.. • , ... . )

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COMPUTER-AIDED ESTIMATION OF STEEL HARDENABILITY, . ) . l

ON THE BASIS OF ALLOY CHEMICAL C~POSITIOO

t

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-.,

" COof'UTER-AIDED ESTIMIITION Of· STEEL HAADEK'BILIlY. . I . . .. -

(

ON ~ BASIS Of ALLOY CHBHCAL C04PQSITION"

I ,

• by

Gregorlos 0'. PIIzlonls. II.Sc. . \ . ( •

• .. (

A Thesis

, Submitted to 'the Faculty of .Grad.;ate Studies

In PIIrtlal Fu~fll"lP"t\Of the. Requlr~ts

for the Degree .' I' ,

Master ot En!;Jlneerlng

, J ~ -' . .. ". .

. . ,'.

Mc:MIIster Un I vers I ty

\ May 1974

o Gregori'os O. Pazioni s 1974

~ . , ~

(' I

! ,d v

-" • \.

\ / \ J

"-

TO MY PARENTS ~

" Q

\ OOYSSEA ANO EIPIDA

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,

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,

M::f4MiTER ,UN IVERS IlY

Hamilton, Ontario

,

MASTER OF ENGINEERING (1974)

(Metallurgy and Mc!terla)S ~c,ence) ! "

TITLE: Computer-A I ded EST I mat I on of Stee,l. Hardenab IIITy

On.the Basis, of Alloy-Chemlcal Composition

)

AUTHOR: Gregorlos O. Pazlonls, ptyhlon (M; .Sc.) Arl'STOTellan

SUPERVISOR:

'NUMBER OF

PAGES:

•

UniversiTy, 'Salonlca, Greece. /

M. ,Sc. Dalhousie UniversiTy, Halifax, Nova ScoTia,

Canada.

Prof~s50r J. S. Klrkaldr j

"" .. , . . ,'

X, 126

SCOPE AND CONTENTS: ,

A numertcal' SoluTion 1'0 the one dimensional unsteady sTat'e heat , transfer problem Includrng the latent"heaT m(oiutlon of the pearlite

~ -' " .

• 0

reaction has boon developed 1'0 predict thermaJ condiTions In a Jomlny ·bar.

This has boon ~blned WiTh a model for pearlite nucleation and, growth . developed by Klrkaldy and used 1'0 predict the TIT' curves for The -

austen~te + pearlite transformation and the pearlite volume fraction as I -

a function of time and Jomlny number~ The conversion of The TIT to CCT I , ,

curves, which 15 a flecesserystep In the latTer calculatlon, has boon . . . . . Investigated using The additiviTy rule and the approximation to IT.due 1'0

Qrange aDd Kieffer. It was concluded that The latter approximation will

of1'on not be Justified. Inc~n of, 'The latenT heat evolu~loll In T~e heat I •

transfer calculaTions was proven to be significant and as such should be

Included In any' accurate algoriThm_for predicting Jomlay curves.

"

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ACKNOWLEDGE~1ENTS

•

It Is' a ~reat pleasure to acknowled~e

the ass I stance of Dr. J. S. KI rka I dy, my research

director, for his comprehensive direction of my

research and for his encoura~ement and advice

·throu~hout,:he course of thl5 work.

r

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, , ,

. TABLE OF' CONTENTS

'~

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CHAPTER I THE IRON-CARac>N SYSTEM I • I. . I ntroduct Ion .. 1.2 Jhe Iron-c8rbon'DI8grem

jr ,

1.3 P88rllte Nucle8tlon 8nd Growth 2 .-

CHAPTER 2 TIME~TEMPERATURE-TRANSFOR~T'ON

DIAGRAMS 9

2.1 I sothenm8 I Tr8nsform8tlons. 9

2.2 The Complet~ T-T-T DI8gr8m of 8n Eutecto I d Stee I ( ·12

2.3 ContInuous CoolIng Tr8nsfor.atlon .)

\ DI8gr8ms 12 ,

CHAPTER , ~ 3 HARDENABllI'PI' 19

3.1 DefInItIon of H8rden8blllty , 19

3.2 CrltlC81 DIMIOter 19

3.3 ",

Id881 CrltlC8hAllMlOter 20 ,

3.4 Janlny.,J&st, or ~nd-Quench Test 21

3.5 Effect of HI ghe!1 Quench I ng Temper8tures y 24

3.6 The Effect of AustenitIc GraIn 'SIze on H8rden8blllty 24

3.7 The Influence of C8rbonContent on H8rden8blllty. 25

3'8 The Influence'of AlloyIng '\ E I_nts on H8rden8b III ty . 25 "",:;"" \

CHAPTER"4 4 UNSTEADY STATE CONDUCTION CF HEAT ,

IN A J()4INY BAR 34

4.1 . GeOeral Formul8tlon of He8t Conduction Problems 34

"

'Ii

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~# ,QiAPTER 5 o IVATION Of CCT F~ TIT \ . . .\ DIAGRAMS 42

c~ "

I

5.1 Introductory . 42

5.2 The Grange-Kieffer Approximation 42

~3 The Rule of Additivity 44 '-..J •

d- . 5.14 Computer-Alded OOrlvatlon of t!M' CCT from the TIT Diagrams 46 ,

\

" ( • "

CHAPTER 6 C<M'UTER-AIDED ESTIMATION OF

HARDE~BILlrY.FOR LOw-ALLdY STEELS 53 , 6.1 Introductl.on 53

r- 6.2 " Calculation of the Velocity 55.-

6.3 Caleuplon of the IjUbatlon Tlma , 59

6.4 Computer-Aided Estimation 60

6.5 The Klrkzlldy model for Pear,llte-Nucleation and G'rowth 61

, . 6.6, Heat Transfer In a Jomlny Bar 64

6.7 Sample Calculations ,'72

, ... '

CHAPTER 7 CALCUVtTION OF THE ~ENABILITIES ."

Of A SEI.ECTION Of EXPERIMENTAL At{) ,

C<J.t.lERC IAL STEELS 90 ~ -' 7.1 Calculation and Discussion of

I) the Results 90

CHAPTER _ 8 Cone I us Ions 96

APPOOIX ~rlcal Techniques '" ,99 (

I .. APPENDIX " Prbgram Structure for Calculating

a CCT from' a TIT Curve 108

\ Iv

APPENDIX III·

REFEREOCES '

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•

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\

.-

J . Computer-aldod EstllrRitlon:of v(T) 'lind ,«T~

\ (

\ , -

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y

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111

120

., --124

• ,

•

-~

',!, J

3.1

, 7.1

••

~.

, LI ST OF TABLES

Severity of Quench Values

Chemical Composition of the Number,and References for the ,,,,., of ,t .. lf .... I'.d

vi

.29

93

,

..

I • I

1.2

1.3

1.4

1.5

1.6

2. I

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

2.10

LIST OF ILWSlRATiOOS

The metllstllble system F~Fe3C

I sotherma I growth h!) I1!te of p88rllte ~s a function of time

Rate of growth (~) of p88rllte ~s a function of rMc1-lon tmperatura.

Isothermal rate of nUCI88tlo~ (N) of p88rllte ~s a function' of time

V~rl~tlon of N ~nd G with temper~ture In a eutectold steel

PMrllte fraction ~s ~ function of time

Reaction curve for Isothermal formation of p88rllte

Tlme-Temperature-Tr~nsfonnatlon dl~gram obt~lned from'r88ctlon curves

The ~rtl~1 Isothermal tr~nsfonnatlon dl~gram for ~n eutectold steel

~Inlte fraction as ~ function' of time

~Inlte fr~ctlon ~s ~ function of temper~ture

o

Mertens~e fr~ctlon ~s ~ function of tempe ture

...':.J

V~rl~tlon of lois ~nd Mf with ~rbon concentl1!t Ion In stee I

Vorl~tlon of lois with am~nese content

V~rl~tlon of h~rdness of amrtenslte ~s a function of carbon content

\ V~rl~tlon of ~rdness of~rtenslte ~s o function of carbon content '

vII

6

,,6

7

7

7

7

14

14

14

15

. - 15

15

I 16

'I 16 ,I

·0

16

J7

\ i'< .. •

'V

'.

:-2.11 The camp I ete I sothel'1lllll d lag".

for an eutectold sfeel 17

2.12 The relatlonsh!p of the continuous \ cooling diagram to·the Isothermal G d I ztgram for an outecto Id steel 18

3.1 The .varlatlon of mlcros1Tucture as a fUnction of cooling rate for an eutectOld steel 28

3.2 .. , U - curves 28 Il

3.3 Relationship Crt 'the crltlCllI diameter 0 to the I d 8111 cdt I cal diameter D:r 29

3.4 ,Stllndard form of Jomlny End-quench T es:t-p I ece 29

3.5 .Hardenab Illty, curve 30 ..

3.6 Reproduc I bill ty !>f hlIrd_b IIlty curves 30 , . t

3.7 \..oClItlon on Jaml ny bar at wh Ich the , cooling rate Is equlwlent to the center of a c I rcu lar bar quenched I n an. I dOlI I

, . . quench I ng med I um 31

',' 3.8 . Effect of quenching t .. , ..... ature on

hlIrdenab III ty , 31

3.9 I dOlI I critical diameter as a function of carbon content and austenite grain size' for I~rbon alloys 32r -

3.10 Effect of !"I'nganese on hlIrd_blllty 32 '.

3.11 Effect of silicon on hardenability 33 , '

4. I Heat balance In a cuble .elelnent. 41 ,----j 4.2 Error functions ~') 41

ISothenMl tnnsfonrratlon dl~"; 5.1 50 . '.

5.2 Experlmentlll eel' dlag". for S.A.E. 4340 .steel . , 51

5.3 - Derived (Grange-Kieffer) CCT diagnllll for S.A.E.' 4340 steel .51

" , viII

6.1

6.2

6.3-6.9

6.10

6.11

6.12

6.13

6.14

6.15

. 6.16 ,

6.17

6.18

7.1

7.2

8.1 .

TTl and derived CCT dlagnllllS (The Gra~ge-Kleffer and the additivity" ru Ie .appl"OXhllltlons) 52

The quadratic effect of Mn on _ hardenability (Rate of Growth). 75

The quadratic effect of Mn on -. hardenabllity (Multiplying Factor) 75

.),.

Derived (Klrkaldy's theorY.) and experiment. I m d I agra",s for a selection of steels 76-82

The Klrkaldy model for nucleation sites 83

Jomlny slab 84

Two dimensional time-space grid 84

Der lved pear Ilt'8 fraction curves 85

Derived cooling curves 86

Derived pearlite fraction curves as a function of grain size (this thesis) 87 .

Oer.lved pelSrllte fr:ac:tlon curws 115 a function of grain size (Klrkaldy(14)) 88

p

HIIrdenabllltY- curves of S.A.E. 4068 (one heat) 89 - \

"Effect of SIIIIIfl .additions of nlckal and chromlwa on depth' hardening of S."~E. 4068 89

Predicted vs. experl .... tal hIIrdenablllty values for a selection of shels 94

. Predicted vs. experimental values for high menganese steels .•. Quadnt'tlc term Included In the calculatlons.- :. " 95

Regression analyses predicted herdenability values vs. experl .... tal ones 98

Ix (

-"

1.1 Two dimenslolllli time-space grid 106

1.2 ~lcuilltion of temperature dlstrl but Ion 106

1.3 Finite difference grid 101

II.I X. Y pillne 11.5.

II.2 Computer progrlllll f I ~Mrt (DerlVlltlon of CCT from TIT and coo I I ng rates). 116

III.I Computer program f I ClW-CMrt (Derivation of TIT-d lagram from

~. Chem I cal Canpos I t Ion) 119 . IV Computer program f low-CMrt ..

(Prediction of Mrdenability from

~ the TIT diagram) 123

.

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CIIAPTER I •

(

THE I RON-CAROON SYSTEM

1.1 Introduction

I '

The development of reliable correlations between hardenability •

and chemical composition Is bf economic Importance to the steel .Industry.' I

Current hardenability bands are very wide due to specification

latitude' and tramp elements, wh I ch resu I tin a waste of ~ II oy I ng elements

and higher steel costs, both to the producer and user.

In this 'thesis we Il'lVestlgate the possibility of computer' alde,d

prediction of hardenability on the basis of chemical composition and a so~l-, empirical description of the metallurgical processes I~volved, developed by

Klrkaldy •

This Is an att7ctlve alternative to the empfrlcal methods of

Grossman or the regression analyses current,ly being used •

The theoretical and experimental background of a semi-empirical

approach starts with the' Iron-carbOn dlagram.(

I ~2 The Iron-carbon diagram

In Fig. 1.1* we give the'lroiarbon diagram for carbon concentrations

less than 6.67 wt.% carbon, which corresponds to the composition of, ~ementlte,

* *

All references are to be found on pages ,12"'-126.· Figures are to be found at the end of each chapter. I

·-Fe3C. In this diagram there are three Invariant points: a peritectic 'point

0" . 0 at .16 wt ... carbon and ,493 C', an eutectic point at 4.3 wt.% carbon and

1147oe-, and a eutectsJ-d pol nt at O.S wt.% and 7230 C.

If austenite' <,f.c.c. Iron) Is allowed to traniform Isothermally ~)

at temperatures Just below 7270 C, the reactions product Is p mixture of the

stable phase~ below the eutect~ld temperature, ferrite and cementite. This

mixture Is called pearlite and consists of alternate plates of Fe3C and

ferrite, wLth ferrite being the continuous phase. . , J '

Pearlite of eutectold.composltlon consists of 87.5% ferrite,

12.5% cementite, and since the density of ferrite a the density of cefTlentlte

the lame(l~e of ferrite and Fe3C have'a width ratio of about 7: I.

1.3 Pearlite nucleation and growth (-

Austenite transforms to pearlite by nucleation and growth of the

new phases. If the austenite Is hQlOOgeneous, nucleation occurs.alqo,st

2 ,

exclusively at grain boundaries. When It Is not hOlOOgeneous, and has concentra-

tlon gradients and contalns,resldual Iron carbide particles, nucleation of

pearlite can occur both at the graln boundaries and In the centers of austenite

gral ns (I ) ~ Th I s can happen when an austen I,te ,transformed to p,:!ar II te Is re-

austenltlzed for an Insufficient time for

pearlite,' once nucleated~ grows

the carb'ldes to dissolve.

by the steady advance at the ends ~

of the already nucleated lamellae. Further nucleation occurs 'also at the

:-_Htterracesbetween pearlite and auste.n14e. Overal'l pearlite grows as a group

of continuous colonies or nodules which are usually spherical In shape. ,

Pearlite colonies grow unhnpeded.untl I they Impinge on adjacent colonies. ,

, \

During this period, the rate of growth, as determined. experimentally,'

fs constant (see Fig. 1.2). After Impingement, the nodules can only grow

Intq austenite remaining between nodules. The rate of growth of pearlite (G)

Is a strong fun~tlon of temperature, (see Fig. 1.3). ,

The rate of n,ucleatlon N Is the number of ~~clel that form In a

unit volume per second. Unlike the Isothermal growth rate, the Isothermal

3

\

)

rate of nucleatlml Is a functlon.of time, (see~ •. 1.4). To compare nucleation

. ates at d,lfferent temporatures, It Is a~essary to consider the average

. cleatlon rate for each temperature, ,(see Fig. 1.5). Fig. 1.5 also shows

the rate of growth, as a function of tempe,ratura. Apprecl ab Ie I nformat l'On

about the changes In microstructure that, occur as a function of temperature •

~an be deduced from a study of these curves.·

. At temperatures s.llghtly below the eutectold temperature, (say 7DOoC), ,.

the rate' of nucleation Is very small, approaching zeno; the growth rate,

-3 -4 . ·though, has a finite value between 10 and 10 mm per sec. Only a few

pearlite nucle1 form and, because of the relatively high growth rate, the

nuclei grow Into large pearlite nodules. These nodul'es grow larger than the

original austenite grains crossing lIustenlte grain boundaries. In th.ls ·0 . ',1

.case of very few, p~arllte nuclei, the nuclei .may be considered as a random

distribution In the austenite ma.trlx, though they actually form lit grain

boundaries. JOhn~on and Mehl(2) have shown that If we assume: (a) constant \

rate- of nucleation N, (b) spherical growth of nodules and (c) constant

growth rate G, then the pearlite fraction Is given as II function of time by

the equation

I • I

•

where f(t) Is the pearlite, fraction and t Is the time·. A typical plot

of f(t) vs t Is given In Fig. 1.6.

At lower temperatures. the rate of nucleation Increases much

taster .than the rate.ot growth. so the n~mber of nucleated pear~lte colonies

Increases. In this case. early In the transformation. the austenite grain

boundaries become outlined by a large number of.pearllte colonies. There

are a number'of~pearllte nodules growing Into a single austenite gralQ and'

It Is not'posslble to think anymore In terms of random n~cleatlon. W~ have

to think. Instead. tn terms of grain boundary nucleation.

tOl'lowlng two reaction equations for this' sase:

f(t) =, I - exp n 3 4 (~G ~GSOGst)

and f(t) c I - exp (- 20GSG t)

Cahn(3) gives the

•

1.2

"'" (

1.3

. corresponding to small and large nucleation rates)respectlvely. The first

Implies sucn a small nucleation rate that the nucleation can be considered

as homogereous. I.e., NH .=NGS'OGS

where , ,

G = growth rate •

NGS = nucleation rate/grain boundary surface unit

0GS = grain boundary surface/unit volume /

,

4

"

Equation 1.2 applies to small undercoollngs, TE -,T. or to a material .\.-:-l-

with suffic!ent alloying addl.tlons to delay t~e diffusion of carbon.~

Equation 1.3 Is for a large enough nucleation rate to fill up the

grain boundaries with nuclea~es at the very beginning of the transformation.

This will apply to pure-carbon steels or low temperature transformation. In

•

',.I"

" Intenmc~late cases, grain-boundary saturation ·may be

transformation rather than at the beginning.

,

,

..

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.200 T Pho ••

1100 (~"'~I'.I 1000

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fig. 1.1 I I I I

fig.

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400

l >00

200

'00

EullClood poot'Il I

.....

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•• h,e (1.,,11. + celMtltlt.) I I

• I CftMnfit. 6.7 ...

1 4.3"'­, o 0-' lD ,~ z.o 2" lO 3~ 40 III!! ~ :..~ 6.0 6.~

•• The metastable system Fe. Fc,C. (Fror:n COfUlilllriOIl 0/ Binary A/loys.

by Hlnsen, Max. and Andcrko. K un)

...

~ .; 11 ,. ,

1.2 I ~ E -. 0

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0.340

0.320

0.300

,00BO

0260

0240

0220

0.200 22

•

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V 1/

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o 230 240 2~O 260 210 28 o Tfonsformohon IirM.nc:

FlO( nC data showio& the liMaril), Off' powlh ralc of pearlite. Eutec· laid h=ah·puriIY :r.tecllram(ormcd at 708' C. Iff n Frye. J. ii .... Jr .• Stansbury. E. E .•

and McElroy, D. L .. T,am. AI • 191, 19~). £.119) . . . ' .~ .

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~40r---.-------~------~

!" 700 ,; 2 660 o

~620

,~ 560

fig.1.3 540 L----�0~·,~-----,-1-6~2~------,Jd

fig.

Grow~Ole.mm per $ec.

