Machinability and Surface Integrity Part 9 docx

181, the resultant force ‘R’ is depicted acting at the tool’s cutting edge, being resolved into components ‘N’ and ‘F’ in direc-tions along and normal to the tool’s face respectively, a

Figure 180 A Rotating Cutting-force Dynamometer (RCD), utilising piezoelectric sensor systems

[Courtesy of Kistler Instrumente AG]

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bration certificate, to ensure that the results obtained

are both valid and sound A cautionary note: if the

dy-namometer has been inadvertently dropped, or it has

possibly collided with an obstruction when in use on

the machine tool, it should be sent back to the

manu-facturer for servicing and recalibration, otherwise,

spurious cutting force data may be the result

7.9 Machining Modelling

and Simulation

Introduction

Previously, it was been widely accepted that most

cut-ting tool modelling technqiues are somewhat

incom-plete, in both their analysis of the process and their

accompanying derived mathematics Early, but worthy

attempts at analysing the chip formation mechanics of

the orthogonal cutting process were undertaken

ini-tially by Ernst and Merchant (1941) – shown

schemat-ically depicted in Fig 181, followed by further work

concerning the analytical graphical interpretation of

the orthogonal cutting action which was presented

by Merchant and Zlatin (1945) and later work by Lee

and Shaffer (1951) – not shown This earlier work

was then followed by Zorev’s (1963) interpretation of

an ‘idealised cutting model’ (Fig 182) In all of these

above modelling cases and others not mentioned – for

brevity’s sake!, the very complex nature of the cutting

process, is a vast subject ‘straddling’ many engineering

and physical disciplines Such modelling involves

as-pects of: the tool’s geometry, chip/tool contact lengths

and pressures, chip formation, cutting forces,

fric-tional and thermal factors and so on, making it

vir-tually impossible to obtain close agreement between

with any truly meaningful results between each

pro-posed model This lack of correlation of these

model-ling processes, is to be expected, as in reality a shear

zone, rather than a shear plane exists, but for

mathe-matical treatment, a shear plane allows some degree of

geometrical association Due to the complex nature of

the inter-related variables that occur in any dynamic

cutting situation for just simply the orthogonal cutting

process, let alone for oblique machining modelling,

this has meant that the ‘optimum modelling solution’

has as of now, not yet been fully addressed

Many ‘learned tomes’ have been written in the past,

concerning the ‘mechanics of machining’ and it is not

the intention to fully discuss them here, in this book which is principally concerned with ‘current practice’ concerning machining applications However, a brief resumé of just one of these ‘orthogonal models’ shown

in Fig 181 will be mentioned below, together with a concise review of friction in metal cutting operations (Fig 182) will be presented, to attempt to show why the subject of ‘theoretically modelling’ the cutting pro-cess is so complicated

Ernst and Merchant’sComposite Cutting Force Circle

If a continuous chip formation is produced when ma-chining ductile materials in an orthogonal cutting process, such as that found when cylindrically turn-ing a component’s periphery with an undeformed chip

thickness (‘t  ’), this will cause a chip compression (‘t ’) – Fig 181(top) The cutting forces can be obtained by employing a cutting force dynamometer as discussed

in the previous section, to typically measure the forces

‘F C ’ and ‘F T’ and so on By utilising such cutting tool dynamometry, Ernst and Merchant (1941) were able

to classify the forces acting in the vicinity of metal cut-ting which gave rise to both local plastic deformation and frictional effects In Ernst and Merchant’s theory

which is often termed the so-called ‘shear-angle solu-tion’ , it is assumed that the cutting edge is always per-fectly sharp and that a continuous-type chip without BUE occurs, this former assumption in practice does

not actually occur Moreover, another assumption in their analysis it was that the chip would behave like

a ‘rigid body’ , which is held in equilibrium by the ac-tion of the applied forces transmitted across the tool/ chip interface and transversely over the shear plane Boothroyd (1975), offered a reasonably ‘elegant solu-tion’ to Ernst and Merchant’s numerical and geometri-cal analysis and, this has been somewhat modified and further simplified below In order to abridge the ‘shear-angle solution model’ shown in Fig 181, the resultant

force ‘R’ is depicted acting at the tool’s cutting edge, being resolved into components ‘N’ and ‘F’ in

direc-tions along and normal to the tool’s face respectively,

as well as into components ‘F N ’ and ‘F S’ – once again, along and normal to the shear plane correspondingly

