Bauschinger effect in age-hardened Inconel X-750 alloy

Bauschinger effect in age-hardened Inconel X-750 alloy

Materials Science and Engineering A311 (2001) 100– 107 www.elsevier.com/locate/msea Bauschinger effect in age-hardened Inconel X-750 alloy J.A. del V...

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Materials Science and Engineering A311 (2001) 100– 107 www.elsevier.com/locate/msea

Bauschinger effect in age-hardened Inconel X-750 alloy J.A. del Valle a,b,*, R. Romero a,c, A.C. Picasso a,c a

Instituto de Fı´sica de Materiales Tandil, Uni6ersidad Nacional del Centro de la Pro6incia de Buenos Aires, Pinto 399, Tandil, Pro6incia de Buenos Aires, Argentina b Consejo Nacional de In6estigaciones Cientı´ficas y Te´cnicas (CONICET), Argentina c Comisio´n de In6estigaciones Cientı´ficas de la Pro6incia de Buenos Aires (CICPBA), Argentina Received 14 July 2000; received in revised form 18 December 2000

Abstract The influence of the g%-precipitates on the Bauschinger effect, in the Inconel X-750 Nickel-base superalloy, has been investigated for: solution heat treated, underaged, and overaged samples. It is found that the internal stresses increase with the aging time, providing evidence on the appearance of unshearable particles in the microstructure, the results are compared with the yield stress data (del Valle, et al., Scripta Mater., 1999, 41, 237). In the overaged samples, the dependence of the internal stresses on strain behaves qualitatively in accordance with the theories of relaxed deformation in dispersion hardened metals; however, their magnitude is smaller than the predicted, producing a contribution to work hardening of about 15%. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Bauschinger effect; Back stress; Nickel alloys; Aging treatment

1. Introduction In a previous study [1] the age-hardening behavior of the Inconel X-750 superalloy has been investigated, it was shown that order hardening from the g%-precipitates produces the main contribution to strengthening. Depending on the mean radius of the g%-precipitates (r) the plastic deformation occurs by shear of the precipitates, r B 15 nm, while in samples beyond approximately r= 40 nm the deformation mechanism involves the Orowan looping between unshearable precipitates. In the intermediate region, both mechanisms should be operative in a certain amount. In the present work we have studied the influence of these precipitates on the Bauschinger effect, on both underaged and overaged samples. To the authors’ knowledge, the available experimental data on the Bauschinger effect in engineering superalloys appear to be scarce. Most investigations have been reported on dispersion hardened metals where it is expected that, under monotonic tensile deformation, * Corresponding author. Tel.: + 54-2293442821; fax: + 5402293444190. E-mail address: [email protected] (J.A. del Valle).

the Orowan loops accumulated around the precipitates give rise to an increase in the flow stress partly due to the development of long range internal stresses into the material. Moreover, it is well known that the mean internal stress opposes the forward deformation, but aids deformation in the reverse sense, leading to a significant Bauschinger effect. Successively, after the plastic strain reaches values of the order of percent, the deformation becomes relaxed with a consequent reduction in the rate of internal stress accumulation. In this situation, the work hardening is supposed to be mainly given by the forest dislocation contribution, plus a minor contribution of the internal stress. Accordingly, in Ref. [2] we have analyzed the Stage II of Work Hardening (WH) in the Inconel X-750 alloy, on a basis of theoretical models considering full plastic relaxation conditions, that is neglecting the internal stresses. In this study, the dependence of the Bauschinger effect on prestrain and g%-precipitate size has been analyzed, and the results compared with the existing models. Moreover, a qualitative description of the relaxation mechanism is given. Therefore, these results allow us to estimate the contribution of the internal stresses to the flow stress. It is shown that to consider work hardening models based on full plastic

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relaxation, in the age-hardened Inconel X-750 alloy, can be a reasonable first approximation. Another important aspect is that, in dispersion hardened alloys, the internal stress is expected to be directly related to the volume fraction of the unshearable precipitates. Consequently, the measurements of the Bauschinger effect give rise to a method for detecting the appearance of unshearable particles in the microstructure and, in this study, they have been used as a tool to investigate the transition of particle shearing to looping as the precipitates grow with the aging time.

other features of the second phase such as the particle strength and misfit strains [8–10]. The back stress acts on the glide dislocations in linear addition to the isotropical friction stress, tfr. Therefore, it is generally considered that the forward flow stress, tf, could be written as: ~f = ~fr + ~b

