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j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 8 ( 2 0 0 8 ) 450456

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / j m a t p r o t e c

Residual stress during heat treatment of steel grinding balls

C. Camurri a, C. Carrasco a,, J. Dille b

a Universidad de Concepci on, Department of Materials Engineering, Casilla 53-C, Concepci on, Chileb Universite Libre de Bruxelles, Faculte des Sciences appliqu ees/ ecole polytechnique, Avenue F. Roosvelt 50CP194/03, Brussels, Belgium

a r t i c l e i n f o

Article history:

Received 24 April 2007

Received in revised form

25 October 2007

Accepted 8 January 2008

Keywords:

Grinding balls

Heat treatment

Martensite

Residual stress

a b s t r a c t

This work aimed to model the temperature distribution, phase transformation and resid-

ual stress induced during the heat treatment of grinding balls of 3 and 5 in. diameter. The

temperature model considered factors such as the heating of the water and the formation

of a steam layer that surrounds the balls at the start of the quenching. The model of the

residual stress field considered: the temperature distribution, the force equilibrium equa-

tions and the constitutive thermo-elastic relationships, including the expansion due to the

austenitemartensite transformation.

A good agreement between the experimental and theoretical values for the temperature

distribution was obtained with the differences at the end of the quenching being no higher

than 0.5%. The experimental behavior of the balls in a mill simulator, as well as the residual

stress measured by X-ray diffraction, agreed satisfactorily with the theoretical predictions.

2008 Elsevier B.V. All rights reserved.

1. Introduction

Moly-Cop at Talcahuano, Chile, produces steel grinding balls

mainly of 3 and5 in.(7.62and 14.70cm, respectively) diameter,

using rolling or forging processes (see Table 1 for details of

their composition).

Immediately after the metal working the balls, at 800 C,

were quenched in water at an entry temperature of 50 C and

for 80 and 180s for the 3 and 5in. diameter 2R of the balls,

respectively. After quenching, the balls were cooled in air for

temperature homogenization for 40 and 120 s, respectively, a

process known as equalization. The 3 in. diameter balls were

then stored in boxes in order to allow them to slowly reach

ambient temperature. When the 5 in. diameter balls had been

equalized a four-step annealing process, each step lasting for

1h, was performed: cooling in the furnace by air pressure,

then heating the balls to 230 C by hot air, then slow cooling

down to 220 C, and finally cooling from 220 C to the ambient

temperature.

Corresponding author. Tel.: +56 41 2207170; fax: +56 41 2203391.E-mail address: [email protected] (C. Carrasco).

During the heat treatment, and as a result of the ther-

mal and structural gradients, residual stresses are generated

which can jeopardize the working life of the balls and, in

some cases, cause fracture before they are used. The aim of

this work was to model and resolves in an analytical way

the residual stress field induced during the quenching and

equalization of the steel balls and thus determine their possi-

ble cracking or fracture points. Also, and for further works,

the model can be useful for predict the optimal operating

conditions during quenching, for instance, the initial temper-

ature and flow of the water, in order to reduce the residual

stresses.

The work was carried out in two main stages: first, the

radial distribution of the temperature of the balls during the

quenching and equalization processes was modeled using

the explicit finite difference method and validated experi-

mentally. Also, the radial distribution of the martensite was

calculated. Second, the residual stress field was simulated

using the data obtained in the previous stage.

0924-0136/$ see front matter 2008 Elsevier B.V. All rights reserved.

doi:10.1016/j.jmatprotec.2008.01.007
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Nomenclature

Cp caloric capacity

E elasticity modulus of the mix

austenitemartensite

h heat transfer coefficient ball ambient

hconvective convective heat transferhradiant radiant heat transfer

K thermal conductivity

Ms temperature of beginning of martensitic trans-

formation

R radius of the balls

r radial coordinate

ra radius of a sphere of 1g of austenite used as

reference for calculus

rma radius of a sphere of 1 g of the mix

austenitemartensite

T radial temperature of the balls.

Ts surface temperature of the balls

Tam ambient (steam, water, air) temperatureu radial displacement

Greek letters

thermal lineal expansion coefficient

lineal expansion coefficient due to phase trans-

formation

rr, radial and circumferential strain, respectively

Poisson modulus

i density of i phase (i = a, austenite; i = m,

martensite)

rr, , ef, 0, R radial, circumferential, effective or

equivalent, yield and rupture stress, respec-

tivelyM volumetric fraction of martensite (or annealed

martensite)

2. Models

The thermal and residual stress models are as follows.