Further, the cutting force ‘F C ’ and the thrust ‘F T’ com-ponents of the resultant force, are also shown Here, it can also be assumed that the entire resultant force is transmitted across the tool/chip interface and that on both the tool’s flank and edge no force occurs, mean-ing that a zero ‘ploughmean-ing-force’ is present – see Fig

184 which illustrated this ‘nose-rounding effect’

The foundation purported by Ernst and Merchant’s

theory, was the proposition that the shear angle ‘φ’

would acquire such a value, thereby reducing the

ac-tual work done to a minimum In view of the fact that

that for preselected cutting conditions, the work done

during cutting was comparative to that of ‘F C’ in terms

of ‘φ’ , hence allowing one to obtain the value of ‘φ’ when ‘F C’ is at a minimum Thus, from Fig 181:

Figure 181 ‘Merchant’s’ composite metal cutting circle for an orthogonal cutting model, where:

• FT = thrust force component,

• FC = cutting force component,

• FN = normal force component on the shear plane,

• FS = shear force component on shear plane,

• R = resultant tool force component,

• N = normal force component on tool face,

• Φ = shear angle,

• α = working normal rake angle,

• τ = mean friction angle on tool face,

• t1 = undeformed chip thickness,

• t2 = deformed/compressed chip thickness,

• A0 = cross-sectional area of uncut chip,

• Ac = cross-sectional area of deformed/compressed chip.

NB: Force arrow vector directions have been reversed and some terms have been modified from the original

work [Source: Ernst & Merchant, 1941]

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FS = R cos(φ + τ – α) (i)

and

Where:

τS  = Workpiece material’s shear strength – on the

shear plane,

AS  = Shear plane’s area,

AC  = Uncut chip’s cross-sectional area,

τ = Mean friction angle, between too and chip [i.e

arctan (FT/FN)],

α = Normal rake angle (i.e working).

From equations (i) and (ii):

R = τ s Ac

sin ϕ



By geometry:

Hence, from equations (iii) and (iv):

Fc= τ s Ac

sin ϕ cos(τ − α)

Now, equation (v) can be differentiated with respect

to ‘φ’ , then equated to zero to obtain a value of ‘φ’

when ‘F C’ is at a minimum Hence, the requisite value

is specified by:

In comparative analysis undertaken by Merchant

(1945), he found close correlation with experimental

results was obtained when machining synthetic

plas-tics, but a somewhat poor theoretical correlation

oc-curred when steel had been machined with cemented

carbide tooling It needs to be mentioned that when

Merchant differentiated with respect to ‘φ’ , it was

as-sumed that ‘A C ’ , ‘α’ and ‘τ S’ would be independent of

‘φ’ , but on further consideration, Merchant decided to

offer in a ‘modified theory’ the relationship of :

Here, Merchant brought in a modification to the shear

strength of the workpiece material ‘τ S’ which now

in-creased linearly with an increase in normal stress ‘σ S’

on the shear plane, where zero normal stress ‘τ S’ is

equal to ‘τ So’ This new assumption by Merchant was confirmed by previous work undertaken in the litera-ture published by Bridgman (1935, ’37 and ’43), where the shear strength when experimental machining of polycrystalline metals was shown to be dependent on the normal stress on the ‘plane of shear’

Now from Fig 181, we can obtaing the following relationship:

and:

F Ns = σSAS= σSAC

From equations (viii) and (ix):

σS= sin ϕ A

Combining equations (iii) and (x):

and, from equations (vii) and (xi):

This equation (i.e xii), explains why the value of ‘τ S’ may be influenced by modifications in the shear angle

(‘φ’), which is now inserted into equation (v), to ob-tain a new equation for ‘F C ’ in terms of ‘φ’ , therefore,

the resulting expression becomes:

sin ϕ cos(ϕ + τ − α)[ − k tan(ϕ + τ − α)] (xiii).