(4)

On the other hand, when the sense of the deformation is reversed the flow stress is expected to occur at the stress tr, given by: ~r = ~fr − ~b

1.1. Internal stresses on dispersion hardened alloys

101

(5)

Hence As mentioned previously, when dispersion hardened alloys are deformed, the mean internal stress in the matrix, called the back stress, opposes the forward flow but aids the reverse flow; in the Brown and Stobbs [3] uniform shear model the back stress on the active slip plane is given by: ~b = 2kGfDmsp

(1)

where g is the Eshelby accommodation factor, G is the shear modulus, f is the volume fraction of unshearable particles, D is the modulus correction factor, and osp is the symmetrical plastic shear strain (osp is half the engineering resolved shear strain). As deformation proceeds, the local stress increases around the particles and, after plastic strains of the order of percent, some of several possible relaxation processes can occur. In this case, it is generally considered that the back stress is given by an equation of the same form as Eq. (1), where the shear strain, osp, is replaced [4] by the unrelaxed shear strain, o* sp: ~b = 2kGfDm*sp

(2)

Any attempt for obtaining a functional relationship between o*sp and osp, should be based on the knowledge of the structures of dislocations, as well as of the relaxation mechanisms that originate them. In most studies carried out in single crystals [5– 8], the dislocation structures have been roughly classified into two groups: ’A’ structures, formed by a double cross-slip process of the screw parts of the Orowan loops, and ’B’ structures, formed by slip in secondary systems. In the latter case Brown and Stobbs [9] obtained an expression for o*sp, related to osp by: m*sp :0.7

  bmsp r

~b = (~f − ~r )/2 =D~b /2

(6)

Therefore, in an ideal case, the back stress results in a constant difference, Dtb, called the permanent softening, between the forward and the reverse flow stress curves, if they are plotted in terms of the absolute stress against the absolute cumulative strain (Wilson’s construction [11]). Actually, the permanent softening is gradually approached through a gradual yielding or ‘roundedness’, called the transient softening, of the reverse curve. In dispersion-hardened systems, a correlation between permanent and transient softening has been observed. Atkinson et al. [10] have found that, for small strain reversals, the transient softening can be described by the phenomenological relationship: ~r /~f = im 1/2 sr

(7)

where osr is the reverse symmetrical shear strain, and the parameter b characterizes the transient softening. Moreover, they have found that b is related to the permanent softening by a reciprocal relationship: Dtb 8 G/b. The transient softening has been attributed by Orowan [12] to the statistics of the obstacle sampling by the dislocations, in other words, the dislocations moving in the forward direction see an array of obstacles harder than the one they see immediately when their motion is reversed. Then, in the relaxed case, it has been suggested that these obstacles could be the plastic zones around the particles; in this way the permanent and transient softenings would be increasing functions of the unrelaxed deformation.

1/2

(3)

where b is the Burgers vector. It must be emphasized, that the above descriptions do not include the influence of multiple slip and the second phase volume fraction, both factors are expected to change the geometry of the plastic relaxation, but also the stacking fault energy, lattice friction, and

2. Experimental The Inconel X-750 alloy (INCO trademark) used in this study was supplied in the form of 12.7 mm diameter bar, its previous heat treatment is described by AMS 5667. The chemical composition is shown in Table 1. From the received material cylindrical test

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Table 1 Chemical composition of the studied alloy (wt.%) Ni

