study of through-thickness residual stress by numerical and experimental techniques

To validate the quenching modelling, the incremental hole drilling and neutron diffraction methods are used to measure the residual stress field in the studied parts.. Keywords: residual

Study of through-thickness residual stress by

numerical and experimental techniques

S Rasouli Yazdi, D Retraint and J Lu

Lasmis (Mechanical Systems and Concurrent Engineering Laboratory) Troyes, France

Abstract: The quenching process of aluminium alloys is modelled using the finite element method The

study of residual stress field induced by quenching is divided into two: the thermal and mechanical aspects

In the thermal problem, the general heat conduction equation is solved and the temperature field during

quenching is calculated In the mechanical problem, the calculated temperature field and mechanical

proper-ties are used to predict the residual stress field

In this paper, the two different boundary conditions used in the thermal problem are examined The first is

surface convection using the appropriate heat transfer coefficient The second is the temperature variation

measured at the surface of the part These boundary conditions are compared, and the advantages and the

drawbacks of each are shown

The influence of different quenching parameters on the level of residual stress is studied To validate the

quenching modelling, the incremental hole drilling and neutron diffraction methods are used to measure the

residual stress field in the studied parts The hole drilling technique has been adapted to measure the residual

stress through a larger thickness of the part The aim of this paper is the combination of numerical and

experimental techniques for the investigation of the through-thickness residual stress field

Keywords: residual stress, quenching, neutron diffraction, incremental hole drilling, aluminium

NOTATION

A surface area of specimen (m2)

A sn , B sn calibration coefficients for geometry n and

layer s

b plate thickness (m)

C is constants at layer s with i ¼ 1; :::; 5

C p specific heat capacity (J/kg⬚C)

d i , d0 interreticular spacing of the diffracting planes (m)

e internal energy (J)

˙e time derivative of internal energy (J/s)

E Young’s modulus (MPa)

h heat transfer coefficient (W/m2⬚C)

k conductivity matrix (W/m⬚C)

K coefficient of the Ramberg–Osgood law (MPa)

p number of time steps

q heat flux (W/m2)

r coefficient of the Ramberg–Osgood law

t time (s)

Dt time interval (s)

T temperature (⬚C)

T0 fluid temperature (⬚C)

x position from the centre of the plate (m)

a angle between the gauge and the principal

direction 1 (deg)

dij Kronecker delta

␧ strain component

␧p plastic strain

␧r radial strain

d␧el

ij elastic strain increment related to the stress

increment by Hooke’s law

d␧p

i j plastic strain increment

d␧t

ij total strain increment

d␧Th thermal strain related to the temperature

incre-ment by the thermal expansion coefficient

v1, v0 Bragg angle (deg)

l monochromatic wavelength of incident neutrons (m)

n Poisson’s ratio

r density (kg/m3)

j stress component (MPa)

jel yield stress (MPa)

j1K , j 2K principal stresses (MPa)

jr, jt radial and tangential stresses respectively (MPa)

trt shearing stress (MPa)

Subscripts

BC boundary condition

E experimental

The MS was received on 11 February 1998 and was accepted after revision

for publication on 1 October 1998.

i normal orientation along the i direction

j principal directions 1, 2 or 3

n time interval number

s layer number

0 quantities measured in the stress-free material

Superscripts

TC thermocouple position

Heat treatments can improve the mechanical properties of

different alloys The general heat treatment for aluminium

alloys is quenching with different quenchants such as air,

water and polymer solutions Each of these quenchants

has a different cooling rate If the cooling rate is rapid, the

mechanical properties obtained are very interesting but the

level of residual stress and distortion can be great For a

slow cooling rate, the levels of residual stress and distortion

are lower but the mechanical properties obtained may not be

very useful Problems with quench distortion, distortion

induced by machining, and residual stress are common,

affect-ing castaffect-ings, forged products, extrusions and rolled plates The

residual stress does not always have harmful effects as it is

known a compressive residual stress can improve fatigue

life [1] Therefore it would be interesting to optimize all

the quenching conditions to obtain the best mechanical

properties, the least distortion and the best fatigue life

Fatigue life prediction can be deduced from the residual

stress field, but the residual stress level is modified by cyclic

loads [2] To predict the exact fatigue life, it is necessary to

know the stabilized level of residual stress Figure 1 shows the flow diagram for integrating the residual stress in a fati-gue life prediction This study can be divided into three: residual stress field calculation or measurement, residual stress relaxation and fatigue life calculation

