Mechanics of Metal Cutting

CHAPTER – 2

THEORY OF METAL CUTTING

1.1 Introduction:

Metal cutting process forms the basis of engineering industry & is involved either

directly or indirectly in the manufacture of nearly every product of our modern civilization. The

theory of metal cutting is of vital importance and a basic knowledge of fundamentals of

machining of materials and of the theory of metal cutting will help to develop scientific approach

in solving problems encountered in machining.

A metal cutting tool is the part of a metal cutting machine tool that, in the cutting process,

acts directly on the blank from which the finished part is to be made. The metal cutting process

accompanied by deformation in compression, tension & shear by a great deal of friction & heat

generation is governed by definite laws. Metal cutting operation involves three basic

requirements. (1) There must be a cutting tool that is harder and wear resistant than the work

piece material, (2) there must be interference between the tool & the work piece as designated by

the feed and depth of cut, and (3) There must be relative motion or cutting velocity between the

tool & the work piece with sufficient force and power to overcome the resistance of work piece

material. As long as above three conditions exist, the portion of the material being machined that

interferes with free passage of the tool will be displaced to create a chip.

1.2 Classification of production process:

The metals are given different usable forms by various processes. These processes may

be classified as under. Metal Forming

Chip-forming Process Chip-less Process

(Metal Cutting)

Continuous-contact Intermittent Continuous Impact or

Cutting cutting (Rolling, Spinning Intermittent

Etc.) Contact

(Forgoing,

Drop-stamping)

Single-edge Double Sizable Ground Chips

Cutting edged Swarf (Honing, Grinding,

(Turning, cutting (Milling) etc.)

Shaping, (Drilling)

Boring)

In chip removal processes the desired shape and dimensions are obtained by separating a layer

from the parent work piece in the form of chips. During the process of metal cutting there is a

relative motion between the work piece & cutting tool. Such a relative motion is produced by a

combination of rotary and translatory movements either of the work piece or of cutting tool or of

both. These relative motions depend upon the type of metal cutting operation. The following

table indicates the nature of relative motion for various cutting processes. In chip less processes

the metal is given the desired shape without removing any material from the parent work piece.

Table 1.1

Sr.

No. Operation Motion of work piece Motion of cutting tool

01 Shaping Fixed Translatory

02 Turning Rotary Translatory

03 Drilling Fixed Rotary & Translatory

04 Milling Translatory Rotary

05 Hobbing Rotary & translatory Rotary

06 Honing Fixed Rotary

07 Grinding (surface) Translatory Rotary

08 Grinding (Cylindrical & Center less) Rotary & translatory Rotary

1.3 Basic elements of cutting tools: The cutting tool consists of three basic elements (1) cutting element or Principle

element – This is the element, which is actually fed into the material of work piece to cut the

chips ex. In drilling lips (or cutting edges) are cutting elements. (2) Sizing element – The part,

which serves to make up any deficiencies of cutting element after sharpening, is sizing element.

It imparts final shape to the machined surface and also provides guidance in tool operation ex. In

drill sizing element; (flute portion) immediately follows the lips). (3) Mounting element – It

serves for securing the tool in machine or holding it in hand of worker ex. In the twist drill the

shank is mounting element. The cutting & sizing element taken together is referred as working

element of the tool.

1.4 Machining parameters:

1.4.1 Cutting Speed (V) – It is the travel of a point on cutting edge relative to surface of cut in

unit time in process of accomplishing the primary cutting motion. It is denoted by ‘V’. The unit

of cutting speed is m/min.

In lathe work for turning a blank of diameter ‘D’ mm, (The diameter of machined surface

is ‘Do’ mm.) rotating at a speed ‘N’ (rpm) the cutting speed at periphery (maximum) is given by.

V = π D N /1000, m/min ........………………………….. 1.4.1

Fig. 1.1 Elements of cutting process in turning

FEED

SPEED

DEPTH OF CUT

From this formula it is easy to find rotational speed

N = 1000 V / Π From figure 1.1. it is evident that the cutting speed varies along the cutting edge from

maximum at point ‘m’ to minimum at point ‘K’ though the rotational speed is same.

In drilling a work piece with a drill of diameter ‘D’ mm., rotating at a speed ‘N’ (rpm) the

cutting speed will vary from zero at center to maximum at periphery given by eq

1000

πV =

Similarly in facing the cutting speed varies from zero at center to maximum at periphery.

1.4.2 Feed (Feed rate) (f, fm)

It is the travel of the cuttin

surface in unit time. The feed may be expressed as distance traveled by the tool in one minute

(fm) or distance traveled by the tool in one revolution (f). The terms ‘f’ and f

f = fm / N, mm/rev

In lathe work, distinction is made between

direction parallel to work axis, cross feed when tool travels in a direction perpendicular to the

work axis, and angular feed when tool travels at an angle to work axis (for example, in turning

tapered surface.)

1.4.3 Depth of cut: (d)

It is the thickness of the layer of metal removed in one cut or pass; measured in direction

perpendicular to machined surface. The depth o

feed motion and, in external longitudinal turning; it is half the difference between the work

diameter and the diameter of machined surface obtained after one pass.

d = (D –

1.4.4 Machining time:

The machining time is calculated by dividing the length of cut, or the length of stroke, by

the feed of the tool in mm/min (f

Tm = L/fm., min. or Tm = L/f.N, min., Where, L = Length of cut.

1.4.5 Metal Removal Rate: (w)

It is the volume of metal removed in unit time expressed as mm

calculate time required to remove specified quantity of material from the work piece.

w = Ac × V = ( b.t) ×Where, Ac = Cross sectional area of chips.