. .. . ~",IC of growth (G) of pearlite as a function of reaction" ~c;m~raturc in a high-purity Iron-carbon alia), of CUICClOid compo::..ili.>n. (From h)~. J. Ii., Jr.,

Stan,bury. E. E .• and McElroy. D. L.. T,on,. A/ME. 197. 19~). p. 219) . .

1400 "

~ 1200

X ,1000

1 800

• \ ! 600 1.4 .

~ ~

D 400 }

200 -.-0

0 2 4 6 6 '0 12 14 Tim.,'" second,

Rate of nuc1c3tio~ (N) of pearlite as a function of time. EUlcctoid steel transformed ill 680 J C. (Mch!. R.obert F.. and Dubc. Auhur. Phast Trans­

/o,moliollJ in Solid,. John Wiley and Son,. Inc .• New YOlk. 19~1. p. ~~~)

. 'I'~o

72~

,.. .

U 700 ~ 67~ ~ , 6~

i! l:. 625 E

fig~ 1.5 o!! 600

575

-- -~RQI' of growth

Rot. of nueltOt~~ Corbon: 0.78 "10 " Monoon ••• : 0 63 oro

~ ASTM oraln I'ZI,5'/4

\, \-

-10" ,0-' '0" Ralt of growth (rnm'"e)

, , I

to""" 10-1 100' 'O~ to" Rol, of nutleollon (nIlCStI/rn",J/ltc )

Variation of Nand G with temper.llure in an eUleeloid steel. (Mehl. R. F., and Dubc, A .• Phau Trans/ormations in Solids, John Wiley and Sons. Jnc .•

New York. 19SI. p. S-4S)

,

7

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~ <.' ~

• c

N" 1000/em'I", I G 1. 3 X!O·'cm/se,

7 , , fig.lo6 0.6 ~ ,

j u e ... ~

-=

0.4

O.~

o 100

/

/ ,

. 7 i/

./ 200 400 600 800 1000

Tltn. t In ucol\Cl,

Theoretical reaction CUf\'C' obt3ined from Johnson and Mehl eQuation. (Mchl. R. F .• and Dube. A.: Phau Tram/ormations in Solids, -John Wiley and Sons.

. Inc., New York. 19~I, p. S4~) .

, /

, •

8

•

\,

, . CHAPTER 2

TIME-TEMPERATURE-TRANSFORMATION DIAGRAMS

2.1 Isothermal Transformations'

If we isothermally react a number of specimens for different· . ,

lengths of time and determine the fraction of the transformation product

In each specimen an.d then plot these data as a function of the reaction time

we obtain a curve as in Fig. 2.1.

A plot of the time required to transform 1% (start) and 99% (end)

of au5'tenlte to pearlite, for a series of temperatures, gives Flg~ 2.2 (

The T-T-1 (tlme-temperatu~transformatIOn) diagram of Flg~

correspQl'lds only to the lsothermar reaction of austenJte .to pearlite.

2.3

It

Is not complete because the low temperature «5500 C) transformations of

austenite are not shown. These are:

•

(a)' The Bainite reaction. This has a dual. nature,-partly pearlltlc and

partly martensltic (see below). This reaction Is difficult to Investiga;te ,I In simple Iron-carbon alloys because I,; overlaps the region of the pe~rll{e tra~sfonnation at temper~turEis around 5000 C •. Steels transformed in this ,~

( temperature range. have structures containing both pearlite and bainite. '-----

Bainite, like pearl Ite, Is ~ mixture of ferrite and carbide. The c~bon,ls •

concentrated Into localized regions"the caHblde particles, leaving an

effectively carbon~ree ferrltlc matrix. Since the carbon Is uniformly

9

i

)

p~

dl strlbuted In \he auste~ I te, the ba I n Ite react Ion I nvo Ives compos I-tl on

changes and requires diffusion of carbon. In that aspect It Is similar to ,

toe pearlite reaction and differs fr()(l1 tile. martensite reaction. Another

difference with the latter Is that the baInite reaction Is not athermal.

The formation of bainite requires time and when austenite transforms to , .

10

bainite Isothermally, a typical ~-shaped reaction curve Is ob~alned (Fig. 2.4).

Bainite grows as pl,ates, a typical martensltlc characteristic.

\ ' When observed on a metallographlc section, bainite has a characterl,stlc

aCicular (needle-like) ap'pearance, similar to that of deformation twins and

martensite plates.' Surface distortions, surface tilts arid accommodation kinks

accompany the formation qf bainite plates, Indicating that lattice shear may . !

be Involved In the formation of the plates. Whl Ie martensite plates, In /-

most cases, form under conditions of .tllgh driving force and grow to their

\ final size In a smal I fraction of a second, bainite plates ~row slowly amt,

continuously. This Is because of the time required for the diffusion processes

'that accompany tho bainite reaction. '- , EaCh/bainite plate Is composed of a

~ volume of ferrite In which carbide particles are embedded. This Is an extremely

fine structure, and use of an electron microscope Is required to resolve Its

components. '

Bainite does not form untl L the austenite temperature fa'ils below

" a d!lf I nl te temperature, des I gnated as B s, (ba I n Ite sta rt temperature). This

temperature Is-sensitive to the presence of externally applied stress~s.

Below Bs' austenite does not. transform complete,ly to I)alnlte, bJJt the amount

"' of bainite formed Increases as the Isothermal reaction tempecature Is lowered

belowBs

' (see Fig. 2.5) •. At temperatures below Bf (bainite finish), austenite

transforms completely to bainite. The cl9Se analogy between the temperature

dependence of 'the bainite rea'ctlon and the temperature dependence of a martens Itlc

1

reaction Is apparent.

(b) The Martensltl.c transformation. The Marrnsltlc crystals are body­

centered tetragonal and can be regarded as an IntermedIate structure between

the normal phases of Iron, face-centered cubIc and body-centered cubic.

The martensite transformation In steels Is prlmarl Iy ~~rmal. Fig. 2.6

shows the formation of athermal martensite as a fUnction of temperature.

'The martens It l"c react I on In stee I sis not revers I b Ie. I ron-carbon martens I te

represents an extremely unstable structure with a high "free energy relative

to the more stable phases, cementite and ferrite. Carbon ~toms are trapped

In what should be a b.c.c. structure, capable 9f holdl,ng at equilibrium an 4' ~.

11

Infinitesimal amount of carbon, and that results In ~ high Internally-stressed

lattice. Moderate reheating promotes decomposition towards equlllljrlum

(temperl ng).

Both the Ms (martensite start) and Mf (martensIte. finish) temperatur~s

In steel ar(l tunctlon~of the carbon content (see Fig. 2.7). The Mf temperature

should'be interpreted as the temperature at whIch the reaction Is completed ,

as far as one can detennlne by visual means; this Is because, theoretically,

the martensite reaction can never be complete even ~t absolute zero temperature.

Substitutional al.loylng elements In steels affect the martensltlc

transformation points. This 1$ shown for manganese In FIg. 2.8. VJ

composlt!o~p containing 1% carbon and more than 6~ manganese, we

For lhose,

see I n the

diagram that M Is so far below room temperature that a quench to room temperature s ,

(avoiding the pearlite reaction) produces a structure th~t can remain as

austenitic Indefinitely.' Alloying elements (Substltutlpnal) lower. the M s

( th t ' t effect, foll-ed by chromium) except *-':r temperature man~anese has e s ~onges V"

for cobalt and aluminum whIch raise It.

12

.' I

The hardness of martensite- In steels results' from the ,presence , ' ,

To obtain a substantially hardened steel, we need: fo

(I) an adequate carbon concentration (see Fig. 2.9 ,and 2.10).

(Ill rapid cooling to produce a martensite structure (to avoid the

pearlite reaction). /

, In low ,alloy steels (Ies~ th~n a~ut 5 percent total alloying elements)

the hardness of martensite can be regarded as a function of ,carbon concentra-

fj' on (;>n I y • \ .' -'

2.2 The complete T-T-T diagram of an Eutectold Steel ';,

The complete .T-T-T diagram Of an Eutectold steel Is ~hown In

Fig. 2.11. a a Above approximately 550 C to 600 C, austenite transforms completely o .

to pearlite • .At, lower temperatures to approximately 450 C. both pearlite , 0.0 0

and bainite are formed. Finally, between 450 C and 210 C, the reaction'

product Is bainite only. The rate at which bainite froms de~reases with

lowered temperature and consequently, ve~ long o

bainite Just above the Ms tanperature. Y. r

2.3 Continuous Cooling Transformation Diagrams

times are required to form

j

Very few conrnerclal heat treatments of steel 'are Isothermal.

In most cases, the metal Is heated Into the austenite, range and thentt contlnuQusly

cooled to room tel1'4>erature. with the cool I ng rate varyln?! with the type of \

treatment and the size and shape of the specimen. The 'difference b~tween

, , '

.,

/ 13

Isothermal transformation diagrams and conti nUQus cooling transformation

diagrams can be understoOd eas I Iy 'by comparl ng these two forms for 1I

steel of eutectold composition, as given In Fig. 2.12. Two different'

cooling curves are also shown In Fig. 2.12.

Consider the cooling curve marked I. At the end 'of approximately

6 seconds this curve crosses the line representing the beg~nnlng of the

pearlite transformation. The I~tersectlon Is marked on the diagram 'as

point a. Point a represents the time required to nucleatope~rllte Isothermal·ly '0

at 650 e, the temperaturo co-ordlnate of poInt a. ~t a specimen cooled

along line I reached the 6500 e Isothermal/at the end of 6 seconds and has o '

been at temperatures above 650 e for the entire 6-second Interval. Because

the time requ I red to start pear II te transformation I s longer at temperatures

above 6500e, the continuously cooled specimen Is not ready'to form pearlite , '

at the ond of 6 seconds. More time Is needed before transformation begins

and since the temperature continuously drops, the point at Which transformation

actually starts lies to the. right and below point a, {,s~ below for a computer­

aided derivation of eeT from T-T-T diagrams); this position Is designated

by tho po In t b. I n the same way, I t can be shown that the fin I sh of the

pearlite transformation on continuous coolln~, point d, Is depressed below

and to the right of point e, the point where the continuous cooling curve

crosses the II no represent I n9 the t I nlsh of the Isothermal transformation.

. ,

fig.2.1

fig.2.2

L

• • ~ tOO

H ~ " n II

u •

f !

" TrQ",formal,on ,.' "- 99 ..... l~fO'ur. ,

Iron"'Ot~ltlon 00 680·C (AI

2~ hon

0 0.' 10, .00 .10' '0' 10' 10'

Time. In MConds • I

I

" .00c-~'---~~.--'--7,,-~>r---r---, . __ ~.!.-.:...:d........

100

600

500

400

300

Trond()fmo"Qn t.mPf"'Q'III"_

99.,. ptorhle

200L.._-_--_-_-_-_-_-_-_-----_-_-I ------------ ----'00

(81

OL-~ __ ~~~~~~~~~ Of ~ tOO 10' 10" 10" .10-

TUne. in MCOnds

, ..

(Al Reaction curve (schematic) for i~othcrmal rorm3ti~n of pc.:Jrlilc. (8) Timc-Icmpcralure-Ir;:nsformation diagr;lm obt:-incd from rcactlon cunes. (Adapted from ",/as ollJolht:rmaJ Tram/ornr(J/;on Diaa:ranu. United St31~ Slccl

Corporo'i<'n. Pin,buran. 1951)

fig.2.3

.'C ooo,----.,;..;.------~--~~~..., . [utectotd ~ture

700~'--- . . -----

600 ~"·~I. :-' t;: p r Peorl

S~ e 400

fXlO i! 200 M, -----------~

--~-"'to -------- - --..: tOO ---ltI

ftO-----------

oL-~~~--~~~~ 0_' 10 100' 101 t04 to' d

T ..... "'.econch The partial isothermal tr3nsform:ation di;Jgram for an cutcctoid steel:

0.79 pcr ccnt Carbon. 0.76 pcr e<:nt Man~""'. (Adopted from Arlo, of IsorhrrmoJ TranI/ormation Dia"anu. Unilcd Stales Slccl Corporation. Piusburah. 19SJ)

14

/

•

-

.fig.2.4

Tim" In ucOf\d.

A charOlctcri"tic of the i~othcrm:11 bOlin,itt Iran~rormOllion i1 Ihal It m~ly nOI go (0 complCli\)n. (After Ikhcnunn. R. F .. ;lnd TrOiano. r\. R. t ,\f,'11.l1 Prucr.'Jl,

70, No. 2, I ?56. p. 97)

f

•

fig.2.S

Temperatur.

Effect of ((rllperature on the" amount of bainite formed in an isolttcrmal .transformation (schematic). (Hehemann. R. F .• nnd Troiano, A. R.o !tltla/ Proluss.

70, No. 2, 1956, p. 97) (

,\ \ ,

I

: ,I i

! I fig.2.6 J .: ~ I

or, ,» .,. , .. '" T ......... _.-C

The rormation or martensite in a 0.40 per "'pt carbon low alloy "~I (2340) as a runction or temperature. (After Grange, R. A., and Stewart, H. M.,

Trans. A/ME, 167, 1946, p. 467)

15

·'

....

-\

fig. 2.7

Variation of 1.(, and Af, with carbon concentration in ~t('c1. I After Tro:ano, A. R., and Greninger, A. 8:, M<I.I ProK"", 50, 1946, p, 303)

>00 ". o~, .......

zoo , , p

, ,

i.., fig.2.S , I ,~ \ to'" ( .. ~. ., ". \ \010 ... ' ..... . .., \

\ \

0 l • • • '" or ..,... ,., c-of ... _

Variation of Mr with manganese content in three series of slccls with different carbon concentrations. (Arter Russell. J. V .• and McG!Jirc, F. T.; TrQfU.

fig. 2".9 ,

1100

1000

• 900 • ! 800 ~ . r 700

i= £ '<00

1l 300. > ZOO'

100

AS,." 33, 1944, p. l03) .

00 0< 08 1.20 F. 'NttQhl ~ cent c.ortx:wt

Variation of hardness of martensite as I function of carbon content IS measured on the Vieke" scale, (After Bain, E. c., Functions of Ih. Alloyilll

E/","nlS in SUt/s, ASM, Oe .. land, 1939)

16

•

,

fig.

10 •

:60 l ~!IO l! :0 40

I

2.1 0 =30 , r , , po • , 0:

.0

0 0 04 06 08 12 r.

Wtq,1 pet' cent corban

Va:ri;uion of lKc.h::lrdnen of m.lrtcm,ilc as a (unelion of carbon conlent as measured 0Tl' the. Rock.~cll·C $c.1.lc. (After Burn~. J. l.. ~1oore. T. 'L .• and

""reher. R. S .• T,D";. AS.II. 26. 1938, p. I) (.

u • .. ~

2 400

Eut-.;to.c;i ~otUf'e ._-- '---'---'-----"""---••

(

fig.2.11 !

200

100

.<;;- -.... --MIO---------------------

--MIO - ------------------

:OOJ~--..l..---IO.l:--~-IOOL:---IO~·---IO.J..,.·-..,......JIO·

. The complcle isothcrm:ll tnnsformation di::lgnm (or .. n CUlec10id stul. Soticc that thjs stttl is nol 3 hith-purity iron-carbon altoy •. but a commcrcI31,~IC'~1 (AISI 1080) conl~ining 0.79 pcr cenl c .. bon ~nd 0.76 pcr cenl nunganese. Thc (!!ttI of Ihe nunii~n= "ill be discun"d in Ch3p.~r 17. (Adop.ed from Alia. of '1OIA~,mtJl rrolU!o'in,ation Dialranu. United States Steel COfJY.)ralion. Plusbureh.

19SI)

17

,

-

~

~ ;;

fig.2.12 ~ ~

E ~

800.-----~----1r-----._----~----_r----_.--_,

600

500 ,

400

,500

zoo

Moo 100

0 01

.. -

£"'~CIOOd tempero'u'.

"

'--" , ,

------ ------------- --------;..---------End of pCQrl,le "0""01-mOI,on on (O"""uOI.'

'" COOlonQ

SIOII of pC'Jtl,l. t,ondOtll\Oloon DR con"nv~, cooling

, , , , ..............

, , ,

\0

, \

\ \ ,

100 , 10' 10'

The relationship .of.lhe continuous cooling di3~r3m to, the isothermJ.1 diagram for an (.ulC'Ctoid su:cI (schematic). (After Alias (1/ JJolh~rm,,1 Trans!ormat;o"

Diuxrams. Unu.d Stot .. St«1 Corporation. Pitt.bur"'. 1951)

,

16

-

,

•

CHAPTER 3

HARDENAB I L ITY

3.1 Definition of Hardenability

When a piece of ~teel Is hardened by quenching, It wi II

generally be found 'to hardon not only at the surface but also to an

appreclablo depth In the piece. That depth differs for different kinds

of stoel. The depth of hardening reflecting the "hardening capaclty" of

the s;;ee I Is ca I I ed Its "har;denab I I I ty".

In Fig, 3. I the CCT diagram of an outectold steel Is shown.

A number of qualitative cooling curves are also shown on tho diagram.

The dashed line Is tho critical-cooling-rate .curve~ Any rate of cooling

faster than this produces a martensltlc only' structure, slower rates

produce a mixture of pearll~e and martensite or pearlite only. In a steel ., specimen of appreciable size, th!3 cooling rates at the surface and In thE! , '

center are nol the same. The difference In, these rates Increases with

the sevorlty of the quench, or the speed of the cooling process.

of the difference of cooling rates It Is possible to have entirely

microstructures at the. surface and the center of the specimen.

3.2 Critical dl~ter

Because

d(f~~rilO~

Saln and Davenport InTroduced the custom of showing the depth ~

of hardening of a quenched round bar by use of a symmetrical U-curve. If

a round bar, hardened by quenching, Is explored across only one diameter

'" for Its extent of hardening, It wi I I shO'tl an asymmetrical hardered pattern

19

)

as In the left side of Fig. 3.2.

The depth at which the 50 per cent martensite

obta~ In a bar of steel Is a function of a nuJber of

structure Is

variables,

Including the composition and grain size of the parent austenitic phase,

the severity of the quench, and the size of the bar. For 'a certain

steel there Is a unique diameter called the "critical diameter", such

that a bar of this diameter harc;lens so that It has 50.% pearlite, 50%

martens I te at I ts center. The va I ue of the ,\:; rl tOea I diameter" D, depends

on the steel and the mean~ of quenching, and It Is a measure of the

hardenability of this particular steel •

..1 3.3 Ideal Critical Diameter

• D depends on the cooling rate, and so to eliminate tlrls non-

material variable, It Is the practice to refer D values to a standard

cooling medium. This standard Is the so-called Ideal quench, and supposes

a hypothetical cooling medium which brings the surface of a piece of steel

Instantly to the temperaturEi of the quenching bath, and m~lntalns· It at

this temperature. The crltlc!!1 dlamet,er corresponding to an. Ide!!1 quench

Is ca I I ed the I dea I cr I t I ca I diameter and. I s des I gnated D1 •

The Ideal quenching medium Is assumed to remove the heat from

" the surface as fast lIS It can flow out from the Inside of the bar. While

such a quenching medium does not exist, Its cooling action on steels Is

c!!pable of computation and comparfson with those of ordinary commerlcal

quenching media, such as water, 011, and brine. Information of this type ~ . fj

Is frequently presented .In the form of curves such lIS shown In Fig. 3.3

I

20

"

wher? the I dea I cr I t I ca I d I amete,r Dr Is p lotted as the absc I s?a, and

the critical diameter D Is plotted as the ordinate. A number of ~

different curves are plotted on this chart, each corresponding to a

dl He rent rate of cooling. In each case, the rate of cooling Is measured

by a number known as Its H value, pr the severity of the quench. Some

21

of the values of this nu~ber for commercial quenches are given In Table 3.1.,

As an example, assume that for a'llrlne quench D Is found to be 1.0 In.