Cr

Fe

Ti

Al

Ta+Nb

Co

Mn

Si

S

Cu

C

Bal

15.4

6.3

2.55

0.77

1.03

0.04

0.03

0.08

B0.001

0.02

0.04

pieces were machined, the samples had a gauge section of 12 mm, 6 mm diameter, and threaded ends. These samples were solid solution heat treated at 1363 K for 3 h and water quenched. To get different g%-precipitate sizes, the samples were aged at 1010 K for different times, the mean precipitate radius was calculated from the law of coarsening published in another work [1], these were r=10, 26, 35, and 53 nm, the volume fraction of the g%-phase was estimated to be f = 0.15, and remains constant during the aging treatment. Moreover, tests on solution heat treated and quenched samples were performed to be compared with those on the aged alloy. The average grain size of the solution treated alloy was d :100 mm, and did not suffer any noticeable modifications during the subsequent aging treatment. As depicted in Fig. 1, the optical metallography shows that there is an MC carbide precipitation inside the grains and M23C6 carbides decorating the grain boundaries, these features of microstructure are well known in the literature [13]. The volume fraction of the carbides inside the grains was measured from the optical micrographs after the solution treatment, yielding the value 0.015 approximately, and there were no noticeable changes with the aging treatment. The tension-compression tests were performed in a Shimadzu testing machine at room temperature and at a constant crosshead speed, the strain rate was 7× 10 − 4 s − 1, the alignment of this machine was sufficiently precise so that the samples could be compressed to a true strain of at least o= 0.05 without noticeable differences (related to buckling) when compared with tensile curves. Test was normally started in tension to prestrains up to 10%, some tests carried out using compressive prestrains revealed no differences in the measured permanent softening, within the experimental scatter. Strain measurements were made with an extensometer attached to the ends of the gauge section of the sample. The load-elongation (plus time) curve was measured using a data acquisition board, subsequently the data were converted into true stress– strain curves, the plastic strain, o, was obtained by means of o =ot − s/E, where ot is the total true strain and E is the Young modulus

3. Results Fig. 2 shows some typical Bauschinger curves for r= 10, 53 nm and solid solution. The curves exhibit a transient softening, given by a gradual yielding, which starts in the unloading curve and continues during the compression curve, this characteristic in the shape of

Fig. 1. Optical micrograph, showing Carbides in the solution heat treated Inconel X-750 alloy.

Fig. 2. Stress – strain curves for a sample solution heat treated, and two aged samples with mean g%-precipitate radii of r =10 and 53 nm.

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corresponding to a point in Fig. 4 for the same precipitate size. It can be noted that sf − sr − Dsb behaves qualitatively such as an exponential decay, which decay constant decreases as the prestrain (that is the permanent softening) increases. Therefore, we have found that from or : 0.2% up to the permanent softening regime, the transient softening can be empirically described by the following relation: |f − |r − D|b = Aexp(− umr )

Fig. 3. A diagram showing the Wilson’s construction for samples aged to r= 35 nm: the upper curve is the tensile reference curve, below the reflections of the compressive parts of the tests. Relevant parameters are also shown.

Fig. 4. Measured values of sb as a function of the plastic prestrain oF, for the alloy in solid solution and aged up to a g%-precipitate radii of r= 10, 26, 35, and 53 nm. In solid lines, the fit of Eq. (11) to the solid solution data and Eq. (13) to r= 53 nm. In dashed line the fit of Eq. (13) to the data of r = 10 nm.

the transition in the forward-reverse curves occurs in all cases, even in the solid solution samples. In Fig. 3 the true stress against true cumulative plastic strain curves, for r= 35 nm, are shown, and the parameters used to describe the Bauschinger experiment are defined. As can be seen the curves become quite parallel, the permanent softening, Dsb, was measured as the minimum difference between the curves in the reverse deformation and the continued forward deformation curve. The values of sb =Dsb/2 obtained are shown in Fig. 4, plotted against the forward plastic prestrain, oF, for the solid solution and four mean g%-precipitate sizes. Fig. 5 shows plots of sf −sr −Dsb as a function of the reverse plastic strain, or, for r =35 nm, each curve

(8)

where A and l are adjustable parameters. The fit of Eq. (8) to the experimental data gives the parameters A and l, which characterize the transient softening. The parameter A remains equal to : 4/3(sF − Dsb) in all cases and, as illustrated in Fig. 6, l has a reciprocal dependence on the permanent softening.

Fig. 5. A plot showing the gradual approach to the permanent softening, it can be noted that the magnitude of the reverse strain involved in the transient softening is an increasing function of the plastic prestrain.

Fig. 6. Measured values of the permanent softening as a reciprocal function of the parameter l, which characterizes the transient softening. The symbols are correspondent with those of Fig. 4.