Fig. 1.2 Sketches Showing V, f and d

From this formula it is easy to find rotational speed

Π D ................... 1.4.2

is evident that the cutting speed varies along the cutting edge from



eed will vary from zero at center to maximum at periphery given by eq

m/min, 1000

N D


It is the travel of the cutting edge in the direction of feed motion relative to the machined


) or distance traveled by the tool in one revolution (f). The terms ‘f’ and f

/ N, mm/rev . . . . . . . . . 1.43

In lathe work, distinction is made between longitudinal feed, when tool travels in a


feed when tool travels at an angle to work axis (for example, in turning


perpendicular to machined surface. The depth of cut is always perpendicular to the direction of


diameter and the diameter of machined surface obtained after one pass.

– Do)/2 mm ............ 1.4.4


the feed of the tool in mm/min (fm). Thus

., min. or Tm = L/f.N, min., Where, L = Length of cut.

(w)

It is the volume of metal removed in unit time expressed as mm


V

Cross sectional area of chips.

is evident that the cutting speed varies along the cutting edge from



eed will vary from zero at center to maximum at periphery given by eqn 1.41.


g edge in the direction of feed motion relative to the machined


) or distance traveled by the tool in one revolution (f). The terms ‘f’ and fm are related by

feed, when tool travels in a


feed when tool travels at an angle to work axis (for example, in turning


f cut is always perpendicular to the direction of



It is the volume of metal removed in unit time expressed as mm3/min. It helps to


b = Width of chip.

t = thickness of chip.

To reduce machining cost machining time should be less i.e. the metal removal rate should be

high. To achieve this following facts should be considered.

1) Proper cutting tool material should be selected.

2) Correct tool (angle) geometry should be produced or ground on tool

3) The tool should be rigidly held to avoid vibrations.

4) Depending on the rigidity of machine – tool system maximum values of speed & feed

should be selected.

A process, which removes metal at a faster rate, may not be the most economical process,

since the power consumed & cost factors must be taken into account. Due to this, to compare

two processes, the amount of metal removed per unit of power consumed in unit time is

determined. This is called “ Specific metal removal rate” and is expressed as, mm3/w/min, if

the power is measured in watts.

1.5 Basic shape of cutting tools: Wedge.

Almost all cutting tools used in metal cutting operations consist of basic form of a wedge,

which is defined as one form of inclined plane in shape of a triangular prism. Assume that a

wedge under the action of force P is penetrating into another body at a constant speed as shown

in Fig.1.3. N

β K

M β N N

N P

P

L

Fig. 1.2 Force acting on an indenting wedge Fig. 1.4 Force triangle at the wedge check

In fig.1.3. The body resists the motion of the wedge. The reaction N.N. appear at the cheeks of

the wedge. The forces N.N. are perpendicular to the cheeks in absence of friction. From the

equilibrium of forces (fig.1.4)

2sin2

1

KM

2/KL2

1

KL

KM

P

N

β=

==

Work surface

K

N

β

N P β

N L 900

α P

Fig.1.4 Orientation of the wedge during Fig.1.5 Orientation of the wedge

the parting or cutting by the indentation process during the separation of chips

Thus, the mechanical advantage in force is dependent on the wedge angle 'B'. The smaller the

angle of wedge, the greater will be the gain in force. In other words, the wedge angle 'β'

determines the resisting force of the cutting edge.

The cutting edge must be oriented at certain required angles with the work surface

depending on nature of operation to be performed. Fig.1.5 shows that the wedge must be set at

right angles to the work surface, so that the driving force "P" is in the direction of parting.

Fig.1.6 shows during chipping the wedge must be set at an angle inclined to work surface so that

separation of chip can be done.

Thus for the wedge two geometric parameters can be defined i.e. (1) The wedge angle 'β'

and (2) the axis of symmetry along which 'P' acts. In addition to above, two more parameters

are introduced to confirm conditions of chipping action. These parameters are set with respect to

velocity Vector, 'V' and are defined as (3) cutting angle 'δ' and (4) clearance angle;, as shown in

fig.1.7. The sign convention for describing these angles are set wr.t. left handed cork screw rule

with "Z" axis coinciding with the direction of the velocity vector, V, and the cutting edge lying

along 'Y' axis. Hence, 'δ' & 'α' are measured positive, when moving from 'Z' to 'X' axis as shown

in fig.1.7. The parameter ' γ ' defines the inclination of the top face of the wedge (called Rake

face) w.r.t. velocity vector V, while the parameter 'α' describes the relief provided from the

bottom face of the wedge (called flank), often another derived parameter, called (5) Rake angle

'γ', is used to describe the inclination of the top face of the wedge. This is derived parameter

given by

γ = 900 - δ .

However if δ > 90, then ' γ ' is negative. Thus from this equation it may be seen that

while 'δ' is always positive the rake angle can become positive or negative depending an value of

angle 'δ'.

v

δ

δ β

β

However in Fig.1.7 (b) the cutting edge of the wedge has been set at right angle to

velocity vector, V, along Y-axis. A new situation arises when the cutting edge. Shifts from Y

axis and another parameter called (6). Inclination angle is needed to describe the orientation of

the wedge with respect to velocity vector "V". The angle " " is measured positive when it lies

in the direction or rotation of left hand cork crew rule in the x-y-z system as shown.

When wedge shaped tool is set w.r.t. the work-place the actual values of the rake angle 'ϒ'

and clearance angle 'α' depend on the actual direction of velocity vector V with respect to the

wedge. The effect of setting the wedge has been shown in fig.1.8. It the wedge is set high w.r.t.

line of centers, the rake angle increases to (χ + ϒ) from ϒ & clearance angle decreases to ( α - χ )

from 'α' opposite is the case when wedge is set low.

1.6 Types of metal cutting processes:

The metal cutting processes are classified in to two types, on the basis of angular

relationship between cutting velocity vector V, & the cutting edge of the tool.