Now, it can be assumed that both ‘k’ and ‘τ So’ are con-stants for the specific workpiece material, further, that

‘A C ’ and ‘α’ are also constants for the machining

op-eration Hence, equation (xiii) can now be

differenti-ated to obtain a new value of ‘φ’ , with the resulting

expression:

2φ + τ – α = C

Where:

C = Can be obtained from ‘arccotk’ , which is a

con-stant for the workpiece material

NB  In further more recent experimental work, it has been shown that ‘τ S’ will remain constant for a speci-fied workpiece material, across a diverse range of

cut-ting conditions, as such, the value of ‘k’ can be equated

to zero

When the shear-angle values are compared for the

plotted linear relationships of the earlier theoretical

and experimental work undertaken by Ernst and

Mer-chant, to that of the later comparative work by Lee and

Shaffer, then these shear-angle relationships diverge

somewhat However, one area that these researchers

both agreed upon, was the fact that friction on the

tool’s face was the most important factor during metal

cutting In the following discussion, the influence that

frictional behaviour has at the tool/chip interface will

be briefly mentioned

Frictional Effects During Machining

For simplicity’s sake, friction between dry sliding

sur-faces will be concisely reviewed In 1699 Amontons

‘laws of friction’8 were formulated, then verified by

Coulomb in 1785, with Bowden and Tabor (1954)

contributing greatly to an explanation of these

empiri-cal laws

If two apparent flat surfaces are placed together, the

larger asperities (i.e peaks) on each mating face will

only establish contact With just normal loading, the

top of these asperities will yeild, creating a real area

of contact until such a time, that they are capable of

supporting an applied load In the main, for the vast

majority of engineering applications this real contact

area (‘A r’), is normally only a very minute portion of

the apparent real of contact (‘Aa’) – even after the

con-tacting asperities of the softer material has plastically

deformed (i.e yielded) Thus:

 ‘Law(s) of friction’ , Amontons in 1699 stated in the main,

that: ‘Friction is independent of the apparent area of contact

and proportional to the normal load between two [mating]

sur-faces.’

 Coulomb (1785), confirmed these ‘frictional laws’ with the

observation that: ‘The coefficient of friction* is substantially

in-dependent of the speed of sliding.’

*Coefficient of friction (µ) = F/N

Thus, the friction force is proportional to the perpendicular

force between contacting surfaces and is independent of the

surface area, or its ‘rubbing speed’.

Where:

Fn = Normal force,

σy = Yield pressure of the softer material

Thus, for most metallic materials, this close asperity contact between mating surfaces creates localised ‘cold

welding’ Prior to any sliding with respect to these two contacting surfaces, a force is necessary to continually shear potentially contacting and reforming asperity tips of these ‘welded junctions’ Hence, the total

fric-tional force ‘F f’ is given by:

Where:

τf  = Shear strength of softer contacting material,

Ar = Real contact area

From equations (i) and (ii), the actual coefficient of

friction (‘µ’) between these contacting surfaces will

be:

In majority of machining operations typified by the continuous turning of metallic workpieces, the

coef-ficient of friction (‘µ’) at the chip/tool interface can

vary quite considerably Its frictional variation, being influenced by any changes in: cutting speed, feedrate, rake angle contact regions – this latter factor is par-ticularly relevant for multi-functional tooling, where the cutting insert’s top face is not planar (i.e flat) In modern machining practice, the normal pressures ex-erted at the tool/chip interface are exceedingly high, typically when machining commercial grades of me-dium carbon plain steels these pressures are >3.5 GPa This high normal pressure at the ‘interface’ causes the real contact area to approach that of the entire area,

where: A r/ A a = unity! This means that under these cir-cumstances the ordinary laws of friction no longer

ap-ply, and the frictional force ‘F f’ is still represented by equation (ii), but is now independent of the normal

force ‘F n’ This acute change in the frictional behaviour during machining, means that the shearing action is not confined only to the interface asperities, but in-cludes workpiece material in the local substrate

