Pisspanen's Card Model

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A. G. AtkinsDepartment of Engineering,

University of Reading,

Whiteknights, Reading RG6 6AY, UK

e-mail: [email protected]

Toughness and ObliqueMetalcuttingThe implications of whether new surfaces in cutting are formed just by plastic flow past the tool or by some fracturelike separation process involving significant surface work, arediscussed. Oblique metalcutting is investigated using the ideas contained in a new alge-braic model for the orthogonal machining of metals (Atkins, A. G., 2003, “Modeling

Metalcutting Using Modern Ductile Fracture Mechanics: Quantitative Explanations for Some Longstanding Problems,” Int. J. Mech. Sci., 45 , pp. 373–396) in which significant surface work (ductile fracture toughnesses) is incorporated. The model is able to predict

explicit material-dependent primary shear plane angles and provides explanations for

a variety of well-known effects in cutting, such as the reduction of at small uncut chipthicknesses; the quasilinear plots of cutting force versus depth of cut; the existence of a

positive force intercept in such plots; why, in the size-effect regime of machining, anoma-lously high values of yield stress are determined; and why finite element method simula-tions of cutting have to employ a “separation criterion” at the tool tip. Predictions fromthe new analysis for oblique cutting (including an investigation of Stabler’s rule for the

relation between the chip flow velocity angle C and the angle of blade inclination i)compare consistently and favorably with experimental results.DOI: 10.1115/1.2164506

1 Introduction

Cook et al. 1, p. 156, in 1954, pointed out that the Piispanen

“deck of cards” single shear plane model for cutting cannot oper-ate in plane plastic strain at constant plastic volume without therebeing a “new surface” at the tip of the tool. That is, there had to bea gap of at least the thickness of the shear band in order to releasematerial at the tip of the tool to be sheared at constant volume. In

Fig. 1, unless a gap is continually formed in the region of XY ,

there is an increase in plastic volume represented by ZWV . Thispercipient observation is rarely mentioned in monographs on ma-chining when basic cutting mechanics are introduced. Astakov 2remarked that there is a major difference between machining andother metalforming processes, in that there must be physical sepa-

ration of the layer to be removed from the work material and thatthe process of separation forms new surfaces.There are two views as to how new surfaces are formed in

cutting: either the traditional view i that they occur simply be-cause of “plastic flow” around the tool tip; or the controversialview ii that they are produced by progressive formation of in-

cremental gaps like X Y as the tool moves forward. Whatever theprecise manner of formation, the production of new surfaces re-quires energy, and Shaw 3 at MIT in the 1950s considered whatwork was likely to be involved in the machining of engineeringmetals under typical conditions. He argued 3 that there were fourcontributions to the energy required for cutting which are i plas-tic flow, ii friction, iii chip momentum change, and iv forma-tion of new surfaces. Their calculations for common engineeringmetals concluded that the contributions from plasticity and fric-tion far exceeded the other two components. In this way they gavesupport to the earlier modeling approach of Merchant 4 for cut-ting who had assumed from the outset that only plasticity andfriction mattered.

Since that time, there has been considerable progress in ma-chining theory, with improved algebraic models involving ever-more-complicated primary and secondary flow fields that betterrepresent experimental observations of chip deformation, and with

improved modeling of friction. For example, Oxley 5 has devel-

oped workhardening slip line field analyses of chip formation, andalso parallel-sided primary and secondary shear zone models inwhich rate and temperature effects are taken into account. A sum-mary of this and related work may be found in 5 and recent

modifications to Oxley’s machining theory may be found in, forexample, Adibi-Sedeh and Madhavan 6; Fang and Jawahir 7have studied machining with the Klopstock restricted contacttool in this way.

In Piispanen/Ernst-Merchant single shear plane modeling of or-

thogonal cutting, predictions for the inclination of the primary

shear plane are material independent that is “ = / 4 −0.5 − ” where is the friction angle and is the tool rake angle.

This expression involves no material properties yet it is wellknown that experimental data fall below the line given by this

relation when plotted on axes of versus − . Predictions for

given by the parallel-sided Oxley model are implicitly materialdependent in the sense that the shear plane angle involves factorscontrolling material yield strength workhardening, rate, and tem-

perature effects. Calculations for each case are complicated, how-ever, and require extensive knowledge of constitutive relations. Itis not known whether there has ever been a systematic investiga-tion that shows quantitative agreement between this theory and

experiment for , , and , nor for cutting forces, for the widerange of metals for which data exist. As discussed later, it is be-lieved that “plasticity and friction only” models, however sophis-ticated, cannot give the material dependence required.

In all this improved modeling, there is no consideration of thework of surface formation. It not clear whether neglect of surfacework is i because researchers do not believe in the necessity of a

gap at the tool tip for plane strain plastic flow in orthogonal ma-chining and that the new surfaces are produced just by plasticflow; or ii they do believe in its requirement but argue, as Shawand coworkers did, that the associated energy is very small and

therefore can be neglected.Since “gaps” at the tool tip suggest “cracks” and “fracture,” and

since cracks are not seen at the tips of tools in continuous-chipmachining of common metals, there has been a reluctance to be-lieve that fracture can have anything to do with continuous-chipmachining of ductile metals; we are not here considering cracksin discontinuous chip formation, nor the formation of the “tear

Contributed by the Manufacturing Engineering Division of ASME for publication

in the JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript receivedJanuary 19, 2005; final manuscript received November 8, 2005. Review conducted

by W. J. Endres.