2.1. Temperature distribution of the balls

For modeling the radial temperature of the balls T, the follow-

ingassumptions were made (Camurri et al., 1997; Kreith, 1973;Incropera and Dewit, 1999; Mills, 1995; Welty, 1979; Farlow,

1982):

The balls are completely spherical and homogeneous, i.e.,

they only have radial temperature gradients.

Fig. 1 Micrography of the center zone of a 5 in. diameter

ball.

The heat losses by radiation and the heats of reaction of thephases formed during the process are neglected.

The thermal conductivity K and caloric capacity of the

steel Cp depends linearly on the temperature. Addition-

ally, the densities of the only two possible phases presents

(see Fig. 1), austenite and or martensite, were assumed as

8.03g/cm3 and 7.75 gr/cm3, respectively (Totten et al., 1992).

The heat losses from the water used for quenching are neg-

ligible.

With this assumption, the heat equation to resolve is

K2TR2

+

2

r

T

R =

t (TCp) (1)

With the following border conditions

T

r(0, t) = 0 and K

T

r(R, t) = h(Ts Tam)

where h is the coefficient of heat transfer between the surface

of the ball at T= Ts and the media at T= Tam.

In a previous work (Camurri et al., 2003; Garca, 2000), the

temperature of the quenching water was assumed to be con-

stant and the vapor layer formed around the balls at the

beginning of the process was neglected. These hypotheses

yielded a poor correlation between theory and experience, so

both effects were included in the present model.Hence, in a first cooling stage during the quenching, until

the surface temperature Ts of the balls is around 220 C, a

radiation and convection heat transfer mechanism across the

vapor layer surrounding the balls is considered. In this case,

Tam =100 C and h = hconvective +0.75 hradiant. The formula for

determines the coefficient hconvective as a function of the tem-

Table 1 Composition (wt.%) of the steel of 3 and 5 in. diameter balls

Diameter C Mn P S Si Cr Mo Al Ti Nb V Cu Ni

3 1.13 0.93 0.01 0.01 0.23 0.76 0.05 0.02

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452 j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 8 ( 2 0 0 8 ) 450456

perature can be found in Incropera and Dewit (1999). This

formula involves the ball diameter and physics and thermal

properties of the liquid and vapor water such as density, ther-

mal conductivity and cinematic viscosity; also the formula

has a constant which depend of the geometry of the bodies,

in this case 0.67 for spheres. For the radiation coefficient an

emissivity of the steel of 0.9 was supposed.

In the second cooling stage during the quenching, untilTs 105 C, the cooling of the balls is produced by vapor

transport from its surface, and the temperature is considered

as Tam =100C. In this stage the coefficient h can be found

in Incropera and Dewit (1999) and depends of the thermo-

physics properties of saturated liquid and/or vapor such as

density, specific heat, cinematic viscosity and superficial ten-

sion. There are also two constant, C and n, in the formula

for obtain h. These constants are related with the system

surface-fluid, and the values for the studied steelwater are

0.013 for C and 1.0 for n (Incropera and Dewit, 1999). Finally,

the third cooling stage begins when the surfaces of the balls

reach theboiling temperature of thewater,and the heat trans-

fer mechanism is pure convection. In this case Tam is equal tothe instantaneous temperature of the water, which is calcu-

lated by main of a heat balance among the heat loosed for

the ball and transfer to the water, with initial temperature of

50 C.

During the equalization and annealing, the heat losses

from the surface of the balls to the air are only by convec-

tion. In all cases, h is function of the surface temperature of

the balls.

Eq. (1) was solved by the explicit finite difference method,

using algorithms for the internal, central and superficial

nodes. Performing with the stability conditions, in the case

of 3in. diameter balls, the radial increments were R/20 and

the temporal increments were 0.02 s and 0.05 s in quenchingand equalization, respectively. For 5 in. diameter balls, those

increments were R/40, 0.05 s in quenching and 0.025 in both

equalization and annealing. The temperature distribution was

validated by inserting thermocouples in three different posi-

tions in the balls: the center, R/2 and R0.7mm, where R is

the radius of the balls. Additionally, the temperature of the

quenching water was continuously measured during the pro-

cess in order to compare it with the model results.

