e l s . N o a t t e m p t has b e e n m a d e to s t u d y w h e t h e r t h e a s s u m e d e l a s t i c c o n t a c t c o n d i t i o n s a r e c o m p a t i b l e w i t h t h e flow o f t h e c h i p a n d in t h a t s e n s e t h e w o r k is t h e o r e t i c a l l y i n c o m p l e t e . T h e m a t h e m a t i c a l m e t h o d o f this p a p e r is a simplified d e v e l o p m e n t o f t h e m a t r i x a n a l y s i s u s e d b y D e w h u r s t to s t u d y f r e e - c h i p f o r m a t i o n [4]. A s e c o n d a r y p u r p o s e o f this p a p e r is to d e s c r i b e the simplification. T h i s is p a r t l y t h e s u b j e c t o f t h e n e x t s e c t i o n . A l s o in that s e c t i o n o t h e r r i g i d - p e r f e c t l y p l a s t i c slip-line field t h e o r i e s o f c h i p f o r m a t i o n a r e briefly r e v i e w e d a n d all are c o m p a r e d w i t h e x p e r i m e n t , so t h a t t h e y m a y later b e c o n t r a s t e d with the e l a s t i c - p e r f e c t l y p l a s t i c a n a l y s i s . 2. RIGID-PERFECTLY PLASTIC SLIP-LINE FIELD THEORIES (2.1) The field variables The development of slip line field theories of free chip formation is summarised in Fig. I. Fig. l(a) in which various machining parameters are also defined shows the earliest field, proposed by Lee and Shaffer[2]. Its boundaries AC, BE and CE are straight lines. AB and BE are required to be equal for equilibrium reasons. The hydrostatic pressure is uniform throughout the field and is equal, for a free chip, to k. The only variable field angle is ~ which is determined by the rake face friction stress ratio (r/k), assumed to be constant over CE: ('c/k) = cos 2~'. Thus, for this field, each of the non-dimensional machining parameters such as (l/f), (t/f) and )t is uniquely determined by ('c/k) and a. An extension of Fig. l(a), introduced by Kudo[3], is shown in Fig. l(b). The slip line AB of Fig. l(a) is replaced by ABD, a centred-fan field of angular width 8. The boundary BC is constructed by the application of Hencky's theorems and the retained assumption of a uniform friction stress over CE. Equilibrium of the free chip still requires AD and DE to be of equal length and the hydrostatic pressure on them to equal k. However the extra field variable ~5removes the unique relationship between the non-dimensional machining parameters and ('c/k) and a. Kudo[3] also introduced the distortion, Fig. l(c), of Fig. l(a) to describe the formation of a curved chip caused by a variable friction stress on the rake face. Compatibility requires AB to be a circular arc, equilibrium that BE is identical to AB and compatibility again that the shape of ABE and its image in the hodograph plane differ only by a scale factor. Analysis in the physical and hodograph planes shows that the
• / A
/ ~
E
A
cA/0
Al
A
E
0
'~
FIG. 1. Metal cutting slip-line fields: a, after Ref. [2]; b-d, after Ref. [3]; c, after Ref. [4].