The fatigue relaxation of residual stress due to quenching

in aluminium alloy 7075 is shown in Fig 2 [3] The

relaxa-tion has been modelled using finite element methods As shown, the relaxation level depends on the level of applied loading The studied alloy is a cyclic hardening material in which, after a few cycles, the residual stress level was stabi-lized A three-dimensional program has been developed in order to calculate the fatigue life of different parts with dif-ferent types of applied loading and consideration of the

resi-dual stress [4].

In this paper the first part of the global study is developed The residual stress induced by quenching is studied This process is modelled by numerical methods using less com-plex boundary conditions The residual stress field in the quenched part has been measured by the modified incremen-tal hole drilling method and the neutron diffraction method The modified hole drilling method has been used because it gives rapid results The neutron diffraction method is the only technique by which to obtain the complete residual stress field However, globally as in the future the measured

or calculated residual stress will be integrated in the fatigue life calculation, measuring the compressive residual stress

in the critical zone near the surface will be sufficient

The thermal and mechanical problems are considered as uncoupled during modelling in the sense that (a) the internal energy depends on only the temperature and (b) the heat flux

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Fig 1 Residual stress integration in the fatigue life calculation

per unit area of the body, flowing into the body, and the heat

supplied externally to the body per unit volume do not

depend on the strains or displacements of the body In

heat-treatable aluminium alloys, precipitation hardening

during quenching does not induce changes in volume

Figure 3 shows the necessary procedure for residual stress

prediction The physical and mechanical data obtained

from the literature are included in the program

2.1 Temperature field calculation

As the thermal and mechanical problems are not coupled,

the equation of energy conservation is as follows [5]:

Heat conduction is assumed to be governed by Fourier’s law

[6]:

The conductivity can be fully anisotropic, orthotropic or iso-tropic In the present case the conductivity is considered as

isotropic; therefore the matrix k is reduced to the scalar k.

Equation (1) together with Fourier’s law [equation (2)]

give the general equation of heat [7]:

rC p

∂T

To obtain the temperature field during quenching, the general heat equation is solved by numerical methods For the time integration, the backward-difference algo-rithm is used The non-linear system obtained is solved by

a modified Newton method [8].

2.1.1 Boundary conditions

In the case of quenching at the part surface there is heat transfer between the part and the quenchant To define this heat transfer, boundary conditions must be known For the temperature field calculation, the boundary condi-tions may be specified as the prescribed temperature

T ¼ Tðx ; tÞ, the prescribed surface heat flux per area, the

prescribed volumetric heat flux per volume and surface

convection q ¼ hðT ¹ T0Þ

The heat transfer coefficient h depends on the geometry,

quenchant, quenching temperature and material This para-meter cannot be determined by pure numerical methods It

is determined by experimental measurement of tempera-tures at different points in the quenched material After the inverse resolution of the heat transfer conduction equa-tion for one dimension, using the measured temperatures,

the expression for the heat transfer coefficient is [9, 10]:

h ¼ mC p

p Dt Aln



Ta¹ ðT nTCÞE

Ta¹ ðTTC

nþpÞE



ð4Þ

In equation (4), the time and the position appear, although

Fig 2 Residual stress relaxation in a quenched cylinder

Fig 3 Modelling diagram

the heat transfer coefficient does not depend directly on the

time and the position The coefficient h depends on the

tem-perature; as the temperature depends on the time and the

position, therefore h depends on them too This boundary

condition requires temperature measurement at different

points of the sample and complex calculations

Another possible boundary condition is the prescribed

temperature T ¼ Tðx ; tÞ The best solution is to measure

the temperature variation during quenching using

thermo-couples, but measuring the temperature at the surface is

very difficult Generally it is preferable to measure the

sub-surface temperature However, applying this measured

tem-perature as a boundary condition does not represent reality

since it is not the exact temperature variation at the part

sur-face Although in the case of quenching of aluminium

alloys, the heat transfer coefficient and the heat conductivity

are very high, the temperatures at the surface or at a slight

distance from the surface are not very different Later in

the work these two boundary conditions are applied

sepa-rately and the results obtained are compared

There is a way to find out the exact temperature variation

at the surface of the part This consists in measuring the

tem-perature at other points of the part and by extrapolation

obtaining the temperature variation at the part surface All

these methods introduce errors into the final results It is

necessary to mention that none of the numerical methods

is 100 per cent accurate

To obtain the temperature variation, accurate

measure-ment is needed but, to obtain the heat transfer coefficient,

both temperature measurement and complex calculations are

needed Using surface temperature variation as a boundary

condition is easier because its determination is less complex

2.2 Thermal results

During quenching there are three phenomena First, a thin

film of vapour is formed at the surface of the part During this time the heat transfer between the part and the quench-ant is very low; therefore the temperature variation is not very rapid and the heat transfer coefficient is quite low Sec-ond, this film starts to disappear and the heat transfer increases At this stage the temperature variation is very fast and the heat transfer coefficient very high Third, the temperature difference between the quenchant and the part

is less; thus the heat transfer decreases, resulting in a low temperature and variation in heat transfer

The studied parts are an aluminium alloy 7075 cylinder of

50 mm diameter, an aluminium alloy 7075 plate (500 mm (length)×500 mm (width)×70 mm (height)) and an alumi-nium alloy 7175 plate (126 mm (length)×53 mm (width)×

24 mm (height)) Considering the dimensions of the parts, they can be considered as infinite Therefore, in the case

of the plates, the heat flow is just through the thickness and, in the case of the cylinder, it is through the radius Figure 4 shows the geometry and the heat flow direction

in the parts The initial temperature of the parts was

467⬚C The aluminium alloy 7075 parts were quenched in

cold water (20⬚C) and the aluminium alloy 7175 part was

quenched in water at 65⬚C

Figure 5 shows the heat transfer coefficient as a function

of time in the first plate (thickness, 70 mm) The three stages explained before are evident To calculate the heat transfer coefficient, thermocouples are used Four are placed in the

plate thickness as follows: at x ¼ 0 mm, x ¼ 17:5 mm,

x ¼ 26 mm and x ¼ 34 mm where x is the position from

the centre plate The measured temperatures allow the cal-culation of the heat transfer coefficient by inverse resolution

of the heat conduction equation

Figure 6 shows the measured temperatures at four points

in the plate of thickness 70 mm In the same figure the cal-culated temperature by extrapolation at the part surface is shown The extrapolated temperature at the part surface

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Fig 4 Geometry and measurement directions in the parts studied

is obviously not different from the temperature measured

1 mm below the surface

Two different boundary conditions are applied

sepa-rately: the first is the heat transfer coefficient and the second

is the surface temperature variation during quenching The

heat transfer coefficient is obtained as explained before

and the surface temperature variation is measured accurately

at the part surface Figure 7 shows the temperature variation

calculated at the centre of the plate of aluminium alloy 7075

using these two boundary conditions The results obtained by

each boundary condition are similar They have been compared

with the measured temperature at the plate centre Using

mea-sured surface temperature variation is less complex than the

heat transfer calculation; therefore it is more interesting to

use the surface temperature variation as the boundary condition

2.3 Residual stress field calculation

The temperature field in the first calculation is recorded and

used in the second calculation The geometry and meshing are the same as in the first calculation The procedure used

in the finite element program is based on an incremental approach This means that the total strain consists of elastic, plastic and thermal strains The basic equation to be used is