Journal of Manufacturing Science and Engineering AUGUST 2006, Vol. 128 / 775

Copyright © 2006 by ASME

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chip” at deep uncut chip thicknesses observed by Rosenhain andSturney 8 where the chip is formed by a crack running parallelwith the machined surface. Articles on whether fracture does playa role in regular chip formation have appeared in the literatureover the years e.g., 2,9,10. Back in 1900, Reuleux 11 sug-gested that chip formation in metal cutting was by bending andsplitting like an axe into wood: That is not true for the tool rake

angles commonly found in metal cutting, but it is true in thecutting of other materials with slender knife-like tools.

When they calculated the work of new surface formation, Shaw

employed the “chemical surface free energy” of a few J/ m2,which is related to the unbalanced surface bonds of freshly formedsurfaces 12. That parameter was used by Griffith 13 for energyof surface formation in glass a very brittle solid in his seminalwork on fracture mechanics. In later developments of fracturemechanics it was realized that for more ductile solids the associ-ated specific surface energy was at least 1000 times as great as thechemical surface energy, because the mechanism of formation of surfaces involved boundary layers of deformed ductile materialextending below the free surfaces, rather than just unmatchedbonds on the surface itself. Even in brittle fracture of ferrousalloys, cleavage usually occurs on a number of planes to give

“river marks” and requires far more work/area to produce than thesimple surface free energy 14. It is in these finite thickness,highly strained, layers that mechanisms of void growth and coa-lescence take place that permit cracks to initiate, propagate, andform new surfaces in ductile metals for a review, see 14. Such

microprocesses require specific surface energies of kJ/m 2 whichare typical of the order of magnitude of the fracture toughness of

ductile solids employed in elastoplastic fracture mechanics J C

values and so on.The present author believes from mechanics arguments that

new surfaces are formed by fracture processes which require sig-nificant surface work, and Trent 15 argues that machined sur-faces are formed by controlled fracture from metallographic ob-servations and scanning electron microscopy SEM pictures. Buteven if one believes that new surfaces in cutting are produced by

plastic flow and not by “cracking,” it is legitimate to ask whatwould have been the effect on Shaw’s conclusions 3 had heincorporated large specific surface energies in his calculationsrather than the miniscule values of surface free energy. The con-clusions would certainly have been different, for recent calcula-tions 16 reveal that surface work can be as large a percentage of total cutting work as the plasticity and friction components, de-pending on cutting conditions tool angles, uncut chip thickness,material properties, friction. Those calculations are contained in arecently proposed algebraic model of orthogonal cutting with asharp tool 16, which incorporates significant surface work aswell as the usual components of plasticity and friction. To keepmatters simple, the single shear plane Piispanen model with Cou-

lomb friction was employed, although it was acknowledged that

there are greatly improved models of chip flow and friction see

above which could have been employed.

Despite its simplicity, the new model predicts a number of well-

known features of experimental machining. In particular, the in-

clination of the primary shear plane becomes explicitly material

dependent owing to the inclusion of the independent second ma-

terial parameter R the specific energy of surface formation as

well as y the shear yield stress which can still include all the

variations with workhardening, rate and temperature considered

by Oxley and co-workers see 5. It was demonstrated that an

important factor in machining is the material toughness/strengthratio R / y. The R parameter is, loosely speaking, an indicator of

material ductility. While, in broad terms, it is accepted for metals

that increased hardness results in reduced ductility, the ratio will

change in complicated detailed ways with rate, temperature,

chemistry, and thermomechanical treatments, and may go up or

down after processing. It was suggested in 16 that changes in the

ratio between hard and soft versions of the same alloy explained

why it can sometimes be easier to cut harder materials than softer.

The ratio, formed into a nondimensional parameter Z using the

uncut chip thickness t 0, i.e., Z = R / yt 0, was shown to control

ductile cutting mechanics 16. For R in kJ/m2 and y in MPa,

10−5 R / y10−3 m; and for t in fractions of millimeters, 1

Z 30, say.

The inclination was shown to be constant above a sufficiently

thick uncut chip thickness strictly a sufficiently small Z ; the

primary shear strain in this region is consequently constant too,

and cutting force varies linearly with uncut chip thickness. Below

the critical uncut chip thickness at larger Z , decreases, in-

creases, and force plots curve down toward the origin; in prac-

tice, data are not always available at these small uncut chip thick-

nesses and the curved part of the plot is absent. Significantly, in

both cases, the new model predicts that plots of cutting force

versus uncut chip thickness do not pass through the origin and

that there should be positive force intercepts one from a linear

back extrapolation of data obtained at large t 0, the other from the

curving-down part of the plot at small t 0. According to the new

theory the two intercepts are measures of the surface work; the

same specific work value may be calculated from the two inter-

cepts via different conversion factors given by the theory; see Eq.2 later for the oblique cutting version.

Experimental cutting force plots have long been known to ex-

hibit force intercepts at zero uncut chip thickness. This is true not

only for metals, but also for plastics and wood 17. Now all

traditional “plasticity and friction” theories of machining predict

that plots of cutting force versus uncut chip thickness should pass

through the origin. In cutting analyses, the positive intercept is

sometimes ignored or explained away in terms of tool bluntness,rubbing/ploughing on the clearance face of the tool, or simply as

“the force unavailable for cutting” 18. In data reduction, differ-ent answers are obtained depending upon whether force interceptsare included or not 19. It is true that for blunt tools, rubbing of the edge does cause a force intercept. But even with the sharpest

tools, the intercepts do not disappear 20 and, it was argued in16 must be caused by something else. The analysis in the presentpaper assumes a sharp tool, or at least a tool whose bluntness issmaller than the natural crack-tip opening at which a crack in the

material will propagate i.e., the so-called crack tip opening dis-placement CTOD 14. Without inclusion of surface work, theintercept would not exist; see Eq. 2 later for oblique cutting.Comparison was made 16 with a wide range of orthogonal ex-

perimental results for , for cutting force plots and so on, all withquite good agreement. The new model also showed that theanomalously high values for yield strength obtained at very small

depths of cut in machining could be explained by not accountingfor the force intercept in data reduction; see Eq. 4 later for theequivalent relation for oblique cutting. That is, the yield strength

Fig. 1 Piispanen’s “deck of cards” model for cutting with asingle shear plane. Plastic slip in plane strain along a finite-width primary shear band is impossible under constant volumeunless a gap occurs in the region of XY . Otherwise ZWV is anincrease in plastic volume „adapted from Cook et al. †1‡….