The H value for a brine quench with no agitation Is 2.0. The Intersection

of the curve for th I s severity of quench wi th the' 1.0 In. ord I nate occurs

at a DI va I ue (absc.l ssa) of 1.4 In.

The Grossman ,method of determining the Ideal critical diameter DI ,

outlined above, Is too time consuming to be of wide practical application.

A 'more convenient and widely used method of determining hardenabillty, ..-'

employs the Jomlny End Quench Test.

$ 3.4 Jomlny Test, or End-Quench Test •

This was first Introduced by Jomlny and Boegehold as. a means

of measuring the hardenabl Iity of the Case on'carburlzed steel. ,Soon

after that, Jomlny(4) showed that the same principle of' end uenchlng could

be extended to Inclute the non~caruburl~lng'steels, Ind ed, the whole

class of heat-treata Ie steels. By far the most extensive use of the test ,

Is now In such st,eels, for which It has been thoroughly standarlzed. (5)

In Its present .torm (Jomlny ) !he test-piece has the shape

shown In Fig. 3.4.' This Is the standard form used for all steels, with

\ '"

the exception of the very shallow hardening ones.

The test has been standardized by the ASTM, SAE and AISI.

In conducting this test, a I In. round specimen 4 In. long, Is heated

uniformly to the proper a'ustenltlzlng temperature (specified for varlolls

stee I cI asses). I tis then rllmoved f rom the furnace and p I,aced on a

fixture where a ~et of water Impinges on the bottom face of the specimen

(Fig. 3.4>.. The size of the orifice,' the distance from the orifice to

the bottom'of the specimen, and the temperature and circulation of the

22

water are all standardized, so that every specimen quenched In this fixture

receives the same rate of cooling. After 10 minutes on the fixture,

the specimen Is removed and two parallel flat surfaces are gro~nd 10n~l­

tudlnally to a depth of 0.015 Inches. , Rockwell C scale' hardness readingS,

are tufe'n at 1/16 Inch Intervals from the quenched end. A typical hardenID II\'ty

curve Is shown In rig. 3.5. When the test Is run In the standardized

manner, the reproducibility In different laboratories Is very good as seen •

In Fig. 3.6.

Of particular Importance Is the relationship between the size,

or diameter of a steel bar quenched In an Ideal quenching medium, which "-

has the same cooling rate at Its center as a given position along the

surface of a ~omlny bar, (see Figs. 3.5 ~nd 3.7). Thus, If we know the

50S martens I te pol,l)t Qn the Joml ny bar, we can- determl ne the I dea I diameter.

Each I ocat Ion on thE! J om I ny test piece, quenched, I n a standard manner,

represents a certain cooling rate, and since the thermal conductlvlty'of

all steels may be assumed to be app~xlmately the same, ,thIs cooling rate •

Is the same for a glve~ position on the test piece regardless of tho

'composition of the steel from which the te9't ~\ece Is made. Each specimen

'- '

I

I

,thus subjected 'to a series of cooling rates varying continuously

from very rapid at the quenched end to~ery slow at, the alr-cooled,ehd.

The cooling of the Jomlny bar during the quench Is duo to •

<..' Two effects, the water quench at the one end and the cooling due to the

surround I ng a I r. At the water-quenched end, the water Is overwhe Iml n~ily " '

the predominate factor and the effect of air-cooling begins to be,felt

'. only se:'lOral Inches from that end •. The effect of the air cooling In a ,. '

one Inch,dlameter size

Knowlton and

Is nonetheless measurable at the non-quenched end.

(6) Snyder showed that shallow-hardening steels,

whose chara.yterlstlcs would appear' within the first ,1/4 Inch from the

quenched end / cou I d be tested on Joml ny bars much shorter than th,e standard

23

"

4-lnch length (Indeed, shorter than one Inch) without Introducing appreciable

variations In the ;esults, and that the diameter of the 'bar' could be ~

sma II as 3/8' - I nch. 'They a I so showed that the un I mpeded he I ~ht of the

water jet could be r.educed from the usual 2 1/2 Inches to aS,llttle as , '>

I-Inch without any detectable variation, In the result, Indicating that

the severity of quench varies 'but little with thls,chango In veloc!ty.

Some steels, such as deep-hardening SAE 4340, ,harden along practically -

the full length of the 4-lnch Jomlny bar. It might be presumed that such , , .

steels cOuld stili be tested by the end~quench principle If the Jomlny bar

were made longer than the 4-lnch standard. However, In such ,cases If

the diameter Is left as small as I-Inch It Is found that the alr-cooll'ng

effect becomes prominent. The air-cooling may be minimized, and consequently'

the end-quench effect better reta,ned, If, when the Jomlny bar Is lengthened, (

Its diameter Is at the same time Increased.

\

, ' ,

3.5 Effect of Higher Quenchl~Temperatures

, , . (7) Jackson and Christenson eXqmlned a number of steels

I f 0 0 n a range 0 quenching temperatures from 1600 F to'2100 F, and

showed that diminished hardening Is caused by quenching from higher

temperatures In steels having ·Iow hardonabilitles, I.e., steels having,

"critical hardnesses" I!, the range 1/8 to 1/2-lnch Jomlny distance.

,There are many conflicting Influences at high quenching

temperatures. , 0

An au~tenltlzlng temperature of 2100 F Is likely to cause

solution and diffusion of relatively Insoluble alloy carbides, If such

24

are present. It Is also II~ely to cause some austenite graln\growth.

Bot~e Influences would cause greater hardenabl Iity. Opposed to

this Is the Jackson-C~rlste~son effect {Fig. 3.8). Jackson and Chrlstonson

showed'that their theoretical heat-flow curves Indicate for a 21000 F o ' ,

(1150 C) quenah, a,longer period of time to traverse the temperature range

1100 to 9000 F (595 to 4800 C) than for a 16000 F Jomlny quench. The

longer cooling time In this range affords more opportunity for fine pearlite

or' upper bainite to form, thus lessening the amount of austenite available

to form martensite, and thus lessening the extent of hardening (see

Fig. 3.8).

3.6 The Effect of Austenltl~Graln Slze.:on Hardenebility I

This Is explelned on the basis of the heterogeneous manner In

which pearlite nucl~ates at austenitic grain boundaries. While the

'\

•

growth rate G of pearlite Is Independent of the aus~enlte grain size,

the total number of nuclei that form per second varies directly with'

,25

the surface aval lab,e for their formation. In a fln9-gralned steel there

Is a larger total grain boundary area/unit volume than .In a coarse-grain

steel,'and therefore the, formation of pearlite In the flne-~aln steel Is

mor~,'rapld and results In a lower hardenabl II tv. \

3.7 The InfluencJ of Carbon Content on Hardenability ,

The hardenabl Iity of a steel Is strongly Influenced by Its

carbon content. In Fig. 3.9 the variation of the Ideal 'critical diameter

,"

o with carbon content Is plotted for different grain sizes. This hardenabl Iity I '

effect should not be confused with the hardening effect of\ carbon. They "

are qu Ite ' Independent phenomena.

I

'3.8 The Influence of Alloying Elements on Hardenability.

;.

Each of the chemlc~1 elements present In steel affects harden-(8) ,

abl I ltV t,o some degree. Grossmlln " In 1952, reviewed his method for

calcuillting hardenabl I Ity from chemical composition lind grain size. In

this empirlclli method the lIlloylng elements lire considered to have multl­

plyl ng effec'ts on the hardenabl Iity. For, plain carbon and low-lIl loy hellt

'treated steels. the method gives a rellsonable first IIpproxlmatlo,n to the' I

expected hardenabl Iity. It Is most useful when a steel of known composition

lind hllrdenllbility Is to be modified II little to have some other desired

" ,

•

hardenability or In developIng substitution ~lloys of the same harden-

ability. The method Is much less reliable when applied to hlghor ~lloy ,

steels! (hardenabilities over'" 01 c 3 In.).

Within the above limitations, the steel Is considered to h~ve

,a "base hardenabillty" attributable to Its carbon content and grain size

alone" the hardenability being expressed as "Ideal critical diameter",

01' For each alloying element, this "base diameter" Is multiplied by a

'numerical factor, whose'magnltude depends on the, quantity of ,that element

present and on Its effectiveness In Inf luenclng hardenability. ,,'

26

Suppose we have a steel of ASTM grain size 5, 0.4 percent carbon,

0.73 percent manganese and 0.37 percent sll kon. From Fig. 3.9 one finds

that the hardenab III ty of the "base stee I" at thEl given gra Ins I iZe Is'

DI

= 0.25 Inch. If manganese Is added now to the "base st&&l n, the effect

Is to multiply the hardenability by a factor that ,we obtain from Fig. 3.10.

For 0.73% manganese the multlplY,lng factor Is 4.0. From Fig. 3.11 we obtain ,-, the multiplying factor for addition of 0.35% silicon as 1.25. So, the

final hardenabillty, expressed ~s Ideal critical dl~ter 01' of ~ 0.40% C,

0.73% loin, 0~'35% SI,steel would be:

01 c o~ase x 4 x 1.25 ~ 1.25 Inch

The corresponding Jomlny harden~bliity from Fig. 3.1 Is 1/4".

'.

Grossmann's monogr~ph gives multiplying f~ctors for most Import~nt

steel elenlents. , ,

We note that In the case of carbide-forming elements, like Cr ,

and I0I0' the h~rdenability effect Is Influenced by carbides which remain

•

undissolved at the moment of quenching, and that the proportion of

carb I des remlll n I ng und Isso I ved .I s I nf I uenced not on I y by the amount

of 1I110ying elements present but also by the thermal history prior

to quenchlng~ Consequently, It Is understandable that the published

multl~lylng factors vary w.ldely In the case of the carbld~-formlr1g

elements.

,

.,.

27 K .-

fig. 3.1

0 .. • ~

u

fig. 3.2 0 a:

ulcc;tOod 1'"'ptHQlur'

600

500

\ l' .; }

400 f • ~ ~

300

,-

\ .,

Uorl.n.ate Mor' ... ~I. C-.. F_ Df\d pea'liI, ..-01''''' pco',,' •

0 0,1 \0 100 10' 10' 10' ~.1O~

T ....... 1ift "conds

The variation of microstructure as a runcti~n of cooling fate for ~n culectoid llcel

60 I I I I

~O--

I I I I 40 I I I I I I

30'

~2rO:O.~ 1--2 '-0.0-\ tLrn)-"'''M~rk&I Hud_ huftll. of 1M ~

0,,", ~.~ If Onl)' .. Sin;:"" IJla ........ __ "::J:Pt..n.t..

Ut. ... "O-~'_--"kal u_~ •. I • • ~k" t'" T_ HaITft of .... U aN ld..,kal '_'nvr t...pl. Eadt ~ u. Awuaae .t II •• , .... U.

)

28

.. ,

table 3.1

fig.3.3

SEVERITY OF QUENCII VALUU fOR Sm,. TVP ... A .. QLI.~CIII"C; CoNDITIONS

·H Vul., (,.20 O.H O.SO 0.10, 1.00 I.SO 2.00 '.00 . co

QUl'IIC'1Ii1ll Condition

rOOt oil quench-no aHitation Goo4 oil Q\!cnch-moder.:l~c Dail'llion Very lood oil quench-Good agilalilJn Strona oil quench .. \oiolcnl acitation Poor Willer quench - -no OIgllalio" Very lood water quench sironil 3gilalion Brine quench-rio asilalic," Brine qL:cnch·-'vlolcnt agitation Ideal quench

Relationship or the critical diameter D 10 the ideal critical diameter {I,

for sevcral rales of coolinl (H values). (}lrter Grossnun,. M. A., EltmtfflJ ,./ HonkNJblllty. ASM. C1evel&nd. 19'2)

•

fig. 3.4

•

<I

Top of fillure

,

29

,

J r

OAT[ AS T. M f~D OI.(~CH TEST LAE:VRATOR't fOr: t4AR()(NAS'Uf'Y' 1 lIt !'p{(N(N

Of ST,EL IA l~~-.aTl TEST tr.O.

tyPE HEAT NO C.~I·~ C "" p S 5, ~, C, M, c. ~;'~~\\ p;.~~~ ~ .or • , •• .'3 ) .011' r' J , , • " , ,

i A REMARKS'

fig. 3.5· TO

,~

eo ~~

~

.~

• u

" 20

I~

I

fig. 3.6

I

I II N

~W,. ~ ;I~i~,,~r .. ,'!:'I::O ~ '1.'.. - ~~ .~ :II; =1" : . ~ ~ . :: :i .

'\

"' .

-l 4 , • 10 U 14 ~ II ~ U ~ 26 U ~ ~ ~ W ~ ~

OISTAAC( f~. QUENCHED ENO Of SPECIIII:N IN SlXTH;"H' Of INCH

• 8 '!' u

." ' . • c 1! D :z:

i ~ u D 0:

A typical end-quench hardenabllltY, curYe.

70

60

r' Hcrdenob,hty ew .... es at 9loborOIQr·ts . on 0 SA [ 4068 Sitel a End·Cooled ... ,,"od

50 ~

40

30 f--

20

10 o

,

. , ,

.~ I! I

I-- '----'

-"

. . , , 'i 'i 'J 2

Oilronce rrom Oucn,ched End, ,n,

, ,:..r ... _r' 11.,. ......... 111)· af • !tl".~ II"" ., Kl...t. T .... l .... i_ , I ,\1.",.1",_. :U .... i." Mc-p", .. I_­.Wllt,. CJ ••••• ,.

30

fig. 3.7

, .

o 4 B 12 16 20 24 20 3l

o.stotac •• ", it Inch UNls (.Joonon, ~Of)

Location on Jominy bar a: which the cooling rale i, equi"alent to Ihe a circlliar bar quenched in an ideal quenching medium. (Afler LamonI.

J. L., lro" Ag~, Oct. 14. 1943)

51 •• 1196 ChemlcOI Composition

1-;;:--.,---.,--- C 030~. Cr 0 10 ~

02

Mn 0.94 5 0.026 5, 0.16 p 0.012

I Groin Size

A.5.T.M. No. 1.8

0.3 0-4 0.5 0.6 Oistonce from Woter-Cl)oled End. Inchel

Eft'rt't of Jliltll Que-nthln .. Til"1III~r.ta;" OD

H ...... lftC or SAl: 14.1)0 St"". Ute'1-... au C.n.c .... -I

(

31

038

0.36 -l-' 0.34 --t-·

0.32 ~

u "- 0.30 u c .. . a 0.28 , ~

fig', 3.9 u E 0.26 0

0 0 Q.24 u ~

0.22

0.20 1----l---'.4'-1-++-Y--i--+--+--t-.-1

0.18 I-"--lf--¥+lhl-+--t--;--+-+---I

0.16 L-_L--lL---I._--"-_ ........ _...L._.J.-_L-.....J o 0.2 0.4 0.6 0.8

Corbon, ~

Ideal critical diameter as a runction or carbon content and austenite grain size lor iron-carbon alloys. (Arter Grossman, M. A" E/em,nrs of Harden­

abilil)', /'sM. Oevcland. 1952)

fig. 3., 0

150

13.0

~ 11.0 u o ... 9.0 .. c ~

Q. 7.0

:>

::l; 50

f-

e-

f-

f-

i-

" l' Monq'onese , / -

ll' 01

f l/

/L 3.0

Vo, 10

o

I I

I , , , ' , I

1.0 2.0 ~an90~se. ~

3.0

)(ulllpbID" F'art .. r. fur C.kulation of F.ft'«1 ~ M.n"an"'t1W' u .• UarcintabiJi17. (C"'/I ••• 11 LA .. .,)

32

, .

(

.. ,

fig. 3.11

2.40

2.20

• 2.00 2 u o ... I.BO .,. c ." ~ 1.60

~

~ 1.40

'1.20

1.00

33

I I ! 1/ '.\

I .I , 'Vi l._~

. I I I I V ,

I i /' C-j . I I V

.-.~-

./ 1

I I

i i V I I I I ,

~ X! i I I , I

[,,( V 1_ ! ! I I r ! I I I I I . I I I

, , . ' I

o I 0.40 I O.BO 1.20 160. I 0.20 0.60 1.00 1.40 1.60

Silicon, ~

·Etreet" or SUicon Oft Harden.bUll,..

'\.. "

CHAPTER 4 \

UNSTEADY STATE CONDUCTION OF HEAT

IN A JOMII1Y BAR

4.1 General Formulation of Heat Conduction Problems

Consider an Infinitesimal ,cubical element, dx dy dz (Fig. 4.1)

In a material of density p and specific heat Cpo The statement of

tho heat balance for this element Is: .J

f heat Input conduction [

heat output bY] conduction

8

+ [ heat J ----generation -

This can be expressed mathematically as:

I

[heat eccumulatlon ] ~Ithln the element

~here K , K , K are the thermal coefficients In the x,y,z directions end x y z • q(x,y,z) Is,the rate of heet generetlon per unit volume. In phese trans-

formatlon'problems this Is provided by latent heat evolution.

Equation 4.1 Is the general statement of the conservation of heat ...

within the element, and for many prectlcal applications It can be simplified

to more convenient forms. Th~s, for unidirectional heat flow, ~hlch approximates

the heat flo~ In the end-quench Jomlny 'bar, equation 4.1 becomes:

34

•

"

4.2

If 1;>, C and K are not strongly temperaturo dependont equation -4.2 glvos:­p

or

•

where a = thennliTdiffuslvlty " klt>C -p

4.3

4.4

To solve a particular heat flow problom oquatlon 4.1 must be

• exprossed In a form appropriate to tho assumptlon5made-ln this problem and

to the assoclatod .Initial and boundary conditions •

4.2 Methods of Solving the Equations of Heat Conductfon

We brief Iy present here the techn Iques re I ated to th I s thos Is

(I) Solutions for semi-Infinite systems without sources.

The differential equation can be solved by finding a particular

analytl-c function that satisfies both tho dl fferentlal equation and the

boundary cord Itlons. Since thero are only a few fun_ctlons that satisfy

35

the d If ferent I a I equat I on for the II near f low of holit, (w I thout heat generat I on)

4.5

I

we may be able to solve·the equation by selecting one of these

functions and, by a suitable cholco of the arbitrary constants,

satisfy the boundary conditions of tho particular problem.

As an Illustration, let us consldor the function

TeA, + Berfc. Y 2/Ot

4.6

whero A .and B are constants. We will procoed to show that this equation

36

•

Is a particular solution of equat.lon 4.5 for boundary and Initial conditions

similar to those for a Jomlny bar. It Is easily seon that A Is a solution

as It wll'l<anlsh when differentiated, elthor with respoct to Y or with

rospect to t. The functl~n, Berfc CY/2;-QT) requires more detailed consideration

The orror function orf[fCY)] and the complementary error function

erfc [fCY)] are defined as:

arf [fCY)] = ~ .rn- S

Hy) -~ e d5

o

orfc [fCy» 2 = I - erf [fCy» = 7Tt

4.7

4.8

Plots of orf [fCy)] and erfc [fCy)] against fCy) are shown In Fig. 4.2.