[11]

d␧t

ij¼d␧el

ij þd␧p

The total strain is strictly a function of geometry and it must satisfy compatibility The material is considered isotropic; therefore the plastic calculations are based on the classic plasticity theory (the von Mises criterion)

The hardening law is a non-linear isotropic hardening law which means that the yield stress varies as a function of the plastic strain:

j ¼ jelþKð␧p

Equation (6) defines the exact curve of stress as the function

of strain K and r depend on temperature; they are very low

at high temperatures All the mechanical and physical

prop-erties have been taken from previous literature [12–14].

2.4 Mechanical results

For mechanical analysis, the calculated temperature field is transferred The boundary conditions in this part will be of the geometrical type The plates and the cylinder explained above are modelled respectively as two-dimensional and axisymmetrical parts Figure 4 shows the directions of measurement in the plates and in the cylinder

The residual stress field is calculated as explained before The calculated field is compared with the experimental field Figures 8 and 9 show the residual stresses in the plate (thick-ness, 70 mm) and in the cylinder (diameter, 50 mm) In these two cases the measured residual stress field is obtained by

the layer removal method [15, 16] The results for the plate

Fig 5 Variation in the heat transfer coefficient as a function of

time

Fig 6 Measured temperature variation at different points of the

plate (thickness, 70 mm) quenched in water at 20⬚C

Fig 7 Measured and calculated temperatures using two different boundary conditions (BCs) at the plate centre (thickness,

70 mm)

and cylinder are given for only half the depth because of

symmetry of the parts The residual stress field obtained

for the plate of aluminium alloy 7175 (thickness, 24 mm)

is developed further

In the case of the plates the calculated residual stresses

along the X and Y directions are similar and therefore just

one of these stresses is presented The calculated residual

stress along the Z direction is zero In the case of the

cylin-der, the residual stresses along the three directions are

dif-ferent

With regard to the aluminium alloy 7175 plate (thickness,

24 mm) the quenching has been modelled As mentioned

before, in the case of the infinite plates the residual stresses

induced by quenching are similar along the X and Y

directions (Fig 10) (for the directions see Fig 4) In this

plate the residual stress field has been measured by the

incremental large hole drilling method and the neutron

diffraction method In the next section, the bases of these

two experimental methods have been developed

3.1 Neutron diffraction method

3.1.1 Principle

Neutron diffraction is a non-destructive technique enabling the in-depth residual stress to be evaluated, owing to the

penetration of most materials up to a depth z of several

cen-timetres by the neutron beam The principle of this method

is very similar to the well-known X-ray diffraction techni-que which is widely used to determine the surface residual stress

When a monochromatic neutron beam interacts with a crystalline material, incident neutrons are subject to diffrac-tion at the planes of atoms and produce strongly diffracted

beams leaving in directions defined by Bragg’s law [17]:

Assuming that l is constant, the differentiation of Bragg’s law (7) gives the following relationship:

␧i¼d i¹d0

d0 ¼ ¹

1 2

1 tan vð2vi¹2v0Þ ð8Þ Then, assuming that the principal directions are not very far from the natural coordinates of the specimen, the strain components measured by neutron diffraction are converted

to stress by the generalized Hooke’s law:

jj¼ E

1 þ n



␧jþ n

1 ¹ 2n

X

j

␧j



ð9Þ

3.1.2 Results

Neutron diffraction measurements were carried out in the

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Fig 8 Residual stress in the plate (thickness, 70 mm) quenched

in cold water (20⬚C)

Fig 9 Axial residual stress in the cylinder (diameter, 50 mm)

quenched in cold water (20⬚C)