(1) Orthogonal cutting process (two dimensional cutting)

(2) Oblique cutting process (three dimensional cutting)

In orthogonal cutting the cutting edge of the tool is perpendicular to cutting speed direction. In

oblique cutting, the angle between the cutting edge & cutting velocity vector is different from

900. fig 1.9 & fig.1.10

Point Orthogonal Cutting Oblique Cutting

1. Definition - The cutting edge of tool

perpendicular to cutting

speed V;

The Cutting edge of the tool is inclined at an

angle other than 900 to V.

2. Alternative name Two dimensional cutting Three dimensional cutting

3. Volume of metal

removal for a cutting

condition.

Less metal removal due to

square cutting condition.

More metal removal, as greater area of chip is

removal for same depth of cut & other

conditions.

4. Tool life - Shorter Longer

5. Friction & -

Chatter

More Less, as small amount of heat developed due

to friction at the job tool interface.

6. Chip flow &- Shape Chip coils in a tight flat

spiral

Chip flow sideways in along curl.

7. Suitable example Slotting, Parting Turning, Drilling.

1.7 Chip formation (MECHANISM):

The portion of the material that has been cut away from the work material is called the chip.

Observations during metal cutting reveal several important characteristics of chip formation:

1) The cutting process generates heat,

2) The thickness of chip is greater than the thickness of layer from which it came.

3) The hardness of the chip is usually much greater than the hardness of parent material,

Fig1.9 Fig1.10

4) The above relative values are affected by changes in cutting, conditions & in properties of the

material to be machined to give chip that range from small lumps to long continuous ribbons.

These observations indicates that the process of chip formation is one of deformation or

plastic flow of the material with the degree of deformation dictating the type of chip that will be

produced. Fig. 1.11 shows progressive formation of a chip using a wedge shaped (single point)

tool. At “a” tool contacts the work piece material. At “b” compression of material takes place at

point of contact. At “c” the cutting force overcomes the resistance of penetration of tool is

begins to deform by plastic flow. As the cutting force increase, either a rupture or plastic flow in

direction generally perpendicular to face of the tool occurs & the chip is formed as shown at “d”.

tool tool tool tool

a b c d

Fig. 1.11 Progressive formation of a metal chip.

The mechanism of deformation can be seen from fig. 1.12. Generally speaking there is

always deformation of metal lying ahead of the cutting edge by a process of shear. Here with

application of force the metal deforms by shear in a narrow zone extending from cutting edge to

the work surface. This zone is treated as single plane for purpose of mathematical analysis & is

commonly referred to as Shear Plane. The angle, which the shear plane makes with direction of

tool travel, is known as Shear angle.

The process of plastic deformation occurring along the plane elongates the individual

crystals of metal in the general direction indicated by the shear angle. This tends to produce

chip. That is thicker than the layer of the parent metal from which it came. Chip material moves

the tool face in layers of distorted material. Each layer is pushed outward by a fixed amount

w.r.t. Its adjacent layer & retains this position as the whole chip slides up the tool face. The

distorted layers now by means of phenomenon of slip & the layers are called slip planes. The

number of slip planes depends upon the lattice structure of parent workplace material. The

distortion of layers tends to strengthen them (work hardening or strain hardening) & therefore the

hardness of chip is much greater than the hardness of the parent material.

Fig1.12

Thus in simple language the mechanism of chip formation in any machining operation is

a rapid series of plastic flow & slip movements ahead of the cutting edge. The degree of plastic

flow ahead of the cutting tool determines the type of chip that will be produced. If the w/p

material is brittle & has little capacity for deformation before fracture the chip will separate

along the shear plane to form what is known as a discontinuous segmental chip. Material that are

more ductile & have capacity for plastic flow will deform along the shear plane without rupture.

The planes tend to slip & weld to successive shear planes, & the result is a chip that flows in a

continuous ribbon along the face of tool. This is known as a continuous chip & is usually much

harder than the parent material because of its strain hardened conditions.

1.8. Types of Chips:

The tool engineer's handbook lists four different types of chips viz.

1) Segmental chips or Discontinuous chips

2) Continuous chips

3) Continuous chip with BUE or BUE chips.

4) Inhomogeneous chips.

1) Discontinuous Chips: These chips are in the form of small individual segments, which may

adhere loosely to each other to form a loose chip. These chips are formed as result of machining

of a brittle material such as gray cast iron or brass castings, etc. These chips are produced by

actual rupture or fracture of metal ahead of the tool in brittle manner. Since the chips break up

into small segments and also shorter chips have no interference with work surface. The friction

between chip & tool reduces resulting in better surface finish. These chips are convenient to

collect, handle & dispose of during production runs. The conditions favorable for formation of

discontinuous chips are:

1) Brittle & non ductile metals (like cast iron brass castings Beryllium, titanium etc.)

2) Low cutting speed.

3) Small rake angle of the tool.

4) Large chip thickness.

2) Continuous Chips: These chips are in the form of long coils having uniform thickness

throughout. These chips are formed as result of machining of relatively ductile materials where

definite successive raptures do not take place, at high cutting speeds. Due to large deformation

possible with ductile materials longer continuous chips are produced. These are referred to as

“ideal” chips because,

i) Due to stable cutting excellent surface finish is obtained.

ii) Low friction between chip & tool & hence heat generation is low and,

iii) Power consumption is low. On the other hand, these chips are difficult to handle & dispose

off. Chip coils can cause injury to operation. However these problems can be avoided by use of

“chip breakers” behind to cutting edge. The conditions favorable for formation of continuous

chips are

1) Ductile material

2) High cutting speeds.

3) Large rake angle of tool.

4) Small chip thickness.

5) Sharp cutting edge.

6) Efficient cutting fluid.

7) Low friction between chip tool interfaces.

3) BUE Chip (or continuous Chip with BUE): These chips are also produced in the form of

long coils like continuous chips, but they are not as smooth as continuous chips. These chips are

characterized by formation of built up edge on the nose of the tool owing to welding of chip

material on to tool face because of high friction between chip tool interfaces. Presence of this

welded material further increases the friction leading to building up of the edge, layer by layer.