It Is noted that the real· variable on the right-hand side of equations

4.7 and 4.8 lsi not ~ but the IntcgrDtlon limit fCy). It cpn be shown that

the orror functions obey the following relatlonsnlps:

erfCo) cO, erfc Co) =

erf (:)0) = I, crfc (00) = 0

Also

erf[-fCy)] = - erf [fCy)]

and

erfc [-f(V)] c I + erf [f(V)]

Error functions can be d I Herentl ated- and I.ntograted. To

differentiate them we can make use of Lelbnltz's rule fOr differentiating

I ntOgrili s:

4.9

That Is, the derivative of a finite Integral, the upp'er and lower limits

of which are represented bV functions, Is equal to thl derivative of the

upper limit, multiplied bV the argument of the Integral evaluated at the

upper limit, minus the derivative of the lower limit, multiplied bV the

argument of the Integral evaluated at the lower limit. By applvlng this

rule to the second term on the right-hand term cf equation 4.6, we obtarn: , J

~ [Berfc -..:L ] = V 2M

-/14at e Ilfa.t -

4.10 B

and

~2 V • [Bertc --] ;-;z 21at 4.11

The time derivatlve'wlll be:

a t - [Berfc -- ] 4.12 at 21at

37

'~'

By canparlng the right-hand srdes of equations 4.11 and 4'.12 we can

readily see that:

2 a~ [Berfc '--.:L ]' '" 2 [Berfc ~ ]

ay u<rt' at Uat 4.1:5 ,

Therefore the function A + Berfc'y/2at Is Indeed a solution of the

differential equation 4.6.

Three InJtlal and boundary conditions for a Jomlny bar can

be represented approximately by:

and

T = TI at t c 0, y > 0

T = T at y = 0, t > 0 o

T -+ T I at y -+ Of!, t > 0

4.14

4.15

4.16

The last boundary condition expresses the fact that for finite Times,

at an Infinite value of the spatial variable, the temperature may be

" considered to renaln aT Its Initial value. Let us now seek the solution

of equation 4.5 and 4.15 - 4.17 In the form given by equation 4.18.

T = A + Berfc --Y­Uoct

4.17

Fran the boundary equation 4.17 and the stated fact that erfc (:)0) = 0,

I t can be seen that A = T I' By sett I ng 'A = T I' and t c 0 we can read I I Y

show that equation 4.18 also satlsf les the boundary condition of equation

4.15. Finally, DOting that erf~ (0) : I, we can satisfy the remaining

boundary condition (equation 4.16) by setting:

B c T - T 0' I

Therefore, the required 'SolUTion Is

38

<,

.'

'_.

4.19

or

T = To + <T I - T ) erf -L a 21.it.

4.19 a

, In most engineering problems It Is ,necessary to calculate

not only .,the temperature .dlstrlbutlon, but also the heat flow per unit

area. The heat flux In the y direction Is:

or = -K (- )

a f r 4.20·

Substituting for T fromequa~on 4.19 and uti Ilzlng the expression for the

first derivative of erfc If(y)] (equation 2

e -y /4at , CJy " K<To - T I) .::....--­

iwat .

Th'erefore, the heat flux at .. the surface

Jomlny bar) , Y " 0 Is

'1y " 0 = K<Tc - TI)

,'wcrt

of

4.10) we obtain

4.21

the' soil d (quenched end of

c.

4.22

It Is sSen that equation 4.22 predicts an Infinite value of the heat

flux at time t = O. This mathematical. exprbsslon reflects the Impossibility ,.

of I<?werlng the surface temperature 'suddehly by.a step value. The total

The solution of ttie linear conduction equation I.n the form of

39

an error function Is appropriate only for. semi-Infinite media with a uniform

'I

Initial temperature distribution, and only when the temperature Is

malntal~ed constant/at the boundaries.' ,There are a ,few analytic

solutions for p..c:lems with q ~ 0, but' none of these are appropriate ,

,to the problem of present Interest. Accordingly, numerical mothod~

,';ust be used as reviewed In Appendix I.

\

"

40

41 \

.'

I I n

- • I pC~ ~._

I " )...----- ----

<. - __ ' _. _ •• ("[) ". ,.41

I

.u/ I' I

I

I. ~ HUI b.llolnce in " cubic:ll (Iclne-ol.

','

fig. 4.2

,20

I'll

Error (unctao"I,

42 CHAPTER 5 ,

.,. ~ DERIVATION OF CCT FROM TTl DIAGRAMS

5.1 Introductorv

The dlroct experimental Investl'lat,i,on of transformation

on continuous cooling Is difficult, excopt In those stools whoso alloy

content makes the auston I to trans form vory s IO\~ Iy. Tho I sothorrn.~ I d i agrilm,

on the othor hand, can be dotermlned with relatlvo easo for most corrroon

grados of stool. It is consoquently more conveniont to derlvo a cooling

diagram from Isothermal data If a satisfactory method of dorivatlon can

be devoloped. Even so, Isothermal'- diagrams.may be Inaccurate, particularly

with regards to"the beqlnnlng line at the "nose" In fast-roactlnq steels,

and to both the berrlnnlng and' end linos In the rogion'Just bolow the

equil ibrlum transformation temperature (A l. In the derivation of a , el

continuous cool ing dlaqram theso inaccuracies arc neccssari Iy Includo~ and. ,

posslblyampliflod.

5.2 Tho Grange-Kiefer Approximation

Grango and Kleffor(9l In 1940 developed a geometrical methOd

for deriving a CCT from a TTT diagram. They made two 'basic assumptions: ~. ,

•

" tl) that the extent of transformation of tho austonlte at the Instant

It cools to the Intersection point X (see Fig. 5.1) of Its constant

cooling rate curve with the ~~Innlng line of tho TIT diagram,

Is not substantially greater than It would have been If quenchod

Instantly to Tx; In other words, that somo additional cooling tlmo

wi II be required before any measurable transformation occurs In

'all cases of practical Interest.

(Ii) that on cooling through et limited temperature range, for~xample,

'-

43

Tx to To' (see Fig·. 5.1), the amount of transformation Is substantially

equal to the amount given by the Isothermal diagram at the moan

temperature 1/2<T + T ) after a time Interval I - I ,(see Fig. 5.1) x a a x

~Ihen the austen I te has cool ed to po I nt X, It wou I d be necossary·

on the basis of the first assumption to hold It at the temperature T for x

an additional time Interval equal to I In order to form a ~Islble amount c x

of transformation product. But the austenite. Is not held at Tx

' continuing

to cool uniformlY (see Fig. 5.1). The addlttlonal.lJlterval after point X

. rC'1ulred to product visible transformation will. obviously be les.~ than Ix I

since the isothermal beqlnnlng line sll'ows that an even shorter tIme Inter,val

Is required for the beginning of tr~nsformatlorr'at lower temperatures. The • I

point 0 at which visible transformation 'Just begins on cooling can be

• graphically estimated on the basis of the second assumption. On this basis

the amount of transformation resulting from continuous- cooling to point 0

Is' equal to that at the corresponding point 0' on the Isothermal diagram,

the c. o-or~es of 0' being To' = 1/2 (T + T ) ~nd I 1 = I - I • I.. x a - a a x If we select 0 properly by trial. and error so that 0' II~s exactly on the

•

~'i. I; , /'

beginning line of the Isothermal diagram then 0, - point B In Fig. 5.1-,

Is the point of beginning of visible transformation on continuous

cooling for a. specific constant cooling rate (50 degrees Fahr. per

second In this example). Different cooling rates give. different points.

These points Joined together give the CCT- start curve. In a simi lar

way we c~n derive the CCT - end ~rve (see Fig. 5.1).

In Fig. 5.2Jwe present an experimental CCT diagram due to

I

Grange' and Kiefer and In Fig. 5.3 the derived CCT diagram for the same

steel. Both dlagnims aro from thllr paper (ref. 9) and on the whole they

show fal r agreement. However, the location of the curves compriSing

the. experimental cooling ~am were (In a number of Instances) not

determined with precision, and Judgement based on oxperlence was used

In drawing the curves. The mest Important discrepancy, according to

Grange and Kiefor, lies In the location of the curve represontlng the

camp I ot I on of the pear II te reacl1 on; the oxperl menta I curve' I s somewhat o

10lier than title derived curvo although the location of the formor was

confined by exam I nation-" of a number of specimens. This geometrical

construction is stl II widely used.

5.3 The rule of additivity

It has been long known .. that some trallsfonnatlons obey an

additivity rule, ('ref. 10, II). /"

I t t d b C h (26)

·The rule of additivity s 5 a e y an. as:

"Given an Isothermal Tn curle for the' time t, as a function •

44

., "

of temperature' at'whlch the reaction has rea~ed a certain fraction ,

of completion Xo ' then, on continuous cooling, at time ~ and

temperature T, when the Integral

~

equals unity, the fraction completed will be X n. o

5. I

The rule has often been stated for the Initiation of. the

reaction. This Is a quantity which Is difficult to define since It

depends on the method ,of observation. Cahn(26) recommends therefore

'that we define ,the time as ,for about I~. reaction.'

( I I ) Avraml (1940) showed that fO~he case wh~ro the nucleation

rate In untransformod material Is proportional to the growth rate forI

every temperature the reaction Is additive. This Is a very special

condition, and one might not expect to'encounter It tor the case of the

pear II te reaction wh I ch I nvol ves very camp lex nuc leatl on and growth ,

45

pro1sses. Yet, additivity has been demonstrated empirically for this

system(H: Krainer(2». As Cahn(26) argues, this could be due to a

tendency towa~ds early nuclear site saturation, which favours the conditions

for precise additlvlty(26). At the same time, Avraml's special conditions

may also be approximately obeyed as KlrkaldY's(14) seml-emplrical an~lysls

Implies. This wi II be discussed later. '"

,.

J.

5.4 Computer-aided Derivation of the eeT from the TIT diagrams

On the basts of the additivity ru Ie we have developed a

computor program to derive the OCT from the TT1 diagram. Let us

cons I der as T the tl me roqui red at temperaturo T for comp let I on of 1%

of tho isothermal transformation of austenite to pearlite. If the - ,

austenite cools continuously then we wll I have 1% of the austenite trans-

formed to pearlite when the Integral

5.2

where TE Is the eutectold temperaturo.

The cooling rate of the austenite-pearlite system Is at any

tempo rat u ro given by:

R..- = dT . -I dt

\ 5.2a

,

Consequently tho Integral, In equation 5.2, can be written as:

I'-'~' . T TE TE

5.3

~ We next write a computer program .whlch for ~ny given cool ing'

path (cooling rate variable or constant) and given TIT diagram predictS· ,. 1

46

tho t"mperature and the time for 1% transformation on this cooling

path. Note that there I s not such a th I ng as a un I que CCT dl agram.

Rather, 'for any consistent set of cooling pllths there will be a

corresponding CCT diagram. For example, we could cohstruct' a standard

CCT diagram by using constant cooling rates of Increasing order; alter-

natively and perhapS more realistically, we could use cooling curves

of the error fUnction type as a standard.

I n Append I x II we deve I op a computer program for an arb I tra ry

/ sot of cooling curves.

In Fig. 5.4 we give the results of a sample calculation using

the TTT diagram ,for a 4068 stoel along with CCTI derlvod using the

Grange-Kieffer model and CCT2 derived using the additivity rule (this

program); the crltl,cal coollnq curve Is the one th"t corresponds to an •

8 mm distance from the quenm!d-end of the Jomlny bar. 'This cooling curve

has been derived through a numerical solution of the heat ~onduytlon ,

equation tor the Jomlny bar.

If we compare CCTI and CCT2 In Flq. 5.4 we see that T~ ,

p red I cts ,comp I et I pn of the I % of the aus ten i te to pear II te/react I on

at an earlier time. This Is due,to the fact thllt t~~t assumption

of the Grange-Kieffer geometrical construction (see above) Is not valid,

since we do Indeed have part

the cool'lng curve Intersect!;

of the Incubllt10n time consumed before )

(

the TTT ~urvo. Mathematically the dlfferonce

between tho Gran~o-Kleffer construction and the additivity rule can

be expressed as fo'ilows: Let us sp lit the Integral

l

47

\

f E

dt -I'

I nto two parts:

,

dt = T

"

5.4

5.5

where T Is ttie temperature at which the cooling curve Intersects x

the TTT curve. We recognize from 5.5 that the first assumption of

. the Grange-Kieffer construction Is mathematically expressed as:

dt = T

o

and the second assumption as:

,

5.6

T = conrJilnt between Tx and To and equal to T(Tt~) where TM = 112 (Tx + To)'

ThaKs:

f ) r dt_= I dt I (t . t ) 5.7

T (TM) =

T(Ti-\) -T (TM) 0 x

T T x x

..

46

l

So wo soo from o'luatl.ons 5.5 - 5; 7 thllt the Grange-Kieffer construction

Is an approximation ~o tho additivity rule. The first assumption

Introduces an error which varies from very small values up to I. , Because of this the Grange-Kieffer construction (CCTI In Fig. 5.4) falls

, to prodlct nucleation for cooling curves that do not Intersect the

poarllte-start curve In the TIT diagram, for example, .. the cooling curve

at 8 mm distance from the quenched end of a Jomlny bar In Fig. 5.4

From equations 5.5, 5.6 and 5.7 we also see that In their

app roxl mat I on:

JT

0"

t::" '.7 T

x

dt t

= 5.8

\

BeCauso equation 5.8 represents only part of the additivity Integral

the valuo'of T satisfying It will always be smaller than the value o .. of T satisfying the complete Integral. Hence the CCT curve obtained

. 0

via tho' Grange-Kieffer cnnstructlon will "always lie towllrds the right

and 10l<er than that derived according to additivity rule.

Whl Ie we have concluded In the course of this work that t~e

di fferonce botween the Grange-Kieffer CCT and those derl ved viII the

additivity rule has little effect on the hardenability values predicted

<14 ) with the Klrkaldy algorithm , the Grange-Kieffer construction must

seriously fall as a predictor of experimental CCT curves. The additivity , "

rule will accordingly be used In all· our .subsequent calculations.

49 ,.

fig. 5.'

r

l I T. I I i lilii I: I l::' Iii : .... (~. I,. II I!"" ,

.- F";;: ":-1' -; -,., -'r ... _ ...... -_.-.- ..... - ~ ... -+. -" 'l ' 1'1lff~ I ' i:-T-:T, :-~ l~CccJ~':"'0-l i I! Ii::: !

so II i' . III,' I, , II I '" I

1100

1JaJ

ICOO

"-h -1-- -1- 1 ~(I \ I: i ~' , -~ +- -, -fl 1 /1-- -1°; :1 ! , I I 9 ~ I ' i + ..--;-,.,,_ •. ' .!~.-

I I \ .... ,r~.I-."f1".J'

li1l: I (1\i!!'t"j" ,: '\UU

..Jl)!*, ~ f- "'P-v4' . £. I r II i i\ i), I , , I \ I I I

, f" 111v i ! I' ':\ I I . u I L~ 1 /O"-rr-/' -.0- ,-'-rt" r I I I! I I: ~ .. I

(lJ ..

O' C6ed!O .. J'" dlO !s . Tm:"'~

-Cb:t.rt Sltnwlnl: a PlJrtinn o( the h'thcrm.ll T~:n,(.,r. nutiun Di:s:um for 1'1(lin C.ld:~n .t:ul«loid 51(('1 !1n~ .1 C..l:I~I,"'lt

. Coulill:f fbtc nr ~o J»'-.:;rtC'1 1:04ltr. I~r S~,:onJ "ltJlI\:.l (('\Jm .\\:, ... Tcml-=r~lure; l:!~Ii&r:&l(,t the ~t~'I" JU\'oln:J in lilC' ~i('tl'uJ 'I'll

Rtb'in~ 1"un,iorm,uion on Cl)Oillllt 10 Ihe lwtlu:rnlll UCL.r:lIl1.

50

'-,

fig. 5.2

fig. 5.3

IJXJ

q 510

,..,..,

-DUcn_ ~ .\."'_~ Tr_d ... , ...... ~A.F_ 4.1"0 ."tn'1 ... C-.ti_ c..Il" al c.:-~ kaca. lb. ... U,.. l:qcn-.nW tJI,wt'f.at ......

-DiKn. Sh-i-= A .... _it. Tu"d¥1U'" _ c-a ..... at C .... J.IIC ...... s.:A.L 4140 SI .. cL lJorTi~ (,.. l..o.t.w. "loU.

51

~

~

(J)

l')

~

8: 4-" III

co :g ~

W I- ,

I I , I ,

fig.5.4

"' / I /

\ ! / ,\ ! I /

, \ \ ' ~ \ \ \(.( /,1. I \' '_J", .j' (1,'

I I I I

o

I ~,' v " " ...... \ C;-" ....... _,; I /; 'l1 ,<." .. ;, I 1c ... I ,-:, ... ' /" ..... , V".J.... OJ

I- \ ' 0 "I (J -." ./ ,J I ..::-

El" /t;' ~ / :i

E E

o o .....

o o .....

0 ~

0 M

0 ('.I

0 .....

....

52

~ ~

u OJ If) ~

OJ

E .--

\ CHAPtER 6

CXl~IPUTER-A I DED EST I MAT I ON OF HARDENAB I LI TY FOR LOW-ALLOY STEELS,

\ , 6.1 Introduct,lon

As an Introduction to this section we refor the

h, (14) reader to -:rho paper by Klrkllidy "Prediction of Alloy Hllrden-

ability from Thermodynamic and Kinetic D'}ta". He stlltes his algorithm

as follows:

" .I. Calculate the coaling curves f~r II representative set of

~ depths In the Jomlny bar, using the time-dependent Fourier

equation for heat conduction.

2. Transform the time, IIxls of the coaling curves so thllt the

t I me zero of ellch curve corresponds to the 111 loy eutectol d

temperature •

3. Calcuillte the Incubation time (TTT stllrt curve) llS II function

of undercoallng.

4. Transform the TTT curve to a CCT curve and superpose the set

of tlme-trllnsformod cooling curves.

5. Calculate the pearlite velocity as II function of tempe,rature • .. Then, proceeding IIlong each cooling curve, Integrllte the . ' velOCity equation from a ,time zero at the CCT curve llnd for

spherical ~olonles orlglnlltlng at each of the grllin-sizes specified <2;>

53

"

and saturated nucleate sl,tes. This Integration yl elds

the pearlite fraction as a function of temperature, a

quantity which reaches a maximum value at 'low temperatures.

The critical cooling curve, which Is tho tangent to the'

CCT curve below the noso, doflnes tho Jomlny distance at

wh I ch t\lo hardness beg I ns to drop off ...