Elastic effects in metal cutting chip formation
459
friction stress distribution between C and E and the angular extent 0 of A B and B E is determined by the values of ~" at C and E and that the radius ratio (dr) of A B and B E depends in addition on ix. The hydrostatic pressure which varies throughout the field is also determined by ~" at C and E. At A, for example, PA/k varies from 1.0 to 1.013 depending on the value of 0. For this type of field a unique relation therefore exists between the machining parameters, tx and the friction stress distribution over CE. However there is no unique relationship involving the average friction stress on CE because the same average can be obtained from several different distributions of friction stress. Finally, Kudo combined his fields lb and lc to give that of Fig. l(d). Most recently, Dewhurst[4] has returned to the study of chip formation under conditions of constant friction stress along CE and has introduced a further extension and distortion of Lee and Shaffer's field. This is shown in Fig. l(e). A fan element BCD centered on C is seen. A B is required by compatibility to be a circular arc but the curvatures of the remaining elements BC, CD, BD and D E are allowed to vary along their lengths. The angular extent of BC is required by Hencky's first theorem to equal that of CD and that of CD, by the constant rake face friction stress boundary condition, to equal that of DE. However the angular extents 0, $ and )1 of AB, BD and DE are allowed to differ. Dewhurst, developing his and Collins'J1] earlier matrix technique for the numerical solution of slip-line field problems, shows that the stress and compatibility conditions of the machining problem are such that the field shape is completely determined by the four angles ~', 0, $, and 71. Further, for a given value of ~"there exists a range of solutions (0, ~k, ~) each of which together with a particular value of (PA/k) give a force and moment free chip. For these fields, therefore, no unique relation exists between the machining parameters, (*lk) and a. Dewhurst records only a limited number of results in his paper. In the next paragraphs an approximate form of his analysis is presented from which further results are derived and which is later developed to incorporate elastic contact effects. (2.2) An approximate form of Dewhurst's analysis Consider the approximate form of Dewhurst's field shown in Fig. 2 in which the curved elements DE, DC, B C and B D are replaced by circular arcs of radii a, b, c and d respectively and the radius of A B is written r. When 7) and $ are small c~-b
(1)
d ~- ~/b
(2)
D ' E = a)l + (b~2)/2
(3)
D' C = b~ - (a~2)12
(4)
D'CID'E = tan ~'.
(5)
and
b may be related to a by noting that
and that
y, Fy ~~
-:,,.
~
Y/
C FIG. 2. A circular arc approximation to Fig. l(e).
460
T . H . C . CHILDS It follows from (3), (4) and (5) that
b/a = (t + "0/2)/(! - "0t/2)
(6)
where t = tan ~. a may be related to r by noting, after Dewhurst, that both the shape of A B D E and of CDE is the same as its image in the hodograph plane. It may hence be shown that
a/r = g/( l
-
"0~bg~)
(7)
where g has been written for the r.h.s, of equation (6). The shape of the field thus depends only on the friction parameter r and the field angle variables "0, 4, and 0. Approximate force and moment relations in the approximate slip-line field are readily written down. Consider, for example, the forces and moment Fx, F, and M acting on unit width of the chip through the element A B of the field where the cartesian coordinates x and y are chosen, as shown in Fig. 2, respectively parallel and perpendicular to the tangent to A B at A (the so-called Mikhlin coordinate system) and M is the moment about B. Series expansions given by Dewhurst and Collins[I] may be truncated to give
Fx/kr ~ (x/r) + (pB/k )(y/r) + 03/3
(8)
F d k r = ( - y / r ) + (pB/k )(x/r) + 02
(9)
M/kr 2 ~ (p,/k)(y/r) + (5/3)(x/r)(y/r)
(10)
where x and y are the projected lengths of A B in the x- and y-directions:
xlr~ 0
(11)
y/r ~ 02/2
(12)
and p , is the hydrostatic pressure at B which, because A B is a /3 slip-line, may alternatively be written (PA - 2 k O ) where PA is the hydrostatic pressure at A. Similar expressions for the force and moment components acting across B D and D E may also be written down and all may then be suitably combined to obtain expressions for the resultant non-dimensional force and moment on the chip in terms of (pA/k), ~, "0, t# and 0. A computer programme has been written to find the combinations of these parameters which yield a free chip. The programme is identical in principle to Dewhurst's[4], using the Powell optimisation routine in the NAG library. However the approximations which have been described give rise to a drastically shortened programme. The results of the computations are shown graphically as the solid lines in Fig. 3 in which the variations of "0, ~ and 0 with (pa/k) a r e shown separately for values of (r/k) of 0.98, 0.70, 0.36 and 0.01. For each value of ('dk) the values of "0, ~ and O plotted against (pa/k) have a common turning point which defines a minimum possible value of (pa/k). This minimum value varies with (,r/k) from 0-57 when (r/k) = 0.98-0.77 when (z[k)=O.Ol. These results are in close agreement with Dewhurst's who records that minimum possible values of (pA/k) vary from 0.56 to 0.77 when (z[k) varies from 1.0 to 0.