As the built-up edge continuous to grow, the chip flow breaks a portion of it into fragments.

Some of them are deposited on the work piece material while the rest are carried away by the

chips. The hardness of this BUE is two to three times higher than the work piece material. This

is the reason why the cutting edge remains active even when it is covered with built-up edge.

The only point in favor of BUE is that it protects the cutting edge from wear due to moving chips

and the action of heat. This brings about an increase in tool life. These chips normally occur

while cutting ductile materials with HSS tools with low cutting speeds. Chips with BUE are

under desirable as they result in higher power consumption poor surface finish and higher tool

wear. Generally speaking any change in cutting conditions that will eliminate or reduce BUE is

desirable, since high friction between chip & tool face is major cause of BUE. Any means of

reduction of friction such as use of lubricant & adhesion preventing agent is often effective to

reduce BUE, especially when it is necessary to operate at low cutting speeds. Tool material with

inherent low coefficient of friction or a high polish on tool face can also reduce friction & hence

BUE.

The conditions favorable for BUE chip are.

1) Ductile material

2) Low cutting speed.

3) Small rake angle of tool.

4) Dull cutting edge.

5) Coarse feed.

6) Insufficient cutting fluid.

7) High friction at chip tool interface.

4) Inhomogeneous Chip: These chips are produced owing

to non uniform strain set up in material during chip

formation and they are characterized by notches on the free

side of chip, while the side adjoining the tool face is

smooth. The shear deformation which occurs during chip

formation causes temperatures on shear plane to rise which

in turn may decrease the strength of material & cause

further strain if the material is poor conductor. This process

when repeated several times results in a large strain at the

point of initial strain. Then a new shear plane will develop

some distance from first and deformation shifts to this

point. The resultant chip is banded with regions of large

and small strain. This is characteristic of metals suffering

marked decrease in yield strength with temperature and poor thermal conductivity. These chips

are produced while machining some steels and titanium alloys at medium cutting speeds.

Table: Factors responsible for the formation of different types of chips.

Factors Types of chips

Discontinuous Continuous With BUE Inhomogeneous

1. Material Brittle Ductile Ductile Which Shows decreased in Yield

Strength with temp. & Thermal

conductivity medium.

-

-

-

-

-

-

-

2. Cutting speed Low High Low

3. Tool geometry Small rake Large rake Small

4. Friction - Lower Higher

5. Chip thickness Large Small Small

6. Cutting fluid - Efficient Poor

7. Feed - - Coarse

8. Cutting edge - Sharp Blunt

1.9 Cutting Ratio (Chip thickness ratio):

During the cutting the mean chip thickness is always greater than, the underformed chip

thickness which is actually fed (or the thickness of metal from which it came) in orthogonal

cutting. The ratio of chip before removal to its thickness after removal from material being cut is

termed as the "Cutting ratio", the inverse of cutting ratio is known as "Chip compression" factor

or chip reduction coefficient. However even in the orthogonal cutting the cross section of chip is

not always rectangular. The chip has a tendency to move side ways so that the width of chip is

more than width of cut. In addition thickness of chip is not uniform throughout its width. It

tends to be thicker at center and tapers slightly towards sides. However for the purpose of

analysis chip width is taken to be equal to width of cut & thickness is taken to be uniform

throughout its width. The chip thickness ratio is always less than unity.

Thus, Cutting ratio, r = t/tc

Where t = undeformed chip thickness (i.e. before cutting) and

tc = mean thickness of chip ( i.e., after cutting )

Chip reduction coefficient K = 1/r

The following methods can be used to determine cutting ratio

1) The cutting ratio "r" can be obtained by direct measurement of "t" & "tc". However since

underside of chip is rough the correct value of "tc" is difficult to obtain and hence tc can be

calculated by measuring length of chip (1c) and weight of piece of chip "W".

tc = W/ (bc .1c.ρ)

Where, bc = length of chip

1c = width of chip

ρ = Density of material assumed to be unchanged during chip

formation.

2) Alternatively, the length of chip (1c) & length of work (l) can be determined. The length

of work can be determined by using a work piece with slot, which will break the chip for each

revolution of work piece. The length of chip can be measured by string.

It can be shown that r = 1/1c as under. When metal is cut there is no change in volume of

metal cut. Hence volume of chip before cutting is equal to volume of chip after cutting i.e.

1.b.t. = 1c.b.tc

or l.t. = 1c.tc (assuming b = bc)

l/lc = t/tc = r

3) Cutting ratio can also be determined by finding chip velocity (Vc) and cutting speed (V).

The chip velocity (Vc) can be accurately determined by determining length of chip with a string

for a particular cutting time measured with the help of a stopwatch. It can be shown that r =

Vc/V, as under. From the continuity equation, we know that volume of metal flowing per unit

time before cutting is equal to volume of metal flowing per unit time after cutting.

i.e. V.b.t. = Vc .b.tc

or Vc/V = t/tc = r (assuming b = bc)

1.10 Shear Angle:

The shear angle is the angle made by shear

plane with the direction of tool travel. In fig 1.7a it is

the angle made by the line AB with direction of tool

travel. The value of this angle depends on cutting

conditions, tool geometry, tool material & work

material. If the shear angle is small, the plane of shear

is larger, the chip is thicker and therefore higher fore is

required to remove the chip. On the other hand, if the

angle is large, the plane of shear will be shorter, the

chip is thinner; hence less force is required to remove

the chip. The shear angle is therefore important

parameter in metal cutting.