, In tho fol lowing sections, theoretical relations and

d'ata are assembled which allow the achievement of this and a rrodl fiod

program. The algorithm Is rrotll fled by Inclusion of the latent ,

'heat ovol utlon for the au stenlte ... pearlite phase transformation;

honce} the necessary nunerlcal solution of the heat. transfer problem •

•

54

6.2 Colculotlon of tho voloclty

Tho ~rowth voloclty of 0 ploln corbon. P?orllto controllod

by volumo diffusion of corbon Is glvon thoorotlcolly by tho oxprosslon:

(rof. 13, 14, 15)

6.1

whoro liT I s tho undorcoolln~, 0. Is 0 thormodynllmlc constont ond

Dc Is tho effoctlvo tompornture-dependont diffusion coefflclont

for carbon In y-Iron; 0 has beon found by KlrkDldy(14) to be empirically

oqunl to 46.2·cm- 1 deg-2•

As we Introduco the effect of olloylng elemontsc.Wo must

keop In mind the difference between the effect of oustenlte stabilizers

and forrlte stabilizers. Austenite stobllizers are thouqht not to

part I t I on between ferr I te ond Fe3C In pear II to (ref. 13>. Theroforo,

their primary effect on the reoctlon rote Is through tholr thormodynamlc

I nf I uence on the undercool I ng for' II carbon-eon!rollod reaction. In IIccord

with Klrkaldy(14) we tllke the effect to be Iinellr In tho concentrlltlon

dopendence of the undorcoollnq:

LIT = T~ - T " TE(e I,OOOoK) + t I

IlC I I

6.2

whero tho concentrations C lire mellsured\~o weight percent • . ::::; ... Forrlte stllbllizers, on the other hand, tond to pllrtltlon

In the tempora;t:ure rllnge of Interest (ref. 16, 17). Since t.he voll.l!lO

,

55

diffusion rntos of the ferrlto stnbllizors In this tempornturo

rDn~e oro very small, such portltlon must occur by phnse boundnry

d I ffus Ion (ref.. 16, 13). ' Ponr II to growth I n II b I nary systom

occurlng by phoso boundary diffusion follows tho theoretlcnl

<18 ) roltltlon

(6T)3 V " ~ • DB 6.3

6C

whoro ~ Iso thermodynemlc constont nnd 6C Is the percentage III loy

ptlrt I t I on betwoen the product phases. KI rklll dy (14) arsumed that,

for ternary and higher order alloys, volume ond boundary diffusion

procossos combine In serlos of conductivities and writes the genoral

velocity expression llS: ,

6.3

whero:

I I _c_+ o Dc

6.4

•

whoro 0 I 1st he bouno;lary d I ff us Ion cacH I c I ent for ferr I te stab III zer I,

and tho KI are constants.

Ferrlto stabilizers not only affect the overall diffusion

coefflclJmt, but thoy contribute to a Iinoor chango In the eutectotd

temperaturo (rof. 14). Thus, we generalize 6.2 to:

57

6.5

whoro J Is surrrnod ovor 1111 II I loy I n9 e I omonts.

VII luos for tho ~'s ,wore estlmeted by Klrkllldy(l4) to be:

f1.1n Q' - 35 ...

lIN I a - 25

PCr a 15

/-. .~

11.10 Q 73

Those wero modlflod lind oxpandod for prosent purposos on tho basis

of knowlod'lo Cjlllnod In this study to:

1\ Q - 35 '1-'n

(lCr " 15

"

/ (l~1o • 7.5 .. ,

.---..,) Ps I " 15 , ,

/

PCu " 20

Further modifications, Improvemen1s and expansions wi II be discussed

bolow. (14 )

From tho ~me reference we IIssume values for the 01'5:

OCr = exp (-33,OOO/R~) ! and

0Mo c exp (-30,OOO/RT)

,,- ..

the Dol values being absorbed In the kl's, where~

·I\;r " 1,420

)

'1.0.0 c 44,000

An ostlmatod set of valuos for SI Is:

DSI " exp (-33,000/RTl

and

Ksl I<cr 710 = -- 0

2

, /

(

\; 'I

~

58

...

\,

I .

6.3 calculation of the Incubation time. )

Nucleation of a single phase In a binary alloy by a

volume diffusional process yields an expression for the Incubation '. time of tho form (ref. 14, 19):

\ ITT

T = .. 6.6 o (6T)2 v ,

or for a phase boundary diffusion process fref. 14, '20):

6.7

Klrkaldy(14) generalizes 6.6 and 6.7 for a complex alloy to:

, efT t O"T • _I (I

o KI T = -= + t...£. 6.6

(6T)2 0' (6T)2 0 I 0 1 c v

.' / where I Is summe~/dver all ferrite stabilizers. He also gives the value

/ -7 2 of a " constant" 2 x 10 em deg.

o = ex.p (-33,000/RT) .' Cr

~ = exp (-30,000/RT)

th 0 I b I b b d I n the K" e va ues now e ng a sor e I s. I OT

I

, He also glv~s Kcr = 633,

.~ = 3,600. An estimated set of values for 51 Is:

051 " exp (-33,OOO/RT)

.....' 59

and , Kcr

c -" 416.5 2

, ~Sl

In order to expand the above formulae to stoels of

hypoeutectold composition we have added a torm of tho form ~ C c c

to Include the carbon effect on changfnq TE when the carbon

concentration In the steel Is not that of eutectold composition

" (0.76% wt.C); the actual form of the carbon correction term Is:

An stlmatod set of values for A and B Is:

A = 112.0

B = - 147.0

" " While this ad roc procedure has only a tenuous theoretical Justification , It proves to be very effective In predicting the effect of carbon on

l

Changing the TE and consequently the predicted hardenability values.

6.4 Computer-aided astlmatlon of v(T) ~nd T(T).

60

In Appendix III we have developed a computer program .Incorporatlng

the material discussed In sections 6.2 and 6.3 and thls.glves us the ~

VeT) and T(T) values of a low-alloy steel on the basis of Its chemical

composition. The final version of that program has yielded good results

,

for outocto I d and for some hypoeutectol d stoel s. In th I s vors lon"

we havo Includod a quadratic term In AT for Mn In accord with

exporlmontal data (ref. 27) as Is Indicated by the Intercopts to

tho loft of Fig. 6.1. Also from Fig. 6.2 (rof. 8) we see that

the hardonability factors ,tor Mn Include 11 str6h'1 quadratic form

and since the main effect of Mn on hl1rdenablllty Is due to th9

chan~e of TE we can conclude that this effect Is also quadratic.

We have used the program based on Klrkaldy's model to

prodlct the T(T) and v(T) values of 22 different al ley steel

compositions. In Fig. 6.3 throu'lh 6.9 we give some of ,the derived

TTT diagrams plofted together with the corresponding experimental

,

ones. These were derived with our original coefficients, as given

above. It is seen that for low-alloy steels and eute~old composition

the a~reement w.lth experimental values Is only fo'ir. (For a reference

table with the specifications and chemical composition of al I steel

ai loys, see end of Chapter 7).

6.5 The Klrkaldy model for pearlite nucleation and growth.

Klrkald/ 14 ) gives a simple model for pearlite nucleation and

growth. This model has been used In this theSis for our calculation

concerning the pearl ite fraction along the Jomlny bar.

He assumed that the austenite grains are 11-gons (see Fi9 . 6.10)

and that at the CCT start curve each grain corner and the center of each

61

hexagonal grain face contains a pearlite nucleus whlch,subsequently grows-T,

as a spherical nodule untl I the start af Impingement • Since there are

. ~. - . "

il 22 groin corners shared by 4 grains and 8 hexegonel faces shared

by 2 grains, ho attributes .9.5 complete nodules to oach greln.

Tho len[lth of a grain edge Is obtained by equating the

nrOD of <In equivalent sphere of diameter d to the face aroa, 1.0. . ,

or

.1 = 0.34d 6.9 ~.

Tho vol umo of I mp I nflod .nodu I es Is:

4 (1)3 vi =(9.5)' 3" 2 6. 10

Tho oqu I va I ant sphere of untransformed matorl a I at first Impl ngomont

accordln[lly has a volumo of:

4 - - " 3

and 11 d I amoto r

" d "0.056 d

4 " - " 3

d" 3 (-) 2 .

6.12

6. II

It Is assumod that the reaction can then be completod by collapse of this

~qulvalent sphere. The volume fraction of pearlite formed along any

cooling curve Is thus approxl·mately obtained via tho Integrals:

,3 T . r

, J v(Tlet 9.5 r " 6.13 x =

(d/2)3 T .

c

62

•

,',

63

~ '-, , for r < t/2 .' f'

l

lind

" x = 9.5 (t/d)3 + (d ll /d)3 _ (~)3.

d •

\'I'Il(! ro \ " -r V(TJdt

" d r = -2

rq/2)

6.14

, > V.2 for r

•

· ) 6.6 Heat transfer In 8'Jomlnv bar.

r " slab of length L (Jomlny bar> Is assumed to have <In Initial

uniform temperature equal to To' "t t= to the end" Is

InstilntilJ)eously quenched to T = T" < To (see Fig. 6.11).' End 0 and

tho sld<,ls ilre assumed~to be Insul.ated so that heat f lows from the slab Into

" th" fluench medium only through the cooling contact,surface II which

Is kept at constant temperature T".

This Is a one dimensional heat flow problem and ,the general

equiltlon Is:

.2.. (K aT ) + q (x) = ax a x

6.15

where q Is the heat generat I on term. When the thorma I conduct I v I ty

not change with x (homogeneous material) equation 6.15 becomes:

a2T + q (x) pC K- = ai p

or 2 •

aT dU +.JL = ,2 pCp

at ax

, where Cl = therma I d I ffus I v I ty =

aT at

k/ pC p

, 6.16 ",/

6.17

Klrkaldy(14) In his previous work had assumed the heat

'leneration torm as not affectln,) substan'tlally the hardenability.

Part of this Investigation was designed to check this assumption.

64

Wo will opply numorlcol tochnlques to solve oquotlon 6.17.

First wo doflno tho following ~Imenslonloss poramoters:

6.16

ny us I ng those d Imons Ion loss parameters, the different I a I equation 6.17

bocomes:

or

/

qT o.

7

a20 l2 --+ --aa2 , aT

o

\

L= ao at

6.19 '.'

6.20

~Ie now construct a two-dimensional grid (seo Fig. 6.12),

\

the grid points of which are uniformly spaced and ~oparated by distances

Oa and OT on tho spaco and tlmo axis, respectively. From the calculus

65

of finite differences we can express the dlfferentla~ terms In equation 6.20

I n the form of fin I te differences and I n terms of the temperatures at

adjacent grid points and the distances between those points, that Is:

6.21

\

2 where (~ ) = second derivative of temperature wlth,respect to distance,

31'12

at grid point m. 0m+I' Om' .;'-. = temperat~re (in dimensionless form)

at grid points m+I, m, and so forth. Similarly, the time derivative of

temperature at point m Is

/ a* - a m m

OT ' 6.22

* whoro Om donotos tho valuo of Om at grid point m of tor a lapso

~ono tlmo

wo bbtaln:

Incromont. substituting from 6.21 and 6.22 at 6.20

.

Sinco a= 6.23 bocomos:

° I - 20 + Om_I L2 _m:::..:.+.!-_-,;..m~_.:::::.!. + __ (80)2 kTo

• Q a m

• o - e m m 6.23

* o - a m m 6.24

(Wo note that built Into the doflnltlon of dlmonslonlo~s

time T = «~ In relation 6.18 Is the assumption that a does not L

chango significantly with temp.uro and phase; In othor words ,

wo tako average valuos).

To satisfy the crltorlo for ~onvorgonce, the ratio bT/(bO,2

must be choson to bo loss than 1/2. We here choose that ratio oqual

to 1/4

* So equation 6.24, solving for ° gives: m

6.25

, 2, • o )..: L * /; f * 9m m-I

k T av 0

6.26

66

"

•

I .

\ . from the definition of dlmenslonles5 time and

, 2 -dlstanco,' for this particular ratio ST/(OP) c 1/4, the rool

, '\

t I mo I ncroment ot 15 .

I • 2 ~t " 4a (Ox) 6.27

and for Cl. a 0.054 and &x = 0.1 em

fit = 0.046 soc 6.213

For a· slab of length L =, 10.4 ~m we wi II have 105

divisions In tho space grid. The numbors of the bounding surfacos

w I I I be an d 105 corrospond I ng I y.

He now examlno tho boundary conditions of the problem:

(n) The cnd correspond I nq to a = I -(x = L, m = 105)

Is Insulated, that Is

or

or

aT 0 at axe x = L,

ao = 0 at e = 1 ae , "-

ao (aB) a=1 = 0

-6.29

In terms of finite differences equation 6.29 gives:

ao 0 m-I - °m+1 (as) a=! =

, 6.30

206 t m=105

J

-"

\

"

, This oxpresslon Is not Irrmodlately usoablo, bocauso tho quantity'

0m+1 falls outsldo the domain of,. computation. From 6.26we havo:

o * ° = I /4 (() + 20 ' + 0 ) + m m+1 m m-I

• q ~ m

1/4(() +1 + 20 + e I) 6.31-m m m- '1"

From equation 6.31 we substitute for ()m+1 In equation 6.30 and

we ol>taln:

or

* () = 1/2 (() I + () ) m m- m 6.32

(2) ~o '

The end 'correspond I ng to 6 = 0 (m= I) has temperature To at

t = 0 and T ~ TA < To at t ~ o.

condition becomes:

T 0 = H= .....2.) at T: 0 T

'T 0

e A at T > 0 = To'

In dlmonslonless f~rm this boundary

,

or If we shiff the temperature scale so that TA = 0

06=0 = I at T ~ 0 6.33

o =0 = 0 at T > 0 6

•

Noxt wo oXllmlno tho torm q. That torm Is tho hoat , m

qonoratlon rato por unit volumo Ilt thl' grid point m (surfaco ml.

I\!; such roto for oach t Imo stop. wo toko tho rllto llt tho ond of

tho provlous tlmo stop. If tho poarllto volumo fraction In tho

* hoqlnnlnq of n stop Is F lind ot tho ond F , tho rnto of chango // p P

In tho poorl Ito volumo frllctlon Is:

, * ' t* -F

F !~ p P 6.34 P . lit + rool tlmo ~ncroment \

( 'md 1ho hOut flonerlltlon rate equals to: ,

, ' . = F qm p

. 6.35

" whoro H

P l-s tbo I atont of phase tranpformatlon whon I unit

volume' of austenlto changos Isothormally to poa~llte •

. We now go back to equation 6.26 llnd,wQ examlne,thq temperatura

corroctlon tarm

LL

K ' T av 0

flocausa of6.3!i,. and 6.34 w~ have: .,.

L? K T av a

? K T av 0

* F - F P T

0

.r-

P

'1m =

K lit

pC LL p

Ii ....P. a

C p

F

LL

K T av a

* - F 2

lI1' ' *. F F P

T 0

P "

p ... "

H = P

(95.0)

* F - F' -p"----'p 'II =

p

6.36

69 ..

/

(h P

H 1C --R r:

p

1 Intont hont of phnso tronsformntlon whon l'lr of

llu~tollito trnnsfor,ns Isothormnlly to l!lr. of ponrlltO).

Wo now substltuto from 6.361nlo (,.31 for tho tompornturo

corr-oct I on torm nnd wo obtlll n :

• F F c 1/4(Om+1 + /'0m + Om_I) + -,,-P_-",-P

To (9',.0) 6.31

[quatlon 6.3} !llvos tho tom[1oroturo nt 0 glvon !lrld point m ns II

function of tho tomporoturo ot m-I, m, m+1 Dttho provlous tlmo r .

~1ep nnd tho ponrllto fr~ctlon IncrOllso at grId po Int 1'1 durln'l thnt

tlmo stop.

* r p

(28 r.) (r' )/.. V(T)· "t F p " :;;-~'c.::J!-C-..:..!--!-_.---,-,-,-"--.;.0-,-

(d/2)3

•

6.38

II

From equation 6.14 wo obtain for F - F for d"/I' < r <0 p P

, ~~ r "0.0,

II 2 V (T) • h • (3.0) • (r ) •

F - F 6.39 c

P P

F~~S that Is

(d/2) 3

snortor than tho ono corrospondlng to

-:--,..'\ -

(for t c t ) o

10

.. '

'-_.t.

M '. WO havo r - F ~ 0.0, (F a 0.0). Also, for cool InC] tlmos

p p p

" lonqor thnp t c t ' corrospondln'l to r D 0.0, that 15 IIftor

tho POllrll~c:lon hils boon c~mplotod, wo havo FM - F D 0.0,

(F e 1.0). p !

P. P

1->-_1 \

•

71

' ..

6.7 Samplo Calculations / We first oxam I nod lawai loy stoo I 4060, wh I,ch w~s

, (5) ex~mlnod oxporlmontal Iy by Jomlny and thoorotlcal Iy by Klrkaldy(14)

Starting

'lIven by

9.6 mm.

from tho predlctod TTT and pearl Ito velocl'ty dla'lrllms as

.( 14) Klrkl1ldy wo predlctod a hardonability (50% pearlite) 'of

This calculation Includod tho Illtont hoat ovolu·tlon correction

term, assumed a 'lraln size of 7 and a total cool In~ time of 92 soc;

tho position of the' 50% point did not chango significantly after a

4(; soc cooling tlrno. In Fig. 6.13 we give the pearl Ito distribution

within the Jomlny bar for this particular calculation Dnd ,for threo o

dlff'i'rent cooling times. In Fig. 6.14 we glvo tho cooling curvos for

a numbor of po I nts a I on~ the Jam I ny bar. Tho comput~r output fnr

this calculation Is Iistod In our files as: run 4068/E45. . ,

Wo repoatod this sam~ calculation neglectln'l the latont

72

heat evolution (run 4060/E40). The 50% point was now found at approximately

11.4 rml.

Oy.comparlng tho above results we see that the latont hoat

evolution has a softening eff~ct on hardonabillty, moving tho 50% ['lInt closer

to the quenched pnd or the Jam I ny bar. Th I s e~tect for th I s p~rt I cu I ar

caso Is about 20%·. Th I s 1 s due, to "the fact that once pearlite starts

transforming, the latent heat evolution slows down the cooling rate and

thus'the remaining austenite stays longer In the austenite + pearlite

transformation temperature area and a larger part of It transforms to

pearl ito.

• . \

If tho 50~ point Ilos

for stool A06B th Is eHnct flots !

closor to the qu~nched end than

sma I I er s I nco tho coo I I n~ ratos

nre hl'lhor and tho slowln'l effect of tho latont hoat-goneratlon

b"comos loss promlnont. At smalle,. coolin,] rates (hi!]hor prodlcted

hilrdonilbilitles) the offoct should become moro prominent. It appoilrs, \

thorefore, that -for accurilte predictions of. Jomlny curves, the liltont

hOilt effects w 1.1 I hilve to bo taken Into llccount. \~o have. accord In'll y

rci'alned It In all the calculations pr~s~nted In this thesis. ,

lie next repeated t~e calculations for steel 4068 with assumed

73

'lnlin SilOS of 3 and 10. For grilln slz~ 10 the 50~~,'r;lllt >:as found at· :;.J rnm

(~un 4061:l/[(5) ilnd for grain SIlO 3, tho 50% point was found at ~ 12 mm

(,"un 4068/[105). In FI'l. 6.15 wo plot the martensite frilctlon alon!]

the Jomlny bar for tho throe grain sizes 3, 7 and 10. If we compar~

theso with the curvos derlvod by Klrkald/14

) In Fig. 6.16 wo noto that

ours are smoother overall and do not ha'ie the step found In Klrkaldy's

curve,. This Is due to the uso of the additivity rule In tho CCT calcu-

lations which permits furthor nucleation towards the left 'of tho stop

part of tho hardonability curves In Fig. 6.16.

It Is evident, when comp1lrlng tho predicted Jomlny curves for

stoel 406£1 In Figs. 6.15 and 6.16 with the -experimental ones for the - "

samo steel in Figs. 6.17 and 6.18, thatth I s a I oor i thm I s a fa I r quant Itat I ve _ ~. . . . ,'. \

(e.g. Inflection point X W Fig. 6.17l predictor of the mean Jomlny depth

and a qualitative predictor of the shape and grain size dependence of. tho

the Jomlny curve. However, it Is clear that ttie model is not sufficiently

comprohon51vo 115 a shape prodlctor. It wll I Ilccordln~ly bo used

In tho romolnlng Investigation a5 a prodlctor for an approprlDto

mOlln Jomlny dopth. On the experlmontal side we define tho Jomlny

dopth 1\5 tho Infloctlon point of the experlmontal Jomlny curvo

ood compuro this with tho 50% poarllte point gonoratod by tho

thooretlcal predictor. Tho Inltllli comporlson for steol 4068 ilnd

'lr"ln SilO 7 15 botwoon tho experlmontal valuo of 9.2 - 9.7 rrm ...,-

<Seo FI~. 6.17> and the theoretlc,,1 value of 9.6 Iml (seo Fig. 6.15).