776 / Vol. 128, AUGUST 2006 Transactions of the ASME




should be determined from the actual slope of the force versus

uncut chip thickness, rather than from lines joining the origin to

individual data points.

Part of the spur for constructing the new algebraic model was

the realization that when workers began to simulate metalcutting

using the finite element method FEM from the 1980s onward,

they could not get the tool to move appreciably unless a so-called

separation criterion was employed at the tool tip. That require-

ment remains to this day in FEM cutting analyses however so-

phisticated the code, even though the same codes are able to

simulate other metalforming processes without use of the extra

feature of a separation criterion. The problem people were solvingwithout the separation criterion was oblique indentation by awedge the tool at the corner of a block, and that is why the toolwould not move very much. Of course, the separation criterioncould be just a computational fix to overcome the singularity at

the sharp tool tip, but calculations 16 for the specific surfacework associated with various physically meaningful separation

criteria gave values of kJ / m2 not the free surface energy of J/m2

assumed quite understandably in the 1950s by Shaw at MIT.What FEM workers were rediscovering was what Cook et al. 1had said in 1954, namely, that in FEM parlance nodes at the tooltip had to be released to permit tool travel. The critical question iswhether release of such nodes consumes negligible, or appre-ciable, work and whether release of nodes constitutes “fracture.”

Put another way, if separation is crucial in FEM analyses of cut-

ting, why is it absent in traditional algebraic analyses?For those who favor separation by plastic flow around the tip of

the tool, it is worth remembering that in plastic flow, elements of

material that are neighbors before permanent deformation are thesame neighbors after flow. In machining this means that elements

just above, and just below, the putative parting line at the tool tip

are supposedly still to be neighbors afterward. That is, elementson the underside of the chip are still “joined” to elements on themachined surface, however far away from one another they mayhave travelled. This is implausible and suggests that simple plastic

flow cannot be the manner in which chip and machined surfaceseparate. Separation is not a consideration in most steady-stateworking processes: The only occasion it may arise is when weenquire what really takes place at the interface between dead

metal zones and material flowing plastically through an extrusion

die 21 or in the operation of bridge/porthole dies. Of course,limits in forming are often connected with fracture and the con-nection, between old-established empirical relations for stress/

strain states at fracture in forging, extrusion, and so on, and po-rous plasticity modeling of crack tip zones in ductile fracturemechanics, has been pointed out 22. An investigation of crop-

ping and guillotining 23 discussed how separation takes place insuch processes and whether cracks should be visible: Cropping isno different in principle from cutting except that the bottom farface of the workpiece is much closer to the tool, so that it is

energetically favorable for flow and fracture to occur in a shearband between the tool and the bottom of the cropped workpiece,along the cropped face rather than, as in machining, on an inclinedshear band from the tip of the tool to the free surface. Consider-

ation of cropping and guillotining can explain why cracks are partand parcel of the process of separation in ductile solids and theformation of new surfaces, yet may not be seen as free-standingcracks ahead of the tool tip. The point is whether the crack just

keeps pace with the cutting tool or whether it is faster, and this isa question of crack stability rather than crack formation 23–25.

In Ref. 16 it was argued that machining was from the class of ductile fracture problems where there is complete plastic collapsein the formation of the chip in which the specific work of surfaceseparation is not negligible. Furthermore it was argued that thefailure of the simple Merchant 4 line of attack is less to do with

problems of uniqueness in plasticity and more to do with theproblem not being properly posed, i.e., not including surfacework. It is noteworthy that McClintock 26 has recently intro-

duced a new field of study called slip line field fracture mechanicsto which the new model is related. It seems clear that a new lineof attack for cutting theories would be to take those slip line fieldsproven to represent curling chip flow and to include in the analy-sis significant surface work.

The present paper extends the new theory in 16 to obliquecutting and provides material-dependent oblique cutting force pre-dictions from minimization of total work done. A number of un-certainties in oblique cutting mechanics, such as how good

Stabler’s law 27 is at relating obliquity angle i to the chip ve-

locity angle C , are investigated.

Even if some readers remain to be convinced that surfaces inmachining are produced by a fracture process, the present paperdemonstrates that when significant surface work is included ineven the traditional simple Ernst-Merchant analysis of obliquecutting, the predictions agree well with well-known experimental

observations, in particular that the shear plane angle dependsexplicitly upon the material being cut, and that positive force in-tercepts should exist in plots of cutting force versus uncut chip

thickness. Furthermore, the inclusion of a parameter R whichquantitatively represents ductility in addition to plastic flow andfriction, makes physical sense in that the machining behaviors of

materials with the same y but different R, or the same R but

different y, are different.