~[ /IT/kl,0.98 ,1,
//
~
4,
).98
0-7 a,O**
/__o, ~ro
0
/iI/0"010"6
0.6
0.70 O'36 0"01
./,;//
/ 0-4 |
0.70
0.2 ~
2
0.1
~
o
o:7
0-4
O,4
0.3 O'2 0.1
.1
o:6
o'-7 IpA/kl
° 0"6
0.7
0.8
0"9
FIG. 3. The dependence of '0, ~band 0 on pA/k for sr = 0.1 (r/k = 0.98) to sr = 0.78, (r/k = 0.01 t, also showing by hatching the solution limits for ct = 0 and 30°.
Elastic effects in metal cutting chip formation
461
Dewhurst gives detailed results for the variation of field angles with (pA/k) when (r/k) = 0.5. These are included as the dotted lines in Fig. 3. It may be inferred by interpolation that the approximate and exact analyses are in good agreement over the lower branches of the curves but as rl, 4, and 0 increase with (pa/k) along the upper branches the approximate analysis, while remaining good in its calculation of ~O, increasingly overestimates 77 and underestimates 0, by up to 10% as the bounds of the values included in Fig. 3 are approached. It is not surprising that the approximate analysis deteriorates as rl, 4, and 0 increase as the basis of the approximations is that these angles are small. (2.3) The ranges of validity of the 1ields For all the fields of Fig. I the regions between the slip-lines A B and the free surface of the workpiece and between the slip-lines A B or, for fields lb and ld, the slip-lines A D and the free surface of the chip are required to be rigid. This imposes limits on the allowable values of the field angles. For the fields of Figs. l(a), (c) and (e), following the work of Hill[9], it can be shown that
(pA/k)<~2y + 1 -- 3 r / 2
(13)
2 cos (/3 - ~'/4) - 1 ~<(pa/k) <~2/3 + 1 - ~r/2
(14)
and
for the fields to be valid where y is the angle between the free surface of the workpiece and the tangent to A B at A and /3 is that between the chip free-surface and the tangent to A B at A. If (13) is violated the workpiece becomes overstressed and if (14) is violated, the chip becomes overstressed. For the fields of Figs. l(b) and (d), equation (14) remains valid provided PA is regarded as the hydrostatic stress on A D at A and /3 as the angle between the chip free-surface and the tangent to A D at A. In that case equation (13) must be modified to
(pA/k) <- 2y -- 28 + 1 - 3~r]2.
(15)
Calculations in the next section indicate that for a given rake face mean friction stress extreme values of the machining parameters are associated with Kudo's straight chip fields, Fig. l(b), and Dewhurst's fields. Kudo's fields are always limited by overstressing of the workpiece. Remembering that pA/k = 1.0 for Kudo's field equation (15) gives the limit to 8 as 8 ~<0.393 - (r + a)/2.