Tho two va I uos agr~';;;we II. /" - , /­

/

\

~ \

'-

)

74

.. ,r----------r------------,------------,

fill, 5"

:' 'v, I---..,..-----::O\'<>-.;::::-;,.::.:::......."f----'" r.

·IL,h' tol ~r""th uf IWllfUI" .... h'nl- . 1','I'.ltUro' III ",,'t\l'Ul for ht~h-I,urlty Iron- i , ... ,l...Jn .nd Iror -cllrhon-ml'lir nGlo ILo.I •. ;: 110 I-:-,"~.~.~, ~,-_:-:;-,~ .. ~,~,~,= .. =.~.-; .• t,~.~,~---'C---'-"'I---'"

-'

("",I •• OU,( c,,"'_1 %I _ 0 It ... C,I 0 .... ~

t" .. ~( I'll' 0"""."'''' .~~'~~.:'~'--~'~'.~':'~.~'.:'~.t------t-----J

•• ,~c_---------!~-------~<-~~-'<--+----_: ., • '0' I '0., 00"

'Ol--il---tl-t·--+-f­I---j-...,-t-+--j--l .... -

.01--+--+ __ +--7/_ 1-) ·L i i t--t--I-~I -11---:-t '0 '/ :---;----

II I J~ __ ~_j ... __ I: i I .ol--!--++-!---__ L-. /: LI

1---1-,,/'-+---'-- 1-

/' ~ i I .~~--~----~----~ o 10 20 )0

_N ........ r.UI', Llfo« ." )I.~.~~ .... • ,.. ....... 1~,r4 •• 4 1t' .. ~.1

75

•

700 /

500

o o

i== 400

300

200 -1 10

I ..

I ig. 6.3

_ .. - ----------~ - -._---.. __ l\el~~ x p ...

, I I I , -,

\ \

-.-

100

--~ . ~

~ ,... ~

t(see)

alloy 5 T T T diagram '.

76

(

I

fig, 6,4

/-I=== .. ~, = ... = .... :-'.':":. ·=··=-::l7;:"·~:-:1"=·e=x~p=,~:':·=··-=···-:':T;:::c=a=1=~.::: .. ==== .. ==-== .. :,:,:_ .. = .. ~ .. =._::::.=~.= .. .::: .. _,=1-=-

700 _------ -~~---i .,.",.,. -- ... - ... ,.- - _.-

, 600

500

u o

i== 400

300

'"

...

"

,. ,.,.,,;'. ( . , .

" , '...... " .... .... .... --,

I 1

,I I I

'- .................. --\ 4", I:-, , , , , ,

\~+ '(? , ,

-\ , , , ,

--~

alloy 18 'T T Tdiagram

700

GOO

500

u o

~ '.00 .

. 300

(f

J

fig, G, 5

\ \

/!. c"~~\~_~ _____ .-.:.L~':"""':""'....1.,...I.-=-...u..~~-r....!:'t.M I eo w'''' ". 'r a..JL:!...::..a-... tAn, {'xp, .- - .• -

" -- ---- - - -= - --:;~----------".­

~ .­( , , ( .......... I , ,

alloy

()~/J ................ . . '\ ...........

9

.

..

, -\ -... -\ -- .... \ \ \ \

\ "l.,<' \ \

T T T diagram

,./

"

70

, '

700

600

500

.P ~ 400

300

fig. G.6

~ ~

/ ~ I .... , ....

I" . I

I I \ \ , " " ,

" " f?.t'" -....: P. - .... _

"

alloy 16

, " ,

"-" " ,

"

r T T diagram

\,

700

600

500 ~.

~

'U 0

~ 400

300

fig. 6.7 '

\.. - --- ----Ae3 expo

. -;_ ... _-- -, ..

-----. --,,-----_ .. --_ .... ,--- ------.----~----- .. -_.-------

AC?j (> xp. '-T calc. _ ---

t>. • r -,.' ----, -/ // I / " / I /

II

l \ , \ ,

"-"

.... aUoy.6 --,

, t;lxp'-_ . ,

TTT . diagram

.. .

00

,

.,

700

600

500

p (::::

400

300

fig.6.8

\==~A.lr=~~==~====~~~~~~· n, t?xp. T ----II;" I.calcs.------_

...... . ---.".,./ ----

1"/,,;' , ..............

I I , , ~~

\ , \ -<1. ... ........... \,(>.. , , ~\ ~,

"-, , "­, , " , ,

100 t (sec)

t ,alloy 19

,

101

TTT r

diagram

~-'

,

,

i

102

•

',",

.>

..

700

600

500 ,

u • ;::: ~O

> <

. I

300

fig. 6.9

T. calc. I -,.--

",. .,---.... />..~

,,"" .. ~ ,,' .... y ....

( "-,

.... " Cot/.

.. ... ..... ,C". (}x ........ .P • ................ -. -. --

alloy 21 T T T diagram

82

/l (

..... ' . fjg~6.'O

,

•

I

\ ,

.',\ 84

fig.6.ll 'A f B

L r

;

..

'r •

t ,

(

, •• 3

I •• 2 . ..

fig.6.12 \ •• 1

, I' M

0 --' f' __ ~ ... , ,

• / '. _t I. I. _I', L -- -- _ ...J

.-1 . I

TOftC' dimension:!1 limc"sratc t;riJ .•

(

r

.. • .' t

65

I

" fig, 6.1 3

u (II III

N • (")

00 0 (")

u E III E l

III

"': ~

~ '0 Ll) C

III

" " 01 0 C N -0 0

III '0 III 0

C'! E ex> N 0 ... -

21 c

'" 0 ... ~

III

'0

J

•

,. ~ J

. , r • ,

~ - ' 86 .,

:..; . ,,-. j

fig. " 6.14

."/ .. 1 E E E -F 1 E E E E 0 ,

oi ''\.. I' U') co. LO "-t' ( ...,. ~ . , , I ..

itT C")

.) f

. . • 0 00 ( ~I C") ....... , 1 . , • 1 .") I ~ ,

100 Wi , <:'I<l> 1-1 . .!!l

I <11 , E I ..... ,

0 1 ("oJ -, .

L I , 1 1 1 10

~ 1 I' ~

1 1

0 -

\

J \ \

\ -

.

, )

(.

fig. 6.15

I

,

. ------

"

\

, \

, \ \ \

'1

\

'0 C Qi

CI C

0 0 U

E 0-.!:E

E Qj~

u C IU ..-III

(

\ \

67

•

Q

I

d:

. )

\. )

(

:

'.

,

..

f i ~ . . 6 . .16

1.01--7· --=::::---__ ~.

W I­en Z W I- '. ~.0.5 ::2

I:-z w u a: W· a...

o

I

"'-\ --. . \

EXPERIM,ENTAL,

. ,

, \ \.

'\ \

I JOMINY DISTANCE (CM.)

2

Prodicfod JOminy curves fo',. steel 40013 as a t","'\~·' ;I~" .H

orc I n (i 18. 4bShQd. II,M is 8l1CptHO

I I"I'II!tnta I curve ( 5)

(

.'

8B ,

)

-. .. , .

J

L

--

fig.6.17

r

fig. 6.18

\

,.

,

~ Il:rct:rolJIhly l!.r>t:r «9/J!xra!cri..'S -I I'~ O?SAE~IilM

-+-. End axMxJ Ml:l1aJ

1- -1,- 1-. '0. - . -• l-i--

'ft I . -;- t- o

I ~ I - : .. oc

...

.I

0

£lrw Ikd AtJ.~

1 :.

CnrW Sh.d No.. '-Hardonabllllr of On. H ... , 01 S.AE. 4{}M 5, .. 1 br BM Cool.., 1IIIhod. A.orag. ddcrmlnallon. of HC'h 0( IIl1t. laboratorl" Gr. plotlftl

1 ~ ~ ~ lbib'1dlillly anr:. if 8/alxntrriis _

00. SI:anf SAL: ~SImI

R..~, .£nJ=A=

\~ 0

1\\\\ ! I

~ - 1 ,

I '" ~ ~ T-o.

. -.0 c= I

I i ,. , , 0

t- -- . ~:~Atjy .

Cur.,. Shett .\'0. 5 - EUed 01 Small AddUion. o( ~·icJ. .. rl and Cl,romiam on Deptl, Iku..,:lrnlng of S.AE. ,sQ6,':

.1. Sh~n by Comparing CU",~ Shu~ .No. ~ lVitl • .\'0. t

89

r

.,'. ,

/ .

I'

I

"

CHAPTI':R 7

CALCU~TlON OF THE HARDENAfjlllTfES OF A

SELECTION OF EXPERIMENTAL AND CQl.1MERCIAL STEELS

/

7.1 Calculations and discussion ,of the results.'

(

We next exam I ned a ser les of, stee I s us I ng the parameters

)~ ,

\ ' and coefficients established above. We first predicted the TTT

diagram frqrp the chemical composition and then derived the 50%

pearlite fraction point of a Jomlny bar. In all the calculations

we Included the latent heat evolution correction term. In Table 7.1 .

we list these stoelswlth t~ composition, file number and reference.

tilth the \lxceptlon of the higher alloy steels, (for ,which

our predicted 50% poln}' hardenabilities were very lowl, "the predicted

, ,

TIT diagrams were acceptabIe within the expectations of such prellmlriary

work,' (soe, for'~xample, Figs. 6.3 - 6.91. In Fig. 7.1 we have plotted (

, \ . the predicted vS,' the experimental hardenabilities for these steels as

estimated to the nearest 1/2 mm. It Is Immediately ob,served that for'

low al'loy, low hardenability (H<IO mml steels'the Klrkaldy theory works

fa i r I y accurate I y, I tis a I'so a cons I stent pred letor for t-he higher

alloy stee Iss I nee the deY i atlon .frqrn the experlmenta I va lue' Is

always In the same direction, This Indicates that we have underestimated

" ,90

,

I'

~he Influence of the a!l'tifelement Interactions. We' have also \ •

cons I dered up to tl) I s po I nt on I y II near coeff I c I ants for such

Qffects when the actual, I nf I uences can be of a quadrat I c or

•

I;

higher polynqmlal nature, for example,

/-' The effects of such extra terms 'for smj! II concen:trat'j ons.

ar~-vanlshlngly small, hence' the. good, agre/ment of the predlctrons

with experiments.

, . An examination of the compo~ltlon of the high hardenabl Iity

steels,e.g.) steels 2,3,'4,11,12 (see Table 7.1) ,suggests' the manganese

Is much more -potent, for concentrations> 1..0 wt.% than the linear •

"' 'algorithm allows. Th~s was to be expected on the basis of obse'rvatlons'

_pres~nted In F 1 gures 6 • I and 6 • 2.' By contrast, our linear estimatIon

for the eHect of NI seems to'w~rk,wel'l up to high concentrations

(3.9 wt. %) as - I s seen for stee I 21.

We attemPtea an Initial allowance for the Increased potency

of manga~se with ~oncontraflon by.~es~bstltutjon

" + flMn

wher,e Co I s a base I~n composition chosen so that the low manganese

(.

91

, f

(

j

..

. effectIveness Is \ncMng~. An InItIal opt)mlzatlon 'ylelded

the va·1 ues: r

'.'

,;. . , _ ' The recal,culated M[denabl'J1tles for_ steets 2,3,4,1~ and 12

·are plotted In FIg. 7.2 In. relatIon to the orIgInal, values. WhIle

the correctIons Mve,'lmp~~ the ~~Iatlons" I,t Is lIpparentfrom" , . , . , [

a re-examInatIon of' the composItIons In relatIon to the~ardenablllt.es

" .-·that there remaIn some strong I4n - Me InteractIons and NI - Cr Inter-

" ,

" ,

/ .., ..... ycrlonswhlch have yet to be accounted for. The trIal and error ( .

'search for' .su~h Interac:tlon ~s' Is beyond the" scope of thIs thesIs. \

"

.J

\

• \

/

,)

\

•

1('

'j."

, l

.... - ,

TAfJL[ 7.1"

1,1n IH Cr' ~,lq Si 'Cu C Ii Ie tlo. wt.% wt.~ wt.~ wt.~ wt.% wt.~ wt.1

I

2

3

4

7

8

10

II

12

13

14

15

16

17

18

19

20

21

4'068

.22

I • '1

.25 .23

.0 .0

1.4 .0 .0

1.4.0.0

.76 .0 .0

. 1.88 .0 .0

.01 .13 .0

.03 .3 .. 0

.72

.39

.49 .3 .0 .39

.26 .3 '.0 .39

.0 .,1 .0 .76 -

.0'.1 .0 .60

.89 .53 .64 .22 .0 .0 .54

1'.65 .0 .0 .36 .29 .0 .43

0.86 .0 .33 .0 2.13 .0 .62

.• 9 .45 .54

2. I " .p '.0

1.56 .0 .0

.87 .0 .0

.94 • 10 • 13

.6 .01.0

.91 .0' .0

.86 .0 .33

.95 .0 .15

.82 .0 .07

.28 2.02 ~.28 b.25 3.9 .0

.22 .0 .0

.0 .01 .0

.0 .04 .0

.0 .22 .0

.02 .0 •• 0

.01 .0 .0

.0 .1 .0

.0 2.13 .0

.44

.47

.51

.63

.48

.36

.50

.62

.0 2.01

.0 2.01

.48 O. II

.0 .0

.0, ,62

.0 .62

.0, '0.61

.0 .59

~.78 0.030.03 0.25 0.28 0.0 0.65 t

\ 1

Reforoncos

Ref. 28

29

29

'29

30a

30b

-oJ

8a

8a

30c

8a

8a

8a

8a

32

32

30d

30e

30f

30g

31

30h

5

93

• )

.

-

••

fig. 7.1

...J. ...L

./"-/ /I)

o ~ N

...J. ...L

@ ®

. ,

/I)

, " " ...::

I

I

o rN

( 94

E E 0..

.)(

.01

/

/

~I

/ {

.. , , , ., .. , .. ,

~

,

.

fig. 7.2

.. , , , , , @)<', . ..

•

"'.'

, . , . " ... O~· . 't( -~. ---~~--: ~ , I .. 'f ,. ,- , I , , .. .. ..

In~ ~'O· In .... .... (\UW) ':ll~:I

, .

o N

In ....

'0 ....

In

"

95

•

~

E E -0-

.f. , )(

Go

\

OMPlER 8

(. "

CONCLUSIONS , '

Dosplte the fact that this W'lIsa ,preliminary work and used

o,xporl menta I hardenab I Iity cu'rves of samet lmas questl onab 10 accuracy,

,it Is nonetheless evident that the present model for predicting harden­L....-.

ability compares very favourably with other similar efforts. '

We refer here particularly to the work of Brown and Jan-es(21)

who produced about 1000 laboratory steels, tested them for hardenability

and chemical analyses, and used,thls excellent data to correlate harden-

ability and chemica'i composition through computer regression analyses. '

Fig. 8. I (Flq. 12 In their p~per) compares their Initial predictions with

the hardonabl litles of an ~rbltrary set of commercial steels. The correlation

obtained here is no better than that obtained with a non-optimized form

of the KI rkaldy algorithm {Fig. 7.1, 0 - 10 mm). While thoir corre!atl~s

are Improved by narra..lng the range of ~ppilcabillty, the number of

correlations required Increases proportionately. By contr;-ast Klrkaid,'s

algorlthm,'based on theory and physical prlnclples,has the potential to ~ , '

be optimized so that the entire range of steels can be treated by a single

sot of parameters within the' formula.

Furthermore, It should not be forgotten that this formul,a ' ...

also\(lCnera~ln and CCT curves which are themselves useful heat trcat-

• mont data. It'ls our conclusion ttl~t the ~ddltlvlty rule Is the first

96 " ,

•

choice for goneratlng the ~-CCT transformation. The Grange--Kieffer procedure"ls a reasonably good substitution but there Is

"-little to be gained In time economy by using It.

From the polnt-of-vlew of accurately predicting Jomlny curves,

we note that the nucleation and growth mode I shou I d al so I nc I udo tho

time dependence of nucleation and a first step towards this end would I .

be the Inclusion of a linear time dependence of nucleation rate. This '""

would certainly make the predicted pearlite fraction vs. Jomlny distance

curves (Figs. 6.15 and 6.16) less steep. Also,graln sizes could be

assumed with a • Gaussian distribution around an appropriate average.

We have already emphasized above that the latent heat evolution .'

has a significant effect on ·the hardenabl Iity and on the shape of the I' ~ Jomlny curves and so must be Included within any detailed predictor for

Joml ny curves.

Care shoulQ be taken In planning the next steps to Improve

the· present model and curront techniques of predicting hardenability for

chemical composition. This Is necessary In order to avoid the high

costs of both laboratory tests and computer time. Unnecessary repetition

should be avoided by having rBS~archers communicate their results and

laboratory tests and by setting up a data bank of accurate laboratory.

(Jomlny) tests and chemical analyses. Needless-te-say, the expected

economic rewards both to producer. and steel userwoald IrOre than Justl fy

that kl nd of effort and reseerch. o

97

/ , fig. I 8.1 .. ,

'.

0 I 0 I

04 ~ • 1

I ~o I

0 1 0

0 0

QOO 0

o aPt:' OJ o 0 0

'b • 0

01 02 03 04 O~

PREDIC TEO [I",ftnl

-Hi~h jlll:'it) steel ~l(\;;ttlun :l!)f,1 i~'d tv L'.1r't'~1 ',\ ..:i : •• ,l!,l"·1 oi c:ommerqal slet!l:;.

(

l

98

)

, '

APPENDIX I . '" "

Numerical Techniques (24, 25) ..

,/

r

and 'access I bill ty of The advent digital computers has led ~ '\

to the development of a very powerful tool for the solution of-differential

• oquatlons by numerical teChniques. Our discussion here will be I'estrlctecl

to the presentation of a few Introducto~y concepts.

Let· us consider the following heat conduction ~quatlon lind

,_ boundary conditions:

a2T J ex - a for O~ y ~ 1 I • 1

J •

. . c~-at ,T ~ 0 at t

T = 0 aty

c 0 1.2 (,

= 1 \~"'" '~"

,', .. , "-Ii:;

" . and

T = T ai' y ~ 0, t > 0, 0 1.4

\ '

A worthwhl Ie first step In all numerical calculations Is to put the

equations In a dimensionless torm~. Accordingly" let us define the following,

dimensionless parameters:

a = !.- , 6= Y. , I .5

T 1 o (

, ' parameters, the differential equation lind ~ , By usln'g these dimensionless

"the boundary conditions shown above may be written as follows: I

" I

. ,

100

The main advantage of the dimensionless representation Is-'that the final I

solution Is mare general; furthermore, the numerical range of the dimensionless

~rlables ~s restricted" that Is, o~ a ~.I, o~ 11\ ~",so that

'~putatlonal errors are. more easily spotted.

, {.