The shear angle can be determined by various methods. It can be obtained by direct

measurement from the photomicrograph of a partially formed chip. This is done by suddenly

withdrawing the tool during the course of cutting action with a quick stop mechanism. The

section of metal in the vicinity of partially formed chip is cut from work piece, ground, polished

& etched for study. This method is not very convenient. The shear angle is generally

determined from the cutting ratio "rc" by the equation.

tanΦ = γ−

γ

sinr1

cosr

c

c

where γ = rake angle

The derivation of the above equation is as follows. from fig 1.7 a

t1 = AB sin φ

t2 = AB sin cos (φ - γ )

φ

γφ+γφ=

φ

γ−φ==

sin

sin sin cos cos

sin

)( cos

r

1

t

t

c1

2

γ+γφ= sin cos cotr

1

c

γ

γ−=

γ

γ−

=φcosr

sinr1

cos

sinr

1

cotc

cc

γ−

γ=φ

sinr1

cosrtan

c

c

1.11 Velocity relationships in orthogonal cutting

There are three velocities in orthogonal cutting process, namely

(i) Velocity of chip (Vf) which is defined as the velocity with which the chip moves over the rake

face of the cutting tool.

(ii) Velocity of shear (Vs) is the velocity with which the work piece metal shears along the shear

plane.

(iii) Cutting velocity (Vc) is the velocity of tool relative to the work piece.

Cutting velocity Vc and rake angle α are always known Vf and Vs can be calculated with

the help of following relations, which refer to the velocity diagram of Fig.30.15.

)( cos

sin,VV cf α−φ

φ=

Vs = Vc, )( cos

cos

α−φα

where α is the rake angle,

φ is the shear angle.

From the principle of kinematics, the relative velocity of two bodies (tool and chip) is

equal to the vector difference between their velocities relative to the reference body (here the

work piece). The vectors of these three velocities - Vc, Vs and Vf - should form a close velocity

diagram (Fig.30.15)

and

Thus Vc = Vs + Vf

Refer Fig. 30.15(b)

From right-angled ∆ ACE

φ=φ=φ= sin.Vsin.AEAC or sinAE

ACc

From right -angled ∆ ABC

)( cos,V)( cos,ABAC or )( cosAB

ACf α−φ=α−φ=α−φ=

From Eqs. (a) and (b)

φ=α−φ sin.V)( cos.V cf

or )( cos

sin.VV cf α−φ

φ=

Consider ∆ ADE

α=α= cos.VDE or cosAE

DEc

Consider ∆ BDE

)( cosBE

DEα−φ=

DE= Vs )( cos α−φ

or

From Eqs. (c) and (d)

Vs. cos (φ -α) = Vc.cos α

or )( cos

cos.VV cs α−φ

α=

1.12 Shear Strain:

During the process of chip formation, each undeformed layer of material passes through the

shear plane and undergoes considerable plastic deformation. Shear strain "εεεε" can be defined as

the ratio of displacement of the layer ∆∆∆∆S along the shear plane to the thickness of layer '∆∆∆∆x'.

Thus shear strain can be related to the shear angle φφφφ and rake angle "γ" by the following

equation:

εεεε = x

)tan(xcotx

x

s

∆γ−φ∆+φ∆

=∆∆

= Cot φ + tan (φ - γ)

or εεεε = )cos(sin

cos

γ−φφγ

This relation can be obtained from the pack of inclined cards model suggested by Prof.

Pushpanen. In this model the formation of chip and its motion along the tool face can be

visualized from an idealized model in which a stack of inclined (playing) cards is pushed against

the tool (fig.1.16 a). As the tool advances, segments, which had been part of the work place,

become part of the chip.

From this figure it can be

seen that card closest to

the tool point slips to a

finite distance relative to

the uncut material as tool

point slips to a finite

distance relative to the

uncut material as tool advances. When the tool point reaches the next card, the previously lipped

card moves up along the tool face as a part of the chip.

BA = BE + AE

BA = ∆x cot φ +∆x cot {90 - (φ -γ)}

= cot ( )γ−φ+φ Tan

but from velocity relations

γ−φγ

=cos

cos

v

vs

φ

=εsinv

v s

1.13 Undeformed chip thickness:

The underformed chip thickness "t" can be estimated by referring fig. 1.17. Where two

consecutive cuts have been shown and various parameters such as feed f, depth of cut d, width of

cut b, thickness of undeformed chip t & chip thickness tc have been marked. It can be easily seen

that the following relations exist.

t= f sin φφφφp b=

psinφd

p

φp

( )( )γ−φ−+φ==ε 90cotcotCE

BA

φφφφp=900

It is clear that the uncut chip thickness depends upon the primary cutting edge angle as

shown in fig.1.18. In fig. 1.18 (e) a where φp = 900, the uncut chip thickness, t = feeds "f"

(mm/rev ) & width of cut b = depth of cut "d".

1.14 Cutting forces:

The force system in general case of conventional turning process is shown in Fig.1.19 a.

The resultant cutting force "R" is expressible by its components: "Px" known as the "feed force"

in the direction of tool travel. "Py" called as "thrust force" in the direction perpendicular to the

produced surface; and "Pz" the "cutting force" or "main force" acting in the direction of cutting

velocity vector. These directions have been chosen for their suitability of being determined by

properly designed tool force dynamometers.

After determining the individual components Px, Py & Pz the resultant force, "R" can be

evaluated as

R = (Px + Py + Pz )1/2

= 222 ZYXPPP ++ ........ 1.14.1

This three-dimensional force system can be reduced to a two-dimensional force system if in

orthogonal plane 0π the forces are considered in such a way that the entire force system is

contained in the considered state, when

R = 2y x

2z PP + ..... . . . 1.14.2

Pxy = 2y

2x PP + ..... . . . 1.14.3

This is possible only when Pxy is contained in plane π0 which is possible only under conditions of

free orthogonal cutting. This corresponds to 'orthogonal system of first kind' for which

conditions are:

i) 0<φ< 90

ii) λ = 0

iii) The chip flow direction lies on the plane π0.

Fig. 4.10 shows the cutting forces for the case of orthogonal system of the first kind.

An orthogonal two-dimensional system of second kind can be obtained by choosing λ and φ

in such a manner that either Px or Py can be made zero.

