Theory of Metal Cutting-mechanics of Metal Cutting

LECTURE-09THEORY OF METAL CUTTING

- Mechanics of Metal Cutting

NIKHIL R. DHAR, Ph. D.DEPARTMENT OF INDUSTRIAL & PRODUCTION

ENGINEERINGBUET

Px

Py

Pz

18/2Department of Industrial & Production Engineering

Mechanics of Metal Cutting

The force acting on a cutting tool during the process of metal cutting are the fundamental importance in the design of cutting tools. The determination of cutting forces necessary for deformation the work material at the shear zone is essential for several important requirements:

to estimate the power requirements of a machine tool to estimate the straining actions that must be resisted by the

machine tool components, bearings, jigs and fixtures to evaluate the role of various parameters in cutting forces to evaluate the performance of any new work material, tool

material, environment, techniques etc. with respect to machinability (cutting forces)


The force system in the general case of conventional turning process is shown in the following Figure.

Px

Py

Pz

Pz

Px

Py Pxy

R

Px = feed force in the direction of the tool travel

Py = thrust force in the direction perpendicular to the produced surface

Pz = cutting force or main force acting in the direction of the cutting velocity.


Pz

Px

PyPxy

X

Y

Πo

φ

PxyR

ZXY

R

.......[2]..........cosPP

]........[1..........sinPP

xyy

xyx


Several forces can be defined relative to the orthogonal cutting model. Based on these forces, shear stress, coefficient of friction, and certain other relationships can be defined.

R

Ps

R2

Workpiece Tool

Chip

γ

β

Pz

Pn

Pxy

NF

R1


Merchant Circle Diagram (MCD)

The following relationships suggest a circle representation of forces as done by Merchant and indicated in the following Figure.

Pxy

Workpiece

PzPn

PsF

N R

γo

γo

η β

η-γo

Tool

Chip

.[7].................... cosPβsin PP

..[6]....................sin Pβ cosPP

]5[....................sin γP γcosPN

]4.[.................... γcosPsin γPF

]3.......[xy

Pz

Pn

Ps

PNFR

xyzn

xyzs

oxyoz

oxyoz

.....[8]tansinγPcosγP

cosγPsinγP

N

Fμ

[5] and [4]Equation From

oxyoz

oxyoz

Where, μ = kinetic coefficient of friction η = mean angle of friction at the rake surface


From the geometry of force relations of MCD circle

Pxy

Workpiece

PzPn

PsF

N R

γo

γo

η β

η-γo

Tool

Chip

]11[..........)γη(β cos

)γ(η cosPP

[10] and [9]Equation Fron

..[10]).........γηcos(βRP

.......[9]).........γcos(ηRP

o

0sz

0s

0z

Based on the shear force, the shear stress (τs) which acts along the shear plane between the work and the chip is:

]12[..........tS

sinβPτ

sinβ

tSplaneshear theof area As where,

A

Pτ

o

ss

o

s

ss


].......[14)γηcos(βsinβ

)γsin(ηtSτP

Similarly,

.[13]..........)γηcos(βsinβ

)γcos(ηtSτP

[12] [11]andEquation From

o

00sxy

o

00sz

Pxy

Workpiece

PzPn

PsF

N R

γo

γo

η β

η-γo

Tool

Chip

In metal cutting one of the main problem is to evaluate the cutting forces Pz and Pxy from the given cutting conditions and initial properties of work material and it is necessary to determine τs, β and η by suitable relationships.


Earnest-Merchant Theory

Ernst and Merchant extended their analysis and studied the relationship between the shear angle and the cutting conditions. They suggested that the shear angle always takes the value that reduces the total energy consumed in cutting to a minimum. Because the total work done in cutting is dependent upon and is a direct function of the component Pz of the cutting force, they developed an expression for Pz in terms of β and the constant properties of the workpiece material. Condition for maximum cutting force (Pz) from Equation [13]

...[16]..........cottSτ2P

[15] and [13]Equation Combining

]15........[..........

2

πcos0)γηβcos(βor 0,)γηsin(βsinβ)γηcos(βcosβ

0γηcos(βsinβ

)γηsin(βsinβ)γηcos(βcosβ)γcos(ηtSτ

0)γηcos(β

)γcos(η.

sinβ

tSτ

dβ

d

dβ

dPor, 0,

dβ

dP

0sz

000

20

0000s

o

00szz

20γ

2η

4π

β


Merchant Theory

Merchant modified the relationship derived by Earnest-Merchant, by assuming that the shear stress along the shear plane varies linearly with normal stress (σn). It is given as (from the following Figure).