Let uS'npw construct a two-dlme~slonal grid (Fig. 1.\), the

grid polnts~of which are unIformly spaced anj separated by. distances

~B and b-t ~n the space and time axes, respectively. It may be shown,

~y applying the calculus of finite dlfferenc?s (or by the Taylor series

expansion) that the differential terms appearing the equation I. 6 can be

-expressed In the following from of f1n.lte differences, In terms of the

temperatures at adjacent grid points and the distances between these points,

320 (a t-)-O ) - (9 -a I) ( ) = m m m m- = ~II

0m+1 - 20m + am-I ., (£B)2

1.10

where (320/3B2) .-second(derlvatlve of temperature with ~es'pect to m

distance, at grld.polnt m; 0m+I; am" ••••• temperature (Indlmenslonless

f~rm). at grid points m+1. m and· so forth. Similarly, the time derivative ,

.of temperature at point m Is

•

I • 11

r

whore e*m denotes the value of em at, grid 1 t ft ' po n m" a er the lapse

of one t I me I ncremcnt ~'t. Subst I tut I n9 from 1.10 and 1.11 1 n equat I on

1.6" we obtain:

e*m - Om = em+1 - 20m + em-I

ST (8a )2 I . 12

.. and therefore

I. 13

" ,

It can be seen that equation 1.13 relatos the value of the

101

.' - r ,

temperature at a given grid point to the values of, temperature at '

, , "" the same and adjacent grid points on the previous time step, as Illustrated

In Fig. I .1. It follows t,hat If the Initial temperature dlstrl6utlon over

the Whole space domain Is known, the tllfl'lperatures on successive time'

steps ,are readily evaluated by the repeated appllclltlon of equation 1.13.

It should be.noted'thllt the finite difference representation

of dl'fferentlal eq~atlons Is only approximate and that the IIcc~racy of

the results 'will depend on'the.nuplber of grid sl!bdlvlslons used; In gerierlll, , ,

the finer the meSh size, the more IIccurllte w III be the resu Its, but lit

the ~ame time, the computational work (or the use of computer time) will

In'~rease 'correspondingly. It ~n be shown thllt In order )0 satisfy the

, 2' criterion' for convergence, the rlltlo 6T/(06) must be chosen to be less

than 1/2. Inspection of equation 1.13 shows tha~ the actual relationship - '

between e* and e depends on the numerlCIII value chosen for this rlltlo; m m

102

Thus, fQr' St/(&~)2 " 1/4, we have:

I • 14

From the definition of dimensionless time and distance, for this

particular ratio' bt/(b6)2, the real time Increment'St Is:

St'= 1/2 I • 15 4cx

As an example, consider the computation of the .numerICal rates

of the temperature prof I Ie for the system of equations 1.1 a and 1.2. - 1.3 ,'" ~, • " -.

Let us assign five grid points to the space 'domain, as Illustrat':ld In ,~: •

Fig. 1.2. The first grid point, m - I, will corresponQ to the y D 0

(6 = 0) plane, and the fifth, m + 4, wi I I. represent y = 1 (6=1). If , , St·

we asslng the value 2 ~ 1/4, our working equation will b~ equation '.1~. (06 )

The boundary conditions specify that the temperatures at the

grid points m - I and m + 4 are constant and of respective values I and O.

After the ~Irst time step, the temperature will be unity at 1l0lnt m = I and

zero at 'a II other p.o I nts. We IMY now evafu ate the temperature d I str I but I on

after the second time step by using equation 1.1't for successive spatial

grid points; It Is seen that after the second time step the profile penetrates

only up to th~ second grid pOint., The calculation Is repeated for subsequent

time steps:

The reader will recall from equation 1.1S that, In terms of

time", each dimensionless time step corresponds to:

• .. 4cx

,

•

o •

Thus, for a slab of I width subdivided Int~.four parts

( ,,"0. five grid pol nts.l we have: ,

6t 12

" --64((

"

As a 50cond example, consider the development o,f a numerical /

representation for the solution of equation L 1 a for the following

boundary conditions:

(i) rThe face corresponding to a " 1 (ya \) is insulated.

(ii) The' faco corresponding to a "O'(y." 0) receives heat by convection

from,a fluid at temperature Tf •

Let us detl ne the d i mens Ion I ess temperature as 8'" TIT f"

The boundary conditions corresponding to (I) and (2) may be expressed as

aT _c ay

.. o at y c l. that Is

ae "0 at Q 1 aa ,," .' 1.16

and h <T f - T) " - k aT at y a O. ay

as ,at ~=O or (1 - 8) ,,-~-~ ~

1.17 4.38

I

As In the previous example, we Sh~ now consider a grid

ana I ogous to f~a1; shown' In Figure \.\; howejer, In th Is case we sha I I

pay particular attention to the grid points In the vicinity of the

bounding surfaces (Fig. 1.3). The sltua:lon 1\ depicted in Figure 1.3

where a number of Intermediate grid points are shown. denoted by m-I •

103

..

I

•

m, m+I, and so forth; In addition, we have the grid point corresponding

to the B = O'plane, ~enoted by 5, and the grid point corresponding to

the B = I plane, denoted bye. For a glven'lnltlal temperature

distribution, we could proceed, by applJrlg equation Lilt td the l/lter­

mediate points (m, mtl, and so forth)" to compute the val.ue of 0*, m

at a given time step, from values of Om-I' Om' 8m+1 on the previous ,

tl~ step. However, In the present case, where the specified temperature ., at the boundary Is not constant but depends on the rate of convection,

It Is necessary to take a different approach.

Let us proceed by putting equation LlG In a finite '()

dlff.erence form:

= 0 I • 18

This expression Is not Immediately usable, because the quantity 0e+1

falls outside the domain of computatl~. W,: may, however, eliminate I

00+1 from ~he above equation by relating equation I,I~ for co-ordinate

point e:

0* e ~ 1/4 [8 +1 + 20 + 8 I] e e e-1.19

Substltutlng the value of 00+1 from this equation Into eq~atlon 4.40

we obtain

8* = 1/4 [28 + 28 I] e e e-1.20

This equation relates the value of ° at e, at a given time

step, to Its val'ues at e and e-I, on the previous time step. , A somewhat ana logOU5 prcX:edure may be app·1 fed for the

transcription ot the boundary condition contained In equation 1.17 •

104

L ,

(I - e ) s = - I .21 '

Also, the corresponding form of equation' 1.1'1 for the grid point s Is

1.22

Substituting :for Bs_ 1 from equation 1.'2.7. In 1.11 and rear~anglng,

(f - es )2Ss e* = -----­s • 4y

+ 1/2 :<es+I' ,+ es) ',-'<; ,] I '

G,~---- ,. ""-" ," _ j ,\ ThLs equation relates'the value of e a~ s,l at a

) ~ ;i3 , >

glve~ time step, to Its

105

va I es at sand s-I, on~ the prey I ous t I me stiP. • ' ,

We are now r~ady to proceed with ~he numerical calculation of ,

the' probJem by using equat!on I'. 23to estab,l;lsh the temperature d1strlbutlon

at the s boundary~ equation I. 2a,jor e boundary, and tho general equation , , 1.14 for the Intermediate point of the grid. /' '-"-",

• This procedure may be modified to be used for the solutlnn of . -

'unsteady state heat conduction problems In which the thermal dltfuslvlty a

I:; qependont on temperature and where q I- O.

fig. 1.1

• fig. 1.2

(

"

j -j

f

• '3

• 5, .-2 t-

• + 1 • , • •

• -1

'''umber or time Slrps

J

2

---,.)

'6 r- ,~

,

. • .

. : -- 1"---,

f ' ... -, '. I •. , " L --- -- - ....J .

. ..

T .... o dimc,)sional limc:-spicc grid.

I '

~ 0.75 0.5 0.25

- - - .

LI.O O.4S~ O.I~5 0.015

I 1.0 0.3;5- 0.06.:!5 0

1.0 O.~S 0 0

1.0 0 0 0

... - I .. m+1 m+:! m+)

Spao:

~ Cakubl~ of Icmorr.Jlurc di~ribulton in E,amplc

(

\ '.-I ••

106

-

•

"

.

fig. 1.3 (

\ \

.~ :: ~"!-",'t: o a III

-i --

! -'

--

~

'2' .- :: ~ .. 5't: lila

'W1.L

)

,

'-

-

,.... +

" .. I ..

M

+ E ,.. + E

-+ E

E

-I E

-+ .. ..

..

107

c-

APPENDIX II

.~

~rogr~m structure for calculating a OCT from a TTT curve

To formulate the problem In a form appropriate to

tho computer we proceed as fo I lows:

(al ." We 'approxlmate the pearlite-start curve by a series of ,

points Ti' t J = ~I; the closer the points the more .accura~ t~r

f I na I resu Its. Espec I a I I Y at the "nos;" reg I on of the pear I fte-

start curve the points should lie .closer together.

(bl Next we determine a sequence of points TJI t J on. the cooling

curve. The cooling rate at each point Is given approximately by:

," '- ;

and we name the Inverse of this as:

106

II.

11.2

'.

.. ' .,

\ /

\

" 1 ~Io a I so na, the va lues T I' tj' . Xland Y I rospocti vel y

Th I s transfonnatlon l!f 'carrl ed out I n the program. Cons I der

now (Fig. 11.1) on an X, Y plot the sequenceofpolnts

PI(X I , YI ), P(X,Y) and PI+ I (X I +

I, Y

I+

I). From the similar

triangles (approximately) PI PI +I

Band PIPA we obtain:

or

or

,

X - XI Y - Y I ----'-- ~ ---'---XI +1 XI

y - y I

the latter ratio being named SLOP.

""

11.3

• • The area of the trap'ezold (PtI+IPI+I P ; PI) = Ai 'i+l

Is given by:

AI'I+I = 11.4

2

and the area of the trapezoid (PIPP'PI'P j ) = A Is given by:

11.5 2

If we specify XI' Y I' XI+ I ' Y 1+1' (we know thorn as datal . . . and A (we shal I show below how to calculate A) then we can from equation

. '.

109

(

"

11.3 and 11.5 calculate X and Y. From equation 11.3 we obtalrr:

11.6

From 11.6 we substitute Into 11.5

2A (XI - Xl = --­

y + Y I

= SLOP * (Y I - Yl 11.7

or

11.8

or

11.9

and~

2 2A Y = SQRT [Y I - STIiP ] II, 10

'.

c·

Substltutlng for Y from 11.10 In equation 11.6 ;'eobtaln the Nalue:

2 2A X = XI + SLOP * [(SQRT (Y I - SLOP II - YI] 11.11

We are now ready to proceed with the Integration and to

calculate the temperature'at which

)

11.12

In the foregoing n~tatlon this Is:

110 I

11.13

The express' on on the right of equat I on II. 13 Is given the

name SUM and Initially asslgnmhthe value 0.0, , ,;

i1 ,/

SUM" 0.0 11.14

Thon the AI,Z value Is calculated and multiplied by MI

11.15

and SUI·\ I s ass I gned a new va I ue

SUf~ = SUM + PROO 11.16

If SUM < 1.0 then the AZ 3 value IS'calculated and multiplied by MZ . ~

.PROO=AZ3

*'M •

11.17

and a new~¥alue Is given to SUM

11.18

The procedure Is repeated end the.value of SUM Is checked

at each step against 1.0. We finally have steps m for which SUf~ ~, 1.0

and step m+l for which SUM)1.0

111

I

•

Then the value T (T < T < m . JI' .,

Tmtl ) Is sought for which SUM ~ 1.0.

The corresponding point on theP - curve/Is found as previously

Shown, the area A <.A· Is calculated as follows (proof ) m,m+1 sequence :

and

PROD' = A * M m I .

= SUM + PROD > 1.0 m m

SUM + PROD' = 1.0 m

PROD' = 1.0 SUMm·= 1.0 - SUMmtl + PROD .

A = PROD' (I.O SUMm+1 + PRODm)' M . = --~-:M""'.'=;'-'----"'-m m

Note that A cOuld also be calculated from (

/ I. a - SUMm A£;- P,ROO' =

\

M M • m m

•

•

but we have to use the 11.18 form as the last retained value in

the computer for SUM Is SU~I' From 11.16 and 11.11 we obtain

X and T=X.

We now have to calculate t, the time It took to cool down . .

to T. Between temperatures TI and TI+I the cooling rate Is RI nnd ,

the time to cool from TI to TI+I Is given by:

112

or

11.2Q ).

Furtl~ermore the time It takes to cool ·down to t'1"peraturo :r Is

g i'ven by:

m-I

CT= L (XI - XI+I> if MI + (Xm - Xl· Mm 11.21

I = I

•.

The program proceeds. as ·follows: CT Is Initially assigned the value 0.0

CT = 0.0 I

Then after the first re-eval~atlon of

SUM = SUM + PROD

CT Is assigned a new value:

and this proceeds In parallel with ""-

assignment of a new value to·CT Is

11.22

SUM Increasing. However, this

done only after we have checked

that SUI~ Is stili less than one. When SUM becomes greater than 1.0 . .

then X Is calc~lated (as shown beforel·and the last value Is assigned

to CT, that Is

) 11.23 .,

Equations 11.11 and 11.23 give 'one point of the CCT - diagram.

113

)

I tis poss I b I e that we may exhaust a II of the X I '

Y'I> ~II values and SUM Is stili less, tlliln 1.0; this mean's that

we are coo~lng'fast enough to avoid entering the austenite +~

pearlite tra;sformatlon area. ' The' ~llest cooling r/lte for

which that happens Is the crltk<'ll coollr\'CJ rate.

In Fig. 11'.2 we, havo presented the flow·dlagram of the

, program and' follow It 'the program (program 11.1) printed In

a genera I 'form, Th Is' program Is' for a max I mum nu'mber of po I nts

(TI.T I) of the .pearl ite-~tar: curve equal to 100~ There Is als,O . .

a maximum number of cooling paths (100), each one given by the .... . . . , . «"

cool inCl rates correspon~lng' to the -pre'1lously mentioned TI values. (

In another ve~sion of the program (the one used In the hardenability

estimation prOCjram) the coollnCl rate Is calculated from the T I , t'l values

of the cooling path.

The program Is very.easy to understand and use. It can, also

be modified to fit specific needs, as ,we have done when we Incorporate

It In our hardenability estimation program (see nex'tAppendlx).

114

115

. ,

/

fig.II.1

p; _____ ~_~R

11" ------- --,til' P , --- ---., .. ~ _______ I

.• H. \ . ---- ___ ~J__ Ft+1.

X

,

. y----.

.;..

fig. 11.2

,I

. ,

e a Ie u la te Ai,i+l

PROD=A .. , .M·I'j' . '1 1+.,

SUM=SUM+PROD

no' .

es

A=( 1.0+PRciD-SUMl/ Mi,j ,

1· X(= Tl !

116

no

. ,

. yes"

no

program I 1.1 or. PAZIONIS ,1(0=':\ l

F"II) ()F RECORD nDor.D~~ TST II"!PUT.()UTPUT,T~PF~.INPUT.TAPF6=OUTPUTI

n~Q'V~TION O~ rrT FROM TTT nIAGR~M

X cT~""c. ~()O T~"DFO'TlIPF,y C;TA"!nc; FOO ,."/TT"F,T "TANI'c; F()P I.T. n, .. " .. C T '''' X ~ on I • T I , "''II • Y I , N'l ,0 I , (In ., II" I ,W ( , "10 I • ~ , "" I

v~ln '--- 1] ... ·=7° n"l ,,,CO 1=1.1(

,or'p,,~" (".If.I' XITI.TITI !-,< ""p"'~T (?Fln.ol

n" ?"OO J=, .... r-.f'\ .,/..,n." 1=' ,'C

.,', ""0" (<;.?f.1 RII.JI' " ""Ip','oT IR~ln.OI

t."" \("I-"TY""111F "'1"\ -:2(\r" T=',~

'~o vlll=,."IITIIll I~n

" J=J+l o ql-..t:o.n

1=0 fJ, 0< 1=1+1

..I

"

1"1 I .r:F.1( I ,"r,o TO 17 " , , = I I Y I T 1+ Y IT.' I I * I X I T I-X I I.' I I In. 0

1,i"l

-'=IOll/1 0 IltJll q , .... : <:.t IV+Q

I" IC;II .... LT.I1. 0 )1 GO TO AS '=II."",-SU".""*IRII.Jl, "I "1'1 = I X I.T I-X I I.' III I Y I 1 1-Y I I +l I I ,?=(Yllll**7-7.0*rISLOP, ,~=C;0QTIA71

·'IJl=Xlll-c.Lnp*IYIII-A~1

c!'1,~ .. =n.n I=n 1= 1 +1 c I '"~ = C I '''. I X I TI- X I I.' I I /I R I 1 • J 1 1 I~I(X(I.711.r:F.I~IJIII GO TO 103 C I J I = c;( I". I X I I. ll-W I J I II t"R I I 1 + 1 I • J I I ~PTT~ 1f..f.71 J.WIJ1.SIJI

<~ ""O"AT I, 110.7F'0.", I~IJ.LT."" r:n TO 17

J

"

"~ln ("IF 'RFCORJ) "ATA DATA DATA DATA DATA

F,,!n ('IF FILf

• . .

, ,

. ,

'.

APPENQI X I 1.1

Computer aided estimation of V(n and TIT).

(a) Assignment of variable names~

i

BMN = 6Mn ' BNI • 6NI ' ~R = 6Cr ' BMa = 6Mb ' OSI = 6S1

' BCU = 6Cu

WPOIN = C.~ , WPCN I = r:. WPCCR C ''I' -NI ' = Cr

WPCCU=C Cu

BKCR = K , Cr

DKCR = K' Cr

.~

DC= D , A c

,

=

BKMO = 1).\0 ' BKS I =

DKI~ = K' DKSI Mb ,

a , AS = a

UC = 6T , V = v , TINC = T

=

DCR=

(b)

DCr ' (J.O = rvo( , OS I " DS I

Assignment of value·s for. III )~

, . K SI

,

WPCMJ = r:. ~o

WPCSI =,CSI

, ..

In the final version of the program which we give In

Program. III ~ 1 we have Included the quadratic term In liT for t-n. S . ..,

Note th~t the C"'" va I ue used I n that case I s an adjusted va I ue G:in '

117 ,

• \

C' being a base quantity. We also have included a carbon Ml

correction term. The corresponding variable names are:

BMl quadratic term = BMNI

With the above cxpl~nations concerning the variable

names used in the program and the help of the flow diagram given in

Fig. III . I, program 111.1 Is readily understood.

116

. , ,

• I

I

"\'

( ,

fig. II 1.1

,

set initial v·alues

calculate T~ ,

f,

yes

calculate D

calculate v(T),'t(T>

wri t e, punch T~.