(16)
Dewhurst's fields may be limited by either equation (13) or (14). In Fig. 3 the hatched regions show the limits to the field angles calculated from the approximate analysis for rake angles of 0 and 30*. For each value of ('dk), the lower limit to the field angles is zero (i.e. Lee and Shaffer's solution), unless a > (¢r/4 - r) in which case (see Dewhurst) the lower limit is determined by equation (13). The upper limit is usually determined by the left-hand inequality of equation (14) but for large values of ('dk) and small values of a volume conservation can impose a more severe constraint. Volume conservation requires that point A of the slip-line field lies above the tool tip C: in Fig. 3, the lower branch of the limiting boundary for a = 0 is determined by volume conservation. A comparison of the limiting values shown in Fig. 3 with those published in Ref. [4] again shows good agreement between the approximate theory and Dewhurst's analysis. (2.4) Non-dimensional machining parameters For all the fields of Fig. !, non-dimensional machining parameters can be calculated from (z/k), a, the field angles and (pa/k). In the most simple case, Fig. l(a), they depend only on ('dk) and a ; in addition it can readily be shown that three of them, (t[l), (da - a) where ~b is the shear plane angle, and ;t depend only on ('dk) while the other parameters such as (t/f), (Ill) Fc/(fk) and F74(fk) can be expressed in terms of them and a :
(t/l) = cos ~r(cos r + sin ~') / (~-a)=¢ ,~ = ~-/4- ~r j
(t/f)
=
(17)
cos (~ - a ) / s i n (d' - a + a )
(I/f) = (t/f)/(t/l) Fc/(fk) = cos (A - a)/[sin (4' - a + a ) cos (d' - a + ~t)] Fr/(fk) = [Fc/(fk)] tan (A - a )
(18)
Equations (18) depend on the field variables only through equations (17) and are valid for all the fields of Fig. I hut equations (17) vary with the type of field: for Dewhurst's field for example ((t[l), ((a- a) and h. depend on rl, 4,, 0 and (PA/k) as well as on (r/k) and are computed from the composition of equations (8), (9), (I 1) and (12) and the equivalent equations for the elements BC, BD and DE of the field. However for all the fields (t/I), (~b- t~) and A remain independent or almost independent of a ; the calculation of the values of these for the various slip-line fields most compactly enables the predictions of the field theories to be considered. Figs. 4(a), 5(a) and 6(a) are a partial record of such calculations in which the range of values of (t/I), (d~ - a) and A allowed by equations ( 1 3 ) , ( 1 4 ) o r ( 1 5 ) are shown by cross-hatching for two values of
462
T . H . C . CHILDS
II1
2"0 k
2"0
1.6
1.6
1.2
1.2
0.8
~
0-4
•
-0-5
£-0"1
0"8
db qb~Sreee~ e
0"4
i,I. -i n=5 2
O
l
i
0"2
0"4
i
i
i
0-6
0"8
1"0
5
1 i
o:2
0"6
' 0"4
{T/k]
0"8
1.'0
[rVk]
FIG. 4. The allowable ranges of (t/l) and (r/k): a, for a rigid-plastic material; b, allowing for elastic contact between the chip and tool: • and O, my own data; [] from Ref. [10]; [] from Ref. [ll]. a
b
!
/
"~. ". .
• o~ /
.4
""
/ •
o
"'y
• ~
20 t
%. :o
.
b
40
"H
20
, , ~ ~°o~.~' .. . "
, /'b~.,:"
SIIIII sis~ / 0 ,, "
•
~6
0~8
1:0
0
~ ,,0-5
0.2'
0.4 '
[~/k]
0.1
0.2
0---6
0-8 '
1-0 '
l~k]
FIG. 5. The allowable ranges of )t and "r/k: a and b, hatching and symbols as for Fig. 4.
II I,p-al
•o ~
~.o.
3O
30
0
20
0.1
,, 0-5 0-2
40
I
•
10 rig2
I QO
•
°o°q6 1 rim2
-%
1'o
~o 3'0 ,o .t °
' 50
-10
0
~o
0
~6
36
) 40
50
20
FIG. 6. The allowable range of (d~ - a ) and A: a and b, hatching and symbols as for Fig. 4.
Elastic effects in metal cutting chip formation
463
rake angle, 0 and 30°. Figs. 4(b), 5(b) and 6(b) are the subject of Section 3 of this paper, while the experimental results are considered in Section (2.5). Fig. 4(a) shows the variation of the allowable range of (t/l) with (z/k). The dashed line is the unique relation predicted by Lee and Shaffer. For both rake angles, the upper limit to (tll) is provided by Dewhurst's fields, constrained by equation (14). For a = 0 the lower limit to (t]l) is provided by Kudo's field Fig. l(b) constrained by equation (15) while for c~ = 30O the lower limit is also provided by Kudo's field while that limit is less than the Lee and Shaffer prediction (i.e. for (z/k) > 0.87) but for (r/k) < 0.87 the lower limit is obtained from Dewhurst's field constrained by equation (13). Fig. 5(a) shows the allowable variation of ;t with (r/k) subject to the same constraints as those which led to Fig. 4(a): a relation much closer to unique is predicted between A and (r/k) than between (t[I) and
(r/k).