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4. Discussion

4.1. The Bauschinger effect in o6eraged samples According to Fig. 4, the measurements of the back stresses suggest a quite parabolic dependence on the plastic prestrain, such as the Brown and Stobbs (BS) model predicts in the case of relaxed deformation [5]. Combining Eq. (2) and Eq. (3) the back stress arising from the g%-precipitates is predicted to be given by: ~ gb´ =1.4kGfD

  bmsp r

1/2

(9)

In an attempt to link the macroscopic curve, sb(oF), with the single crystal predicted curve, tb(osp), of Eq. (9) we use the Taylor effective orientation factor, M= 3.06: tb =sb/M, 2osp =MoF (more properly the product Mo is G, the algebraic sum of shears). Thus, Eq. (9) was rewritten as: | gb´ = 5.3kGfD

  bm F r

1/2

$CfG

  bm F r

1/2

(10)

where C$ 2 is a constant calculated with the factors g = (7− 5n)/20(1 −n) =0.41, for spheres in a matrix deformed with multiple slip orientations, and D= G g%/ [G g% − g(G g% − G)] $1, taking into account that the shear moduli of the g%-precipitate and matrix are nearly identical in superalloys. An estimate of G and n of Inconel X-750 can be obtained by assuming they are the same as for pure Nickel: G =63.4 GPa and n = 0.373 corresponding to {111} slip planes [14]. Another difficulty in the data analysis is that the back stress in the samples in solid solution is not negligible when these are compared with the back stress in the aged samples. In these samples it was empirically found that sss b could be characterized through the relationship: −3 | ss G(m F)0.44 b =1.7× 10

(11)

The origin of the permanent softening in the solid solution treated samples is probably owing to two causes: Firstly, it is expected that some heterogeneity in the forest dislocation substructure, as it have been reported by Pedersen et al. [15] and Christodoulou et al. [16] in pure copper, gives rise to the development of back stress, however the magnitude of the measured −4 back stress, sss (at oF :0.1) is much b /G  7 ×10 −5 higher than the value 3 × 10 estimated from the Pedersen et al. data. Secondly, and more possible, the back stress should be given by the influence of the carbides which volume fraction is quite considerable. It can be noted that the exponent of Eq. (11) is near 0.5, which is expected in particle hardened alloys. According to our observations there are no noticeable changes in the volume fraction of the carbides with the aging treatment, and it can be expected that these

particles also produce back stresses in the g%-precipitate hardened alloy. Then, if we assume that these two sources of back stresses act in an independent way in the aged samples, we can add both contributions, sg% b and sss b , obtaining: g´ |b = | ss b +|b

(12)

and inserting Eq. (10) and Eq. (11) into Eq. (12), the back stress should be given by: |b = 1.7× 10 − 3G(m F)0.44 + CfG

  bm F r

1/2

(13)

Therefore, Eq. (13) was used to fit the experimental data in Fig. 3, for overaged samples with mean g%-precipitate radii of r= 35 and 53 nm, using C as an adjustable parameter. In this case, we assume that the majority of the g%-precipitates behaves as unshearable particles, then the volume fraction of Eq. (13) takes the value f= 0.15. For samples with precipitate radius of r= 35 and 53 nm we obtained C= 0.18 and 0.20 respectively. Clearly, these values are one order of magnitude lower than the predicted by the BS model. In summary, it seems that the BS model describes well the dependence of the back stress with the forward strain but underestimates the plastic relaxation in this alloy. From Fig. 4, it seems that the back stress has a maximum for the points corresponding to r=26 nm; the points corresponding to r= 35 and 53 nm appear to be below those of r=26 nm. However, we recognize that the scattering of the experimental data precludes to confirm the dependence of the unrelaxed strain on the precipitate radius, given by Eq. (3). If the back stress were independent on the precipitate radius, the volume fraction of unshearable precipitates would be the most important microstructural parameter. The plastic relaxation in this alloy could be increased by factors such as shearing of particles, interaction of plastic zones of neighborhood particles, the misfit strains of the g%-precipitates [1,17], and multiple slip aided relaxation [18]. For instance, it is well known that dislocation loops around the g%-precipitates can shrink, shearing the particle, if the selfstress of the loop plus the applied shear stress exceed the stress required to create an antiphase boundary inside the particle. The previous study of the critical resolved shear stress (CRSS) [1] with the order hardening theories leads to values of the antiphase boundary energy within 0.2B gapb B 0.27 J/m2. Taking this into account, we have analyzed the stability of Orowan loops, and found that the g%-precipitates probably can not support two coplanar loops. Fig. 7 shows a TEM micrograph of an Inconel X-750 tested sample, previously aged to r=24 nm; as can be seen, Orowan loops are formed around the precipitates, and sometimes around two neighbor precipitates, the smallest Orowan loops have a radius of

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particle, n0, by: m*sp =

n0b 4r

(14)

From Fig. 8, the slip plane spacing is about 100b, then for particles of radius r= 53 nm the number of Orowan loops is likely to be n0 : 3. Therefore, it is expected that these loops generate a back stress of the order of sb : M2gGfD(n0b/4r)=80 MPa, which is in good agreement with the experimental results.