σn

τs

τ0

k

]20...[.................... )

0γηtan(βk1

0τ

sτ

[19] and [17]Eqation From

[19].................... )0

γηtan(βs

τn

σ

)0

γηtan(βs

As

P

sA

nP

)0

γηtan(βs

Pn

P

......[18].......... )0

γηRsin(βn

P and )0

γηcos(βRs

P

MCD of relations force ofgeometry theFrom

]17......[..........n

σk0

τs

τ


Combining Equation [13] and [20]

]23[..........β)tan(ccotβt0

Ss

τz

P


]22.......[....................085to080c(k)1cot0

γη2β

k)0

γηcot(2β

0)0

γηsin(2βk)0

γηcos(2β

0)sin(sin)cos(sin

)cos(sin)sin(cos

)sin(sin)cos(cos)cos(

0)tan(1)cos(sin

)cos(or 0,

dβz

dP

[21]Equation from (Pz) force cutting maximumfor Condition

]21.....[..........)tan(1)cos(sin

)cos(

200

00

0000

00

00

00

00

k

kktS

or

k

tS

d

d

d

dP

k

tSP

s

oz

oz


Lee and Shaffer Theory

According to this theory the shear occurs on a single plane. So for a cutting process according to this theory, the following are supposed to hold good: The material ahead of the cutting tool behaved as ideal plastic material The chip does not get hardened The chip and parent work material are separated by a shear plane.

..[25]..........1tanγξtSτP

γtanξγcos

γsinξ

βtan

1βcot

Where,

....[25]..........1βcot tSτP


..[24]....................4

πγηβ

00sz

oo

o

0sz

0

Lee and Shaffer derived the following relationship as:


Thermal Aspect of Chip Formation

Machining is inherently characterized by generation of heat and high cutting temperature. At such elevated temperature the cutting tool if not enough hot hard may lose their form stability quickly or wear out rapidly resulting in increased cutting forces, dimensional inaccuracy of the product and shorter tool life. The magnitude of this cutting temperature increases, though in different degree, with the increase of cutting velocity, feed and depth of cut, as a result, high production machining is constrained by rise in temperature. This problem increases further with the increase in strength and hardness of the work material. Knowledge of the cutting temperature rise in cutting is important, because increases in temperature: adversely affect the strength, hardness and wear resistance of the cutting

tool cause dimensional changes in the part being machined, making control of

dimensional accuracy difficult and can induce thermal damage to the machined surface, adversely affecting its

properties and service life.


In addition, the machine tool itself may be subjected to temperature gradients, causing distortion of the machine. The main sources of heat in metal cutting are shown in the following Figure. These three distinct heat sources are: the shear zone (q1), where the main plastic

deformation takes place the chip-tool interface zone (q2), where secondary

plastic deformation due to friction between the heated chip and the tool takes place

the work tool interface (q3), at flanks where frictional rubbing occurs.

Workpiece Tool

Chipq1

q2q3

air gsurroundin theinto radiatedheat of

Amount workpiece theinto passingheat ofAmount toolcutting

in the remainingheat ofAmount chipsin away heat ofAmount

generatedheat of

amount Total

:as written becan formation chipin balanceheat The


Various studies have been made of temperatures in cutting, based on heat transfer and dimensional analysis, using experimental data. A simple and approximate expression for the mean temperature for orthogonal cutting is

333..04.0

K

tV

C

UT c

where,

T = mean temperature rise at the tool-chip interface (oC)

U = specific energy in the operation (N-m/mm3)

Vc = cutting velocity (m/sec)

t = depth of cut (mm)

pC = volumetric specific heat of the workpiece (J/mm2-C)

K = thermal diffusivity (ratio of thermal conductivity to volumetric specific heat) of the workpiece material (m2/sec).


Exercise

The dynamometer recorded the following, feed force 200 kg, cutting force 300 kg. The rake angle of the tool used was 10o. The chip thickness ratio 0.35. Find

Shear angle (β) Shear force (Ps) Co-efficient of friction at the chip-tool interface (μ) and the friction angle (η) Compressive force at the shear plane (Pn).

A seamless tube 3cm outside diameter is reduced in length on a lathe with the help of a single point cutting tool. The cutting speed is 40 m/min, the depth of cut is 0.125mm. The length of continuous chips, for one revolution of the tube, on measurement comes to be 17.77cm. The cutting force is 200 kg and the feed force is 75 kg. the rake angle of the tool is 35o.Calculate,

Co-efficient of friction Chip thickness ratio Shear plane angle Velocity of the chip along the tool face Velocity of shear along the shear plane


During the machining of AISI-1025 steel, with 0-10-6-6-8-90-1 (mm) ORS shaped tool the following observations were taken:

Feed 0.50 mm/rev Depth of cut = 2.0 mm Cutting speed = 40 m/min The shear angle = 20o

The power consumed while machining= 3kW The power consumed while running idle = 0.50 kWCalculate: The shear force Chip thickness ratio Normal pressure on the chip Chip thickness


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Theory of Metal Cutting-mechanics of Metal Cutting