2 Oblique Cutting

In the orthogonal cutting of ductile materials, material issheared along the primary shear plane and spiral chip curl is pro-duced by secondary shear, the axis of rotation of chip curl beingparallel to the cutting edge of the tool. In the oblique cutting of ductile materials, primary and secondary shear also take place butnow the chip comes off as a helix, again with the axis of rotationapproximately parallel to the tool cutting edge, Fig. 2a. In asimple single shear plane model of cutting where secondary shearis, of course, absent there are three velocity components, namely,

the workpiece approach velocity V W , the shear velocity V S in the

shear plane, and the chip velocity V C in the plane of the tool face.In orthogonal machining, all three velocities and the hodograph liein the plane of cutting, and the direction of shear and direction of chip flow are both along lines of steepest slope in the primaryshear plane and along the rake face of the tool respectively. In

oblique cutting, both the primary shear direction in the shearplane, and the chip flow direction across the tool face, are no

longer in the directions of steepest slope: V S is now at an angle S

the shear flow angle to the normal to the cutting edge in the

shear plane, Fig. 2b; the shearing action at angle S produces the

final skewed direction of the chip which is defined by the angle C

the chip flow angle to the normal to the cutting edge in the rake

face, Fig. 2a. Since all three velocities V W , V S , and V C form ahodograph in one plane they are related by geometry; see, forexample, 28, p. 80, and also Eq. A8 in the Appendix of the

present paper. This also produces a relationship between S and

C that is given by Eqs. 4-12 in 28, viz.

tan s =tan i cos n − n − tan c sin n

cos n1

where n and n are explained as follows.The inclination of the cutting blade to the workpiece approach

velocity vector means that a number of alternative definitions of tool rake angle, and of shear plane angle, in oblique machining arepossible; for a discussion see 3,28. There is a the “normal,”

“oblique,” or “primary” rake angle n, which is the rake anglemeasured in the plane normal to the cutting edge; b the “veloc-

ity” or “true” rake angle V , measured in the plane parallel to thecutting velocity vector and perpendicular to the machined surface;

and c the effective rake angle e, measured in the plane contain-ing the cutting velocity vector and the chip flow velocity vector.

We shall use the normal rake angle n in what follows, but since





Fig. 2 „a … Permanent skewing of chip in oblique cutting into a helix as opposed to into a spiral in orthogonalcutting. Rake angle n =10 deg, depth of cut 0.13 mm; material steel „from Shaw †3‡…; „b … chip formation inoblique cutting showing the three velocity components, namely, the workpiece approach velocity V W , the shearvelocity V S in the shear plane, and the chip velocity V C in the plane of the tool face „from Amarego and Brown†28‡…; and „c … the resultant force F res has components F P parallel with the velocity approach vector V W ; F Q perpendicular to the finished work surface; and F R perpendicular to the other two. F P is the “power” force, F Q is the “thrust” force, and F R is the “radial” force which are the forces usually measured by a dynamometer. C

is the chip velocity angle; C is the angle of inclination of the friction force F to the normal to the cutting edgein the rake face; S is the shear flow angle; and S is the angle of inclination of the shear force F S to the normalto the cutting edge in the shear plane „from Amarego and Brown †28‡….





all the different are related, other expressions are possible see

Eqs. 8 and 9 later. Similarly we shall also employ the normal

shear plane angle n to define the inclination of the primary shear

plane, Fig. 1c, rather than the alternative effective shear angle

e. As with the different definitions of , the different definitions

of are related 28.

The resultant force F res has components F P parallel with the

velocity approach vector V W ; F Q perpendicular to the finished

work surface; and F R perpendicular to the other two, Fig. 2c. F Pis the “power” force, F Q is the “thrust” force, and F R is the “ra-

dial” sideways force. These are the forces usually measured by a

dynamometer. They are related by force resolution; see, for ex-

ample, pp. 82–83 of 28. The shear force in the shear plane is

denoted by F S which is inclined at the angle S to the normal to

the cutting edge in the shear plane, and the friction force between

tool and chip is denoted by F which is at the angle C to the

normal to the cutting edge in the rake face. Note that S and C

are permitted at this stage to be different from S and C , that is

the shear and friction force vectors are not necessarily colinear

with their respective displacement vectors see later.

Experiments 29 on cold-rolled 1015 steel with tools orientated

up to i =30 deg obliquity, and on cold rolled 1008 steel 19 with

i up to 40 deg obliquity, show that n is essentially independent of

the angle of obliquity other things being equal so that for a given

tool rake angle and friction, the quasilinear plots of F P and F Qversus uncut chip thickness are also approximately independent of i, and thus follow the well-known trend in orthogonal cutting of

lower forces for greater n. The power required for cutting over

this same range of obliquity is also approximately constant. How-

ever, the corresponding quasilinear plots of the sideways force F Rversus uncut chip thickness do depend on i and, for given tool

rake angle, increase as i increases see Figs. 3a–3c later. If,

instead of varying i at constant n, n is varied at constant i, plots

of F P and F Q versus uncut chip thickness now depend upon n,

but this time results in 19 show that F R is apparently indepen-

dent of n. There is, of course, no theory yet to predict any of

these plots because there is no means of predicting the material-

dependent n. That will be an outcome of this paper.

In oblique cutting the resultant cutting force is not in the plane

perpendicular to the finished surface as it is in orthogonal cutting;nor is it necessarily in the plane of the three velocities. In the

plane of the finished surface, the components of the resultant force

and resultant velocity are coincident: it is what happens out of that

plane where there is uncertainty; for a discussion see, for example,

3,28. Uncertainties also arise about whether the force and veloc-

ity vectors are colinear. Experiments to establish the angles are

often difficult to interpret, and whether it is C or C that is being

determined depends upon the method employed, i.e., chip dis-

placements give C and forces give C 3,19,28,29; note that

different answers are given depending on whether the positive

force intercept is subtracted from the total forces, as illustrated in

Appendices B and C in 19. Shaw et al. 29 performed experi-

ments where it was shown that the deviation between the shear

velocity vector and the resultant force vector in the shear plane

was insignificant in all their tests. There were deviations however

between C and C in some of their experiments. Note that Shaw

and co-workers use the symbol S for S and employ S = S

− S = S − S ; likewise C is used for C and C = C − C

= C − C .The simplest assumption is to say that all force and velocity

components are coincident, i.e., S = S and C = C . When this

assumption is made, C is known in terms of n, n, i, and nEq. 7, and S is also known from Eq. 1. Alternatively, only

one of the pair may be taken as colinear. A general theory assumes

that neither pair is equal.