Fig 10 Residual stress in the quenched plate (thickness, 24 mm)

obtained by the neutron diffraction method and the numerical method

diffractometer of residual stress and texture measurement

(REST) of the Studsvik Neutron Research Laboratory

(NFL) in Sweden Strain scans were made in the

longitudi-nal, transverse and normal directions (Y, X and Z directions

respectively) across the (24 mm) thickness of the sample

The (311) reflection of aluminium, with a 2 mm×2 mm×

20 mm gauge volume, was used for transverse and normal

measurements For longitudinal measurements, the gauge

height was reduced to 15 mm because of geometric

prob-lems The stress-free interplanar spacing d0was obtained

by studying three small samples cut out of the same

speci-men Young’s modulus E and Poisson’s ratio n were

calcu-lated for the (113) crystallographic orientation from

aluminium single-crystal constants using the Kro¨ner [18]

model They were 66 GPa and 0.357 respectively

The residual stress distribution is plotted in Fig 10 The

mid-plane located at a depth of 12 mm is the symmetry

plane The longitudinal (Y direction) and transverse (X

direction) stresses reach as high as 80 MPa in the

mid-thick-ness but are slightly lower in magnitude near both surfaces;

they become tensile at around 7 mm under each surface The

normal stress does not fluctuate very much and remains near

to a zero value To validate the quenching modelling, the

numerical results have been compared with the

experimen-tal results obtained from the neutron diffraction method in

Fig 10

3.2 Incremental large hole drilling method

3.2.1 Principle

The classic incremental hole drilling method is

semidestruc-tive [19] It consists in drilling a small hole (diameter, from

1 to 5 mm) in the sample and at each depth measuring the

strain in the hole plane The hole diameter is chosen

accord-ing to the part thickness and the residual stress gradient

Generally the hole can be drilled to a depth of 50 per cent

of the final hole diameter to measure the residual stress

dis-tribution The greater the hole diameter, the further one can

drill into the part In the quenching case the residual stress is

distributed over the depth of the whole part, which means

that there is high compressive residual stress at the part

sur-face and a very high tensile residual stress in the centre of

the part; therefore a large drilling diameter is necessary

The large hole drilling method is carried out in two faces

of the aluminium alloy 7175 plate (thickness, 24 mm)

Using materials equilibrium laws before and after

remov-ing a layer and if just the layer s is considered, the reaction

stresses at the part surface in the zone where gauges are

placed after hole drilling can be obtained from the following

equations:

jrs¼C 1sðj1ksþj2ksÞ

C 2sðj1ks¹j2ksÞ

2 cos ð2asÞ ð10Þ

jts

C 3sðj1ksþj2ksÞ

C 4sðj1ks¹j2ksÞ

2 cosð2asÞ ð11Þ

trts¼C 5s sinð2asÞ



j2ks¹j1ks

2



ð12Þ

where C 1s , C 2s , C 3s , C 4s and C5s are the constants which

depend on the gauge positions, hole diameters, layer s

loca-tions and the total hole depths:

␧rs¼1

Equation (14) is obtained from equations (10) to (13):

␧rsðasÞ ¼A snðj1ksþj2ksÞ þB snðj1ks¹j2ksÞcosð2asÞ

ð14Þ

The A sn and B sn coefficients are called calibration coeffi-cients and they depend on the geometry of the hole diameter

gauges, the location of layer s and the hole depth.

These coefficients are calculated by numerical methods

based on the finite element method [20] The radial strains

are measured by gauges; therefore as, j1ksand j2ks can be calculated

3.2.2 Results

In the quenched plate case, the part thickness is about

24 mm The chosen diameter is about 10 mm The hole

posi-tion is in the XY plane (Fig 4) as far as possible from the part

edges because the plate has been obtained from an infinite plate and, when cutting the original plate, the residual stres-ses were relaxed near the edges of the obtained part For the chosen diameter there is no existing rosette As is known each classic rosette is made of three gauges In this case, six gauges are placed around the drilled hole at a distance equal to approximately the hole diameter from the hole cen-tre The angle between two gauges is about 45⬚ Each rosette

