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.

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