2.2. Residual stress of the balls

To model the residual stress field, the following hypotheseswere made:

Throughout the cooling there are only two phases in the

balls: austenite and/or martensite, as Fig. 1 shows.

The induced residual stresses are only normal stresses.

The residual stressesinduced in the austenite are negligible

due to the low yield stress of this phase at a temperature

above Ms, the temperature at which start the martensitic

transformation. For example, at 400 and 600 C the yield

stress of austenite is 120 and 80 MPa, respectively, while the

yield stressof martensite at 200C is800MPa (Okamura and

Kawashima, 1988).

In general, the modeling of coupled thermo mechanical

problems with also phases transformations normally use

numerical techniques such as finite elements for its resolu-

tion. For example, in Hossain et al. (2004), this methodology

is addressed in the modeling with a posterior experimental

validation by means of neutrons diffraction, of the resid-

ual stresses induced during the quenching of stainless steel

spheres of 30 mm diameter. In the present work, a simpleana-lytical solution for the residual stresses field in grinding balls

is presented based on the following equations.

2.2.1. Equation of Equilibrium of radial forcesrr

r+ 2

(rr )

r= 0 (2)

where rr and are the normal, radial and circumferential

stresses, respectively.

2.2.2. Equations relating deformations and displacements

rr =u

rand =

u

r(3)

where u, rr and are the radial displacements, and the radial

and circumferential strain, respectively.

2.2.3. Constitutive thermo-elastic-phase transformation

equations

These include the thermal expansion coefficient of the

steel and the linear coefficient of expansion due to the

austenitemartensite phase transformation.

rr =(rr 2)

E+ F(r) and =

(1 )

E

Err + F(r)

(4)

and

F(r) = T(r) + M(r)

where E and are the elasticity and Poisson modulus,

respectively of the austenitemartensite mix at temperatures

below Ms, obtained from reference data for their individual

modulus (Okamura and Kawashima, 1988) and, in the case of

the Youngs modulus E, were also obtained experimentally in

this research from tensile curvesof standard samples of 5 mm

in diameter and 25mm gage length L0. These samples were

quenched at temperatures below Ms and subjected to tensile

tests by an Instron machine at those temperatures. The slope

of the linear part of the stressstrain curve corresponded tothe E values (Rodrguez, 2006).

The symbol corresponds to the thermal expansion coef-

ficient of this mix, corresponds to the linear expansion

coefficient due to the phase transformation from austenite to

martensite during quenching and equalization or the linear

contraction coefficient due to the phase transformation from

martensite to annealed martensite during the annealing of

5 in. diameter balls. Finally, T(r) and M(r) are the changes

in the temperature and the volumetric fraction of marten-

site formed at each radial position for temperatures below

from Ms, respectively. Determined the temperature of the balls

for each time and knowing the CCT curves of the steels for

obtain Ms, it is possible determines the volumetric fraction

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of martensite formed (M(r)) using the KoistinenMarrburger

equation (Krauss, 1990). Obviously, M = 0 if the temperature

in some radial portion is higher than Ms. As the densities of

the austenite (a) and the martensite (m) are known as well

as the fraction of formed martensiteM(r), the coefficient (r)

is determined as

= rma rara

(5)

where rma and ra are the radius of the 1g of spherical shape

(base of the calculus) of the mix austenitemartensite or only

austenita, respectively, and which are calculated as

ra =

3

4aand rma =

3

4(1 M)a +Mm)(6)

If must be calculated in the transformation martensite

to annealing martensite, the procedure is analogous, and in

this case M represents the fraction of annealing martensite

and should be determined from an annealing parameter as is

shown in Shi et al. (2001).

By combining formulas (2)(4), the radial equation of the

force equilibrium can be expressed in terms of the radial dis-

placement u as

r

1

r2

r(r2u)

=

1+

1

F(r)

r(7)

Note that F(r) is known because the radial distribution of tem-

perature T and the coefficient (r) have been determined as

was described previously.

Then by a double integration the radial displacement u is

obtained

u(r) =(1 + )

r2F(r)dr

r2(1 )+ C1

r

3+

C2

r2(8)

Using the constitutive thermo-elastic-phase transforma-

tion Eq. (4), finding the stresses in terms of the strains

expressed as a function ofu and u/r, theradial normal stress

is

rr =E

(1 + )(1 )

(1 )

u

r+ 2

u

r (1 + )F(r)

(9)

where C1 and C2 are constants which must be determined

from the following border conditions

Ifr = R (radius of the ball), rr = 0.IfT(r =0)< Ms, then rr/r = 0.