4.2. Transition from the cutting process to the Orowan process

Fig. 7. TEM bright field micrograph, showing Orowan loops around the g%-particles in a sample aged to r = 24 nm and deformed.

The results of Fig. 4 suggest that the permanent softening presents a transition between the samples in solid solution and the overaged samples which can be attributable to the transition among the mechanisms of cutting to looping of g%-precipitates. The reason for this behavior is that the precipitate coarsening produces a gradual increase in the quantity of unshearable precipitates in the sample, and the back stress depends on the volume fraction of these precipitates. Therefore a correlation between the data on critical yield stress and the back stress is expected. Fig. 9 shows the 0.2% yield stress (s0.2) minus the 0.2% yield stress of the alloy solid solution heat treated, sss 02 = 283 MPa, as a function of the mean precipitate radius, the back stress measured at oF : 10% is also shown (the point corresponding to r= 53 nm is an average). These data of s0.2 are extracted from [1], sss 02 is only used as a reference value in Fig. 9 to show the increase of the yield stress with the aging treatment. As mentioned earlier, it was shown [1] that samples of Inconel X-750 containing g%-precipitates larger than

Fig. 8. TEM bright field micrograph, showing sheared g%-particles in a sample aged to r =50 nm and deformed to o = 1.5%.

about rloop 15 nm, and double loops are not observed. On the other hand, in a sample aged to r= 50 nm and strained to o =1.5%, some precipitates appear to be sheared, Fig. 8. Of course, this does not prove that these particles can not store geometrically necessary dislocations, in earlier observations of Ref. [2] it has been shown that a tangle of dislocations is finally formed around all the g%-precipitates in overaged samples strained to o= 5%. Also, shearing events of ordered d%-precipitates larger than the critical radius have been observed in an age-hardened Al– Li alloy [19]. Therefore, it seems that the stresses carried elastically by the g%-precipitates can not overcome the stresses produced by one Orowan loop per slip plane. In the simplest model, [10] the unrelaxed strain o* sp can be related to the number of Orowan loops around the

Fig. 9. Measured values of the g%-precipitate stress contribution to the yield stress, Ref. [1](), and the back stress measured at oF = 10% ( ). In solid line, the back stress calculated with Eq. (17) using C =0.19 and the critical radius rloop =15 nm.

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Taking into account only the unshearable g%-precipitates and assuming that the measured back stress, sb, has the dependence on size and volume fraction of Eq. (13), we can rewrite the back stress formula as: |b = 1.7× 10 − 3G(m F)0.44 + Cf

Fig. 10. Bauschinger effect parameter (BEP) as a function of the prestrain oF. The symbols are correspondent with those of Fig. 4.

a mean radius of approximately 40 nm deform by the Orowan mechanism. While for samples with mean particle radius lower than about 15 nm, the plastic deformation occurs by shearing the g%-precipitates by two coupled dislocations. In order to characterize the transition in yield stress and back stress, it is convenient to use the concept of the critical particle radius, generally defined from the critical condition when the force required to shear the precipitate, by one dislocation, is equal to the force required to loop it. A good estimation of the critical particle radius is the smallest radius of the Orowan loops of Fig. 7, rloop 15 nm. According to Ardell and Huang [20], this critical radius is given by the g%-mean radius of the peakaged samples, r peak. Recently, Jeon and Park [21] have investigated the transition from shearing to looping in Al–Li single crystals, their results suggest that the critical radius is close to the mean planar particle radius at the onset of the peak strength condition, according to the Ardell and Huang model. As can be seen in Fig. 9, r peak coincides approximately with rloop of Fig. 7 in agreement with the Ardell and Huang proposal. With this information, the volume fraction and the mean radius of the unshearable precipitates can be calculated theoretically as a function of rloop, r, and the precipitate size distribution P(r,r) with: f

orw

(r) =f

r orw(r) =

&

&

1.85r

z loop 1.85r

z loop

z 3P(z,r) dz

zP(z,r)dz

,&

,&

1.85r

z 3P(z,r) dz

(15)