3 Model

The internal work done in our model has three components,namely, plasticity on the single shear plane, friction on the rakeface of the tool, and fracture toughness work at the tool tip. Asexplained in 16, traditional force resolution methods using theMerchant circle can only apply when there is no separation work.As soon as separation work is included, part of the work done bythe external forces provides that work, but since separation isquantified in terms of energy/area and not a force or stress, it isimpossible using force resolution to identify how much goes toseparation and how much to plasticity. Force resolution can still

be used, however, to determine the friction force F along the rakeface of the tool. This is permissible because friction occurs at theboundary between the external work and the internal plasticityand surface separation works.

The Appendix contains the algebraic expressions for frictionforce and increment of friction work; the increment of plasticwork; the increment of surface formation work fracture tough-ness work ; and the increment of external work.

By equating the internal and external work rates an expression

is obtained for the power force F P namely,

F P

Rw= oblique / Z + 1

Qoblique

2

in which Z = R / yt 0 is the nondimensional term incorporating

material properties and uncut chip thickness, and Qoblique is thefriction factor given by

Qoblique = 1 − F

F P sin n cos i cos C − C

cos n − ncos C

3

The ratio F / F P in Eq. 3 may be eliminated using Eqs. A1 to

A3 but it is algebraically cumbersome. Without the fracture term

in Eq. 2, i.e., when F P / Rw = oblique / ZQoblique, the analysis re-verts to an Ernst-Merchant solution for oblique cutting. However,as demonstrated in 16 the fracture toughness term should not benegligible in practical metalcutting. Equation 2 says that therewill be a positive force intercept in plots of cutting force versusuncut chip thickness, and that it is a measure of the material

toughness R. As in traditional analyses, the slope of the plots gives

the material shear yield strength y; the difference now is whether

the line passes through the origin or has an intercept.Blunt tools will rub on the clearance face of the tool and this

will add to the intercept predicted by the sharp-tool analysis givenin Eq. 2; see Sec. 4.

For given material, tool rake angle, friction and uncut chip

thickness, the primary shear plane angle n is obtained by mini-mizing the total work done or, equivalently, by minimizing Eq.

2. Once the optimum F P has been established from the minimi-

zation of Eq. 2, F Q and F R are obtained from Eqs. A2 andA3.

In Eq. 2, 1 / Z = yt 0 / R represents the uncut chip thickness

for a given material R / y ratio. In orthogonal cutting, to which

Eqs. 2, 3, and A6 apply with i =0 and S = C = 0, it is foundboth experimentally 5 and predicted by the new theory 16 that

above a limiting uncut chip thickness, is constant and so is

constant whence Eqs. 2 and 3 predict that a plot of F P versust 0 will be linear with slope yw / Q and intercept Rw / Q. At

smaller t 0 greater Z , is predicted to become smaller, so

becomes greater and F P versus t 0 curves downward, but still does

not pass through the origin, and has an intercept of Rw since Q

=1 at zero t 0 when =0. It will be found that similar things occur

for oblique cutting, although the constancy of oblique is slightly

affected by the cos S factor in the denominator of Eq. A6. Even

so, quasilinear plots of component forces versus t 0 are predictedfor oblique cutting and, as discussed later, are confirmed by ex-

periment. Calculations show that F R increases at greater i , mainly

because Q oblique decreases.





The specific cutting pressure “unit power” given by F P / wt 0becomes

F P / wt 0 = 1/ Qoblique oblique y + R / t 0 = y / Qoblique oblique + Z

4

which is expected to rise disproportionately at small t 0 owing tothe inverse-dependent final term on the right-hand sides of therelations. This is called the “size effect” in cutting and has beendiscussed in 16 in terms of the new model where it is shown that

the anomalously high values of y derived from “plasticity and

friction only” analyses in this range of small t 0 are erroneous and

are caused by neglect of the intercept in the force plots; that is, yshould be determined from the slope of the plot and not from lines

joining the origin to data points. If the uncut chip thicknesses are

such that Z is above the critical value at which decreases and increases see above, so that the force plot bends down towardthe origin, another discrepancy will arise. It was shown 16 thatexperimental variations in specific cutting pressure with uncutchip thickness given by Kopalinsky and Oxley 30 could be ex-

plained with a constant y when the orthogonal version of Eq. 4was employed.

The area of the shear plane is given by

As = wt 0 /sin n cos i 5

and from the force F s along the shear plane we obtain

F s = oblique sin n cos i cos n + n − n

Qoblique cos n − ncos s y As

+ cos i cos n + n − n

Qoblique cos n − ncos s Rw 6

where tan n =tan cos C , Eq. A4.

Equation 6 suggests a linear plot of F s versus As in obliquecutting having slope

oblique sin n cos i cos n + n − n

Qoblique cos n − ncos s y

and intercept

cos i cos n + n − n / Qoblique cos n − ncos s Rw

When i =0, Eq. 6 reduces to the corresponding relation for or-thogonal cutting given in 16 and analyzed there against datagiven by Thomsen and co-workers 18. The important features

predicted by Eq. 6 for oblique cutting are that all F s versus As

data should fall into a single line having a positive force intercept,

for all t 0, all n and all i, providing that the slope first square-bracketed term remains constant see later.