For the orthogonal system of second kind either

i) "Py" is made zero by having λ = 0 and

ii) φ = 90 when two dimensional force system is

R = 2 x

2z PP +

... . . . . 1.14.3

Fig. 4.11 shows the disposition of cutting forces in plane

orthogonal turning with λ = 0 and φ = 90.

Another, alternative way of having an orthogonal system of the second kind is to have Px=0

during radial turning or facing operation, when

R = 2y

2z PP +

Fig. 4.12 shows the disposition of cutting forces in plane orthogonal radial turning or facing with

λ = 0 and φ = 0.

However out of all the above cases shown in fig 4.10 4.11 and 4.12 the cutting in the first

two cases is "non free" or 'restricted" type where the auxiliary cutting edge is also active in

causing deviation of chip flow direction from the orthogonal plane.

The contribution of auxiliary cutting edge is to deviate Pxy from the orthogonal plane.

This deviation is small & neglected if the depth of cut is very large compared to feed, such

process is called "Restricted Orthogonal cutting.

However during cutting of a thin pipe with a cutting edge whose length is

considered to be very large compared to the width of cut, a "pure" orthogonal cut of first or

second kind could be obtained. The principal schemes of metal cutting shall be based on pure

orthogonal cutting from which schemes for oblique or other continuous and intermittent cutting

processes like drilling, milling, etc., can be derived by similarly principles.

FIG1.22 FORCES IN METAL CUTTING

In the lathe tool dynamometer the two components of the resultant force can be measured

by selecting suitable orthogonal cutting set up as shown in fig. 1.22. The resultant cutting force

is carried by the shear plane as well as by chip tool interface on tool face "R" can be resolved

into friction force "F" & normal force "N" on the shear plane. "R" can be resolved into shear

force, "Fx" inclined at an angle φ with direction of tool travel or along the shear plane and

backing up force "E"N setup by material normal to "F".

1.15 Merchant's Analysis (Theory) : Earnest & Merchant (1941) analysed the mechanics of metal cutting in order to develop

mathematical relationship connecting the variable in metal cutting the model is based on the

minimization or rate of energy dissipation. The simplify the mathematical relationship he made

following assumptions:

1) The chip behaves as a free body in stable equilibrium under the action of two equal,

opposite and collinear resultant forces viz. R & R.

2) The tool edge is sharp.

3) The work material suffers deformation across a thin shear plane.

4) This is no side spread (or the deformation is two-dimensional).

5) There is uniform distribution of normal & shear forces on the shear plane &

6) The work material is rigid, perfectly plastic (or behaves like ideal plastic)

7) As, (shear plane area). Ts (shear stress) & "B" (Friction angle), are constant & are

independent of shear angle 'φ'

Forces on the chip (Merchant’s Analysis, theory)

From the concept of chip formation and measuring force Ft and Ff with a cutting tool

dynamometer, Merchant was able to build up a picture of forces acting in the region of cutting

which give rise to plastic deformation and sliding of the chip down the tool rake face.

See fig 30.16(a)

The forces exerted by the work piece on the chip are

Fc - Compressive force on the shear plane.

Fs - Shear force on the shear plane.

The force exerted by the tool on the chips are

N - Normal force at the rake face of tool.

F - Frictional force along the rake face of tool.

The forces acting on the tool and measured by dynamometer are

Ft - tangential or cutting force

Ff - feed force

Angle α is tool rake angle, φ is shear plane angle and

β is the angle of friction

(a) Graphical Treatment

Using the concept explained in fig. 30.16(a) it is now possible to find graphically the magnitude

of force Fc , Fs, N and F.

The vector diagram of forces is constructed as follows [fig 30.16(b)] Draw Ff and Ft to

some convenient scale and joint AB to obtain their resultant. Bisect AB and draw a circle having

the resultant force as its diameter. Set off BE, making angle φ with force Ft, to cut circle at E.

Join EA. The magnitudes of Fs and Fc are now known. Set off a line BG at an angle (90-α) with

Ft (Ft is vertical and BB' is horizontal, ∠DBB' is 900) Join GA. The magnitude of force N and F

are thus known, as also the coefficient of friction at the chip tool interface (F/N). Angle BAG is

the angle of friction between chip and tool.

Tan β = F/N

(b) Analytical Treatment [See fig 30.16 (b)]

F = GH + HB = AI + HB

Or F = Ff. cos α + Ft. sin α ………….30.17

N = AG = DH - DI

= Ft.cos α - Ft. sin α …………..30.18

Now α−α

α+α=

sinFcosF

sinFcosF

N

F

ft

tf

Dividing R.H.S. by cos α

α−

α+=

tanFF

tanFF

N

F

ft

tf …………..30.19

The resultant tool force, R (eq 30.2) can be resolved into two components N and F normal to and

along the rake surface respectively, Fig. 30.16(b). Since F must be the friction force due to the

existence of the normal load N, as per usual convention.

F/N = µ …………..30.20

Where µ is the average coefficient of friction between the chip and the tool. From eqs 30.16,

30.19 and 30.20.

α−

α+=β==µ

tanFF

tanFFtan

N

F

ft

tf …………….30.21

Fs = Ft. cos φ - OD

Or Fs = Ft. cos φ - Ff. sin φ

Fc = AO + OE

Fc = Ff. cos φ + Ft. sin φ

In ∆ AGB, ∠GBA = 180 – 90 - β = 90 - β

Hence ∠ABD = 90 – α - (90 - β) = 90 – α - 90 + β = β - α

Now Ft = BD

From ∆ ABD )( cosR)(or AB

Ft)(or BDα−β=

Thus Ft = R. cos (β - α) ………… 30.24

Ft = R. cos (β - α) ………… 30.25

Also from ∆ ABE )( cosR

Fs α−β+φ=

Now from Eqs. (30.24) and 30.26)

)cos(R

)cos(R

F

F

s

t

α−β+φ

α−β=

or Ft = Fs. )cos(

)cos(

α−β+φ

α−β ………….30.27

Assumption Made

The above derived relations (eqs) are based upon the following assumptions

1. The tool is perfectly sharp and it does not make any flank contact with the job.

2. The cutting velocity remains constant.

3. A continuous chip with no built up edge is produced

4. The chip does not flow to either side

5. Chip shears continuously across the shear plane as the shear stress reaches the value of

shear flow stress.