In work published by Zorev in 1963, his ‘model’ considered the frictional behaviour in continuous metal cutting operations where no BUE was present (Fig 182) Under these machining conditions, Zorev

noted that the normal stresses at the tool/chip

inter-face were sufficiently high enough to cause ‘A r/ A a’ to

approach unity, over the zone denoted given by length

‘l st’ (Fig 182) – this being termed the ‘sticking region’ 80

– also see magnified quick-stop photomicrograph in

Fig 184(top) showing the affect on the resultant chip

The ‘sliding region’ , extends from from the end of the

‘sticking region’ to a point where the chip loses contact

with the rake face (i.e ‘l f ’-‘l st ’), here, the ‘A r /A a’ ratio is

less that unity – meaning that the coefficient of friction

is constant In 1964, Wallace and Boothroyd produced

evidence that the ‘sticking manner’ observed on the

underside of the chip had abruptly stopped They

ob-served that in an adjacent vicinity to that of the tool’s

cutting edge, grinding marks on the rake face were

im-printed onto the chip’s underside This phenomenon

indicated that no relative motion between the chip and

the tool had occurred, suggesting that here, the ‘real’

and ‘apparent’ areas of contact were ‘equalised’ in this

region Therefore, under ‘sticking friction’ conditions,

the ‘mean angle of friction’ on the rake face, depends

on the:

1 Form of the normal stress distribution,

2 Tool/chip contact length (i.e ‘l f’),

3 Mean shear strength of the chip material – in the

‘sticking region’ ,

4 Coefficient of friction – in the ‘sliding region’.

NB  It seems apparent that a simple numerical value

for the mean angle of friction, is inadequate when

at-tempting to completely describe the frictional

circum-stances on the rake face

In Fig 182, is schematically depicted the ‘frictional

model’ purported by Zorev (1963) Here, Zorev

sug-gested that the normal stress distribution (‘σ f’) on the

tool’s rake face, could be represented by the following

simple expression:

0 ‘Sticking region’ at the tool/chip interface is often termed the

‘stagnation zone’ , but in this case it appears on the formed/

sheared chip – depicted in Fig 184 (top) Here, is shown a

magnified and etched view of a quick-stop for an insert

cut-ting carbon steel at 150 m min– A ‘stagnation zone’ is present,

that follows the insert’s profile, with ‘softened’ workpiece

ma-terial protecting the tool, by a sticking/sliding action A ‘flow

zone’ occurs after ‘shear plane’ , visibly dividing undeformed/

deformed material.

Where:

‘x’ = Distance along the rake face – from the position where the chip losses contact with the tool’s face,

‘q’ and ‘y’ = are constants.

The maximum normal stress ‘σ fmax ’ occurs when ‘x’ equals ‘l f’ , so:

σfmax = qlf y

or, transposing with respect to ‘q’:

Substituting for ‘q’ in equation (iv), we get:

Thus, in the ‘sliding region’ , ranging from: x=0 to

x = lf – lst the coefficient of friction ‘µ’ is constant and, the distribution of shear stress ‘τ f’ along this region, is represented by:

τf = σf µ = µ σfmax (x/lf)y (vii)

So, in the ‘sticking region’ , the shear stress becomes a

maximum (‘τ st’), therefore from: x= lf – lst to x=lf:

By integrating equation (vi) to obtain the normal force

‘F n’ acting on the rake face (i.e from Fig 181), gives:

F n = aw�l f

 σ f max(x�l f)y dx

Where:

aw = Chip width (i.e width of cut)

The friction force ‘F f’ on the tool’s rake face, is ob-tained by:

F f = aw [τst l st+ l f�−lst

 µ σ f max(x�l f)y dx]

= τst aw lst + µ σfmax aw (lf – lst)+y/lf y(1+y) (x)

At position: ‘x = lf-lst ’ , the normal stress ‘σ f’ is given by:

‘τ st  /µ’ Moreover, from equation (vi), it is given by:

σfmax (lf-lst/lf)y

Therefore:

τst = µ σfmax (lf – lst/lf)y (xi)

By substituting equation (xi) in equation (x), it

simpli-fies the expression for ‘F f’ , as follows:

Ff = τst aw lst + τst aw (lf – lst)/1+y (xii)