The most general expression for F P given by Eq. 2 is in terms

of three unknowns, namely, n, C , and S . When the assumptionis made that both the chip velocity and friction force vectors, andthe shear velocity and shear force vectors, are colinear that is,

C = C and S = S , Stabler 27 showed that the only unknown

is n since C and S are both expressible in terms of n. His

relation for C is given by Eq. 4-26 in 28, viz.

tan n + n =

tan i cos n

tan C − sin n tan i 7

and S is given by Eq. 1. The solution then follows the same

procedure as that for orthogonal machining in 16 in which n isobtained by minimization of total work done. Results are obtained

for given tool rake angle n, friction angle , inclination angle i ,

and nondimensional material-geometry parameter Z = R / yt 0.

Progressively more complicated solutions are obtained by al-

lowing either i S = S but C C and searching for the n,

C pairs that give the absolute minimum for work done; or ii C = C but S S and searching for the n, S pairs that givethe absolute minimum for work done; or iii allowing all three to

vary and finding the absolute minimum. In this paper we shall

limit ourselves to predictions based upon S = S and C = C

,since the aim is to show that oblique cutting may be sensiblyanalyzed in terms of the new model incorporating significant sur-face work. Further work will investigate options i–iii in duecourse.

4 Solution With S= S and C =

C and Comparison

With Experimental Data

Nondimensionalized plots of oblique cutting forces versus un-

cut chip thickness derived from the optimised Eq. 2 are easilyshown to have the right sort of shapes given by experiment, but

predictions by the new analysis require the material properties yand R both of which are affected by rate, temperature, and envi-

ronment. While some knowledge of y is often available indirectly

through hardness, independent measures of R are usually not

available, although use of the fracture mechanics J C parameterwould not be far wrong. Indeed it has been suggested that machin-

ing could be a method of simultaneously determining y and R inthe intermediate strain rate range 31. In 16 it was shown how

to determine the best fit values of y and R , for a given materialundergoing given machining conditions over a wide range of vari-ables, from the intercepts and slopes of experimental plots of cut-ting forces versus uncut chip thickness. The same procedures canbe applied in oblique cutting.

Brown and Amarego 19 report an extensive series of resultsfor oblique cutting of SAE 1008 cold drawn steel of 240 VPN in

the form of a 4 in. 102 mm outside diameter tube having a wall

thickness of 0.126 in. 3.2 mm. Three sets of HSS tools with a

6 deg clearance angle and 0 deg, 10 deg, 20 deg, 30 deg, and

40 deg angles of obliquity were employed, with uncut chip thick-

nesses in the range 0.0025 in. to 0.008 in. 0.064 mm to

0.203 mm. Linear plots of the cutting force components vs. uncutchip thickness were obtained, all having definite positive forceintercepts.

The friction angle can be determined from the ratio of the F Pand F Q plots in the usual way from Eq. A3, and is found to be

some 0.8 rad or 46 deg over most of the range of results reportedin 19. Calculations using the new model of metal cutting showthat agreement for cutting forces, tool rake angles, and obliquity

angles over the whole range of uncut chip thickness is obtainedwhen 510−3

R / y610−3 in. 1.310−4 R / y1.5

10−4 m with individual values of R 300 lb/in.2 54 kJ/ m2and y =56, 000 psi 390 MPa. For a nonmartensitic steel, a hard-

ness of 240 VPN corresponds with an ultimate tensile stress of

3.45240 =828 MPa 32. The tensile flow stress predicted

from the y derived from the machining data will be some 2 yTresca yield criterion, i.e., 2390=780 MPa, which is sensible

for a UTS of 828 MPa. The value of the fracture toughness R isalso reasonable in terms of independently determined values 16.The lines in Fig. 3 are the predictions of the new analysis and thedata points are those of Brown and Amarego 19. We note that

although there is separation between the plots for F P and F Q at allobliquities, the differences are not marked within the range of obliquities employed experimentally, which is why Brown and

Amarego 19 considered that F P and F Q were independent of

obliquity angle i. However, at tool obliquities greater than those

employed in the experiments, the predicted F P and F Q force com-

ponents separate out, with F Q changing sign. F R versus t 0 is bothexperimentally and theoretically always different for all obliqui-ties.

Brown and Amarego 19 also present other results for obliquecutting forces versus uncut chip thickness for various different

tool rake angles n all at constant angle of obliquity. This time, F Pand F Q lie on distinct lines for each rake angle, exactly as for

orthogonal machining, but F R bunch up into an apparently singleline. The new analysis is capable of reproducing such results not





shown here noting, however, that separate lines are predicted for

F R, albeit closely spaced within the range of 0 degi40 degover which experiments were conducted.

Shaw et al. 29 performed 0.248 in. 6.3 mm wide obliqueplaning experiments on 1015 steel using CCl 4 as the cutting fluid.The range of parameters is not as wide as in 19 and only a single

uncut chip thickness was employed, namely 0.005 in.0.125 mm. We note also that the friction angles derived by Shaw

et al. for oblique cutting vary quite a bit. All calculations in this

paper are presented employing the same for all angles of obliq-uity. Calculations determined by the new model using their ex-

perimental results suggest that R 800 lb/in. 105 kJ/m2 and

y 20,000 psi 140 MPa, giving R / y 0.045 in. 110−3 mand therefore Z =9. Although Shaw et al. state that their 1015 steelis “cold rolled,” no hardness value is given for us to make a

comparison between the yvalues derived from the cutting model

and from the quasistatic hardness, as done above for Brown and

Amarego’s results above 19.