The earlier discussed theoretical analysis of mechanics of metal is of use only if the value of the

shear angle φ is known. Machining is an unconstrained process and the shear angle (or chip

thickness) has no obvious value as, for example, has the exit thickness in rolling.

Apart from the time involved in determining the magnitude of the shear angle experimentally,

the understanding of the machining process is clearly incomplete if one cannot formulate a

satisfactory criterion for the orientation of the shear plane.

1. Theory of Ernst and Merchant

According to this hypothesis, the shear plane orientates itself so that

(a) the work done in cutting is a minimum, or

(b) the maximum shear stress occurs on the shear plane.

Refer Fig.30.16(b), ∆ ABE

)( cosR

Fs φ+α−β=

or )( cos. RFs α−β+φ=

and ,A.F sss τ= (from eqn.30.29)

or φ

τ=sin

A.F 1

ss (30.35) [Refer article 30.16]

From Eqs. (30.34) and (30.35)

)( cos

1.

sin

A.R

1s

α−β+φφ

τ= (30.36)

Also, from Fig.30.16(b)

F1 = R . cos (β -α) (30.37)

Now from equation (30.36) and (30.37)

)( cos

)( cos

sin

AF

1s1 α−β+φ

α−β×

φ

τ= (30.38)

Eq. (30.38) may be differentiated w.r.t. φ and equated to zero to find the value of shear

angle, φ for which F1 is a minimum.

).0( zero)(cos.sin

)( sin)( cos. cos).( cosA

d

F d

221s1 =

α−β+φφ

α−β+φ−α−β+φφα−βτ−=

φ

or cos φ .cos (φ +β - α ) – sin (φ + β - α ) = 0

or cos ( φ + φ + β - α ) = 0

cos (2 φ + β + α ) = 0

2 φ + β - α = 2

π (30.39)

or )(2

1

4224α−β−

π=

α+

β−

π=φ

∴ Shear angle, )(2

1

4α−β−

π=φ (30.40)

-Merchant found that the above theory agreed well with experimental results obtained when

cutting synthetic plastics but agreed poorly with experimental results obtained for steel machined

with a sintered carbide tool.

-It should be noted that in differentiating equation (30.38) with respect to φ, it was assumed that

A1, α and ι should be independent of φ. On reconsidering these assumptions, Merchant decided

to include in a new theory the relationship.

sss ko

σ+τ=τ (30.41)

Power and energy Relationship:

The power or the total energy per unit time or the rate of energy consumption is the product of

cutting speed "V" and cutting force Fc i.e. E = Fc x V, K.g. mm/min.

The energy consumed during cutting process is primarily utilized at the shear plane,

where plastic deformation takes place and at chip tool interface where friction resists the flow of

chip. The total energy per unit time (E) is approximately equal to the sum of shear energy (Es),

Friction energy (Ef) and negligible amount of energy required to curl the otherwise straight chip,

kinetic energy required to accelerate the chip, surface energy required to produce new surface

etc.

Thus, E = Es + Ef

The energy required per unit time per unit volume of metal removed per unit time is called

specific energy (e)

Thus total specific energy

e = E/b.t.v.

e = Fc V/b.t.v., (Kg/mm/min)/(mm3/min.)

e = Fc/b.t. kg/mm2

Similarly specific shear energy (es) & specific friction energy (ef) can be defined by the

following relations.

eS = ES/b.t.v. = FS . VS/b.t.v. = FS . cosν/b.t.cos (φ-ν), Kg/mm2

and

ef = Ef/b.t.v. = FS . VS/b.t.v. = F/b.tc ,Kg/mm2

SOLVED PROBLEMS :

Example :

1) In an orthogonal cutting operation, following date have been observed :

Uncut chip thickness, t = 0.125 mm.

chip thickness, tc = 0.250 mm

Width of cut, b = 6,500 mm.

V = 100 m/min.

Rake angle, ν = 100

Cutting force, Fz = 70 Kg.

Trust force, Ft = 25 kg.

Determine : Shear angle, the friction angle, shear and normal stress on shear plane, shear strain,

shear strain rate, cutting power, specific shear energy, friction energy, cutting energy.

Solution :

(i) Shear angle : φ

(ii) Friction angle

(iii) Shear and normal stress on shear plane

Shear foce = FS = Fc .cos φ - Ft sin φ = 70 . (.88) - 25 (.47) = 44.85 Kg

Normal force = Fn = Fc sin φ + Ft cos φ = 70.(.47) + 25 (.88) = 54.9 Kg.

Area of shear plane = AS = b.t/sinφ = 6.5 (0.125)/.47 = 1.73 /mm2

Shear stress, TS = FS/AS = 49.85/1.73 = 28.82, Kg/mm2

Normal Stress σn = Fn/AS = 54.90/1.73 = 31.74, Kg/mm2

(iv) Shear strain

∈= cot φ + tan (φ-ν ) = 2.18

(v) Shear strain rate 's = νS/ts

Where, shear velocity Vs = V.cos ν/cos (φ-ν)

= 100.cos (10)/cos (28.33-10)

= 103.75 m/min.

Shear plane thickness "ts" is assumed equal to one tenth of shear plane length i.e.

Thus, shear rate = 103.75/0.026 = 3938.755-1

(vi) Cutting power

E = Fe V/4500 = 70.000/4500 = 1.55 H.P.

vii) Specific shear energy E's

viii) Specific cutting energy 'e' = Fc x c/b.t.v.