If T(r = 0)Ms, then rr(r r*)=0, where r* is the radius for

which T(r = r*) = Ms.

Additionally, if for some radius r** the effective or equiva-

lent stress ef defined in this case as ef= |rr |, with the circumferential stress, results equal or greater than the

yield stress of the steel 0, then ef(r=r**) =0, ifef

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Fig. 4 Experimentaltheoretical comparison of quenching

water temperatures.

Fig.5 shows the numerical radial distribution of martensite

in 3 in. diameter balls at the end of quenching and equaliza-

tion.

Fig. 5(a) indicates that at the end of quenching and for a

radius smaller than 1.9 cm, the temperature of the balls was

higher than Ms (Ms =220 C for the steel used), and conse-

quentlyno martensite wasobservedin thatzone. Thisimplies,

according to the previousdiscussion, thatthe residual stresses

at that radius are negligible.

Figs. 3 show that at the end of the equalization the centers

of theballshave a temperature closely of 150C, which implies

that at that moment there is 54% of martensite when r = 0. As

a result, when the equalization concludes, all the balls haveresidual stress.

Fig. 6 shows the modulus of elasticity as a function of the

temperature of the austenitemartensite mix obtained from

thereference data(Okamuraand Kawashima, 1988) and exper-

imentally in this investigation (Rodrguez, 2006).

Fig. 6 shows a very good agreement between both data

groups up to temperatures of around 170C. For higher

temperatures, the experimental dataobtainedfrom thisinves-

tigation are lower than the data from the literature for

martensite alone. This is probably due to the expansion effect

of the pull roads used to subject the sample during the test,

which implies a longer displacement for eachtensile strength.

Since a temperature extensometer was not available, the elon-gationof thesample was obtained from thedisplacement data

recorded by the Instron machine minus the effect of the elon-

gation of the equipment, including the pull roads, which was

determined from previous traction tests at room temperature

made on steel samples of known modulus of elasticity.

Figs. 7 and 8 show the theoretical radial distribution of

circumferential stress for balls of 3 and 5 in. diameter, respec-

tively, at the end of the quenching and equalization. Some

experimental measurements of residual stresses on the sur-

face of industrial 3 and 5 in. balls, i.e., after the slow cooling

in boxes after the equalization (3in. diameter balls) or after

annealing (5in. diameter balls) are included. These values

were obtained by X-ray diffraction (XRD) with a diffrac-

Fig. 5 Numerical martensite distribution in balls of 3 in.diameter: (a) at the end of the quenching and (b) at the end

of the equalization.

Fig. 6 Variation in modulus of elasticity with temperature

of the austenitemartensite mix. Comparison between

reference and experimental data.

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Fig. 7 Theoretical radial distribution of circumferential

stress for 3 in. diameter balls at the end of quenching

(dotted line) and equalization (continuous line) and

experimental (spots in black).

tometer (Siemens D 500). Also, as a reference, in depth

experimental values of the residual stresses were obtained

by measures on 1.0cm1.0cm6.5 cm parallelepipeds cut

from the 3 in. diameter balls, or from 1.0 cm1.0cm12.0cm

parallelepipeds from the 5 in. diameter balls.

Note from Figs. 7 and 8 that after the quenching, in zones

where there are only austenite, the residual stresses are cero,

while after the equalization, the totality of the ball as the mix

austenitemartensite and, as a consequence, all the mate-

rial has residual stresses. In the other hand, it is observed

that the absolute values of the residual stresses induced dur-

ing the quenching, are greater than the induced during the

equalization, just because that treatment has the function of

Fig. 8 Theoretical radial distribution of circumferential

stress for 5 in. diameter balls at the end of quenching

(dotted line) and equalization (continuous line) and

experimental (spots in black).