Rather than illustrating the allowable relations between (~b- a) and (z/k) Fig. 6(a) shows the solution range for ( ~ b - a ) and A because this approaches the familiar manner of recording machining data of plotting ~b against (3. - a). The non-uniqueness between (• - a) and A is marked, the solution ranges being limited by the same constraints that gave Fig. 4(a). (2.5) Comparisons with experiment The shortcomings of the rigid-perfectly plastic theories are most readily illustrated by referring to experiments on the machining of steels at speeds above those which give built-up-edge formation, for which much data exists. The experimental results in Figs. 4, 5 and 6 have been collected from my own work, Zorev's[10] and that of Eggleston, Herzog and Thomsen [l l], (other worker's results have also been reviewed and are in agreement with those presented here). The solid symbols refer to experiments on non-free cutting carbon and alloy steels and the open symbols to tests on re-sulpherised, so called free-cutting, steels. In my own tests (circular symbols) the contact length I has been assessed from series of experiments with restricted contact tools, noting the longest restricted length for which small reductions in length influenced chip formation. Zorev and Eggleston and his co-workers observed the length over which chip material adhered to the rake race of the tool. The data encompasses tests for which 0 < a < 30°. It is clear that only with respect to the relationship between (th - c~) and ,~ do the experimental results lie mainly within the rigid-perfectly plastic solution ranges. When machining parameters which involve the contact length are considered (z as well as (t/l) is such a parameter as its calculation requires an estimate of I) the rigid-perfectly plastic theory is inadequate. 3. ELASTIC-CONTACT EFFECTS Fig. 7(a) shows the main model used to introduce elastic effects into the analysis of chip formation. The contact length CF of total length (1 + n)l is considered to be composed of a plastic portion CE of length l and an elastic part EF of length nl. (z/k) is assumed constant over CE and the normal pressure (pn/k) is governed by the Dewhurst-type slip-line field ABCDE. Over EF, (z/k) and (pn/k) are arbitrarily assumed to decrease parabolically from their values at E to zero at F, giving rise to a resultant force R. (The parabolic variation accords roughly with observation but the analysis to be described could readily be repeated with other assumed distributions.) The programme written to find PA/k, r/, ~ and 0 for free chip formation (Section 2.2) was adapted to find, for a range of r and n, the values of pA]k, 77, $ and 0 which would give rise to a resultant force over ABDE equal and opposite to the elastic-generated R. As an example, to be compared with Fig. 3(b), the solid lines of Fig. 8 show the calculated variation of ~b with (pa]k) for r = 0.1 and for values of n from 0 to 20 while the hatched areas define the allowable solution ranges for a = 0 and 30°. The incorporation of elastic effects
Pn
FIG. 7. a, the main model incorporating elastic contact effects; b, a further model when ~b= 0.
T. H. C. CHILDS
464
0.;t. |n 0
2 3.255
lilt/
8
12
20
::O:oo
0.1
0 0-5
0"7
0-9
a 1.1 •
1.3
1.5
1.7
1.9
2-1
IpA/kl FIG. 8. The dependence of ~b on pA/k for r = 0' ! and values of the elastic parameter n from 0 to 20, also showing by hatching the solution limits for a = 0 and 30 °.