uses three gauges; thus from these six gauges it is possible to form different rosettes which are similar by simply changing the orientation In this way the strains which relax during cutting can be measured at more points on the part surface and the uniformity verified for the residual stresses calcu-lated from the measured strains at each depth The part has been drilled up to 5 mm To obtain more information about the residual stress level, another hole was drilled at exactly the same place as the first but on the other part

face (parallel to the XY plane) As regards the hole depth,

it was possible to go deeper but the chosen diameter was quite large and, the more the part is drilled, the more the gauge sensitivity decreases and the more difficult it is to detect the strains Our chosen hole diameter is larger than the usual hole diameters The finite element calculation method for the calibration coefficients which are required

to obtain the residual stress field from the measured strains

is the technique generally applied for small holes In the case studied, we applied the calculation method to a hole diameter of 10 mm Figure 11 shows the measured residual stress using the large incremental hole drilling method

and the neutron diffraction method The Y and X residual

stresses are obtained; the difference between them is not

very great With the incremental hole drilling technique,

the normal (Z direction) residual stress cannot be measured

but in the case of our sample geometry this is not important

because the normal residual stress is nearly zero As was

expected, there is compressive residual stress at the surface

and it is about 70 MPa

ON THE LEVEL OF RESIDUAL STRESS

After the validation of our model, the effects of the

quench-ing parameters were studied The level of residual stress

changes with different quenching parameters These

para-meters are generally defined by the quenchant, the

quench-ing temperature and the quenched zones (with controlled

cooling methods in quenching)

The residual stress field due to quenching has never been

integrated in a fatigue life calculation For a given fatigue

life, it is possible to define the necessary residual stress field

[21] Thus it can be interesting to change quenching

condi-tions so as to obtain the residual stress field required for

improved fatigue life Figure 12 shows the influence of

the quenching temperature on the residual stress level A

low quenching temperature introduces a high residual stress

into the part, and a high quenching temperature introduces a

low residual stress and a lower distortion Generally the

quenchant and the quenching temperature influence the

cool-ing speed; therefore, to vary the residual stress level, both the

quenchant and the quenching temperature can be varied

The compressive residual stress is known to increase the

fatigue life From the fatigue life calculation it is possible to

determine the part of the sample in which the compressive

residual stress is required so as to define the quenched

zone as a function of the fatigue life [21].

In Fig 11 the residual stress field obtained by the modified incremental hole drilling method and the neutron diffraction method are compared In this figure the results are shown to

be not very different The existing difference is not very great considering the errors introduced by the measurement techniques Estimated errors are⫾20 MPa for the

incremen-tal hole drilling method and⫾10 MPa for the neutron

dif-fraction method The level of the measured residual stresses is not very high Considering the errors of each method and the level of the measured residual stress, the results of each method seem to be acceptable In Fig 10 the calculated residual stress is compared with the experi-mental data The part has been quenched in water at

65⬚C Considering the quenching temperature and the plate

thickness (24 mm), the induced residual stress is not very high The maximum compressive stress and the maximum tensile stress obtained by calculation are about 75 and

55 MPa respectively These maxima are very similar to experimental values The only difference between them is that the calculated value changes sign (compressive to ten-sile) at a lower depth than the experimental value does It is necessary to mention that, the higher the level of the induced residual stress, the more accurately the residual stress can be calculated (Figs 8 and 9) This may be due to mechanical data such as the yield stress, which in the calculation is sup-posed to be temperature dependent Yield stress measure-ments at different temperatures are not very accurate, and thus errors can be introduced in the calculation Another possible source of error is the residual stress measurements Globally the maximum tensile and compressive stresses have been predicted correctly The differences obtained between numerical and experimental results are similar to

the differences obtained in the previous studies [22] For

high residual stress fields even the distribution throughout the thickness is correct, whereas for low residual stress