Fig. 9 In time distribution of circumferential stress for 3 in.

diameter balls at 0.07 mm below the surface at the end of

quenching (dotted line) and equalization (continuous line).

homogenizer the temperature of the balls. It is also interest-

ing to note that the part of the ball with traction stress at the

end of the equalization coincide with the volume with mix

austenitemartensite during the quenching. The circumferen-

tial stresses in this volume at the end of the equalization are

positivebecausethe transformed martensite during thistreat-

ment has less rigidity than the martensite formed during the

quenching, and for this reason the central zones of the balls

have more trends to expand circumferentially. Additionally,

Figs. 7 and 8 show the good agreement between the experi-

mental and theoretical results at the end of equalization of

the circumferential residual stresses on the surface. Also, that

those greater tensile values occur at a certain depth below the

surface of the balls, which coincides with the position of their

fracture when occurs during the heat treatment. The values

of the residual stresses inside the balls show an acceptable

agreement between the theoretical (end of equalization) and

experimental values, the latter being only a reference due to

the effect of cutting the samples from the balls for the XRD

measurements, which introduces distortions and also stress

release.

Fig. 9 shows the time evolution of the circumferential

stress of 3in. diameter ball at 0.07mm below the surface after

quenching and equalization. From that figure can be seen the

variation from compressive stresses at the end of the quench-ing to traction stresses at the end of the equalization.

4. Conclusions

This simple model of the heat transfer gives the temperature

distribution of the balls during quenching and equalization,

with good experimental agreement.

The model of the residual stresses duly predicts the

observed experimental fact in a mill simulator that, when the

balls fracture during the equalization, the fracture begins at a

certain depth from the surface.

The modeled and experimental residual stresses in the

balls compare well. It should be noted that the experimen-

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tal values measured inside the balls are only intended as a

reference, due to the effects of distortion and stress release

when cutting the samples for the XRD measurements.

Future improvement to the model can be making, such to

include the release of residual stresses in the balls due to plas-

tic micro-deformations during their heat treatment and the

local deformation hardening associated to this phenomena.

Finally, to use the model as a predictor of the better oper-ational conditions during the heat treatment, such as initial

temperature and flow and agitation of water for reduce resid-

ual stresses of the balls, mayor accuracy in thein depthvalues

of the residual stresses must be obtained. For this, techniques

such as hole drilling or neutrons diffraction must be used.

Acknowledgements

This work has been supported by The National Council

of Research in Science and Technology of Chile, CONI-

CYT (FONDECYT project no. 1050078). The authors gratefully

acknowledge this support.

r e f e r e n c e s

Camurri, C., Carrasco, C., Canete, P., 1997. Etude de lamodification du procede de fabrication des systemes debroyage. Revue de Metallurgie 3, 949954.

Camurri, C., Garca, A., Canete, P., 2003. Mathematical model forthe residual stresses induced during the heat treatment ofgrinding balls. Mater. Sci. Forum 426432, 39333938.

Farlow, S., 1982. Partial Differential Equations for Scientists andEngineers. Elsevier.

Garca, A., 2000. Esfuerzos residuales durante el temple demedios de molienda. Memoria de Ttulo, Facultad deIngeniera, Universidad de Concepcion.

Hossain, S., Daymond, M.R., Truman, C.E., Smith, D.J., 2004.Prediction and measurements of residual stresses inquenched stainless-steel spheres. Mater. Sci. Eng. A, 339349.

Incropera, P., Dewit, D., 1999. Fundamentos de Transferencia deCalor. Prentice Hall.

Krauss, G., 1990. Steels Heat Treatment and Processing Principles.ASM Publications.

Kreith, F., 1973. Principios de Transferencia de Calor. HerrerosHnos.

Mills, A., 1995. Transferencia de Calor. Irwin.Okamura, K., Kawashima, H., 1988. Proceedings of the 32nd Japan

Congress Materials Research, p. 323329.Rodrguez, P., 2006. Esfuerzos residuales en medios de molienda:

determinacion experimental del modulo de elasticidad de lamezcla austenitamartensita entre Ms y la temperaturaambiente. Memoria de Ttulo, Facultad de Ingeniera,Universidad de Concepcion.

Rodrguez, D., 2007. Modelacion matematica del campo de

esfuerzos residuales inducido durante el tratamiento termicode bolas de molienda, M.Sc. Tesis, Universidad deConcepcion.

Shi, W., Liu, C., Liu, Z., 2001. Numerical simulation of internalstresses and microstructure distribution of heavy forgingsduring quenching and tempering processes. In: The FirstSino-Korean Conference on Advanced ManufacturingTechnology, China.

Totten, G., Bates, C., Clinton, N., 1992. Handbook of Quenchantsand Quenching Technology. ASM International, Hardbound.

Welty, J., 1979. Transferencia de Calor Aplicada a la Ingeniera.Limusa.

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