is seen to displace the relation between qJ and (pA/k) to both higher th and higher (pA/k) values. Qualitatively similar shifts occur in the values of rl and 0. The boundary of the solution range, for example abcde for a =30 °, is determined by various constraints: over ab, n = 0; over bc the left-hand inequality and over cd the right-hand inequality of equation (14) limits the solutions; the position of de is determined by equation (13); the requirement that ~b/>0 gives rise to the part ea of the boundary. Further slip-line fields can be developed from the solutions for which q~= 0 by allowing the slip-line A B C (Fig. 7a) to develop into a fan, as shown in Fig. 7(b). These solutions are then limited by equation (15). The ranges of the machining parameters calculated from these new solutions are shown in Figs. 4(b), 5(b) and 6(b). In these the contact length I' is the sum of the plastic and elastic parts and the friction stress • ' is the average value over I'. In Fig. 4(b) the solution ranges for a = 0 and 30° are shown for r = 0-1 and 0.5 (i.e. for ('rlk) over the plastic contact length equal to 0.98 and 0-54). The allowable variations of A with (r'/k) depend only slightly on a: in Fig. 5(b) the dashed lines bound the entire solution range and in particular the hatched regions define the solution ranges for ~"= 0.1,0.2 and 0-5. In Fig. 6(b) the solution ranges for a = 0 and 30° are again shown for ~"= 0.1 and 0.5. In addition the upper boundary of the range for ~"= 0 and the lower boundary for r = 0-2 are also shown. In all three figures an indication is given by the lightly dashed lines of the relationships between the machining parameters and the elastic contact parameter n. The incorporation of elastic effects is seen in Fig. 4(b) to extend the solution range to lower values of (t/l') and in Fig. 5(b) to higher values of )t for each value of ('r'/k). The experimental results then fall within the allowable ranges. The only extension to the solution range in Fig. 6(b) occurs at high values of A where little experimental data lies. It is interesting to note however that a maximum value of A of 55° is predicted.
4. DISCUSSION AND CONCLUSIONS T h e e x i s t e n c e of an elastic c o n t a c t r e g i o n b e t w e e n the chip a n d tool has long b e e n r e c o g n i s e d e x p e r i m e n t a l l y b u t i g n o r e d in m e c h a n i c a l t h e o r i e s of chip f o r m a t i o n . This p a p e r s h o w s h o w elastic c o n t a c t f o r c e s c a n be i n c o r p o r a t e d into p e r f e c t l y plastic slipline field t h e o r i e s of chip f o r m a t i o n b y r e g a r d i n g t h e m as e x t e r n a l l y a p p l i e d f o r c e s r e l a t e d to the a s s u m e d length of elastic c o n t a c t . T h e m a i n a s s u m p t i o n is that the friction stress m a y be a s s u m e d to be c o n s t a n t o v e r the plastic c o n t a c t b e t w e e n the chip a n d tool a n d to d e c r e a s e p a r a b o l i c i a l l y o v e r the elastic c o n t a c t part a n d that the ratio b e t w e e n the n o r m a l p r e s s u r e o n the rake face a n d the f r i c t i o n stress is c o n s t a n t o v e r the elastic c o n t a c t region. G i v e n this, it is s h o w n that for a n y g i v e n c o n s t a n t f r i c t i o n stress o v e r the plastic c o n t a c t length the a d d i t i o n of a v a r i a b l e elastic c o n t a c t length e x t e n d s the m a c h i n i n g p a r a m e t e r s o l u t i o n r a n g e in a m a n n e r in a g r e e m e n t with e x p e r i m e n t (see Figs. 4-6). T h e a n a l y s i s is a p p r o x i m a t e . C i r c u l a r arc a p p r o x i m a t i o n s to the field c h a r a c teristics are m a d e in b o t h the p h y s i c a l p l a n e a n d the h o d o g r a p h b u t t h e s e h a v e b e e n s h o w n in S e c t i o n (2.2) to i n t r o d u c e little e r r o r into the c a l c u l a t i o n s . T h e e x i s t e n c e of a n elastic c o n t a c t l e n g t h implies a small c o m p o n e n t of plastic flow of the chip into the
Elastic effectsin metal cuttingchip formation
465