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Fig 11 Residual stress in the quenched plate (thickness, 24 mm)

obtained by the neutron diffraction method and the

incre-mental large hole drilling method

Fig 12 Residual stress in the plate (thickness, 70 mm) quenched

in cold (20⬚C) and hot (80 ⬚C) water

fields the calculation predicts fewer thickness effects from

the compressive residual stress As this calculated residual

stress field is required in a fatigue life calculation, a smaller

depth for the compressive residual stress does not create a

problem because the estimated fatigue life will be shorter

than the real value, thus giving greater safety

In this study, the hybrid approach of numerical and

experi-mental techniques is developed for a residual stress field

study of quenched parts This is the first procedure in the

global approach for residual stress integration in fatigue

life prediction Quenching has been modelled using the

finite element method Both thermal and mechanical data

are necessary for this modelling The most important

thermal parameter is the heat transfer coefficient which

enables the boundary conditions in the thermal problem to

be defined This coefficient is obtained from an

experi-mental temperature field The heat transfer coefficient is

obtained by inverse resolution of the heat conduction

equa-tion Different numerical methods can be applied to

deter-mine this coefficient but all of them need the experimental

temperature fields To reduce the difficulty at this point,

instead of using the heat transfer coefficient to define the

boundary conditions, the measured temperature as near as

possible to the part surface has been used From these two

different boundary conditions the same temperature field

is obtained Thus, in the case of materials and quenchants

with a high conductivity, the temperature measured exactly

at the part surface can be used as the boundary condition in

the thermal problem It is true that using this method can

introduce errors into the calculation but these errors are

small and they are less than the errors obtained from heat

transfer coefficient calculation

The calculated residual stress field has been compared

with the measured residual stress field The numerical

resi-dual stress field is close to the experimental value; therefore

the quenching model has been validated Using the same

model, the quenching has been modelled for different

quenching temperatures The lower the quenching

tempera-ture, the higher is the residual stress obtained

The measurement techniques used were the neutron

dif-fraction method and the incremental large hole drilling

method The incremental large hole drilling method is an

extension of the classic incremental hole drilling method

This technique enables more rapid measurement of the

residual stress at a greater depth to be made The residual

stress obtained by this method has been compared with

the residual stress field obtained by the neutron diffraction

method The residual stress levels in these two cases are

close considering the errors due to each technique; therefore

the incremental large hole drilling method can be taken as

valid With this modified technique it is possible to measure

the through-thickness residual stress field induced by heat

treatments or surface treatments of different types of alloy

The next stage of this study is to integrate the residual stress field due to quenching in a fatigue life calculation Before this, calculation of the relaxation of residual stress has to be taken into account

ACKNOWLEDGEMENTS

The authors are grateful to the Studsvik Neutron Research Laboratory for their help in the measurement of residual stress by neutron diffraction method The authors are also grateful to Mr G Houset and Mr A Voinier at the Univer-site´ de Technologie de Troyes for their technical help The authors are also grateful to ‘Pole de Mode´lisation’ for its financial support

REFERENCES

1 Lu, J., Flavenot, J F and Lieurade, H P Inte´gration de la

notion des contraintes re´siduelles dans les bureaux d’e´tudes, les contraintes re´siduelles au bureau d’e´tudes CETIM, Senlis, France, 1991, pp 9–34

2 Lu, J., Flavenot, J F and Turbat, A Prediction of residual

stress relaxation during fatigue In Mechanical Relaxation of

Residual Stresses, ASTM special technical publication 993

(Ed L Mordfin), 1988, pp 75–90 (American Society for Test-ing and Materials, Philadelphia, Pennsylvania)

3 Rasouli Yazdi, S and Lu, J Simulation of quenching and

fatigue relaxation of residual stresses in aluminum parts In

Proceedings of the Fifth International Conference on Residual

Stresses (Eds T Ericsson, M Ode´n and A Andersson),

Insti-tute of Technology, Linko¨ping Universitet, Linko¨ping, Sweden, 1997, pp 490–495

4 Akrache, R and Lu, J Fatigue life prediction for 3D

struc-tures In Proceedings of the Fifth International Conference

on Computational Plasticity (Eds D R J Owen, E Onate

and E Hinton), Barcelona, 1997, pp 1021–1026 (CIMNE, Barcelona, Spain)

5 Lemaitre, L and Chaboche, J.L Me´canique des Mate´riaux

Solides, 1984 (Dunod, Paris).