,

"-

119

"

yes

•

~ ,

C. IR=3) program II 1.1

END OF RECORD PHOGHAM TST IIN~UT.OUTPUT.TAPE5·INPUT'.TAPE6.0UTPUT.PUNCHI

Ii) KtAD 15 .lOli) tlMthl)~ 1.I)CI< .I)MO·.I)SlotlCU .tlC.tlMNl vO FORMAT 18F10.01 2u READ 15.10u" WPCMN.~PCNl.wPCCI<.~PCMU.WPCSI.WPCCU.WPCC 30 UUIN=tlM"'*WPCMN+tlNfIIl * I WPCMIII**, 2.0 I I '" 31 UCl\cl =tlNI *WPCNI .... / 3LL!C(R=tlCR*WPCCR ' B UCMO=tlMO*WPCMO 34 UCS I =US 1 *WPCS 1 35 UC'CO=tlCU*WPCCU 36 l,lCC=BC *WPCC ',' • . 4u ·flA=lUUv~U+UCMN~UCNI+UCCR+UCMO+UC~l+UCLU+ULL+lll.o ?v ktAU 15,£vUJ b~LR.I)~MU.b~~l.u~(K.u~MU.V~Sl

"v,FlJI-:M"T 16FIO.al . 6u ,AKCR=bKCR*wPCCR W AKMO=UKMO*WPCMO 62 AKSI=BKSI*WPCSI . 63 CKCR=UKCR*WPCCR b4CK~'0=DKMO*WPCMO 65 CKSI=UKSI*WPCSI 79 k=I.Y81l~ 60 Ht::AD 15,5001 T.DC "0 FOt'MAT 12Fl5.01 01 UC=TtA-T &4 ·1 F WC. LE. IOU) I GO TO 80 91 UCR=E!WI-330UO. R*T I I 92 DMO=t:XPI-3uuOU.O/l TIl 93 USI=EXPI-330UO.oi,R*T' I ,I PkuU,=AK(KI (l)C~*UC I "~ PKuHO=A,,",OI I UMo*lic I .3 PHuSI=AKSIIIDSI*UCI lu Tc)T 1 :PROCR+PROMO+PROS"i 30 'TOT2=1.0/DC 4" T()T=TOTl+TOT2 5u D=l.U/TOT 6U A=46.2 7v V=A*()*IIUCI**i.OI .v AS=~.v*111U.01**1-7.0'1 .1 PRRC,,=CKCR/IOCR*UCI ,i PkRII,U=C!<.MUI IDMO*UC) 03 PRRS I =CKS 11 I OSl *UC f ,,5 TOT3=PRRCR~PRRMO+PRRSI J6 TOT4=TOT2+T0T3 10 TINC=IIAS*TI*ITOT411/1IUCI**2.0

' 5. TC=T-273.0 'i9 wRITE 16.9001 T.UC.TC.V.TlNC .J FORMAT IIP5E20.61 ,

PUNCH 3333.T.T1NC.V 33 FURMAT IIP3EIO.41

, ,

vi IF IT.Gr.162u.011 GO TO 80 • Iv '~RITE 16.9601 wPCMN.~PCNI.wPCCR.WPCMO.wPCSI.WPCCu.wPCC Lv ... RlTE 16.9601 tlMN.bHI.tlCR'tlMO.I:lSI.bCU.bC.~l o. FUKMAT IIP8E.l5.41 jv .. RIlt. 16.9701 tlII.CR.IiI\.I'IO.tl~Sl.IJII.LR.)JJUoIU,Uf,.Sl 7" FORMAT I1P6E15.41

I

Pt\llu,dS

!

76132' • ,

.' .

~1 wRITE (6.4444) TEA , " :j'

4444 FORMAT(lE20.4) PUNCH 2222.TEA

2222 FOR~AT (lPIE10.4) STOP

,END END OF RECORD

-8.lJ " -25.0 +15.u +7.:' +15.0 -20.0 -147.0 -24·0 2.8 0.0 0.0 0.0 0.01 0.0 0.47

+1420.0 +25000.0 +710.0 +833.0 +1000.0 +416.5 ,-lUOO.O 0.OUUOOO026 965.0 0.000000026 '165.0 0.000000025

. 955.0 u.UUUOOO024 946.0 U.00000002~

942.u u.uOu\i00U22 937.0 0.000000021 934.u 0.000000021.1 931.0 0.OU1.I000019 923.0 0.000000017 915.0 0.000000015

.J' 9u7.0 0.000000013 tl95 .. 0 0.Ov0060u11 BB3.u 0.00U1.I00U09 761 :, 676.0 U.OI.IOUO,OI.lOtl 668.0 0.001.1000007 660.0 U.ouOOOOO06 650.0 0.00uOOOO05 840.0 0.0001.100004 825.u iJ.OuuOOOO03 8u6.0 u.UUuOOOOO..! 775.U " O.oouOOOOOl 770.0 0.OOUOOOOu09 765.0 0.0000000008 760.u 0.0000000007 745.0 0.0000000005 724.0 U:0000000003 710.0 ".0000000002 684.0 .... ooouoooool 678.U U.oouOOOOOO08 668.0 0.00000000006 654.0 0.00000000004 • 633.0' 0.00000000002 613.U 0.0000000000'1

END OF FILE ' , CD TOT 0104

APPENDIX IV

In Fig. IV.I we give the flow diagram of the program

that estimates hardemlbil ity on the 'basis of the TIT diagram

(predicted from chemical composition) and the Klrkaldy model for

pearlite nucleation and growth. In program IV.I we give this

program.

We have assigned variable names as follows:"

The radius r' of the pearlite nodules at grid point m Is

r' = R(M)

The radius r" of the equivalent collapsing sphere (see previous

description) at grid point m is

r" = RR(M)

The dimensionless temperature at g.p. m Is

e = TO(M) m

The pearlite velocity at g.p. m and temperature TO(M) Is

v = VTO(M)

The Incubation time at temperature TO(M) Is

T = TIN(M)

1'20

\

I

The fraction of the Incubatlon·tlme passed at g.p. m Is CCT(MI

* The em values are given by;

* e = TON(MI m

The pearlite fraction at g.p. m Is

F c-PF(MI p

-There Is also the list of temperatures and corresponding "

Incubation times and pearlite velocities from tho TTl diagram that

qoes Into the deck of data. For a certain temperature, (Jth 1n the ..

list of datal·the corresponding Incubation time and the pearlite

velocity we assign the names:

ATO(JI, ATIN(JI, VATO(JI ..

The real time Incremebt St = Dr. . • I

The d, d" v~lue,s. are, :0'100.

The eutectold temperature Is TE. "-

The;' IC va I ue Is AA. P P . The t/2 value Is AL.

The Initial temperature Is TO. "

- " The temperature values are fed .. Into the program In 'degrees

Kelv·ln, the program au:tomatlcally changes the base (quench tempensture . I

Is put _equl!!l ,to zerol I!!nd expresse, them In dlni!lnslonless form •. The

~rogram' Is flnl!!l Iybrought Into such a form thl!!t can be easily used and

th~ output Is In such I!! form thl!!t extensive informatron can be derived

from .• ~n, I~e., the OCT curve,'coollng curves, the critical cooling

"

12·1

.. ,

..

curve" the pearlite fraction versus time (5 - curves), etc.)and

most Important, the pearlite and martensite distribution along

the Jomlny bar.

An updated form of the program Is now In use where one

needs to feed Into the program only one card with the chemical

composlt'lon and grain size of the elloy steel and this yields

In the output TIT and CCT diagram, the pearlite d'lstrlbutlon along

the J om I ny bar .. etc. Th I s program has been used for the pred I ct I on

of the pearlite distribution for 160 a'lloy steel composltlons.~

122

•

"

o

RRtIl. d/2.0 R(I).O.O CCTCIl.O.O

stop

writ. m,K, TCm),CCnm).~(m)

T.;..0.25'( Tm'1.2.0'Tm'~)' CT

I ,

v

'~I ., 'P"!p' 3.0' (R R(m))2' VTCm )"y(d/ 2.0)2

I T;;,.0.25'(~~Tm·r",..1) I<-~ -f!.J--------, •

~. Fp<'U5'(R(m))2. ~T(m)"0d1.o)2 +

IHm). R~.VT(m)'"

TIN(mhATlfW) Vf(m).VATtd

no

f

'" ~

c

N ~

. ,

~ogram IV~1 r

e:""'f\ ,,~ QJ='("("'IQf'\

OD",:Q"" T~T (T"'O\lT."tJTPIlT.TAP~C;=TNPlIT.TAP~~"nIlTOI)T' n,vr.,.,nu DI,n~,.oo(,n~,.Tn~ln~,.TnN('~~'.ftTn('nn'.ftT,,,,(,nn,.

T I" I , nr. , • r r T ( , n~ , • VT"I ( , nr. , • V ~ Tn ( ,n n , • P~ ( 1 no; , • ft T ( ,nn , . n"U"Lf pp~Cr5lnN P.oR.DP~.CT~TD.TDN.D.AA

(11 DC',.," (c..~h,01 r).nn

C'("oQUI'r (f"''l''.''.J:"'".''' ~'= ("I ....

V'=I"

,..n[:;', T='1,1()r:; 01 r 1='[\. n DO I 1 I = C r)r) , I ( ? .0' rrTlf,="'.("\ ."r'\(,,~,."l

OC",~,"'.,( c. ."''' 1 Te'" ~"D" or I 1 FIr. r'l orfl" ((:;,177) nT

'"r."hT ClFIO.O' Dt:"J\n (1;.1"7) AL

,"o""T 11 ~ln.o, rH·~~ (c:,lQ?l db

(""'7'''.'\T (r"\'''.''l 1''''' ,c.l T=1.47

.\

oq" lo.~"1 AT(f,.~TIM(Il'VATI))' C''''')'·'f\T ("1:~l('.O) ...

(1"I'..IT T "'l!J:" "r~"" {!:' .. -'''7''70 1 T'" :-rV)"flT (1':"1".("'1)

T .... r.-::TI')_"1:'7"1:.n

T,..r~(Tr_""'''1:.01/TnQ

"'''1 "'17(')(,\ T=1,47

.'T"III = (hTC' 1-~"7~.;OI/TOR ,I"HIT' "1I'~ "'n -,1'"\(''-'' V=1.10C; e" T" 1 00 ""'H.' { Of 1 -= '1." ( .. r:f" T'" "lnnf"

'C" f' •• CO ...... 1 1 r:.n Tn ·An

Ir I".cn.ln~, r.n Tn 71(1) 'r II T" I" 1\. "T. T';·~' r.n TI') 1 ('100 I""'~'T T ~III~

T =("l. 0

'=f·' ."

,.

'r IITnC"'I.LT.(ATI)(JII,Gn Tn 1!'" " ,. I .. 1 '; ft T T N ( T I , ....... , II 1 -= \"\ T n r T ,

rrTl"l=rrTc v l.nT/(TTOl(OIIll .. II r r T r ", 1 • (~ • (1 • ° 11 roo TO - 1 ('Ion

!~ 110 CV".L T.II.L'· (:0 Tn ,0(10 r",., T T "'''. II='

OO(·.'I=OD( .... '-(VTO("1"·"T 'c (("DCvll.LT.(n.,,,, r-n·Tn 000

."

·,c_-.~.((OD(V" •• ?"'.(VTnr¥"."IT/frn/,.nl •• ~.n,

512 PAZIONIS

. ,

.1

..

. :

,\.

t .'

I I

I - .'

( "

\..

" (T="PF*~A/ITn~"'7"'.n' 0' T"" I')' =il. ""* I I Tn I ".' ".'.0* I T .... I~' ,. IT .... 1"'-1' , ,.I'T

r.'l TI' ~OOo

00 POI!'):n.o o T""III1=1"\ • .,,,* I I T"I"'.' I , ... ,.0* I T .... I'" , '.IT .... I~-'" I

r:.(\ T" ":I""" " r,...",T T"'1I'~

'oOI ... ,=PI"'+IVTnl' .. Il* .... T " "or=~."*"."*1 10-1'" ,**".n'*IVTr'llu,,*nT/1 Ir'I/?(H**"'';OI n rT="'DF*~AIITn-~""'.OI .

Trw I", =1"\ • .,"* I I Tnl".' II.? 0* ITI)I"'I.I Tnl~-' 111.("T . " r:." T'" "1 no" o T"" I " , = ~ • "* I I Tn r ,",-1 1 I ... I TO I ... ,',., " (".f" Tn ":);000

" r""lTT~lt'F

t., "" '"nf'l T=,,'f"r; , T" I!' = Tn 'I IT' (\ rrH.!T,! "" II='

') "="'+1 v:V""1

, IF 1"l'nl""n).Ff'J.O) 1 T~ '{vr",'I(v., (jl.N~ .0) '" :'"'I" .,.,0n J=l"nc;

r.n TO "10;00 .1') Tn "I",n1

o '''''ITF 1<" .. ,On) T.IC.Tf\ITI.("rTIT' r"f"'I_'" T (? T"", .,,.,?n.Ft'

1'1 rl"'l"'TT~I"C

, (""TI"IIF O,F IIT"I,oll.LT.I('I.", r.1') TO ",o;on ,.. r.f" Tr"'\ "70

o (n"T I "\iF " 1"">" lln('ln "'=1., ('Iii

r Tr ({rrT!Vl,.r:T.(,."'" r:n T" ~~rH'

rFI",=n.o , c," T" 4 nno

IF IIRI~".r.T.ALI r.n Tn "IRnn ',01'1"1=","*1 IPIM, )**"."'/I In/::>."I**"'.O) " r.r" Tn 4 (1()n

J

, 010 1"'=".". I 1"."*1 IIL/r'lll**".nl.1 I Inr'l/?"I**".nl-I IQQI"'ll**?Oll/1 ~( (1""I/"."l**'),."l

1'1 r""'~'TT~"IC'.

,."" ~<n" '1=1,10<; ""O'lF ".,l."O'l) T.IC,T"ITI,("rTlTl.PF(YI ~"o~~T l'r,'l,,",n.~.E'n.~)

, (""n I NUF , n' IITr:I'''Il.LT.IO.::>ll r,n TO 4Ron "\ r.f"I T'" 7"

rf'HIT T~" Ie

'r Iv.1 T.'"On 1 r.n Tn 70 crno

. ~"'ln .. ,," " .. QF('mm

".f'I'l"'tf'l ().~""O t"~ .• ..'"'1?

f"l.fl4A

"I"'" ., r:: c • "

"'o"~'I.~F.C;"·.O""'.01\l"f'-o6 "O .. ~'1 .4<,?c;C'.n'4oF]J4F.,.no; . • ~t" ... .r',c. • .,.c;c:..,s:- ... "" LJ;f,c;7r:'-n4

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51

,n0n~ ~(l?l, ,0".' ~F +01 1 ." 1 ""F -04 "ocr~C?", .,"'. 'F+fll 1. """'1'-04 ~r",,~.r."II.I.."lL.-rf='+rq 1 .40I..c F-f"\4

11r:.'v .. r ... r"/l."'''OQ~+-('\"' 1 .4 Q A?F-n4 .,!,,,('Ir:+f'\"'I.~?t:.~~+Ol' .':IQItoAE'-04, l"')"r.+,....,~. 1 O"7(,r'+n, 1.' o' .... J:'-t"I4 O{,(' '"I C ... " "r,. ~ ('? I-l=' +('\ 1 R. "74 7Q~ _or; "onr ~G?", ., ~4 ~ F ... Ql "7. F. ?'iQF -Oil .onor~O?O ,oOQ?F+Ol F..4F.Of,F-OII '4ncr~"?' " ,,,.,,,+,,;>o;.n4F.7F-0'i "rnr~(\?' ".,~, 0F ... n2"1.1 "4"1'-00; , .,,,r<r ... /"1"". n "noC"+"" • Q,,",R6C'_nc:;

,,,"'1"1:' +"')? • a? r PF +"? 1 • R'"7 OAF' -~~ '1""'1:'+('''''. a ,f')~f.+n" • R'"04t:'_nI) ,'ln0C"+".,.,.P?n o F+()'1. R '7P6E-OC. 11.,,"'r+"'''''.O?(\O~+''7' .R'7R6F-n" "", ... r??,O?"R,, ... n2-J .R7R4F-Oo; , ., "" I" C ... r- ., ., • 0 ., (' 0 I=" +" ., , • R'" Q AF _n ~ • .," 1"\ r ... !"\ ., ., • 0 "'I ('\ 0 C' .... ,... ., " • Q'" Q" C' _" r:. , ., '\" co + " ... ., • 0 ., (' Q r. .... "7' • Q .., c:t 4':-_n c.

'1""'1='" +I)'"}" • a 7('1 a F +()? 1 • ~"'Q6J=' -nc; '" ,,('\ C ... ,,", ., ? • 0 , () p C' + ')', • Q'" QA F - n'" ''If'''ic+,,,,.a,0RF+(I7 1 • R1' R4E-nC; ,,;0",+,,?,,"'I',," ... 1'71."71'4F-05 ,"r'I"'~+"""".n?('QF+n?' .Q7 R6F-nc; • 1""t' +'"'''? 0 .,no!=' .... "?' • R7"4,:-n~ ,.,r.I"IC +"""'. 0 .,,,o~ .... n? 1 • Q'706F _nl\ , 'l"""C'+('\ ?.,'. Q .,·('\o!="~n-,6,. Q"'7D4F"_nc.

11,"'\ lie +(1"? 0 ?no!=" +0? 1 • Q7A4F -os; 1 :l('I"\C .... n?? 0 .,nQf:"+f)?1. 0'7J:14F-OC; ,",", ~r?? 0 ?nAF ... n71 • R?"4F ~Oo; '7""C" +,,",.,..,.0 ?0oJ:" .... n? 1 • Q~04F_(\C;

''1'" '" r ... "'" ~ • 0 ? ('\ 0 F +"? , • Q '7 Q4,F _n~

,11"1",C"+f].,.,.Q.,001='+"" .p"'P6~-nc;

,.,,,,..~+('I"'''. o.,noF+0" ;O-,Q41=_nc. , ., 1"\ I'" r .... ('\ ., ., • 0 ., rH) co .... n ." • 0 '7 06 F _" c;

l1r'1r"'C .... "''''? a? nA F +n?1 • A7Q6~ ~OC; ""''''C ... "?? t:f?OOF +021. R7 A4F-nc; , ., ,.., r'I C .... f' .., ., • 0 ., () 0 F + n." • Q'" 06 F' _n" ,."'1 I"\c +""? 0 ?Oo r= .... n' 1. P7P4F -or; ""'('C'+0?,.Q?npF'+o" .A7Q&.E'-OC.' .,_"('"'It' ... I .... ?" • 0"00 F +'1? 1. Q7Q6F -nc;

C1"0~n '('\"".0 ~.n~nnnn

"',n lnno.O n.OOOOOO ,1'l'l..'l

nH) nl'" "tLF

I i

CD TOT 0164

, , ,

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B. A. M. Grossman, Elements of HIIrdenablllty, American Society for Me,llIls (1952).

Sa. Ibid, p. 57.

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13~

H. Kralner, Arklv. Elsenhutenw., ~19 (1936).

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17.

lB.

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, '

124'

I

. I .

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"

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I· 26: J. W. Gahn, Acta Met. ~ 572 (1956).

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'. ~

28. C. F. Jatczak, MGt. Trans. 4. 2267 (1973).

29. W. W. Clas, Phase Transformation Kinetics and Herdenability of Medlum-Cerbon Alloy Steels, Fig. 20, CLIMAX Molybdenum Company.

3011. Atlas of I Soth8F11lll I TransfOrmation Diagrams, p. 42, United States Steel. "

lOb. Ibid, p. .48.

3Oc. Ibid. p. 88

3Od. Ibid, p. 39.

308. Ib,/.d, p. 88.

3Of. Ibid, p. 87.

3Og. Ibid, p. 86

3Oh. Ibl d, p. 54

125 \ .

126

31. D. Brown, N.'Rldley, Journal ot 1.5.1., August 1966. 811. (

32. Met!! I S Hl!ndbcjok, Vo I LIII8 I, p. 207.

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35. R. A. Grange; Met. Trllns. ~ 2231 (1973).

36. No P. Puis, J. S. Klrkllldy, Met. Trllns.l, 2TI7 (1972) •

•