Hence, the mean coefficient of friction on the rake face,

can now be found from both equations (ix) and (xii):

Mean friction angle = Ff/Fn = τst/σfmax (1+ylst/lf ) (xiii)

From equation (xi), the mean normal stress on therake

face (σ fav) is given by:

σfav = Fn/awlf = σfmax/1+y

Thus:

Substituting for ‘σ fmax’ in equation (xiii), produces:

Mean friction angle = arctan{τst/σfav [1+y(lst/lf )]} (xv)

However, from Zorev’s experimental work, he found

that the term:

τst [1+y(lst/lf )]/1+y

will remain relatively constant for a specified

work-piece material, across a diverse range of unlubricated

cutting conditions and, as a result, the equation for the

mean friction angle becomes:

Where:

‘K’ = a constant.

What Zorev’s equation (xvi) indicates, is that the mean

friction angle is somewhat dependent on the mean

normal stress on the rake face, allowing the result to

explain the effect of modifications in the working

nor-mal rake to that of the mean friction angle Thus, as the

value of ‘α’ (i.e working normal rake angle) increases,

the resultant component tool force being normal to

the rake face will decrease, so that the mean normal

stress will be reduced Moreover, from equation (xvi),

an increase in ‘α’ could be expected to increase the

mean friction angle This observation, has been

con-firmed by experimental work undertaken by Pugh

(1958), where an increase in ‘α’ was shown to result in

a a complementary increase with repect to the mean

friction angle – across an extensive range of workpiece materials

Since the passage of time when both of these ‘theo-retical models’ for the mechanics of machining and the frictional work produced by Ernst and Merchant and that of Zorev, respectively, were produced Consider-able effort and progress has been made into advances in

our present understanding of: ‘slip-line field modelling’ ,

‘thermal and frictional modelling of the cutting process’ ,

‘chip-flow analysis’ , ‘examination of the primary and secondary shear zones’ , non-orthogonal (three-dimen-sional) machining processes, advances in tool geometries with their accompanying force and shear relationships

and the advent of applications to machining

opera-tions employing finite element analysis (FEA) If all of

these subjects and other not listed, were only concisely mentioned, then the book would be of epic propor-tions! defeating the object of reviewing cutting appli-cations and trends in current practice However, the role of FEA, in both machining research and for the

‘dynamic and geometric modelling’ of applied residual stresses occurring in both the tool and workpiece in the anticipated cutting processes is worthy of discus-sion The topic of ‘computer simulation’ utilising an FEA approach, has become of increasing importance

of late and several companies have produced products

in this field However, to obtain some degree of con-sistency in the dialogue only one particular company’s FEA product will now be described

Computer Simulation of Machining Processes – an Introduction

By the application of computers to simulate and anal-yse machining processes, this provides tool manufac-turers and users alike, assistance in improving machin-ing efficiency and will also predict the likely cuttmachin-ing response to real-time machinability environments Most of the early two-dimensional simulation models for the mechanics of machining, focussed attention on shear-zone modelling These previous ‘models’ ignored vital factors such as the frictional conditions along and between the rake’s tool/chip interface, furthermore, any potential work-hardening and temperature effects

in this vicinity were also tended to be disregarded

Over the last two decades, the application of finite

ele-ment analysis (FEA) to that of three-dimensional

ma-chining applications, has been successfully applied to

the cutting process Major advantages of using FEA,

are its ability to accurately compute: complex material

property definitions; tool/chip interactions; non-linear

geometric boundary conditions – typified by the chip’s

surface; prediction of local variables – such as stress

and temperature distributions in the cutting locality

Several approaches for the numerical modelling

of machining operations and in particular, for that of metal cutting have been employed, such as ‘Lagrang-ian and Euler‘Lagrang-ian’ techniques In the former case which

has been utilised for over two decades, ‘Lagrang-ian methods’ utilise the tracking of discrete material

points Here, the technique is uses a predetermined line of separation at the tool’s point, which propagates

a fictitious crack ahead of the tool’s tip Previously, this

Figure 182 An idealised orthogonal cutting model of chip-tool friction, where:

• σf = normal stress,

• τf = shear stress,

• τst = shear strength of chip material in the sticking region,

• If = chip-tool contact,

• Ist = length of sticking region.