In comparison with the data in 19 it would appear that theShaw et al. steel was not fully work hardened, and was tougher

and softer. This would explain the rather high values for y some

60– 80,000 psi or 420–560 MPa given in 29 which come about

because of neglect of the positive force intercept in Eqs. 2 or 6.

As explained in 16, y should be determined from the slope of

such plots instead of presuming that the plot passes through the

origin: when R is relatively high as here the discrepancy be-

comes worse. These remarks take into account that rate and tem-

perature may produce different values from quasistatic properties

derived from hardness, i.e., the pair of values for toughness from

the intercept and for shear yield stress from the slope of the

Fig. 3 Experimental results given in †19‡ for „a … F P , „b … F Q , and „c … F R versus depth of cut for 1008 cold drawn steel; w =0.126 in. „3.24 mm… and n =20 deg for all. Inclination angles in experiments are i =10 deg, 20 deg, 30 deg, and 40 deg. Fulllines are predictions of new theory employing É0.8 radians or 46 deg, R É300 lb/in.2 „54 kJ/m2…, and y É56,000 psi„390 MPa…. Although there is separation between the plots for F P and F Q at all obliquities, the differences are not markedwithin the range of obliquities employed experimentally. At tool obliquities greater than those employed in the experiments,the predicted F P and F Q force components separate out, with F Q changing sign. F R versus t 0 is always different for allobliquities.





force varied with obliquity. Within the range of obliquity they

employed i40 deg, their Fig. 2 shows that at constant depth of

cut F P begins immediately to rise with i for constant e— up to

nearly twice the value for zero obliquity; for constant V , F P also

increases but at a far lower rate; but at constant n, F P is practi-cally constant and independent of obliquity. The same pattern of

behavior was shown by the thrust force F Q; the radial force F Rincreased with i as would be expected. The various definitions of

are related trigonometrically, viz.

tan V = tan n /cos i 8

and

sin e = sin i sin C + cos i cos C sin n 9

Since, for small i, cos i 1, there will be only minor differences

between tools with constant V and n when the obliquity is nottoo big. This is confirmed by various experimental data 19,29.

The effective rake angle e, on the other hand, has C in itsdefinition which is why apparent departures from the orthogonalbehavior appear at lower obliquities.

Because the power force was constant with obliquity when the

n definition of tool rake angle was employed, Brown and Ama-

rego 19 concluded that the n definition was the “controlling

rake angle.” However, for constant n, Figs. 4a–4c show that

while F P / Rw is predicted to be practically constant at small i , it

decreases at large i particularly at small Z , i.e., at large uncut chip

thickness for a given material; the theory predicts that F Q / Rw isalso essentially constant at small i as found by Brown and Ama-

rego 19 but decreases at large i; and that F R / Rw is almost linear

with i at small i, but increases at a faster rate at larger i. In all

these predictions n is still the same as orthogonal. All predictionsdepend, of course, upon rake angle, friction, and material

toughness/strength R / y ratio combined with the uncut chip

thickness in the non-dimensional Z parameter, but calculationsshow that the same trends are visible for all.

Brown and Amarego 19 plotted other data to justify their

choice of n as the “controlling rake angle.” For example, in

experiments with −10 deg n +20 deg they found that F P and

F Q both decreased linearly with n, data from all i falling on tothe same two lines. Calculations by the new theory not shownhere predict this behavior but additionally demonstrate that the

negative slope depends upon Z : At large Z 100 small uncut chipthicknesses the slope is almost zero no change in F P or F Q with

n, but the slopes progressively become more negative as Z de-creases, that is, a with the same material when the uncut chipthicknesses is increased, or b at the same uncut chip thickness

with “more brittle,” i.e., lower R / y ratio, materials.

Why the power force F P in oblique cutting, and the inclination

n of the primary shear plane, are essentially independent of the

angle of obliquity of the tool at least up to i 40 deg or so maybe explained as follows:

The minimization of the internal work rate to give n, is a

minimization of oblique / Qoblique where oblique and Qoblique are

given by Eqs. A6 and 3, respectively, in both of which, in this

paper, S = S and C = C . For n to be independent of i, the

effect of n on changes in oblique in the numerator must be com-

pensated by the effect of n on changes in Qoblique in the denomi-nator. For given n and given , oblique depends on S : But S

is uniquely determined from C when the assumption is made that

S = S and C = C . Since the values of n from minimization are

independent of i in this range, it follows that values for S pre-

dicted from minimization must be in the range where cos S 1.

Also the effect of variations of n on Qoblique given by Eq. 3 inthis range must not be marked. Both observations are confirmedby the theoretical predictions. The answers depend upon the non-

dimensional parameter Z = R / yt 0 but, again, for the particular

metals investigated, depths of cut employed, and range of angles

of obliquity, S is never more than about 20 deg, and

cos 20 deg=0.94. This is why oblique is, for practical purposes,

“constant” above a limiting depth of cut exactly as in orthogonal

machining, despite the cos S factor in the denominator of Eq.

A6, and why F S versus AS data are expected to follow the usual

linear plot predicted by Eq. 6. That is not to say, however, that at

greater angles of obliquity and different Z , the values of oblique

and Qoblique may not be different from values found in orthogonal

cutting and Fig. 4 demonstrates that this is indeed the case, even

with the assumption that S = S and C = C which gives n= orthogonal.

