8/18/2019 Artículo Just, Calculating hardenability Curves.

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INTERNATIONAL REPORT

ew ormulas fo r

Calculating Hardenability Curves

y

RWIN

UST

By correlat ing average composi-

tions with Jominy band hardnesses

for

a

variety of carbon and

al loy steels, the author

developed equations for calculating

Jominy curves from compositions.

To

SIMPLIFY

the determina-

tion of hardenability, we have de-

vised a method for calculating the

Jominy curve. In developing the

equations, we used the multiple re-

gression analysis technique to deter-

mine quantitative effect of the alloy-

ing elements. (Multiple balancing

calculations are used to keep all but

one variable constant to determine

the effect of that variable alone.) As

a warning, we should not expect the

formulas to predict a steel's

harden-

ability precisely. Our formulas are

intended mainly to assist the de-

signer in determining the steel to

select, and to help the metallurgist

in correcting the melt.

Performing the Calculations

If we plot a curve of average

values for each Jominy band and

correlate these mean hardenability

values to the average values of the

composition ranges of the grades

concerned, we can ascertain the

quantitative effects of the alloying

elements. To start out, we deter-

mined linear hardenability models.

Multiple regression calculations car-

ried out for different distances from

the quenched end of a Jominy bar

give us hardenability formulas, three

examples of which follow:

J1

=

52 ( C) + 1.4 ( Cr)

+

1.9

(

Mn)

+

Rc 33

Jo

=

89 C + 23 Cr 7.4 Ni +

24 Mn 34 Mo+ 4.5 Si c 30

J2,

=

74

C

+ 18 Cr + 5.2 Ni +

33 Mo + 16 Mn + 21 V Rc 29

In these equations, the Rc hard-

ness of the Jominy specimen is indi-

cated by J, and the subscripts show

distances to the end face in

1/16

in.

Ranges are 0.10 to 0.64 C, 0.15 to

1.95 Si, 0.45 to 1.75 Mn, 0 to 5.0

Ni, 0 to 1.55 Cr, 0 to 0.52 Mo, and

0 to 0.2 V. Only the significant ele-

ments are included in the equations.

Nonsignificant factors are eliminated,

step by step.

If we extrapolate the values from

the equations for zero distance from

the quenched end, we obtain

rule

of thumb for calculating the surface

hardness of a quenched bar from its

carbon content:

Jo

=

h C) (lo2)+RC 36.

The formula is accurate for steels

with 0.2 to 0.6% C.

A Nonlinear Model

Next, we wished to determine

whether this linear hardenability

model could be further refined by

introducing nonlinear terms. (The

nonlinear terms are radicals because

experience shows that the specific

effect of an alloying element is re-

duced as its content rises.) When

the regression calculation was per-

formed with the square roots of the

variables, we found that the only

factor of constant significance was

\/C, and therefore used it instead

of C. With this change, the carbon

transformation increases the multiple

correlation coefficient in the neigh-

borhood of the quenched end; data

scatter is reduced accordingly. The

following equations apply:

J,

=

6 0 n + 1.6 Cr

+ 1.5

Mn

+Rc 16

JR

=

100

v T+

7.5 Ni

+

22 Cr

+

22 Mn 33 Mo+ 6.2 Si 22 V

- C 56

J z a = 8 5 -\/C+ 1 9 C r

+

5 . 7 N i +

34 Mo + 16 Mn

+

25 V + 2.1 Si

- C 53.

Again, we can extrapolate the

equations to determine a formula for

calculating hardness at the quenched

end:

J, = 60fl c 20

In the instance, the formula works

for steels with 0.1 to 0.6% C. It

should be used if the hardenability

calculation yields higher values.

Deriving a Comprehensive Formula

The approximately constant effect

of the alloying factors outside the

full-hardened region enabled us to

perform a comprehensive regression

analysis, the object being to produce

a single formula for predicting the

Jominy curve. For this purpose, we

introduced another factor, E, de-

fined as the distance (in

1/16 in.) from

the quenched end into the calcula-

tion. The following equation results:

J4.2,

=

98 \/C .025 E2

fl

20 Cr + 6.4 Ni+ 19 Mn + 34 Md

+ 28V- 24-+ 2.8 6E- Re7

The correlation coefficient of this

formula is high, only 4% of the vari-

ance in hardenability being unac-

counted for.

When using the over-all formula,

we must exclude the region close to

the quenched end face (E

< 4/16

in.)

because the numerical coefficients of

the terms are not constant. Hard-

nesses in the region up to the end,

however, can be easily ascertained

by using the preceding equation.