0 1.85r

P(z,r)dz

(16)

z loop

In Ref. [1] we have found that the Davies et al. coarsening theory [22], called the Lifshitz– Slyozov Encounter Modified (LSEM), gives a probability distribution of precipitate radii for f = 0.15 (which ends in 1.85r) that describes satisfactorily the experimental data.

orw

G

  bm F r orw

1/2

(17)

Eq. (17) is plotted in solid line in Fig. 9, where we have used the average value C= 0.19; and f orw and r orw calculated with Eq. (15) and Eq. (16), taking rloop =15 nm and the LSEM distribution. The plot of Eq. (17) is in reasonable agreement with the extracted results from the Bauschinger effect data. As can be seen, the data scatter precludes to confirm the dependence of the back stress on the precipitate radius beyond r= 26 nm. However, the transition of shearable/unshearable precipitates, which occurs for sizes down r= 26 nm, clearly has a counterpart in the increase of the back stress, owing to the linear dependence on the volume fraction of unshearable precipitates.

4.3. The transient softening As mentioned earlier, Orowan [12] suggests that a forward moving dislocation sees an array of obstacles harder than the one it sees immediately after its motion is reversed; on this basis, in the relaxed case these obstacles can be given by the plastic zones around g%-precipitates plus carbides in the aged alloy, and only around the carbides in the solid solution treated samples. The correspondence between transient softening and permanent softening, shown in Fig. 6, holds even in both cases.

4.4. The contribution of the back stresses to the WH When plastic relaxation has been placed the WH can be considered to be due to forest hardening plus the residual back stresses. The fractional contribution of the back stresses to the WH is better described by the Bauschinger effect parameter (BEP) defined as: | BEP = F b (18) | − |0.2 In Fig. 10 it can be seen that the BEP decreases with the prestrain, reaching an almost constant value of 15% whatever the precipitation stage. Accordingly, the WH data of the earlier paper [2] have been reanalyzed. The flow stress equation in tension was modified, combining the stress components arising from several glide obstacles plus the back stress and the forest dislocation contribution, sd. The sd(dsd/do) versus sd diagrams show that, despite the inclusion of the back stress in the analysis of the data, the behavior observed is only slightly modified, and the agreement with the theory discussed in Ref. [2] is still satisfactory.

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5. Conclusions

Acknowledgements

The influence of the g%-precipitates on the Bauschinger effect, in the Inconel X-750 Nickel-base superalloy, has been investigated by means of measurements of the permanent softening in tension– compression tests, from which the magnitude of the internal stresses has been estimated for solution heat treated, underaged, and overaged samples as a function of the forward strain. It is found that the internal stresses increase with the aging time, providing evidence for the appearance of unshearable particles in the microstructure. Our results indicate that measurements of internal stresses can be used as a tool to investigate the transition between shearable to unshearable precipitates in order hardened alloys. Moreover, it is found that our results are in agreement with the information arising from the analysis of the critical resolved shear stress [1]. In the overaged samples, the dependence of the internal stresses with strain behaves qualitatively in accordance with the theories of relaxed deformation in dispersion hardened metals, but the magnitude of the back stress developed is smaller than the predicted and the plastic relaxation appears to be given by plastic deformation of the precipitates. Moreover, the dependence of the unrelaxed strain on the precipitate radius is an open question. The fractional contribution of the back stress to the work hardening reaches a saturation level of  15%, both in solution heat treated samples and in aged samples. It seem, therefore, that the consideration of work hardening models based on full plastic relaxation, in the Inconel X-750 alloy can be a reasonable first approximation. However, despite forest hardening dominates the work hardening quantitatively, we believe that the back stress contribution should be taken into account in order to give an accurate description of the plastic behavior and the unloading and reverseloading effects.

The authors acknowledge the financial support of: Consejo Nacional de Investigaciones Cientı´ficas y Te´ cnicas (PIP/BID No 4318/97), Agencia Nacional de Promocio´ n Cientı´fica y Tecnolo´ gica (PICT No 0192/98), CICPBA and SeCyT-UNCentro, Argentina. Moreover, they wish to thank the assistance of Teresita Maldonado and Osvaldo Toscano.

.

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