Shaw et al. 29 state that there is a paradox in their results in

that their derived y increase with i even though oblique definitelydecreases with i. The predictions of the new theory agree that

oblique decreases especially at larger i the absolute values depend

on Z and , but the analysis is capable of explaining the results of

Shaw et al. with a constant mean flow stress y. In a companion

paper employing a rotary cutting tool, Shaw et al. 33 explainedtheir paradox in terms of a size effect with uncut chip thickness,together with another size effect associated with the thickness of the shear plane. As explained in Sec. 3 above, there is doubt about

the values of y quoted by Shaw et al. because of problems asso-ciated with the influence of the positive force intercept in plots of

cutting force versus uncut chip thickness on the calculated y.

Shaw et al. used only one uncut chip thickness in their experi-

ments, so they could not have derived y from a slope but onlyfrom the line joining the origin with the one data point for each

obliquity. Such a procedure will always overestimate the shearyield stress, as explained in 16.

The chip flow direction C given from Eq. 7 is automatically

predicted by the new model, once n is predicted; and S also is

then given by Eq. 1. Since tan n in Eq. 7 is given by

tan cos C according to Eq. A4, C has to be solved numeri-

cally once n is known. Some predictions from the new model for

how C is expected to vary with n at different obliquities are

shown in Fig. 5a. It is seen that values of C depend upon Z and

i, and that C increases with n, the rate of increase being greater

at greater values of i .

Historically, a number of empirical relations between C and

inclination angle have been put forward 27,34. Stabler 27 pro-

posed that C i for all n which corresponds with plane strainchip formation, i.e., the width of cut and width of chip are the

same. Some predictions for C versus i given by the new model

for the Brown and Amarego experimental conditions are shown inFig. 5b together with the machining data. The results confirmwhat has long been known from experiments, namely, that while

Stabler’s rule is not too bad an approximation for small obliqui-

ties, the rake angle has an effect on C . The new analysis alsopredicts that the relationship should additionally depend upon un-cut chip thickness through the parameter Z.

For completeness, Figs. 6a and 6b show the predictions for

S in terms of n, i, and Z corresponding to the predictions for C

in Fig. 5. It transpires that the value for S increases somewhat as

Z increases but is virtually independent of n; and rises almost

linearly with i .One of the referees of this paper remarked that the old experi-

mental data used in this paper to justify the new model were fromslow speed cutting with HSS tools where built-up edges are pos-sible, and that modern carbide/diamond/ CBN tools use verysmall even negative inclination angles. That is true. The question

of positive or negative obliquity is of great interest when inter-preted in terms of the relative motion between tool and cut mate-rial, that is, to the ratio of velocity parallel to the cutting edge tothe velocity perpendicular to the edge. In cutting with knivesvery large rake angles this ratio is known as the slice/push ratio

and is given the symbol 35. In the case of cutting thin slices of floppy materials, i.e., materials which do not store or dissipate

strain energy and for which the only work required concernstoughness and friction, the other end of the spectrum of machin-





ing mechanics from metalcutting, it was shown that the larger the

, the smaller the cutting forces. An explanation was thus givenfor the common experience that, however, sharp a kitchen knife,mere pressing down alone does not achieve much of a cut, but thatcutting is “much easier” as soon as sideways motion is introduced.The effect is noticeable because there is a nonlinear coupling be-tween the vertical and horizontal forces, so that the slightest side-ways motion disproportionately reduces the vertical force 35.Reduced cutting forces are important in the food industry in orderto minimize damage to cut slices. A cutting blade inclined at angle

i has =tan i. The sign of i is immaterial when the tool has noindependent motion the “sideways” force simply changes direc-tion but when the tool can move independently e.g., Napier’s

rotary tool 33, milling cutters, may increase or decrease andthis is why “uphill” and “downhill” cutting are different. For ex-ample, in a bacon slicer where the material is cut on the centerline

of the wheel, is given approximately by the ratio of the periph-eral speed of the disk blade to the feed speed of the material;

when cut above or below the centerline, is different because the

tan i term comes into play and has a different sign above and

below the centerline. The analysis for floppy materials 35 shows

that cutting “uphill” negative i is more difficult than when cut-ting “downhill.” The action of a cylinder grass lawnmower has

recently been investigated in this way, together with the optimumplanform for flat disk cutters 36.

6 Conclusions

Oblique cutting has been investigated in terms of a recentmodel of machining 16 which, by incorporating significant spe-cific works of separation ductile fracture toughnesses, was able

to predict a material-dependent primary shear plane angle n andhence remove most of the shortcomings of the traditional “plas-ticity and friction” approaches to cutting. It has been shown thatthe new analysis describes oblique machining well and gives atheoretical basis for a number of well-known experimental resultsin oblique cutting such as the near independence of the power

force F P and thrust force F Q with angle of tool obliquity i when

the normal rake angle n

of the tool is constant, and the quasilin-ear plots of cutting force components with uncut chip thickness at

either constant n or constant i. However, at larger angles of obliquity outside the range of current experimental results, thenew analysis predicts departures from this simple picture.

As for orthogonal machining, an important parameter in the

new approach is Z = R / yt 0 which combines the toughness-tostrength material property with the uncut chip thickness. The ex-

plicit predictions for the primary shear plane angle n, and all

other predictions, are controlled by Z . It has been demonstrated

that the primary shear strain oblique is essentially constant above alimiting uncut chip thickness—at least over the range of obliquityangles for which experimental results are available—which ex-

Fig. 5 „a … Theoretical variation of chip velocity angle C withtool rake angle n for two tool obliquity angles „i =40 deg and20 deg… and two values of Z „10 and 1…. „b … A “Stabler” plot of C versus i for various n . Full lines are present theory, pointsare the Brown and Amarego results †19‡.

Fig. 6 Predictions for the shear flow angle S „a … as it dependson i at different n and different Z ; and „b … as it depends on n at different i and different Z „ n has only minor influence inthese plots, the main influence is from Z …



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