An Interaction Model

In making the calculations per-

formed up to now, we have assumed

that the variables are independent

of each other. This assumption, how-

ever, only approximates the truth

since we must expect interaction

between carbon and the alloying

elements. If we postulate that a

parabolic relationship exists between

carbon and other elements, the re-

gression calculation is performed

with the products CL, C2L, and

C C . The resulting equations follow:

J, = 7.6 Mn + 138 C2Mn 8

C W n

+

4.6 Cr

+

21 CCr) + 129

C T r 73 C T r

+

9.6 MO+ 214

C2Mo

-

195 C3Mo)

+

5.3 Ni

(36 CNi) + 214 C2Ni - 65 C3Ni

+ 11 V + 7.6 Si+Rc 5.5

JZ2= 148 C2Mn - 98 C3Mn +

Cr

+

101 C2Cr 39 C3Cr

+

14 Mo

+

238 C2Mo

-

74 C3Mo

+ 2.9 Ni

+

50 C3Ni

+

Rc

Terms in parentheses are not con-

Mr. Just

is

department head, Metal-

lurgical Laboratory, Volkswagenwerk

AG,

Wolfsburg, West Germany.

8/18/2019 Artículo Just, Calculating hardenability Curves.

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0

4 8 12

16 20 0

4 8

12 16 20

Distance From uenched End

1 \ 1 6

In.

Fig.

1

alculated determinations of points for hardenability

curves (based on midpoints of ranges for elements that go into

these standard SAE grades) indicate that the formulas work satis-

factorily. Most calculated curves fall within the standard bands.

sidered significant.

In Fig.

1

we see calculated Jominy

curves of eight typical steels plotted

on scatter bands. The interactions

led us to split the steels into two

groups, carburizing (C

<

0.28 ) and

hardenable (C

>

0.29 ) grades.

When we performed regression anal-

ysis separately for case-hardening

and heat-treatable steels, the fol-

lowing equations resulted:

Case-hardenable steel

J4 25= 87

f

14 Cr

+

5.3 Ni

29 Mo

+

16 Mn 21.2 fl

2.21 E Rc 22

Hardenable steel

J4-25

=

78

C

+ 22 Cr

+

21 Mn

f

6.9 Ni

+

33 Mo 20.3

fl +

1.86

E +

Rc 18

As expected, the formula for the

hardenable grades has the higher

mefficients.

Grain Size Is Considered

Next, we determined a formula

based on the compositions and grain

sizes of the 37 steels listed

in

the

U S. Steel Atlas. It is shown below:

J1.4n == 88

\/iS

.0135 E2 VC

19 Cr 6.3 Ni

16

Mn

35 Mo

+

5 Si 0.82 KASTM

20*+ 2.11E-Rc2

The formula is applicable for steels

with the following ranges: 0.08 to

0.56 C, 0.20 to 1.88 Mn, 0 to 8.94

Ni, 0 to 1.97 Cr, 0 to 0.53 Mo, 0 to

3.8 Si, and

1.5

to

11

ASTM grain

size

K).

pplying

Hardenability Formulas

These formulas can be used to

calculate compositions. Let's say the

designer requires a fatigue strength

of 85,000 psi (60 kg per sq mm). To

achieve this value, we need a hard-

ness of Rc 40 after heat treatment

(HQT). Then, quenched hardness

(H,) must be high enough above

this hardness to obtain satisfactory

tempering (HQ

=

35 0.5 HQT).

For our example, this will be Rc 55.

From this value, we determine the

required carbon content as follows:

t

Hardness = 60fl Rc

15

for

95 martensite. Solving for carbon,

we obtain 0.44 .

Now, we need to determine the

required alloy content. If the steel

is to be without molybdenum or

nickel (for economic or strategic

reasons), we use the remaining ele-

ments that affect hardenability; these

are chromium, manganese, and sili-

con. To apply the formula and de-

termine the required additions, we

must first know the appropriate dis-

tance

(E)

from the quenched end.

Let's assume, for example, that the

radius of the component is

1

in. and

that the fatigue strength is required

to be present at

3 4

radius. Applying

the formula:

in in,, 0.254

R2

0.15

R

.0445 R3

+

0.118 in.,

we find the distance from the

quenched end to be

A6

in. If, for

metallurgical reasons (freedom from

slag), we need 1

Mn and 0.3 Si, we

insert these values into the formula

and solve for chromium as follows:

55

=

78 (0.44) 22 Cr 21

20.3

+

1.86 (8) 18.

Then, Cr

= 1 1

Thus the re-

quired composition of the steel is

0.45 C, 1.0 Mn, and 1 1Cr.

If we find tha t such a composition

is not in the list of standard steels,

we repeat the calculation with other

fixed values (for example, with 1.0 Si

in place of 0.3 Si). Otherwise, a

corresponding specification will have

to be established, and this step

normally involves ex t ra expense.

Therefore, reworking the calcula-

tions in attempting to find a standard

steel should not be dropped at too

early a stage.

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