(70)/(6x5x0.125) = 86.15 kg/mm2

ix) Specific friction energy = E - es = 86.15 - 63.65 = 22.5 kg/mm2

Example 2 : During machining of a C-30 steel with 0-10-6-7-8-80-0.5 mm (ORS) shaped

tungsten carbide tool, the following observations, have been made, depth of cut, d = 2 mm, feed f

= 0.2 mm/rev. speed V = 200 m/min. chip thickness tc 0.40 mm.

Calculate shear angle width of chip

Solution :

d = 2mm, 0p = 800, tc = 0.40, V = 10

Now, thickness of uncut chip t = f.sin 0P = 0.197 mm

Chip thickness ratio, r = t/tc = 0.49

Shear angle, 0 = tan-1

(r.cos v/(1-r sin v) = 23.820

Width of Chip, b = d/sin 0P = 2/Sin 80 = 2.03 mm

Example 3. During machining of C-20, steel with a triple carbide cutting tool 0-8-7-10-70-

1mm(ORS) shape the following data was obtained.

Feed = 0.18 mm/rev., Depth of cut = 2.0 mm.

Cutting speed = 120 mpm, Chip thickness = 0.4 mm.

Determine chip reduction coefficient & shear angle.

Solution : = 8, = 70

Uncut chip. thickness = sin

= 0.18 sin 70 = 0.169 mm.

chip reduction coefficient k = 0.42

Now = tan-1

( r.cos / ( 1 - r sin ), = 23.980

Example 4 : In orthogonal turning process the feed is 0.25 mm/rev. at 50 rpm. The thickness

of chip removed is 0.5 mm.

(a) What is the cip thickness ratio ?

(b) If the wok diameter is 50 mm before the cut is taken what is the approximate length of chip

removed in the minute. Assume a continuous chip is produced in process.

Solution : Uncut chip thickness, t = f = 0.25 mm & ctc = 0.5 mm

Therefore r = mt/tc = 0.5

Length of chip before cutting = D.N.= 50.50 = 7854 mm/min.

Length of chip after cutting, Lc = r.L. = 0.5.7854 = 3927 mm/min.

1. Show clearly by means of neat sketches only the meaning of the terms (1) Cutting speed, (2)

Feed & (3) Depth of cut as applied to turning process.

2. Explain the basic wedge action in metal cutting. Why is the cutting angle "δ" always positive?

3. Explain the mechanism of chip formation in metal cutting.

4. Explain why built up edge on cutting tool is under desirable ?

5. Classify & explain different types of chips produced in metal cutting.

6. Why are the discontinuous chips preferred over the continuous type ?

7. Differentiate between the orthogonal cutting & oblique cutting process.

8. What is cutting ratio (or chip thickness ratio) and chip compression factor (or chip reduction

coefficient) ?

9. What are the various methods of estimating cutting ratio?

10.What is shear angle ? How it can be measured ?

11. Prove that

tan φ = r cos ν/(1 - r sin ν) Where, φ = Shear angle.

r = Cutting ratio.

ν = Rake angle.

12. Prove that

Vc = Vsin φ/cos (φ - ν ) and Vs = Vcosν/cos (φ - ν )

Where V, Vc, Vs are cutting, chip & shear velocities respt.

"φ" is shear angle & ν is rake angle.

13. Prove that shear strain "∈" in orthogonal cutting is given by ∈= tan (φ - ν ) + cosφ, where

φ is the shear angle and ν is the rake angle.

14. How is the thickness & width of undeformed chip estimated in turning operation

15. What is metal removal rate ? What is specific metal removal rate ? How can "MRR" be

increased ?

16. What is meant by the orthogonal cutting system of first & second kind? Illustrate with

neat sketches.

17. What are the components of resultant force in an oblique cutting operation?

18. What are the assumptions of Merchant's theory?

19 Prove that φ = π/4 + ν/2 - β/2 where φ is shear angle, ν is rake angle & β is friction angle.

20. What is the modified Merchant's theory of metal cutting?

21. What is meant by power consumed in metal cutting? What are its various components?

22. What are total specific energy, specific shear energy, and specific friction energy?

23. How can the resultant force in orthogonal cutting be estimated by graphical method?

24. What is metal cutting? What are the basic requirements for metal cutting? How are the

metal cutting processes classified?

25. What is the effect of setting up of the cutting edge on rake & clearance angle?

26. In an orthogonal cutting operation the following data is obtained.

1) Cutting force = 180 Kg.

2) Feed force = 100 Kg.

3) Chip thickness ratio = 0.32 Kg.

Find graphically or otherwise shear force on shear plane, normal force on shear plane, Frictional

force, Normal force, and resultant Force.

27. Find the values of F.N, F, FN, R, and µ for an orthogonal cutting process if cutting force

is 170 kg., thrust force is 90 Kg, shear angle is 300.

28. In an orthogonal cutting operation following data have been observed.

Cutting speed = 15 m/min.

Uncut chip thickness = 0.06 mm

Width of cut = 3.00 mm.

Chip thickness ratio = 0.50

Rake angle = 200

Cutting force = 38 Kgf.

Thrust force = 14 Kgf.

Determine: shear angle, friction angle, shear stress along shear plane, chip velocity, shear

strain in chip, cutting power and specific cutting power.

29. A tool making an orthogonal cut has a rake angle of - 100. The feed is 0.10 mm, the

width of cut 6.5 mm. the speed 160 mpm, and a dynamometer measures the cutting force to be

180 kg and normal thrust force to be 140 kg. A high speed photograph shows a shear angle of

200. Estimate,

(a) Chip thickness (b) coefficient of friction. (c) Shear and normal stress on shear plane (d)

shearing strain, (e) H.P. to shear the metal (f) H.P. lost in friction.

Mechanics of Metal Cutting

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