tool and hence a slight modification of the velocity boundary conditions in the plastic-rigid hodograph. This has been ignored as being likely to have only a small influence on forces and flow in the slip-line field compared to that of the circular arc approximations first mentioned. Variable friction stress over the plastic contact length has not been analysed mainly because it is not known how to incorporate such variation into Dewhurst's analysis but also because it is thought that it would not lead to any extension of the solution range: in the absence of elastic contact the solution range associated with the variable friction stress field of Fig. l(c) is much less than that associated with Dewhurst's field, Fig. l(e). The allowable elastic contact lengths have been found to be limited by overstressing criteria: for example in Fig. 8 no solution exists for n 1>20 if a = 30°, whereas for ~"= 0-5 and o~ = 0, n ~<5 (see Figs. 4b or 6b). In a complete theory there would be additional compatability limits on n: a tightly curled chip is likely to be associated with smaller lengths of elastic contact than straight chips. It is not known at present whether values of n as large as 20 ever exist in practice. For values of n < 2, Fig. 6(b) shows that the relationship between (~b- a) and A depends but little on n whereas Figs. 4(b) and 5(b) show n strongly influences the relationship between A and (r'/k) and (t[l') and (r'/k). This is important to the experimentalist. Good estimates of (r/k) over the plastic contact length can, if the theory is correct and n < 2 , be made from measurements of (d~-c~) and A but estimates made by dividing the rake face friction force by the measured total contact area are likely to give very different answers. Some of the experimental points in Fig. 5(b) are associated with n values between 2 and 5 and a number, mostly Zorev's, with values still greater. To some extent this may be the result of overestimating the contact length; this point is returned to later. However, it is clearly of the greatest interest to look more closely at what determines elastic contact length, to determine for example whether the values of ~ and n associated with the data of Figs. 4-6 vary understandably with speed, feed, material properties or with the machine tool structure. Only then will it be established if the present model is adequate or whether variable plastic friction stresses, non-parabolic elastic contact stress distributions or work-hardening effects as discussed by Oxley [5] or Childs and Rowe[12] still are significant. Nevertheless some preliminary comments on the data in Figs. 4-6 may be made. The interpretation of machining data may be made more or less easy by the manner of its display. Historical reasons have made it most usual to present the results of tests as the variation of ~b with (A - a ) . In Section (2.4) it is argued that the variations of (~b- a) and A as well as of (t/l) with (r/k) are more fundamental. The experimental results of Figs. 4-6, when interpreted in the light of the theory developed here, give weight to this. The data for non-free-cutting steels is seen to be clustered in regions of higher ~" values than that for free-cutting steels; generally the non-free-cutting steels are associated with plastic contact friction stresses > 0.9 k while equivalent values for the free-cutting steels are between 0.7 and 0-9k. This is qualitatively as might be expected. More detailed comments must await further work. It will only be noted here that the experimental interpretation of what is the total contact length is of central importance to the comparison of the theory and experiment. In Figs. 4--6 the data points fall into clusters not only depending on the type of steel cut but also on whether contact length was determined by restricted contact cutting tests or by visual inspection of the tool face. Restricted contact tests give shorter estimates of contact length than do visual inspections. Finally a comment may be made on the use of the numerical optimisation technique with circular arc slip-line field approximations for analysing other slip-line field problems. The method will be useful for those mixed problems like the present in which the equilibrium and velocity boundary conditions are coupled. Where they can be uncoupled simpler methods of solution retaining the circular arc approximation can be and have been used[13].
466 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
T . H . C . CHILDS REFERENCES P. Dewhurst and I. F. Collins, Int. J. Num. Meth. Engng 7, 357-78 (1973). E. H. Lee and B. W. Shaffer, J. Appl. Mech. 18, 405-13 (1951). H. Kudo, Int. J. Mech. Sci. 7, 43-55 (1965). P. Dewhurst, Proc. R. Soc. Lond. A360, 587-610 (1978). P. L. B. Oxley and W. F. Hastings, Phil. Trans. R. Soc. Lond. A252, 565-84 (1976). H. Chandrasekaran and D. V. Kapoor, Trans. A S M E Bg'/, 495-502, (1965). S. Kato, K. Yamaguchi and M. Yamada, Trans. A S M E B94, 683-9, (1972). T. H. C. Childs, Proc. I. Mech. E. Lond. 186 Ptl, 717-27 (1973). R. Hill, J. Mech. Phys. Solids 2, 278--85 (1954). N. N. Zorev, Metal Cutting Mechanics. Pergamon Press, Oxford (1966). D. M. Eggleston, R. Herzog and E. G. Thomsen, Trans. A S M E B81,263-79 (1959). T. H. C. Childs and G. W. Rowe, Rep. Prog. Phys. 36, 223-88 (1973). F. A. A. Crane and J. M. Alexander, J. Inst. Met. 96, 289-300 (1968).