# 5.1 can be expressed as, (5.11) Where R

5.1 Machining theory has been
described to find out the optimized value of cutting force during machining on
D2 steel.

FIGURE 5.1:
Model of chip formation used in Oxley’s analysis of orthogonal machining.

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Oxley’s
machining theory has been employed incorporating necessary changes with respect
to substitution of values of stress and strain hardening index of Holloman’s
equation. This theory can allow the prediction of optimized machining responses
which can further help to control machine tool operations. The following steps
have been considered for finding out optimal solutions for forces, shear angles
etc.

Step 1: Specify
initial cutting conditions e.g. cutting speed (v), uncut chip thickness (t1),
cutting width (w), initial work piece temperature (TW), and rake
angle (?0).

Step 2: Consider
shear thickness ratio from a set of values, ?i.

Step 3: a)
Assume values of strain rate constant ‘C0’.

b) Assume values of shear angle (?0).

Step 4: To
calculate shear velocity

(5.1)

And shear plane
length,

(5.2)

Step 5: To
calculate shear strain rate along shear plane (?AB) shear strain rise up to shear plane (?AB).

(5.3)

Step 6: To
calculate effectively strain along shear plane

(5.4)

Step 7: Iteration
for shear plane temperature

Assume shear plane temperature TAB=Work
piece temperature, Tw

(5.5)

where A, B, C,
n, m are model parameters.

? =Strain

= Strain rate

0= Reference strain rate

t=
Temperature in ?C

tr=
Room temperature ?C

tm=Melting
temperature ?C

Shear flow
stress (Kchip) in the chip
adjacent to interface can be given as

(5.6)

Angle of
inclination (?) of resultant cutting force R to shear plane AB is given by

(5.7)

And the angle
between resultant and normal force can be expressed as,

(5.8)

Shear flow
stress along shear plane AB

(5.9)

And shear force
on shear plane AB

(5.10)

Step 8:
Iterative step for temperature within the PDZ,

The proportion
of shear zone heat conducted into the work piece ‘?’ can be expressed    as,

(5.11)

Where R =Thermal
number

(5.12)

Where K =Thermal
conductivity

Temperature rise
in the primary shear zone is

(5.13)

And average
temperature along the shear plane AB is given by

(5.14)

This
evaluation temperature was compared with old temperature and on finding
significant difference, the new temperature was again evaluated by repeating
the step 7 every time and again the new values of temperature were stored in
separate matrix. When the difference between the new temperature values and the
preceding one was found to be insignificant, then step 9 takes place.

Step 9:
Resultant force (R) can be expressed as

(5.15)

Feed force (FX)
can be given by

(5.16)

Friction force
(F) can be given as

(5.17)

Normal force (N)
on rake angle is given by

(5.18)

Step 10:
Calculate the thickness t2 as

(5.19)

To calculate the
chip velocity ‘V’

(5.20)

Where v is the cutting velocity

The tool chip
contact length ‘h’ can be expressed as

(5.21)

Shear stress at
interface (?int) can be
determined by

(5.22)

Shear strain
rate (?) can be expressed as

(5.23)

Effective strain
rate at interface (

)

(5.24)

Step 11: Assume
chip mean temperature

(5.25)

Consider
specific heat of chip (S) as

(5.26)

And the thermal
conductivity (K) as

(5.27)

Average chip
temperature (

) can be determined as

(5.28)

New chip
temperature is given by

(5.29)

Difference
between old TC and the new TC is determined during
iterative procedure and continued till the difference between the TC and
the new TC values is insignificant. Finally new TC value
can be determined. During each iterative step, new specific heat and new
thermal conductivity “K” for the chip were incorporated.

Step 12: Tool
chip interface

The maximum
temperature rise

in the chip is calculated from,

(5.30)

Interface
temperature can be determined from

(5.31)

Flow stress can
be determined employing Johnson Cook constitutive model. The Johnson Cook
constitutive model has been shown below:

(5.32)

Where A, B, C,
n, m are model parameters.

? =Strain

= Strain rate

0= Reference strain rate

t=
Temperature

tr=
Room temperature

tm=Melting
temperature

Shear flow
stress (Kchip) in the chip
adjacent to interface can be given as

(5.33)

Step 13: To
check for the shear angle ?0=45?

(5.34)

During
successive iteration, 0.1? is considered as increment to old value of ?0 and the calculation is
repeated from step 3(b) with the new value of ?0.

When the new
value of ?0 is 45, then
the iteration step 14 is continued.

Step 14: To find
?intand KchipVS?0
and to select solution point for ?0
at which ?int=Kchip.

Step 15: To find
out the normal stress, at tool chip interface as

(5.35)

The normal
stress ?N is found from
the boundary conditions at B for the normal stress distribution along AB, ?N’ can be
determined as

(5.36)

?N and
?N’ are
compared during each successive iterations and if found significant, the new
value C0 is assigned and next calculation is repeated from step
3(a).

Step 16: When
the difference between ?Nand
?N’ is
significant, then to ensure whether ‘?’ is minimized or not. If (?=?min),
to predict shear angle and main cutting forces with this condition.

If
(???min) then the new value of ?
is considered and calculation is repeated from step 2.Entire calculation is
repeated till minimum value of ? is obtained and subsequently corresponding
shear angle and mean cutting force are obtained. Flow chart of the
computational algorithm has been shown in figure 5.2.

Step
1

Step
2

Step
3a

Step
3b

Step
4

Step
5

Step
6

Step
7

Step
8

Difference
between new TAB & old TAB

Step
9

Step
10

Step
11

Difference
between new TC & old TC

Step
13

Step
12

?
=45?

Step
14

Step
15 Difference between ?N & ?N’

?
=?final

Step
16 ? =?min?

Plot
FC VS ? & determine ? =?min for
minimum FC

Print
?0,FC

Tc=New
Tc

YES

NO

YES

NO

YES

NO

Significant

Insignificant

Insignificant

Significant

?
=?min

Estimate       new C

?=?+0.1

FIGURE 5.2:
Flow chart of the computational algorithm

When
the program was developed for simulation work, initially the testing data
were obtained from source literature for verification with the obtained data
from the simulation work. The following plot shows the nature of variation of
experimental data from that of the data obtained from simulation work using JC
model in Oxley’s predictive modelling theory. Since there is good agreement
between two sets of data as shown in the figure 5.3, it was decided to
utilize the model for finding out the nature of variation of forces for D2 cold
worked steel.

EN19
Steel Composition and properties:

TABLE 5.1:
Composition of EN19 steel

C

S

P

Mo

Ni

Si

V

Cr

Mn

Nb

Ti

Al

Cu

0.37

0.027

0.016

0.23

1.51

0.22