6 Landau, L and Lifchitz, E Physique The´orique, Vol 6, 2nd

edition, 1989, pp 273–334 (Mir, Moscow)

7 Fletcher, A J Thermal Stress and Strain Generation in Heat

Treatment, 1989 (Elsevier Applied Science, Barking, Essex,

and The Universities Press, Belfast)

8 ABAQUS Theory Manual, 1996 (Hibitt, Karlsson and

Sorensen, Incorporated, USA)

9 Price, R.F and Fletcher, A J Generation of thermal stress

and strain during quenching of low alloy steel plates Metals

Technol (Lond.), 1981, 8, 427–446.

10 Fletcher, A J and Nasseri, M Effect of plate orientation on

quenching characteristics Mater Sci Technol., 1995, 11,

375–381

11 Beck, G and Ericsson, T Prediction of Residual Stresses due

to Heat Treatment, 1987, pp 27–40 (Deutsche Gesellschaft

fu¨r Metallkunde Informationsgesellschaft mbH, Oberursel)

12 Hatch, J E Aluminum—Properties and Physical Metallurgy,

1984 (American Society for Metals, Metals Park, Ohio)

13 Ledbetter, H M Temperature behaviour of Young’s moduli

of forty engineering alloys In Materials Studies for Magnetic

Fusion Energy Applications at Low Temperature—IV, 1981,

pp 257–269

14 Takeuti, Y G., Komori, S., Noda, N and Nyuko, H

Ther-mal stress problems in industry 3: Temperature dependency

of elastic moduli for several metals at temperatures from

–196 to 1000⬚C J Thermal Stresses, 1979, 2, 233–250.

15 Jeanmart, P H and Bouvaist, J Finite element calculation

and measurement of thermal stresses in high strength

alumi-num alloys In Advances in Surface Treatments Technology

Applications—Effects, Vol 4, Residual Stresses, 1987, pp.

327–340 (Pergamon, Oxford)

16 Habachou, R Mode´lisation de la trempe et du

de´tensionne-ment par de´formation plastique des barres et plaques

d’al-liages d’aluminium The`se de doctorat, Institut National des

Sciences Applique´es de Lyon, 1983

17 Holden, T M and Roy, G The application of neutron diffraction

to the measurement of residual stress and strain In Handbook of

Measurement of Residual Stresses (Ed J Lu), 1996, pp 133–

148 (Society for Experimental Mechanics, New York) (Fairmont Press and Prentice-Hall, Englewood Cliffs, New Jersey)

18 Kro¨ner, E Berechnung der elastischen Konstanten des

Vielk-ristalls aus den Konstanten des einkVielk-ristalls Z Physik, 1958,

151, 504–505.

19 Schajer, G S., Roy, G., Flaman, M T and Lu, J Hole

dril-ling and ring core methods In Handbook of Measurement of

Residual Stresses (Ed J Lu) 1996, pp 5–34 (Society for

Experimental Mechanics, New York) (Fairmont Press and Prentice-Hall, Englewood Cliffs, New Jersey)

20 Lu, J De´veloppement de la me´thode de mesure de contraintes

re´siduelles par le perc¸age pas a` pas The`se de doctorat, Univer-site´ de Technologie de Compie`gne, 1986

21 Akrache, R Pre´vision de la dure´e de vie en fatigue des

struc-tures 3D par la me´thode des e´le´ments finis The`se de doctorat, Universite´ de Technologie de Compie`gne, 1998

22 Becker, R., Karabin, M E., Liu, J C and Smelser, R E.

Distortion and residual stress in quenched aluminum bars J.

Appl Mechanics, 63, 1996, pp 699–705.

S00598 䉷 IMechE 1998

JOURNAL OF STRAIN ANALYSIS VOL 33 NO 6