449 Study of through-thickness residual stress by numerical and experimental techniques S Rasouli Yazdi, D Retraint and JLu 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 (m 2 ) 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 ЊC) d i , d 0 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/m 2 ЊC) k conductivity matrix (W/m ЊC) K coefficient of the Ramberg–Osgood law (MPa) m mass (kg) p number of time steps q heat flux (W/m 2 ) r coefficient of the Ramberg–Osgood law t time (s) Dt time interval (s) T temperature (ЊC) T 0 fluid temperature (ЊC) x position from the centre of the plate (m) a angle between the gauge and the principal direction 1 (deg) d ij 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 ij plastic strain increment d␧ t ij total strain increment d␧ Th thermal strain related to the temperature incre- ment by the thermal expansion coefficient v 1 , v 0 Bragg angle (deg) l monochromaticwavelengthofincidentneutrons(m) n Poisson’s ratio r density (kg/m 3 ) j stress component (MPa) j el yield stress (MPa) j 1K , j 2K principal stresses (MPa) j r , j t radial and tangential stresses respectively (MPa) t rt shearing stress (MPa) Subscripts BC boundary condition E experimental S00598 ᭧ IMechE 1998 JOURNAL OF STRAIN ANALYSIS VOL 33 NO 6 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 a ambient TC thermocouple position 1 INTRODUCTION 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 castings, 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. 2 NUMERICAL MODEL DESCRIPTION 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 S00598 ᭧ IMechE 1998JOURNAL OF STRAIN ANALYSIS VOL 33 NO 6 Fig. 1 Residual stress integration in the fatigue life calculation 450 S RASOULI YAZDI, D RETRAINT AND J LU 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]: ¹r ˙ e ¹ div q ¼ 0 ð1Þ Heat conduction is assumed to be governed by Fourier’s law [6]: q ¼ k: grad T ð2Þ 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 ∂t ¼ div½k gradðTÞÿ ð3Þ 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 T 0 Þ. 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 DtA ln T a ¹ðT TC n Þ E T a ¹ðT TC nþp Þ E ! ð4Þ In equation (4), the time and the position appear, although S00598 ᭧ IMechE 1998 JOURNAL OF STRAIN ANALYSIS VOL 33 NO 6 Fig. 2 Residual stress relaxation in a quenched cylinder Fig. 3 Modelling diagram 451STUDY OF RESIDUAL STRESS BY NUMERICAL AND EXPERIMENTAL TECHNIQUES 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 (126mm (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 ЊC. The aluminium alloy 7075 parts were quenched in cold water (20 ЊC) and the aluminium alloy 7175 part was quenched in water at 65 ЊC. 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 0mm, x ¼ 17:5 mm, x ¼ 26mm and x ¼ 34mm 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 S00598 ᭧ IMechE 1998JOURNAL OF STRAIN ANALYSIS VOL 33 NO 6 Fig. 4 Geometry and measurement directions in the parts studied 452 S RASOULI YAZDI, D RETRAINT AND J LU 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 ij ¼ d␧ el ij þ d␧ p ij þ d ij d␧ Th ð5Þ 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 j el þ Kð␧ p Þ r ð6Þ 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 S00598 ᭧ IMechE 1998 JOURNAL OF STRAIN ANALYSIS VOL 33 NO 6 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 ЊC Fig. 7 Measured and calculated temperatures using two different boundary conditions (BCs) at the plate centre (thickness, 70 mm) 453STUDY OF RESIDUAL STRESS BY NUMERICAL AND EXPERIMENTAL TECHNIQUES 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 EXPERIMENTAL RESULTS 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]: l ¼ 2d sinv ð7Þ Assuming that l is constant, the differentiation of Bragg’s law (7) gives the following relationship: ␧ i ¼ d i ¹ d 0 d 0 ¼¹ 1 2 1 tan v ð2v i ¹ 2v 0 Þð8Þ 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: j j ¼ E 1 þ n  ␧ j þ n 1 ¹ 2n  j ␧ j  ð9Þ 3.1.2 Results Neutron diffraction measurements were carried out in the S00598 ᭧ IMechE 1998JOURNAL OF STRAIN ANALYSIS VOL 33 NO 6 Fig. 8 Residual stress in the plate (thickness, 70 mm) quenched in cold water (20 ЊC) Fig. 9 Axial residual stress in the cylinder (diameter, 50 mm) quenched in cold water (20ЊC) Fig. 10 Residual stress in the quenched plate (thickness, 24 mm) obtained by the neutron diffraction method and the numerical method 454 S RASOULI YAZDI, D RETRAINT AND J LU 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 × 2mm × 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 d 0 was 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 7mm 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: j rs ¼ C 1s ðj 1ks þ j 2ks Þ 2 þ C 2s ðj 1ks ¹ j 2ks Þ 2 cos ð2a s Þð10Þ j ts C 3s ðj 1ks þ j 2ks Þ 2 ¹ C 4s ðj 1ks ¹ j 2ks Þ 2 cosð2a s Þð11Þ t rts ¼ C 5s sinð2a s Þ  j 2ks ¹ j 1ks 2  ð12Þ where C 1s , C 2s , C 3s , C 4s and C 5s are the constants which depend on the gauge positions, hole diameters, layer s loca- tions and the total hole depths: ␧ rs ¼ 1 E ðj rs ¹ n ¬ j ts Þð13Þ Equation (14) is obtained from equations (10) to (13): ␧ rs ða s Þ¼A sn ðj 1ks þ j 2ks ÞþB sn ðj 1ks ¹ j 2ks Þ cosð2a s Þ 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 a s , j 1ks and j 2ks 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 45Њ. 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 5mm. 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 S00598 ᭧ IMechE 1998 JOURNAL OF STRAIN ANALYSIS VOL 33 NO 6 455STUDY OF RESIDUAL STRESS BY NUMERICAL AND EXPERIMENTAL TECHNIQUES 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. 4 INFLUENCE OF THE QUENCH PARAMETERS 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 thequenching temperature influencethecool- ing speed; therefore, to varythe 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]. 5 DISCUSSION 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 Ϯ20 MPa for the incremen- tal hole drilling method and Ϯ10 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 ЊC. 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 S00598 ᭧ IMechE 1998JOURNAL OF STRAIN ANALYSIS VOL 33 NO 6 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 ЊC) and hot (80ЊC) water 456 S RASOULI YAZDI, D RETRAINT AND J LU 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. 6 CONCLUSIONS 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. 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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 ᭧ IMechE 1998JOURNAL OF STRAIN ANALYSIS VOL 33 NO 6 458 S RASOULI YAZDI, D RETRAINT AND J LU . 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, . 449 Study of through-thickness residual stress by numerical and experimental techniques S Rasouli Yazdi, D Retraint and JLu Lasmis (Mechanical Systems and Concurrent Engineering. Measured and calculated temperatures using two different boundary conditions (BCs) at the plate centre (thickness, 70 mm) 45 3STUDY OF RESIDUAL STRESS BY NUMERICAL AND EXPERIMENTAL TECHNIQUES and cylinder