[Source: Zorev, 1963]

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routine precluded the resolution of the cutting edge

radius and an accurate resolution of the secondary

shear zone – due to severe mesh distortion In an

at-tempt to alleviate any form of element mesh distortion

‘adaptive remeshing techniques’ have been employed

to resolve the tool’s cutting edge radius Whereas the

‘Eulerian approach’ tracks volumes rather than

mate-rial particles, having the advantage of not needing to

rezone any distorted meshes Moreover the ‘Eulerian

technique’ , requires ‘steady-state free-surface tracking

algorithms’ and relied upon a particular bur

unrea-sonable assumption that a uniform chip thickness

oc-curred, further this method precluded the modelling

of either a segmented chip formation, or that of the

milling process The former technique of a ‘Lagrangian

FEA machining model’ will be reviewed (Fig 183), as it

has the integrating ability to achieve ‘adaptive

remesh-ing’ with explicit dynamics and tightly coupled

tran-sient thermal analysis, allowing it to ‘model’ the

com-plex interactions between the cutting tool’s geometry

and that of the workpiece

Lagrangian FEA Simulation

Machining Modelling

In Fig 183, just a few images of the Lagrangian FEA

machining model are depicted for several

applica-tions of machining operaapplica-tions This simulation

mod-elling technique contains a tightly coupled

mechanical material response capability, this being a

vital factor for any elevated temperatures that occur

at the tool/chip interface, furthermore, having a fully

adaptive mesh generation ability Hence, the material

modelling facility forms an intergral part when

at-tempting to predict the workpiece’s material

behav-iour, under high strain and increased stress

condi-tions This advanced modelling capability ensures the

accurate capture of any strain-hardening, or thermal

softening effects coupled to rate sensitivity properties,

for a given set of material conditions Many of today’s

ferrous and non-ferrous materials, together with

‘ex-otic materials’ such as nickel and titanium alloys can

also be successfully simulated, across a diverse range

of single- and multi-point cutting tools (e.g turning,

milling, drilling, broaching, sawing, etc.)

Machining Simulation – Validation

Not only can simulation techniques of the type shown

in Fig 183 be utilised for workpiece material

machin-ing modellmachin-ing: work-hardenmachin-ing; thermal-softenmachin-ing effects; machining-induced residual stresses; induced temperature effects and heat-flow analyses with the

‘adaptive’ ‘Lagrangian FEA machining model’ The

‘model’ can also reasonably accurately predict: both

two- and three-dimensional cutting force magnitudes Invariably, the individual cutting force components can be closely validated to actual machining working practice, the same can also be said for a comparison

of a ‘dynamically-modelled chip’ to that of an actual chip’s morphology – including both its chip-curling tendency and any chip segmentation occurring These validated simulation capabilities enable the cutting process to be improved, by using the:

overall production cycle times,

to improve tool life and part quality,

flank wear and how this wear land influences sub-sequent: temperatures, pressures and forces,

onset and magnitude of these unwanted effects,

poten-tial machined component fatigue and aids in part deformation analysis

NB The influence that tool coatings have on the

dynamic machining efficiency can also be reviewed, plus the capability to customise the tooling with chip-breakers to improve chip-curl, or chip evacu-ation abilities

Computer machining simulation of the type illus-trated in Fig 183, can be integrated into an overall CAD/CAM package, enabling a range of significant advantages to accrue, without having to operate a costly and time-consuming task of undertaking an extensive machinability trial This off-line machining simulation facility, allows: realistic cycle-time calcula-tions including cut-and non-cut timings; visualisation

of tool paths to high-light and then avoid any localised

‘power-spikes’ occurring during a machining opera-tion and many other useful producopera-tion features The use of a dynamic FEA machining simulation package similar to the one mentioned and illustrated

in Fig 183, adds a scientific element to the understand-ing of the overall machinability of specific workpiece materials and associated tooling Such machining simulation offers not only a visual interpretation to

Figure 183 Simulated machining processes [Courtesy of Third Wave Advant Edge]

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